[    {        "title": "2005.04572v1.Anisotropic_Magnetocaloric_Properties_of_The_Ludwigite_Single_Crystal_Cu2MnBO5.pdf",        "content": "1 \n Anisotropic Magnetocaloric Properties of The Ludwigite S ingle Crystal  Cu 2MnBO 5 \nA.G. Gamzatov1,*, Y.S. Koshkid’ko2, D. C. Freitas3, E. Moshkina4, L. Bezmaternykh4 \nA.M. Aliev1, S.-C. Yu5, and M.H. Phan6 \n1Amirkhanov Institute of Physics, DSC of RAS, 367003, Makhachkala, Russia  \n2Institute of Low Temperature and Structure Research, PAS, 50 -950, Wroclaw, Poland  \n3Instituto de Física, Universidade Federal Fluminense, Campus da Praia Vermelha, 24210 -346 \nNiterói, RJ, Brazil  \n4L.V. Kirensky Institute of Physic s SB RAS, 660036 Krasnoyarsk, Russia  \n5Department of Physics , Ulsan National Institute of Science and Technology, Ulsan 44919, \nSouth Korea  \n6Department of Physics, University of South Florida, 4202 East Fowler Avenue Tampa, Florida 33620, \nUSA \nAbstract  \nWe present the results of a thorough study of the specific heat and magnetocaloric properties of a \nludwigite crystal Cu 2MnBO 5 over a temperature range of 60 – 350 K and in magnetic fields up to \n18 kOe. It is found that at temperatures below the Cu rie temperature ( TC ~ 92 K), CP(T)/T possesses \na linear temperature -dependent behavior, which is associated with the predominance of two -\ndimensional antiferromagnetic interactions of magnons. The temperature independence of \nCP/T=f(Т) is observed in the temperature range of 95 – 160 K, which can be attributed to the \nexcitation of the Wigner glass phase. The magnetocaloric effect (i.e. the adiabatic temperature \nchange, Tad (T,H)) was assessed through a direct measurement or an indirect method using the \nCP(T,H) data. Owing to its strong magneto crystalline anisotropy, an anisotropic MCE or the \nrotating MCE  (Tadrot (T)) is observed in Cu 2MnBO 5. A deep minimum in the Tadrot (T) near the \nTC is observed and may be associated with the anisotropy of the paramagnetic susceptibility . \nKeywords:  Ludwigite,  Magnetocaloric effect,  Specific heat, Anisotropy.  \n*Corresponding authors: gamzatov_adler@mail.ru  (A.G)  2 \n Ludwigites belong to quasi -low-dimensional transition metal oxyborates, which are \nprominent repr esentatives of systems with strongly correlated properties [1 -3]. The oxyborate \nCu2MnBO 5 has a monoclinic -distorted structure of ludwigite and is characterized by P21/c space \ngroup due to the presence of Cu2+ and Mn3+ cations in Jahn -Teller cations. Macros copic magnetic \nand specific heat studies have shown that this compound undergoes a magnetic phase transition \ninto the ferrimagnetic phase at TC ~ 92  K [1 -3]. The crystal -orientational dependences of \nmagnetization and magnetic susceptibility revealed the anisotropic magnetic characteristic both \nbelow the TC and in the paramagnetic regime (associated with the anisotropy of g -tensor due to \nmonoclinic dist ortions introduced by Jahn -Teller Cu2+ and Mn3+ ions [3]). The microscopic \nmagnetic structure of this ludwigite was experimentally studied using powder neutron diffraction  \n[2]. It was determined that out of four nonequivalent positions occupied by magnetic  ions, three \npositions are predominantly occupied by Cu2+ ions, and one by Mn3+ ions. It was also found that \nin the compound under study there was only a partial ordering of the magnetic moments in the \nferrimagnetic phase due to the small moment of Cu2+ ion in position 2a. The magnetic moments \nof Cu2+ and Mn3+ ions in the ferrimagnetic phase are antiparallel and their directions do not \ncoincide with the main crystallographic directions in the crystal [2]. The structural, magnetic and \nthermodynamic propertie s of Cu 2MnBO 5 were studied [1 -3]. However, the magneto -thermal \nresponse of the material has not been investigated to date.  \nThe utilizing of the magnetocaloric effect (MCE) in magnetic cooling technology is of \ncurrent interest as it has the potential to re place conventional gas compression techniques [4, 5]. \nIn addition to its perspective cooling application, the MCE has recently been used as a useful \nresearch tool for the analysis and interpretation of competing magnetic phases and the collective \nmagnetic phenomena in a wide range of exotic magnetic materials [6 -10]. In this regard, we present \nhere results of the first comprehensive study of the anisotropic and frequency -dependent \nmagnetocaloric properties of the ludwigite crystal Cu 2MnBO 5.  3 \n The Cu 2MnBO 5 single crystals were synthesized by the flux method with the ratio of the \ninitial components Bi 2Mo 3O12:1.3B 2O3:0.7Na 2CO 3:0.7Mn 2O3:2.1CuO by spontaneous nucleation \n[3]. The Cu 2MnBO 5 sample has a plate shape, with dimension of 3x3x0.3 mm3 in the a*b*c \nconfi guration.  The specific heat was measured by an ac -calorimetry. Direct measurements of the \nadiabatic temperature change Tad were carried out by the modulation method [ 11] in the direction \nof the magnetic field in the ab-plane . The application of an alternating magnetic field to the sample \ninduced a periodic change in the temperature of the sample,  due to the MCE. This temperature \nchange was recorded by a differential thermo couple. The frequency of the alternating magnetic \nfield in this experiment was 0.2 Hz. An alternating magnetic field with amplitude of up to 4 kOe \nwas generated using an electromagnet and an external control power unit. The control alternating \nvoltage was supplied to the power unit from the output of the (Lock -in) amplifier SR 830. An \nalternating magnetic field of 18 kOe was created by a source of permanent magnetic field of an \nadjustable intensity manufactured by AMT & CLLC.  \nFig. 1 shows the temperature de pendence of the specific heat Cp(T) for Cu 2MnBO 5. The \ninset of Fig.1 demonstrates the temperature dependence of CP/T in the range of 70 –150 K and at \nН = 0, 6.2, and 11 kOe. The Cp(T) dependence of Cu 2MnBO 5 was reported in [2], and our results \nare in good agreement with these data. Note that CP(Т) dependence exhibits a pronounced lambda \nanomaly at ТC = 92 K, associated with the ferromagnetic -paramagnetic (FM -PM) phase transition, \nwhich is suppressed upon the  application of a magnetic field, while the maximum of the heat \ncapacity shifts by 4 K toward a higher temperature at H = 11 kOe. Two features of the Cp(T,H) \nbehavior are worthy of note. Below the ТС, the CP/T(T) is linear and unchanged with temperature \nin the temperature range of 95 -160 K. A similar behavior was observed for a number of \nferroborates [ 12, 13]. At T < TC, the CP/T = f(Т) is well described by the expression 𝐶𝑃=𝛼𝑇2 (the \ndashed line in Fig. 1). T his behavior has been explained by the predominance of two -dimensional \nantiferromagnetic interactions of magnons [ 12]. 4 \n Using the expressions, 𝐶𝑃=𝛼𝑇2, 𝛼=7.2𝑘𝐵32𝜋𝐷(𝑁𝐴⁄) ⁄ , where α is determined from the \nexperimental data (Fig. 1) α = 0.01118 J/mol K3, (N/A) = 3 10-9 mole/cm2 – the number of \nmagnetic ions per unit area [ 12], we can estimate the spin wave stiffness 𝐷=\n7.2𝑘𝐵32𝜋𝛼(𝑁𝐴⁄)=8.9810−59𝐽2𝑐𝑚2⁄ . Then using √𝐷=𝑧𝐽𝑆𝑎 , we have obtained the numerical \nvalue of the exchange integral from the Cp(T) data at T<TC as 𝐽=√𝐷𝑧𝑆𝑎=5.2610−23𝐽 ⁄ or J/k B = \n3.8 K. Here we suppose S = 2 for Mn3+ ions, z = 3 and a = 3 Å as the medium distance of local \nions. We have neglected little variations in distance between the local ions and the magnetic \narrangements of the walls . This value of J/k B = 4 K, which is compared to that reported for Fe 3BO 5 \n(J/k B = 2.1 K), which orders near 70 K [ 12]. We can also obtain the magnons s peed in the walls of \nCu2MnBO 5, 𝑣=8.99102𝑚𝑠⁄, by considering a linear dispersion relation with k ħ𝑤=𝑐𝑘. Just to \ncompare, 2D antiferromagnetic magnons were also found in another oxyborate called hulsite with \n𝑣=1.18103𝑚𝑠⁄ [14]. This reinforces the presence of 2D antiferromagnetic magnons in our \ncompound.  \nThe temperature -independent CP/T=f(Т) behavior is observed in the temperature range of \n95-160 K (Fig. 1), which is associated  with precursor effects of the charge ordering in a Wigner  \nglass phase [12]. Another possibility is that it may arise from short range magnetic ordering in a \nlow dimensional structure persisting just above the magnetic transition [13]. It is interesting to \npoint out that the coefficients of these linear terms, C/T = 773 mJ/mol K2 obtained for Fe 3BO 5 [12] \nand C/T = 843 mJ/mol K2 for the present work are of similar magnitude.  \nThe inset of Fig. 1 also shows the temperature dependence of the magnetic contribution to \nthe specific heat, ΔСр(Т)=CP - CPh (CPh is the background contribution/lattice contribution). The \nvalue of the heat capacity jump ΔСр in the phase transition region is ≈8 J/mol  K. The temperature \ndependence of ΔS is also shown in Fig. 1, associated with the disordering of the magnetic system \nduring the phase transition and determined using the expression: ∆𝑆(𝑇)=∫(𝛥𝐶𝑃𝑇⁄)dT. \nAccording to our data, the ΔS is 0.44 J/mol K, which is slightly smaller than that obtai ned by the \nauthors of work [2] ( ΔS ~ 0.6 J/mol K). Both values are much smaller compared with the 5 \n theoretical value S* = SMn+SCu = nMnRln(2S(Mn3+)+1)+ nCuRln(2S(Cu2+)+1) = 25.2 J/mol K \n[2]. Such difference in the experimental and theoretical estimates of Δ S* is characteristic of many \nmagnetic materials, due to various reasons such as magnetic inhomogeneity [15 -17] or this \ndifference is indicative of the absence of complete ordering of the magnetic moments at the \nmagnetic phase transition, whic h agrees with the results from the neutron magnetic scattering data \n[2]. \nDirect measurements of MCE ( Tad) were carried out in magnetic fields up to 18 kOe. To \nour best knowledge, no information on direct MCE measurements was reported in ludwigites. In \nwork [1 8], for Co 4.76Al1.24(O2BO 3)2, the maximum value of the MCE was estimated  from the \nCP(T,H) data. According to the data [1 8], at T = 80 K (far from the phase transition temperature \nTN = 57 K), the maximum ΔS of 7.5 J/kg K was observed  for a field change of 9 T.  \nFigure 2(a) compares values of ∆Tad in a magnetic field of 18 kOe  for two orientations of \nthe crystal (see the inset of Fig. 3 (a)). The maximum value of ∆Tad, according to direct \nmeasurements in a magnetic field of 18 kOe, is ΔT|| = 0.45 K at Т ≈ 95 К. A rotation of 90 degrees \nleads to a decrease to ΔT⊥ = 0.30 К, i.e. the magnitude of ∆Tad decreases by 1.5 times. In addition, \nthe maximum of ∆Tad also shifts toward a low temperature T = 93 K. As can be seen from Fig. \n3(b), the strong anisotropy of the MCE appears below the TC. So, at T = 82 K, the ratio ΔT||/ΔT⊥ = \n6.6, while at Т = ТС the ratio ΔT||/ΔT⊥ = 1.5. The observed anisotropy of the MCE is in good \nagreement with the anisotropy of the magnetization of Cu 2MnBO 5 [2]. \nFigure 2(b) shows the change in temperature ΔTrot of Cu 2MnBO 5 due to a 90o rotation of \nthe crystal within the ab plane, which is defined as the rotating MCE (RMCE ). These values of \nΔTrot are obtained by subtracting the curves shown in Fig. 2(a), measured along the two directions \nof the magnetic field at H||a and at H⊥a, i.e. ΔTrot = ΔT||a - ΔT⊥a. As one can see from Fig. 2(b), the \nΔTrot(Т) dependence shows a minimum near the ТС, and the ΔTrot reaches a maximum at T = 82 K, \nwhich is 0.24 K at H = 18 kOe. Although the ΔTrot value is not gigantic, it is comparable with the \nΔTH⊥a value for this sample, as well as with that of the RMCE for Gd (ΔTrot ~ 0.3 K at 15 kOe) 6 \n [19, 20]. The inset of Fig. 3b shows the angular dependence of ΔTrot at T = 80 K in a field of 18 \nkOe. At present, the record values of ΔTrot are -4.4 K at Н = 20 kOe for DyNiSi [ 21], 5.2 K at  H = \n50 kOe for HoMn 2O5 [22], and – 1.6 K at  Н = 13 kOe  for NdCo 5 [23]. It is also important to note \nthat the RMCE is reversible (see the inset of Fig. 3 (b)), which is a necessary condition for a \nmagnetocaloric material to be used in magnetic cooling technology. Balli et al.  proposed [ 22, 24] \na rotating magnetic refrigerat or in which the regenerator consists of single -crystal blocks of \nHoMn 2O5 or TbMn 2O5, and is rotated in a constant magnetic field. The idea of creating a \nthermomagnetic generator based on the RMCE was proposed at an earlier time though [2 5].  \nThe observed a nomaly in ΔTrot(Т) near the ТС on is also worthy of note. Such anomalous \nbehavior was not observed previously when studying the anisotropy of MCE in magnetic single \ncrystals. According to [2 6], an equation for determining the adiabatic change in temperature of a \nferrimagnet taking into account the contribution of two magnetic sublattices can be written in the \nfollowing form:  \n                                                  ∆𝑇𝑎𝑑=𝑇\n𝐶𝑃,𝐻(𝜕𝑀1⃗⃗⃗⃗⃗⃗ \n𝜕𝑇𝑑𝐻⃗⃗ +𝜕𝑀2⃗⃗⃗⃗⃗⃗ \n𝜕𝑇𝑑𝐻⃗⃗ ), \nwhere M1 is the magnetization of the first sublattice, M2 is the magnetization of the second \nantiparallel sublattice, d H is the magnetic field increment, and CP,H is the heat capacity. For \nCu2MnBO 5, these are the Mn3+ and Cu2+ sublattices, which determine positive  and negative \nexchange interactions in the system. It is suggested that the minimum in ΔTrot(Т) near the ТС is a \ncombination of several mechanisms. Firstly, magnetostriction can also make a significant \ncontribution to the MCE along with the paraprocess [2 6]. It was shown in [5] that the \nmagnetostriction anisotropy of Cu 2MnBO 5 has an unusual form; for a parallel configuration of \nH||с the crystal in the fields under consideration up to 20 kOe is compressed along the c-axis. At \nthe same time, fo r the H⊥c configuration, the magnetostriction has a usual quadratic form (the \ncrystal expands). It is known that magnetostrictive contributions to MCE, depending on whether \nthe crystal is compressed or expanded, will have opposite signs [ 27]. Another mecha nism causing \nthe anomaly in ΔTrot(Т) near the TC could be related to the temperature shift of the MCE maxima 7 \n at different orientations of the crystal in a magnetic field (see Fig. 3a), due to the competition of \npositive and negative exchange interactions i n Cu 2MnBO 5. \nAs is known, the energy of magnetocrystalline  anisotropy (MCA) is determined through \nthe MCA constants, which determine the RMCE. According to [2 5], the magnetic contribution to \nentropy caused by the rotation of the magnetization vector is equal to:  \n∆𝑆𝑎𝑛𝑖𝑠=−𝜕∆𝐸𝑎𝑛𝑖𝑠\n𝜕𝑇      (1) \nwhere ∆Eanis - change in the anisotropy energy upon rotation of the sample in a magnetic field from \nthe b-axis to the a-axis. For the case of rotation of the magnetization vector in the ab-plane near \nCurie temperature, taking into account the first anisotropy constan t, the change in the anisotropy \nenergy can be written in the following form:  \n( )02 2\n1 sin sin Θ Θ K EH anis − =\n      (2) \nwhere ΘH - the angle between the a-axis and the magnetization vector in the field, Θ0 - the angle \nbetween the a-axis and the magnetization vector without the field.  \nThus, the magnetic contribution to entropy caused by the process of rotation of the magnetization \nvector in the prismatic plane is:  \nΔ𝑆𝑎𝑛𝑖𝑠=−𝜕𝐾1\n𝜕𝑇(𝑠𝑖𝑛2𝛩𝐻−𝑠𝑖𝑛2𝛩0).    (3) \nAs a rule, in the temperature r egion near Curie temperature, the anisotropy field is small and the \nmagnetization vector completely rotates in the direction of the magnetic field, therefore, in our \ncase, from equation (3) we can get an equation for RMCE:  \nΔ𝑆𝑟𝑜𝑡=−𝜕𝐾1\n𝜕𝑇       (4) \nAs can be seen from equation (4), the value of rotational entropy is determined by the temperature \nbehavior of the first anisotropy constant. For this reason, the observed minimum cannot be \nexplained within the framework of the classical theory of magnetoc rystalline anisotropy. Earlier \nstudies RMCE near the Curie temperature was carried out in [ 28]. One broad maximum of the \nRMCE was observed much lower than the Curie temperature. An increase in the magnetic field 8 \n led to an expansion of the maximum of the RM CE. In this case, the existence of a complex \ntemperature dependence of the RMCE can be associated with the anisotropy of the paramagnetic \nsusceptibility. In work [3] at temperatures above the Curie temperature, it was shown that the \nmagnetic susceptibility  has a strong dependence on the orientation of the single crystal in a \nmagnetic field. In work [ 29], the influence of the orientation of the monoclinic Nd 2Ti2O7 single \ncrystal in the magnetic field on the value of the paramagnetic Curie point was found. Th is \nphenomenon itself could lead to the appearance of two maxima in the temperature dependence of \nthe RMCE in our sample.  \nTo use expression (4), it is necessary to know the value of the magnetocrystalline \nanisotropy constants, which can be obtained from the  magnetization curves along the hard axis of \nmagnetization. In this paper, by definition, we consider:  \n∆𝑇𝑟𝑜𝑡=−𝑇\n𝐶𝑃(𝑇,𝐻)∆𝑆𝑟𝑜𝑡        (5) \nTherefore, we can use the MCE data obtained from direct measurements (Fig. 2) and the CP(T) \ndata measured at t he same magnetic field configurations (Fig. 3). Figure 3 shows the CP(T)/T \ndependence at H = 0 and at 18 kOe  for two orientations of the crystal with respect to the applied \nmagnetic field. It can be seen that the application of a 18 kOe field almost completely suppresses \nthe anomaly of ΔTrot(Т). The orientation of the magnetic field in the direction of H⊥a cause d a \nchange in the temperature dependence and a shift of the maximum temperature by 4 K towards \nlow temperatures in comparison with the curve at Ha, i.e. the slight anisotropy is observed. By \ntaking into consideration of the data shown in Fig. 3, 𝛥𝑆𝑟𝑜𝑡(𝑇) can be written in the following \nexpression:  \n∆𝑆𝑟𝑜𝑡(𝑇)=∆𝑆𝐻∥𝑎(𝑇)−∆𝑆𝐻⊥𝑎(𝑇)=1\n𝑇(𝐶𝑃𝐻∥𝑎(𝑇,𝐻)⋅∆𝑇𝐻∥𝑎(𝑇)−𝐶𝑃𝐻⊥𝑎(𝑇,𝐻)⋅∆𝑇𝐻⊥𝑎(𝑇))        (6) \nThe result of the estimated rotating magnetic entropy change ( Srot) as a function of \ntemperature in a magnetic field of 18 kOe using expression (6) is presented in the inset of Fig. 3. \nThe temperature trend of Srot (the inset of Fig. 3) is similar to that of Trot (Fig. 3(b)).  9 \n We have studied the specific heat and magn etocaloric properties of Cu 2MnBO 5 over the \ntemperature range of 60 –350 K and in magnetic fields up to 18 kOe. At temperatures below the \nTC, CP(T)/T shows a linear temperature -dependent behavior, which is associated with the \npredominance of two -dimensional antiferromagnetic interactions of magnons with a linear \ndispersion relation propagating in the walls of the ludwigite.  The temperature -independent \nbehavior CP/T=f(Т) observed in the temperature range of 95 –160 K is likely associated with the \nexcitation of the Wigner glass phase. Due to its strong magnetocrystalline anisotropy, the MCE \nanisotropy, and consequently the RMCE, is observed in Cu 2MnBO 5. The deep minimum in RMCE \nnear the TC is also observed and suggested to be associated with the anisotropy of the paramagnetic \nsusceptibility . \n \nAll authors contributed equally this work  \nThe work was supported by the Russian Science Foundation under Grant No. 18 -12-00415. \nThe research was also carried out as part of the state task of the Ministry of Science of the Russian \nFederation (# AAAA - A17-117021310366 -5). \nThe data that support the findings of this study are available from the corresponding author \nupon reasonable request.  \n \nREFERENCES  \n1. S. Sofronova, E. Moshkina, I. Nazarenko, Yu. Seryotkin, S.A. Nepijko, V. Ksenofontov, K. \nMedjanik, A. Veligzhanin, L. Bezmaternykh, Crystal growth, structure, magnetic properties \nand theoretical exchange interaction calcu lations of Cu 2MnBO 5, Journal of Magnetism and \nMagnetic Materials , 420, 309 (2016).  \n2. E. Moshkina, C. Ritter, E. Eremin, S. Sofronova, A. Kartashev, A. Dubrovskiy, L. \nBezmaternykh. Magnetic structure of Cu 2MnBO 5 ludwigite: thermodynamic, magnetic \nproperties and neutron diffraction study, J. Phys.: Condens. Matter.  29, 245801 (2017).  10 \n 3. A. A. Dubrovskiy, M. V. Rautskii, E. M. Moshkina, I. V. Yatsyk, R. M. Eremina, EPR -\nDetermined Anisotropy of the g -Factor and Magneto striction of a Cu 2MnBO 5 Single Crystal \nwith a Ludwigite Structure, JETP Letters  106, 716 (2017).  \n4. M. Balli, S. Jandl, P. Fournier, and A. Kedous -Lebouc, Advanced materials for magnetic \ncooling: Fundamentals and practical aspects, Applied Physics Reviews  4, 021305 (2017).  \n5. V. Franco, J. S. Blázquez, B. Ingale, A. Conde, The Magnetocaloric Effect and Magnetic \nRefrigeration Near Room Temperature: Materials and Models, Annual Review of Materials \nResearch , 42, 305 (2012).  \n6. M.H. Phan, B. Morales, H. Srikanth, C.L. Zhang and S.W. Cheong , Phase coexistence and \nmagnetocaloric effect in (La,Pr,Ca)MnO 3, Physical Review B   81, 094413(2010) . \n7. P. Lampen, N.S. Bingham, M.H. Phan, H.T. Yi, S.W. Cheong, and H. Srikanth , Macroscopic \nphase diagram and magnetocalori c study of metamagnetic transitions in the spin chain system \nCa3Co2O6, Physical Review B  89, 144414  (2014 ). \n8. A. Biswas, S. Chandra, T. Samanta, B. Ghosh, M.H. Phan, A. K. Raychaudhuri, I. Das, and \nH. Srikanth , Universality in the entropy change for the inve rse magnetocaloric effect , Physical \nReview B  87, 134420  (2013 ). \n9. E. Clements, R. Das, L. Li, Paula J. L. Kelley, M.H. Phan, V. Keppens, D. Mandrus, and H. \nSrikanth , Critical Behavior and Macroscopic Phase Diagram of the Monoaxial Chiral \nHelimagnet Cr 1/3NbS 2, Scientific Reports 7, 6545 (2017 ). \n10. R. P. Madhogaria, R . Das, E .M. Clements, V . Kalappattil, M .-H. Phan, H . Srikanth, N. T. \nDang, D. P. Kozlenko, and N. S. Bingham , Evidence of long -range ferromagnetic order and \nspin frustration effects in the double p erovskite, Physical Review B  99, 104436  (2019 ). \n11. A. M. Aliev, Direct magnetocaloric effect measurement technique in alternating magnetic \nfields, arXiv:1409.6898 (2014).  \n12. J.C. Fernandes, R.B. Guimarães, M.A. Continentino, L. Ghivelder, & R.S. Freitas, Specific \nheat of Fe 3O2BO 3: Evidence for a Wigner glass phase, Physical Review B  61, R850 (2000).  11 \n 13. M.A. Continentino, A.M. Pedreira, R.B. Guimaraes, M. Mir, J.C. Fernandes, R.S. Freitas, & \nL. Ghivelder, Specific heat and magnetization studies of Fe 2OBO 3, Mn 2OBO 3 and \nMgScOBO 3. Physical Review B  64, 014406 (2001).  \n14. D. C. Freitas, R. B. Guimarães, J. C. Fernandes, M. A. Continentino, C. B. Pinheiro, J. A. L. \nC. Resende, G. G. Eslava, and L. Ghivelder , Planar magnetic interactions in the hulsite -type \noxyborate, Co 5.52Sb0.48(O2BO 3)2, Physical Review B  81, 174403 (2010).  \n15. M.R. Lees, O.A. Petrenko, G. Balakrishnan, D. McK. Paul, Specific heat \nof Pr0.6(Ca 1−xSrx)0.4MnO 3 (0<x<1),  Physical Review B  59, 1298 (1999).  \n16. Y. Moritomo, A. Machida, E. Nishibori, M. Takata, M. Sakata, Enhanced specific heat jump \nin electron -doped  CaMnO 3: Spin ordering driven by charge separation, Physical Review B 64, \n214409 (2001).  \n17. A. G. Gamzatov, A. B. Batdalov, I. K. Kamilov, A . R. Kaul, N. A. Babushkina, Resistivity, \nspecific heat, and magnetocaloric effect of La 0.8Ag0.1MnO 3: Effect of isotopic substitution \nof 16O→18O, Applied Physics Letters  102, 032404 (2013).  \n18. C.P.C. Medrano, D.C. Freitas, E.C. Passamani, C.B. Pinheiro, E. Baggio -Saitovitch, M.A. \nContinentino, & D.R. Sanchez, Field -induced metamagnetic transitions and two -dimensional \nexcitations in ludwigite Co 4.76Al1.24(O2BO 3)2, Physical Review B  95, 214419 (2017).  \n19. A. M. Mansanares, F. C. G. Gandra, M. E. Soffner, A. O. Gui marães, E. C. da Silva2 H. \nVargas, and E. Marin, Anisotropic magnetocaloric effect in gadolinium thin films: \nMagnetization measurements and acoustic detection, Journal of Applied Physics  114, 163905 \n(2013).  \n20. S.A. Nikitin , E.V. Talalaeva , L.A. Cher nikova , G.E. Chuprikov , T.I. Ivanova , G.V. \nKazakov , G.A. Yarkho ,Features of the magnetic behavior and of the magnetocaloric effect in \na single crystal of gadolinium,  JETP  47, 105 (1978).  12 \n 21. H. Zhang, Y. Li, E. Liu, Y. Ke, J . Jin, Y. Long, B. Shen, Giant rotating magnetocaloric effect \ninduced by highly texturing in polycrystalline DyNiSi compound,  Scientific reports  5, 11929 \n(2015).  \n22. M. Balli, et al. Anisotropy -enhanced giant reversible rotating magnetocaloric effect in \nHoMn 2O5 single crystals, Applied Physics Letters  104, 232402 (2014).  \n23. S. A. Nikitin, K. P. Skokov, Yu. S. Koshkid’ko, Yu. G. Pastushenkov and T. I. Ivanova , Giant \nrotating magnetocaloric effect in the region of spin -reorientation transition in the NdCo5 \nsingle crystal, Phys. Rev. Lett.  105, 137205 (2010).  \n24. M. Balli, S. Jandl, P. Fournier, D.Z. Dimitrov, Giant rotating magnetocaloric effect at low \nmagnetic fields in multiferroic TbMn 2O5 single crystals,  Applied Physics Letters  108(10), \n102401 (2016).  \n25. J.A. Barclay, A 4 K to 20 K rotational -cooling magnetic refrigerator capable of 1mW to > 1W \noperation, Cryogenics  20, 467 -471 (1980) . \n26. S. V. Vonsovsky, Magnetism (John Wiley, New York, 1974), Vol. 2.  \n27. Vittorio Basso, The magnetocaloric effect at the first -order magneto -elastic phase transition, \nJ. Phys.: Condens. Matter 23, 226004 (2011) . \n28. Nikitin, S.A., Ivanova, T.I., Zvonov, A.I., Koshkid'ko, Y.S., Ćwik, J. & Rogacki, K., \nMagnetization, magnetic anisot ropy and magnetocaloric effect of the Tb 0.2Gd0.8 single crystal \nin high magnetic fields up to 14T in region of a phase transition, Acta Materialia  161, 331 \n(2018) . \n29. Hui Xing, Gen Long, Hanjie Guo, Youming Zou, Chunmu Feng, Guanghan Cao, Hao Zeng \nand Zhu -An Xu, Anisotropic paramagnetism of monoclinic Nd 2Ti2O7 single crystals J. Phys.: \nCondens. Matter  23, 216005 (2011) . \n \n \n 13 \n  \nFig. 1. Temperature dependence of CP/T for Cu 2MnBO 5 at Н = 0, 6.2, and 11 kOe. Inset (i ) shows \nthe temperature dependences of CP (blue) and the entropy change S (red line).  Inset (i i) shows \nthe temperature dependence of an enlarged anomalous part of the CP/T (points).  \n \na) b)  \nFig. 2. a) Temperature dependence of MCE ( ∆Tad) in a magnetic field of 18 kOe for two \norientations of the crystal with respect to the applied magnetic field. The inset of Fig. 3(a) shows \nthe ΔT||/ΔT⊥(T) dependence in a field of 18 kOe  and the orientation of the crystal in a magnetic \nfield. b) The adiabatic temperature change ΔTrot of Cu 2MnBO 5 caused by a 90o rotation of the \n0 50 100 150 200 250 300 350050100150200250\nii)\n80 90100 110 120 130 140 1500.000.020.040.060.080.10CP (J/mol K)\nT (K)0.00.10.20.30.40.5\n S (J/mol K)CP (J/mol K)\nT (K)TC=91 Ki)\n80 90100 110 120 130 140 1500.750.800.850.900.95\n H=0\n 6.2 kOe\n 11 kOeCP/T (J/mol K2)\nT (K)C/T=aT\n80 90 100 110 120 1300.00.10.20.30.40.5\n H IIa\n H ^aTad (K)\nT (K)H=18 kOeCu2MnBO5\n80 90 100 110123456\nTII/T^\nT (K)H\na\nb\n80 90 100 110 120 1300.000.050.100.150.200.25Trot (K)\nT (K)TC~92 K\nH=18 kOe0 90 180 270 3600.050.100.150.200.25Tad (K)\nqT=80 K14 \n crystal as shown in the inset of Fig. 3(a). The inset of Fig. 3(b) shows the angular dependence of \n∆Tad at T = 80 K in a field of 18 kOe.  \n \n \n \nFig. 3. CP(T)/T dependence at Н = 0 and 18 kOe for two orientations of the crystal with respect \nto the magnetic field direction, Ha and H^a. The inset shows the 𝛥𝑆𝑟𝑜𝑡(𝑇) at H = 18 kOe.  \n \n \n80 85 90 95 100 105 110 115 1200.800.850.900.95\n H=0\n Hчч a\n H^ aCP/T (J/mol K2)\nT (K)80 90 100 110 1200.00.20.40.6Srot (J/kg K)\nT (K)"    },    {        "title": "2005.14559v1.Electron_spin_resonance_and_ferromagnetic_resonance_spectroscopy_in_the_high_field_phase_of_the_van_der_Waals_magnet_CrCl__3_.pdf",        "content": "Electron spin resonance and ferromagnetic resonance spectroscopy\nin the high-\feld phase of the van der Waals magnet CrCl 3\nJ. Zeisner,1, 2,\u0003K. Mehlawat,1, 3,\u0003A. Alfonsov,1M. Roslova,4\nT. Doert,5A. Isaeva,1, 2, 3B. B uchner,1, 2, 3and V. Kataev1\n1Leibniz IFW Dresden, D-01069 Dresden, Germany\n2Institute for Solid State and Materials Physics, TU Dresden, D-01062 Dresden, Germany\n3W urzburg-Dresden Cluster of Excellence ct.qmat, Germany\n4Department of Materials and Environmental Chemistry,\nStockholm University, SE-106 91, Stockholm, Sweden\n5Faculty of Chemistry and Food Chemistry, TU Dresden, D-01062 Dresden, Germany\n(Dated: June 1, 2020)\nWe report a comprehensive high-\feld/high-frequency electron spin resonance (ESR) study on sin-\ngle crystals of the van der Waals magnet CrCl 3. This material, although being known for quite a\nwhile, has received recent signi\fcant attention in a context of the use of van der Waals magnets in\nnovel spintronic devices. Temperature-dependent measurements of the resonance \felds were per-\nformed between 4 and 175 K and with the external magnetic \feld applied parallel and perpendicular\nto the honeycomb planes of the crystal structure. These investigations reveal that the resonance line\nshifts from the paramagnetic resonance position already at temperatures well above the transition\ninto a magnetically ordered state. Thereby the existence of ferromagnetic short-range correlations\nabove the transition is established and the intrinsically two-dimensional nature of the magnetism\nin the title compound is proven. To study details of the magnetic anisotropies in the \feld-induced\ne\u000bectively ferromagnetic state at low temperatures, frequency-dependent ferromagnetic resonance\n(FMR) measurements were conducted at 4 K. The observed anisotropy between the two magnetic-\n\feld orientations is analyzed by means of numerical simulations based on a phenomenological theory\nof FMR. These simulations are in excellent agreement with measured data if the shape anisotropy\nof the studied crystal is taken into account, while the magnetocrystalline anisotropy is found to be\nnegligible in CrCl 3. The absence of a signi\fcant intrinsic anisotropy thus renders this material as a\npractically ideal isotropic Heisenberg magnet.\nI. INTRODUCTION\nMagnetic van der Waals materials belong to a class of\nphysical systems that currently receive considerable at-\ntention in solid state and materials research [1{6]. As a\ncommon feature these materials crystallize in a layered\nstructure with the individual layers being separated by\nthe so-called van der Waals gap, see Fig. 1. The weak\nvan der Waals coupling between the adjacent layers re-\nsults in dominating magnetic interactions within the lay-\ners while magnetic couplings between the layers remain\nrelatively weak. Thus, van der Waals magnets can be\nconsidered as quasi-two-dimensional magnetic systems.\nThese systems enable experimental studies of the spe-\nci\fc magnetic properties arising from the interplay of\nan e\u000bectively reduced dimensionality and the respective\nsingle-ion anisotropies determined by the type of mag-\nnetic ions and their local environments, see for instance\n[4]. Moreover, the weak van der Waals bonds between the\nlayers allow mechanical exfoliation of bulk crystals down\nto the few-layer or even monolayer limit [2, 3], thereby\napproaching the experimental systems to the true two\ndimensional (2D) limit. In addition, the combination of\nvarious materials sharing similar layer structures and the\nweak van der Waals couplings between the layers open up\n\u0003These authors contributed equally to this work.numerous possibilities for the creation of (magnetic) van\nder Waals heterostructures [5{7]. These stacks of sev-\neral few-layer crystals of di\u000berent materials are a promis-\ning route towards electronic devices with speci\fcally tai-\nlored magnetic properties, see, e.g., Ref. [6] and refer-\nences therein. In both respects { the fundamental study\nof 2D magnetism and its application in the framework of\nmagnetic heterostructures { the characterization of mag-\nnetic anisotropies in van der Waals materials represents a\nkey task. Magnetic resonance spectroscopies are valuable\ntools to accomplish this task due to their high sensitivity\nwith respect to the presence of magnetic anisotropies in\na system. As an example, in a previous work [8] some\nof the authors quantitatively investigated the magnetic\nanisotropies in the van der Waals magnet Cr 2Ge2Te6by\nmeans of electron spin resonance (ESR) and ferromag-\nnetic resonance (FMR) studies. Here, we report a char-\nacterization of the (e\u000bective) magnetic anisotropy in the\nhigh-\feld phase of the related compound CrCl 3. The last\nis a member of the family of transition-metal trihalides\nwhose magnetic ions (here: Cr3+, 3d3,S= 3=2,L= 3)\nare situated in the center of an octahedron built by the\nhalogen ligands (here: Cl\u0000ions). These octahedra form\nedge-sharing networks in the crystallographic abplane\nwhich e\u000bectively results in a magnetic honeycomb lat-\ntice, see Fig. 1.\nIt is worthwhile to mention that the title compound\nbelongs to the \frst materials studied using the ESR tech-arXiv:2005.14559v1  [cond-mat.mtrl-sci]  29 May 20202\nFIG. 1. Crystal and magnetic structures of CrCl 3at low temperatures. (a) View along the b-axis of the unit cell (non-primitive\nhexagonal) of the rhombohedral low-temperature structure (space group R3 [9]). The magnetic Cr3+ions (red spheres) are\noctahedrally coordinated by the Cl ligands (green spheres). (b) These CrCl 6octahedra build an edge-sharing network in the\nabplane which e\u000bectively leads to a honeycomb-like arrangement of the magnetic ions (illustrated by solid red lines). Each\nof these honeycomb layers in the abplane is well separated from its neighbors along the caxis by the van der Waals gap,\nas shown in (a). At temperatures below the transition into a magnetically long-range ordered state, spins in the honeycomb\nlayers are coupled ferromagnetically and oriented within the ab-plane, while spins in neighboring layers are coupled by a weaker\nantiferromagnetic interaction [10, 11]. The resulting spin structure in the magnetically ordered state at zero magnetic \feld\n[10, 11] is schematically illustrated by the arrows at the Cr sites. Crystallographic data are taken from Ref. [9].\nnique by E. K. Zavoisky, the pioneer of this spectroscopy\n(see, for instance, [12, 13]). However, in the context of\nmagnetic van der Waals materials the magnetic prop-\nerties of CrCl 3came back to the focus of current re-\nsearch interest [11, 14{23]. Basic magnetic properties\nof bulk CrCl 3crystals have been reported in the past\ndecades, see, e.g., Refs. [11, 16, 24{28]. In particular,\na two-step transition into a magnetically long-range or-\ndered state was observed [11, 16, 28]. Below a temper-\natureT2D\nc\u001817 K, spins within the honeycomb layers\norder ferromagnetically and align parallel to the honey-\ncomb plane, i.e., the abplane [cf. Fig. 1(b)], while spins\nof individual Cr layers are not coupled to spins in the\nneighboring layers [11, 16, 28]. Consequently, the order-\ning at around 17 K is of an e\u000bectively 2D nature. The\nferromagnetic character of the dominant exchange inter-\nactions between spins in the abplane is also evidenced\nby the positive Curie-Weiss temperatures \u0002 CWbetween\n27 and 43 K derived from measurements of the static sus-\nceptibility [11, 16, 17, 25, 29, 30]. Below a temperature\nT3D\nNof about 14 K [11, 16] (15.5 K in Ref. [28]) CrCl 3en-\nters into a three-dimensional (3D) antiferromagnetically\nordered state at zero magnetic \feld. In this magnetic\nphase, ferromagnetically ordered spins in the honeycomb\nlayers are coupled by antiferromagnetic interactions be-\ntween neighboring layers [10], as illustrated in Fig. 1(a).\nHowever, the long range antiferromagnetic order can be\nsuppressed already in relatively small magnetic \felds of\nabout 0.6 T (external \feld applied perpendicular to thehoneycomb planes, i.e., Hkc) and 0.25 T (external \feld\napplied in the honeycomb planes, i.e., H?c) at 2 K [11],\nrespectively. Above these \felds, a saturation of the mag-\nnetization at around 3 \u0016B/Cr was observed [11, 16]. The\nlow values of the saturation \felds thus con\frm the (rel-\native) weakness of the antiferromagnetic interlayer cou-\nplings. Moreover, if demagnetization e\u000bects are taken\ninto account, the saturation \felds become almost identi-\ncal for both orientations of the external \feld with respect\nto the honeycomb layers [11, 16]. Therefore, the exper-\nimentally observed magnetic anisotropies appear to be\ndominated by dipole-dipole interactions which are at the\norigin of demagnetization \felds and the so-called shape\nanisotropy [11, 16]. The apparent size of the magnetic\nanisotropy thus depends strongly on the dimensions of\nthe studied samples whereas the intrinsic magnetocrys-\ntalline is expected to be much weaker, if it exists at all.\nIn order to disentangle and quantify the two contribu-\ntions to an anisotropic magnetic response, which could\nbe of relevance in the case of CrCl 3, we studied in this\nwork the details of the magnetic anisotropies in the \feld-\ninduced ferromagnetic-like phase of the title compound\nby means of high-\feld/high-frequency (HF) FMR over\na wide range of frequencies at magnetic \felds exceeding\nthe low-temperature saturation \felds of CrCl 3.3\nII. SAMPLES AND EXPERIMENTAL\nMETHODS\nSingle-crystalline CrCl 3samples used in this study\nwere synthesized by a chemical vapor transport reac-\ntion between Cr metal and Cl 2gas as described in detail\nin Ref. [29]. The results of single-crystal X-ray di\u000brac-\ntion studies at room temperature con\frming the mon-\noclinic (space group C2=m) high-temperature modi\fca-\ntion of the title compound are also reported there. More-\nover, magnetic properties of the samples were studied\nin Ref. [16] by means of speci\fc heat as well as static\nand dynamic magnetization measurements. These are\nconsistent with the \fndings reported in previous studies\n[11, 26, 28], in particular, they con\frm the presence of\ntwo successive magnetic phase transitions mentioned in\nthe previous section. This does not only demonstrate the\nhigh quality of the used CrCl 3crystals but also ensures\nthe comparability of the magnetic properties reported in\nthis work and in the literature [11, 23, 26, 28]. Additional\ncharacterization details can be found in the Appendix.\nThe compound CrCl 3is known to undergo a struc-\ntural phase transition at temperatures around 240 K from\nthe high-temperature monoclinic phase to an rhombo-\nhedral phase (space group R3) at lower temperatures\n[9, 11]. These two structural modi\fcations di\u000ber mainly\nin the stacking sequence of the honeycomb layers while\nthe structure within the layers is very similar in both\nphases [11]. The crystal structure of the rhombohedral\nmodi\fcation is shown in Fig. 1 as the focus of the present\nstudy lies on the magnetic properties at lower tempera-\ntures. In this phase, individual honeycomb layers are\nstacked along the caxis in an -ABC-sequence. Conse-\nquently, each unit cell contains three layers along the c\naxis, see Fig. 1(a). The strong chemical bonding within\ntheabplane and the comparatively weak van der Waals\ncouplings between the layers result in \rat, platelet-like\nsingle crystals allowing an easy identi\fcation of the caxis\nas the direction perpendicular to the platelet plane.\nThe ESR and FMR measurements were carried out us-\ning two di\u000berent setups. For continuous wave (cw) HF-\nESR/HF-FMR studies a homemade spectrometer was\nemployed. The spectrometer consists of a network vec-\ntor analyzer (PNA-X from Keysight Technologies) for\ngeneration and detection of microwaves in the frequency\nrange from 20 to 330 GHz, oversized waveguides, and a\nsuperconducting solenoid (Oxford Instruments) provid-\ning magnetic \felds up to 16 T. The magnetocryostat is\nequipped with a variable temperature insert that en-\nables measurements in the temperature range between\n1.8 and 300 K. All HF measurements presented in the\nfollowing section were carried out in transmission ge-\nometry employing the Faraday con\fguration. In addi-\ntion, cw ESR/FMR measurements at a \fxed microwave\nfrequency of about 9.6 GHz and in magnetic \felds up\nto 0.9 T were performed using a commercial X-band\nspectrometer (EMX from Bruker). This spectrome-\nter is equipped with a helium \row cryostat (ESR900from Oxford Instruments) and a goniometer, allowing\ntemperature-dependent measurements in the range 4 -\n300 K and angle-dependent studies, respectively.\nIII. RESULTS AND DISCUSSION\nIn the following, results of systematic ESR measure-\nments are presented and discussed. These measurements\nwere carried out with the external magnetic \feld Hap-\nplied parallel and perpendicular to the crystallographic\ncaxis, respectively. Since the focus of the present study\nlies on a detailed investigation of the (e\u000bective) magnetic\nanisotropies in the \feld-polarized ferromagnetic state of\nCrCl 3at low temperatures, this work is mainly con-\ncerned with the behavior of the resonance \feld Hresas a\nfunction of temperature and microwave frequency over a\nbroad frequency range. Further aspects of the magnetic\nproperties of CrCl 3derived from ESR measurements at\nlower microwave frequencies, such as the excitations of\nthe 3D antiferromagnetic state and the spin dynamics\nabove the magnetic phase transitions, were reported in\nRefs. [23, 31].\nA. Temperature dependence\nThe temperature dependence of the resonance shift\n\u000eH(T) =Hres(T)\u0000Hres(100K) measured at low and\nhigh microwave frequencies \u0017and in both magnetic \feld\ncon\fgurations is shown in Fig. 2. This quantity is a mea-\nsure of the deviation of the resonance \feld at a given\ntemperature Tfrom the ideal paramagnetic resonance\nposition. This position is determined by the standard\nresonance condition of a paramagnet [32]\n\u0017=g\u0016B\u00160Hres=h : (1)\nHere,gdenotes the gfactor of the resonating spins and\n\u0016B,\u00160, andhare Bohr's magneton, the vacuum per-\nmeability, and Planck's constant, respectively. In the\npresent case, the resonant shift was determined with re-\nspect to the expected resonance \feld at 100 K which was\ncalculated according to Eq. (1) using the gfactor de-\nrived from frequency-dependent measurements at 100 K,\nsee below. Upon lowering the temperature below \u001875 K,\nthe resonance position is shifted progressively to smaller\n\felds when the external magnetic \feld is applied perpen-\ndicular to the caxis, resulting in a negative shift \u000eH.\nForHkcthis trend is reversed yielding a positive res-\nonance shift. Thus, based on the qualitative behavior\nof the temperature-dependent resonance shift, it can be\nconcluded that the experimentally observable (e\u000bective)\nmagnetic easy direction lies within the abplane which is\nin agreement with the proposed spin structure [10] and\nprevious magnetization measurements [11, 16, 26]. More-\nover, the\u000eH(T) curves obtained for HkcandH?c\nare asymmetric with respect to the \u000eH= 0 line of an4\nideal paramagnet, see Fig. 2. This asymmetry is a direct\nconsequence of the di\u000berent frequency-\feld dependencies\n\u0017(Hres) expected for the two di\u000berent \feld orientations in\na ferromagnetically ordered system [33, 34]. In particu-\nlar, a shift of the resonance line should be larger when the\nexternal magnetic \feld is oriented parallel to the mag-\nnetic anisotropy axis which in the case of CrCl 3is the\nmagnetic hard axis parallel to the crystallographic caxis.\nThe onset of a \fnite resonance shift already at temper-\natures signi\fcantly above the ordering temperature T2D\nc\nprovides clear evidence for the low-dimensional character\nof the spin correlations in this material as it was discussed\nin the context of one-dimensional systems, for instance in\nRefs. [35{37]. Moreover, the 2D nature of magnetic cor-\nrelations above T2D\ncwas reported in Ref. [31] based on\nmeasurements of the resonance shift and the linewidth\nangular dependence at various temperatures and at low\nmicrowave frequencies of about 9.4 GHz. While our mea-\nsurements of \u000eH(T) at 9.6 GHz are consistent with this\nprevious study [31], we observed the onset of a \fnite\nresonance shift at higher temperatures when employing\nfrequencies of about 90 GHz. Thus, the higher exter-\nnal magnetic \felds associated with the higher microwave\nfrequencies strengthen the ferromagnetic correlations re-\nsponsible for the shift of the resonance line, similar to\nthe situation found in the related van der Waals magnet\nCr2Ge2Te6[8]. Finally, it is worthwhile mentioning that\nthe appearance of short-range correlations at tempera-\ntures far above the magnetic ordering temperature is con-\nsistent with the deviation of the temperature-dependent\nstatic susceptibility from a Curie-Weiss behavior already\nbelow\u0018125 K, which was reported in Ref. [29]. Taken to-\ngether, temperature-dependent measurements of the res-\nonance shift at two di\u000berent frequencies and \feld orien-\ntations demonstrate, \frst, the apparent easy-plane type\nmagnetic anisotropy and, second, the 2D character of dy-\nnamic spin correlations in CrCl 3far above the long-range\nordering temperatures.\nB. Frequency dependence\nTo shed light on the details of the magnetic\nanisotropies, frequency-dependent investigations were\nconducted at 4 and 100 K, i.e., at temperatures deep in\nthe magnetically ordered state and well above the mag-\nnetic phase transition, respectively. Exemplary spectra\nare presented in the insets of Fig. 3. At both tem-\nperatures, ESR/FMR spectra consist of a single nar-\nrow resonance line with typical linewidths (full width\nat half maximum) of about 25 mT at 100 K. The small\nlinewidths allow an easy and precise determination of\nthe resonance \felds which correspond to the positions\nof the minima in the microwave transmission. The re-\nsulting frequency-\feld diagrams are shown in the main\npanels of Fig. 3. At 100 K a linear frequency-\feld de-\npendence\u0017(Hres) is observed for both orientations of the\nexternal magnetic \feld. Such a behavior is expected in\n/s48 /s50/s53 /s53/s48 /s55/s53 /s49/s48/s48 /s49/s50/s53 /s49/s53/s48 /s49/s55/s53/s45/s50/s48/s48/s45/s49/s48/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48\n/s32/s72 /s32/s124 /s124 /s32 /s99 /s32/s58/s32 /s110 /s32/s61/s32/s57/s48/s46/s48/s32/s71/s72/s122\n/s32/s72 /s32/s124 /s124 /s32 /s99/s32 /s58/s32 /s110 /s32/s61/s32/s57/s46/s54/s32/s71/s72/s122\n/s32/s72 /s32/s94 /s32/s99 /s32/s58/s32 /s110 /s32/s61/s32/s57/s48/s46/s49/s32/s71/s72/s122\n/s32/s72 /s32/s94 /s32/s99/s32 /s58/s32 /s110 /s32/s61/s32/s57/s46/s54/s32/s71/s72/s122/s109\n/s48/s40/s72\n/s114/s101/s115/s40/s84 /s41/s32/s45/s32 /s72\n/s114/s101/s115/s40/s49/s48/s48/s32/s75/s41/s41/s32/s40/s109/s84/s41\n/s84 /s32/s40/s75/s41/s84/s32/s50/s68\n/s67/s84/s32/s51/s68\n/s78FIG. 2. Temperature dependence of the resonance shift\n\u000eH(T) =Hres(T)\u0000Hres(100K) at microwave frequencies\nof about 9.6 and 90 GHz (open and \flled symbols, respec-\ntively). In these measurements the external magnetic \feld\nwas oriented parallel (red circles) as well as perpendicular\n(blue squares) to the caxis. The dashed horizontal line rep-\nresents the zero shift \u000eH= 0 expected for an uncorrelated\nparamagnet. Vertical dashed lines indicate the zero-\feld tran-\nsition temperatures T2D\ncandT3D\nNof the transition into the 2D\nferromagnetically ordered phase and the 3D antiferromagnet-\nically ordered state [11], respectively.\nthe paramagnetic regime of CrCl 3and can be well de-\nscribed by the standard resonance condition of a param-\nagnet [Eq. (1)] [32]. Fits to the data according to Eq. (1)\nare shown in Fig. 3(a) as solid lines and yielded gfactors\ngk= 1:970\u00060:005 andg?= 1:990\u00060:005 forHkc\nandH?c, respectively. The experimentally determined\ngfactors are in good agreement with the values antici-\npated for Cr3+ions (3d3,S= 3=2,L= 3) in an octa-\nhedral crystal \feld [32]. The merely small deviation of g\nfrom the free-electron gfactor of 2 as well as the slight\nanisotropy observed in the measurements indicate that\nthe magnetism in CrCl 3is largely dominated by the spin\ndegrees of freedom while the orbital angular momentum\nis practically completely quenched in the \frst order. The\nmentioned small deviations from the ideal spin-only be-\nhavior result, most likely, from second-order spin-orbit\ncoupling e\u000bects [32]. The gfactors derived in this work\nfrom frequency-dependent measurements are, moreover,\nconsistent with the saturation magnetization of about\n3\u0016B/Cr which was experimentally observed [11, 16] and\nis theoretically expected for spins S= 3=2 and agfac-\ntor of 2. These observations already suggest that spin-\norbit coupling and, consequently, the intrinsic magne-\ntocrystalline anisotropies are very weak (or even negligi-\nble) in CrCl 3, as it was also mentioned in the literature\n[11, 16, 26].5\n/s48 /s49 /s50 /s51 /s52 /s53 /s54 /s55 /s56 /s57 /s49/s48 /s49/s49/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48/s51/s53/s48/s52/s48/s48/s110 /s32/s40/s71/s72/s122/s41\n/s109\n/s48/s32/s72\n/s114/s101/s115/s32/s40/s84/s41/s32/s72 /s32/s124/s124/s32 /s99\n/s32/s72 /s32/s94 /s32/s99/s84 /s32/s61/s32/s52/s32/s75/s49/s46/s53 /s49/s46/s54 /s53/s46/s52 /s53/s46/s54 /s57/s46/s54 /s57/s46/s57 /s49/s48/s46/s50\n/s78/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32\n/s115/s105/s103/s110/s97/s108/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121\n/s109\n/s48/s32/s72 /s32/s40/s84/s41/s51/s52/s32/s71/s72/s122 /s49/s52/s52/s32/s71/s72/s122 /s50/s54/s50/s32/s71/s72/s122\n/s48 /s49 /s50 /s51 /s52 /s53 /s54 /s55 /s56 /s57 /s49/s48 /s49/s49 /s49/s50 /s49/s51/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48/s51/s53/s48\n/s32/s72 /s32/s124/s124/s32 /s99/s32 /s58/s32 /s103 /s32/s61/s32/s49/s46/s57/s55/s48/s32 /s177 /s32/s48/s46/s48/s48/s53/s32\n/s32/s72 /s32/s94 /s32/s99/s32 /s58/s32 /s103 /s32/s61/s32/s49/s46/s57/s57/s48/s32 /s177 /s32/s48/s46/s48/s48/s53/s110 /s32/s40/s71/s72/s122/s41\n/s109\n/s48/s32/s72\n/s114/s101/s115/s32/s40/s84/s41/s84 /s32/s61/s32/s49/s48/s48/s32/s75\n/s51/s46/s48 /s51/s46/s53 /s55/s46/s48 /s55/s46/s53 /s49/s48/s46/s48 /s49/s48/s46/s53/s109\n/s48/s32/s72 /s32/s40/s84/s41/s78/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32\n/s115/s105/s103/s110/s97/s108/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s57/s48/s32/s71/s72/s122\n/s49/s57/s55/s32/s71/s72/s122/s50/s55/s53/s32/s71/s72/s122/s40/s97 /s41 /s40/s98 /s41\nFIG. 3. Frequency dependence of the resonance \feld Hresat 100 K (a) and 4 K (b), respectively. To quantitatively investigate\nthe magnetic anisotropies in CrCl 3, measurements were carried out with the external magnetic \feld applied parallel (red circles)\nand perpendicular (blue squares) to the crystallographic caxis, respectively. Open symbols correspond to the resonance \felds\nmeasured at \u0017= 9:6 GHz. Solid lines in (a) are linear \fts to the \u0017(Hres) dependencies according to the standard resonance\ncondition of a paramagnet given in Eq. (1) [32]. At 4 K, the measured frequency-\feld dependencies were simulated using\nEq. (2) which describes the resonance frequency in the case of FMR. Corresponding simulations are shown in (b) as solid\nlines. Exemplary spectra recorded with Hkcare presented for both temperatures in the respective insets. For comparison, all\nspectra shown were normalized (note the di\u000berent breaks in the \feld axes). Arrows in the \u0017(Hres) denote the positions of the\nspectra given in the insets.\nFor a detailed analysis of the relevant anisotropies\npresent in the title compound, frequency-dependent mea-\nsurements were conducted at 4 K, i.e., in the magnetically\nordered state. We emphasize that the lowest frequency\nused in the systematic HF measurements corresponds to\na resonance \feld of about 0.75 T [cf. Fig. 3(b)] which\nexceeds the reported saturation \felds [26]. Therefore,\nthe \feld-polarized, e\u000bectively 2D ferromagnetic state of\nCrCl 3is probed by our HF magnetic resonance measure-\nments allowing us to describe the obtained \u0017(Hres) de-\npendencies in the framework of a theory of FMR that\nis applicable in a single-domain ferromagnetic material,\nas will be discussed in the following section. In contrast\nto the measurements in the paramagnetic state, the 4 K\ndata show a clear anisotropy regarding the two magnetic-\n\feld orientations which is consistent with the resonance\nshifts of opposite sign observed in the temperature-\ndependent studies (see Fig. 2). If the external magnetic\n\feld is applied within the abplane (H?c), the resonance\npositions are systematically shifted towards smaller mag-\nnetic \felds (with respect to the paramagnetic resonance\npositions) for all measured frequencies. For Hkca neg-\native intercept with the frequency axis of the \u0017(Hres)\ndiagram is observed due to the shift of the resonance po-\nsitions to higher magnetic \felds. Qualitatively, such a\nbehavior is expected in the case of a ferromagnetically\nordered system with an easy-plane anisotropy [34, 38].\nIn a semi-classical picture, these shifts of the resonanceposition result from anisotropic internal \felds in the mag-\nnetic crystal. These \felds are caused, in turn, by dipole-\ndipole interactions between the magnetic moments asso-\nciated with the spins or an intrinsic magnetocrystalline\nanisotropy, i.e., by spin-orbit coupling. In the following\nsection the resonance \felds in the ferromagnetic state\nwill be simulated employing a theory based on this semi-\nclassical description of magnetic resonance. This allows\nto determine quantitatively the contributions of the two\npossible sources of magnetic anisotropy in CrCl 3.\nC. Analysis of the magnetic anisotropies\nAs mentioned in the previous sections, it is possi-\nble to describe the experimentally observed frequency-\n\feld dependence in the ferromagnetic state at 4 K in the\nframework of a semi-classical, phenomenological theory\nof FMR [34, 38, 39]. Conceptually, the di\u000berence between\nESR and FMR lies in the fact, that ESR is the resonant\nexcitation of individual paramagnetic spins within a mag-\nnetic system, while FMR describes the resonance of the\ntotal magnetization Min a ferromagnetically ordered\nmaterial. Thus, the resonance frequencies expected for a\ncorrelated ferromagnetic spin system can be calculated\nby considering the classical vector of the macroscopic\nmagnetization and the appropriate free energy density\nF[34, 38, 39]. In this case, the resonance frequency is6\ngiven by the following expression [34, 38, 39]\n\u00172\nres=g2\u00162\nB\nh2M2ssin2\u0012\u0012@2F\n@\u00122@2F\n@'2\u0000\u0010@2F\n@\u0012@'\u00112\u0013\n;(2)\nwhereMsis the saturation magnetization and 'and\u0012\ndenote the spherical coordinates of the magnetization\nvectorM(Ms;';\u0012). Note that in the present case the\nspherical coordinate system is chosen such that the z\naxis coincides with the crystallographic caxis in the low-\ntemperature structure of CrCl 3. For a calculation of the\nresonance position under given experimental conditions,\ni.e., a speci\fc orientation of the external magnetic \feld\nrelative to the studied sample and a \fxed microwave fre-\nquency, Eq. (2) has to be evaluated at the equilibrium po-\nsition ('0;\u00120) of the macroscopic magnetization vector.\nThe orientation of Min the equilibrium state is found\nnumerically by minimizing Fwith respect to the spheri-\ncal coordinates, taking into account the particular exper-\nimental conditions. Since the qualitative considerations\nregarding the temperature dependence of the resonance\nshift as well as the \u0017(Hres) dependencies at 4 K suggest\nan easy-plane type anisotropy, a uniaxial magnetocrys-\ntalline anisotropy was included in the free energy density\nterm as an initial approach to describe the anisotropies\nin CrCl 3. The free energy density is then given (in SI\nunits) by\nF=\u0000\u00160H\u0001M\u0000KUcos2(\u0012)+\n1\n2\u00160M2\ns(Nxsin2(\u0012) cos2(')+\nNysin2(\u0012) sin2(') +Nzcos2(\u0012)):(3)\nThe \frst term is the Zeeman-energy density describing\nthe coupling between the magnetization vector Mand\nthe external magnetic \feld H. The second term repre-\nsents the uniaxial magnetocrystalline anisotropy whose\nstrength is parametrized by the energy density KU. Fi-\nnally, the third contribution to Fis the shape anisotropy\nenergy density which is characterized by the demagneti-\nzation factors Nx,Ny, andNz[40, 41]. These factors are\ndetermined by the dimensions of the crystals under study.\nIn the present case, the shape of the measured platelet-\nlike CrCl 3crystal was described by the demagnetization\nfactors of an extended \rat plate ( Nx=Ny= 0;Nz= 1\n[42]), whose xyplane corresponds to the crystallographic\nabplane and the zaxis lies parallel to the caxis. This\napproximation to the real sample shape is justi\fed by\nthe platelet-like shape of the studied CrCl 3crystal whose\nlateral dimensions in the abplane (of about 400 \u0016m\u0002\n450\u0016m) are much larger than the thickness of the crys-\ntal along the caxis. Due to experimental reasons, the\nsample thickness could not be measured precisely which\nhampered a determination of the demagnetization factors\nsolely based on the sample dimensions. However, the de-\nviations between the approximated and the true demag-\nnetization factors can be expected to be small. In addi-\ntion to the sample shape, the value of the saturation mag-\nnetizationMsenters into the simulation of the frequency-\n\feld dependence. For the simulations presented in thefollowing, the reported saturation magnetization of about\n3\u0016B/Cr at 1.8 K [16] was used which corresponds to\nMs\u0019314:97\u0002103J/Tm3. Furthermore, the gfactors\ndetermined independently from the frequency-dependent\nmeasurements at 100 K and for both \feld orientations\n(see Sec. III B) were set as \fxed parameters in the sim-\nulations. Thus, the only free parameter, which was ad-\njusted to match simulated with measured data, was the\nmagnetocrystalline anisotropy energy density KU.\nThe \fnal results of the simulations are presented in\nFig. 3(b) as solid lines and show an excellent agree-\nment between the simulated and the measured resonance\npositions. Most importantly, this agreement could be\naccomplished by solely taking into account the shape\nanisotropy, i.e., by setting KUto zero. Thus, it is ulti-\nmately proven that the anisotropy observed in dynamic\nand static magnetic investigations of CrCl 3[11, 16, 23,\n26] is, indeed, due to the shape anisotropy caused by\nlong-range dipole-dipole interactions, whereas the intrin-\nsic magnetocrystalline anisotropy can be neglected. The\npresent work therefore could verify by means of highly\nsensitive HF magnetic resonance investigations the con-\nclusions of these previous studies [11, 16, 23, 26] and\nsupports the results of the recent \frst-principle calcu-\nlations of the magnetic anisotropy of CrCl 3monolayers\n[43]. It can be concluded that, intrinsically, CrCl 3is\nmagnetically isotropic while the apparent anisotropy ob-\nserved in experiments can be tuned (within certain lim-\nits) by choosing a particular sample shape. Moreover,\nCrCl 3might serve as a valuable reference system for fu-\nture magnetic resonance studies of other van der Waals\nmagnets, since it illustrates the pure e\u000bect of the shape\nanisotropy on, e.g., the frequency-\feld dependence. As\nsimilar sample shapes can be expected for this large fam-\nily of layered materials, it follows that it is very important\nto take into account this source of magnetic anisotropy\nwhen aiming at a precise quanti\fcation of magnetocrys-\ntalline anisotropies in van der Waals magnets.\nIV. CONCLUSIONS\nIn summary, we studied the details of the mag-\nnetic anisotropies in the van der Waals magnet CrCl 3\nby means of systematic HF ESR and FMR measure-\nments. By extending the frequency range of previ-\nous works [23, 31], the \feld-polarized, e\u000bectively ferro-\nmagnetic low-temperature state of the title compound\nwas investigated. Numerical simulations of the mea-\nsured frequency-\feld dependence at 4 K revealed that\nthe anisotropy observed experimentally is governed by\nthe shape anisotropy of the studied CrCl 3crystal. In\ncontrast, the intrinsic magnetocrystalline anisotropy is\nfound to be negligible in this compound, thus supporting\nthe conclusions drawn in previous studies [11, 16, 23, 26].\nConsidering the large current scienti\fc interest in mag-\nnetic van der Waals materials, CrCl 3may serve as a\nreference for future quantitative analyses of magnetic7\nanisotropies in related layered magnets, since it provides\nan excellent example for the impact of the particular sam-\nple shape on the apparent magnetic anisotropy. Finally,\ntemperature-dependent measurements of the resonance\nshift showed the onset of a \fnite shift and, thus, the\nexistence of short-range spin correlations already at tem-\nperatures well above the magnetically ordered state. This\nobservation provides further evidence for the intrinsically\n2D nature of the magnetism in the van der Waals magnet\nCrCl 3.\nConsidering a continuously increasing number of quasi-\n2D van der Waals compounds exhibiting a rich variety\nof intriguing magnetic properties, it is appealing to ap-\nply systematically the frequency-tunable high-\feld ESR\nspectroscopy to investigate the spin dynamics and in par-\nticular the magnetic anisotropy of these materials. The\nlatter appears to be a key factor for determining the type\nof a magnetically ordered ground state. For example,\nthe members of the family of the van der Waals layered\nmetal phosphorous trichalcogenides MPS 3(M = Mn, Fe,\nNi) [44] feature di\u000berent types of magnetic anisotropy\n[45] and exhibit dissimilar antiferromagnetically ordered\nground states, as, e.g., was illustrated in Ref. [46]. Quan-\nti\fcation of the parameters of magnetic anisotropy with\nESR spectroscopy may shed more light on a possible rela-\ntionship between the anistropy and the type of magnetic\norder in these compounds.\nACKNOWLEDGMENTS\nThis work was \fnancially supported by the Deutsche\nForschungsgemeinschaft (DFG) within the Collabora-\ntive Research Center SFB 1143 \\Correlated Magnetism\nFrom Frustration to Topology (project-id 247310070)\nand the Dresden-Wrzburg Cluster of Excellence (EXC\n2147) \\ct.qmat - Complexity and Topology in Quantum\nMatter\" (project-id 39085490). K. M. acknowledges the\nHallwachsRntgen Postdoc Program of ct.qmat for \fnan-\ncial support. A. A. acknowledges \fnancial support by\nthe DFG through Grant No. AL 1771/4-1.\nAppendix: Details on the sample characterization\nA typical x-ray di\u000braction (XRD) pattern of a powder\nsample of CrCl 3is shown in Fig. 4. It was collected\nusing a PANalytical XPert Pro MPD di\u000bractometer with\nCu-K\u000b1radiation (\u0015= 1:54056 \u0017A) in the BraggBrentano\ngeometry at room temperature in the 2 \u0012range between\n10 and 90\u000e. The CrCl 3sample displays strongly preferred\norientation leading to only (00 l) peaks visible.\nA typical scanning electron microscopy (SEM) image\nof CrCl 3crystallites is shown in Fig. 5. It was made with\na Hitachi SU8020 microscope equipped with an Oxford\nSilicon Drift X-MaxN energy dispersive x-ray spectrome-ter (EDX) at an acceleration voltage of 20 kV. Results of\nthe EDX analysis at selected areas indicated in Fig. 5 are\nIntensity (arb. units)\n2q (°)\nFIG. 4. XRD powder pattern of a sample of CrCl 3. Red ticks\ndisplay the peak positions of the reference CrCl 3from the\nInorganic Crystal Structure Database (22080-ICSD, SCXRD,\n298 K). Note that only (00 l) peaks are visible due to the strong\ntexturing of the powder sample.\nSpectrumSpectrum\nSpectrum\nSpectrum\nFIG. 5. SEM image of CrCl 3crystallites. Rectangles indicate\nthe regions where EDX spectra were collected.\npresented in Table I. They evidence a very homogeneous,\nclose to the ideal composition Cr : Cl = 1 : 3 of the single\ncrystal.\nTABLE I. Results of the analysis of the EDX spectra of the\nCrCl 3crystal taken at selected areas indicated in Fig. 5.\nSpectrum 826 827 828 829 Calculated\nCl (%) 74.77 74.69 74.49 74.87 75\nCr (%) 25.23 25.31 25.51 25.13 258\n1J.-G. Park, \\Opportunities and challenges of 2D magnetic\nvan der Waals materials: magnetic graphene?\" J. Phys.:\nCondens. Matter 28, 301001 (2016).\n2Cheng Gong, Lin Li, Zhenglu Li, Huiwen Ji, Alex Stern,\nYang Xia, Ting Cao, Wei Bao, Chenzhe Wang, Yuan\nWang, Z. Q. Qiu, R. J. Cava, Steven G. Louie, Jing Xia,\nand Xiang Zhang, \\Discovery of intrinsic ferromagnetism\nin two-dimensional van der Waals crystals,\" Nature 546,\n265 (2017).\n3Bevin Huang, Genevieve Clark, Efr\u0013 en Navarro-Moratalla,\nDahlia R. Klein, Ran Cheng, Kyle L. Seyler, Ding Zhong,\nEmma Schmidgall, Michael A. McGuire, David H. Cobden,\nWang Yao, Di Xiao, Pablo Jarillo-Herrero, and Xiaodong\nXu, \\Layer-dependent ferromagnetism in a van der Waals\ncrystal down to the monolayer limit,\" Nature 546, 270\n(2017).\n4Kenneth S. Burch, David Mandrus, and Je-Geun Park,\n\\Magnetism in two-dimensional van der Waals materials,\"\nNature 563, 47 { 52 (2018).\n5M. Gibertini, M. Koperski, A. F. Morpurgo, and K. S.\nNovoselov, \\Magnetic 2D materials and heterostructures,\"\nNat. Nanotechnol. 14, 408 { 419 (2019).\n6Cheng Gong and Xiang Zhang, \\Two-dimensional mag-\nnetic crystals and emergent heterostructure devices,\" Sci-\nence363, 706 (2019).\n7A. K. Geim and I. V. Grigorieva, \\Van der Waals het-\nerostructures,\" Nature 499, 419 (2013).\n8J. Zeisner, A. Alfonsov, S. Selter, S. Aswartham, M. P.\nGhimire, M. Richter, J. van den Brink, B. B uchner, and\nV. Kataev, \\Magnetic anisotropy and spin-polarized two-\ndimensional electron gas in the van der Waals ferromagnet\nCr2Ge2Te6,\" Phys. Rev. B 99, 165109 (2019).\n9Bruno Morosin and Albert Narath, \\XRay Di\u000braction\nand Nuclear Quadrupole Resonance Studies of Chromium\nTrichloride,\" J. Chem. Phys. 40, 1958{1967 (1964).\n10J.W. Cable, M.K. Wilkinson, and E.O. Wollan, \\Neutron\ndi\u000braction investigation of antiferromagnetism in CrCl 3,\"\nJ. Phys. Chem. Solids 19, 29 { 34 (1961).\n11Michael A. McGuire, Genevieve Clark, Santosh KC,\nW. Michael Chance, Gerald E. Jellison, Valentino R.\nCooper, Xiaodong Xu, and Brian C. Sales, \\Magnetic be-\nhavior and spin-lattice coupling in cleavable van der Waals\nlayered CrCl 3crystals,\" Phys. Rev. Materials 1, 014001\n(2017).\n12E. Zavoisky, \\Spin magnetic resonance in the decimetre-\nwave region,\" Journal of Physics USSR 10, 197 (1946).\n13B. I. Kochelaev and Y. V. Yablokov, The Beginning of\nParamagnetic Resonance (World Scienti\fc, 1995).\n14A. Glamazda, P. Lemmens, S.-H. Do, Y. S. Kwon, and K.-\nY. Choi, \\Relation between Kitaev magnetism and struc-\nture in\u000b\u0000RuCl 3,\" Phys. Rev. B 95, 174429 (2017).\n15Omar Besbes, Sergey Nikolaev, Noureddine Meskini, and\nIgor Solovyev, \\Microscopic origin of ferromagnetism in\nthe trihalides CrCl 3and CrI 3,\" Phys. Rev. B 99, 104432\n(2019).\n16G. Bastien, M. Roslova, M. H. Haghighi, K. Mehlawat,\nJ. Hunger, A. Isaeva, T. Doert, M. Vojta, B. B uchner, and\nA. U. B. Wolter, \\Spin-glass state and reversed magnetic\nanisotropy induced by Cr doping in the Kitaev magnet\n\u000b\u0000RuCl 3,\" Phys. Rev. B 99, 214410 (2019).\n17Mykola Abramchuk, Samantha Jaszewski, Kenneth R.Metz, Gavin B. Osterhoudt, Yiping Wang, Kenneth S.\nBurch, and Fazel Tafti, \\Controlling Magnetic and Op-\ntical Properties of the van der Waals Crystal CrCl 3\u0000xBrx\nvia Mixed Halide Chemistry,\" Adv. Mater. 30, 1801325\n(2018).\n18Wei-Bing Zhang, Qian Qu, Peng Zhu, and Chi-Hang Lam,\n\\Robust intrinsic ferromagnetism and half semiconductiv-\nity in stable two-dimensional single-layer chromium tri-\nhalides,\" J. Mater. Chem. C 3, 12457{12468 (2015).\n19Lucas Webster and Jia-An Yan, \\Strain-tunable magnetic\nanisotropy in monolayer CrCl 3, CrBr 3, and CrI 3,\" Phys.\nRev. B 98, 144411 (2018).\n20Xinghan Cai, Tiancheng Song, Nathan P. Wilson,\nGenevieve Clark, Minhao He, Xiaoou Zhang, Takashi\nTaniguchi, Kenji Watanabe, Wang Yao, Di Xiao,\nMichael A. McGuire, David H. Cobden, and Xiaodong Xu,\n\\Atomically Thin CrCl 3: An In-Plane Layered Antiferro-\nmagnetic Insulator,\" Nano Lett. 19, 3993{3998 (2019).\n21N. Bykovetz, A. Hoser, and C. L. Lin, \\Critical region\nphase transitions in the quasi-2D magnet CrCl 3,\" AIP\nAdv. 9, 035029 (2019).\n22Dahlia R. Klein, David MacNeill, Qian Song, Daniel T.\nLarson, Shiang Fang, Mingyu Xu, R. A. Ribeiro, P. C.\nCan\feld, Efthimios Kaxiras, Riccardo Comin, and Pablo\nJarillo-Herrero, \\Enhancement of interlayer exchange in an\nultrathin two-dimensional magnet,\" Nat. Phys. 15, 1255{\n1260 (2019).\n23David MacNeill, Justin T. Hou, Dahlia R. Klein, Pengxi-\nang Zhang, Pablo Jarillo-Herrero, and Luqiao Liu, \\Giga-\nhertz Frequency Antiferromagnetic Resonance and Strong\nMagnon-Magnon Coupling in the Layered Crystal CrCl 3,\"\nPhys. Rev. Lett. 123, 047204 (2019).\n24Wilford N. Hansen and Maurice Gri\u000bel, \\Heat Capacities\nof CrF 3and CrCl 3from 15 to 300\u000eK,\" J. Chem. Phys.\n28, 902{907 (1958).\n25Wilford N. Hansen and Maurice Gri\u000bel, \\Paramagnetic\nSusceptibilities of the Chromium (III) Halides,\" J. Chem.\nPhys. 30, 913{916 (1959).\n26H. Bizette, C. Terrier, and A. Adam, \\Aimantations prin-\ncipales du chlorure chromique,\" C. R. Acad. Sci. 252, 1571\n(1961).\n27Albert Narath and H. L. Davis, \\Spin-Wave Analysis of\nthe Sublattice Magnetization Behavior of Antiferromag-\nnetic and Ferromagnetic CrCl 3,\" Phys. Rev. 137, A163 {\nA178 (1965).\n28B. Kuhlow, \\Magnetic Ordering in CrCl 3at the Phase\nTransition,\" Phys. Status Solidi (a) 72, 161{168 (1982).\n29Maria Roslova, Jens Hunger, Ga el Bastien, Darius Pohl,\nHossein M. Haghighi, Anja U. B. Wolter, Anna Isaeva, Ul-\nrich Schwarz, Bernd Rellinghaus, Kornelius Nielsch, Bernd\nB uchner, and Thomas Doert, \\Detuning the Honeycomb\nof the\u000b-RuCl 3Kitaev Lattice: A Case of Cr3+Dopant,\"\nInorg. Chem. 58, 6659{6668 (2019).\n30C. Starr, F. Bitter, and A. R. Kaufmann, \\The Magnetic\nProperties of the Iron Group Anhydrous Chlorides at Low\nTemperatures. I. Experimental,\" Phys. Rev. 58, 977 { 983\n(1940).\n31S. Chehab, J. Amiell, P. Biensan, and S. Flandrois, \\Two-\ndimensional ESR behaviour of CrCl 3,\" Physica B: Con-\ndensed Matter 173, 211 { 216 (1991).\n32A. Abragam and B. Bleaney, Electron Paramagnetic Reso-9\nnance of Transition Ions (Oxford University Press, Oxford,\n2012).\n33E. A. Turov, Physical Properties of Magnetically Ordered\nCrystals , edited by A. Tybulewicz and S. Chomet (Aca-\ndemic Press, New York, 1965).\n34Michael Farle, \\Ferromagnetic resonance of ultrathin\nmetallic layers,\" Rep. Prog. Phys. 61, 755 (1998).\n35Kazukiyo Nagata and Yuichi Tazuke, \\Short Range Order\nE\u000bects on EPR Frequencies in Heisenberg Linear Chain\nAntiferromagnets,\" J. Phys. Soc. Jpn. 32, 337{345 (1972).\n36Kiichi Okuda, Hirao Hata, and Muneyuki Date, \\Elec-\ntron Spin Resonance in One Dimensional Antiferromagnet\nCu(C 6H5COO) 23H2O,\" J. Phys. Soc. Jpn. 33, 1574{1580\n(1972).\n37Kokichi Oshima, Kiichi Okuda, and Muneyuki Date, \\g-\nShift in Low Dimensional Antiferromagnets at Low Tem-\nperatures,\" Journal of the Physical Society of Japan 41,\n475{480 (1976).\n38G. V. Skrotskii and L. V. Kurbatov, \\Phenomenological\nTheory of Ferromagnetic Resonance,\" in Ferromagnetic\nResonance , edited by S. V. Vonsovskii (Pergamon Press\nLtd., Oxford, 1966) Chap. Phenomenological Theory of\nFerromagnetic Resonance, pp. 12{77.\n39J. Smit and H. G. Beljers, \\Ferromagnetic Resonance Ab-\nsorption in BaFe 12O19,\" Philips Research Reports 10, 113(1955).\n40J. A. Osborn, \\Demagnetizing Factors of the General El-\nlipsoid,\" Phys. Rev. 67, 351 (1945).\n41D. C. Cronemeyer, \\Demagnetization factors for general\nelipsoids,\" J. Appl. Phys. 70, 2911 (1991).\n42Stephen Blundell, Magnetism in Condensed Matter (Ox-\nford University Press, Oxford, 2001).\n43Feng Xue, Yusheng Hou, Zhe Wang, and Ruqian Wu,\n\\Two-dimensional ferromagnetic van der Waals CrCl 3\nmonolayer with enhanced anisotropy and Curie temper-\nature,\" Phys. Rev. B 100, 224429 (2019).\n44Fengmei Wang, To\fk A. Shifa, Peng Yu, Peng He, Yang\nLiu, Feng Wang, Zhenxing Wang, Xueying Zhan, Xiaod-\ning Lou, Fan Xia, and Jun He, \\New Frontiers on van\nder Waals Layered Metal Phosphorous Trichalcogenides,\"\nAdv. Funct. Mater. 28, 1802151 (2018).\n45P. A. Joy and S. Vasudevan, \\Magnetism in the layered\ntransition-metal thiophosphates MPS 3(M=Mn, Fe, and\nNi),\" Phys. Rev. B 46, 5425{5433 (1992).\n46Hao Chu, Chang Jae Roh, Joshua O. Island, Chen Li,\nSungmin Lee, Jingjing Chen, Je-Geun Park, Andrea F.\nYoung, Jong Seok Lee, and David Hsieh, \\Linear Mag-\nnetoelectric Phase in Ultrathin MnPS 3Probed by Opti-\ncal Second Harmonic Generation,\" Phys. Rev. Lett. 124,\n027601 (2020)."    },    {        "title": "2006.02118v2.Superconductivity_induced_change_in_magnetic_anisotropy_in_epitaxial_ferromagnet_superconductor_hybrids_with_spin_orbit_interaction.pdf",        "content": "Superconductivity-induced change in magnetic anisotropy in epitaxial\nferromagnet-superconductor hybrids with spin-orbit interaction\nC\u0013 esar Gonz\u0013 alez-Ruano,1Lina G. Johnsen,2Diego Caso,1Coriolan Tiusan,3, 4\nMichel Hehn,4Niladri Banerjee,5Jacob Linder,2and Farkhad G. Aliev1,\u0003\n1Departamento F\u0013 \u0010sica de la Materia Condensada C-III,\nInstituto Nicol\u0013 as Cabrera (INC) and Condensed Matter Physics Institute (IFIMAC),\nUniversidad Aut\u0013 onoma de Madrid, Madrid 28049, Spain\n2Center for Quantum Spintronics, Department of Physics,\nNorwegian University of Science and Technology, NO-7491 Trondheim, Norway\n3Department of Physics and Chemistry, Center of Superconductivity Spintronics and Surface Science C4S,\nTechnical University of Cluj-Napoca, Cluj-Napoca, 400114, Romania\n4Institut Jean Lamour, Nancy Universit\u0012 e, 54506 Vandoeuvre-les-Nancy Cedex, France\n5Department of Physics, Loughborough University,\nEpinal Way, Loughborough, LE11 3TU, United Kingdom\nAbstract\nThe interaction between superconductivity and ferromagnetism in thin \flm superconductor/ferromagnet heterostructures\nis usually re\rected by a change in superconductivity of the S layer set by the magnetic state of the F layers. Here we\nreport the converse e\u000bect: transformation of the magnetocrystalline anisotropy of a single Fe(001) layer, and thus its preferred\nmagnetization orientation, driven by the superconductivity of an underlying V layer through a spin-orbit coupled MgO interface.\nWe attribute this to an additional contribution to the free energy of the ferromagnet arising from the controlled generation of\ntriplet Cooper pairs, which depends on the relative angle between the exchange \feld of the ferromagnet and the spin-orbit \feld.\nThis is fundamentally di\u000berent from the commonly observed magnetic domain modi\fcation by Meissner screening or domain\nwall-vortex interaction and o\u000bers the ability to fundamentally tune magnetic anisotropies using superconductivity - a key step\nin designing future cryogenic magnetic memories.\nSuperconductivity (S) is usually suppressed in the\npresence of ferromagnetism (F) [1{5]. For example, in\nF/S/F spin-valves the transition temperature TCof the\nS layer is di\u000berent for a parallel alignment of the F layer\nmoments compared to an anti-parallel alignment [6{9].\nInterestingly, for non-collinear alignment of the F layer\nmoments in spin-valves [10{12] or Josephson junctions\n[13{22], an enhancement in the proximity e\u000bect is found\ndue to the generation of long-range triplet Cooper pairs,\nimmune to the pair-breaking exchange \feld in the F lay-\ners. So far, the reciprocal modi\fcation of the static prop-\nerties of the ferromagnet by superconductivity has been\nlimited to restructuring [23] and pinning of magnetic do-\nmains walls (DWs) by Meissner screening and vortex-\nmediated pinning of DWs [24{27].\nModi\fcation of the magnetization dynamics in the\npresence of superconductivity has been studied in [28{\n36]. Recently, theoretical and experimental results have\nindicated an underlying role of Rashba spin-orbit cou-\npling (SOC), resulting in an enhancement of the prox-\nimity e\u000bect and a reduction of the superconducting TC,\nalong with enhanced spin pumping and Josephson cur-\nrent in systems with a single F layer coupled to Nb\nthrough a heavy-metal (Pt) [37{43]. In this context,\nV/MgO/Fe [44] has been shown to be an e\u000bective sys-\ntem to study the e\u000bect of SOC in S/F structures with\nfully epitaxial layers.\nAt \frst glance, altering the magnetic order in S/F\nheterostructures leading to a change in the direction ofmagnetization appears non-trivial due to the di\u000berence\nin the energy scales associated with the order parame-\nters. The exchange splitting of the spin-bands and the\nsuperconducting gap are about 103K and 101K, re-\nspectively. However, this fundamentally changes if one\nconsiders the possibility of controlling the magneto crys-\ntalline anisotropy (MCA) by manipulating the compet-\ning anisotropy landscape with superconductivity, since\nthe MCA energy scales are comparable to the supercon-\nducting gap energy. Interestingly, emergent triplet su-\nperconducting phases in S/SOC/F heterostructures o\u000ber\nthe possibility to observe MCA modi\fcation of a F layer\ncoupled to a superconductor through a spin-orbit coupled\ninterface, triggered by the superconducting phase [45].\nIn this communication, we present evidence that cubic\nin-plane MCA in V/MgO/Fe(001) system is modi\fed by\nthe superconductivity of V through SOC at the MgO/Fe\ninterface [46]. Our detailed characterization of the coer-\ncive \felds of the rotated soft Fe(001) and sensing hard\n(Fe/Co) ferromagnetic layers by tunnelling magnetoresis-\ntance e\u000bect (TMR) [47] along with numerical simulations\ndismisses the Meissner screening and DW-vortex interac-\ntions as a source of the observed e\u000bects.\nThe magnetic tunnel junction (MTJ) multilayer stacks\nhave been grown by molecular beam epitaxy (MBE) in a\nchamber with a base pressure of 5 \u000210\u000011mbar following\nthe procedure described in [48]. The samples were grown\non [001] MgO substrates. Then a 10 nm thick seed of\nanti-di\u000busion MgO underlayer is grown on the substrate\nTypeset by REVT EX 1arXiv:2006.02118v2  [cond-mat.supr-con]  26 Jun 2020(a)\nV\n+-\nCr (2nm)MgO (2nm)z\nθ\nFMFM\nFMFM\n0π/2π3π/2 2πT=10 K\nFMFMFMFM\nFMFMFMFM\nFMFM\nH[100] (kOe)TMR (%)[100][100]-\n[100][100][100]-\n-[010] [010]Fe[100]\nFe[010]\nKC\nΦH(b)\nTMR (%)\n-3 -2 -1 0 1 2 3010203040\n010203040\n(c) (d)\nT=10 KFIG. 1: (a) Sketch of the junctions under study. Fe(10\nnm)Co(20 nm) is the hard (sensing) layer while Fe(10 nm)\nis the soft ferromagnet where spin reorientation transitions\nare investigated. (b), Sketch showing the top view without\nthe hard Fe/Co layer, with the 4-fold in plane magnetic en-\nergy anisotropy expected for the Fe(001) atomic plane of the\nmagnetically free layer, for temperatures above TC(yellow\nline) and well below TC(dashed cyan). Note that during\nthe epitaxial growth, the Fe lattice is rotated by 45 degress\nwith respect to MgO. Parts (c) and (d) show in-plane spin re-\norientation transitions between parallel (P), perpendicular in\nplane (PIP) and antiparallel (AP) relative magnetization al-\nlignments of the soft and hard F layers for a 30 \u000230\u0016m2junc-\ntion at T=10 K (above TC). Indices above the inset sketches\nindicate the direction of the soft layer. The in-plane rotation\nhas been carried out with the angle \b Hof the magnetic \feld\nrelative to the Fe[100] axis going from \u000030 to 390 degrees.\nto trap the C from it before the deposition of the Fe (or\nV). Then the MgO insulating layer is epitaxially grown\nby e-beam evaporation, the thickness approximately \u00182\nnm and so on with the rest of the layers. Each layer is an-\nnealed at 450 C for 20 mins for \rattening. After the MBE\ngrowth, all the MTJ multilayer stacks are patterned in\n10-40 micrometre-sized square junctions (with diagonal\nalong [100]) by UV lithography and Ar ion etching, con-\ntrolled step-by-step in situ by Auger spectroscopy. The\nmeasurements are performed inside a JANISR\rHe3cryo-\nstat. The magnetic \feld is varied using a 3D vector mag-\nnet. For the in-plane rotations, the magnetic \feld mag-\nnitude was kept at 70-120 Oe, far away from the soft\nFe(001) and hard Fe/Co layers switching \felds obtained\nfrom in-plane TMRs (see Supplemental Material S1,S2\n[49]). This way, only the soft layer is rotated and the\ndi\u000berence in resistance can be atributted to the angle\nbetween the soft and hard layers.\nFigure 1a shows the device structure with the Fe/Cohard layer sensing the magnetization alignment of the 10-\nnm thick Fe(001) soft layer. A typical TMR plot above\nTCis shown in Figure 1c. The resistance switching shows\na standard TMR between the P and AP states. However,\nthe epitaxial Fe(001) has a four-fold in-plane anisotropy\nwith two ortogonal easy axes - [100] and [010] - (Fig-\nure 1b). These MCA states could be accessed by an\nin-plane rotation of the Fe(001) layer with respect to the\nFe/Co layer using \feld greater than the coercive \feld of\nthe Fe(001) layer without disrupting the Fe/Co magne-\ntization (see also Supplemental material S1 [49] for the\nmagnetic characterization of the Fe/Co layer). This is\nshown in Figure 1d, where TMR is plotted as a function\nof the in-plane \feld angle with respect to the [100] direc-\ntion angle \b H. This gives rise to four distinct magnetiza-\ntion states with P, perpendicular in-plane (PIP) and AP\nstates re\rected by the TMR values. Supplemental Ma-\nterial S3 [49] discusses the weak magnetostatic coupling\nbetween the two FM layers (detected through resistances\nin-between the P and AP states in the virgin state of\ndi\u000berent samples), showing that it does not a\u000bect the ca-\npability to reorient the soft layer independently of the\nhard one. It also demonstrates that the soft layer retains\ndi\u000berent magnetic directions at zero \feld.\nFigure 2 analyzes the most probable in-plane magne-\ntization orientations of the Fe(001) layer through mag-\nnetic \feld rotations at \fxed temperatures from above to\nbelowTC. Typically, no qualitative changes in TMR are\nobserved above and below TCin the 0\u0000\u0019\feld rotation\nangle (\bH) span (Figure 2a). However, in the \u0019\u00002\u0019\nrange, the TMR qualitatively changes below TC=2, pos-\nsibly indicating new stable magnetization states along\ndi\u000berent directions to the ones stablished by the princi-\npal crystallographic axes (Figure 2a).\nTo ascertain the exact angle \b FMbetween the two\nF layers, we have calibrated the magnetization direc-\ntion of the soft layer with respect to the hard Fe/Co us-\ning the Slonczewski formula (Supplemental Material S4\n[49]). The aplicability of the macrospin approach to de-\nscribe TMRs and magnetization reorientation resides in\nthe high e\u000bective spin polarization obtained ( P= 0:7)\n[47], approaching to the values typically reported for\nFe/MgO in a fully saturated state [50, 51]. Figure 2b\nis a histogram representing the probability of obtaining\na speci\fc \b FMas temperature is lowered from above to\nbelowTC. We observe that the most probable Fe(001)\ndirections are oriented along the [100] and [010] princi-\npal axes above TC=2, while below TC=2 it splits in three\nbranches roughly oriented along \u0019=4 angles. The split of\nthe [0 10] state into three branches is also visualized in\nFig.2e, with a plot of the counts vs. temperature around\nthe [110];[010] and [1 10] magnetization directions.\nInterestingly, once the rotation is initiated in the AP\ncon\fguration, the magnetization apparently locks in the\n(\u0019+\u0019=4) (or [ 110]) state (Figures 2b,d,f). This probably\narises due to the improved initial macrospin alignment,\n20 π π/2 3π/2 2π3π/2 5π/4 7π/4 2π π\n3π/2 5π/4 7π/4 2π π \nΦFM(a) (d)\n 5 K\n 4 K\n 3 K\n 2.3 K\n 2.2 K\n 2 K\n 1.5 K\n 1.2 K\n 0.8 K\n 0.6 K\n 0.3 K\n 0.25 K03640444852\n  \n 0.3 K\n 1.2 K\n 2.2 K\n 3 K\n 4 K\nΦHR (kΩ)\n04080120160Nº of points\n0.3 K0.5 K0.7 K1 K1.2 K1.6 K2.1 K2.5 K3 K3.9 K4 K4.5 K5.5 K6.5 K90100110120130\n  0.3 K\n 1.2 K\n 2.5 K\n 4.5 K\n150\n100\n50\n0(b)π π/2 3π/2 2π\n ΦH\nΦFMNº of points R (kΩ)\nHMHL\nMSLΦFMΦH[100]\n0 1 2 3 4 5051015202530\nT (K)Total countTC[110] (5π/4)\n[110] (7π/4)[010] (3π/2)\n0 1 2 3 4 5 6 7010203040\nTC[110] (5π/4)\n[110] (7π/4)[010] (3π/2)Total count\nT (K)(e)\n(c) (f)[100] [010] [100] [010] [100] [100] [110] [010] [110] [100]FIG. 2: Typical angular dependence of the resistance of a\nV/MgO/Fe/MgO/Fe/Co junction on the orientation of the\nin plane \feld with respect to the main crystalline axes from\nabove to below TCwhen the rotation is initialted from P (a-c)\nand from AP state (d-f). The inset sketches the experimental\ncon\fguration, showing the angles between the ferromagnetic\nlayers (\b FM) and of the external magnetic \feld (\b H). Parts\n(b,e) correspondingly represent the experimental data shown\nin (a,d) in form of histograms, dividing the 0-2 \u0019interval in\n36 zones. Parts (c,f) plot the histograms in (b,e) as counts\nvs temperature for the intermediate states (AP+ \u0019=4 or the\n[110] axis, AP+ \u0019=2 =PIP or [010], and AP+3 \u0019=4 or [110])\nfor the second half of the rotation.\nwhich is not fully achieved in the AP state with a pre-\nceeding P-AP rotation. We believe that with the full\n2\u0019\feld rotation, magnetization inhomogeneities or local\nDWs created during the P-AP state rotation help to over-\ncome MCA energy barriers more easily. The suggested\nsuppresion of the local DWs with the magnetization rota-\ntion initated from the AP state can be indirectly inferred\nfrom the broadening of the [ 100] to [0 10] transition in the\nnormal state detected as a small (extrinsic) number of\ncounts around [ 110] (Figure 2f).\nFor a more systematic analysis, we performed a series\nof in-plane TMR measurements along di\u000berent directions\nrelative to the symmetry axes. The \frst experiment (i)\nwas performed with an initial saturation \feld of \u00061 kOe\nin the [100] direction, followed by a TMR in the [210]\ndirection (between [100] and [110]). The second (ii) ini-\ntially saturates both the hard and soft layers along the\n[100] direction. Then, a minor loop is performed starting\n \n-400 -200 0 200 400\nH (Oe)0 1 2 3 4 5 6050100150200250300\nπ/8 switch field (Oe)\n Negative field\n Positive fieldHSwitch (Oe)(a)\n51.251.652.052.452.853.2\n  \n-1.0 -0.5 0.0 0.5 1.03540455055\n  \nH (kOe) 0.3 K\n 1.5 K\n 2.2 K\n 4 K\n0 50 100 150420440460480500520\nHπ/4 (Oe) 6 K\n 1.6 K\nP (θ=0)θ=π/4θ=π/2θ=3π/4R (kΩ)\nT (K)R (kΩ)\n1020406080100\nTransition Probability (%)Msat≈MSLMsat=MSL\nMSLMsat\n ε≈kBT ε≫kBT\n ε'≫kBTFe[100]\nFe[010]\nKCπ/8\nHTMR\n2 34 0.5R (kΩ)\nT (K)(b)\n(c) (d)FIG. 3: (a) TMR measurements on a S/F/F 30 \u000230\u0016m2\njunction with H oriented along [210] (inset in (b)), for various\ntemperatures. The increase in R is associated with a tran-\nsition from the [110] magnetization orientation to a forced\n[210] direction of the soft layer. (b), Variation of the transi-\ntion \feld with T for the positive and negative \feld branches.\nInset: exchange energy anisotropy and direction of the ap-\nplied \feld (H TMR). (c), Two TMRs performed on a 10 \u000210\n\u0016m2junction in the [110] direction at T=6 K and 1.6 K, after\napplying 1 kOe in the [100] direction. The 6 K TMR starts in\nP state, while the 1.6 K TMR starts already in a tilted state.\nRight axis: estimation of the angle \u0012between the two F layers\nbased on the Slonczewski formula. (d), Probability of \fnding\na tilted state at H= 0 (triangular points in (b)) vs T (in log\nscale), averaged with 7 experimental points around each T.\nThe line is a guide for the eye. Insets: sketch of the magnetic\nanisotropy below and above TC, with the saturation magneti-\nzation (M sat) and the zero \feld magnetization state measured\nfor the soft layer (M SL).\"and\"0represent the energy barrier\nseparating the [100] magnetization direction from the closest\nminimum below and above TC, respectively.\nfrom zero \feld and going up to 150 Oe along the [110]\naxis.\nBoth experiments further suggest the possibility of\nsuperconductivity-induced changes of MCA. The inset\nof Figure 3a shows the full \feld sweep range in the \frst\n(i) con\fguration, and Figure 3a zooms in close to the AP\ncon\fguration. When we sweep the \feld in the [210] di-\nrection, we detect a weak but robust resistance upturn\nat temperatures below approximately TC=2 (Figure 3).\nThis additional TMR increase (shown by the arrows in\nFigure 3a) roughly corresponds to an 8-10 degree rota-\ntion in the relative spin direction between the soft and\nhard layer towards their AP alignment (see Supplemental\nMaterial S4 [49] for an analysys of the calculated angle\nerror). Within the proposed macrospin approximation,\nthis could be understood as a redirection of the soft layer\nmagnetization forced by the external \feld, from the ini-\ntially blocked [110] direction towards the external \feld\n[210] direction. A strong increase of the characteristic\n3\feld,Hswitch , required to reorient the soft layer from\n[110] towards [210] when T decreases below TC=2, could\nre\rect the superconductivity-induced MCA energy min-\nimum along the [110] direction.\nThe minor TMR loops along [110] (Figure 3c) realized\nafter saturation along [100] point on a thermally induced\nmagnetization reorientation from [100] towards [110] even\nat zero \feld, in a temperature range below TCwhere the\nbarrier between adjacent energy minima is comparable\ntokBT. The zero-\feld reorientation becomes less prob-\nable when the thermal energy is insu\u000ecient to overcome\nthe barrier (Figure 3d). An estimation of the in-plane\nnormal-state MCA energy barrier done through magne-\ntization saturation along [100] and [110] provides a value\nof only a few \u0016eV/atom (Supplemental Material S5 [49]).\nHowever, the real barrier is determined by the nucleation\nvolume, which depends on the exchange length in the ma-\nterial. With a DW width of about 3 nm for Fe(001) we\nestimate the MCA barrier to be at least 100\u0000101mV.\nBefore describing our explanation of the MCA modi-\n\fcation of Fe(001) in the superconducting state of V(40\nnm)/MgO(2 nm)/Fe(10 nm) system, we discard alter-\nnative interpretations of the observed e\u000bects. Meissner\nscreening [24, 25], if present, would introduce about a\n10% correction to the actual magnetic \feld independently\nof the external \feld direction (see Supplemental Material\nS2 [49]). The reason for the weak in-plane \feld screening\ncould be the small superconductor thickness (40 nm),\nonly slightly exceeding the estimated coherence length\n(26 nm). On the other hand, intermediate multidomain\nstates are expected to be absent in when magnetization\nis directed along [110] (Supplemental material S6 [49]).\nIndeed, our experiments show that magnetization, when\nlocked below TCin the (\u0019+\u0019=4) angle, hardly depends\non the absolute value of the external \feld along [110]\nvaried between 0 and 100 Oe. Moreover, simulations\nof the vortex-DW interaction using MuMax3 [52] and\nTDGL codes [59] discard the vortex mediated DW pin-\nning [26, 27] scenario including when interfacial magnetic\ndefects created by mis\ft dislocations [54] are considered\n(see Supplemental material S6 [49]). The vortex pinning\nmechanism also contradicts that only the (0 \u0000\u0019) \feld\nrotation span (Figure 2a) gets a\u000bected below TC=2. The\nobserved irrelevance of the junction area (Supplemental\nmaterial S7 [49]) contradics the importance of the vortex-\nedge DWs interaction. The shape and vortex-DWs in-\nteraction e\u000bects, if relevant, would strengthen magneti-\nzation pinning along [100], but not [110] (Supplemental\nmaterial S6, S7 [49]). Finally, we also indicate that the\nMCA modi\fcation from singlet superconductivity would\nnot enable any zero \feld rotation to non-collinear mis-\nalignment angles, in contrast to our data (Fig. 3d).\nTo explain our results, we consider the possibility\nin which the invariance of the superconducting prox-\nimity e\u000bect to magnetization rotation is broken in the\npresence of SOC. It has been predicted that triplet-\n0.0000.0050.0100.0150.0200.0250.030\nF-FAP\n TC+\n 0.8TC\n 0.7TC\n 0.6TC\n 0.5TC\n 0.4TC\nπ π/2 3π/2 π/4 3π/4 5π/4 7π/4 2π 0\nΦFM\nφFM\nCondensate TripletsMgO V\nSOC assisted\nconversionFe\nCondensate\nstrengthenedFewer triplets\nSOC assisted\nconversion weakened\nCondensation energy gain\nT > TcFe[100]\nFe[010]\nT < TcFe[100]\nFe[010]MSL\nMSL\nMgO V Fe(a)\n(b)FIG. 4: Numerical modelling. (a), Free energy Fvs in-plane\nmagnetization angle \b FMfor temperatures below the super-\nconducting critical temperature and just above the critical\ntemperature ( T+\nC). The free energy is plotted relative to the\nfree energy in the AP con\fguration FAPand has been nor-\nmalized to the hopping parameter tused in the tight-binding\nmodel. (b). Illustration of the physical origin of the change\nin magnetic anisotropy induced by the superconducting layer.\nAboveTC, V is a normal metal and the soft Fe layer has a 4-\nfold in-plane magnetic energy anisotropy (yellow line). Below\nTC, V is superconducting and in\ruences the soft Fe layer via\nthe proximity e\u000bect: a leakage of Cooper pairs into the ferro-\nmagnet. Due to the SOC at the interface, a magnetization-\norientation dependent generation of triplet Cooper pairs oc-\ncurs. The generation of triplets is at its weakest for a mag-\nnetization pointing in the [ 110] direction, giving a maximum\nfor the superconducting condensation energy gain. This mod-\ni\fes the magnetic anisotropy of the soft Fe layer (cyan line),\nenabling magnetization switching to the [ 110] direction (blue\narrow). The magnetic anisotropy does not show the weak\nAP coupling between the two Fe layers, causing an absolute\nminimum in \b FM=\u0019(a).\nsuperconductivity is e\u000bectively generated even for weakly\nspin-polarized ferromagnets with a small spin-orbit \feld\n[55]. In addition to generating triplet pairs, the SOC also\nintroduces an angle-dependent anisotropic depairing \feld\nfor the triplets [43, 45]. In V/MgO/Fe, the Rashba \feld is\ncaused by a structural broken inversion symmetry at the\n4MgO interfaces [44]. We model our experimental results\nusing a tight-binding Bogolioubov-de Gennes Hamilto-\nnian on a lattice and compute the free energy (Sup-\nplemental material S8 [49]). The Hamiltonian includes\nelectron hopping in and between the di\u000berent layers, a\nRashba-like SOC at the MgO/Fe interface, an exchange\nsplitting between spins in the Fe layers, and conventional\ns-wave superconductivity in the V layer. The free energy\ndetermined from this Hamiltonian includes the contribu-\ntion from the superconducting proximity e\u000bect, and an\ne\u000bective in-plane magnetocrystalline anisotropy favoring\nmagnetization along the [100] and [010] axes. Experi-\nmentally, we see a weak anti-ferromagnetic coupling be-\ntween the Fe(100) and Fe/Co layers (which does not af-\nfect the capability to reorient the soft layer independently\nof the hard one) described by an additional contribution\nfAFcos(\bFM) with a constant parameter fAF>0.\nFigure 4 shows the total free energy of the system as a\nfunction of the IP magnetization angle \b FMfor decreas-\ning temperatures. Due to the increase in the supercon-\nducting proximity e\u000bect, additional local minima appear\nat \bFM=n\u0019=2 +\u0019=4, wheren= 0;1;2;:::(i.e.[110],\n[\u0016110], [ \u00161\u001610], and [1 \u001610], respectively). This is a clear signa-\nture for the proximity-induced triplet correlations. These\nare most e\u000eciently generated at angles \b FM=n\u0019=2\n(i.e.[100], [010], [ \u0016100], and [0 \u001610]) for a heterostructure\nwith a magnetic layer that has a cubic crystal structure\nlike Fe [45]. As a result, the decrease in the free energy\nis stronger at angles \b FM=n\u0019=2 +\u0019=4 where more\nsinglet Cooper pairs survive. Our numerical results thus\ncon\frm that the experimentally observed modi\fcation of\nthe anisotropy can be explained by the presence of SOC\nin the S/F structure alone, without including supercon-\nducting proximity e\u000bects from misalignment between the\nFe(100) and Fe/Co layers. Moreover, Figure 4 illustrates\nwhy the \b FM=n\u0019=2 +\u0019=4 states only appear experi-\nmentally when the external \feld is rotated from an AP to\nP alignment (Figure 2). Because of the weak AP coupling\nbetween the ferromagnetic layers, the energy thresholds\nfor reorienting the magnetization from one local mini-\nmum to the next are higher under a rotation from AP to\nP alignment.\nIn conclusion, we present experimental evidence\nfor superconductivity-induced change in magnetic\nanisotropy in epitaxial ferromagnet-superconductor hy-\nbrids with spin-orbit interaction. We believe that this\nmechanism is fundamentally di\u000berent from the previous\nreports of magnetisation modi\fcation arising from Meiss-\nner screening and vortex induced domain wall pinning,\neven though the spin-triplet mechanism and performed\nsimulations require many assumptions. Our results es-\ntablish superconductors as tunable sources of magnetic\nanisotropies and active ingredients for future low dissipa-\ntion superspintronic technologies. Speci\fcally, they could\nprovide an opportunity to employ spin-orbit proximity\ne\u000bects in magnetic Josephson junction technology andappoach it to Fe/MgO-based junctions that are widely\nused in commercial spintronic applications.\nAcknowledgements\nWe acknowledge Mairbek Chshiev and Antonio Lara\nfor help with simulations, Yuan Lu for help in sample\npreparations and Igor Zutic and Alexandre Buzdin for\nthe discussions. The work in Madrid was supported\nby Spanish Ministerio de Ciencia (MAT2015-66000-P,\nRTI2018-095303-B-C55, EUIN2017-87474) and Conse-\njer\u0013 \u0010a de Educaci\u0013 on e Investigaci\u0013 on de la Comunidad\nde Madrid (NANOMAGCOST-CM Ref. P2018/NMT-\n4321) Grants. FGA acknowledges \fnancial support from\nthe Spanish Ministry of Science and Innovation, through\nthe Mara de Maeztu Programme for Units of Excel-\nlence inR&D(MDM-2014-0377, CEX2018-000805-M).\nNB was supported by EPSRC through the New In-\nvestigator Grant EP/S016430/1. The work in Nor-\nway was supported by the Research Council of Nor-\nway through its Centres of Excellence funding scheme\ngrant 262633 QuSpin. C.T. acknowledges \\EMERSPIN\"\ngrant ID PN-IIIP4-ID-PCE-2016-0143, No. UEFISCDI:\n22/12.07.2017. The work in Nancy was supported by\nCPER MatDS and the French PIA project \\Lorraine\nUniversit\u0013 e d'Excellence\", reference ANR-15-IDEX-04-\nLUE.\nSUPPLEMENTARY MATERIAL\nIn the supplementary material, the section S1 presents\na magnetic characterization of the hard Fe/Co layer of\nthe junctions under study. Section S2 presents a mag-\nnetic characterization of the soft Fe(001) layer and stud-\nies the possible in\ruence of the Meissner screening on\nthe coercive \felds of the soft and hard layers. Section S3\nestimates the strength of the weak antiferromagnetic cou-\npling between magnetically soft and hard electrodes. Sec-\ntion S4 provides details about the calibration of the angle\nbetween the soft and hard layers using the Slonczewski\nformula, as well as discussing the possible sources of error\nfor this calibration and their magnitude. Section S5 pro-\nvides an estimation for the magneto-anisotropic energy\nbarrier between the [110] and [100] magnetization direc-\ntions, normalized per volume or per atom. Section S6\nnumerically evaluates the possible domain walls pinning\nby superconducting vortices. Section S7 discusses the\ncontribution of the shape to the magnetic anisotropy. Fi-\nnally, section S8 provides details on the theoretical mod-\nelling of the observed e\u000bects.\n5S1. Magnetic characterizarion of the hard Fe/Co\nlayer\nFigure S5 shows the magnetic characterization of the\nhard Fe/Co bilayer, determined from a typical spin-\nvalve M-H loop on a standard Fe/MgO/Fe/Co single\ncrystal MTJ system (continuous layers, unpatterned).\nThe nominal thickness of the layers on this sample,\nMgO(100)/Fe(30 nm)/MgO(2 nm)/Fe(10 nm)/Co(20\nnm), has been chosen to optimize the magnetic properties\nof the MTJ stack [48]. The TMR measurements of the\ncoercive \felds of the hard (H C;Hard ) and soft (H C;Soft )\nlayers in MTJs under study show that they are well sep-\narated from the external \feld values used to rotate the\nsoft layer. Figure S6 shows that the hard layer switching\n\felds obtained from TMRs along [100], [010] and [110]\nmeasured in our junctions remain far above the typical\nrange of 70-120 Oe which is used to rotate the soft layer.\nMoreover, Figure S7 also shows the typical temperature\ndependence of H C;Hard , demonstrating its independence\nwith temperature from well above to well below TC.\n-600 -400 -200 0 200 400 600-1.0-0.50.00.51.0\nhard Fe/Co bilayer\n  M/MS\nH (Oe)soft Fe\nFIG. 5: Magnetic characterization of a Fe(30\nnm)/MgO/Fe(10 nm)Co(20 nm) structure, at room temper-\nature, along the [100] direction.\nS2. Magnetic characterizarion of the soft Fe(001)\nlayer and estimation of the Meissner screening\nThe magnetostatic Meissner screening has been dis-\ncussed mainly in studies with perpendicular magnetiza-\ntion [24]. In the case of the experiments with in-plane\n\feld rotation which we carry out, such \feld expulsion\ncould induce some screening of the external magnetic\n\feld applied to invert or rotate the magnetization of the\nsoft Fe(001) layer (which is the closest to the supercon-\n0.0 22.5 45.00100200300400500\n \n[110] [210]HC,Hard (Oe)\nθ[100] (deg)Field range used to rotate the soft layer\n[100]FIG. 6: Coercive \feld of the hard Fe/Co layer for magnetic\n\feld oriented along di\u000berent crystallographic directions [100],\n[110] and [210], above the superconducting critical tempera-\nture (T= 5 K). The grey band shows the typical \feld range\nused to manipulate the magnetization of the soft Fe(100) layer\nin the rotation experiments.\nductor), and with less probability a\u000bect the switching of\nthe more distant hard Fe/Co layer.\nFigure S8 shows the typical variation of the coercive\n\feld of the soft Fe(001) ferromagnetic layer with temper-\nature from above to below the critical temperature. We\nobserve some weak increase of the coercive \feld below 10\nOe, which could be due to spontaneous Meissener screen-\ning and/or vortex interaction with domain walls. These\nchanges, however, are an order of magnitude below the\ntypical magnetic \felds applied to rotate the Fe(001) layer\n(70-120 Oe). As we also show in Figure S7, the coer-\ncive \feld of the hard FeCo layer (typically above 400-500\nOe) shows practically no variation (within the error bars)\nwithin a wide temperature range, from 3 TCto 0:1TC,\ndiscarding the in\ruence of the Meissner screening on the\nhard layer.\nAs the superconducting layer is much larger in area\nthan the ferromagnetic one, these experiments point out\nthat the possible existing Meissner screening would intro-\nduce about a 10% correction to the actual external \feld\nacting on the soft ferromagnet, regardless of the external\n\feld direction.\nS3. Estimation of the weak antiferromagnetic cou-\npling of the two ferromagnetic layers.\nIn order to quantify the unavoidable weak antiferro-\nmagnetic magnetostatic coupling between the rotated\nsoft Fe(001) and the practically \fxed hard FeCo layer, we\nshow low \feld TMR measurements where the AP state is\nachieved and then maintained at zero \feld (Figure S9a).\nOne clearly observes that the AP and P states can be\n6FIG. 7: Typical temperature dependence of the coercitive\n\feld of the hard Fe/Co ferromagnetic layer along the [100]\ndirection. The critical temperature is marked with a dashed\nred line. We relate the excess scatter observed in the hard\nlayer with the extra structural disorder at the Fe/Co interface,\nproviding an enhanced coercive \feld for the Fe/Co layer. The\nblue line is a guide for the eye. The inset shows the method for\ndetermining the coercive \feld: it's the \feld of the \frst point\nafter the hard layer transition from in each TMR experiment.\nFIG. 8: Typical temperature dependence of the coercitive\n\feld of the soft Fe(001) ferromagnetic layer measured along\n[100] direction. The critical temperature is marked with a\ndashed red line. Blue lines are guides for the eye. The inset\nshows the method for determining the coercive \feld: a logistic\n\ft was performed for the transition, and the coercive \feld was\nde\fned as the mid-height value of the \ft.\nobtained as two di\u000berent non-volatile states, and there-\nfore the antiferromagnetic coupling is not su\u000ecient to\nantiferromagnetically couple the two layers at zero \feld.\nThe stability of the P state against the antiferromagnetic\ncoupling is con\frmed by the temperature dependence of\nthe resistance in the P and AP states. The P state shows\nstable resistance values at least below 15 K (Figure S9b).\nThis means that the antiferromagnetic coupling energy\nis well below 2 mV.\nFIG. 9: Two experiments demonstrating the stability of the\nP and AP states at zero \feld. (a) TMR to AP state before a\ncritical temperature measurement: the sample was \frst satu-\nrated in the P state with H= 1000 Oe in the [100] direction,\nand then a negative \feld sweeping was performed to -200 Oe\nand back to 0 Oe in the same direction in order to switch the\nsoft layer into the AP state, where it remained at zero \feld.\n(b) Two critical temperature measurements: the sample was\nsaturated in the P state, and then switched to AP state as\ndescribed in (a) for the AP measurement. After this, the\ntemperature was risen to 15 K and let to slowly cool down to\nT\u00182 K. The increase in resistance below 4 K corresponds to\nthe opening and deepening of the superconducting gap, since\nthe voltage used was only a few microvolts in order to dis-\ntinguish the superconducting transition from its appearance.\nBoth experiments show no sudden changes in resistance, as\nwould happen if any magnetic transition took place.\nS4. Calibration of the angle between the two fer-\nromagnetic layers\nIn order to estimate the angle between the two ferro-\nmagnets for the TMR measurements and rotations, we\nused the Slonczewski model [56]. By using values of the\nresistance in the AP, P and PIP states established above\nTC, we can calculate the desired angle \u0012with the follow-\n7ing expression:\nG\u00001=G1\u00001+\u0002\nG2\u0000\n1 +p2cos\u0012\u0001\u0003\u00001: (1)\nHere,Gis the total conductance of the sample, G1and\nG2are the conductances of each of the two tunnel bar-\nriers, andpis the spin polarization in the ferromagnets,\nfor which we obtain values between 0.7 and 0.8 depending\non the sample (the value being robust for each individual\none).\nIn order to ascertain the precission of this calibration\nmethod, an analysis of the di\u000berent errors has been per-\nformed. First, an standard error propagation calculation\nwas done to estimate the uncertainty in the resistance\nvalues, taking typical values for the current and voltage\nof 100 nA and 5 mV, respectively, which gives us a typical\nresistance value of 50 k\n. The current is applied using a\nKeithley 220 Current Source, which has an error of 0 :3%\nin the operating range according to the user manual. The\nvoltage is measured using a DMM-522 PCI multimeter\ncard. In the speci\fcations, the voltage precision is said\nto be 5 1=2 digits. With all this, the resistance error ob-\ntained is \u0001R=75.08 \n or a 0.15% of relative error. Using\nthis value, the error bars in the measurements shown\nin the main text would be well within the experimental\npoints.\nFor the calculated angle, the error propagation method\nis not adequate. It gives errors bigger than 360 degrees\nfor some angles, and in general over 30 degrees. This is\nclearly not what it is observed in reality: the performed\n\fts are quite robust, showing little variance in the esti-\nmated angle when changing the input parameters all that\nis reasonable. Instead, we have used a typical rotation\nperformed on a 30 \u000230\u0016m2sample. The \ftting to the\nSlonczewski formula needs three input values: the resis-\ntance in the P state ( RP), the resistance in the AP state\n(RAP) and the resistance in the PIP state ( RPIP). Using\nthese, a numerical algorithm calculates the spin polariza-\ntion (p), the resistance of the F/F barrier ( RFIF), and\nthe resistance of the F/S (F/N) barrier ( RNIF). These\ngive us the total resistance of the sample as a function\nof the angle \b FMbetween the two ferromagnets or, re-\nciprocally, the angle as a function of resistance. For our\nestimation, we have varied the value of the RPIPinput\nparameter from the lowest to the highest possible in the\nPIP state of the rotation, as well as taking an interme-\ndiate value which would be used in a normal analysis\n(the P and AP resistance values are always taken as the\nminimum and maximum resistance values in the rotation\nrespectively). The calculated parameters for the resis-\ntance of each barrier and the polarization may slightly\nvary from one \ftting to another, but the overall \ftting\nremains remarkably stable, as shown in Figure 10.\nAs expected, the di\u000berence is higher for the PIP state,\nand minimum in the P and AP state that are \\\fxed\".\n3.8 4 4.2 4.4 4.6 4.802468\n3.8 4 4.2 4.4 4.6 4.804590135180 ϕFM,max  - ϕFM,min  (deg)ϕFM (deg)\nR (Ω)x104x104(a)\n(b)RPIP,min\nRPIP\nRPIP,max\nR (Ω)FIG. 10: (a) \b FMas a function of resistance for the \fttings\nwith maximum, usual, and minimum RPIP used, in the P-\nAP resistance range. (b) di\u000berence of calculated angle vs\nresistance (in the P-AP resistance range) for the \fttings with\nmaximum and minimum RPIPused.\nThe di\u000berence doesn't exceed 7 degrees, and it keeps be-\nlow 2 degrees near the P and AP states.\nS5. Saturation magnetization for thin Fe(001) \flms\nin [100] and [110] directions\nDi\u000berent M vs H measurements were performed at\nroom temperature on a 10 nm thick Fe \flms, both for the\neasy [100] and hard [110] crystallographic axes, in order\nto estimate the magnetocrystalline anisotropy (MCA) en-\nergy. The results are depicted in Fig. S11. Using the\nsaturation \feld for the two directions, the anisotropy\nenergy can be estimated as KFe=MFeHSat=2 =\n5:1\u0002105erg\u0001cm\u00003, whereMFe= 1714 emu/cm3is\nused. The anisotropy energy per unit cell is therefore\nMAE= 6:674\u0016eV, or 3.337 \u0016eV per atom. The obtained\nenergy barrier is similar to the one measured using fer-\nromagnetic resonance [57].\nAs shown in Fig.S12, the experimental MCA energy\n8-50 -25 0 25 50-1.0-0.50.00.51.0\n  M/Msat\nH (Oe)[100] in plane easy axis\n-1 -0.5 0 0.5 1-101\n[110] in plane hard axis\nHsat \nH (kOe)(a)\n(b)\nM/MsatFIG. 11: M vs H measurements on a 10 nm thick Fe \flm for\nthe easy [100] (a) and hard [110] (b) crystallographic axis.\nThe saturation \feld ( Hsat) for the easy axis is around 10 Oe,\nwhile for the hard direction it reaches up to 600 Oe.\nvalues have been theoretically confronted with theoreti-\ncal/numerical calculations of the angular in-plane varia-\ntion of magnetic anisotropy, using the ab-initio Wien2k\nFP-LAPW code [58]. The calculations were based on a\nsupercell model for a V/MgO/Fe/MgO slab similar to the\nexperimental samples. To insure the requested extreme\naccuracy in MCA energy values ( \u0016eV energy range), a\nthoroughly well-converged kgrid with signi\fcantly large\nnumber ofk-points has been involved. Within these cir-\ncumstances, our theoretical results for the Fe(001) thin\n\flms show standard fourfold anisotropy features and rea-\nsonable agreement with the experimentally estimated\n\fgures with a maximum theoretical MAE of 4.9 \u0016eV\nper atom (expected theoretical under-estimation of the\nmagnetocristalline energy within the GGA approach).\nNote that the superonducting-V induced MCA modu-\nlation features cannot be described within the ab-initio\nFP-LAPW approach, describing the V in its normal\nmetallic state. Therefore, the below TCexperimentally\nobserved MCA energy modulations have to be clearly\nrelated to the proximity e\u000bect in the superconducting\nV/MgO/Fe(001) system and not to any speci\fc MCA\nfeature of Fe(001) in the V/MgO/Fe(001)/MgO complex\nstacking sequence.\n0.0 0.4 0.8 1.2 1.6012345\n  MAE\n 4.7sin2(2θ)\nθ (radians)MAE (μeV/Atom)\n[100][110]Fe\nθFIG. 12: Ab-initio calculation of magnetocryscalline\nanisotropy energy (MAE) as a function of the in-plane ori-\nentation angle \u0012, de\fned in the inset. Solid line is a phe-\nnomenological \ft to a sin2(2\u0012) function.\nS6. Evaluation of the vortex induced pinning of\ndomain walls\nUsing MuMax3 [52], we have compared numerically the\nDWs formation along the [100] and [110] magnetization\ndirections. The simulations took place in samples with\n3\u00023\u0016m2lateral dimensions (100 nm rounded corners\nwere used as the devices have been fabricated by optical\nlithography), with 512 \u0002512\u000216 cells, atT= 0. The rest\nof the parameters used were Aex= 2:1\u000210\u000011J/m for\nthe exchange energy, Msat= 1:7\u0002106A/m for the sat-\nuration magnetization, a damping parameter \u000b= 0:02,\nand crystalline anisotropy parameters KC1= 4:8\u0002104\nJ/m3andKC3=\u00004:32\u0002105J/m3. The goal of the\nsimulations was to evaluate the DW formation and their\ninteraction with the superconducting vortices induced by\nthe vertical component of the stray \felds at a 2-3 nm from\nthe Fe(001) surface. We observed that, depending on the\nexternal \feld, in the range of 70-1000 Oe both edge-type\nand inner-type DWs are formed when the \feld is directed\nalong [100], and mainly edge type DWs are formed with\n\feld along [110] (Figure S13a).\nWe have also calculated the interaction Ibetween the\nDW related excess exchange energy Eexand the vertical\ncomponent of the stray \felds, Be\u000b(Figure S13b):\nI=ZNx\n0ZNy\n0jBe\u000bjEexFdxdy (2)\nWhereNxandNyare the total number of cells in each\ndimension of the simulation, and Fis a \flter \\Vortex gen-\neration function\" that takes into account the simulated\ndependence of the number of vortices on the vertically\napplied \feld (Figure S13c). The vortices were simulated\nusing the Time Dependent Ginzburg Landau code devel-\n90.0 0.5 1.0051015\nDW-vortex eff. interaction (arb. u.)\nH (kOe) H[110]\n H[100]\n0 0.2 0.40200400\nNumber of vorticesT=2 K(c)\nH/HC2H[110]|H|=1000 OeH[100]|H|=70 Oe\nx103\nH[110](b) (a)\nH[100]\nOrder parameter01\n0\nMZ/MFIG. 13: (a) Typical DW formation mapped by MuMax3 simulations for the [110] and [100] applied \feld directions in the\nnon-saturated (70 Oe) and saturated (1000 Oe) \feld regimes. The color map represents the out of plane component of the\nmagnetization, while the red arrows indicate the in plane direction. (b) Values of the 2D integral Ibetween the local exchange\nenergy (DWs) and the perpendicular component of the stray \felds at a distance of 2-3 nm from the ferromagnet, taking into\nacount the vortex generation function F. (c) Vortex generation function F, represented as number of vortices formed in a 5 \u00025\n\u0016m2square 40 nm thick superconducting Vanadium \flm as a function of the applied perpendicular \feld (normalized by the\nsecond critical \feld Hc2), simulated at T= 2 K by using the TDGL code described in [59]. The insert shows a typical image\nof the vortices at H=0.15 Hc2\noped in Madrid described in [59]. The TDGL simulations\ntook place in 5\u00025\u0016m2Vanadium samples with 200 \u0002200\ncells, atT= 2 K, with a coherence length \u00180= 2:6\u000210\u00008\nbased on our experimental estimations for the studied de-\nvices,\u0014= 3 andTC= 4 K. A uniform \feld was applied in\nthe perpendicular direction, its magnitude varying from\n0:1HC2to 0:6HC2, and the number of vortices generated\nin the relaxed state were counted.\nThe second critical \feld in the vertical direction ( Hc2=\n3 kOe) was determined experimentally. The estimated\ninteraction shows that in the weakly saturated regime,\nwhen the inner DWs could emerge and the DW-vortex in-\nteraction increases, such interaction should pin the mag-\nnetization along the [100] direction, corresponding to\nthe MCA already present in the normal state, therefore\nblocking any magnetization rotation towards the [110] di-\nrection, contrary to our experimental observations. The\npossible reason for the irrelevance of the DW-vortex in-\nteraction in our system is that inner DWs are expected\nto be of Neel-type for the thickness considered [60].\nFinally we mention that our numerical evaluations\nshow that, if present, the vortex-DW interaction should\nremain dominant for the magnetization directed along\n[100] respect [110] and for the magnetic \feld range 70-\n1000 Oe also without KC3parameter providing the MCA\nenergy minima along [110]. In these simulations the\nFe(001) layer has been considered to be smooth. In or-\nder to further approach simulations to the experiment,\nwe have also veri\fed that the conclusions above are not\na\u000bected by the introduction of interfacial magnetic dis-\norder due to mismatch defects (every 30 lattice periods)\nwith 25% excess of Fe moment at the Fe/MgO interface\n[61]. More detailed simulations involving also interface\nroughness could be needed to further approach the realexperimental situation.\nS7. Magnetization alignment along [110] and irrel-\nevance of the junction area for the superconductivity\ninduced MCA modi\fcation\nAs we mentioned in the main text, our experiments\npoint that Fe(001) layers are close to a highly saturated\nstate when the magnetization is directed along [100] or\nequivalent axes. On the other hand, micromagnetic simu-\nlations (Figure S13a) show that the magnetization align-\nment is more robust in the [110] direction (or equivalent)\nrather than in the [100] direction (or equivalent). So,\nif we indeed reach a highly saturated state in the [100]\ndirection, this should also be the case for the [110] direc-\ntion. Therefore, the emergent stable tunneling magne-\ntoresistance states we observe experimentally below Tc,\ncannot be explained in terms of the intermediate multi-\ndomain states but rather correspond to the dominant\n[110] magnetization alignment of the Fe(001) layer.\nAs shown in Figure S14, our experiments shows that\nthe observed e\u000bects remain qualitatively unchanged when\nthe junction area is varied about an order of magnitude.\n10051015202530\nTMR (%)10x10  μm2\n 1.5 K\n 5 K\n010203040\n20x20  μm2\n 1.5 K\n5 KTMR (%)\n01020304050\n30x30  μm2\n 1.5 K\n 5 KTMR (%)\nΦH\nΦFM Nº of points\nT=5 K\nT=1.5 K10x10  μm2\n20x20  μm2\n30x30  μm2(a) (c)\nπ3π/2 2π 5π/4 7π/4 π3π/2 2π 5π/4 7π/4 π3π/2 2π 5π/4 7π/4(b)\n(d)ΦH ΦH\n40\n040\n0\nπ3π/2 5π/4\n0 2 4 6 8010203040\n0 2 4 6 80102030(e) (f)Nº of points\nNº of points\nT (K) T (K)[110] (5π/4)[010] (3π/2)\n[110] (5π/4)\n[110] (7π/4)[010] (3π/2)30x30  μm2\nTC TC10x10  μm2FIG. 14: In-plane \feld rotation experiments with H= 70 Oe below (blue) and above (red) TCfor 10\u000210 (a), 20\u000220 (b) and\n30\u000230 (c)\u0016m2junctions. (d) Histograms of the calculated angle between the two FM layers \u001eFMfor these same rotations,\nabove and below TC, showing the [ 110] states for low temperatures. (e) and (f) show the evolution of the [110], [ 110] and [010]\n(PIP) states with temperature for the same 10 \u000210 and 30\u000230\u0016m2junctions (qualitatively simular evolution is shown in\nFigure 2f in the main text for the 20 \u000220\u0016m2junction).\nS8. Modelling\nWe describe the V/MgO/Fe structure by the Hamilto-\nnian [45]\nH=\u0000tX\nhi;ji;\u001bcy\ni;\u001bcj;\u001b\u0000X\ni;\u001b(\u0016i\u0000Vi)cy\ni;\u001bci;\u001b\n\u0000X\niUini;\"ni;#+X\ni;\u000b;\fcy\ni;\u000b(hi\u0001\u001b)\u000b;\fci;\f\n\u0000i\n2X\nhi;ji;\u000b;\f\u0015icy\ni;\u000b^n\u0001(\u001b\u0002di;j)\u000b;\fcj;\f(3)\nde\fned on a cubic lattice. The \frst term describes\nnearest-neighbor hopping. The second term includes the\nthe chemical potential and the potential barrier at the\ninsulating MgO layers. The remaining terms describes\nsuperconducting attractive on-site interaction, ferromag-\nnetic exchange interaction, and Rashba spin-orbit inter-\naction, respectively. These are only nonzero in their re-\nspective regions. In the above, tis the hopping integral,\n\u0016iis the chemical potential, Viis the potential barrier\nthat is nonzero only for the MgO layer, U > 0 is the\nattractive on-site interaction giving rise to superconduc-tivity,\u0015iis the local spin-orbit coupling magnitude, ^ nis\na unit vector normal to the interface, \u001bis the vector of\nPauli matrices, di;jis a vector from site ito sitej, andhi\nis the local magnetic exchange \feld. The number opera-\ntor used above is de\fned as ni;\u001b\u0011cy\ni;\u001bci;\u001b, andcy\ni;\u001band\nci;\u001bare the second-quantization electron creation and an-\nnihilation operators at site iwith spin\u001b. The supercon-\nducting term in the Hamiltonian is treated by a mean-\n\feld approach, where we assume ci;\"ci;#=hci;\"ci;#i+\u000e\nand neglect terms of second order in the \ructuations \u000e.\nWe consider a system of size Nx\u0002Ny\u0002Nzsetting\nthe interface normals parallel to the xaxis and assuming\nperiodic boundary conditions in the yandzdirections.\nTo simplify notation in the following, we de\fne i\u0011ix,\nj\u0011jx,ijj= (ix;iy) andk\u0011(ky;kz). We apply the\nFourier transform\nci;\u001b=1p\nNyNzX\nkci;k;\u001bei(k\u0001ijj)(4)\nto the above Hamiltonian and use that\n1\nNyNzX\nijjei(k\u0000k0)\u0001ijj=\u000ek;k0: (5)\n11We choose a new basis\nBy\ni;k= [cy\ni;k;\"cy\ni;k;#ci;\u0000k;\"ci;\u0000k;#] (6)\nspanning Nambu\u0002spin space, and rewrite the Hamilto-\nnian as\nH=H0+1\n2X\ni;j;kBy\ni;kHi;j;kBi;k: (7)\nAbove, the Hamiltonian matrix is given by\nHi;j;k=\u000fi;j;k^\u001c3^\u001b0+\u000ei;jh\ni\u0001i^\u001c+^\u001by\u0000i\u0001\u0003\ni^\u001c\u0000^\u001by\n+hx\ni^\u001c3^\u001bx+hy\ni^\u001c0^\u001by+hz\ni^\u001c3^\u001bz\n\u0000\u0015isin(ky)^\u001c0^\u001bz+\u0015isin(kz)^\u001c3^\u001byi\n;(8)\nwhere \u0001iis the superconducting gap which we solve for\nself-consistently, ^ \u001ci^\u001bj\u0011^\u001ci\n^\u001bjis the Kronecker product\nof the Pauli matrices spanning Nambu and spin space,\n^\u001c\u0006\u0011(^\u001c1\u0006i^\u001c2)=2, and\n\u000fi;j;k\u0011\u00002t[cos(ky) + cos(kz)]\u000ei;j\n\u0000t(\u000ei;j+1+\u000ei;j\u00001)\n\u0000(\u0016i\u0000Vi)\u000ei;j:(9)\nThe constant term in Eq. (7) is given by\nH0=\u0000X\ni;kf2t[cos(ky) + cos(kz)] +\u0016i\u0000Vig\n+NyNzX\nij\u0001ij2\nUi:(10)\nWe absorb the sum over lattice sites in Eq. (7) into the\nmatrix product by de\fning a new basis\nWy\nk= [By\n1;k;:::;By\ni;k;:::;By\nNx;k]: (11)\nEq. (7) can then be rewritten as\nH=H0+1\n2X\nkWy\nkHkWk; (12)\nwhere\nHk=2\n64H1;1;k\u0001\u0001\u0001H1;Nx;k\n.........\nHNx;1;k\u0001\u0001\u0001HNx;Nx;k3\n75 (13)\nis Hermitian and can be diagonalized numerically. We\nobtain eigenvalues En;kand eigenvectors \b n;kgiven by\n\by\nn;k= [\u001ey\n1;n;k\u0001\u0001\u0001\u001ey\nNx;n;k];\n\u001ey\ni;n;k= [u\u0003\ni;n;kv\u0003\ni;n;kw\u0003\ni;n;kx\u0003\ni;n;k]:(14)\nThe diagonalized Hamiltonian can be written on the form\nH=H0+1\n2X\nn;kEn;k\ry\nn;k\rn;k; (15)where the new quasi-particle operators are related to the\nold operators by\nci;k;\"=X\nnui;n;k\rn;k;\nci;k;#=X\nnvi;n;k\rn;k;\ncy\ni;\u0000k;\"=X\nnwi;n;k\rn;k;\ncy\ni;\u0000k;#=X\nnxi;n;k\rn;k:(16)\nThe superconducting gap is given by \u0001 i\u0011Uihci;\"ci;#i.\nWe apply the Fourier transform in Eq. (4) and use\nEq. (16) in order to rewrite the expression in terms\nof the new quasi-particle operators. Also using that\nh\ry\nn;k\rm;ki=f\u0000\nEn;k=2\u0001\n\u000en;m, we obtain the expression\n\u0001i=\u0000Ui\nNyNzX\nn;kvi;n;kw\u0003\ni;n;k[1\u0000f(En;k=2)] (17)\nfor the gap, that we use in computing the eigenenergies\niteratively. Above, f\u0000\nEn;k=2\u0001\nis the Fermi-Dirac distri-\nbution.\nUsing the obtained eigenenergies, we compute the free\nenergy,\nF=H0\u00001\n\fX\nn;kln(1 +e\u0000\fEn;k=2); (18)\nwhere\f= (kBT)\u00001. The preferred magnetization di-\nrections are described by the local minima of the free\nenergy. In the main body of the paper, we use this to\nexplain the possible magnetization directions of the soft\nferromagnet when rotating an IP external magnetic \feld\nover a 2\u0019angle starting at a parallel alignment with the\nhard ferromagnet.\nOther relevant quantities to consider in modelling\nthe experimental system is the superconducting coher-\nence length and the superconducting critical temper-\nature. In the ballistic limit, the coherence length is\ngiven by\u0018=~vF=\u0019\u00010, wherevF=1\n~dEk\ndk\f\f\nk=kFis the\nFermi velocity related to the normal-state eigenenergy\nEk=\u00002t[cos(kx) + cos(ky) + cos(kz)]\u0000\u0016, and \u0001 0is the\nzero-temperature superconducting gap [62]. The critical\ntemperature is found by a binomial search, where we de-\ncide if a temperature is above or below Tcby determining\nwhether \u0001 NSx=2increases towards a superconducting so-\nlution or decreases towards a normal state solution from\nthe initial guess under iterative recalculations of \u0001 i. We\nchoose an initial guess with a magnitude very close to\nzero and with a lattice site dependence similar to that of\nthe gap just below Tc.\nIn the main plot showing the free energy under IP ro-\ntations of the magnetization, we have chosen parameters\n12t= 1,\u0016S=\u0016SOC =\u0016F= 0:9,V= 2:1,U= 1:35,\n\u0015= 0:4,h= 0:8,NS\nx= 30,NSOC\nx = 3,NF\nx= 8, and\nNy=Nz= 60. All length scales are scaled by the lattice\nconstanta, all energy scales are scaled by the hopping pa-\nrametert, and the magnitude of the spin-orbit coupling\n\u0015is scaled by ta. In order to make the system com-\nputationally manageable, the lattice size is scaled down\ncompared to the experimental system, however the re-\nsults should give qualitatively similar results as long as\nthe ratios between the coherence length and the layer\nthicknesses are reasonable compared to the experimental\nsystem. For this set of parameters, the superconduct-\ning coherence length is approximately 0 :6NS\nx. Since the\ncoherence length is inversely proportional to the super-\nconducting gap, Uhas been chosen to be large in order to\nallow for a coherence length smaller than the thickness\nof the superconducting layer. Although this results in\na large superconducting gap, the modelling will qualita-\ntively \ft the experimental results as long as the other pa-\nrameters are adjusted accordingly. We therefore choose\nthe local magnetic exchange \feld so that h\u001d\u0001, as in\nthe experiment. For this parameter set, h\u001920\u0001. The\norder of magnitude of \u0015is 1 eV \u0017A, given that t\u00181 eV and\na\u00184\u0017A. This is realistic considering Rashba parameters\nmeasured in several materials [63]. The Rashba spin-\norbit \feld at the interfaces of V/MgO/Fe is caused by\na structural inversion asymmetry across the MgO layer,\nand breaks the inversion symmetry at the MgO interfaces\n[44]. This causes generation of triplet-superconductivity\neven for weakly spin-polarized ferromagnets with a small\nspin-orbit \feld [64]. We are therefore not dependent upon\na strong magnetic exchange \feld and a strong spin-orbit\n\feld for realizing the observed e\u000bects. For the AF cou-\npling contribution to the free energy, we set fAF= 0:01\nin order to \ft the anisotropy of the experimental system\njust aboveTC.\n\u0003e-mail: farkhad.aliev@uam.es\n[1] V.L. Ginzburg, Ferromagnetic superconductors, Zh.\nEksp. Teor. Fiz. 31, 202 (1956); Sov. Phys. JETP 4, 153\n(1957).\n[2] B. T. Matthias, H. Suhl and E. Corenzwit, Spin exchange\nin superconductors, Phys. Rev. Lett. 1, 152 (1958).\n[3] A. I. Buzdin, L. N. Bulaevskii, M. L. Kulich and S. V. Pa-\nnyukov, Magnetic superconductors, Sov Phys. Uspekhi,\n27, 927 (1984).\n[4] J. Y. Gu, C.-Y. You, J. S. Jiang, J. Pearson, Y. B.\nBazaliy and S. D. Bader, Magnetization-orientation de-\npendence of the superconducting transition temperature\nin the ferromagnet-superconductor-ferromagnet system:\nCuNi/Nb/CuNi, Phys. Rev. Lett. 89, 267001 (2002).\n[5] I. C. Moraru, W. P. Pratt, Jr. and N. O. Birge, Ob-\nservation of standard spin-switch e\u000bects in ferromag-\nnet/superconductor/ferromagnet trilayers with a strong\nferromagnet, Phys. Rev. B 74, 220507(R) (2006).[6] L. R. Tagirov, Low-\feld superconducting spin switch\nbased on a superconductor/ferromagnet multilayer,\nPhys. Rev. Lett. 83, 2058 (1999).\n[7] A. I. Buzdin, A. V. Vedyayev and N. V. Ryzhanova, Spin-\norientation dependent superconductivity in F/S/F struc-\ntures, Europhys. Lett., 48(6), 686-691 (1999).\n[8] I. Baladi\u0013 e, A. Buzdin, N. Ryzhanova and A. Vedyayev,\nInterplay of superconductivity and magnetism in su-\nperconductor/ferromagnet structures,Phys. Rev. B 63,\n054518 (2001).\n[9] P. V. Leksin, N. N. Garif'yanov, I. A. Garifullin,\nJ. Schumann, V. Kataev, O. G. Schmidt and B.\nB uchner, Manifestation of New Interference E\u000bects in\na Superconductor-Ferromagnet Spin Valve, Phys. Rev.\nLett. 106, 067005 (2011).\n[10] X. L. Wang, A. Di Bernardo, N. Banerjee, A. Wells, F.\nS. Bergeret, M. G. Blamire and J. W. A. Robinson, Gi-\nant triplet proximity e\u000bect in superconducting pseudo\nspin valves with engineered anisotropy, Phys. Rev. B 89,\n140508(R) (2014).\n[11] A. Singh, S. Voltan, K. Lahabi and J. Aarts, Colossal\nProximity E\u000bect in a Superconducting Triplet Spin Valve\nBased on the Half-Metallic Ferromagnet CrO 2, Phys.\nRev. X 5, 021019 (2015).\n[12] P.V. Leksin, N. N. Garifyanov, I. A. Garifullin, Ya.V.\nFominov, J. Schumann, Y. Krupskaya, V. Kataev, O.\nG. Schmidt and B. B uchner, Evidence for triplet super-\nconductivity in a superconductor-ferromagnet spin valve,\nPhys. Rev. Lett., 109, 057005 (2012).\n[13] R. S. Keizer, S. T. B. Goennenwein, T. M. Klapwijk, G.\nMiao, G. Xiao and A. A. Gupta, A spin triplet supercur-\nrent through the half-metallic ferromagnet CrO 2, Nature\n439, 825 (2006).\n[14] T. S. Khaire, M. A. Khasawneh, W. P. Pratt, Jr. and N.\nO. Birge, Observation of spin-triplet superconductivity\nin Co-based Josephson junctions, Phys. Rev. Lett. 104,\n137002 (2010).\n[15] M. S. Anwar, F. Czeschka, M. Hesselberth, M. Porcu and\nJ. Aarts, Long-range supercurrents through half-metallic\nferromagnetic CrO 2, Phys. Rev. B 82, 100501(R) (2010).\n[16] J. W. A. Robinson, J. D. S. Witt and M. G. Blamire,\nControlled injection of spin-triplet supercurrents into a\nstrong ferromagnet, Science 329, 59 (2010).\n[17] C. Visani, Z. Sefrioui, J. Tornos, C. Leon, J. Bri-\natico, M. Bibes, A. Barth\u0013 el\u0013 emy, J. Santamar\u0013 \u0010a and J.\nE. Villegas, Equal-spin Andreev re\rection and long-\nrange coherent transport in high-temperature supercon-\nductor/halfmetallic ferromagnet junctions, Nature Phys.,\n8, 540 (2012).\n[18] N. Banerjee, J. W. A. Robinson and M. G. Blamire, Re-\nversible control of spin-polarized supercurrents in ferro-\nmagnetic Josephson junctions, Nature Comm. 5, 4771\n(2014).\n[19] A. Iovan, T. Golod and V. M. Krasnov, Controllable gen-\neration of a spin-triplet supercurrent in a Josephson spin\nvalve, Phys. Rev. B 90, 134514 (2014).\n[20] J. Linder and J. W. A. Robinson, Superconducting spin-\ntronics, Nature Phys. 11, 307 (2015).\n[21] M. Eschrig, Spin-polarized supercurrents for spintronics:\na review of current progress, Rep. Prog. Phys. 78, 104501\n(2015).\n[22] M. G. Flokstra, N. Satchell, J. Kim, G. Burnell, P. J. Cur-\nran, S. J. Bending, J. F. K. Cooper, C. J. Kinane, S. Lan-\ngridge, A. Isidori, N. Pugach, M. Eschrig, H. Luetkens,\n13A. Suter, T. Prokscha and S. L. Lee, Remotely induced\nmagnetism in a normal metal using a superconducting\nspin-valve, Nature Phys., 12, 97 ( 2016).\n[23] A.I. Buzdin and L. N. Bulaevskii, Ferromagnetic \flm on\nthe surface of a superconductor: Possible onset of inho-\nmogeneous magnetic ordering, Zh. Eksp. Teor. Fiz. 94,\n256-261 (1988), Sov. Phys. JETP 67, 576-578 (1988).\n[24] L. N. Bulaevskii and E. M. Chudnovsky, Ferromagnetic\n\flm on a superconducting substrate, Phys. Rev. B 63,\n012502 (2000).\n[25] S. V. Dubonos, A. K. Geim, K. S. Novoselov and I. V.\nGrigorieva, Spontaneous magnetization changes and non-\nlocal e\u000bects in mesoscopic ferromagnet-superconductor\nstructures, Phys. Rev. B 65, 220513R (2002).\n[26] J. Fritzsche, R. B. G. Kramer and V. V. Moshchalkov,\nVisualization of the vortex-mediated pinning of ferromag-\nnetic domains in superconductor-ferromagnet hybrids,\nPhys. Rev. B 79, 132501 (2009).\n[27] P. J. Curran et al., Irreversible magnetization switching\nat the onset of superconductivity in a superconductor fer-\nromagnet hybrid, Appl. Phys. Lett. 107, 262602 (2015).\n[28] X. Weintal and P. W. Brouwer, Magnetic exchange in-\nteraction induced by a Josephson current, Phys. Rev. B\n65, 054407 (2002).\n[29] Y. Tserkovnyak and A. Brataas, Current and\nspin torque in double tunnel barrier ferromagnet-\nsuperconductorferromagnet systems, Phys. Rev. B 65,\n094517 (2002).\n[30] C. Bell, S. Milikisyants, M. Huber and J. Aarts, Spin dy-\nnamics in a superconductor-ferromagnet proximity sys-\ntem, Phys. Rev. Lett. 100, 047002 (2008).\n[31] B. Braude and Ya. M. Blanter, Triplet Josephson e\u000bect\nwith magnetic feedback in a superconductor-ferromagnet\nheterostructure, Phys. Rev. Lett. 100, 207001 (2008).\n[32] E. Zhao and J. A. Sauls, Theory of nonequilibrium spin\ntransport and spin-transfer torque in superconducting-\nferromagnetic nanostructures, Phys. Rev. B 78, 174511\n(2008).\n[33] F. Konschelle and A. Buzdin, Magnetic moment manip-\nulation by a Josephson current, Phys. Rev. Lett. 102,\n017001 (2019).\n[34] J. Linder and T. Yokoyama, Supercurrent-induced mag-\nnetization dynamics in a Josephson junction with two\nmisaligned ferromagnetic layers, Phys. Rev. B 83, 012501\n(2011).\n[35] J. Linder, A. Brataas, Z. Shomali and M. Zareyan, Spin-\ntransfer and exchange torques in ferromagnetic supercon-\nductors, Phys. Rev. Lett. 109, 237206 (2012).\n[36] N. G. Pugach and A. I. Buzdin, Magnetic moment ma-\nnipulation by triplet Josephson current, Appl. Phys. Lett.\n101, 242602 (2012).\n[37] F. S. Bergeret and I. V. Tokatly, Singlet-triplet\nconversion and the long-range proximity e\u000bect in\nsuperconductor-ferromagnet structures with generic spin\ndependent \felds, Phys. Rev. Lett., 110, 117003 (2013).\n[38] N. Banerjee, J. A. Ouassou, Y. Zhu, N. A. Stelmashenko,\nJ. Linder and M. G. Blamire, Controlling the supercon-\nducting transition by spin-orbit coupling, Phys. Rev. B\n97, 184521 (2018).\n[39] K.-R. Jeon, C. Ciccarelli, A. J. Ferguson, H. Kure-\nbayashi, L. F. Cohen, X. Montiel, M. Eschrig, J. W. A.\nRobinson and M. G. Blamire, Enhanced spin pumping\ninto superconductors provides evidence for superconduct-\ning pure spin currents, Nat. Mater. 17, 499 (2018).[40] K.-R. Jeon, C. Ciccarelli, A. J. Ferguson, H. Kure-\nbayashi, L. F. Cohen, X. Montiel, M. Eschrig, S. Komori,\nJ. W. A. Robinson and M. G. Blamire, Exchange-\feld en-\nhancement of superconducting spin pumping, Phys. Rev.\nB99, 024507 (2019).\n[41] N. Satchell and N. O. Birge, Supercurrent in ferromag-\nnetic Josephson junctions with heavy metal interlayers,\nPhys. Rev. B 97, 214509 (2018).\n[42] N. Satchell, R. Loloee and N. O. Birge, Supercurrent\nin ferromagnetic Josephson junctions with heavy-metal\ninterlayers. II. Canted magnetization, Phys. Rev. B 99,\n174519 (2019).\n[43] S. H. Jacobsen, J. A. Ouassou and J. Linder, Crit-\nical temperature and tunneling spectroscopy of\nsuperconductor-ferromagnet hybrids with intrinsic\nRashba-Dresselhaus spin-orbit coupling, Phys. Rev. B\n92, 024510 (2015).\n[44] I. Mart\u0013 \u0010nez, P. H ogl, C. Gonz\u0013 alez-Ruano, J. P. Cascales,\nC. Tiusan, Y. Lu, M. Hehn, A. Matos-Abiague, J. Fabian,\nI. Zutic and F. G. Aliev, Interfacial spin-orbit coupling:\na platform for superconducting spintronics, Phys. Rev.\nAppl. 13, 014030 (2020).\n[45] L. G. Johnsen, J. Linder and N. Banerjee, Magnetization\nreorientation due to superconducting transition in heavy\nmetal heterostructures, Phys. Rev. B 99, 134516 (2019).\n[46] H. X. Yang, M. Chshiev and B. Dieny, First-principles\ninvestigation of the very large perpendicular magnetic\nanisotropy at Fe/MgO and Co/MgO interfaces. Phys.\nRev. B 84, 054401 (2011).\n[47] I. Mart\u0013 \u0010nez, C. Tiusan, M. Hehn, M. Chshiev and F.\nG. Aliev, Symmetry broken spin reorientation transi-\ntion in epitaxial MgO/Fe/MgO layers with competing\nanisotropies, Sci. Rep. 8, 9463 (2018).\n[48] C. Tiusan, M. Hehn, F. Montaigne, F. Greullet, S. An-\ndrieu and A. Schuhl, Spin tunneling phenomena in single\ncrystal magnetic tunnel junction systems, J. Phys.: Con-\ndens. Matter 19, 165201 (2007).\n[49] See Supplemental Material at XXX for more details.\n[50] S. S. P. Parkin, C. Kaiser, A. Panchula, P. M. Rice, B.\nHughes, M. Samant, and S.-H. Yang, Giant tunnelling\nmagnetoresistance at room temperature with MgO (100)\ntunnel barriers, Nat. Mater. 3, 862 (2004).\n[51] S. Yuasa, T. Nagahama, A. Fukushima, Y. Suzuki and\nK. Ando, Giant room temperature magneto-resistance\nin single-crystal Fe/MgO/Fe magnetic tunnel junctions,\nNat. Mater. , 868 (2004).\n[52] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen,\nF. Garcia-Sanchez and B. Van Waeyenberge, The design\nand veri\fcation of MuMax3, AIP Advances 4, 107133\n(2014).\n[53] A. Lara, C. Gonz\u0013 alez-Ruano and F.G.Aliev, Time-\ndependent Ginzburg-Landau simulations of supercon-\nducting vortices in three dimensions, Low Temperature\nPhysics 46, 316 (2020).\n[54] D. Herranz, F. Bonell, A. Gomez-Ibarlucea, S. Andrieu,\nF. Montaigne, R. Villar, C. Tiusan, and F.G.Aliev,\nStrongly suppressed 1/f noise and enhanced magnetore-\nsistance in epitaxial FeV/MgO/Fe magnetic tunnel junc-\ntions, Applied Physics Letters 96, 202501 (2010).\n[55] T. Vezin, C. Shen, J. E. Han and I. \u0014Zuti\u0013 c, Enhanced spin-\ntriplet pairing in magnetic junctions with s-wave super-\nconductors, Phys. Rev. B 101, 014515 (2020).\n[56] J. C. Slonczewski, Conductance and exchange coupling of\ntwo ferromagnets separated by a tunneling barrier. Phys.\n14Rev. B 39, 6995 (1989).\n[57] A. N. Anisimov, M. Farle, P. Poulopoulos, W. Platow,\nK. Baberschke, P. Isberg, R. Wppling, A. M. N. Niklas-\nson and O. Eriksson, Orbital Magnetism and Magnetic\nAnisotropy Probed with Ferromagnetic Resonance, Phys.\nRev. Lett. 82, 2390 (1999).\n[58] P. Blaha, K.Schwarz, F. Tran, R. Laskowski, G.K.H.\nMadsen and L.D. Marks, An APW+lo program for cal-\nculating the properties of solids, J. Chem. Phys. 152,\n074101 (2020).\n[59] A. Lara, C. Gonz\u0013 alez-Ruano and F.G.Aliev, Time-\ndependent Ginzburg-Landau simulations of supercon-\nducting vortices in three dimensions, Low Temperature\nPhysics 46, 316 (2020).[60] A. Hubert and R. Schfer, Magnetic domains. The analysis\nof magnetic microstructures (Springer, Berlin, 1999).\n[61] E. Jal et al; Interface Fe magnetic moment enhance-\nment in MgO/Fe/MgO trilayers, Appl. Phys. Lett. 107,\n092404 (2015).\n[62] J. Bardeen, L. N. Cooper and J. R. Schrie\u000ber, Theory of\nSuperconductivity, Phys. Rev. 108, 1175 (1957).\n[63] A. Manchon, H. C. Koo, J. Nitta, S. M. Frolov, and R. A.\nDuine, New perspectives for Rashba spinorbit coupling,\nNat. Mater. 14, 871 (2015).\n[64] T. Vezin, C. Shen, J. E. Han, and I. \u0014Zuti\u0013 c, Enhanced\nspin-triplet pairing in magnetic junctions with s-wave su-\nperconductors, Phys. Rev. B, 101, 014515 (2020).\n15"    },    {        "title": "2006.04192v1.Magnetocrystalline_anisotropy_of_the_easy_plane_metallic_antiferromagnet_Fe__2_As.pdf",        "content": "* d-cahill@illinois.edu  \n†\nzhu.diao@famu.edu  Magnetocrystalline  anisotropy of the easy -plane metallic antiferromagnet Fe 2As \n \nKexin Yang ,1,2  Kisung Kang ,2,3  Zhu Diao ,2,4,5,\n†  Manohar H. Karigerasi ,2,3 Daniel P. Shoemaker,  2,3 \nAndré Schleife ,2,6  David G. Cahill  1,2,3,*  \n \n1Department of Physics, University of Illinois at Urbana -Champaign, Urbana, Illinois 61801, USA  \n2Materials Research Laboratory, University of Illinois at Urbana -Champaign, Urbana, Illinois 61801, USA  \n3Materials Science and Engineering, University of Illinois at Urbana -Champaign, Urbana, Illinois 61801, USA  \n4Department of Physics,  AlbaNova University Center, Stockholm University, SE -106 91 Stockholm, Sweden  \n5Department of Physics, Florida A&M University, Tallahassee, Florida 32307, USA  \n 6National Center for Supercomputing Applications, University of Illinois at Urbana -Champaign, Urbana, Illinois 61801, USA  \nABSTRACT  \nMagnetocry stalline anisotropy is a fundamental property of magnetic materials that determin es the \ndynamics of magnetic precession, the frequency of spin waves, the thermal stability of magnetic \ndomains, and the efficiency of spintronic devices.  We combine  torque magnetometry  and density \nfunction al theory calculation s to determine the  magnetocrystalline  anisotropy of the metallic \nantiferromagnet Fe 2As. Fe2As has a tetragonal crystal structure  with the Néel vector l ying in the \n(001) plane. We report  that the  four-fold magnetocrystalline anisotropy in the (001) -plane of Fe 2As \nis extremely small , \n3\n22 150 J/m K=−  at T = 4 K,  much smaller than perpendicular magnetic \nanisotropy of ferromagnetic structure widely used in spintronics  device .  K22 is strongly \ntemperature dependent and close to  zero at T > 150 K. The anisotropy  \n1K in the (010) plane  is too \nlarge to be measure d by torque magnetometry  and we determine  \n1K= -830 kJ/m3 using  first-\nprinciples density function al theory . Our simulations show that the  contribution to the anisotropy \nfrom classical magnetic dipole -dipole  interactions is  comparable to the contribution  from spin -\norbit coupling.  The calculated four-fold anisotropy  in the (001) plane  \n22K  ranges from \n3290 J/m−  \nto \n3280 J/m , the same order of magnitude as the  measured value . We use d K1 from theory to \npredict  the frequency and polarization of the  lowest frequency  antiferromagnetic resonance mode \nand find that the mode is linearly polarized in the (001) -plane with \nf=  670 GHz .  \n2 \n  \nINTRODUCTION  \nAntiferromagnets (AFs) have potential advantages over ferromagnets for spintronic device s. \nCollinear  AFs are relatively insensitive to external field s because the net magnetization  is zero . \nAFs typically have much higher antiferromagnetic resonance (AFMR) frequency than  \nferromagnets  and therefore pr ocessional switching can occur in AFs at a faster rate than in \nferromagnets.  \nThe recent discovery  of electrical  manipulation and detection of spin configurations  in metallic \nAFs ha s led to a rapidly expanding scientific literature  on this class of magnetic materials.  \nTetragon al crystals with easy -plane magnetic anisotropy are preferred because the  two degenerate \norientations of the Néel vector can store binary information. In crystals with globally \ncentrosymmetric but locally non -centrosymmetric magnetic structure s—e.g., CuMnAs and \nMn 2Au—an electrical current exerts a torque on the N éel vector and the domain structure can \npotentially be switched electrically  [1][2][3][4].  \nA small value of t he in -plane magnetocrystalline  anisotropy facilitates e lectrical switching of the \ndomain orientation since a smaller torque is needed to overcome the energetic barrier that separates \nthe two orientations. Thermal stability of the domain requires, however, a  large v alue of the in -\nplane anisotropy. The N éel-Arrhenius law provides an estimate of t he rate of thermal fluctuations  \nof a single domain [5]: \n \n01exp\nBEfkT =− \n     , (1) \nwhere 𝜏 is the average time between thermally -activated changes in the direction of the \nmagnetization , \n0f is the resonance frequency,  \nEis the energy barrier between two degenerate \nmagnetic state s and  \nBkT  is the thermal energy. \nE  is given by the product of an anisotropy \nparameter  K and the volume of the domain  V; \nE KV= . Stable data storage typically requires  \n/ 40BE k T\n to meet the criteria that data must be retained for 10 years [6].     \n3 \n For the media of  conventional  hard drive s, the anisotropy parameter K is controlled by the \nperpendicular magnetocrystalline  anisotropy of ordered intermetallic alloys . In the emerging \ntechnology of magneti c random access memory (MRAM), K is controlled by the interfacial  \nmagnetic  anisotropy of  a ferromagnetic layer adjacent to the oxide barrier in a magnet ic tunnel \njunction . The perpendicular magnetic anisotropy \n1K  of MRAM materials is  typically  \n6 7 -3\n1 10 10  J m K\n [7]. \nMagnetocrystalline anisotropy Eani is described by a phenomenological expansion of the energy \nas a function of  direction cosines for the orientation of magnetization  of a ferromagnet or the \nsublattice magnetization of an antiferromagnet (AF) .  For a tetragonal crystal, the expansion to \nfourth order gives 3 coefficients \n1K , \n2K and \n22K  [8]. \n1K is a second order coefficient; \n2K  and \n22K\n are fourth order coefficients .  \n \n()2 4 4\n1 2 22 / sin sin sin cos 4aniE V K K K     = + +    , (2)  \nwhere \n   is the angle of the magnetization relative to the <001> direction  and \n  is the angle of \nthe magnetization relative to the <100> direction  (Fig. 2 ). The coefficient \n1K  describes the two -\nfold anisotropy  in (010) plane and \n2K  represents  the higher order fou r-fold symmetry  of the (010) \nplane . Because the effect of  \n2K is usually much smaller than \n1K , K2 will be neglected in the \nfollowing discussion. A crystal with an easy -plane anisotropy is described by \n10 K . The \ncoefficient \n22K  describes the four -fold anisotropy of the (001) plane and determines the thermal \nstability of an easy -plane domain structure.  \nAn external magnetic field  applied to an antiferromagnet (AF)  produces  a small induced magnetic \nmoment . The induced moment is small  because tilting of the orientation of sublattice \nmagnetization  is constrained by strong exchange interaction between the magnetic sublattices  that \nfavors a parallel alignment of the sublattices .  In general , however , the induced  magnetic moment \nis not parallel to the applied field  because magnet ocrystalline  anisotropy  favors an  orientation of \nthe sublattice magnetization along an easy axis  [9][10].    \n4 \n The lack of alignment between the induced moment  m and the applied field  B produces a \nmacroscopic torque on the sample , \n=τ m B .  A torque magnetometer measures this torque .  Data \nfor the torque as a function of applied field is sensitive to  magnetocrystalline anisotropy as long \nas the anisotropy is n either  too small nor too large. If the anisotropy is small, then the angle \nbetween m and B is small and the torque becomes difficult to detect .   If the anisotropy is large, \nthen the direction of m is fixed with respect to the crystallographic axis and the torque does not \nprovide information about the magnitude of the anisotropy. We can measure the in -plane four-fold \nanisotropy  of a mm -size bulk crystal of Fe 2As by torque magnetometry but the out -of-plane two-\nfold anisotropy  is not accessible to this technique  because the external field  is too small compare \nto anisotropy field  to extract information  about the anisotropy in the out -of-plane direction .  We \ninstead employ first -principles calculation s based on density functional theory  to dete rmine K1. \nWhen  magnetic energy is larger than the anisotropy energy , the amplitude of the torque in the \n(001) -plane sa turates and the four-fold magnetic anisotropy, \n22K  can be directly determined from \nthe amplitude of the torque . We measured three samples extracted from the same growth run , and \nthe \n22K  value of all three samples is  comparable to  -150 J/m3 at 4 K. The magnitude of \n22K  drops \nquickly as temperature increase s and reaches a small value above 150 K . The temperature \ndependence of magnetic anisotropy f or antiferromagnets is similar to  ferromagnets , follow ing a \npower law of sublattice magnetization [11][12][13].  \nStrikingly, t orque data for the applied field rotating in the (010) -plane reveal the motion of domain \nwalls.  An applied field in the (010) plane of 1 T is sufficient to orient the N éel vector fully \nperpendicular to the applied field.  Domain wall mot ion occurs even at T = 4 K and, therefore, is \nnot thermally activated.  \nIn the final section , we derive  the lowest -frequency, zone -center AFMR  frequency for easy -plane \nAFs, \n22 1 1 2 ( ) 2EEH H H H H   = −  − , where \n is the gyromagnetic ratio,  and \nEH, \n1H , \n22H\n are the exchange field, out -of-plane anisotropy field and in -plane anisotropy field , respectively. \nThe anisotropy fields are calculated with anisotropy energy and sublattice magnetization: \n11 / H K M=\n and \n22 22 / H K M= . With \n1K  calculated by DFT as  \n1K = -830 kJ/m3, the AFMR \nfrequency is \nf  = 670 GHz  at 4 K .  \n5 \n METHODS  \nFe2As crystallizes in the Cu 2Sb tetragonal crystal structure. Based on the corresponding magnetic \nsymmetry (mmm1' magnetic point group), the Néel vector of Fe 2As has two degenerate \norientations in the (001) -plane [14][15][16].  \nThe Fe2As crystal was synthesized by mixing Fe and As powders in a 1.95:1 ratio and vacuum \nsealing inside a quartz tube. The vacuum tube was heated at 1  ℃/min up to 600 ℃ and held for 6 \nhours  in a furnace. The temperature was then ramped to 975  ℃ at 1 ℃/min and held for 1 hour \nbefore cooling down slowly to 900  ℃ at 1 ℃/min. Finally, the quartz tube was kept at 900  ℃ for \n1 hour and allowed to cool down to room temperature in the furnace at 10  ℃/min.   We obtained  a \nlarge silver -hued  crystal ingot of Fe 2As and it easily detached from the quartz tube. Part of the \ningot was crushed into powder for powder XRD characterization and the  data showed phase pure \nFe2As. But the sample is slightly off -stochiometry as described in reference [17]. The remaining \nportion of the ingot was then fractured and the fractured surface revealed  a smooth facet . Laue \ndiffraction was carried out after polishing this fractured surface. A four -fold symmetry pattern was \nobserved indicating the fractured surface is  the (001) plane.   \nWe used a wire saw to cut the sample  into smaller pieces for magnetic property characterization \nand torque measurement s. One of the pieces was measured on the superconducting quantum \ninterference device vibrating sample magnetometer (SQUID -VSM, see below) , the other three \npieces were used in torque magnetometry measurements . We name these three samples measured  \nby torque magnetometry sample A , sample B , sample C  and the one for SQUID -VSM is named \nsample D .  \nThe temperature -dependent magnetic  susceptibility  was measured with a SQUID -VSM in a \nQuantum Design Magnetic Properties Measurement System (MPMS). The susceptibility of the  \nsample was measured while cooling from 398 K to 4 K in a 10 mT  field.  \nTorque measurements were performed in a Quantum Design Physical Property Measurement \nSystem (PPMS). We mounted the sample on a standard torque sensor chip , P109A from Quantum \nDesign with a sensitivity of 1 × 10-9 N·m. The PPMS horizontal  sample rotator was used to control \nthe angle between th e crystal and the applied field.  During the measurement, the external field  \n6 \n rotated in either the (010) plane or the (001) plane, while the field -induced moment  reside d in the \nsame plane as the rotating applied field . We detect ed the torque component, \nmB , that is \nperpendicular to th is plane . \nWe performed first -principles DFT simulations using the Vienna Ab Initio  Simulation Package \n(VASP) [18][19] ,  to calculate  the two -fold anisotropy\n1K  and obtain an estimate for  the four-fold \nanisotropy  \n22K . The projector augmented wave (PAW) method [20] is used to describe the \nelectron -ion interaction. Kohn -Sham states are expanded into plane waves up to a kinetic energy \ncutoff of 600 eV . The Brillouin zone is sampled by  a \n21 21 7  Monkhorst -Pack [21] (MP) k-\npoint grid and the total energy is converged  self-consistently to within \n910−  eV. The local density \napproximation (LDA) [22] and the generalized -gradient approximation developed by Perdew, \nBurke, and Ernzerhof (PBE) [23] are used to describe the exchange -correlation  energy function , \nand results from the two different computational strategies  are compared . \nAchieving the extremely high accuracy for total energies that is required to compute the  (001) \nplane  magnetocrystalline anisotropy  that is  on the order of \neV  per magnetic unit cell is \nnumerically challenging; the required convergence parameters render it computationally too \nexpensive to perform such calculations fully self -consistently.  Instead , we use the convergence \nparameters quoted above to compute Kohn -Sham states, electron density, and relaxe d atomic \ngeometries for collinear magnetism and take non -collinear magnetism and spin -orbit coupling [24] \ninto account without self -consistency of the Hamiltonian, as described in Ref. [25]. \n \nRESULTS AND DISCUSSION  \n1. Magneti c susceptibility and domain wall motion  \nWe use d ata for the magnetic susceptibility as input for modeling the torque magnetometry data \nand to provide insight into the reorientation of antiferromagnetic domains in a n external  magnetic \nfield. Figure 1(a) summarizes the results  for the magnetic susceptibility  in the limit of small field . \nWe fixed the applied field at 10 mT , along the <100> or <001> direction, measured the induced  \n7 \n magnetic moment  while cooling from T = 398 K, and calculated the susceptibility,  \n/MH= , \nwhere M is the magnetization . The measured susceptibility is similar  to that in  a prior report  [14].  \nWhen the applied field is along an easy ax is, we must  take domain wall motion  into account . We \nassume  that a 10 mT external field is too weak to  significant ly affect the  domain  structure . We \nfurther assume that the magnetic moment generated by an  applied field  along  the <100> direction \n(the a-axis of the crystal)  has equal  contribut ions from two types  of domains that we label as D1 \n(Néel vector along <100>) and D2 (Néel vector along <010>) as illustrated  in Fig. 2 . For an applied \nfield in the (001) -plane, we define the susceptibility parallel to the Néel vector as \n\n  while that \nperpendicular to the Néel vector as \n⊥ . On the other hand, t he susceptibility  for an applied fie ld \nin the <001> direction is define d as \n⊥ . We expect\n⊥ and \n⊥  to be similar but d ue to the \ntetragonal symmetry of the crystal structure , \n⊥ and \n⊥   are not necessarily equal. We sho w \nbelow that the difference between \n⊥  and \n⊥  is less than 5%.  \nMeasurements of the magnetization as a function of field , see Fig. 1 (b), show that \nc  is constant \nfor H applied along the c-axis. For H along the a-axis, \na increases with field  at low field, and is \napproximately constant for an applied field > 1 T.  We attribute the field dependence of \na  to \ndomain wall motion  and the consequent evolution of the  populations  of domains with N éel vectors \nparallel and perpendicular to the applied field.   \nThe populations of the two degenerate  domains  can be estimated from the M vs H curve in Fig.  \n1(a) by expressing the field-induced magnetization as  \n|| || aM H H    ⊥⊥ =+  and \ncMH ⊥= ,  \nwhere \n⊥ and \n||  are the normalized domain fraction  perpendicular and parallel to the a-axis, \nrespectively, and \n||1 ⊥+= . We made three assumptions: (1) \n||0.5 ⊥==  at zero field; (2) \n1 ⊥\n at high field;  and (3) domain wall motion is reversible.  The field -dependent distribution of \ndomains parallel and perpendicular to the external field along the a-axis at T = 4 K is shown in \nFig. 1(c)  and are treated as free parameters ; we find  \n\n = 0.008 and \n⊥ anisotropy in the = 0.018 . \nFor an ideal co llinear antiferromagnet, we expect  \n\n = 0 at low temperatures  [9]. This is not what \nwe obser ved in our measurements. The reason is that there is a background contribution to the  \n8 \n magnetic susceptibility that we do not yet understand.  We assume that the background \nsusceptibility is isotropic.  \n2. Torque magnetometry  \nThe field -induced torque is  the cross product of the field -induced magnetic moment and the \napplied field, \n=mB . The direction of the induced magnetic moment m is given by the \nminimum in the total energy:\ntot m ani exE E E E= + + , where \nmE  is the magnetic energy; \naniE  is the \nmagne tocrystalline anisotropy energy ; and \nexE  is the exchange energy that couples the two \nsublattices. We refer to the condition\nm aniEE\n  as the low field limit and the condition\nm aniEE  \nas the intermediate field  regime [26]. We ignore a separate\nexE  term when we analyze th e torque \ndata in the (001) -plane  because we assume that the exchange interaction stays the same and can \nbe represented by susceptibility . For torque data in the (010) plane, our analysis is based on the \nanisotropy in the susceptibility, \n|| ⊥− , which is also related to the strength of the exchange \ninteraction.  \nFig. 2(a) shows the experimental geometry when the applied field is rotating in  the (010) -plane. \n\n is the angle between  the c-axis and the applied field. The induced moment s are also in the \n(010) -plane . Thus, the torque is along the <010> direction . In the  low temperature limi t, \n||0 =  if \nthe isotropic background is igonored ; the induced  moment of domain D1 is  therefore  along <001>  \nand the induced moment of D2  lies in the (010) plane between <001>  and <100> . The direction of  \nthe induced moment of D2 is determined by \n⊥  and \n⊥. \nFig. 2(b) shows the experimental geometry  when  the applied  field is rotating in the (001) -plane . \nIn this case, because of the relatively small magnetocrystalline  anisotropy, the tilt of the sublattice \nmagnetization away from the crystal axes is significant . \n is the angle between the external field \nand the crystal axes ; \n1, \n2 are the angle s between  the direction s of the sublattice magnetization  \nof domain s D1, D2 and the  crystal axes , respectively  (\n1 and \n2  are not necessar ily equal) . \n  \n9 \n 2.1 Torque magnetometry in the (010) -plane in the low  field limit  \nThe torque is zero when  the applied  field is along the easy or hard axis  of a sample , because the \ninduced magnetization is in the same direction as the field. Here , we refer  to the easy axis as the \nlowest energy orientation of the N éel vector , and define orientations of the  hard axis as \nperpendicular to the easy axis . When the applied field is  oriented  away from an easy or hard axis, \nthe direction of the induced magnetization shifts toward a hard axis because  \n|| ⊥ . In an AF, \nthe slope of  the torque  as a function of field orientation  has opposite signs when  the field passes \nthrough the orientation of an easy axis and when  it passes through the orientation of a  hard axis.   \nIn our sign convention, torque with a negative slope as a function of angle indicates a hard axis; \ntorque with a positive slope as a function of angle indicates a n easy axis. In a single magnetic \ndomain of Fe2As, there are two hard axes : the c-axis perpendicular to the (001) plane  and the axis \nperpendicular to the Néel vector  in the (001) plane .  \nTorque data at 4 K with the field rotating in the (010) -plane are shown in Fig.  3(a). The slope when \nthe field is along the a-axis (\n=  ) is positive at B = 0.5 T and changes to negative  at B > 0.5 \nT. Therefore,  when the 0.5 T field is oriented along  the a-axis, the a-axis is  an easy axis  but when \nB is greater than  0.5 T, the a-axis becomes  a hard axis . This interpretation is consistent with the \nanalysis of the domain  distribution discussed above and displayed in Fig. 1(c). When the applied \nfield along the a-axis is larger than 1 T, the majority of  domains are in the D2 configuration  and \nthe a-axis becomes a  hard axis .   \nThe slope of torque data when the applied  field is along  the c-axis (\n=  ) is negative because  the \nc-axis is a hard axis. However, when  \n1=   and B > 0.5 T , the sign of the  torque changes abruptly . \nThis dramatic  change  in the sign of the torque  is periodic ; the periodicity  indicates that  domain \nwall motion is reversible. When the applied field is aligned along the c-axis, the  populations  of \ndomain  D1 and D 2 are equal. As  the field rotates away from the c-axis, the projection of the applied \nfield in the (001) -plane changes the domain distribution as described by Fig. 1(c). When  the field \nreturns  to the c-axis, the populations  of domain  D1 an d domain D2 become equal again.   \n10 \n To model the torque data , we first calculate the direction  and magnitude of the induced \nmagnetization M by describing  the suscept ibility as a tensor  \ni ij j jMH  = [8].  As shown in Fig. \n2(a), there are two degenerate domains with their Néel vector s perpendicular to each other. For  \ndomains  of type D1, the Néel vector  is along the a-axis and the susceptibility tensor is  \n \n||\n100\n0 ' 0\n00D\n\n⊥\n⊥\n=   . (3) \nFor domains  of type D2, the Néel vector  is along the b-axis and the susceptibility tensor is  \n \n2 ||' 0 0\n00\n00D\n\n⊥\n⊥\n=   . (4) \nThe external field in the (010) -plane is \n( )0sin( 0 cos(THH  = ) )  , where 𝜉 is the angle \nbetween the c-axis and the external field as depicted in Fig. 2(a) . We consider  the effect of the \nprojection of the applied  field along  the a-axis 𝐻0sin(𝜉) on the domain distribution as described \nby the data of Fig. 1(c) . The torque signal of two types of domains are\n1 1 1/D D DV=  HB  and \n2 2 2/D D DV=  HB\n, while the total torque is the sum  \n12DD+  . To evaluate this model , we \nuse the measured magnetic susceptibility as shown in Fig. 1. The free parameter s are \n/⊥⊥  and\n||\n.  \nFig. 3(b) and (c) show the  calculated value s of \n1D and \n2D; the solid line s in Fig. 3(a) are  the \nsummation of  the two. The good correspondence  between the model and  the data supports our \nassertion that  domain wall motion is reversible.  \nThe difference in the sign of  \n1D and \n2D  contributes to the abrupt change in the torque signal near \nan angle of 10°. For D1 domain s, when the applied  field is rotating in  the (010) -plane, the induced \nmoment is always close to the c-axis because  \n|| is small . For D2  domain s, the field -induced \nmagnetization is \n( )0 sin( 0 cos(TMH    ⊥⊥ = ) ) . If \n⊥⊥= , the induced magnetization   \n11 \n \n( )0sin( 0 cos(TMH   ⊥= ) ) , would  be parallel to the applied field \nH. This would infer  that \nthere is no  torque signal from D2  domains and t he total to rque signal would be generated only by \nD1 domains ( Fig. 3(b) ). \nHowever, the total torque signal w e observe  is obviously different from what is depicted in Fig. \n3(b) (D1 domains only) . The torque signal  resembles a combination of two domains , hence, we \ncan conclude \n⊥⊥ . On the other hand , the dramatic change  in the sign of the total torque signal \nat \n1=   and T = 4 K indicates  that \n1D  and \n2D have opposite sign s. Since the induced moment \nof D1 domains is along the c-axis, the induced moment of D2  domains  must be  between B and the \na-axis. \nThe difference between  \n⊥ and\n⊥  is also observed in the dependence of  M on H (Fig. 1(b) ).  \nFigure 4(a) shows the difference between  \ncM  and \naM  as a function of applied field and \ntemperature. The bump of  \ncaMM−  at room temperature and below is due to domain wall motion; \nthe difference  at high fields is a result of  \n⊥⊥− .  \nThe magnetization of  D2 domains  in the (010) -plane experiences an anisotropic environment that \noriginates  from the difference of the a- and c-axis of the crystal . By fitting the model to the torque \ndata, we determine \n⊥⊥−  as a function of temperature ( Fig. 4(b) ).  We observe  a change in sign \nof \n⊥⊥−  near 200 K that is consistent with the anisotropy in the dependence of M on H (Fig. \n4(a)).  This change in sign indicates that the field-induced magnetization of D2  domains  is between  \nthe applied magnetic field  and the a-axis below 200 K, and between  the applied magnetic field  and \nthe c-axis above 200 K. The anisotropy in the perpendicular susceptibility, \n⊥⊥− , is  always \nsmall compare d to its absolute value :\n0.05\n⊥⊥\n⊥− . \n2.2 Torque magnetometry  in the (001) -plane under  intermediate field  \nFigure 5(a) shows torque data acquired  with the applied field rotating in the (001) plane . As \nexpected, the data show  four-fold symmetry . We attribute the  small  two-fold symmetry to a \nbackground  that comes from misalignment of the sample . (The c-axis is not precisely   \n12 \n perpendicular to the field direction. ) The amplitude of the four -fold contribution to the torque as a \nfunction of applied field  of three samples from the same growth run is plotted in Fig. 5(b) . \nTo quantify the  magnetic anisotropy  in the (001) -plane of Fe 2As, we analyze the torque data by \nminimiz ing the total energy  for D1 domains and D2 domains , respectively , then add \n1Dτ  and \n2Dτ  \ntogether to compare it to the data . When\nmE  is comparable to or larger than  \naniE  (\n(001)\nm aniEE ), \nNéel vector s start tilting away from  crystal axes as shown in Fig. 2(b) . We assume that the two \nsublattice magnetization s are approximately parallel to each other, so the exchange interaction is \nconsidered in magnetic energy . With the applied field rotating in the (001) -plane of Fe 2As, the \ntotal energy is   \n \n22 1 21/ ( ) ( , ) ( ) cos(4 )2T\ntot totE V H H K K K       = − + + +  , (5) \nwhere \n11 tot D D  = , \n1 =   for D1 domains and \n22 tot D D  = , \n2 =  for D2 domains . \nTo obtain  an accurate value of 𝐸𝑚, we rotate the susceptibility tensor together with the  Néel vector\n( ) ( ) ( )RTRR      =\n[8], where  \n is the angle between the Néel  vector  and the crystal axis, and\n()R\n is the rotation matrix:  \n \ncos( ) sin( ) 0\n( ) sin( ) cos( ) 0\n0 0 1R\n  − \n=   (6) \nDuring the rotation, the field component along  \n1DM  and \n2DM  result in a change in the populations \nof D1 and D2  domains . Thus, \n1D  and \n2D  are determined by the angle between the applied \nfield and the spin axis, \n()+ .  \n \nThe analogous behavior of a  uniaxial antiferromagnet  (AF) [10] provides a point  of comparison. \nIn a uniaxial AF , the critical field for the spin -flop transition is \n122( ) / ( )cH K K ⊥ = + −\n . For \nFe2As, as shown in  Fig. 1(c) and  Fig. 5(b), we do not observe a sudden change in the domain \npopulations that would  be characteristic of a spin-flop transition. Furthermore, in a perfect crystal   \n13 \n that is free from disorder , the single domain structure created by an ap plied field would  persist \nafter the field is removed . (In ferromagnets, domains form  to reduce t he contribution of the \nmagnetic energy of  stray fields to the total energy . In AF s, this driving force for domain formation \nis absent.) We attribute gradual and reversible domain movement in Fe 2As to random strain fields \ncreated by static disorder in the crystal that create local variations in the anisotropy energy .  \nThe midpoint of the chang e in the populations of the D1 and D2 domains  as a function of field  is, \nhowever, close to what is expected for the characteristic field of a spin -flop transition of an  ideal \nsingle domain  easy plane antiferromagnetic crystal . The total energy of D2 when \n90=\n  is [8] \n \n2 2 2\n1 2 22 2 || 211cos(4 ) ( ) cos ( ) .22totEK K K H HV     ⊥⊥ = + + + − + −  (7) \nBoth anisotropy and magnetic energy are function of \n2  but with different p eriodicity. For an \napplied field along the <010> direction ( 𝜓 = 0º), \n2( 90 )totE=\n  is always a global minimum while \n2( 0 )totE=\n is at a local minimum under low fields and it becomes the global maximum when the \napplied field is larger than a characteristic field \ncH . We can derive  this critical field from \n2\n||\nmax\n22() 1arccos2 16 | |H\nK⊥−=\n in which  \nmax  represents the spin orientation when total energy is at \nglobal maximum under a known applied field. When \nc HH , \nmax 0 ( )c HH   =\n . Therefore, \n0 22 16 / ( ) 560 mTcHK  ⊥ = − =\n using the value of \n3\n22 150 J/m K=−  from our torque \nmagnetometry measurement as  described below .  \nWe attribute our observations that  domains begin to move in a n applied  field smaller than \ncH  and  \ndomain wall motion is not complete until  the applied field is  larger than \ncH to an inhomogeneous \ndistribution of local values of the magnetocrystalline anisotropy. We speculate that random strains \nin the crystal cause d by point and extended defects create a relatively broad distribution of \nanisotropy at different locations in the sample.  When the external  field is removed, the random \nstrain field  controls the  energy of the domain orientation  and the volume fraction of domains  \nrecovers its initial stat es.  \n14 \n The torque induced by the applied  field is the  derivative of the total energy  \n/totdE d= . In the  \nintermediate field  regime , i.e., \nm aniEE   or \nc HH , we assume that the sample is a single \ndomain and the sublattice magnetization is always nearly perpendicular to the applied  field, i.e. , \n/2   +\n. The “ intermediate field” regime refers to  an applied  field larger than the \ncharacteristic  field, but not large enough to significantly change the exchange interaction.  An \nimportant assumption here is that the tilt of the two sublattice magnetization s in the external field \nis small enough to  be neglected . With this approximation, the magnetic energy is nearly \nindependent of  \n and \n, \n2\n01\n2mEH ⊥ − . Then, the torque can be easily related to the anis otropy:  \n224 sin(4 )tot anidE dEKdd=  = −\n, where  \n no longer depends on the m agnitude of the applied \nfield.  \nA graduation reorientation of domains  of an easy -plane antiferromagnet  as a function of applied \nfield was also recently observed in  50 nm thick  CuMnAs epitaxial layers grown on GaP  [27]. \n(CuMnAs and Fe 2As have essentially the same crystal structure wit h Cu and Mn atoms in CuMnAs \noccupying the same lattice sites as the two crystallographically distinct Fe atoms in Fe 2As.) The \nstrength of the field needed to reorient antiferromagnetic domains in CuMnAs epitaxial layers is \nsimilar to what we observe in Fe 2As bulk crystals .  Thinner, 10 nm  thick,  CuMnAs layers show a \npronounced in -plane uniaxial anisotropy and a more abrupt transition in domain structure as a \nfunction of field than 50 nm thick layers.  X-ray magnetic linear dichroism (XMLD) measurements \nof CuMnAs epitaxial layers  reveals that the domain reorientation is not fully reversible and \nhysteretic  for fields less than 2 T.  X -ray photoelectron microscopy (XPEEM) images acquired \nafter applying 7 T fields in orthogonal directions also show that the domain structure does not \nrevert to a fixed configuration in zero field.  \nBase d on our observation in Fig 5(b), as field increases, \n  increases quickly then saturate s. At \nhigher field s, \n is slightly smaller than the saturation value, rather than staying the same until 9  \nT.  At higher field s, the tilting of spins caused by the external field cannot be neglected, so \nexchange interaction is no longer a constant . In our model, however, the torque  stays the same \nafter saturat ion based on our assumption of constant susceptibility and exchange interaction. This  \n15 \n assumption is no longer valid in higher field. While induced magnetization is smaller than \nH , \nthe torque is  also smaller than the saturation value.  \nAs our model  predicts , the experimentally measured  torque amplitude saturates as the applied field \napproaches 1 T for sample B, and 3 T for samples  A and  C. Hence, i t is safe to  select  1 T and 3 \nT as the “intermediate field” regime for sample B  and for sample s A and C , respectively . The \nmeasured  \n22K value of  sample A  is \n3-150 J/m . The field -dependen ce of \n22K  in all three samples \nfollow the same trend ; however, individual data points  do not overlap perfectly . We attribute this \ndiscrepancy  to variations in the defect microstructures and stoichiometries of the three samples.   \nWith a temperature -dependent measurement of torque in the ab-plane at an intermediate field,  we \nobtain  the temperature -dependence of \n22K  as shown in Fig . 6. The overall temperature  dependen ce \nis the similar for all three samples  with relatively minor differences . As temperature increases, the \nmagnitude of  \n22K decreases and becomes close to  zero at T > 150 K. \n22K  of sample A  becomes  \nslightly positive  for T > 150 K. From Eq. (2), the total energy reaches a minimum when the Néel \nvector is along the crystal a- and b-axis (\n0= or \n90 ) for \n220 K  at zero field . When  \n220 K , \nthe N éel vector lies in direction s with \n45 =    [8]. \n \n3. First -principles calculations of magnetocrystalline anisotropy  \nMagnetocrystalline anisotropy of Fe 2As has two contributions, one from spin -orbit interaction \n(SOI) and one from classical magnetic dipole -dipole interaction (MDD) : anisotropy  from SOI is \ncalculated using DFT for noncollinear magnetism with spin-orbit coupling ,  by rotating the Néel \nvector both within the easy plane (001) and out of the plane towards the hard axis (010).  The \ncorresponding total -energy changes are visualized in Fig.  S2 and the anisotropy energies are then \nobtained by fitting the energy change vs. Néel vector orientation to Eq. (2). This leads to a two -\nfold symmetric SOI anisotropy energy for the Néel vector in the (010) plane and a four -fold \nsymmetric one for the (001) plane. In DFT -LDA , the (010) plane ani sotropy energy is K\n1= -320 \nkJ/m\n3  and K\n22= -290 J/m\n3  for the (001) plane.  A non-zero \n22K  indicates the existence of two local  \n16 \n energy minima. In DFT -PBE, the corresponding values are K\n1= -530 kJ/m\n3  for the (010) plane \nand \n22K = 280 J/m\n3  for the (001) plane.  The sign of \n22K  differing between DFT -LDA and DFT -\nPBE implies that the energetic ordering of these two minima is inverted. Negative \n22K  means that \nthe energy minimum occurs for a Néel vector  along a <100>  equivalent direction and positive \n22K  \nfor a Néel vector   along a <110> equivalent direction.  \nThe MDD contribution is computed using a classical model that is parametrized using the chemical \nand magnetic ground -state structure from DFT . We use the ground -state chemic al and magnetic \nstructure from DFT -LDA as well as DFT -PBE, to evaluate the following expression for the \nclassical magnetic dipole -dipole interaction and to compare the influence of exchange and \ncorrelation:  \n \n2\n0\n53 ( ) ( ) ( ) ( ) 1\n24i ij i ij i j ij\nd\nij ijm r r m r r m r m r r\nEr\n       −    =−    (8) \nTo obtain the anisotropy energy for bulk Fe 2As from this expression, we use an interaction shell \nboundary \nijr  of 180  Å, which converges the result to within \n710−  eV. This leads to a two -fold \nsymmetric MDD contribution to the anisotropy energy in the (010) plane of K\n1= -220 kJ/m\n3  for \nLDA. For PBE, the corresponding value is  K\n1= -300 kJ/m\n3 . The MDD contribution in the (001) \nplane is less than 1 neV and, thus, negligible.  \nTherefore , we find a total out -of-plane anisotropy energy of -540 kJ/m\n3  and -830 kJ/m\n3  from LDA \nand PBE, respectively . We  attribute ~2/3 of the total out -of-plane anisotropy energy  to the SOI \ncontribution and ~1/3 to the MDD contr ibution. Both terms show two -fold symmetry with the hard \naxis along the <001>  direction.  Torque magnetometry can only measure a lower bound of \n1K\n36 kJ/m\n3 for the out -of-plane anisotropy energy  and does not contradict our DFT results.  \nThe in -plane anisotropy energy is computed as  \n22K= -290 J/m\n3  (DFT -LDA ) and \n22K = 280 J/m\n3  \n(DFT -PBE), while the measured result is \n22K  = -150 J/m\n3 .   \n17 \n  \n4. Antiferromagnetic resonance of easy plane antiferromagnets  \nWithout anisotropy, the magnon dispersion of energy in antiferromagnet is zero at the center of \nthe Brillouin zone. Anisotropy introduce s a band gap at the zone center. The antiferromagnetic \nresonance (AFMR) mode we refer to in this work describes this precessional magnetization motion  \nat the zone center.  With the anisotropy value s we determined by theory and experiment , we can \nmake an est imation of the AFMR frequency.  \nWe start from equation s of motion under the ‘macrospin’ approximation of the two magnetic \nsublattice s in domain D1  [28][29]: \n \n1 1 1\n1 22 2 22 2 1ˆ ˆˆ ()bc\nbc\nexdM M MM H H i M H j M H kdT M M      =  + + − + + − +            (9)\n \n   \n \n2 2 2\n2 22 1 22 1 1ˆ ˆˆ ()bc\nbc\nexdM M MM H H i M H j M H kdT M M      =  − − + − + + − +            (10) \nWhere  \n is the gyromagnetic ratio, \n1M  and \n2M  are sublattice magnetization s of domain D1, \n1H  \nand \n22H  are out -of-plane and in -plane anisotropy field s, respectively , which  can be written as \n11 / H K M=\n and \n22 22 / H K M= . \nexHM =  and \n is the inter -sublattice exchange interaction. \n12aaM M M= − \n, \nbM  and \ncM  are magnetization component s along the b- and the c-axis, \nrespectively,  during  spin process ion. \nBecause the two sublattice s along the a-axis are aligned antiparallel  to each other, the anisotropy \nfields along  the a-axis are of opposite sign s. Along  the b- and c-axes, the sign of the effective \nanisotropy  field is determined by the signs of \n1,2bM  and \n1,2cM . In domain D1, although the \nsublattice magnetization s stay along the a-axis, the in -plane anisotropy is of four-fold symmetry, \nso there is equivalent anisotropy  energy contribution along the a- and b-axes. The effective  \n18 \n anisotropy field s along the a- and b-axes are determined by the projection o f magnetization on \nthese axes.  \nThe only non -zero solution of the equation of motion requires  \n12bbMM=   and \n12ccMM=− , as \nshown in Fig. S2. The corresponding angular frequency can be expressed as  \n22 1 | | 2 ( )exH H H =−\n.  \nIn easy -plane AFs, \n10 K  and its absolute value is usually much larger  than \n22K , thus the \nfrequency is always real. Besides, the AFMR frequency is smaller with smaller \n22 1HH−  value , \nbecause the system is more isotropic. For easy -plane materials with  \n22 1 0 KK−\n  , the AFMR \nfrequency is dominated by  the anisotropy in the direction perpendicular to the easy plane.  \nFor the exchange field, we use sublattice magnetization  \n5\n14 10  A/mDM =  and an exchange \nintegral  \n1/⊥   with calculated \n⊥ = 0.003 6. We obtain  an exchange field \n140 TexH . With \ncalculated  \n1K value from  DFT -PBE, K\n1= -830 kJ/m\n3 , the AFMR frequency is \nf  = 670 GHz .  \nFor tetragonal antiferromagnets  like Fe 2As, the AFMR is dominated by \n1K  because\n1 22KK\n  . \nThe same relation is also valid for Mn 2Au where a previous  calculation  [30] shows that the \nmagnitude of the out-of-plane anisotropy is also much larger than the in-plane anisotropy. It is \nimportant to determine \n1K  to estimate AFMR frequency , and both \n1K  and \n22K  value are needed \nfor thermal stability of spintronics materials.  \nAs discussed in Ref s. [2] and [31], the electrical current typically  switch es only  a small number  of \nantiferro magnetic domains . As the anisotropy energy scales with sample volume, the total in-plane \nanisotropy energy is determined by the volum etric difference of the two kinds of domains, \n1 2 22()DD E V V K = −\n. \n12DDVV− depends on temperature, current density and the pulse width  [32]. \nIf the attempt frequency  is not high enough and \n12DDVV−  is small, the magnetic state is more \nsusceptible to  thermal fluctuations . \n  \n19 \n CONCLUSION  \nWe performed torque magnetometry measurement s on an easy-plane antiferromagnet Fe 2As. The \nmeasurement results prove that the domain wall motion  in the single -crystalline sample is \nreversible , and allow us  to extract  the in -plane anisotropy  when the magnetic energy  \nmE  is \ncomparable to magnetocrystalline anisotropy energy\naniE . The in -plane anisotropy of Fe 2As is K22 \n= -150 J/m3 at 4 K . \n22K is strongly temperature -dependent  and its magnitude decreases as a function \nof temperature.  This means that the domain structure  in Fe2As may be  easily perturbed  by a small \napplied field at room temperature . With \n1K = -830 kJ/m\n3   calculated from DFT, we derived the \nAFMR frequency \n22 1 2 ( )2ex f H H H\n=−  = 670 GHz . Our analysis of torque magnetometry \ndata suggest s that the in -plane magnetic anisotropy of some  candidate materials for \nantiferromagnetic spintronic applications, such as Fe2As, can be very small at room  temperature . \nA field smaller than 1 T is sufficient  to significantly alter its domain structure. The measurement \nof \n22K  in Fe2As provides a baseline value for further studies of magnetic anisotropy of easy -plane \nantiferromagnets and the motion of antiferro magnetic domain  walls . \n \nACKNOWLEDGEMENTS  \n \nThis work was undertaken as part of the Illinois Materials  Research Science and Engineering \nCenter, supported by the  National Science Foundation MRSEC program under NSF  Award No. \nDMR -1720633. This work made use of the Illinois  Campus  Cluster, a computing resource that is \noperated by  the Illinois Campus Cluster Program (ICCP) in conjun ction  with the National Center \nfor Supercomputing Applications  (NCSA) and which is supported by funds from the University  \nof Illinois at Urbana -Champaign. This research is part of  the Blue Waters sustained -petascale \ncomputing project, which  is supported by  the National Science Foundation (Awards  No. OCI -\n0725070 and No. ACI -1238993) and the state of  Illinois. Blue Waters is a joint effort of the \nUniversity of  Illinois at Urbana -Champaign and its National Center for  Supercomputing  \n20 \n Applications. Z.D. acknowled ges support from the Swedish  Research Council (VR) under Grant \nNo. 2015 -00585 , co-funded by Marie Skłodowska -Curie Actions (Project INCA  600398) . \n \nREFERENCES  \n[1] P. Wadley, B. Howells, J. Železný, C. Andrews, V. Hills, R.P. Campion, V. Novák, K. Olejník, F. \nMaccherozzi, S.S. Dhesi, S.Y. Martin, T. Wagner, J. Wunderlich, F. Freimuth, Y. Mokrousov, J. Kuneš, \nJ.S. Chauhan, M.J. Grzybowski, A.W. Rushforth, K.W. Edm onds, B.L. Gallagher, and T. Jungwirth, \nScience. 351, 587 (2016).  \n[2] M.J. Grzybowski, P. Wadley, K.W. Edmonds, R. Beardsley, V. Hills, R.P. Campion, B.L. Gallagher, \nJ.S. Chauhan, V. Novak, T. Jungwirth, F. Maccherozzi, and S.S. Dhesi, Phys. Rev. Lett. 118, 057701 \n(2017).  \n[3] K. Olejník, T. Seifert, Z. Kašpar, V. Novák, P. Wadley, R.P. Campion, M. Baumgartner, P. \nGambardella, P. Nemec, J. Wunderlich, J. Sinova, P. Kužel, M. Müller, T. Kampfrath, and T. Jungwirth, \nSci. Adv. 4, eaar3566 (2018).  \n[4] M. Meinert, D. Graulich, and T. Matalla -Wagner, Phys. Rev. Appl. 9, 064040 (2018).  \n[5] H.J. Richter, J. Phys. D. Appl. Phys. 32, R147 (1999).  \n[6] J.J. Nowak, R.P. Robertazzi, J.Z. Sun, G. Hu, J.H. Park, J. Lee, A.J. Annunziata, G.P. Lauer, R. \nKothandar aman, E.J. O’Sullivan, P.L. Trouilloud, Y. Kim, and D.C. Worledge, IEEE Magn. Lett. 7, 1 \n(2016).  \n[7] A. V. Khvalkovskiy, D. Apalkov, S. Watts, R. Chepulskii, R.S. Beach, A. Ong, X. Tang, A. Driskill -\nSmith, W.H. Butler, P.B. Visscher, D. Lottis, E. Chen, V.  Nikitin, and M. Krounbi, J. Phys. D. Appl. \nPhys. 46, 074001 (2013).  \n[8] M. Herak, M. Miljak, G. Dhalenne, and A. Revcolevschi, J. Phys. Condens. Matter 22, 026006 \n(2010).  \n[9] Coey, Magnetism and Magnetic Materials  (Cambridge University Press, 2009).  \n[10] U. Gäfvert, L. Lundgren, B. Westerstrandh, and O. Beckman, J. Phys. Chem. Solids 38, 1333 (1977).  \n[11] C. Zener, Phys. Rev. 96, 1335 (1954).  \n[12] J.C. Burgiel and M.W.P. Strandberg, J. Phys. Chem. Solids 26, 877 (1965).  \n[13] P. Pincus, Phys. Rev. 113, 769 (1959).  \n[14] H. Katsuraki and N. Achiwa, J. Phys. Soc. Japan 21, 2238 (1966).  \n[15] H. Katsuraki and K. Suzuki, J. Appl. Phys. 36, 1094 (1965) . \n[16] N. Achiwa, S. Yano, M. Yuzuri, and H. Takaki, J. Phys. Soc. Japan 22, 156 (1967).  \n[17] K. Yang, K. Kang, Z. Diao, A. Ramanathan, M.H. Karigerasi, D.P. Shoemaker, A. Schleife, and \nD.G. Cahill, Phys. Rev. Mater. 3, 124408 (2019).  \n[18] G. Kresse and J.  Furthmüller, Phys. Rev. B 54, 11169 (1996).   \n21 \n [19] G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).  \n[20] P.E. Blöchl, Phys. Rev. B 50, 17953 (1994).  \n[21] H.J. Monkhorst and J.D. Pack, Phys. Rev. B 13, 5188 (1976).  \n[22] D.M. Ceperley and B.J. Alder, Phys. Rev. Lett. 45, 566 (1980).  \n[23] J.P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).  \n[24] D. Hobbs, G. Kresse, and J. Hafner, Phys. Rev. B 62, 11556 (2000).  \n[25] A. Schrön, C. Rödl, and F. Bech stedt, Phys. Rev. B 86, 115134 (2012).  \n[26] T. Nagamiya, K. Yosida, and R. Kubo, Adv. Phys. 4, 1 (1955).  \n[27] M. Wang, C. Andrews, S. Reimers, O.J. Amin, P. Wadley, R.P. Campion, S.F. Poole, J. Felton, \nK.W. Edmonds, B.L. Gallagher, A.W. Rushforth, O. Makar ovsky, K. Gas, M. Sawicki, D. Kriegner, K. \nOlejník, V. Novák, T. Jungwirth, M. Shahrokhvand, U. Zeitler, S.S. Dhesi, and F. Maccherozzi, Phys. \nRev. B 101, 094429 (2020).  \n[28] C. Kittel, Introduction to Solid State Physics  (Wiley, 1953).  \n[29] C. Kittel, Phy s. Rev. 82, 565 (1951).  \n[30] A.B. Shick, S. Khmelevskyi, O.N. Mryasov, J. Wunderlich, and T. Jungwirth, Phys. Rev. B 81, \n212409 (2010).  \n[31] P. Wadley, S. Reimers, M.J. Grzybowski, C. Andrews, M. Wang, J.S. Chauhan, B.L. Gallagher, R.P. \nCampion, K.W. Edmon ds, S.S. Dhesi, F. Maccherozzi, V. Novak, J. Wunderlich, and T. Jungwirth, Nat. \nNanotechnol. 13, 362 (2018).  \n[32] T. Matalla -Wagner, M.F. Rath, D. Graulich, J.M. Schmalhorst, G. Reiss, and M. Meinert, Phys. Rev. \nAppl. 12, 064003 (2019).  \n \n \n \n \n \n \n \n \n \n \n \n  \n22 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n (a) (b) (c) \nFigure 1. (a) Temperature dependence of the  magnetic susceptibility of Fe 2As in the low field \nlimit as measured using 10 mT  field applied along the a-axis (blue data points)  and c-axis  (black \ndata points) of the crystal . (b)  Dependence of Fe 2As magnetization M on applied field  H at T \n= 4 K. With H along the c-axis (red line), M is a linear function of H. With H along the a-axis \n(black curve), the non -linear dependence of M on H is due to the rotation of antiferromagnetic \ndomains. (c) The population of domain s with Néel vectors parallel and perpendicular to the \napplied field estimated from the dependence of M on H. The assumptions are: 1) in zero field, \nthe population of domains with Néel vectors in the a and b directions are equal; and 2) in the \nhigh field limit, the Néel vector is perpendicular to the applied field.  \n0.0 0.5 1.0 1.5 2.0051015202530\nH || cH || aM (kA m-1)\nField (T)\n0.0 0.5 1.0 1.5 2.00.00.20.40.60.81.0\nField (T)⊥\n||\n0 100 200 300 4000.0000.0050.0100.015\nTemperature (K)ac \n23 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Figure 2. Geometry of the torque magnetometry experiments. The a-b-c coordinate s are the \ncrystal axes.  MD1 and MD2 are the sublattice magnetization s of the two types  of domains  labeled \nas D1 and D2. (a) The magnetic unit cell of Fe 2As. (b) Three -dimensional perspective of the \nmeasurement with the magnetic field rotating in the ac-plane. The magnetic field makes an \nangle \n  with the c-axis of the crystal. MD1 and MD2 are assumed to stay along the a- and the b-\naxis, respectively. T he torque is along the b-axis. ( c) Plan-view of the measurement with the \nmagnetic field rotating in the ab-plane. The magnetic field makes an angle \n  with the b-axis \nof the crystal.  MD1 and MD2 tilt away  from a- and b-axis by  \n1 and \n2 , respectively (\n1 and \n2  \nare not necessarily the same) . The torque is along  the c-axis (normal to the plane of the drawing) . (a) (b) \n (c) \n \n24 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n (a)  (b)  (c) \nFigure 3. (a) Torque magnetometry measurement s in the  ac-plane of Fe 2As at  T = 4 K. Open \nsymbols are measured data; solid lines are fits to the data. The legend gives the magnitude of \nthe applied field labeled by color.  (b) Calculated torque generated by  domains of type D1 as a \nfunction of applied field. (c) Calculated torque generated by  domains of type D2 as a function \nof applied field.  \n0 90 180 270 360-4-3-2-101234\n3 T\n2.5 T\n2 T\n1.5 T\n1 T\n0.5 TTorque (kN m-2)\n (deg)\n0 90 180 270 360-4-3-2-101234Torque (kN m-2)\n (deg)3 T\n2.5 T\n2 T\n1.5 T\n1 T\n0.5 T\n0 90 180 270 360-4-3-2-101234\n3 T\n2.5 T\n2 T\n1.5 T\n1 T\n0.5 T\n (deg)Torque (kN m-2) \n25 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Figure 4 . (a) The field dependence of the difference \ncaMM−  between the magnetization with \nan applied field along the c direction  \ncM and the field applied along the a direction \naM . Each \ncurve is labeled by the measurement temperature.  When the applied field along the a-axis is \nlarger than 1.5 T , all domains can be treated as equivalent to  MD2. Therefore , the slope of the \ndata for magnetic fields larger than 1.5 T  is the difference in the susceptibility \n⊥⊥− , where \n⊥\n is the susceptibility in ab -plane perpendicular to MD2 and \n⊥  is the susceptibility along c-\naxis perpendicular to MD2. (b) Comparison of the temperature dependent of \n⊥⊥−  value from \ndirect  measurements  of the type shown in panel (a) and from fitting  the torque data . (a) (b) \n0.0 0.5 1.0 1.5 2.0-2.0-1.5-1.0-0.50.00.51.0Mc-Ma (kA m-1)\nField (T)4 K77 K200 K300 K\n360 K\n0 50 100 150 200 250 300 350 400-6-4-2024681012\nTorque fittingLinear fitting Mc- Ma \nat high field'⊥-⊥ (10-4)\nTemperature (K) \n26 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Figure 5 . (a) Torque magnetometry measurement s of Fe 2As (sample A ) in the ab-plane at T=4 \nK. (b) The  amplitude of the  four-fold component of the torque  extracted from measurements of \nthe type shown in panel (a) at T = 4 K  and compared to an analytical model  (see text) . When \namplitude of the four -fold component of the torque saturates at the value \n0  , the in -plane \nanisotropy is 𝜏0 ≈ 4𝐾22.  \n (a) (b) \n0 2 4 6 8 100200400600800Torque amplitude (N m-2)\nField (T)sample B\nsample A\nsample C\nModel\n0 90 180 270 360-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0Torque (kN m-2)\n (deg)3 T\n2 T\n1.5 T\n1 T\n0.5 T \n27 \n  \n \n \n \nFigure 6. Temperature dependence of in-plane magnetocrystalline anisotropy 𝐾22 of Fe 2As. The torque data \nwith the external field rotating in the (001) plane were measured with an intermediate field strength (3 T for \nsample A  and sample C , and 1 T for sample B ) and th e amplitude of the four-fold symmetry was extracted to \nobtain  the in -plane anisotropy with 𝜏0 ≈ 4𝐾22. Intermediate field is  defined as a field  strength under which  the \ntorque amplitude of four-fold symmetry saturates at 4 K.  The error bars represent 20% uncertainty in \ndetermining the saturation value of the torque amplitude. All three samples have the same uncertainty, and we \nonly plot error bar for sample A to ensure that the plot is free from cluttering.   \n0 100 200 300 400-200-160-120-80-400K22 (J m-3)\nTemperature (K) sample A\n sample B\n sample C"    },    {        "title": "2007.08624v1.Anisotropic_magnetocaloric_effect_and_critical_behavior_in_CrSbSe__3_.pdf",        "content": "arXiv:2007.08624v1  [cond-mat.str-el]  15 Jul 2020Anisotropic magnetocaloric effect and critical behavior in CrSbSe 3\nYu Liu,1Zhixiang Hu,1,2and C. Petrovic1,2\n1Condensed Matter Physics and Materials Science Department ,\nBrookhaven National Laboratory, Upton, New York 11973, USA\n2Materials Science and Chemical Engineering Department,\nStony Brook University, Stony Brook, New York 11790, USA\n(Dated: July 20, 2020)\nWe report anisotropic magnetocaloric effect and critical be havior in quasi-one-dimensional ferro-\nmagnetic CrSbSe 3single crystal. The maximum magnetic entropy change −∆Smax\nMis 2.16 J kg−1\nK−1for easy aaxis (2.03 J kg−1K−1for hard baxis) and the relative cooling power RCPis 163.1\nJ kg−1for easy aaxis (142.1 J kg−1for hard baxis) near Tcwith a magnetic field change of 50\nkOe. The magnetocrystalline anisotropy constant Kuis estimated to be 148.5 kJ m−3at 10 K,\ndecreasing to 39.4 kJ m−3at 70 K. The rescaled ∆ SM(T,H) curves along all three axes collapse\nonto a universal curve, respectively, confirming the second order ferromagnetic transition. Further\ncritical behavior analysis around Tc∼70 K gives that the critical exponents β= 0.26(1), γ=\n1.32(2), and δ= 6.17(9) for H/bardbla, whileβ= 0.28(2), γ= 1.02(1), and δ= 4.14(16) for H/bardblb. The\ndetermined critical exponents suggest that the anisotropi c magnetic coupling in CrSbSe 3is strongly\ndependent on orientations of the applied magnetic field.\nI. INTRODUCTION\nLow-dimensional ferromagnetic (FM) semiconductors,\nholding both ferromagnetism and semiconducting char-\nacter, form the basis for nano-spintronics application.\nRecently, the two-dimensional (2D) CrI 3and Cr 2Ge2Te6\nhave attracted much attention since the discovery of in-\ntrinsic 2D magnetism in mono- and few-layer devices.1–4\nIntrinsic magnetic order is not allowed at finite temper-\nature in low-dimensional isotropic Heisenberg model by\nthe Mermin-Wagner theorem,5however, a large magne-\ntocrystalline anisotropy removes this restriction, for in-\nstance,thepresenceofamagneticallyorderedstateinthe\n2D Ising model. The enhanced fluctuations in 2D limit\nmake symmetry-breaking order unsustainable, however\nby gapping the low-energy modes through the introduc-\ntion of anisotropy, order could be established by provid-\ning stabilization of long-range correlations in 2D limit.\nGiven the reduced crystal symmetry in low-dimensional\nmagnets, an intrinsic magnetocrystalline anisotropy can\nbe expected and points to possible long-range magnetic\norder in atomic-thin limit.\nIn ternary chromium trichalcogenides, Cr(Sb,Ga)X 3\n(X = S, Se, Te) displays a pseudo-one-dimensional crys-\ntal structure. This is different from Cr(Si,Ge)Te 3that\nfeatures layered structure and a van der Waals bonds\nalong the c-axis. In Cr(Sb,Ga)X 3, the CrX 6octahedra\nform infinite, edge-sharing, double rutile chains. The\nneighboring chains are linked by Sb or Ga atoms. The\nFM semiconductor CrSbSe 3has attracted considerable\nattention.6–9A band gap of0.7 eV was determined by re-\nsistivity and optical measurements.8The Cr in CrSbSe 3\nexhibits a high spin state with S= 3/2, and orders fer-\nromagnetically below the Curie temperature Tcof 71 K.8\nFM state in CrSbSe 3is fairly anisotropic with the aaxis\nbeing the easy axis and the baxis being the hard axis.\nThe critical analysis where magnetic field was applied\nalong the aaxis suggests that the ferromagnetism inCrSbSe 3cannot be simply described by the mean-field\ntheory.10,11This invites the detailed investigation on its\nanisotropic critical behavior.\nThe magnetocaloric effect (MCE) can give additional\ninsight into the magnetic properties, and it could be also\nused to assess magnetic refrigeration potential.12–20Bulk\nCrSiTe 3exhibits anisotropic entropy change ( −∆Smax\nM)\nwith the values of 5.05 and 4.9 J kg−1K−1at 50 kOe\nfor out-of-plane and in-plane fields, respectively, with the\nmagnetocrystalline anisotropy constant Kuof 65 kJ m−3\nat 5 K.14The values of −∆Smax\nMare about 4.24 J kg−1\nK−1(out-of-plane) and 2.68 J kg−1K−1(in-plane) at\n50 kOe for CrI 3with a much larger Kuof 300±50 kJ\nm−3at 5 K.21The large magnetocrystalline anisotropy\nis important in preserving FM in the 2D limit.\nInthepresentworkweinvestigatetheanisotropicmag-\nnetic properties of pseudo-one-dimensional CrSbSe 3sin-\ngle crystals. The magnetocrystalline anisotropy constant\nKuis strongly temperature-dependent.It takes a value of\n∼148.5 kJ m−3at 10 K and monotonically decreases\nto 39.4 kJ m−3at 70 K for the hard baxis. The Ku\nof CrSbSe 3is much larger than that of Cr 2(Si,Ge) 2Te6\nbut comparable with that of Cr(Br,I) 3. This results in\nanisotropicmagneticentropychange∆ SM(T,H)andrel-\native cooling power (RCP), as well as in magnetic criti-\ncal exponents β,γ, andδthat point to the nature of the\nphase transition. The anisotropic magnetic coupling of\nCrSbSe 3is strongly dependent on orientations of the ap-\nplied magnetic field, providing an excellent platform for\nfurthertheoreticalstudiesoflow-dimensionalmagnetism.\nII. EXPERIMENTAL DETAILS\nCrSbSe 3single crystals were fabricated by the self-flux\ntechniquestartingfrom anintimate mixtureofrawmate-\nrials Cr (99.95%, Alfa Aesar) powder, Sb (99.999%, Alfa\nAesar) pieces, and Se (99.999%, Alfa Aesar) pieces with2\nFIG. 1. (Color online). (a) Crystal structure of CrSbSe 3and\n(b) representative single crystals on a millimeter-grid pa per.\n(c) Powder x-ray diffraction (XRD) pattern of CrSbSe 3. (d)\nTemperature-dependent magnetization M(T) measured in H\n= 1 kOe with zero field cooling (ZFC) and field cooling (FC)\nmodes along all three axes (left axis) and inverseaverage ma g-\nnetization 1 /Mave= 3/(Ma+Mb+Mc) (right axis) fitted by\nthe Curie-Weiss law. Inset shows the field-dependent magne-\ntization M(H) at 2 K.\na molar ratio of 7 : 33 : 60. The starting materials were\nsealed in an evacuated quartz tube and then heated to\n800◦C and slowlycooled to 680◦C with a rate of2◦C/h.\nNeedle-like single crystals with lateral dimensions up to\nseveral millimeters can be obtained. The element analy-\nsis was performed using an energy-dispersive x-ray spec-\ntroscopy in a JEOL LSM-6500 scanning electron micro-\nscope (SEM), confirming a stoichiometric CrSbSe 3single\ncrystal. The powder x-ray diffraction (XRD) data were\ntaken with Cu K α(λ= 0.15418 nm) radiation of Rigaku\nMiniflex powder diffractometer. The anisotropy of mag-\nnetic properties were measured by using one single crys-\ntal with mass of 0.32 mg and characterized by the mag-\nnetic property measurementsystem (MPMS-XL5, Quan-\ntum Design). The applied field ( Ha) was corrected as\nH=Ha−NM, whereMis the measured magnetization\nandNis the demagnetization factor. The corrected H\nwas used for the analysis of magnetic entropy change and\ncritical behavior.\nIII. RESULTS AND DISCUSSIONS\nFigure 1(a) displays the CrSbSe 3crystal structure.\nThe material crystalizes in an orthorhombic lattice with\nthe space group of Pnma. That is a pseudo-one-\ndimensional structure with double rutile chains of CrSe 6\noctahedra that are aligned parallel to the baxis. As\nshown in Fig. 1(b), the baxis is along the long crystal\ndimension in single crystals. Within the double chain,\nthe Cr cations form an edge-sharing triangular arrange-\nment, while the Sb atoms link the adjacent chains. Inthe powder XRD pattern [Fig. 1(c)], all peaks can be\nwell indexed by the orthorhombic structure (space group\nPnma) with lattice parameters a= 9.121(2) ˚A,b=\n3.785(2) ˚A andc= 13.383(2) ˚A, in good agreement with\nprevious report.8\nFigure1(d)showsthetemperaturedependenceofmag-\nnetization M(T) along all three axes measured at H= 1\nkOe. There is no bifurcation seen between the zero-field-\ncooling (ZFC) and field-cooling (FC) curves, indicating\nthe high quality of single crystal. The M(T) curves are\nnearly isotropic at high temperature but show an obvi-\nous anisotropic magnetic response for field applied along\ndifferent axes at low temperature. For H/bardbla, a rapid in-\ncrease near 70 K in M(T) on cooling corresponds well to\nthe reported paramagnetic (PM) to FM transition.8For\nH/bardblbandH/bardblc, an anomalous peak feature is observed,\nwhich is also seen in Cr 2(Si,Ge) 2Te6and Cr(Br,I) 3with\na large magnetocrystalline anisotropy.22–24The inverse\naverage magnetization 1 /Mave= 3/(Ma+Mb+Mc) can\nbe well fitted from 200 to 300 K by using the Curie-\nWeiss law, which generates an effective moment of 4.6(1)\nµB/Cr and a positive Weiss temperature of 125(1) K, in\nline with the values reported for CrSbSe 3polycrystals.6,7\nThe anisotropic magnetization isotherms measured at T\n= 2 K [inset in Fig. 1(d)] show a similar saturated mag-\nnetization Msat 3µB/Cr, consistent with expectation of\nS= 3/2 for Cr3+, but different saturated fields Hsof 1\nkOe, 18kOe, and 12kOefor a,b, andcaxis, respectively,\nclose to the values in the previous reports.8,9\nTo further characterize the anisotropic magnetic prop-\nertiesofCrSbSe 3,theisothermalmagnetizationwithfield\nup to 50 kOe applied along each axis from 10 to 100 K\nare presented in Figs. 2(a)-2(c). At high temperature,\nthe curves have linear field dependence. With decreasing\ntemperature, the curves bend with negative curvatures,\nindicating dominant FM interaction. Based on the clas-\nsical thermodynamics and Maxwell’s relation, the mag-\nnetic entropy change ∆ SM(T,H) is given by:25,26\n∆SM=/integraldisplayH\n0[∂S(T,H)\n∂H]TdH=/integraldisplayH\n0[∂M(T,H)\n∂T]HdH,\n(1)\nwhere [∂S(T,H)/∂H]T= [∂M(T,H)∂T]His based on\nthe Maxwell’s relation. For magnetization measured at\nsmall temperature and field intervals,\n∆SM=/integraltextH\n0M(Ti+1,H)dH−/integraltextH\n0M(Ti,H)dH\nTi+1−Ti.(2)\nThe calculated −∆SM(T,H) are presented in Figs. 2(d)-\n2(f). All the curves exhibit a peak feature near Tc.\nPeaks broads asymmetrically on both sides with increas-\ning field. The maximum of −∆SM(T,H) reach 2.16 J\nkg−1K−1, 2.11 J kg−1K−1, and 2.03 J kg−1K−1fora,\nb, andcaxis, respectively. It should be noted that all\nthe values of −∆SM(T,H) for easy aaxis are positive,\nhowever, the values for hard bandcaxes are negative at\nlow temperatures in low fields stemming from a strong\ntemperature-dependent magnetic anisotropy.183\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s48/s49/s50\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s48/s49/s50\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s48/s49/s50/s77/s32/s40/s101/s109/s117/s47/s103/s41\n/s72/s32/s40/s107/s79/s101/s41/s49/s48/s32/s75\n/s49/s48/s48/s32/s75\n/s72/s32/s47/s47/s32/s97/s40/s97/s41\n/s49/s48/s48/s32/s75/s49/s48/s32/s75\n/s40/s98/s41/s77/s32/s40/s101/s109/s117/s47/s103/s41\n/s72/s32/s40/s107/s79/s101/s41/s72/s32/s47/s47/s32/s98\n/s49/s48/s32/s75\n/s49/s48/s48/s32/s75/s40/s99/s41/s77/s32/s40/s101/s109/s117/s47/s103/s41\n/s72/s32/s40/s107/s79/s101/s41/s72/s32/s47/s47/s32/s99/s53 /s32/s32/s32/s32/s32/s32/s32 /s32/s49/s48\n/s49/s53 /s32/s32/s32 /s32/s50/s48\n/s50/s53 /s32/s32/s32 /s32/s51/s48\n/s51/s53/s32 /s32/s52/s48\n/s52/s53 /s32/s32/s32 /s32/s53/s48/s72/s32/s47/s47/s32/s97/s40/s100/s41/s45 /s83\n/s77/s40/s74/s47/s107/s103/s45/s75/s41\n/s84/s32/s40/s75/s41/s107/s79/s101\n/s107/s79/s101/s72/s32/s47/s47/s32/s98/s40/s101/s41/s45 /s83\n/s77/s40/s74/s47/s107/s103/s45/s75/s41\n/s84/s32/s40/s75/s41/s53 /s32/s32/s32/s32/s32/s32/s32 /s32/s49/s48\n/s49/s53 /s32/s32/s32 /s32/s50/s48\n/s50/s53 /s32/s32/s32 /s32/s51/s48\n/s51/s53/s32 /s32/s52/s48\n/s52/s53 /s32/s32/s32 /s32/s53/s48\n/s107/s79/s101\n/s53 /s32/s32/s32/s32/s32/s32/s32 /s32/s49/s48\n/s49/s53 /s32/s32/s32 /s32/s50/s48\n/s50/s53 /s32/s32/s32 /s32/s51/s48\n/s51/s53/s32 /s32/s52/s48\n/s52/s53 /s32/s32/s32 /s32/s53/s48/s72/s32/s47/s47/s32/s99\n/s40/s102/s41/s45 /s83\n/s77/s40/s74/s47/s107/s103/s45/s75/s41\n/s84/s32/s40/s75/s41\nFIG. 2. (Color online). (a-c) Typical initial isothermal ma g-\nnetization curves measured along all three axes with temper -\nature ranging from 10 to 100 K. (d-f) The corresponding cal-\nculated magnetic entropy change −∆SM(T) at various fields\nchange.\nBased on a generalized scaling analysis,27the normal-\nized magnetic entropy change, ∆ SM/∆Smax\nM, estimated\nfor each constant field, is scaled to the reduced tempera-\nturetas defined in the following equations:\nt−= (Tpeak−T)/(Tr1−Tpeak),T < T peak,(3)\nt+= (T−Tpeak)/(Tr2−Tpeak),T > T peak,(4)\nwhereTr1andTr2arethe lowerand upper referencetem-\nperatures at half-width full maximum of ∆ SM/∆Smax\nM.\nAs we can see, the normalized ∆ SM/∆Smax\nMnearTccol-\nlapses onto a universal curve at the indicated fields for\nall three axes [Figs. 3(a)-3(c)], indicating the second-\norder PM-FM transition in CrSbSe 3. The ineligible devi-\nation at low temperatures along the hard baxis is mostly\ncontributed by the magnetocrystalline anisotropy effect.\nBasedon the Stoner-Wolfarthmodel,28the magnetocrys-\ntalline anisotropy constant Kucan be estimated from\nthe saturation regime in the isothermal magnetization\ncurves. Within this model the value of Kuin single do-\nmain state is related to the saturation magnetization Ms\nand the saturation field Hswithµ0is the vacuum per-\nmeability:\n2Ku\nMs=µ0Hs. (5)/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s49/s50\n/s48/s49/s50/s45/s54 /s45/s52 /s45/s50 /s48 /s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s45/s56 /s45/s54 /s45/s52 /s45/s50 /s48 /s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s45/s56 /s45/s54 /s45/s52 /s45/s50 /s48 /s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48/s48/s52/s48/s56/s48/s49/s50/s48/s49/s54/s48\n/s48 /s52/s48 /s56/s48/s50/s48/s51/s48/s52/s48/s48 /s52/s48 /s56/s48/s53/s49/s48/s49/s53/s48 /s50/s53 /s53/s48/s52/s48/s56/s48/s49/s50/s48\n/s48 /s50/s53 /s53/s48/s52/s48/s56/s48/s49/s50/s48\n/s48 /s50/s53 /s53/s48/s52/s48/s56/s48/s49/s50/s48\n/s45 /s83/s109 /s97/s120\n/s77/s61/s32/s97/s72/s110\n/s72/s47/s47/s97\n/s72/s47/s47/s98\n/s72/s47/s47/s99/s45 /s83/s109/s97/s120 /s77\n/s40/s74/s47/s107/s103/s45/s75/s41\n/s72/s32/s40/s107/s79/s101/s41/s40/s101/s41\n/s82/s67/s80/s61/s32/s98/s72/s109 /s72/s47/s47/s97\n/s72/s47/s47/s98\n/s72/s47/s47/s99\n/s82/s67/s80/s32/s40 /s49/s48/s50\n/s74/s47/s107/s103/s41/s32/s32/s32/s83\n/s77/s47 /s83/s109/s97/s120 /s77\n/s116/s32/s32/s32/s107/s79/s101\n/s32/s49/s48/s32 /s32/s50/s48/s32 /s32/s51/s48\n/s32/s52/s48/s32 /s32/s53/s48/s40/s97/s41\n/s72/s32/s47/s47/s32/s97/s72/s32/s47/s47/s32/s98/s40/s98/s41\n/s32/s32/s32\n/s116/s72/s32/s47/s47/s32/s99/s40/s99/s41\n/s32/s32/s32\n/s116\n/s40/s100/s41\n/s32/s32/s75\n/s117/s32/s40/s107/s74/s47/s109/s51\n/s41\n/s84/s32/s40/s75/s41/s72/s32/s47/s47/s32/s98/s77\n/s115/s32/s40/s101/s109/s117/s47/s103/s41\n/s84/s32/s40/s75/s41\n/s72\n/s115/s32/s40/s107/s79/s101/s41\n/s84/s32/s40/s75/s41/s72/s32/s40/s107/s79/s101/s41/s32/s84\n/s50\n/s32/s84\n/s49/s32/s84\n/s50\n/s32/s84\n/s49\n/s72/s32/s40/s107/s79/s101/s41/s32/s84\n/s50\n/s32/s84\n/s49\n/s72/s32/s40/s107/s79/s101/s41\nFIG. 3. (Color online). (a-c) Normalized magnetic entropy\nchange ∆ SMas a function of the reduced temperature t\nalong all three principal crystallographic axes of CrSbSe 3. In-\nsets show the evolution of the reference temperatures T1and\nT2. (d) Temperature dependence of the magnetocrystalline\nanisotropy constant Ku, the saturation field Hs, and the sat-\nuration magnetization Ms(insets) estimated from the hard\nbaxis below Tcfor CrSbSe 3. (e) Field dependence of the\nmaximum magnetic entropy change −∆Smax\nMand the rela-\ntive cooling power (RCP) with power-law fitting in solid line s\nalong all three axes for CrSbSe 3.\nWhenH/bardblb, the anisotropy becomes maximal. We esti-\nmated the Msby usinga linearfit of M(H) above20kOe\nforH/bardblb, which monotonically decreases with increasing\ntemperature[insetinFig. 3(d)]. Thenwedeterminedthe\nHsas the intersection point of two linear fits: one being\na fit to the saturated regime at high field, and the other\nbeing a fit of the unsaturated linear regime at low field.\nThe values of Hssharea similartemperature dependence\n[inset in Fig. 3(d)], resulting in a strongly temperature-\ndependent Ku[Fig. 3(d)]. The calculated Kuis∼148.5\nkJ m−3at 10 K for CrSbSe 3, which is much larger than\nthat of Cr 2(Si,Ge) 2Te6,18and comparable with that of\nCr(Br,I) 3.21\nAnother parameter to characterize the potential mag-\nnetocaloric effect of materials is the relative cooling\npower (RCP):29\nRCP=−∆Smax\nM×δTFWHM, (6)\nwhere the FWHM denotes the full width at half max-\nimum of −∆SMcurve. The RCPcorresponds to the4\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s50/s52/s54/s56/s49/s48\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s50/s52/s54/s56/s49/s48\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s50/s52/s54/s56/s49/s48/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53/s48/s49/s50/s51/s52/s53/s54\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53/s48/s49/s50\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53/s48/s49/s50/s51/s52/s53/s54/s77/s50\n/s32/s40 /s49/s48/s50\n/s101/s109/s117/s50\n/s47/s103/s50\n/s41\n/s72/s47/s77/s32/s40/s107/s79/s101/s45/s103/s47/s101/s109/s117/s41/s40/s97/s41/s72/s32/s47/s47/s32/s97\n/s72/s32/s47/s47/s32/s98/s40/s98/s41/s77/s50\n/s32/s40 /s49/s48/s50\n/s101/s109/s117/s50\n/s47/s103/s50\n/s41\n/s72/s47/s77/s32/s40/s107/s79/s101/s45/s103/s47/s101/s109/s117/s41\n/s72/s32/s47/s47/s32/s99/s40/s99/s41/s77/s50\n/s32/s40 /s49/s48/s50\n/s101/s109/s117/s50\n/s47/s103/s50\n/s41\n/s72/s47/s77/s32/s40/s107/s79/s101/s45/s103/s47/s101/s109/s117/s41/s72/s32/s47/s47/s32/s97/s40/s100/s41\n/s32/s32/s77/s49/s47\n/s32/s40 /s49/s48/s53\n/s40/s101/s109/s117/s47/s103/s41/s49/s47\n/s41\n/s40/s72/s47/s77/s41/s49/s47\n/s32/s40/s107/s79/s101/s45/s103/s47/s101/s109/s117/s41/s49/s47\n/s72/s32/s47/s47/s32/s98/s40/s101/s41\n/s32/s32/s77/s49/s47\n/s32/s40 /s49/s48/s53\n/s40/s101/s109/s117/s47/s103/s41/s49/s47\n/s41\n/s40/s72/s47/s77/s41/s49/s47\n/s32/s40/s107/s79/s101/s45/s103/s47/s101/s109/s117/s41/s49/s47\n/s72/s32/s47/s47/s32/s99/s40/s102/s41\n/s32/s32/s77/s49/s47\n/s32/s40 /s49/s48/s53\n/s40/s101/s109/s117/s47/s103/s41/s49/s47\n/s41\n/s40/s72/s47/s77/s41/s49/s47\n/s32/s40/s107/s79/s101/s45/s103/s47/s101/s109/s117/s41/s49/s47\nFIG. 4. (Color online). The Arrott plot of M2vsH/M(a-c)\nand the modified Arrott plot of M1/βvs (H/M)1/γ(d-f) with\nindicated βandγfora,bandcaxis, repsectively.\nTABLE I. The values of magnetic entropy change ( −∆Smax\nM)\nand relative cooling power (RCP) at field change of 50 kOe.\nCritical exponents of CrSbSe 3alonga,b, andcaxis, recpec-\ntively. The MAP, KFP, and CI represent the modified Arrott\nplot, theKouvel-Fisherplot, and thecritical isotherm, re spec-\ntively.\n−∆Smax\nMRCPTcβ γ δ\n(J/kg-K) (J/kg)\nH/bardbla2.16 163.1\nMAP 70.2(1) 0.26(1) 1.32(2) 6.08(12)\nKFP 70.2(1) 0.26(1) 1.33(2) 6.12(12)\nCI 6.17(9)\nH/bardblb2.03 142.1\nMAP 70.1(2) 0.28(2) 1.02(1) 4.64(22)\nKFP 70.3(2) 0.32(2) 1.03(1) 4.22(17)\nCI 4.14(16)\nH/bardblc2.11 154.1\nMAP 69.8(2) 0.26(1) 1.14(5) 5.38(2)\nKFP 69.8(2) 0.25(1) 1.19(4) 5.76(2)\nCI 5.35(17)/s49 /s49/s48/s49/s48/s50/s48/s51/s48\n/s54/s48 /s54/s50 /s54/s52 /s54/s54 /s54/s56 /s55/s48 /s55/s50 /s55/s52 /s55/s54 /s55/s56 /s56/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s32\n/s84/s32/s40/s75/s41/s77\n/s115/s32/s40/s101/s109/s117/s47/s103/s41/s40/s97/s41\n/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53\n/s32 /s32/s40/s107/s79/s101/s45/s103/s47/s101/s109/s117/s41\n/s54/s48 /s54/s50 /s54/s52 /s54/s54 /s54/s56 /s55/s48 /s55/s50 /s55/s52 /s55/s54 /s55/s56 /s56/s48/s45/s52/s48/s45/s51/s48/s45/s50/s48/s45/s49/s48/s48\n/s32/s72/s32/s47/s47/s32/s97 /s32\n/s32/s72/s32/s47/s47/s32/s98 /s32\n/s32/s72/s32/s47/s47/s32/s99 /s32/s40/s98/s41/s32/s32/s72/s32/s47/s47/s32/s97 /s32\n/s32/s72/s32/s47/s47/s32/s98 /s32\n/s32/s72/s32/s47/s47/s32/s99 /s32\n/s84/s32/s40/s75/s41/s77\n/s115/s40/s100/s77\n/s115/s47/s100/s84/s41/s45/s49\n/s32/s40/s75/s41\n/s48/s50/s52/s54/s56/s49/s48\n/s40/s100/s45/s49 /s48/s47/s100/s84/s41/s45/s49/s32/s40/s75/s41/s77/s32/s40/s101/s109/s117/s47/s103/s41\n/s72/s32/s40/s107/s79/s101/s41/s32/s72/s32/s47/s47/s32/s97\n/s32/s72/s32/s47/s47/s32/s98\n/s32/s72/s32/s47/s47/s32/s99/s84\n/s99/s32/s61/s32/s55/s48/s32/s75\nFIG. 5. (Color online). (a) Temperature dependence of the\nspontaneous magnetization Ms(left) and the inverse initial\nsusceptibility χ−1\n0(right)withsolidfittingcurvesfor CrSbSe 3.\nInset shows the log 10Mvs log 10Hat 70 K with linear fitting\ncurve. (b) Kouvel-Fisher plots of Ms(dMs/dT)−1(left) and\nχ−1\n0(dχ−1\n0/dT)−1(right) with solid fittingcurves for CrSbSe 3.\namount of heat which could be transferred between cold\nand hot parts of the refrigerator in an ideal thermody-\nnamic cycle. The calculated RCP values of CrSbSe 3\nreach maxima at 50 kOe of 163.1 J kg−1, 142.1 J kg−1,\nand 154.1 J kg−1fora,b, andcaxis, respectively [Fig.\n3(e)]. In addition, the field dependence of −∆Smax\nMand\nRCP can be well fitted by using the power-law relations\n−∆Smax\nM=aHnandRCP=bHm[Fig. 3(e)].30\nFor a second-order PM-FM phase transition, the\nspontaneous magnetization ( Ms) below Tc, the initial\nmagnetic susceptibility ( χ−1\n0) above Tc, and the field-\ndependent magnetization (M) at Tccan be characterized\nby a set of critical exponents β,γ, andδ, respectively.31\nThemathematicaldefinitionsofthe exponentsfrommag-\nnetization measurement are given below:\nMs(T) =M0(−ε)β,ε <0,T < T c, (7)\nχ−1\n0(T) = (h0/m0)εγ,ε >0,T > T c,(8)\nM=DH1/δ,T=Tc, (9)\nwhereε= (T−Tc)/Tc;M0,h0/m0andDare the critical\namplitudes.32For the original Arrott plot, β= 0.5 and5\n/s48/s46/s49 /s49 /s49/s48/s53/s49/s48/s49/s53/s50/s48\n/s48/s46/s49 /s49 /s49/s48/s53/s49/s48/s49/s53/s50/s48\n/s48/s46/s49 /s49 /s49/s48/s53/s49/s48/s49/s53/s50/s48\n/s84/s32/s62/s32/s84\n/s99\n/s32/s32/s77/s124 /s124/s45\n/s40/s101/s109/s117/s47/s103 /s41\n/s72/s124 /s124/s45/s40 /s41\n/s32/s40/s107/s79/s101/s41/s40/s97/s41/s32/s32/s72/s32/s47/s47/s32/s97\n/s32/s61/s32/s48/s46/s50/s54\n/s32/s61/s32/s49/s46/s51/s51/s84/s32/s60/s32/s84\n/s99\n/s84/s32/s62/s32/s84\n/s99/s84/s32/s60/s32/s84\n/s99\n/s32/s61/s32/s48/s46/s51/s50\n/s32/s61/s32/s49/s46/s48/s51/s40/s98/s41/s32/s32/s72/s32/s47/s47/s32/s98\n/s32/s32/s32\n/s72/s124 /s124/s45/s40 /s41\n/s32/s40/s107/s79/s101/s41/s84/s32/s62/s32/s84\n/s99/s84/s32/s60/s32/s84\n/s99\n/s32/s61/s32/s48/s46/s50/s53\n/s32/s61/s32/s49/s46/s49/s57/s40/s99/s41/s32/s32/s72/s32/s47/s47/s32/s99/s32/s32\n/s72/s124 /s124/s45/s40 /s41\n/s32/s40/s107/s79/s101/s41\nFIG. 6. (Color online). Scaling plots of renormalized m=\nM|ε|−βvsh=H|ε|−(β+γ)above and below Tcfor CrSbSe 3.\nγ= 1.0.33Based on this, the magnetization isotherms\nM2vsH/Mshould be a set of parallel straight lines.\nThe isotherm at the critical temperature Tcshould pass\nthrough the origin. As shown in Figs. 4(a-c), all curves\nin the Arott plot of CrSbSe 3are nonlinear, with a down-\nward curvature, indicating that the Landau mean-field\nmodel is not applicable to CrSbSe 3. From the Banerjee′s\ncriterion,34the first (second) order phase transition cor-\nresponds to a negative (positive) slope. Thus, the down-\nwardslope confirmsits a second-orderPM-FM transition\nin CrSbSe 3. In the more general case, the Arrott-Noaks\nequation of state provides a modified Arrott plot:35\n(H/M)1/γ=aε+bM1/β, (10)\nwhereε= (T−Tc)/Tcandaandbare fitting constants.\nFigures 4(d-f) present the modified Arrott plots for all\nthree axes with a self-consistent method,36,37showing a\nset of parallel quasi-straight lines at high field.\nTo obtain the anisotropic critical exponents β,γand\nδ, the linearly extrapolated Ms(T) andχ−1\n0(T) against\ntemperature are plotted in Fig. 5(a). According to Eqs.\n(7) and (8), the solid fitting lines give that β= 0.26(1)\nandγ= 1.32(2) for easy aaxis, close to the reported\nvalues (β= 0.25 and γ= 1.38).8For the baxis,\nβ= 0.28(2) and γ= 1.02(1), while for the caxis,\nβ= 0.26(1) and γ= 1.14(5). This lies between the\nvalues of theoretical tricritical mean field model ( β=\n0.25 and γ= 1.0) and 3D Ising model ( β= 0.325 and\nγ= 1.24).38,39The value of βis outside of the win-\ndow 0.1≤β≤0.25 for a 2D magnet,40suggesting a\n3D magnetic behavior for quasi-1D CrSbSe 3. Accord-\ning to Eq. (9), the M(H) atTcshould be a straight\nline in log-log scale with the slope of 1 /δ. Such fitting\nyieldsδ= 6.17(9), 4.14(16), and 5.35(17), for a,b, andc\naxis, respectively, which agrees well with the calculated\nvalues from βandγbased on the Widom relation δ=\n1+γ/β.41Theself-consistencycanalsobecheckedviathe\nKouvel-Fisher method,42whereMs(T)/(dMs(T)/dT)−1andχ−1\n0(T)/(dχ−1\n0(T)/dT)−1plotted against tempera-\nture should be straight lines with slopes 1 /βand 1/γ,\nrespectively. The linear fits to the plots [Fig. 5(b)] yield\nβ= 0.26(1) and γ= 1.33(2) for aaxis,β= 0.32(2) and\nγ= 1.03(1) for baxis,β= 0.25(1) and γ= 1.19(4) for c\naxis, respectively, very close to the values obtained from\nmodified Arrott plot. All the critical exponents obtained\nfrom different methods are listed in Table I. It seems that\nthe critical behavior of CrSbSe 3is much different along\nthe different axis and cannot be described by any sin-\ngle model. However, it is clear that 3D critical behavior\ndominates in quasi-1D CrSbSe 3and the strong magne-\ntocrystalline anisotropy in CrSbSe 3plays an important\nrole in the origin of anisotropic critical exponents.\nScaling analysis can be used to estimate the reliability\nof the obtained critical exponents. According to scaling\nhypothesis, the magnetic equation of state in the critical\nregion obeys a scaling relation can be expressed as:\nM(H,ε) =εβf±(H/εβ+γ), (11)\nwheref+forT > T candf−forT < T c, respec-\ntively, are the regular functions. In terms of the variable\nm≡ε−βM(H,ε) andh≡ε−(β+γ)H, renormalized mag-\nnetization and renormalized field, respectively, Eq.(10)\nreduces to a simple form:\nm=f±(h). (12)\nIt implies that for a true scaling relation with the proper\nselection of β,γ, andδ, the renormalized mversush\ndata will fall onto two different universal curves; f+(h)\nfor temperature above Tcandf−(h) for temperature be-\nlowTc. Using the values of βandγobtained from the\nKouvel-Fisher plot, we have constructed the renormal-\nizedmvshplots in Fig. 6. It is clear seen that all the\nexperimental data collapse onto two different branches:\none above Tcand another below Tc, confirming proper\ntreatment of the critical regime.\nIV. CONCLUSIONS\nIn summary, we have studied in details the anisotropic\nmagnetocaloric effect and critical behavior of CrSbSe 3\nsingle crystal. The second-orderin nature of the PM-FM\ntransition near Tc= 70 K has been verified by the scal-\ning analysis of magnetic entropy change ∆ SM. A large\nmagnetocrystalline anisotropy constant Kuis estimated\nto be 148.5 kJ m−3at 10 K, comparable with that of\nCr(Br,I) 3. A set of critical exponents β,γ, andδalong\neach axis estimated from various techniques match rea-\nsonably well and follow the scaling equation, indicating a\n3D magnetic behavior in CrSbSe 3. Further neutron scat-\ntering and theoretical studies are needed to shed more\nlight on the anisotropic magnetic coupling in low dimen-\nsions.6\nACKNOWLEDGEMENTS\nThis work was supported by the US DOE-BES, Divi-\nsion of Materials Science and Engineering, under Con-\ntract No. de-sc0012704 (BNL).\n1B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein,\nR. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A.\nMcGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-\nHerrero, and X. D. Xu, Nature 546, 270 (2017).\n2M. A. McGuire, G. Clark, KC. S., W. M. Chance, G. E.\nJellison, Jr., V. R. Cooper, X. Xu, and B. C. Sales, Phys.\nRev. M1014001 (2017).\n3C. Gong, L. Li, Z. L. Li, H. W. Ji, A. Stern, Y. Xia, T.\nCao, W. Bao, C. Z. Wang, Y. Wang, Z. Q. Qiu, R. J. Cava,\nS. G. Louie, J. Xia, and X. Zhang, Nature 546, 265 (2017).\n4M. A. McGuire, H. Dixit, V. R. Cooper, and B. C. Sales,\nChem. Mater. 27, 612 (2015).\n5N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133\n(1966).\n6V. Volkov, G. V. Tendeloo, J. V. Landuyt, S. Amelinckx,\nE. Busheva, G. Shabunina, T. Aminov, and V. Novotort-\nsev, J. Solid State Chem. 132, 257 (1997).\n7D. A. Odink, V. Carteaux, C. Payen, and G. Ouvrard,\nChem. Mater. 5, 237 (1993).\n8T. Kong, K. Stolze, D. Ni, S. K. Kushwaha, and R. J.\nCava, Phys. Rev. Mater. 2, 014410 (2018).\n9Y. Sun, Z. Song, Q. Tang, and X. Luo, J. Phys. Chem. C\n10.1021/acs.jpcc.0c02101 (2020).\n10Y. Liu and C. Petrovic, Phys. Rev. B 97, 014420 (2018).\n11G. T. Lin, X. Luo, F. C. Chen, J. Yan, J. J. Gao, Y. Sun,\nW. Tong, P. Tong, W. J. Lu, Z. G. Sheng, W. H. Song,\nX. B. Zhu, and Y. P. Sun, Appl. Phys. Lett. 112, 072405\n(2018).\n12Y. Liu and C. Petrovic, Phys. Rev. B 97, 174418 (2018).\n13X. Yu, X. Zhang, Q. Shi, S. Tian, H. Lei, K. Xu, and H.\nHosono, Front. Phys. 14, 43501 (2019).\n14W. Liu, Y. Dai, Yu E. Yang, J. Fan, L. Pi, L. Zhang, and\nY. Zhang, Phys. Rev. B 98, 214420 (2018).\n15J. Yan, X. Luo, F. C. Chen, J. J. Gao, Z. Z. Jiang, G. C.\nZhao, Y. Sun, H. Y. Lv, S. J. Tian, Q. W. Yin, H. C. Lei,\nW. J. Lu, P. Tong, W. H. Song, X. B. Zhu, and Y. P. Sun,\nPhys. Rev. B 100, 094402 (2019).\n16Y. K. Fu, Y. Sun, and X. Luo, J. Appl. Phys. 125, 053901\n(2019).\n17Y. Sun and X. Luo, Phys. Status Solidi B 1900052 (2019).\n18Y. Liu and C. Petrovic, Phys. Rev. Mater. 3, 014001\n(2019).\n19Y. Liu, J. Li, J. Tao, Y. Zhu, and C. Petrovic, Sci. Rep. 9,\n13233 (2019).20W. Liu, Y. Wang, J. Fan, L. Pi, M. Ge, L. Zhang, and Y.\nZhang, Phys. Rev. B 100, 104403 (2019).\n21N. Richter, D. Weber, F. Martin, N. Singh, U. Schwingen-\nschl¨ ogl, B. V. Lotsch, and M. Kl¨ aui, Phys. Rev. Mater. 2,\n024004 (2018).\n22Y. Liu, L. Wu, X. Tong, J. Li, J. Tao, Y. Zhu, and C.\nPetrovic, Sci. Rep. 9, 13599 (2019).\n23L. D. Casto, A. J. Clune, M. O. Yokosuk, J. L. Musfeldt,\nT. J. Williams, H. L. Zhuang, M. W. Lin, K. Xiao, R. G.\nHennig, B. C. Sales, J. Q. Yan, and D. Mandrus, APL\nMater.3, 041515 (2015).\n24S. Selter, G. Bastien, A. U. B. Wolter, S. Aswartham, and\nB. B¨ uchner, Phys. Rev. B 101, 014440 (2020).\n25V. Pecharsky, K. Gscheidner, J. Magn. Magn. Mater. 200,\n44 (1999).\n26J. Amaral, M. Reis, V. Amaral, T. Mendonc, J. Araujo,\nM. Sa, P. Tavares, J. Vieira, J. Magn. Magn. Mater. 290,\n686 (2005).\n27V. Franco and A. Conde, Int. J. Refrig. 33, 465 (2010).\n28E. C. Stoner and E. P. Wohlfarth, Philos. Trans. R. Soc.\nLond. Ser. Math. Phys. Sci. 240, 599 (1948).\n29K. A. Gschneidner Jr., V. K. Pecharsky, A. O. Pecharsky,\nand C. B. Zimm, Mater. Sci. Forum 315, 69 (1999).\n30V. Franco, J. S. Blazquez, and A. Conde, Appl. Phys. Lett.\n89, 222512 (2006).\n31H. E. Stanley, Introduction to Phase Transitions and Crit-\nical Phenomena (Oxford U. P., London and New York,\n1971).\n32M. E. Fisher, Rep. Prog. Phys. 30, 615 (1967).\n33A. Arrott, Phys. Rev. B 108, 1394 (1957).\n34S. K. Banerjee, Phys. Lett. 12, 16 (1964).\n35A. Arrott, and J. Noakes, Phys. Rev. Lett. 19, 786 (1967).\n36W. Kellner, M. F¨ ahnle, H. Kronm¨ uller, and S. N. Kaul,\nPhys. Status Solidi B 144, 387 (1987).\n37A.K. Pramanik, andA. Banerjee, Phys. Rev.B 79, 214426\n(2009).\n38K. Huang, Statistical Machanics, 2nd ed. (Wiley, New\nYork, 1987).\n39J. C. LeGuillou, and J. Zinn-Justin, Phys. Rev. B 21, 3976\n(1980).\n40A. Taroni, S. T. Bramwell, and P. C. W. Holdsworth, J.\nPhys.: Condens. Matter 20, 275233 (2008).\n41B. Widom, J. Chem. Phys. 41, 1633 (1964).\n42J. S. Kouvel, and M. E. Fisher, Phys. Rev. 136, A1626\n(1964)."    },    {        "title": "2008.05075v1.Metastable_skyrmion_lattices_governed_by_magnetic_disorder_and_anisotropy_in__β__Mn_type_chiral_magnets.pdf",        "content": "arXiv:2008.05075v1  [cond-mat.str-el]  12 Aug 2020Metastable skyrmion lattices governed by magnetic disorde r and\nanisotropy in β-Mn-type chiral magnets\nK. Karube,1,∗J. S. White,2,∗V. Ukleev,2C. D. Dewhurst,3R. Cubitt,3\nA. Kikkawa,1Y. Tokunaga,4H. M. Rønnow,5Y. Tokura,1,6and Y. Taguchi1\n1RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198 , Japan.\n2Laboratory for Neutron Scattering and Imaging (LNS),\nPaul Scherrer Institute (PSI), CH-5232 Villigen, Switzerland .\n3Institut Laue-Langevin (ILL), 71 avenue des Martyrs,\nCS 20156, 38042 Grenoble cedex 9, France\n4Department of Advanced Materials Science,\nUniversity of Tokyo, Kashiwa 277-8561, Japan.\n5Laboratory for Quantum Magnetism (LQM), Institute of Physics ,\n´Ecole Polytechnique F´ ed´ erale de Lausanne (EPFL), CH-1015 L ausanne, Switzerland.\n6Tokyo College and Department of Applied Physics,\nUniversity of Tokyo, Bunkyo-ku 113-8656, Japan.\nMagnetic skyrmions are vortex-like topological spin textu res often observed in\nstructurally chiral magnets with Dzyaloshinskii-Moriya i nteraction. Among them,\nCo-Zn-Mn alloys with a β-Mn-type chiral structure host skyrmions above room tem-\nperature. In this system, it has recently been found that sky rmions persist over a\nwide temperature and magnetic field region as a long-lived me tastable state, and\nthat the skyrmion lattice transforms from a triangular latt ice to a square one. To\nobtain perspective on chiral magnetism in Co-Zn-Mn alloys a nd clarify how various\nproperties related to the skyrmion vary with the compositio n, we performed sys-\ntematic studies on Co 10Zn10, Co9Zn9Mn2, Co8Zn8Mn4and Co 7Zn7Mn6in terms of\nmagnetic susceptibility and small-angle neutron scatteri ng measurements. The ro-\nbust metastable skyrmions with extremely long lifetime are commonly observed in\nall the compounds. On the other hand, preferred orientation of a helimagnetic prop-\nagation vector and its temperature dependence dramaticall y change upon varying\nthe Mn concentration. The robustness of the metastable skyr mions in these materi-2\nals is attributed to topological nature of the skyrmions as a ffected by structural and\nmagnetic disorder. Magnetocrystalline anisotropy as well as magnetic disorder due\nto the frustrated Mn spins play crucial roles in giving rise t o the observed change in\nhelical states and corresponding skyrmion lattice form.\nI. INTRODUCTION\nNon-collinear and non-coplanar spin textures have recently attra cted much attention as\na source of various emergent electromagnetic phenomena. Magne tic skyrmions, vortex-like\nspin textures characterized by an integer topological charge, ar e a prototypical example of\nsuch non-coplanar magnetic structures[1–4], and are anticipated to be applied to spintronics\ndevices since they can be treated as particles and driven by an ultra -low current density[5–\n7]. Thus far, skyrmions have been observed in various magnets due to several microscopic\nmechanisms, such as competition between the Dzyaloshinskii-Moriya interaction (DMI) and\nferromagneticexchangeinteraction[3,4,8–19], magneticdipoleint eraction[20–23], andmag-\nnetic frustration or Ruderman-Kittel-Kasuya-Yosida (RKKY) inte raction[24–26]. Among\nthem, the DMI arises from relativistic spin-orbit interaction and bro ken inversion symmetry\neither at interfaces of thin-film layers[8–12] or in bulk materials with n oncetrosymmetric\ncrystal structures[3, 4, 13–19].\nIn the structurally chiral magnets as represented by B20-type c ompounds (e.g., MnSi[3],\nFe1−xCoxSi[4], FeGe[13]) and Cu 2OSeO3[14] with the space group of P213, the DMI grad-\nually twists ferromagnetically coupled moments to form a long-period helimagnetic state\ndescribed by a magnetic propagation vector ( qvector). The magnitude of qis given by\nq∝D/J, whereDandJcorrespond to the DMI constant and the exchange stiffness, re-\nspectively, and the propagation direction is determined by magnetic anisotropy. Near the\nhelimagnetic transition temperature Tc, magnetic fields induce a triangular-lattice skyrmion\ncrystal (SkX) as illustrated in Fig. 1(c), which is often described as a triple-qstructure\nwith the qvectors displaying mutual 120◦angles perpendicular to the magnetic field. In\ngeneral, SkX is stabilized by thermal fluctuations and thus its therm odynamical equilib-\n∗These authors equally contributed to this work3\nrium state is confined to a narrow temperature and magnetic field re gion just below Tc, and\ntopologically-trivial helical or conical states are the thermodynam ically most stable states\nat lower temperatures.\nRecently, Co-Zn-Mn alloys have been identified as a new class of chira l magnets based on\nbulk DMI which host skyrmions above room temperature[15]. The mat erials crystallize in\naβ-Mn-type chiral cubic structure with the space group of P4132 (defined as right-handed\nstructure) or P4332 (left-handed structure), where 20 atoms per unit cell are dist ributed\nover two Wyckoff sites (8 cand 12d) as illustrated in Fig. 1(a). The 8 csites are mainly\noccupied by Co atoms while the 12 dsites are mainly occupied by Zn and Mn[27–30]. The\nβ-Mn-type structure forms in all the solid solutions of (Co 0.5Zn0.5)20−xMnxfrom Co 10Zn10\nto Mn 20(β-Mn itself)[27, 31].\nMagnetic phase diagram on the temperature ( T) - Mn concentration ( x) plane is repro-\nduced from Ref. [31] and displayed in Fig. 1(b) with additional informa tion obtained in\nthe present study. One end member Co 10Zn10shows a helimagnetic ground state with Ref.\n[15] reporting a magnetic periodicity λ∼185 nm below Tc∼460 K.Tcrapidly decreases as\npartial substitution of Mn proceeds, and Co 8Zn8Mn4withTc∼300 K exhibits a thermally\nequilibrium SkX state at room temperature under magnetic fields. Th e magnitude of the\nDMI constant in Co 8Zn8Mn4has been experimentally evaluated to be D∼0.53 mJ/m2,\nwhich is several times smaller than that in FeGe[32]. The DMI critically de pends on band\nstructure and electron band filling as demonstrated in Fe-doped Co 8Zn8Mn4, where even a\nsign change in the DMI, namely, reversal of skyrmion helicity, occur s as the Fe concentration\nis increased[33].\nAlthough the thermodynamical equilibrium SkX phase in Co 8Zn8Mn4exists only in a\nnarrow temperature and magnetic field region, it has been demonst rated that a once-created\nSkX can persist over the whole temperature region below room temp erature and a wide\nmagnetic field region as a long-lived metastable state via a convention al (a few K/min) field\ncooling (FC)[34]. Moreover, the lattice form of the metastable SkX u ndergoes a reversible\ntransition, accompanied by large increase in q, from a conventional triangular lattice to\na novel square one, which is described as an orthogonal double- qstate, as illustrated in\nFig. 1(e)[35]. Similar robust metastable SkX has been observed in Co 9Zn9Mn2withTc∼\n400 K, where the metastable SkX persists even at zero field above r oom temperature[36].\nBy means of Lorentz transmission electron microscopy (LTEM) for thin-plate specimens,4\nvarious exotic skyrmion-related structures, such as I- or L-sha ped elongated skyrmions[37]\n[Fig. 1(g)], a smectic liquid-crystalline structure of skyrmions[38] an d a meron-antimeron\nsquare lattice[39], have been observed.\nThe other end member β-Mn is well known as a spin liquid, while the lightly doped β-Mn\nalloys with slight disorder exhibit a spin glass, due to geometrical frus tration among anti-\nferromagnetically coupled Mn spins in the hyper-kagome network of the 12dsites[40–45].\nTherefore, (Co 0.5Zn0.5)20−xMnxpossesses both magnetic frustrations inherent to β-Mn and\nmagnetic disorder due to the mixture of ferromagnetic Co spins and antiferromagnetic Mn\nspins, which give rise to a spin glass phase below Tgover a wide range of the Mn concen-\ntration (3 ≤x≤19)[31]. In particular, the spin glass phase invades the helical phase for 3\n≤x≤7 displaying reentrant spin glass behavior[46] as similarly reported fo r a number of\nferromagnets[47–52], antiferromagnets[53, 54] and helimagnets [55] with chemical and mag-\nnetic disorder. The spin glass nature was confirmed by previous fre quency-dependent ac\nsusceptibility measurements for Co 7Zn7Mn6(Tc∼160 K,Tg∼30 K) [29, 31, 56]. It has\nbeen discovered that a novel equilibrium phase of disordered skyrm ions exists just above Tg\nin Co7Zn7Mn6, which is thermodynamically disconnected from the conventional eq uilibrium\nSkX phase just below Tc, and presumably stabilized by a cooperative interplay between the\nchiral magnetism with DMI and the frustrated magnetism[31].\nDespite these extensive studies for Co-Zn-Mn alloys, it remains elus ive how the\n(meta)stability and lattice form of skyrmions vary with the Mn conce ntration from the\nend member Co 10Zn10, and how the two magnetic elements of Co and Mn contribute to\nthe skyrmion formation and the transformation of skyrmion lattice . In order to obtain a\nmore complete perspective on the skyrmion states in the Co-Zn-Mn alloys, here we present\nnew data sets obtained by small-angle neutron scattering (SANS) a nd magnetic suscepti-\nbility that have not been reported previously. From the new data we report: (i) Identifi-\ncation of the equilibrium skyrmion phase, and temperature and field d ependence of helical\nand metastable skyrmion states in Co 10Zn10, (ii) an off-axis magnetic field experiment for\nmetastable skyrmions in Co 8Zn8Mn4to reveal the role of anisotropy, (iii) temperature and\nfield dependence of metastable skyrmions in Co 9Zn9Mn2below ambient temperature down\nto the lowest temperature to clarify the effect of dilute Mn moments that becomes signif-\nicant only at low temperatures, (iv) temperature and field evolution of heavily disordered\nmetastable skyrmions in Co 7Zn7Mn6, and (v) lifetime of metastable skyrmions in Co 10Zn105\nandCo 8Zn8Mn4. Takingthenewresultstogetherwithourpreviousones, weshows ystematic\nchanges in lattice forms and lifetimes of the metastable skyrmions, a s well as the key roles\nof magnetic disorder and anisotropy, as functions of temperatur e and Mn concentration. As\nsummarized in Fig. 2, the metastable skyrmion state prevails in a wide t emperature and\nfield range for all the materials. In Co 10Zn10, triangular lattice of skyrmions transforms to\nrhombic-like [Fig. 1(d)] asthe temperature is lowered in a low field. This triangular-rhombic\nstructural transformationisgoverned byenhanced magnetocr ystalline anisotropythat favors\nq∝bardbl<111>. On the other hand, in the Mn-doped compounds, q-vector orientation changes\nfrom<111>to<100>, and structural transformation from triangular lattice to squar e one\noccurs at low temperatures and low fields. As the Mn concentration is increased, and as the\nhelimagnetic Tcthus falls, the antiferromagnetic correlations of Mn spins start to develop on\ncooling from higher temperatures. The resulting magnetic disorder drives a large increase\ninqvalue, thereby triggering the transformation to the square lattic e state, in cooperation\nwith the enhanced magnetic anisotropy toward q∝bardbl<100>at low temperatures.\nThe format of this paper is as follows. After we describe experiment al methods in Sec-\ntion II, results and discussion are presented in Section III, and co nclusion is given in Section\nIV. In the section III, we first overview helical and skyrmion state s in all the compounds\n(Subsections A, Figures 2, 3, Table 1). Then, we show detailed resu lts of magnetic suscepti-\nbility and SANS measurements for each composition: Co 10Zn10(Subsection B, Figures 4-6),\nCo8Zn8Mn4(Subsection C, Figure7), Co 9Zn9Mn2(Subsection D, Figure8), and Co 7Zn7Mn6\n(Subsection E, Figure 9-11), and summarize all of them (Subsectio n F). Finally, we present\ncomposition dependence of lifetime of metastable skyrmions (Subse ction G, Figures 12, 13).\nWith all these results, we discuss the roles of magnetic disorder and anisotropy in leading\nto the robust metastable skyrmions and their novel lattice forms in Co-Zn-Mn alloys.\nII. EXPERIMENTAL METHODS\nA. Sample preparation\nSingle-crystalline Co 10Zn10was grown by a self-flux method in an evacuated quartz tube.\nThesinglecrystalswerecutalongthe(110),(-110)and(001)plan eswitharectangularshape\nfor magnetization and ac susceptibility measurements as well as SAN S measurement after6\nthe crystalline orientation was determined by the X-ray Laue diffrac tion method. Single\ncrystals of Co 9Zn9Mn2, Co8Zn8Mn4and Co 7Zn7Mn6were grown by the Bridgman method\nas described in our previous papers[31, 34, 36], and were cut along t he (110), (-110) and\n(001) planes for Co 9Zn9Mn2, and along the (100), (010) and (001) planes for Co 8Zn8Mn4\nand Co 7Zn7Mn6, respectively.\nB. Magnetization and ac susceptibility measurements\nMagnetization and ac susceptibility measurements were performed with a supercon-\nducting quantum interference device magnetometer (MPMS3, Qua ntum Design). High-\ntemperature measurements above 400 K for Co 10Zn10were performed by using an oven\noption. In the ac susceptibility measurements, the ac drive field was set asHac= 1 Oe\nfor all the compounds, and the ac frequency was selected to be f= 17 Hz for Co 10Zn10\nandf=193 Hz for Co 9Zn9Mn2, Co8Zn8Mn4and Co 7Zn7Mn6. Magnetic fields were applied\nalong the [110] direction for Co 10Zn10and Co 9Zn9Mn2, and along the [100] direction for\nCo8Zn8Mn4and Co 7Zn7Mn6, respectively. Due to the difference in the shape between the\nsamples used intheacsusceptibility (field parallel totheplate) andth eSANSmeasurements\n(field perpendicular to the plate), their demagnetization factors a re different. To correct for\nthis difference so that a common absolute magnetic field scale is used t hroughout this paper,\nthe field values for the ac susceptibility measurements are calibrate d asHc=NHwhereN\n= 3.0, 2.0, 3.7 and 2.7 for Co 10Zn10, Co9Zn9Mn2, Co8Zn8Mn4and Co 7Zn7Mn6, respectively.\nC. Small-angle neutron scattering (SANS) measurements\nSANSmeasurements forCo 10Zn10, Co9Zn9Mn2andCo 8Zn8Mn4were performedusing the\nSANS-I instrument at the Paul Scherrer Institute (PSI), Switze rland. For high-temperature\nmeasurements above 300 K for Co 10Zn10and Co 9Zn9Mn2, a bespoke oven stick designed\nfor SANS experiments was used. SANS measurements of Co 7Zn7Mn6were done using the\nD33 instrument at the Institut Laue-Langevin (ILL), France. Th e neutron wavelength was\nselected to be 10 ˚A with a 10% full width at half maximum (FWHM) spread in all the\nmeasurements. For all the SANS data shown here, nuclear and inst rumental background\nsignals have been subtracted by using data taken either well above the helimagnetic ordering7\ntemperature Tc, or well above the polarized ferromagnetic transition field.\nFor Co 10Zn10, the mounted single-crystalline sample was installed into a horizontal field\ncryomagnet so that the incident neutron beam ( ki) and the magnetic field ( H) were parallel\nto the [110] direction. The cryomagnet was rotated (‘rocked’) tog ether with the sample\naround the vertical [001] direction, and the rocking angle ( ω) between kiandHwas scanned\nfrom−20◦to 20◦by 2◦step. Here, ω= 0◦corresponds to the ki∝bardblHconfiguration. The\nobserved FWHM of the rocking curves (scattering intensity versu sω) was always broader\nthan 19◦. Therefore, to deduce accurately the relative intensities and pos itions of all of the\nBragg spots in single images, all the SANS images displayed in this paper are obtained by\nsumming multiple SANS measurements taken over −8◦≤ω≤8◦.\nAs detailed in our previous papers[31, 34, 36], similar SANS measureme nts were per-\nformedwith the ki∝bardblH∝bardbl[110] configurationforCo 9Zn9Mn2, andki∝bardblH∝bardbl[001]configuration\nfor Co 8Zn8Mn4and Co 7Zn7Mn6.\nIII. RESULTS AND DISCUSSION\nA. Overview of state diagrams\nFirst, we briefly overview the results of basic magnetic properties in helimagnetic states\nand metastable skyrmion states for all the compounds.\n1. Helical state\nFigure 3(a-d) shows temperature ( T) dependence of magnetization ( M) under a small\nmagnetic field of 20 Oe. In Co 10Zn10,Mshows a sharp increase due to a helimagnetic\ntransition at Tc∼414 K, and then stays almost independent of temperature both fo r the\nfield cooling (FC) and the zero-field-cooled field-warming (ZFC-FW) p rocesses. The other\ncompounds exhibit gradual decrease in Mupon cooling at some temperature region ( TL≤T\n≤TH). Co9Zn9Mn2shows almost temperature-independent Min a wide temperature range\nbelowTc∼396 K, while the gradual decrease is observed below TH∼50 K. In Co 8Zn8Mn4,\nT-independent behavior below Tc∼299 K is followed by the gradual decrease in Mupon\ncooling from TH∼120 K down to TL∼40 K. A sharp drop of MatTg∼9 K observed\nonly in the ZFC-FW process is due to a reentrant spin-glass transitio n. Co7Zn7Mn6exhibits8\nthe gradual decrease in MbelowTH∼120 K down to TL∼50 K below the helimagnetic\ntransition at Tc∼158 K. The reentrant spin glass transition is observed around Tg∼26\nK as manifested in a large deviation between FC and ZFC-FW processe s. The reentrant\nspin glass transition temperature Tgdetermined by the M(T) curve is very close to the\nzero frequency limit of Tgdetermined by ac susceptibility curves with various frequencies\nas previously reported[56]. Tgdoes not show significant dependence on the magnetic field\nbelow the field-induced ferromagnetic region as plotted in the phase diagrams in Fig. 2(d,\ne). These characteristic temperatures ( Tc,TH,TLandTg) are plotted in the T-xphase\ndiagram in Fig. 1(b).\nMagnetization curves at 2 K that characterize the ground states are shown in Supple-\nmentary Fig. S1[57]. The values of Tcand saturation magnetization Msat 2 K and at 7 T\nin all the compounds are summarized in Table 1. From the value of Ms∼11.3µB/f.u. in\nCo10Zn10, magnetic moment at Co site is estimated to be ∼1.1µB. WhileTcmonotonically\ndecreases with increasing Mn concentration, Msdisplays a maximum in Co 9Zn9Mn2.\nThe characteristic T-dependence of Mat 20 Oe is compared with that of the magnitude\nof helical wavevector ( q) from SANS measurements in Fig. 3(e-h). It turns out that the\ngradual decrease in MfromTHtoTLcorresponds to the large increase in the value of q,\ni.e. the decrease in the helimagnetic periodicity λ(= 2π/q). The largest value of λat\nhigh temperatures ( λmax) and the smallest value of λat low temperatures ( λmin) for each\ncompound are summarized in Table 1. We also present the preferred orientation of qin\nTable 1. While the preferred qdirection is <111>for Co 10Zn10, it is parallel to <100>in\nCo9Zn9Mn2, Co8Zn8Mn4and Co 7Zn7Mn6, as detailed in the following sections.\n2. Metastable skyrmion state\nFigure 2 shows the T-Hdiagrams determined by ac susceptibility and SANS mea-\nsurements for each composition. In these diagrams, the equilibrium SkX phase and the\nmetastable SkX state are presented together, as determined by field scans after ZFC and by\nfield scans after a FC via the equilibrium phase, respectively, as sche matically illustrated in\nFig. 2(a). The robust metastable skyrmion state that exists over a very wide temperature\nand field region is commonly observed from Co 10Zn10to Co7Zn7Mn6. In Co 10Zn10the trian-\ngular lattice of metastable skyrmions (M-T-SkX) distorts and tran sforms to a rhombic one9\n(M-R-SkX) at low fields during the FC. On the other hand, the lattice form changes to a\nsquareone(M-S-SkX)atlowtemperatures andlowfields intheothe r Mn-dopedcompounds.\nAs the partial Mn substitution proceeds, the phase space of the m etastable triangular SkX\nstate (M-T-SkX) is squeezed, accompanied by the suppression of Tc, while the region occu-\npied by square SkX state (M-S-SkX) expands toward higher tempe ratures. The transition\nto the square SkX is accompanied by a large increase in qas similarly observed in the helical\nstates [Fig. 3(f-h)].\nB. Co 10Zn10\nInthissection, wepresent detailedresultsofSANSandacsuscept ibility measurements for\nCo10Zn10and show that thepreferred orientation of qvector is <111>direction, and thaton\nfield cooling metastable skyrmion lattice distorts and undergoes a st ructural transformation\nfrom a conventional triangular lattice to a rhombic one.\n1. Identification of equilibrium SkX\nFirst, we identify an equilibrium SkX phase (Fig. 4). Photographs of s ingle-crystalline\nsamples used for SANS and ac susceptibility measurements are show n in Fig. 4(a). Figure\n4(b) shows the field variation of SANS images at 410 K. The scatterin g signal observed at\n0 T is attributed to a helical multi-domain state. Here, 4 broad spots are observed but\nthe peak positions are not aligned clearly to any unique set of high-sy mmetry crystal axes.\nThis peculiar scattering distribution near Tcmay be attributed to rather flexible nature of q\nvector due to the negligible anisotropy at high temperatures under an influence of possible\nresidual local strain of the sample. Under magnetic fields parallel to the incident neutron\nbeam, the SANS signal diminishes at 0.02 T most likely due to the transit ion to a conical\nstate whose qvector is parallel to the field and thus not detected in this configura tion. At\n0.03 T, a broad 6-spot pattern, a hallmark of triangular lattice of sk yrmions (triple- q⊥H),\nappears. As the magnetic field is further increased to 0.06 T, the co nical state is stabilized\nagain and the signal disappears. SANS intensity perpendicular to th e field is plotted against\nfield in Fig. 4(c). The SANS intensity exhibits a peak at 0.03 T, where th e volume fraction\nof skyrmions is maximized.10\nThe field dependence of the real- ( χ′) and imaginary-part ( χ′′) of the ac susceptibility at\n410 K is shown in Fig. 4(d). A dip structure is observed in χ′between 0.02 T and 0.05 T\nwhere in addition χ′′exhibits a clear double peak structure. These features correspo nd to a\nSkX state surrounded by a conical state, and show a good agreem ent with the region of the\nenhanced SANS intensity. Therefore, the phase boundaries of Sk X state are determined as\nthe peak positions at the both sides of the dip structure in χ′. Contour plots of χ′andχ′′\nwith the phase boundaries on the T-Hplane are presented in Fig. 4(e) and (f), respectively.\nThe equilibrium SkX phase exists in a narrow temperature region from Tc∼412 K down\nto 407 K, below which the field-induced conical state is the most stab le.\n2. Temperature evolution of helical state in zero field\nNext, we discuss temperature variation of the helical state (Fig. 5 ). Figure 5(b) shows\nthe SANS patterns at selected temperatures on zero-field cooling from 410 K to 1.5 K. As\ntemperature is lowered from Tcdown to 300 K, the SANS intensity gradually accumulates\nalong the [1-11] direction due to enhanced magnetic anisotropy. Up on further decreasing\ntemperature down to 1.5 K, the SANS intensity splits into 4 spots: 2 s pots with stronger\nintensity along the [1-11] direction, and the other 2 spots with weak er intensity along the\n[-111] direction. This 4 spot pattern at 1.5 K is expected for the mult i-domain helical state\nwithq∝bardbl<111>as schematically illustrated in Fig. 5(a). Notably, the preferred orie ntation\nofqvector in Co 10Zn10is different from those of the other (Co 0.5Zn0.5)20−xMnxwithx≥2,\nwhich is found to be <100>by our previous studies[31, 34, 36].\nFigure 5(c) shows the temperature dependence of the SANS inten sity integrated for the [-\n111]and[1-11]directions( φ=55◦,125◦, 235◦and305◦), whereconventionalorder-parameter\nlike evolution of the intensity for q∝bardbl<111>is clearly observed. The radial qdependence of\nthe SANS intensity for the direction close to [1-11]is displayed in Fig. 5 (d). The peak center\nslightly shifts toward higher qregion and the peak width slightly increases on cooling. The\nhelicalqvalue, which is determined as the peak center of a Gaussian function fitted to the\nintensity vs qcurve [solid line in Fig. 5(d)], and its FWHM are plotted against temperat ure\nin Fig. 3(e) and (i), respectively. Both the qvalue and the FWHM gradually increases\non cooling but the variation is much smaller than that in the Mn-doped c ompounds. The\nhelical periodicity estimated as λ= 2π/qvaries a little from 156 nm to 143 nm on cooling11\nas summarized in Table 1. Note that the values of Tc= 414 K and λ= 156 nm for a single\ncrystal of Co 10Zn10in the present study are smaller than the values ( Tc= 462 K and λ\n= 185 nm) for a polycrystalline sample in the previous study[15], proba bly due to a slight\ndifference in the composition.\nAsdetailedinSupplementary Fig. S4[57], wealsofindthatthehelicalst ateat1.5Kforms\na chiral soliton lattice[59] under magnetic fields due to the enhanced magnetocrystalline\nanisotropy which enforces the qvector along <111>, and perpendicular to the field.\n3. Temperature dependence of metastable SkX\nNext, we discuss temperature variation of the metastable SkX sta te (Fig. 6). The SANS\npatternsinaFCat0.03Tfrom412Kto1.5Karedisplayed inFig. 6(b). I nordertogainthe\nsufficient cooling rate across the boundary between the equilibrium S kX and conical phases,\nthe magnetic field was applied at 412 K (slightly higher than presented in Fig. 4), where\nthe observed SANS pattern is ring-like, indicating that the triangula r lattice of skyrmions\nis orientationally disordered. As the temperature is lowered, the rin g-like SANS pattern\ngradually changes to broad 4 spots along the [-111] and [1-11] direc tions. The scattering\nintensity is plotted as a function of azimuthal angle φin Fig. 6(c). Clearly, 4 peaks around\nφ= 55◦, 125◦, 235◦and 305◦emerge out of a rather featureless profile at 412 K as the\ntemperature is lowered down to 1.5 K[58].\nThe 4-spot SANS pattern observed at low temperatures is expect ed for a rhombic lattice\nof skyrmions with double- qvectors∝bardbl<111>that are rotated by 110◦from each other\nas schematically illustrated in Fig. 6(a) and Fig. 1(d). The SANS intens ity integrated\nover the region close to the [-111] and [1-11] directions under the F C is plotted against\ntemperature together with the intensity integrated around the [0 01] and [-110] directions in\nFig. 6(d). Below 360 K the intensity around the [-111] and [1-11] dire ctions becomes higher\nthan that around the [001] and [-110] directions. This temperatur e corresponds to the onset\nof triangular-rhombic lattice structural transition and is plotted w ith a purple diamond\n(right one) in the state diagram in Fig. 2(b). The broad SANS patter n and the gradual\ntemperature evolution indicate coexistence of the triangular SkX s tate and the rhombic one\nover a wide temperature region.\nThe transformation to the rhombic lattice is probably driven by signifi cant enhancement12\nof magnetocrystalline anisotropy at low temperatures that favor sq∝bardbl<111>as found in the\nhelical state (Fig. 5). It is also noted that the absolute value of qslightly increases upon\nloweringtemperatureasplottedinFig. 3(e). Therelationbetweent helatticetransformation\nand the change in the qvalue is further discussed in the summary section of metastable SkX\n(Subsection F-2).\n4. Metastability of skyrmions against field variation\nWe also investigated the metastability and the lattice form of skyrmio ns against field\nvariation. The detailed field-dependent SANS data at 300 K after th e FC (0.03 T) is\npresented in Supplementary Fig. S2[57]. In the field sweepings towar d positive direction,\nthe broad 4-spot pattern with stronger intensity at the [-111] an d [1-11] directions changes\nto a uniform ring above 0.1 T as observed in the equilibrium SkX phase at 412 K and 0.03\nT. This field variation corresponds to the change in skyrmion lattice f orm from the rhombic\none to the original disordered triangular one. Importantly, such a ring-like pattern is never\nobserved upon the field sweeping at 300 K after ZFC. This result ens ures that the broad\n4-spot pattern appearing after the FC is attributed to the metas table rhombic SkX (M-R-\nSkX) and distinct from helical multi-domain state. The SANS intensity originating from the\ndisordered triangular SkX persists up to the region close to the field -induced ferromagnetic\nphase in contrast to the helical state. In the field sweeping toward negative direction, on\nthe other hand, the scattering intensity around the [-111] and [1- 11] directions is further\nincreased and the 4-spot pattern becomes clearer. This indicates that the rhombic lattice is\nmore stable than the triangular one at zero or negative fields. As th e field is swept further\nnegative, finally the SANS intensity from the rhombic SkX disappears upon the transition\nto the equilibrium conical state.\nThe field variation at 1.5 K after the FC (0.03 T) is displayed in Supplemen tary Fig.\nS3[57], where similar change in the SANS pattern was observed while th e 4 broad spots\npersist over a wider field region than that observed at 300 K.\nThefieldvariationsofSANSat300Kand1.5Kareconsistent withthos eofacsusceptibil-\nity, where smaller values (as compared with conical state) corresp onding to the metastable\nSkX are observed, accompanied by a characteristic asymmetric hy steresis. Similar field-\ndependent ac susceptibility measurements after FC processes we re performed at different13\ntemperatures, and boundaries of the metastable SkX state are p lotted in the T-Hphase\ndiagram in Fig. 2(b). The metastable SkX state persists over a wide T-Hregion, including\nroom temperature and zero field. Within the metastable state, the lattice form of skyrmions\nundergoes the transition from triangular to rhombic at a low- Tand low-Hregion where q\nvector anisotropy along <111>is significant.\nC. Co 8Zn8Mn4\nIn this section, we describe detailed results of SANS measurements with off-axis field con-\nfiguration in Co 8Zn8Mn4and provide evidence for increased magnetocrystalline anisotropy\nthat favors q∝bardbl<100>at low temperatures.\n1. Temperature-driven structural transition of metastabl e skyrmion lattice\nFirst, we review the temperature variation in metastable SkX under a field parallel to\nthe [001] direction, as reported in our previous paper[34] (see Sup plementary Fig. S5 for\nthe details[57]). In a FC process at 0.04 T, SANS pattern transform s from 12 spots to 4\nspots below 120 K. The 12-spot pattern at high temperatures cor responds to a 2-domain\ntriangular SkX state, in which one of triple- qis parallel to the [010] or [100] direction.\nThe 4-spot pattern at low temperatures is attributed to a square SkX state with double- q\nvectors parallel tothe[010] and[100]directions. This temperatur e variationisreversible and\nthe triangular SkX revives already at 200 K in the subsequent re-wa rming process, which\nrules out the possibility of relaxation to a helical multi-domain state. I t is noted that the\ntransformation to the square SkX below 120 K is accompanied by a lar ge increase in qas\npresented in Fig. 3(g).\n2. Temperature-dependent magnetic anisotropy as revealed by an off-axis field measurement\nNext, we show how qvector orientation changes during the triangular-square struct ural\ntransition of skyrmion lattice under an off-axis magnetic field (Fig. 7) . The experimental\nconfiguration for the SANS measurement is illustrated in Fig. 7(a). T he magnetic field of\n0.03 T was applied along the direction tilted away from the [001] directio n by 15◦. While14\nkeeping this configuration, the cryomagnet and the sample were ro tated together around the\nvertical [010] direction. Here, ωis defined as a rocking angle between the incident neutron\nbeam (ki) and the applied magnetic field ( H), namely ω= 0◦forki∝bardblHandω=−15◦for\nki∝bardbl[001].\nRocking curves (SANS intensity versus ω) at selected temperatures are presented in Fig.\n7(b). Here, integrated intensity at φ=90◦and270◦isplottedagainst ω. Fortheequilibrium\ntriangular SkX state, a rocking curve is expected to take a maximum atω= 0◦because\ntriple-qare usually perpendicular to the applied field regardless of the cryst al orientation.\nHowever, the observed rocking curve at 295 K [red symbols in Fig. 7( b)] shows a broad\nmaximum around ω∼20◦. This indicates that the effective magnetic field inside the sample\n(Heff) is further tilted from the external field to the opposite side of the [001] axis as shown\nin Fig. 7(a), which can be understood in terms of a demagnetization e ffect in the present\nrectangular-shaped sample[60]. As the temperature is lowered, th e peak position of the\nrocking curve shifts to lower angle and eventually locates at −15◦at 40 K.\nTemperature variations of the SANS patterns for ki∝bardblHeffandki∝bardbl[001], averaged over\nthe rocking angles of 14◦≤ω≤20◦and−20◦≤ω≤ −10◦, respectively, are shown in\nFig. 7(c) and (d). For ki∝bardblHeff, a 6-spot pattern is observed at 295 K. This 6-spot pattern\ncorresponds to a single-domain triangular SkX state, in which triple- qare perpendicular to\nHeffand one of them is parallel to the vertical [010] direction. The 6 spot s are still discerned\nat 200 K as the triangular SkX persists as a metastable state. At 12 0 K, the side 4 spots\nbecome much weaker as compared with the vertical 2 spots, and fin ally the 6-spot pattern\nis not discerned at 40 K. For ki∝bardbl[001], on the other hand, only 2 vertical spots out of the 6\nspots are observed at 295 K. Below 120 K, intensities around the ho rizontal region increase,\nand finally a 4-spot pattern is observed at 40 K.\nThe SANS intensity for the two different configurations is plotted ag ainst temperature\nin Fig. 7(e), clearly showing the change in intensity distribution below 1 20 K from the side\n4 spots perpendicular to Heff(blue circles) to the side 2 spots parallel to the [100] direction\n(red squares), in good accord with the large shift of the peak posit ion in the rocking curves\npresented in Fig. 7(b). Therefore, the double- qin the square SkX at low temperatures are\naligned to the [010] and [100] direction, and the latter is not perpend icular to Heff. This is in\nmarked contrast to the triangular SkX state at high temperature s where all the triple- qare\nperpendicular to Heff. Thus, the triangular-square SkX transition under the off-axis fie ld15\nis accompanied by a reorientation of skyrmions as schematically illustr ated in Fig. 7(f).\nThis result indicates that the magnetic anisotropy favoring q∝bardbl<100>is strongly enhanced\nduring the transformation to the square SkX as the temperature is reduced.\nD. Co 9Zn9Mn2\nInthis section, we present how the qvector evolves in terms of magnitude and orientation\nas a function of temperature in Co 9Zn9Mn2. In brief, the change in qdirection, and the\nvariation of qvalue, are qualitatively similar to those found in Co 8Zn8Mn4but take place\nat lower temperatures.\n1. Temperature dependence of metastable SkX\nFirst, we show temperature variation of the metastable SkX state (Fig. 8). Selected\nSANS patterns in a FC process at 0.04 T from 390 K to 10 K are displaye d in Fig. 8(b). In\nthis measurement, the magnetic field and the incident neutron beam are parallel to the [110]\ndirection. At 390 K within the equilibrium SkX phase, the SANS pattern shows 6 spots,\ncorresponding to a triangular SkX with one of triple- q∝bardbl[001] as illustrated in the right\npanel of Fig. 8(a). The triangular SkX persists down to 100 K as a me tastable state during\nthe FC. Below 50 K, the signal from the side 4 spots becomes weaker as compared with the\nvertical 2 spots, as similarly observed in the off-axis field measureme nt for Co 8Zn8Mn4(Fig.\n7(b)).\nFigure 8(d) presents the temperature dependence of the SANS in tensity for the side 4\nspots and the vertical 2 spots, clearly showing the cross-correla tion between the reduced\nintensity for the side 4 spots and the increased intensity for the [00 1] direction below 50 K.\nRocking curves during the FC are displayed in Fig. 8(c). While the rock ing curve exhibits\na peak around ω= 0◦above 100 K, the intensity around ω= 0◦decreases below 50 K and\nfinally the rocking curve forms a concave shape around ω= 0◦at 10 K. These results are\nsimilar to those observed in the SANS pattern for Co 8Zn8Mn4under the off-axis field as\nshown in Fig. 7. Therefore, we attribute the change in the SANS pat tern below 50 K in\nCo9Zn9Mn2to theqvector reorientation and associated transition from a triangular S kX to\na square SkX. Namely, one of the double- qin the square SkX is aligned to the [001] direction16\nbut the other is aligned to [100] or [010] directions that are out of th e (110) plane, resulting\nin 2 vertical spots on the (110) plane as illustrated in the left panel o f Fig. 8(a).\nThe radial qdependence of the SANS intensity for the [001] direction is shown in F ig.\n8(e). While the peak center ( q∼0.05 nm−1) is almost temperature independent above\n100 K, a large shift up to q∼0.07 nm−1is observed below 50 K, accompanied by a slight\nbroadening of the width. Thus, the transition from the triangular S kX to the square SkX in\nCo9Zn9Mn2is also accompanied by the increase in the magnitude and width of the qvector,\nagain similarly as observed in Co 8Zn8Mn4.\n2. Field dependence of metastable SkX\nNext, we discuss field variation in the qvector in the metastable SkX state at 10 K after\nthe FC under 0.04 T (see Supplementary Fig. S6 for the details[57]). W ith increasing field\nfrom 0.04 T to 0.3 T, the 2-spot SANS pattern gradually changes to a ring-like one, in which\nthe scattering intensity lies in the (110) plane as similarly observed at high temperatures.\nThis indicates that the square SkX with q∝bardbl<100>at low temperatures transforms to an\norientationally disordered triangular SkX with q⊥Hat high fields.\nTaking the above results into account, we summarize the state diag ram of the metastable\nSkX in Co 9Zn9Mn2in Fig. 2(c), which is characterized by a large region of the triangular\nSkX state and a small region of the square SkX state existing only at low temperatures\nbelow∼50 K and low fields.\nE. Co 7Zn7Mn6\nIn this section, we discuss how the magnetic state is affected by the increased Mn con-\ncentration in Co 7Zn7Mn6.\n1. Heavily-disordered square lattice within the metastabl e state\nFirst, we discuss the temperature variation of the metastable SkX (Fig. 9). Figure 9(b)\nshows the change in the SANS patterns during a FC at 0.025 T from 14 6 K to 1.5 K. In\nthis measurement, the magnetic field and the incident neutron beam are parallel to the17\n[001] direction. At 146 K within the equilibrium SkX phase, a 12-spot pa ttern correspond-\ning to a 2-domain triangular SkX with one of triple- q∝bardbl[010] or [100], as schematically\nillustrated at right panel in Fig. 9(a), is observed. However, the sp ots display a signifi-\ncant azimuthal broadening as compared with those observed from the equilibrium SkX in\nCo8Zn8Mn4sample. The 12-spot pattern changes to a pattern of 4 broadene d spots at 100\nK, which corresponds to a metastable square SkX with double- q∝bardbl[010] and [100] as shown\nat left panel in Fig. 9(a). With further decreasing temperature do wn to 60 K, the 4 spots\nbecome even broader. This pattern of 4 very broad spots remains almost unchanged across\nthe reentrant spin glass transition ( Tg∼30 K) down to 1.5 K.\nThe SANS intensity from the metastable SkX is plotted as a function o f temperature\nin Fig. 9(c). Above 130 K, the intensities integrated for directions c lose to<100>and\n<110>show similar values as expected for a 12-spot pattern. Below 120 K, the intensity for\n<100>becomes larger than that for <110>due to the change in the pattern from 12 spots\nto 4 spots. The intensities for both regions significantly decrease b elow 120 K, and become\nsimilar again below 60 K, which corresponds to the broadening of the 4 spots. Therefore,\nwhile the triangular-square SkX transition occurs below 120 K in comm on with Co 8Zn8Mn4,\nthe square SkX is severely disordered below ∼90 K. This is in accord with the disordering of\nthe helical state of Co spins below ∼90 K on ZFC[31] due to the development of short-range\nantiferromagnetic correlation of Mn spins.\nThe radial qdependence of the SANS intensity for <100>is presented in Fig. 9(d). From\n120 K to 60 K, where the triangular-square SkX transition occurs, the peak center exhibits\na large shift from 0.06 nm−1to 0.08 nm−1, and the magnitude and width of the peak\nsignificantly decreases and increases, respectively. Thus, the tr iangular-square structural\ntransformation of SkX in Co 7Zn7Mn6is also accompanied by the large increase in qas well\nas in its width. This indicates that the metastable SkX is heavily disorde red, similarly to\nthe helical state during ZFC [Fig. 3(l)].\n2. Field dependence of metastable SkX\nNext, we discuss how the metastable SkX state varies with field swee ping in the well-\nordered region at 100 K (Fig. 10) and in the disordered region at 60 K (Fig. 11).\nThe field variation of the metastable SkX state at 100 K is presented in Fig. 10. Figure18\n10(a) shows the field variation in the SANS patterns at 100 K after a field cooling (FC)\nat 0.025 T. The detailed field dependence of the SANS intensity integr ated over the region\naround<100>and<110>is displayed in Fig. 10(c). With increasing the field to the\npositive direction, the initial 4-spot pattern changes to a uniform r ing above 0.055 T, where\nthe scattering intensities for <100>and<110>completely overlap while showing a clear\nshoulder, and the signal persists up to 0.1 T. This change is ascribed to the transformation\nfrom the square SkX to an orientationally disordered triangular SkX similarly to the case\nof Co8Zn8Mn4[34]. In the field sweeping to the negative direction, the 4-spot patt ern again\nchanges to the ring-like one at −0.055 T but the intensity is much weaker than that observed\nat 0.055 T. In the returning process from 0.15 T to 0 T, clear SANS sig nal is not observed\nbecause the metastable SkX is completely destroyed at a high-field r egion and a conical\nstate is stabilized instead. The large and asymmetric hysteresis of t he intensity plot in Fig.\n10(c) is attributed to the existence of the metastable SkX state. For comparison, the field\ndependence oftheSANSpatternandtheintegratedintensity at1 00Kafterzero-fieldcooling\nis shown in Fig. 10(b) and Fig. 10(d), respectively. In this case, 4 sp ots corresponding to\nthe helical multi-domain state with q∝bardbl<100>is observed up to 0.05 T, but the signal\ndisappears at 0.1 T without exhibiting a clear ring-like patten. This field variation for\nhelical state after ZFC is totally different from that observed afte r the FC. Therefore, it can\nbe concluded that the metastable SkX created by the FC survives o ver a wide field region\nat 100 K, and the square SkX changes to the triangular SkX at high fi elds.\nWe also show the field variation of the metastable SkX state at 60 K in F ig. 11, and\ndiscuss another transition to the second equilibrium skyrmion phase . The field-dependent\nSANS patterns at 60 K after the FC (0.025 T) are presented in Fig. 1 1(a). The SANS\nintensity integrated over the region around <100>and<110>in this process is plotted\nagainst magnetic field in Fig. 11(c). As the field is increased to the pos itive direction, the\ninitial pattern with broad 4 spots originating from the disordered sq uare SkX changes to\na broad ring pattern above 0.07 T, resulting in the almost equal scat tering intensities for\n<100>and<110>. In the field sweeping to the negative field region, the broad ring pat tern\nwith similar intensity is observed at −0.07 T and −0.1 T. Therefore, the field variation after\nthe FC is less asymmetric between the positive fields and the negative fields as compared\nwith the result at 100 K. In the returning process from the field-ind uced ferromagnetic\nphase (±0.2 T) down to 0 T, the broad ring pattern appears again and remains down to19\nzero field. Note that the intensity of the ring pattern around zero field is relatively large and\ncomparable to that of the initial broad 4-spot pattern just after the FC. For comparison,\nwe show SANS patterns observed at the same magnetic fields after ZFC in Fig. 11(b),\nand the field dependence of the SANS intensity is plotted in Fig. 11(d) . In this process,\nthe broad 4-spot pattern (disordered helical state) changes to a broad ring above 0.07 T\nsimilar to the FC case. The field dependence commonly observed for t he FC (both for\npositive and negative field directions) and ZFC processes is in accord with the existence\nof another field-induced equilibrium phase at low temperatures. In o ur previous study on\nCo7Zn7Mn6, the broad ring pattern at low temperatures has been identified to be three-\ndimensionally disordered skyrmions, which are stabilized by frustrat ed magnetism of Mn\nspins and exist as an equilibrium phase that is distinct from the conven tional SkX phase\njust below Tc[31]. Therefore, theobserved change fromthebroad4-spotpat terntothebroad\nring pattern above 0.07 T in the FC case corresponds to the transit ion from the metastable\nsquareSkXstate(originatingfromthehigh-temperatureequilibriu mSkXphase)totheother\nequilibrium disordered skyrmion phase. The ring pattern remaining at zero field after the\nfield decreasing process is explained in terms of the metastable skyr mions that are created\ninitially in the high field region and persist down to zero field.\nOn the basis of the above results, the state diagram of the metast able SkX in Co 7Zn7Mn6\nissummarizedintheFig. 2(e). ThemetastablesquareSkX(M-S-SkX )stateexistsbelow120\nK and at low fields inside the metastable triangular SkX (M-T-SkX) sta te. In addition, the\nequilibrium disordered skyrmion (E-DSk) phase is stabilized at low temp eratures just above\nthe reentrant spin glass transition around 30 K. Note that the E-D Sk phase is discerned also\nin a negative-field region, reflecting its thermodynamical equilibrium n ature.\nF. Summary of helical state and metastable skyrmion state\nTaking the SANS results for all the compounds into account, we com e back to Fig. 2 and\nFig. 3 again and discuss how the helical and the metastable skyrmion s tates change with\nMn concentration.20\n1. Correspondence between magnetization and qvector in helical state\nTable 1 summarizes the several quantities that characterize the h elical state. One im-\nportant effect of Mn substitution is the change in qvector direction: While the preferred\norientations of qvectors are <111>directions in Co 10Zn10, they are <100>directions in all\nthe Mn-doped compounds. The helical structure is formed by the f erromagnetically coupled\nCo spins accompanied with DMI, but as described in Table 1 (see also Su pplementary Fig.\nS1), saturationmagnetization Msmeasured at2Kand7Ttakes amaximum atCo 9Zn9Mn2.\nThis suggests that Co and Mn are ferromagnetically coupled at least in the low Mn concen-\ntration region, as also demonstrated recently by X-ray magnetic c ircular dichroism (XMCD)\nmeasurements[61].\nMn substitution produces another important effect: Figures 3(e- h) summarize the tem-\nperature dependence of the helical qvalue measured for all the compounds. The qvalue was\ndetermined as the peak center of Gaussian function fitted to the a zimuthal-angle-averaged\nSANS intensity as a function of q. In Co 10Zn10,qgradually and slightly increases by only\n∼10% upon cooling from Tcto low temperatures. On the other hand, Co 9Zn9Mn2shows\na large increase in qbelow 50 K by ∼45% while for Co 8Zn8Mn4and Co 7Zn7Mn6,qalso\ndisplays a large increase of ∼50% below 120 K. Notably, the large observed increases in q\nalmost coincide with the gradual decrease in MfromTHtoTLas shown in Fig. 3(a-d). The\ntemperature region where qsignificantly varies is indicated in the T-xphase diagram in Fig.\n1(b).\nFigures 3(i-l) show the temperature dependence of the full width a t half maximum\n(FWHM) of the Gaussian function fitted to the SANS intensity versu sq, which provides\na measure of the spatial coherence of the helimagnetic order. The FWHM increases upon\ncoolinginallthecompositions whileshowing strongcorrelationwiththe increase inthemag-\nnitude of qvalue [Figs. 3(e-h)]. Among them, Co 7Zn7Mn6exhibits a significant increase\nbelow∼90 K, which indicates that the helical state in Co 7Zn7Mn6becomes severely dis-\nordered at low temperatures. The increase in the FWHM for Co 9Zn9Mn2and Co 8Zn8Mn4\nalso indicates the evolution of magnetic disorder to some extent, alt hough they are less\nsignificant as compared with Co 7Zn7Mn6.\nThe above temperature and Mn concentration dependence of helic al states is well ex-\nplained by the interplay between the ferromagnetically coupled Co sp ins forming the helical21\nstate and the antiferromagnetically coupled Mn spins. Namely, the s ignificant increase in\ntheqvalue (or decrease in the helical periodicity) at low temperatures is c aused by the ef-\nfective decrease in the ratio of the ferromagnetic exchange inter action to the DMI, which is\nattributed to short-range antiferromagnetic correlations of th e Mn spins. These short-range\ncorrelations act as a source of disorder for helimagnetic Co spins, a nd start to develop at\nincreasingly higher temperature (higher T/Tc) as the Mn concentration is increased. The\nincrease of the qvalue finally saturates below TLprobably because of the slowing of the\nantiferromagnetic fluctuations of Mn spins and resulting in a quasi-s tatic disorder for the\nhelimagnetic Co spins. As the temperature is further reduced, Mn s pins eventually freeze\nand undergo the reentrant spin glass transition as observed in the magnetization measure-\nments for Co 8Zn8Mn4and Co 7Zn7Mn6[Figs. 3 (c) and (d)]. The freezing temperature Tg\nalso increases with the Mn concentration due to the associated enh ancement of the antifer-\nromagnetic Mn spin correlations.\n2. Summary of metastable skyrmion state\nFigure 2 is the summary of the equilibrium and metastable skyrmion pha se diagrams\non theT-Hplane determined by SANS and ac susceptibility measurements in Co 10Zn10,\nCo9Zn9Mn2, Co8Zn8Mn4and Co 7Zn7Mn6. In all the compounds, the metastable SkX state\nis realized by a conventional field cooling via the equilibrium SkX phase ju st below Tcand\nprevailsover averywidetemperatureandfieldregion. Inaddition, w hiletheequilibriumand\nmetastable skyrmions at high temperatures form a conventional t riangular lattice described\nwith triple- qvectors, the lattice structure transforms to the novel double- qstates at low\ntemperatures as follows. In Co 10Zn10[Fig. 2(b)], the lattice form of the metastable SkX\ntransforms from triangular to a rhombic one (M-R-SkX) with double -q∝bardbl<111>below∼\n360 K with a broad coexistence region (a purple region). At all tempe ratures, the triangular\nlattice is restored as the field is increased. In Co 9Zn9Mn2[Fig. 2(c)], while the triangular\nlattice of the metastable skyrmions (M-T-SkX) persists over a wide temperature region, it\ntransforms to square one (M-S-SkX) with double- q∝bardbl<100>below∼50 K as shown with a\npink region. The triangular lattice is also recovered at high fields. In C o8Zn8Mn4[Fig. 2(d)]\nandCo 7Zn7Mn6[Fig. 2(e)], the triangular-squaretransition ofskyrmion lattice occ urs below\n∼120 K, and the square SkX persists below the reentrant spin glass t ransition temperatures22\nas plotted with yellow circles. In Co 7Zn7Mn6, the metastable square SkX state originating\nfrom the conventional SkX phase undergoes another transition t o the other frustration-\ninduced equilibrium skyrmion phase (E-DSk; orange region) as the fie ld is increased at\ntemperatures below ∼60 K.\nThe structural transitions of the skyrmion lattice to the rhombic o ne and to the square\noneareperhapsattributedtotheenhancedmagnetocrystallinea nisotropytoward q∝bardbl<111>\nandq∝bardbl<100>, respectively. However, the qvector anisotropy alone is not sufficient to\ndrive the lattice structural transition. Importantly, these tran sformations are accompanied\nby an increase in absolute value of q: the triangular-rhombic transition in Co 10Zn10is\nobserved while qvalue slightly increases over a broad temperature range [Fig. 3(e)], and the\ntriangular-square transition in Mn-doped compounds occurs only w henqvalue significantly\nincreases below a specific temperature of TH[Fig. 3(f-h)]. As discussed quantitatively in the\nfollowing, the increase in qvalue is crucial for the transformation of the skyrmion lattice in\nterms of skyrmion density.\nThe ratio of skyrmion density (number of the skyrmion per area) in t he rhombic lattice\n(nR) to that in the triangular one ( nT) is given by nR/nT=3√\n3\n4√\n2(aT/aR)2=3√\n3\n4√\n2(qR/qT)2.\nHere,aT(qT) andaR(qR) are the lattice constant (the qvalue) of the triangular lattice\nand the rhombic one, respectively. Since the total number of meta stable skyrmions that are\ntopologically protected is conserved during the transition from the triangular lattice to the\nrhombic one ( nR=nT), the lattice constant should decrease (the qvalue should increase) by\nthe factor of/radicalBig\n4√\n2\n3√\n3∼1.043 as the temperature is lowered. The qRvalue that is consistent\nwith this condition is presented with an dashed purple line in the q(T) plot for Co 10Zn10\n[inset of Fig. 3(e)]. Here, qTis taken as the qvalue at 412 K. It is found that the observed\nsmall increase in qvalue during the FC in Co 10Zn10easily satisfies the above condition\naround∼200 K. This result is in accord with the fact that the transformation to rhombic\nlattice starts at high temperatures.\nIn the case of the transition to square lattice in the Mn-doped comp ounds, on the\nother hand, the aforementioned condition is more difficult to fulfill. Th e ratio of the\nskyrmion density in the square lattice ( nS) to that in the triangular one is given by\nnS/nT=√\n3\n2(aT/aS)2=√\n3\n2(qS/qT)2, where aS(qS) is the lattice constant (the qvalue)\nof the square lattice. To keep the skyrmion density constant ( nS=nT), the lattice constant\nshould decrease (the qvalue should increase) by the larger factor of/radicalBig\n2√\n3∼1.075. The23\nvalue ofqSsatisfying this condition is denoted with an dashed pink line in the q(T) plot in\nCo9Zn9Mn2, Co8Zn8Mn4and Co 7Zn7Mn6[Fig. 3(f-h)]. Here, qTis taken as the qvalue at\nthe equilibrium SkX phase. In Co 9Zn9Mn2and Co 8Zn8Mn4,qvalue is almost T-constant\nat high temperatures. However, qvalue significantly increases and exceeds the necessary\nqSvalue below TH(∼50 K for Co 9Zn9Mn2and∼120 K for Co 8Zn8Mn4and Co 7Zn7Mn6),\nwhich would describe the transformation to the square lattice. This result well explains\nthe observed onset temperature of the transformation to the s quare lattice. Nevertheless,\nthe observed increase in qaroundTLis as large as qS/qT∼1.5, and the skyrmion density\nbecomes too large if we assume a uniformly and perfectly ordered sq uare SkX as shown in\nFig. 1(e).\nAlternatively, the large increase in qunder the conserved skyrmion density can be rec-\nonciled with either of the following two scenarios: (i) a microscopic pha se separation into\na square SkX state and a helical phase as schematically illustrated in F ig. 1(f), or (ii) the\nonset of skyrmion deformation along the <100>directions [Fig. 1(g)]. It should be noted\nthat, in both (i) and (ii), the total number of skyrmions is identical t o the original triangular\nSkX. Although the latter deformed skyrmions have been observed in a thin-plate sample by\nLTEM[37] and reproduced by a micromagnetic simulation[61], it is difficult to experimen-\ntally distinguish the two scenarios from SANS studies on a three-dime nsional bulk sample.\nIn any case, the large increase in qvalue at low temperatures and the enhanced qvector\nanisotropy toward q∝bardbl<100>cooperatively drive the skyrmion lattice transformation as\nthe metastable triangular SkX phase is cooled down to low temperatu res.\nIt is also noted that the square lattice of (elongated) skyrmions in t he present case is\ndistinct from a meron-antimeron square lattice with vanishing topolo gical charge (genuine\nsuperposition of orthogonal double qvectors) as recently observed in a thin-plate sample of\nCo8Zn9Mn3by LTEM[39]. The meron-antimeron state appears as an equilibrium st ate just\nbefore entering the equilibrium triangular SkX phase near Tc. While the meron-antimeron\nsquare lattice is stabilized by the in-plane shape anisotropy charact eristic of a thin-plate\nspecimen as theoretically predicted[62, 63], the transition to the sq uare SkX in the present\ncase probably originates from magnetocrystalline anisotropy (loca l magnetization ∝bardbl<100>,\nwhich in turn results in qvector anisotropy along <100>) as well as the large increase in q,\nas discussed above.24\nG. Lifetime of metastable skyrmion\nIn the former sections, we revealed that the SkX state can persis t at low temperatures\nby a field cooling process. Since the low temperature states are only metastable, the SkX\nshould relax to the more stable conical state with some lifetime. In ou r previous study\n(Ref. [36]),weinvestigatedtemperature-dependent relaxationt imesofmetastableskyrmions\nin Co9Zn9Mn2and observed extremely long lifetimes even at high temperatures. F or a\nsystematic understanding of behavior, in this section we extend th e lifetime measurement\nto the other compounds Co 10Zn10and Co 8Zn8Mn4, and discuss how the metastable SkX\nlifetime varies with temperature and Mn concentration (Fig. 12 and F ig. 13).\nFigures 12(a-c) show H-Tphase diagrams of equilibrium states near Tcin Co10Zn10,\nCo9Zn9Mn2and Co 8Zn8Mn4, respectively. For the purpose of better comparison between\nthe three compounds, Tranges are selected in the respective panels in such a way that nor-\nmalized temperature T/Tcis from 0.90 to 1.01, as indicated on the upper abscissa. Clearly,\nthe equilibrium SkX phase (green region) expands upon increasing th e Mn concentration\ndue to the increased chemical and magnetic disorder. After a FC via the equilibrium SkX\nphase, the temporal variation of ac susceptibility χ′(t) was measured at a fixed temperature,\nand this experiment was repeated at several different temperatu res, as denoted with the\ncolored circles.\nThe normalized ac susceptibility χ′\nN(t)≡[χ′(∞)−χ′(t)]/[χ′(∞)−χ′(0)] is plotted as a\nfunction of time in Fig. 12(d-f). Here, χ′(0) andχ′(∞) correspond to an initial value for\nthe metastable SkX state and the value for the equilibrium conical st ate as a fully relaxed\nstate, respectively, and hence χ′\nN= 1 fort= 0 and χ′\nN= 0 fort→ ∞. In Co 10Zn10[Fig.\n12(d)], a clear relaxation from the metastable SkX state to the equ ilibrium conical state is\nobserved at 404 K, just below the equilibrium SkX phase, and for whic h the relaxation\ntime is the order of 104s (several hours). The relaxation time further increases upon\nlowering the temperature. In Co 9Zn9Mn2[Fig. 12(e)], the observed relaxation in the χ′\nN(t)\ncurve within the measurement time ( ∼1 day) is less than 40% even at 380 K, just below\nthe equilibium SkX phase, and the relaxation becomes even longer as t he temperature\nis lowered. In Co 8Zn8Mn4[Fig. 12(f)], only a few % of relaxation is observed in the\nχ′\nN(t) even at 280 K, just below the equilibrium SkX phase. Therefore, th e lifetime of\nmetastable SkX at temperatures close to the equilibrium SkX phase b oundary, for example25\nthe temperatures indicated with the red circles in Fig. 12(a-c), bec omes relatively longer as\nthe Mn concentration is increased.\nTodiscuss morequantitatively, the χ′\nN(t)curves arefittedtoastretched exponential func-\ntion, exp/braceleftbig\n−(t/τ)β/bracerightbig\n. The obtained βvalues are in the range of 0.3 - 0.5, indicating a highly\ninhomogeneous distribution of relaxation times. The relaxation time τin all the compounds\n(Co10Zn10, Co9Zn9Mn2and Co 8Zn8Mn4) is plotted against T/Tcin Fig. 13(a). The τvalue\nexponentially increases as the temperature is lowered, and become s virtually infinite when\nT/Tcis less than ∼0.9. Remarkably, the data points from the three different compoun ds\ncollapse onto a single curve althoughthe applied magnetic field as well a s the respective tem-\nperature regions are different. Following the arguments described in Ref. [64], all the data\npoints are fitted to a modified Arrhenius law (pink solid line), τ=τ0exp{a(Tc−T)/T}.\nHere, the activation energy for the relaxation from the metastab le SkX state to the equilib-\nrium conical state, as schematically illustrated in Fig. 13(b), is assum ed to be T-dependent\nasEg=a(Tc−T) nearTc, instead of constant Egfor standard Arrhenius law. The obtained\nfitting parameters are a= 215 and τ0= 63 s. For comparison, reported values of aand\nτ0for several materials[64, 65] are summarized in Table 2. Recently, W ildet al. reported\nthatτ0sensitively depends on the applied magnetic field in their LTEM studies o f a thin\nplate of Fe 1−xCoxSi[66]. For the present ac susceptibility measurements on bulk cryst als of\nCo-Zn-Mn alloys, we used the applied magnetic field for each compoun d where the SANS\nintensity from the equilibrium SkX is strongest in the field sweeping mea surement. The fact\nthat theτvalues from three different compounds collapse onto a single curve ( namely,τ0are\nthe same) may originate from the fact that the applied field values ar e close to the optimal\nones in the respective samples.\nFromthe obtained value of a, the activation energy Egis estimated to be larger than8000\nK at a temperature T= 0.9Tcfor Co 10Zn10and Co 9Zn9Mn2. This energy scale protecting\nthe metastable skyrmion state is much larger than the ferromagne tic exchange interaction\n(several 100 K), and thus attributed to the topological nature o f the skyrmion with a large\ndiameter ( ∼100 nm) that involves a great number of spins.\nThe coefficient of relaxation time τ0is inversely correlated to the critical cooling rate[67]\nfor quenching the SkX phase to lower temperatures. Since the obt ained value of τ0in Co-\nZn-Mn alloys is the order of a minute, the conventionally slow cooling ra te (dT/dt∼ −1\nK/min) is sufficient to quench the SkX phase. This is quite different fro m the value of τ0∼26\n10−4s in MnSi[64], where an ultra-rapid cooling rate ( dT/dt∼ −100 K/s) is necessary to\nquench the SkX phase. The large difference in τ0that depends on the material is probably\nattributed to randomness in the system. In the case of Co-Zn-Mn alloys, there are random\nsite occupancies in the crystal structure; the 8 csite is randomly occupied by Co and Mn,\nand the 12 dsite is occupied by Co, Zn, and Mn[27–30]. This gives rise to “weak pinnin g” in\nthe terminology of density wave physics[68], which may play an importa nt role in the robust\nmetastability of the skyrmion.\nFrom a microscopic viewpoint, the destruction of metastable skyrm ions in the bulk takes\nplace through the creation of a pair of Bloch points, or equivalently a n emergent magnetic\nmonopole-antimonopole pair, from a singularity point of a skyrmion st ring, followed by their\npropagation[69, 70], as schematically illustrated in Fig. 13(c). Coeffic ientsaandτ0are\nroughly governed by the creation and the propagation processes of monopole-antimonopole\npairs, respectively. In a clean system like MnSi, the monopole and ant imonopole can easily\nmove, which leads to skyrmion string destruction after the pair cre ation. In a dirty system\nlike Co-Zn-Mn alloys, the movement of monopole and antimonopole is hin dered by magnetic\nimpurities or defects, and consequently τ0of the metastable skyrmion string is significantly\nincreased. A similar long-lived metastable SkX that is accessible by a mo derate cooling rate\nhas been reported in Fe 1−xCoxSi alloys[66, 69, 71], which may bear some resemblance to the\npresent case. More recently, the increased lifetime of metastable SkX has been observed also\nin Zn-doped Cu 2OSeO3, whereτ0in a Zn 2.5% doped sample is 50 times larger than that\nin a non-doped sample while avalue is unchanged[65]. In the present case of Co-Zn-Mn\nalloys, the good scaling of the τvsT/Tcplot in Fig. 13(a) indicates that both aandτ0are\nalmost independent of the Mn concentration. This is probably becau se Co10Zn10already\npossesses substantial randomness in the site occupancy at 12 dsite (2 Co and 10 Zn per unit\ncell), and thus the attempt time τ0is already sufficiently long and not further increased by\nadditional randomness due to the Mn substitution. Nevertheless, Mn substitution expands\nthe equilibrium SkX phase toward lower temperature as seen in Fig. 12 (a-c) and Fig. 13(a),\nand hence the relaxation time just below the equilibrium SkX phase bec omes longer. It\nis also interesting to note that the equilibrium skyrmion phase is expan ded by the static\nmagnetic disorder while thermal fluctuation is considered to be impor tant for the realization\nof the equilibrium phase close to Tc.27\nIV. CONCLUSION\nIn the present study, to provide the perspective on the chiral ma gnetism in β-Mn-type\nCo-Zn-Mn alloys with bulk Dzyaloshinskii-Moriya interaction (DMI), we have performed\nmagnetization, ac susceptibility and small-angle neutron scattering measurements on single-\ncrystal samples of (Co 0.5Zn0.5)20−xMnxwithx= 0, 2, 4 and 6. The Mn-free end member\nCo10Zn10exhibits a helimagnetic ground state (periodicity λ∼156 nm) below the transition\ntemperature Tc∼414 K, where the helical propagation vector qis aligned to <111>at low\ntemperatures (Fig. 5). Upon applying magnetic fields, Co 10Zn10exhibits an equilibrium\nSkX phase above 400 K in a narrow temperature and magnetic field re gion (Fig. 4), which\nis quenched down to lower temperatures as a metastable state by a conventionally-slow field\ncooling (Fig. 6). The lifetime of the metastable SkX is extremely long an d virtually infinite\nbelow 380 K (Figs. 12 and 13). The metastable SkX state is highly robu st and persists\nover the whole temperature range below Tcand a wide magnetic field region, including\nroom temperature and zero field (Fig. 2). The lattice of metastable skyrmions distorts and\ntransforms from a conventional triangular one to a rhombic one at low temperatures and\nlow magnetic fields (Figs. 2 and 6).\nAs the partial substitution with Mn proceeds, Tcdecreases and the preferred qorienta-\ntion switches to <100>. The saturation magnetization is the largest for Co 9Zn9Mn2(Table\n1). At low temperatures, the helical qvalue significantly increases, or equivalently λsignif-\nicantly decreases, below TH∼50 K for Co 9Zn9Mn2and below TH∼120 K for Co 8Zn8Mn4\nand Co 7Zn7Mn6(Fig. 3). In Co 7Zn7Mn6, the helical state is severely disordered at low\ntemperatures. Upon further decreasing temperature, a reent rant spin glass transition oc-\ncurs atTg∼10 K for Co 8Zn8Mn4andTg∼30 K for Co 7Zn7Mn6while such a transition\nis not observed for Co 10Zn10and Co 9Zn9Mn2down to 2 K (Figs. 1 and 3). In common\nwith Co 10Zn10, a long-lived metastable SkX state is realized in the Mn-doped materia ls by a\nmoderate FC through the equilibrium SkX phase, and persists over a wide temperature and\nfield region. On the other hand, the lattice form of the metastable S kX changes to a square\none at low temperatures (Figs. 2, 7, 8 and 9). While the triangular Sk X is dictated by the\napplied field as q⊥H, the square SkX is governed by the magnetocrystalline anisotropy as\nq∝bardbl<100>regardless of the applied field direction (Figs. 7 and 8). During the tr ansition\nto the square lattice, the periodicity of SkX significantly shrinks belo wTHsimilar to the28\nhelical periodicity in zero-field cooling (Fig. 3).\nFrom these results, we conclude the followings: The helical and skyr mion states in Co-\nZn-Mn alloys are basically formed by ferromagnetic Co spins in the pre sence of the DMI.\nWhile Co and Mn are ferromagnetically coupled at least in the low Mn conc entrations, an-\ntiferromagnetic Mn-Mn correlations become increasingly significant in the higher Mn con-\ncentrations and start to develop at higher temperatures. As the temperature is lowered, the\ndevelopment of antiferromagnetic Mn-Mn correlation leads to a diso rdering of the helical\nstate and simultaneously decreases helical pitch, and ultimately und ergoing a spin freezing\ntransition at very low temperatures. The robust metastability of s kyrmions is attributed to\nthe topological protection by a large number of involved Co spins as w ell as weak pinning\nfrom the magnetic disorder. The structural transformations be tween metastable triangular\nSkX and either a rhombic one or a square one are driven by a decreas e in the distance\nbetween the skyrmions under the influence of the magnetocrysta lline anisotropy that favors\nq∝bardbl<111>andq∝bardbl<100>in undoped and Mn-doped compounds, respectively. These\nfindings unveil the complex interplay between chiral magnetism and t he frustrated Mn spins\nthat is also greatly affected by magnetic disorder and anisotropy, a nd provide a significant\nunderstanding of the topological phases and properties in this clas s ofβ-Mn-type chiral\nmagnets.\nAcknowledgments\nWe are grateful to T. Arima, T. Nakajima, D. Morikawa, X. Z. Yu, L. C. Peng and N.\nNagaosa for fruitful discussions. We thank M. Bartkowiak for sup port of SANS experiments\nabove room temperature at Paul Scherrer Institute (PSI), Swit zerland. This work was\nsupported by JSPS Grant-in-Aids for Scientific Research (Grant N o. 24224009 and No.\n17K18355), JST CREST (Grant No. JPMJCR1874), the Swiss Nation al Science Foundation\n(SNSF)Sinergianetwork‘NanoSkyrmionics(GrantNo. CRSII5 171003)’,theSNSFprojects\n200021153451, 200021 188707 and 166298, and the European Research Council project\nCONQUEST.\nThe datasets for the SANS experiments done on D33 at the ILL are avail-\nable through the ILL data portal (http://doi.ill.fr/10.5291/ILL-DA TA.5-42-410 and29\nhttp://doi.ill.fr/10.5291/ILL-DATA.5-42-443).\n[1] A. N. Bogdanov and D. A. Yablonskii, Sov. Phys. JETP 68, 101 (1989).\n[2] N. Nagaosa and Y. Tokura, Nat. Nanotechnol. 8, 899 (2013).\n[3] S. M¨ uhlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosc h, A. Neubauer, R. Georgii and P.\nB¨ oni, Science 323, 915 (2009).\n[4] X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han, Y. Mat sui, N. Nagaosa and Y.\nTokura, Nature 465, 901 (2010).\n[5] F. Jonietz, S. Muhlbauer, C. Pfleiderer, A. Neubauer, W. M ¨ unzer, A. Bauer, T. Adams, R.\nGeorgii, P. B¨ oni, R. A. Duine, K. Everschor, M. Garst and A. R osch, Science 330, 1648\n(2010).\n[6] X. Z. Yu, N. Kanazawa, W. Z. Zhang, T. Nagai, T. Hara, K. Kim oto, Y. Matsui, Y. Onose\nand Y. Tokura, Nat. Commun. 3, 988 (2012).\n[7] J. Iwasaki, M. Mochizuki and N. Nagaosa, Nat. Nanotechno l.8, 742 (2013).\n[8] S. Heinze, K. von Bergmann, M. Menzel, J. Brede, A. Kubetz ka, R. Wiesendanger, G.\nBihlmayer and S. Bl¨ ugel, Nat. Phys. 7, 713 (2011).\n[9] N. Romming, C. Hanneken, M. Menzel, J. E. Bickel, B. Wolte r, K. Bergmann, A. Kubetzka\nand R. Wiesendanger, Science 341, 636 (2013).\n[10] J. Matsuno, N. Ogawa, K. Yasuda, F. Kagawa, W. Koshibae, N. Nagaosa, Y. Tokura and M.\nKawasaki, Sci. Adv. 2, e1600304 (2016).\n[11] S. Woo, K. Litzius, B. Kr¨ uger, M.-Y. Im, L. Caretta, K. R ichter, M. Mann, A. Krone, R.\nReeve, M. Weigand, P. Agrawal, P. Ischer, M. Kl¨ aui, G. S. D. B each, I. Lemesh and M.-A.\nMawass, Nat. Mater. 15, 501 (2016).\n[12] A. Soumyanarayanan, M. Raju, A. L. G. Oyarce, A. K. C. Tan , M.-Y. Im, A. P. Petrovic, P.\nHo, K. H. Khoo, M. Tran, C. K. Gan, F. Ernult and C. Panagopoulo s, Nat. Mater. 16, 898\n(2017).\n[13] X. Z. Yu, N. Kanazawa, Y. Onose, K. Kimoto, W. Z. Zhang, S. Ishiwata, Y. Matsui and Y.\nTokura, Nat. Mater. 10, 106 (2011).\n[14] S. Seki, X. Z. Yu, S. Ishiwata and Y. Tokura, Science 336, 198 (2012).\n[15] Y. Tokunaga, X. Z. Yu, J. S. White, H. M. Rønnow, D. Morika wa, Y. Taguchi and Y. Tokura,30\nNat. Commun. 6, 7638 (2015).\n[16] W. Li, C. Jin, R. Che, W. Wei, L. Lin, L. Zhang, H. Du, M. Tia n and J. Zang, Phys. Rev. B\n93, 060409(R) (2016).\n[17] I. K´ ezsm´ arki, S. Bord´ acs, P. Milde, E. Neuber, L. M. E ng, J. S. White, H. M. Rønnow, C. D.\nDewhurst, M. Mochizuki, K. Yanai, H. D. Ehlers, V. Tsurkan an d A. Loidl, Nat. Mater. 14,\n1116 (2015).\n[18] T. Kurumaji, T. Nakajima, V. Ukleev, A. Feoktystov, T. A rima, K. Kakurai and Y. Tokura,\nPhys. Rev. Lett. 119, 237201 (2017).\n[19] A. K. Nayak, V. Kumar, T. Ma, P. Werner, E. Pippel, R. Saho o, F. Damay, U. K. Rosler, C.\nFelser and S. S. P. Parkin, Nature 548, 561 (2017).\n[20] X. Z. Yu, Y. Tokunaga, Y. Kaneko,W. Z. Zhang, K. Kimoto, Y . Matsui, Y. Taguchi and Y.\nTokura, Nat. Commun. 5, 3198 (2014).\n[21] W. Wang, Y. Zhang, G. Xu, L. Peng, B. Ding, Y. Wang, Z. Hou, X. Zhang, X. Li, E. Liu, S.\nWang, J. Cai, F. Wang, J. Li, F. Hu, G. Wu, B. Shen and X.-X. Zhan g, Adv. Mater. 28, 6887\n(2016).\n[22] Z. Hou, W. Ren, B. Ding, G. Xu, Y. Wang, B. Yang, Y. Zhang, E . Liu, F. Xu, W. Wang, G.\nWu, X. Zhang, B. Shen and Z. Zhang, Adv. Mater. 29, 1701144 (2017).\n[23] R. Takagi, X. Z. Yu, J. S. White, K. Shibata, Y. Kaneko, G. Tatara, H. M. Rønnow, Y. Tokura\nand S. Seki., Phys. Rev. Lett. 120, 037203 (2018).\n[24] M. Kakihana, D. Aoki, A. Nakamura, F. Honda, M. Nakashim a, Y. Amako, S. Nakamura, T.\nSakakibara, M. Hedo, T. Nakama and Y. ¯Onuki, J. Phys. Soc. Jpn. 87, 023701 (2018).\n[25] T. Kurumaji, T. Nakajima, M. Hirschberger, A. Kikkawa, Y. Yamasaki, H. Sagayama, H.\nNakao, Y. Taguchi, T. Arima and Y. Tokura, Science 365, 914 (2019).\n[26] M. Hirschberger, T. Nakajima, S. Gao, L. Peng, A. Kikkaw a, T. Kurumaji, M. Kriener, Y.\nYamasaki, H. Sagayama, H. Nakao, K. Ohishi, K. Kakurai, Y. Ta guchi, X. Z. Yu, T. Arima\nand Y. Tokura, Nat. Commun. 10, 5831 (2019).\n[27] T. Hori, H. Shiraish and Y. Ishii, J. Magn. Magn. Mater. 310, 1820 (2007).\n[28] W. Xie, S. Thimmaiah, J. Lamsal, J. Liu, T. W. Heitmann, D . Quirinale, A I. Goldman, V.\nPecharsky and G. J. Miller, Inorg. Chem. 52, 9399 (2013).\n[29] J. D. Bocarsly, C. Heikes, C. M. Brown, S. D. Wilson, and R . Seshadri, Phys. Rev. Mater. 3,\n014402 (2019).31\n[30] T.Nakajima, K.Karube,Y. Ishikawa, M. Yonemura, N. Rey nolds, J.S.White, H. M.Rønnow,\nA. Kikkawa, Y. Tokunaga, Y. Taguchi, Y. Tokura, and T. Arima, Phys. Rev. B 100, 064407\n(2019).\n[31] K. Karube, J. S. White, D. Morikawa, C. D. Dewhurst, R. Cu bitt, A. Kikkawa, X. Z. Yu,\nY. Tokunaga, T. Arima, H. M. Rønnow, Y. Tokura, and Y. Taguchi , Sci. Adv. 4, eaar7043\n(2018).\n[32] R. Takagi, D. Morikawa, K. Karube, N. Kanazawa, K. Shiba ta, G. Tatara, Y. Tokunaga, T.\nArima, Y. Taguchi, Y. Tokura, and S. Seki, Phys. Rev. B 95, 220406(R) (2017).\n[33] K. Karube, K. Shibata, J. S. White, T. Koretsune, X. Z. Yu , Y. Tokunaga, H. M. Rønnow,\nR. Arita, T. Arima, Y. Tokura, and Y. Taguchi, Phys. Rev. B 98, 155120 (2018).\n[34] K. Karube, J. S. White, N. Reynolds, J. L. Gavilano, H. Oi ke, A. Kikkawa, F. Kagawa, Y.\nTokunaga, H. M. Rønnow, Y. Tokura, and Y. Taguchi, Nat. Mater .15, 1237 (2016).\n[35] As explained in Section III-F-2, there are two possibil ities for the observed double- qstate\nwith increased qvalues: One is a coexistence of the square lattice of skyrmio ns and the helical\nstate [Fig. 1(f)], and the other is elongated-skyrmion stat e on the square lattice [Fig. 1(g)].\nHereafter, we refer to the double- qstate as “square lattice of skyrmions (or square SkX)” for\nthe purpose of simplicity throughout the paper.\n[36] K. Karube, J. S. White, D. Morikawa, M. Bartkowiak, A. Ki kkawa, Y. Tokunaga, T. Arima,\nH. M. Rønnow, Y. Tokura, and Y. Taguchi, Phys. Rev. Mater. 1, 074405 (2017).\n[37] D. Morikawa, X. Z. Yu, K. Karube, Y. Tokunaga, Y. Taguchi , T. Arima and Y. Tokura, Nano\nLett.17, 1637 (2017).\n[38] T. Nagase, M. Komatsu, Y. G. So, T.Ishida, H. Yoshida, Y. Kawaguchi, Y. Tanaka, K. Saitoh,\nN. Ikarashi, M. Kuwahara, and M. Nagao, Phys. Rev. Lett. 123, 137203 (2019).\n[39] X. Z. Yu, W. Koshibae, Y. Tokunaga, K. Shibata, Y. Taguch i, N. Nagaosa and Y. Tokura,\nNature564, 95 (2018).\n[40] H. Nakamura, K. Yoshimoto, M. Shiga, M. Nishi, and K. Kak urai, J. Phys. Condens. Matter\n9, 4701 (1997).\n[41] J. R. Stewart, B. D. Rainford, R. S. Eccleston, and R. Cyw inski, Phys. Rev. Lett. 89, 186403\n(2002).\n[42] J. R. Stewart, K. H. Andersen, and R. Cywinski, Phys. Rev . B78, 014428 (2008).\n[43] J. R. Stewart and R. Cywinski, J. Phys.: Condens. Matter 21, 124216 (2009).32\n[44] J. R. Stewart, A. D. Hillier, J. M. Hillier, and R. Cywins ki, Phys. Rev. B 82, 144439 (2010).\n[45] J. A. M. Paddison, J. R. Stewart, P. Manuel, P. Courtois, G. J. McIntyre, B. D. Rainford,\nand A. L. Goodwin, Phys. Rev. Lett. 110, 267207 (2013).\n[46] It should be noted that the helimagnetic order formed by Co spins and the spin glass state of\nfrustrated Mn spins coexist. Therefore, it may also be calle d “embedded spin glass” as used\nin Ref. [51].\n[47] B. R. Coles, B. V. B. Sarkissian, and R. H. Taylor, Phil. M ag. B37, 489 (1978).\n[48] K. Motoya, S. M. Shapiro, and Y. Muraoka, Phys. Rev. B 28, 6183 (1983).\n[49] H. Maletta, G. Aeppli, and S. M. Shapiro, J. Magn. Magn. M ater.31, 1367 (1983).\n[50] N. Hanasaki, K. Watanabe, T. Ohtsuka, I. K´ ezsm´ arki, S . Iguchi, S. Miyasaka, and Y. Tokura,\nPhys. Rev. Lett. 99, 086401 (2007).\n[51] J. O. Piatek, B. Dalla Piazza, N. Nikseresht, N. Tsyruli n, I.ˇZivkovi´ c, K. W. Kr¨ amer, M.\nLaver, K. Prokes, S. Mataˇ s, N. B. Christensen, and H. M. Rønn ow, Phys. Rev. B 88, 014408\n(2013).\n[52] I.Mirebeau, N. Martin, M.Deutsch, L.J.Bannenberg, C. Pappas, G. Chaboussant, R. Cubitt,\nC. Decorse, and A. O. Leonov, Phys. Rev. B 98, 014420 (2018).\n[53] H. Yoshizawa, S. Mitsuda, H. Aruga, and A. Ito, J. Phys. S oc. Jpn. 58, 1416 (1989).\n[54] C. J. Leavey, J. R. Stewart, B. D. Rainford, and A. D. Hill ier, J. Phys.: Condens. Matter 19,\n145288 (2007).\n[55] T. Sato and K. Morita, J. Phys.: Condens. Matter 11, 4231 (1999).\n[56] In Ref. [31] we reported for Co 7Zn7Mn6that the real part of the ac susceptibility exhibits a\ndrop, the imaginary part a peak at Tg∼30 K, and the frequency-dependent Tgincreases (29\nK→34 K) as the ac frequency fis increased (0.1 Hz →1000 Hz). The frequency-dependence\nofTgsatisfies a power law f−1=τ0[Tg(f)/Tg(0)−1]−zν(τ0= 5.3×10−8s,Tg(0) = 24.6\nK,zν= 10.7). The slow spin flipping time τ0∼10−8s is a typical value for reentrant spin\nglass[50]. A similar measurement for Co 7Zn7Mn6was also reported in Ref. [29].\n[57] See Supplementary Materials at [URL will be inserted by publisher] for details.\n[58] The asymmetric peak broadenings around the [-111] and [ 1-1-1] directions at 1.5 K are at-\ntributed to imperfect summations of SANS patterns along the rocking direction.\n[59] Y. Togawa, T. Koyama, K. Takayanagi, S. Mori, Y. Kousaka , J. Akimitsu, S. Nishihara, K.\nInoue, A. S. Ovchinnikov, and J. Kishine, Phys. Rev. Lett. 108, 107202 (2012).33\n[60] On the basis of demagnetization coefficients in the recta ngular-shaped sample and a magne-\ntization value at 295 K, the extra tilt angle αof the effective magnetic field Hefffrom the\nexternal field Hwas calculated to be α= 23◦, showing a good agreement with the observed\npeak position of the rocking curve at 295 K.\n[61] V. Ukleev, Y. Yamasaki, D. Morikawa, K. Karube, K. Shiba ta, Y. Tokunaga, Y. Okamura,\nK. Amemiya, M. Valvidares, H. Nakao, Y. Taguchi, Y. Tokura, a nd T. Arima, Phys. Rev. B\n99, 144408 (2019).\n[62] S. D. Yi, S. Onoda, N. Nagaosa, and J. H. Han, Phys. Rev. B 80, 054416 (2009).\n[63] S.-Z. Lin, A. Saxena and C. D. Batista, Phys. Rev. B 91, 224407 (2015).\n[64] H. Oike, A. Kikkawa, N. Kanazawa, Y. Taguchi, M. Kawasak i, Y. Tokura and F. Kagawa,\nNat. Phys. 12, 62 (2016).\n[65] M. T. Birch, R. Takagi, S. Seki, M. N. Wilson, F. Kagawa, A .ˇStefanˇ ciˇ c, G. Balakrishnan, R.\nFan, P. Steadman, C. J. Ottley, M. Crisanti, R. Cubitt, T. Lan caster, Y. Tokura, and P. D.\nHatton, Phys. Rev. B 100, 014425 (2019).\n[66] J. Wild, T. N. G. Meier, S. Pollath, M. Kronseder, A. Baue r, A. Chacon, M. Halder, M.\nSchowalter, A. Rosenauer, J. Zweck, J. Muller, A. Rosch, C. P fleiderer, and C. H. Back, et\nal., Sci. Adv. 3, e1701704 (2017).\n[67] F. Kagawa and H. Oike, Adv. Mater. 29, 1601979 (2017).\n[68] G. Gr¨ uner, Rev. Mod. Phys. 60, 1129 (1988).\n[69] P. Milde, D. K¨ ohler, J. Seidel, L. M. Eng, A. Bauer, A. Ch acon, J. Kindervater, S. M¨ uhlbauer,\nC. Pfleiderer, S. Buhrandt, C. Sch¨ utte and A. Rosch, Science 340, 1076 (2013).\n[70] F. Kagawa, H. Oike, W. Koshibae, A. Kikkawa, Y. Okamura, Y. Taguchi, N. Nagaosa, and\nY. Tokura, Nat. Commun. 8, 1332 (2017).\n[71] W. M¨ unzer, A. Neubauer, T. Adams, S. M¨ uhlbauer, C. Fra nz, F. Jonietz, R. Georgii, P. B¨ oni,\nB. Pedersen, M. Schmidt, A. Rosch and C. Pfleiderer, Phys. Rev . B81, 041203(R) (2010).34\nFIG. 1. (a) Schematics of β-Mn-type chiral crystal structures as viewed along the [111]\ndirection. Two enantiomers with the space group P4332 (left-handed structure) and P4132\n(right-handed structure) are shown. Blue and red circles repres ent 8cand 12dWyckoff\nsites, respectively. The network of 12 dsites forms a hyper-kagome structure composed\nof corner-sharing triangles. (b) Temperature ( T) - Mn concentration ( x) phase diagram\nin (Co 0.5Zn0.5)20−xMnx(0≤x≤10) at zero field, determined by magnetization ( M)\nmeasurements. Closed symbols are data of polycrystalline samples r eproduced from our\nprevious work in Ref. [31]. Copyright 2018, American Association for the Advancement of\nScience. Open symbols are data of single-crystalline samples ( x= 0, 2, 4, 6) in the present\nstudy [see Fig. 3(a-d) for the detailed determination of the phase b oundaries]. A shaded\nregion with red indicates the temperature range TL≤T≤THwhereMat 20 Oe and the\nmagnitude of helical wavevector ( q) vary (decreases and increases respectively on cooling)\nsignificantly. (c-g) Schematic illustrations of various skyrmion lattic es: (c) triangular\nlattice, (d) rhombic lattice, (e) square lattice, (f) coexistence of square lattice of skyrmions\nand helical state, (g) I- or L-like deformed skyrmions on a square la ttice. Corresponding35\nhybridized qvectors are also presented in each panel.\nFIG. 2. Summary of equilibrium and metastable skyrmion states in tem perature ( T) -\nmagnetic field ( H) plane in (b) Co 10Zn10, (c) Co 9Zn9Mn2, (d) Co 8Zn8Mn4(reproduced from\nour previous work in Ref. [34]. Copyright 2016, Springer Nature) an d (e) Co 7Zn7Mn6. The\nmeasurement processes for the state diagrams are schematically illustrated in panel (a). The\nequilibrium skyrmion crystal (SkX) phase just below Tc(green area) and the equilibrium\ndisordered skyrmion (DSk) phase (orange area) are determined b y field-sweeping measure-36\nments of ac susceptibility and SANS after zero-field cooling (ZFC). T he metastable SkX\nstate [triangular lattice (light-blue area), rhombic lattice (purple ar ea) and square lattice\n(pink area)] are determined by field-sweeping measurements towa rd positive and negative\ndirections after the field cooling (FC) via the positive-field equilibrium S kX phase. Here,\nwe use the following notations; H: helical, C: conical, F: ferromagnet ic, E: equilibrium, M:\nmetastable, T: triangular, R: rhombic, S: square, SkX: skyrmion c rystal, DSk: disordered\nskyrmions, and RSG: reentrant spin glass. The phase boundaries a re determined by ac\nsusceptibility ( χ′). For the purpose of comparison with SANS measurements, calibra ted\nvalues (Hc) are used for the magnetic field. The details about the determinatio n of the\nphase boundaries of the equilibrium SkX state and the metastable Sk X state in Co 10Zn10\nare described in the captions of Fig. 4(d) and Supplementary Fig. S2 (c), respectively.\nThe boundaries between the metastable triangular SkX and the met astable rhombic SkX\n(purple diamonds) are determined by SANS results [see Fig. 6(d) and Supplementary Figs.\nS2(d) and S3(d) for the details]. The boundary between the metas table triangular SkX and\nthe metastable square SkX (red squares) are determined as inflec tion points of χ′(H) as\ndetailed in Ref. [34]. The reentrant spin glass transition temperatur es (yellow open circles)\nare determined as inflection points of a sharp drop in χ′(T), as also observed in M(T) in\nthe zero-field-cooled field-warming process [Fig. 3(c, d)]. The equilib rium DSk phase in\nCo7Zn7Mn6, determined as the region where a spherical SANS pattern is obser ved, was\nreproduced from our previous work in Ref. [31]. Copyright 2018, Am erican Association for\nthe Advancement of Science.37\nFIG. 3. Temperature dependence of magnetization and qvector in Co 10Zn10(red),\nCo9Zn9Mn2(green), Co 8Zn8Mn4(blue) and Co 7Zn7Mn6(purple). (a-d) Magnetization\n(M) under a magnetic field of 20 Oe is plotted against temperature. Solid lines show\nthe data collected during field cooling (FC), and broken lines represe nt those taken in a\nfield-warming run after a zero-field cooling down to 2 K (ZFC-FW). Th e helimagnetic\ntransition temperature ( Tc) indicated with closed triangles is determined as an inflection\npoint of a sharp increase in Mon the FC process. THandTLare the temperatures at\nwhich a gradual decrease in Mon cooling starts and ends, respectively. These temperatures\nare determined as inflection points with a broad peak in dM/dTon the FC process and\npresented with open triangles. The reentrant spin glass transition temperature ( Tg) denoted\nwith closed triangles is determined as an inflection point of a sharp incr ease inMon the\nZFC-FW process. These temperatures are also plotted in the T-xphase diagram in Fig.\n1(b). (e-h) Temperature variation of the magnitude of qin the helical state at zero field\n(open squares), and qin equilibrium and metastable SkX states under magnetic fields\n(closed circles). The qvalues are determined as the peak center of Gaussian function fitt ed\nto SANS intensity as a function of q. The inset of panel (e) shows an enlarged view of q(T)\nfor Co 10Zn10. The purple dashed line in the inset of (e) and the pink dashed line in (f- h)38\nindicate the expected qvalues for rhombic and square lattice of skyrmions (= 1.043 qTand\n1.075qT, whereqTare the values in the equilibrium triangular SkX state), respectively, with\nthe assumption of constant skyrmion density from the high-tempe rature triangular lattice\n(see discussion in Section III-F-2 for the details). (i-l) Temperatu re dependence of full\nwidth at half maximum (FWHM) of the Gaussian function fitted to SANS intensity versus q\nComposition Co10Zn10Co9Zn9Mn2Co8Zn8Mn4Co7Zn7Mn6\nTc(K) 414 396 299 158\nMs(µB/f.u.) 11.3 14.3 12.7 7.97\nλmax(nm) 156 132 110 112\nλmin(nm) 143 91 73 74\nPreferred qdirection <111> < 100> < 100> < 100>\nTABLE. 1. Summary of several physical parameters for helimagne tic state in Co 10Zn10,\nCo9Zn9Mn2, Co8Zn8Mn4and Co 7Zn7Mn6. Helimagnetic transition temperature Tcis\ndetermined from the temperature variation in magnetization under 20 Oe. Saturation\nmagnetization Msis defined as a magnetization value at 2 K and 7 T. Helimagnetic\nperiodicity λ(maximum value λmaxat high temperature and minimum value λminat low\ntemperature) is calculated from the qvalue obtained from SANS measurements in zero\nfield. The preferred orientation of the helical qvector is also determined from SANS\nmeasurements.39\nFIG. 4. Identification of equilibrium SkX phase in Co 10Zn10. (a) Photos of single-crystalline\nbulk samples of Co 10Zn10used for SANS and ac susceptibility measurements. Crystal axes\nand the direction of applied magnetic field ( H) are also indicated. (b) SANS images at\nselected fields in the field increasing run at 410 K. The intensity scale o f the color plot is\nfixed for the 4 panels. (c) Field dependence of SANS intensity at 410 K. The intensity40\nwas integrated over the region close to <111>(red area in the inset) as detailed in Fig.\n5(c). (d) Field dependence of the real part ( χ′, red line) and the imaginary part ( χ′′, blue\nline) of the ac susceptibility in the field increasing run at 410 K. Since th e demagnetization\nfactor is different from that in the SANS measurement, the field valu es are calibrated as\nHc= 3.0×H. The phase boundary between the helical and conical states ( HH−C, black\ntriangle) is determined as the inflection point of χ′. The phase boundaries between the\nequilibrium SkX and conical states ( HESkX1(2), green triangles) are determined as the peak\npositions at both sides of the dip structure in χ′. The phase boundary between the conical\nand induced-ferromagnetic states ( HC−F, black triangle) is determined as the inflection\npoint ofχ′. The equilibrium SkX region is indicated with the light-green shading in pa nels\n(c) and (d). (e) Contour plot of χ′on theT-Hcplane. The region with χ′≤42 emu/mol\nis displayed with white color. (f) Contour plot of χ′′on theT-Hcplane. Phase boundaries\ndetermined by χ′[panel (d)] and similar data points at different temperatures are plo tted\nas open symbols in panels (e) and (f).41\nFIG. 5. Temperature dependence of helical state in Co 10Zn10in zero field. (a) Schematic\nSANS pattern on the (110) plane expected for a helical state with q∝bardbl<111>. The helical\nstate forms four domains with q∝bardbl[-111] (2 red spots), [1-11] (2 blue spots), [11-1] (out of\nthe plane) and [111] (out of the plane), respectively, resulting in 4 s pots on the (110) plane\natφ= 55◦, 125◦, 235◦, 305◦. Here, φis defined as the clockwise azimuthal angle from\nthe vertical direction. (b) SANS images observed at 410 K, 360 K, 3 00 K and 1.5 K. The\nintensity scale of the color plot varies between each panel. The [-111 ] and [1-11] directions\nare indicated with broken arrows. (c) Temperature dependence o f the SANS intensity\nintegrated over the azimuthal angle area at φ= 55◦, 125◦, 235◦, 305◦with the width of ∆ φ\n= 30◦(green area in the inset). (d) Radial |q|dependence of SANS intensity, integrated over\nthe azimuthal angle area at φ= 125◦, 305◦with the width of ∆ φ= 30◦(green area in the\ninset), atseveral temperatures. The datapointsarefittedtoa Gaussianfunction(solidline).42\nFIG. 6. Metastable SkX state in Co 10Zn10during a field cooling process at 0.03 T. (a)\nSchematic SANS pattern on the (110) plane, perpendicular to the fi eld, expected for\na rhombic SkX with q∝bardbl<111>. (b) SANS images observed at 412 K, 300 K, 100 K\nand 1.5 K during the FC process at 0.03 T. The intensity scale of the co lor plot varies\nbetween each panel. The magnetic field was applied at 412 K after ZFC . (c) Azimuthal\nangle (φ) dependence of SANS intensity at 412 K, 300 K and 1.5 K. The intensit y data\nfor 1.5 K is vertically shifted by 2 for clarity. The angles correspondin g to the [-111]\nand [1-11] directions ( φ= 55◦, 125◦, 235◦and 305◦) are indicated by dashed lines. (d)\nTemperature dependence of the SANS intensity. Blue symbols deno te the intensity\nintegrated over the region close to the [-111] and [1-11] directions (φ= 55◦, 125◦, 235◦\nand 305◦with the width of ∆ φ= 30◦; blue area in the inset). The intensity integrated\nover the region around the [001] and [-110] directions ( φ= 0◦, 90◦, 180◦and 270◦with\nthe width of ∆ φ= 30◦; red area in the inset) are denoted by red symbols. The tempera-\nture regions of the equilibrium triangular SkX, metastable triangular SkX and metastable\nrhombic SkX are indicated with the light-green, light-blue and purple s hadings, respectively.43\nFIG. 7. Reorientation of metastable SkX at low temperatures in Co 8Zn8Mn4under an\noff-axis field. (a) Schematic top view of the experimental configura tion. Rocking angle ( ω)\nis defined as the angle between the incident neutron beam ( ki) and the applied magnetic\nfield (H). In this experiment, magnetic field was tilted by 15◦away from the [001] direction.\nDue to the demagnetization effect from the tilted magnetic field applie d to the rectangular-\nshaped sample (gray object), the effective magnetic field ( Heff, pink arrow) inside the sample\nis tilted by more than 15◦from [001] direction. The additional tilting angle ( α) ofHefffrom\nHis estimated to be α∼20◦according to both the calculation of the demagnetization\nfield[60] and the peak position of rocking curve at 295 K shown in pane l (b). (b) Rocking\ncurves at selected temperatures in the field cooling (FC) process. SANS intensities are44\nintegrated over the azimuthal angle area at φ= 90◦, 270◦with the width of ∆ φ= 90◦(red\narea in the inset). (c, d) SANS images at 295 K, 200 K, 120 K and 40 K d uring a FC process\nat 0.03 T. These SANS images are averaged over the limited rocking an gles: (c) 14◦≤ω≤\n20◦(aroundHeff) and (d) −20◦≤ω≤ −10◦(around [001]). The intensity scale of the color\nplot varies between each panel. (e) Temperature dependence of t he SANS intensities during\nthe FC process. Blue closed circles show the SANS intensity integrat ed over the azimuthal\nangle area at φ= 60◦, 120◦, 240◦, 300◦with the width of ∆ φ= 30◦in the rocking angle\nrange of 14◦≤ω≤20◦(blue area in the inset). Red open squares represent the SANS\nintensity integrated over the azimuthal angle area at φ= 90◦, 180◦with the width of ∆ φ=\n30◦in the rocking angle range of −20◦≤ω≤ −10◦(red area in the inset). The temperature\nregions of the equilibrium triangular SkX, metastable triangular SkX a nd metastable\nsquare SkX are indicated with the light-green, light-blue and pink sha dings, respectively.\n(f) Schematic illustration of transformation from a triangular SkX t o a square SkX\nunder the off-axis effective field Hefftilted from [001]. While the skyrmions in the triangu-\nlar SkX areparallel to Heff, those inthe square SkX at low temperatures areparallel to [001].45\nFIG. 8. Metastable SkX state in Co 9Zn9Mn2. (a) Schematic SANS pattern on the (110)\nplane, perpendicular to the field, expected for a triangular SkX sta te (right panel) and\nfor a square SkX state (left panel). The triangular SkX state with o ne of triple- q∝bardbl[001]\nshows 6 spots at φ= 0◦, 60◦, 120◦, 180◦, 240◦, 300◦. Here,φis defined as the clockwise\nazimuthal angle from the vertical [001] direction. The square SkX s tate with double- q∝bardbl\n[001] and ∝bardbl[100] or [010] (out of the plane) exhibits 2 spots at φ= 0◦, 180◦. (b) SANS\nimages observed at 390 K, 100 K, 50 K and 10 K in the field cooling (FC) p rocess at 0.04\nT. The intensity scale of the color plot varies between each panel. Th e magnetic field was46\napplied at 390 K after zero field cooling. (c) Rocking curves at select ed temperatures in the\nFC process. SANS intensities are integrated over the azimuthal an gle area at φ= 90◦, 270◦\nwith the width of ∆ φ= 90◦(red area in the inset). The origin of the rocking angle ( ω= 0◦)\ncorresponds to ki∝bardblH∝bardbl[110]. (d) Temperature dependence of the SANS intensities. Blue\nclosed circles show the SANS intensity integrated over the azimutha l angle area at φ= 60◦,\n120◦, 240◦, 300◦with the width of ∆ φ= 30◦(blue area in the inset) and finally divided by\n2. Red open squares represent the SANS intensity integrated ove r the azimuthal angle area\natφ= 0◦, 180◦with the width of ∆ φ= 30◦(red area in the inset). The temperature regions\nof the equilibrium triangular SkX, metastable triangular SkX and meta stable square SkX\nare indicated with the light-green, light-blue and pink shadings, resp ectively. (e) Radial |q|\ndependence of SANS intensity, integrated over the azimuthal ang le area at φ= 0◦, 180◦\nwith the width of ∆ φ= 30◦(red area in the inset), at several temperatures during the FC\nprocess. The data points are fitted to a Gaussian function (solid line ).47\nFIG. 9. Temperature dependence of metastable SkX state in Co 7Zn7Mn6. (a) Schematic\nSANS pattern on the (001) plane, perpendicular to the field, expec ted for a triangular\nSkX state (right panel) and for a square SkX state (left panel). Th e triangular SkX state\nforms two domains with one of the triple- q∝bardbl[100] (6 blue spots) and ∝bardbl[010] (6 red spots),\nrespectively, resulting in 12 spots at every 30◦fromφ= 0◦. Here, φis defined as the\nclockwise azimuthal angle from the vertical [010] direction. The squ are SkX state with\ndouble-q∝bardbl[100] and [010] shows 4 spots at φ= 0◦, 90◦, 180◦, 270◦. (b) SANS images\nobserved at 146 K, 100 K, 60 K and 1.5 K during the field cooling (FC) pr ocess at 0.025\nT. The intensity scale of the color plot varies between each panel. Th e magnetic field\nwas applied at 146 K after zero field cooling. (c) Temperature depen dence of the SANS\nintensities. Blue closed circles show the SANS intensity integrated ov er the azimuthal\nangle area at φ= 45◦, 135◦, 225◦, 315◦with the width of ∆ φ= 30◦(blue area in the\ninset). Red open squares represent the SANS intensity integrate d over the azimuthal angle\narea atφ= 0◦, 90◦, 180◦, 270◦with the width of ∆ φ= 30◦(red area in the inset). The\ntemperature regions of the equilibrium triangular SkX, metastable t riangular SkX and48\nmetastable square SkX are indicated with the light-green, light-blue and pink shadings,\nrespectively. (d) Radial |q|dependence of SANS intensity, integrated over the azimuthal\nangle area at φ= 0◦, 90◦, 180◦, 270◦with the width of ∆ φ= 30◦(red area in the inset), at\nseveral temperatures in the FC process. The data points are fitt ed to a Gaussian function\n(solid line).49\nFIG. 10. Field effect on metastable SkX state and helical state in Co 7Zn7Mn6at 100 K. (a)\nSANS images at selected fields during field scans to the positive and ne gative directions at\n100 K after a field cooling (FC) at 0.025 T. (b) SANS images observed a t 0 T, 0.05 T and\n0.1 T at 100 K after a zero-field cooling (ZFC) (reproduced from our previous work in Ref.\n[31]. Copyright 2018, American Association for the Advancement of Science). The intensity\nscale of the color plot is fixed for all the panels in (a) and (b). (c) Field dependence of the\nSANS intensities at 100 K after the FC defined similarly as in Fig. 9(c). T he field regions\nof the metastable triangular SkX and the metastable square SkX ar e indicated with the\nlight-blue and pink shadings, respectively. (d) Field dependence of t he SANS intensities at\n100 K after the ZFC (reproduced from our previous work in Ref. [31 ]. Copyright 2018,\nAmerican Association for the Advancement of Science). Green clos ed circles show the\nSANS intensity integrated over the azimuthal angle area at φ= 45◦, 135◦, 225◦, 315◦with\nthe width of ∆ φ= 30◦(green area in the inset). Orange open squares represent the SA NS\nintensity integrated over the azimuthal angle area at φ= 0◦, 90◦, 180◦, 270◦with the width\nof ∆φ= 30◦(orange area in the inset).50\nFIG. 11. Metastable and low-temperature equilibrium skyrmion stat es in magnetic fields in\nCo7Zn7Mn6at 60 K. (a) SANS images at selected fields during field scans to the po sitive\nand negative directions at 60 K after a FC at 0.025 T. (b) SANS images observed at 0 T,\n0.07 T and 0.1 T at 60 K after a ZFC. The intensity scale of the color plot is fixed for all\nthe panels in (a) and (b). (c) Field dependence of the SANS intensitie s at 60 K after the\nFC, integrated over the azimuthal angle areas defined similarly as in F ig. 9(c). The field\nregions of the metastable square SkX and the equilibrium disordered skyrmions (DSk) are\nindicated with the pink and orange shadings, respectively. (d) Field d ependence of the\nSANS intensities at 60 K after the ZFC (reproduced from our previo us work in Ref. [31].\nCopyright 2018, American Association for the Advancement of Scie nce), integrated over the\nazimuthal angle areas defined similarly as in Fig. 10(d). The field region of the equilibrium\nDSk phase is indicated with the orange shading.51\nFIG.12. (a-c)Temperature ( T)-magnetic field( H)phase diagramsnear Tcin(a)Co 10Zn10,\n(b) Co 9Zn9Mn2and (c) Co 8Zn8Mn4, determined by ac susceptibility ( χ′) measurements\nwithH∝bardbl[110]. For the magnetic field, calibrated values Hcare used. The displayed range of\nT/Tc(upper horizontal axis) is fixed to be 0.90 - 1.01 for the three panels . Time-dependent\nχ′measurements after a field cooling (FC, pink arrow) via the equilibrium SkX phase (green\nregion) are performed at several temperatures denoted with cir cles, whose color corresponds52\nto the color of data points in panels (d-f). (d-f) Time dependence o f the normalized ac\nsusceptibility, defined as χ′\nN(t)≡[χ′(∞)−χ′(t)]/[χ′(∞)−χ′(0)], for (d) Co 10Zn10, (e)\nCo9Zn9Mn2(reproduced from our previous work in Ref. [36] with permissions. C opyright\n2017, American Physical Society) and (f) Co 8Zn8Mn4, measured in the processes described\nin panels (a-c). Here, χ′(0) is an initial value (metastable SkX state), and χ′(∞) is the value\nfor the equilibrium conical state which, as a fully relaxed state is assu med to be the value\nofχ′at the same magnitude of field after a field decreasing run from a fer romagnetic phase.\nThe data points are fitted to stretched exponential functions (d otted lines), exp/braceleftbig\n−(t/τ)β/bracerightbig\n.53\nFIG. 13. (a) Relaxation time ( τ) of metastable SkX states plotted against normalized\ntemperature T/Tcin Co 10Zn10(red squares), Co 9Zn9Mn2(green triangles, reproduced\nfrom our previous work in Ref. [36] with permissions. Copyright 2017 , American Physical\nSociety) and Co 8Zn8Mn4(blue circles). The τvalues are determined by the fits in\nFigs. 12(d-f). The solid arrows and dotted lines indicate temperatu re ranges and lower\nboundaries of equilibrium SkX phases, respectively. The data points from the three different\ncompositions, showing a good scaling, are fitted to a modified Arrhen ius law (pink solid\nline),τ=τ0exp{a(Tc−T)/T}, with an assumption of a temperature-dependent activation\nenergyEg=a(Tc−T) as described in Ref. [36, 64, 65]. The obtained parameters are a=\n215 and τ0= 63 s. (b) Schematic illustration of free energy landscape with a met astable\nSkX state and the most stable conical state. (c) Schematic illustra tion of the destruction\nprocess for a skyrmion string induced by movement of an emergent monopole-antimonopole\npair for a clean system and a dirty system. In the dirty system, the propagation of the\nmonopole and antimonopole are hindered by magnetic disorder while les s so in the clean54\nsystem.\nMaterial a τ 0(s) Reference\nCo-Zn-Mn 215 63 This work\nMnSi 65 2.7 ×10−4[64]\nCu2OSeO3 96 3 [65]\nCu2OSeO3(Zn 2.5% doped) 94 150 [65]\nTABLE. 2. Summary of relaxation parameters for metastable SkX in various materials.arXiv:2008.05075v1  [cond-mat.str-el]  12 Aug 2020Supplementary materials for\n“Metastable skyrmion lattices governed by magnetic disord er and\nanisotropy in β-Mn-type chiral magnets”\nK. Karube,1,∗J. S. White,2,∗V. Ukleev,2C. D. Dewhurst,3R. Cubitt,3\nA. Kikkawa,1Y. Tokunaga,4H. M. Rønnow,5Y. Tokura,1,6and Y. Taguchi1\n1RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198 , Japan.\n2Laboratory for Neutron Scattering and Imaging (LNS),\nPaul Scherrer Institute (PSI), CH-5232 Villigen, Switzerland .\n3Institut Laue-Langevin (ILL), 71 avenue des Martyrs,\nCS 20156, 38042 Grenoble cedex 9, France\n4Department of Advanced Materials Science,\nUniversity of Tokyo, Kashiwa 277-8561, Japan.\n5Laboratory for Quantum Magnetism (LQM), Institute of Physics ,\n´Ecole Polytechnique F´ ed´ erale de Lausanne (EPFL), CH-1015 L ausanne, Switzerland.\n6Department of Applied Physics and Tokyo College,\nUniversity of Tokyo, Bunkyo-ku 113-8656, Japan.\n∗These authors equally contributed to this work2\nA. Magnetization curve at 2 K\nIn Fig. S1(a), we show field dependence of magnetization at 2 K meas ured up to 7\nT in Co 10Zn10, Co9Zn9Mn2, Co8Zn8Mn4and Co 7Zn7Mn6. The saturation magnetization\n(Ms), which is defined as the value at 2 K and 7 T and summarized in Table 1 in t he\nmain text, takes a maximum at Co 9Zn9Mn2(Ms= 14.3µB/f.u.), while the helimagnetic\ntransition temperature Tcmonotonically decreases from Co 10Zn10as the Mn concentration\nis increased. This indicates that Co spin and Mn spins are ferromagne tically coupled at\nleast in the low Mn concentration region. As seen in the magnified view a round the low-field\nregion [Fig. S1(b)], hysteresis is observed for −0.3 T≤µ0H≤0.3 T for Co 8Zn8Mn4and for\n−0.7 T≤µ0H≤0.7 T for Co 7Zn7Mn6. The hysteresis is observed only below the reentrant\nspin glass transition temperature Tgin Co8Zn8Mn4(Tg∼10 K) and Co 7Zn7Mn6(Tg∼30\nK). The hysteresis is not discernible in Co 10Zn10and Co 9Zn9Mn2, neither of which exhibits\nthe reentrant spin glass transition down to 2 K. Similar results of mag netization have been\nreported in Ref. [S1].\nB. Field dependence of metastable SkX at 300 K in Co 10Zn10\nHere, we show the detailed results of field variation in metastable sky rmion crystal (SkX)\nat 300 K in Co 10Zn10(Fig. S2).\nSmall-angle neutron scattering (SANS) patterns for several mag netic fields at 300 K\nafter a field cooling (FC) with 0.03 T are shown in Fig. S2(a). In the field sweeping process\ntoward positive direction (pink arrows), the SANS pattern with str onger intensity along the\n[-111] and [1-11] direction changes to a uniform ring at 0.1 T similar to t hat observed in the\nequilibrium SkX phase at 412 K at 0.03 T. This field variation correspond s to the change in\nlattice form of skyrmions from rhombic-like one to the original triang ular one. The ring-like\nSANS signal corresponding to the triangular SkX persists up to 0.2 T . In the field sweeping\nrun toward negative direction (light-blue arrows) down to −0.06 T, the intensity along the\n[-111] and [1-11] directions increases and clear 4 spots are observ ed, indicating that lattice\nform of skyrmions approaches ideal rhombic one. Upon further inc reasing the field in the\nnegative direction, the SANS intensity from the rhombic SkX disappe ars at−0.1 T. For\ncomparison, SANS patterns at the same magnetic fields at 300 K aft er a zero-field cooling3\n(ZFC) are presented in Fig. S2(b). The SANS pattern with broad int ensity around the\n[1-11] direction originating from a helical state disappears at 0.1 T, w hich is totally distinct\nfrom the field variation in the metastable SkX after the FC.\nThe real-part of ac susceptibility ( χ′) and the SANS intensities at 300 K after the FC\nare plotted against applied field in Fig. S2(c) and S2(d), respectively . Here, ZFC data are\nalso plotted in the same graph for comparison. The small values of χ′and the large SANS\nintensities observed for −0.1 T≤µ0H≤0.2 T in the initial field scans, as compared to those\nin the returning runs, are attributed to the metastable SkX state . Thus, the metastable SkX\nstate survives over a wide field region up to the conical-ferromagne tic phase boundary and\ndown to the negative fields as marked with blue triangles in Fig. S2(c). The boundary\nbetween the rhombic SkX and the triangular SkX is determined as the magnetic field where\nthe SANS intensity around the [-111] and [1-11] directions becomes higher than that around\nthe [001] and [-110] directions [purple triangle in Fig. S2(d)], and plott ed in the state\ndiagram of Fig. 2(b) in the main text.\nC. Field dependence of metastable SkX at 1.5 K in Co 10Zn10\nWe also discuss field variation in metastable SkX in Co 10Zn10at 1.5 K (Fig. S3).\nFigure S3(a) shows several SANS patterns at 1.5 K for selected fie lds after the FC with\n0.03 T. In the field sweeping toward positive direction, the broad 4-s pot pattern originating\nfrom the rhombic lattice of skyrmions persists up to 0.10 T and finally t ransforms to a ring-\nlike pattern with strong intensity at 0.16 T. In the field sweeping proc ess toward negative\ndirection, the 4-spot pattern is preserved down to −0.16 T before Bragg spots disappear at\n−0.20 T. This result demonstrates that the metastable rhombic SkX s urvives over a wider\nfield region than that observed at 300 K but finally original triangular lattice is restored at\nhigh magnetic fields. For comparison we also show field variation in the S ANS pattern at 1.5\nK after ZFC in Fig. S3(b). The 4 spots along the [-111] and [1-11] dire ction, corresponding\nto the helical multi-domain state with q∝bardbl<111>, are observed up to 0.16 T with gradual\ndecreases inscattering intensity and qposition. However, aring-likepattenisnever observed\nat any fields, which is distinct from the result after the FC.\nWe show the field dependence of χ′at 5 K and SANS intensities at 1.5 K after the FC in\nFig. S3(c) and S3(d), respectively. The both measurements exhib it large and asymmetric4\nhysteresis, corresponding to the metastable SkX state as indicat ed with blue triangles. A\nboundary of rhombic-triangular skyrmion lattices [purple triangle in F ig. S3(d)] is deter-\nmined similarly to the previous section. The field dependence of χ′is complex as compared\nwith the result at 300 K [Fig. S2(c)], and the hysteresis subsists to −0.3 T in the negative\nfield region, while the SANS intensity originating from the metastable S kX state disappears\nat−0.2 T. These results indicate that a helical state with qvector along the [11-1] or [111]\ndirection, which is out of the (110) plane and thus not detectable in t he present SANS\nconfiguration, persists up to the field-induced ferromagnetic pha se boundary after complete\ndestruction of the metastable SkX state around −0.2 T. For comparison, the field depen-\ndence of χ′at 5 K and SANS intensities at 1.5 K after ZFC are also displayed in the sa me\nfigures. In this case, a hysteresis in χ′corresponding to the helical multi-domain state is\nobserved between −0.3 T and 0.3 T, which is wider than the hysteresis region (between −0.2\nT and 0.2 T) in the SANS intensity that represents the contribution f rom helical states with\nq∝bardbl[-111] and [1-11]. These ZFC results also indicate that a helical state withq∝bardbl[11-1]\nand [111] remains above 0.2 T. The complex field dependence is attribu ted to the strong\nmagnetocrystalline anisotropy that favors q∝bardbl<111>. As described in the next section, the\nhelical state forms a chiral soliton lattice under magnetic fields due t o the strong anisotropy\nofqvector. In the field returning process from high fields both after t he FC and ZFC, a\nclear dip structure in χ′and relatively large SANS intensity are observed between −0.1 T\nand 0.1 T. This is probably because helical qvector flops from the field direction to within\nthe plane perpendicular to the field.\nD. Chiral soliton lattice in Co 10Zn10\nIn this section, we describe possible formation of chiral soliton lattic e in Co 10Zn10[Fig.\nS4]. As presented in Fig. S3(b, d), the 4-spot SANS pattern, which corresponds to the\n2-domain helical state with q∝bardbl[-111] and [1-11], persists up to high fields near the ferro-\nmagneticregion. Inthisfieldvariation, weakhigher-harmonicscatt eringsignalsareobserved\nalong the [1-11] direction away from the strong main spots, which is m ore clearly discerned\nin the SANS image at 0.02 T plotted in logarithmic scale in Fig. S4(a). For q uantitative\ndiscussion, the radial qdependence of SANS intensity along the [1-11] direction is plotted\nin Fig. S4(b). At zero field, the Bragg peak is observed at q0∼0.044 nm−1. As the field5\nis increased to 0.02 T, scattering intensity above 0.07 nm−1is enhanced, and the second,\nthird, and even fourth harmonic peaks ( q2,q3andq4) are clearly observed in addition to\nthe first main peak ( q1). The higher harmonic peaks are still discerned at 0.16 T while the\npeak positions shift to lower qregion. These peaks are fitted to Gaussian functions and the\npeak center of qn(n= 1, 2, 3) normalized by nq0is plotted against field in Fig. S4(c). The\nsecond and third harmonic peak positions show a good scaling with the first peak position\nas expected as qn=nq1. With increasing field, the value of qn/nq0decreases down to 0.8\nbefore SANS signals disappear at 0.24 T.\nThese results are reminiscent of a chiral soliton lattice (CSL): one- dimensional chain\nof nonlinear helimagnetic structures as schematically illustrated in Fig . S4(d), which has\nbeen observed in monoaxial chiral magnets as represented by CrN b3S6[S2]. Following the\ntheoretical analysis by Togawa et al.[S2], field variation in qvalue for the CSL is expressed\nasq(H)/q(0) =π2/[4K(κ)E(κ)], where K(κ) andE(κ) are the elliptic integrals of the\nfirst and second kinds. Here, the elliptic modulus κ(0< κ <1) is determined by energy\nminimization and given by κ/E(κ) =/radicalbig\nH/Hc, whereHcis the ferromagnetic saturation\nfield. This theoretical result for the CSL is plotted with pink dashed lin e in Fig. S4(c).\nHere, we set µ0Hc= 0.24 T, above which the SANS intensity disappears, and found that\nthe observed field variation in qn/nq0agrees with the theoretical curve.\nA CSL is stabilized under magnetic fields rather than a conventional c onical state if q\nvector is strongly anisotropic and thus perpendicular to the fields. The CSL has been also\nreported in a cubic chiral magnet Cu 2OSeO3under a uniaxial strain which fixes the qvector\nalong the strain direction[S3]. In the present study for a cubic chira l magnet Co 10Zn10\nwithout uniaxial strain, the CSL behavior is observed probably beca useqvector anisotropy\nalong the <111>direction becomes strong at low temperatures, to which the magne tic field\nis applied perpendicularly.\nE. Transformation of metastable skyrmion lattice in Co 8Zn8Mn4\nHere, we review the metastable SkX state induced by FC and struct ural transition of\nthe skyrmion lattice within the metastable state in Co 8Zn8Mn4, which was reported in our\nprevious paper[S4] (Fig. S5). Selected SANS patterns in a FC at 0.04 T from 295 K to 40 K\nare shown in Fig. S5(b). Here, the magnetic field and the incident neu tron beam are parallel6\nto the [001] direction. At 295 K and 0.04 T, inside the equilibrium SkX pha se, clear 12 spots\nareobserved. This 12-spotpattern corresponds to a 2-domaint riangular SkX state, in which\none of triple- qis parallel to the [010] or [100] direction as illustrated in the right pane l of Fig.\nS5(a). The triangular SkXpersists down to200K asametastable st ateduring theFC. From\n120 K to 40 K, the 12-spot pattern gradually transforms to a 4-sp ot pattern. This 4-spot\npattern corresponds to a square SkX state with double- qvectors parallel to the [010] and\n[100] directions as illustrated in the left panel of Fig. S5(a). In the s ubsequent re-warming\nprocess, the triangular SkX revives already at 200 K. This reversib ility of the metastable\nSkX can rule out the possibility that the 4-spot pattern is due to a he lical multi-domain\nstate with zero topological charge.\nSANS intensity is plotted as a function of temperature in Fig. S5(c). Above 120 K,\nthe intensities for the <100>and<110>directions show similar values as expected for a\n12-spot pattern. Below 120 K, the intensity ratio of <100>to<110>becomes larger due\nto the gradual transformation to 4 spots. The SANS intensities sh ow maximum at 250 K\nand decrease below 200 K, indicating evolution of magnetic disorder in the metastable SkX,\nespecially in the square SkX.\nThe radial qdependence of the SANS intensity for <100>is shown in Fig. S5(d). From\n120 K to 40 K, where the triangular-square SkX transition occurs, the peak center shows a\nlarge shift from 0.055 nm−1to 0.09 nm−1and the peak intensity decreases. The temperature\ndependence of qin the metastable skyrmion state during the field cooling is similar to tha t\nof the helical state during zero-field cooling as shown in Fig. 3(g) in th e main text.\nF. Field dependence of metastable SkX in Co 9Zn9Mn2\nInthissection, wedescribefieldvariationinthemetastableSkXstat eatlowtemperatures\nin Co9Zn9Mn2(Fig. S6). Selected SANS patterns at 10 K after the FC (0.04 T) are shown\nin Fig. S6(a). The 2-spot pattern at 0.04 T corresponds to the squ are SkX as discussed\nin Fig. 8 in the main text. With increasing field up to 0.3 T, the 2-spot pat tern gradually\nchanges to a ring-like one. The detailed field dependence of the SANS intensity is presented\nin Fig. S6(c). While the large intensity from the 2 vertical spots is rap idly suppressed with\nincreasing field, the weak intensity from the side region hardly decre ases, and eventually the\nintensities from the two regions become similar at 0.3 T.7\nAs shown in Fig. S6(b), the rocking curve at 0.3 T clearly shows a peak structure around\nω= 0◦. This indicates that the scattering intensity from the ring-like patt ern lies in the\n(110) plane, namely qvector is perpendicular to the field, similar to the observation at high\ntemperatures. Thus, the field variation in the SANS pattern is inter preted as the change\nfrom the square SkX with q∝bardbl<100>to an orientationally disordered triangular SkX with\nq⊥H.8\n[S1] J. D. Bocarsly, C. Heikes, C. M. Brown, S. D. Wilson, and R . Seshadri, Phys. Rev. Mater. 3,\n014402 (2019).\n[S2] Y. Togawa, T. Koyama, K. Takayanagi, S. Mori, Y. Kousaka , J. Akimitsu, S. Nishihara, K.\nInoue, A. S. Ovchinnikov, and J. Kishine, Phys. Rev. Lett. 108, 107202 (2012).\n[S3] Y. Okamura, Y. Yamasaki,2 D. Morikawa, T. Honda, V. Ukle ev, H. Nakao, Y. Murakami, K.\nShibata, F. Kagawa, S. Seki, T. Arima, and Y. Tokura, Phys. Re v. B96, 174417 (2017).\n[S4] K. Karube, J. S. White, N. Reynolds, J. L. Gavilano, H. Oi ke, A. Kikkawa, F. Kagawa, Y.\nTokunaga, H. M. Rønnow, Y. Tokura, and Y. Taguchi, Nat. Mater .15, 1237 (2016).9\nFIG.S1. (a)Magneticfield( H)dependenceofmagnetization( M)at2Kinsingle-crystalline\nsamples of Co 10Zn10, Co9Zn9Mn2, Co8Zn8Mn4and Co 7Zn7Mn6. Magnetic fields are applied\nalong the [110] direction for Co 10Zn10and Co 9Zn9Mn2, and along the [100] direction for\nCo8Zn8Mn4andCo 7Zn7Mn6, respectively. (b)Magnifiedviewoflowfieldregioninpanel (a).10\nFIG. S2. Field dependence of metastable SkX state and helical stat e in Co 10Zn10at 300 K.\n(a) SANS images observed at selected fields during field scans to the positive and negative\ndirections at 300 K after a field cooling (FC) at 0.03 T. The intensity sc ale of the color plot\nis fixed through the panels. (b) SANS images observed at 0 T, 0.06 T a nd 0.1 T at 300\nK after a zero-field cooling (ZFC). The intensity scale of the color plo t is fixed for these\npanels. For panels (a) and (b), the [-111] and [1-11] directions are presented with broken\narrows. (c) Field dependence of the real part of the ac susceptib ility (χ′) at 300 K after FC\nwithµ0Hc= 0.03 T (red line) and ZFC (green line). The blue triangles ( HMSkX+( −)) denote\nthe boundaries of metastable SkX that are plotted in the state diag ram in Fig. 2(b) in the\nmain text. The boundary at the positive field side is determined as the field where the\nhysteresis in χ′vanishes, and the boundary at the negative field side is determined a s the\ninflection point. (d) Field dependence of the SANS intensities at 300 K after the FC at 0.03\nT and ZFC. For the FC data, SANS intensity integrated over the azim uthal angle areas at\nφ= 55◦, 125◦, 235◦, 305◦with the width of ∆ φ= 30◦(blue area in the inset) and at φ=\n0◦, 90◦, 180◦, 270◦with the width of ∆ φ= 30◦(red area in the inset) are presented by blue\nclosed circles and red open squares, respectively. The purple trian gle shows the boundary11\nbetween the metastable rhombic SkX and the triangular one and is de termined as the field\nwhere the former intensity (blue circle) becomes higher than the lat er (red square). For the\nZFC data, SANS intensity integrated over the region at φ= 55◦, 125◦, 235◦, 305◦with the\nwidth of ∆ φ= 30◦(green area in the inset) is indicated with green closed triangles. Her e,\nφis defined as the clockwise azimuthal angle from the vertical directio n. In panels (c) and\n(d), the regions of the metastable rhombic SkX and the triangular o ne are indicated with\npurple and light-blue shadings, respectively.12\nFIG. S3. Field dependence of metastable SkX state and helical stat e in Co 10Zn10at 1.5 K.\n(a) SANS images observed at selected fields during field scans to the positive and negative\ndirections at 1.5 K after a FC (0.03 T). The intensity scale of the color plot is fixed through\nthe panels. (b) SANS images observed at 0 T, 0.1 T and 0.16 T at 1.5 K af ter a ZFC.\nThe intensity scale of the color plot is fixed for these panels. (c) Field dependence of χ′\nat 5 K after the FC with µ0Hc= 0.03 T (red line) and ZFC (green line). The boundaries\nof metastable SkX ( HMSkX+( −), blue triangles) are determined similarly as in Fig. S2(c)\n(d) Field dependence of the SANS intensities at 1.5 K after the FC (0.0 3 T) and ZFC,\nintegrated over the azimuthal angle areas defined similarly as in Fig. S 2(d). The boundary\nbetween the rhombic SkX and the triangular one (purple triangle) is d etermined similarly\nas in Fig. S2(d). In panels (c) and (d), the regions of the metastab le rhombic SkX and the\ntriangular one are indicated with purple and light-blue shadings, resp ectively.13\nFIG. S4. Chiral soliton lattice formed under magnetic fields at 1.5 K in C o10Zn10. (a)\nSANS image measured at 1.5 K and 0.02 T after ZFC. The intensity of th e color plot is\ndisplayed in a logarithmic scale. (b) Radial |q|dependence of SANS intensities (logarithmic\nscale) measured at 0 T (black), 0.02 T (red) and 0.16 T (blue) at 1.5 K a fter ZFC. The\nintensities are integrated over the azimuthal angle area at φ= 125◦, 305◦with the width of\n∆φ= 30◦[white frames in panel (a)]. The black arrow ( q0) indicates the peak position at\n0 T. The red arrows ( q1,q2,q3andq4) for 0.02 T indicate the main peak position, second,\nthird and fourth harmonic peak positions, respectively. (c) Norma lizedqvalues,q1/q0,\nq2/2q0andq3/3q0, are plotted as a function of magnetic field. Dashed pink line indicates\na theoretical curve for the chiral soliton lattice (see text for det ails). (d) Schematic spin\nconfiguration of chiral soliton lattice.14\nFIG. S5. Temperature dependence of metastable SkX state in Co 8Zn8Mn4. (a) Schematic\nSANS pattern on the (001) plane, perpendicular to the field, expec ted for a triangular\nSkX state (right panel) and for a square SkX state (left panel). Th e triangular SkX state\nforms two domains with one of triple- q∝bardbl[100] (6 blue spots) and ∝bardbl[010] (6 red spots),\nrespectively, resulting in 12 spots at every 30◦fromφ= 0◦. Here, φis defined as the\nclockwise azimuthal angle from the vertical [010] direction. The squ are SkX state with\ndouble-q∝bardbl[100] and [010] shows 4 spots at φ= 0◦, 90◦, 180◦, 270◦. (b) SANS images\nobserved at 295 K, 200 K, 120 K and 40 K on the field cooling (FC) proc ess at 0.04 T\n(reproduced from our previous work in Ref. [S4]. Copyright 2016, S pringer Nature). The\nintensity scale of the color plot varies between each panel. The magn etic field was applied\nat 295 K after zero-field cooling. (c) Temperature dependence of the SANS intensities. Blue\nclosed circles show the SANS intensity integrated over the azimutha l angle area at φ= 45◦,\n135◦, 225◦, 315◦with the width of ∆ φ= 30◦(blue area in the inset). Red open squares15\nrepresent the SANS intensity integrated over the azimuthal angle area atφ= 0◦, 90◦, 180◦,\n270◦with the width of ∆ φ= 30◦(red area in the inset). The temperature regions of the\nequilibrium triangular SkX, the metastable triangular SkX and the met astable square SkX\nare indicated with the light-green, light-blue and pink shadings, resp ectively. (d) Radial\n|q|dependence of SANS intensity, integrated over the azimuthal ang le area at φ= 0◦, 90◦,\n180◦, 270◦with the width of ∆ φ= 30◦(red area in the inset), at several temperatures on\nthe FC process. The data points are fitted to a Gaussian function ( solid line).16\nFIG. S6. Field dependence of metastable SkX state in Co 9Zn9Mn2at 10 K. (a) SANS\nimages at 0.04 T, 0.2 T and 0.3 T measured at 10 K after a field cooling (FC ) at 0.04\nT. The intensity scale of the color plot varies between each panel. (b ) Rocking curves at\nseveral fields in the field-increasing process after the FC. SANS int ensities are integrated\nover the azimuthal angle area at φ= 90◦, 270◦with the width of ∆ φ= 90◦(red area in the\ninset). Here, φis defined as the clockwise azimuthal angle from the vertical [001] dir ection.\n(c) Field dependence of the SANS intensities. Blue closed circles show the SANS intensity\nintegrated over the azimuthal angle area at φ= 60◦, 120◦, 240◦, 300◦with the width of ∆ φ\n= 30◦(blue area in the inset) and finally divided by 2. Red open squares repr esent the\nSANS intensity integrated over the azimuthal angle area at φ= 0◦, 180◦with the width of\n∆φ= 30◦(red area in the inset). The field regions of the metastable triangula r SkX and the\nmetastable square SkX are indicated with the light-blue and pink shad ings, respectively."    },    {        "title": "2008.05125v1.Prediction_on_Properties_of_Rare_earth_2_17_X_Magnets_Ce2Fe17_xCoxCN___A_Combined_Machine_learning_and_Ab_initio_Study.pdf",        "content": "arXiv:2008.05125v1  [cond-mat.mtrl-sci]  12 Aug 2020Prediction on Properties of Rare-earth 2-17-X Magnets Ce 2Fe17−xCoxCN : A Combined\nMachine-learning and Ab-initio Study\nAnita Halder,1Samir Rom,1Aishwaryo Ghosh,2and Tanusri Saha-Dasgupta1,∗\n1Department of Condensed Matter Physics and Material Scienc es,\nS. N. Bose National Centre for Basic Sciences, JD Block,\nSector III, Salt Lake, Kolkata, West Bengal 700106, India.\n2Department of Physics, Presidency University, Kolkata 700 073, India\n(Dated: August 13, 2020)\nWe employ a combination of machine learning and first-princi ples calculations to predict magnetic properties\nof rare-earth lean magnets. For this purpose, based on train ing set constructed out of experimental data, the\nmachine is trained to make predictions on magnetic transiti on temperature (T c), largeness of saturation magne-\ntization (µ0Ms), and nature of the magnetocrystalline anisotropy ( Ku). Subsequently, the quantitative values of\nµ0MsandKuof the yet-to-be synthesized compounds, screened by machin e learning, are calculated by first-\nprinciples density functional theory. The applicability o f the proposed technique of combined machine learning\nand first-principles calculations is demonstrated on 2-17- X magnets, Ce 2Fe17−xCoxCN. Further to this study,\nwe explore stability of the proposed compounds by calculati ng vacancy formation energy of small atom intersti-\ntials (N/C). Our study indicates a number of compounds in the proposed family, offers the possibility to become\nsolution of cheap, and efficient permanent magnet.\nPACS numbers:\nINTRODUCTION\nPermanent magnets are a part of almost all the most im-\nportant technologies, starting from acoustic transducers , mo-\ntors and generators, magnetic field and imaging systems to\nmore recent technologies like computer hard disk drives,\nmedical equipment, magneto-mechanics etc.[ 1]The search\nfor efficient permanent magnets is thus everlasting. In this\nconnection, the family of rare-earth (RE) and 3dtransition\nmetal (TM) based intermetallics has evolved over last 50\nyears or so, and has transformed the landscape of permanent\nmagnets.[ 2,3] Two most prominent examples of RE-TM per-\nmanent magnets, that are currently in commercial produc-\ntion, together with hard magnetic ferrites, are SmCo 5, and\nNdFe14B.\nWhile SmCo 5and NdFe 14B provide reasonably good solu-\ntions, keeping in mind the resource criticality of RE elemen ts\nlike Nd and Sm, a significant amount of effort has been put\nforward in search of new permanent magnets without criti-\ncal RE elements or with less content of those. The idea is to\noptimize the price-to-performance ratio.[ 2] This has lead to\ntwo routes, (a) search for potential magnets devoid of rare-\nearth elements,[ 4] and (b) designing of rare-earth lean inter-\nmetallics using abundant RE elements such as La and Ce in-\nstead of Sm and Nd.[ 5–7] As stressed by Coey,[ 8] the demand\nin hand is to seek for new, low-cost magnets with maximum\nenergy product bridging the ferrites and presently used RE\nmagnets. Following the route (b), cheap, new ternary and\nquartnary RE-lean RE-TM intermetallics need to be explored ,\nas binaries have been well explored. In parallel, Co being ex -\npensive, it may be worthwhile to focus on intermetallic com-\npounds containing Fe.\nStarting from the simplest binary RE-TM structure of\nCaCu5, by replacing nout ofmRE (R) sites with a pair ofTM (M) sites, R m−nM5m+2nstructures are obtained. This\ncan give rise to several possible binary structures of diffe r-\nent chemical compositions, listed in order of RE-leanness;\nRM13(7.1%), RM12(7.7%), R2M17(10.5%), R2M14(12.5\n%), RM5(16.7%), R6M23(20.7%), R2M7(22.2%), RM3\n(25%), RM2(33%) etc. Judging by the rare-earth content,\n1:13, 1:12, 2:17, 2:14 compounds may form examples of rare-\nearth lean materials. It is desirable to modify the known bi-\nnary compounds containing low cost RE’s belonging to these\nfamilies to achieve best possible intrinsic magnetic prope r-\nties, namely (i) high spontaneous or saturation magnetizat ion\n(µ0Ms), at least around 1T, (ii) a Curie temperature (T c) high\nenough for the contemplated devise use, 600 K or above, and\n(iii) a mechanism for creating sufficiently high easy-axis c o-\nercivity ( Ku). The synthesis and optimization of properties\nof real materials in experiment is both time-consuming and\ncostly, being mostly based on trial and error. Computationa l\napproach in this connection is of natural interest to screen\ncompounds, before they can be suggested and tested in lab-\noratory. Typical computational approaches in this regard a re\nbased on density functional theory (DFT) calculations. A de -\ntailed calculation estimating all required magnetic prope rties,\ni.eMs, Tc,Kufrom first-principles is expensive and also not\ndevoid of shortcomings. For example, estimation of T cre-\nlies on parametrization of DFT or supplemented Ucorrected\ntheory of DFT+ Utotal energies to construct spin Hamilto-\nnian and solution of spin Hamiltonian by mean field or Monte\nCarlo method. While this approach would work for local-\nized insulators, its application to metallic systems with i tin-\nerant magnetism is questionable, as it fails even for elemen tal\nmetals like Fe, Co and Ni.[ 9] A more reasonable approach of\nDFT+dynamical mean field (DMFT)[ 10] is significantly more\nexpensive. An alternative approach would be to use machine\nlearning (ML) technique based on a suitable training datase t.2\nFIG. 1: (Color online) Steps of Machine learning combined DF T approach for predictions of properties in Ce 2Fe17−xCoxCN permanent\nmagnets.\nThis approach has been used for RE-TM permanent magnets\nbased on DFT calculated magnetic properties database of M s\nandKu.[5,11] Creation of database based on calculations,\neven with high throughput calculations is expensive, and re -\nlies on the approximations of the theory. It would be far more\ndesirable to built a dataset based on experimental results, and\nthen train the ML algorithm based on that. However, the size\nand availability of the experimental data in required forma t\ncan be a concern. Focusing on the available experimental\ndata on RE lean intermetallics, the set of T cis largest, fol-\nlowed by that for Ku, and M s. While the quantitative values\nof Tc’s in Kelvin or degree Celsius are available in literature,\nfor magnetocrystalline anisotropy often only the informat ion\nwhether they are easy-axis or easy-plane are available. Sim -\nilarly, the µoMsvalues are reported either in µB/f.u. or in\nemu/gm or in Tesla, conversion from µB/f.u. and emu/gm to\nTesla requiring information of the volume and density, whic h\nmay introduce inaccuracies up to one decimal point. Restric t-\ning experimental data to those containing values of Ku, and\nµoMsvalues in the same format (either Tesla or µB/f.u. or\nemu/gm) reduces the dataset of Kuand Mssignificantly, mak-\ning application of ML questionable. We thus use a two-prong\napproach, as illustrated in Fig. 1. We first create a database ofTc, MsandKufrom available experimental data on RE-lean\nintermetallics, and use ML for prediction of T cvalues, for pre-\ndicting whether µ0Mssatisfies the criteria of being larger than\n1 Tesla, and for predicting the sign of Ku. For M sandKu, ML\nthus serves the purpose of initial screening. We next evalua te\nMsand the magnetic anisotropy properties based on elaborate\nDFT calculations. Calculation of the magnetic anisotropy e n-\nergy (MAE) is challenging due to its extremely small value.\nHowever, since the pioneering work of Brooks,[ 12] several\nstudies[ 6,13–15] have shown that Ucorrected DFT generally\nreproduces the orientation and the right order of magnitude of\nthe MAE.\nWe demonstrate applicability of our proposed approach on\nCe and Fe based 2:17 RE-TM intermetallics, Ce 2Fe17−xCox\ncompounds ( x= 1,..., 7). Our choice is based on follow-\ning criteria, (a) the compounds contain rare earth Ce which i s\nthe cheapest one among the RE family having market price\nof∼5 USD/Kg.[ 16] The cost of other components Fe, C\nand N are all <1 USD/Kg. The price of Co is higher than\nFe,[16] being less abundant metal. The Co:Fe ratio is thus\nrestricted within 0.4. (b) Co substitution in place of Fe has\nbeen reported[ 17,18] to be efficient in simultaneous enhance-\nments of K uas well as T cin several TM magnets. This is3\nin sharp contrast to other TM substitutes, such as Ti, Mo, Cr,\nand V , where magnetic anisotropy as well as T care gener-\nally suppressed. (c) the search space belongs to 2:17 family ,\nwhich is the family in which most of the instances in our train -\ning set belongs to. (d) this class of compounds is found to\nbe more stable than the well explored 1:12 compounds. (e)\nfor large saturation magnetization it is desirable to use Fe -\nrich compounds, which is also less expensive compared to\nCo. (f) although Ce has negative second order Stefan’s fac-\ntor which favors in-plane MAE, experimental findings suppor t\nthat the nitrogenation and carbonation can switch the MAE\nfrom easy plane to easy axis.[ 19] (g) though R 2Fe17com-\npounds display large magnetization value due to high Fe con-\ntent, these compounds are disadvantageous as they exhibit l ow\nCurie temperature.[ 20] Presence of Co, as well as C/N inter-\nstitials help in increasing T c. (h) while magnetic properties of\ncarbo-nitrides are expected to be similar to that of nitride s for\nsufficiently high concentration of N, carbo-nitride compou nds\nhave been proven to show better thermal stability.[ 21]\nOur study suggests that Fe-rich Ce 2Fe17−xCoxCN com-\npounds may form potential candidate materials for low-cost\npermanent magnets, satisfying the necessary requirements of\na permanent magnet with T c>600 K,µ0Ms>1 Tesla and\neasy-axis K u>1 MJ/m3. The calculated maximal energy\nproduct and estimated anisotropy field, which are technolog i-\ncally interesting figures of merit for hard-magnetic materi als,\nturn to be within the reasonable range. Some of the studied\ncompounds may possibly bridge the gap between low maxi-\nmal energy product and high anisotropy field for SmCo 5and\nvice versa for Nd 2Fe14B.\nMACHINE LEARNING APPROACH\nDatabase construction & Training of Model\nAiming to search new candidates for permanent magnets\nwe use supervised machine learning (ML) algorithm which\nhelps us to screen compounds with high T c(Tc/greaterorsimilar600 K),\nhigh M s(µ0Ms>1 Tesla), and easy axis anisotropy ( Ku>\n0) among the huge number of possible candidates of unex-\nplored RE-TM intermetallics. The first step of any ML algo-\nrithm is to construct a dataset. We construct three datasets\nof existing RE-TM compounds for T c, Msand Kusepa-\nrately using the following sources: ICSD,[ 22] the handbook\nof magnetic materials,[ 23] the book of magnetism and mag-\nnetic materials,[ 24] and other relevant references.[ 19,21,25–\n78] The datasets are presented as supplementary materials\n(SM)[ 79] as easy reference for future users. To construct the\ndatabase of rare-earth lean compounds, RE percentage in the\nintermetallic compounds is restricted to 14% which includes\nthe four different binary RE-TM combinations namely RM 12,\nRM13, R2M17and R2M14along with their interstitial and de-\nrived compounds. We discard RM 13from the dataset as only\nfew candidates are available from this series with known ex-\nperimental T c, MsandKu.Attribute Type Attribute Notation Value range\nStoichiometric CW absolute deviation <∆Z > 1.70-16.74\nof atomic no.\nCW av. of < ZTM> 10-33.30\natomic no. of TM\nCW av. of < ZLE> 0-9.79\natomic no. of LE\nCW av. Z < Z > 21.08-37.71\nCW electronegativity ∆ǫ 0.61-1.84\ndiff. of RE &TM\nCW RE percentage RE% 4.76-14.29\nCW TM percentage TM% 38.46-95.24\nCW LE percentage LE% 0-53.85\nElement Atomic no. of RE ZRE 58-71\nPresence of NTM yes/no\nmore than one TM\nPresence of LE NLE yes/no\nElectronic Total no. of f electrons fn1-28\nTotal no. of f electrons dn30-136\nTABLE I: List of 13 different attributes with description, n ota-\ntion and range used in the ML algorithm. Here ”CW” stands for\n”composition-weighted”.\nWe list a total of 565 compounds with reported experimen-\ntal Tc, among which majority of the compounds (about 55%)\nbelong to R 2M17series. The minimum contribution to the\ndataset comes from R 2M14(about10%) family. The high-\nest Tcin the dataset belongs to R 2M17class of compounds\nnamely Lu 2Co17[25] with T c∼1203 K and the compound\nwith lowest T cis NdCo 7.2Mn4.8(∼120 K),[ 23] a member\nfrom RM 12family. In the dataset all three compositions with\nRE to TM ratio 2:17, 2:14 and 1:12 show a large variation\nin Tchaving the difference between maximum and minimum\nvalues as 1051, 775 and 991 K respectively. There exists few\ncompounds in the dataset with more than one reported value\nof Tc. For example T cof SmFe 10Mo2has been reported with\ntwo different values of 421 K[ 80] and 483 K.[ 81] There are\nother examples of such multiple T c.[82–86] The quality of the\nsample, their growth conditions, coexistence of compounds\nin two or multiple phases and accuracy of the measurements\nmay lead to the multiple values of T creported for a particular\ncompound. In such cases, we consistently consider the large st\namong the reported values of T c. Notably in majority of cases\nwe find little variation in reported values of T c(∼20-50 K).\nThe dataset of M sis relatively smaller than T c, contain-\ning only 195 entries. The majority of the compounds in this\ndataset belong to 2:17 composition similar to the database o f\nTc. The relatively smaller dimension of M sdataset is primar-\nily due to fact that experimental reports available for M sare\nmuch less than T c. Secondly M shas been mostly reported\nat room temperature, in some cases at low temperature. To\nmaintain uniformity of the dataset we consider M sreported at\nroom temperature, resulting in a lesser number of compounds\nin the M sdataset.\nReports with quoted values of anisotropy constant are even\nmore rare. Our exhaustive search resulted in only 73 data4\npoints. This pushes the dataset size to the limit of ML algo-\nrithms, for which predictive capability becomes questiona ble\ndue to large bias masking the small variance.[ 87] On other\nhand, if we allow for also experimental data reporting only\nsign of Ku, this dataset gets expanded to a reasonable size of\n258.\nAfter constructing the dataset, we carry out preprocessing\nof the data, as outlined in Ref.[ 88]. It comprises of removal of\nnoisy data, outliers and correlated attributes. For details see\nAppendix.\nThe next and the most crucial step is to construct a set\nof simple attributes, which are capable of describing the in -\nstances (in this case RE-TM compounds) and then deploy ML\nalgorithm to map them to a target (in this case T c, Msand\nKu). The attributes considered in this study are summarized\nin Table. I, which can be divided into three broad categories ,\nnamely, stoichiometric attributes, element properties an d elec-\ntronic configuration attributes. The stoichiometric attri butes\nmay contain the information of both elemental and composi-\ntional properties as suggested by Ward et al.[ 89] This is based\non taking compositional weights (CW) of elemental proper-\nties.\nIn the third step, we train different popular machine learn-\ning algorithms with the constructed dataset for prediction . We\nuse ML algorithm in three different problems; (a) to predict\nthe compounds with T cmore than 600 K, (b) compounds\nwithµ0Ms>1 Tesla, and (c) compounds with easy-axis\nanisotropy. Regression is used in the former case, whereas\nlatter two cases are treated as classification problems. We\nuse five different ML algorithms for regression in case of\nTcnamely Ridge Regression (RR),[ 90] Kernel Ridge Re-\ngression (KRR),[ 91] Random Forest (RF),[ 92,93] Support\nVector Regression (SVR)[ 94] and Artificial Neural Network\n(ANN).[ 95]The details can be found in Appendix. Out of the\nfive different ML algorithms, it is seen that random forest pe r-\nforms best, which has been also successfully used for predic -\ntion of Heusler compounds,[ 96] half-Hausler compounds,[ 97]\ndouble perovskite compounds,[ 88] half-Heusler semiconduc-\ntor with low-thermal-conductivity,[ 98] zeolite crystal struc-\nture classification[ 99] etc. Results presented in the following\nare based on random forest method.\nModel evaluation\nThe final step is to employ the trained algorithm on yet-\nto-be synthesized RE-TM compounds, and thus to explore\nnew compositions with targeted properties. We choose\nCe2Fe17−xCoxCyNz(y,z= 0/1;x= 0...8) as the explo-\nration set for application of the trained ML algorithm. This\nresults in a set of 36 compounds among which 8 compositions\n(Ce2Fe17−xCoxCN,x= 1,..., 8) have neither been synthe-\nsized experimentally nor studied theoretically, to the bes t of\nour knowledge. We apply our trained ML algorithms on all\nof these 36 compounds and the results are summarized in Fig.\nFIG. 2: (Color online) ML predictions of Curie temperature ( Tc)\nfrom regression model, and saturation magnetization (M s) and\nanisotropy constant ( Ku) from classification model. The upper\n(middle/lower) panel shows the results of T c(Ms/Ku). The ex-\nploration set is Ce 2Fe17−xCoxCyNzwhereyandzcan have val-\nues either 0 or 1, and x= 0...8, acronymed as xyz. In the\ntop panel, non-interstitial compounds, carbonated, nitro genated and\ncarbo-nitrogenated compounds are symbolized by circle, di amond,\nsquare and upper triangle. Different colors specify compou nds with\ndifferentxvalues. The middle panel shows the ML prediction con-\nfidence for M s. In the lower panel, ML prediction confidence for\nKuis illustrated. Here the upper (lower) half having bars with no-\nfill (shaded) shows the confidence for the compounds with posi tive\n(negative) Ku.\n2. The top panel of Fig. 2 shows the predicted T cof all the\ncompounds. It is seen that the nitrogenation or carbonation\nincreases the T cwith respect to their respective parent com-\npound Ce 2Fe17−xCox. Our ML model predicts that the ni-\ntrides have higher T cthan that of the carbides. For x≤5,\nthe enhancement of T cis maximum for the compounds where\nboth carbon and nitrogen are present. For x > 5, Tcshows\nslight decrease compared to only nitrogenated case. It is al so\nnoted that the relative rise in T cin interstitial compounds com-\npared to parent compounds, decays gradually with Co concen-\ntration. The increase in T cvaries from ∼200 K to 10 K as\nxvaries from 0 to 8 for carbides and nitrides whereas intro-\nduction of both nitrogen and carbon shows the variation from5\nFIG. 3: (Color online) Crystal structure of Ce 2Fe17−xCoxCN magnets. The Ce, Fe/Co and C/N atoms are shown with large, m edium and\nsmall balls, respectively. Four transition metal sublatti ces 9d, 18f, 18hand 6care shown in black, green, magenta and yellow colored balls,\nrespectively. Left panel shows the crystal structure viewe d with c-axis pointed vertically up and the right panel shows the crystal structure\nviewed along the c-axis.\n∼310 K to 30 K. Our result reproduces the trend of exper-\nimental findings in a qualitative manner. The experimental\nresults for x= 0 (Ce 2Fe17),[100,101] concluded that the en-\nhancement in T cis highest in presence of both carbon and\nnitrogen[ 102,103] (Tc∼721 K), followed by nitrogenated\ncompound[ 104,105] (Tc∼700 K) and lowest for carbonated\ncompound[ 102,103] (Tc∼589 K). Though it is not possible\nto compare the results quantitatively as the stoichiometry of\nthe experimentally studied carbonated and nitrogenated co m-\npounds are not the same as in our exploration dataset, but the\noverall trend is similar. We also find that our ML model under-\nestimates the T cof the pure binary compound Ce 2Fe17.[20]\nThis is expected, as already discussed, our model is less pre -\ncise for the prediction of low T ccompounds.\nSwitching to the M spart, the middle panel of Fig. 2 shows\nthe confidence of classification of compounds with µ0Ms\nmore than 1 T. The confidence value closer to 1 implies that\nthe prediction is viable to be more accurate. All the com-\npounds are classified in favor of forming permanent magnets\nwithµ0Ms>1 T. For compounds like Ce 2Fe17−xCoxthe pre-\ndiction confidence varies from 0.6 to 0.8 with increasing Co\nconcentration, whereas the carbon and nitride compounds ar e\nalways classified with high prediction confidence.\nThe predictions from classification model on Kuis\nshown in bottom panel of Fig. 2. We find while\nthe anisotropy of Fe 17−xCoxcompounds without inter-\nstitial C/N ( x= 2,...7) atoms are predicted to be\neasy-plane, their carbonated/nitrogenated/carbo-nitro genated\ncounterparts show easy-axis anisotropy. For pure Fe com-\npounds, apart from carbo-nitrogenated compound, all are pr e-\ndicted to be easy-plane, while for Fe 16Co compounds carbon-\nated as well as carbo-nitrogenated compounds are predicted to\nbe easy-axis. This in turn, highlights the effectiveness of Cosubstitution on making Kupositive. We note the prediction\nconfidence of the carbo-nitrogenated compounds are around\n0.75.\nOn basis of the above ML analysis, we pick up seven yet-to-\nbe synthesized compounds, Ce 2Fe17−xCoxCN,x= 1,..., 7.\nThis choice is guided by the compounds satisfying T c>600\nK from regression model, and µ0Ms>1 Tesla with easy-axis\nanisotropy from classification models, and being Fe-rich. I n\nfollowing, we describe their crystal structure, and presen t re-\nsults of DFT calculated electronic structure, anisotropy p rop-\nerties, and stability properties.\nDFT CALCULATED PROPERTIES OF PREDICTED\nCOMPOUNDS\nCrystal Structure\nThe Ce 2Fe17compounds crystallize in the rhombohedral\nTh2Zn17-type structure (space group R ¯3m), derived from the\nCaCu5-type structure with a pair (dumbbell) of Fe atoms for\neach third rare earth atom in the basal plane and the substi-\ntuted layers stacked in the sequence ABCABC .... As shown\nin Fig. 3, the transition metal atoms are divided into four su b-\nlattices, 9 d, 18f, 18hand 6c, having 3 (9), 6(18), 6 (18), and\n2 (6) multiplicity in the one (three) formula unit primitive -\nrhombohedral (hexagonal) unit cell. The TM atoms occupy-\ning the 6csites, referred as dumbbell sites, form the ...-TM-\nTM-RE-RE- ...chains running along the c-axis of the hexag-\nonal cell. The 18 fTM atoms form a hexagonal layer, which\nalternates with the hexagonal layer formed by 9 dand 18hTM\natoms. The 6 cTM-TM doumbells pass through the hexagons\nformed by 18 fTM’s. For the interstitial C and N atoms, neu-6\ntron powder diffraction,[ 106] EXAFS experiments confirmed\nthat they fill voids of nearly octahedral shape formed by a rec t-\nangle of 18 fand 18hTM atoms and two RE atoms at opposite\ncorners, which are the 9 esites of Th 2Zn17-type structure, and\nhaving the shortest distance from the RE sites among all avai l-\nable interstitial sites. All our calculations are thus carr ied out\nwith C/N atoms in 9 epositions. The RE atoms in 6 cposition\nas well as light elements C/N in 9 einterstitial sites belong to\nthe same layer as 18 fTMs. As the 9 esites are in the same\nc-plane with the RE sites, having RE atoms at neighbors, in-\ntroduction of interstitials like C and N, is expected to have a\nprofound influence on the the electronic environment of RE\natom, thereby altering the magneto-crystalline anisotrop y.\nAlthough the R ¯3m symmetry is lowered upon Co substitu-\ntion and the spin-orbit coupling (SOC) in the anisotropy cal -\nculation, for the ease of identification, we will still use th e the\nnotations 9 d, 18f, 18hand 6c. Our total energy calculations\nshow that Co preferentially occupy sites in the sequence 9 d>\n18h > 6c >18f. Out of available 17 TM sites we have con-\nsidered Co substitution up to 7 sites, which result in Fe-ric h\nphases of compositions Ce 2Fe17−xCoxCN withx= 1, 2,...,\n7. Following the site preference we consider Co atoms in 9 d\nand 18hsites.\nWe expect the lattice parameters not to change much upon\nCo substitution, as Fe and Co, being neighboring elements in\nperiodic table, has similar atomic radii. Nevertheless, to check\nthe influence of Co substitution on lattice structure, we opt i-\nmize the lattice constant and the volume for all xvalues. Fol-\nlowing our expectation, the results show only a marginal de-\ncrease in lattice parameter and volume (with a maximum devi-\nation of 1 %) upon increasing Co content, in line with the find-\nings by Odkhuu et al.[ 18] for 1:12 compounds, and the exper-\nimental findings by Xu and Shaheen on 2:17 compounds.[ 19]\nThis minimal change is found to have no appreciable effect on\nmagnetic properties, as explicitly checked on representat ive\ncompounds with x= 1, 4 and 7. We thus choose the lattice\nstructure as the optimized lattice structure of x= 0 (see Ap-\npendix), with lattice constant = 6.59 ˚Aand angle β= 83.3oof\nthe rhombohedral unit cell[ 107] in subsequent calculations.\nMagnetic Moment and Electronic Structure\nIn the following we present the DFT results for the mag-\nnetic moments and density of states (DOS), as given in\nGGA+U+SOC calculations. The details of the DFT calcu-\nlations are presented in the Appendix. Importance of applic a-\ntion of supplemented Hubbard Uon RE sites within LDA or\nGGA+Uformalism is considered as one of the possible means\nto deal with localized forbitals of RE ions, and have shown to\nprovide reasonable description.[ 13,14] Previous calculations\nin compounds containing Ce, showed variation of Uwithin\n3 eV to 6 eV , keeps the results qualitatively same.[ 6,108] In\nthe following, we present results for Uapplied on Ce atoms\nchosen to be 6 eV .\nFig. 4 shows the calculated total magnetic moments of the1.61.71.81.9Total moment (Tesla)\nFe   Co\n16  Fe   CoFe   CoFe   CoFe   CoFe   CoFe   Co\n15     2 14     313     4 12     511     610     7\nFIG. 4: (Color online) Calculated total moment (black cir-\ncles),µ0M in Tesla plotted for increasing Co concentrations of\nCe2Fe17−xCoxCN compounds. Shown are also experimental\nresults[ 19] (red, square) for Ce 2Fe17−xCoxNycompounds measured\nat room temperature. For comparison between T = 0 K calculate d\nmoments, and experimental data measured at room temperatur e, the\nexperimental data has been scaled by a factor of 1.3.\nFIG. 5: (Color online) Calculated spin (top) and orbital (bo ttom)\nmoments at Ce, Fe(9 d), Fe(18f), Fe(18h), Fe(6c) and Co sites in the\nrepresentative case of Ce 2Fe15Co2CN compound.\nseven mixed Fe-Co compounds, Ce 2Fe17−xCoxCN (x= 1, 2,\n...7). The total magnetic moment shows a decreasing trend\nwith increase of Co concentration, arising from the fact that\nCo moment is smaller that of Fe. However, it is reassuring to\nnote that even for compound with largest Co concentration,\nCe2Fe10Co7CN, the calculated moment is more than 1.657\nFIG. 6: (Color online) Left: Density of states of Ce 2Fe15Co2CN compound, projected onto Ce f(brown), Ce d(shaded green), Fe d(blue),\nCod(shaded red) and CN p(shaded orange) characters. Right: Density of states of Ce 2Fe15Co2CN compound projected to different Fe d’s:\nFe(9d) (shaded indigo), Fe(18 h) (magenta), Fe(18 f) (green) and Fe(6 c) (brown). The zero of the energy is set at Fermi energy.\nTesla. This is in agreement with ML prediction, which pre-\ndictsµ0Msof all the considered compounds to be larger than\n1 Tesla, though it is to be noted the ML predictions are made\nfor room temperature moments while the DFT calculated mo-\nments are at T = 0 K. The measured values of total moment in\ncorresponding nitrogenated compounds show good compari-\nson (cf Fig. 4) with our calculated moments. In particular,\nbarring the data on x ≈2, the other two data point show good\nmatching with the trend of theoretical results. We note that the\nexperimentally determined moments are for Ce 2Fe17xCoxNy\ncompounds, which contains only N as interstitial atom, and\nthe value of yis not mentioned, which may even vary depend-\ning on value of x.\nFig. 5 shows the spin and orbital moments projected to Ce,\nFe(9d), Fe(18f), Fe(18h), Fe(6c) and Co atoms for the rep-\nresentative case of Ce 2Fe15Co2CN compound. The results\nfor other Co concentrations are similar. In presence of larg e\nSOC coupling at Ce site, a substantial orbital moment devel-\nops, which is oppositely aligned to its spin moment followin g\nHund’s rule. Considering 3+ nominal valence of Ce, it would\nbe in 4f1state, with S=1/2 and L=3. While the calculated\nvalue of Ce spin moment is close to 1 µB(≈0.95µB) in ac-\ncordance with nominal S=1/2 state, the orbital moment shows\nsignificant quenching with a calculated value of about 0.5 µB.\nThis value of orbital moment is in agreement with DFT calcu-\nlated values of other Ce containing RE-TM magnets.[ 6,109]\nThe 4felectrons are coupled to 5 delectrons at Ce site by\nintra-atomic exchange interaction, following which their spin\nmoments are aligned in parallel direction. The delocalized 5d\nelectrons at Ce site, hybridize with Fe/Co 3 delectrons, favor-\ning antiparallel alignment of Ce and Fe/Co spins, as found in\nFig. 5. The spin magnetic moment at Fe sites show a distribu-\ntion, with Fe at 6 csite having largest moment, followed by Feat 9dand 18hsites while Fe at 18 fsite shows the lowest mo-\nment. We notice that Fe (6 c) atoms occupying the dumbbell\nsites, have less connectivity compared to Fe(9 d), Fe (18f) and\nFe (18h), and thus possess the largest moment, being of most\nlocalized character. Among Fe (9 d), Fe(18f), Fe(18h) sites\nFe (18f) has smallest moment, driven by the fact that inter-\nstitial C and N atoms are in same plane as Fe (18 f) causing\nenhanced d-phybridization, and reduction in moment. These\nspin moments though are larger than that of bulk Fe ( ≈2.2\nµB). The orbital moment at Fe sites are tiny ( ≈0.05µB). In\ncomparison, Co shows significantly smaller spin moment ( ≈\n1.7µB) and somewhat larger orbital moment ( ≈0.1µB), jus-\ntifying the fall in total moment with increasing concentrat ion\nof Co.\nFig. 6 shows the density of states of Ce 2Fe15Co2CN, pro-\njected to various orbital characters. The Ce 4 fstates are all\nunoccupied in the majority spin channel, partly occupied in\nthe minority spin channel, in accordance with nominal f1oc-\ncupancy. The RE 4 f- TM 3dhybridization through empty RE\n5dstates is visible, making the spin splitting at Fe and Co site s\nantiparallel to that of Ce. The C/N pstates mostly spanning\nthe energy range -7 eV to -4 eV , show non negligible mixing\nwith Fed, Codand Ce characters, justifying their role in in-\nfluencing the magnetic properties. Fe dand Codstates span\nabout the same energy range from -4 eV to 2 eV , with states\nmostly occupied in the majority spin channel and partially o c-\ncupied in the minority spin channel, largely accounting for the\nmetallicity of the compound. Spin splitting of Fe dis larger\nthan that of Co, being consistent with larger magnetic momen t\nof Fe compared to Co. Projection to different inequivalent F e\nsites (cf right panel of Fig. 6), Fe(9 d), Fe(18h), Fe(18f) and\nFe(6c) shows that Fe(6 c) belonging to dumbbell pair is dis-\ntinct from other Fe sites, which also exhibit largest magnet ic8\nFIG. 7: (Color online) Top: Calculated magnetocrytalline anisotropy\nconstant in MJ/m3plotted for increasing Co concentrations of\nCe2Fe17−xCoxCN compounds. The inset shows the anisotropy in\norbital moment (see text for details). Bottom: The GGA+ U+SOC\nDOS projected to Ce fenergy states with magnetization axis\npointed along easy-axis, for Ce 2Fe17(black), Ce 2Fe17CN (red) and\nCe2Fe16CoCN (blue). The zero of the energy is set at Fermi energy,\nwith unoccupied part shown shown as shaded. The arrow indica tes\nthe shift in occupied part.\nmoment among all Fe’s.\nMagneto-crystalline Anisotropy\nHaving an understanding of the basic electronic structure,\nin terms of magnetic moments and density of states, we next\nfocus on calculation of magneto-crystalline anisotropy co n-\nstant, Ku, which is a crucial quantity responsible for coerciv-\nity in a permanent magnet. MAE defines the energy required\nfor turning the orientation of the magnetic moment under ap-\nplied field, expressed as E(θ)≈K1sin2θ+K2sin4θ+\nK3sin4θcos4φ, where K 1, K2, and K 3are the magnetic\nanisotropy constants, θis the polar angle between the mag-\nnetization vector and the easy axis ( c-axis), and φis the\nazimuthal angle between the magnetization component pro-\njected onto the abplane and the a-axis. In most cases, the\nhigher order term K 3is relatively small compared with K 1\nand K2. Forθ=π/2, one may thus write Ku≈K1+K2.It’s positive and negative values indicate the easy axis and\neasy plane anisotropy, respectively. To satisfy the criter ia of a\ngood permanent magnet, it should have easy axis anisotropy\nwith value larger than 1 MJ/m3.[2,8] The MAE in RE-TM\narises from two contributions, (i) MAE of the RE sublattice\ndue to strong spin-orbit coupling and crystal field effect an d\n(ii) MAE of TM sublattice. The interplay of the two decides\nthe net sign and magnitude. In particular, in the proposed\ncompounds, presence of Co with significant value of orbital\nmoment, makes the contribution of TM sublattice important.\nWhile 2:17 compounds, primarily show easy plane anisotropy ,\nswitching to easy axis anisotropy for interstitial compoun ds\nhave been reported. In particular, upon nitrogenation, eas y\nplane anisotropy has been reported for Ce containing mixed\nFe-Co compounds.[ 19] As mentioned already, the interstitial\natoms occupy the same plane as the RE atoms, significantly\ninfluencing their properties. With predicted high T cand large\nsaturation moment of our proposed compounds with carbon-\nation and nitrogenation, it remains to be seen whether they\nwould exhibit easy axis anisotropy of reasonable values, as\nrequired for a legitimate candidate for permanent magnet. F or\nthis purpose, we carry out calculations within GGA+ U+SOC\nwith magnetization axis pointing along the crystallograph ic c-\naxis and perpendicular to it. The importance of application of\nUon proper description of MAE in terms of its sign and order\nof magnitude has been stressed upon by several authors.[ 6,13]\nIn order to establish our method on calculation of MAE in-\nvolving small energy difference, we first apply our method\nto known and well studied case of SmCo 5, with choice of U\n= 6 eV on Sm, and obtained a MAE value of 24.4 meV/f.u,\nwhich agrees well with GGA+ U+SOC calculated value of\n21.6 meV/f.u., reported in literature[ 13] as well as experimen-\ntally measured values of 13-16 meV/f.u.[ 110] The calculated\nresults for the proposed Ce 2Fe17−xCoxCN are shown in top\npanel of Fig. 7. We found that MAE shows site-dependence\non the Co substitution. We consider configurations with Co\natoms substituting Fe(9 d) and Fe(18 h) sites , configurations\ninvolving other substituting sites being energetically mu ch\nhigher. We consider configurations which are energetically\nclose (within 600 K) and calculate the Co-composition de-\npendent MAE using the virtual crystal approximation. Speci f-\nically, for x= 1 we consider configurations Co@Fe(9 d) and\nCo@Fe(18 h), the latter being 3.58 meV higher compared to\nformer. Similarly for x= 2, we consider Co@ 2 ×Fe(9d)\nand Co@ 2 ×Fe(18h), the latter being 4.43 meV higher com-\npared to former. For x= 3, the configurations considered are,\nCo@ 2×Fe(9d)+ Fe(18h); Co@ 3 ×Fe(9d); Co@Fe(9 d) +\n2×Fe(18h), the energies being 0 meV (set as zero of energy),\n12.37 meV and 47.66 meV , respectively. For x= 4, the con-\nfigurations considered are, Co@ 2 ×Fe(9d) + 2×Fe (18h);\nCo@ 3×Fe(9d) + Fe(18h), the energies being 0 meV (set as\nzero of energy) and 36.5 meV , respectively. For x= 5 , 6 and\n7, only one configuration is considered, others being energe t-\nically much higher, namely, Co@3 ×Fe(9d) + 2×Fe(18h),\nCo@3×Fe(9d)+ 3×Fe(18h) and Co@3 ×Fe(9d) + 4×\nFe(18h), respectively.9\nFIG. 8: (Color online) Calculated anisotropy field in Tesla (left) a nd maximal energy product in kJ/m3(right) plotted for increasing Co\nconcentrations of Ce 2Fe17−xCoxCN compounds.\nConsidering spin-orbit effect only on Ce atom, it is found\nto account for about 60 %of the calculated MAE. We find all\nthe calculated MAE is positive, in good agreement with ML\nprediction on mixed Fe-Co carbo-nitride compounds. Furthe r\nMAE values show non-monotonic dependence on Co concen-\ntration. Such non-monotonic trend upon varying TM con-\ntent has been also reported in context of R(Fe 1−xCox)11TiZ\n(R = Y and Ce; Z= H, C, and N)[ 7] and R-TM systems in\ngeneral.[ 111] In the inset of top panel of Fig. 7, we show\nthe calculated orbital magnetic anisotropy ( ∆ML) defined as\n∆ML= ML(a) - M L(c), as employed in Ref. 18, ML(c) and\nML(a) being the orbital moment along the c-axis and a-axis,\nrespectively. We find a correlation between ∆MLand Ku,\nqualitatively satisfying Bruno’s expression[ 112] for itinerant\nferromagnets given as, K u= (ξ\n4µB)∆ML, whereξis the\nstrength of SOC.\nMost of the easy-axis Kuvalues are found to be larger than\n1 MJ/m3, except Fe 14Co3and Fe 13Co4for which it is found\nto be 0.74 and 0.91 MJ/m3, respectively. Few of the con-\ncentrations exhibit easy-axis Kuvalues larger than 2 MJ/m3,\ne.g.Fe15Co2(3.54 MJ/m3), Fe12Co5(3.39 MJ/m3), Fe11Co6\n(3.39 MJ/m3), Fe10Co7(9.10 MJ/m3), being comparable to\nNd2Fe14B (4.9 MJ/m3).[113]\nTo obtain microscopic understanding of the role of\nCo substitution and doping by C, N on magnetocrys-\ntalline anisotropy, we further calculate the magnetocrys-\ntalline anisotropy of Fe-only compounds Ce 2Fe17, Ce2Fe17C,\nCe2Fe17N and Ce 2Fe17CN. This results in negative K uval-\nues for Ce 2Fe17, and Ce 2Fe17C (-2.12 MJ/m3and -1.35\nMJ/m3), a tiny positive value for Ce 2Fe17N (0.26 MJ/m3)\nand positive value for co-doped compound Ce 2Fe17CN (1.27\nMJ/m3). We further plot the GGA+ U+SOC density of states\n(cf bottom panel, Fig. 7) with magnetization axis along c-\naxis projected to Ce fstates for Ce 2Fe17, Ce2Fe17CN and\nCe2Fe16CoCN, which is expected to reveal the mechanism of\nuniaxial anisotropy. We find that a lowering of occupied Cefenergy states and increase in band width occur upon intro-\nduction of light elements C and N. This gets further helped\nby substitution of Co, caused by hybridization between Ce f\nstates and Co dand C,Npstates. This gain in hybridization\nenergy stabilizes easy-axis magnetization (cf. Ref. 114) as ob-\nserved experimentally.[ 19]\nMaximal energy product and Anisotropy Field\nWhile, the estimates of Kuandµ0Msare useful infor-\nmation to access the effectiveness of the suggested materi-\nals as permanent magnets, technologically interesting figu res\nof merit of hard magnetic materials, are the maximal energy\nproduct (BH) max and anisotropy field H a. These can be esti-\nmated from the knowledge of µ0MsandKuas follows,\n(BH)max=(0.9µ0Ms)2\n4µ0\nHa=2Ku\nµ0Ms\nThe factor 0.9 in the expression for (BH) maximplies the com-\nmon assumption that ideally out 10 %of a processed bulk hard\nmagnet consists of non-magnetic phases.[ 115] The estimated\n(BH)max and Hais shown in Fig. 8. The (BH) max value\nis found to range from 444 to 540 kJ/m3, in comparison to\nexperimentally measured values 516 kJ/m3and 219 kJ/m3\nfor Nd 2Fe14B[116] and SmCo 5,[116] respectively. The H a\nshows a strong variation with Co concentration, ranging fro m\n≈1 Tesla to 14 Tesla. [117]\nWe further note that the hardness parameter, defined as κ\n=/radicalBig\nKu\nµ0M2s, turns out to be greater than 1 for Ce 2Fe15Co2CN,\nCe2Fe12Co5CN, Ce 2Fe11Co6CN, and Ce 2Fe10Co7CN com-\npounds, employing the calculated T = 0 K values of Kuand\nMs.10\n∆Ef(CN)∆Ef(N)∆Ef(C)\nx= 1 4.32 2.10 0.97\nx= 2 3.99 2.09 0.85\nx= 3 4.16 2.09 0.88\nx= 4 3.98 2.10 0.79\nx= 5 3.82 2.07 0.70\nx= 6 3.91 2.05 0.72\nx= 7 3.78 2.01 0.69\nTABLE II: Vacancy formation energy for carbon ( ∆Ef(C)), ni-\ntrogen (∆Ef(N)) and nitrogen-carbon ( ∆Ef(CN)) in eV in\nCe2Fe17−xCoxCN compounds.\nStability\nUnlike the other RE-TM magnets like 1:12 compounds, one\nof the advantage of 2:17 compounds is their stability. Both\nstable form of Ce 2Fe17and its Co substituted form have been\nreported in literature.[ 19] Calculation of formation enthalpies,\nas given in Ref. 18,Eform=Ecompound −/summationtext\nkNkǫk/summationtext\nkNk, whereNk\nindicate number of different atoms (Ce, Fe, Co, N and C) in\nthe cell, and ǫkdenote energy/atom of bulk Ce in FCC struc-\nture,α−Fe, Co in HCP structure, in molecular nitrogen and\nC in graphite structure, gives values -0.61 to -0.59 eV/atom\nfor the studied Ce 2Fe17−xCoxCN compounds.\nA major challenge with interstitial compounds, though, is\nthe nitrogen diffusion.[ 21] It has been further suggested the\nblockage of nitrogen diffusion by carbon layer is useful in r e-\nduction of nitrogen outgassing in carbo-nitrides. In parti cular,\nheating up Sm 2Fe17carbo-nitrides at a constant rate in a dif-\nferential scanning calorimeter, the onset temperature of n itro-\ngen outgassing was found to be higher by more than 40 K, as\ncompared to nitride counterpart.[ 21] This justifies the choice\nof carbo-nitrides as our exploration set. To this end, we cal -\nculate the vacancy formation energy of the interstitial ato ms\nin our chosen compounds. For this purpose, we calculate the\nformation energy of the N and/or C vacancy ( ∆Ef) defined\nas,\n∆Ef=EN(C)vac−Epristine+EN(C)\nwhereEN(C)vacandEpristinedenote the optimized total en-\nergies of compound containing N and/or C vacancy, and va-\ncancy free compound. The internal positions for defect free\npristine structure and structures containing nitrogen and /or\ncarbon vacancies are performed keeping the lattice parame-\nters fixed. EN(C)is the energy per N or C atom, which is\nobtained from calculation of N 2molecule or graphite. The ob-\ntained results for Ce 2Fe17−xCoxCN compounds in minimum\nenergy configuration of Co is shown in Table II. The vacancy\nformation energies show hardly any variation on chosen con-\nfiguration for a given Co concentration.\nThe vacancy formation energies, listed in Table II, show\nonly small variation between compounds of varying Co\nconcentration, with the general trend ∆Ef(CN)>/summationtext\n(∆Ef(N)+∆Ef(C)). The individual nitrogen vacancy\nformation energy and carbon vacancy formation energy, arein overall agreement with that found for related compound,\nSmCaFe 17C(N)3.[6] The vacancy formation energy for co-\ndoped carbon-nitrogen compounds are found to be enhanced\nby about 35-40 %compared to the sum of the individual\nC and N vacancy formation energies, proving the carbo-\nnitrogenation co-doping to provide better thermal stabili ty.\nWe also check our results by repeating vacancy formation en-\nergy calculations for x= 0 compounds, which however do not\nshow significant difference, suggesting Co doping not havin g\nmajor role in stability, as also indicated by no significant v ari-\nation of results between x= 1, 2, 3, 4, 5, 6 and 7.\nCONCLUSION\nDesigning alternative solutions for permanent magnets, sa t-\nisfying the criteria of low-cost, while keeping the magneti c\nproperties comparable to those of permanent magnets in use,\nis of utmost importance for cost-effective technology. To-\nwards this goal, we use a combined route of machine learning,\nbased on experimental data, and the first-principles calcul a-\ntions. While machine learning has been applied for problem of\nrare-earth magnets,[ 5] those studies have been based on the\ndataset created out of high throughput calculations. Being de-\npendent on calculation-based inputs, creation of such data base\nis not only computationally expensive, but also not devoid o f\napproximations of the theory. Our study, to the best of our\nknowledge, being based on a exhaustive search of experimen-\ntal data, is first of this kind in context of rare-earth magnet s.\nWhile a large volume of experimental data is available with\nnumerical value of T c, the corresponding dataset with numer-\nical values of M sandKuis small. On the other hand, there\nexists sizable dataset with information of Kubeing positive\n(easy axis) or negative (easy plane), and µ0Msbeing larger or\nsmaller than 1 Tesla. We thus employ regression model of ma-\nchine learning training to make predictions on numerical va l-\nues of T c, and classification model to make predictions on sign\nofKu, andµ0MSbeing larger or smaller than 1 Tesla. We ap-\nply the trained machine learning to 2:17 rare-earth transit ion\nmetal compounds with carbon and nitrogen in interstitials. We\nchoose the compounds to contain abundant rare-earth Ce, and\nto be Fe-rich to make them cost-effective. Although nitro-\ngenated version of this series has been investigated,[ 19] the\nsystematic study of the carbo-nitride family to the best of o ur\nknowledge is unavailable. The machine learning predicts T c\nof the chosen carbo-nitride family to be larger than 600 K,\nµ0MS>1 Tesla, and Ku>0, thereby indicating the pos-\nsibility of them to become good solutions for cost-effectiv e,\npermanent magnets. Subsequent first-principles calculati ons,\nshow T=0 K, µ0MSto be larger than 1.65 Tesla, and Ku/greaterorsimilar\n1 MJ/m3for the entire family, Ce 2Fe17−xCoxCN (x= 1,...\n7). Calculated Kuvalues are found to be comparable to the\nstate-of-art permanent magnet Nd 2Fe14B for Ce 2Fe15Co2CN,\nCe2Fe12Co5CN, Ce 2Fe11Co6CN, and Ce 2Fe10Ce7CN. This\nresults in two figure of merits for hard magnets, (BH) max and\nHain range of 444-540 kJ/m3and≈1 - 14 T, respectively .11\nIn spite of good magnetic properties, one of the limitation\nof practical applications of interstitial 2:17 magnets is t he for-\nmation of nitrogen/carbon vacancies at high temperature. By\ncalculating the N-(C)-vacancy formation energy, we show th at\ncarbo-nitrogenation co-doping enhances the vacancy forma -\ntion energy significantly, by 35-40 %compared to sum of in-\ndividual doping. This is likely to improve the thermal stability\nat high temperature condition.\nOur computational exercise based on exhaustive search of\nexperimental database, should motivate future experiment al\nprocesses in making high-performance 2:17 interstitial ma g-\nnets, with cheapest RE element Ce, the most abundant 3d\nmetal, Fe and cheap non-metal interstitial dopings like C an d\nN. The estimated price-to-performance based on calculated\nenergy product, and available market price[ 16] turns out to be\n0.03-0.22 USD/J. The enhanced thermal stability of the carb o-\nnitrides compounds against the vacancy formation of the lig ht\nelements further boosts the promises of the suggested com-\npounds.\nACKNOWLEDGEMENT\nThe authors acknowledge the support of DST Nano-\nmission for the computational facility used in this study.\nAPPENDICES\nDFT details\nDFT calculations for electronic structure, magnetocrys-\ntalline anisotropy are performed using the all-electron de nsity-\nfunctional-theory code in full potential linear augmented\nplane wave (FP-LAPW) basis, as implemented in WIEN2K\ncode.[ 118] For expensive structural optimization calculations,\nthe plane wave based calculations, as implemented in Vi-\nenna Ab-initio Simulation Package (V ASP),[ 119] are carried\nout. The exchange-correlation functional is chosen to be\ngeneralized-gradient approximation (GGA) of Perdew, Burk e,\nand Ernzerhof.[ 120] The localized nature of 4 fstates of Ce is\ncaptured through GGA+ Ucalculations,[ 121] with choice of\nU= 6 eV and J H= 0.8 eV . For light rare earths like Ce the U\nvalue was shown to range from 4 eV to 7 eV , without affecting\nmuch the physical properties.[ 108] The spin-orbit coupling ef-\nfect at Ce, and TM sites are captured through GGA+ U+SOC\ncalculations.\nFor FP-LAPW calculations, APW +lo is used as the ba-\nsis set, and the spherical harmonics are expanded upto l=\n10 and the charge density and potentials are represented upt o\nl=6. The sphere radii are set at 2.5, 1.9, 2.34, 1.56 and\n1.51 bohr for Ce, Fe, Co, N, and C. For good convergence, a\nRKmax value (the product of the smallest sphere radius and\nthe largest plane-wave expansion wave vector) of 7.0 is used .\nWe set the cutoff between core and valence states at −8.0 Ry.\nThe k-space integrations are performed with 112 k-points inirreducible Brillouin zone (BZ), following the report of us e\nof 80 k-points in irreducible BZ in case of SmCo 5to provide\ngood estimate of MAE.[ 13] Nevertheless, the convergence of\nresults on k-space mesh is checked by carrying out calculati on\nwith 260 k-points.\nThe structural optimization in plane wave basis is carried\nout starting with experimental structure of Sm 2Fe17CN, [ 107]\nreplacing Sm with Ce, and relaxing all the internal coordi-\nnates until forces on all of the atoms become less than 0.001\neV/˚A. Upon moving from Sm 2:17 carbide/nitride interstitial\ncompounds to Ce counterpart, the cell volume changes only\nnominally by 0.2 %to 0.4%.[6] For the plane wave calcula-\ntions, energy cut-off of 600 eV and Monkhorst pack k-points\nmesh of8×8×8are used.\nAll the calculations are performed by considering a\ncollinear spin arrangement. The MAE is obtained by calcu-\nlating the GGA+ U+SOC total energies of the system, in FP-\nLAPW basis as Ku= Ea- Ec, where E aand Ecare the ener-\ngies for the magnetization oriented along the crystallogra phic\naandcdirections, respectively. For accurate estimates of va-\ncancy formation energy, we also use FP-LAPW basis.\nData preprocessing in Machine Learning\nWhile constructing the database, we avoid inclusion of\nnoisy data. We do bootstrapping to normalize the data which\nis followed by removal of outliers with the help of violin\nplot. A data is removed if it lies outside of Q1-1.5 ×IQR or\nQ3+1.5×IQR, where IQR is the interquartile range and Q1,\nQ2 and Q3 are lower, median and upper quartile respectively.\nIn the next step we identify correlated attributes using Pea r-\nson’s correlation coefficient which can be defined as,\nr=/summationtexti=n\ni=1(xi−¯x)(yi−¯y)/radicalBig/summationtexti=n\ni=1(xi−¯x)2/radicalBig/summationtexti=n\ni=1(yi−¯y)2\nHerenis the sample size, xiandyiare sample points and\n¯xand¯yare the sample means.\nThe heatmap obtained by using the above mentioned cor-\nrelation is shown in Fig. 9. The correlation between the\nattributes is mapped between 0 and 1, considering the abso-\nlute values. The highly correlated attributes with correla tion\ngreater than 0.75 are as follows:\n1. Electronegativity difference between RE and TM ( ∆ǫ)\nand CW average of atomic no. of TM ( < ZTM>)\n2. CW TM percentage ( TM%) and CW average of atomic\nno. of TM ( < ZTM>).\n3. CW TM percentage ( TM%) and Electronegativity dif-\nference between RE and TM ( ∆ǫ).\n4. Total number of f electrons ( fn) and Atomic no. of RE\n(ZRE).12\nFIG. 9: (Color online) Heatmap indicating the correlation b etween\ndifferent attributes considered to built ML algorithm. The color code\nis shown in the side panel. The boxes with red represent weak o r no\ncorrelation, whereas blue boxes represent strong correlat ion between\nthe attributes.\n5. LE percentage ( LE%) and CW average of atomic no.\nof TM (< ZTM>).\n6. LE percentage ( LE%) and Electronegativity difference\nbetween RE and TM ( ∆ǫ).\n7. LE percentage ( LE%) and CW TM percentage\n(TM%).\nWe thus discard ∆ǫ,LE%,ZREand< ZTM>from the list\nof attributes.\nModel construction for training in ML\nFIG. 10: (Color online) Coefficient of determination R2score of five\ndifferent ML algorithms applied to T cdataset.The performance of a model can be quantified in terms\nof coefficient of determination which can be expressed as\nfollows:[ 122]\nR2= 1−/summationtextN\ni=1[yi−f(xi)]2\n/summationtextN\ni=1[yi−µ]2\nfor predictions f(xi)and a set of actual values yiwith mean\nµ. If the algorithm performs perfectly, R2score is 1. Fig.\n10 shows score R2for five different algorithms. RR al-\ngorithm circumvents issues in ordinary linear regression l ike\nover-fitting or failure in finding unique solution due to mul-\nticollinearity. It develops on least square error by adding an\nextra penalty/regularization term to the loss function of o rdi-\nnary linear regression. KRR builds on the ridge regression\ntechnique by using kernel trick [ 123] so that it can capture the\nnonlinearity present in the feature space. It can fit a non lin ear\nfunction by learning from a linear function spanned by a ker-\nnel which in turn mimics a non-linear function in the origina l\nspace. SVR originated from support vector machines which\nare mainly popular in classification problem. It is based on\nthe idea to search a hyperplane [ 124] by minimizing the er-\nror which is able to separate two different classes. SVR also\nuses kernel trick to map the data into a high dimensional fea-\nture space and then performs linear regression to fit the data .\nThese three models are based on the same principle of linear\nregression and SVR is the best form according to our result.\nR2score is 0.66 for SVR whereas it is found to be poor ( ≈\n0.25) for other two algorithms.\nApart from these we use two other algorithms, ANN and\nRF. The model performance scores are satisfactory for both\nof them. A simple ANN architecture called perceptron imple-\nments a processing element or artificial neuron called Thres h-\nold Logic Unit (TLU) which can have one or more input(s)\nand one output. Each input is related to a weight. The TLU\ncalculates the weighted sum of its inputs, applies a step fun c-\ntion (generally Heaviside or sign function) to it and output s\nthe result. A perceptron [ 125] is simply a layer of TLUs op-\nerating in parallel and connected to all the inputs. Trainin g\nan ANN model is equivalent to learning each weight factor in\nan iterative cycle. A more complex system (Multi-Layer Per-\nceptron) can be built by associating additional interconne cted\nlayers to the architecture. A well functioning system consi sts\nof an input layer, several hidden layers and an output layer.\nIn our case we have one input layer, two hidden layers where\nrectified linear unit (ReLU)[ 126] is used as activation function\nalong with L2 regularization in the kernel, and an output lay er.\nThe constructed ANN model shows 0.80 as R2score.\nRandom forest is an ensemble method which consists of\nmultiple decision trees. Each tree is built on a portion of en -\ntire training data with a subset of total number of attribute s.\nTree algorithm is based on ’top to bottom’ approach, start-\ning from a root node, it consists of many intermediate nodes\nand ends at leaf nodes. At each node of a tree a particular\nattribute classifies the data and helps to grow the tree. The\nprediction is based on accumulating the results from all suc h13\nFIG. 11: (Color online) Model output from RF algorithm for T cof RE-TM intermetallics. The left panel shows the compariso n of Tcobtained\nfrom literature and predicted T c. The distribution of absolute error between predicted T cand actual T cis shown in the upper, right panel, while\nthe lower, right panel presents the distribution of relativ e error for the compounds with T c>600 K.\ntrees, taking ensemble average in case of regression or cons id-\nering votes from majority trees in case of classification. Su ch\nan algorithm can capture the complex and nonlinear interac-\ntion between different attributes and can built a robust and\nsophisticated model. Our random forest consists of 100 tree s\nbuilt by bootstrapped[ 127] sampling of the training set. Each\ntree allows checking a maximum of log2(number of features)\nwhile detecting the best split node. The quality of such a spl it\nis measured by using mean squared error (Gini index) in re-\ngression (classification). The model efficiency is calculat ed\nby running out-of-bag samples down each of the trees. We use\nten-fold cross validation to extract the hyper-parameter a nd to\nconstruct the best model.\nFig. 11 shows the result of the best regression model us-\ning RF algorithm in case of T c. The plot in the left panel\nshows the predicted T cversus T cobtained from experiments.\nThe determination score R2is high enough (0.86), indicating\na good agreement between the predicted T cand experimen-\ntally reported T c. The mean absolute error in this model is 60\nK. Additionally we evaluate absolute error and relative err or\nfor the compounds with T c>600 K (cf Fig. 11, right panel).\nThis analysis helps to determine the model performance for\nthe compounds with T c>600 K as we are interested to pre-\ndict new RE-TM intermetallics with high T c. The distribution\nof absolute error shows that for the most of the compounds\n(≈85%) the absolute error is less than 100 K. For 65 %of the\npredicted cases, the absolute error is less than 50 K. We also\ncheck the absolute error for the compounds with T c<300 K\n(not included in the figure). In this case our model predicts\n≈76%compounds with absolute error less than 100 K and\n50%instances are predicted with absolute error of 50 K. Thisobservation prompts us to conclude that though the model pre -\ndiction is in general good, it is less accurate for low T ccom-\npounds compared to high T ccompounds. The distribution\nof relative error, expressed as ǫrel= (Texp\nc-Tpredicted\nc )/Texp\nc,\nprovides further support to this statement, which is shown i n\nbottom, right panel of Fig. 11. The relative error distribu-\ntion appears Gaussian like with slight asymmetry about the\nmean position. The relative error is less in the right side of\nthe mean position than the left side suggesting the predicti on\nof Tcsuffers less overestimation than underestimation. As\nfound, only 1 %of the instances are having ǫrel>50%, 3%of\nthe instances have 50 %> ǫrel>30%and 2%instances have\n30%> ǫrel>25%, most cases having tiny values of ǫrel.\nThis gives us confidence in accuracy of the predicted T cfor\ncompounds with T cs exceeding 600 K.\nTurning to M s, we use random forest algorithm to classify\nhigh M sfrom low M scompounds. The best model by per-\nforming 10-fold cross validation is built up with 81.53 %ac-\ncuracy. The resultant confusion matrix is shown in Fig. 12.\nFor classification problem, F1 score determines the balance\nbetween precision and recall. In this case F1 score 82.2 %indi-\ncates good anticipation with slight favour towards the pred ic-\ntion of compounds with high M s(µ0Ms>1) (83.8%) com-\npared to the compounds with low M s(µ0Ms<1) (79.2%).\nSimilar to M s, we use random forest algorithm for Ku, to\nclassify positive Kufrom negative Kucompounds. The best\nmodel by performing 10-fold cross validation, in this case,\nis built up with 80.62 %accuracy Like M s, in this case F1\nscore for positive Kuis 83%and for negative is Ku77.5%\nsuggesting slight preference of classification towards pos itive\nKuwhich is also captured in the plot of confusion matrix as14\nFIG. 12: (Color online) Normalized confusion matrix for\nµ0Ms(violet) and Ku(grey) classification using 10-fold cross-\nvalidation. Here positive (negative) class represents eit her com-\npounds with µ0Ms>(<)1T, or compounds with uniaxial\nanisotropy i.e K u>(<)0 MJ/m3. True positive/negative or\nTP/TN are the compounds where their classes are predicted co r-\nrectly. Whereas false positive (FP) and false negative (FN) are the\noff-diagonal terms of the matrix where the classes are incor rectly\nclassified.\nshown in Fig. 12.\n∗Electronic address: t.sahadasgupta@gmail.com\n[1] K. Buschow, Rep. Prog. Phys. 541123 (1991).\n[2] J. M. D. Coey, IEEE Trans. Magn. 47, 4671 (2011).\n[3] Hong Sun, Y . Otani and J.M.D. Coey, Magnetism and Mag-\nnetic Materials 104-107 1439-1440 (1992).\n[4] A. Vishina, O. Y . Vekilova, T. Bj¨ orkman, A. Bergman, H. C .\nHerper, O. Eriksson, Phys. Rev. B 101, 094407 (2020).\n[5] J. J. M¨ oller, W. K¨ orner, G. Krugel, D. F. Urban, Acta Mat eri-\nalia153, 53 (2018).\n[6] T. Pandey, M-H. Du and D. S. Parker, Phys. Rev. Appl. 9\n034002 (2018).\n[7] L. Ke and D. D. Johnson, Phys. Rev. B 94, 024423 (2016).\n[8] J. M. D. Coey, Scr. Mater. 67, 524 (2012).\n[9] S. V . Halilov, H. Eschrig, A. Y . Perlov, P. M. Oppeneer, Ph ys.\nRev. B 58, 293 (1998).\n[10] A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, R ev.\nMod. Phys. 68, 13 (1996).\n[11] W. K¨ orner, G. Krugel, C. Els¨ asser, Sci. Rep. 624686 92016).\n[12] H. Brooks, Phys. Rev. 58909 (1940).\n[13] P. Larson, I. I. Mazin and D. A. Papaconstantopoulos, Ph ys.\nRev. B 67214405 (2003)\n[14] S. Yehia, S. H. Aly, A. E. Aly, Computational Materials S ci-\nence 41482 (2008).\n[15] H. Ucar, R. Choudhary, D. Paudyal, J. Magn. Magn. Mater,\n496165902 (2020).\n[16] https://en.wikipedia.org/wiki/Prices −of−chemical −elements;\nJ.M.D.Coey Engineering 6, 119 (2020).\n[17] D. Odkhuu, and S. C. Hong, Phys. Rev. Appl. 11, 054085\n(2019).\n[18] D. Odkhuu, T. Ochirkhuyag, S. C. Hong, Phys. Rev. Appl. 13,054076 (2020).\n[19] Xie Xu and S. A. Shaheen, Journal of Applied Physics 73,\n5896 (1993).\n[20] H. Fujii, H. Sun. (1995). Chapter 3 Interstitially modi fied in-\ntermetallics of rare earth and 3D elements. Handbook of Mag-\nnetic Materials, 303-404.\n[21] X. Chen, Z. Altounian and D. H. Ryan, J. Magn. Magn. Mater .\n125169 (1993).\n[22] https://icsd.nist.gov/guide.html\n[23] K.H.J. Buschow, Handbook of Magnetic Materials, V olum es\n6&9 (Elsevier, 1991-1995).\n[24] J. M. D. Coey, Magnetism and Magnetic Materials (Cam-\nbridge University Press, 2010)\n[25] Shiqiang Liu, Chin. Phys. B 28, 017501 (2019).\n[26] S. R. Mishra, Gary J. Long, O. A. Pringle, D. P. Middleton , Z.\nHu et al., J. Appl. Phys. 79, 3145 (1996).\n[27] H. Klesnar, K. Hiebl and P. Rogl, Journal of the Less-Com mon\nMetals 154, 217 (1989).\n[28] R. Guetari, R. Bez, A. Belhadj, K. Zehani, A. Bezerghean u, N.\nMliki, L. Bessais, C.B. Cizmas, Journal of Alloys and Com-\npounds 588, 64 (2014).\n[29] Y . Otani, D. P. F. Hurley, Hong Sun, and J. M. D. Coey, Jour nal\nof Applied Physics 69, 5584 (1991).\n[30] M. Merches, W.E. Wallace and R.S. Craig, Journal of Mag-\nnetism and Magnetic Materials 24, 97 (1981).\n[31] F. Pourarian, R. Obermyer, Y . Zheng, S. G. Sankar, and W. E.\nWallace, Journal of Applied Physics 73, 6272 (1993).\n[32] F. Weitzer, H. Klesnar, K. Hiebl, and P. Rogl, J. Appl. Ph ys.\n67, 2544 (1990).\n[33] X.P. Zhong, R.J. Radwanski, F.R. de Boer, T.H. Jacobs an d\nK.H.J. Buschow, Journal of Magnetism and Magnetic Materi-\nals86, 333 (1990).\n[34] A. T. Pedziwiatr and W. E. Wallace, Journal of Applied\nPhysics 61, 3439 (1987).\n[35] R. van Mens, Journal of Magnetism and Magnetic Material s\n61, 24 (1986).\n[36] Hu Bo-Ping and J. M. D. Coey, Journal of the Less-Common\nMetals 142, 295 (1988).\n[37] H. Y . Chen, S. G. Sankar, and W. E. Wallace, Journal of Ap-\nplied Physics 63, 3969 (1988).\n[38] M. Juczyk and W. E. Wallace, Journal of Magnetism and Mag -\nnetic Materials 59, 182 (1986).\n[39] Y .G. Xiao, G.H. Rao, Q. Zhang, G.Y . Liu, Y . Zhang, J.K.\nLiang, Journal of Alloys and Compounds 407, 1 (2006).\n[40] E. Girt, M. Guillot, I. P. Swainson, Kannan M. Krishnan, Z.\nAltounian, and G. Thomas, Journal of Applied Physics 87,\n5323 (2000).\n[41] Wolfgang K¨ orner, Georg Krugel, and Christian Els¨ ass er, Sci.\nRep6, 24686 (2016).\n[42] A.M. Sch¨ onh¨ obel et al., Journal of Alloys and Compoun ds\n786, 969-974 (2019).\n[43] Y .Z. Wang et al., Journal of Magnetism and Magnetic Mate ri-\nals104-107 , 1132-1134 (1992).\n[44] E.P Wohlfarth and K.H.J. Buschow, Handbook of Magnetic\nMaterials, V ol-4 (Elsevier, 1988).\n[45] Y .Z. Wang et al., Journal of Applied Physics 73, 6251 (1993).\n[46] D. P. F. Hurley and J M D Coey, J. Phys. Condens. Matter 4,\n5573 (1992).\n[47] Yosuke Harashima, Kiyoyuki Terakura, Hiori Kino, Shoj i\nIshibashi, and Takashi Miyake, Phys. Rev. B 92, 184426\n(2015).\n[48] Y .G. Xiao, G.H. Rao, Q. Zhang, J. Luo, G.Y . Liu, Y . Zhang,\nand J.K. Liang, Physica B 369, 56 (2005).\n[49] V .K. Sinha, S.F. Cheng, W.E. Wallace, and S.G. Sankar, J our-15\nnal of Magnetism and Magnetic Materials 81, 227-233 (1989).\n[50] M. Jurczyk, Journal of Magnetism and Magnetic Material s89,\nL5-L7 (1990).\n[51] M. Katter, J. Wecker, C. Kuhrt, L. Schultz, X.C. Kou, and\nR. Gr¨ ossinger, Journal of Magnetism and Magnetic Material s\n111, 293-300 (1992).\n[52] M. Jurczyk and W.E. Wallace, Journal of Magnetism and Ma g-\nnetic Materials 59, L182-L184 (1986).\n[53] Yang Fu-ming, Li Qing-an, Zhao Ru-wen, Kuang Jian-ping ,\nF. R. de Boer, J. P. Liu, K. V . Rao, G. Nicolaides, and K. H. J.\nBuschow, Journal of Alloys and Compounds 177, 93 (1991).\n[54] Bao-gen Shen, Fang-wei Wang, Lin-shu Kong, Lei Cao, and\nHui-qun Guo, Journal of Magnetism and Magnetic Materials\n127, L267-L272 (1993).\n[55] X. C. Kou, T. S. Zhao, R. Grssinger, and F. R. de Boer, Phys .\nRev. B 46, 6225 (1992).\n[56] Zhi-gang Sun et al, J. Phys. Condens. Matter 12, 2495 (2000).\n[57] Z.X. Tang, E.W. Singleton, and G.C. Hadjipanayis, IEEE\nTrans. on Magn., 28, 5 (1992).\n[58] C.H. de Groot, K.H.J. Buschow, and F.R. de Boer, Physica B\n229, 213-216 (1997).\n[59] L. Zhang, D.C. Zeng, Y .N. Liang, J.C.P. Klaasse, E. Bruc k,\nZ.Y . Liu, F.R. de Boer, and K.H.J. Buschow, Journal of Mag-\nnetism and Magnetic Materials 214, 31-36 (2000).\n[60] Bing Liang, Bao-gen Shen, Fang-wei Wang, Tong-yun Zhao ,\nZhao-hua Cheng, Shao-ying Zhang, Hua-yang Gong, and\nWen-shan Zhan, Journal of Applied Physics 82, 3452 (1997).\n[61] Zhi-gang Sun, Shao-ying Zhang, Hong-wei Zhang, and Bao -\ngen Shen, Journal of Alloys and Compounds 322, 6973\n(2001).\n[62] Lin Qin, Sun Yunxi, Lan Jian, Lu Shizhong, and Jiang Hong -\nwei, Chinese Phys. Lett. 8, No. 5 , 267 (1991).\n[63] Jing-Yun Wang, Bao-Gen Shen, Shao-Ying Zhang, Wen-Sha n\nZhan, and Li-Gang Zhang, Journal of Applied Physics 87, 427\n(2000).\n[64] O. Isnard, and M. Guillot, Journal of Applied Physics 87, 5326\n(2000).\n[65] Zhi-gang Sun, Hong-wei Zhang, Shao-ying Zhang, and Bao -\ngen Shen, Physica B 305, 127134 (2001).\n[66] M.V . Satyanarayana, H. Fujii, and W.E. Wallace, Journa l of\nMagnetism and Magnetic Materials 40, 241-246 (1984).\n[67] Zhi-gang Sun, Hong-wei Zhang, Shao-ying Zhang, Jing-y un\nWang, and Bao-gen Shen, J. Phys. D: Appl. Phys. 33, 485491\n(2000).\n[68] Zhi-gang Sun, Hong-wei Zhang, Shao-ying Zhang, Jing-y un\nWang, and Bao-gen Shen, Journal of Applied Physics 87, 8666\n(2000).\n[69] E. A. Tereshina, H. Drulis, Y . Skourski, and I. S. Teresh ina,\nPhys. Rev. B 87, 214425 (2013).\n[70] Yingchang Yang, Qi Pan, Xiaodong Zhang, and Senlin Ge, J .\nAppl. Phys. 72, 2989 (1992).\n[71] Z Altounian, Xu Bo Liu, and Er Girt, J. Phys. Condens. Mat ter\n15, 33153322 (2003).\n[72] Linshu Kong, Jiabin Yao, Minghou Zhang, and Yingchang\nYang, Journal of Applied Physics 70, 6154 (1991).\n[73] O. Isnard, S. Miraglia, J.L. Soubeyroux, D. Fruchart, a nd p.\nL’Heritier, Journal of Magnetism and Magnetic Materials 137,\n151-156 (1994).\n[74] D. P. Middleton, S. R. Mishra, Gary J. Long, O. A. Pringle , Z.\nHu, W. B. Yelon, F. Grandjean, and K. H. J.Buschow, Journal\nof Applied Physics 78, 5568 (1995).\n[75] T. Pandey and David S. Parker, Sci Rep 8, 3601 (2018).\n[76] H. Luo, Z. Hu, W. B. Yelon, S. R. Mishra, G. J. Long, O. A.\nPringle, D. P. Middleton, and K. H. J. Buschow, Journal ofApplied Physics 79, 6318 (1996).\n[77] O. Isnard, S. Miraglia, D. Fruchart, j. Deportes, and P.\nL’Heritier, Journal of Magnetism and Magnetic Materials 131,\n76-82 (1994).\n[78] A.V . Andreev, D. Rafaja, J. Kamarad, Z. Arnold, Y . Homma ,\nY . Shiokawa, Journal of Alloys and Compounds 383, 4044\n(2004).\n[79] The supplementary materials contain the three dataset s for T c,\nµ0Msand Ku.\n[80] V . Psycharis, M. Anagnostou, C. Christides, and D. Niar chos,\nJournal of Applied Physics 70, 6122 (1991).\n[81] K. Ohashi, Y . Tawara, R. Osugi, and M. Shimao, Journal of\nApplied Physics 64, 5714 (1988).\n[82] Satoshi Hirosawa, Yutaka Matsuura, Hitoshi Yamamoto, Set-\nsuo Fujimura, Masato Sagawa et al,J. Appl. Phys. 59, 873\n(1986).\n[83] C. Abache and H. Oesterreicher,J. Appl. Phys. 57, 4112\n(1985).\n[84] Z. X. Tang, G. C. Hadjipanayis and V . Papaefthymiou, Jou rnal\nof Alloys and Compounds 194, 87 (1993).\n[85] Y . Z. Wang, B. P. Hu, X. L. Rao, G. C. Liu, L. Yin, W. Y . Lai,\nW. Gong, and G. C. Hadjipanayis, Journal of Applied Physics\n73, 6251 (1993).\n[86] M. Anagnostou, C. Christides and D. Niarchos, Solid Sta te\nCommunications, 78, 681 (1991).\n[87] Y . Zhang and C. Ling, npj Computational Materials 4, 25\n(2018).\n[88] Anita Halder, Aishwaryo Ghosh, and Tanusri Saha Dasgup ta,\nPhys. Rev. Materials 3, 084418 (2019).\n[89] L. Ward, A. Agrawal, A. Choudhary, and C. Wolverton, npj\nComp. Mat. 2, 16028 (2016).\n[90] Wessel N. van Wieringen, Lecture notes on ridge regress ion,\n2020, arXiv:1509.09169v5.\n[91] Vladimir V ovk, Kernel ridge regression, Empirical Inf erence,\nSpringer Berlin Heidelberg, 2013, ISBN 9783642411366.\n[92] Leo. Breiman, Random forests, Machine learning 45.1,5-32\n(2001).\n[93] Andy Liaw, and Matthew Wiener, ”Classification and regr es-\nsion by randomForest.”, R news 2.3, 18-22 (2002).\n[94] Harris Drucker, CJC Burges, Linda Kaufman, Alex J. Smol a,\nand Vladimir Vapnik, Support vector regression machines,\nAdvances in neural information processing systems, pp. 155 -\n161, 1997.\n[95] Mohamad Hassoun, Fundamentals of artificial neural net -\nworks, MIT press, 1995, ISBN 9780262514675.\n[96] Anton O. Oliynyk, Erin Antono, Taylor D. Sparks, Leila\nGhadbeigi, Michael W. Gaultois, Bryce Meredig and Arthur\nMar ,Chem. Mater. 28, 7324 (2016).\n[97] Fleur Legrain, Jes´ us Carrete, Ambroise van Roekeghem ,\nGeorg K.H. Madsen and Natalio Mingo, J. Phys. Chem. B 122,\n625 (2018).\n[98] Jess Carrete, Wu Li, and Natalio Mingo, Shidong Wang, an d\nStefano Curtarolo, Phys. Rev. X 4, 011019 (2014).\n[99] D. Andrew Carr, Mohammed Lach-hab, Shujiang Yang, Iosi f\nI. Vaisman, Estela Blaisten-Barojas, Microporous and Meso -\nporous Materials 117, 339 (2009).\n[100] Fujii, H., K. Tatami, M. Akayama and K.Yamamoto, 1992a ,\nin: Proc. 6th Int. Conf. on Ferrites, ICF6, Tokyo and Kyoto,\nJapan (The Japan Society of Powder and Powder Metallurgy,\nTokyo) p. 1081.\n[101] Fujii, H., M. Akayama, K. Nakao and K. Tatami, J. Alloys\nComp. 219, 10 (1995).\n[102] Altounian, Z., X. Chen, L.X. Liao, D.H. Ryan and J.O. St rOm-\nOlsen, J. Appl. Phys. 73, 6017 (1993).16\n[103] Chen, X., Z. Altounian and D.H. Ryan, J. Magn. Magn. Mat er.\n125, 169 (1993).\n[104] Buschow, K.H.J., T.H. Jacobs and W. Coene, IEEE Trans.\nMagn. MAG-26, 1364 (1990).\n[105] Liu, J.P., K. Bakker, ER. de Boer, T.H. Jacobs, D.B. de M ooij\nand K.H.J. Buschow, J. Less-Common Met. 170, 109 (1991).\n[106] W.G. Haije, T. H. Jacobs and K. H. J Buschow, J. Less-\nCommon Met. 163353 (1990); T. W. Capehart, R.K. Misra\nand F.E. Pickerton, App. Phys. Lett. 581395 (1991).\n[107] X. C. Kou, R. Grossinger, M. Katter, J. Wecker, L. Schul tz, T.\nH. Jacobs, K. H. J Buschow, Journal of Applied Physics, 70,\n2272(1991).\n[108] I.L.M. Locht, Y .O. Kvashnin, D.C.M. Rodrigues, M. Per eiro,\nA. Bergman, L. Bergqvist, A.I. Lichtenstein, M.I. Katsnels on,\nA. Delin, A.B. Klautau, B. Johansson, I. Di Marco, O. Eriks-\nson, Phys. Rev. B 94, 085137 (2016).\n[109] R. K. Chouhan, A. K. Pathak, D. Paudyal, V .K. Pecharsky ,\narXiv:806.01990.\n[110] B. Szpunar, Acta Phys. Pol. A 60, 791 (1981); T.-S. Zhao,\nH.-M. Jin, R. Grossinger, X.-C. Kou, and H.R. Kirchmayr, J.\nAppl. Phys. 70, 6134 (1991); I.A. Al-Omari, R. Skomski, R.A.\nThomas, D. Leslie-Pelecky, and D.J. Sellmyer, IEEE Trans.\nMagn. MAG-37, 2534 (2001).\n[111] N. Thuy, J. Franse, N. Hong, and T. Hien, J. Phys. Colloq ues\n49, 499 (1988).\n[112] P. Bruno, Phys. Rev. B 39, 865 (1989).\n[113] J. Herbst, Rev. Mod. Phys. 63, 819 (1991).\n[114] T. Miyake and H. Akai, J. Phys. Soc. Jpn 87, 041009 (2018).\n[115] Y . Hirayama, Y . Takahashi, S. Hirosawa, K. Hono, Scrip ta\nMater. 95, 70, (2015).[116] K.H.J. Buschow, Concise Encyclopedia of Magnetic\nand Superconducting Materials, Elsevier, 2005, ISBN\n9780080457659.\n[117] J.M.D. Coey, Magnetism and Magnetic Materials, 2010, ISBN\n0521816149, https://doi.org/10.1017/CBO9780511845000\narXiv:arXiv:1011.1669v3.\n[118] P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka and J .\nLuitz, WIEN2K, An Augmented Plane Wave + Loca Orbitals\nProgram for Calculating Crystal Properties (Technische Un i-\nversitt Wien, Vienna, 2001).\n[119] G. Kresse and J. Hafner, Phys. Rev. B 47, R558 (1993), G.\nKresse and J. Furthmueller, Phys. Rev. B 54, 11169 (1996).\n[120] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Let t.77,\n3865 (1996); 78, 1396(E) (1997).\n[121] A.I. Liechtenstein, V .I. Anisimov, and J. Zaanen, Phy s. Rev.\nB52, R5467 (1995).\n[122] Nico JD Nagelkerke, ”A note on a general definition of th e co-\nefficient of determination.”, Biometrika 78.3 ,691-692 (1991).\n[123] Bernhard Schlkopf, ”The kernel trick for distances”, Advances\nin neural information processing systems, 301-307 (2001).\n[124] Support Vector Regression(SVR),\nhttps://icme.hpc.msstate.edu/mediawiki/images/5/55/ SVR.pdf\n[125] I. Stephen, ”Perceptron-based learning algorithms. ”, IEEE\nTransactions on neural networks 50.2, 179 (1990).\n[126] Di Wei, Anurag Bhardwaj, and Jianing Wei, Deep Learnin g\nEssentials, Packt Publishing, 2018, ISBN 9781785880360.\n[127] Bradley Efron, and Robert J Tibshirani, An Introducti on to the\nBootstrap, CRC Press, 1994, ISBN 9780412042317."    },    {        "title": "2008.12061v1.Different_universality_classes_of_isostructural_U_TX__compounds___T____Rh__Co__Co___0_98__Ru___0_02_____X____Ga__Al_.pdf",        "content": " Different universality classes of isostructural U TX compounds ( T = Rh,  \n    Co, Co 0.98Ru 0.02; X = Ga, Al)  \nPetr Opletal, Vla dimír Sechovský and Jan Prokleška  \nCharles University, Faculty of Mathematics and Physics, Department of Condensed Matter Physics  \nKe Karlovu 5, Prague 2, Czech Republic  \nAbstract  \nMagnetization isotherms of the 5 f-electron ferromagnets URhGa, UCoGa and UCo 0.98Ru0.02Al \nwere measured at temperatures in the vicinity of their Curie temperature in order to investigate \nthe critical behavior near the ferromagnetic phase transition . These compounds adopt the layered \nhexagonal ZrNiAl -type structure and exhibit huge uniaxial magnetocrystalline anisotropy. The \ncritical β, γ and δ exponents were determined by analyzing Arrott -Noakes plots, Kouvel -Fisher \nplots, critical isotherms , scaling theory  and Widom scaling relations. The values obtained for \nURhGa and UCoGa can be explained by the results of the renormalization group theory for a 2D \nIsing system with long -range interactions similar to URhAl reported by other investigators . On \nthe other hand, the critical exponents determined for UCo 0.98Ru0.02Al are characteristic of a 3D \nIsing ferromagnet with short -range interactions suggested in previous studies also for the \nitinerant 5f -electron paramagnet UCoAl situated near a ferromagnetic  transition. The change \nfrom the 2D to the 3D Ising system is related to the gradual delocalization of 5 f electrons in the \nseries of the URhGa, URhAl, UCoGa to UCo 0.98Ru0.02Al and UCoAl compounds and appears \nclose to the strongly itinerant nonmagnetic limi t. This indicates possible new phenomena that \nmay be induced by the change of dimensionality in the vicinity of the quantum critical point.   \n  Introduction  \nCritical phenomena have been one of the most studied issues of physics since the critical \npoints we re discovered by T. Andrews [1]. The continuous (second -order) phase transition was \nfound in systems like ferromagnets [2] to be connected with unified behavior near and at the \ncritical point which can be described by critical exponents. In the case of ferromagnets three \ncritical exponents, γ, β and δ, cha racterize the behavior near and at the critical point. They can be \ndetermined from experimental data using the relations [3]: \n \n 𝑀𝑆(𝑇) ~ |𝑡|𝛽(𝑇<𝑇𝐶)                           (1) \n \n𝜒(𝑇)−1 ~ |𝑡|−𝛾′(𝑇<𝑇𝐶),|𝑡|−𝛾(𝑇>𝑇𝐶)    (2) \n \n                                                𝑀𝑠 ~ (𝜇0𝐻)1𝛿⁄𝑓𝑜𝑟𝑇 =𝑇𝐶,                        (3) \n \nwhere Ms is the spontaneous magnetization, χ is magnetic susceptibility and H is a magnetic \nfield.  \nThe universal behavior near critical points was described by the renormalization group \ntheory first mentioned by L. P. Kadanoff [4] and then fully developed by K. G. Wilson [5–7]. \nThe universality class is determined by dimensionality of the system d, dimensionality of the \norder parameter n and the range of the interaction . There exist several universality classes in \nmagnetism. The most known are 3D Ising ( n = 1), XY model ( n = 2) and 3D Heisenberg ( n = 3).  \nIt is desirable to investigat e how a certain universal critical behavior and magnetic \ndimensionality is related to the particularities of a given material (symmetry of crystal structure, \nhierarchy and anisotropy of magnetic interactions, degree of localization of “magnetic” \nelectrons,  etc.). Large groups of isostructural compounds containing transition -element ions with \none type of “magnetic” d or f electrons provide useful playgrounds for investigation of these \naspects. The UTX compounds ( T -  transition metal, X – p-metal) crystalliz ing in the hexagonal \nZrNiAl -type structure constitute such a suitable group of materials [8]. The crystal structure \nconsi sts of U -T and T-X basal plane layers alternating along the c-axis. The strong bonding of 5 f-\nelectron orbitals within the U -T layer in conjunction with strong spin -orbit interaction leads to a \nhuge uniaxial magnetocrystalline anisotropy that locks the U ma gnetic moments in the c-axis and \nthus makes these materials suitable for investigating Ising systems.     \nThe critical magnetic behavior has so far been studied on two compounds of this \nisostructural group, UCoAl [9] and URhAl [10].  UCoAl, is an itinerant 5 f electron paramagnet \nundergoing , at low temperatures , a metamagnetic transition with a critical field of ~ 0.7 T [11]. \nKarube et al. reported that it behaves near to the critical endpoint as a 3D Ising system with \nshort -range interactions [9]. On the other hand, URhAl  was reported behaving as a 2D Ising \nferromagnet with long -range interactions [10]. The observed difference of magnetic \ndimensionality of UCoAl and URhAl indicates that the common layered hexagonal crystal \nstructure and uniaxial magnetocrystalline anisotropy shared by all compounds of the U TX family \n(T – transition metal, X = Al, Ga, Sn, In) [8,11]  is not a sufficient condition for sharing also a \ncommon magnetic universality class.  To test this aspect we prepared single crystals of three hexagonal U TX  ferromagnets – \nUCoGa [11,13 –15], URhGa [16–18] and UCo 0.98Ru0.02Al (a close analog of UCo 0.99Ru0.01Al \n[19,20] ) and measured thei r magnetization isotherms in the neighborhood of their Curie \ntemperature . The measured data were analyzed by investigating Arrott -Noakes plots  [21,22] , \nKouvel -Fischer plots  [23], critical isotherms , scaling theory  [24] and Widom scaling relations \n[25] in order to determine the critical exponents and  their correspon ding universality classes. The \nfirst two compounds were found behaving as 2D Ising systems similar t o previously reported \nURhAl [10] whereas the critical exponents determined for UCo 0.98Ru0.02Al point to the 3D Ising \nuniversality class similar to UCoAl [9]. It is discussed that the change between these two classes \nof universality is associated with a changing degree of delocalization of 5f electrons . \n \n \nExperimental  \nSingle crystals of URhGa, UCoGa and UCo 0.98Ru0.02Al were prepared from stoichiometric \nmelts by Czochralski method using a triarc furnace. Each single crystal was wrapped in a \ntantalum foil and sealed in a quartz tube in evacuated to 10-6 mbar. The EDX analysis \naccomplished with a scanning electron microscope Tescan Mira I LMH equipped by a \nbackscatter electrons detector confirmed the stoichiometric composition of URhGa, UCoGa and \nUCo 0.98Ru0.02Al. The X -ray Laue method (Laue diffractometer of Photonic Science) shown  good \nquality of the crystals. Samples for magnetization w ere cut by a wire saw to a rectangular prism \nshape. The c-axis magnetization M was measured using an MPMS -7-XL (Quantum Design) in \nfields from 0.1 T to 4 T at different temperatures in the vicinity of Curie temperature.  \n The values of internal magnetic field H were calculated as:  \n \nH = H a - NM              (4), \n \nwhere N is the demagnetization factor. The demagnetization factor was evaluated using the \nformula published by Aharoni [26].   \n \nResults and Discussion   \n \nURhGa and UCoGa  0 2 4 6 8 10 12 14 160246810\n  M1/ 1011 (A/m)1/ \n(H/M)1/T = 0.4 KTC= 41.01 \n  0.02  K\n = 0.40 \n  0.01\n = 1.18 \n  0.01\nT= 39 K\nT= 43 K \n0 5 10 15 20024681012141618\nT = 47.8 KM1/ 1012 (A/m)1/ \n(H/M)1/T = 44.6 KT = 0.4 KTC= 46.29 \n  0.03 K\n= 1.26 \n  0.01\n = 0.37 \n  0.01  \nFig.  1: Modified Arrott plots for URhGa (left) and UCoGa (right). Resulting critical exponents and Curie temperature \ncorresponding to each compound are in the figures. Magnetic curves close to Curie temperature were used for fit of Arrott -\nNoakes equation.  \nThe C urie temperature of a ferromagnet is usually determined by Arrott plot ( M2 vs H/M \nplot) analysis of magnetization isotherms [21] measured at temperatures in the critical region.     \nThe linear Arrott plots are, in fact, a  graphical representation of the Landau equation of state in \nthe theory  of second -order phase transition  [27–29], which has been later specified for \nferromagnets by Ginsburg  [30]. This approach is certainly suitable for investigation of \nhomogenous isotropic ferromagnets. This condition is not met for most real systems, whi ch is \ndocumented by considerable curvature in  Arrott plots. Th e value of Curie temperature TC can be \nthe refined by finding the  and  coefficients for which modified Arrott plots (Arrott -Noakes \nplots)   M1/β vs. H/M1/γ are linear. The Arrott -Noakes plot comes from the analysis of \nmagnetization curves based on the Arrott -Noakes equation [22]: \n \n(𝐻𝑀⁄)1𝛾⁄=(𝑇−𝑇𝐶)\n𝑇1+(𝑀𝑀1⁄ )1𝛽⁄            (5), \n \nwhere M1 and T1 are material constants that are temperature independent in the vicinity of the \nphase transition. Magnetization curves are then fitted by (5) to get the M1/β vs. H/M1/γ plots linear \nand parallel by varying free parameters β and γ while  keeping the T1 and M1 values fixed.  The \nresults for our studied compounds are presented in Fig. 1 and Table 1 together with the best \nvalues of TC and critical exponents β, γ.  \n 0.1 14681012\nT = 43 K\n  M 104 (A/m)\n0H (T)T = 39 KT = 0.4 K  = 3.89 \n  0.01\n0.1 12345678\nT = 0.4 K\nT = 47.8 K M 105 (A/m)\n0H (T)T = 45 K= 4.32 \n  0.02 \nFig.  2: Logarithmic plot of magnetic isotherms of URhGa (left) and UCoGa (right). Critical isotherms for each compound are \nhighlighted by dashed line. The critical exponent δ from fit to critical exponent for each compound are part of the figure.  \nThe critical exponent δ can be calculated from β and γ from  the Widom scaling law [25]. It \ncan be determined from the magnetization curve using Eq. (3), too. In Fig. 2 we display the \nmagnetization curves of both compounds in a log -log plot because only the isotherm at TC is \nlinear in this represen tation, as seen from Eq. (3). A linear function is then fitted to data points \nforming the straightest isotherm.  The resulting δ-values  are shown in Fig. 2 and Table 1. For \nURhGa , UCoGa the δ-value  calculated from the Widom scaling law [25] is 3.95 , 4.4, which is \nclose enough to the value of 3.89 (4.5) shown in Fig. 2. The agreement between values from the \nWidom scaling law and values from cri tical isotherms could be improved as we did not know the \nexact TC during measurement, and therefore we used isotherms at temperature closest to  the \nassumed TC. \nThe β- and γ-values can be  further  refined by analyzing  Kouvel -Fisher plots [23] which are \nbased on the definition of critical exponents in Eqs. (1) and (2). The spontaneous magnetization \nMS is determined from the intersection of the M1/β axis of a Arrott -Noakes plot with straight lines \nand the inverse susceptibility χ-1 is determ ined from the intersections of straight lines with the \n(H/M)1/γ axis of the Arrott -Noakes plot. Kouvel and Fisher  [23] showed  that by dividing by \ntemperature derivatives of Eqs. (3) and (4) one gets a new set of equations  \n \n𝑀𝑆(𝑇)[𝑑𝑀𝑆(𝑇)𝑑𝑇⁄ ]−1=|𝑇−𝑇𝐶|/𝛽(𝑇)   (6) \n \n𝜒−1(𝑇)[𝑑𝜒−1(𝑇)𝑑𝑇⁄ ]−1=|𝑇−𝑇𝐶|/𝛾(𝑇).  (7) \n \nIn the vicinity of TC the β(T) and γ(T) became equal to the corresponding critical β and γ \nvalues. The critical exponents are determined from the slope and the Curie temperature is \ndetermined from the intersection with  the T axis in the Kouvel -Fisher plots shown in Fig. 3. The \nresulting critical exponents for URhGa and UCoGa are presented in Fig. 3 and Table 1.  39 40 41 42 43012345\nT +\nC = 40.90 \n  0.01\n = 1.21 \n  0.01\n fit to MS\n fit -1\nT (K)MS(dMS/dT)-1 (K)T -\nC = 40.92 \n  0.06\n = 0.42 \n  0.02\n0.00.51.01.52.0\n -1 (d-1/dT)-1 (K)\n44.5 45.0 45.5 46.0 46.5 47.0 47.5 48.012345 \nT (K)MS/(dMS/dT) (K)T +\nC= 46.33 \n  0.05 K\nT -\nC= 46.43 \n  0.06 K\nTC= 46.38 \n  0.06 K\n = 1.29 \n  0.07\n = 0.40 \n  0.02\n0.00.20.40.60.81.01.21.4\n -1/(d-1/dT) (K) \nFig.  3: Kouvel -Fisher plot for spontaneous magnetization and susceptibility for URhGa (left) and UCoGa (right). The critical \nexponents are shown in corresponding figures.  \nTo check whether the critical exponents above and below the phase transition ar e equal, we \nhave separately determined the critical exponent γ (T > TC) and γ’ (T < TC) using the scaling \ntheory that predicts a reduced equation of state close to the phase transition [24]: \n \n𝑀(𝜇0𝐻,𝑡)=|𝑡|𝛽𝑓±(𝜇0𝐻|𝑡|𝛽+𝛾⁄ ),  (8) \n \nwhere f+ is for T > TC and f- is for T < TC and both are regular analytical functions. If the correct \nβ, γ and TC values are chosen then the data points in the plot M(µ0H,t)/|t|β versus µ0H/|t|β+γ should \nfall on two universal curves, one for T < TC and second for T > TC and these curves should \napproach each other asymptotically.  Magnetization data for both compounds were fitted by Eq. \n(8). The results are shown in Fig. 4 together with the values of critical exponents and TC, \nrespectively. For both compounds, the difference of the determined critical exponent γ for T \nabove and below TC is very small which corroborates the presumption that it does not change.  \n \n10 100 1000 10000110 M/|t| 105 (A/m)\n0H/|t|(+) (T) T<TC\n T>TCTC= 40.85 K\n = 0.39 \n  0.01\n+=1.18 \n  0.01\n-=1.19 \n  0.02\n0.01 0.1 1 10 100110\n T<TC\n T>TC\n M/|t| 105 (A/m)\n0H/|t|(+) (T)TC= 46.23 \n  0.04\n= 0.37 \n  0.01\n+= 1.28 \n  0.03\n-= 1.26 \n  0.02\n \nFig.  4: The scaled magnetization plotted against the renormalized magnetic field below and above T C. Results of fit of scaling \ntheory from Eq. (8) for both compounds are shown in figures URhGa (left) and UCo Ga (right).  \nThe critical exponents of URhGa and UCoGa can be explained similarly to the URhAl case \n[10] by introducing a weak long -range magnetic exchange interaction in form J(r) ~ r-(d+σ), where \nσ is the range of  the exchange interaction [31]. Fischer et al. [31] applied the renor malization group theory for a system with an interaction J(r) for which  the equation for γ has been obtained \nin the form:  \n \n𝛾=1+4\n𝑑(𝑛+2\n𝑛+8)∆𝜎+8(𝑛+2)(𝑛−4)\n𝑑2(𝑛+8)2[1+2𝐺(𝑑\n2)(7𝑛+20)\n(𝑛−4)(𝑛+8)]∆𝜎2,  (9) \n \nwhere ∆𝜎=(𝜎−𝑑\n2) and 𝐺(𝑑\n2)=3−1\n4(𝑑\n2)2\n. The obtained values of γ were examined using Eq. \n(9) by substituting the possible values  of d (dimension of the system ) = 1, 2, or 3,  and n \n(dimension of the order parameter ) = 1, 2, or 3  in all possible combinations and comparing the \nresulting values of σ for γ and β.  The best agreement for both, URhGa and UCoGa, has been \nfound for a 2D Ising system with LR interactions. The resulting σ value and corresponding β, γ \nand δ values are listed in Table 1. The same universality c lass was reported for URhAl [10]. \n \nUCo 0.98Ru0.02Al \nThe critical exponents γ = (1.27 ± 0.01) and β = (0.32 ± 0.01) determined by Arrott -Noakes \nanalysis of magnetization data collected on the UCo 0.98Ru0.02Al single crystal (see Arrott -Noakes \nplots in Fig. 5) with TC = 22.79 K compare well to the theoretical values γ = 1.241 and β = 0.325 \nfor the 3D Isin g model  [24,32] . \n \n0 5 10 15 20 25012345678\nT = 0.1 K T = 24.5 K \n  M1/ 1014 (A/m)1/\n(H/M)1/T = 22 K  = 1.27 \n  0.01\n = 0.32 \n  0.01\nTC = 22.79 \n  0.03 K\n0.1 13456\nT = 0.1 KT = 23.6 KTC = 22.8 K\n = 4.92 \n  0.01M 105 (A/m)\n0H (T)T = 22 K\n \nFig.  5: Arrott -Noakes plots (left) and logarithmic plots (right) of magnetization isotherms for UCo 0.98Ru0.02Al. The results of the \nfit by Arrott -Noakes equation (Eq. ( 5)) are displayed in left figure while in right figure the result of the fit by critical isotherm can \nbe found.  \nFurther on, the value δ = (4.92 ± 0.01) corresponding to the critical isotherm (see Fig. 5) is \nclose to the value of 4.97 determined from Widom scaling relations using γ and β determined \nfrom Arrott -Noakes equation. The value corresponding to the critical isotherm is close to th e \nvalue of 4.82 known for the 3D Ising model.  \nTo further analyze the UCo 0.98Ru0.02Al magnetization data , the Kouvel -Fisher method (for \nthe respective plot see Fig. 6) was used in a similar way as above. The resulting critical \nexponents γ = (1.24 ± 0.04) and β = (0.33 ± 0.01) with TC = (22.79 ± 0.02) compare to values for \nthe 3D Ising model.  22.2 22.4 22.6 22.8 23.0 23.2 23.40.00.30.60.91.21.51.8\n = 1.24\n 0.04\nT +\nC = 22.79\n 0.01 \n fit to MS\n fit to -1\nT (K)MS/(dMS/dT) (K)\n = 0.33\n 0.01\nT -\nC = 22.79\n 0.02\n0.00.10.20.30.40.5\n -1/(d-1/dT) (K)\n1 10 100 100012345 M/t 105 (A/m)\nH/t(+) 107 (A/m) T<TC\n T>TCTC = 22.74 K\n = 1.241\n = 0.325 \nFig.  6: (Left)  Kouvel -Fisher plot of UCo 0.98Ru0.02Al  with results of fit  of Eqs. (6) and (7). (Right) Plot of the scaled magnetization \nvs. the rescaled magnetic field for uCo0.98Ru0.02Al. Results of fit of scaling theory from Eq. (8) are part of the figure.  \nWe have confirmed that UCo 0.98Ru0.02Al behaves as a 3D Ising system by plotting \nmagnetization data using the reduced equation of state in eq. 8 with values of β and γ for the 3D \nIsing model in Fig. 6. The data points fall on two different curves, for T below and above TC = \n22.74 K, respectivel y, which approach asymptotically to merge. The 3D Ising behavior \nUCo 0.98Ru0.02Al resembles behavior of pure UCoAl reported by Karube et al. in [9]. \nThe values  of critical exponents and other relevant information obtained in the present \nwork on URhGa, UCoGa and UCo 0.98Ru0.02Al single crystals are displayed in Table 1. The \ninformation available for UCoAl [9] is included for comparison and further discussion.  \n \nTable I: Critical exponents and other relevant parameters obtained for  URhGa, UCoGa and \nUCo 0.98Ru0.02Al  by the analysis of Arrott -Noakes plots (A -N plots), Kouvel -Fisher plots (K -F \nplots), scaling relations (Scaling), critical isotherms (Critical i.)and data for UCoAl [9] for \ncomparison.  \n  Method  TC (K)  γ'(T < TC) γ(T > TC) δ σ \nURhGa  A-N plots  41.01 ± 0.02  0.40 ± 0.01  1.18 ± 0.01    \n K-F plots  40.91 ± 0.04  0.42 ± 0.02  1.21 ± 0.01    \n Scaling  40.85 ± 0.02  0.39 ± 0.01  1.18 ± 0.01 1.19 ± 0.02    \n Critical i.     3.89 ± 0.01   \nLR exchange: \nJ(r) ~ r-(d+σ) \nd = 2, n = 1    0.39  1.18  4.03  1.21 \nUCoGa  A-N plots  46.29 ± 0.03  0.37 ± 0.01  1.26 ± 0.01    \n K-F plot  46.38 ± 0.06  0.40 ± 0.02  1.29 ± 0.07    \n Scaling  46.23 ± 0.04  0.37 ± 0.01  1.28 ± 0.03 1.26 ± 0.02    \n Critical i.     4.32 ± 0.01   \nLR exchange: \nJ(r) ~ r-(d+σ) \nd = 2, n = 1   0.36  1.26  4.5 1.28  \nThe table presents the most striking result of our study - although the three studied \nUTX ferromagnets adopt the same type of layered hexagonal crystal structure with strong \nuniaxial magnetocrystalline anisotropy they do not fall in the same universality class. URhGa  and \nUCoGa exhibit  a 2D Ising character similar to URhAl [10] whereas UCo 0.98Ru0.02Al behaves as \nta 3D Ising system , also  reported for UCoAl [9].  \n The 5f -electrons in U intermetallics are known to have a dual character (partially localized, \npartially itinerant)  [33–35]. The localized and itinerant character s appear in different proportions \ndepending on crystallographic and chemical environments of U ion being reflected in a wide \nrange of their magnetic behavior. This is a consequence of the wide exte nsion of  the 5f-wave \nfunctions which allow s for considerable direct overlaps between  5f-wave functions of  the \nnearest -neighbor U ions as well as  hybridization of valence electron states of ligands (5f -ligand \nhybridization). As a result, the original atomic  character of the 5f -wave functions is destroyed \nwhile  the related magnetic moments is washed out and adequately reduced. In the strong 5f -5f \noverlap and 5f -ligand hybridization limits the 5f -electrons are predominantly itinerant, the 5f \nmagnetic moments vanish and the magnetic order is lost (UCoAl in our case).  \nRhodes and Wohlfarth propose d that the ratio µeff /µs between the effective and spontaneous \nmagnetic moment can be taken as a measure of the degree of itinerancy of the magnetic electrons  \n[36,37] . In Table 2, we can see that µeff /µs increases along the series as listed from top to \nbottom. In the Rhodes -Wohlfarth scenario, the degree of itinerancy of 5f -electrons increases \nwhen proceedi ng from URhGa towards UCoAl.  \nWe can also see that the lattice parameter a shown in the same table simultaneously \ndecreases along the series.  The close packing of U and T atoms in the basal plane of the  \nhexagonal ZrNiAl -type structure results both in a non -negligible 5f -5f overlap and a strong 5f -d \nhybridization involving the transition metal d states which compress most of the 5f charge \ndensity towards the basal plane. The reduction of  a is intimately co nnected with the decreasing \nU-U and U -T interatomic distances within the basal plane. The simultaneously enhanced 5f -5f \noverlaps and 5f -d hybridization causes a higher degree of itinerancy, which corroborates the \nRhodes -Wohlfarth scenario.  \nThe change from 2D to three dimensional behavior was described by  Takahashi , who \nconsiders quasi - itinerant ferromagnets  in [38]. In a quasi -2D system  the spin fluctuation in  the z-\ndirection differ s from spin fluctuations in  the xy plane , which  comes from  the uniaxial \nmagnetocrystalline anisotropy of the system. The critical behavior of the system in the  region of  \nthe T-H space centered  on [TC, 0] is described as 3D, whil e outside of this  region  the critical \nbehavior is 2D. For stronger  anisotropy , the region  of 3D behavior is reduced . The l arge \nmagneto crystalline anisotropy observed in U TX compounds crystallizing in the ZrNiAl -type \nstructure  [8,11,15,39]  leads to suppression of  a 3D behavior and  instead  a 2D behavior is UCo 0.98Ru0.02Al A-N plots  22.79 ± 0.03  0.32 ± 0.01  1.27 ± 0.01    \n K-F plots  22.79 ± 0.02  0.33 ± 0.01  1.24 ± 0.04    \n Scaling  22.74 ± 0.02  0.325  1.241    \n Critical i.     4.92 ± 0.01   \nUCoAl  NMR \nmeas.  - 0.26 1.2 5.4  observed as seen  in URhGa, UCoGa and URhAl.  It is important to emphasiz e that the system s \ndescribed by Takahashi show different critical behavior s, but the effect of anisotropy on  the \nnature of  fluctuations near TC is expected to  be similar.  \n \nTable 2: The values of effective and spontaneous magnetic moment,  µeff and µs, respectively, and \nthe µeff /µs ratio and lattice parameters for our studied URhGa, UCoGa and UCo 0.98Ru0.02Al \ncompounds completed by the values for URhAl and UCoAl. The “ µs value” for UCoAl is the \nmagnetic moment in the field just above the metamagnetic transition. The true µs value for \nUCoAl is indeed equal to 0; the ground state is paramagnetic.  \nCompound  µeff \n(µB/f.u.)  µs \n(µB/f.u.)  µeff \n/µs Ref. Universality  \nclass  a \n(pm)  c \n(pm)  Ref. \nURhGa  2.45 1.17 2.11 [40] 2D Ising  700.6  394.5  [41] \nURhAl  2.50 1.05 2.38 [10] 2D Ising  696.5  401.9  [41] \nUCoGa  2.40 0.65 3.69 [15] 2D Ising  669.3  393.3  [41] \nUCo 0.98Ru0.02Al 1.73 0.36 4.81 * 3D Ising  669.1 396.6  * \nUCoAl  1.60 0.30 5.33 [11] 3D Ising  668.6  396.6  [42] \n*) unpublished data   \n \nFurther inspection of Table 2 reveals that the change from 2D Ising to 3D Ising universality \nclass happens when the 5f -electrons become considerably itinerant.  This agrees with  Takahashi’s \ndescription  in [38], in which for more itinerant system s larger anisotropy is nee ded to suppress  \nthe region of  3D behavior compared to more localized systems . This explains the 3D behavior of \nUCoAl and UCo 0.98Ru0.02Al even though the magnetocrystalline anisotropy is comp arable to that  \nin URhGa, UCoGa and URhAl  [8,11,15,39] . Also, we see that the p -metal affects the degree of \nlocalization/itineracy, which in turn affects the universality class of the system. The hierarchy of \nexchange interactions will undoubtedly play  an essential role in controlling dimensionality. The \ninvolvement of theorists in resolving these issues is strongly desirable. It would also be \ninteresting to investigate the evolution of critical exponents and magnetic dimensionality \ntogether with the de velopment of lattice parameters while applying  hydrostatic pressure. A \npossible change of magnetic dimensionality with applied pressure near to a quantum critical \npoint may open new questions in the quantum criticality research.  \n \nConclusions  \nThe magnetization isotherms of their URhGa, UCoGa and UCo 0.98Ru0.02Al were measured at \ntemperatures near to  their ferromagnetic transition s where  we investigated the critical behavior. \nThe values of critical exponents β, γ and δ have been determined by ana lyzing magnetization \ndata presented in Arrott -Noakes plots, Kouvel -Fisher plots, critical isotherms , scaling theory  and \nWidom scaling relations. The results point to the 2D Ising uni versality class for URhGa and \nUCoGa, similar to URhA l reported by other investigators  [10]. Data obtained for \nUCo 0.98Ru0.02Al are c haracteristic of the  3D Ising  universality class  which was suggested in \nprevious studies also for the itinerant 5f -electron paramagnet UCoAl. At the high degree of \nitinerancy of 5f -electrons a change from the 2D to the 3D character is observed  between  UCoGa \nand UCo 0.98Ru0.02Al. Possible new phenomena may be expected when a dimensionality change \nhappens in the vicinity of the quantum critical point.   \nAcknowledgments  \n This research was supported by Grant Agency of Charles University, grant No.  1630218  and by \nthe Czech Science Foundation, grant No.16 -06422S. Experiments were performed in MGML \n(www.mgml.eu), which is supported within the program of Czech Research Infrastructures \n(project no. LM2018096). It was also supported by OP VVV project MATFUN unde r Grant \nCZ.02.1.01/0.0/0.0/15_003/0000487.  We would also like to thank to Dr.  \nRoss Colman for proofreading of the text, and language corrections.  \n \nReferences  \n[1] T. Andrews, XVIII. The Bakerian Lecture . —On the continuity of the gaseous and liquid \nstates of matter, Philos. Trans. R. Soc. London. 159 (1869) 575 –590. \n[2] J. Hopkinson, I. Magnetic properties of alloys of Nickel and iron, Proc. R. Soc. London. \n48 (1891) 1 –13. \n[3] M.E. Fisher, The theory of equilibriu m critical phenomena, Reports Prog. Phys. 30 (1967) \n306. \n[4] L.P. Kadanoff,  Scaling laws for ising models near    T   c    , Phys. Phys. Fiz. Phys. \nФизика. 2 (1966) 263 –272. \n[5] K.G. Wilson, Renormalization group and critical phenomena. I. Renormalization  group \nand the Kadanoff scaling picture, Phys. Rev. B. 4 (1971) 3174 –3183.  \n[6] K.G. Wilson, Renormalization group and critical phenomena. II. Phase -space cell analysis \nof critical behavior, Phys. Rev. B. 4 (1971) 3184 –3205.  \n[7] K.G. Wilson, The renormaliza tion group: Critical phenomena and the Kondo problem, \nRev. Mod. Phys. 47 (1975) 773 –840. \n[8] V. Sechovsky, L. Havela, Chapter 1 Magnetism of ternary intermetallic compounds of \nuranium, in: K.H.J. Buschow (Ed.), Handb. Magn. Mater., Elsevier, 1998: pp. 1 –289. \n[9] K. Karube, T. Hattori, S. Kitagawa, K. Ishida, N. Kimura, T. Komatsubara, Universality \nand critical behavior at the critical endpoint in the itinerant -electron metamagnet UCoAl, \nPhys. Rev. B. 86 (2012).  \n[10] N. Tateiwa, J. Pospíšil, Y. Haga, E. Yama moto, Critical behavior of magnetization in \nURhAl: Quasi -two-dimensional Ising system with long -range interactions, Phys. Rev. B. \n97 (2018) 64423.  \n[11] V. Sechovsky, L. Havela, F.R. de Boer, J.J.M. Franse, P.A. Veenhuizen, J. Sebek, J. \nStehno, A. V Andreev , Systematics across the UTX series (T = Ru, Co, Ni; X = Al, Ga, \nSn) of high -field and low -temperature properties of non -ferromagnetic compounds, Phys. \nB+C. 142 (1986) 283 –293. \n[12] N. Tateiwa, J. Pospíšil, Y. Haga, H. Sakai, T.D. Matsuda, E. Yamamoto, Itinerant \nferromagnetism in actinide 5f -electron systems: Phenomenological analysis with spin \nfluctuation theory, Phys. Rev. B. 96 (2017) 35125.  \n[13] A. V Andreev, A. V Deryagin, R.Y. Yumaguzhin, Crystal structure and magnetic \nproperties of UGaCo and UGaNi  single crystals, Sov. Phys. - JETP. 59 (1984) 1082 –1086.  \n[14] P. Opletal, P. Proschek, B. Vondráčková, D. Aurélio, V. Sechovský, J. Prokleška, Effect \nof thermal history on magnetism in UCoGa, J. Magn. Magn. Mater. 490 (2019) 165464.  \n[15] H. Nakotte, F.R. de Boer, L. Havela, P. Svoboda, V. Sechovsky, Y. Kergadallan, J.C. \nSpirlet, J. Rebizant, Magnetic anisotropy of UCoGa, J. Appl. Phys. 73 (1993) 6554 –6556.  \n[16] V. Sechovsky, F. Honda, K. Prokes, O. Syshchenko, A. V Andreev, J. Kamarad, Pressure -induced phe nomena in U intermetallics, Acta Phys. Pol. B. 34 (2003) 1377 –1386.  \n[17] V. Sechovský, L. Havela, N. Pillmayr, G. Hilscher, A. V Andreev, On the magnetic \nbehaviour of UGaT series, J. Magn. Magn. Mater. 63 –64 (1987) 199 –201. \n[18] V. Sechovsky, L. Havela, F. R. de Boer, P.A. Veenhuizen, K. Sugiyama, T. Kuroda, E. \nSugiura, M. Ono, M. Date, A. Yamagishi, Hybridization and magnetism in U(Ru, Rh)X, \nX=Al, Ga, Phys. B Condens. Matter. 177 (1992) 164 –168. \n[19] A. V Andreev, L. Havela, V. Sechovsky, M.I. Bartashevich,  J. Sebek, R. V Dremov, I.K. \nKozlovskaya, Ferromagnetism in the UCo1 -xRuxAl quasiternary intermetallics, Philos. \nMag. B -Physics Condens. Matter Stat. Mech. Electron. Opt. Magn. Prop. 75 (1997) 827 –\n844. \n[20] P. Opletal, J. Prokleška, J. Valenta, P. Proschek , V. Tkáč, R. Tarasenko, M. Běhounková, \nŠ. Matoušková, M.M. Abd -Elmeguid, V. Sechovský, Quantum ferromagnet in the \nproximity of the tricritical point, Npj Quantum Mater. 2 (2017).  \n[21] A. Arrott, Criterion for ferromagnetism from observations of magnetic i sotherms, Phys. \nRev. 108 (1957) 1394 –1396.  \n[22] A. Arrott, J.E. Noakes, Approximate equation of state for nickel near its critical \ntemperature, Phys. Rev. Lett. 19 (1967) 786 –789. \n[23] J.S. Kouvel, M.E. Fisher, Detailed magnetic behavior of nickel near its  curie point, Phys. \nRev. 136 (1964).  \n[24] V. Privman, P.C. Hohenberg, A. Aharony, Universal Critical -Point Ampl itude Relations, \nin: C. Domb, J.L. Lebowitz (Eds.), Phase Transitions Crit. Phenom. Vol. 14, ACADEMIC \nPRESS LIMITED, London, 1991: p. 367.  \n[25] B. Widom, Equation of State in the Neighborhood of the Critical Point, J. Chem. Phys. 43 \n(1965) 3898 –3905.  \n[26] A. Aharoni, Demagnetizing factors for rectangular ferromagnetic prisms, J. Appl. Phys. \n83 (1998) 3432 –3434.  \n[27] L.D. Landau, On the theory of phase transitions. I., Zh. Eksp. Teor. Fiz. 7 (1937) 19.  \n[28] L.D. Landau, On the theory of phase transitions. I., Phys. Z. Sowjet. 11 (1937) 26.  \n[29] D. TER HAAR, On the Theory of Phase Transitions, in: Men Phys. L.D. Landau, \nElsevier, 1969: pp. 61 –84. \n[30] V.L. Ginzburg, Behavior of ferromagnetic substances in the vicinity of the curie point, Zh. \nEksp. Teor. Fiz. 17 (1947) 833.  \n[31] M.E. Fisher, S.K. Ma, B.G. Nickel, Critical exponents for long -range interactions, Phys. \nRev. Lett. 29 (1972) 917 –920. \n[32] J.C. Le Guillou, J. Zinn -Justin, Critical exponents from field theory, Phys. Rev. B. 21 \n(1980) 3976 –3998.  \n[33] T. Takahashi, N. Sato, T. Yokoya, A. Chainani, T. Morimoto, T. Komatsubara, Dual \nCharacter of 5f Electrons in UPd2Al3 Observed by High -Resolution  Photoemission \nSpectroscopy, J. Phys. Soc. Japan. 65 (1996) 156 –159. \n[34] G. Zwicknagl, 5f Electron correlations and core level photoelectron spectra of Uranium \ncompounds, Phys. Status Solidi. 250 (2013) 634 –637. \n[35] G. Zwicknagl, 3 = 2+1: Partial localiz ation, the dual character of 5f electrons and heavy \nfermions in U compounds, in: J. Magn. Magn. Mater., North -Holland, 2004: pp. E119 –\nE120.  \n[36] P. Rhodes, E.P. Wohlfarth, The effective Curie -Weiss constant of ferromagnetic metals \nand alloys, Proc. R. Soc.  Lond. A. 273 (1963) 247 –258. [37] E.P. Wohlfarth, Magnetic properties of crystallineand amorphous alloys: A systematic \ndiscussion based on the Rhodes -Wohlfarth plot, J. Magn. Magn. Mater. 7 (1978) 113 –120. \n[38] Y. Takahashi, Spin -fluctuation theory of quasi -two-dimensional itinerant -electron \nferromagnets, J. Phys. Condens. Matter. 9 (1997) 10359 –10372.  \n[39] P.A. Veenhuizen, F.R. de Boer, A.A. Menovsky, V. Sechovsky, L. Havela, MAGNETIC -\nPROPERTIES OF URuAl and URhAl SINGLE -CRYSTALS, J. Phys. 49 (1988) 48 5–\n486. \n[40] P. Opletal, Peculiarities of magnetism on the verge of ferromagnetic ordering, Faculty of \nMathematics and Physics, Charles University, 2019.  \n[41] A.E. Dwight, Alloy Chemistry of Thorium, Uranium, and Plutonium Compounds, in: Dev. \nStruct. Chem. Alloy Phases, Springer US, 1969: pp. 181 –226. \n[42] D.J. Lam, J.B. Darby, J.W. Downey, L.J. Norton, Equiatomic ternary compounds of \nuranium and aluminium with group viii transition elements, J. Nucl. Mater. 22 (1967) 22 –\n27. \n "    },    {        "title": "2009.00410v1.Lattice_dynamics_effects_on_the_magnetocrystalline_anisotropy_energy__application_to_MnBi.pdf",        "content": "Lattice dynamics effects on the magnetocrystalline anisotropy energy: application to\nMnBi\nAndrea Urru1and Andrea Dal Corso1,2\n1International School for Advanced Studies (SISSA),\nVia Bonomea 265, 34136 Trieste (Italy).\n2DEMOCRITOS IOM-CNR Trieste (Italy).\n(Dated: September 2, 2020)\nUsing a first-principles fully relativistic scheme based on ultrasoft pseudopotentials and density\nfunctional perturbation theory, we study the magnetocrystalline anisotropy free energy of the fer-\nromagnetic binary compound MnBi. We find that differences in the phonon dispersions due to the\ndifferent orientations of the magnetization (in-plane and perpendicular to the plane) give a differ-\nence between the vibrational free energies of the high-temperature and low-temperature phases.\nThis vibrational contribution to the magnetocrystalline anisotropy energy (MAE) constant, Ku, is\nnon-negligible. When the energy contribution to the MAE is calculated by the PBEsol exchange\nand correlation functional, the addition of the phonon contribution allows to get a T= 0KKuand\na spin-reorientation transition temperature in reasonable agreement with experiments.\nI. INTRODUCTION\nRecently, there has been a significant effort toward the\nrealization of rare-earth-free permanent magnets1,2. Due\nto its magnetic properties, such as a high Curie tempera-\nture, well above room temperature, and a large uniaxial\nmagnetic anisotropy, MnBi2–5has emerged as a promis-\ning candidate among the transition-metal-based materi-\nals.\nBelow the Curie temperature, estimated to be Tc=\n680K6,7, MnBi is a ferromagnet which crystallizes in\nthe NiAs structure. Its magnetocrystalline anisotropy\nenergy (MAE) as a function of temperature is peculiar:\natT= 0KtheMAEconstant Kuisnegative,itsreported\nexperimental value being \u00000:2MJ / m3(\u0019\u00000:12meV /\ncell) with an easy axis in the basal plane8, and increasing\nwithT, unlike most magnetic systems4,8. AtT\u001990K\nKubecomes positive, thus leading to a spin-reorientation\ntransition: from 90K to 140K, the easy axis rotates\noutside the basal plane, and above 140K it is parallel to\nthe c axis9.\nSeveral studies, both experimental and theoretical,\nhave been carried out during the past years to under-\nstand this intriguing property. Experiments studied sev-\neralpropertiesofMnBi, includingthethermalexpansion.\nIn particular, the spin-reorientation transition comes to-\ngether with a small kink in the lattice parameters at\nT\u001990K10,11, which has been interpreted as the sign\nof a phase transition. Theoretical calculations, based on\nDensity Functional Theory (DFT) within the Local Den-\nsity Approximation (LDA) and the Generalized Gradient\nApproximation(GGA)fortheexchange-correlationfunc-\ntional, correctly predict MnBi to be a metal and a ferro-\nmagnet in the low-temperature phase, and to have a neg-\nativeKu, in agreement with experiments. Yet, they are\nbelieved to overestimate the magnitude of Kuby nearly\nan order of magnitude and are often not able to repro-\nduce the correct behavior of Kuas a function of tem-\nperature. Refs. 12 and 13 showed that the treatment ofcorrelation effects by means of the DFT+U approach is\nimportant to get the correct behavior of Kuas a function\nof temperature. In particular, in Ref. 13 the inclusion\nof the thermal expansion effects on Kuallowed to get a\nspin-reorientation temperature in agreement with exper-\niments and a theoretical MAE in good agreement with\nexperimental results, especially in the temperature range\n150-450 K.\nMore recently, in Ref. 14 it was suggested that the\nspin-reorientation phenomenon might be partially due to\nlattice dynamics. Such statement was supported by the\ncalculation of the lattice dynamics contribution to Ku,\nobtained by averaging the MAE over configurations in\nwhich the Mn and Bi atoms were displaced according to\nthe mean square atomic displacements as a function of\ntemperature.\nRecently, we extended Density Functional Perturba-\ntion Theory (DFPT) for lattice dynamics with Fully Rel-\nativistic (FR) Ultrasoft pseudopotentials (US-PPs) to\nmagnetic materials15. The new formulation allows to de-\ntect differences in the phonon frequencies for different\norientations of the magnetization, thus making possible\nto evaluate the vibrational free energy contribution to\nthe MAE.\nIn this paper we study, by means of ab-initio tech-\nniques, the lattice dynamics of ferromagnetic MnBi for\ntwo different orientations of the magnetization: 1. in–\nplane; 2.perpendiculartotheplane. Wefindthatthetwo\nphonon dispersions mainly differ in the high-frequency\noptical branches, where the phase with magnetic mo-\nments pointing in the out-of-plane direction shows, on\naverage, phonon modes of 2 cm\u00001lower in frequency.\nStarting from the difference of the vibrational density\nof states of the two phases we compute the vibrational\ncontribution to MAE. We find that, if the energy con-\ntribution to MAE is computed by the PBEsol exchange-\ncorrelation functional, the phonon contribution is of the\nsameorderofmagnitudeasthegroundstateMAE,hence\nit plays a relevant role in the calculation of Kuand toarXiv:2009.00410v1  [cond-mat.mtrl-sci]  1 Sep 20202\ndetermine the spin-reorientation transition temperature\nTSR.\nII. METHODS\nFirst-principle calculations were carried out by means\nof DFT16,17within the LDA18and the Perdew-Burke-\nErnzerhof optimized for solids (PBEsol)19schemes\nfor the exchange-correlation functional approximation,\nas implemented in the Quantum ESPRESSO20–22and\nthermo_pw23packages. The atoms are described by\nFR US-PPs24, with 3p,4s, and 3delectrons for Mn\n(PPs Mn.rel-pz-spn-rrkjus_psl.0.3.1.UPF and\nMn.rel-pbesol-spn-rrkjus_psl.0.3.1.UPF , from\npslibrary 0.3.125,26) and with 6s,5d, and 6pelectrons\nfor Bi (PPs Bi.rel-pz-dn-rrkjus_psl.1.0.0.UPF\nand Bi.rel-pbesol-dn-rrkjus_psl.1.0.0.UPF , from\npslibrary 1.0.025,26).\nMnBicrystallizesintheNiAsstructure, withanhexag-\nonal lattice described by the point group D6h. The inclu-\nsionofmagnetismdifferentiatesthestructuresintoalow-\nsymmetry phase ( m?chenceforth), below TSR, and a\nhigh-symmetry phase ( mkchenceforth), above TSR.\nIn particular, the mkcphase is described by the mag-\nnetic point group D6h(C6h), compatible with an hexago-\nnalBravaislattice, whilethe m?cphasehasamagnetic\npoint group D2h(C2h), compatible with a base-centered\northorhombic Bravais lattice. We checked the relevance\nof the lattice parameter b, which is not constrained by\nsymmetry in the m?cphase, and concluded that it is\nnot crucial to make the structure more stable than the\nmkcphase, hence in the rest of the paper we use the\nideal value b=p\n3a. In Table I we summarize the data\nrelative to the lattice constants and to the magnetic mo-\nment of Mn atoms, obtained with the LDA and PBEsol\nfunctionals, and compare them with previous theoretical\nresults and with experiments. The LDA geometry is in\ngood agreement with the theoretical results reported in\nRef. 14, but both lattice constants underestimate the\nexperimental values: in particular, ais 2 % smaller than\nexperiment, while cis 8 % smaller than experiment. The\nPBEsol geometry gives lattice constants slightly smaller\nthanPBE(reportedinRef. 14)andexperiments: aandc\nare 0.5 % and 6 % smaller than experiment, respectively.\nThem?candmkcphases have slightly different lat-\ntice constants aandc, but in Table I we report only one\nstructure because the differences in the lattice constants\nare beyond the significative digits reported. In this pa-\nper we use the LDA to compute the phonon frequencies\nand their contribution to the MAE, while the PBEsol\nis used to compute the energy contribution to the MAE\nand to correct it for thermal expansion effects. The LDA\nand the PBEsol (at T= 0K) calculations are performed\nat the geometry reported in Table I. The computed Mn\nmagnetic moment mMnis in agreement with previous\ncalculations reported in literature12,14:mMnis 10 %(20\n%) smaller than experiment within the PBEsol (LDA)Exchange-correlation a(Å)c(Å)mMn(\u0016B)\nfunctional\nLDA (this work) 4.165.57 3.2\nLDA (Ref. 14) 4.205.54 3.29\nGGA-PBEsol (this work) 4.245.67 3.5\nGGA-PBEsol (Ref. 14) 4.285.63 3.56\nGGA-PBE (Ref. 14) 4.355.76 3.69\nGGA-PBE (Ref. 12) 4.315.74 3.45\nGGA-PBE + U (Ref. 12) 4.396.12 3.96\nexp. (Ref. 10) 4.276.053.8-4.2\nTable I. Computed (FR LDA and PBEsol), theoretical refer-\nence (LDA, PBEsol, PBE, and PBE+U), and experimental\nlattice constants and Mn magnetic moments.\napproximation.\nThe pseudo wave functions (charge density) have been\nexpanded in a plane waves basis set with a kinetic energy\ncut-off of 110 (440) Ry. The Brillouin Zone (BZ) integra-\ntions have been done using a shifted uniform Monkhorst-\nPack mesh27of12\u000212\u00028k-points. The presence of\na Fermi surface has been dealt with by the Methfessel-\nPaxton smearing method28, with a smearing parame-\nter\u001b= 0:015Ry. The dynamical matrices have been\ncomputed on a uniform 4\u00024\u00023q-points mesh, and\na Fourier interpolation was used to obtain the complete\nphonon dispersions and the free energy. The latter has\nbeen obtained approximating the BZ integral with a\n300\u0002300\u0002300q-points mesh.\nIII. RESULTS\nMnBi is a magnetic binary compound, in which mag-\nnetism is carried mainly by the Mn atoms, while Bi is re-\nsponsible for a strong spin-orbit interaction. As a conse-\nquence, strong magnetocrystalline anisotropy effects are\nexpected.\nPhonon dispersions\nHere we consider the phonon dispersions of MnBi\nwith two different orientations of the magnetic moments,\nm?c(in-plane) and mkc(out-of-plane), and among\nall the possible in-plane orientations m?c, we choose\nmka,aandcbeing the primitive vectors of the hexag-\nonal Bravais lattice). The presence of a magnetization\nleads to a difference in the Bravais lattice the magnetic\npoint group is compatible with, as discussed in the pre-\nvious Section. In order to compare the phonon disper-\nsions in the same BZ, we choose to set the geometry\nin the base-centered orthorhombic Bravais lattice, which\nis compatible with the low-symmetry phase (magnetic\npoint group D2h(C2h)).\nThe phonon dispersions are illustrated in Fig. 1. The\nphonon modes are split in two groups, separated by a3\n 0 50 100 150 200\nΓXS RA ZΓYX1A1TYFrequency (cm−1)\nm || cm || a\n 0 50 100 150 200\nΓXS RA ZΓYX1A1TYFrequency (cm−1)\n 0 50 100 150 200\nΓXS RA ZΓYX1A1TYFrequency (cm−1)m|| cm⊥ c\nFigure 1. Computed FR LDA phonon dispersions of MnBi\nwith magnetic moments oriented in plane ( m?c, black line),\nand out of plane ( mkc, red line).\ngap. The low-frequency branches (up to \u0018100cm\u00001)\nare dominated by displacements of the heavy element Bi,\nwhile the high-frequency branches (from \u0018150cm\u00001to\n\u0018200cm\u00001) are mainly displacements of the Mn atoms.\nThe main difference between the phonon frequencies of\nthe two phases is a rigid shift: the phonon frequencies\nof the phase with in-plane magnetization are higher than\nthose of the phase with out-of-plane magnetization. The\nshift is about 0:5cm\u00001in the low-frequency branches,\nwhile it is about 2 cm\u00001in the high-frequency branches.\nMoreover, there are differences due to symmetry. The\nsystem with in-plane magnetization has lower symmetry\nand some modes, degenerate when the magnetization is\nalong c, split. As an example, in Table II we report the\nphonon modes at Z and X 1. At Z, in the phase with mk\ncthere are two groups of four-fold degenerate modes,\nwhich become four couples of degenerate modes in the\nphase m?c: the splittings are quite small, in the range\n0.02-0.08 cm\u00001. At X 1, in the configuration mkcthere\nare four two-fold degenerate modes, which split from 0.1\ncm\u00001to 1 cm\u00001. Similar splittings are found also along\nthe other high-symmetry lines.\nMAE\nIn previous works, the spin-reorientation transition,\ndue to the change of Kufrom negative to positive at\nTSR\u001990K, has been explained as an effect of thermal\nexpansion of the crystal parameters aandc.\nIn Refs. 12 and 13 Kth: exp:\nu, the function Kuobtained\naccounting for thermal expansion effects, has been com-\nputed within the LSDA +SO (spin-orbit) +U scheme\nusing the experimental lattice constants as a function\nof the temperature, reported in Refs. 10 and 11 and\nfinding in this way a good agreement with experiments.q mkc m ?c\nZ\u0017(cm\u00001) degeneracy \u0017(cm\u00001) degeneracy\n67.135 467.212 2\n67.231 2\n180.245 4181.263 2\n181.340 2\nX161.427 261.554 1\n62.503 1\n72.461 272.467 1\n72.577 1\n176.735 2178.156 1\n178.556 1\n184.374 2185.195 1\n185.395 1\nTable II. Computed FR LDA phonon frequencies at high-\nsymmetry points Z and X 1for the configurations mkcand\nm?c. Only the degenerate modes are shown for the config-\nuration mkc, and how the degeneracy is lowered or lifted if\nm?c.\nHere instead we compute Kth: exp:\nuwithin the FR PBEsol\nscheme. In Fig. 2(a) we show \u0016Ku, the contribution to Ku\ngiven by the energy difference of the electronic ground-\nstates, for a mesh of geometries and on top of it we indi-\ncate with black and white points the thermal expansion\ndata added to the theoretical PBEsol T= 0K crystal\nparameters. The resulting Kth: exp:\nuis reported in Fig.\n2(b).Kth: exp:\nuis negative and slightly increases with in-\ncreasingaandc. Yet, such an increase is not sufficient to\ncross the value Kth:exp:\nu = 0because the T= 0K energy\ndifference (\u0019\u00000:73meV/cell) is significantly lower than\nthe experimental value ( \u0019\u00000:15meV / cell), similarly to\nwhat found within the LDA and GGA approximations in\nRefs. 12 and 13, and because the energy difference land-\nscape shows a local maximum of about \u00000:4meV / cell.\nIn addition to the thermal expansion effect we con-\nsider also the effect of the lattice dynamics on Ku: in\nfact, since the two phases have slightly different phonon\nfrequencies, they have different vibrational entropies that\ngive a temperature-dependent contribution to the MAE\ndefined as the difference of the vibrational free energies\nof the two phases. We write the constant Kuas29:\nKu=Kth: exp:\nu +Kvib\nu+Kmag\nu; (1)\nwhereKth: exp:\nuis the thermal expansion contribution,\nKvib\nuis the lattice dynamics contribution, and Kmag\nuis\nthemagnoncontribution, whichwedonotconsiderinthe\npresent work. The term Kvib\nucan be computed from the\nphonon frequencies using the harmonic approximation:\nKvib\nu=Z+1\n0d!1\n2~!\u0002\ng?(!)\u0000gk(!)\u0003\n+kBTZ+1\n0d!\u0002\ng?(!)\u0000gk(!)\u0003\nln\u0010\n1\u0000e\u0000~!=kBT\u0011\n;\n(2)\nwhereg?(!)(gk(!)) is the phonon density of states rela-\ntive to the phase with m?c(mkc). In our case Kvib\nu4\n−1−0.9−0.8−0.7−0.6−0.5−0.4−0.3\n 0 50 100 150 200Kuth. exp. (mev / cell)\nTemperature (K)(a)\n(b) 4.22 4.24 4.26 4.28 4.3a (Å) 1.335 1.34 1.345 1.35 1.355c/a−1.6−1.4−1.2−1−0.8−0.6−0.4K−u (meV/cell)\nFigure 2. Energy contribution to the MAE constant, \u0016Ku,\ncomputed in the FR PBEsol scheme. (a) \u0016Kuas a function\nof lattice constants. Black and white dashed lines represent\nthe geometries at different temperatures, obtained from Refs.\n10 and 11, respectively, as explained in the main text. (b)\nKth:exp:\nuas a function of temperature. Red and blue lines\nrepresent the values of Kth:exp:\nuobtained from the black and\nwhite dashed lines of Fig. 2 (a), respectively.\n 0 0.5 1 1.5 2 2.5 3 3.5 4\n 0  100  200  300  400  500  600  700Kuvib (mev / cell)\nTemperature (K)\nFigure 3. Vibrational contribution to the MAE constant Ku,\ncomputed from the phonon frequencies within the harmonic\napproximation (Eq. (2))\nis always positive and increases with increasing temper-\nature, as shown in Fig. 3: its magnitude is comparable\n−0.5 0 0.5 1 1.5 2 2.5 3 3.5\n 0 100 200 300 400 500 600 700Ku (mev / cell)\nTemperature (K)exp. datatotal MAEtotal MAE(a)\n(b)−0.4−0.2 0 0.2 0.4 0.6 0.8\n 0 50 100 150 200Ku (mev / cell)\nTemperature (K)Figure 4. Comparison between computed and experimental\nMAE constant Ku. Black squares represent experimental\ndata8, red and blue lines represent the total Kucomputed,\nwhere theKth:exp:\nucontribution has been obtained from Refs.\n10 and 11, respectively, as explained in the main text. (a)\nDetailed comparison in the temperature range 0\u0000200K, to\nhighlightTSR. (b) Comparison in a wider range of tempera-\ntures ( 0\u0000600K).\nwiththatof Kth: exp:\nu, henceitgivesacrucialcontribution\nin determining the MAE constant Ku.\nIn Figs. 4(a-b) we show the total Kudefined in Eq.\n(1) and compare it with the experimental data. By com-\nparison with Fig. 2(b), it is clear that the addition of\nthe vibrational contribution is important. In particular,\nthe zero-point vibrational free energy difference allows to\nhave aT= 0K value of Kuin good agreement with\nexperiments and, together with the thermal contribution\n(second term of Eq. (2)), to get a transition tempera-\ntureTSRin reasonable agreement with the experiments\n(TSR\u001990K andTSR\u0019110K with the data given\nby Refs. 10 and 11, respectively). Moreover, the vibra-\ntional contribution allows to have a fair agreement with\nexperimental data8in a rather large temperature range,\nas shown in Fig. 4(b). At variance with Ref. 13, we do\nnot find a maximum in Kubecause the T= 0K geome-\ntry corresponds to the theoretical geometry and because\nthe vibrational contribution increases with T. At high\ntemperatures ( T >500K) our results do not agree with\nthe experiment, suggesting that additional contributions\nmay become important in this temperature range, in par-\nticular the term Kmag\nuin Eq. (1), which can result in a5\ndecreasing KuwithT30. Moreover, we mention that in\nthehigh-temperaturelimitadditionaleffectsnotincluded\nin Eq. (1), as the magnon-phonon coupling31,32, might\nbe non-negligible.\nIV. CONCLUSIONS\nBy means of recent developments in first-principles\ntheoretical tools, we have studied the lattice dynamics of\nthe MnBi ferromagnet for two orientations of the mag-\nnetization, m?candmkc. We have shown that\nthe differences in the phonon frequencies give rise to a\ncontribution to the MAE that is comparable with the\nelectronic one. We have found that the vibrational con-\ntribution is relevant to explain the behavior of the MAE\nconstantKuas a function of temperature. We could\nalso get an estimate of the spin-reorientation tempera-\ntureTSRin fair agreement with the experimental value\nusing the PBEsol approximation to evaluate the energy\ncontribution to the MAE. The use of other functionals\nsuchasLDAwouldgiveinsteadalowervalueoftheMAEatT= 0K than that predicted by PBEsol, and would\nlead to predict TSR\u0019500K. The use of the Hubbard\nU parameter would result, instead, in a higher value of\nKuatT= 0K13and, as a consequence, we would get\nKu>0when adding the vibrational contribution, thus\nthe spin-reorientation transition would not be explained.\nIn Ref. 14 a first estimate of the vibrational contribu-\ntion to MAE was given, which led to TSR\u0019450K, far\nfrom the experimental value. In this work we have found\navibrationalcontributionofthesameorderofmagnitude\nas found in Ref. 14, and we have included also the zero-\npoint vibrational free energy difference, which gives an\nimportant contribution and leads to an estimated spin-\nreorientation temperature TSR\u0019100K, when used to-\ngether with the energy MAE given by PBEsol.\nACKNOWLEDGMENTS\nComputational facilities have been provided by SISSA\nthrough its Linux Cluster and ITCS and by CINECA\nthrough the SISSA-CINECA 2019-2020 Agreement.\n1D. J. Sellmyer, B. Balamurugan, W. Y. Zhang, B. Das, R.\nSkomski, P. Kharel, and Y. Liu, Advances in Rare-Earth-\nFree Permanent Magnets. In: Marquis F. (eds) Proceed-\nings of the 8th Pacific Rim International Congress on Ad-\nvanced Materials and Processing. Springer, Cham; DOI:\nhttps://doi.org/10.1007/978-3-319-48764-9_212 (2013).\n2J. Cui, M. Kramer, L. Zhou, F. Liu, A. Gabay,\nG. Hadjipanayis, B. Balasubramanian, D. Sell-\nmyer, Acta Materialia 158, 118-137; DOI:\nhttps://doi.org/10.1016/j.actamat.2018.07.049 (2018).\n3C. Guillaud, J. Phys. Radium 12, 143; DOI:\n10.1051/jphysrad:01951001202014300 (1951).\n4B. W. Roberts, Phys. Rev. 104, 607; DOI: 10.1103/Phys-\nRev.104.607 (1956).\n5J. B. Yang, K. Kamaraju, W. B. Yelon, and W. J. James,\nAppl. Phys. Lett. 79, 1846; DOI: 10.1063/1.1405434\n(2001).\n6T. Chen, J. Appl. Phys. 45, 2358; DOI: 10.1063/1.1663594\n(1974).\n7M. A. McGuire, H. Cao, B. C. Chakoumakos, and\nB. C. Sales, Phys. Rev. B 90, 174425; DOI:\nhttps://doi.org/10.1103/PhysRevB.90.174425 (2014).\n8W. E. Stutius, T. Chen, and T. R. Sandin, Magnetism and\nmagnetic materials - 1973: Nineteenth Annual Conference,\nDOI: 10.1063/1.2947245 (1974).\n9T. Hihara and Y. K \u0016oi, J. Phys. Soc. Jpn. 29, 343; DOI:\nhttps://doi.org/10.1143/JPSJ.29.343 (1970).\n10J. B. Yang, Y. B. Yang, X. G. Chen, X. B. Ma, J. Z.\nHan, Y. C. Yang, S. Guo, A. R. Yan, Q. Z. Huang,\nM. M. Wu et al., Appl. Phys. Lett. 99, 082505; DOI:\nhttps://doi.org/10.1063/1.3630001 (2011).\n11K. Koyama, Y. Mitsui, and K. Watanabe,\nSci. Technol. Adv. Mater. 9, 024204; DOI:\nhttps://doi.org/10.1088/1468-6996/9/2/024204 (2008).12N. A. Zarkevich, L.L. Wang, and D. D. Johnson, APL\nMater. 2, 032103; DOI:https://doi.org/10.1063/1.4867223\n(2014).\n13V. P. Antropov, V. N. Antonov, L. V. Bekenov, A.\nKutepov, and G. Kotliar, Phys. Rev. B 90054404; DOI:\n10.1103/PhysRevB.90.054404 (2014).\n14K. V. Shanavas, D. Parker, and D. J. Singh, Sci. Reports\n4, 7222; DOI: 10.1038/srep07222 (2014).\n15A.UrruandA.DalCorso, Phys.Rev.B 100, 045115; DOI:\nhttps://doi.org/10.1103/PhysRevB.100.045115 (2019).\n16P. Hohenberg and W. Kohn, Phys. Rev. 136, B864; DOI:\nhttps://doi.org/10.1103/PhysRev.136.B864 (1964)\n17W. Kohn and L. J. Sham, Phys. Rev. 140, A1133; DOI:\nhttps://doi.org/10.1103/PhysRev.140.A1133 (1965)\n18J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048; DOI:\nhttps://doi.org/10.1103/PhysRevB.23.5048 (1981).\n19J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A.\nVydrov, G. E. Scuseria, L. A. Constantin, X. Zhou,\nand K. Burke, Phys. Rev. Lett. 100, 136406; DOI:\nhttps://doi.org/10.1103/PhysRevLett.100.136406 (2008).\n20P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car,\nC. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni,\nI. Dabo, et al., J. Phys. Condens. Matter 21, 395502;\nDOI: 10.1088/0953-8984/21/39/395502 (2009) (See http:\n//www.quantum-espresso.org ).\n21P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. B.\nNardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli,\nM. Cococcioni, et al., J. Phys. Condens. Matter 29,\n465901; DOI: doi.org/10.1088/1361-648X/aa8f79 (2017).\n22P. Giannozzi, O. Baseggio, P. Bonfá, D. Brunato, R. Car,\nI. Carnimeo, C. Cavazzoni, S. de Gironcoli, P. Delugas, F.\nFerrari Ruffino, et al., J. Chem. Phys. 152, 154105; DOI:\nhttps://doi.org/10.1063/5.0005082 (2020).\n23thermo_pw is an extension of the Quantum ESPRESSO (QE)\npackage which provides an alternative organization of the6\nQE work-flow for the most common tasks. For more infor-\nmation see https://dalcorso.github.io/thermo_pw .\n24A. Dal Corso and A. Mosca Conte,\nPhys. Rev. B 71, 115106; DOI:\nhttps://doi.org/10.1103/PhysRevB.71.115106 (2005).\n25A. Dal Corso, Comp. Mat. Sci. 95, 337; DOI:\nhttps://doi.org/10.1016/j.commatsci.2014.07.043 (2014).\n26Seehttps://dalcorso.github.io/pslibrary .\n27H. J. Monkhorst and J.D. Pack, Phys. Rev. B 13, 5188;\nDOI: https://doi.org/10.1103/PhysRevB.13.5188 (1976).\n28M. Methfessel and A.T. Paxton, Phys. Rev. B 40, 3616;\nDOI: https://doi.org/10.1103/PhysRevB.40.3616 (1989).29F. Körmann, A. Dick, B. Grabowski, B. Hallstedt, T.\nHickel, and J. Neugebauer, Phys. Rev. B 78, 033102; DOI:\nhttps://doi.org/10.1103/PhysRevB.78.033102 (2008).\n30J. B. Staunton, S. Ostanin, S. S. A. Razee, B.\nL. Gyorffy, L. Szunyogh, B. Ginatempo, and\nE. Bruno, Phys. Rev. Lett. 93, 257204; DOI:\nhttps://doi.org/10.1103/PhysRevLett.93.257204 (2004).\n31F. Körmann, A. Dick, B. Grabowski, T. Hickel,\nand J. Neugebauer, Phys. Rev. B 85, 125104; DOI:\nhttps://doi.org/10.1103/PhysRevB.85.125104 (2012).\n32J. Hellsvik, D. Thonig, K. Modin, D. Iu-\nsan, A. Bergman, O. Eriksson, L. Bergqvist,\nand A. Delin, Phys. Rev B 99, 104302; DOI:\nhttps://doi.org/10.1103/PhysRevB.99.104302 (2019)."    },    {        "title": "2009.00920v1.Phase_transition_in_the_magnetocrystalline_anisotropy_of_tetragonal_Heusler_alloys__Rh__2T_Sb___T___Fe__Co.pdf",        "content": "arXiv:2009.00920v1  [cond-mat.mtrl-sci]  2 Sep 2020G. H. F.; Rh 2CoSb 2020\nPhase transition in the magnetocrystalline anisotropy of t etragonal Heusler alloys:\nRh2TSb,T=Fe, Co\nGerhard H. Fecher,∗Yangkun He, and Claudia Felser\nMax-Planck-Institute for Chemical Physics of Solids, D-01 187 Dresden, Germany\n(Dated: September 3, 2020)\nThis work reports on first principles calculations of the ele ctronic and magnetic structure of\ntetragonal Heusler compounds with the composition Rh 2FexCo1−xSb (0≤x≤1). It is found that\nthe magnetic moments increase from 2 to 3.4 µBand the Curie temperature decreases from 500 to\n464 K with increasing Fe content x. The 3dtransition metals make the main contribution to the\nmagnetic moments, whereas Rh contributes only approximate ly 0.2µBper atom, independentof the\ncomposition. The paper focuses on the magnetocrystalline a nisotropy of the borderline compounds\nRh2FeSb, Rh 2Fe0.5Co0.5Sb, and Rh 2CoSb. A transition from easy-axis to easy-plane anisotropy is\nobserved when the composition changes from Rh 2CoSb to Rh 2FeSb. The transition occurs at an\niron concentration of approximately 40%.\nKeywords: Electronic structure, Magnetocrystalline anis otropy, Intermetallic compounds, Rh 2FeSb,\nRh2CoSb\nI. INTRODUCTION\nPermanent or hard magnets are made of bulk ma-\nterials with strong anisotropy, which may be based on\nmagnetocrystalline, shape anisotropy, or both. In mag-\nnets with magnetocrystalline anisotropy, there should be\nonly one easy crystal axis of magnetisation so that the\nanisotropy is uniaxial. Such an uniaxial magnetocrys-\ntalline anisotropy is found, for example, in tetragonal or\nhexagonal systems. Heusler alloys are compounds with\nformula T2T′M, where TandT′are transition met-\nals, and Mis a main group element. Some of these\ncompounds and alloys crystallise in tetragonal structure;\nhowever, most of them have a cubic crystal structure.\nOne advantage of Heusler compounds is that most of\nthem do not contain rare earth elements; rather, the\nmagneticpropertiesareprovidedby3 dtransitionmetals.\nMany tetragonal Heusler alloys are Mn-based, and sev-\neral exhibit structural martensite–austenitephase transi-\ntions. In particular, in the inversestructures with space\ngroupI4m2, the magnetic moments of the Mn atoms ex-\nhibit antiparallel coupling. Thus, these alloys are gener-\nallyferrimagnetswithlowsaturationmagnetisation. The\nRh2TMalloys (T′= V, Mn, Fe, Co; M= Sn, Sb) crys-\ntallise in a regular tetragonal structure with space group\nI4/mmmandareexpectedtoexhibituniaxialanisotropy\nwhen the 3 dtransition metals have large moments.\nExperiments on the crystal structure and magnetic\nproperties of Rh 2-based Heusler compounds were re-\nported by Dhar et al.[1], who observed a tetragonal\nstructure and a magnetic moment of 1.4 µBin the prim-\nitive cell. A Curie temperature of approximately 450 K\nwas measured. Further, Fallev et al.recently reported\nab initio calculations for many tetragonal Heusler com-\npounds (including Rh 2FeSb and Rh 2CoSb) [2]. This\nwork proposed that thin films of Rh 2CoSb exhibit uni-\n∗fecher@cpfs.mpg.deaxial, perpendicular anisotropy with the easy direction\nalong the c([001]) axis. Experiments and calculations\nboth suggest that Rh 2CoSb might be suitable hard mag-\nnetic material with uniaxial anisotropy. However, the\nconstituent elements, in particular Rh, might be too ex-\npensive for applications where bulk materials are needed,\nfor example, permanent magnets in electric engines.\nHowever, the cost of the materials is not as important\nfor thin film applications, for example, magnetic record-\ning media or magnetoelectronic memory devices.\nWe recently reported experiments on the magnetic\nproperties of Rh 2CoSb[in print, will be added later] . It\nwasfoundthat Rh 2CoSbhasuniaxialanisotropy,where c\nis theeasyaxis. The present work describes theoretically\nthe magnetic properties of Rh 2CoSb, its sister compound\nRh2FeSb, and alloys with mixed Co 1−xFexcomposition.\nII. DETAILS OF THE CALCULATIONS\nThe electronic and magnetic structures of Rh 2TSb\n(T= Fe, Co) were calculated using Wien2k [3–5] and\nSprkkr [6, 7] in the local spin density approxima-\ntion. In particular, the generalised gradient approxima-\ntion of Perdew, Burke, and Ernzerhof [8] was used to\nparametrise the exchange correlation functional. A k-\nmesh based on 126 ×126×126 points of the full Bril-\nlouin zone was used for integration when the total en-\nergies were calculated to determine the magnetocrys-\ntalline anisotropy (see also Appendix C). The calcula-\ntions are described in greater detail in References [9, 10].\nThe spin spirals and magnons were calculated accord-\ning to the schemes described in References [11] and [12],\nrespectively. Calculations for the disordered or off-\nstoichiometric compounds with mixed site occupations\nwere performed using Sprkkr and the coherent poten-\ntial approximation (CPA) [13] in the full potential mode.\nThe CPA allows the simulation of random site occupa-\ntion by different elements. Complications arising in the2\ncalculation of the magnetic anisotropy energies are dis-\ncussedin detail byKhan et al.[14], who comparedresults\nobtained using Wien2k andSPRKKR .\nThe basic crystal structure of the tetragonal Heusler\ncompounds [prototype, Rh 2VSn;tI8;I4/mmm(139)\ndba] is shown in Figure 1(a). The atoms are located in\nthe ferromagnetic structure on the 4d, 2b, and 2a Wyck-\noff positions of the centred tetragonal cell. The magnetic\norder changes the symmetry, and the resulting magnetic\nspace group for collinear ferromagnetic order with mo-\nments along the caxis isI4/mm′m′(139.537), where′\nis the spin reversal operator [15]. The symmetry is re-\nduced to that of space group Im′m′m(71.536) when the\nmagnetisation /vectorMis along the aaxis ([100]) or F m′m′m\n(69.524) for /vectorM/bardbl[110].\na)   ordered ( regular )                            b)   disordered ( T\u0003 Sb) c)  ordered ( inverse )\nFIG. 1. Crystal structure of Rh 2TSb (T= Fe, Co).\nIn the well-ordered regular structure (a), the sites of the l at-\ntice with space group I4/mmm(139) are occupied as follows:\n4d (0 1/2 1/4), Rh; 2b (0 0 1/2), T; and 2a (0 0 0), Sb. In the\ndisordered structure (b), the Tand Sb atoms are randomly\ndistributed on the 2b and 2a sites. The inversetetragonal\nstructure is shown in (c) for comparison.\nThe electronic structure and magnetic properties were\ncalculated using the optimised lattice parameters. As\nstarting point, the lattice parameters of two alternative\nstructures were optimised using Wien2k. In addition\nto the regular Heusler structure described above, the in-\nverse structure with space group I4m2 (119)dbcawas\nassumed. In this structure, the positions of the Co atom\nand one of the Rh atoms areinterchanged. Spin–orbitin-\nteractionwasconsideredowingtothehigh ZvaluesofRh\nand Sb. Note that the spin–orbit interaction is an intrin-\nsic property in the fully relativistic Sprkkr calculations,\nwhichsolvetheDiracequation. Theresultsoftheoptimi-\nsationaresummarisedinTableI.Theregularstructureis\nfound to have lower energy; it thus describes the ground\nstate. The energy difference compared to the inverse\nstructure is approximately 430 meV. The formation en-\nthalpy is calculated as∆ Hf=Etot−(2ERh+ECo+ESb),\nthat is, the difference between the total energy of the\ncompound in different structures and the sum of the en-\nergies of the elements in their ground state structure.\nThe formation enthalpy is clearly lower for the regular\nstructure than for the inverse tetragonal structure. Note\nthat the formation enthalpy is even lower (-220 meV) for\nthe cubic L21structure. The calculated lattice parame-ters are in good agreement with experimental values [1];\nhowever, the calculated cvalue and c/aratio are approx-\nimately 4% larger. This finding might be explained by\neither a temperature effect or some disorder in the ex-\nperiment.\nTABLE I. Structural properties of Rh 2CoSb.\nCalculations are performed for the regular (139) and invers e\n(119) Heusler structures. The lattice parameters ( a,c,c/a),\nformation enthalpy (∆ Hf), and spin magnetic moment msof\nthe primitive cell (total experimental magnetic moment) ar e\nlisted. Experimental values from Reference [1] are shown fo r\ncomparison. Note that the magnetic moment in this reference\nis not saturated.\nCalculated Exp.\n139 119 here [1]\na[˚A] 4.0104 3.95 4.0393 4.04\nc[˚A] 7.3628 7.56 7.1052 7.08\nc/a 1.836 1.91 1.759 1.75\n∆Hf[meV] -754 -325\nms[µB] 2.04 1.79 2.36 1.4\nTC[K] 450 450\nIII. RESULTS AND DISCUSSION\nA. Electronic and magnetic structure of Rh 2CoSb\nThe calculated electronic structure of Rh 2CoSb in the\nregular tetragonal Heusler structure is illustrated in Fig-\nure 2 in terms of the band structure and density of states\n(n(E)). The relativistic bands, spin-resolved total den-\nsity of states, and its atomic contributions are shown.\nThe electronic structure is calculated in the full relativis-\ntic mode by solving the Dirac equation. The band struc-\nture from semi-relativistic calculations is shown in the\nAppendix.\nBoth rhodium and cobalt contribute to the magnetic\nmoment of the compound. The spin and orbital mag-\nnetic moments are mCo\ns= 1.656µBandmCo\nl= 0.139µB\nfor cobalt and mRh\ns= 0.206µBandmRh\nl= 0.007µB\nfor rhodium, respectively. The overall magnetic moment\n(spinplusorbital)oftheprimitivecellis mtot= 2.188µB.\nThe orbitalmoment of the Co atoms makes a remarkably\nlarge contribution.\nThe real space chargeand spin distributions are shown\nin Figure 3. The charge density ( σ(r)) of the atoms\nhas no striking shape. It appears to be nearly spheri-\ncal but still reflects the two- or fourfold symmetry. As\nexpected, most of the electrons are close to the ion cores.\nBy contrast, the spin or magnetisation density ( σ(r)) has\na much more pronounced shape depending on the plane.\nIn particular, in the (110) plane, it has a distinct butter-\nfly shape. The spin densityis positive at both the Coand\nRh atoms. It is clearly higher near the Co atoms than3\n/s77 /s78 /s83\n/s48/s88 /s77\n/s48/s45/s54/s45/s52/s45/s50/s48/s50\n/s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54/s69/s110/s101/s114/s103/s121/s32/s32/s32 /s69 /s40/s107 /s41/s32/s45/s32\n/s70/s32/s91/s101/s86/s93\n/s69/s108/s101/s99/s116/s114/s111/s110/s32/s109/s111/s109/s101/s110/s116/s117/s109/s32/s32/s32 /s107/s109 /s105/s110/s111/s114/s105/s116/s121 \n/s68/s101/s110/s115/s105/s116/s121 /s32/s111/s102/s32/s115/s116/s97/s116/s101/s115/s32/s32/s32 /s110 /s40/s69 /s41/s32/s91/s101/s86/s45/s49\n/s93/s109 /s97/s106/s111/s114/s105/s116/s121 \n/s32/s116/s111/s116/s97/s108\n/s32/s82/s104\n/s32/s67/s111\nFIG. 2. Electronic structure of Rh 2CoSb (I).\nShownis thefullyrelativistic bandstructuretogetherwit hthe\ntotal and site (Rh and Co) specific, spin-resolved densities of\nstates.\nnear the Rh atoms, which ultimately gives Co a higher\nmagnetic moment. The magnetisation density of the Rh\natoms is aligned along the magnetisation direction and\npoints somewhat toward the nearest Co atoms.\n1. Magnetic anisotropy\nFurther, the directional dependence of the magnetisa-\ntion was investigated to explain the collinear magnetic\norder in detail. In particular, the total energy was cal-\nculated for cases where the magnetisation points along\ndifferent crystallographic directions. The obtained en-\nergy differences make it possible to determine the mag-\nnetocrystalline anisotropy (see also Appendix C).\nInthemagneticanisotropyofRh 2CoSb,the easyaxisis\nalong the c([001]) axis. The simple second-order uniax-\nial anisotropy constant is Ku= 1.37 MJ/m3(see Equa-\ntions (C1) and (C2) in Appendix C1). This results in\nan anisotropy field of µ0Hu≈2.4 T. A more detailed\nanalysis reveals that the simple second-order anisotropy\nconstant Kuisnotsufficienttodescribethemagnetocrys-\ntalline anisotropy, as discussed in Section IIID.\nFurther, thedipolarmagnetocrystallineanisotropywas\ncalculated as described in Appendix C3 and was found\nto be ∆Edipaniso= 0.09µeV. The positive value indicates\nan easy dipolar direction along the [001] axis. The dipo-\nlar anisotropyis rather small compared to the anisotropy\ncalculated from the total energy. Here, it was calcu-\nlated for a sphere with a radius of 30 nm. The results\nfor other shapes will be different, resulting in a distinct\nshape anisotropy. In particular, in thin films, the di-\nmension perpendicular to the film is much smaller than\nthe dimensions in the film plane. Therefore, the sum-\nmation in Equation (C16) becomes a truncated sphere\nthat is strongly anisotropic, and a pronounced thin film\nanisotropyappears. This thin film anisotropywill alsobe\naffected by the magnetic moments, which are different at/s48/s99\n/s82/s104/s67/s111/s67/s104/s97/s114/s103/s101/s32/s100/s101/s110/s115/s105/s116/s121 /s32/s32/s32 /s114 /s40 /s114 /s41/s91/s48/s48/s49/s93\n/s83/s98\n/s48 /s97/s48/s99/s77/s97/s103/s110/s101/s116/s105/s115/s97/s116/s105/s111/s110/s32/s100/s101/s110/s115/s105/s116/s121 /s32/s32/s32 /s115 /s40 /s114 /s41/s91/s48/s48/s49/s93\n/s91/s49/s48/s48/s93/s48/s46/s48\n/s49/s46/s48\n/s49/s48/s48/s46/s48\n/s83/s98/s67/s111\n/s48 /s97 /s32/s214/s50\n/s91/s49/s49/s48/s93/s48/s46/s48\n/s48/s46/s49\n/s49/s46/s48\nFIG. 3. Electronic structure of Rh 2CoSb (II).\nFully relativistic charge ( ρ(r)) and spin ( σ(r)) distributions\nfor the (001) and (110) planes are shown. The calculation\nis form/bardblc, that is, the magnetisation points along [001] ac-\ncording to the easy axis behaviour of the magnetic anisotropy.\n(Note: colour bars are in atomic units.)\ninterfaces and surfaces from that at the centre layers of\nthe film.\n2. Spiral spin order\nThe energy of the spin spirals was calculated to search\nfor non-collinear magnetic order. The spin spirals were\ncalculated for different directions and different cones. In\nplanar spirals, the spins are perpendicular to the propa-\ngation direction. Figure 4 compares the energies of pla-\nnar spirals along the high-symmetry directions.\nThe spirals along [100] or [110] propagate in the four-\nfold plane, whereas the spiral in the [001] direction prop-\nagates along the caxis. In all cases, the lowest energy\nis observed at q= 0. The magnetic moment of the Co\natoms varies by approximately 17% at maximum. The\nmagnetic moment of the Rh atoms decreases with in-\ncreasing qand vanishes at the border, independent of\nthe propagation direction of the spiral.\nThe spin direction was assumed to be perpendicular4\n/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48\n/s48 /s189 /s49/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s48 /s189 /s49 /s48 /s189 /s49/s69/s110/s101/s114/s103/s121/s32/s32/s32 /s69 /s40/s113/s41/s32/s91/s109/s101/s86/s93/s113 /s32/s124 /s124 /s32/s91/s48/s48/s49/s93\n/s32/s32\n/s113 /s32/s124 /s124 /s32/s91/s49/s48/s48/s93\n/s32/s32\n/s113 /s32/s124 /s124 /s32/s91/s49/s49/s48/s93/s32/s32/s32/s32/s32/s32 /s109 /s40/s113 /s41/s32/s91\n/s66/s93/s67/s111\n/s82/s104\n/s83/s112/s105/s114/s97/s108/s32/s118/s101/s99/s116/s111/s114/s32/s32/s32 /s124/s113/s124 /s32/s32/s32/s91 /s47/s100\n/s104/s107/s108 /s93\n/s32/s32 /s32\nFIG. 4. Planar spin spirals in Rh 2CoSb.\nThe spiral energies are given with respect to q= 0, that\nis, ∆E(q) =E(q)−E(0). The behaviour of the magnetic\nmoment m(q) is shown for Rh and Co.\nto theqvector in the above calculations for planar spi-\nrals. Thus, the angle between /vector qand the local magnetic\nmoment /vector miwas set to Θ = π/2. Next, the spirals were\nassumed to be conical with 0 <Θ< π/2 to allow for a\nmore detailed analysis. The calculations were performed\nforqalong [001]. Figure 5 displays the results for coni-\ncal spirals with various cone angles. The highest energies\nappear for the planar spiral. The energy at the border\nof the Brillouin zone ( q=π/c) exhibits a sine depen-\ndence. Thus, it vanishes in the antiferromagnetic state.\nThe behaviour of the local magnetic moments suggests\nmore localised behaviour at the Co atoms and induced\nbehaviour at the Rh atoms.\n/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48\n/s48 /s189 /s49/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48\n/s48 /s49 /s189/s48/s176 /s51/s48/s176 /s54/s48/s176 /s57/s48/s176/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s113 /s32/s124 /s124 /s32/s91/s48/s48/s49/s93\n/s32/s61/s32/s57/s48/s176\n/s32/s32/s69/s110/s101/s114/s103/s121/s32/s32/s32 /s69 /s40/s113 /s41/s32/s91/s109/s101/s86/s93/s32/s61/s32/s55/s53/s176\n/s32/s32\n/s32/s61/s32/s54/s48/s176\n/s32/s32\n/s83/s112/s105/s114/s97/s108/s32/s118/s101/s99/s116/s111/s114/s32/s32/s32 /s124/s113/s124 /s32/s32/s32/s91 /s47/s99 /s93/s32/s61/s32/s51/s48/s176\n/s32/s32\n/s69/s110/s101/s114/s103/s121/s32/s32/s32 /s69 /s40 /s41/s32/s91/s109/s101/s86/s93\n/s67/s111/s110/s101/s32/s97/s110/s103/s108/s101/s32/s32/s32/s32/s32/s32 /s113\n/s48/s48/s49/s32/s61/s32 /s47/s99\nFIG. 5. Conical spirals in Rh 2CoSb.\nSpiral energies for different cone angles and the wave vector\nalong the caxis (q/bardbl[001]) are shown. The angular dependence\natq=π/cis also shown.\nThe calculatedspiralenergiesindicate that this type of\nmagnetic order is rather improbable. The spiral energies\nincrease monotonously with the wave vector and cone\nangle, rather independent on the /vector qdirection. The mono-tonic behaviour suggests that a canted magnetic order is\nalso very unlikely [16].\n3. Exchange coupling and magnons\nThe exchange coupling energies were calculated using\nthe scheme of Liechtenstein et al.[17, 18] to estimate\nthe Curie temperature, spin stiffness, and presence of\nmagnons [12]. The exchange coupling parameters are\nplotted in Figure 6(a). The most dominant parameters\nfor Co–Co and Co–Rh interactions are shown; all the\nothers are comparatively small. The largest interaction\nappears for Co atoms in the centre and nearest to the Co\nin the neighbouring plane. From the calculated exchange\ncoupling energies, the Curie temperature was found to\nbeTC= 498 K, which is close to the experimental value\n(450 K) [1]. The calculated spin wave stiffness constant\nisDij= 866 meV ·˚A2, and the interpolation scheme of\nPadjaet al.[19] yields an extrapolated spin wave stiff-\nness ofD0= 864 meV ·˚A2.\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48/s48/s50/s52/s54\n/s77 /s71 /s88/s77\n/s48/s71/s48/s53/s48/s49/s48/s48/s49/s53/s48\n/s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48/s77/s71/s88 /s77/s71/s69/s120/s99/s104/s97/s110/s103/s101/s32/s99/s111/s117/s112/s108/s105/s110/s103/s32/s112/s97/s114/s97/s109/s101/s116/s101/s114/s32/s32/s32 /s74\n/s105/s106/s40/s100 /s41/s32/s91/s109/s101/s86/s93\n/s82/s101/s108/s97/s116/s105/s118/s101/s32/s100/s105/s115/s116/s97/s110/s99/s101/s32/s32/s32 /s100 /s32/s47/s32 /s97/s67/s111/s32/s97/s116/s32/s99/s101/s110/s116/s101/s114\n/s32/s67/s111/s32/s45/s32/s67/s111\n/s32/s67/s111/s32/s45/s32/s82/s104/s40/s97/s41\n/s32/s32/s77/s97/s103/s110/s111/s110/s32/s101/s110/s101/s114/s103/s121/s32/s32/s32 /s69 /s40/s113 /s41/s32/s91/s109/s101/s86/s93\n/s77/s97/s103/s110/s111/s110/s32/s109 /s111/s109 /s101/s110/s116/s117/s109 /s32/s32/s32 /s113/s32/s32/s98/s111/s116/s104\n/s32/s32/s67/s111/s32/s111/s110/s108/s121 /s40/s98/s41/s32\n/s32\n/s110 /s40/s69 /s41/s32/s91/s109 /s101/s86/s45/s49\n/s93/s40/s99/s41\nFIG. 6. Exchange coupling parameters of Rh 2CoSb.\n(a) Exchange coupling parameters for Co–Co and Co–Rh in-\nteraction as functions of distance. Lines are drawn for bett er\ncomparison. (b) Magnon dispersion and (c) density of states .\nCalculations were performed with and without accounting fo r\nthemagnetic momentoftheRhatoms (thatis, for bothatoms\nand for Co only).5\nThe magnon dispersion was calculated by Fourier\ntransformation of the real space exchange coupling pa-\nrameters. The result is presented in Figure 6(b), and\nthe magnon density of states is shown in Figure 6(c).\nTwo calculations were made; in one calculation, only the\nCo–Co interaction was considered, and in the other, the\nmoments of the Rh atoms, which result in additional Co–\nRh and Rh–Rh coupling, were included. The latter cal-\nculation yields flat dispersion curves and a high density\nof states. A comparison of the two calculations reveals\nthat the magnons are dominated by the Co–Co interac-\ntion. Note that the Curie temperature in only 10 K lower\nwhen the Rh moments and the corresponding exchange\nparameters are ignored.\nB. Results for Rh 2FeSb\nThe calculations for Rh 2FeSb were performed in the\nsame way as for Rh 2CoSb. The regular structure with\nspace groupno. 139wasfound to be more stable than the\ninverse structure with space group no. 119. In addition,\nas in the case of Rh 2CoSb, the calculated clattice pa-\nrameter, and thus c/a, are considerably larger than the\nexperimental values (see Table II).\nTABLE II. Structural properties of Rh 2FeSb.\nCalculations are performed for the regular tetragonal Heus ler\nstructures. Lattice parameters ( a,c,c/a) and spin magnetic\nmoment msof the primitive cell are listed. Experimental\nvalues from Reference [1] are shown for comparison. Note\nthat the experimental moments in [1] are not saturated.\nExperiment\nCalculated This work Ref. [1]\na[˚A] 4.0418 4.0671 4.07\nc[˚A] 7.3995 7.0161 6.96\nc/a 1.8308 1.7251 1.71\nms[µB] 3.4 3.8 2.8\nTC[K] 510 510\nThe electronic structure of Rh 2FeSb is illustrated in\nFigure 7. The fully relativistic band structure and the\nspin- and site-resolved densities of states are shown. The\ncalculated spin and orbital magnetic moments are mFe\ns=\n2.978µBandmFe\nl= 0.080µBforironand mRh\ns= 0.228µB\nandmRh\nl= 0.006µBfor rhodium, respectively. The over-\nall magnetic moment (spin plus orbital) of the primitive\ncell ismtot= 3.488µB. The magnetic moment of the\nFe atoms is strongly localised, which is typical of Heusler\ncompounds with high magnetic moments. It clearly ex-\nceeds the value for elemental iron.\nThe real space charge and spin distributions of\nRh2FeSb are shown in Figure 8. As in the Co-containing\ncompound, σ(r) does not havea pronouncedshape (com-\npare Figure 3). The magnetisation density ( σ(r)) around\nthe Fe atoms has a less distinct shape compared to Co/s77 /s78 /s83\n/s48/s88 /s77\n/s48/s45/s54/s45/s52/s45/s50/s48/s50\n/s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54/s69/s110/s101/s114/s103/s121/s32/s32/s32 /s69 /s40/s107 /s41/s32/s45/s32\n/s70/s32/s91/s101/s86/s93\n/s69/s108/s101/s99/s116/s114/s111/s110/s32/s109/s111/s109/s101/s110/s116/s117/s109/s32/s32/s32 /s107/s109 /s105/s110/s111/s114/s105/s116/s121 \n/s68/s101/s110/s115/s105/s116/s121 /s32/s111/s102/s32/s115/s116/s97/s116/s101/s115/s32/s32/s32 /s110 /s40/s69 /s41/s32/s91/s101/s86/s45/s49\n/s93/s109 /s97/s106/s111/s114/s105/s116/s121 \n/s32/s116/s111/s116/s97/s108\n/s32/s82/s104\n/s32/s70/s101\nFIG. 7. Electronic structure of Rh 2FeSb (I).\nFully relativistic band structure is shown, along with the\ntotal- and site-specific spin-resolved densities of states for Rh\nand Fe.\nin Rh2CoSb; it is also not greatly affected by changes in\nthe magnetisation direction. The main difference is the\nmagnetisation density around the Rh atoms, which is ro-\ntated and appears to be aligned along the magnetisation\ndirection.\nTable III compares the calculated magnetic data of\nRh2CoSb and Rh 2FeSb. Rh 2FeSb clearly has a smaller\norbitalmagneticmomentthanRh 2CoSb,whereasitsspin\nmagnetic moment is higher because of the effect of the Fe\natoms. The induced magnetic moments of the Rh atoms\nare similar in both compounds.\nTABLE III. Calculated magnetic properties of Rh 2FeSb,\nRh2Fe0.5Co0.5Sb, and Rh 2CoSb.\nSpinmsand orbital mlmagnetic moments per atom (Rh, T\n= Co, Fe with m/bardblcin all cases) of the primitive cell ( total) are\nlisted, as well as Curie temperature TC, spin stiffness D0, and\nanisotropy parameters. (Note that the dipolar anisotropy i s\nthree orders of magnitude lower than the magnetocrystallin e\npart.)\nFe Fe 0.5Co0.5 Co\nmRh\ns[µB] 0.237 0.239 0.204\nmRh\nl[µB] 0.006 0.008 0.006\nmFe\ns[µB] 3.006 2.977 -\nmFe\nl[µB] 0.080 0.084 -\nmCo\ns[µB] - 1.747 1.674\nmCo\nl[µB] - 0.132 0.137\nmtotal\ns[µB] 3.44 2.81 2.04\nmtotal\nl[µB] 0.09 0.12 0.15\nTC[K] 465 480 500\nD0[meV˚A2] 590 700 870\nKu[MJ/m3] -1.21 -0.23 1.37\n|µ0Ha|[T] 1.34 0.31 2.43\n∆Edipaniso[kJ/m3] 1.9 2.06\n/s48/s99\n/s82/s104/s70/s101/s67/s104/s97/s114/s103/s101/s32/s100/s101/s110/s115/s105/s116/s121 /s32/s32/s32 /s114 /s40 /s114 /s41/s91/s48/s48/s49/s93\n/s83/s98\n/s48 /s97/s48/s99/s77/s97/s103/s110/s101/s116/s105/s115/s97/s116/s105/s111/s110/s32/s100/s101/s110/s115/s105/s116/s121 /s32/s32/s32 /s115 /s40 /s114 /s41/s91/s48/s48/s49/s93\n/s91/s49/s48/s48/s93/s48/s46/s48\n/s49/s46/s48\n/s49/s48/s48/s46/s48\n/s83/s98/s70/s101\n/s48 /s97 /s32/s214/s50\n/s91/s49/s49/s48/s93/s48/s46/s48\n/s48/s46/s49\n/s49/s46/s48\nFIG. 8. Electronic structure of Rh 2FeSb (II).\nFully relativistic charge ( ρ(r)) and spin ( σ(r)) distributions\nfor different planes. Magnetisation is perpendicular to cwith\nmalong [100] in accordance with the easy plane behaviour\nof the magnetic anisotropy. (Note: colour bars are in atomic\nunits.)\nThe calculated Curie temperatures are of the same or-\nder of magnitude as the experimental values. In contrast\ntothe calculatedresults, however,the experimentalvalue\nof the Fe compound is higher than that of the Co com-\npound. A possible reason is differences in the variation\nof the lattice parameters with temperature, which affect\nthe exchange coupling parameters and thus TCand also\nthe spin stiffness. Note that a much lower Curie tem-\nperature is obtained for the Co compound when it is off-\nstoichiometric (see Appendix A2), whereas the TCvalue\nof the off-stoichiometric Fe compound is slightly higher.\nThe anisotropy has the hardaxis along the z([001])\ndirection, and the easy plane is the the basal plane.\nBy contrast, for Rh 2CoSb, the zdirection is the easy\naxis. The simple uniaxial anisotropy constant is Ku=\n−1.21 MJ/m3. Consequently, the anisotropy field is\n|µ0Ha|= 1.34 T. The appearance of the ”hard”axis\nalongzisoppositetoRh 2CoSbwhere zisthe”easy”axis.\nThe dipolar magnetocrystalline anisotropy of Rh 2FeSb\nis ∆Edipaniso= 0.09µeV, indicating that the easy dipo-\nlar direction is along the [001] axis, like that of the Co-containing compound. This behaviour is caused by the\nstrong magnetic moments of the 3 dtransition metals, in\naddition to the elongation of the tetragonal crystal struc-\nture along the caxis.\nThe dynamic magnetic properties of Rh 2FeSb are\nshown in Figure 9. The spin spirals and magnons are\nsimilar to those of Rh 2CoSb; however, their energies ex-\ntend to higher values. The behaviour of the spin spirals\nrules out the presence of non-collinear magnetic struc-\nture [16].\n/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48\n/s48 /s189 /s49/s48/s49/s50/s51\n/s48 /s189 /s49 /s48 /s189 /s49\n/s77 /s88 /s77\n/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48/s51/s53/s48\n/s48/s46/s48/s48 /s48/s46/s48/s53/s77 /s88 /s77/s69/s110/s101/s114/s103/s121/s32/s32/s32 /s69 /s40/s113/s41/s32/s91/s109/s101/s86/s93/s113 /s32/s124 /s124 /s32/s91/s48/s48/s49/s93\n/s32/s32\n/s113 /s32/s124 /s124 /s32/s91/s49/s48/s48/s93\n/s32/s32\n/s113 /s32/s124 /s124 /s32/s91/s49/s49/s48/s93/s32/s32/s32/s32/s32/s32 /s109 /s40/s113 /s41/s32/s91\n/s66/s93/s70/s101\n/s82/s104\n/s83/s112/s105/s114/s97/s108/s32/s118/s101/s99/s116/s111/s114/s32/s32/s32 /s124/s113/s124 /s32/s32/s32/s91 /s47/s100\n/s104/s107/s108/s93\n/s32/s32 /s32/s32/s32/s77/s97/s103/s110/s111/s110/s32/s101/s110/s101/s114/s103/s121/s32/s32/s32 /s69 /s40/s113 /s41/s32/s91/s109/s101/s86/s93\n/s77/s97/s103/s110/s111/s110/s32/s109/s111/s109/s101/s110/s116/s117/s109/s32/s32/s32 /s113/s32/s32/s98/s111/s116/s104\n/s32/s32/s70/s101/s32/s111/s110/s108/s121 /s32\n/s32\n/s103 /s40/s69 /s41/s32/s91/s109/s101/s86/s45 /s49\n/s93\nFIG. 9. Dynamic magnetic properties of Rh 2FeSb.\nSpiral energies with correspondingmagnetic momentsandth e\nmagnon dispersion are shown, along with the magnon density\nof states g(E). Magnon calculations were performed with and\nwithout the magnetic moment of the Rh atoms; in the latter\ncase, all Fe–Rh interactions are neglected.\nC. Results for Rh 2FexCo1−xSb\nOwing to the differences in magnetic anisotropy be-\ntween the Fe- and Co-based compounds, it is interest-\ning to investigate a mixed system containing both Fe\nand Co. Therefore, calculations were also performed\nfor Rh 2FexCo1−xSb using Sprkkr and the CPA. The\nCPA enables the simulation of random occupation of Fe\nand Co atoms at a single site (here 2 b). The obtained\nmagnetic properties of Rh 2Fe0.5Co0.5Sb are shown in Ta-\nble III. The uniaxial anisotropy constant is negative, like\nthat of Rh 2FeSb; however, its absolute value is consider-\nably lower (by a factor of 35) than that of Rh 2CoSb.7\nThedependenceofthemagneticpropertiesonthecom-\nposition is shown in Figure 10. The total magnetic mo-\nment increases with increasing Fe content, mainly be-\ncause Fe has a higher spin magnetic moment ( ≈3µB)\nthan Co( ≈1.7µB). The individualmagneticmomentsof\nthe atoms are nearly unaffected by the composition. The\ncalculated Curie temperature decreases with increasing\nFe content.\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s52/s46/s48\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s49/s48/s48/s52/s54/s48/s52/s55/s48/s52/s56/s48/s52/s57/s48/s53/s48/s48/s77/s97/s103/s110/s101/s116/s105/s99/s32/s109/s111/s109/s101/s110/s116/s32/s32/s32 /s109 /s40/s120 /s41/s32/s91\n/s66/s93\n/s70/s101/s32/s99/s111/s110/s116/s101/s110/s116/s32/s32/s32 /s120/s32/s109\n/s116/s111 /s116\n/s32/s109\n/s115\n/s32/s109\n/s108 \n/s84\n/s67/s40/s120 /s41/s32/s91/s75/s93\nFIG. 10. Magnetic properties of Rh 2FexCo1−xSb.\nTotal (mtot), spin (ms), and orbital ( ml) magnetic moments\nas functions of Fe content xare shown. The inset shows the\nCurie temperature ( TC).\nD. Magnetocrystalline anisotropy of\nRh2FexCo1−xSb\nThus far, only the simplest case of uniaxial magne-\ntocrystalline anisotropy has been considered. The equa-\ntions for extending the calculations to more detailed\ncases are given in Appendix C. These equations were\nused to calculate the fourth-order uniaxial and tetrag-\nonal anisotropy constants, which were used to obtain the\nmagnetocrystalline anisotropy energy distributions.\nThe calculated uniaxial energy distributions Eu′(θ,φ)\n(see Equations (C4) and (C18) in the Appendix) of\nRh2FeSb, Rh 2Fe0.5Co0.5Sb, and Rh 2CoSb are plotted in\nFigure 11 for comparison.\nThe different behaviour of the anisotropy is clearly re-\nvealed in Figure 11. Rh 2FeSb has an easyplane, and cis\nthehardaxis; Rh 2Fe0.5Co0.5Sb has an easyplane as well,\nbut ahardcone, and in Rh 2CoSb, the cdirection is the\neasyaxis. Rh 2Fe0.5Co0.5Sb has a much lower anisotropy\nthan the pure compounds, and the differences between\nthe energies of the abplane and the caxis are very small.\nA hard cone appears with its maximum at an angle of\nθ3,4=±35.7◦(see Equation (C9) in Appendix C1).\nThe calculated anisotropy constants for uniaxial and\ntetragonalsymmetry arecomparedin Table IV. The sim-\npleKufromEquation(C2)(seeAppendix C) clearlycan-D\f\u00035K \u0015)H6E\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003 5K \u0015)H \u0013\u0011\u0018 &R \u0013\u0011\u0018 6E\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003F\f\u00035K \u0015&R6E \nFIG. 11. Uniaxial magnetic anisotropy of Rh 2TSb com-\npounds.\nEnergy distributions Eu′(θ,φ) ofT= Fe (a), Fe 0.5Co0.5(b),\nand Co (c). The energies Ex,y,zare given in µeV. (Please\nnote the different energy scales.)\nnot describe the magnetic anisotropy correctly.\nTABLE IV. Comparison of the anisotropy constants of\nRh2TSb,T= Fe, Fe 0.5Co0.5, and Co.\nRh2FeSb Rh 2Fe0.5Co0.5Sb Rh 2CoSb\nuniaxial\nKu[MJ/m3] -1.21 -0.23 1.37\nK0[MJ/m3] 1.31 0.39 0.0\nK2[MJ/m3] -2.19 0.50 3.62\nK4[MJ/m3] 0.98 -0.73 -2.25\ntetragonal\nK0,0[MJ/m3] 1.31 0.39 0.0\nK2,0[MJ/m3] -2.19 0.50 3.62\nK4,0[MJ/m3] 0.93 -0.81 -2.40\nK4,4[MJ/m3] 0.05 0.08 0.15\nThe dependence of the uniaxial anisotropy constants\non the composition is illustrated in Figure 12. The uni-\naxial anisotropy constant Kudecreases with increasing\niron content and exhibits a zero-crossing at x0≈0.4. At\nintermediate iron contents, more complex behaviour ap-\npears, as shown by the composition dependence of K2i\nand the results in Figures 11 and 13.\nThe calculated tetragonal energy distributions\nEa′(θ,φ) (see Equations (C12) and (C18) in the Ap-\npendix) of Rh 2FeSb, Rh 2Fe0.5Co0.5Sb, and Rh 2CoSb\nare shown in Figure 13. As in the plot of the uniaxial\nanisotropy in Figure 11, the differences in the anisotropy\nare easily observed. In Rh 2FeSb, the hardaxis is along\nthez([001]) direction, and the anisotropy exhibits\nweak variation in the basal plane, which is close to\ntheeasyplane. Closer examination of the basal plane\nshows biaxial behaviour with easyaxes along the [110]\nand [110] axes, but the energy difference between these\ndirections and the [100] or [010] axes is very small. The\nanisotropy of Rh 2CoSb is still almost uniaxial, with the\neasyaxis along the c([001]) axis, and varies weakly\nin the basal plane. Rh 2Fe0.5Co0.5Sb has much lower\nanisotropy than the pure compounds and exhibits more\ncomplicated directional behaviour.\nThe directional dependence of the orbital magnetic\nmoments was analysed to clarify the role of the spin–8\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s45/s50/s45/s49/s48/s49/s50\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s49/s48/s49/s65/s110/s105/s115/s111/s116/s114/s111/s112/s121/s32/s99/s111/s110/s115/s116/s97/s110/s116/s115/s32/s32/s32 /s75\n/s50/s105/s40/s120 /s41/s32/s91/s77/s74/s47/s109/s51\n/s93\n/s70/s101/s32/s99/s111/s110/s116/s101/s110/s116/s32/s32/s32 /s120/s32/s32/s32 /s75\n/s48/s40/s120 /s41\n/s32/s32/s32 /s75\n/s50/s40/s120 /s41\n/s32/s32/s32 /s75\n/s52/s40/s120 /s41/s75\n/s117/s40/s120 /s41/s32/s91/s77/s74/s47/s109/s51\n/s93\nFIG. 12. Anisotropy constants of Rh 2FexCo1−xSb com-\npounds.\nThe inset shows the uniaxial anisotropy constant obtained\nusing Equation (C3) in Appendix C1.\nD\f\u00035K \u0015)H6E\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003 5K \u0015)H \u0013\u0011\u0018 &R \u0013\u0011\u0018 6E\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003F\f\u00035K \u0015&R6E \nFIG. 13. Tetragonal magnetic anisotropy of Rh 2TSb com-\npounds.\nEnergy distributions Ea′(θ,φ) ofT= Fe (a), Fe 0.5Co0.5(b),\nand Co (c). The energies Ex,y,zare given in µeV. (Please\nnote the different energy scales.)\norbit interaction. The magnetic moments for m/bardblcare\nlisted in Table III. The ratio of the total orbital moment\nto the total spin moment, ml/ms, was used owing to the\nlarge differences between the magnetic moments for dif-\nferent compositions. Figure 14 shows the ratio ml/ms\nas a function of the difference in the energies in several\nmagnetisation directions [ hkl]. For both Rh 2CoSb and\nRh2FeSb, the ratio is largest for magnetisation along the\ncaxis ([001]) and lowest in the basal plane. This finding\ninvolves not only the ratio but also the orbital momenta\nthemselves, indicating that the orbital moment is not al-\nways largest when the magnetisation is along the easy\naxis (or in the easy plane). Here it depends at least par-\ntially on the angle between the magnetisation and caxis,\nas shown by the values for other directions.\nTo further examine the nature of the anisotropy, the\ncharge and spin density distributions were analysed with/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53/s48/s46/s48/s50/s48/s46/s48/s51/s48/s46/s48/s52/s48/s46/s48/s53/s48/s46/s48/s54/s48/s46/s48/s55/s48/s46/s48/s56\n/s91/s49/s48/s48/s93/s91/s49/s49/s49/s93/s91/s49/s48/s49/s93/s77/s111/s109/s101/s110/s116/s32/s114/s97/s116/s105/s111/s32/s32/s32 /s109\n/s108/s32/s47/s32 /s109\n/s115\n/s69/s110/s101/s114/s103/s121/s32/s100/s105/s102/s102/s101/s114/s101/s110/s99/s101/s32/s32/s32 /s69\n/s104/s107/s108/s32 /s32/s69\n/s109/s105/s110/s32/s91/s109 /s101/s86/s93/s91/s48/s48/s49/s93\n/s91/s49/s49/s48/s93/s91/s49/s49/s49/s93/s91/s49/s48/s49/s93/s91/s48/s48/s49/s93/s32/s32/s32/s82/s104\n/s50/s67/s111/s83/s98\n/s32/s32/s32/s82/s104\n/s50/s70/s101/s83/s98\nFIG. 14. Directional dependence of the orbital moments of\nRh2T′Sb,T′= Co, Fe.\nNote that the orbital moments are given relative to the spin\nmoments for better comparison.\nrespect to the magnetisationdirection (comparealsoFig-\nures 3 and 8). As mentioned above, the symmetry\nchangeswhenthemagnetisationisappliedalongdifferent\ncrystallographic directions. The point group symmetry\nof the 2b sites occupied by Fe and Co is D4handD2dfor\nRh on 4d. Applying the magnetisation along one of the\nhigh-symmetry axes, i.e., the c([001]) or a([100]) axis,\nchanges the symmetry of the 2b sites to C4horC2h, re-\nspectively. As a result, the irreducible representations\nand basic functions depend on the magnetisation direc-\ntion. For C4h, they are ag,bg, andegwith the l= 2\nbasic functions dz2, (dx2−y2,dxy), and (dxz,dyz). For\nC2h, they are agandbgwith (dz2,dx2−y2,dxy) and (dxz,\ndyz). Similar differences appear for the 4d sites. The\ncharge and spin density distributions for different mag-\nnetisation directions are compared in Figure 15 for the\ncompounds containing only Fe or Co.\nAs mentioned above, the details of the charge density\nare not easily observed directly from the graph when the\nmagnetisation direction is changed, because the graph\nshows mainly the positions of the atoms. However, they\ncan be observed if one investigates the difference in the\ncharge distribution, which is plotted as ∆ ρ(r). It was\ncalculated for both compounds as the difference between\nthe chargedensities obtained assuming that the magneti-\nsation is parallel ( m/bardbl[001]) or perpendicular ( m/bardbl[100]) to\nthecaxis. Inbothcompounds, themagnetisationhasthe\nsame effect on ∆ ρ(r) near the Rh atoms. That is, the\ncharge distribution is rotated with the direction of the\nmagnetisation. In the same way, the Rh-based spin den-\nsities are affected by the magnetisation direction. They\nchange from [001]-aligned when m/bardbl[001] to [100]-aligned\nwhenm/bardbl[100], regardless of which 3 dtransition metal is\nused. The situation is different near the 3 dtransition\nmetals Fe and Co, where ∆ ρ(r) andσ(r) are affected\nvery differently by the magnetisation direction. The rea-9\n/s48/s99\n/s82/s104/s70/s101/s114\n/s48/s48/s49/s40 /s114 /s41/s91/s48/s48/s49/s93/s48/s46/s48\n/s48/s46/s51\n/s56/s46/s48\n/s50/s50/s54/s46/s52\n/s49/s48/s48/s48/s46/s48\n/s83/s98/s68/s114 /s40 /s114 /s41\n/s45/s48/s46/s50\n/s48/s46/s48\n/s48/s46/s50\n/s48 /s97/s48/s99\n/s82/s104/s67/s111\n/s91/s49/s48/s48/s93/s91/s48/s48/s49/s93/s48/s46/s48\n/s48/s46/s51\n/s56/s46/s48\n/s50/s50/s54/s46/s52\n/s49/s48/s48/s48/s46/s48\n/s83/s98\n/s82/s104\n/s50/s67/s111/s83/s98/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s82/s104\n/s50/s70/s101/s83/s98\n/s48 /s97/s91/s49/s48/s48/s93/s45/s48/s46/s50\n/s48/s46/s48\n/s48/s46/s50/s115 /s40 /s114 /s41/s59/s32 /s109 /s32/s124/s124/s32/s91/s48/s48/s49/s93 /s115 /s40 /s114 /s41/s59/s32 /s109 /s32/s124/s124/s32/s91/s49/s48/s48/s93\n/s48/s46/s48\n/s48/s46/s49\n/s49/s46/s48\n/s48 /s97/s91/s49/s48/s48/s93/s48 /s97/s91/s49/s48/s48/s93/s48/s46/s48\n/s48/s46/s49\n/s49/s46/s48\nFIG. 15. Electronic structure of Rh 2FeSb and Rh 2CoSb.\nFully relativistic charge ( ρ(r)) and spin ( σ(r)) density distributions in a (100)-type plane for magnetis ation parallel and\nperpendicular to the caxis are shown. ρ001(r) is the charge density for m/bardbl[001], and ∆ ρ=ρ001−ρ100is the difference between\nthe charge densities for m/bardbl[001] and m/bardbl[100]. (Note: colour bars are in atomic units.)\nson is the different occupation of 3 dvalence electrons of\nFe (nFe\nd= 6.6) and Co ( nCo\nd= 7.8), which are responsible\nfor the different spin moments. The overall differences in\nthe charge and spin densities at different magnetisation\ndirections result in different total energies.\nFinally, the gain or loss of energy with changes in the\nmagnetisation direction results in the magnetocrystalline\nanisotropy. The electronic structure of the two com-\npounds, Rh 2FeSb and Rh 2CoSb, differs depending onthe\nmagnetisation direction, which is reflected in the change\nin the anisotropy from the easy plane to the easy axis\nwhen Fe is replaced with Co.\nIV. CONCLUSIONS\nThe electronic and magnetic structure of tetrag-\nonal Heusler compounds with the composition\nRh2FexCo1−xSb were investigated by ab initio cal-\nculations. The calculations revealed that the magnetic\nmoment increases and the Curie temperature decreases\nwith increasing Fe content x. The Rh atoms have onlysmall, composition-independent magnetic moments.\nThe magnetic properties are determined by those\nof the Fe and Co atoms and thus depend strongly\non the composition. The total energies for various\nmagnetisation directions were calculated to determine\nthe magnetic anisotropy. The analysis is described in\ndetail in an extended Appendix. For bulk materials,\nthe magnetocrystalline anisotropy is found to be much\nstronger (by three orders of magnitude) than the dipolar\nanisotropy. Special attention was given to the borderline\ncompounds, Rh 2FeSb and Rh 2CoSb. The most striking\nresult was that a composition-dependent transition from\neasy-axis to easy-plane anisotropy occurs at an iron\nconcentration of approximately 40%.\nAppendix A: Disorder\n1. Rh 2CoSb with Co–Sb-type antisite disorder\nSupplementary calculations were performed for dis-\nordered Rh 2CoSb and Rh 2FeSb using Sprkkr us-10\ning the coherent potential approximation. For ex-\nample, the disordered compound may be written as\nRh2(Co1−x/2Sbx/2)(Cox/2Sb1−x/2), where xis the disor-\nder level. The result for x= 1, which denotes complete\nCo–Sb disorder, is illustrated in Figure 1(b). Alterna-\ntively, it can be assumed that disorder between the Co\nand Rh atoms decreases the magnetic moments, which is\nconsistent with the results of calculations of the inverted\nstructure in space group 119, but not with those when\nCo–Sb disorder is assumed, as shown below.\nThe evolution of the magnetic moments of Rh 2CoSb\nand Rh 2FeSb with increasing disorder is shown in Fig-\nure 16. The total magnetic moment in the fully disor-\ndered state is approximately 20% larger for Rh 2CoSb\nand approximately 10% larger for Rh 2FeSb than those\nof the compounds in the completely ordered state. The\norbital moments are nearly constant in both compounds\nand are independent of the degree of disorder ( x). The\ndecrease in the total moments with decreasing xis at-\ntributed to the decrease in the spin magnetic moments\nof both compounds.\n/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s52/s46/s48\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s65/s110/s105/s115/s111/s116/s114/s111/s112/s121/s32/s32/s32 /s75\n/s117/s40/s120 /s41/s32/s91/s77/s74/s47/s109/s51\n/s93\n/s82/s104\n/s50/s40/s67/s111\n/s49/s45/s120/s83/s98\n/s120/s41/s40/s67/s111\n/s120/s83/s98\n/s49/s45/s120/s41/s82/s104\n/s50/s40/s70/s101\n/s49/s45/s120/s83/s98\n/s120/s41/s40/s70/s101\n/s120/s83/s98\n/s49/s45/s120/s41/s77/s97/s103/s110/s101/s116/s105/s99/s32/s109/s111/s109/s101/s110/s116/s32/s32/s32 /s109 /s40/s120 /s41/s32/s91 /s109\n/s66/s93\n/s68/s105/s115/s111/s114/s100/s101/s114/s32/s108/s101/s118/s101/s108/s32/s32/s32 /s120/s32/s109\n/s116/s111/s116\n/s32/s109\n/s115\n/s32/s109\n/s108/s32/s109\n/s116/s111/s116\n/s32/s109\n/s115\n/s32/s109\n/s108\nFIG. 16. Disorder-induced changes in magnetic moments and\nanisotropy of Rh 2(T1−x/2Sbx/2)(Tx/2Sb1−x/2)T= Co, Fe.\nTotal, spin, and orbital magnetic moments are shown, along\nwith the uniaxial anisotropy constant Kuas a function of\ndisorder level x.\nFigure 16 shows that the type of anisotropy (easy axis\nfor Rh 2CoSb and easy plane for Rh 2FeSb) is retained\neven in the completely disordered state. However, the\nabsolute value of the second-order uniaxial anisotropy\nconstant Kudecreases. That is, the anisotropy becomes\nweaker with increasing disorder. The Kuvalues of both\ncompounds are approximately 70% higher in the com-\npletely disordered state. No direct correspondence is\nobserved between the behaviour of Kuand that of thespin, orbital, or total magnetic moments. The effects of\ndisorder and composition on the magnetic anisotropy of\nthe Co–Fe system were investigated using first principles\nCPA calculations by Turek et al.[20], who also observed\na decrease in anisotropy with increasing disorder.\n2. Off-stoichiometric alloys\nIn many experiments, 2:1:1stoichiometrywasnot fully\nreached, but an excess of Fe or Co and a deficiency of Sb\nwas obtained. In particular, the magnetic properties of\nRh2T1+xSb1−x, withT= Fe, Co, were calculated for\nx= 0.12. As in the study of disorder, the calculations\nwere performed using the CPA. The 2 asite is assumed\nto be occupied by 12% with Fe (or Co) and by 88% with\nSb, whereas the occupations of the 4 dand 2bsites are\nunchanged.\nThe calculated magnetic properties of the off-\nstoichiometric alloys are listed in Table V. The magnetic\nmoments and spin stiffness D0are enhanced in both al-\nloys, and the values are higher than those of the stoi-\nchiometric compound. In particular, the excess Co and\nFe atoms on the 2 asite contribute a large spin moment.\nThe total magnetic moment, ms+ml= 2.627µB, of\nRh2Co1.12Sb0.88is very similar to the experimentally ob-\nserved value of 2 .6µB. The Curie temperature of the\noff- stoichiometric Co-containing compound is slightly\nlower,whereasthatoftheFecompoundisslightlyhigher.\nThese findings, along with the spin stiffness results,\nsuggest that the exchange coupling parameters of the\nstoichiometric compounds differ from those of the off-\nstoichiometric alloys.\nThe type of anisotropy (easy plane or easy axis) is the\nsame in the off-stoichiometric alloys as in the stoichio-\nmetric compounds. The uniaxial anisotropy constants\ndiffer, however. They are enhanced in the Fe alloy and\nreduced in the Co alloy.\nAppendix B: Semi-relativistic band structures\nThe semi-relativistic band structures of Rh 2FeSb and\nRh2CoSb arecompared in Figure 17to illustrate the spin\ncharacteristics of the bands. The band structures are\nsimilar; the main differences result from the larger band\nfilling in the Co-based compound, which has one more\nvalence electron than the Fe compound. Further, the\nlarger spin splitting in the Fe compound clearly results\nin a large spin magnetic moment.\nAppendix C: Magnetocrystalline anisotropy\nIn this Appendix, the discussion of the magnetocrys-\ntalline anisotropy is extended beyond simple uniaxial ap-\nproximations. The magnetocrystalline energy of uniaxial11\nTABLE V. Calculated magnetic properties of off-\nstoichiometric Rh 2T1.12Sb0.88.\nSpinmsand orbital mlmagnetic moments per atom (Rh,\nCo, Fe) and those of the primitive cell ( total) are listed, as\nwell as the Curie temperature TCand spin stiffness D0.m2b\ns,l\nrepresents the magnetic moments at the original position,\nandm2a\ns,lrepresents the moments of the excess Fe and Co\natoms at the initial Sb position.\nRh2T1.12Sb0.88 T= Fe T= Co\nmRh\ns[µB] 0.308 0.244\nmRh\nl[µB] 0.012 0.009\nm2b\ns[µB] 2.988 1.691\nm2b\nl[µB] 0.092 0.140\nm2a\ns[µB] 3.564 2.521\nm2a\nl[µB] 0.072 0.155\nmtotal\ns[µB] 4.010 2.453\nmtotal\nl[µB] 0.124 0.174\nTC[K] 490 480\nD0[meV˚A2] 690 1100\nKu[MJ/m3] -1.667 0.826\n/s45/s54/s45/s52/s45/s50/s48/s50\n/s77 /s78 /s83\n/s48/s88 /s77\n/s48/s45/s54/s45/s52/s45/s50/s48/s50/s82/s104\n/s50/s70/s101/s83/s98/s69/s110/s101/s114/s103/s121/s32/s32/s32 /s69 /s40/s107 /s41/s32/s45/s32\n/s70/s32/s91/s101/s86/s93\n/s82/s104\n/s50/s67/s111/s83/s98/s69/s110/s101/s114/s103/s121/s32/s32/s32 /s69 /s40/s107 /s41/s32/s45/s32\n/s70/s32/s91/s101/s86/s93\n/s69/s108/s101/s99/s116/s114/s111/s110/s32/s109 /s111/s109 /s101/s110/s116/s117/s109 /s32/s32/s32 /s107\nFIG. 17. Semi-relativistic band structures of Rh 2FeSb and\nRh2CoSb.\nRed and blue indicate majority and minority states, respec-\ntively.\nsystems can be derived from the first principles total en-\nergiesfordifferentmagnetisationdirections(quantisation\naxes).\nTextbooks give different descriptions of the magnetic\nanisotropy, in particular, different equations for the\nanisotropy constants [21–26]. Therefore, care must be\ntaken when comparing the results of this workwith those\nof other studies or comparing other studies with eachother.\n1. Uniaxial magnetic anisotropy\nIt is often assumed that the magnetocrystalline\nanisotropy in tetragonal or hexagonal systems is simply\ndescribed by a second-order dependence on the angle θ\nbetween the caxis and the magnetisation direction, that\nis,\nKusin2(θ), (C1)\nwhereKuis the uniaxial anisotropy constant. In that\ncase,\nKu=E100−E001(C2)\nissimplycalculatedfromthedifferencebetweentheen-\nergies for magnetisation along the principal axes, c/bardbl[001]\nanda/bardbl[100]. For Ku>0, theeasyaxis is along the c\naxis, whereas Ku<0 describes an easyplane where c\nis thehardaxis. For a distinct magnetic anisotropy in\ntheabplane, it would be more accurate to use the lowest\nenergy of the two in-plane directions along the princi-\npal axis and the diagonal, which are the [100] and [110]\ndirections, respectively:\nKu= min(E100,E110)−E001. (C3)\nEquation (C1) has another serious drawback; namely,\nthe anisotropy is completely independent of the crystal\nlattice, and the anisotropic energy distribution always\nhas the same shape regardless of the c/aparameter and\nwhether the crystal has tetragonal, hexagonal, or some\nother structure. That is, Equation (C1) is ultimately\nuseful only for distinguishing between easyandhardc\naxes.\nNow we consider only tetragonal systems. By using\nthe series expansion/summationtextK2ν,0sin2ν(θ) up to the fourth\norder in sin( θ), the uniaxial magnetocrystalline energy is\nexpressed as\nEuniaxial\ncrys=K0+K2sin2(θ)+K4sin4(θ).(C4)\nThe equations for the sixth-order uniaxial anisotropy\nare discussed by Jensen and Bennemann [27], for exam-\nple. In the following, the subscript ”crys”is omitted, and\nthe energies are indexed only by direction or by ”uni”.\nFor the high-symmetry directions [ h,k,l] and the low-\nest indices ( h,k,l= 0,1), the energies depend on the\nanisotropy coefficients as follows:12\nE001=K0, (C5)\nE100=K0+K2+K4,or\nE110=K0+K2+K4,and\nE101=K0+K2sin2(θ101)+K4sin4(θ101),or\nE111=K0+K2sin2(θ111)+K4sin4(θ111).\nNote that the energiesfor the [100]and [110]directions\nare identical only when uniaxial anisotropy is assumed.\nThe energies for the [101] and [111] directions, however,\nhave different angles with respect to the caxis. From\nEquations (C4 and C5), K2andK4may be obtained,\nfor example, from the differences:\nE100−E001=K2+K4and (C6)\nE101−E001=K2sin2(θ)+K4sin4(θ).\nForz=c/a, the angle θis found using θ101=θ011=\narctan(1/z). FromEquation(C5)or(C6), theanisotropy\nconstants Kiare given by\nK0=E001, (C7)\nK2= (E101−E001)(z2+2)+(E101−E100)1\nz2,\nK4= (E001−E101)(z2+1)+(E100−E101)(z2+1)\nz2.\nAlternatively, E111andθ111= arctan(√\n2/z) may be\nused, but the resulting equations will have a different\ndependence on c/a. The uniaxial magnetocrystalline\nanisotropyenergy( Eu)isthedifferencebetweenthemag-\nnetocrystalline energy (here Euni) and the isotropic con-\ntribution, which is the spherical part K0:\nEu=Euni−K0. (C8)\nAccording to this equation, the uniaxial magnetocrys-\ntalline anisotropy energy may be positive or negative,\ndepending on the directions and values of K(see also\nAppendix C4).\nEquation (C4) has four extremal values at\nθi= 0,π\n2,and±arcsin/parenleftBigg/radicalbigg\n−K2\n2K4/parenrightBigg\n,(C9)\nwhere the first derivative of the fourth-order equation\n[Equation (C4)] vanishes, that is, for dEuniaxial/dθ= 0.\nThesolutions θ3,4arerealonlyiftheanisotropyconstants\nobey the relation 0 ≤−K2\n2K4≤1, that is, K2K4≥0,\n|K2| ≤2|K4|. ForK2=−2K4, one has θ3,4=±90◦. For\na realθ3,4, one has an easyor ahardcone. The resulting\nextremal energies areE(0) =K0, (C10)\nE(π/2) =K0+K2+K4,\nE(θ3,4) =K0−K2\n2\n4K4.\nThe minima or maxima are obtained using the second\nderivatives of the energy at the extremal angles:\nd2E\ndθ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n0= 2K2, (C11)\nd2E\ndθ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nπ/2=−2(K2+2K4),\nd2E\ndθ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nθ3,4=−2K2(K2+2K4)\nK4.\nThe minima appear for positive 2ndderivatives\n(d2E/dθ2/vextendsingle/vextendsingle\nθi>0)anddefinetheeasydirection(s)ofmag-\nnetisation. Indeed, one has to search for the absolute\nminimum and maximum to find the correct easy and\nhard axes, planes, or cones. An easy cone appears for\nK2<0,K4>−K2/2, and the corresponding cone angle\nis given by θ3,4. A special hard cone exists for K2>0,\nK4=−K2, where both the caxis and the abplane have\nthe same (lowest) energy. The energy barrier at the hard\ncone must be overcome, however, to change the magneti-\nsation direction from the easy axis to the easy plane and\nvice versa. In the range −∞< K4<−K2/2, the solu-\ntions are metastable when K2>0. The complete fourth-\norder uniaxial anisotropy phase diagram is presented in\nTable VI.\nTABLE VI. Uniaxial anisotropy phase diagram.\nabstands for basal plane, cstands for c-axis. Cones may have\nan opening angle θorπ/4 with respect to the caxis.\nK2 K4 easy hard\n>0 −∞···− K2 ab cone (θ)\n>0 −K2 ab,c cone (45◦)\n>0−K2···−K2/2 c cone (θ)\n>0 −K2/2···∞ c ab\n= 0 = K2= 0 undefined, spherical\n<0 −K2···∞ cone (θ) ab\n<0 −K2 cone (45◦) ab,c\n<0−K2/2···−K2 cone (θ) c\n<0−∞···− K2/2 ab c\nThe magnetic anisotropy phase diagram for fourth-\norder uniaxial anisotropy is displayed in Figure 18. It is\nsimilartothegraphicalrepresentationsreportedinRefer-\nences[27,28]. Thedifferentphasesaredistinguished. For\nK4=−K2<0, there is a distinct metastable case with\nequal energies for magnetisation along the caxis and in13\ntheabplane. At this line, a transition occurs from easy-\naxistoeasy-planebehaviour. Inthemetastableregionfor\nK4<−K2/2<0, easy-axis behaviour appears, whereas\neasy- plane behaviour appears for K4<−K2<0 (see\nTable VI). In both cases, the sizes of the anisotropy con-\nstants determine how easily one state can switch to the\nother and the stability of the state with lower energy.\nThe energy barrier to cross the hard cone has a size of\n−K2\n2\n4K4, as mentioned above.\n/s75\n/s52\n/s75\n/s50/s101/s97/s115/s121/s32/s112/s108/s97/s110/s101/s101/s97/s115/s121/s32/s99/s111/s110/s101 /s101/s97/s115/s121/s32/s97/s120/s105/s115\n/s109/s101/s116/s97/s32/s115/s116/s97/s98/s108/s101\nFIG. 18. Magnetic anisotropy phase diagram.\nIn themetastability range,hard-cone -type anisotropy occurs.\nIn the sketches of the E(θ) distributions, it is assumed that\nK0= 0.\n2. Tetragonal magnetic anisotropy\nThe uniaxial magnetic anisotropy does not reflect the\nsymmetry of the crystal structure. The symmetry of the\nanisotropy should generally be the same as the symme-\ntry of the crystal potential; thus, it is given by the fully\nsymmetric irreducible representation of the point group,\nthat is,a,a1,ag, or similar. Again, by using a series\nexpansion up to the fourth order in sin( θ), the magne-\ntocrystalline energy of a tetragonal system is expressed\nas\nEtetragonal\ncrys =2/summationdisplay\nν=0K2ν,0sin2ν(θ)+K4,4sin4(θ)f(φ),\n=Euniaxial\ncrys+K4,4sin4(θ)f(φ),\nf(φ) = cos(4 φ).\nf(φ) has an azimuthal dependence on 4 φ, which re-\nsults in the expected fourfold symmetry. Some worksusedf′(φ) = sin4(φ)+cos4(φ), which results in different\nequations and Kvalues. Higher-order approximations\nwill include terms with K6,0,K6,4,K8,0,K8,4,K8,8, and\nso on. Subtracting K0from Equation (C12) yields\nEa(/vector r) =K2,0sin2(θ)+[K4,0+K4,4cos(4φ)]sin4(θ).\n(C12)\nIn the following, the subscript ”crys” is omitted, and\nthe energies are indexed only by direction or by ”tet”.\nFor the high-symmetry directions [ h,k,l] and the lowest\nindices (h,k,l= 0,1), Equation (C12) gives\nE001=K0,0, (C13)\nE100=K0,0+K2,0+K4,0+K4,4,\nE110=K0,0+K2,0+K4,0−K4,4,and\nE101=/summationdisplay\nν=0,2K2ν,0sin2ν(θ101)+K4,4sin4(θ101),or\nE111=/summationdisplay\nν=0,2K2ν,0sin2ν(θ111)−K4,4sin4(θ111).\nForz=c/a, the angle θ101is found using θ101=\nθ011= arctan(1 /z). Alternatively, E111withθ111=\narctan(√\n2/z) may be used. From the first four ener-\ngies of Equation (C13), the anisotropy constants Kl,m\nare found to be\nK0,0=E001, (C14)\nK2,0= (E101−E001)(z2+2)\n+(E101−E100)1\nz2,\nK4,0= (E001−E101)(z2+1)\n+(E100−E101)1\nz2\n+1\n2(E100+E110−2E101),\nK4,4=1\n2(E100−E110).\nThe magnetocrystalline anisotropy energy ( Ea) is the\ndifference between the magnetocrystalline energy (here\nEtet) and the isotropic contribution, which is the spher-\nical part K0:\nEa=Etet−K0. (C15)\n3. Dipolar magnetic anisotropy\nIn non-cubic systems, the dipolar anisotropy does not\nvanish and also contributes to the magnetocrystalline\nanisotropy. It is calculated from a direct lattice sum\nyielding the dipolar energy:14\nEdip(/vector n) =µ0\n8π/summationdisplay\ni/negationslash=j/bracketleftBigg\n/vector mi·/vector mj\nr3\nij−3(/vector rij·/vector mi)(/vector rij·/vector mj)\nr5\nij/bracketrightBigg\n,\n(C16)\nwhere/vector n=/vectorM/Mis the magnetisation direction, and\nrijrepresents the distance vectors between the magnetic\nmoments miandmj. The individual magnetic moments,\n/vector miand/vector mj, do not necessarily have to be collinear in\ngeneral.\nIn a simplified picture, only the 3 dtransition elements\nTcarry a significant magnetic moment in the Rh 2TSb\ncompoundsinvestigatedhere. Inallcasesofasinglemag-\nneticionwhereallthemagneticmomentsinthestructure\nare collinear along /vector n, the equation simplifies to\nEdip(/vector n) =µ0m2(/vector n)\n8π/summationdisplay\ni/negationslash=j1\nr3\nij/bracketleftBigg\n1−3r2\nn,ij\nr2\nij/bracketrightBigg\n,\n=µ0m2(/vector n)\n8π/summationdisplay\ni/negationslash=j1−3cos2(θij)\nr3\nij,(C17)\nwherern,ij=rn,ij(/vector n) is a projection of the position\nvector onto the direction of the magnetic moment, and\nθijis the angle between them. In Equation (C17), the\nsign of the energy is completely defined by the crystal\nstructure when the summation is over a spherical parti-\ncle. Note that the size of the magnetic moment, m(/vector n),\ndepends on the magnetisation direction when the spin–\norbit interaction is taken into account.\nFinally, the dipolar anisotropy is given by the differ-\nence between the energies for two different directions,∆Edipaniso=E(/vector n2)−E(/vector n1). Again, the two well- dis-\ntinguished directions are the /vector n1= [001] and /vector n2= [100]\ndirections, which are along the caxis and in the basal\nplane along a, respectively. Positive values indicate an\neasy dipolar direction that is along the [001] axis. It has\na second-order angular dependence.\n4. Plotting the magnetic anisotropy\nAccording to Equations (C8) and (C15), the magne-\ntocrystalline anisotropy energy may be positive or neg-\native, depending on the direction of ( θ,φ) and the K\nvalues. Consequently, it is difficult to visualise the\nanisotropy energy by plotting the three-dimensional dis-\ntribution of Ea(/vector r) =Ea(θ,φ). Therefore, the alternative\nanisotropy energy Ea′with respect to the lowest energy\nis generally plotted, where\nEa′=Ea−min(Ea), (C18)\nwhich is still positive even when Ea<0. The easy\ndirections or planes are identified as those for which\nEa′= 0.Ea′is used to plot the magnetocrystalline\nanisotropy in the main text.\nACKNOWLEDGMENTS\nWe thank the groups of P. Blaha (Vienna) and H.\nEbert (Munich) for providing their computer codes.\n[1] S. K. Dhar, A. K. Grover, S. K. Malik, and R. Vija-\nyaraghavan, Peaks in low field a.c. susceptibility of ferro-\nmagnetic Heusler alloys, Sol. St. Comm. 33, 545 (1980).\n[2] S. V. Faleev, Y. Ferrante, J. Jeong, M. G. Samant,\nB. Jones, and S. S. P. Parkin, Heusler compounds with\nperpendicular magnetic anisotropy and large tunnel-\ning magnetoresistance, Phys. Rev. Materials 1, 024402\n(2017).\n[3] P.Blaha, K.Schwarz, P.Sorantin,andS.B.Trickey,Full -\npotential, linearized augmented plane wave programs for\ncrystalline systems, Comput. Phys. Commun. 59, 399\n(1990).\n[4] K. Schwarz and P. Blaha, Solid state calculations using\nWIEN2k, Comput. Mater. Sci. 28, 259 (2003).\n[5] P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka,\nand J. Luitz, WIEN2k: An Augmented PlaneWave + Lo-\ncal Orbitals Program for Calculating Crystal Properties\n(Wien, 2013).\n[6] H. Ebert, Fully relativistic band structure calculatio ns\nfor magnetic solids - formalism and application, in Elec-\ntronic Structure and Physical Properties of Solids. The\nUse of the LMTO Method , Lecture Notes in Physics, Vol.\n535, edited by H. Dreysse (Springer-Verlag, Berlin, Hei-delberg, 1999) pp. 191 – 246.\n[7] H. Ebert, D. K¨ odderitzsch, and J. Minar, Calculat-\ning condensed matter properties using the KKR-Greens\nfunction method – recent developments and applications,\nRep. Prog. Phys 74, 096501 (2011).\n[8] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized\nGradient Approximation Made Simple, Phys. Rev. Lett.\n77, 3865 (1996).\n[9] H. C. Kandpal, G. H. Fecher, and C. Felser, Calculated\nelectronic and magnetic properties of the half-metallic,\ntransition metal based Heusler compounds, J. Phys. D:\nAppl. Phys. 40, 1507 (2007).\n[10] G. H. Fecher, S. Chadov, and C. Felser, Theory of the\nhalf-metallic heusler compounds, in Spintronics , edited\nbyC.FelserandG.H.Fecher(SpringerVerlag,Dordrecht\nHeidelberg New York London, 2013) Book section 7, p.\n115.\n[11] S. Mankovsky, G. H. Fecher, and H. Ebert, Electronic\nstructure calculations in ordered and disordered solids\nwith spiral magnetic order, Phys. Rev. B 83, 144401\n(2011).\n[12] J. Thoene, S. Chadov, G. Fecher, C. Felser, and\nJ. K¨ ubler, Exchange energies, Curie temperatures and15\nmagnons in Heusler compounds, J Phys. D: Appl. Phys.\n42, 084013 (2009).\n[13] P. Soven, Coherent-Potential Model of Substitutional\nDisordered Alloys, Phys. Rev. 156, 809 (1967).\n[14] S. A. Khan, P. Blaha, H. Ebert, J. Minar, and O. Sipr,\nMagnetocrystalline anisotropy of FePt: A detailed view,\nPhys. Rev. B 94, 144436 (2016).\n[15] S. J. Joshua, Symmetry principles and magnetic symme-\ntry in solid state physics. (Adam Hilger, IOP Publishing\nLtd., Bistol, Philadelphia, New York, 1991).\n[16] R. Lizarraga, L. Nordstr¨ om, L. Bergqvist, A. Bergman,\nE. Sj¨ ostedt, P. Mohn, and O. Eriksson, Conditions for\nNoncollinear Instabilities of Ferromagnetic Materials,\nPhys. Rev. Lett. 93, 107205 (2004).\n[17] A.I.Liechtenstein, M. I.Katsnelson, andV.A.Gubanov ,\nExchange interactions and spin-wave stiffness in ferro-\nmagnetic metals, J. Phys. F: Met. Phys. 14, L125 (1984).\n[18] A.I.Liechtenstein, M.I.Katsnelson, V.P.Antropov,a nd\nV. A. Gubanov, Local Spin Density Functional Approach\nto the Theory of Exchange Interactions in Ferromagnetic\nMetals and Alloys, J. Magn. Magn. Mater. 67, 65 (1987).\n[19] M. Pajda, J. Kudrnovsky, I. Turek, V. Drchal, and\nP. Bruno, Ab initio calculations of exchange interactions,\nspin-wave stiffness constants, and Curie temperatures of\nFe, Co, and Ni, Phys. Rev. B 64, 174402 (2001).[20] I. Turek, J. Kudrnovsky, and K. Carva, Mag-\nnetic anisotropy energy of disordered tetrago-\nnal Fe-Co systems from ab initio alloy theory,\nPhys. Rev. B 86, 174430 (2012).\n[21] R. Skomsky and J. M. D. Coey, Permanent Magnetism ,\nStudiesinCondensed Matter Physics(Taylor andFrancis\nGroup, New York, 1999).\n[22] J. K¨ ubler, Theory of Itinerant Electron Magnetism\n(Clarendon Press, Oxford, 2000).\n[23] B. D. Cullity and C. D. Graham, Introduction to Mag-\nnetic Materials; 2nd Ed. (JohnWileyandSons,Hoboken,\n2009).\n[24] J.M.D.Coey, Magnetism and Magnetic Materials (Cam-\nbridge University Press, Cambridge, 2010).\n[25] N. A. Spaldin, Magnetic Materials, 2nd Ed. Fundamen-\ntals and Applications (Cambridge UniversityPress, Cam-\nbridge, 2011).\n[26] K. M. Krishnan, Fundamentals and Applications of Mag-\nnetic Materials (Oxord University Press, Oxord, 2016).\n[27] P. J. Jensen and K. H. Bennemann, Magnetic structure\nof films: Dependence on anisotropy and atomic morphol-\nogy, Srf. Sci Rep. 61, 129 (2006).\n[28] R. Skomski, H.-P. Oepen, and J. Kirschner, Unidi-\nrectional anisotropy in ultrathin transition-metal films,\nPhys. Rev. B 58, 11138 (1998)."    },    {        "title": "2009.01034v1.New_highly_anisotropic_Rh_based_Heusler_compound_for_magnetic_recording.pdf",        "content": "1 \n New highly -anisotropic Rh-based Heusler compound  for \nmagnetic recording   \nYangkun He1,*, Gerhard H. Fecher1,*, Chen guang Fu1, Yu Pan1, Kaustuv Manna1, Johannes \nKroder1, Ajay Jha2, Xiao Wang1, Zhiwei Hu1, Stefano Agrestini3, Javier Herrero -Martí n4, \nManuel Valvidares4, Yurii Skourski5, Walter Schnelle1, Plamen Stamenov2, Horst Borrmann1, \nLiu Hao Tjeng1, Rudolf Schaefer6,7, S. S. P. Parkin8, J. M. D. Coey2 and Claudia Felser1 \n \n1Max-Planck -Institute for Chemical Physics of Solids, D -01187 Dresden, Germany  \n2School of Physics, Trinity College, Dublin 2, Ireland  \n3 Diamond Light Source, Harwell Science and Innovation Campus, Didcot, OX11 0DE, \nUK \n4ALBA Synchrotron Light Source, Cerdany ola del Valles, 08290 Barcelona, Catalonia, \nSpain  \n5 Dresden High Magnetic Field Laboratory (HLD -EMFL), Helmholtz -zentrum \nDresden–Rossendorf, 01328 Dresden, Germany  \n6Leibniz Institute for Solid State and Materials Research (IFW) Dresden, Helmholtz  \nstrasse  20, D -01069 Dresden, Germany  \n7Institute for Materials Science, TU Dresden, D -01062 Dresden, Germany  \n8Max Planck Institute of Microstructure Physics, Halle, Germany  \nKey words: magnetocrystalline anisotropy, tetragonal Heusler  alloy , magnetic \nhardness parameter , 4d magnetism , magnetic recording  \n \nThe development of high -density magnetic recording media is limited by the \nsuperparamagnetism in very small ferromagnetic c rystals . Hard magnetic \nmaterials with strong perpendicular anisotropy offer stability and high  recording  \ndensity . To overcome the difficulty of writing media with a large coercivity,  heat \nassisted magnetic recording (HAMR) has been developed , rapidly heat ing the \nmedia  to the Curie temperature Tc before  writing , followed by rapid c ooling. \nRequirements are a suitable Tc, coupled with anisotropic thermal conductivity  \nand hard magnetic properties . Here we introduce Rh 2CoSb  as a new hard \nmagnet  with potential  for thin film magnetic recording . A magnetocrystalline \nanisotropy of 3.6 MJm-3 is combined with  a saturation magnetization of  μ0Ms = \n0.52 T at 2 K (2.2 MJm-3 and 0.44 T at room -temperature) . The magnetic \nhardness parameter of 3.7 at room temperature is  the highest observed for a ny \nrare-earth free  hard magnet . The anisotropy is related to an unquenched orbital \nmoment of 0.42 µB on Co, which is hybridized with neighbouring  Rh atoms with  \na large spin -orbit interaction.  Moreover, the pronounced \ntemperature -dependence of the anisotropy that follows from its Tc of 450 K, \ntogether with a  high  thermal conductivity of 20 Wm-1K-1, makes Rh 2CoSb  a 2 \n candidate for development for heat assisted writing with a recording density in \nexcess of 10 Tb/in2. \n \nThe pace of doubling of information density on magnetic recording media has \nslackened  in recent years, as the effective  size of the perpendicularly recorded grains \napproached  the superparamagnetic blocking diameter , which is the low er size  limit for \nstable  ferromagnetism , directly related to the magnetocrystalline anisotropy  energy  K1. \nTo resist demagnetization by random thermal fluctuation, the volume  V of a magnetic \nmaterial must satisfy  the empirical condition that K1V/kBT > 60 (K1V > 1.5 eV), where  \nkB and T are the Boltzmann constant and ambient temperature , respectively1-3. For \nhigh-density magnetic recording  media , strong  perpendicular  uniaxial anisotropy is  \nrequired . To create a film for magnetic recording or a permanent magnet that remains \nfully magnetized regardless of its shape, K1 should exceed µ 0Ms2, where Ms is the \nspontaneous saturation magnetization. Therefore , the magnetic hardness parameter κ \n= \n , a convenient figure of merit for permanent magnets4, should be \nlarger than 1 . This is  difficult to achieve in rare-earth free  materials , other than CoPt \nor FePt  with L1 0 structure.  \nThe development of modern  magnetic recording media spanned three  \ngenerations. The first generation for tapes  and discs depended on the s hape anisotropy \nof acicular fine particles of ferrimagnetic or ferromagnetic oxides, γFe2O3 or CrO 25, \nwhich were magnetized  in-plane. The second generation  was based on  hexagonal  \nCo-Cr-Pt thin films  with perpendicular magnetization , which is used in current  hard \ndiscs6. But Cr decreases  the K1 of Co-Pt alloy , limiting the recording density . An ideal 3 \n material for recording should be hard for storage and soft for writing , because of the \nlimited fields that can be generated by the miniature writing electromagnet. L1 0 FePt, \na hard magnet with K1 = 6.6 MJm-3 and κ = 2.02,  is the basis of new , third generation  \nof heat-assisted magnetic recording  (HAMR ) media7,8, where writing  is realized by \nheating the material close  to its Curie point  with a laser -powered near -field \ntransducer9. However, its high Tc of 750 K makes  rapid  heating and cooling  \nproblem atic and t akes time10. To achieve  a large temperature -variation  in K1, 10 % Cu \nis doped to obtain  a 100 K reduction in Tc, but this is accompanied by a big drop of K1 \nto 0.8 MJm-33,11. Additionally , unavoidable disordered A1 FePt impurities with a \nsmall er K1 and a lower Tc make the stabili zation of its  properties difficult12. Therefore, \nit is useful  to look for new materials with  a strong K1, a fairly suitable  Tc and good \nthermal conductivity  for development as new thin film media  \nTo achieve sufficient anisotropy, a non-cubic crystal structure (for example \ntetragonal or hexagonal) and a large spin -orbit interaction  with the right sign  is \nneeded . When looking  for new materials with  uniaxial magnetocrystalline anisotropy , \na significant deviation of the  c/a ratio from 1 for tetragonal or \n  for hexagonal  \nstructures , is sought . Moreover, it is important to pair a 3 d metal (Mn, Fe, Co) that \nprovide s a substantial magnetic moment with a heavy atom that enhances the \nspin-orbit interaction.  Rh is a 4 d element where the  spin-orbit interaction  is large r \nthan in  3d elements , and it acquires a small induced magnetic moment when paired \nwith a 3 d metal . Therefore, it is worthwhile looking for new opportunities in  magnetic \nrecording among  Rh-based alloys  with a tetragonal or hexagonal structure . 4 \n Heusler alloys are a large family of compounds with formula X 2YZ, where X \nand Y are  usually transition metals and Z is a main group element13,14. The abundant \nchoice of elements provides great scope to search for new materials with specific \nproperties. Rh2CoSb has a c/√2a ratio of 1.24, the largest among all reported \nRh-based magnetic Heusler compounds ( c/√2a is used to describe the tetragonal \ndistortion relative to a cubic lattice). A magnetic moment  per formula  of 1.4 μB in an \n0.7 T applied field and a Tc of about 450 K were reported  for polycrystalline samples \nmany years ago15. Recent  ab-initio  calculations proposed uniaxial anisotropy with an \neasy c-axis16. Both experiments and calculations suggest that Rh 2CoSb might be an \ninteresting hard magnetic material .  \nHere we study the  anisotropic  magnetic , thermal and transport  properties , as well \nas the  temperature -dependent anisotropy of single crystals of Rh 2CoSb and investigate \nthe contribution of rhodium to the magnetism.  \nResults  \nCrystal structure  \nMagnetic Heusler alloys with a  tetragonal  structure are often  Mn-based  with \nspace group I\nm217-22. However, the Mn-Mn magnetic coupling is  usually \nantiferromagnetic  at the nearest -neighbo ur separation  in this structure , leading to \nferrimagnetic order  with relatively low magnetization  (Figure 1a). Some tetragonal \nRh-based Heusler alloys including  Rh2Mn 1.12Sn0.88, Rh 2FeSn, and Rh 2CoSn  and \nRh2CoSb15 with 3 d transition element s Mn, Fe and Co, crystallize in a tetragonal \nstructure  with space group I4/mmm (D0 22 structure).  The 4 d element Rh provides a 5 \n strong spin orbit interaction  that contributes to the magnetocrystalline anisotropy in \nthe tetragonal lattice23. \nBoth XRD  and TEM  experiments  show  that Rh 2CoSb has a well -ordered \ntetragonal D022 structure with a = 4.0393 (6)  Å and c = 7.1052 (7)  Å. Here,  the 4d (0 \n0 ¼) site is occupied by Rh, 2b (0 0 ½) by Co, and 2 a (0 0 0) by Sb as shown in \nFigure 1c. The tetragonal distortion c/\na = 1.2436(3)  agree s with the published  \nvalue15, being the largest among all reported Rh -based magnetic Heusler compounds.  \nA sketch of the  easy-axis energy surface  is included in Fig ure 1c. \nUnlike many  Mn-based Heusler compounds, which are often  cubic at high \ntemperature and may undergo a martensitic phase transition to become tetragonal, \nRh2CoSb does not have a first -order transition at high temperature and remains in the \ntetragonal phase at  all temperatures  up to the melting point of 1482 K. A differential \nscanning calorimeter (DSC) measurement showing only the melting transition is \npresented in Supplementary Information . Since the phase melts congruently, single \ncrystals can be grown by the  Bridgeman method. They were cut along the c and a \naxes for further measurements and fully characterized by x-ray diffraction (XRD), \nwavelength dispersive x -ray spectroscopy (WDX) and transmission electron \nmicroscopy (TEM) in Supplement al Information . WDX establishes a homogenous \ncomposition of Rh 50.3Co25.6Sb24.1(see Supplement).  Errors in composition range from \n0.1 – 0.2 % . 6 \n \nRhSb\nE\nZ\nEc\nEb EaCo\nabc\n1.0 1.1 1.2 1.301234\nMs (B/f.u.)\nc / 2aRh2CoSb Rh2CoSnRh2Mn1.12Sb0.88\nRh2FeSbRh2FeSn\nMn2RhSn\nMn2NiGa Mn2FeGa\nMn3GaMn2RhSn\nMn2PtIn\nMn2PtGa\n(a) (b)\n(c) \nFigure. 1. Tetragonal  structure of Rh 2CoSb. a, Tetragonal distortion versus magnetic moment  \nfor Mn -based and Rh -based  Heusler  compound s. Rh-based compound s show  both large \ndistortion and  large  magnetization.  Note that some compounds do not saturate  in the  applied \nmagnetic  fields of 7 T  for Mn 2PtGa, 10 T for Mn 2FeGa,  6.6 T for Rh2CoSn  and Rh2FeSn, and \n0.7 T for Rh2FeSb  and Rh2Mn 1.12Sb0.88. b, Comparison of the magnetic hardness parameter κ \nwith other  hard magnets. The light grey plane marks the threshold κ = 1. c, Unit cell of \ntetragonal Heus ler compound Rh 2CoSb, whose magnetocrystalline anisotropy surface shows  \neasy-axis anisotropy  along c. \n \nMagnetic properties  \nMagnetic properties were measured along  the c and a axes of single crystals as \nillustrated  in Figure 2. A magnet o-optical Kerr microscopy study24 shows surface \ndomains of a two -phase branched domain pattern of higher generation on the (001) 7 \n surface, which is commonly  observed in uniaxial magnets. At 2 K , the magnetization \nalong c saturates easily , at 37.5 Am2kg-1. Taking the density of 11 ,030 kg m-3 into \nconsideration , the saturation magnetization  μ0Ms is 0.52 T, which is similar to  that of  \npure nickel . It correspond s to a moment of  2.6 μB per Rh2CoSb formula  unit. The \nnon-integer magnetic moment per unit cell indicates that Rh2CoSb is not a half metal. \nHowever, a field  μ0Ha of 17. 5 T is required to saturate  the sample  along  the a axis, \nwhere  Ha is the anisotropy field.  The magnetocrystalline anisotropy  constant  K1 = ½ \nμ0HaMs is 3.6 MJm-3, which is larger than  for any other  rare-earth free  compound \nexcept for CoPt and FePt . Its relatively small Ms makes   larger than for these \nmaterials and  coercivity may exceed that of FePt3. Both μ0Ms and K1 decrease with \nincreasing temperature, as shown in Fig ure 2b, but at room  temperature they are still  \n0.44 T and 2.2 MJm-3, respectively.  Big Barkhausen jumps  observed during c axis \ndemagnetization , together with a high initial susceptibility after thermal \ndemagnetization indicat e that single -crystal Rh2CoSb i s a nucleation type magnet25,26. \nDetailed measurements of the magnetization along c and a at different temperatures \nare reported in the Supplement al information . \nThe Curie temperature  is deduced to be  450 K  from the temperature scans of \nmagnetization in a 10 mT field.  The M-T curve  measured  along the hard axis in a \nmagnetic field  of 0.5 T exhibit s a sharp peak at 440 K , indicating a rapid built -up of \nanisotropy  just below Tc. The slope d K1/dT is ~ 20 kJm-3K-1 (twice as large as for \nFePt27). The ac -susceptibility  χ near Tc is shown in Figure 2d. It exhibit s a typical \nHopkinson peak  along  the c axis15,28. Since χ ∝ Ms2/K1, during cooling the rate of  8 \n increase of  Ms and K1 vary at different temperatures near Tc, giving a trough in ac \nsusceptibility below the Hopkinson peak . There is nothing unusual in  our M-T and \nM-H measurement s at this temperature.    \nWe compare in Fig ure 1b the magnetic hardness parameter of Rh 2CoSb with \nother materials that exhibit  μ0Ms > 0.4 T, which is taken as a threshold necessary for \nuseful stray fields.  κ is a practical  figure of merit for hard magnet ic materials  that \nmust be  be greater  than 1 if the material is to resist  self-demagnetization when \nfabricated into any desired  shape4. Rh 2CoSb has κ = 4.1 at 2 K and 3. 7 at room \ntemperature, which is more than any other rare-earth free  magnet. Only SmCo 5 has a \nlarger value.  \nAll the indications are  that Rh2CoSb has the potential to be a good  hard magnet \nwith strong uniaxial anisotropy.  There is no  structural  transition, avoiding the \nproblem s of decomposition ( MnBi decomposes at 628 K) or twinning (like many  \nnearly -cubic  rare-earth free  magnets such as  MnGa, MnAl, FePt  and CoPt29). Unlike \nmany rare -earth magnets, the sample is stable  in air  and its magnetic properties have  \nbeen found to  remain unchanged for a year.  \nThanks to the strong K1 at 300 K , Rh2CoSb promise s a significant reduction in \nthermally stable grain size from diameter D=7–9nm in today's perpendicular \nCo-Cr-Pt media down to D=4–5nm in future Rh 2CoSb media, resulting in a potential  \nstorage density of more than 10 Terabit /inch2 (for D = 5 nm  with half area occupied ). \nDetailed calculation can be found in Supplemental information . HAMR media with \ngrain diameters of only a few nm are hard to fabricate because the grains musts be 9 \n exchange decoupled by a n intergranular material.  Powder XRD data  revealed  the \npresence of a few percent of a RhSb secondary phase in our polycrystalline samples. \nNonmagnetic RhSb appears at the grain boundar ies, pinning the domain s and creating \nhysteresis . It is  a candidate for separating the nanocrystal grains in Rh2CoSb thin film \nmedia .  \n0 5 10 15 20 25 30010203040\n0 100 200 300 400 5000510152025303540\n300 350 400 450 5000.000.010.020.030.040.050.060.07\n25 μmMagnetization (A m2 kg-1)\nMagnetic field (T)2 K H//a\n300 K H//a300 K H//c2 K H//c(a)\n(c) (d)(b)\n0 100 200 300 400 50001234\nTemperature (K)K1 (MJm-3)\n01234\n Magnetization (A m2 kg-1)\nTemperature (K)   0.5 T  0H // c\n 0.01 T  0H // c\n   0.5 T  0H // a\n 0.01 T  0H // a\n\nH // cH // a\n    \n '\nTemperature (K)Tc \n \nFigure 2. Magnetic properties of Rh 2CoSb.  a, Magnetization curve s at 2 K and 300 K with \nthe field along  the a or c axes.  The insert shows a pattern of  branch ed domain s at the surface \nperpendicular  to the c axis, typical of  a strong uniaxial ferro magnet . b, Magnetocrystalline \nanisotropy calculated from  magnetization curves at different temperatures. The blue line 10 \n shows  the magnetic hardness parameter κ at different temperatures . c, Magnetization at \ndifferent temperatures with the field along a or c axes. For better visibility s ome values are \nenlarged by a factor of 10 or 100 , depend ing on the field direction and the value  of H. d, \nAC-susceptibility near Tc, showing a typical Hopkinson peak , due to the rapid  increase of \nanisotropy  just below Tc. \nTransport properties.  \nTransport properties  of Rh 2CoSb are presented in Figure  3. The resistivity is  \nvery anisotropic. With increasing temperature, the  a-axis resistivity increases \nmonoton ically from 53 µΩ cm at 2 K  to 192 µΩ cm at Tc, after which the resistivity , \ndominated by  spin disorder scattering , tends to saturate30. The  c-axis resistivity is  less \nthan half as large  at 2 K , 21 µ Ω cm, but it increas es with  temperature and show s a \nsimilar trend  to the a-axis resistivity . The difference is due to intrinsic mobility and \nextrinsic d omain wall scattering perpendicular to c axis31. Magnetoresistance and Hall \nmeasurements are also very anisotropic (see Supplement) . \nThe Seebeck coefficient is about -10 WK-1m-1 at 300 K  along both axes (c and a) \nwith an error of ±10%. The opposite sign s of the Hall effect (positive) and Seebeck \ncoefficient  (negative) indicate  the co -existence of  both light holes and heavy electrons \nat the Fermi energy32. Detailed data are presented  in the Supplement . The spin \npolarization P of the electrons at the Femi level was deduced from  a point contact \nAndreev ref lection measurement at 2 K (see Supplement). The measured value of P is \n13 %  and agree s with well with the calculated spin polarization of the density of states \nat the Fermi energy (see Supplement). The transport polarization will be different due 11 \n to different effective masses of minority and majority electrons33. \nThe measured thermal transport properties in Fig ures. 3b and 3c along c and a \naxes are also highly anisotropic. The total thermal conductivity along c is about twice \nas large as along a, and it is mainly explained  by the carrier contribution , following \nthe Wiedemann –Franz law34. The remaining, almost isotropic, part is dominated by \nthe phonon cont ribution.  The slight upturn of the thermal conductivity at temperature s \nabove 200 K may be  due to uncertainty in the radiative heat losses , which is estimated \nto be about  ±10%. In addition, magnon s or electron -magnon interactions  might \ninfluence the thermal conductivity . The limiting  Lorenz number generally depends a \nlittle on temperature34. The anisotropy r eflects the anisotropic electronic structure, \nwhich is the origin of the giant magnetocrystalline anisotropy. The c-axis thermal \nconductivity of 20 Wm-1K-1 at room temperature is roughly twice  that of unsegregated  \nL10 FePt (11-13 Wm-1K-1 35,36), or A1 FePt  (9 Wm-1K-1 36). The anisotropic transport \nproperties, including magnetoresistance , anomalous Hall effect , and Seebeck effect , \nresult from  the anisotropic  electronic structure , that leads to an anisotropic mobility of \nthe charge carriers.  \n \nFigure. 3. Transport properties of Rh 2CoSb. a, Longitudinal resistivity along the c (red curve) \nand the a axis (blue curve).  b and c,  Thermal conductivity along  the c and a axes and the 12 \n charge carrier contribution (estimated from  the Wiedemann –Franz law as LT/ρxx, where L = \n2.44 WΩK−2 is the Lorenz number) . The remaining part is mainly  the phonon  contribut ion. \n \nDiscussion  \nLittle information  is available  about the magnetism of hard magnetic 3 d-4d \nintermetallic compounds, other than  FePd, which has the tetragonal L 10 structure with \nK1 = 1.8 MJm-3, and YCo 5 which has the hexagonal CaCu 5 structure5 with K1 = 6.5 \nMJm-3. A number of  ternary , tetragonal Rh -based intermetallics are known to order \nferro magnetically  with a Curie point above room temperature15. Only one atom out of \nfour in the Rh2CoSb formula is cobalt,  but the high Tc of 450 K indicates a strong \nexchange interaction . We see from  our ab-initio  calculations (Supplement al \ninformation ) that Rh is clearly in a spin -polarized state , and that it contributes to the \nferromagnetism.  The measured  magnetic  moment  per formula  of 2.6 μB is much \nhigher than  the ordinary  ~1.6 μB moment  of Co  in the elemental  state or in metallic \nalloys with other 3 d elements . The difference should  be attributed , at least partially , to \nRh, or else to enhanced Co spin and orbital moment s, as there is no magnetic \ncontribution  from  Sb.  \nWe also prepared polycrystalline Rh2FeSb with the same crystal structure, which \nexhibits easy-plane  magnetization . At 300 K under 7 T, the incomplete ly saturated \nmoment reaches 3.8 μB per formula , far greater than the 2.2 μB of metallic iron.  The \nCurie temperature  of 510 K is  even  higher than  that of  Rh2CoSb , which is in contra st \nwith other  isostructural Fe and Co intermetallics . A detailed comparison of the spin 13 \n and orbital contributions for  Rh2FeSb and Rh 2CoSb  deduced  from ab-initio  \ncalculation s is provided  in the Supplement al Information . All the evidence illustrate s \nthat Rh plays an important role in the ferromagnetism  of the se ternary compounds . In \nfact, it has already been shown  to carry an induced moment  in binary Fe -Rh and \nCo-Rh alloys that is roughly one third of the  3d moment37,38.  \nTo reveal the site specific magnetic moments of Rh and Co , X-ray magnetic \ncircular dichroism (XMCD) measurements were performed at the L2,3 edges of Co and \nRh, respectively . The spin and orbital moments of each element39-41 are determined \nfrom XMCD using sum rule analysi s42,43. The same number  of 7.8 electrons in the \nvalence d shell is assumed for both Co and Rh. This value was found in fully \nrelativistic ab-initio  calculations. The measured XMCD spectra are shown in Figure 4. \nThe XMCD  signal  has the same sign in the spectra obtained at the Co and Rh L2,3 \nedges , which proves that the coupling between Co and Rh is ferromagnetic . \nThe XMCD signal at the L3 edge for Co is considerably larger than that at the \nL2 edge, which indicates a sizeable orbital  contribution . Sum rule analysis reveals that \nspin and orbital moments for Co are 1.53 ±  0.15 µB and 0.42 ±  0.04 µB, respectively. \nThe orbital moment of Co in R h2CoSb far surpasses  that of elemental Co,  where the \nvalue is 0.15 μ B44. Therefore, the orbital moment of Co makes a sizeable contribution \nto the overall magnetization of 1. 95 ±  0.1 9 µB. The presence of a large orbital moment \nis also evident in Figure 2a from  the 0.2 µ B difference of saturation magnetization \nbetween the c and a. \nThe spin and orbital moments of Rh obtained from the sum rule analysis are 14 \n 0.28± 0.03 µ B and 0.02 0± 0.002 µ B, respectively, resulting in a total moment of \n0.30± 0.03 µ B. These values account  for the reduced photon polarization at high \nenergies. Using these moments , together with those of Co, the total magnetic moment \nof Rh 2CoSb should amount to 2.55± 0.22 µB, which is  in good agreement with the \nmagnetic measurement of 2. 6 µB.  \n \nFigure. 4. XAS and XMCD  spectra  of Rh 2CoSb. a, Co L2,3 and b, Rh L2,3 XAS and XMCD \nspectra. Spectra were taken at 25 K temperature and 6 T induction field. The photon helicity is \noriented parallel ( µ+) in black solid line or antiparallel ( µ−) in red solid line to the magnetic \nfield. The background is shown in the black dash line with edge jumps.  \nThe Co atoms have a substantial  orbital magnetic moment in Rh 2CoSb, both \nfrom experiments and ab-initio  calculations. The orbital moment was 27% of the spin \nmoment, compared to  5% in elemental -Co (hcp) and 3% for Fe in calculations of \nthe sister compound Rh 2FeSb. The larger orbital moment  for cobalt reflects the 15 \n hybridization of Co 3 d states with 4 d states of Rh, which has a stronger spin-orbit \ninteraction than Co.  \nIn many Heusler compound s, the martensitic transition from a cubic to a \ntetragonal structure is explained by  a band  Jahn-Teller effect23 that results in  a \nsplitting of energy levels by a modification of the ir width. We also calculate d \nRh2CoSb in the higher energy cubic L21 structure , but saw no sign of Jahn-Teller type \nsplitting  in the electr onic structure . The formation enthalpies for our compound in  the \ntetragonal  I4/mmm  (space group 139), inverse tetragonal I\nm2 (space group 119) and \ncubic L21 (space group 225)  structures  are -753, -325 and -221 meV , respectively. T he \nenergy difference between  the cubic  and tetragonal structure s is 532 meV, \nsignificantly  more than in other Rh -based Heusler  compound s. Thus, the hypothetical  \ncubic  to tetragonal  phase  transition  temperature ( Tcub-tet) should be above  the melting \npoint . Among the reported Rh2CoZ Heusler  compounds , only Rh 2CoSb and Rh 2CoSn \nare tetragonal at room temperature ; the others are cubic. However, Rh 2CoSn has a \nmartensitic transition at around 600 K with a cubic structure at higher temperature23. \nBy avoid ing any  such transition, Rh 2CoSb is superior to other Rh 2CoZ alloys for \nHAMR  media . \nOur work will help  to design new rare-earth free  materials with strong uniaxial \nmagnetocrystalline anisotropy , for which we  need :  \n1) a low-symmetr y crystal structure  (tetragonal or hexagonal );  \n2) heavy atoms with  a large spin -orbit interaction  (4d, 5d or 5f elements );  \n3) another possible element with  the right electronegativity to help stabilize the 16 \n structure and help produce an advantageous electronic structure.  \nThe magnetic moments of heavy non-rare-earth elements are small  at best . \nTherefore, the design rule for  rare-earth free  metallic  magnet s is to pair a 3d metal \n(Mn, Fe, Co) that can provide most of the  magnetization  with a  heavy atom  that \nsupplies a large spin -orbit interaction . Rh, Ir, Pd and Pt are all possible choice s, since \nthey are in the same group as Co or Ni and they may exhibit  a significant induced \nmagnetization37,38,45. In fact, Pd is already  ferromagnetic with Tc = 17 K if lightly \ndoped  with 0.5% of Co46 and when  it is alloyed with 50% Co , it has the highest Tc \namong  the binary  compounds  CoRh (550 K), CoIr ( 40 K), CoPd (950 K) and CoPt \n(840 K)47,48. Our study outlines the design principles for new rare-earth free  hard \nmagnets , but i t is unrealistic to imagine that Rh -based alloys will ever be used as bulk \nfunctional materials, in view of its high cost . This is much less important in functional \nthin films, because  the quantities used  are tiny . Strong perpendicular anisotropy is of \nincreasing importance in spintronics and  especially in  HAMR ; there thermal \nconductivity is also an important co nsideration.  \nRef. 16 report ed the preparation of Rh2CoSb film s by sputtering  or ion -beam \ndeposition  between 373 -873 K . A large perpendicular anisotropy was observed. \nDisorder is common in sputter ed films . Our experiment s indicate  that the Rh is well \nordered, but there might be some disorder between Co and Sb atoms. From ab-initio  \ncalculation s, the most stable crystal structure for both  ordered and disordered phase s \nis the tetragonal D0 22 structure  and the lattice constant is very similar. T he total \nmagnetic moment in the fully Co -Sb disordered state is about 20% larger compared to 17 \n the completely ordered state. The orbital moment is nearly constant and independent \nof the degree of disorder , so the increase of the total moment is attributed to the spin \nmoment . \nWe compar e Rh2CoSb with L1 0 FePt5 for HAMR in Table 1. The magnetic \nproperties for Rh 2CoSb at 2 K are shown with  bracket s, those at 300 K without. The \nmagnetic properties for L1 0 FePt do not vary much between 2 K and 300 K.  The \nwriting speed is limited by the cooling time wh en the total heat produced by laser \ntransfer s to the substrate. Since Rh 2CoSb has a much lower Curie point and higher \nthermal conductivity, its writing speed will be roughly 6.8 times faster than L1 0 FePt \n(see Supplementary information) . \n \nTable 1.  Comparison of Rh 2CoSb with L10 FePt for HAMR  at room temperature . The properties \nfor Rh 2CoSba at 2 K is show n in bracket.  \n \nConclusion  \nIn summary, Rh2CoSb is a uniaxial  ferro magnet with a remarkable  \nmagnetocrystalline anisotropy of 3. 6 MJm-3, due to a large unquenched orbital \nmoment  of 0.42 µB on Co that arises from hybridization with the surrounding  Rh, \nwhere spin orbit coupling is strong . The magnetic hardness parameter of κ = 3. 7 at Materials Crystal \nstructure  K1  \n(MJm-3) μ0Ms \n(T) μ0Ha (T) κ  Tc \n(K) Tcub-tet Thermal \nstability  Thermal \nconductivity \n(Wm-1K-1) Writing \nspeed  \nRh2CoSb  I4/mmm  2.2 \n(3.6) 0.44 \n(0.52) 12 \n(17) 3.7 \n(4.1) 450 > melting  \npoint  Stable  20 in c axis \n12 in a axis Fast \nL10 FePt P4/mmm  6.6 1.43 11 2 750  1573 K  Transition to \nA1 FePt  11 Slow 18 \n room temperature is the highest observed so far in any rare-earth free  magnet. The \nrelatively low Curie point of Tc = 450 K leads to a large temperature -dependen ce of  \nK1 from a high base at room  temperature, which is an asset for data writing . The \nanisotropic  thermal conductivity, especial its large c axis value of 20 Wm-1K-1, which \nis much larger than  that of the current FePt HAMR material , is important for cooling, \nwhich  would  lead to a 6.8 times faster writing speed than FePt.  Unlike FePt with its \norder/disorder phase transition, Rh 2CoSb  is a stable phase without any structural \ntransition below  the melting point  and its properties are stable  in air.  All these features \ncommend Rh2CoSb as a candidate for HAMR  media with a recording density of more  \nthan 10 Tb/in2 and high writing speed . \n \nMethods  \nSingle crystal growth.  The single crystals of Rh 2CoSb were grown by the Bridgeman \nmethod. First, the high purity (> 99.99%) elements Rh, Co and Sb were cut into small \npieces and arc-melted together to prepare  polycrystalline samples.  The initial atomic \nratio of Rh, Co and Sb was 2:1:1.03.  Additional Sb was added due to compensate for \nits high vapor pressure. From powder XRD data, there were always a few percent of a \nRhSb secondary phase in the polycryst alline samples. Hence the polycrystalline \nmaterials were heated up to  1600 K (above the melting point of RhSb  at 1583 K) for 3 \ndays to achieve a  homogeneous liquid  before beginning the single crystal growth, \nwhich took about 5 days during  cool down to 1273 K.  The composition  of the crystals \nwas checked by wavelength dispersive X-ray spectroscopy that show ed a 19 \n homogenous composition of Rh 50.3Co25.6Sb24.1. The crystals were characterized by \npowder X -ray diffraction as single -phase  with a tetragonal structure.  The orientation  \nof the single crystals  was confirmed  by the Laue method . \nMagnetization measurements.  These  were conducted  on single crystals with the \nmagnetic field applied along either  the a or c axis using a vibrating sample  \nmagneto meter (MPMS 3, Quantum Design).  The sample size was 3.80\n 0.78\n 1.30 \nmm3.  For measurement with the field along  the a axis, the sample was carefully \nimmobilized  with glue in view of the stron g torque . High -field m agnetization \nmeasurements were performed in the Dresden High Magnetic Field Laboratory using \na pulsed magnet . \nMagnet o-optical Kerr microscopy.  Domain images were performed by using the \npolar Kerr effect in a wide -field magneto -optical Kerr microscope at room \ntemperature with a  polished  1 mm thick  single crystal with (001) surface . Detailed \ndescription of the method can be found elsewhere49. \nElectrical transport measurements.  The longitudinal and Hall resistivities were \nmeasured on a Quantum Design PPMS 9 using the low -frequency alternati ng current \n(ACT) option  for data below 320 K . The longitudinal resistivity was measured with \nstandard four -probe method, while for the Hall resistivity measurements, the \nfive-probe method was used with a balance protection meter to  eliminate possible \nmagnetoresistance signals50. Longitudinal resistivity measurement above 320 K, were \nmeasured on a home -made device using a  four-probe method and careful calibration. \nThe accuracy of resistivity measurement is ± 5%.  20 \n Thermal transport measurements.  The thermal  conductivity and Seebeck  \nthermopower were measured adiabatically  in the Quantum Design  PPMS using \nthermal transport option (TTO). The uncertainty of the radiative heat losses at high \ntemperature and the uncertainty of the geometry is estimated as ± 10%. \nAndreev reflection measurements. Measurements are performed on a polished (001) \ncrystal surface in a flow of helium vapor, using a mechanically  sharpened Nb tip in \nthe absence of an external magnetic field.   Data are analy zed using the modified BTK \nmodel, as detailed elsewhere51. The best fit to spectrum is obtained with barrier \nparameter Z~ of 0.35, an electron temperature of  2 K and a spin polarization of  13%. \nXMCD measurements. X-ray magnetic circular dichroism (XMCD) was measured at \nthe beamline BL29 (BOREAS) of the synchrotron ALBA in Barcelona (Spain). \nXMCD spectra at the L2,3 absorption edges of Co and Rh were taken at a temperature \nof 25 K in a vacuum chamber with a pressure  of 10−9 mbar. The x -ray absorption \nspectra (XAS) were measured using circular polarized light with photon helicity \nparallel ( µ+) or antiparallel ( µ−) to the fixed magnetic field in the sequence \nµ+µ−µ−µ+µ−µ+µ+µ− to disentangle the XMCD.  An induction field of  6 T was applied \nalong  the c axis. The polarization delivered by the Apple II -type elliptical undulator \nwas close to 100% for the Co L2,3 edges and 70% for the Rh L2,3 edges. The spectra \nwere recorded using the total yield mode.  \n \nSupporting Information  \nSupporting Information is available from the Wiley Online Library or from the author.  21 \n  \nAcknowledgement  \nThis work was financially supported by the European Research Council Advanced \nGrant (No. 742068) “TOPMAT”, the European Union’s Horizon 2020 research and \ninnovation programme (No. 824123) “SKYTOP”, the European Union’s Horizon \n2020 research and innovation programme (No. 766566) “ASPIN”, the Deutsche \nForschungsgemeinschaft (Project -ID 258499086) “SFB 1143”, the Deutsche \nForschungsgemeinschaft (Project -ID FE 633/30 -1) “SPP Skyrmions”, the DFG \nthrough the Wü rzburg -Dresden Cluster of Excellence on Complexity an d Topology in \nQuantum Matter ct.qmat (EXC 2147, Project -ID 39085490) and the DFG through \nSFB 1143. We acknowledge the support of the High Magnetic Field Laboratory \nDresden (HLD) at HZDR, members of the European Magnetic Field Laboratory \n(EMFL).  \n \nConflict of  Interest  \nThe authors declare no conflict of interest.  \n \nAuthor contributions  \nSingle crystals were grown by Y .H. and K.M. The characterization of the crystal, \nmagnetic and transport measurement were performed by Y .H. with the help of C.Fu, \nY .P., J.K., A.J., Y .S.,W.S., P.S., H.B., R.S. and S.S.P.P. XMCD was measured by X.W., \nZ.H., S.A., J.H., M.V . and L.H.T. First principle calculations were carried out by 22 \n G.H.F. All the authors discussed the results. The paper was written by Y .H., J.M.D.C. \nand G.H.F. with fee dback from all the authors. The project was supervised by C.  \nFelser.  \n \nReferences\n \n1. C. Chappert , A. Fert, and F. N. V. Dau, Nat. Mater.  2007 , 6, 813 –823. \n2. M.T. Kief and R.H. Victora , MRS Bulletin  2018 , 43, 87-92. \n3. D. Weller, G. Parker, O. Mosendz, A. Lyberatos, D. Mitin, N. Y . Safonova, & M. \nAlbrecht, J. Vac. Sci. & Technol. B  2016 , 34, 060801 . \n4. R. Skomski, & J. M. D.  Coey, Scr. Mater.  2016 , 112, 3–8. \n5. J. M. D. Coey,  Magnetism and Magnetic Materials , Cambridge Univ. Press, \nCambridge, 2010 , Ch. 4 and 8. \n6. S.I.  Iwasaki, & K. Ouchi,  IEEE T. Magn . 1978 , 14, 849 -851. \n7. L. Pan, and D. B. Bogy , Nat. Photonics  2009 , 3, 189 –190. \n8. J. M. Hu, L .Q. Chen, and C .W. Nan, Adv. Mater.  2016 , 28, 15–39. \n9. W. A.  Challener , C. Peng , A. V .  Itagi, D. Karns , W. Peng , Y. Peng , X. Yang , X. Zhu, \nN. J. Gokemeijer, Y . -T. Hsia, G. Ju, R. E. Rottmayer , M. A. Seigler , and E. C.  Gage , \nNat. Photonics  2009 , 3, 220-224. \n10. A. V. Kimel , and M . Li, Nat. Rev. Mater.  2019 , 4, 189-200. \n11. D. Weller, O.  Mosendz, G.  Parker, S.  Pisana, and T.S.  Santos, Phys.Status Solidi A  \n2013 , 210, 1245.  \n12. C.‐B.  Rong, D.  Li, V .  Nandwana, N.  Poudyal, Y .  Ding, Z.L.  Wang , H. Zeng, J.P. \nLiu, Adv. Mater.  2006 , 18, 2984.  \n13. A. K. Nayak, A. K. Nayak, M . Nicklas, S . Chadov, P . Khuntia, C . Shekhar, A . \nKalache, M . Baenitz, Y . Skourski, V . K. Guduru, A . Puri, U . Zeitler, J. M. D. Coey & 23 \n                                                                                                                                                   \nC. Felser  Nat. Mater.  2015 , 14, 679–684. \n14. H. Kurt, K. Rode, P. Stamenov, M. Venkatesan, Y. -C. Lau, E. Fonda,  and J. M. D. \nCoey , Phys. Rev. Lett.  2014 , 112, 027201 . \n15. S. K.  Dhar, A. K.  Grover, S. K.  Malik, R.  Vijayaraghavan, Solid State Commun.  \n1980 , 33, 545. \n16. S. V . Faleev, Y . Ferrante, J . Jeong, M . G. Samant, B . Jones, and S . S. P. Parkin , \nPhys. Rev. Mater.  2017 , 1, 024402.  \n17. T. Gasi, A . K. Nayak, J . Winterlik , V. Ksenofontov, P . Adler , M. Nicklas, and C . \nFelser , Appl. Phys. Lett.  2013 , 102, 202402.  \n18. P. J. Brown,  T. Kanomata, K . Neumann , K. U. Neumann , B. Ouladdiaf , A. Sheikh \nand K . R. A. Ziebeck , J. Phys.: Condens. Matter.  2010 , 22, 506001.  \n19. B. Balke,  G. H.  Fecher, J. Winterlik  and C. Felser , Appl. Phys. Lett.  2007 , 90, \n152504 . \n20. A. K.  Nayak, C.  Shekhar, J.  Winterlik, A. Gupta, & C.  Felser, Appl. Phys. Lett.  \n2012 , 100, 152404.  \n21. A. K. Nayak , M. Nicklas, S. Chadov, C. Shekhar, Y . Skourski, J. Winterlik, and C. \nFelser , Phys. Rev. Lett. 2013 , 110, 127204.  \n22. O. Meshcheriakova, S. Chadov, A.K. Nayak, U.K. Rößler, J. Kübler, G. André, \nA.A. Tsirlin, J. Kiss, S. Hausdorf, A. Kalache, W. Schnelle , M. Nicklas, and C. Felser , \nPhys. Rev. Lett. 2014 , 113, 087203.  \n23. J. C.  Suits, Solid State Commun.  1976 , 18, 423. \n24. A. Hubert, R.  Schä fer, Magnetic Domains, Springer, Berlin,  1998 , p.p. 314, 379.  \n25. Y . Otani, J. M. D. Coey , J. Appl. Phys.  1990 , 67, 4619.  \n26. T. Nishioka, G.  Motoyama, & N. K.  Sato, J. Magn. Magn. Mater.  2004 , 272, \nE151 . 24 \n                                                                                                                                                   \n27. S. Okamoto, N.  Kikuchi, O.  Kitakami, T.  Miyazaki, Y .  Shimada, & K.  Fukamichi, \nPhys. Rev. B  2002 , 66, 024413.  \n28. M. Balanda, Acta Phys. Pol. A  2013 , 124, 964 -976. \n29. J. Cui, M. Kramer, L. Zhou , F. Liu, A. Gabay , G. Hadjipanayis , B. \nBalasubramanian , D. Sellmyer , Acta Mater.  2018 , 158, 118 -137. \n30. C. Hass, Phys. Rev. 1968 , 168, 531.  \n31. D. Elefant, R. Schä fer, Phys. Rev. B  2010 , 82, 134438.  \n32. J. Dijkstra, H. H. Weitering, C. F. Van Bruggen, C. Haas, R. A. De Groot, J. Phys.: \nCondens. Matter  1989 , l, 9141 -9161.  \n33. G. H. Fecher, S . Chadov  and C . Felser , Theory of the Half -Metallic Heusler \nCompounds  Spintronics, Springer, Dordrecht , 2013,  115-165. \n34. J. M.  Ziman, Electrons and Phonons, Oxford University Press , 1960 . \n35. A. Chernyshov, D.  Treves, T.  Le, F.  Zong, A.  Ajan , R. Acharya, J. Appl. Phys. \n2014 , 115, 17B735.  \n36. A. Giri, S. H.  Wee, S.  Jain, O.  Hellwig, P. E.  Hopkins, Sci. Rep.  2016 , 6, 32077.  \n37. G. R.  Harp, S. S. P.  Parkin, W. L.  O' Brien , B. P. Tonner, Phys. Rev. B  1995 , 51, \n12037 . \n38. G. R.  Harp , S. S. P.  Parkin , W. L. O' Brien , B. P. Tonner, J. Appl. Phys. 1994 , 76, \n6471.  \n39. S. Agrestini , Z. Hu, C. -Y. Kuo, M.W.  Haverkort, K. -T. Ko, N. Hollmann, Q.  Liu, \nE. Pellegrin , M. Valvidares, J.  Herrero -Martin, P.  Gargiani , P. Gegenwart, M.  \nSchneider , S. Esser , A. Tanaka , A. C.  Komarek, L. H.  Tjeng, Phys. Rev. B 2015 , 91, \n075127 . \n40. S. Das, A. D.  Rata, I. V .  Maznichenko, S.  Agrestini, E.  Pippel,  N. Gauquelin, J.  25 \n                                                                                                                                                   \nVerbeeck, K.  Chen, S. M.  Valvidares, H.  Babu Vasili, J.  Herrero -Martin, E.  Pellegrin, \nK. Nenkov, A.  Herklotz, A.  Ernst, I.  Mertig, Z.  Hu, and K.  Dö rr, Phys. Rev. B 2019 , \n99, 024416.  \n41. N. Hollmann, S.  Agrestini , Z. Hu, Z.  He, M.  Schmidt, C.-Y. Kuo, M.  Rotter, A. A.  \nNugroho, V .  Sessi, A.  Tanaka, N. B.  Brookes, and L. H.  Tjeng, Phys. Rev. B 2014 , 89, \n201101(R).  \n42. B. T.  Thole, P.  Carra, F.  Sette,  G. van der Laan , Phys. Rev. Lett.  1992 , 68, 1943 . \n43. P.B. Carra, T.  Thole , M. Altarelli,  X. Wang,  Phys. Rev. Lett.  1993 , 70, 694.  \n44. C. T.  Chen , Y .U. Idzerda , H. -J. Lin, N.V . Smith , G. Meigs , E. Chaban , G.H.  Ho, E. \nPellegrin , F. Sette , Phys. Rev. Lett.  1995 , 75, 152.  \n45. V . V .  Krishnamurthy,  D. J.  Singh , N. Kawamura,  M. Suzuki,  T. Ishikawa , Phys.  \nRev. B  2006 , 74, 064411.  \n46. A. Pilipowicz,  H. Claus,  Phys. Rev. B  1987 , 36, 773.  \n47. H. Masumoto , K. Watanabe,  K. Inagawa,  J. Jpn. I. Met.  1976 , 17, 592.  \n48. B. Predel,  H. Landolt,  R. Bö rnstein,  Phase equilibria, crystallographic and \nthermodynamic data of binary alloys (V ol. 5). Berlin, Heidelberg, Springer , 1991.  \n49. I.V .  Soldatov,  R. Schä fer , Rev. Sci. Instrum . 2017 , 88, 073701.  \n50. S.D. Pinzon, Magnetic properties and interface scattering contribution to the \nanomalous Hall effect in cobalt/palladium multilayers, California State University, \nLong Beach, 2016 , p16. \n51. P. Stamenov,  J. Appl. Phys. 2013 , 113, 17C718.  \n 1 \n New highly -anisotropic Rh -based Heusler compound  for \nmagnetic recording   \n \nSupplementary Information  \n \nYangkun He1,*, Gerhard H. Fecher1,*, Chenguang Fu1, Yu Pan1, Kaustuv Manna1, Johannes \nKroder1, Ajay Jha2, Xiao Wang1, Zhiwei Hu1, Stefano Agrestini3, Javie r Herrero -Martí n 4, \nManuel Valvidares4, Yurii Skourski5, Walter Schnelle1, Plamen Stamenov2, Horst Borrmann1, \nLiu Hao Tjeng1, Rudolf Schaefer6,7, S. S. P. Parkin8, J. M. D. Coey2 and Claudia Felser1 \n \n1Max-Planck -Institute for Chemical Physics of Solids, D -01187 Dresden, Germany  \n2School of Physics, Trinity College, Dublin 2, Ireland  \n3 Diamond Light Source, Harwell Science and Innovation Campus, Didcot, OX11 0DE, \nUK \n4ALBA Synchrotron Light Source, Cerdanyola del Valles, 08290 Barcelona, Catalonia, \nSpain  \n5 Dresden High Magnetic Field Laboratory (HLD -EMFL), Helmholtz -zentrum \nDresden–Rossendorf, 01328 Dresden, Germany  \n6Leibniz Institute for Solid State and Materials Research (IFW) Dresden, Helmholtz \nstrasse 20, D -01069 Dresden, Germany  \n7Institute for Materials Sc ience, TU Dresden, D -01062 Dresden, Germany  \n8Max Planck Institute of Microstructure Physics, Halle, Germany  \n \nKey words: magnetocrystalline anisotropy, tetragonal Heusler alloy , magnetic \nhardness parameter, 4 d magnetism, magnetic recording  2 \n  \n1. Single crystal growth and c omposition  \nThe composition of Rh 2CoSb single  crystal has been identified by both \nenergy -dispersive X -ray spectroscopy (EDX) and wave length dispersive X -ray \nspectroscopy (WDX ), which shows a homogenous composition . The average \nvalues and standa rd deviation are shown in the Table S1.  The sample has a \nhomogenous composition of Rh 50.3Co25.6Sb24.1. \n \nFig. S 1. EDX curve of Rh 2CoSb single crystal. The insert is the image of the single \ncrystal.  \n \nTable S1. WDX data of Rh 2CoSb single crystal at ten diffe rent points.  \nPoint  Rh(at.%)  Co(at.%)  Sb(at.%)  \n1 50.7 25.2 24.1 \n2 50.7 25.4 24.0 \n3 50.3 25.7 24.0 \n4 49.9 26.1 24.1 \n5 50.0 25.9 24.1 \n6 50.4 25.6 24.0 \n7 50.4 25.6 24.1 \n8 50.7 25.4 24.0 \n9 50.3 25.5 24.2 \n10 50.1 25.8 24.0 \nAverage  50.3±0.29 25.6±0.27 24.1±0.07 \n \n \n2. Orientation  \nThe quality and orientation of the single crystal were confirmed by Laue method, 3 \n which shows clear spots in both c and a axes, fitting well with the simulation . \n \nFig. S 2. Laue pattern of Rh 2CoSb single crystal from c (a) and a (b) axes.  \n \n3. Crystal structure  \nThe tetragonal structure was identified by both powder X -ray diffraction and \nhigh-resolution transmission electron microscopy (TEM).  The data show  that \nRh2CoSb has a well -ordered tetragonal D0 22 structure with a = 4.0393 (6) Å a nd c \n= 7.1052 (7) Å. Here, the 4 d (0 0 ¼) site is occupied by Rh, 2 b (0 0 ½) by Co, and \n2a (0 0 0) by Sb. The tetragonal distortion  is c/\na = 1.2436(3).  \n20 30 40 50 60 70 80 90 100(103)\n(211)(004)(200)(112)Intensity (a.u.) Iobs\n Ical\n Iobs - Ical\n Bragg position\n2 (degree)(101)\n \nFig. S 3. Powder X -ray diffraction of Rh 2CoSb  at room temperature . \n 4 \n \n \nFig. S 4.  High -resolution TEM image of the [110] plane, showing a tetragonal lattice \nwith c/√2a = 1.24. The insert is the selected area electron diffraction image, whose \nsuperlattice reflection indicates a  well-ordered structure.  \n \n4. Thermal analysis  \nDifferential scanning calorimeter (DSC) and thermogravimetric (TG) analysi s \n(see Fig. S 5) have been performed to investigate the phase transitions at high \ntemperature. Unlike many Mn -based Heusler compounds which are often cubic at \nhigh temperature and may experience a martensitic transition to become tetragonal, \nRh2CoSb does no t have first order transition at high temperature and is a single \nphase at all temperature range  up to the melting point of about 1482 K, according \nto DSC and TG (see Fig. S 5). Therefore, the common problem of twinning in \nmany permanent magnets such as MnA l, MnGa, FePt and CoPt does not exist and \nsingle crystals have been successfully grown the by Bridgeman method.  5 \n \n200 400 600 800 1000 1200 1400 1600 1800\nTemperature (K)DSC (a.u.)\nT = 1482 K\n-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0\n TG (%) \nFig. S 5. DSC and TG curve of Rh 2CoSb cooling from the melt. The TG curve (blue) is \nflat, indicating little Sb loss d uring heating. The peak in the DSC curve (black) is referred \nas the freezing point of 1482 K. No other peaks are detected, excluding any other first \norder transition. The measurement rate of thermal analysis was set as 10 K/min.  \n \n5. Magnetization measurement  \nThe single crystal magnetization curve along  the c axis is shown in Fig. S6 (a), \nwhose enlarged part at low field is shown in (b). During the demagnetization, the \nsudden drop of the magnetization, namely Barkhausen jumps, together with a high \nsusceptibility  after thermal demagnetization, sugges t single crystal Rh2CoSb as a \nnucleation type permanent magnet. The magnetization curve along  the a axis in (c) \nis not saturated at 7 T, indicating strong magnetocrystalline anisotropy. μ0Ms2 /K1 \nat different temperatures is shown in (d). The peak near Tc is responsible for the \nHopkinson effect in Figure 2d. A small piece of single crystal was ground into \npowder and then annealed at 800 K for 5 h to remove the inner strain before \nbonded i n epoxy at 400 K to make a n isotropic  bonded magnet. Coercivity, as an \nextrinsic property, is determined mainly by additional structural features such as \nlattice defects, grain boundaries, sample or particle size, and surface irregularities. \nThe coerciviti es of bonded isotropic magnet s with grain size s of ~2 0 μm and 1 -3 \nμm show a significant increase from 0.2 T to 0.9 T at room temperature. The  B-H \ncurve is shown in ( f) for the isotropic magnet (1 -3 μm grain size). The energy \nproduct ( BH)max is about 10 kJm-3, which is only a quarter of the theoretical value.  \nTo find a secondary phase and good alignment remain to be done for further study . \nFor polycrystalline samples produced by arc -melting shown in (g), the coercivity \nis about 0.2 T.   6 \n \n \nFig. S 6. Magnetization curves of Rh 2CoSb. Single crystal data from the c axis at different \ntemperatures are shown in (a), whose enlarged part at low field is shown in (b). Single crystal \ndata from the a axis at different temperatures are shown in (c). (d) μ0Ms2 /K1 at different 7 \n temperatures.  M-H curve (e) of isotropic b onded  magnet s with a grain size of about 1 -3 μm  \n(red) and ~20 μm (blue)  at 300 K . The insert shows their distribution histogram  (unit: μm) . \nThe B-H curve of the magnet with grain size of 1 -3 μm at 300 K  is shown in (f).  \nMagnetization curve of polycrystalline sample is shown in (g).  \n \n6. Electrical Transport measurement  \nTransport properties including the resistivity and Hall effect of Rh 2CoSb are \nshown in Fig. S 7 and S8 . The resistivity is very ani sotropic. With increasing \ntemperature, the a-axis resistivity increases monotonically from 53 µΩ cm at 2 K to \n192 µΩ cm at Tc, after which the resistivity is then dominated by spin disorder \nscattering  and increas es slowly1. The resistivity above Tc, which is close to the \nminimum metallic conductivity (~200 µΩ cm)-1, indicates that the mean free path is \nclose to the interatomic spacing when the moments are disordered. The c axis \nresistivity is half as large, 21 µΩ cm at 2 K but it increases with temperature and \nshows a similar trend to the a axis resistivity. Domain walls along the c axis in most \nof the volume, but not in the surface region as shown in the insert of Fig. 2a and \nSupplementary Fig. S 11, influence the resistivity. When the charge carriers travel  \nalong c the current is parallel to the moment direction; but when the carriers move in \nthe basal plane they tend to follow helical paths due to the Lorenz force, leading to a \nlarger resistivity2. Like other ferromagnets, Rh 2CoSb shows a negative transvers e \nmagnetoresistance (MR), which increases with increasing temperature, reaching \n-1.75 % at 300 K under a 7 T field. However, when the field is applied along a (taking \ncare to immobilize the crystal) and the current is passed in a perpendicular b direction,  \nthe MR is positive (0.39%) at 300 K during the hard axis magnetization process, as \nshown in Fig. S7c. \nThe results of the Hall measurements are shown in Fig. S7d. When the field is \nparallel to the c axis and the current is along a, the normal Hall effect d ue to the \nLorenz force is positive and increases linearly with applied field, indicating mainly \nhole-type charge carriers with a density of 1.5\n 1022 cm-3 corresponding to 0.86 per \nformula. The mobility is 2 -7 cm2V-1s-1 between 2 -300 K. Detailed  data can be seen in \nSupplementary Fig. S 8. The Hall effect was also measured with the field along the a \naxis and the current along b at 300 K. The magnetization at a field of 9 T is far from \nsaturation, hence the data are a mixture of a normal and an anom alous Hall effect. \nTheir values could be estimated by linear extrapolation of the curve to 12.4 T, the \nsaturation field from Fig. 2a, and extrapolation from the normal Hall effect. The \nsignificantly larger anomalous Hall resistivity of 1.33 µΩ cm measured along the a \naxis than the value of 0.75 µΩ cm measured along the c axis is attributed to the \nanisotropic crystal and electrical structure . 8 \n \n \nFig. S 7.  Transport properties of Rh 2CoSb. a, Longitudinal resistivity along the c (red \ncurve) and the a axis (blue  curve). b, Magnetoresistance along a with field along c. c, \nAnisotropic magnetoresistance. d, Anomalous Hall effect with field along a. \n \nThe electrical transport measurement was measured both along c and a axes. The \nHall conductivity was calculated by:  \nσxy = -ρxy / (ρ xy2 +ρxx2)                 \nwhere ρ xx and ρ xy are the longitudinal resistivity (along a) at zero field and the Hall \nresistivity (along a), respectively3. Since ρ xx >> ρ xy and the anomalous Hall resistivity \nρxyA is proportional to the square of  ρxx as shown in the insert of Fig. S8b, the \nanomalous Hall conductivity is almost unchanged with variation of the temperature, \nindicating a side-jump effect or an intrinsic mechanism due to the Berry curvature \nmechanism4 as there are a lot of band crossin gs in the ferromagnetic state ( see \nSupplementary Fig.  S13). The σ xx is of order 104 Ω-1 cm-1, which is also in the region \nof intrinsic AHE3.   \n The anomalous Hall resistivity versus temperature is shown in Fig. S8c, while \nthe charge carrier density as well as the mobility deduced from the normal Hall effect \nis shown in Fig. S 8d. The charge carrier density is deduced by n = 1/(e RH), where R H \nis the slope of the ρxy. The mobility is calculated  by μ = RH / ρxx. 9 \n \n \nFig. S8. (a) Hall effect with field along c at different temperatures. (b) Hall conductivity \nat different temperatures. The insert shows the linear fitting of anomalous Hall \nresistivity ρAxy versus ρxx2. (c) The anomalous Hall resistivity versus temperature.  (d) The \ncharge carrier density as well as the mobility at different temperatures.  \n \nAlthough some quasi -two-dimensional metals5,6 and superconductors7-9 show a \ngiant resistivity anisotropy (the ratio of the resistivities measured along c and a can be \nmore than 10 to 100), t he anisotropic resistivity in Rh 2CoSb is among the largest \nfound in three -dimensional bulk materials. For the tetragonal Heusler compound \nMn 1.4PtSn, the resistivity along c is only about 7% larger than along a10. \nOrthorhombic UFe 2Al10 shows a less than 10% difference between a, b and c11. \nHexagonal Mn 3Ge exhibits a 2.5 times larger resistivity along a at 2 K, but the \ndifference decreases with increasing temperature, and vanishes around room \ntemperature12. The anisotropic transport propertie s, including magnetoresistance and \nanomalous Hall effect, indicate that the magnetism leads to an anisotropic mobility of \nthe charge carriers.  \n \n7. Seebeck  coefficient measurement  \nThe Seebeck  coefficient  was measured both along c and a axes. From Figure S9 it is \nseen that the absolute value of the Seebeck coefficient is slightly  higher along the \nc-axis for T > 10 K . The behavior of the anisotropic Seebeck coefficients is reciprocal 10 \n to the conductivity as expected from the Onsager relation . This reflects the ani sotropy \nof the Onsager coefficients that are tensors.  In tetragonal materials one has Sxx = Syy \nSzz. It should further be noted that the elements of the Onsager tensors depend on \nthe magnetization of the sample or the applied magnetic field, in general13,14.      \n0 50 100 150 200 250 300 350-12-10-8-6-4-202Seebeck coefficient (VK-1)\nTemperature (K) c axis\n a axis\n \nFig. S 9. Seebeck  coefficient measurement along a and c axes a t different temperatures.  \n \n8.  Spin polarization  \nThe spin polarization P of the electrons at the Femi level was deduced from point \ncontact Andreev ref lection measurement at 2 K  that is shown in Fig S10.   Data are \nanalysed using the modified BTK model, as described in detail elsewhere15. The best \nfit to the spectrum is obtained with a barrier parameter Z~ of 0.35, an electron \ntemperature of  2 K and a spin polarization of  13%.  11 \n \n-6 -4 -2 0 2 4 61.01.11.21.31.41.5G/Gn Data\n Fit\nD1* = 1.24(1)  meV\nD2 = 1.50 meV\nZ* = 0.35(1)\nP* = 0.13(3)\nTe* = 3.3(3) K\nGn = 786(1)  G0\nT = 2.3 K\nUa (meV) \nFig. S1 0. The spin polarization P of the electrons at the Fe mi level was deduced from \npoint contact Andreev refection measurement at 2 K.  The fit parameters for the BTK \nmodel are included in the figure (see also Ref.15) \n \n9. Magneto -optical Kerr  microscopy  \nMagneto -optical K err microscopy study shows the surface domains of a \ntwo-phase branched domain pattern of higher generation on the (001) surface, which \nis usually observed in uniaxial magnets.  For such a pattern, the domain walls are \naligned parallel to the c-axis only in the volume of the bulk crystal. When \napproaching the surface, the magnetization strictly follows the c-axis, but the domain \nwalls does not follow,  as shown in Fig. S1 1. Further explanation can be found in Ref. \n16. \n \nFig. S1 1. The surface domain pattern of a  strong uniaxial magnet  adapted from Ref.  16. \n \n10. Electron structure calculations  12 \n 1) Ab-initio calculations.  \nThe electronic and magnetic structures of Rh 2CoSb were calculated by means \nof the first principles computer  programs Wien2k17-20 and SPKKR21,22 in the \nlocal spin density approximation. In particular, the generalized gradient \napproximation (GGA) of Perdew, Burke and Ernzerhof23 was used for the \nparametrization of the exchange correlation functional. A k -mesh based on 126 ×  \n126 ×  126 points of the full Brillouin zone was used for integration when \ncalculating the total energies for determination of the magnetocrystalline \nanisotropy.  For more details see [arxiv .] \n \n2) Results.  \nThe electronic and magnetic structure calculat ions confirm the uniaxial \nmagnetocrystalline anisotropy  with Ku 1.4 MJ/m³ . A total moment of 2.19 μB is \nfound from the GGA calculations using SPRKKR, the spin and orbital moments \nhave values of 2.04 μ B and 0.15 μB, respectively.  \nFrom the spin and site r esolved data , it is found that Co contributes per atom a \nmagnetic moment of 1.81 μB of which about 0. 14 μB is the orbital moment, and \nthe major part of 1.67  μB is the spin moment. A small spin moment of about 0.2 μB \nand negligible orbit moment is carried b y Rh.  \nBoth, anisotropy constant and magnetic moments are smaller than the \nexperimental values.  Indeed, Ku does not include any temperature or macroscopic \nmagnetic  effects  e.g.: domains and domain walls. The orbital magnetic moment of \nCo is larger in the XM CD measurements. However, these measurements observe \nan excited state. Using the orbital corrected potential of Brooks24, the value is \nincreased to about 0.4 μ B without changing the spin moment.  \nFigures S 12 and S 13 show results of the ab -initio calculations  using SPRKKR . \nFigure S 12 shows the spin and site resolved density of states and Figure S 13 the \nspin resolved band stru cture.  13 \n \n \nFig. S12. The valence band density of states.  \n \nFig. S13. Semi -relativistic band structures of Rh 2CoSb. Majority states are \ndrawn in red and minority states in blue.  \n \n3) Comparison of Rh 2FeSb and Rh 2CoSb  \nTable S2 shows the calculated occupation of th e 3d orbitals of the Co and Fe \natoms (number of electrons in a sphere within muffin tin radius) in Rh2FeSb and \nRh2CoSb. Part of the d-electron density is delocalized and found in between the atoms 14 \n and part is more localized on Rh. At both atoms the five ma jority -spin d-states are \nalmost fully occupied (about 4.5 out of 5) resulting in a nearly spherical distribution. \nThe occupation of the minority -spin states is quite different. In particular, the \noccupation of the minority dz2 orbital of Co modifies the sp in of the charge density of \nCo compared to Fe. The different occupancy of the minority orbitals is responsible for \nthe difference in anisotropy —easy plane for Fe and easy axis for Co.  The calculations \nwere performed with Wien2k . \n \nTable S2. Occupation of th e orbitals at the Fe and Co atoms in Rh 2FeSb and \nRh 2CoSb.  \nPoint  Rh2FeSb  Rh2CoSb  \nMajority  Minority  Majority  Minority  \ndz2 0.946  0.106  0.934  0.862  \ndxy 0.915  0.538  0.909  0.823  \ndx2-y2 0.917  0.271  0.912  0.426  \ndxz 0.910  0.377  0.899  0.385  \ndyz 0.910  0.377  0.899 0.385  \ntotal 4.594  1.668  4.553  2.881  \nmoment  2.926  1.672  \n \nThe Co atoms have a rather large orbital magnetic moment in Rh 2CoSb, from \nboth experiments and ab-initio  calculations. The calculated value for Co in Rh 2CoSb \nwas found to be about 8% of the spi n moment, whereas it is 4.8% in elemental -Co \n(hcp) and 3% for Fe in the sister compound Rh 2FeSb. The larger orbital moments for \nCo reflect hybridization of Co 3 d bands with 4 d bands of Rh, which has a strong \nspin-orbit interaction.  \n \n11. Grain size and data s torage density  \nThe superparamagnetic blocking radius25 for a particle is calculated as Rb = \n(6kBT/K1)1/3, where kB and T are Boltzmann constant and  temperature respectively. \nFor Rh 2CoSb, it  is Rb = 2.3 nm at 300 K. For modern perpendicular recording in \nCo-Cr-Pt, the magnetic media are not particles, but tall slim grains. The aspect ratio is  \nabout 3, in order to increase the shape anisotropy. For h ard magnets like FePt and \nRh2CoSb one does not need to further increase the anisotropy  by large r aspect ratio s. \nHowever, it is difficult to control a uniform grain height  in short  grains in order to get  \na stable Curie temperature . As a result, the idea l aspect ratio is about 1.5 to 226. To \nresist demagnetization by random thermal fluctuation, the volume V of the magne tic \nmaterial must satisfy the empirical condition that K1V/kBT > 60. Therefore, V must be \nlarger than 113 nm3 for each grain and the diameter is calculated as 4.2 nm for \nRh2CoSb at 300 K  for an aspect ratio of 2. In-plane heat transfer is also another thin g \nwe must pay attention to. To avoid the heating by a neighbour ing grain  during writing \nas well as to increase the signal -to-noise ratio , it is better to slightly increase the \ncentre -to-centre distance of grains. This increased distance for Rh 2CoSb can still be 15 \n smaller than for FePt, due to the much lower Curie temperature (450 K vs 750 K)  that \nallows to use lower temperatures . A potential storage density of more than 10 \nTerabit/inch2 can be realized27 when assuming that the centre -to-centre distance of \ngrains is 5 nm with half area occupied .  \n \n12. Writing speed estimation  \nRh2CoSb has a mass density  of ρ = 11\n103 kg m-3 and its  formula weight is \n386.5  g mol-1. We calculate the heat capacity to be  Cp = 4\n 3R=100 J  mol-1K-1 or a \nvolume heat c apacity of Cv = 2.85\n 106 J m-3K-1. The time for each grain to cool down \nis calculated to be τV = Cv∙A/λ. That is τV = 3.56\n 10-12 s, when u sing a thermal \nconductivity  of λ = 20 Wm-1K-1 and assuming an area for each grain  (5 nm in \ndiameter)  of A = 2.5\n 10-17m2,  \nThe a ngular velocity  of a hard disk  is ω = 7000 r min-1 and t he radi us for a \none-inch disk  is r = 1 inch =  0.0254 m. Therefore , one has a  scanning velocity  of v = \nω∙2πr = 18.56 m s-1. The time of heating for each gra in is τh = \n /v = 2.7\n10-10 s and \nthe number of heated grains  becomes  ngrains= τh/τv = 75.8.  \nTo complete a magnetization process,  \n \nwhere Hc is the coercivity, T is temperature, t is time, x is the position that satisfies \ndx/dt = V  and Δx =\n , fFMR is the ferromagnetic resonance frequency (for large angle \nprecession) at the temperature point of the maximal thermal gradient. It is a measure \nof how fast the switching proceeds within a grain in the trailing edge . No critical \ndifferences in fFMR are expected between the materials. The field of the write head  Hw \nshould satisfy \n . The writing speed is proportional to \n . \nSimilarly  one has for FePt  Cv = 3.1\n 106 J m-3K-1 and gets τV = Cv∙A/λ = \n7.0\n 10-12 s, which is twice as large as the value for Rh 2CoSb. Though it is unknown \nfor the coercivity in films  for Rh 2CoSb , we a ssume that Hc is proportional to the \nanisotropy field Ha, hence \n  is proportion al to (Ha(300K) -0)/(Tc-T300K). The Ha for 16 \n Rh2CoSb and FePt are 12.4 T and 11.0 T at room temperature, and their Tc are 450 K \nand 750 K , respectively. Therefore , \n for Rh2CoSb  is about 3.4 times larger tha n \nthat for FePt . The writing speed ratio for Rh 2CoSb and FePt , propo rtional to \n , \nis therefore 6.8. 17 \n  \nReference s \n \n1. C. Hass, Spin -Disorder Scattering and Magnetoresistance of Magnetic \nSemiconductors, Phys. Rev . 168, 531 (1968).  \n2. D. Elefant, R. Schä fer, Giant negative do main wall resistance in iron, Phys. Rev. B  \n82, 134438 (2010).  \n3. N. Nagaosa,  J. Sinova, S. Onoda, A. H. MacDonald, & N. P. Ong, Anomalous hall \neffect. Rev. Mod. Phys.  82, 1539 (2010).  \n4. E. Liu  et al. Giant anomalous Hall effect in a ferromagnetic kagome -lattice \nsemimetal. Nat. Phys. 14, 1125 (2018).  \n5. Y . Wang et al. Anisotropic anomalous Hall effect in triangular itinerant ferromagnet \nFe3GeTe 2. Phys. Rev. B  96, 134428 (2017).  \n6. J. Y . Liu et al. A magnetic topological semimetal Sr 1-yMn 1-zSb2 (y, z < 0.1).  Nat. \nMater.  16, 905 (2017).  \n7. Y . J. Song et al. Synthesis, anisotropy, and superconducting properties of LiFeAs \nsingle crystal. Appl. Phys. Lett.  96, 212508 (2010).  \n8. Y . Eltsev et al. Anisotropic resistivity and Hall effect in MgB 2 single crystals. Phys . \nRev. B  66, 180504(R) (2002).  \n9. S. W. Tozer,  A . W.  Kleinsasser,  T. Penney,  D. Kaiser  & F. Holtzberg. Measurement \nof Anisotropic Resistivity and Hall Constant for Single -Crystal YBa 2Cu3O7-x. Phys. \nRev. Lett. 95, 1768 (1987).  \n10. P. Vir, et al. Anisotropic  topological Hall effect with real and momentum space \nBerry curvature in the antiskyrmion -hosting Heusler compound Mn 1.4PtSn. Phys. Rev. \nB 99, 140406(R) (2019).  \n11. R. Troc et al. Electronic, magnetic, transport, and thermal properties of \nsingle -crystalline  UFe 2Al10. Phys. Rev. B  92, 104427 (2015).  \n12. N. Kiyohara et al. Giant Anomalous Hall Effect in the Chiral Antiferromagnet \nMn3Ge. Phys. Rev. Appl.  5, 064009 (2016).  \n13. J. M. Ziman “Electrons and Phonons”, Oxford UniversityPress (1958) . \n14. J. F. Nye “Physic al Properties of crystals”, Oxford Science Publications (1955) . \n15. P. Stamenov, Point contact Andreev reflection from semimetallic bismuth —The \nroles of the minority carriers and the large spin -orbit coupling. J. Appl. Phys. 113, 18 \n                                                                                                                                                   \n17C718 (2013).  \n16. A. Hubert , R. Schä fer, Magnetic Domains, Springer, Berlin, p.p. 314, 379 (1998).  \n17. P. Blaha, K.  Schwarz , P. Sorantin , S. B.  Trickey,  Full-potential, linearized \naugmented plane wave programs for crystalline systems. Commun. Comput. Phys  59, \n399-415 (1990).  \n18. K. Schwarz & P. Blaha, Solid state calculations using WIEN2k. Comput. Mater. \nSci. 28, 259 -273 (2003).  \n19. P. Blaha, K.  Schwarz , G. K.  Madsen , D. Kvasnicka & J. Luitz, wien2k: An \naugmented plane wave+ local orbitals program for calculating crystal properties \n(Wie n, 2013).  \n20. P. Blaha , K. Schwarz , F. Tran, R. Laskowski , G.K.H.  Madsen  & L.D. Marks,  \nWIEN2k: An APW+lo program for calculating the properties of solids, J. Chem. Phys.  \n152, 074101 (2020).  \n21. H. Ebert, Fully relativistic band structure calculations for mag netic \nsolids -formalism and application. In Electronic Structure and Physical Properties of \nSolids (pp. 191 -246). Springer, Berlin, Heidelberg (1999).  \n22. H. Ebert , D. Koedderitzsch & J. Minar, Calculating condensed matter properties \nusing the KKR -Green's function method —recent developments and applications.  Rep. \nProg. Phys. 74, 096501 (2011).  \n23. J. P.  Perdew , K. Burke & M.  Ernzerhof , Generalized Gradient Approximation \nMade Simple , Phys. Rev. Let t. 77, 3865 (1996).  \n24. M. S.S.  Brooks , Calculated ground state properties of light actinide metals and \ntheir compounds, Physica B  130, 6 (1985).  \n25. J. M. D. Coey, Magnetism and Magnetic Materials Ch. 4 and 8 (Cambridge Univ. \nPress, Cambridge, 2010).  \n26. D. Weller , G. Parker , O. Mosendz , A. Lyberatos , D. Mitin , N. Y .  Safonova  & M. \nAlbrecht, FePt heat assisted magnetic recording media , J. Vac. Sci. & Technol. B  34, \n060801 (2016).  \n27. C. V ogler , C. Abert , F. Bruckner , D. Suess , D. Praetorius , Heat -assisted magne tic \nrecording of bit -patterned media beyond 10 Tb/in2, Appl. Phys. Lett. 108, 102406 \n(2016).  "    },    {        "title": "2009.01638v3.MAELAS__MAgneto_ELAStic_properties_calculation_via_computational_high_throughput_approach.pdf",        "content": "MAELAS: MAgneto-ELAStic properties calculation\nvia computational high-throughput approach\nP. Nievesa,\u0003, S. Arapana, S. H. Zhangb,c, A. P. K ˛ adzielawaa, R. F. Zhangb,c,\nD. Leguta\naIT4Innovations, VŠB - Technical University of Ostrava, 17. listopadu 2172/15, 70800\nOstrava-Poruba, Czech Republic\nbSchool of Materials Science and Engineering, Beihang University, Beijing 100191, PR China\ncCenter for Integrated Computational Materials Engineering, International Research Institute\nfor Multidisciplinary Science, Beihang University, Beijing 100191, PR China\nAbstract\nIn this work, we present the program MAELAS to calculate magnetocrys-\ntalline anisotropy energy, anisotropic magnetostrictive coefficients and magnetoe-\nlastic constants in an automated way by Density Functional Theory calculations.\nThe program is based on the length optimization of the unit cell proposed by Wu\nand Freeman to calculate the magnetostrictive coefficients for cubic crystals. In\naddition to cubic crystals, this method is also implemented and generalized for\nother types of crystals that may be of interest in the study of magnetostrictive ma-\nterials. As a benchmark, some tests are shown for well-known magnetic materials.\nKeywords: Magnetostriction, Magnetoelasticity, High-throughput computation,\nFirst-principles calculations\nPROGRAM SUMMARY\nProgram Title: MAELAS\nDeveloper’s respository link: https://github.com/pnieves2019/MAELAS\nLicensing provisions: BSD 3-clause\nProgramming language: Python3\nNature of problem: To calculate anisotropic magnetostrictive coefficients and magnetoe-\nlastic constants in an automated way based on Density Functional Theory methods.\nSolution method: In the first stage, the unit cell is relaxed through a spin-polarized cal-\n\u0003Corresponding author.\nE-mail address: pablo.nieves.cordones@vsb.cz\nPreprint submitted to Computer Physics Communications February 9, 2021arXiv:2009.01638v3  [cond-mat.mtrl-sci]  7 Feb 2021culation without SOC. Next, after a crystal symmetry analysis, a set of deformed lat-\ntice and spin configurations are generated using the pymatgen library [1]. The energy of\nthese states is calculated by the first-principles code V ASP [3], including the SOC. The\nanisotropic magnetostrictive coefficients are derived from the fitting of these energies to a\nquadratic polynomial [2]. Finally, if the elastic tensor is provided [4], then the magnetoe-\nlastic constants are calculated too.\nAdditional comments including restrictions and unusual features: This version supports\nthe following crystal systems: Cubic (point groups 432, ¯43m,m¯3m), Hexagonal (6 mm,\n622, ¯62m, 6=mmm ), Trigonal (32, 3 m,¯3m), Tetragonal (4 mm, 422, ¯42m, 4=mmm ) and\nOrthorhombic (222, 2 mm,mmm ).\nReferences\n[1] S. P. Ong, W. D. Richards, A. Jain, G. Hautier, M. Kocher, S. Cholia, D. Gunter, V .\nL. Chevrier, K. A. Persson, and G. Ceder, Comput. Mater. Sci. 68, 314 (2013).\n[2] R. Wu, A. J. Freeman, Journal of Applied Physics 79, 6209–6212 (1996).\n[3] G. Kresse, J. Furthmüller, Phys. Rev. B 54 (1996) 11169.\n[4] S. Zhang and R. Zhang, Comput. Phys. Commun. 220, 403 (2017).\n1. Introduction\nA magnetostrictive material is one which changes in size due to a change\nof state of magnetization. These materials are characterized by magnetostrictive\ncoefficients ( l). In many technical applications such as electric transformers, mo-\ntor shielding, and magnetic recording, magnetic materials with extremely small\nmagnetostrictive coefficients are required. By contrast, materials with large mag-\nnetostrictive coefficients are needed for many applications in electromagnetic mi-\ncrodevices as actuators and sensors [1–4]. Typically, elementary Rare-Earth (R)\nmetals (under low temperature and high magnetic field) and compounds with R\nand transition metals exhibit a high magnetostriction ( l>10\u00003). In particular,\nthe highest magnetostrictions were found in the RFe 2compounds with Laves\nphase C15 structure type (face centered cubic) [5]. For instance, Terfenol-D\n(Tb 0:27Dy0:73Fe2) is a widely used magnetostrictive material thanks to its giant\nmagnetostriction along [111] crystallographic direction ( l111=1:6\u000210\u00003) un-\nder moderate magnetic fields ( <2 kOe) at room temperature [6]. Beyond cubic\n2systems, the research of magnetostrictive materials has been focused on hexagonal\ncrystals like RCo 5(space group 191), hexagonal and trigonal R 2Co7and R 2Co17\nseries, and tetragonal R 2Fe14B [7, 8]. More recently, the problem of R avail-\nability [9] has also motivated the exploration of R-free magnetostrictive materials\nlike Galfenol (Fe-Ga), spinel ferrites (CoFe 2O4), Nitinol (Ni-Ti alloys), Fe-based\nInvars, and Ni 2MnGa [10–12].\nConcerning the theory of magnetostriction, the basic equations for cubic (I)\ncrystals were developed by Akulov [13] and Becker et al. [14] in the 1920s and\n30s. In the next three decades, great advances took place due to the outstanding\nworks of Mason [15], Clark et al. [16], and Callen and Callen [17], as well as\nmany others, where the theory was extended to other crystal symmetries. Over\nthe last decades, modern electronic structure theory based on Density Functional\nTheory (DFT) has been successfully applied to describe magnetostriction of many\nmaterials [1, 10, 11, 18–29]. Nowadays, a common method to calculate magne-\ntostrictive coefficients is based on the optimization of the unit cell length pro-\nposed by Wu and Freeman for cubic crystals [18, 19]. In this work, we present the\nMAELAS program where this methodology is implemented and generalized for\nthe main crystal symmetries in the research field of magnetostriction. The paper\nis organized as follows. In Section 2, we review some theoretical concepts and\nequations of magnetostriction. In Section 3, we explain in detail the methodol-\nogy and workflow of the program, while some examples are shown in Section 4.\nThe paper ends with a summary of the main conclusions and future perspectives\n(Section 5).\n2. Theory of magnetostriction\nThe magnetostrictive response is mainly originated by two kind of sources: (i)\nisotropic exchange interaction and (ii) strain dependence of magnetocrystalline\nanisotropy [8]. The magnetostriction due to isotropic exchange leads to frac-\ntional volume changes, and doesn’t depend on the magnetization direction [30].\nOn the other hand, the strain dependence of magnetocrystalline anisotropy is\nresponsible for the magnetostriction that depends on the magnetization orienta-\ntion (anisotropic), and is originated by the spin-orbit coupling (SOC) and crystal\nfield interactions [8, 31]. The current version of the program MAELAS calcu-\nlates the magnetostrictive coefficients and magnetoelastic constants related to the\nanisotropic magnetostriction.\nLet’s consider l0the initial length of a demagnetized material along the di-\nrection bbb(jbbbj=1), and lthe final length along the same direction bbbwhen the\n3Figure 1: Magnetostriction of a single crystal under an external magnetic field ( aaakHHH) perpendicu-\nlar to the measured length direction ( bbb?HHH). Symbols MandMsstand for macroscopic magnetiza-\ntion and saturation magnetization, respectively. Dash line on the right represents the original size\nof the demagnetized material. The magnetostriction effect has been magnified in order to help to\nvisualize it easily, in real materials it is smaller ( Dl=l0\u001810\u00003\u000010\u00006).\nsystem is magnetized along the direction aaa(jaaaj=1). The relative length change\n(l\u0000l0)=l0=Dl=l0can be written as [8]\nDl\nl0\f\f\f\f\faaa\nbbb=å\ni;j=x;y;zeeq\ni j(aaa)bibj; (1)\nwhere eeq\ni jis the equilibrium strain tensor. This equation describes the Joule effect\n[32], once it is rewritten in terms of the magnetostrictive coefficients ( l) conve-\nniently. Fig.1 shows a sketch of magnetostriction.\nThe deformation of a solid can be described in terms of the displacement vec-\ntoruuu(rrr) =rrr000\u0000rrrthat gives the displacement of a point at the initial position rrr\nto its final position rrr000after it is deformed. For small deformations (infinitesimal\nstrain theory), the strain tensor ( ei j) can be expressed in terms of the displacement\nvector as[33]\nei j=1\n2\u0012¶ui\n¶rj+¶uj\n¶ri\u0013\n; i;j=x;y;z (2)\nwhere ¶ui=¶rjis called the displacement gradient (second-order tensor). The equi-\nlibrium strain tensor is obtained through the minimization of both the elastic ( Eel)\n4and magnetoelastic ( Eme) energies [5, 8]\n¶(Eel+Eme)\n¶ei j=0; i;j=x;y;z (3)\nwhere the total energy must be invariant under the symmetry operations of the\ncrystal lattice [5]. Let’s write general equations for EelandEme. The elastic energy\ndepends on the fourth-order elastic stiffness tensor ci jklthat links the second-order\nstrain and stress ( si j) tensors through the generalized Hooke’s law\nsi j=å\nk;l=x;y;zci jklekl;i;j=x;y;z: (4)\nTaking advantage of the symmetry of stress and strain tensors, the Hooke’s law\ncan be written in matrix notation as\n0\nBBBBBB@sxx\nsyy\nszz\nsyz\nsxz\nsxy1\nCCCCCCA=0\nBBBBBB@cxxxx cxxyy cxxzz cxxyz cxxzx cxxxy\ncyyxx cyyyy cyyzz cyyyz cyyzx cyyxy\nczzxx czzyy czzzz czzyz czzzx czzxy\ncyzxx cyzyy cyzzz cyzyz cyzzx cyzxy\nczxxx czxyy czxzz czxyz czxzx czxxy\ncxyxx cxyyy cxyzz cxyyz cxyzx cxyxy1\nCCCCCCA0\nBBBBBB@exx\neyy\nezz\n2eyz\n2exz\n2exy1\nCCCCCCA(5)\nTo facilitate the manipulation of this equation it is convenient to define the follow-\ning six-dimensional vectors (V oigt notation)\n˜sss=0\nBBBBBB@˜s1\n˜s2\n˜s3\n˜s4\n˜s5\n˜s61\nCCCCCCA=0\nBBBBBB@sxx\nsyy\nszz\nsyz\nsxz\nsxy1\nCCCCCCA; ˜eee=0\nBBBBBB@˜e1\n˜e2\n˜e3\n˜e4\n˜e5\n˜e61\nCCCCCCA=0\nBBBBBB@exx\neyy\nezz\n2eyz\n2exz\n2exy1\nCCCCCCA; (6)\nand replace ci jklbyCnmcontracting a pair of cartesian indices into a single integer:\nxx!1,yy!2,zz!3,yz!4,xz!5 and xy!6. Using these conversion rules\nthe Hooke’s law is simplified to\n˜si=6\nå\nj=1Ci j˜ej;i=1;:::;6 (7)\n5where in matrix form reads\n0\nBBBBBB@˜s1\n˜s2\n˜s3\n˜s4\n˜s5\n˜s61\nCCCCCCA=0\nBBBBBB@C11C12C13C14C15C16\nC21C22C23C24C25C26\nC31C32C33C34C35C36\nC41C42C42C44C45C46\nC51C52C53C54C55C56\nC61C62C63C64C65C661\nCCCCCCA0\nBBBBBB@˜e1\n˜e2\n˜e3\n˜e4\n˜e5\n˜e61\nCCCCCCA: (8)\nwhere Ci j=Cji. Then the elastic energy up to second-order in the strain can be\nwritten as\nEel=E0+V0\n26\nå\ni;j=1Ci j˜ei˜ej+O(˜e3); (9)\nwhere E0andV0are the equilibrium energy and volume, respectively. The magne-\ntoelastic energy Emecomes from the strain dependence of the magnetocrystalline\nanisotropy energy (MAE) EK[34, 35]. Performing a Taylor expansion of EKin\nthe strain we have\nEK=E0\nK+6\nå\ni=1\u0012¶EK\n¶˜ei\u0013\n0˜ei+1\n26\nå\ni;j=1\u0012¶2EK\n¶˜ei¶˜ej\u0013\n0˜ei˜ej+O(˜e3); (10)\nwhere E0\nKcorresponds to the MAE of the undeformed state that contains the mag-\nnetocrystalline anisotropy constants K. The third term in the right hand side of\nEq.10 is the second-order magnetoelastic energy that leads to a very small ad-\nditional contribution to the second-order elastic energy given by Eq.9, so that is\nusually neglected [34, 36]. The first-order magnetoelastic energy\nEme=6\nå\ni=1\u0012¶EK\n¶˜ei\u0013\n0˜ei (11)\nis obtained by taking the direct product of the symmetry strains and direction co-\nsine polynomial for each irreducible representation, multiplying by a constant,\ncalled the magnetoelastic constant and finally summing over the different rep-\nresentations [5, 8, 16, 17]. Frequently, the first-order magnetoelastic energy is\nconsidered up to second-order of the direction cosine polynomial a. In cartesian\ncoordinates, it may be written as\nEme=3\nå\ni=1gi(a0)˜ei+6\nå\ni=1fi(a2)˜ei+O(a4); (12)\n6Table 1: Number of independent second-order elastic constants of each crystal system. Number of\nindependent magnetoelastic and magnetostrictive coefficients up to second-order of the direction\ncosine polynomial in the first-order magnetoelastic energy. In the last column we specify which\ncrystal systems are supported by the current version of MAELAS.\nCrystal system Point groupsSpace\ngroupsElastic\nconstants\n(Ci j)Magnetoelastic\nconstants\n(b)Magnetostriction\ncoefficients\n(l)MAELAS\nTriclinic 1 ;¯1 1\u00002 21 36 36 No\nMonoclinic 2 ;m;2=m 3\u000015 13 20 20 No\nOrthorhombic 222 ;2mm;mmm 16\u000074 9 12 12 Yes\nTetragonal (II) 4 ;¯4;4=m 75\u000088 7 10 10 No\nTetragonal (I) 4 mm;422;¯42m;4=mmm 89\u0000142 6 7 7 Yes\nTrigonal (II) 3 ;¯3 143\u0000148 7 12 12 No\nTrigonal (I) 32 ;3m;¯3m 149\u0000167 6 8 8 Yes\nHexagonal(II) 6 ;¯6;6=m 168\u0000176 5 8 8 No\nHexagonal (I) 6 mm;622;¯62m;6=mmm 177\u0000194 5 6 6 Yes\nCubic (II) 23 ;m¯3 195\u0000206 3 4 4 No\nCubic (I) 432 ;¯43m;m¯3m 207\u0000230 3 3 3 Yes\nwhere functions giand ficontain the magnetoelastic constants ( b). In the fol-\nlowing subsections, we show the form of Eqs.1, 9 and 12 for the main crystal\nsymmetries studied in magnetostriction, which are implemented in the program\nMAELAS. The remaining crystal systems not discussed here might be included\nin the new versions of the code. In Table 1, we present a summary of the crystal\nsystems supported by MAELAS. Here, we use the notation of Wallace [37, 38]\n(I/II) to distinguish Laue classes within the same crystal system.\nBefore analyzing each crystal system, we must make an important remark\nabout the notation for the strain tensor ei j. In previous works discussing magne-\ntostriction like Refs.[5, 8, 35], the V oigt definition of the strain tensor was used\n(eV\ni j;i;j=x;y;z) [39], which is related to the one defined in the present work as\neii=eV\nii, 2ei j=eV\ni j;i6=j. Consequently, the following elastic and magnetoelastic\nenergies (in terms of the strain tensor with two cartesian indices) contain numeri-\ncal factors different to those given in Kittel and Clark works [5, 35] for the terms\nwith non-diagonal elements of the strain tensor ( ei j;i6=j). The following expres-\nsions for the relative length change ( Dl=l0) are the same as in Kittel and Clark\nworks [5, 35] because the sum in Eq.1 runs over all possible values of indices\ni;j=x;y;z, while in Kittel and Clark works [5, 35] the sum runs up to i>j.\nIn the present work, the equations of magnetostrictive coefficients expressed in\nterms of the elastic and magnetoelastic constants are also the same to those given\nin Kittel and Clark works [5, 35].\n72.1. Cubic (I)\n2.1.1. Single crystal\nFor cubic (I) systems (point groups 432, ¯43m,m¯3m) the elastic stiffness tensor\nreads\nCcub=0\nBBBBBB@C11C12C12 0 0 0\nC12C11C12 0 0 0\nC12C12C11 0 0 0\n0 0 0 C44 0 0\n0 0 0 0 C44 0\n0 0 0 0 0 C441\nCCCCCCA; (13)\nso there are three independent elastic constants C11,C12andC44. Hence, the\nelastic energy Eq.9 becomes\nEcub\nel\u0000E0\nV0=C11\n2(˜e2\n1+˜e2\n2+˜e2\n3)+C12(˜e1˜e2+˜e1˜e3+˜e2˜e3)\n+C44\n2(˜e2\n4+˜e2\n5+˜e2\n6)\n=cxxxx\n2(e2\nxx+e2\nyy+e2\nzz)+cxxyy(exxeyy+exxezz+eyyezz)\n+2cyzyz(e2\nxy+e2\nyz+e2\nxz);(14)\nwhere C11=cxxxx,C12=cxxyyandC44=cyzyz. On the other hand, the first-order\nmagnetoelastic energy up to second-order direction cosine polynomial contains 3\nmagnetoelastic constants [17]. From the symmetry strains and direction cosine\npolynomial for each irreducible representation, it is possible to obtain the follow-\ning magnetoelastic energy in cartesian coordinates [5, 8, 10]\nEcub(I)\nme\nV0=b0(˜e1+˜e2+˜e3)+b1(a2\nx˜e1+a2\ny˜e2+a2\nz˜e3)\n+b2(axay˜e6+axaz˜e5+ayaz˜e4)\n=b0(exx+eyy+ezz)+b1(a2\nxexx+a2\nyeyy+a2\nzezz)\n+2b2(axayexy+axazexz+ayazeyz);(15)\nwhere b0is the volume magnetoelastic constant, and b1andb2are the anisotropic\nmagnetoelastic constants. Next, replacing Eqs.14 and 15 into Eq.3, we find the\n8following equilibrium strains\neeq\ni j=\u0000b2aiaj\n2C44; i6=j; i;j=x;y;z\neeq\nii=\u0000b1a2\ni\nC11\u0000C12\u0000b0\nC11+2C12+b1C12\n(C11\u0000C12)(C11+2C12);i=x;y;z(16)\nInserting these equilibrium strains into Eq.1 gives\nDl\nl0\f\f\f\f\faaa\nbbb=la+3\n2l001\u0012\na2\nxb2\nx+a2\nyb2\ny+a2\nzb2\nz\u00001\n3\u0013\n+3l111(axaybxby+ayazbybz+axazbxbz);(17)\nwhere\nla=\u0000b0\u00001\n3b1\nC11+2C12;\nl001=\u00002b1\n3(C11\u0000C12);\nl111=\u0000b2\n3C44:(18)\nThe coefficient ladescribes the volume magnetostriction, while l001andl111are\nthe anisotropic magnetostrictive coefficients that give the fractional length change\nalong the [001] and [111] directions when a demagnetized material is magnetized\nin these directions, respectively. The superscript ainlastands for one irreducible\nrepresentation of the group of transformations which take the crystal into itself [5,\n8], so it should not be confused with the direction of magnetization aaa. The MAE\nin an unstrained cubic crystal up to sixth-order of direction cosine polynomial is\n[35, 40]\nE0\nK\nV0=K0+K1(a2\nxa2\ny+a2\nxa2\nz+a2\nya2\nz)+K2a2\nxa2\nya2\nz; (19)\nwhere K0,K1andK2are the magnetocrystalline anisotropy constants.\n2.1.2. Polycrystal\nThe theory of magnetostriction for polycrystalline materials is more complex\nthan for single crystals. A widely used approximation is to assume that the stress\ndistribution is uniform through the material. In this case the relative change in\n9length may be put into the form [8, 13, 34, 41]\nDl\nl0\f\f\f\f\faaa\nbbb=3\n2lS\u0014\n(aaa\u0001bbb)2\u00001\n3\u0015\n; (20)\nwhere\nlS=2\n5l001+3\n5l111: (21)\nThis result is analogous to the Reuss approximation used in the elastic theory of\npolycrystals to obtain a lower bound of bulk and shear modulus [8, 42–44]. A\ndiscussion about the limitations of this approximation can be found in Ref.[45].\n2.2. Hexagonal (I)\n2.2.1. Single crystal\nThe elastic stiffness tensor for hexagonal (I) system (point groups 6 mm, 622,\n¯62m, 6=mmm ) reads\nChex=0\nBBBBBB@C11C12C13 0 0 0\nC12C11C13 0 0 0\nC13C13C33 0 0 0\n0 0 0 C44 0 0\n0 0 0 0 C44 0\n0 0 0 0 0C11\u0000C12\n21\nCCCCCCA; (22)\nso that it has five independent elastic constants C11,C12,C13,C33andC44. As a\nresult, the elastic energy Eq.9 is\nEhex\nel\u0000E0\nV0=1\n2C11(˜e2\n1+˜e2\n2)+C12˜e1˜e2+C13(˜e1+˜e2)˜e3+1\n2C33˜e2\n3\n+1\n2C44(˜e2\n4+˜e2\n5)+1\n4(C11\u0000C12)˜e2\n6\n=1\n2cxxxx(e2\nxx+e2\nyy)+cxxyyexxeyy+cxxzz(exx+eyy)ezz+1\n2czzzze2\nzz\n+2cyzyz(e2\nyz+e2\nxz)+(cxxxx\u0000cxxyy)e2\nxy(23)\nwhere C11=cxxxx,C12=cxxyy,C13=cxxzz,C33=czzzz, and C44=cyzyz. The first\norder magnetoelastic energy up to quadratic direction cosine polynomial contains\n106 magnetoelastic constants [17]. In cartesian coordinates it can be written as [5]\nEhex(I)\nme\nV0=b11(exx+eyy)+b12ezz+b21\u0012\na2\nz\u00001\n3\u0013\n(exx+eyy)+b22\u0012\na2\nz\u00001\n3\u0013\nezz\n+b3\u00141\n2(a2\nx\u0000a2\ny)(exx\u0000eyy)+2axayexy\u0015\n+2b4(axazexz+ayazeyz):\n(24)\nOnce the equilibrium strains are calculated by minimizing Eqs.23 and 24 through\nEq.3 and inserted into Eq.1, one finds [5, 8, 16]\nDl\nl0\f\f\f\f\faaa\nbbb=la1;0(b2\nx+b2\ny)+la2;0b2\nz+la1;2\u0012\na2\nz\u00001\n3\u0013\n(b2\nx+b2\ny)\n+la2;2\u0012\na2\nz\u00001\n3\u0013\nb2\nz+lg;2\u00141\n2(a2\nx\u0000a2\ny)(b2\nx\u0000b2\ny)+2axaybxby\u0015\n+2le;2(axazbxbz+ayazbybz);(25)\nwhere\nla1;0=b11C33+b12C13\nC33(C11+C12)\u00002C2\n13;\nla2;0=2b11C13\u0000b12(C11+C12)\nC33(C11+C12)\u00002C2\n13;\nla1;2=\u0000b21C33+b22C13\nC33(C11+C12)\u00002C2\n13;\nla2;2=2b21C13\u0000b22(C11+C12)\nC33(C11+C12)\u00002C2\n13;\nlg;2=\u0000b3\nC11\u0000C12;\nle;2=\u0000b4\n2C44:(26)\nThese magnetostrictive coefficients are related to the normal strain modes for a\ncylinder [5, 8]. The equation of the relative length change in the form of Eq.25\nwas proposed by Clark et al. [16]. In literature there are different arrangements of\nthe right hand side of Eq.25 that leads to other definitions of the magnetostrictive\ncoefficients, like those defined by Mason [15], Birss [34], and Callen and Callen\n[17]. The conversion formulas between Eq.25 and all these other conventions can\n11be found in Appendix A. These conversion formulas are implemented in the pro-\ngram MAELAS, so that the magnetostrictive coefficients are also given according\nto these definitions. Note that in some works [40, 45] the magnetostrictive coef-\nficients lg;2andle;2in Eq.25 are named as le;2andlz;2, respectively, which is\nmore consistent with the Bethe’s group-theoretical notation [45]. The MAE in an\nunstrained hexagonal crystal up to fourth-order of areads [40]\nE0\nK\nV0=K0+K1(1\u0000a2\nz)+K2(1\u0000a2\nz)2: (27)\n2.2.2. Polycrystal\nUnder the assumption of uniform stress, the relative change in length for poly-\ncrystalline hexagonal (I) systems can be written as [34]\nDl\nl0\f\f\f\f\faaa\nbbb=x+h(aaa\u0001bbb)2; (28)\nwhere his, in both easy axis and easy plane MAE, given by\nh=\u00002\n15Q4+1\n5Q6+7\n15Q8: (29)\nThe quantity xis different for easy axis and easy plane. In the case of easy axis, x\nis given by\nx=2\n3Q2+4\n15Q4\u00001\n15Q6+1\n15Q8; (easy axis) (30)\nwhile for easy plane is\nx=\u00001\n3Q2\u00001\n15Q4\u00001\n15Q6\u00004\n15Q8: (easy plane) (31)\nThe quantities Qi(i=2;4;6;8) are the anisotropic magnetostrictive coefficients in\nBirss’s convention [34], and are related to the magnetostrictive coefficients defined\nin Eq. 25 through Eq. A.4. We have implemented these formulas in MAELAS,\nso that it also calculates handx.\n2.3. Trigonal (I)\n2.3.1. Single crystal\nThe elastic stiffness tensor for trigonal (I) system (point groups 32, 3 m,¯3m)\nhas 6 independent elastic constants C11,C12,C13,C33,C44andC14, and it is given\n12by\nCtrig(I)=0\nBBBBBB@C11 C12 C13 C14 0 0\nC12 C11 C13\u0000C14 0 0\nC13 C13 C33 0 0 0\nC14\u0000C14 0 C44 0 0\n0 0 0 0 C44 C14\n0 0 0 0 C14C11\u0000C12\n21\nCCCCCCA: (32)\nHence, inserting this tensor into Eq.9 we have the following elastic energy\nEtrig(I)\nel\u0000E0\nV0=1\n2C11(˜e2\n1+˜e2\n2)+C12˜e1˜e2+C13(˜e1+˜e2)˜e3+1\n2C33˜e2\n3\n+1\n2C44(˜e2\n5+˜e2\n4)+1\n4(C11\u0000C12)˜e2\n6+C14(˜e6˜e5+˜e1˜e4\u0000˜e2˜e4):\n=1\n2cxxxx(e2\nxx+e2\nyy)+cxxyyexxeyy+cxxzz(exx+eyy)ezz+1\n2czzzze2\nzz\n+2cyzyz(e2\nxz+e2\nyz)+(cxxxx\u0000cxxyy)e2\nxy+4cxxyz(exyexz+exxeyz\u0000eyyeyz):\n(33)\nwhere C11=cxxxx,C12=cxxyy,C13=cxxzz,C14=cxxyz,C33=czzzz, and C44=cyzyz.\nOn the other hand, the magnetoelastic energy contains 8 independent magnetoe-\nlastic constants [17]. In cartesian coordinates it can be written as [8]\nEtrig(I)\nme\nV0=b11(exx+eyy)+b12ezz+b21\u0012\na2\nz\u00001\n3\u0013\n(exx+eyy)+b22\u0012\na2\nz\u00001\n3\u0013\nezz\n+b3\u00141\n2(a2\nx\u0000a2\ny)(exx\u0000eyy)+2axayexy\u0015\n+2b4(axazexz+ayazeyz)\n+b14\u0002\n(a2\nx\u0000a2\ny)eyz+2axayexz\u0003\n+b34\u00141\n2ayaz(exx\u0000eyy)+2axazexy\u0015\n:\n(34)\n13Next, we obtain the equilibrium strains via Eq.3. Replacing them into Eq.1 leads\nto [8]\nDl\nl0\f\f\f\f\faaa\nbbb=la1;0(b2\nx+b2\ny)+la2;0b2\nz+la1;2\u0012\na2\nz\u00001\n3\u0013\n(b2\nx+b2\ny)\n+la2;2\u0012\na2\nz\u00001\n3\u0013\nb2\nz+lg;1\u00141\n2(a2\nx\u0000a2\ny)(b2\nx\u0000b2\ny)+2axaybxby\u0015\n+lg;2(axazbxbz+ayazbybz)+l12\u00141\n2ayaz(b2\nx\u0000b2\ny)+axazbxby\u0015\n+l21\u00141\n2(a2\nx\u0000a2\ny)bybz+axaybxbz\u0015\n;(35)\nwhere\nla1;0=b11C33+b12C13\nC33(C11+C12)\u00002C2\n13;\nla2;0=2b11C13\u0000b12(C11+C12)\nC33(C11+C12)\u00002C2\n13;\nla1;2=\u0000b21C33+b22C13\nC33(C11+C12)\u00002C2\n13;\nla2;2=2b21C13\u0000b22(C11+C12)\nC33(C11+C12)\u00002C2\n13;\nlg;1=C14b14\u0000C44b3\n1\n2C44(C11\u0000C12)\u0000C2\n14;\nlg;2=1\n2b4(C11\u0000C12)\u0000b34C14\n1\n2C44(C11\u0000C12)\u0000C2\n14;\nl12=C14b4\u0000C44b34\n1\n2C44(C11\u0000C12)\u0000C2\n14;\nl21=1\n2b14(C11\u0000C12)\u0000b3C14\n1\n2C44(C11\u0000C12)\u0000C2\n14:(36)\nThe MAE in an unstrained trigonal crystal up to fourth-order in ais the same to\nthe hexagonal case (Eq.27).\n142.4. Tetragonal (I)\n2.4.1. Single crystal\nThe tetragonal (I) crystal system (point groups 4 mm, 422, ¯42m, 4=mmm ) has\nthe following elastic stiffness tensor\nCtet(I)=0\nBBBBBB@C11C12C13 0 0 0\nC12C11C13 0 0 0\nC13C13C33 0 0 0\n0 0 0 C44 0 0\n0 0 0 0 C44 0\n0 0 0 0 0 C661\nCCCCCCA; (37)\nHence, it has six independent elastic constants C11,C12,C13,C33,C44andC66.\nThe elastic energy is given by\nEtet(I)\nel\u0000E0\nV0=1\n2C11(˜e2\n1+˜e2\n2)+C12˜e1˜e2+C13(˜e1+˜e2)˜e3+1\n2C33˜e2\n3\n+1\n2C44(˜e2\n4+˜e2\n5)+1\n2C66˜e2\n6\n=1\n2cxxxx(e2\nxx+e2\nyy)+cxxyyexxeyy+cxxzz(exx+eyy)ezz+1\n2czzzze2\nzz\n+2cyzyz(e2\nyz+e2\nxz)+2cxyxye2\nxy\n(38)\nwhere C11=cxxxx,C12=cxxyy,C13=cxxzz,C33=czzzz,C44=cyzyzandC66=cxyxy.\nOn the other hand, there are 7 independent magnetoelastic constants [17]. The\nmagnetoelastic energy can be written as [8, 10]\nEtet(I)\nme\nV0=b11(exx+eyy)+b12ezz+b21\u0012\na2\nz\u00001\n3\u0013\n(exx+eyy)+b22\u0012\na2\nz\u00001\n3\u0013\nezz\n+1\n2b3(a2\nx\u0000a2\ny)(exx\u0000eyy)+2b0\n3axayexy+2b4(axazexz+ayazeyz):\n(39)\n15After the equilibrium strains are calculated by minimizing Eqs.38 and 39 through\nEq.3 and replaced into Eq.1, we have [8]\nDl\nl0\f\f\f\f\faaa\nbbb=la1;0(b2\nx+b2\ny)+la2;0b2\nz+la1;2\u0012\na2\nz\u00001\n3\u0013\n(b2\nx+b2\ny)\n+la2;2\u0012\na2\nz\u00001\n3\u0013\nb2\nz+1\n2lg;2(a2\nx\u0000a2\ny)(b2\nx\u0000b2\ny)+2ld;2axaybxby\n+2le;2(axazbxbz+ayazbybz);(40)\nwhere\nla1;0=b11C33+b12C13\nC33(C11+C12)\u00002C2\n13;\nla2;0=2b11C13\u0000b12(C11+C12)\nC33(C11+C12)\u00002C2\n13;\nla1;2=\u0000b21C33+b22C13\nC33(C11+C12)\u00002C2\n13;\nla2;2=2b21C13\u0000b22(C11+C12)\nC33(C11+C12)\u00002C2\n13;\nlg;2=\u0000b3\nC11\u0000C12;\nld;2=\u0000b0\n3\n2C66;\nle;2=\u0000b4\n2C44:(41)\nMason derived an equivalent equation to Eq.40 using a different arrangement of\nthe terms and definitions of the magnetostrictive coefficients [15]. The conversion\nformulas between the magnetostrictive coefficients in Eq.40 and those defined by\nMason are shown in Appendix B. The MAE in an unstrained tetragonal crystal\nup to fourth-order in ais the same to the hexagonal case (Eq.27).\n2.5. Orthorhombic\n2.5.1. Single crystal\nThe orthorhombic crystal system (point groups 222, 2 mm,mmm ) has 9 inde-\npendent elastic constants C11,C12,C13,C22,C23,C33,C44,C55andC66, its elastic\n16stiffness matrix reads[38, 44]\nCortho=0\nBBBBBB@C11C12C13 0 0 0\nC12C22C23 0 0 0\nC13C23C33 0 0 0\n0 0 0 C44 0 0\n0 0 0 0 C55 0\n0 0 0 0 0 C661\nCCCCCCA; (42)\nHence, inserting it into Eq.9 leads to the following expression for the elastic en-\nergy\nEortho\nel\u0000E0\nV0=1\n2C11˜e2\n1+1\n2C22˜e2\n2+C12˜e1˜e2+C13˜e1˜e3+C23˜e2˜e3+1\n2C33˜e2\n3\n+1\n2C44˜e2\n4+1\n2C55˜e2\n5+1\n2C66˜e2\n6\n=1\n2cxxxxe2\nxx+1\n2cyyyye2\nyy+cxxyyexxeyy+cxxzzexxezz+cyyzzeyyezz\n+1\n2czzzze2\nzz+2cyzyze2\nyz+2cxzxze2\nxz+2cxyxye2\nxy:(43)\nwhere C11=cxxxx,C22=cyyyy,C12=cxxyy,C13=cxxzz,C23=cyyzz,C33=czzzz,\nC44=cyzyz,C55=cxzxzandC66=cxyxy. The magnetoelastic energy contains 12 in-\ndependent magnetoelastic constants [17]. Mason derived the following expression\nof the relative length change [15]\nDl\nl0\f\f\f\f\faaa\nbbb=la1;0b2\nx+la2;0b2\ny+la3;0b2\nz+l1(a2\nxb2\nx\u0000axaybxby\u0000axazbxbz)\n+l2(a2\nyb2\nx\u0000axaybxby)+l3(a2\nxb2\ny\u0000axaybxby)\n+l4(a2\nyb2\ny\u0000axaybxby\u0000ayazbybz)+l5(a2\nxb2\nz\u0000axazbxbz)\n+l6(a2\nyb2\nz\u0000ayazbybz)+4l7axaybxby+4l8axazbxbz+4l9ayazbybz:\n(44)\nNote that we added the terms that describes the volume magnetostriction ( la1;0,\nla2;0andla3;0), which were not included in the original work of Mason [15]. The\nexpression of the magnetoelastic energy and the relations between magnetostric-\ntive coefficients, elastic and magnetoelastic constants were not shown by Mason\neither. For completeness, here we deduce it from Eqs. 43 and 44. To do so, we\n17aim to find the unknown functions giandfiin the general form of the magnetoe-\nlastic energy in cartesian coordinates given by Eq. 12. Firstly, we minimize Eqs.\n43 and 12 via Eq. 3. This gives a set of equations that links the unknown functions\ngiandfiwith the equilibrium strains. Next, we extract the equilibrium strains by\ndirect comparison between Eqs. 1 and 44. Finally, we substitute the equilibrium\nstrains into the set of equations that relates giandfiwith the equilibrium strains,\nfrom which we obtain giandfi. Inserting the calculated giandfiinto Eq. 12 we\nhave\nEortho\nme\nV0=b01exx+b02eyy+b03ezz+b1a2\nxexx+b2a2\nyexx+b3a2\nxeyy+b4a2\nyeyy\n+b5a2\nxezz+b6a2\nyezz+2b7axayexy+2b8axazexz+2b9ayazeyz;\n(45)\nwhere\nb01=\u0000C11la1;0\u0000C12la2;0\u0000C13la3;0\nb02=\u0000C12la1;0\u0000C22la2;0\u0000C23la3;0\nb03=\u0000C13la1;0\u0000C23la2;0\u0000C33la3;0\nb1=\u0000C11l1\u0000C12l3\u0000C13l5\nb2=\u0000C11l2\u0000C12l4\u0000C13l6\nb3=\u0000C12l1\u0000C22l3\u0000C23l5\nb4=\u0000C12l2\u0000C22l4\u0000C23l6\nb5=\u0000C13l1\u0000C23l3\u0000C33l5\nb6=\u0000C13l2\u0000C23l4\u0000C33l6\nb7=C66(l1+l2+l3+l4\u00004l7)\nb8=C55(l1+l5\u00004l8)\nb9=C44(l4+l6\u00004l9):(46)\nAlternatively, one can deduce the magnetoelastic energy using the general ap-\nproach based on the symmetry strains and direction cosine polynomial for each\nirreducible representation [5, 8, 17]. This approach may lead to different defi-\nnitions of the magnetoelastic constants and magnetostrictive coefficients, as we\nhave discussed for the hexagonal (I) and tetragonal (I) systems in Appendix A\nand Appendix B, respectively. A generalization of the approach taken by Becker\nand Doring [14] for orthorhombic crystals can be found in Ref. [46]. The MAE\n18in an unstrained orthorhombic crystal up to fourth-order in ais[15]\nE0\nK\nV0=K0+K1a2\nx+K2a2\ny: (47)\n3. Methodology\n3.1. Calculation of magnetostrictive coefficients and magnetoelastic constants\nThe methodology implemented in the program MAELAS to calculate the\nanisotropic magnetostrictive coefficients is a generalization of the approach pro-\nposed by Wu and Freeman for cubic crystals [18, 19]. In this method, one measur-\ning length direction bbbiand two magnetization directions ( aaai\n1andaaai\n2) are chosen\nfor each magnetostrictive coefficient ( li) in such a way that\nDl\nl0\f\f\f\f\faaai\n1\nbbbi\u0000Dl\nl0\f\f\f\f\faaai\n2\nbbbi=rili; (48)\nwhere riis a real number. In Table 2 we show the selected set of bbbi,aaai\n1andaaai\n2\nin MAELAS that fulfils Eq.48 for each ri. Next, the left hand side of Eq.48 is\nwritten as\nDl\nl0\f\f\f\f\faaai\n1\nbbbi\u0000Dl\nl0\f\f\f\f\faaai\n2\nbbbi=l1\u0000l0\nl0\u0000l2\u0000l0\nl0=2(l1\u0000l2)\n(l1+l2)h\n1\u0000l1+l2\u00002l0\nl1+l2i\n=2(l1\u0000l2)\nl1+l2\u0014\n1+l1+l2\u00002l0\nl1+l2+:::\u0015\n\u00192(l1\u0000l2)\nl1+l2;(49)\nwhere in the last approximation we assume jl1(2)\u0000l0j=l0\u001c1. This assumption\nis reasonable for all known magnetostrictive materials. For instance, a very large\nvalue of Dl=l0is about 4 :5\u000210\u00003found in TbFe 2(Laves phase C15) along direc-\ntion [111] at T=0K [6], where this approximation is fine. This approximation\nallows to get rid of l0(length along bbbin the macroscopic demagnetized state)\nwhich can’t be calculated with DFT methods easily. Combining Eqs.48 and 49\none can write the magnetostrictive coefficients as\nli=2(l1\u0000l2)\nri(l1+l2); (50)\n19where the value of rifor each liis given in Table 2. The quantities l1andl2cor-\nrespond to the cell length along bbbwhen the magnetization points to aaa1andaaa2, re-\nspectively, and are calculated through an optimization of the energy. Namely, a set\nof deformed unit cells is firstly generated using the deformation modes described\nin Appendix C. For each deformed cell, the energy is calculated constraining the\nspins to the directions given by aaa1andaaa2. Next, the energy versus the cell length\nalong bbbfor each spin direction aaa1(2)is fitted to a quadratic polynomial\nE(aaaj;l) =Ajl2+Bjl+Cj;j=1;2 (51)\nwhere Aj,BjandCj(j=1;2) are fitting parameters. The minimum of this func-\ntion for spin direction aaa1(2)corresponds to l1(2)=\u0000B1(2)=(2A1(2)). Once l1and\nl2are determined, one obtains the magnetostrictive coefficients using Eq.50. The\nmagnetostrictive coefficients can also be written in terms of the derivative of the\nenergy with respect levaluated at l=l2as [20]\nli\u0019\u00001\nhiB1\u0001¶[E(aaa2;l)\u0000E(aaa1;l)]\n¶l\f\f\f\f\f\nl=l2(52)\nwhere B1is always negative. In Table 2, we see that our choice of bbbandaaa1(2)\nmakes rdepend on some magnetostrictive coefficients for l7,l8andl9in or-\nthorhombic crystals. For instance, working out the coefficient l7via Eq.50 we\nhave\nl7=(a2+b2)(l1\u0000l2)\nab(l1+l2)\u0000(a\u0000b)(a[l1+l2]\u0000b[l3+l4])\n4ab; (53)\nwhere aandbare the relaxed (not distorted) lattice parameters of the unit cell.\nHere, MAELAS makes use of the values of l1,l2,l3andl4calculated pre-\nviously in order to compute l7. Note that a simpler expression for l7can be\nachieved choosing the measuring length direction bbb=\u0010\n1p\n2;1p\n2;0\u0011\n. However,\nfrom a computational point of view, it is easy to extract the cell length lalong\nbbb=\u0010\nap\na2+b2;bp\na2+b2;0\u0011\nof each deformed cell generated with the deformation\ngradients discussed in Appendix C. Similarly, one can deduce the explicit equa-\ntion for l8andl9.\nLastly, if the elastic tensor is provided in the format given by the program\nAELAS [44], then the magnetoelastic constants ( bk) are also calculated from the\nrelations bk=bk(li;Cnm)given in Section 2.\n20Table 2: Selected cell length ( bbb) and magnetization directions ( aaa1,aaa2) in MAELAS to calculate\nthe anisotropic magnetostrictive coefficients according to Eq.48. The first column shows the crys-\ntal system and the corresponding lattice convention set in MAELAS based on the IEEE format\n[44]. The second column presents the equation of the relative length change that we used in Eq.48\nfor each crystal system. In the last column we show the values of the parameter rthat is defined in\nEq.48. The symbols a;b;ccorrespond to the lattice parameters of the relaxed (not distorted) unit\ncell.\nCrystal systemDl\nl0Magnetostrictive\ncoefficientbbb a aa1 aaa2 r\nCubic (I) Eq.17 l001 (0;0;1) ( 0;0;1) ( 1;0;0)3\n2\naaakˆxxx,bbbkˆyyy,ccckˆzzz l111\u0010\n1p\n3;1p\n3;1p\n3\u0011 \u0010\n1p\n3;1p\n3;1p\n3\u0011 \u0010\n1p\n2;0;\u00001p\n2\u0011\n3\n2\nHexagonal (I) Eq.25 la1;2(1;0;0)\u0010\n1p\n3;1p\n3;1p\n3\u0011 \u0010\n1p\n2;1p\n2;0\u0011\n1\n3\naaakˆxxx,ccckˆzzz la2;2(0;0;1) ( 0;0;1) ( 1;0;0) 1\nbbb=\u0010\n\u0000a\n2;p\n3a\n2;0\u0011\nlg;2(1;0;0) ( 1;0;0) ( 0;1;0) 1\na=b6=c le;2 (a;0;c)p\na2+c2\u0010\n1p\n2;0;1p\n2\u0011 \u0010\n\u00001p\n2;0;1p\n2\u0011\n2ac\na2+c2\nTrigonal (I) Eq.35 la1;2(1;0;0) ( 0;0;1)\u0010\n1p\n2;1p\n2;0\u0011\n1\naaakˆxxx,ccckˆzzz la2;2(0;0;1) ( 0;0;1) ( 1;0;0) 1\nbbb=\u0010\n\u0000a\n2;p\n3a\n2;0\u0011\nlg;1(1;0;0) ( 1;0;0) ( 0;1;0) 1\na=b6=c lg;2 (a;0;c)p\na2+c2\u0010\n1p\n2;0;1p\n2\u0011 \u0010\n1p\n2;0;\u00001p\n2\u0011\nac\na2+c2\nl12(a;0;c)p\na2+c2\u0010\n0;1p\n2;1p\n2\u0011 \u0010\n0;1p\n2;\u00001p\n2\u0011\na2\n2(a2+c2)\nl21(a;0;c)p\na2+c2\u0010\n1p\n2;1p\n2;0\u0011 \u0010\n1p\n2;\u00001p\n2;0\u0011\nac\na2+c2\nTetragonal (I) Eq.40 la1;2(1;0;0)\u0010\n1p\n3;1p\n3;1p\n3\u0011 \u0010\n1p\n2;1p\n2;0\u0011\n1\n3\naaakˆxxx,bbbkˆyyy,ccckˆzzz la2;2(0;0;1) ( 0;0;1) ( 1;0;0) 1\na=b6=c lg;2(1;0;0) ( 1;0;0) ( 0;1;0) 1\nle;2 (a;0;c)p\na2+c2\u0010\n1p\n2;0;1p\n2\u0011 \u0010\n\u00001p\n2;0;1p\n2\u0011\n2ac\na2+c2\nld;2\u0010\n1p\n2;1p\n2;0\u0011 \u0010\n1p\n2;1p\n2;0\u0011 \u0010\n\u00001p\n2;1p\n2;0\u0011\n1\nOrthorhombic Eq.44 l1 (1;0;0) ( 1;0;0) ( 0;0;1) 1\naaakˆxxx,bbbkˆyyy,ccckˆzzz l2 (1;0;0) ( 0;1;0) ( 0;0;1) 1\nc<a<b l3 (0;1;0) ( 1;0;0) ( 0;0;1) 1\nl4 (0;1;0) ( 0;1;0) ( 0;0;1) 1\nl5 (0;0;1) ( 1;0;0) ( 0;0;1) 1\nl6 (0;0;1) ( 0;1;0) ( 0;0;1) 1\nl7(a;b;0)p\na2+b2\u0010\n1p\n2;1p\n2;0\u0011\n(0;0;1)(a\u0000b)(a[l1+l2]\u0000b[l3+l4])+4abl7\n2(a2+b2)l7\nl8(a;0;c)p\na2+c2\u0010\n1p\n2;0;1p\n2\u0011\n(0;0;1)(a\u0000c)(al1\u0000cl5)+4acl8\n2(a2+c2)l8\nl9(0;b;c)p\nb2+c2\u0010\n0;1p\n2;1p\n2\u0011\n(0;0;1)(b\u0000c)(bl4\u0000cl6)+4bcl9\n2(b2+c2)l9\n213.2. Program workflow\nThe program MAELAS has been designed to read and write files for the Vi-\nenna Ab initio Simulation Package (V ASP) code [47–49]. The workflow of MAE-\nLAS can be splitted into 5 steps: (i) cell relaxation, (ii) test of MAE, (iii) genera-\ntion of distorted cells and spin directions, (iv) calculation of the energy with V ASP,\nand (v) calculation of magnetostrictive coefficients and magnetoelastic constants.\nIn Fig.2 we show a diagram with a summary of the MAELAS workflow. In the\nfirst step, it performs a full cell relaxation (ionic positions, cell volume, and cell\nshape) of the input unit cell. If one wants to use non-relaxed lattice parameters\n(like experimental ones), then this step can be skipped. In the next step, it is rec-\nommended to check if it is possible to obtain a realistic value of MAE for the\nground state (not distorted cell). To do so, MAELAS generates the V ASP input\nfiles to calculate MAE for two spin directions given by the user which should be\ncompared with available experimental data. In the third step, MAELAS performs\na symmetry analysis with pymatgen library [50] to determine the crystal system,\nredefine the structure using the same IEEE lattice convention as in AELAS [44],\nand lastly generates a set of volume-conserving deformed unit cells and spin di-\nrections according to Table 2. Alternatively, the crystal symmetry of the system\ncan also be imposed by the user manually, which could be useful in some cases. In\nthe fourth step, one should run the V ASP calculations using these inputs. In order\nto help in this task, MAELAS generates some bash scripts to run all calculations\nwith V ASP automatically. Lastly, MAELAS analyzes the calculated energies and\nfits them to a quadratic function (see Section 3.1) in order to obtain the magne-\ntostrictive coefficients. If the elastic tensor is provided in the format given by the\nprogram AELAS [44], then the magnetoelastic constants ( bk) are also calculated\nfrom the relations bk=bk(li;Cnm)given in Section 2. The obtained magnetostric-\ntive coefficients and MAE can be further analyzed through the online visualization\ntool called MAELASviewer that we have also developed [51–53].\nDetecting possible calculation failures on fly is a very important feature of\nan automated high-throughput code. For instance, MAELAS prints a warning\nmessage when the R-squared of the quadratic curve fitting is lower than 0.98. It\nalso automatically generates figures showing the quadratic curve fitting and the\nenergy difference between states with spin directions aaa2andaaa1versus the cell\nlength along bbb, so that the users can check the results easily.\n3.3. Computational details\nThe MAELAS code is written in Python3, and its source and documentation\nfiles are available in GitHub repository [54]. The DFT calculations are performed\n22Figure 2: Workflow of the program MAELAS and its connection with the program AELAS [44].\nwith V ASP code, which is an implementation of the projector augmented wave\n(PAW) method [55]. We use the interaction potentials generated for the Perdew-\nBurke-Ernzerhof (PBE) version [56] of the Generalized Gradient Approximation\n(GGA). We follow the recommended procedure to determine MAE with SOC in-\ncluded non-self-consistently. Namely, a first collinear spin-polarized job (without\nSOC) is performed to calculate the wavefunction and charge density, and then\na second non-collinear spin-polarized job is performed in a non-self-consistent\nmanner, by switching on the SOC, reading the wavefunction and charge density\ngenerated in the collinear job, and defining the spin orientation through the quanti-\nsation axis (SAXIS-tag) [57]. The number of bands included in the non-collinear\njob is set twice as large as the number of bands in the collinear job. By de-\nfault, the code sets the tetrahedron method with Blöchl corrections for smearing\n23in the calculation of MAE. The energy convergence criterion of the electronic\nself-consistency was chosen as 10\u00009eV/cell, while the force convergence crite-\nrion of ionic relaxation was used, with all forces acting on atoms being lower than\n10\u00003eV/Å. MAELAS also generates the input file with the set of k-points in the\nreciprocal space for V ASP by using an automatic Monkhorst–Pack k-mesh [58]\ngamma-centered grid with length parameter Rkgiven by the user in the command\nline. Note the default settings generated by MAELAS for V ASP might not work\nwell for some materials, so that the user should check and tune them accordingly.\nFor instance, in Section 4 we show some specific V ASP settings and tests for few\nknown materials. It is possible to use MAELAS with other codes instead of V ASP,\nafter file conversion to V ASP format files. Although, this process might require\nsome extra work for the user. For instance, we have recently made an interface\nbetween MAELAS and LAMMPS (spin-lattice simulations) [59–61].\n4. Examples\nIn this section, we present some examples of the calculation of MAE, anisotropic\nmagnetostrictive coefficients, elastic and magnetoelastic constants using MAE-\nLAS combined with AELAS [44] for a set of well-known magnetic materials.\nAELAS determines second-order elastic constants from the quadratic coefficients\nof the polynomial fitting of the energies versus strain relationships efficiently. For\neach material, we split the analysis into two parts. Firstly, we perform a cell relax-\nation, evaluate MAE and compute magnetostriction with MAELAS. In the second\nstage, we calculate the elastic constants with AELAS, and we use them as inputs\nto compute the magnetoelastic constants with MAELAS. To do so, we follow the\nworkflow discussed in Section 3.2 (see Fig. 2). A summary of the results obtained\nin the following tests is shown in Tables 3, 4 and 5. All calculations correspond\nto zero-temperature.\n24Table 3: Anisotropic magnetostrictive coefficients and MAE calculated using the program MAE-\nLAS and measured in experiment ( T\u00190 K) for a set of magnetic materials. In parenthesis we\nshow the magnetostrictive coefficients with Mason’s definitions obtained using the relations given\nby Eq. A.2. These data correspond to the simulations with the largest number of k-points discussed\nin the main text.\nMaterial Crystal system MethodMagnetostriction\ncoefficientMAELAS\n(\u000210\u00006)Expt.\n(\u000210\u00006)MAEMAELAS\n(µeV/atom)Expt.\n(µeV/atom)\nFCC Ni Cubic (I) DFT GGA l001 -78.4 -60aE(110)\u0000E(001) 0.03 -2.15b\nSG 225 l111 -46.1 -35aE(111)\u0000E(001) 0.34 -2.73b\nSD-MD l001 -61.9hE(110)\u0000E(001) -2.14h\nl111 -35.4hE(111)\u0000E(001) -2.86h\nBCC Fe Cubic (I) DFT GGA l001 25.7 26aE(110)\u0000E(001) 0.24 1.0b\nSG 229 l111 17.2 -30aE(111)\u0000E(001) 0.32 1.34b\nSD-MD l001 25.9hE(110)\u0000E(001) 0.99h\nl111 -30.3hE(111)\u0000E(001) 1.33h\nHCP Co Hexagonal (I) DFT SCAN la1;2(lA) 85 (-78) 95 (-66)cE(100)\u0000E(001) 53 61b\nSG 194 la2;2(lB) -115 (-92) -126 (-123)c\nlg;2(lC) 15 (115) 57 (126)c\nle;2(lD) -19 (-1) -286 (-128)c\nDFT LSDA+U la1;2(lA) 111 (-109) E(100)\u0000E(001) 58\nJ=0:8eV la2;2(lB) -251 (-114)\nU=3eV lg;2(lC) 4 (251)\nle;2(lD) -51 (10)\nYCo 5 Hexagonal (I) DFT LSDA+U la1;2-90jla1;2j<100dE(100)\u0000E(001) 365 567e\nSG 191 la2;2115jla2;2j<100d\nlg;276\nle;2141\nFe2Si Trigonal (I) DFT GGA la1;2-9 E(100)\u0000E(001) -38\nSG 164 la2;215\nlg;18\nlg;228\nl12 -3\nl21 -13\nL10FePd Tetragonal (I) DFT GGA la1;2-21 E(100)\u0000E(001) 106 181f\nSG 123 la2;279\nlg;231\nle;228\nld;2106\nla1;0\u0000la1;2\n3+lg;2\n2100g\nYCo Orthorhombic DFT LSDA+U l1 -11 E(100)\u0000E(001) 22\nSG 63 l2 32 E(010)\u0000E(001) -23\nl3 70\nl4 -74\nl5 -30\nl6 7\nl7 36\nl8 -20\nl9 35\naRef.[40],bRef.[62],cRef.[63],dRef.[7],eRef.[64],fRef.[65],gRef.[66],hRef.[59]\n25Table 4: Elastic and magnetoelastic constants calculated using the interface between AELAS and\nMAELAS codes. The third column shows the DFT method used to compute the elastic constants.\nThe experimental elastic constants of FCC Ni and BCC Fe were measured at T\u00190 K, while\nfor HCP Co T\u0019300 K. The experimental magnetoelastic constants were estimated using the\nexperimental elastic constants (seventh column) and the experimental magnetostrictive coefficients\nin Table 3 via the relations given in Section 2. The sixth column presents calculations of the elastic\nconstants available in the Materials Project database [67, 68].\nMaterial Crystal system MethodElastic\nconstantAELAS\n(GPa)Mat.Proj.\n(GPa)Expt.\n(GPa)Magnetoelastic\nconstantMAELAS\n(MPa)Expt.\n(MPa)\nFCC Ni Cubic (I) DFT GGA C11 298 276g261hb1 15.5 9.9\nSG 225 C12 166 159g151hb2 19.4 13.9\nC44 140 132g132h\nSD-MD C11 264fb1 10.4\nC12 152fb2 14.1\nC44 133f\nBCC Fe Cubic (I) DFT GGA C11 288 247a243bb1 -5.2 -4.1\nSG 229 C12 152 150a138bb2 -5.3 10.9\nC44 104 97a122b\nSD-MD C11 230fb1 -3.7\nC12 134fb2 10.6\nC44 116f\nHCP Co Hexagonal (I) DFT LSDA+U C11 327 307db21 -21.3 -31.9\nSG 194 J=0:8eV C12 157 165db22 48.3 25.5\nU=3eV C13 130 103db3 -0.7 -8.1\nC33 308 358db4 7.1 42.9\nC44 69 75d\nDFT SCAN C11 648 b21 -51.3\nC12 212 b22 40.5\nC13 189 b3 6.4\nC33 633 b4 8.9\nC44 239\nDFT GGA C11 358c\nC12 165c\nC13 114c\nC33 409c\nC44 95c\nYCo 5 Hexagonal (I) DFT GGA C11 208 192eb21 14.9\nSG 191 C12 103 123eb22 -10.4\nC13 114 113eb3 -8.0\nC33 270 262eb4 -13.6\nC44 49 48e\nDFT LSDA+U C11 -63 b21 13.9\nC12 363 b22 -7.9\nC13 115 b3 32.5\nC33 249 b4 -12.4\nC44 44\naRef.[69],bRef.[70],cRef.[71],dRef.[72],eRef.[73],fRef.[59],gRef.[74],hRef.[75]\n26Table 5: Elastic and magnetoelastic constants calculated using the interface between AELAS and\nMAELAS codes. The third column shows the DFT method used to compute the elastic constants.\nThe experimental elastic constants of L1 0FePd were measured at T\u0019300 K. The sixth column\npresents calculations of the elastic constants available in the Materials Project database [67, 68].\nMaterial Crystal system MethodElastic\nconstantAELAS\n(GPa)Mat.Proj.\n(GPa)Expt.\n(GPa)Magnetoelastic\nconstantMAELAS\n(MPa)Expt.\n(MPa)\nFe2Si Trigonal (I) DFT GGA C11 428 415ab21 3.1\nSG 164 C12 164 169ab22 -4.2\nC13 133 133ab3 -0.7\nC14 -27 -25ab4 3.3\nC33 434 428ab14 -1.4\nC44 118 107ab34 -0.4\nL10FePd Tetragonal (I) DFT GGA C11 324 293b214cb21 -2.4\nSG 123 C12 67 62b143cb22 -15.2\nC13 133 125b143cb3 -7.9\nC33 264 254b227cb0\n3 -7.9\nC44 101 99b92cb4 -5.6\nC66 37 38b93c\nYCo Orthorhombic DFT GGA C11 76 94db1 -0.9\nSG 63 C12 45 61db2 0.6\nC13 48 44db3 -5.0\nC22 102 93db4 5.7\nC23 55 56db5 0.9\nC33 141 121db6 1.5\nC44 40 38db7 -5.0\nC55 27 29db8 1.1\nC66 39 41db9 -8.2\nDFT LSDA+U C11 101 b1 -1.7\nC12 65 b2 1.2\nC13 58 b3 -3.8\nC22 94 b4 4.3\nC23 70 b5 -0.1\nC33 138 b6 2.3\nC44 42 b7 -4.4\nC55 29 b8 1.1\nC66 35 b9 -8.7\naRef.[76],bRef.[77],cRef.[78],dRef.[79]\n4.1. FCC Ni\nIn the first example, we consider FCC Ni, which is described by Eq.17 since\nit is a cubic (I) system.\n4.1.1. Cell relaxation, MAE and magnetostrictive coefficients\nFirstly, we perform a cell relaxation using an automatic k-point mesh with\nlength parameter R k=160 centered on the G-point (46\u000246\u000246), 16 valence\nstates, and energy cut-off 520 eV with PAW method and GGA-PBE. The re-\nlaxed lattice parameter is a=3:50419 Å. Next, we analyze the dependence of\n27MAE and magnetostrictive coefficients on the k-point mesh (R k=80, 100, 120,\n140 and 160) for this relaxed unit cell using the same V ASP settings as in the\ncell relaxation. The results are shown in Fig. 3. We observe that E(1;1;1)\u0000\nE(0;0;1)>0 for all calculations up to 93150 k-points, while the experimental\nvalue is\u00002:7µeV/atom [80]. This deviation may be due to the fact that we have\nnot used a sufficiently large number of k-points, as Halilov et al. pointed out\n[80, 81]. Our results are in good agreement with the calculations performed by\nTrygg el at. where a similar number of k-points were used [82]. We also see\nthatE(1;1;1)\u0000E(0;0;1)is approaching to negative values as the number of k-\npoints is increased. Interestingly, the calculated magnetostrictive coefficients are\nin quite good agreement with the experimental ones [40] despite the deviation\nof MAE for the unstrained unit cell. One possible reason for this result may be\nthat the calculation of the magnetostrictive coefficients involves larger energy dif-\nference between magnetization directions than the determination of MAE for the\nunstrained unit cell (which might be close to the accuracy limit of V ASP \u0018µeV),\nsee Fig.4. Consequently, a k-point mesh with about 105k-points may be sufficient\nto obtain reliable magnetostrictive coefficients for FCC Ni using GGA, although\na much more dense k-mesh would be needed to obtain a reliable MAE for the\nunstrained unit cell [80, 81]. Note that these two properties come from the SOC,\nso that in general it would be highly desirable that the used method to calculate\nthe energies describes well both MAE and magnetostriction. Recently, we devel-\noped a spin-lattice model within the framework of coupled spin and molecular\ndynamics (SD-MD) that reproduces accurately the experimental elastic and mag-\nnetoelastic energies at zero-temperature [59]. We obtained very good results by\napplying MAELAS to this coarse-grained model of SOC, see Tables 3 and 4.\n4.1.2. Elastic and magnetoelastic constants\nTo compute the elastic constants we make use of AELAS code [44]. As inputs,\nwe use the same relaxed cell and V ASP settings as in the calculation of magne-\ntostriction, but with lower number of k-points R k=60 (17\u000217\u000217 for the not\ndistorted cell) and not including SOC. Once we have the elastic constants, we\nuse them as inputs to derive the magnetoelastic constants with MAELAS. The re-\nsults are shown in Table 4, where we also include calculations of elastic constants\navailable in the Materials Project database [67, 68] and experimental data [75].\nWe observe that the value of C11=298 GPa obtained with GGA is higher than\nthe one in the Materials Project C11=276 GPa and in the experiment C11=261\nGPa. Concerning the magnetoelastic constants, we see that both b1andb2are in\nfairly good agreement with the experiment.\n28Figure 3: Calculation of (left) MAE of the unstrained unit cell and (right) magnetostrictive coeffi-\ncients for FCC Ni as a function of k-points.\nFigure 4: Calculation of l001for FCC Ni using MAELAS. (Left) Quadratic curve fit to the energy\nversus cell length along bbb= (0;0;1)with spin direction aaa1= (0;0;1). (Right) Energy difference\nbetween states with spin directions aaa2= (1;0;0)andaaa1= (0;0;1)against the cell length along\nbbb= (0;0;1).\n4.2. BCC Fe\nIn this example, we consider BCC Fe, which is described by Eq.17 since it is\na cubic (I) system.\n4.2.1. Cell relaxation, MAE and magnetostrictive coefficients\nIn the first stage, we we perform a cell relaxation for the conventional cubic\nunit cell of the BCC (2 atoms/cell) using a 57 \u000257\u000257 k-mesh with 185193\nk-points in the Brillouin zone. The interactions were described by a PAW po-\ntential with 14 valence electrons within the PBE approximation to the exchange-\ncorrelation, and the PW basis was generated for an energy cut-off of 380 eV (30%\n29larger than the default value). The relaxed lattice parameter is a=2:82509 Å.\nIn Fig. 5 we show the dependence of MAE and magnetostriction on the k-points\nfor this relaxed cell using the same exchange-correlation and energy cut-off as\nin the cell relaxation. The calculated values of MAE with the largest number of\nk-points (262144) are E(110)\u0000E(001) =0:32µeV/atom and E(111)\u0000E(001) =\n0:24µeV/atom which are a bit lower than the experimental values 1 µeV/atom and\n1:34µeV/atom, respectively [62]. Concerning the magnetostrictive coefficients,\nwe obtained l001=25:7\u000210\u00006andl111=17:2\u000210\u00006, while the experimental\nvalues at T=4:2K are l001=26\u000210\u00006andl111=\u000030\u000210\u00006[40]. We see\nthatl001is quite close to the experimental result, while l111is in good agreement\nwith previous DFT calculations [21, 25, 83] but it has the opposite sign as the\nexperimental value. The calculation of l111is presented in Fig.6. We observe\nthat the sign of the derivative of the energy difference between states with spin di-\nrections aaa2=\u0010\n1p\n2;0;\u00001p\n2\u0011\nandaaa1=\u0010\n1p\n3;1p\n3;1p\n3\u0011\nwith respect to the cell length\nalong bbb=\u0010\n1p\n3;1p\n3;1p\n3\u0011\nevaluated at l=l2is equal to the sign of the calculated\nl111(>0), as expected from Eq.52. However, the experimental l111is negative.\nThis deviation might be due to a possible failure of GGA related to the location of\nthe Fermi level in a region of majority band t2g density of states [84, 85]. Aim-\ning to clarify the influence of MAELAS in this result, we applied MAELAS to a\nspin-lattice model for BCC Fe, that reproduces accurately the experimental elastic\nand magnetoelastic energies, obtaining almost the same experimentally observed\nmagnetostriction [59], see Tables 3 and 4.\nFigure 5: Calculation of (left) MAE of the unstrained unit cell and (right) magnetostrictive coeffi-\ncients for BCC Fe as a function of k-points.\n30Figure 6: Calculation of l111for BCC Fe using MAELAS. (Left) Quadratic curve fit to the energy\nversus cell length along bbb=\u0010\n1p\n3;1p\n3;1p\n3\u0011\nwith spin direction aaa1=\u0010\n1p\n3;1p\n3;1p\n3\u0011\n. (Right) Energy\ndifference between states with spin directions aaa2=\u0010\n1p\n2;0;\u00001p\n2\u0011\nandaaa1=\u0010\n1p\n3;1p\n3;1p\n3\u0011\nagainst\nthe cell length along bbb=\u0010\n1p\n3;1p\n3;1p\n3\u0011\n.\n4.2.2. Elastic and magnetoelastic constants\nAs inputs for AELAS code [44], we use the same relaxed cell and V ASP set-\ntings as in the calculation of magnetostriction, but with lower number of k-points\nRk=60 (21\u000221\u000221 for the not distorted cell) and not including SOC. Once\nwe have the elastic constants, we use them as inputs to derive the magnetoelastic\nconstants with MAELAS. The results are shown in Table 4, where we also in-\nclude calculations of elastic constants available in the Materials Project database\n[67, 68] and experimental data [70]. We observe that the value of C11=288 GPa\nobtained with AELAS is significantly higher than the one in the Materials Project\nC11=247 GPa and in the experiment C11=243 GPa. Regarding the magnetoelas-\ntic constants, we see that the value for b1=\u00005:2 MPa generated with MAELAS\nis close to the estimated experimental value b1=\u00004:1 MPa. However, we obtain\na negative sign for b2=\u00005:4 MPa, while in the experiment it is positive b2=10:9\nMPa. This deviation is due to the positive sign of l111given by DFT that we have\nmentioned above, see Eq.18 [25, 84, 85].\n4.3. HCP Co\nAs a first example of hexagonal (I) system, we consider HCP Co.\n4.3.1. Cell relaxation, MAE and magnetostrictive coefficients\nFor this material, we set the length parameter R k=160 for the generation\nof the automatic k-point mesh, which for the relaxed (not distorted) cell, results\n31in a 75\u000275\u000240 k-point grid with 250000 points in the Brillouin zone. All cal-\nculations were done with an energy cut-off 406 :563 eV (50% larger than the\ndefault one), 15 electrons in the valence states, and the meta-GGA functional\nSCAN [86], with aspherical contributions to the PAW one-centre terms. The\nrelaxed lattice parameters are a=b=2:4561 Å and c=3:9821 Å. The calcu-\nlated MAE for the relaxed cell is E(100)\u0000E(001) =53µeV/atom which is quite\nclose to the experimental value 61 µeV/atom [62]. As it is shown in Table 3, the\ncalculated magnetostrictive coefficients are also close to the experimental ones,\nexcept for le;2. Similarly, converting them into Mason’s definitions via the rela-\ntions given by Eq.A.2, we see that only lD=\u00001\u000210\u00006is significantly deviated\nfrom the experiment ( \u0000128\u000210\u00006) [63]. Aiming to clarify this result, we per-\nformed a direct calculation of lDusing bbb=\u0010\n1p\n2;0;1p\n2\u0011\n,aaa1=\u0010\n1p\n2;0;1p\n2\u0011\nand\naaa2= (0;0;1)finding lD=\u00009\u000210\u00006, which is consistent with the indirect cal-\nculation through Clark’s definition but still far from the experimental value. Fig.7\nshows the quadratic curve fit to the energy versus cell length along bbb= (1;0;0)\nwithaaa1=\u0010\n1p\n3;1p\n3;1p\n3\u0011\nto calculate la1;2.\nFigure 7: Calculation of la1;2for HCP Co using MAELAS with the meta-GGA functional\nSCAN. (Left) Quadratic curve fit to the energy versus cell length along bbb= (1;0;0)with spin\ndirection aaa1=\u0010\n1p\n3;1p\n3;1p\n3\u0011\n. (Right) Energy difference between states with spin directions\naaa2=\u0010\n1p\n2;1p\n2;0\u0011\nandaaa1=\u0010\n1p\n3;1p\n3;1p\n3\u0011\nagainst the cell length along bbb= (1;0;0).\nWe have also performed a second test using the rotationally invariant LSDA+U\napproach introduced by Liechtenstein et al. [87] fixing J=0:8eV and varying U\non the d-electrons [64]. In all calculations we use the same pseudopotential and\n32number of k-points as in the tests performed with SCAN. The energy cut-off is set\nto 380 eV . In this case the relaxed lattice parameters are a=b=2:48896 Å and\nc=4:02347 Å. The analysis of MAE and magnetostriction for different values of\nUis shown in Fig.8. We see that MAE approximates the experimental value at\nU=3eV , so that we might expect a reliable description of SOC for this value of\nU. Increasing Uup to 3eV has a significant effect on la1;2andla2;2making them\nto approach the experimental values. On the other hand, lg;2andle;2don’t change\ntoo much within the range of values used for U. The sign of all magnetostrictive\ncoefficients are in good agreement to the experimental ones. However, as in the\ncase with SCAN, le;2is significantly underestimated. Possible reasons for this\nsystematic deviation might be a failure of DFT [85], the applied deformations\n(we used the default value 0 :01 for the tag\u0000sthat sets the maximum value of\nparameter sin the generation of the deformed unit cells, see Eq.C.4), the used\nV ASP settings (k-point mesh, exchange-correlation functional, smearing method,\nlattice parameters, ...) or higher order corrections in the equation of the relative\nlength change Eq.25 [88]. This issue should be further investigated to clarify its\npossible causes.\nFigure 8: Calculation of (left) MAE of the unstrained unit cell and (right) magnetostrictive coeffi-\ncients for HCP Co using the LSDA+U approach with different values of parameter U.\n4.3.2. Elastic and magnetoelastic constants\nFor the calculation of the elastic constants we use the same relaxed cell and\nV ASP settings as in the calculation of magnetostriction, but without SOC and\nlower number of k-points R k=60 (28\u000228\u000215 for the not distorted cell). In\naddition to SCAN, we also run calculations with LSDA+U setting J=0:8eV\n33andU=3eV . In Table 4, we see that LSDA+U and GGA (Materials Project\ndatabase [74]) give better results than SCAN for both the elastic and magnetoe-\nlastic constants. The magnetoelastic constants obtained with LSDA+U are mod-\nerately good, except for b3andb4which are one order of magnitude lower than in\nthe experiment, mainly due to the deviations coming from lg;2andle;2given by\nMAELAS, see Table 3.\n4.4. YCo 5\nIn this example we study the hexagonal (I) system YCo 5with prototype CaCu 5\nstructure (space group 191).\n4.4.1. Cell relaxation, MAE and magnetostrictive coefficients\nWe use the simplified (rotationally invariant) approach to the LSDA+U intro-\nduced by Dudarev et al. [89] with parameters U=1:9 eV and J=0:8 eV for\nCo, and U=J=0 eV for Y given in Ref.[64]. For the calculation of the re-\nlaxed cell, MAE and magnetostrictive coefficients we used an automatic k-point\nmesh with length parameter R k=100 centered on the G-point (23\u000223\u000225 for\nthe not distorted cell), 11 and 9 valence states for Y and Co, respectively, and en-\nergy cut-off 375 eV . The cell relaxation leads to lattice parameters a=b=4:9253\nÅ and c=3:9269 Å. The calculated MAE is E(100)\u0000E(001) =365µeV/atom\nwhich is lower than the experimental value 567 µeV/atom [64]. Andreev measured\nthe magnetostriction along a and c axis, finding that the magnitude of jla1;2jand\njla2;2jcan not be greater than 10\u00004[7]. We obtained la1;2=\u000090\u000210\u00006and\nla2;2=115\u000210\u00006which are quite close to the experimental upper limit. In\nFig.9 we present the quadratic curve fit to the energy versus cell length along\nbbb= (0;0;1)withaaa1= (0;0;1)to calculate la2;2.\n4.4.2. Elastic and magnetoelastic constants\nThe calculation of the elastic constants is performed using the same relaxed\ncell and V ASP settings as for magnetostriction, but without SOC and lower num-\nber of k-points R k=60 (14\u000214\u000215 for the not distorted cell). In addition\nto LSDA+U, we also run calculations with GGA. In Table 4, we observe that\nLSDA+U leads to an unstable phase ( C11\u0000C12<0), while GGA gives better\nresults.\n4.5. Fe 2Si\nTo illustrate the application of MAELAS to trigonal (I) systems, we apply it\nto Fe 2Si (space group 164) [90].\n34Figure 9: Calculation of la2;2for YCo 5using MAELAS. (Left) Quadratic curve fit to the energy\nversus cell length along bbb= (0;0;1)with spin direction aaa1= (0;0;1). (Right) Energy difference\nbetween states with spin directions aaa2= (1;0;0)andaaa1= (0;0;1)against the cell length along\nbbb= (0;0;1).\n4.5.1. Cell relaxation, MAE and magnetostrictive coefficients\nFor the calculation of the cell relaxation, MAE and magnetostrictive coeffi-\ncients we used an automatic k-point mesh with length parameter R k=80 centered\non the G-point (24\u000224\u000217 for the not distorted cell), 14 and 4 valence states for\nFe and Si, respectively, and energy cut-off 520 eV with PAW method and GGA-\nPBE. The relaxed lattice parameters are a=3:9249 Å and c=4:8311 Å. The\ncalculated MAE is E(100)\u0000E(001) =\u000038µeV/atom (easy plane). Sun et al. re-\nported MAE values with the screened hybrid Heyd-Scuseria-Ernzerhof (HSE06)\nfunctional smaller than with PBE for 2D Fe 2Si [91, 92]. Chi Pui Tang et al. calcu-\nlated some electronic properties for bulk Fe 2Si finding that the densities of states\nin the vicinity of the Fermi level is mainly contributed from the d-electrons of Fe\n[93]. In Table 3, we observe that the overall anisotropic magnetostriction given\nby MAELAS is rather small, which makes this material interesting for high-flux\ncore applications because it can reduce hysteresis loss [94].\n4.5.2. Elastic and magnetoelastic constants\nAs inputs for AELAS, we use the same relaxed cell and V ASP settings as in the\ncalculation of magnetostriction, but without SOC and lower number of k-points\nRk=60 (18\u000218\u000212 for the not distorted cell). In Table 5, we see that AELAS\ngives similar elastic constants as in the Materials Project [76]. The derived mag-\nnetoelastic constants are small which is consistent with the low magnetostrictive\ncoefficients that we obtained previously.\n35Figure 10: Calculation of la2;2for Fe 2Si using MAELAS. (Left) Quadratic curve fit to the energy\nversus cell length along bbb= (0;0;1)with spin direction aaa1= (0;0;1). (Right) Energy difference\nbetween states with spin directions aaa2= (1;0;0)andaaa1= (0;0;1)against the cell length along\nbbb= (0;0;1).\n4.6. L1 0FePd\nAs an example of tetragonal (I) system, we calculate the anisotropic magne-\ntostrictive coefficients of L1 0FePd (space group 123).\n4.6.1. Cell relaxation, MAE and magnetostrictive coefficients\nFor the calculation of the cell relaxation, MAE and magnetostrictive coeffi-\ncients we used an automatic k-point mesh with length parameter R k=100 cen-\ntered on the G-point (37\u000237\u000227 for the not distorted cell), 8 and 10 valence\nstates for Fe and Pd, respectively, and energy cut-off 375 eV with PAW method\nand GGA-PBE. The relaxed lattice parameters are a=2:6973 Å and c=3:7593\nÅ. We obtained a MAE E(100)\u0000E(001) =106µeV/atom which is lower than in\nthe experiment 181 µeV/atom [65]. The values of the obtained anisotropic mag-\nnetostrictive coefficients are shown in Table 3. Shima et al. reported a relative\nlength change equal to 100 \u000210\u00006along a-axis under a magnetic field in the same\ndirection ( bbb=aaa= (1;0;0)) [66]. According to Eq.40, this measurement corre-\nsponds to la1;0\u0000la1;2\n3+lg;2\n2. For the anisotropic part of this quantity, we obtained\n\u0000la1;2\n3+lg;2\n2=22:5\u000210\u00006. In Fig.11 we show the quadratic curve fit to the en-\nergy versus cell length along bbb= (1;0;0)withaaa1= (1;0;0)to calculate lg;2.\n36Figure 11: Calculation of lg;2for L1 0FePd using MAELAS. (Left) Quadratic curve fit to the\nenergy versus cell length along bbb= (1;0;0)with spin direction aaa1= (1;0;0). (Right) Energy\ndifference between states with spin directions aaa2= (0;1;0)andaaa1= (1;0;0)versus the cell\nlength along bbb= (1;0;0).\n4.6.2. Elastic and magnetoelastic constants\nThe calculation of the elastic constants with AELAS is performed using the\nsame relaxed cell and V ASP settings as for magnetostriction, but without SOC\nand lower number of k-points R k=60 (22\u000222\u000216 for the not distorted cell). As\nwe see in Table 5, we obtain similar results as in the Materials Project database\n[77].\n4.7. YCo\nFor the case of orthorhombic systems, we study the compound YCo (space\ngroup 63) [95].\n4.7.1. Cell relaxation, MAE and magnetostrictive coefficients\nIn this case, we use the simplified (rotationally invariant) approach to the\nLSDA+U [89] with parameters U=1:9 eV and J=0:8 eV for Co, and U=J=0\neV for Y in the same way as in YCo 5[64]. For the calculation of the relaxed cell,\nMAE and magnetostrictive coefficients we used an automatic k-point mesh with\nlength parameter R k=90 centered on the G-point (22\u00029\u000223 for the not distorted\ncell), 11 and 9 valence states for Y and Co, respectively, and energy cut-off 375\neV . The cell relaxation leads to lattice parameters a=4:0686 Å, b=10:3157 Å\nandc=3:8957 Å. As we see in Table 3, both MAE and magnetostriction are quite\nsmall for this material.\n37Figure 12: Calculation of l5for YCo using MAELAS. (Left) Quadratic curve fit to the energy\nversus cell length along bbb= (0;0;1)with spin direction aaa1= (1;0;0). (Right) Energy difference\nbetween states with spin directions aaa2= (0;0;1)andaaa1= (1;0;0)against the cell length along\nbbb= (0;0;1).\n4.7.2. Elastic and magnetoelastic constants\nThe calculation of the elastic constants is performed using the same relaxed\ncell and V ASP settings as for magnetostriction, but without SOC and lower num-\nber of k-points R k=60 (15\u00026\u000215 for the not distorted cell). In addition to\nLSDA+U, we also run calculations with GGA. In Table 5, we observe that both\nLSDA+U and GGA lead to similar elastic and magnetoelastic constants.\n5. Conclusions and future perspectives\nIn summary, the program MAELAS offers computational tools to tackle the\ncomplex phenomenon of magnetostriction by automated first-principles calcula-\ntions. It could potentially be used to discover and design novel magnetostric-\ntive materials by a high-throughput screening approach. In particular, materials\nwith giant magnetostriction (beyond conventional cubic and hexagonal systems),\nisotropic or very low magnetostriction (like FeNi alloys) might be of technological\nimportance.\nThe preliminary tests of the program show quite encouraging results, although\nthere is still room for improvement. First principle calculations are still quite chal-\nlenging for materials with very low MAE or localized 4f-electrons [64]. In this\nsense, MAELAS could also be a useful tool to understand, test and improve the\nDFT methods to compute induced properties by SOC and crystal field interactions\nlike the MAE of unstrained systems and anisotropic magnetostriction.\n38Presently, we are working on new features of MAELAS and online visual-\nization tools [51]. We are also considering to increase the number of supported\ncrystal systems, as well as to implement more computationally efficient methods\nto calculate magnetoelastic constants and magnetostrictive coefficients. These ex-\ntensions might be included in new versions of the code.\nAcknowledgement\nThis work was supported by the ERDF in the IT4Innovations national super-\ncomputing center - path to exascale project (CZ.02.1.01/0.0/0.0/16-013/0001791)\nwithin the OPRDE. This work was supported by The Ministry of Education, Youth\nand Sports from the Large Infrastructures for Research, Experimental Develop-\nment, and Innovations project “e-INFRA CZ - LM2018140”. This work was sup-\nported by the Donau project No. 8X20050 and the computational resources pro-\nvided by the Open Access Grant Competition of IT4Innovations National Super-\ncomputing Center within the projects OPEN-18-5, OPEN-18-33, and OPEN-19-\n14. DL, SA, and APK acknowledge the Czech Science Foundations grant No. 20-\n18392S. P.N., D.L., and S.A. acknowledge support from the H2020-FETOPEN\nno. 863155 s-NEBULA project.\nAppendix A. Conversion between different definitions of magnetostrictive\ncoefficients for hexagonal (I)\nThe magnetostrictive coefficients for hexagonal (I) shown in Eq.25 were de-\nfined by Clark et al. in 1965 [16]. However, one can find other definitions like\nthose given by Mason [15], Birss [34], and Callen and Callen [17]. In this ap-\npendix, we show the conversion formulas between these definitions and those\nprovided by Clark et al. [16] (Eq.25).\nAppendix A.1. Mason’s form\nIn 1954, based on a general thermodynamic function with stresses and in-\ntensity of magnetization as the fundamental variables [96], Mason derived the\n39following form of the relative length change [15]\nDl\nl0\f\f\f\f\faaa\nbbb=la1;0\nMason(b2\nx+b2\ny)+la2;0\nMasonb2\nz+lA[(axbx+ayby)2\u0000(axbx+ayby)azbz]\n+lB[(1\u0000a2\nz)(1\u0000b2\nz)\u0000(axbx+ayby)2]\n+lC[(1\u0000a2\nz)b2\nz\u0000(axbx+ayby)azbz]+4lD(axbx+ayby)azbz:\n(A.1)\nThese magnetostrictive coefficients are related to those defined in Eq.25 as [16]\nla1;0\nMason =la1;0+2\n3la1;2\nla2;0\nMason =la2;0+2\n3la2;2\nlA=\u0000la1;2+1\n2lg;2\nlB=\u0000la1;2\u00001\n2lg;2\nlC=\u0000la2;2\nlD=1\n2le;2\u00001\n4la1;2+1\n8lg;2\u00001\n4la2;2:(A.2)\nNote in the original work of Mason [15] the terms that describes the volume mag-\nnetostriction were not included. Here we added these terms ( la1;0\nMason ,la2;0\nMason ) in\norder to fully recover the Eq.25.\nAppendix A.2. Birss’s form\nIn 1959 Birss derived an equivalent equation of relative length change in this\nform [34]\nDl\nl0\f\f\f\f\faaa\nbbb=Q0+Q1b2\nz+Q2(1\u0000a2\nz)+Q4(1\u0000a2\nz)b2\nz+Q6(axbx+ayby)azbz\n+Q8(axbx+ayby)2:\n(A.3)\n40These magnetostrictive coefficients are related to those defined in Eq.25 as [16]\nQ0=la1;0+2\n3la1;2\nQ1=la2;0+2\n3la2;2\u0000la1;0\u00002\n3la1;2\nQ2=\u0000la1;2\u00001\n2lg;2\nQ4=la1;2+1\n2lg;2\u0000la2;2\nQ6=2le;2\nQ8=lg;2:(A.4)\nAppendix A.3. Callen and Callen’s form\nIn 1965 Callen and Callen obtained other equivalent form of the equation of\nrelative length change by including two-ion interactions into the theory of magne-\ntostriction arising from single-ion crystal-field effects [17]. It reads\nDl\nl0\f\f\f\f\faaa\nbbb=1\n3la\n11+1\n2p\n3la\n12\u0012\na2\nz\u00001\n3\u0013\n+2la\n21\u0012\nb2\nz\u00001\n3\u0013\n+p\n3la\n22\u0012\na2\nz\u00001\n3\u0013\u0012\nb2\nz\u00001\n3\u0013\n+lg\u00141\n2(a2\nx\u0000a2\ny)(b2\nx\u0000b2\ny)+2axaybxby\u0015\n+2le(axazbxbz+ayazbybz);\n(A.5)\nThese magnetostrictive coefficients are related to those defined in Eq.25 as [17]\nla\n11=2la1;0+la2;0+2la1;2+la2;2\nla\n12=4p\n3la1;2+2p\n3la2;2\nla\n21=\u00001\n2la1;0+1\n2la2;0\nla\n22=\u00001p\n3la1;2+1p\n3la2;2\nle=le;2\nlg=lg;2:(A.6)\n41Appendix B. Conversion between different definitions of magnetostrictive\ncoefficients for tetragonal (I)\nIn 1994 Cullen et al. [8] derived the equation of relative length change given\nby Eq.40 for tetragonal (I) system. In 1954 Mason obtained an equivalent equation\nthat reads [15]\nDl\nl0\f\f\f\f\faaa\nbbb=la1;0\nMason(b2\nx+b2\ny)+la2;0\nMasonb2\nz+1\n2l1[(axbx\u0000ayby)2\u0000(axby+aybx)2\n+(1\u0000b2\nz)(1\u0000a2\nz)\u00002azbz(axbx+ayby)]+4l2azbz(axbx+ayby)\n+4l3axaybxby+l4[b2\nz(1\u0000a2\nz)\u0000azbz(axbx+ayby)]\n+1\n2l5[(axby\u0000aybx)2\u0000(axbx+ayby)2+(1\u0000b2\nz)(1\u0000a2\nz)]:\n(B.1)\nThese magnetostrictive coefficients are related to those defined by Eq.40 in the\nfollowing way\nla1;0\nMason =la1;0+2\n3la1;2\nla2;0\nMason =la2;0+2\n3la2;2\nl1=\u0000la1;2+1\n2lg;2\nl2=1\n2le;2\u00001\n4la2;2\u00001\n4la1;2+1\n8lg;2\nl3=1\n2ld;2\u0000la1;2\nl4=\u0000la2;2\nl5=\u0000la1;2\u00001\n2lg;2:(B.2)\nNote in the original work of Mason [15] the terms that describes the volume mag-\nnetostriction were not included. Here we added these terms ( la1;0\nMason ,la2;0\nMason ) in\norder to fully recover the Eq.40.\nAppendix C. Generation of the deformed unit cells\nIn this appendix we present the procedure to generate the deformed unit cells\nfor the calculation of each magnetostrictive coefficient. The deformed unit cells\n42are generated by multiplying the lattice vectors of the initial unit cell aaa= (ax;ay;az),\nbbb= (bx;by;bz),ccc= (cx;cy;cz)by the deformation gradient Fi j[97]\n0\n@a0\nxb0\nxc0\nx\na0\nyb0\nyc0\ny\na0\nzb0\nzc0\nz1\nA=0\n@FxxFxyFxz\nFyxFyyFyz\nFzxFzyFzz1\nA\u00010\n@axbxcx\naybycy\nazbzcz1\nA (C.1)\nwhere a0\ni,b0\niandc0\ni(i=x;y;z) are the components of the lattice vectors of the\ndeformed cell. In the infinitesimal strain theory, the deformation gradient is re-\nlated to the displacement gradient ( ¶ui=¶rj) asFi j=di j+¶ui=¶rj, where di jis the\nKronecker delta. Hence, according to Eq.5, the strain tensor ei jcan be written in\nterms of the deformation gradient as\neee=0\n@exxexyexz\neyxeyyeyz\nezxezyezz1\nA=1\n20\n@2(Fxx\u00001)Fxy+Fyx Fxz+Fzx\nFxy+Fyx2(Fyy\u00001)Fyz+Fzy\nFxz+Fzx Fyz+Fzy2(Fzz\u00001)1\nA: (C.2)\nIn MAELAS, we consider deformation gradients to optimize the unit cell in the\nmeasuring directions bbbgiven by Table 2. Additionally, we also constrain the\ndeterminant of the deformation gradients to be equal to one ( det(FFF) =1) in order\nto preserve the volume of the unit cells (isochoric deformation) [21, 98]. We point\nout that there are other possible variants of the following deformation modes [85].\nAppendix C.1. Cubic (I) system\nFor cubic (I) systems MAELAS generates two set of deformed unit cells with\ntetragonal deformations along bbb= (0;0;1)and trigonal deformations along bbb=\n(1=p\n3;1=p\n3;1=p\n3)to calculate l001andl111, respectively (see Table 2). The\ndeformation gradients for these two deformation modes are\nFFFl001\nbbb=(0;0;1)(s) =0\nB@1p1+s0 0\n01p1+s0\n0 0 1 +s1\nCA;FFFl111\nbbb=\u0010\n1p\n3;1p\n3;1p\n3\u0011(s) =z0\n@1s\n2s\n2s\n21s\n2s\n2s\n211\nA\n(C.3)\nwhere z=3p\n4=(4\u00003s2+s3). The parameter scontrols the applied deformation,\nand its maximum value can be specified through the command line of the program\nMAELAS using tag \u0000s. The total number of deformed cells can be chosen with\ntag\u0000n.\n43Appendix C.2. Hexagonal (I) system\nIn the case of hexagonal (I), MAELAS generates 4 sets of deformed cells using\nthe following deformation gradients\nFFF\f\f\fla1;2\nbbb=(1;0;0)(s) =0\nB@1+s 0 0\n01p1+s0\n0 01p1+s1\nCA;FFF\f\f\fla2;2\nbbb=(0;0;1)(s) =0\nB@1p1+s0 0\n01p1+s0\n0 0 1 +s1\nCA\nFFF\f\f\flg;2\nbbb=(1;0;0)(s) =0\nB@1+s 0 0\n01p1+s0\n0 01p1+s1\nCA;FFF\f\f\fle;2\nbbb=(a;0;c)p\na2+c2(s) =w0\n@1 0sc\n2a\n0 1 0\nsa\n2c0 11\nA\n(C.4)\nwhere w=3p\n4=(4\u0000s2),aandcare the lattice parameters of the relaxed (not\ndeformed) unit cell. The fractions c=aanda=cwere introduced in the deformation\ngradient elements Fle;2\nxzandFle;2\nzx, respectively, in order to generate deformations\nthat meet the property bbb=aaa+ccc\njaaa+cccj=aaa000+ccc000\njaaa000+ccc000j, where aaa000andccc000are the lattice vectors\nof the distorted unit cell, see Fig. C.13. This deformation mode makes it easy\nto compute the cell length lin the measuring direction bbbsince it is just l=jaaa000+\nccc000j. It is inspired by the trigonal deformation for the cubic (I) case FFFl111where\ndeformations meet the property bbb=aaa+bbb+ccc\njaaa+bbb+cccj=aaa000+bbb000+ccc000\njaaa000+bbb000+ccc000j.\nFigure C.13: Sketch of the deformation generated by the deformation gradient FFFle;2given by\nEq.C.4 to calculate le;2. The purple line represents the relaxed cell with lattice parameters (a;b;c),\nwhile the red line stands for the deformed cell with lattice parameters (a0;b0;c0). This deformation\nmeets the property bbb=aaa+ccc\njaaa+cccj=aaa000+ccc000\njaaa000+ccc000j.\n44Appendix C.3. Trigonal (I) system\nIn the case of trigonal (I), MAELAS generates 6 sets of deformed unit cells\nusing the following deformation gradients\nFFF\f\f\fla1;2\nbbb=(1;0;0)(s) =0\nB@1+s 0 0\n01p1+s0\n0 01p1+s1\nCA;FFF\f\f\fla2;2\nbbb=(0;0;1)(s) =0\nB@1p1+s0 0\n01p1+s0\n0 0 1 +s1\nCA\nFFF\f\f\flg;1\nbbb=(1;0;0)(s) =0\nB@1+s 0 0\n01p1+s0\n0 01p1+s1\nCA\nFFF\f\f\flg;2\nbbb=(a;0;c)p\na2+c2(s) =FFF\f\f\fl12\nbbb=(a;0;c)p\na2+c2(s) =FFF\f\f\fl21\nbbb=(a;0;c)p\na2+c2(s) =w0\n@1 0sc\n2a\n0 1 0\nsa\n2c0 11\nA:\n(C.5)\nAppendix C.4. Tetragonal (I) system\nIn the case of tetragonal (I), MAELAS generates 5 sets of deformed cells using\nthe following deformation gradients\nFFF\f\f\fla1;2\nbbb=(1;0;0)(s) =0\nB@1+s 0 0\n01p1+s0\n0 01p1+s1\nCA;FFF\f\f\fla2;2\nbbb=(0;0;1)(s) =0\nB@1p1+s0 0\n01p1+s0\n0 0 1 +s1\nCA\nFFF\f\f\flg;2\nbbb=(1;0;0)(s) =0\nB@1+s 0 0\n01p1+s0\n0 01p1+s1\nCA;FFF\f\f\fle;2\nbbb=(a;0;c)p\na2+c2(s) =w0\n@1 0sc\n2a\n0 1 0\nsa\n2c0 11\nA\nFFF\f\f\fld;2\nbbb=\u0010\n1p\n2;1p\n2;0\u0011(s) =w0\n@1s\n20\ns\n21 0\n0 0 11\nA:\n(C.6)\n45Appendix C.5. Orthorhombic system\nFor orthorhombic crystals MAELAS generates 9 sets of deformed cells using\nthe following deformation gradients\nFFF\f\f\fl1\nbbb=(1;0;0)(s) =0\nB@1+s 0 0\n01p1+s0\n0 01p1+s1\nCA;FFF\f\f\fl2\nbbb=(1;0;0)(s) =0\nB@1+s 0 0\n01p1+s0\n0 01p1+s1\nCA\nFFF\f\f\fl3\nbbb=(0;1;0)(s) =0\nB@1p1+s0 0\n0 1 +s 0\n0 01p1+s1\nCA;FFF\f\f\fl4\nbbb=(0;1;0)(s) =0\nB@1p1+s0 0\n0 1 +s 0\n0 01p1+s1\nCA\nFFF\f\f\fl5\nbbb=(0;0;1)(s) =0\nB@1p1+s0 0\n01p1+s0\n0 0 1 +s1\nCA;FFF\f\f\fl6\nbbb=(0;0;1)(s) =0\nB@1p1+s0 0\n01p1+s0\n0 0 1 +s1\nCA\nFFF\f\f\fl7\nbbb=(a;b;0)p\na2+b2(s) =w0\n@1sb\n2a0\nsa\n2b1 0\n0 0 11\nA;FFF\f\f\fl8\nbbb=(a;0;c)p\na2+c2(s) =w0\n@1 0sc\n2a\n0 1 0\nsa\n2c0 11\nA\nFFF\f\f\fl9\nbbb=(0;b;c)p\nb2+c2(s) =w0\n@1 0 0\n0 1sc\n2b\n0sb\n2c11\nA:\n(C.7)\nReferences\n[1] M. Gibbs, Modern trends in magnetostriction, Springer, 2001.\n[2] N. Ekreem, A. Olabi, T. Prescott, A. Rafferty, M. Hashmi, An overview of\nmagnetostriction, its use and methods to measure these properties, Journal of\nMaterials Processing Technology 191 (1) (2007) 96 – 101, advances in Ma-\nterials and Processing Technologies, July 30th - August 3rd 2006, Las Vegas,\nNevada. doi:https://doi.org/10.1016/j.jmatprotec.2007.03.064 .\nURL http://www.sciencedirect.com/science/article/pii/\nS0924013607002889\n[3] F. T. Calkins, A. B. Flatau, M. J. Dapino, Overview of magnetostric-\ntive sensor technology, Journal of Intelligent Material Systems and Struc-\ntures 18 (10) (2007) 1057–1066. arXiv:https://doi.org/10.1177/\n461045389X06072358 ,doi:10.1177/1045389X06072358 .\nURL https://doi.org/10.1177/1045389X06072358\n[4] V . Apicella, C. S. Clemente, D. Davino, D. Leone, C. Visone, Review of\nmodeling and control of magnetostrictive actuators, Actuators 8 (2). doi:\n10.3390/act8020045 .\nURL https://www.mdpi.com/2076-0825/8/2/45\n[5] A. Clark, Chapter 7 magnetostrictive rare earth-Fe 2compounds, V ol. 1\nof Handbook of Ferromagnetic Materials, Elsevier, 1980, pp. 531 – 589.\ndoi:https://doi.org/10.1016/S1574-9304(05)80122-1 .\nURL http://www.sciencedirect.com/science/article/pii/\nS1574930405801221\n[6] G. Engdahl, Handbook of giant magnetostrictive materials, Academic Press,\n1999.\n[7] A. Andreev, Chapter 2 Thermal expansion anomalies and spontaneous\nmagnetostriction in rare-earth intermetallics with cobalt and iron, V ol. 8\nof Handbook of Magnetic Materials, Elsevier, 1995, pp. 59 – 187.\ndoi:https://doi.org/10.1016/S1567-2719(05)80031-9 .\nURL http://www.sciencedirect.com/science/article/pii/\nS1567271905800319\n[8] J. R. Cullen, A. E. Clark, K. B. Hathaway, Materials, science and technology,\nVCH Publishings, 1994, Ch. 16 - Magnetostrictive Materials, pp. 529–565.\n[9] S. Massari, M. Ruberti, Rare earth elements as critical raw materials: Focus\non international markets and future strategies, Resources Policy 38 (1)\n(2013) 36 – 43. doi:https://doi.org/10.1016/j.resourpol.2012.\n07.001 .\nURL http://www.sciencedirect.com/science/article/pii/\nS0301420712000530\n[10] D. Fritsch, C. Ederer, First-principles calculation of magnetoelastic coef-\nficients and magnetostriction in the spinel ferrites CoFe 2O4and NiFe 2O4,\nPhys. Rev. B 86 (2012) 014406. doi:10.1103/PhysRevB.86.014406 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.86.014406\n[11] H. Wang, Y . N. Zhang, R. Q. Wu, L. Z. Sun, D. S. Xu, Z. D. Zhang, Under-\nstanding strong magnetostriction in Fe 100\u0000xGaxalloys, Scientific Reports\n473 (1) (2013) 3521. doi:10.1038/srep03521 .\nURL https://doi.org/10.1038/srep03521\n[12] M. Dapino, Encyclopedia of Smart Materials, John Wiley and Sons, Inc.,\nNew York, 2000.\n[13] N. S. Akulov, Z. Physik 52, 389.\n[14] R. Becker, W. Doring, Ferromagnetismus, Verlag. Julius Springer, Berlin,\n1939.\n[15] W. P. Mason, Derivation of magnetostriction and anisotropic energies for\nhexagonal, tetragonal, and orthorhombic crystals, Phys. Rev. 96 (1954) 302–\n310. doi:10.1103/PhysRev.96.302 .\nURL https://link.aps.org/doi/10.1103/PhysRev.96.302\n[16] A. E. Clark, B. F. DeSavage, R. Bozorth, Anomalous Thermal Expansion\nand Magnetostriction of Single-Crystal Dysprosium, Phys. Rev. 138 (1965)\nA216–A224. doi:10.1103/PhysRev.138.A216 .\nURL https://link.aps.org/doi/10.1103/PhysRev.138.A216\n[17] E. Callen, H. B. Callen, Magnetostriction, forced magnetostriction, and\nanomalous thermal expansion in ferromagnets, Phys. Rev. 139 (1965) A455–\nA471. doi:10.1103/PhysRev.139.A455 .\nURL https://link.aps.org/doi/10.1103/PhysRev.139.A455\n[18] R. Wu, A. J. Freeman, First principles determinations of magnetostriction\nin transition metals (invited), Journal of Applied Physics 79 (8) (1996)\n6209–6212. arXiv:https://aip.scitation.org/doi/pdf/10.1063/\n1.362073 ,doi:10.1063/1.362073 .\nURL https://aip.scitation.org/doi/abs/10.1063/1.362073\n[19] R. Wu, L. Chen, A. Freeman, First principles determination of\nmagnetostriction in bulk transition metals and thin films, Journal\nof Magnetism and Magnetic Materials 170 (1) (1997) 103 – 109.\ndoi:https://doi.org/10.1016/S0304-8853(97)00004-8 .\nURL http://www.sciencedirect.com/science/article/pii/\nS0304885397000048\n[20] R. Wu, L. Chen, A. Shick, A. Freeman, First-principles determinations of\nmagneto-crystalline anisotropy and magnetostriction in bulk and thin-film\n48transition metals, Journal of Magnetism and Magnetic Materials 177-181\n(1998) 1216 – 1219, international Conference on Magnetism (Part II).\ndoi:https://doi.org/10.1016/S0304-8853(97)00382-X .\nURL http://www.sciencedirect.com/science/article/pii/\nS030488539700382X\n[21] T. Burkert, O. Eriksson, P. James, S. I. Simak, B. Johansson, L. Nordström,\nCalculation of uniaxial magnetic anisotropy energy of tetragonal and trigonal\nFe, Co, and Ni, Phys. Rev. B 69 (2004) 104426. doi:10.1103/PhysRevB.\n69.104426 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.69.104426\n[22] G. Petculescu, R. Wu, R. McQueeney, Chapter three - magne-\ntoelasticity of bcc Fe-Ga Alloys, V ol. 20 of Handbook of Mag-\nnetic Materials, Elsevier, 2012, pp. 123 – 226. doi:https:\n//doi.org/10.1016/B978-0-444-56371-2.00003-9 .\nURL http://www.sciencedirect.com/science/article/pii/\nB9780444563712000039\n[23] Y . N. Zhang, J. X. Cao, R. Q. Wu, Rigid band model for prediction of mag-\nnetostriction of iron-gallium alloys, Applied Physics Letters 96 (6) (2010)\n062508. arXiv:https://doi.org/10.1063/1.3318420 ,doi:10.1063/\n1.3318420 .\nURL https://doi.org/10.1063/1.3318420\n[24] Y . Zhang, R. Wu, Mechanism of large magnetostriction of galfenol, IEEE\nTransactions on Magnetics 47 (10) (2011) 4044–4049.\n[25] Y . Zhang, H. Wang, R. Wu, First-principles determination of the rhombohe-\ndral magnetostriction of Fe 100\u0000xAlxand Fe 100\u0000xGaxalloys, Phys. Rev. B 86\n(2012) 224410. doi:10.1103/PhysRevB.86.224410 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.86.224410\n[26] S. C. Hong, W. S. Yun, R. Wu, Giant magnetostriction of Fe 1\u0000xBexalloy:\nA first-principles study, Phys. Rev. B 79 (2009) 054419. doi:10.1103/\nPhysRevB.79.054419 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.79.054419\n[27] V . I. Gavrilenko, R. Q. Wu, Magnetostriction and magnetism of rare\nearth intermetallic compounds: First principle study, Journal of Applied\n49Physics 89 (11) (2001) 7320–7322. arXiv:https://doi.org/10.1063/\n1.1356038 ,doi:10.1063/1.1356038 .\nURL https://doi.org/10.1063/1.1356038\n[28] R. Wu, Magnetism and magnetostriction in GdFe 2and GdCo 2, Journal of\nApplied Physics 85 (8) (1999) 6217–6219. arXiv:https://doi.org/10.\n1063/1.370226 ,doi:10.1063/1.370226 .\nURL https://doi.org/10.1063/1.370226\n[29] M. Werwi ´nski, W. Marciniak, Ab initio study of magnetocrystalline\nanisotropy, magnetostriction, and Fermi surface of L1 0FeNi (tetrataen-\nite), Journal of Physics D: Applied Physics 50 (49) (2017) 495008. doi:\n10.1088/1361-6463/aa958a .\nURL https://doi.org/10.1088/1361-6463/aa958a\n[30] E. Wasserman, Chapter 3 Invar: Moment-volume instabilities in\ntransition metals and alloys, V ol. 5 of Handbook of Ferromag-\nnetic Materials, Elsevier, 1990, pp. 237 – 322. doi:https:\n//doi.org/10.1016/S1574-9304(05)80063-X .\nURL http://www.sciencedirect.com/science/article/pii/\nS157493040580063X\n[31] R. Skomski, J. M. D. Coey, Permanent magnetism, Institute of Physics Pub-\nlishing, 1999.\n[32] J. Joule, On a new class of magnetic forces, Annals of Electricity, Mag-\nnetism, and Chemistry 8 (1842) 219–224.\n[33] L. D. Landau, E. M. Lifshitz, Theory of elasticity / by L. D. Landau and\nE. M. Lifshitz ; translated from the Russian by J. B. Sykes and W. H. Reid,\nPergamon London, 1959.\n[34] R. Birss, The saturation magnetostriction of ferromagnetics, Advances in\nPhysics 8 (31) (1959) 252–291. arXiv:https://doi.org/10.1080/\n00018735900101198 ,doi:10.1080/00018735900101198 .\nURL https://doi.org/10.1080/00018735900101198\n[35] C. Kittel, Physical theory of ferromagnetic domains, Rev. Mod. Phys. 21\n(1949) 541–583. doi:10.1103/RevModPhys.21.541 .\nURL https://link.aps.org/doi/10.1103/RevModPhys.21.541\n50[36] H. Mueller, Properties of rochelle salt. iv, Phys. Rev. 58 (1940) 805–811.\ndoi:10.1103/PhysRev.58.805 .\nURL https://link.aps.org/doi/10.1103/PhysRev.58.805\n[37] D. C. Wallace, Thermodynamics of Crystals, Wiley, New York, 1972.\n[38] F. Mouhat, F. m. c.-X. Coudert, Necessary and sufficient elastic stability\nconditions in various crystal systems, Phys. Rev. B 90 (2014) 224104. doi:\n10.1103/PhysRevB.90.224104 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.90.224104\n[39] L. A.E.H., A Treatise on the Mathematical Theory of Elasticity, Dover pub-\nlisher, New York, 1944.\n[40] R. C. O’Handley, Modern magnetic materials, Wiley, 2000.\n[41] E. W. Lee, Magnetostriction and magnetomechanical effects, Reports on\nProgress in Physics 18 (1) (1955) 184–229. doi:10.1088/0034-4885/18/\n1/305 .\n[42] A. Reuss, Berechnung der Fließgrenze von Mischkristallen auf Grund\nder Plastizitätsbedingung für Einkristalle, ZAMM - Journal of Ap-\nplied Mathematics and Mechanics 9 (1) (1929) 49–58. arXiv:https:\n//onlinelibrary.wiley.com/doi/pdf/10.1002/zamm.19290090104 ,\ndoi:10.1002/zamm.19290090104 .\nURL https://onlinelibrary.wiley.com/doi/abs/10.1002/zamm.\n19290090104\n[43] R. Hill, The elastic behaviour of a crystalline aggregate, Proceedings of\nthe Physical Society. Section A 65 (5) (1952) 349–354. doi:10.1088/\n0370-1298/65/5/307 .\n[44] S. Zhang, R. Zhang, AELAS: Automatic ELAStic property deriva-\ntions via high-throughput first-principles computation, Computer\nPhysics Communications 220 (2017) 403 – 416. doi:https:\n//doi.org/10.1016/j.cpc.2017.07.020 .\nURL http://www.sciencedirect.com/science/article/pii/\nS0010465517302400\n51[45] E. du Trémolet de Lacheisserie, D. Gignoux, M. Schlenker, Magnetism,\nSpringer New York, New York, NY , 2002, Ch. 12 Magnetoelastic Effects,\npp. 351–398.\n[46] W. J. Carr, Chapter 10 Magnetostriction, Magnetic properties of metals and\nalloys, American Society for Metals, Cleveland, 1959.\n[47] G. Kresse, J. Hafner, Ab initio molecular dynamics for liquid metals, Phys.\nRev. B 47 (1993) 558–561. doi:10.1103/PhysRevB.47.558 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.47.558\n[48] G. Kresse, J. Furthmüller, Efficiency of ab-initio total energy cal-\nculations for metals and semiconductors using a plane-wave ba-\nsis set, Computational Materials Science 6 (1) (1996) 15 – 50.\ndoi:https://doi.org/10.1016/0927-0256(96)00008-0 .\nURL http://www.sciencedirect.com/science/article/pii/\n0927025696000080\n[49] G. Kresse, J. Furthmüller, Efficient iterative schemes for ab initio total-\nenergy calculations using a plane-wave basis set, Phys. Rev. B 54 (1996)\n11169–11186. doi:10.1103/PhysRevB.54.11169 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.54.11169\n[50] S. P. Ong, W. D. Richards, A. Jain, G. Hautier, M. Kocher, S. Cholia,\nD. Gunter, V . L. Chevrier, K. A. Persson, G. Ceder, Python materials\ngenomics (pymatgen): A robust, open-source python library for mate-\nrials analysis, Computational Materials Science 68 (2013) 314 – 319.\ndoi:https://doi.org/10.1016/j.commatsci.2012.10.028 .\nURL http://www.sciencedirect.com/science/article/pii/\nS0927025612006295\n[51] P. Nieves, S. Arapan, A. P. K ˛ adzielawa, D. Legut, MAELASviewer: An\nonline tool to visualize magnetostriction, Sensors 20 (22). doi:10.3390/\ns20226436 .\nURL https://www.mdpi.com/1424-8220/20/22/6436\n[52] https://maelasviewer.herokuapp.com .\n[53] https://github.com/pnieves2019/MAELASwiewer .\n[54] https://github.com/pnieves2019/MAELAS .\n52[55] G. Kresse, D. Joubert, From ultrasoft pseudopotentials to the projector\naugmented-wave method, Phys. Rev. B 59 (1999) 1758–1775. doi:10.\n1103/PhysRevB.59.1758 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.59.1758\n[56] J. P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approxima-\ntion made simple, Phys. Rev. Lett. 77 (1996) 3865–3868. doi:10.1103/\nPhysRevLett.77.3865 .\nURL https://link.aps.org/doi/10.1103/PhysRevLett.77.3865\n[57] D. Hobbs, G. Kresse, J. Hafner, Fully unconstrained noncollinear mag-\nnetism within the projector augmented-wave method, Phys. Rev. B 62 (2000)\n11556–11570. doi:10.1103/PhysRevB.62.11556 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.62.11556\n[58] H. J. Monkhorst, J. D. Pack, Special points for brillouin-zone integrations,\nPhys. Rev. B 13 (1976) 5188–5192. doi:10.1103/PhysRevB.13.5188 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.13.5188\n[59] P. Nieves, J. Tranchida, S. Arapan, D. Legut, Spin-lattice model for cubic\ncrystals (2020). arXiv:2012.05076 .\n[60] J. Tranchida, S. Plimpton, P. Thibaudeau, A. Thompson, Massively parallel\nsymplectic algorithm for coupled magnetic spin dynamics and molecular\ndynamics, Journal of Computational Physics 372 (2018) 406 – 425.\ndoi:https://doi.org/10.1016/j.jcp.2018.06.042 .\nURL http://www.sciencedirect.com/science/article/pii/\nS0021999118304200\n[61] S. Plimpton, Fast parallel algorithms for short-range molecular dy-\nnamics, Journal of Computational Physics 117 (1) (1995) 1 – 19.\ndoi:https://doi.org/10.1006/jcph.1995.1039 .\nURL http://www.sciencedirect.com/science/article/pii/\nS002199918571039X\n[62] M. Getzlaff, Fundamentals of magnetism, Springer, Berlin, Heidelberg,\n2008.\n[63] A. Hubert, W. Unger, J. Kranz, Messung der magnetostriktionskonstanten\ndes kobalts als funktion der temperatur, Z. Physik 224 (1) (1969) 148–155.\n53doi:10.1007/BF01392243 .\nURL https://doi.org/10.1007/BF01392243\n[64] M. C. Nguyen, Y . Yao, C.-Z. Wang, K.-M. Ho, V . P. Antropov, Magnetocrys-\ntalline anisotropy in cobalt based magnets: a choice of correlation parame-\nters and the relativistic effects, Journal of Physics: Condensed Matter 30 (19)\n(2018) 195801. doi:10.1088/1361-648x/aab9fa .\nURL https://doi.org/10.1088/1361-648X/aab9fa\n[65] H. Shima, K. Oikawa, A. Fujita, K. Fukamichi, K. Ishida, A. Sakuma, Lat-\ntice axial ratio and large uniaxial magnetocrystalline anisotropy in L1 0-type\nFePd single crystals prepared under compressive stress, Phys. Rev. B 70\n(2004) 224408. doi:10.1103/PhysRevB.70.224408 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.70.224408\n[66] H. Shima, K. Oikawa, A. Fujita, K. Fukamichi, K. Ishida, Magnetic\nanisotropy and mangnetostriction in L1 0FePd alloy, Journal of Mag-\nnetism and Magnetic Materials 272-276 (2004) 2173 – 2174, pro-\nceedings of the International Conference on Magnetism (ICM 2003).\ndoi:https://doi.org/10.1016/j.jmmm.2003.12.898 .\nURL http://www.sciencedirect.com/science/article/pii/\nS0304885303018316\n[67] M. de Jong, W. Chen, T. Angsten, A. Jain, R. Notestine, A. Gamst,\nM. Sluiter, C. Krishna Ande, S. van der Zwaag, J. J. Plata, C. Toher, S. Cur-\ntarolo, G. Ceder, K. A. Persson, M. Asta, Charting the complete elastic\nproperties of inorganic crystalline compounds, Scientific Data 2 (1) (2015)\n150009. doi:10.1038/sdata.2015.9 .\nURL https://doi.org/10.1038/sdata.2015.9\n[68] A. Jain, S. Ong, G. Hautier, W. Chen, W. Richards, S. Dacek, S. Cholia,\nD. Gunter, D. Skinner, G. Ceder, K. Persson, The materials project: A mate-\nrials genome approach to accelerating materials innovation, APL Materials\n1 (2013) 011002.\n[69] K. Persson, Materials data on Fe (SG:229) by materials project (1 2015).\ndoi:10.17188/1189317 .\n[70] J. A. Rayne, B. S. Chandrasekhar, Elastic constants of iron from 4.2 to 300\nk, Phys. Rev. 122 (1961) 1714–1716. doi:10.1103/PhysRev.122.1714 .\nURL https://link.aps.org/doi/10.1103/PhysRev.122.1714\n54[71] K. Persson, Materials data on Co (SG:194) by materials project (2 2016).\ndoi:10.17188/1263614 .\n[72] H. J. McSkimin, Measurement of the elastic constants of single crystal\ncobalt, Journal of Applied Physics 26 (4) (1955) 406–409. arXiv:https:\n//doi.org/10.1063/1.1722007 ,doi:10.1063/1.1722007 .\nURL https://doi.org/10.1063/1.1722007\n[73] K. Persson, Materials data on YCo5 (SG:191) by materials project (2 2016).\ndoi:10.17188/1202401 .\n[74] K. Persson, Materials data on Ni (SG:225) by materials project (2 2016).\ndoi:10.17188/1199153 .\n[75] B.-J. Lee, J.-H. Shim, M. I. Baskes, Semiempirical atomic potentials for the\nfcc metals Cu, Ag, Au, Ni, Pd, Pt, Al, and Pb based on first and second\nnearest-neighbor modified embedded atom method, Phys. Rev. B 68 (2003)\n144112. doi:10.1103/PhysRevB.68.144112 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.68.144112\n[76] K. Persson, Materials data on Fe2Si (SG:164) by materials project (2 2016).\ndoi:10.17188/1198977 .\n[77] K. Persson, Materials data on FePd (SG:123) by materials project (2 2016).\ndoi:10.17188/1202438 .\n[78] A. Al-Ghaferi, P. Müllner, H. Heinrich, G. Kostorz, J. Wiezorek,\nElastic constants of equiatomic L1 0-ordered FePd single crys-\ntals, Acta Materialia 54 (4) (2006) 881 – 889. doi:https:\n//doi.org/10.1016/j.actamat.2005.10.018 .\nURL http://www.sciencedirect.com/science/article/pii/\nS1359645405006129\n[79] K. Persson, Materials data on YCo (SG:63) by materials project (2 2016).\ndoi:10.17188/1307912 .\n[80] J. Kübler, Theory of itinerant electron magnetism, Oxford University Press,\nNew York, 2009.\n[81] S. V . Halilov, A. Y . Perlov, P. M. Oppeneer, A. N. Yaresko, V . N. Antonov,\nMagnetocrystalline anisotropy energy in cubic fe, co, and ni: Applicability\n55of local-spin-density theory reexamined, Phys. Rev. B 57 (1998) 9557–9560.\ndoi:10.1103/PhysRevB.57.9557 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.57.9557\n[82] J. Trygg, B. Johansson, O. Eriksson, J. M. Wills, Total energy calculation\nof the magnetocrystalline anisotropy energy in the ferromagnetic 3 dmetals,\nPhys. Rev. Lett. 75 (1995) 2871–2874. doi:10.1103/PhysRevLett.75.\n2871 .\nURL https://link.aps.org/doi/10.1103/PhysRevLett.75.2871\n[83] M. Fähnle, M. Komelj, R. Q. Wu, G. Y . Guo, Magnetoelasticity of Fe: Pos-\nsible failure of ab initio electron theory with the local-spin-density approx-\nimation and with the generalized-gradient approximation, Phys. Rev. B 65\n(2002) 144436. doi:10.1103/PhysRevB.65.144436 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.65.144436\n[84] N. J. Jones, G. Petculescu, M. Wun-Fogle, J. B. Restorff, A. E. Clark,\nK. B. Hathaway, D. Schlagel, T. A. Lograsso, Rhombohedral magnetostric-\ntion in dilute iron (Co) alloys, Journal of Applied Physics 117 (17) (2015)\n17A913. arXiv:https://doi.org/10.1063/1.4916541 ,doi:10.1063/\n1.4916541 .\nURL https://doi.org/10.1063/1.4916541\n[85] M. Fähnle, M. Komelj, R. Q. Wu, G. Y . Guo, Magnetoelasticity of fe: Pos-\nsible failure of ab initio electron theory with the local-spin-density approx-\nimation and with the generalized-gradient approximation, Phys. Rev. B 65\n(2002) 144436. doi:10.1103/PhysRevB.65.144436 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.65.144436\n[86] J. Sun, A. Ruzsinszky, J. P. Perdew, Strongly constrained and appropriately\nnormed semilocal density functional, Phys. Rev. Lett. 115 (2015) 036402.\ndoi:10.1103/PhysRevLett.115.036402 .\nURL https://link.aps.org/doi/10.1103/PhysRevLett.115.036402\n[87] A. I. Liechtenstein, V . I. Anisimov, J. Zaanen, Density-functional theory and\nstrong interactions: Orbital ordering in Mott-Hubbard insulators, Phys. Rev.\nB 52 (1995) R5467–R5470. doi:10.1103/PhysRevB.52.R5467 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.52.R5467\n56[88] A. Mishima, H. Fujii, T. Okamoto, Anomalous Saturation Magnetostric-\ntion of Gd Single Crystal, Journal of the Physical Society of Japan 40 (4)\n(1976) 962–967. arXiv:https://doi.org/10.1143/JPSJ.40.962 ,doi:\n10.1143/JPSJ.40.962 .\nURL https://doi.org/10.1143/JPSJ.40.962\n[89] S. L. Dudarev, G. A. Botton, S. Y . Savrasov, C. J. Humphreys, A. P. Sut-\nton, Electron-energy-loss spectra and the structural stability of nickel ox-\nide: An lsda+u study, Phys. Rev. B 57 (1998) 1505–1509. doi:10.1103/\nPhysRevB.57.1505 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.57.1505\n[90] H. Kudielka, Die Kristallstruktur von Fe 2Si, ihre Verwandtschaft zu den\nOrdnungsstrukturen des a-(Fe,Si)-Mischkristalls und zur Fe 5Si3-Struktur,\nZeitschrift für Kristallographie - Crystalline Materials 145 (1-6) (01 Dec.\n1977) 177 – 189. doi:https://doi.org/10.1524/zkri.1977.145.16.\n177.\nURL https://www.degruyter.com/view/journals/zkri/145/1-6/\narticle-p177.xml\n[91] Y . Sun, Z. Zhuo, X. Wu, J. Yang, Room-temperature ferromagnetism in two-\ndimensional Fe 2Si nanosheet with enhanced spin-polarization ratio, Nano\nLetters 17 (5) (2017) 2771–2777. doi:10.1021/acs.nanolett.6b04884 .\nURL https://doi.org/10.1021/acs.nanolett.6b04884\n[92] A. V . Krukau, O. A. Vydrov, A. F. Izmaylov, G. E. Scuseria, Influence of the\nexchange screening parameter on the performance of screened hybrid func-\ntionals, The Journal of Chemical Physics 125 (22) (2006) 224106. arXiv:\nhttps://doi.org/10.1063/1.2404663 ,doi:10.1063/1.2404663 .\nURL https://doi.org/10.1063/1.2404663\n[93] C. P. Tang, K. V . Tam, S. J. Xiong, J. Cao, X. Zhang, The structure\nand electronic properties of hexagonal Fe 2Si, AIP Advances 6 (6) (2016)\n065317. arXiv:https://doi.org/10.1063/1.4954667 ,doi:10.1063/\n1.4954667 .\nURL https://doi.org/10.1063/1.4954667\n[94] T. Hikosaka, F. Nakamura, S. Oikawa, A. Takeo, Y . Tanaka, Low noise\nFeAlSi soft magnetic under-layer for CoPtCrO double-layered perpendicular\nrecording media, IEEE Transactions on Magnetics 37 (4) (2001) 1586–1588.\n57[95] P. Villars, K. Cenzual, R. Gladyshevskii, Handbook, De Gruyter, Berlin,\nBoston, 27 Nov. 2015. doi:https://doi.org/10.1515/9783110334678 .\nURL https://www.degruyter.com/view/title/304631\n[96] W. P. Mason, A phenomenological derivation of the first- and second-order\nmagnetostriction and morphic effects for a nickel crystal, Phys. Rev. 82\n(1951) 715–723. doi:10.1103/PhysRev.82.715 .\nURL https://link.aps.org/doi/10.1103/PhysRev.82.715\n[97] E. B. Tadmor, R. E. Miller, Modeling Materials, Cambridge University\nPress, 2009. doi:10.1017/cbo9781139003582 .\nURL https://doi.org/10.1017/cbo9781139003582\n[98] G. Guo, Orientation dependence of the magnetoelastic coupling\nconstants in strained FCC Co and Ni: an ab initio study, Jour-\nnal of Magnetism and Magnetic Materials 209 (1) (2000) 33 – 36.\ndoi:https://doi.org/10.1016/S0304-8853(99)00639-3 .\nURL http://www.sciencedirect.com/science/article/pii/\nS0304885399006393\n58"    },    {        "title": "2009.06440v1.Multifunctional_Antiperovskites_driven_by_Strong_Magnetostructural_Coupling.pdf",        "content": "Multifunctional Antiperovskites driven by Strong Magnetostructural Coupling\nHarish K. Singh, Ilias Samathrakis, Nuno M. Fortunato, Jan Zemen, Chen Shen, Oliver Gut\reisch, and Hongbin Zhang\nInstitute of Materials Science, Technical University Darmstadt,\nOtto-Berndt-Strasse 3, 64287 Darmstadt, Germany\nFaculty of Electrical Engineering, Czech Technical University in Prague,\nTechnicka 2, Prague 166 27, Czech Republic and\nDepartment of Physics, Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom\nBased on density functional theory calculations, we elucidated the origin of multifunctional prop-\nerties for cubic antiperovskites with noncollinear magnetic ground states, which can be attributed\nto strong isotropic and anisotropic magnetostructural coupling. 16 out of 54 stable magnetic an-\ntiperovskite M 3XZ (M = Cr, Mn, Fe, Co, and Ni; X = selected elements from Li to Bi except for\nnoble gases and 4f rare-earth metals; and Z = C and N) are found to exhibit the \u0000 4g/\u00005g(i.e.,\ncharacterized by irreducible representations) antiferromagnetic magnetic con\fgurations driven by\nfrustrated exchange coupling and strong magnetocrystalline anisotropy. Using the magnetic defor-\nmation as an e\u000bective proxy, the isotropic magnetostructural coupling is characterized, and it is\nobserved that the paramagnetic state is critical to understand the experimentally observed negative\nthermal expansion and to predict the magnetocaloric performance. Moreover, the piezomagnetic\nand piezospintronic e\u000bects induced by biaxial strain are investigated. It is revealed that there is\nnot a strong correlation between the induced magnetization and anomalous Hall conductivities by\nthe imposed strain. Interestingly, the anomalous Hall/Nernst conductivities can be signi\fcantly tai-\nlored by the applied strain due to the \fne-tuning of the Weyl points energies, leading to promising\nspintronic applications.\nI. INTRODUCTION\nSmart materials like multiferroic materials with en-\nhanced coupling between di\u000berent degrees of freedom\n(e.g., mechanical, electronic, and magnetic) are promis-\ning for engineering devices for future applications such\nas sensors, transducers, memories, and spintronics.1{3\nCubic antiperovskite (APV) compounds host the two\nmost appealing aspects of multiferroics, e.g., magne-\ntoelectric coupling and piezomagnetic e\u000bect (PME).3,4\nIn APV materials, the strong magnetoelectric coupling\nis achieved by combining piezoelectric and piezomag-\nnetic heterostructure composites.5{7A signi\fcant PME\nis reported for Mn-based nitrides like Mn 3SnN, mak-\ning such compounds a suitable component for fabricat-\ning magnetoelectric composite.8,9The PME in APVs can\nbe attributed to the strong magnetostructural coupling,\nwhich manifests itself also as giant negative thermal ex-\npansion (NTE)10{12and magnetocaloric/barocaloric ef-\nfect.9,13{17From the materials perspective, many Mn-\nbased APV carbides go through a \frst-order magnetic\nphase-transition and possess a large magnetocaloric ef-\nfect.14,18For instance, Mn 3GaC exhibits a huge mag-\nnetic entropy change (\u0001S M) of 15 J/kgK under an ap-\nplied magnetic \feld of 2T.16The strong magnetostruc-\ntural coupling in APVs is driven by the cubic-to-cubic\n\frst-order transition wherein a change in the crystal vol-\nume brings about a change in the frustrated magnetic\nstates. Last but not least, APVs have been investigated\nrecently due to the presence of a treasury of multifunc-\ntionality such as superconductivity,19thermoelectric,20\nmagnetostriction.21\nParticularly, from the topological transport properties\npoint of view, the existence of \fnite anomalous Hall con-ductivity (AHC) in noncollinear antiferromagnets has at-\ntracted noticeable attention due to possible applications\nin AFM spintronics for information storage and data pro-\ncessing.22{25The spin-dependent transport phenomena\ncan provide spin-polarized charge current and large pure\nspin current, which could be achieved premised on two\nfundamental properties, i.e., AHC and spin Hall con-\nductivity (SHC). The kagome lattice turns out to be\nan elementary model to host giant AHC.26{28Recently,\nMn-based APV nitrides have been proposed to exhibit\nlarge AHC in frustrated AFM kagome-lattice.29{32It\nis observed that Mn 3GaN exhibits vanishing and non-\nvanishing AHC for two di\u000berent magnetic ordering \u0000 5g\nand \u0000 4g, respectively.29,30In this regard, for magnetic\nmaterials with noncollinear AFM ground states, the non-\nvanishing AHC is only feasible with speci\fc magnetic\nspace group symmetry, i.e., the AHC tensor depends\non the magnetic group symmetry.33For instance, Mn 3X\n(X= Ga, Ge, and Sn) and Mn 3Z (Z= Ir, Pt, and Rh) have\nbeen reported to have a di\u000berent form of AHC tensor\nas a result of di\u000berent magnetic ordering.22,34{36A pos-\nsible phase transition (\u0000 5g$\u00004g) attained by strain or\nchemical modi\fcation could make these materials suit-\nable for novel AFM spintronic applications. The spin-\npolarized current could also be generated by temperature\ngradient instead of the applied electric \felds, resulting\nin anomalous Nernst conductivity (ANC), also termed\nas spin caloritronics.37;38A large ANC in noncollinear\nAFMs could be useful for establishing spin caloritronics\ndevices that exhibit useful prospects in energy conversion\nand information processing. Noticeably, a large ANC of\n1.80 AK\u00001m\u00001has been reported for APV Mn 3NiN at\n200 K,39which is slightly less than half of the highest\nreported ANC of 4.0 AK\u00001m\u00001in Co 2MnGa.40;41En-arXiv:2009.06440v1  [cond-mat.mtrl-sci]  14 Sep 20202\nFIG. 1: The list of abbreviations used in the manuscript.\nhancing the AHC and ANC by applying strain could be\ncrucial for realizing AFM spintronic devices.\nIn this work, we carried out a systematic analysis of\n54 cubic APV systems (Pm \u00163m) with chemical formula\nM3XZ (see Fig. S1) to determine their magnetic ground\nstates, their tunability via biaxial strain, and the re-\nsulting spintronic properties. Explicit calculations were\nperformed to obtain the energies of four phases, i.e.,\n\u00004g, \u00005g, non-magnetic (NM), and ferromagnetic (FM)),\nwhere the magnetic anisotropy energy (MAE) de\fned as\nthe energy di\u000berence between \u0000 5gand \u0000 4gis examined to\nunderstand the origin of the noncollinear magnetic states\nwith the help of spin-orbit coupling energy. Moreover, a\ndetailed analysis of the lattice constant variation with\nrespect to the magnetic states reveals that the paramag-\nnetic (PM) state is critical in the magnetic phase tran-\nsition, enabling us to predict potential NTE and mag-\nnetocaloric materials. Last but not least, the PME was\nstudied by introducing biaxial strains (compressive and\ntensile), which causes possible phase transitions between\n\u00005g$\u00004g, and leads to a signi\fcant modi\fcation in the\nAHC and ANC, dubbed as a piezospintronic e\u000bect.30,42\nIn-depth analysis on symmetry analysis and electronic\nstructure suggests that the piezospintronic e\u000bect is orig-\ninated from the existence of Weyl nodes whose position\ncan be tailored by strain, resulting in promising applica-\ntions for future spintronic devices.\nII. COMPUTATIONAL DETAILS\nOur density functional theory (DFT) calculations are\nperformed using the projector augmented wave (PAW)\nmethod as e\u000bectuated in the VASP package.43Theexchange-correlation functional is approximated using\nthe generalized gradient approximation (GGA) as param-\neterized by Perdew-Burke-Ernzerhof (PBE).44We used\nan energy cuto\u000b of 500 eV for the plane-wave basis set,\nand a uniform k-points grid of 13 \u000213\u000213 within the\nMonkhorst-pack scheme for the Brillouin zone integra-\ntions. The Methfessel-Paxton scheme is used to deter-\nmine the partial occupancies of orbitals with a smearing\nwidth of 0.06 eV. The spin-orbit coupling (SOC) is con-\nsidered in all the calculations. In order to verify the\nmagnetic ground state of various compounds, the more\naccurate total energy calculations are performed using a\nhigher energy cuto\u000b of 600 eV and a dense k-mesh of\n25\u000225\u000225.\nThe magnetic ground states are obtained by comparing\nthe total energies of \fve con\fgurations (see Table S1)\n\u00004g, \u00005g, FM, PM (see Fig. 2, and NM states, where the\nlattice constants are fully optimized for each magnetic\ncon\fguration (see Table S2). The PM state is modeled\nbased on the disordered local moment (DLM) picture\n(see Fig. 2(d)), where a special quasi-random structure\nmodeled using a 2 \u00022\u00022 supercell by imposing zero to-\ntal magnetization.45That is, a supercell with the same\nnumber of up and down moments is considered with a\npair correlation function same as A 50B50random alloys,\ngenerated using the alloy theoretic automated toolkit\ncode.46Moreover, to investigate the piezomagnetic and\npiezospintronic e\u000bects, biaxial in-plane strain is applied,\nwhich reduces the crystalline symmetry from cubic to\ntetragonal. The crystal structures are fully relaxed where\nthe optimal lattice constants along c-direction are eval-\nuated by the polynomial \ftting of the energies from a\nseries of calculations.\nThe AHC is evaluated using the WannierTools code,47,\nwhere the required accurate tight-binding models are\nobtained by the maximally localized Wannier functions\n(MLWFs) using the Wannier90 code.48The s, p, and d\norbitals of M and X atoms and the s and p orbitals for N\natom are considered, resulting in 80 MLWFs in total for\nevery noncollinear APVs. The AHC is computed by inte-\ngrating the Berry curvature on a uniform 401 \u0002401\u0002401\nk-points mesh to guarantee good accuracy, which can be\nexpressed as:49\n\u001bxy=\u0000e2\n~Zdk\n(2\u0019)3X\nnf\u0002\n\u000f(k)\u0000\u0016\u0003\n\nn;xy(k) (1)\n\nn;xy(k) =\u00002ImX\nm6=nh knjvxj kmih kmjvyj kni\n\u0002\n\u000fm(k)\u0000\u000fn(k)\u00032(2)\nwhere e is elementary charge, \u0016is the chemical potential,\n n=mdenotes the Bloch wave function with energy eigen-\nvalue\u000fn=m,vx=yis the velocity operator along Cartesian\nx=ydirection, and f[\u000f(k)\u0000\u0016] is Fermi-Dirac distribution\nfunction. The ANC \u000bxyis evaluated based on the Mott3\nFIG. 2: Possible magnetic structures of antiperovskites: (a) \u0000 5g, (b) \u0000 4g, (c) FM and, (d) PM. The spin chirality for \u0000 4gand\n\u00005gis assumed to be +1.\nrelation, which yields:\n\u000bxy=\u0000\u00192k2\nBT\n3ed\u001bxy\nd\u000f\f\f\n\u000f=\u0016(3)\nIn this work, we evaluated only the derivative of AHC at\nthe Fermi level d \u001bxy/d\u000f, which provides a quantitative\nmeasurements of the ANC.\nIII. RESULTS AND DISCUSSION\nA. Validation and prediction of magnetic ground\nstate\nAs the magnetic transition metal atoms are located at\nthe face centers of the cubic cell for magnetic APVs, it\nleads to a frustrated kagome-lattice in the (111)-plane.\nCorrespondingly, noncollinear magnetic structures are\nexpected when the interatomic exchange interaction is\nAFM for the nearest neighbors. As shown in Fig. 2, \u0000 4g,\nand \u0000 5gare the two most common magnetic con\fgura-\ntions reported for APVs, resulting in 120\u000emagnetic angle\ncon\fgurations within the (111)-plane between three mag-\nnetic moments. The \u0000 4gstate can be obtained from \u0000 5g\nby simultaneously rotating the moments of three metal\natoms within the (111)-plane by 90\u000e. The APVs with the\nnoncollinear magnetic ground state are listed in Table. I,\nin comparison to the available experimentally measured\nresults. Interestingly, all the APVs with noncollinear\nmagnetic ground states are nitrides including Cr and Mn,\nwhile all carbides end up with the FM ground state ex-\ncept Mn 3SnC, which exhibits the \u0000 5gstate (see Table\nS1). It is noted that there are also other magnetic con-\n\fgurations such as ferrimagnetic and canted states for\nMn3XC (X = Ga, Sn, and Zn),50{52which will be saved\nfor future investigations.\nThe magnetic ground states of noncollinear-APVs are\nin good agreement with the experimental measurements.\nFor instance, both Mn 3GaN13and Mn 3ZnN53,54have the\n\u00005gmagnetic ground state, which are consistent with our\nDFT calculations. Interestingly, many Mn-based APVsexhibit a mixed \u0000 4g+ \u0000 5gmagnetic ordering, with a\npossible meta-magnetic transition to the other magnetic\nphases.53For instance, Mn 3AgN is characterized by two\ndistinct magnetic phase transitions. A mixed \u0000 4g+ \u0000 5g\nphase exists below 55 K, whereas pure \u0000 5gstate persists\nat intermediate temperature range (55 K <T<290 K),\nand lastly, it undergoes the magnetic transition to PM\nstate at 290 K.53Likewise, Mn 3NiN displays such mixed\nmagnetic phases.53Mn3SnN shows four di\u000berent mag-\nnetic and crystallographic phases.55\nThe mixed magnetic ordering can be attributed to the\nMAE between the \u0000 5gand \u0000 4gstates, which can be ex-\npressed as\nMAE = E \u00005g\u0000E\u00004g; (4)\nwhere E indicates the total energy of the correspond-\ning magnetic con\fguration. For Mn 3AgN, Mn 3NiN, and\nMn3SnN with experimentally observed mixed magnetic\nstates, the corresponding MAE is 0.063 meV, 0.144 meV,\nand 0.190 meV, respectively. Nevertheless, it is still un-\nclear why Mn 3AgN is stabilized in the \u0000 5gstate between\n55K and 290K. We suspect that those compounds with\nMAE greater than 0.2 meV between \u0000 4gand \u0000 5gshould\nhave pure noncollinear magnetic states. In our study, the\nMn-based APVs Mn 3XN with X = Ag, Co, Ir, Ni, Pd,\nRh, and Sn exhibit \u0000 4gwhile those with X = Au, Ga, Hg,\nIn, Pt, and Zn display the \u0000 5gas the magnetic ground\nstate (see Table I and S1). This is consistent with the re-\ncent DFT calculations for Mn 3XN (X = Ni, Zn, Ga, Sn,\nand Pt)57, except for Mn 3XN (X= In, Pd, and Ir). In\nRef. 51, Mn 3InN exhibit \u0000 4gstate while Mn 3XN (X= Ir\nand Pd) display \u0000 5gstate. The MAE for Mn 3InN calcu-\nlated by Huyen et al. is 74.6 meV, which is much larger\nthan our MAE (-1.84 meV). As detailed below, a large\nMAE could be expected for the compound with strong\nSOC; thus, it is questionable for Mn 3InN with such a\nlarge MAE of 74.6 meV.57To clarify the discrepancy, we\nperformed calculations of the MAE using the Quantum\nEspresso (QE) code58with the same parameters consider\nin Huyen et al. study. The QE and VASP calculations\nprovide the same magnetic ground state and MAE for4\nTABLE I: Compilation of the APVs (M 3XZ) with noncollinear magnetic ground state. All the units are in meV. For each\ncompound, the calculated magnetic ground state (\\magGS\") is listed and the change in the atomically resolved SOC energy\n(\u000eESOC) between \u0000 5gand \u0000 4gstate are enumerated for M and X atoms. The calculated MAE and \u0001E SOCare summarized as\ngiven by equation 4 and 5, respectively. The last column lists the available experimental magnetic ground state.\nM3XZ magGSM(\u000eESOC)X(\u000eESOC)\u0001ESOC MAE Refs.\nCr3IrN \u00004g -0.367 4.532 4.164 2.187\nCr3PtN \u00004g -0.936 5.639 4.703 2.868\nCr3SnN \u00004g -0.568 1.158 0.589 0.471\nMn3AgN \u00004g 0.3987 -0.2486 0.151 0.063 \u00004g+\u00005g53\nMn3AuN \u00005g 1.205 -4.353 -3.148 -1.671\nMn3CoN \u00004g 0.005 1.716 1.721 1.218 AFM11\nMn3GaN \u00005g -1.111 1.10 -1.110 -0.291 \u00005g53,56\nMn3HgN \u00005g -0.746 -3.691 -4.438 -1.551\nMn3InN \u00005g -0.734 -0.224 -0.958 -1.849\nMn3IrN \u00004g 0.599 19.705 20.305 10.548\nMn3NiN \u00004g 1.216 -0.87 0.346 0.144 \u00004g+\u00005g53\nMn3PdN \u00004g 1.292 -1.115 0.176 0.357 AFM11\nMn3PtN \u00005g 3.872 -13.877 -10.005 -4.969\nMn3RhN \u00004g -0.021 3.482 3.460 1.702\nMn3SnN \u00004g 0.099 0.479 0.578 0.190 \u00004g+\u00005g53\nMn3ZnN \u00005g -0.756 -0.105 -0.861 -1.452 \u00005g53;54\nMn3XN (X= Ir and Pt) systems, which is in contradic-\ntion to the Huyen study performed using QE.57\nLastly, based on our high-throughput calculations, three\nunreported Cr-based APVs are stable, our calculations\nreveal that Cr 3XN (Cr 3IrN, Cr 3SnN, and Cr 3PtN) have\n\u00004gas the lowest energy magnetic con\fguration. While\namong APV nitrides, Mn 3AuN and Mn 3HgN are the\nother two new noncollinear systems with \u0000 5gmagnetic\nground state. To the best of our knowledge, there is no\nexperimental neutron di\u000braction measurement available\ntil now for such APVs, making them interesting for fu-\nture studies. To shed more light on the origin of MAE,\nwe performed detailed calculations of the SOC energy\nin order to obtain atomically resolved contributions to\nthe MAE. Like MAE, the change in the total SOC en-\nergy (\u0001E SOC) of M 3XN can be de\fned as the di\u000berence\nbetween the SOC energy (E SOC) of the \u0000 5gand \u0000 4gmag-\nnetic states.59\n\u0001ESOC= E SOC(\u00005g)\u0000ESOC(\u00004g); (5)\nIn the same way, the atomically resolved change of SOC\nenergy (\u000eESOC) can also be de\fned for each atomic\nspecies, as shown in Table I. The positive (negative) val-\nues of MAE and \u0001E SOCindicate the \u0000 4g(\u00005g) magnetic\nground state. On the one hand, it is found that the sign\nof MAE is the same as that of \u0001E SOC (see Table I).\nThat is, both the \u0001E SOCand MAE are equally valid to\ncharacterize the magnetic ground states, with a linear\nscaling behavior observed (see Fig. S2). For example,\nCr3PtN has \u0000 4gmagnetic ground state with an MAE of\n2.868 meV and \u0001E SOCof 4.70 meV. The atomic resolved\n\u000eESOCof Pt and Cr are 5.639 meV and -0.938 meV. Ev-\nidently, the \u000eESOCof Pt is larger than Cr \u000eESOCorigi-\nnating in the strong SOC of Pt. As a result, the \u000eESOC\nof Pt is a determining factor for the MAE and \u0001E SOCsign. On the other hand, for most APVs, the contribu-\ntion of the X elements to MAE is more signi\fcant than\nthat of the magnetic elements M, as indicated by the\natomic resolved \u000eESOC(see Table I). Taking the Mn 3XN\nwith X = Co, Rh, and Ir as examples, the MAE are\n1.22 meV, 1.70 meV, and 10.55 meV, corresponding to\nthe dominant \u000eESOCof X as 1.72 meV, 3.46 meV, and\n20.31 meV, respectively. It is noted that \u000eESOCof Mn is\nsmaller than 3% of \u0001 ESOCfor such compounds. This can\nbe attributed to the enhanced strength of atomic SOC of\nthe X elements, e.g., the strength of atomic SOC for Co,\nRh, and Ir atoms is about 0.065 eV, 0.152 eV, and 0.452\neV, respectively.60Therefore, the enhanced atomic SOC\nstrength of the X elements is favorable for the strong\nMAE of the APV compounds.\nB. NTE and barocaloric\nTurning now to the magnetostructural coupling, which\ncan be best represented for APVs by the negative thermal\nexpansion (NTE), magnetocaloric, and PME. NTE ma-\nterials exhibit contraction in the lattice parameters with\nrespect to temperature in contrast to most materials.61\nTo study NTE of APVs, we evaluated the relative change\nin the lattice constants \u0001a/a 0on the transition from the\nnoncollinear magnetic ground state to PM state (where\na0is the lattice constant of \u0000 4gor \u0000 5gnoncollinear state\nand \u0001a is lattice constant di\u000berence between PM and \u0000 4g\nor \u0000 5gstate) and compared with the \u0001a/a 0values ob-\ntained from the experimental measurement at their tran-\nsition temperature (see Fig. 3(a)).11,12,61The average of\nthe evaluated \u0001a/a 0is in good agreement with the exper-\nimental observation except for Mn 3GaN (see Fig. 3(a)).\nTogether with the mismatch for the other cases, we sus-5\npect it may be due to the \fnite temperature e\u000bect, as\nour DFT calculations are done at zero Kelvin.\nImportantly, it is found that the PM state is critical and\ncannot be approximated as the NM state, e.g., the \u0001a/a 0\nbetween the noncollinear and NM states is about \fve\ntimes as large as the experimental value (see Fig. 3(a)).\nAn ameliorated estimation could be achieved by de\fning\nthe PM phase as a collinear AFM arrangement gener-\nated by using the disordered local moment (DLM) the-\nory.62The \u0001a/a 0of Cr 3IrN, Cr 3PtN, and Cr 3SnN on\nthe transition from \u0000 4gmagnetic ground state to PM are\n-0.0053, -0.0067, and -0.0070, respectively. That is, Cr-\nbased APVs undergo the lattice contraction with \u0001a/a 0\ncomparable to the reported NTE in Mn-based APV ma-\nterials such as Mn 3GaN and Mn 3ZnN (see Fig. 3(a)).\nFurthermore, the equilibrium lattice constants decrease\nand increase for the Mn 3XN (X=Ga, Hg, In, Ni, Pd,\nPt Sn, and Zn) and Mn 3XN APVs (X = Au, Co, Ir,\nand Rh) on the transition from a noncollinear magnetic\nground state to PM state, respectively (see Fig. 3(a)).\nAgAu CoGa Hg In Ir Ni Pd Pt Rh SnZn-0.04-0.03-0.02-0.010∆a/a0(a)NC-NM\nNC-PM\nExperiment\nIrPtSn\nAgAu CoGa Hg In Ir Ni Pd Pt Rh SnZn00.61.21.82.43ΣM (%)NC-PM\nNC-NM\nFM-NM\nFM-PM\nIrPtSn(b)\nFIG. 3: (a) The relative change in the lattice constant\n(\u0001a/a 0) and (b) the magnetic deformationP\nMfor the\nCr3XN and Mn 3XN antiperovskites (X and X are the ele-\nments present on the x-axis in violet and black, respectively).\nThe lattice constants of noncollinear, FM, PM, and NM states\nare used to determine \u0001a/a 0andP\nM.\nRecently, Bocarsly et al. proposed a magnetic defor-\nmation proxyP\nMto predict the magnetocaloric ma-\nterials as the theoretical evaluation of (\u0001S M) is verychallenging.63This proxy measures the volume deforma-\ntion of the NM and magnetic structure, which correlates\nwell with the experimentally measured \u0001S Mvalues. In\ntheir study, they screened 167 ferromagnetic materials,\nincluding seven APVs. For the APVs, it is noted thatP\nMscales linearly with the percentage volume di\u000berence\n\u0001VMbetween the FM and NM unit cell (see Fig. S3).\nInspired by their work, we evaluatedP\nMfor the APVs\nwith FM and noncollinear magnetic ground state in order\nto estimate the potential magnetocaloric e\u000bect. For FM\nAPVs, it is found that theP\nMis most signi\fcant for the\nFe-based APVs within the 1.2-1.7% range (see Table S5).\nFor example, theP\nMof Fe 3RhN, Fe 3PtN, and Fe 3NiN\nare 1.67%, 1.59%, and 1.54%, respectively. Whereas Co-,\nMn-, and Ni-based APVs haveP\nM<1.0% except for\nMn3ZnC (1.33%) (see Table S5). Thus, we expect a sig-\nni\fcant magnetocaloric e\u000bect in Fe 3RhN, Fe 3PtN, and\nFe3NiN, based on the argument that theP\nMapproxi-\nmate to 1.5% could exhibit a considerable magnetocaloric\ne\u000bect.63\nAs to the noncollinear APVs, we evaluated theP\nMby\nconsidering four \u0001V Mcombinations, namely, the non-\ncollinear magnetic ground state with PM and NM states\n(NC-PM and NC-NM), the FM with PM and NM states\n(FM-PM and FM-NM), see Fig. 3(b)). It is observed\nthat theP\nMvalue is small by using the \u0001V Mof the\nNC-PM and FM-PM states, in comparison to the other\ntwo combinations (see Fig. 3(b)). Nevertheless, a small\nvalue ofP\nMdoes not mean a low \u0001S M.63For instance,\nthe experimentally measured \u0001S Mof Mn 3GaN is very\nsigni\fcant (22.3 Jkg\u00001K\u00001), and it also exhibits giant\nbarocaloric e\u000bect.13Overall, theP\nMof Mn 3GaN should\nbe largest among all the APVs. The calculatedP\nM\nof Mn 3GaN is largest 0.805% for the \u0000 5gand PM state\ncombination, which veri\fes the experimental observation.\nWhereas, theP\nMof Mn 3GaN for the other three combi-\nnations is not the largest among the APVs (see Fig. 3(b)).\nTherefore, it can be concluded for the noncollinear sys-\ntems, that the approach of Bocarsly et al. proxy does\nnot provide reasonableP\nMobtained from the \u0001V M\ncombination of NC-NM, FM-NM, and FM-PM states.\nThe implicit correlation of \u0001S MwithP\nMindicates the\nstrong magnetostructural coupling in APVs with signif-\nicant \u0001V M(NC-PM). This conduces to the prediction\nof possible NTE and magnetocaloric materials with siz-\nableP\nMby virtue of signi\fcant \u0001V M(see Table S6).\nFor example, the Cr 3XN APVs (X= Ir, Pt, and Sn) with\nappreciableP\nMcould exhibit potential applications as\nNTE and magnetocaloric materials.\nC. Piezomagnetism\nPME provides another e\u000bective characterization of the\nmagnetostructural coupling,6which manifests itself as\nthe response of magnetization to strain. The origin\nof PME is explained based on symmetry together with\nan illustration of the magnetic spin directions (see Fig.6\nS4 and the Piezomagnetism section in the supplemen-\ntary information). For APVs with noncollinear magnetic\nground states, the total bulk magnetization is vanish-\ning, but a net magnetization can be induced under \f-\nnite compressive/tensile strain, as reported for Mn-based\nAPVs.6,8Our calculations con\frm their results on the\nMn-based APVs except for the Mn 3CoN and Mn 3RhN\n(see Fig. S5), where the resulting net magnetization\nfrom our (Zemen et al. )8calculations are 0.646 (0.305)\nand 0.214 (-0.143) \u0016B/f.u., respectively. Interestingly,\nfor Mn 3CoN, a net magnetic moment of 0.362 \u0016B/f.u.\nis induced at Co atoms under 1% tensile strain, whereas\nthe local magnetic moment of Co is zero in the cubic\nunstrained state. The magnetic moment direction of Co\nis aligned in (111) plane similar to Mn atom direction\nin the \u0000 4gstate. As a result, Mn 3CoN could be an in-\nteresting material for elastocaloric and magneto-elastic\napplications. Also, newly predicted Cr 3PtN exhibits sig-\nni\fcant PME as large as 0.21 \u0016B/f.u. at 1% compressive\nstrain. Surprisingly, the PME e\u000bect is asymmetric with\nrespect to the applied compressive and tensile strains.\nMost APVs exhibit more signi\fcant PME with tensile\nstrain except Mn 3SnN and Cr 3PtN (see Fig. S6 and S7),\ne.g., the net magnetization for Mn 3SnN at 1% compres-\nsive strain is as large as 0.73 \u0016B/f.u. in comparison to\n0.47\u0016B/f.u. at 1.0% tensile strain.\nThe magnitude of the magnetoelectric e\u000bect is minimal\nin the intrinsic bulk AFM materials, such as Cr 2O3.64.\nThe two-phase heterostructure materials consolidate the\nmagnetoelectric e\u000bect. A recent study corroborates APV\nas a potential material for magnetoelectric composite.6,65\nThe piezoelectric perovskite could be one suitable sub-\nstrate for such composite heterostructure as they also\nhave comparable lattice parameters with APVs.5,66The\nheterostructure amalgamate the piezoelectric and piezo-\nmagnetic properties, which are coupled by an interfacial\nstrain. Based on our PME analysis, we propose Cr- and\nMn-based noncollinear systems with signi\fcant PME as\npotentials candidates for magnetoelectric composite.\nInterestingly, the biaxial strain can induce phase transi-\ntion between di\u000berent magnetic con\fgurations, i.e., \u00004g\nand \u0000 5g), which can further lead to a signi\fcant change\nin the transport properties as discussed below. It is found\nthat a few APV materials undergo a magnetic phase\ntransition due to the imposed biaxial strain (see Table\nS4). For instance, cubic Cr 3SnN has \u0000 4gas the mag-\nnetic ground state, which preserves under compressive\nstrain. However, a transition into a distorted \u0000 5gstate is\nobtained by applying a tensile strain of 0.5% and 1.0%.\nSimilarly, there is a phase transition for Mn 3AuN from\n\u00005gto \u0000 4gdriven by compressive strain, whereas the \u0000 5g\ncon\fguration is retained under the tensile strain (see Ta-\nble S4). Such transitions can be attributed to signi\fcant\nchanges in the MAE with respect to the biaxial strain\n(see Table S4), but there is no consistent MAE trend as\nthe absolute value of MAE is mostly determined by the\natomic SOC strengths.D. Anomalous Hall Conductivity (Unstrained)\nIt is well known that FM compounds commonly exhibit\nAHC due to the presence of a net magnetization,67where\nthe broken time-reversal symmetry ( T) and SOC are two\nessential prerequisites. Noncollinear AFMs also display\nnon-zero AHC, as observed in Mn 3X (X = Ir, Ge, and\nSn).22,34As a linear response property, the occurrence of\nAHC can be elucidated based on the symmetry analysis.\nFor cubic APVs, the magnetic space group of \u0000 5gand\n\u00004gareR\u00163m(166.97) and R\u00163m0(166.101), respectively.\nFor \u0000 5g, the local magnetic moment spin directions of M\natoms are invariant under the mirror symmetry transfor-\nmationM(0\u001611),M(10\u00161), andM(\u0016110), leading to vanishing\nAHC (see Table S8).29,30Whereas in the \u0000 4gstate, the\nabove-mentioned mirror symmetries are broken, but the\nproduct of mirror ( M) and time-reversal symmetry Tre-\ntain the \u0000 4gcon\fguration. This results in \fnite AHCs\nwith all three o\u000b-diagonal components non-zero but of\nthe same amplitude (see Table S8).29\nExplicit calculations con\frm the above symmetry argu-\nments. For instance, the AHC is zero for Mn 3XN (X =\nAu, Hg, In, Pt, and Zn) with the \u0000 5gground state (see\nFig. S8(b) and S9). Whereas, the signi\fcant AHC is\nobserved for the APVs with the \u0000 4gground state, such\nas the AHC of Cr 3IrN and Cr 3PtN are 414.6 and 278.6\nS/cm, respectively. The AHC of APVs is summarized\nin the supplementary information (see Fig. S8, S9, and\nS10). In comparison to the previous studies on an indi-\nvidual or a few compounds,29,32our results are in good\nagreement (see Table S9). For instance, the AHC of\nMn3SnN from our and calculations by Gurung et al. are\n106.5 and 133 S/cm29, respectively. Whereas the value\n(-73.9 S/cm) obtained by Huyen et al.57is of opposite\nsign in addition to the di\u000berence in the absolute values.\nIn this regard, for Mn 3NiN, the sign of experimentally\nobserved AHC is con\frmed by our DFT calculations,31\nconsistent with that obtained by Zhou et al.32,39This\nmight be due to opposite magnetization direction while\nmaintaining the noncollinear con\fgurations or chirality\nof noncollinear spin con\fgurations speci\fed in the calcu-\nlations, subject to further investigations. Moreover, it is\nobserved that the absolute values of AHC for the same\ncompound in the same magnetic ground state are scat-\ntered as well (see Table S9). We suspect that this is due\nto the di\u000berent lattice constants and numerical param-\neters used in di\u000berent calculations. For instance, Zhou\net al. performed calculations using the experimental lat-\ntice constants on Mn 3NiN and other APV compounds,\nwhich are on average about 1.0-1.3% smaller than the\nfully relaxed lattice constants in this work.\nE. Strain induced AHC (Piezospintronics)\nAs discussed above, that all cubic APVs with non-\ncollinear magnetic ground states display signi\fcant\nPMEs where a net magnetization can be induced by ap-7\n050100150200250300|∆ AHC| (S/cm)σx\nσy\nσz\n00.10.20.30.40.50.6\nMnet(µB/f.u.)\n MnetΓ4g〈−−〉 Γ5gΓ4gΓ5g\nAgIr PdNi Pt Ir SnRhSnCoIn Zn SnPdZnIn Au PtHg\nFIG. 4: The piezospintronic e\u000bect is illustrated by the change in AHC ( j\u0001AHC j) at 1% tensile strain and cubic phase. The\nj\u0001AHC jand net magnetization is shown for the Cr 3XN and Mn 3XN antiperovskites (X and X are the elements present on the\nx-axis, respectively). The left, middle, and right panels correspond to the \u0000 4g, phase transition between \u0000 4gand \u0000 5g, and \u0000 5g\nstate, respectively.\n.\nplying biaxial strain; an interesting question is whether\nthe AHC can be tailored as well. From the symmetry\npoint of view, for the \u0000 5gstate, the magnetic space group\nchanges from R\u00163m(166.97) to C2=m(12.58) with biaxial\nstrain, resulting in \fnite \u001bxand\u001bywith the same ampli-\ntude but opposite sign, while the \u001bzcomponent remains\nzero.30This is con\frmed by our calculations on those\ncompounds with the \u0000 5gground state. For instance, the\nAHC induced by 1% biaxial strain is as large as -71.2\nS/cm for Mn 3PtN (see Fig. S8(b)), comparable to that in\nMn3HgN (see Fig. S9). The Mn 3InN and Mn 3ZnN have\nquite low AHC due to weak spin-orbit coupling strength\n(see Fig. S9).\nSimilarly, the AHC of APVs in the \u0000 4gstate can also be\ntailored by the biaxial strain. In this case, the magnetic\nspace group is reduced from R\u00163m0(166.101) to C20=m0\n(12.62), the resulting \u001bxand\u001bycomponents are the same\nwhile the\u001bzcomponent possesses a distinct value (see\nFig. S8(a), and Table S8). Explicit evaluations of the\nAHC for APVs in distorted \u0000 4gstates verify the symme-\ntry arguments, where the \u001bxand\u001bycomponents behave\nthe same with respect to strain (see Fig. S8(a)). Interest-\ningly, it is observed that more signi\fcant changes occur\nin AHC for the \u0000 4gcompounds than the \u0000 5gcases. For\ninstance, the \u001bzcomponent of AHC in Cr 3IrN attains\nthe largest AHC of 693.1 S/cm at 1% tensile strain, with\nan increase of 278 S/cm compared to the value in the\ncubic geometry (see Fig. S8(a)). Moreover, the moder-\nate strain can even lead to a sign change of the AHC, asobserved on the \u001bzcomponent of Mn 3AgN and Mn 3CoN\nwith 0.5% compressive and 1% tensile strain, respectively\n(see Fig. S10).\nOne interesting question is whether the piezospintronic\ne\u000bect ( i.e., the variation of AHC induced by biaxial\nstrain) correlates with the PME, as summarized in Fig. 4\nfor the variation of such quantities comparing the cases\nwith 1% tensile strain and cubic geometries. Obvi-\nously,j\u0001AHCjcannot be directly inferred from the in-\nduced net magnetization, particularly exempli\fed by the\nAPVs in the \u0000 4gmagnetic ground state. For example,\nthej\u0001AHCjand net magnetization of Cr 3IrN are 278.5\nS/cm (\u001bzcomponent) and 0.082 \u0016B/f.u., respectively,\nwhile for Mn 3SnN, thej\u0001AHCjand net magnetization\nare 35.5 S/cm ( \u001bzcomponent) and 0.47 \u0016B/f.u., respec-\ntively. Moreover, for Mn 3CoN, the net magnetization\nandj\u0001AHCjof\u001bxare 250.7 S/cm and 0.646 \u0016B/f.u. It\nsigni\fes that thej\u0001AHCjcould be large for the small net\nmagnetization and vice-versa (see Fig. 4). Interestingly,\nthej\u0001AHCjand net magnetization are smaller for those\ncompounds in the \u0000 5gmagnetic state, in comparison to\nthose of such materials with the \u0000 4gmagnetic state, e.g.,\nthej\u0001AHCjand net magnetization of Mn 3ZnN are 9.17\nS/cm and 0.029 \u0016B/f.u., respectively.\nParticularly, the APVs undergoing a phase transition\nbetween \u0000 4gand \u0000 5gstates driven by the epitaxial strain\nexhibit strong piezospintronic e\u000bect while the magnitude\nof the PME is marginal. For instance, Mn 3ZnN has a \u0000 5g\nmagnetic ground state in the cubic phase with vanishing8\nFIG. 5: (a) The calculated ANC of Mn 3PdN in the \u0000 4gmagnetic state. The Weyl points (0.43, -0.35, -0.49 and equivalent\ncoordinates) shift across the Fermi level with strain (inset above) and the Berry curvature for the 1.4% compressive biaxial\nstrain (inset below). (b) The band structure of Mn 3PdN for unstrained (black) and 1.4% compressive biaxial strained (blue).\nThe presence of Weyl point near the Fermi energy is indicated by black (cubic) and blue dashed line (-1.4% biaxial strain).\nAHC. A phase transition into the \u0000 4gstate by 1% tensile\nstrain results in a considerable j\u0001AHCjof 152.2 S/cm for\nthe\u001bycomponent (see Fig. 4). On the other hand, the\nphase transition from the \u0000 4gto \u0000 5gstates for Cr 3SnN\nleads to aj\u0001AHCjas large as 127.6 S/cm (Fig. 4). This\nis comparable to the j\u0001AHCjof 241.8 S/cm obtained as-\nsuming the \u0000 4gstate with 1% tensile strain. Therefore,\nwe suspect that those compounds with \u0000 4gstates are\nmore promising to host a strong piezospintronic e\u000bect.\nF. Anomalous Nernst e\u000bect\nThe intriguing behavior of AHC under biaxial strain\ncan be attributed to the \fne-tuning of the electronic\nstructure. To further illustrate such sensitivity, we in-\nvestigated the ANC, which shares the same symmetry\nas AHC but is proportional to the derivative of AHC\nwith respect to energy following the Mott formula.68Tak-\ning Mn 3NiN as an example, the corresponding ANC is\nas large as 16569 S/cm.eV. This is consistent with 1.80\nAK\u00001m\u00001at 200K obtained by Zhou et al.39, up to a\nfactor of two but with alike dependence with respect\nto the energy around the Fermi energy (see Fig. S15).\nThe numerical discrepancy might be due to the di\u000berent\nlattice constants used in the calculations, as discussed\nabove. Such a large ANC of Mn 3NiN is comparable to the\nlargest values observed experimentally in Co 2FeGe (3.16\nAK\u00001m\u00001), Co 2MnGa (4.0 AK\u00001m\u00001), and Fe 3Ga (3.0\nAK\u00001m\u00001).40,41,69,70Interestingly, the strain also has a\nstronger in\ruence on the ANC. For the tetragonally dis-torted Mn 3NiN in the \u0000 4gstate, the ANC becomes as\nlarge as 20035 S/cm.eV at -0.5% compressive strain for\nthe\u000bxand\u000bycomponents, i.e., enhanced by 21% com-\npared to that of the cubic case. Moreover, the ANC of\nCr3SnN is enhanced signi\fcantly, resulting in a value of\n-16321 S/cm.eV at 0.5% tensile strain for the \u000bzcompo-\nnent, which is comparable to that in Mn 3NiN. Last but\nnot least, a sign change can be induced in the ANC by\nthe biaxial strain. For instance, the ANC correspond-\ning to the\u000bxand\u000bycomponents of Cr 3XN changes to\n-3196.9 S/cm.eV at 1% compressive strain starting from\n6823.4 S/cm.eV for the cubic case in the \u0000 4gstate. This\nis also observed for other APVs such as Cr 3XN (X=Ir\nand Sn) and Mn 3XN (X=Co, Rh, Ag, Sn, and Ir) (see\nFig. S11).\nTo explicate the origin of the piezospintronic e\u000bects on\nboth AHC and ANC, our detailed analysis of the elec-\ntronic structure reveals that the tunability of AHC and\nANC by strain can be attributed to the presence of Weyl\npoints close to the Fermi energy. Taking Mn 3PdN as an\nexample, as shown in Fig. 5(a), the ANC can be tuned be-\ntween 7037.3 S/cm.eV at 1% compressive strain to -14599\nS/cm.eV at 1.5% compressive strain, with a sign change\nat 1.3% strain. Such an ANC is comparable to that ob-\nserved in Mn 3NiN and Cr 3SnN, awaiting further experi-\nmental validation. The band structure shows no obvious\nchanges due to the applied strain (see Fig. 5(b)). How-\never, it is found that there are 12 Weyl points at (0.43,\n-0.35, -0.49) and equivalent k-points with an energy of 20\nmeV below the Fermi energy in the cubic phase. Such\nWeyl points will be shifted across the Fermi energy upon9\napplying compressive strain, e.g., reaches to -1.75 meV\nfor -1.3% biaxial strain, and 1.87 meV above the Fermi\nenergy for 1.4% strain (see above inset of Fig. 5(a)). The\nAHC is mostly enhanced when the Weyl points are lo-\ncated at the Fermi energy, due to the singular behavior\nof the Berry curvature at the Weyl nodes (see below in-\nset of Fig. 5(a)). This is consistent with Ref. 51 and our\nobservation in Mn 3GaN.30,57For magnetic materials, the\nWeyl nodes with opposite chiralities ( i.e., Berry curva-\ntures with opposite sign) are located at the same energy,\nbut the total contributions to the AHC can be signi\fcant,\nparticularly if the Weyl nodes are within \\several\" or \\a\nfew\" meV around the Fermi energy. Correspondingly,\nthe sign of ANC, which is determined by the derivative\nof AHC can be tuned as the Weyl points are shifted across\nthe Fermi level. We suspect such tunability of AHC and\nconcomitant ANC by manipulating the Weyl points with\nthe biaxial strain or other possible stimuli is promising for\nfuture applications, such as Fe 3Al with giant transversal\nthermoelectric e\u000bects.70\nIV. CONCLUSION\nIn summary, we carried out a systematic analysis of 16\ncubic antiperovskite M 3XZ compounds with noncollinear\nmagnetic ground states, focusing on the magnetic prop-\nerties driven by isotropic and anisotropic magnetostruc-\ntural coupling. It is found that there exists a strong\ncompetition between di\u000berent noncollinear magnetic con-\n\fgurations where a large MAE clearly de\fnes the non-\ncollinear magnetic ground state, while small MAE leads\nto the mixed \u0000 4gand \u0000 5gcon\fgurations. For such ma-\nterials with mixed magnetic ions, the MAE cannot be\nunderstood based on the perturbation theory, whereas\nthe SOC energy can be used to get a reliable atomic\nresolved contribution, which is mostly determined by\nthe strength of atomic SOC. The magnetic ground state\nanalysis resulted in the prediction of \fve novel APVs\nwith noncollinear ground state, especially Cr 3XN APVs\n(X=Ir, Pt, and Sn) with the \u0000 4gmagnetic con\fgura-\ntion. The isotropic magnetostructural coupling indicated\nby the magnetic deformation \u0001a/a 0can be considered\nas an e\u000bective descriptor for the magnetocaloric e\u000bect.However, we observed that the magnetic deformation is\nbetter measured comparing the paramagnetic and mag-\nnetically ordered states, resulting in better agreement\nwith the experimental NTE results, rather than com-\nparing the NM and ordered state. Therefore, we sug-\ngest the recently proposed proxy for predicting magne-\ntocaloric e\u000bects based on the magnetic deformation could\nbe improved by using quasi-random approximation of the\nparamagnetic state. More interestingly, biaxial strain not\nonly causes a signi\fcant PME in such materials but also\nleads to a strong in\ruence on MAE, AHC, and ANC after\nconsidering spin-orbit coupling. Based on detailed sym-\nmetry analysis, we performed an explicit evaluation of the\nAHC and found that those compounds with the \u0000 4gmag-\nnetic ground state have large AHC, which is susceptible\nto the epitaxial strain. Nevertheless, there is no strong\ncorrelation between the net magnetization and induced\nAHC when \fnite strain is applied. Detailed analysis re-\nveals that the sensitivity of AHC and derived ANC with\nrespect to strain can be attributed to the \fne-tuning of\nenergies for the Weyl nodes, opening up further possibil-\nities for engineering spintronic devices in the future.\nV. ACKNOWLEGMENTS\nThe authors are grateful and acknowledge TU Darm-\nstadt Lichtenberg high-performance computer support\nfor the computational resources where the calculations\nwere conducted for this project. The authors thank Prof.\nManuel Richter of IFW Dresden for providing the SOC\nstrength data and discussion. This project was supported\nby the Deutsche Forschungsgemeinschaft (DFG, German\nResearch Foundation)-Project-ID 405553726-TRR 270.\nNuno Fortunato acknowledges European Research Coun-\ncil (ERC) funding for \fnancial support under the Eu-\nropean Unions Horizon 2020 research and innovation\nprogramme (Grant No. 743116 project Cool Innov).\nThe work of Jan Zemen was supported by the Ministry\nof Education, Youth and Sports of the Czech Repub-\nlic from the OP RDE program under the project In-\nternational Mobility of Researchers MSCA-IF at CTU\nNo.CZ.02.2.69/0.0/0.0/18 \u0000070/0010457.\n1M. Fiebig, T. Lottermoser, D. Meier, and M. Trassin, \\The\nevolution of multiferroics,\" Nat. Rev. Mater, vol. 1, no. 8,\npp. 16046{16059, 2016.\n2M. Trassin, \\Low energy consumption spintronics using\nmultiferroic heterostructures,\" J.Phys. Condens. Matter,\nvol. 28, no. 3, pp. 033001{ 033016, 2015.\n3J. Ma, J. Hu, Z. Li, and C.-W. Nan, \\Recent progress in\nmultiferroic magnetoelectric composites: from bulk to thin\n\flms,\" Adv. Mater., vol. 23, no. 9, pp. 1062{1087, 2011.\n4N. A. Spaldin and M. Fiebig, \\The renaissance of magneto-\nelectric multiferroics,\" Science, vol. 309, no. 5733, pp. 391{392, 2005.\n5C. X. Quintela, K. Song, D.-F. Shao, and et al. , \\Epi-\ntaxial antiperovskite/perovskite heterostructures for ma-\nterials design,\" Sci.Adv, vol. 6, no. 30, pp. eaba4017{\neaba4023, 2020.\n6P. Lukashev, R. F. Sabirianov, and K. Belashchenko,\n\\Theory of the piezomagnetic e\u000bect in mn-based antiper-\novskites,\" Phys. Rev. B, vol. 78, pp. 184414{184418, Nov\n2008.\n7D.-F. Shao, G. Gurung, T. R. Paudel, and E. Y. Tsym-\nbal, \\Electrically reversible magnetization at the antiper-10\novskite/perovskite interface,\" Phys. Rev. Mater., vol. 3,\nno. 2, pp. 024405{024413, 2019.\n8J. Zemen, Z. Gercsi, and K. G. Sandeman, \\Piezomag-\nnetism as a counterpart of the magnetovolume e\u000bect in\nmagnetically frustrated mn-based antiperovskite nitrides,\"\nPhys. Rev. B, vol. 96, pp. 024451{024458, Jul 2017.\n9D. Boldrin, A. P. Mihai, B. Zou, J. Zemen, and et al. ,\n\\Giant piezomagnetism in mn 3nin,\" ACS Appl. Mater.\nInterfaces, vol. 10, no. 22, pp. 18863{18868, 2018.\n10J. Lin, P. Tong, X. Zhou, H. Lin, Y. Ding, Y. Bai, L. Chen,\nX. Guo, C. Yang, B. Song, Y. Wu, S. Lin, W. H. Song,\nand Y. P. Sun, \\Giant negative thermal expansion cover-\ning room temperature in nanocrystalline gan xmn3,\"Appl.\nPhys. Lett, vol. 107, no. 13, pp. 131902{131906, 2015.\n11K. Takenaka, M. Ichigo, T. Hamada, A. Ozawa,\nT. Shibayama, T. Inagaki, and K. Asano, \\Magnetovolume\ne\u000bects in manganese nitrides with antiperovskite struc-\nture,\" Sci.Technol. Adv. Mater., vol. 15, no. 1, pp. 015009{\n015019, 2014.\n12K. Takenaka and H. Takagi, \\Giant negative thermal ex-\npansion in ge-doped anti-perovskite manganese nitrides,\"\nAppl. Phys. Lett, vol. 87, no. 26, pp. 261902{261904, 2005.\n13D. Matsunami, A. Fujita, K. Takenaka, and M. Kano, \\Gi-\nant barocaloric e\u000bect enhanced by the frustration of the\nantiferromagnetic phase in mn 3gan,\" Nat. Mater, vol. 14,\nno. 1, pp. 73{78, 2015.\n14T. Peng, W. Bo-Sen, and S. Yu-Ping, \\Mn-based antiper-\novskite functional materials: Review of research,\" Chinese\nPhys. B, vol. 22, no. 6, pp. 067501{067512, 2013.\n15K. Shi, Y. Sun, J. Yan, S. Deng, L. Wang, H. Wu, P. Hu,\nH. Lu, M. I. Malik, Q. Huang, and et al. , \\Baromag-\nnetic e\u000bect in antiperovskite mn 3ga0:95n0:94by neutron\npowder di\u000braction analysis,\" Adv. Mater., vol. 28, no. 19,\npp. 3761{3767, 2016.\n16T. Tohei, H. Wada, and T. Kanomata, \\Negative mag-\nnetocaloric e\u000bect at the antiferromagnetic to ferromag-\nnetic transition of mn 3gac,\" J.Appl. Phys., vol. 94, no. 3,\npp. 1800{1802, 2003.\n17D. Boldrin, E. Mendive-Tapia, J. Zemen, and et al. , \\Mul-\ntisite exchange-enhanced barocaloric response in mn 3nin,\"\nPhys. Rev. X, vol. 8, pp. 041035{041041, Nov 2018.\n18B. Wang, P. Tong, Y. Sun, X. Luo, X. Zhu, G. Li, X. Zhu,\nS. Zhang, Z. Yang, W. Song, and J. Dai, \\Large magnetic\nentropy change near room temperature in antiperovskite\nsncmn 3,\"EPL, vol. 85, no. 4, pp. 47004{47008, 2009.\n19T. He, Q. Huang, A. Ramirez, Y. Wang, K. Regan, N. Ro-\ngado, and et al. , \\Superconductivity in the non-oxide per-\novskite mgcni 3,\"Nature, vol. 411, no. 6833, pp. 54{56,\n2001.\n20S. Lin, P. Tong, B. Wang, J. Lin, Y. Huang, and\nY. Sun, \\Good thermoelectric performance in strongly\ncorrelated system sncco 3with antiperovskite structure,\"\nInorg. Chem., vol. 53, no. 7, pp. 3709{3715, 2014.\n21T. Shibayama and K. Takenaka, \\Giant magnetostriction\nin antiperovskite mn 3cun,\" J.Appl. Phys, vol. 109, no. 7,\npp. 07A928{07A930, 2011.\n22H. Chen, Q. Niu, and A. H. MacDonald, \\Anomalous hall\ne\u000bect arising from noncollinear antiferromagnetism,\" Phys.\nRev. Lett., vol. 112, pp. 017205{017209, Jan 2014.\n23T. Jungwirth, X. Marti, P. Wadley, and J. Wunder-\nlich, \\Antiferromagnetic spintronics,\" Nat. Nanotechnol.,\nvol. 11, no. 3, pp. 231{241, 2016.\n24V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and\nY. Tserkovnyak, \\Antiferromagnetic spintronics,\" Rev.Mod. Phys., vol. 90, pp. 015005{015061, Feb 2018.\n25I.\u0014Zuti\u0013 c, J. Fabian, and S. Das Sarma, \\Spintronics: Fun-\ndamentals and applications,\" Rev. Mod. Phys., vol. 76,\npp. 323{410, Apr 2004.\n26E. Liu, Y. Sun, N. Kumar, L. Muechler, A. Sun, L. Jiao,\nS.-Y. Yang, D. Liu, A. Liang, Q. Xu, and et al. , \\Giant\nanomalous hall e\u000bect in a ferromagnetic kagome-lattice\nsemimetal,\" Nat. Phys., vol. 14, no. 11, pp. 1125{1131,\n2018.\n27G. Xu, B. Lian, and S.-C. Zhang, \\Intrinsic quantum\nanomalous hall e\u000bect in the kagome lattice cs 2limn 3f12,\"\nPhys. Rev. Lett., vol. 115, pp. 186802{186805, Oct 2015.\n28Q. Wang, S. Sun, X. Zhang, F. Pang, and H. Lei, \\Anoma-\nlous hall e\u000bect in a ferromagnetic fe 3sn2single crystal with\na geometrically frustrated fe bilayer kagome lattice,\" Phys.\nRev. B, vol. 94, pp. 075135{075139, Aug 2016.\n29G. Gurung, D.-F. Shao, T. R. Paudel, and E. Y. Tsymbal,\n\\Anomalous hall conductivity of noncollinear magnetic an-\ntiperovskites,\" Phys. Rev. Materials, vol. 3, pp. 044409{\n044416, Apr 2019.\n30I. Samathrakis and H. Zhang, \\Tailoring the anomalous\nhall e\u000bect in the noncollinear antiperovskite mn 3gan,\"\nPhys. Rev. B, vol. 101, pp. 214423{214428, Jun 2020.\n31D. Boldrin, I. Samathrakis, J. Zemen, and et al. , \\Anoma-\nlous hall e\u000bect in noncollinear antiferromagnetic mn 3nin\nthin \flms,\" Phys. Rev. Materials, vol. 3, pp. 094409{\n094415, Sep 2019.\n32X. Zhou, J.-P. Hanke, W. Feng, F. Li, G.-Y. Guo, Y. Yao,\nS. Bl ugel, and Y. Mokrousov, \\Spin-order dependent\nanomalous hall e\u000bect and magneto-optical e\u000bect in the\nnoncollinear antiferromagnets mn 3xn with x= ga, zn, ag,\nor ni,\" Phys. Rev. B, vol. 99, pp. 104428{104440, Mar 2019.\n33S. V. Gallego, J. Etxebarria, L. Elcoro, E. S. Tasci, and\nJ. M. Perez-Mato, \\Automatic calculation of symmetry-\nadapted tensors in magnetic and non-magnetic materials:\na new tool of the bilbao crystallographic server,\" Acta\nCrystallogr AFound Adv., vol. 75, no. 3, pp. 438{447,\n2019.\n34G.-Y. Guo and T.-C. Wang, \\Large anomalous nernst and\nspin nernst e\u000bects in the noncollinear antiferromagnets\nmn3x (x= sn, ge, ga),\" Phys. Rev. B, vol. 96, pp. 224415{\n224423, Dec 2017.\n35J. K ubler and C. Felser, \\Non-collinear antiferromagnets\nand the anomalous hall e\u000bect,\" EPL, vol. 108, no. 6,\npp. 67001{67005, 2014.\n36A. K. Nayak, J. E. Fischer, Y. Sun, B. Yan, J. Karel, A. C.\nKomarek, C. Shekhar, N. Kumar, W. Schnelle, J. K ubler,\nC. Felser, and S. S. P. Parkin, \\Large anomalous hall ef-\nfect driven by a nonvanishing berry curvature in the non-\ncolinear antiferromagnet mn 3ge,\" Sci.Adv., vol. 2, no. 4,\npp. e1501870{e1501874, 2016.\n37G. E. Bauer, E. Saitoh, and B. J. Van Wees, \\Spin\ncaloritronics,\" Nat. Mater., vol. 11, no. 5, pp. 391{399,\n2012.\n38S. R. Boona, R. C. Myers, and J. P. Heremans, \\Spin\ncaloritronics,\" Energy Environ. Sci., vol. 7, no. 3, pp. 885{\n910, 2014.\n39X. Zhou, J.-P. Hanke, W. Feng, S. Bl ugel, Y. Mokrousov,\nand Y. Yao, \\Giant anomalous nernst e\u000bect in non-\ncollinear antiferromagnetic mn-based antiperovskite ni-\ntrides,\" Phys. Rev. Materials, vol. 4, pp. 024408{024415,\nFeb 2020.\n40A. Sakai, Y. P. Mizuta, A. A. Nugroho, R. Sihombing, and11\net al. , \\Giant anomalous nernst e\u000bect and quantum-critical\nscaling in a ferromagnetic semimetal,\" Nat. Phys., vol. 14,\nno. 11, pp. 1119{1124, 2018.\n41S. N. Guin, K. Manna, J. Noky, S. J. Watzman, and et\nal., \\Anomalous nernst e\u000bect beyond the magnetization\nscaling relation in the ferromagnetic heusler compound\nco2mnga,\" NPG Asia Mater., vol. 11, no. 1{9, pp. 11{16,\n2019.\n42Z. Liu, Z. Feng, H. Yan, X. Wang, X. Zhou, P. Qin, H. Guo,\nR. Yu, and C. Jiang, \\Antiferromagnetic piezospintronics,\"\nAdv. Electron. Mater., vol. 5, no. 7, pp. 1900176{1900184,\n2019.\n43G. Kresse and J. Furthm uller, \\E\u000ecient iterative schemes\nfor ab initio total-energy calculations using a plane-wave\nbasis set,\" Phys. Rev. B, vol. 54, pp. 11169{11186, Oct\n1996.\n44J. P. Perdew, K. Burke, and M. Ernzerhof, \\Generalized\ngradient approximation made simple,\" Phys. Rev. Lett.,\nvol. 77, pp. 3865{3868, Oct 1996.\n45A. Zunger, S.-H. Wei, L. G. Ferreira, and J. E. Bernard,\n\\Special quasirandom structures,\" Phys. Rev. Lett.,\nvol. 65, pp. 353{356, Jul 1990.\n46A. van de Walle, M. D. Asta, and G. Ceder, \\The Alloy\nTheoretic Automated Toolkit: A user guide,\" Calphad,\nvol. 26, pp. 539{553, 2002.\n47Q. Wu, S. Zhang, H.-F. Song, M. Troyer, and A. A.\nSoluyanov, \\Wanniertools : An open-source software\npackage for novel topological materials,\" Comput. Phys.\nCommun., vol. 224, pp. 405{416, 2018.\n48A. A. Mosto\f, J. R. Yates, Y.-S. Lee, I. Souza, D. Vander-\nbilt, and N. Marzari, \\wannier90: A tool for obtaining\nmaximally-localised wannier functions,\" Comput. Phys.\nCommun., vol. 178, no. 9, pp. 685{699, 2008.\n49D. Xiao, M.-C. Chang, and Q. Niu, \\Berry phase e\u000bects on\nelectronic properties,\" Rev. Mod. Phys., vol. 82, pp. 1959{\n2007, Jul 2010.\n50M.-H. Yu, L. Lewis, and A. Moodenbaugh, \\Large\nmagnetic entropy change in the metallic antiperovskite\nmn3gac,\" J.Appl. Phys., vol. 93, no. 12, pp. 10128{10130,\n2003.\n51E. Dias, K. Priolkar, A. Das, G. Aquilanti, O. C \u0018 akir,\nM. Acet, and A. Nigam, \\E\u000bect of local structural distor-\ntions on magnetostructural transformation in mn 3snc,\" J.\nPhys. D:Appl. Phys., vol. 48, no. 29, pp. 295001{295007,\n2015.\n52V. N. Antonov, B. N. Harmon, A. N. Yaresko, and A. P. Sh-\npak, \\Electronic structure, noncollinear magnetism, and x-\nray magnetic circular dichroism in the mn 3znc perovskite,\"\nPhys. Rev. B, vol. 75, pp. 165114{165120, Apr 2007.\n53D. Fruchart and E. F. Bertaut, \\Magnetic studies of the\nmetallic perovskite-type compounds of manganese,\" J.\nPhys. Soc. Jpn., vol. 44, no. 3, pp. 781{791, 1978.\n54S. Deng, Y. Sun, L. Wang, Z. Shi, H. Wu, Q. Huang,\nJ. Yan, K. Shi, P. Hu, A. Zaoui, and et al. , \\Frustrated\ntriangular magnetic structures of mn 3znn: applications in\nthermal expansion,\" J.Phys. Chem. C, vol. 119, no. 44,\npp. 24983{24990, 2015.\n55D. Fruchart, E. Bertaut, J. Senateur, and R. Fruchart,\n\\Magnetic studies on the metallic perovskite-type com-\npound mn 3snn,\" J.Phys. Lett., vol. 38, no. 1, pp. 21{23,1977.\n56E. Bertaut, D. Fruchart, J. Bouchaud, and R. Fruchart,\n\\Di\u000braction neutronique de mn 3gan,\" Solid State\nCommun., vol. 6, no. 5, pp. 251{256, 1968.\n57V. T. N. Huyen, M.-T. Suzuki, K. Yamauchi, and\nT. Oguchi, \\Topology analysis for anomalous hall e\u000bect\nin the noncollinear antiferromagnetic states of mn 3an (a=\nni, cu, zn, ga, ge, pd, in, sn, ir, pt),\" Phys. Rev. B, vol. 100,\npp. 094426{094434, Sep 2019.\n58P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car,\nC. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni,\nI. Dabo, and et al. , \\Quantum espresso: a modular and\nopen-source software project for quantum simulations of\nmaterials,\" J.Phys.: Condens. Matter., vol. 21, no. 39,\npp. 395502{395520, 2009.\n59V. Antropov, L. Ke, and D. \u0017Aberg, \\Constituents of mag-\nnetic anisotropy and a screening of spin{orbit coupling in\nsolids,\" Solid State Commun., vol. 194, pp. 35{38, 2014.\n60K. H. J. Buschow, Handbook ofmagnetic materials,\nvol. 13. Elsevier, 2003.\n61S. Deng, Y. Sun, L. Wang, Z. Shi, H. Wu, Q. Huang,\nJ. Yan, K. Shi, P. Hu, A. Zaoui, and C. Wang, \\Frustrated\ntriangular magnetic structures of mn 3znn: Applications in\nthermal expansion,\" J.Phys. Chem. C, vol. 119, no. 44,\npp. 24983{24990, 2015.\n62I. A. Abrikosov, A. Ponomareva, P. Steneteg, S. Baran-\nnikova, and B. Alling, \\Recent progress in simulations of\nthe paramagnetic state of magnetic materials,\" Curr Opin\nSolid State Mater Sci, vol. 20, no. 2, pp. 85{106, 2016.\n63J. D. Bocarsly, E. E. Levin, C. A. C. Garcia, K. Schwen-\nnicke, S. D. Wilson, and R. Seshadri, \\A simple compu-\ntational proxy for screening magnetocaloric compounds,\"\nChem. Mater., vol. 29, no. 4, pp. 1613{1622, 2017.\n64M. Date, J. Kanamori, and M. Tachiki, \\Origin of mag-\nnetoelectric e\u000bect in cr 2o3,\"J.Phys. Soc. Jpn., vol. 16,\nno. 12, pp. 2589{2589, 1961.\n65J. Ryu, S. Priya, K. Uchino, and H.-E. Kim, \\Magnetoelec-\ntric e\u000bect in composites of magnetostrictive and piezoelec-\ntric materials,\" J.Eelectroceram., vol. 8, no. 2, pp. 107{\n119, 2002.\n66H. Tashiro, R. Suzuki, T. Miyawaki, K. Ueda, and\nH. Asano, \\Preparation and properties of inverse per-\novskite mn 3gan thin \flms and heterostructures,\" J.Korean\nPhys. Soc., vol. 63, no. 3, pp. 299{301, 2013.\n67N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and\nN. P. Ong, \\Anomalous hall e\u000bect,\" Rev. Mod. Phys.,\nvol. 82, pp. 1539{1592, May 2010.\n68D. Xiao, Y. Yao, Z. Fang, and Q. Niu, \\Berry-phase e\u000bect\nin anomalous thermoelectric transport,\" Phys. Rev. Lett.,\nvol. 97, pp. 026603{026606, Jul 2006.\n69J. Noky, J. Gooth, C. Felser, and Y. Sun, \\Characteri-\nzation of topological band structures away from the fermi\nlevel by the anomalous nernst e\u000bect,\" Phys. Rev. B, vol. 98,\npp. 241106{241110, Dec 2018.\n70A. Sakai, S. Minami, T. Koretsune, T. Chen, T. Higo,\nY. Wang, T. Nomoto, and et al. , \\Iron-based binary ferro-\nmagnets for transverse thermoelectric conversion,\" Nature,\nvol. 581, no. 7806, pp. 53{57, 2020."    },    {        "title": "2009.13465v1.Electronic_and_magnetic_characterization_of_epitaxial_CrBr__3__monolayers.pdf",        "content": "Electronic and magnetic characterization of epitaxial CrBr 3\nmonolayers\nShawulienu Kezilebiekea,1,\u0003Orlando J. Silveiraa,1Md N. Hudaa,1Viliam\nVa\u0014 no,1Markus Aapro,1Somesh Chandra Ganguli,1Jouko Lahtinen,1Rhodri\nMansell,1Sebastiaan van Dijken,1Adam S. Foster,1, 2and Peter Liljeroth1\n1Department of Applied Physics, Aalto University, FI-00076 Aalto, Finland\n2Nano Life Science Institute (WPI-NanoLSI),\nKanazawa University, Kakuma-machi, Kanazawa 920-1192, Japan\nThe ability to imprint a given material property to another through proximity\ne\u000bect in layered two-dimensional materials has opened the way to the creation of de-\nsigner materials. Here, we use molecular-beam epitaxy (MBE) for a direct synthesis\nof a superconductor-magnet hybrid heterostructure by combining superconducting\nniobium diselenide (NbSe 2) with the monolayer ferromagnetic chromium tribromide\n(CrBr 3). Using di\u000berent characterization techniques and density-functional theory\n(DFT) calculations, we have con\frmed that the CrBr 3monolayer retains its ferro-\nmagnetic ordering with a magnetocrystalline anisotropy favoring an out-of-plane spin\norientation. Low-temperature scanning tunneling microscopy (STM) measurements\nshow a slight reduction of the superconducting gap of NbSe 2and the formation of\na vortex lattice on the CrBr 3layer in experiments under an external magnetic \feld.\nOur results contribute to the broader framework of exploiting proximity e\u000bects to\nrealize novel phenomena in 2D heterostructures.\naThese authors contributed equally.arXiv:2009.13465v1  [cond-mat.mtrl-sci]  28 Sep 20202\nTwo-dimensional magnetic materials constitute an ideal platform to experimentally access\nthe fundamental physics of magnetism in reduced dimensions1{3. Furthermore, because\nof the ease of fabricating heterostructures, ferromagnetic van der Waals (vdW) materials\npresent attractive opportunities for designer 2D magnetic4,5, magnetoelectric6, and magneto-\noptical arti\fcial heteromaterials7. As the di\u000berent components in a vdW heterstructure only\ninteract via weak vdW forces, the properties of the constituent materials are not strongly\nmodi\fed. This means that one can { for example { imprint magnetic properties of 2D\nmagnets to the other layers without modifying their intrinsic properties and create novel\nspintronic and magnonic devices8{10.\nThis designer concept can be utilized in systems combining magnetism with superconduc-\ntivity to realize topological superconductivity11,12. It is currently attracting intense attention\ndue to its potential role in building blocks for Majorana-based qubits for topological quantum\ncomputation12{14. While there are very few potential real materials exhibiting topological\nsuperconductivity15{18, in a designer material the desired physics emerges from the engi-\nneered interactions between the di\u000berent components. For topological superconductivity,\none needs to combine s-wave superconductivity with magnetism and spin-orbit coupling\nto create an arti\fcial topological superconductor12,19. However, the coupling between the\ncomponents is highly sensitive to the interfacial structure and electronic properties2,20and\nthus, vdW materials with atomically sharp and highly uniform interfaces are an attrac-\ntive platform with which to realize and harness exotic electronic phases arising in designer\nmaterials.\nLayered materials that remain magnetic down to the monolayer (ML) limit have been\nrecently demonstrated4,5,21. While the \frst reports relied on mechanical exfoliation for the\nsample preparation, related materials CrBr 3and Fe 3GeTe 2have also been grown using\nmolecular-beam epitaxy (MBE) in ultra-high vacuum (UHV)22,23, which is essential for real-\nizing clean edges and interfaces. The inherent lack of surface bonding sites due to the layered\nnature of these materials prevents chemical bonding between the layers and results in a better\ncontrol of the interfaces. Recently, we have successfully fabricated a superconducting ferro-\nmagnetic hybrid system based on vdW heterostructures using MBE24,25. More importantly,\nby combining spin-orbit coupling, 2D ferromagnetic CrBr 3, and superconducting niobium\ndiselenide (NbSe 2), we have demonstrated the existence of the one-dimensional Majorana\nedge modes using low-temperature scanning tunneling microscopy (STM) and spectroscopy3\n(STS)25. However, for future applications, further systematic studies are desired for a better\nunderstanding of the electronic and magnetic properties of the 2D ferromagnetic CrBr 3/\nNbSe 2heterostructures.\nIn this work, we report a detailed characterisation of the CrBr 3/NbSe 2hybrid system\nusing low-temperature STM/STS and X-ray photo-electron spectroscopy (XPS). Combin-\ning magneto-optical Kerr e\u000bect (MOKE) measurements and density functional calculations\n(DFT), we unambiguously con\frm the ferromagnetism of the CrBr 3islands on NbSe 2sub-\nstrate. Our results give further experimental information on the magnetic properties of\nCrBr 3, and demonstrate a clean and controllable platform for creating superconducting-\nmagnetic hybrid systems with a great potential for integration into future electronic devices\nthat could be controlled externally through electrical6, mechanical26, chemical27, or optical\nmeans28.\nFigure 1a shows the top and side views of the most stable CrBr 3-NbSe 2heterostructure\ngeometry obtained by density functional theory (DFT) calculations (details are given in\nthe Methods section), where the single-layer CrBr 3consists of Cr atoms that are arranged\nin a honeycomb lattice structure and each atom is surrounded by an octahedron of six Br\natoms. The energetically most favourable stacking has one Cr atom located on top of a Se 2\npair, while the other Cr atom is on top of the hole site of the NbSe 2. The fully relaxed\nlattice parameters of the heterostructures reveal that the CrBr 3is strained by about 7 %,\nwhile the NbSe 2is compressed by less than 2 %, which are not su\u000ecient to drastically\na\u000bect their electronic and magnetic properties29,30(see Supporting Information). X-ray\nphotoelectron spectroscopy (XPS) (Figure 1b) displays sharp characteristic peaks for Cr\nand Br, demonstrating the purity of the CrBr 3\flm as well as consistency with previous\nreports on bulk samples31,32. Figure 1c shows a large scale STM image illustrating the\ntypical morphology of the single-layer CrBr 3\flms grown on NbSe 2. The CrBr 3islands are\natomically \rat and up to 200 nm in size, while clean areas of the bare NbSe 2substrate are\nclearly exposed. Figure 1d shows an atomically resolved STM image of the CrBr 3monolayer,\nrevealing periodically spaced triangular protrusions. These features are formed by the three\nneighbouring Br atoms as highlighted in the Fig. 1e showing the simulated STM topographic\nimage obtained through DFT calculations. Finally, the STM topography exhibits a clear,\nwell-ordered superstructure with 6.3 nm periodicity. This superstructure is explained by the\nfact that when the CrBr 3and NbSe 2lattices are overlaid, 19 NbSe 2unit cells accommodate4\nFIG. 1. Growth of CrBr 3on NbSe 2. (a) Computed top and side views of the CrBr 3-NbSe 2\nheterostructure. (b) Core-level spectra of CrBr 3-NbSe 2heterostructure. The pass energy was 80\neV for the black curve and 20 eV for colored curves (c) STM image of a monolayer thick CrBr 3\nisland grown on NbSe 2using MBE (STM feedback parameters: Vbias= +1 V, I= 10 pA, scale\nbar: 39 nm). (d) Atomically resolved image on the CrBr 3layer. The moir unit cell is denoted by\nthe large rhombus (STM feedback parameters: Vbias= +2 V, I= 0:5 nA, scale bar: 3 nm). (e)\nCalculated constant-current STM image of a monolayer of CrBr 3on NbSe 2:Vbias= +0:5 V, scale\nbar: 3 nm. The inset in (e) shows a zoom in the calculated STM images, with the atomic structure\nof the CrBr 3superimposed.\n10 unit cells of CrBr 3, thus forming a 6.3 nm \u00026.3 nm superstructure (a moir\u0013 e pattern) as\nobserved in the STM images of CrBr 3as shown in Figure 1d. The rotationally misaligned\nCrBr 3domains have a slight e\u000bect the on moir periodicity (supporting info).\nWe experimentally determined the electronic structure of single-layer CrBr 3-NbSe 2using\nscanning tunneling spectroscopy (STS) measurements and compared the results directly5\nFIG. 2. (a) Typical long-range experimental d I/dVspectra on a ML CrBr 3on the NbSe 2substrate.\n(b) Simulated spin polarized projected density of states (PDOS) on the CrBr 3layer of the CrBr 3-\nNbSe 2heterostructure (continuous lines) and on pristine CrBr 3layer (dashed lines). (c) Band\nstructure and PDOS of the pristine CrBr 3layer (dashed lines) and CrBr 3-NbSe 2heterostructure\n(continuous lines). The black continuous line in (c) is the PDOS on the NbSe 2substrate in the\nCrBr 3-NbSe 2heterostructure.\nwith DFT calculations. Figure 2a shows a typical STM d I/dVbspectrum of single-layer\nCrBr 3acquired over a large bias range. In the \flled state regime (bias voltage Vb<0), the\nspectrum is relatively \rat and featureless until Vb\u0018 \u00001 V is reached, where the d I/dVb\nslightly increases, and a steep rise is seen at Vb\u0018-1.5 V. The dominant feature in the empty\nstate regime ( Vb>0) of the dI/dVbspectrum starts with a small bump around \u00180:5 V and\na steep rise in d I/dVbat\u00181:2 V, as shown in Figure 2a. Those features, apart from a hard\nshift of \u00180:5 eV, are consistent with the DFT spin polarized projected density of states\n(PDOS) on the CrBr 3layer of the heterostructure shown in Figure 2b. The steep rises in the\ndI/dVbspectrum are around 3 V apart and mostly arise from the spin up bands of the CrBr 3\nlayer: its electronic properties are well preserved in the heterostructure as compared to the\npristine CrBr 3layer, as can be seen in Figure 2b and c. The band structures and PDOS\nshown in Figure 2c reveal that the bands in the [-2.0,-1.0] eV and [-1.0,1.0] eV windows have\na majority contribution from the NbSe 2layer, where the NbSe 2'sd-band in the [-1.0,1.0] eV\nwindow is completely preserved and slightly spin polarized due to the proximity with the6\nmagnetic CrBr 3. However, the PDOS on the CrBr 3layer in the [-2.0,-1.0] eV and [-1.0,1.0]\neV windows are non-zero and consistent with the bumps in the d I/dVbspectrum, which we\nattribute to charge recon\fguration in the heterostructure as shown in Figure 3a.\nFIG. 3. (a) Di\u000berential charge density of the CrBr 3-NbSe 2heterostructure, where the yellow and\nblue colors indicate charge accumulation (+) and depletion (-), respectively. (b) Magnetic hysteresis\nin monolayer CrBr 3on NbSe 2at several di\u000berent temperatures (indicated in the \fgure). (c) The\ntemperature dependence of the coercive \feld of CrBr 3on NbSe 2. The coercive \feld decreases as\nthe temperature increases until it vanishes at 16 K.\nAfter the electronic characterization of the samples, we will next focus on their magnetic\nproperties. The isolated CrBr 3layer has a predicted out-of-plane magnetic moment of 6.000\n\u0016Bper unit cell, where each Cr atom has three unpaired electrons. Our DFT calculations\nshow that the magnetism of the CrBr 3layer is well-preserved in the heterostructure, which\nshows a slightly larger magnetic moment of 6.097 \u0016Bper unit cell due to induced magne-\ntization on the NbSe 2layer. The PDOS on the CrBr 3layer of the heterostructure reveals\nthat the majority spin up channel has a band gap of 3 eV, while the minority spin down\nchannel has a band gap of around 5 eV, both shown in blue and red, respectively, in Figure\n2c.\nWe study the truly 2D itinerant ferromagnetism in CrBr 3monolayer on NbSe 2using\nmagneto-optical Kerr e\u000bect (MOKE) microscopy (details of the experimental procedures\nare given the Methods section). The magnetic \feld was applied perpendicular to the sam-7\nple. Figure 3b shows the MOKE signal of a CrBr 3monolayer on NbSe 2as a function of\nthe external magnetic \feld at several di\u000berent temperatures. Notably, monolayer CrBr 3is\nferromagnetic, as evidenced by the prominent hysteresis seen here. MOKE measurements\nfurther reveal that as the temperature is increased, the hysteresis loop shrinks and eventually\ndisappears at a transition temperature Tcof about 16 K. The Tccan also be extracted from\na temperature dependence of the coercivity Hc(blue squares in Figure 3c), where the onset\nofHcin the CrBr 3monolayer clearly occurs at around 16 K. The same behaviour was ob-\nserved for mechanically exfoliated monolayer CrBr 3\rake, where the transition temperature\nTcwas around 20 K33. Moreover, the out-of-plane coercive \feld Hcfor exfoliated monolayer\nCrBr 3\rake is 4 mT28,33,34while in our MBE grown monolayer CrBr 3on NbSe 2substrate\nthe out-of-plane coercive \feld increases up to \u00182:5 mT at the lowest temperatures we can\nreach in our MOKE setup as shown in Figure 3c.\nAfter con\frming the existence of ferromagnetism on the monolayer CrBr 3on NbSe 2\nsubstrate, it is interesting to see the e\u000bects of the superconducting substrate NbSe 2on the\nmonolayer CrBr 3. Isolated CrBr 3is a ferromagnetic insulator; however, due to the monolayer\nthickness, it is possible to tunnel through such a structure with STM. In addition, due to\nthe charge transfer at the interface between CrBr 3and NbSe 2, the heterostructure itself\nhas a metallic nature. Therefore, one can expect a measurable interaction between CrBr 3\nand NbSe 2. The superconductivity of CrBr 3/NbSe 2heterostructure was studied by STM at\nT= 350 mK. Figures 4a,b show experimental d I/dVbspectra (raw data) taken on a bare\nNbSe 2and on monolayer CrBr 3, respectively. The d I/dVbspectrum of bare NbSe 2(Figure\n4a) has a hard gap with an extended region of zero di\u000berential conductance around the Fermi\nenergy. dI/dVbspectra were \ftted by the McMillan two-band model35, with parameters\n\u00011= 1:28 meV,\r1= 0:38 and \u0001 2= 0:74 meV,\r1= 0:13 meV, respectively. In contrast, the\nspectra taken in the middle of the CrBr 3island have small but distinctly non-zero di\u000berential\nconductance inside the gap of the NbSe 2substrate. We observe pairs of conductance onsets\nat\u00060:3 mV around zero bias. This feature results from the formation of Shiba-bands in the\nNbSe 2caused by the induced magnetization from the CrBr 3island25. These extra features\nare not reproduced by the two-band model (Figure 4b). Nevertheless, a double gap s-wave\nBCS-type \ftting gives a slight reduction of both gap parameters (\u0001 1= 1:20 meV,\r1= 0:40\nmeV and \u0001 2= 0:73 meV,\r2= 0:50 meV respectively).\nTo obtain more detailed insight into the superconductivity on CrBr 3/NbSe 2heterostruc-8\nFIG. 4. (a,b) Experimental d I/dVspectroscopy (black) on the NbSe 2substrate (a) and in the\nmiddle of a CrBr 3island (b) measured at T= 350 mK. We also show a \ft to a double gap s-wave\nBCS-type model (red). (c) STM topography image (STM feedback parameters: Vbias= +1 V,\nI= 10 pA, scale bar: 22 nm). (d-f) Vortex imaging on CrBr 3/NbSe 2heterostructure. We have\nrecorded grid spectroscopy (100 \u0002100 spectra) over an area of 110 \u0002110 nm2atT= 350 mK under\nan applied out-of-plane magnetic \feld of 0.65 T. The maps are obtained from full spectroscopic\nscans from -3 mV to 3 mV at each pixel at the indicated bias voltages. (g) Line scan of the d I/dVb\nsignal along the white line marked in panel (d).\nture, we investigate the dependence of our d I/dVbspectra under an applied out-of-plane\nmagnetic \feld. In a type-II SC such as NbSe 2, we would expect to observe an Abrikosov\nvortex lattice in a d I/dVbmap acquired near the energy of the superconducting gap36.\nFigs. 4d-f show d I/dVbgrid maps at 0, 0.6 and 0.8 mV bias voltages, respectively. The\nmaps are recorded under 0.65 T out-of-plane magnetic \feld on an area with both CrBr 3\nislands and bare NbSe 2surface (Figure 4c). It is seen from Figures 4d-f that the vortices\nexhibit a highly ordered hexagonal lattice similar to those observed on the clean NbSe 2\nsurface. This is the \frst time vortices have been clearly observed in a hybrid ferromagnet-9\nsuperconductor-system. We measured the spatial variation of the d I/dVbspectra as a\nfunction of distance away from the vortex center (along the dashed line in Figure 4d). The\nresults are given in Figure 4g, which shows the measured d I/dVbas functions of distance\nand sample bias V on a color scale. One can see that only one peak appears at zero-bias\nin the dI/dVbspectra near the vortex center, and this peak splits into two away from the\nvortex core. The splitting energy increases linearly with distance. One of the intriguing\nproperties of a topological superconductor is that vortices on its surface are expected to host\nMajorana zero modes37. This mode results in a peak in the local density-of-states at the\nFermi energy in the center of the vortex. In contrast to bound states in vortices on conven-\ntional superconductors, a Majorana mode should not split in energy away from the vortex\ncenter37{39. Due to the broadening of the resonance in the vortex spectra, we cannot resolve\nthe individual components in the d I/dVbspectra and the zero bias feature persists up to\n5 nm from the vortex core before the clearly split features can be observed. This is within\nthe range of spatial distributions of the Majorana zero mode reported in the literature (a\nfew nm up to 30 nm)38{41. In addition, it is important to note that Majorana zero modes in\nvortex cores have only been observed at low magnetic \feld (e.g. 0.1 T for Bi 2Se3/NbSe 238,41\nand its amplitude is expected to decay exponentially as the external \feld is increased. In\norder to con\frm whether the vortices in our system host Majorana zero modes at their\ncores, further experiments, in particular using spin-polarized tunneling measurements42, are\nclearly necessary.\nIn summary, we have provided experimental evidence for the realisation of a 2D ferromagnet-\nsuperconductor van der Waals heterostructure. More importantly, we have experimentally\ncon\frmed that the CrBr 3monolayer retains its ferromagnetic ordering with a magnetocrys-\ntalline anisotropy favoring an out-of-plane spin orientation on NbSe 2. Our DFT calculations\nshowing an induced moment in Nb from hybridization with Cr d-orbitals con\frm the im-\nprinting of magnetic order on NbSe 2from a 2D vdW magnetic insulator. Our results provide\na broader framework for employing other proximity e\u000bects to tailor materials and realize\nnovel phenomena in 2D heterostructures.10\nMETHODS\nMolecularbeam epitaxy (MBE) sample growth: The CrBr 3thin \flm was grown on a freshly\ncleaved NbSe 2substrate by compound source MBE. The anhydrous CrBr 3powder of 99 %\npurity was evaporated from a Knudsen cell. Before growth, the cell was degassed up to the\ngrowth temperature 350\u000eC until the vacuum was better than 1 \u000210\u00008mbar. The sample was\nheated by electron beam bombardment and temperatures were measured using an optical\npyrometer. The growth speed was determined by checking the coverage of the as-grown\nsamples by STM. The optimal substrate temperature for the growth of CrBr 3monolayer\n\flms was \u0018270\u000eC. Below this temperature, CrBr 3forms disordered clusters on the NbSe 2\nsurface. The NbSe 2crystal is directly mounted on a sample holder using a two component\nconducting silver epoxy which only allows us to heat the sample up to \u0018300\u000eC.\nScanning tunneling microscopy (STM) and spectroscopy (STS) measurements: After\nthe sample preparation, it was inserted into a low-temperature STM (Unisoku USM-1300)\nhoused in the same UHV system and all subsequent experiments were performed at T= 350\nmK. STM images were taken in constant-current mode. d I/dVbspectra were recorded by\nstandard lock-in detection while sweeping the sample bias in an open feedback loop con\fg-\nuration, with a peak-to-peak bias modulation of 30-50 \u0016V for a small bias range and 10 mV\nfor a lager bias range, respectively at a frequency of 707 Hz. Spectra from grid spectroscopy\nexperiments were normalized by the normal state conductance, i.e. d I/dVbat a bias voltage\ncorresponding to a few times the superconducting gap.\nSample transfer: After the sample growth, the sample was transferred to the load lock\n(\u001810\u00009mbar) and then to the directly connected glove bag. The load lock is slowly vented\nwith pure nitrogen to the glove bag atmosphere, and the sample is then transferred into the\nglove bag with a magnetic transfer rod for XPS and MOKE measurements.\nX-Ray photoelectron spectroscopy (XPS) experiments: The XPS spectra were measured\nusing a Kratos Axis Ultra system, equipped with monochromatic Al K \u000bX-ray source. All\nmeasurements were performed using an analysis area of 0 :3mm\u00020:7mm. The black curve\nin Fig 1b was measured using 80 eV pass energy and 1 eV energy step whereas the colored\ncurves were taken with 20 eV pass energy and 0.1 eV energy step. The energy calibration\nwas done using the C 1s peak at 284.8 eV.\nMagneto-optical Kerr e\u000bect (MOKE) measurements: MOKE was carried out using an11\nEvico Magnetics system based on a Zeiss Axio Imager D1 microscope with a Hamamatsu\nC4742-95 digital camera. The sample was placed in a Janis research ST-500 cold \fnger\ncryostat with optical access and cooled using liquid He. Imaging of the sample was carried\nout in a polar Kerr con\fguration. Hysteresis loops were constructed from images taken\nusing a long working distance 100x lens after background image subtraction and corrected\nfor a linear background slope due to the Faraday e\u000bect on the lens.\nDFT calculations: Calculations were performed with the DFT methodology as imple-\nmented in the periodic plane-wave basis VASP code43,44. Atomic positions and lattice pa-\nrameters were obtained by fully relaxing all structures using the spin-polarized Perdew-\nBurke-Ernzehof (PBE) functional45including Grimme's semiempirical DFT-D3 scheme for\ndispersion correction46, which is important to describe the van der Waals (vdW) interactions\nbetween the CrBr 3and the NbSe 2layers. The stacking of the layers used in our calculations\nis the same used in a previous work25. The convergence criterion of self-consistent \feld\n(SCF) computation was set to 10\u00005eV and the threshold for the largest force acting on the\natoms was set to less than 0.012 eV/ \u0017A. A vacuum layer of 12 \u0017A was added to avoid mirror\ninteractions between periodic images. Further calculations of band structures and density\nof states were realized using the hybrid Heyd-Scuseria-Ernzerhof (HSE06) functional47{49,\nwhich improves the description of the band structure as compared to the PBE functional.\nThe interactions between electrons and ions were described by PAW pseudopotentials, where\n4sand 4pshells were added explicitly as semicore states for Nb and 3 pshells for Cr. An\nenergy cuto\u000b of 550 eV is used to expand the wave functions and a systematic k-point con-\nvergence was checked, where the total energy was converged to the order of 10\u00004eV. Spin\npolarization was considered by setting an initial out-of-plane magnetization of 3 \u0016Bper Cr\natom and zero otherwise.\nACKNOWLEDGMENTS\nThis research made use of the Aalto Nanomicroscopy Center (Aalto NMC) facilities and\nwas supported by the European Research Council (ERC-2017-AdG no. 788185 \\Arti\fcial\nDesigner Materials\") and Academy of Finland (Academy professor funding no. 318995 and\n320555, Academy postdoctoral researcher no. 309975). Computing resources from the Aalto\nScience-IT project and CSC, Helsinki are gratefully acknowledged. ASF has been supported12\nby the World Premier International Research Center Initiative (WPI), MEXT, Japan.\n\u0003shawulienu.kezilebieke@aalto.\f\n1K. S. Burch, D. Mandrus, and J.-G. Park, Magnetism in two-dimensional van der Waals mate-\nrials, Nature 563, 47 (2018).\n2M. Gibertini, M. Koperski, A. F. Morpurgo, and K. S. Novoselov, Magnetic 2D materials and\nheterostructures, Nat. Nanotechnol. 14, 408 (2019).\n3C. Gong and X. Zhang, Two-dimensional magnetic crystals and emergent heterostructure de-\nvices, Science 363, 10.1126/science.aav4450 (2019).\n4C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao, W. Bao, C. Wang, Y. Wang, Z. Q.\nQiu, R. J. Cava, S. G. Louie, J. Xia, and X. Zhang, Discovery of intrinsic ferromagnetism in\ntwo-dimensional van der Waals crystals, Nature 546, 265 (2017).\n5B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, R. Cheng, K. L. Seyler, D. Zhong,\nE. Schmidgall, M. A. McGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-Herrero, and X. Xu,\nLayer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit, Nature\n546, 270 (2017).\n6S. Jiang, J. Shan, and K. F. Mak, Electric-\feld switching of two-dimensional van der Waals\nmagnets, Nat. Mater. 17, 406 (2018).\n7D. Zhong, K. L. Seyler, X. Linpeng, R. Cheng, N. Sivadas, B. Huang, E. Schmidgall,\nT. Taniguchi, K. Watanabe, M. A. McGuire, W. Yao, D. Xiao, K.-M. C. Fu, and X. Xu, Van der\nWaals engineering of ferromagnetic semiconductor heterostructures for spin and valleytronics,\nSci. Adv. 3, e1603113 (2017).\n8X. Lin, W. Yang, K. L. Wang, and W. Zhao, Two-dimensional spintronics for low-power elec-\ntronics, Nature Electronics 2, 274 (2019).\n9M.-C. Wang, C.-C. Huang, C.-H. Cheung, C.-Y. Chen, S. G. Tan, T.-W. Huang, Y. Zhao,\nY. Zhao, G. Wu, Y.-P. Feng, H.-C. Wu, and C.-R. Chang, Prospects and opportunities of 2D\nvan der Waals magnetic systems, Ann. Phys. 532, 1900452 (2020).\n10V. P. Ningrum, B. Liu, W. Wang, Y. Yin, Y. Cao, C. Zha, H. Xie, X. Jiang, Y. Sun, S. Qin,\nX. Chen, T. Qin, C. Zhu, L. Wang, and W. Huang, Recent advances in two-dimensional magnets:\nPhysics and devices towards spintronic applications, Research 2020 , 1768918 (2020).13\n11X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83,\n1057 (2011).\n12M. Sato and Y. Ando, Topological superconductors: a review, Rep. Prog. Phys. 80, 076501\n(2017).\n13C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Non-abelian anyons and\ntopological quantum computation, Rev. Mod. Phys. 80, 1083 (2008).\n14R. M. Lutchyn, E. P. A. M. Bakkers, L. P. Kouwenhoven, P. Krogstrup, C. M. Marcus, and\nY. Oreg, Majorana zero modes in superconductor-semiconductor heterostructures, Nat. Rev.\nMater. 3, 52 (2018).\n15P. Zhang, K. Yaji, T. Hashimoto, Y. Ota, T. Kondo, K. Okazaki, Z. Wang, J. Wen, G. D.\nGu, H. Ding, and S. Shin, Observation of topological superconductivity on the surface of an\niron-based superconductor, Science 360, 182 (2018).\n16D. Wang, L. Kong, P. Fan, H. Chen, S. Zhu, W. Liu, L. Cao, Y. Sun, S. Du, J. Schneeloch,\nR. Zhong, G. Gu, L. Fu, H. Ding, and H.-J. Gao, Evidence for Majorana bound states in an\niron-based superconductor, Science 362, 333 (2018).\n17S. Zhu, L. Kong, L. Cao, H. Chen, M. Papaj, S. Du, Y. Xing, W. Liu, D. Wang, C. Shen,\nF. Yang, J. Schneeloch, R. Zhong, G. Gu, L. Fu, Y.-Y. Zhang, H. Ding, and H.-J. Gao, Nearly\nquantized conductance plateau of vortex zero mode in an iron-based superconductor, Science\n367, 189 (2020).\n18Z. Wang, J. O. Rodriguez, L. Jiao, S. Howard, M. Graham, G. D. Gu, T. L. Hughes, D. K.\nMorr, and V. Madhavan, Evidence for dispersing 1D Majorana channels in an iron-based su-\nperconductor, Science 367, 104 (2020).\n19A. I. Buzdin, Proximity e\u000bects in superconductor-ferromagnet heterostructures, Rev. Mod.\nPhys. 77, 935 (2005).\n20D. Zhong, K. L. Seyler, X. Linpeng, N. P. Wilson, T. Taniguchi, K. Watanabe, M. A. McGuire,\nK.-M. C. Fu, D. Xiao, W. Yao, and X. Xu, Layer-resolved magnetic proximity e\u000bect in van der\nWaals heterostructures, Nat. Nanotechnol. 15, 187 (2020).\n21B. Huang, M. A. McGuire, A. F. May, D. Xiao, P. Jarillo-Herrero, and X. Xu, Emergent\nphenomena and proximity e\u000bects in two-dimensional magnets and heterostructures, Nat. Mater.\n(2020).14\n22W. Chen, Z. Sun, Z. Wang, L. Gu, X. Xu, S. Wu, and C. Gao, Direct observation of van der\nWaals stacking-dependent interlayer magnetism, Science 366, 983 (2019).\n23S. Liu, X. Yuan, Y. Zou, Y. Sheng, C. Huang, E. Zhang, J. Ling, Y. Liu, W. Wang, C. Zhang,\nJ. Zou, K. Wang, and F. Xiu, Wafer-scale two-dimensional ferromagnetic Fe 3GeTe 2thin \flms\ngrown by molecular beam epitaxy, npj 2D Mater. Appl. 1, 30 (2017).\n24S. Kezilebieke, M. N. Huda, P. Dreher, I. Manninen, Y. Zhou, J. Sainio, R. Mansell, M. M.\nUgeda, S. van Dijken, H.-P. Komsa, and P. Liljeroth, Electronic and magnetic characterization\nof epitaxial VSe 2monolayers on superconducting NbSe 2, Commun. Phys. 3, 116 (2020).\n25S. Kezilebieke, M. N. Huda, V. Va\u0014 no, M. Aapro, S. C. Ganguli, O. J. Silveira, S. G lodzik, A. S.\nFoster, T. Ojanen, and P. Liljeroth, Topological superconductivity in a designer ferromagnet-\nsuperconductor van der Waals heterostructure, arXiv:2002.02141 .\n26Z. Wu, J. Yu, and S. Yuan, Strain-tunable magnetic and electronic properties of monolayer\nCrI3, Phys. Chem. Chem. Phys. 21, 7750 (2019).\n27S. Jiang, L. Li, Z. Wang, K. F. Mak, and J. Shan, Controlling magnetism in 2D CrI 3by\nelectrostatic doping, Nat. Nanotechnol. 13, 549 (2018).\n28Z. Zhang, J. Shang, C. Jiang, A. Rasmita, W. Gao, and T. Yu, Direct photoluminescence\nprobing of ferromagnetism in monolayer two-dimensional CrBr 3, Nano Lett. 19, 3138 (2019).\n29L. Webster and J.-A. Yan, Strain-tunable magnetic anisotropy in monolayer CrCl 3, CrBr 3, and\nCrI3, Phys. Rev. B 98, 144411 (2018).\n30Y. Xu, X. Liu, and W. Guo, Tensile strain induced switching of magnetic states in NbSe 2and\nNbS 2single layers, Nanoscale 6, 12929 (2014).\n31J. C. Carver, G. K. Schweitzer, and T. A. Carlson, Use of xray photoelectron spectroscopy to\nstudy bonding in cr, mn, fe, and co compounds, J. Chem. Phys. 57, 973 (2020).\n32I. Pollini, Multiplet splitting of core-electron binding energies in chromium compounds, PRB\n67, 155111 (2003).\n33M. Kim, P. Kumaravadivel, J. Birkbeck, W. Kuang, S. G. Xu, D. G. Hopkinson, J. Knolle, P. A.\nMcClarty, A. I. Berdyugin, M. Ben Shalom, R. V. Gorbachev, S. J. Haigh, S. Liu, J. H. Edgar,\nK. S. Novoselov, I. V. Grigorieva, and A. K. Geim, Micromagnetometry of two-dimensional\nferromagnets, Nat. Electron. 2, 457 (2019).\n34H. H. Kim, B. Yang, S. Li, S. Jiang, C. Jin, Z. Tao, G. Nichols, F. S\fgakis, S. Zhong, C. Li,\nS. Tian, D. G. Cory, G.-X. Miao, J. Shan, K. F. Mak, H. Lei, K. Sun, L. Zhao, and A. W. Tsen,15\nEvolution of interlayer and intralayer magnetism in three atomically thin chromium trihalides,\nProc. Natl. Acad. Sci. 116, 11131 (2019).\n35Y. Noat, J. A. Silva-Guilln, T. Cren, V. Cherkez, C. Brun, S. Pons, F. Debontridder,\nD. Roditchev, W. Sacks, L. Cario, P. Ordejn, A. Garca, and E. Canadell, Quasiparticle spectra\nof 2H\u0000NbSe 2: Two-band superconductivity and the role of tunneling selectivity, Phys. Rev.\nB92, 134510 (2015).\n36H. F. Hess, R. B. Robinson, R. C. Dynes, J. M. Valles, and J. V. Waszczak, Scanning-tunneling-\nmicroscope observation of the Abrikosov \rux lattice and the density of states near and inside a\n\ruxoid, Phys. Rev. Lett. 62, 214 (1989).\n37H.-H. Sun and J.-F. Jia, Detection of majorana zero mode in the vortex, npj Quantum Materials\n2, 34 (2017).\n38J.-P. Xu, M.-X. Wang, Z. L. Liu, J.-F. Ge, X. Yang, C. Liu, Z. A. Xu, D. Guan, C. L. Gao,\nD. Qian, Y. Liu, Q.-H. Wang, F.-C. Zhang, Q.-K. Xue, and J.-F. Jia, Experimental detection of\na majorana mode in the core of a magnetic vortex inside a topological insulator-superconductor\nbi2te3=nbse 2heterostructure, Phys. Rev. Lett. 114, 017001 (2015).\n39J.-X. Yin, Z. Wu, J.-H. Wang, Z.-Y. Ye, J. Gong, X.-Y. Hou, L. Shan, A. Li, X.-J. Liang, X.-X.\nWu, J. Li, C.-S. Ting, Z.-Q. Wang, J.-P. Hu, P.-H. Hor, H. Ding, and S. H. Pan, Observation\nof a robust zero-energy bound state in iron-based superconductor fe(te,se), Nature Physics 11,\n543 (2015).\n40Q. Liu, C. Chen, T. Zhang, R. Peng, Y.-J. Yan, C.-H.-P. Wen, X. Lou, Y.-L. Huang, J.-P. Tian,\nX.-L. Dong, G.-W. Wang, W.-C. Bao, Q.-H. Wang, Z.-P. Yin, Z.-X. Zhao, and D.-L. Feng,\nRobust and clean majorana zero mode in the vortex core of high-temperature superconductor\n(li0:84fe0:16)OHFeSe, PRX 8, 041056 (2018).\n41D. Wang, L. Kong, P. Fan, H. Chen, S. Zhu, W. Liu, L. Cao, Y. Sun, S. Du, J. Schneeloch,\nR. Zhong, G. Gu, L. Fu, H. Ding, and H.-J. Gao, Evidence for majorana bound states in an\niron-based superconductor, Science , eaao1797 (2018).\n42H.-H. Sun, K.-W. Zhang, L.-H. Hu, C. Li, G.-Y. Wang, H.-Y. Ma, Z.-A. Xu, C.-L. Gao, D.-\nD. Guan, Y.-Y. Li, C. Liu, D. Qian, Y. Zhou, L. Fu, S.-C. Li, F.-C. Zhang, and J.-F. Jia,\nMajorana zero mode detected with spin selective andreev re\rection in the vortex of a topological\nsuperconductor, PRL 116, 257003 (2016).16\n43G. Kresse and J. Furthm uller, E\u000ecient iterative schemes for ab initio total-energy calculations\nusing a plane-wave basis set, Phys. Rev. B 54, 11169 (1996).\n44G. Kresse and J. Furthmller, E\u000eciency of ab-initio total energy calculations for metals and\nsemiconductors using a plane-wave basis set, Computational Materials Science 6, 15 (1996).\n45J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple,\nPhys. Rev. Lett. 77, 3865 (1996).\n46S. Grimme, J. Antony, S. Ehrlich, and H. Krieg, A consistent and accurate ab initio parametriza-\ntion of density functional dispersion correction (DFT-D) for the 94 elements H-Pu, J. Chem.\nPhys. 132, 154104 (2010).\n47J. Heyd and G. E. Scuseria, E\u000ecient hybrid density functional calculations in solids: Assessment\nof the HeydScuseriaErnzerhof screened Coulomb hybrid functional, J. Chem. Phys. 121, 1187\n(2004).\n48J. Heyd, G. E. Scuseria, and M. Ernzerhof, Erratum: Hybrid functionals based on a screened\nCoulomb potential [J. Chem. Phys. 118, 8207 (2003)], J. Chem. Phys. 124, 219906 (2006).\n49J. Heyd, G. E. Scuseria, and M. Ernzerhof, Hybrid functionals based on a screened Coulomb\npotential, J. Chem. Phys. 118, 8207 (2003)."    },    {        "title": "2011.00394v1.Tunable_magneto_optical_effect__anomalous_Hall_effect_and_anomalous_Nernst_effect_in_two_dimensional_room_temperature_ferromagnet__1T__CrTe__2_.pdf",        "content": "Tunable magneto-optical e\u000bect, anomalous Hall e\u000bect and anomalous Nernst e\u000bect in\ntwo-dimensional room-temperature ferromagnet 1T-CrTe 2\nXiuxian Yang,1, 2Xiaodong Zhou,1Wanxiang Feng,1,\u0003and Yugui Yao1\n1Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurement,\nMinistry of Education, School of Physics, Beijing Institute of Technology, Beijing 100081, China\n2Kunming Institute of Physics, Kunming 650223, China\n(Dated: November 3, 2020)\nUtilizing the \frst-principles density functional theory calculations together with group theory\nanalyses, we systematically investigate the spin order-dependent magneto-optical e\u000bect (MOE),\nanomalous Hall e\u000bect (AHE), and anomalous Nernst e\u000bect (ANE) in a recently discovered two-\ndimensional room-temperature ferromagnet 1 T-CrTe 2. We \fnd that the spin prefers an in-plane\ndirection by the magnetocrystalline anisotropy energy calculations. The MOE, AHE, and ANE\ndisplay a period of 2 \u0019=3 when the spin rotates within the atomic plane, and they are forbidden if\nthere exists a mirror plane perpendicular to the spin direction. By reorienting the spin from in-plane\nto out-of-plane direction, the MOE, AHE, and ANE are enhanced by around one order of magnitude.\nMoreover, we establish the layer-dependent magnetic properties for multilayer 1 T-CrTe 2and predict\nantiferromagnetism and ferromagnetism for bilayer and trilayer 1 T-CrTe 2, respectively. The MOE,\nAHE, and ANE are prohibited in antiferromagnetic bilayer 1 T-CrTe 2due to the existence of the\nspacetime inversion symmetry, whereas all of them are activated in ferromagnetic trilayer 1 T-CrTe 2\nand the MOE is signi\fcantly enhanced compared to monolayer 1 T-CrTe 2. Our results show that\nthe magneto-optical and anomalous transports proprieties of 1 T-CrTe 2can be e\u000bectively modulated\nby altering spin direction and layer number.\nI. INTRODUCTION\nAlthough two-dimensional (2D) materials have been\nexplored for more than a decade, the magnetic order\nrarely survives in atomically thin \flms due to thermal\n\ructuations [1]. The realization of 2D magnets is a big\nchallenge [2] and has attracted extensive attention [3, 4].\nThe 2D magnetic van der Waals (vdW) materials are es-\npecially expected to open up a wide range of possibilities\nfor spintronics [5{7]. Thanks to the improvement of theo-\nretical methods and experimental capabilities, more and\nmore 2D magnetic vdW materials have been discovered,\nwhich indicates that the \feld of 2D magnets is advancing\nrapidly [8]. In recent years, for example, tens of 2D vdW\nmaterials with stable magnetic orders have been observed\nin layered FePS 3[9, 10], Cr 2Ge2Te6[11], CrX3(X=I, Br,\nCl) [12{21], Fe 3GeTe 2[22{24],MX 2(M=V, Mn;X=Se,\nTe) [25{27], MnSn [28], PtSe 2[29], and CrTe 2[30{32].\nAimed at the applications of 2D spintronics, detect-\ning spontaneous magnetization is the primary step. The\nstandard techniques, such as superconducting quantum\ninterference device (SQUID) magnetometer and neu-\ntron scattering, are challenging to use for 2D magnetic\nvdW materials [3, 4]. Instead, the magneto-optical ef-\nfects (MOE), represented by the Kerr [33] and Fara-\nday [34] e\u000bects, are considered to be a powerful and\nnon-contact (non-destructive) probe of magnetism in 2D\nmaterials [11, 12]. The magneto-optical Kerr and Fara-\nday e\u000bects are de\fned as the rotation of the polarization\nplanes of re\rected and transmitted light beams when a\n\u0003wxfeng@bit.edu.cnlinearly polarized light hits the magnetic materials [35].\nIn condensed matter physics, the MOE and the anoma-\nlous Hall e\u000bect (AHE) [36], where the latter is character-\nized by a transverse voltage generated by a longitudinal\ncharge current in the absence of external magnetic \felds,\nare two fundamental phenomena that usually coexist in\nferromagnets and antiferromagnets. There are two dis-\ntinct contributions to the AHE, that is the extrinsic AHE\n(i.e., side jump and skew scattering) depending on the\nscattering of electron o\u000b impurities or due to disorder,\nand the intrinsic AHE expressed in term of Berry cur-\nvatures in a perfect crystal [36]. According to the Kubo\nformula [37, 38], the intrinsic anomalous Hall conduc-\ntivity (AHC) can be straightforwardly extended to the\noptical Hall conductivity, which is intimately related to\nthe magneto-optical Kerr and Faraday e\u000bects [39]. Be-\ncause of the inherent relationship between the intrinsic\nAHE and MOE, they are often studied together. More-\nover, the transverse charge current can also be generated\nby a longitudinal temperature gradient, called anoma-\nlous Nernst e\u000bect (ANE) [40], which has attracted enor-\nmous interest mainly due to its promising applications\non the thermoelectric aspects. The giant ANE has been\nrecently discovered in chiral magnets [41, 42] and topo-\nlogical semimetals [43, 44].\nThe catalog of 2D magnetic vdW materials is rich;\nhowever, the ferromagnetic candidates with high Curie\ntemperatures ( TC) are still limited, hindering enormously\nthe development of 2D spintronics. Fortunately, a 2D\ndichalcogenide with the 1 Tpolytype, 1 T-CrTe 2[see\nFigs. 1(a) and 1(b)], has been recently synthesized with\nan exceptionally high TC(>300 K) [30{32]. In this\nwork, based on the \frst-principles density functional the-\nory calculations and group theory analyses, we system-arXiv:2011.00394v1  [cond-mat.mtrl-sci]  1 Nov 20202\natically investigate the electronic, magnetic, magneto-\noptical, and anomalous charge and thermoelectric trans-\nports properties of monolayer and multilayer 1 T-CrTe 2\n(hereafter, we use CrTe 2for simpli\fcation). We \fnd\nthat monolayer CrTe 2is a ferromagnetic metal with the\nin-plane magnetization direction. By calculating mag-\nnetocrystalline anisotropy energy (MAE), the magneti-\nzation direction is \fnely identi\fed along the y-axis [see\nFig. 1(a)], and the maximal value of MAE between in-\nplane and out-of-plane magnetization directions reaches\nto 82.9\u0016eV/cell, which is much smaller than that of\nfamous 2D ferromagnets CrI 3(1.37 meV/cell) [45] and\nFe3GeTe 2(2.76 meV/cell) [46]. It indicates that the spin\ndirection of monolayer CrTe 2can be easily tuned by an\nexternal magnetic \feld. The MOE, AHE, and ANE dis-\nplay a period of 2 \u0019=3 by rotating the spin within the xy\nplane, and they disappear if there exists a mirror plane\nperpendicular to the spin direction. We then show that\nchanging the spin from in-plane to out-of-plane direction\ncan enhance the MOE, AHE, and ANE by around one\norder of magnitude. Additionally, the layer-dependent\nmagnetic properties for multilayer CrTe 2are studied, and\nantiferromagnetism and ferromagnetism for bilayer and\ntrilayer CrTe 2are predicted, respectively. For antiferro-\nmagnetic bilayer CrTe 2, the MOE, AHE, and ANE are\nfully suppressed due to the existence of the spacetime\ninversion symmetry TP(TandPare time-reversal and\nspatial inversion operations, respectively). In contrast,\nthe MOE, AHE, and ANE are activated in ferromagnetic\ntrilayer CrTe 2, and the MOE is signi\fcantly enhanced\ncompared to that of monolayer CrTe 2. Our results show\nthat the magneto-optical and anomalous transports pro-\nprieties of 2D CrTe 2are tunable by altering the magne-\ntization direction and the number of layers.\nII. METHODOLOGY\nThe \frst-principles calculations were performed by\nVienna ab initio simulation package ( vasp ) [47, 48]\nwithin the framework of density functional theory. The\nprojector augmented wave method (PAW) [49] was\nemployed to model the ion cores and the exchange-\ncorrelation functional of generalized gradient approxima-\ntion (GGA) with the Perdew-Burke-Ernzerhof parame-\nterization (PBE) [50] to simulate the valence electrons.\nSpin-orbit coupling was included in the calculations for\nthe MAE, MOE, AHE, and ANE. The plane-wave cut-\no\u000b energy was set to be 500 eV. The structures were\nrelaxed until the maximum force on each atom is less\nthan 0:0001 eV/ \u0017A and the energy convergence criterion\nis 10\u00007eV. The Brillouin zone integration was carried\nout by 16\u000216\u00021k-points sampling. A vacuum layer\nwith the thickness of at least 15 \u0017A was used to avoid the\ninteractions between adjacent layers and the vdW cor-\nrection was adopted by DFT-D2 method in multilayer\nstructures. The optical conductivity, AHC, and ANC are\nscaled by a factor of Z=d 0to exclude the vacuum region,whereZis the cell length normal to the atomic plane\nandd0= 6.23 \u0017A, 12.46 \u0017A, and 18.69 \u0017A are the e\u000bective\nthicknesses of monolayer, bilayer, and trilayer CrTe 2, re-\nspectively. Since the dorbitals of Cr atom are not fully\n\flled, the LDA+U method [51, 52] was used to account\nfor the Coulomb correlation with U = 2.0 eV [53].\nTo obtain the MOE, such as the Kerr and Faraday\nspectra, the optical conductivity should be primarily cal-\nculated. Here, we constructed the maximally localized\nWannier functions (MLWFs) in a non-self-consistent pro-\ncess by projecting onto s,p, anddorbitals of Cr atom as\nwell as onto sandporbitals of Te atom, using a uniform\nk-mesh of 16\u000216\u00021 points in conjunction with the wan-\nnier90 package [54]. The optical conductivity was then\ncalculated by integrating the dipole matrix elements (un-\nder the MLWFs basis) over the entire Brillouin zone using\na very dense k-points of 300\u0002300\u00021. The absorptive\nparts of optical conductivity are given by [37, 38, 55],\n\u001b1\nxx(!) =\u0015\n!X\nk;jj0[j\u0005+\njj0j2+j\u0005\u0000\njj0j2]\u000e(!\u0000!jj0);(1)\n\u001b2\nxy(!) =\u0015\n!X\nk;jj0[j\u0005+\njj0j2\u0000j\u0005\u0000\njj0j2]\u000e(!\u0000!jj0);(2)\nwhere the superscripts 1 and 2 indicate the real and imag-\ninary parts, \u0015=\u0019e2\n2~m2Vis a material speci\fc constant ( e\nandmare the charge and mass of an electron, ~is re-\nduced Planck constant, and Vis volume of unit cell),\njandj0denote occupied and unoccupied states at the\nsamek-point, \u0005\u0006\njj0are the dipole matrix elements rel-\nevant to right-circularly (+) and left-circularly ( \u0000) po-\nlarized lights, ~!is the photon energy, ~!jj0is the en-\nergy di\u000berence between jandj0states. Utilizing the\nKramers-Kronig transformation, the dispersive parts can\nbe obtained as,\n\u001b2\nxx(!) =\u00002\n\u0019PZ1\n0\u001b1\nxx(!0)\n!02\u0000!2d!0; (3)\n\u001b1\nxy(!) =2\n\u0019PZ1\n0!\u001b2\nxy(!0)\n!02\u0000!2d!0; (4)\nwherePis the principal integral.\nThe Kerr e\u000bect is characterized by the rotation angle\n(\u0012K) and ellipticity ( \"K), which are usually combined into\nthe complex Kerr angle,\n\u001eK=\u0012K+i\"K=i2!d\nc\u001bxy\n\u001bsxx; (5)\nwherecis the speed of light in vacuum, dis the thin-\n\flm thickness, and \u001bs\nxxis the optical conductivity of a\nnonmagnetic substrate. Similarly, the complex Faraday\nangle is given by,\n\u001eF=\u0012F+i\"F=i!d\n2c(n+\u0000n\u0000); (6)\nwheren2\n\u0006= 1 +4\u0019i\n!(\u001bxx\u0006i\u001bxy) are eigenvalues of di-\nelectric tensor. By considering the fact that j4\u0019i\n!(\u001bxx\u00063\n03060901\n201\n501\n802\n102\n402\n7030033003060901\n201\n501\n802\n102\n402\n70300330\nyx\ny1\n0 µeV(d)TeTe(a)y\n/s106\n (degree)abx\nCr(b)z\n8\n2.9 µeVyz\n(c)/s113\n (degree)\nFIG. 1. (Color online) (a,b) Top and side views of monolayer 1 T-CrTe 2. The blue spheres represent Cr atoms, whereas dark-\ngray and silver-white spheres represent Te atoms in the upper and lower sublayers. The pink dashed lines draw up the 2D\nprimitive cell, and the red arrows indicate the directions of spin magnetic moments. The top and bottom panels in (b) present\nthe spin directions along the y- andz-axis, respectively. (c,d) The magnetocrystalline anisotropy energy of monolayer 1 T-CrTe 2\nby rotating the spin magnetic moment within the yzandxyplanes. The spin along the y-axis is set to be the reference state.\ni\u001bxy)j\u001c1, the complex Faraday angle can be approxi-\nmately written as,\n\u001eF=\u0012F+i\"F'\u00002\u0019d\nc\u001bxy; (7)\nFrom Eqs. (5) and (7), one can see that the o\u000b-diagonal\nelements of optical conductivity ( \u001bxy), also known as the\noptical Hall conductivity, is determinative to both Kerr\nand Faraday e\u000bects. It should be mentioned here that\nEqs. (5){(7) are the expressions for 2D systems with a\npolar geometry [56], that is, the incident light propagates\nalong the\u0000zdirection.\nPhysically speaking, the MOE is closely related to the\nAHE. For example, the dc limit of the real part of the\no\u000b-diagonal element of optical conductivity, i.e., \u001b1\nxy(!!\n0), is nothing but the intrinsic AHC, which can also be\ncalculated from the Berry-phase formula [57],\n\u001bA\nxy=\u0000e2\n~VX\nn;kfnk\nn\nxy(k); (8)\nwheren,k, andfnkare band index, crystal momen-\ntum, and Fermi-Dirac distribution function, respectively.\nn\nxy(k) is the band-resolved Berry curvature, given by,\n\nn\nxy(k) =\u0000X\nn06=n2Im[h nkj^vxjh n0kih n0kj^vyjh nki]\n(\"nk\u0000\"n0k)2\n(9)\nwhere ^vx;yis the velocity operator along the xorydi-\nrection, nkand\"nkare the eigenvector and eigenvalue\nat band index nand crystal momentum k, respectively.\nThe intrinsic anomalous Nernst conductivity (ANC) can\nbe written as [58, 59]\n\u000bA\nxy=e\n~TVX\nn;k\nn\nxy(k)\u0002[(\"nk\u0000\u0016)fnk\n+kBTln(1 +e\u0000(\"nk\u0000\u0016)=kBT)] (10)\nwhereT,\u0016, andkBis temperature, chemical potential,\nand Boltzmann constant, respectively. Thus, the ANC\ncan be related to the AHC by the Mott formula [58].4\n-2-1012-\n2-1012-50 5 0\n5 E (eV) ↑ \n↓Γ\nK M Γ (a)S\n || yE (eV)Γ\nK M Γ (b) ↑ \nTotal \nCr 3d \nTe 5p↓ \nFIG. 2. (Color online) (a) The spin-polarized band struc-\nture and density of states (in the unit of states/eV/cell)\nfor monolayer 1 T-CrTe 2. (b) The relativistic band struc-\nture and orbital-decomposed density of states (in the unit\nof states/eV/cell) for monolayer 1 T-CrTe 2when the spin is\nalong they-axis.\nIII. RESULTS AND DISCUSSION\nIn this section, we successively present the results of\nmonolayers and multilayer CrTe 2. The magnetic ground\nstates are \frst established by calculating the MAE. The\ncorresponding electronic band structures are explicitly\ncalculated. The magnetic group theory is then used to\ndetermine the nonzero elements of optical conductivity,\nwhich is the critical ingredient to evaluate the magneto-\noptical Kerr and Faraday spectra. Finally, the anomalous\nHall and anomalous Nernst conductivities are evaluated\nby using the Berry-phase formulas. The dependence of\nthe MOE, AHE, and ANE on the magnetization direction\nand layer number will be detailedly discussed.\nA. Monolayer CrTe 2\n1. Crystal, magnetic, and electronic structures\nThe top and side views of monolayer CrTe 2(space\ngroup P 3m1, No. 164) are depicted in Figs. 1(a) and 1(b).\nEach primitive cell contains one Chromium (Cr) atomand two Tellurium (Te) atoms, forming a sandwich struc-\nture Te-Cr-Te. The optimized lattice constant of mono-\nlayer CrTe 2isa= 3:722\u0017A.\nTo con\frm the magnetic ground state, we compared to-\ntal energies among the nonmagnetic, antiferromagnetic,\nand ferromagnetic states using a supercell of 2 \u00022\u00021,\nand the results show that ferromagnetic state is more\nstable than nonmagnetic and antiferromagnetic states\nby 2:90 eV and 55 :32 meV, respectively. Additionally,\nthe magnetic ground state can be determined via the\nsuper-exchange mechanism [60{62]. The magnetic ex-\nchange interactions depend on the \flling of the dorbitals\nof the cations and on the angle formed by the chemical\nbonds connecting the ligand and magnetic atoms, and\nparticularly, when the angle equals to 90\u000ethe ferromag-\nnetic interactions are optimal. In the case of CrTe 2, the\nbond angle between Cr-Te-Cr is 87\u000e, which accounts for\nferromagnetic interactions. Furthermore, the MAE, de-\n\fned as MAE( \u0012;') =E(\u0012;')\u0000E(\u0012= 90\u000e;'= 90\u000e)\n[here,E(\u0012;') is the total energy when the spin mag-\nnetic moment ( S) orients to the polar angle \u0012and az-\nimuthal angle '], is computed by rotating the spin mag-\nnetic moment on the xyandyzplanes, respectively, as\nshown in Figs. 1(c) and 1(d). The positive values of\nMAE suggest a preferred magnetization along the y-axis\n(\u0012= 90\u000e;'= 90\u000e) rather than along other directions.\nFigure 1(c) shows that the out-of-plane magnetization\n(along thezaxis) is not prior due to the positive MAE\nof 82:9\u0016eV/cell. Figure 1(d) further indicates a small in-\nplane magnetocrystalline anisotropy (0 \u0016eV/cell\u0014MAE\n\u001410\u0016eV/cell), in good agreement with experimental ob-\nservation [31, 32]. Therefore, the magnetic ground state\nof the system is con\frmed and the spin magnetic moment\nshould be along the y-axis, i.e.,S(\u0012= 90\u000e;'= 90\u000e) or\nSky[see top panel of Fig. 1(b)], which is consistent\nwith previously theoretical calculation [63]. Thus, the\nspin direction of CrTe 2can be easily tuned by applying\nan external magnetic \feld as the maximal MAE of CrTe 2\n(82:9\u0016eV/cell) is much smaller than that of Fe 3GeTe 2\n(2:76 meV/cell) [46], which has been realized experimen-\ntally. This provides a technical basis for us to reorient\nthe spin magnetic moment from in-plane to out-of-plane\ndirection [see bottom panel of Fig. 1(b)].\nWe then discuss the electronic structures of monolayer\nCrTe 2. Fig. 2(a) plots the spin-polarized band structures\nand density of states, in which the red and blue lines rep-\nresent the spin-up ( \") and spin-down ( #) bands, respec-\ntively. The spin-polarized band structures combined with\ndensity of states clearly show that monolayer CrTe 2is a\nferromagnetic metal. After including spin-orbit coupling,\nthe relativistic band structures and orbital-decomposed\ndensity of states with the magnetization Skyare illus-\ntrated in Fig. 2(b). The band structure is good consistent\nwith a recently theoretical calculation [64]. For the den-\nsity of states, we only present the dominant components,\ni.e., the 3dorbitals of Cr atom (the orange pattern) and\n5porbitals of Te atoms (the green pattern), which have\nnearly equal contributions around the Fermi energy.5\nTABLE I. The magnetic space group (MSG) and magnetic point group (MPG) of monolayer 1 T-CrTe 2as a function of azimuthal\n(') and polar ( \u0012) angles when the spin rotates within the xy(\u0012=\u0019=2, 0\u0014'\u0014\u0019) andyz(0\u0014\u0012\u0014\u0019,'=\u0019=2) planes.\n0\u000e15\u000e30\u000e45\u000e60\u000e75\u000e90\u000e105\u000e120\u000e135\u000e150\u000e165\u000e180\u000e\nMSG(')C2=m P \u00161C20=m0P\u00161C2=m P \u00161C20=m0P\u00161C2=m P \u00161C20=m0P\u00161C2=m\nMPG(') 2=m \u00161 20=m0 \u00161 2=m \u00161 20=m0 \u00161 2=m \u00161 20=m0 \u00161 2=m\nMSG(\u0012)P\u00163m01C20=m0C20=m0C20=m0C20=m0C20=m0C20=m0C20=m0C20=m0C20=m0C20=m0C20=m0P\u00163m01\nMPG(\u0012)\u001631m020=m020=m020=m020=m020=m020=m020=m020=m020=m020=m020=m0 \u001631m0\n2. Magnetic group theory\nThe group theory is a powerful tool for identifying the\nnonvanishing elements of the optical Hall conductivity,\nwhich is the key factor in predicting the MOE. Addi-\ntionally, the AHE and ANE have the same symmetry\nrequirements with the MOE due to their physical rela-\ntions [refer to Eqs. (1){(10)]. Hence, we take the optical\nHall conductivity as an example, and the results of sym-\nmetry analyses are applicable to the MOE, AHE, and\nANE. The magnetic space and point groups for mono-\nlayer CrTe 2are calculated by using the isotropy soft-\nware [65]. Table I lists the results when the spin rotates\nwithin thexyandyzplanes. Since the optical Hall con-\nductivity is translationally invariant, it is su\u000ecient to\nrestrict the analysis to magnetic point group. Moreover,\nthe vector-form notation of the optical Hall conductiv-\nity, given by \u001b(!) = [\u001bx;\u001by;\u001bz] = [\u001byz;\u001bzx;\u001bxy], is used\nfor convenience as it can be regarded as a pseudovector,\njust like spin. Thus, for a 2D system, there always has\n\u001bx=\u001by= 0 and only \u001bzis potentially nonzero.\nLet us start with the situation that rotating the spin\nwithin the xyplane. The magnetic point group has a\nperiod of\u0019=3: 2=m!\u00161!20=m0!\u00161!2=m, and three\nnonrepetitive elements are 2 =m,\u00161, and 20=m0. First, the\ngroup 2=m(when'=n\u0019=3 withn2N) has a mirror\nplane that is parallel to the z-axis and is perpendicular to\nthe spin direction. Such a mirror operation reverses the\nsign of\u001bz, and thus indicating \u001bz= 0. It results in the\nvanishing optical Hall conductivity, i.e., \u001b(!) = [0;0;0].\nOn the other hand, all mirror symmetries are broken if\n'6=n\u0019=3. The group 20=m0contains a combined sym-\nmetryTM, whereTis the time-reversal symmetry and\nMis a mirror plane that parallels to both the z-axis and\nspin direction. Both TandMoperations reverse the\nsign of\u001bz, and hence \u001bzis even underTM symmetry.\nIt gives rise to the nonvanishing optical Hall conductivity,\n\u001b(!) = [0;0;\u001bz]. Finally, for the group \u00161 =fE;Pg, none\nof its elements (unit operation Eand spatial inversion P)\ncan a\u000bect\u001bz, and hence the optical Hall conductivity are\nabsolutely allowed.\nWe next turn to the case that the spin lies within the\nyzplane. The evolution of magnetic point group exhibits\na period of \u0019, and only two groups \u001631m0and 20=m0are\nneeded to analyze. If \u0012= 0 or\u0019, the group \u001631m0contains\nthreeTM symmetries with Mkz. The nonvanishingoptical Hall conductivity can be expected since any one\nof the threeTM symmetries a\u000bords \u001bz6= 0. Once the\nspin cants away from the z-axis (\u00126= 0) or from the \u0000z-\naxis (\u00126=\u0019), the magnetic point group changes to be\n20=m0, in which only one of the three TM symmetry\nleaves (here,Mis just the yzplane) but still ensures\n\u001bz6= 0. To summarize, the optical Hall conductivity is\nnonzero when the spin lies within the yzplane, that is,\n\u001b(!) = [0;0;\u001bz].\n3. Optical and magneto-optical properties\nAfter obtaining the electronic structures and magnetic\ngroups of monolayer CrTe 2with di\u000berent magnetization\ndirections, we now focus on the optical conductivity,\nwhich is prerequisite to evaluate the MOE.\nWe \frst discuss the results of in-plane magnetization\nwhenS(90\u000e;0\u000e) (Skx),S(90\u000e;30\u000e), andS(90\u000e;90\u000e)\n(Sky), shown in Figs. 3(a-d). According to Eqs. (1)\nand (2), the absorptive parts of optical conductivity, \u001b1\nxx\nand\u001b2\nxy, have direct physical interpretations, which mea-\nsures the average and di\u000berence in absorptions of the left-\nand right-circularly polarized light, respectively. The \u001b1\nxx\nplotted in Fig. 3(a) exhibits two sharp absorption peaks\nat 0.6 and 2.2 eV. Since \u001b1\nxxis directly related to the\ninterband transition probability and jointed density of\nstates, it is not a\u000bected by the spin direction, similarly\nto Mn 3XN (X= Ga, Zn, Ag, or Ni) [39]. On the other\nhand, the\u001b2\nxyplotted in Fig. 3(d) oscillates drastically\nin the low-energy region and tends to zero above 5.5 eV.\nThe positive and negative values of \u001b2\nxyindicate that\nthe interband transitions are dominated by the excita-\ntions caused by the left- and right-circularly polarized\nlight, respectively. The signs of \u001b2\nxyfor the states of\nS(90\u000e;30\u000e) andS(90\u000e;90\u000e) are opposite, which has the\nsame physical mechanism of the intrinsic AHC for mono-\nlayer LaCl [66]. It should be further noticed that for the\nstate ofS(90\u000e;0\u000e),\u001b2\nxyis suppressed due to the pres-\nence of the mirror plane Mthat is perpendicular to S,\nwhich is consistent with the previous group theory anal-\nyses. Utilizing the Kramers-Kronig transformation, the\ndispersive parts of optical conductivity, \u001b2\nxxand\u001b1\nxy, can\nbe obtained from the corresponding absorptive parts ac-\ncording to Eqs. (3) and (4). The dependence of \u001b2\nxxand\n\u001b1\nxyon the magnetization direction, featured in Figs. 3(b)6\n02 4 6 050\n2 4 6 -20240\n2 4 6 -0.050.000.050\n2 4 6 -0.10.00.10\n2 4 6 050\n2 4 6 -20240\n2 4 6 -0.20.00.20\n2 4 6 0.00.5Photon Energy (eV)/s1151x\nx (1015 s-1)S\n || xS\n (90°, 30°)S\n || y(a)P\nhoton Energy (eV)/s1152x\nx (1015 s-1)(b)P\nhoton Energy (eV)/s1151x\ny (1015 s-1)(c)P\nhoton Energy (eV)/s1152x\ny (1015 s-1)(d)P\nhoton Energy (eV)/s1151x\nx (1015 s-1)S\n || yS\n (45°, 90°)S\n || z(e)P\nhoton Energy (eV)/s1152x\nx (1015 s-1)(f)/s115\n1x\ny (1015 s-1)P\nhoton Energy (eV)(g)/s115\n2x\ny (1015 s-1)P\nhoton Energy (eV)(h)\nFIG. 3. (Color online) The real diagonal (a,e), imaginary diagonal (b,f), real o\u000b-diagonal (c,g), and imaginary o\u000b-diagonal\n(d,h) elements of optical conductivity for monolayer 1 T-CrTe 2with in-plane and out-of-plane magnetization, respectively. For\na better comparison, the curves when spin points along y-axis (Sky) are replotted in (e-h).\n02 4 6 -0.20.00.20\n2 4 6 -0.50.00.50\n2 4 6 -2020\n2 4 6 -2020\n6 0120180-0.30.00.30\n6 0120180-0.60.00.60\n6 0120180-3030\n6 0120180-404Photon Energy (eV)/s113K (deg) \nS || x \nS (90°, 30°) \nS || y(a)P\nhoton Energy (eV)/s101K (deg)(b)P\nhoton Energy (eV)/s113F (105 deg/cm)(c)P\nhoton Energy (eV)/s101F (105 deg/cm)(d)(\ne)/s113 K (deg)/s106\n (deg)0.70 eV(f)/s101K (deg)/s106\n (deg)0.60 eV(g)0\n.79 eV/s113F (105 deg/cm)/s106\n (deg)(h)0\n.62 eV/s101F (105 deg/cm)/s106\n (deg)\nFIG. 4. (Color online) The Kerr rotation angle \u0012K(a), Kerr ellipticity \"K(b), Faraday rotation angle \u0012F(c), and Faraday\nellipticity\"F(d) for monolayer 1 T-CrTe 2with in-plane magnetization. (e-h) The Kerr and Faraday rotation angles and\nellipticities as a function of azimuthal angle 'at selected photon energies.\nand 3(c), resemble that of \u001b1\nxxand\u001b2\nxy.\nThen, we proceed to the out-of-plane magnetization\nby considering the spin within the yzplane, for example\nS(0\u000e;90\u000e) (Skz) andS(45\u000e;90\u000e). As shown in Figs. 3(e)\nand 3(f),\u001b1\nxxhas two absorption peaks at 0.6 and 2.2 eV,\nand meanwhile \u001b2\nxxpresents two valleys at 0.5 and 2.0\neV. This is identical to the situation of in-plane magne-\ntization and further indicates that the diagonal elements\nof optical conductivity are not a\u000bected by the spin di-\nrection. In contrast, the o\u000b-diagonal elements of optical\nconductivity, \u001b1\nxyand\u001b2\nxy[see Figs. 3(g) and 3(h)], obvi-\nously depend on the spin direction. \u001b1\nxyand\u001b2\nxyoscillate\nas a function of photon energy with di\u000berent spin direc-tions and reach the maximal values when the spin points\ntowards the z-axis. It is important to notice that the o\u000b-\ndiagonal elements of optical conductivity with the out-of-\nplane magnetization are enhanced by about one order of\nmagnitude compared to that of in-plane magnetization.\nNow, we present the magneto-optical Kerr and Fara-\nday spectra with in-plane and out-of-plane magnetiza-\ntion, as shown in Figs. 4 and 5, respectively. The Kerr\nand Faraday spectra are rather similar to that of \u001bxy,\nand the reason can be simply attributed to their close\nrelationships [refer to Eqs. (5) and (7)]. For the in-plane\nmagnetization, the Kerr and Faraday angles are vanish-\ning if the spin points along '=n\u0019=3, for example '= 0\u000e7\n02 4 6 -1010\n2 4 6 -2020\n2 4 6 -20-100100\n2 4 6 -20-100100\n9 0180270360-1010\n9 0180270360-2020\n9 0180270360-20-10010200\n9 0180270360-20-1001020Photon Energy (eV)/s113K (deg) \nS || y \nS (45°, 90°) \nS || z(a)P\nhoton Energy (eV)/s101K (deg)(b)/s113\nF (105 deg/cm)P\nhoton Energy (eV)(c)/s101\nF (105 deg/cm)P\nhoton Energy (eV)(d)/s113K (deg)/s113\n (deg)0.66 eV(e)( f)0\n.88 eV/s101K (deg)/s113\n (deg)(g)0\n.82 eV/s113F (105 deg/cm)/s113\n (deg)(h)0\n.91 eV/s101F (105 deg/cm)/s113\n (deg)\nFIG. 5. (Color online) The Kerr rotation angle \u0012K(a), Kerr ellipticity \"K(b), Faraday rotation angle \u0012F(c), and Faraday\nellipticity\"F(d) for monolayer 1 T-CrTe 2with out-of-plane magnetization. For a better comparison, the curves for the spin\npointing along the y-axis (Sky) are also plotted in (a-d). (e-h) The Kerr and Faraday rotation angles and ellipticities as a\nfunction of polar angle \u0012at selected photon energies.\n09 01 80270360-4-20240\n9 01 80270360-10010-10 -4-2024-\n10 -10-505101506 01 201 80-1010\n6 01 201 80-505(c)/s115Ax\ny (×103 S/cm)/s113\n (deg)-1.03 eVT\n = 300K(f)/s97\nAx\ny (A/mK)/s113\n (deg)-1.10 eV/s115Ax\ny (×103 S/cm)E\n (eV) S || x \nS || y \nS || z(a) \nS || x \nS || y \nS || zT\n = 300K/s97Ax\ny (A/mK)E\n (eV)(d)-1.13 eV/s115Ax\ny (×103 S/cm)/s106\n (deg)(b)T\n = 300K-1.34 eV/s97Ax\ny (A/mK)/s106\n (deg)(e)\nFIG. 6. (Color online) (a,d) The intrinsic anomalous Hall ( \u001bA\nxy) and anomalous Nernst ( \u000bA\nxy) conductivity for monolayer 1 T-\nCrTe 2as a function of the Fermi energy when the spin points along the x-,y-, andz-axis.\u000bA\nxyis calculated at the temperature\nof 300 K. The black arrows indicates the maximal values of \u001bA\nxyand\u000bA\nxy. (b,e)\u001bA\nxyand\u000bA\nxyas a function of azimuthal angle '\nwhen the Fermi energies are set to be -1.13 and -1.34 eV, respectively. (c,f) \u001bA\nxyand\u000bA\nxyas a function of polar angle \u0012when\nthe Fermi energies are set to be -1.03 and -1.10 eV, respectively.\n[see Figs. 4(a-d)], due to the symmetry restriction. When\nthe spin rotates to '=\u0019=6+n\u0019=3, the Kerr and Faraday\nangles reach their maximums, that is, \u0012max\nK= 0:24 deg\nand\u0012max\nF= 3:00\u0002105deg/cm at the photon energies of\n0.70 and 0.79 eV, respectively. The \u0012max\nKof CrTe 2is com-\nparable with the Kerr rotation angles of monolayer CrI 3\n(0.286 deg) [12] and of blue phosphorene (0.12 deg) [67].\nMoreover, Figs. 4(e-h) show that the maximal Kerr andFaraday angles of monolayer CrTe 2exhibits a period of\n2\u0019=3 when the spin rotates within the xyplane.\nOn the other hand, the Kerr and Faraday spectra with\nthe out-of-plane magnetization are illustrated in Fig. 5.\nInheriting from the o\u000b-diagonal elements of optical con-\nductivity, the Kerr and Faraday spectra with out-of-plane\nmagnetization are signi\fcantly stronger than that with\nin-plane magnetization [Figs. 5(a-d)]. The maximal Kerr8\n(a)( b)( c)( d)(\ne)( f)( g)( h)yzz\ny\nCrTeT\ne\nFIG. 7. (Color online) Side views of bilayer (a-d) and trilayer (e-h) 1 T-CrTe 2with in-plane and out-of-plane ferromagnetic and\nantiferromagnetic con\fgurations. The red arrows label the spin directions.\nand Faraday rotation angles appear when \u0012= 0\u000e(Skz),\nthat is,\u0012max\nK =\u00001:34 deg and \u0012max\nF =\u000017:30\u0002105\ndeg/cm at the photon energies of 0.66 and 0.82 eV, re-\nspectively. Moreover, the Kerr and Faraday angles have\na period of 2 \u0019as a function of polar angle \u0012, as shown in\nFigs. 5(e-h), demonstrating again that the MOE can be\ne\u000bectively modulated by tuning the spin direction.\n4. Anomalous Hall and anomalous Nernst e\u000bects\nAs mentioned above, the dc limit of real o\u000b-diagonal el-\nement of optical conductivity, i.e., \u001b1\nxy(!!0), is nothing\nbut the AHC ( \u001bA\nxy), which can be alternatively evaluated\nby integrating the Berry curvature over the entire Bril-\nlouin zone [see Eq. (8)] [57]. Figure 6(a) plots the AHC\nas a function of the Fermi energy when the spin points\nalong thex-,y-, andz-axis.\u001bA\nxyis vanishing when Skx\nand turns to appear if Skyandz, indicating the same\nsymmetry requirements for the MOE. In analogy to the\nKerr and Faraday angles, the \u001bA\nxywith out-of-plane mag-\nnetization ( Skz) is signi\fcantly larger than that with in-\nplane magnetization ( Sky). Thus, at the actual Fermi\nenergy, the \u001bA\nxywith both in-plane and out-of-plane mag-\nnetization are relatively small, which is adverse to mea-\nsure experimentally. Nevertheless, the pronounced peaks\nof AHC arise after appropriate holes are introduced. Forexample,\u001bA\nxycan increase up to -4539.23 S/cm at -1.03\neV whenSkzand up to 1168.12 S/cm at -1.13 eV when\nSky, respectively. When the spin rotates within the xy\nandyzplanes,\u001bA\nxyexhibits the periods of 2 \u0019=3 and 2\u0019,\nrespectively [depicted in Figs. 6(b) and 6(c)], which are\nidentical to the behaviors of Kerr and Faraday angles.\nThe ANE, being regarded as the thermoelectric coun-\nterpart of the AHE, is a celebrated e\u000bect from the realm\nof spin caloritronics [68, 69]. The conclusions of symme-\ntry analyses for the AHE are also applicable to the ANE,\ncomparing with Eqs. (8) and (10). That is, the ANC \u000bA\nxy\nis forbidden when the spin is along '=n\u0019=3, e.g.,Skx,\nand turns to be nonzero if Skyorz, as clearly shown in\nFig. 6(d). Due to the high Curie temperature of CrTe 2\n(TC>300 K) [30{32], the \frst-principles calculations\nof the ANC are carried out at the room-temperature of\n300 K. Similarly to the AHC \u001bA\nxy, the ANC \u000bA\nxywith\nout-of-plane magnetization ( Skz) is evidently larger\nthan that with in-plane magnetization ( Sky). For both\nin-plane and out-of-plane magnetization, \u000bA\nxyare almost\nzero at the actual Fermi energy and give rise to pro-\nnounced peaks by hole doping. For example, \u000bA\nxyreaches\nup to 13.61 A/mK at -1.10 eV when Skzand up to 5.30\nA/mK at -1.34 eV when Sky, respectively. Moreover,\n\u000bA\nxydisplays a period of 2 \u0019=3 (2\u0019) when the spin rotates\nwithin thexy(yz) plane, as shown in Figs. 6(e) and 6(f),\njust like the AHC as well the Kerr and Faraday angles.9\n-2-1012-2-1012E (eV) S || yT\nrilayerΓ\nK M Γ (b)E  (eV) S || yB\nilayerΓ\nK M Γ (a)\nFIG. 8. (Color online) Relativistic band structures of bilayer\nand trilayer 1 T-CrTe 2when the spin is along the y-axis.\nB. Multilayer CrTe 2\nIn this subsection, we shall discuss the layer-dependent\nmagnetic properties as well as the MOE, AHE, and ANE\nof multilayer CrTe 2. The bilayer and trilayer CrTe 2with\nthe AA-stacking pattern, which could be directly exfo-\nliated from the bulk structure, are considered here. To\ndetermine the magnetic ground states, we calculate the\ntotal energy ( Etot) of in-plane and out-of-plane ferro-\nmagnetic and antiferromagnetic structures, as depicted\nTABLE II. The total energy Etot(in the unit of eV) per unit\ncell of bilayer (BL) and trilayer (TL) 1 T-CrTe 2with in-plane\nand out-of-plane ferromagnetic (i-FM and o-FM) and antifer-\nromagnetic (i-AFM and o-AFM) con\fgurations. The super-\nscriptsa\u0000hcorresponds to the magnetic structures presented\nin Fig. 7. The energies of nonmagnetic (NM) structures are\ngiven for the reference. The relaxed lattice constant a(in the\nunit of \u0017A) is also listed.\ni-FMa;eo-FMc;gi-AFMb;fo-AFMd;hNM\nBLEtot -31.996 -31.983 -32.021 -32.017 -28.885\na 3.763 3.754 3.784 3.786 3.469\nTLEtot -48.216 -48.208 -48.238 -48.237 -43.528\na 3.778 3.778 3.791 3.793 3.478in Fig. 7. It should be stressed here that all the trilayer\nCrTe 2are actually ferromagnetic with \fnite net magne-\ntization, while we mention the \\antiferromagnetic\" tri-\nlayer structures [Figs. 7(f,h)] just because of the inter-\nlayer antiferromagnetic order. The energy results are\nsummarized in Tab. II, from which one can \fnd that\nthe antiferromagnetic structures for both bilayer and tri-\nlayer CrTe 2[Figs. 7(b,d) and 7(f,h)] are energetically fa-\nvorable. Thus, the in-plane antiferromagnetic structures\n[Figs. 7(b) and 7(f)] are most stable with the slightly\nlower energies of \u00184 and\u00181 meV/cell than the out-of-\nplane ones for bilayer and trilayer, respectively. In the\nfollowing, we only focus on the bilayer and trilayer CrTe 2\nwith the in-plane antiferromagnetic structures. The elec-\ntronic band structures plotted in Fig. 8 demonstrate the\nmetallic nature of both bilayer and trilayer CrTe 2.\nUsing the group theory, we analyze whether the MOE,\nAHE, and ANE can exist in bilayer and trilayer CrTe 2.\nThe magnetic point group of in-plane antiferromagnetic\nbilayer [Fig. 7(b)] is 2 =m0, which contains the space-\ntime inversion symmetry TPthat forbids any signals of\nmagneto-optical responses as well as anomalous charge\nand thermoelectric transports. In contrast, the mag-\nnetic point group of in-plane antiferromagnetic trilayer\n[Fig. 7(f)] is the same as that of monolayer CrTe 2, i.e.,\n20=m0, which allows the presence of the MOE, AHE, and\nANE.\nThe layer number can in\ruence the MOE, AHE, and\nANE of multilayer CrTe 2. The magneto-optical Kerr and\nFaraday rotation angles of in-plane antiferromagnetic bi-\nlayer and trilayer CrTe 2are plotted in Figs. 9(a) and 9(b),\nin which the results of monolayer CrTe 2are given for\ncomparison. As expected, the \u0012Kand\u0012Fof bilayer struc-\nture are zero due to the presence of TPsymmetry. For\nthe trilayer structure, the largest \u0012Kand\u0012Fare -1.76\ndeg and 4.60\u0002105deg/cm at the photon energies of 0.38\nand 3.04 eV, respectively. One can \fnd that the Kerr\nand Faraday e\u000bects of trilayer structure are generally\nstronger than that of monolayer structure. Moreover, the\nAHC and ANC of monolayer, bilayer, and trilayer CrTe 2\nwith in-plane magnetization are presented in Figs. 9(c)\nand 9(d), respectively. Clearly, the \u001bA\nxyand\u000bA\nxyof bilayer\nstructure are vanishing due to the symmetry restriction.\nAlthough the \u001bA\nxyand\u000bA\nxyof trilayer structure are very\nsmall at the actual Fermi energy, they can be signi\fcantly\nenhanced by hole doping. The largest \u001bA\nxyand\u000bA\nxyare\n1147.03 S/cm at -1.18 eV and -4.81 A/mK at -1.12 eV,\nrespectively. The AHE and ANE of trilayer structure are\nslightly smaller than that of monolayer structure.\nIV. SUMMARY\nIn summary, using the \frst-principles density func-\ntional theory calculations and group theory analyses, we\nhave systematically investigated the electronic, magnetic,\nmagneto-optical, anomalous charge and thermoelectric\ntransport properties of monolayer, bilayer, and trilayer10\n02 4 6 -2-1010\n2 4 6 -404-10 -2-101-\n10 -10-50510Photon Energy (eV)/s113K (deg) \nMonolayer \nBilayer \nTrilayer(a)(\nb)/s113 F (105 deg/cm)P\nhoton Energy (eV)(c)E\n (eV)/s115Ax\ny (×103 S/cm)(\nd)/s97 Ax\ny (A/mK)E\n (eV)\nFIG. 9. (Color online) The Kerr (a) and Faraday (b) rotation angles of bilayer and trilayer 1 T-CrTe 2. The anomalous Hall\nand anomalous Nernst conductivities of bilayer and trilayer 1 T-CrTe 2as a function of the Fermi energy. The magnetization\ndirection is along the y-axis. For a better comparison, the curves of monolayer 1 T-CrTe 2are also plotted.\n1T-CrTe 2. The monolayer is a ferromagnetic metal with\nthe in-plane magnetization along the y-axis. The in-plane\nmagnetocrystalline anisotropy energy is as small as 10\n\u0016eV/cell, indicating that the spin can be easily rotated\nwithin thexyplane. The magneto-optical Kerr and Fara-\nday rotation angles as well as anomalous Hall and Nernst\nconductivities exhibit a period of 2 \u0019=3 when the spin ro-\ntates within the xyplane, and their maximums of \u0012K=\n0.24 deg,\u0012F= 3.00\u0002105deg/cm,\u001bA\nxy= 1168.12 S/cm,\nand\u000bA\nxy= 5.30 A/mK (300 K) appear at '=n\u0019=3+\u0019=6\nwithn2N. At the azimuthal angle '=n\u0019=3, the\nmirror planes that are normal to the spin direction sup-\npress the magneto-optical, anomalous Hall, and anoma-\nlous Nernst e\u000bects. If the spin cants from in-plane to out-\nof-plane direction, the magneto-optical, anomalous Hall,\nand anomalous Nernst e\u000bects are signi\fcantly enhanced,\nand particularly they reach to the maximal values of \u0012K\n= -1.34 deg, \u0012F= -17.30\u0002105deg/cm,\u001bA\nxy= -4539.23\nS/cm, and \u000bA\nxy= 13.61 A/mK (300 K) when the spin is\nalong thez-axis (i.e., polar angle \u0012= 0\u000e). The bilayer\n1T-CrTe 2prefers an in-plane antiferromagnetic structure\nwith the magnetization along the y-axis, which has the\nspacetime inversion symmetry TPthat prohibits the sig-\nnals of magneto-optical responses as well as anomalous\nHall and Nernst transports. The trilayer 1 T-CrTe 2is also\ninclined to the in-plane antiferromagnetic order between\ntwo adjacent layers, but has \fnite net magnetization dueto the odd number of layers. The magnetic point group of\ntrilayer structure with in-plane antiferromagnetic order is\nidentical to that of monolayer structure and thus allows\nthe presence of all the physical phenomena mentioned\nabove. In particular, the magneto-optical Kerr and Fara-\nday rotation angles (anomalous Hall and Nernst conduc-\ntivities) of trilayer structure are obviously larger (slightly\nsmaller) than that of monolayer structure with the mag-\nnetization along the y-axis. For example, the maximal\nvalues of\u0012K= -1.76 deg, \u0012F= 4.60\u0002105deg/cm,\u001bA\nxy=\n-1147.03 S/cm, and \u000bA\nxy= -4.81 A/mK (300 K) are found\nin the trilayer structure . Our results suggest that the\nmagneto-optical, anomalous Hall, and anomalous Nernst\ne\u000bects for two-dimensional room-temperature ferromag-\nnet 1T-CrTe 2can be e\u000bectively modulated by altering\nmagnetization direction and layer number.\nACKNOWLEDGMENTS\nW.F. and Y.Y. acknowledge the support from the Na-\ntional Natural Science Foundation of China (Grants No.\n11874085 and No. 11734003) and the National Key R&D\nProgram of China (Grant No. 2016YFA0300600). X.Z.\nacknowledges the support from the Graduate Technolog-\nical Innovation Project of Beijing Institute of Technology\n(Grant No. 2019CX10018).11\n[1] C. Gong and X. Zhang, Two-dimensional magnetic crys-\ntals and emergent heterostructure devices, Science 363,\n706 (2019).\n[2] N. D. Mermin and H. Wagner, Absence of ferromag-\nnetism or antiferromagnetism in one-or two-dimensional\nisotropic Heisenberg models, Phys. Rev. Lett. 17, 1133\n(1966).\n[3] F. Hellman, A. Ho\u000bmann, Y. Tserkovnyak, G. S. D.\nBeach, E. E. Fullerton, C. Leighton, A. H. MacDon-\nald, D. C. Ralph, D. A. Arena, H. A. D urr, P. Fischer,\nJ. Grollier, J. P. Heremans, T. Jungwirth, A. V. Kimel,\nB. Koopmans, I. N. Krivorotov, S. J. May, A. K. Petford-\nLong, J. M. Rondinelli, N. Samarth, I. K. Schuller, A. N.\nSlavin, M. D. Stiles, O. Tchernyshyov, A. Thiaville, and\nB. L. Zink, Interface-induced phenomena in magnetism,\nRev. Mod. Phys. 89, 025006 (2017).\n[4] K. S. Burch, D. Mandrus, and J.-G. Park, Magnetism\nin two-dimensional van der Waals materials, Nature 563,\n47 (2018).\n[5] B. Sachs, T. O. Wehling, K. S. Novoselov, A. I. Licht-\nenstein, and M. I. Katsnelson, Ferromagnetic two-\ndimensional crystals: single layers of K 2CuF 4, Phys. Rev.\nB88, 201402(R) (2013).\n[6] J.-G. Park, Opportunities and challenges of two-\ndimensional magnetic van der Waals materials: magnetic\ngraphene?, J. Phys.: Condens. Matter 28, 301001 (2016).\n[7] D. Zhong, K. L. Seyler, X. Linpeng, R. Cheng,\nN. Sivadas, B. Huang, E. Schmidgall, T. Taniguchi,\nK. Watanabe, M. A. McGuire, W. Yao, D. Xiao, K.-\nM. C. Fu, and X. Xu, Van der Waals engineering of fer-\nromagnetic semiconductor heterostructures for spin and\nvalleytronics, Sci. Adv 3, e1603113 (2017).\n[8] N. Samarth, Condensed-matter physics: Magnetism in\n\ratland, Nature 546, 216 (2017).\n[9] J.-U. Lee, S. Lee, J. H. Ryoo, S. Kang, T. Y. Kim,\nP. Kim, C.-H. Park, J.-G. Park, and H. Cheong, Ising-\ntype magnetic ordering in atomically thin FePS 3, Nano\nLett. 16, 7433 (2016).\n[10] X. Wang, K. Du, Y. Y. F. Liu, P. Hu, J. Zhang,\nQ. Zhang, M. H. S. Owen, X. Lu, C. K. Gan, P. Sen-\ngupta, C. Kloc, and Q. Xiong, Raman spectroscopy\nof atomically thin two-dimensional magnetic iron phos-\nphorus trisul\fde (FePS 3) crystals, 2D Mater. 3, 031009\n(2016).\n[11] C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao,\nW. Bao, C. Wang, Y. Wang, Z. Q. Qiu, R. J. Cava, S. G.\nLouie, J. Xia, and X. Zhang, Discovery of intrinsic fer-\nromagnetism in two-dimensional van der Waals crystals,\nNature 546, 265 (2017).\n[12] B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein,\nR. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A.\nMcGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-\nHerrero, and X. Xu, Layer-dependent ferromagnetism\nin a van der Waals crystal down to the monolayer limit,\nNature 546, 270 (2017).\n[13] D. R. Klein, D. MacNeill, J. L. Lado, D. Soriano,\nE. Navarro-Moratalla, K. Watanabe, T. Taniguchi,\nS. Manni, P. Can\feld, J. Fern\u0013 andez-Rossier, and\nP. Jarillo-Herrero, Probing magnetism in 2D van der\nWaals crystalline insulators via electron tunneling, Sci-\nence360, 1218 (2018).[14] S. Jiang, L. Li, Z. Wang, K. F. Mak, and J. Shan, Con-\ntrolling magnetism in 2D CrI 3by electrostatic doping,\nNat. Nanotechnol. 13, 549 (2018).\n[15] B. Huang, G. Clark, D. R. Klein, D. MacNeill,\nE. Navarro-Moratalla, K. L. Seyler, N. Wilson, M. A.\nMcGuire, D. H. Cobden, D. Xiao, W. Yao, P. Jarillo-\nHerrero, and X. Xu, Electrical control of 2D magnetism\nin bilayer CrI 3, Nat. Nanotechnol. 13, 544 (2018).\n[16] S. Jiang, J. Shan, and K. F. Mak, Electric-\feld switching\nof two-dimensional van der Waals magnets, Nat. Mater.\n17, 406 (2018).\n[17] Z. Wang, I. Guti\u0013 errez-Lezama, N. Ubrig, M. Kroner,\nM. Gibertini, T. Taniguchi, K. Watanabe, A. Imamo\u0015 glu,\nE. Giannini, and A. F. Morpurgo, Very large tunnel-\ning magnetoresistance in layered magnetic semiconductor\nCrI3, Nat. Commun. 9, 2516 (2018).\n[18] N. Sivadas, S. Okamoto, X. Xu, C. J. Fennie, and\nD. Xiao, Stacking-dependent magnetism in bilayer CrI 3,\nNano Lett. 18, 7658 (2018).\n[19] M. Kim, P. Kumaravadivel, J. Birkbeck, W. Kuang,\nS. G. Xu, D. G. Hopkinson, J. Knolle, P. A. McClarty,\nA. I. Berdyugin, M. Ben Shalom, R. V. Gorbachev, S. J.\nHaigh, S. Liu, J. H. Edgar, K. S. Novoselov, I. V. Grig-\norieva, and A. K. Geim, Micromagnetometry of two-\ndimensional ferromagnets, Nat. Electron. 2, 457 (2019).\n[20] D. R. Klein, D. MacNeill, Q. Song, D. T. Larson, S. Fang,\nM. Xu, R. A. Ribeiro, P. C. Can\feld, E. Kaxiras,\nR. Comin, and P. Jarillo-Herrero, Enhancement of inter-\nlayer exchange in an ultrathin two-dimensional magnet,\nNat. Phys. 15, 1255 (2019).\n[21] H. H. Kim, B. Yang, S. Li, S. Jiang, C. Jin, Z. Tao,\nG. Nichols, F. S\fgakis, S. Zhong, C. Li, S. Tian, D. G.\nCory, G.-X. Miao, J. Shan, K. F. Mak, H. Lei, K. Sun,\nL. Zhao, and A. W. Tsen, Evolution of interlayer and\nintralayer magnetism in three atomically thin chromium\ntrihalides, PNAS 116, 11131 (2019).\n[22] Y. Deng, Y. Yu, Y. Song, J. Zhang, N. Z. Wang, Z. Sun,\nY. Yi, Y. Z. Wu, S. Wu, J. Zhu, J. Wang, X. H. Chen,\nand Y. Zhang, Gate-tunable room-temperature ferro-\nmagnetism in two-dimensional Fe 3GeTe 2, Nature 563,\n94 (2018).\n[23] Z. Fei, B. Huang, P. Malinowski, W. Wang, T. Song,\nJ. Sanchez, W. Yao, D. Xiao, X. Zhu, A. F. May, W. Wu,\nD. H. Cobden, J.-H. Chu, and X. Xu, Two-dimensional\nitinerant ferromagnetism in atomically thin Fe 3GeTe 2,\nNat. Mater. 17, 778 (2018).\n[24] J.-M. Xu, S.-Y. Wang, W.-J. Wang, Y.-H. Zhou, X.-L.\nChen, Z.-R. Yang, and Z. Qu, Possible Tricritical Behav-\nior and Anomalous Lattice Softening in van der Waals\nItinerant Ferromagnet Fe 3GeTe 2under High Pressure,\nChin. Phys. Lett. 37, 076202 (2020).\n[25] M. Bonilla, S. Kolekar, Y. Ma, H. C. Diaz, V. Kalappat-\ntil, R. Das, T. Eggers, H. R. Gutierrez, M.-H. Phan, and\nM. Batzill, Strong room-temperature ferromagnetism in\nVSe 2monolayers on van der Waals substrates, Nat. Nan-\notechnol. 13, 289 (2018).\n[26] J. Li, B. Zhao, P. Chen, R. Wu, B. Li, Q. Xia, G. Guo,\nJ. Luo, K. Zang, Z. Zhang, H. Ma, G. Sun, X. Duan, and\nX. Duan, Synthesis of Ultrathin Metallic MTe 2(M= V,\nNb, Ta) Single-Crystalline Nanoplates, Adv. Mater. 30,\n1801043 (2018).12\n[27] D. J. O'Hara, T. Zhu, A. H. Trout, A. S. Ahmed, Y. K.\nLuo, C. H. Lee, M. R. Brenner, S. Rajan, J. A. Gupta,\nD. W. McComb, and R. K. Kawakami, Room tempera-\nture intrinsic ferromagnetism in epitaxial manganese se-\nlenide \flms in the monolayer limit, Nano Lett. 18, 3125\n(2018).\n[28] Q.-Q. Yuan, Z. Guo, Z.-Q. Shi, H. Zhao, Z.-Y. Jia,\nQ. Wang, J. Sun, D. Wu, and S.-C. Li, Ferromagnetic\nMnSn Monolayer Epitaxially Grown on Silicon Substrate,\nChin. Phys. Lett. 37, 077502 (2020).\n[29] A. Avsar, A. Ciarrocchi, M. Pizzochero, D. Unuchek,\nO. V. Yazyev, and A. Kis, Defect induced, layer-\nmodulated magnetism in ultrathin metallic PtSe 2, Nat.\nNanotechnol. 14, 674 (2019).\n[30] D. C. Freitas, R. Weht, A. Sulpice, G. Remenyi, P. Stro-\nbel, F. Gay, J. Marcus, and M. N\u0013 u~ nez-Regueiro, Fer-\nromagnetism in layered metastable 1T-CrTe 2, J. Phys.:\nCondens. Matter 27, 176002 (2015).\n[31] X. Sun, W. Li, X. Wang, Q. Sui, T. Zhang, Z. Wang,\nL. Liu, D. Li, S. Feng, S. Zhong, H. Wang, V. Bouchiat,\nM. Nunez Regueiro, N. Rougemaille, J. Coraux, A. Pur-\nbawati, A. Hadj-Azzem, Z. Wang, B. Dong, X. Wu,\nT. Yang, G. Yu, B. Wang, Z. Han, X. Han, and Z. Zhang,\nRoom temperature ferromagnetism in ultra-thin van der\nWaals crystals of 1T-CrTe 2, Nano Research (2020),\n10.1007/s12274-020-3021-4.\n[32] A. Purbawati, J. Coraux, J. Vogel, A. Hadj-Azzem,\nN. Wu, N. Bendiab, D. Jegouso, J. Renard, L. Marty,\nV. Bouchiat, A. Sulpice, L. Aballe, M. Foerster,\nF. Genuzio, A. Locatelli, T. O. Mentes, Z. V. Han,\nX. Sun, M. N\u0013 u~ nez-Regueiro, and N. Rougemaille, In-\nplane magnetic domains and N\u0013 eel-like domain walls in\nthin \rakes of the room temperature CrTe 2van der Waals\nferromagnet, ACS Appl. Mater. Interfaces 12, 30702\n(2020).\n[33] J. Kerr, On rotation of the plane of polarization by re-\n\rection from the pole of a magnet, Phil. Mag. 3, 321\n(1877).\n[34] M. Faraday, Experimental researches in electricity.-\nNineteenth series, Phil. Trans. R. Soc. 136, 1 (1846).\n[35] V. Antonov, B. Harmon, and A. Yaresko, Elec-\ntronic structure and magneto-optical properties of solids\n(Springer Science & Business Media, 2004).\n[36] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and\nN. P. Ong, Anomalous hall e\u000bect, Rev. Mod. Phys. 82,\n1539 (2010).\n[37] R. Kubo, Statistical-mechanical theory of irreversible\nprocesses. I. General theory and simple applications to\nmagnetic and conduction problems, J. Phys. Soc. Jpn.\n12, 570 (1957).\n[38] C. S. Wang and J. Callaway, Band structure of nickel:\nSpin-orbit coupling, the Fermi surface, and the optical\nconductivity, Phys. Rev. B 9, 4897 (1974).\n[39] X. Zhou, J.-P. Hanke, W. Feng, F. Li, G.-Y. Guo, Y. Yao,\nS. Bl ugel, and Y. Mokrousov, Spin-order dependent\nanomalous Hall e\u000bect and magneto-optical e\u000bect in the\nnoncollinear antiferromagnets Mn 3XN with X= Ga, Zn,\nAg, or Ni, Phys. Rev. B 99, 104428 (2019).\n[40] W. Nernst, Ueber die electromotorischen Kr afte, welche\ndurch den Magnetismus in von einem W armestrome\ndurch\rossenen Metallplatten geweckt werden, Ann.\nPhys. 267, 760 (1887).\n[41] N. Hanasaki, K. Sano, Y. Onose, T. Ohtsuka, S. Iguchi,\nI. K\u0013 ezsm\u0013 arki, S. Miyasaka, S. Onoda, N. Nagaosa,and Y. Tokura, Anomalous Nernst e\u000bects in pyrochlore\nmolybdates with spin chirality, Phys. Rev. Lett. 100,\n106601 (2008).\n[42] M. Ikhlas, T. Tomita, T. Koretsune, M.-T. Suzuki,\nD. Nishio-Hamane, R. Arita, Y. Otani, and S. Nakat-\nsuji, Large anomalous Nernst e\u000bect at room temperature\nin a chiral antiferromagnet, Nat. Phys. 13, 1085 (2017).\n[43] T. Liang, J. Lin, Q. Gibson, T. Gao, M. Hirschberger,\nM. Liu, R. J. Cava, and N. P. Ong, Anomalous Nernst\ne\u000bect in the dirac semimetal Cd 3As2, Phys. Rev. Lett.\n118, 136601 (2017).\n[44] C. Wuttke, F. Caglieris, S. Sykora, F. Scaravaggi,\nA. U. B. Wolter, K. Manna, V. S uss, C. Shekhar,\nC. Felser, B. B uchner, and C. Hess, Berry curvature un-\nravelled by the anomalous Nernst e\u000bect in Mn 3Ge, Phys.\nRev. B 100, 085111 (2019).\n[45] W.-B. Zhang, Q. Qu, P. Zhu, and C.-H. Lam, Ro-\nbust Intrinsic Ferromagnetism and Half Semiconductiv-\nity in Stable Two-dimensional Single-layer Chromium\nTrihalides, J. Mater. Chem. C 3, 12457 (2015).\n[46] H. L. Zhuang, P. R. C. Kent, and R. G. Hennig, Strong\nanisotropy and magnetostriction in the two-dimensional\nStoner ferromagnet Fe 3GeTe 2, Phys. Rev. B 93, 134407\n(2016).\n[47] G. Kresse and J. Hafner, Ab initio molecular dynamics\nfor liquid metals, Phys. Rev. B 47, 558 (1993).\n[48] G. Kresse and J. Furthm uller, E\u000ecient iterative schemes\nfor ab initio total-energy calculations using a plane-wave\nbasis set, Phys. Rev. B 54, 11169 (1996).\n[49] P. E. Bl ochl, Projector augmented-wave method, Phys.\nRev. B 50, 17953 (1994).\n[50] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized\ngradient approximation made simple, Phys. Rev. Lett.\n77, 3865 (1996).\n[51] V. I. Anisimov, J. Zaanen, and O. K. Andersen, Band\ntheory and Mott insulators: Hubbard U instead of Stoner\nI, Phys. Rev. B 44, 943 (1991).\n[52] S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J.\nHumphreys, and A. P. Sutton, Electron-energy-loss\nspectra and the structural stability of nickel oxide: An\nLSDA+ U study, Phys. Rev. B 57, 1505 (1998).\n[53] X. Sui, T. Hu, J. Wang, B.-L. Gu, W. Duan, and M.-\ns. Miao, Voltage-controllable colossal magnetocrystalline\nanisotropy in single-layer transition metal dichalco-\ngenides, Phys. Rev. B 96, 041410(R) (2017).\n[54] A. A. Mosto\f, J. R. Yates, Y.-S. Lee, I. Souza, D. Van-\nderbilt, and N. Marzari, wannier90: A tool for obtaining\nmaximally-localised Wannier functions, Comput. Phys.\nCommun. 178, 685 (2008).\n[55] J. Callaway, Quantum theory of the solid state (Aca-\ndemic Press, 2013).\n[56] W. Feng, G.-Y. Guo, and Y. Yao, Tunable\nmagneto-optical e\u000bects in hole-doped group-IIIA metal-\nmonochalcogenide monolayers, 2D Mater. 4, 015017\n(2017).\n[57] Y. Yao, L. Kleinman, A. H. MacDonald, J. Sinova,\nT. Jungwirth, D.-s. Wang, E. Wang, and Q. Niu, First\nprinciples calculation of anomalous Hall conductivity in\nferromagnetic bcc Fe, Phys. Rev. Lett. 92, 037204 (2004).\n[58] D. Xiao, Y. Yao, Z. Fang, and Q. Niu, Berry-phase e\u000bect\nin anomalous thermoelectric transport, Phys. Rev. Lett.\n97, 026603 (2006).\n[59] X. Zhou, J.-P. Hanke, W. Feng, S. Bl ugel, Y. Mokrousov,\nand Y. Yao, Giant anomalous Nernst e\u000bect in non-13\ncollinear antiferromagnetic Mn-based antiperovskite ni-\ntrides, Phys. Rev. Materials 4, 024408 (2020).\n[60] P. W. Anderson, Antiferromagnetism. Theory of superex-\nchange interaction, Phys. Rev. 79, 350 (1950).\n[61] J. B. Goodenough, An interpretation of the mag-\nnetic properties of the perovskite-type mixed crystals\nLa1\u0000xSrxCoO 3\u0000\u0015, J. Phys. Chem. Solids 6, 287 (1958).\n[62] J. Kanamori, Superexchange interaction and symmetry\nproperties of electron orbitals, J. Phys. Chem. Solids 10,\n87 (1959).\n[63] H. Y. Lv, W. J. Lu, D. F. Shao, Y. Liu, and Y. P. Sun,\nStrain-controlled switch between ferromagnetism and an-\ntiferromagnetism in 1 T-CrX 2(X = Se, Te) monolayers,\nPhys. Rev. B 92, 214419 (2015).\n[64] S. Li, S.-S. Wang, B. Tai, W. Wu, B. Xiang, X.-L. Sheng,\nand S. A. Yang, Tunable anomalous Hall transport inbulk and two-dimensional 1T-CrTe 2: A \frst-principles\nstudy, arXiv:2006.10795.\n[65] H. T. Stokes, D. M. Hatch, and B. J.\nCampbell, ISOTROPY Software Suite.,\nhttps://stokes.byu.edu/iso/isotropy.php .\n[66] Z. Liu, G. Zhao, B. Liu, Z. F. Wang, J. Yang, and F. Liu,\nIntrinsic quantum anomalous Hall e\u000bect with in-plane\nmagnetization: searching rule and material prediction,\nPhys. Rev. Lett. 121, 246401 (2018).\n[67] X. Zhou, W. Feng, F. Li, and Y. Yao, Large magneto-\noptical e\u000bects in hole-doped blue phosphorene and gray\narsenene, Nanoscale 9, 17405 (2017).\n[68] G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Spin\ncaloritronics, Nat. Mater. 11, 391 (2012).\n[69] S. R. Boona, R. C. Myers, and J. P. Heremans, Spin\ncaloritronics, Energy Environ. Sci. 7, 885 (2014)."    },    {        "title": "2011.05455v3.Spin_waves_in_the_collinear_antiferromagnetic_phase_of_Mn___bf_5__Si___bf_3__.pdf",        "content": "Spin{waves in the collinear antiferromagnetic phase of Mn 5Si3\nF. J. dos Santos,1, 2,\u0003N. Biniskos,3,yS. Raymond,4K. Schmalzl,5M. dos\nSantos Dias,1P. Ste\u000bens,6J. Persson,7S. Bl ugel,1S. Lounis,1, 8and T. Br uckel7\n1Peter Gr unberg Institut and Institute for Advanced Simulations,\nForschungszentrum J ulich &JARA, D-52425 J ulich, Germany\n2Department of Physics, RWTH Aachen University, 52056 Aachen, Germany\n3Forschungszentrum J ulich GmbH, J ulich Centre for Neutron Science at MLZ, Lichtenbergstr. 1, 85748 Garching, Germany\n4Universit\u0013 e Grenoble Alpes, CEA, IRIG, MEM, 38000 Grenoble, France\n5Forschungszentrum J ulich GmbH, J ulich Centre for Neutron Science at ILL, 71 avenue des Martyrs, 38000 Grenoble, France\n6Institut Laue-Langevin, 71 avenue des Martyrs, 38000 Grenoble, France\n7Forschungszentrum J ulich GmbH, J ulich Centre for Neutron Science (JCNS-2)\nand Peter Gr unberg Institut (PGI-4), JARA-FIT, 52425 J ulich, Germany\n8Faculty of Physics, University of Duisburg-Essen, 47053 Duisburg, Germany\n(Dated: December 10, 2020)\nBy combining two independent approaches, inelastic neutron scattering measurements and density\nfunctional theory calculations, we study the spin{waves in the collinear antiferromagnetic phase\n(AFM2) of Mn 5Si3. We obtain its magnetic ground{state properties and electronic structure. This\nstudy allowed us to determine the dominant magnetic exchange interactions and magnetocrystalline\nanisotropy in the AFM2 phase of Mn 5Si3. Moreover, the evolution of the spin excitation spectrum\nis investigated under the in\ruence of an external magnetic \feld perpendicular to the anisotropy\neasy{axis. The low energy magnon modes show a di\u000berent magnetic \feld dependence which is\na direct consequence of their di\u000berent precessional nature. Finally, possible e\u000bects related to the\nDzyaloshinskii{Moriya interaction are also considered.\nI. INTRODUCTION\nThe study of magnetism at a microscopic level can\nlead to designing cutting{edge technological applications\nfor data process and storage, information transmission,\nand magnetic refrigeration. In recent years, antiferro-\nmagnetic (AFM) materials have attracted great interest\nin the research \feld of spintronics1. Bulk Mn 5Si3is an\nAFM intermetallic compound that is hosting rich physics.\nIts interesting properties, such as the complex magnetic\nstructure2, the anomalous Hall e\u000bect3, and the inverse\nmagnetocaloric e\u000bect4have been attributed to an insta-\nbility of the Mn magnetic moments. It is also worth\nmentioning that in nanoparticle5and nanowire6form,\nMn5Si3is considered to have great potential in future\nelectronic and spintronic devices. Despite the intense re-\nsearch activity over the past decades2{4,7{16, many open\nquestions remain regarding the minimal magnetic model\nHamiltonian, the role of the spin \ructuations in the mag-\nnetically ordered phases and which Mn site is responsi-\nble for them. To address some of these questions, in the\npresent study, we perform inelastic neutron scattering\nexperiments and apply \frst{principles calculations.\nThe crystal and magnetic structure of bulk Mn 5Si3\nhas been established by neutron di\u000braction measure-\nments2,7,8and the magnetic phase diagram as a func-\ntion of temperature and magnetic \feld has been ex-\ntensively studied by magnetization and electrical trans-\nport measurements9{12,17. In the paramagnetic (PM)\nstate, Mn 5Si3crystallizes in the hexagonal space group\nP63=mcm with two distinct crystallographic positions\nfor Mn atoms (sites Mn1 and Mn2)7and undergoes two\nsuccessive \frst order phase transitions towards antiferro-\n(a) (b)\nSi Mn2Mn1J4\nJ2\nJ1\nJ3\nab\nc\nFIG. 1. (a) Projection of the structure of Mn 5Si3in the\nAFM2 phase in the ab{plane of the orthorhombic cell accord-\ning to single{crystal neutron di\u000braction data8. The blue solid\nlines indicate the relevant exchange interactions used in the\nHeisenberg Hamiltonian. (b) Predicted metastable FM phase\nfrom \frst{principles calculations (see details in text). The\ntwo triangles indicate Mn2 atoms located in di\u000berent planes.\nmagnetic phases, which occur at TN2\u0019100 K (AFM2)\nandTN1\u001966 K (AFM1), respectively17. The electric\nresistivity shows a metallic behavior with two anomalies\ncorresponding to these phase transitions10.\nAtTN2\u0019100 K (AFM2), the crystal structure changes\nfrom hexagonal to orthorhombic with space group Ccmm\nand Mn2 divides into two sets of inequivalent positions8.\nIn the orthorhombic cell magnetic re\rections follow the\nconditionh+kodd and the magnetic propagation vec-arXiv:2011.05455v3  [cond-mat.mes-hall]  9 Dec 20202\ntor is \u0014= (0;1;0). In the AFM2 phase, the Mn1\nand one{third of the Mn2 atoms have no ordered mo-\nments and the remaining Mn2 atoms have their mag-\nnetic moments of magnitude 1.48(1) \u0016Baligned almost\nparallel and antiparallel to the b{axis of the orthorhom-\nbic cell8(see Fig. 1(a)). At a lower temperature, at\nTN1\u001966 K (AFM1), a structural distortion occurs to\nan orthorhombic cell without inversion symmetry (space\ngroupCc2m)2. The magnetic moments reorient in a\nhighly noncollinear and noncoplanar arrangement, while\nthe propagation vector remains the same. Mn1 atoms\nacquire a magnetic moment of magnitude 1.20(5) \u0016Band\nstill one{third of the Mn2 atoms have no ordered mo-\nments, just as in the AFM2 phase. The rest of the Mn2\natoms carry a moment of 2.30(9) and 1.85(9) \u0016B, depend-\ning on their site.\nII. EXPERIMENTAL PART\nA. Experimental details\nThe Mn 5Si3single crystal was grown by the Czochral-\nski method4. The sample with a mass of about 7 g was\nmounted on an aluminum sample holder and was oriented\nin the [100]/[010] scattering plane of the orthorhombic\nsymmetry. Inelastic neutron scattering (INS) measure-\nments were carried out on the cold triple{axis spectrom-\neters (TAS) IN1218and ThALES at the Institut Laue\nLangevin (ILL) in Grenoble, France. Both TAS were\nsetup in W con\fguration and inelastic scans were per-\nformed with constant kf, where kfis the wave{vector of\nthe scattered neutron beam.\nUnpolarized INS measurements were performed at\nIN12 and focusing setups were employed. The spectrome-\nter was equipped with pyrolytic graphite (PG(002)) crys-\ntals as the monochromator and analyzer and 40'{open{\nopen collimations were installed. Higher{order contami-\nnation was removed using a velocity selector (VS) before\nthe monochromator and a beryllium (Be) \flter in the\nscattered neutron beam. The sample was cooled below\nroom temperature with a4He \row cryostat. Spin dy-\nnamics investigations with unpolarized neutrons under\nthe magnetic \feld were carried out using a 10 T verti-\ncal \feld magnet. For these measurements, the Be \flter\nwas removed. The single crystal was cooled down from\nthe PM state to T= 80 K (AFM2 phase) without the\npresence of an external magnetic \feld. The \feld was ap-\nplied along the c{axis of the orthorhombic symmetry of\nMn5Si3and the spectra were collected with increasing\n\feld strength.\nAt ThALES longitudinal polarization analysis (LPA)\nwas performed using the CRYOPAD device19to guide\nand orient the neutron beam polarization with a strictly\nzero magnetic \feld in the sample position. The TAS was\nequipped with polarizing Heusler (Cu 2MnAl(111)) crys-\ntals as the monochromator and analyzer. A \ripping ratio\nof 14 was determined from measurements in a graphite\nFIG. 2. Temperature dependence of the dynamical spin\nsusceptibility \u001f00(Q;E) of Mn 5Si3atQ= (0:9;2;0) and\nE= 0:5 meV measured with unpolarized neutrons with\nkf= 1:5\u0017A\u00001. The vertical red dashed lines indicate TN2\u0019\n100 K andTN1\u001966 K. The arrow indicates the temperature\n(T= 80 K) where INS data were collected.\nsample. Fully focusing setups were employed and higher{\norder contamination was removed using a VS and a Be\n\flter before the monochromator and in the scattered neu-\ntron beam, respectively. Inelastic scans were performed\nwith a constant kfof 1.1 \u0017A\u00001. For the polarized INS ex-\nperiments the common Cartesian coordinate system was\nused20: thex{axis parallel to the scattering vector Q21,\nthey{axis perpendicular to Qin the scattering plane and\nthez{axis perpendicular to the scattering plane.\nB. Unpolarized INS measurements\nTo determine the extent of the critical spin \ructua-\ntions related to the two AFM transitions, spectra were\ncollected with an unpolarized neutron beam at small q.\nA (Q,E) position was carefully selected to avoid contri-\nbutions to the measured intensity from the elastic and\ninelastic scattering from the magnetic zone centers and\nlow energy magnon modes, respectively. The obtained\nintensity after background subtraction was corrected by\nthe detailed balance factor so that the \fnal result relates\nto the imaginary part of the dynamical spin susceptibil-\nity\u001f00(Q;E). Fig. 2 shows the extent of the spin \ruc-\ntuations for Q= (0:9;2;0) andE= 0:5 meV in the PM\nstate, as well as, the critical \ructuations due to the two\nAFM phase transitions as seen by the broad tail above\nTN.\u001f00(Q;E) shows two maxima at TN2andTN1, in\nagreement with the established magnetic phase diagrams\nof Mn 5Si3where the AFM transitions occur9,12.\nMagnetic excitations were measured around the mag-\nnetic zone center G= (1;2;0) atT= 80 K, a tempera-\nture selected well inside the AFM2 phase where the in-\ntensity shows a plateau and the combined critical spin\n\ructuations from the PM to the AFM2 ( TN2\u0019100 K)3\nand from the AFM2 to the AFM1 ( TN1\u001966 K) transi-\ntions have minimal intensities (see Fig. 2). The energy\ndependence of the measured excitations at di\u000berent Qh\npositions is shown in Fig. 3(a), where Q= (Qh;2;0). The\nindividual spectra consist of three peaks. The \frst peak\nthat is always centered at E= 0 meV corresponds to the\nelastic line. The two other peaks appear at \fnite Eand\nshift to higher energy transfers as Qhincreases, charac-\nteristic of dispersive spin waves. To analyze the obtained\nspectra, a constant background was assumed and Gaus-\nsian functions were selected to describe the peaks. One\ncould argue that the signal at \fnite EatQ= (1:037;2;0)\nandQ= (1:05;2;0) could be described by a single broad\npeak. However, the use of polarized neutrons (see Sec-\ntion II.D) justi\fes the existence of two peaks for the ex-\ncitations, since the polarized INS cross{sections implore\nstrict \ftting conditions. A typical E{scan collected at\nQ= (1:005;2;0) with the individual \ft for the elastic\nand spin-wave signals is shown in Fig. 3(b).\nThe obtained low energy experimental spin{wave dis-\npersion along the ( h00) symmetry direction in the AFM2\nphase of Mn 5Si3is shown in Fig. 3(c). There are two\ncharacteristic features: (i) two small energy gaps at q= 0\nand (ii) the lowest magnetic excitations can be described\nby the empirical dispersion relation E=p\n\u00012+C2q222,\nwhere \u0001 refers to the spin{gap and Cis a constant.\nThe obtained values for the two magnon modes are:\n\u0001\u000b= 0:408(7) meV, C\u000b= 6:3(5) meV/r.l.u., \u0001 \f=\n0:181(4) meV and C\f= 4:7(3) meV/r.l.u.. The observed\ngaps are indications of two easy{axis anisotropies, an as-\nsumption stemming from the collinear spin arrangement\nin the AFM2 phase of Mn 5Si3. In order to describe the\nmagnon spectrum and to extract the dominant magnetic\nexchange interactions and magnetocrystalline anisotropy\nin the AFM2 phase, theoretical calculations were em-\nployed (see Section III). In what follows, the modes orig-\ninating from \u0001 \u000band \u0001\fwill be referred to as the \u000band\nthe\f-modes, respectively.\nC. Unpolarized INS measurements under magnetic\n\feld\nForHk^c, neutron di\u000braction measurements13per-\nformed in single crystals of Mn 5Si3indicate that no\n\feld{induced transition occurs within the AFM2 phase\nand consistently with the macroscopic data9,11,12the\nPM state is not reached up to H= 8 T due to the\nsteepTN2(H) phase boundary. In order to investigate\nthe magnon spectrum under magnetic \feld, energy spec-\ntra were collected at three di\u000berent Qhpositions (1,\n1.018 and 1.025 r.l.u.) around the magnetic Bragg peak\nG= (1;2;0). Figure 4(a) shows such characteristic scans\natQh= 1 r.l.u. for di\u000berent magnetic \felds. The ob-\ntained spectra were analyzed as described in the previous\nsection. For increasing magnetic \feld, the two peaks that\ncorrespond to the \u000band the\f{modes show di\u000berent be-\nhavior. As the \feld increases, the position and intensity\n\u000b-mode\n\f-mode(a)\n(b)\n(c)\nFIG. 3. (a) Energy spectra of Mn 5Si3atQ= (1 +qh;2;0)\nmeasured at IN12 spectrometer with unpolarized setup with\nkf= 1:05\u0017A\u00001atT= 80 K. (b) Fitting of the E{scan at\nQ= (1:005;2;0). The three dashed lines correspond to Gaus-\nsian functions sitting on top of a \rat background. The solid\nlines indicate the overall \fts as described in the text. (c) Low\nenergy magnon dispersion at T= 80 K along the ( h00) di-\nrection. The solid lines are \fts with the empirical dispersion\nrelationE=p\n\u00012+C2q2.\nof the\u000b{mode is not signi\fcantly a\u000bected in contrast to\nthe\f{mode, which disperses with the \feld and a con-\ntinuous diminution of intensity is observed. It should be\nnoted that at H= 4 T, the two peaks seem to merge.\nThe same observations regarding the behavior of the\ntwo modes stem from measurements at Qh= 1:018 r.l.u.\nandQh= 1:025 r.l.u. and are summarized in Fig. 4(b).\nThe obtained results indicate that the modes behave dif-\nferently under the external magnetic \feld possibly due\nto their di\u000berent polarization. To shed light on this be-\nhavior, spectra were collected using the polarized INS4\n\f-mode\n\u000b-mode(a)\n(b)\nFIG. 4. (a) Inelastic spectra of Mn 5Si3forHk^cobtained\natQ= (1;2;0) atT= 80 K (AFM2 phase) with unpolarized\nbeam withkf= 1:05\u0017A\u00001at IN12. The lines indicate \fts with\nGaussian functions. (b) Energy of the spin excitations as a\nfunction of the external magnetic \feld at three di\u000berent Qh\npositions at T= 80 K (AFM2 phase). Lines are guides for\nthe eyes.\nmethod.\nD. Polarized INS measurements\nAs a general rule, neutron scattering is only sensitive\nto magnetic excitations perpendicular to Q20. With LPA\nit is possible to separate magnetic \ructuations polarized\nalong di\u000berent directions in spin space. The initial polar-\nization was prepared parallel to x{axis, perpendicular to\nQin the scattering plane ( y{axis) and perpendicular to\nthe scattering plane ( z{axis), and the \fnal polarization\nwas analyzed for a scattering process reversing the initial\npolarization by 180\u000e. The corresponding measurement\nchannels are canonically labeled SF xx, SFyy, and SFzz,\nwhere SF stands for \\Spin{Flip\".\nThe neutron scattering double di\u000berential cross{sections for the three SF channels are20:\nSFxx=\u0012d2\u001b\nd\ndE\u0013x\nSF/BGSF+h\u000eMyi+h\u000eMzi(1)\nSFyy=\u0012d2\u001b\nd\ndE\u0013y\nSF/BGSF+h\u000eMzi(2)\nSFzz=\u0012d2\u001b\nd\ndE\u0013z\nSF/BGSF+h\u000eMyi(3)\nwhere BG SFis the background (which includes the nu-\nclear spin scattering) and h\u000eMyiandh\u000eMzithe mea-\nsured magnetic \ructuations. Considering that: (i) Qis\nin theab{plane, (ii) in the AFM2 phase the magnetic\nmoments lie parallel and antiparallel to the b{axis and\n(iii) spin{waves correspond to precession perpendicular\nto the ordered moment, then in the crystal frame the\ncross{sections become:\nSFxx/BGSF+ sin2\u0012h\u000eMai+h\u000eMci (4)\nSFyy/BGSF+h\u000eMci (5)\nSFzz/BGSF+ sin2\u0012h\u000eMai (6)\nwith\u0012the angle between Qand the [100] direction and\ncan be calculated by \u0012= arctan(Qk\nQha\nb).\nTo get further insight regarding the polarization de-\npendence of the two magnon modes, E{spectra at di\u000ber-\nentQpositions were collected in the three SF channels at\nT= 80 K. The magnetic \ructuations h\u000eMaiandh\u000eMci\nwere extracted by taking the di\u000berence of intensities be-\ntween the di\u000berent polarization channels. A typical re-\nsult of such analysis is shown in Fig. 5 at Q= (1:06;2;0)\nwhere the intensity for h\u000eMaiis corrected by the angle\nprefactor. The peak positions of the subtracted spin \ruc-\ntuations spectra are consistent with the established low\nenergy magnon dispersion curves obtained from the un-\npolarized data and shown in Fig. 3(c). Moreover, it is\nevident that the maximum of intensity in h\u000eMaiand in\nh\u000eMcicorresponds to the \u000b{mode and \f{mode, respec-\ntively. This hints that the elliptic polarization of each\nmode is di\u000berent and points along di\u000berent crystal axis.\nIII. THEORETICAL CALCULATIONS\nA. Density functional theory\nFirst{principles calculations were performed to deter-\nmine the ground{state electronic and magnetic proper-\nties of the AFM2 phase of Mn 5Si3. Our study was\nbased on the atomic structure speci\fed in Ref. 8. We\nemployed density functional theory (DFT) using the\nfull{potential Korringa{Kohn{Rostoker Green{function\n(KKR{GF) method including spin{orbit coupling, as im-\nplemented in the JuKKR code23, using the local{spin|\ndensity approximation24. The cuto\u000b for the angular mo-\nmentum expansion of the scattering problem was set to\nlmax= 3. Furthermore, the energy integration was per-\nformed in the upper complex energy plane25with 305\n\f-mode\n\u000b-mode\nFIG. 5. Subtracted spin \ructuations spectra h\u000eMaiand\nh\u000eMciof Mn 5Si3obtained at ThALES and measured at\nQ= (1:06;2;0) atT= 80 K. The intensity for h\u000eMaiwas\ncorrected by the factor sin2\u0012= 0:545. The lines indicate \fts\nwith Gaussian functions.\npoints in a rectangular path and 5 Matsubara frequencies\nat a temperature of T= 473:68 K, and the Brillouin zone\nintegration was performed with 30 \u000215\u000230k{points. The\nmagnetic exchange tensor, which parametrizes the spin\nHamiltonian discussed in the following section, was ob-\ntained through the in\fnitesimal{rotations method26,27.\nIn these calculations, the number of Matsubara frequen-\ncies was increased to 10 with T= 100 K.\nWe explored two possible magnetic con\fgurations with\nself{consistent calculations, the AFM con\fguration that\nis shown in Fig. 1, and a ferromagnetic (FM) phase with\n\fnite magnetic moments in both Mn1 and Mn2 sites. The\nAFM was found the most energetically favorable with the\nFM phase being 146 meV per unit cell higher in energy.\nIn the FM phase, the magnetic moments are 2.6 and\n1.2\u0016Bfor the Mn2 and Mn1 sites, respectively, while the\nSi sites have magnetic moments of 0.1 \u0016Bantiparallel to\nthose of the manganese sites. For the AFM phase, two{\nthirds of the Mn2 site carry magnetic moments of 2.4 \u0016B\nwhile the other Mn and the Si atoms have no magnetic\nmoment.\nCombining the magnetic force theorem with the\nfrozen potential approximation, the magnetocrystalline\nanisotropy was determined from band energy di\u000berences\nbetween states with di\u000berent orientations of the spin\nmagnetic moments. For the AFM phase, we obtained the\nfollowing energy di\u000berences when aligning the magnetic\nmoments along the main crystal axes: Ea\u0000Eb= 0:12,\nEc\u0000Eb= 0:09, andEa\u0000Ec= 0:03 meV per magnetic\natom, which indicates that bandcare the \frst and sec-\nond preferred axis, respectively, and ais the hard{axis.\nB. Model Hamiltonian and spin{wave calculation\nWe mapped the ab initio calculations onto a quantum\nHeisenberg Hamiltonian to study the spin{wave excita-tions in the adiabatic approximation, which reads as\nH=\u0000X\nijJijSi\u0001Sj\u0000X\n\u000bk\u000bX\ni(S\u000b\ni)2: (7)\nThe \frst term is due to the magnetic exchange interac-\ntion whose coupling is given by Jij.Siis the spin for\nwhich we set S= 1. The second term accounts for the\nbiaxial magnetocrystalline anisotropy ( Hani) withkb=\nEa\u0000Eb= 0:12 meV and kc=Ea\u0000Ec= 0:03 meV, both\nbeing positive. From the DFT calculations, we obtained\nthatkb>kc, which makes bthe primary easy{axis, and\ncthe secondary easy{axis. The biaxial anisotropy could\nalso be modeled by a combination of an easy{axis along\nband an easy{plane ( bc{plane) anisotropy.\nThe magnetic exchange interactions were obtained\nfrom \frst{principles calculations for the AFM2 phase.\nThe results for the \frst few Mn2 pairs as indicated in\nFig. 1(a) are: J1=\u000012:23,J2=\u00002:16,J3= 3:98 and\nJ4=\u00002:89 meV. Most interactions, apart from J3, have\nAFM character. This indicates that the AFM2 phase is\nfavored by all those pair interactions, that is, within this\nset of interactions, there is no frustration. J1,J2, andJ3\ncorrespond to couplings between magnetic moments in\nthe same [Mn2] 6octahedra. J1has the highest value and\nis the exchange interaction between the spins located on\na triangle in the ab{plane (distance 2.789 \u0017A).J2andJ3\ncouple spins located on adjacent triangles separated by\nc/2 with distances 2.893 and 4.019 \u0017A, respectively. The\nexchange interaction J4concerns the shortest distance\n(4.364 \u0017A) between spins located on adjacent [Mn2] 6oc-\ntahedra.\nWe employ the linear spin{wave approximation to ob-\ntain the spin{wave excitations of the quantum Heisen-\nberg Hamiltonian using the computed magnetic inter-\naction parameters. The spin{wave excitations are the\neigenstates of the dynamical matrix associated with the\nquantum Heisenberg Hamiltonian in Eq. (7), as explained\nin detail in Ref. 28. We start by constructing a local co-\nordinate system for every magnetic site with the local z{\naxis coinciding with the classical ground{state spin orien-\ntation. In this local frame, we expand the quantum spin\noperators using the linearized Holstein{Primako\u000b trans-\nformation as Si=\u0010p\n2Sai+ay\ni\n2;p\n2Sai\u0000ay\ni\n2i;S\u0000ay\niai\u0011\n,\nwhereay\niandaiare bosonic ladder operators29. Keeping\nonly terms up to second order in the Holstein{Primako\u000b\nbosons, the Hamiltonian can be written as H=H0+H2.\nTheH0term is a constant corresponding to the classical\nground{state energy. The second term\nH2=\u0000X\nkX\n\u0016\u0017ay(k)H\u0016\u0017(k)a\u0017(k) (8)\ncontains the quadratic terms of the Holstein{Primako\u000b\nbosons describing the spin excitations, where H(k) is a\n2n\u00022nmatrix and a\u0016(k) =\u0010a\u0016(k)\nay\n\u0016(\u0000k)\u0011\nwitha\u0016(k) =\n1p\nNP\nme\u0000ik\u0001Rmam\u0016.nandNare the number of atoms6\nin the unit and the number of unit cells under pe-\nriodic Born{von{Karman boundary conditions, respec-\ntively. The spin{wave eigenvalues !(k) and eigenvectors\njkiare then obtained by a Bogoliubov transformation30.\nThis process diagonalizes the system's dynamical ma-\ntrixD=gH2, where gis a diagonal matrix containing\n\u00001 on its \frst half and 1 on the second while ensur-\ning the bosonic character of the diagonalizing basis. The\nspin{wave inelastic scattering spectrum is computed with\nour theory for spin{resolved electron{energy{loss spec-\ntroscopy (SREELS) of noncollinear magnets presented in\nRef. 28, where we employ time{dependent perturbation\ntheory to describe the interaction between the probing\nbeam and the magnetic excitations. The same theory\ncan be applied with little modi\fcation to describe inelas-\ntic neutron scattering. This method has been applied\nto investigate ferromagnetic and antiferromagnetic non-\ncollinear spin textures in Refs. 31{33.\nIV. COMPARING EXPERIMENTAL AND\nTHEORETICAL RESULTS\nA. Results without external magnetic \feld\nAs already mentioned, the DFT calculations found\nthat the collinear AFM2 phase is more stable than the\nFM one, in line with the experimental \fnding that the\ncollinear AFM2 phase is stable in an intermediate tem-\nperature range8. The AFM2 phase has the peculiarity\nthat one{third of the Mn2 sites and all the Mn1 sites\nhave vanishing ordered magnetic moments, with the main\nquestion being whether this is due to a collapse of the lo-\ncal magnetic moment or due to a disordering of their\norientation. In the DFT calculations, we considered the\ncollapse scenario, which did not lead to a strong energetic\npenalty, given that the FM phase was still substantially\nhigher in energy. However, the obtained strong AFM\nexchange interactions between the Mn2 moments in the\n[Mn2] 6octahedra (the stacked triangles in Fig. 1) can\nalso lead to a \fnite temperature scenario where a third\nof the Mn2 moments is orientationally disordered, as pro-\nposed for Mn 3Pt in Ref. 34. The DFT calculations can\nthus also support a hybrid scenario where orientational\ndisorder is mixed with sizable \ructuations of the mag-\nnitude of the moments for the Mn2 sites with vanishing\nordered moments. Concerning the Mn1 sites, the DFT\ncalculations suggest that their magnetic moments are un-\nstable on their own, and come into existence depending\non the Mn2 environment. When this environment is FM\na sizable magnetic moment arises in the Mn1 sites, while\nif this environment is AFM the magnetic moment col-\nlapses. We conclude that the DFT calculations do not\nrule out the existence of strong spin \ructuations either\non the Mn1 or on the Mn2 sites. Previous experimental\nwork4has suggested that the magnetic excitation spec-\ntrum of the AFM2 phase consists of propagating spin{\nwaves and di\u000buse spin \ructuations, originating from the\n(a)\n(b)\nFIG. 6. (a) Theoretical inelastic scattering spectrum for the\nAFM2 phase of Mn 5Si3with parameters obtained from DFT\ncalculations in the absence of an external \feld. (b) Zooming\nin the low energy regions of the spectrum a splitting of the\nspin{wave modes is observed (double spin{gap) due to the\nsystem's biaxial anisotropy.\npresence of magnetic and nonmagnetic Mn sites within\nthis phase, respectively. Further investigation is required\nto enlighten this phenomenon.\nThe INS data obtained in the collinear AFM2 phase\nof Mn 5Si3have demonstrated a spin excitation double\nenergy gap for zero \feld at Q= (1;2;0), see Fig. 4(b).\nThat is, the low energy spectrum is composed of two\nexcitations of nonvanishing and distinct energies. In\na system with uniaxial anisotropy, these two excita-\ntions would be degenerate. Using the parameters ob-\ntained in our \frst{principles simulations and given in\nthe previous sections, we calculated the inelastic scat-\ntering spectrum for this phase of our material, as seen in\nFig. 6(a). In the low energy region around the Brillouin\nzone center G= (1;2;0), we observe an energy gap of\nabout 3 meV. However, a closer look reveals the double\nenergy{gap structure which is due to the second easy{\naxis anisotropy predicted in our calculations, as demon-\nstrated by Fig. 6(b).\nDespite this qualitative agreement, these energy gaps\nare much higher than those in Fig. 4(b) obtained through\nthe neutron scattering measurements. We believe that\nthis discrepancy comes from the di\u000eculties associated\nwith modeling the spin{wave spectrum at \fnite temper-\nature. For example, the computed magnetic moments\n(2.4\u0016B) are higher than the ones obtained experimen-\ntally (1.48\u0016B)8, which could be explained by the ther-\nmal \ructuations of their orientations which have not been\ntheoretically considered. Possible e\u000bects associated with\nthe spin \ructuations of the Mn1 or Mn2 sites are also\nnot taken into account by our model Hamiltonian. An-\nother source of uncertainty is how temperature may a\u000bect\nthe magnetic interactions that de\fne the spin{wave ex-\ncitation spectrum. For instance, the e\u000bective magnetic7\nanisotropy energy was found to be strongly temperature{\ndependent in Mn 5Ge335, the ferromagnetic counterpart\nof Mn 5Si3.\nTo allow a direct comparison between the theoreti-\ncal spin{wave energies and the experimental data, we\nrescaled the DFT parameters to match the experimen-\ntal results. A good agreement was obtained by scaling\ndown uniformly the magnetic exchange interaction and\nthe anisotropy parameters by a factor of ten. For \fne\ntuning, the second easy{axis anisotropy parameter was\nadjusted from the scaled DFT value kc= 0:003 meV to\nkc= 0:009 meV.\nB. Results under external magnetic \feld\nUsing the rescaled parameters, we calculated the spin{\nwave energies as a function of the applied magnetic \feld\nto shed light on the observed behavior of the two low\nenergy magnon modes. To this aim, we included in our\nmodel Hamiltonian the Zeeman term:\nHZ=\u00002\u0016BX\niH\u0001Si; (9)\nto account for an external magnetic \feld applied along\nthecandacrystal axes. It is worth mentioning that pre-\nvious investigations in AFM systems with uniaxial and\nbiaxial anisotropy have described the in\ruence of an ex-\nternal magnetic \feld in the magnon properties36,37. How-\never, the observed behavior from our INS results shown\nin Fig. 4(b) is not su\u000eciently covered.\nThe results of our calculations for an external mag-\nnetic \feld applied along the ccrystal axis are shown in\nFig. 7(a). We observe that the energy of one mode is\ninsensitive to the magnetic \feld ( \u000b{mode), while the en-\nergy of the other mode ( \f{mode) increases monotoni-\ncally for larger \felds. Moreover, the \f{mode is lower in\nenergy than the \u000b{mode for zero \feld, and for increasing\n\feld strength they eventually cross, which is in qualita-\ntive agreement with the INS data shown in Fig. 4(b).\nNo avoided crossing is observed with our model. At zero\n\feld, the energy of the \u000b{mode is solely a function of\nkband the magnetic exchange interactions. The energy\ndi\u000berence between the two modes is determined by kc,\nwhich is zero for kc= 0. When kc=kb, the\f{mode gap\ncloses. These results are made explicit by our analytical\nstudy of a corresponding one{dimensional antiferromag-\nnetic system including nearest{neighbor{only exchange\ninteraction Jand a biaxial anisotropy. In the asymp-\ntotic limit of k\u001cJ38, the energies of two spin{wave\nmodes are E\u000b\u00192Sp\nJkbandE\f\u00192Sp\nJ(kb\u0000kc).\nWe note that kc> kbrepresents an unstable situation\nbecause the c{axis becomes the preferred one, but the\nmagnetic moments are aligned along b. If we apply the\n\feld alonga, the \feld dependence of these modes inverts,\nas shown in Fig. 7(b).\nHkc\nHka\u000b-mode\f-mode\n\u000b-mode\n\f-mode(a)\n(b)\n(c) (d)\n\u000b-mode\nab\nc\n\f-mode\nab\nc\nFIG. 7. Adiabatic spin{wave energies of Mn 5Si3as a function\nof the external \feld applied along (a) the c{axis (perpendicu-\nlar to the preferred axis but parallel to the second anisotropy\neasy{axis). The \f{mode disperses with the \feld because it\nhas linear polarization perpendicular to it. Meanwhile, the\n\u000b{mode has linear polarization parallel to the \feld (along the\nc{axis). (b) External \feld applied along the a{axis (parallel\nto the hard{axis). The \feld couples with the \u000b{mode which\nhas linear polarization perpendicular to it. In this case, no\nenergy crossing occurs. Lines in (a) and (b) are guides for the\neyes. (c) and (d) precessional dynamics of the spins for biaxial\nanisotropy and without an external magnetic \feld for modes\n\u000band\f, respectively. The red and blue arrows indicate an-\ntiparallel spins within a Mn2 triangle precessing around their\nequilibrium direction ( b{axis). The central gray arrows rep-\nresent the dynamics of the net magnetization of the system,\nwhich correspond to linear (longitudinal) precessions.\nC. Precessional nature of the spin{wave modes\nTo understand why the two spin{wave modes react so\ndi\u000berently to the external magnetic \feld, we now dis-\ncuss their precessional nature. For zero \feld and kc= 0\n(i.e. with uniaxial anisotropy), the modes are degener-8\nate as pointed out before. For these modes, the spins\nhave an elliptical precession because in a certain instant\nof their revolution they are perfectly antiparallel but a\nquarter of revolution later, they are noncollinear36. This\nnoncollinear alignment is unfavored by the exchange in-\nteraction, making the precession elliptical. The ellipses\ndescribing the precession of each mode have their major\naxes perpendicular to each other. However, there is no\npreferred orientation for these major axes as long as the\nsystem remains isotropic in the ac{plane. Besides, these\nmodes can also be globally characterized by the dynam-\nics of the net magnetization, which is de\fned by the sum\nof all spins at each instance of time. For these modes,\nthe net magnetization oscillates linearly along the minor\nellipse axis of the corresponding mode, as illustrated by\nthe central gray arrow in Fig. 7(c) and (d). These modes\nare said to be linearly polarized along the oscillation axis\nof the net magnetization. Furthermore, they have no net\nangular momentum and can be thought of as longitudinal\nmodes28.\nA preferred direction of the ellipse major axes can be\nachieved in two ways. Firstly, we can apply a magnetic\n\feld perpendicularly to the preferred axis, i.e., ?b{axis.\nAssuming that the \feld was applied along c, the\u000b{mode\nbecomes linearly polarized along c, and consequently, the\n\f{mode is polarized along a. An oscillating net magne-\ntization parallel to the \feld is not subjected to a torque\ndue to that \feld. Therefore, only the \f{mode with po-\nlarization perpendicular to the magnetic \feld is a\u000bected\nsuch that its energy increases with the \feld.\nSecondly, we can introduce a second anisotropy axis\nperpendicular to the \frst with weaker strength, let us\nsay alongc. The energy of the mode with the major axis\nof the ellipse along c(\f{mode) is reduced because the\nsystem gains energy when the spins tilt in that direction.\nThe energy of the \u000b{mode, which tilts mostly perpendic-\nularly toc, is not much a\u000bected. Adding the magnetic\n\feld also along cincreases the energy of the \f{mode as\nbefore, which eventually leads to the crossing of the en-\nergy of the two modes (see Figure 7(a)). Figure 7(b)\ndemonstrates that if we apply the magnetic \feld along\na, therefore, perpendicular to the linear polarization of\nthe\u000b{mode, then it would be the energy of this mode to\nincrease with the \feld.\nPolarized neutron scattering experiments around an\nAFM zone center capture the elliptic polarization of the\nmagnon modes. Therefore, the proposed behavior re-\ngarding the polarization of the \u000band\f{mode is also\nre\rected in the polarized INS spectra shown in Fig. 5.\nThe ellipses major axis of the processional motion of the\n\u000b{mode is along the a{axis and consistently the intensity\nof this mode will appear in the subtracted spin \ructua-\ntions spectra obtained from LPA in h\u000eMai. Equivalently,\nthe\f{mode shows maximum intensity in h\u000eMci, since the\nmajor axis of the processional motion is along the c{axis.\nHkc\nHka\u000b-mode\f-mode\n\u000b-mode\n\f-mode(a)\n(b)\nFIG. 8. E\u000bects of the DMI on the spin{wave dispersion of\nMn5Si3atQ= (1;2;0). The external \feld was applied along\n(a) thec{axis (perpendicular to the preferred axis but parallel\nto the second anisotropy easy{axis), and along (b) the a{axis\n(parallel to the hard{axis). The DMI increases the energy of\nthe\f{mode and its e\u000bect on the \u000b{mode is almost negligible.\njDijj=Dis in units of meV. Lines are guides for the eyes.\nD. Canting of magnetic moments\nPolarized single{crystal neutron di\u000braction experi-\nments8in the AFM2 phase of Mn 5Si3showed that the\nmagnetic moments are aligned along the b{axis and that\nthere are no components of moments parallel to the c{\naxis. However, a more recent neutron di\u000braction study\nperformed on polycrystalline samples7suggested a devi-\nation from the perfect collinearity, which is temperature{\ndependent increasing up to 8\u000enear 70 K with respect to\ntheb{axis, but with the moments still con\fned in the\nab{plane. Such a spin canting could originate from the\nDzyaloshinskii{Moriya39,40exchange interaction (DMI).\nTo investigate the \feld dependence of the two modes in\nthe presence of DMI, we included the following term into\nour model Hamiltonian:\nHDMI=\u0000X\nijDij\u0001(Si\u0002Sj): (10)\nThe results of the ab initio calculation indicated that\nthe DMI is indeed \fnite and smaller than 10% of the ex-\nchange interaction for the Mn2 pair coupled by J1where\nDijjj^c. The impact of DMI on the \u000band\f{mode for two9\n\feld directions is shown in Fig. 8. The e\u000bect of the DMI\nin the\u000b{mode is almost negligible, independently of the\ndirection of the external magnetic \feld. In contrast, the\nDMI causes the \f{mode energy to increase. For a \feld\napplied along the a{axis, the almost \rat dispersion of\nthe\f{mode shifts to higher energies with increasing D,\nwhich eventually results in a crossing with the \u000b{mode\n(see the blue curves in Fig. 8(b)). When the \feld is ap-\nplied along the c{axis, the increase in energy is accom-\npanied by a change of the \\parabolic\" dispersion of the\n\f{mode, as shown in Fig. 8(a). It is worth noticing that\nat zero \felds, the energy of the \u000b{mode is solely given\nby the exchange interaction and the \frst anisotropy axis,\nwhile the\f{mode energy is determined by the DMI in\ncombination with the second anisotropy axis. Based on\nour results from the INS measurements (see Fig. 4(b)),\nwe can conclude since a crossing of the two modes is\nobserved for Hk^c, that the DM exchange interaction\nshould be very weak in the AFM2 phase of Mn 5Si3.\nV. CONCLUSIONS\nWe investigated the microscopic magnetic properties\nof the collinear antiferromagnetic phase of Mn 5Si3with\ninelastic neutron scattering measurements and density\nfunctional theory simulations. The measurements have\nrevealed two low{energy spin{wave modes that disperse\nwith the wave{vector as expected for gapped antiferro-\nmagnetic magnons. The high resolution of the measure-\nments also allowed us to detect the small energy split-\nting between the two modes of about 0.2 meV. When a\nmagnetic \feld was applied to the sample along the c{\naxis, the two modes showed a very di\u000berent response\nto the \feld. The energy of the lower{energy mode in-\ncreased with increasing \feld, eventually crossing the en-ergy of the other mode, which showed no observable re-\nsponse to the \feld. These modes were further charac-\nterized by polarized neutron scattering measurements,\nwhich showed that their polarization is distinct and likely\nelliptical, with di\u000berent major axes for each ellipse. Our\n\frst{principles calculations capture the main features ob-\nserved so far experimentally, namely, the collinear ar-\nrangement of the magnetic moments, the coexistence\nof magnetic and nonmagnetic Mn sites, and the biax-\nial magnetocrystalline anisotropy. Also, we expanded\nthe existing studies for collinear antiferromagnets with\nbiaxial anisotropy regarding the polarization of the low\nenergy spin{wave modes and their evolution under the\nin\ruence of an external magnetic \feld. Finally, we inves-\ntigated theoretically the impact of the Dzyaloshinskii{\nMoriya interaction in the magnon spectrum and its im-\nplications depending on the strength of the anisotropy.\nOur results could \fnd echo in other similar systems where\ninteresting magnonic phenomena occur in the \feld of an-\ntiferromagnetic spintronics.\nVI. ACKNOWLEDGMENTS\nThe neutron data collected at the Institut Laue\nLangevin for the present work are available at 41{43.\nN.B. acknowledges the support of JCNS through the\nTasso Springer fellowship. This work was also supported\nby the Brazilian funding agency CAPES under Project\nNo. 13703/13-7 and the European Research Council\n(ERC) under the European Union's Horizon 2020 re-\nsearch and innovation program (ERC-consolidator Grant\nNo. 681405-DYNASORE). We gratefully acknowledge\nthe computing time granted by JARA-HPC on the su-\npercomputer JURECA at Forschungszentrum J ulich and\nby RWTH Aachen University.\n\u0003f.dos.santos@fz-juelich.de (currently based at EPFL: \ra-\nviano.dossantos@ep\r.ch)\nyn.biniskos@fz-juelich.de\n1V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono,\nand Y. Tserkovnyak, Rev. Mod. Phys. 90, 015005 (2018).\n2P. J. Brown, J. B. Forsyth, V. Nunez, and F. Tasset,\nJournal of Physics: Condensed Matter 4, 10025 (1992).\n3C. S urgers, G. Fischer, P. Winkel, and H. v. L ohneysen,\nNature Communications 5, 3400 (2014).\n4N. Biniskos, K. Schmalzl, S. Raymond, S. Petit, P. Ste\u000bens,\nJ. Persson, and T. Br uckel, Physical Review Letters 120,\n257205 (2018).\n5B. Das, B. Balasubramanian, P. Manchanda, P. Mukher-\njee, R. Skomski, G. C. Hadjipanayis, and D. J. Sellmyer,\nNano Letters 16, 1132 (2016), publisher: American Chem-\nical Society.\n6Y. Sun, B. Sun, J. He, G. Yang, and C. Wang, Nature\nCommunications 11, 647 (2020), number: 1 Publisher: Na-\nture Publishing Group.7M. Gottschilch, O. Gourdon, J. Persson, C. d. l. Cruz,\nV. Petricek, and T. Brueckel, Journal of Materials Chem-\nistry 22, 15275 (2012), publisher: The Royal Society of\nChemistry.\n8P. J. Brown and J. B. Forsyth, Journal of Physics: Con-\ndensed Matter 7, 7619 (1995).\n9C. S urgers, T. Wolf, P. Adelmann, W. Kittler, G. Fischer,\nand H. v. L ohneysen, Scienti\fc Reports 7, 42982 (2017).\n10L. Vinokurova, V. Ivanov, E. Kulatov, and A. Vlasov,\nJournal of Magnetism and Magnetic Materials 90-91 , 121\n(1990).\n11R. F. Luccas, G. S\u0013 anchez-Santolino, A. Correa-Orellana,\nF. J. Mompean, M. Garc\u0013 \u0010a-Hern\u0013 andez, and H. Suderow,\nJournal of Magnetism and Magnetic Materials 489, 165451\n(2019).\n12S. C. Das, K. Mandal, P. Dutta, S. Pramanick, and\nS. Chatterjee, Physical Review B 100, 024409 (2019), pub-\nlisher: American Physical Society.\n13M. R. Silva, P. J. Brown, and J. B. Forsyth, Journal of\nPhysics: Condensed Matter 14, 8707 (2002), publisher:10\nIOP Publishing.\n14H. Al-Kanani and J. Booth, Journal of Magnetism and\nMagnetic Materials 140-144 , 1539 (1995), international\nConference on Magnetism.\n15A. Irizawa, A. Yamasaki, M. Okazaki, S. Kasai,\nA. Sekiyama, S. Imada, S. Suga, E. Kulatov, H. Ohta, and\nT. Nanba, Solid State Communications 124, 1 (2002).\n16C. S urgers, W. Kittler, T. Wolf, and H. v.\nL ohneysen, AIP Advances 6, 055604 (2016),\nhttps://doi.org/10.1063/1.4943759.\n17D. Songlin, O. Tegus, E. Br uck, J. C. P. Klaasse, F. R.\nde Boer, and K. H. J. Buschow, Journal of Alloys and\nCompounds 334, 249 (2002).\n18K. Schmalzl, W. Schmidt, S. Raymond, H. Feilbach,\nC. Mounier, B. Vettard, and T. Br uckel, Nuclear Instru-\nments and Methods in Physics Research Section A: Accel-\nerators, Spectrometers, Detectors and Associated Equip-\nment 819, 89 (2016).\n19L. P. Regnault, B. Ge\u000bray, P. Fouilloux, B. Longuet,\nF. Mantegazza, F. Tasset, E. Leli\u0012 evre-Berna, S. Pujol,\nE. Bourgeat-Lami, N. Kernavanois, M. Thomas, and\nY. Gibert, Physica B: Condensed Matter Proceedings of\nthe Third European Conference on Neutron Scattering,\n350, E811 (2004).\n20T. Chatterji, ed., Neutron scattering from magnetic mate-\nrials, 1st ed. (Elsevier, Amsterdam ; Boston, 2006) oCLC:\nocm61748467.\n21In this article we use the orthorhombic coordinate system\nand the scattering vector Qis expressed in Cartesian coor-\ndinates Q= (Qh;Qk;Ql) given in reciprocal lattice units\n(r.l.u.). The wave{vector qis related to the momentum\ntransfer through ~Q=~G+~q, where Gis a Brillouin\nzone center and G= (h;k;l ).\n22S. Ibuka, S. Itoh, T. Yokoo, and Y. Endoh, Phys. Rev. B\n95, 224406 (2017).\n23N. Papanikolaou, R. Zeller, and P. H. Dederichs, Journal\nof Physics: Condensed Matter 14, 2799 (2002), publisher:\nIOP Publishing.\n24S. Vosko, L. Wilk, and M. Nusair, Can. J. Phys. 58, 1200\n(1980).\n25K. Wildberger, P. Lang, R. Zeller, and P. H. Dederichs,\nPhysical Review B 52, 11502 (1995), publisher: American\nPhysical Society.\n26A. I. Liechtenstein, M. I. Katsnelson, V. P. Antropov, and\nV. A. Gubanov, Journal of Magnetism and Magnetic Ma-terials 67, 65 (1987).\n27H. Ebert and S. Mankovsky, Phys. Rev. B 79, 045209\n(2009).\n28F. J. dos Santos, M. dos Santos Dias, F. S. M. Guimar~ aes,\nJ. Bouaziz, and S. Lounis, Physical Review B 97, 024431\n(2018).\n29T. Holstein and H. Primako\u000b, Physical Review 58, 1098\n(1940).\n30N. N. Bogoljubov, Il Nuovo Cimento (1955-1965) 7, 794\n(1958).\n31F. J. dos Santos, M. dos Santos Dias, and S. Lounis, Phys-\nical Review B 102, 104401 (2020), publisher: American\nPhysical Society.\n32F. J. dos Santos, M. dos Santos Dias, and S. Lounis, Phys-\nical Review B 102, 104436 (2020), publisher: American\nPhysical Society.\n33F. J. dos Santos, M. dos Santos Dias, and S. Lounis,\nPhysical Review B 95, 134408 (2017), publisher: Amer-\nican Physical Society.\n34E. Mendive-Tapia and J. B. Staunton, Phys. Rev. B 99,\n144424 (2019).\n35N. Maraytta, J. Voigt, C. Salazar Mej\u0013 \u0010a, K. Friese, Y. Sk-\nourski, J. Per\u0019on, S. M. Salman, and T. Br uckel,\nJournal of Applied Physics 128, 103903 (2020),\nhttps://doi.org/10.1063/5.0020780.\n36F. Ke\u000ber and C. Kittel, Physical Review 85, 329 (1952),\npublisher: American Physical Society.\n37S. M. Rezende, A. Azevedo, and R. L. Rodr\u0013 \u0010guez-Su\u0013 arez,\nJournal of Applied Physics 126, 151101 (2019), publisher:\nAmerican Institute of Physics.\n38F. J. dos Santos, First-principles study of collective\nspin excitations in noncollinear magnets , Ph.D. thesis,\nForschungszentrum J ulich GmbH, Zentralbibliothek, Ver-\nlag (2020), iSBN: 9783958064591 Number: RWTH-2020-\n01879.\n39I. Dzyaloshinsky, Journal of Physics and Chemistry of\nSolids 4, 241 (1958).\n40T. Moriya, Phys. Rev. 120, 91 (1960).\n41https://doi.ill.fr/10.5291/ILL-DATA.CRG-2331 (2016).\n42https://doi.ill.fr/10.5291/ILL-DATA.CRG-2620 (2019).\n43https://doi.ill.fr/10.5291/ILL-DATA.4-01-1618 (2019)."    },    {        "title": "2011.05722v2.Characterization_of_room_temperature_in_plane_magnetization_in_thin_flakes_of_CrTe__2__with_a_single_spin_magnetometer.pdf",        "content": "Characterization of room-temperature in-plane magnetization in thin \rakes of CrTe 2\nwith a single spin magnetometer\nF. Fabre,1A. Finco,1A. Purbawati,2A. Hadj-Azzem,2N. Rougemaille,2J. Coraux,2I. Philip,1and V. Jacques1\n1Laboratoire Charles Coulomb, Universit\u0013 e de Montpellier and CNRS, 34095 Montpellier, France\n2Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut NEEL, 38000 Grenoble, France\n(Dated: February 12, 2021)\nWe demonstrate room-temperature ferromagnetism with in-plane magnetic anisotropy in thin\n\rakes of the CrTe 2van der Waals ferromagnet. Using quantitative magnetic imaging with a single\nspin magnetometer based on a nitrogen-vacancy defect in diamond, we infer a room-temperature\nin-plane magnetization in the range of M\u001827 kA/m for \rakes with thicknesses down to 20 nm.\nIn addition, our measurements indicate that the orientation of the magnetization is not determined\nsolely by shape anisotropy in micron-sized CrTe 2\rakes, which suggest the existence of a non-\nnegligible magnetocrystalline anisotropy. These results make CrTe 2a unique system in the growing\nfamily of van der Waals ferromagnets, as it is the only material platform known to date which o\u000bers\nan intrinsic in-plane magnetization and a Curie temperature above 300 K in thin \rakes.\nI. INTRODUCTION\nFerromagnetic van der Waals (vdW) crystals o\u000ber nu-\nmerous opportunities both for the study of exotic mag-\nnetic phase transitions in low-dimensional systems [1] and\nfor the design of innovative, atomically-thin spintronic\ndevices [2, 3]. Since the discovery of a two-dimensional\n(2D) magnetic order in monolayers of CrI 3[4] and\nCr2Ge2Te6[5] crystals, the family of vdW ferromagnets\nhas expanded very rapidly [6{8]. However, most of these\ncompounds have a Curie temperature ( Tc) well below 300\nK, which appears as an important drawback for future\ntechnological applications. An intense research e\u000bort is\ntherefore currently devoted to the identi\fcation of high-\nTc2D magnets [3].\nIn this context, the vdW crystal Fe 3GeTe 2appears as\na serious candidate because it can be grown in wafer-\nscale through molecular beam epitaxy and it exhibits a\nstrong perpendicular magnetic anisotropy [9, 10]. Al-\nthough its intrinsic Tcdrops to 130 K in the monolayer\nlimit [11], it might be raised above room-temperature ei-\nther by ionic gating [12, 13], interfacial engineering [14],\nor by micro-patterning, as demonstrated so far for rather\nthick \flms [15]. In addition, other FeGeTe alloys, such\nas Fe 4GeTe 2and Fe 5GeTe 2, exhibit high Tc, still lower\nthan room temperature but close to it [16, 17]. An-\nother promising strategy consists in incorporating mag-\nnetic dopants into 2D materials to form dilute magnetic\nsemiconductors [18]. This approach was recently em-\nployed to induce room-temperature ferromagnetism in\nWSe 2monolayers doped with vanadium [19, 20]. Finally,\nan intrinsic ferromagnetic order was reported in epitax-\nial layers of VSe 2[21], MnSe 2[22] and VTe 2[23] under\nambient conditions, although the interpretation of these\nexperiments still remains debated [24, 25].\nIn this work, we follow an alternative research direc-\ntion by studying the room-temperature magnetic proper-\nties of micron-sized \rakes exfoliated from a CrTe 2crystal\nwith 1Tstructure. In its bulk form, this layered transi-\ntion metal dichalcogenide is a ferromagnet with in-planemagnetization, i.e.pointing perpendicular to the caxis,\nand aTcaround 320 K [26]. This combination of proper-\nties is unique in the growing family of vdW ferromagnets.\nRecent studies have reported that the magnetic order is\npreserved at room temperature in exfoliated CrTe 2\rakes\nwith thicknesses in the range of a few tens of nanome-\nters [27, 28]. However, obtaining quantitative estimates\nof the magnetization in such micron-sized \rakes remains\na di\u000ecult task, which requires the use of non-invasive\nmagnetic microscopy techniques combining high sensitiv-\nity with high spatial resolution. These performances are\no\u000bered by magnetometers employing a single nitrogen-\nvacancy (NV) defect in diamond as an atomic-size quan-\ntum sensor [29{31]. In recent years, this microscopy tech-\nnique has found many applications in condensed matter\nphysics [32], including the study of chiral spin textures in\nultrathin magnetic materials [33{35], current \row imag-\ning in graphene [36] and the analysis of the magnetic or-\nder in vdW magnets down to the monolayer limit [37{39].\nHere we use scanning-NV magnetometry to infer quan-\ntitatively the in-plane magnetization in exfoliated CrTe 2\n\rakes under ambient conditions. Our measurements con-\n\frm that the ferromagnetic order is preserved in few tens\nof nanometers thick \rakes, although with a low room-\ntemperature magnetization M\u001827 kA/m. This value\nis \fve times smaller that the one measured in a bulk\nCrTe 2crystal. Such a reduction of the magnetization is\nattributed to a decreased Curie temperature in exfoliated\n\rakes. Moreover, our results show that shape anisotropy\nalone does not \fx the in-plane orientation of the mag-\nnetization in micron-sized CrTe 2\rakes, pointing out the\nexistence of a substantial magnetocrystalline anisotropy.\nII. MATERIALS AND METHODS\nA bulk 1T-CrTe 2crystal was synthesized following the\nprocedure described in Ref. [26]. The in-plane magnetiza-\ntion of this layered ferromagnet was \frst characterized as\na function of temperature through vibrating sample mag-arXiv:2011.05722v2  [cond-mat.mtrl-sci]  11 Feb 20212\nnetometry under a magnetic \feld of 500 mT. The results\nshown in Fig. 1(a) indicate a Curie temperature around\n320 K and a magnetization reaching M\u0018120 kA/m\nunder ambient conditions. CrTe 2\rakes with thicknesses\nranging from a few tens to a hundred of nanometers were\nthen obtained by mechanical exfoliation and transferred\non a SiO 2/Si substrate. We note that the probability\nto obtain thin CrTe 2\rakes through mechanical exfolia-\ntion is still very low compared to other layered transi-\ntion metal dichalcogenides, such as MoS 2or WSe 2[28].\nThe thinnest \rake studied in this work has a thickness\nof\u001820 nm. Like all van der Waals ferromagnets known\nto date, CrTe 2\rakes are unstable under oxygen atmo-\nsphere. However, a recent study combining X-ray and\nRaman spectroscopy has shown that oxidation of CrTe 2\n\rakes occurs typically within a day scale under ambient\nconditions and is limited to the very \frst outer layers [28].\nIn this work, CrTe 2\rakes were not encapsulated and all\nthe measurements were done within a day after exfolia-\ntion to mitigate oxidation.\nMagnetic imaging was performed with a scanning-NV\nmagnetometer operating under ambient conditions [31].\nAs sketched in Fig. 1(b), a single NV defect integrated\ninto the tip of an atomic force microscope (AFM) was\nscanned above CrTe 2\rakes to probe their stray mag-\nnetic \felds. At each point of the scan, a confocal optical\nmicroscope placed above the tip was used to monitor the\nmagnetic-\feld-dependent photoluminescence (PL) prop-\nerties of the NV defect under green laser illumination.\nIn this work, we employed a commercial diamond tip\n(Qnami, Quantilever MX) with a characteristic NV-to-\nsample distance dNV= 80\u000610 nm, as measured through\nan independent calibration procedure [40]. Two di\u000berent\nmagnetic imaging modes were used. In the limit of weak\nstray \felds ( <5 mT), quantitative magnetic \feld map-\nping was obtained by recording the Zeeman shift of the\nNV defect electron spin sublevels through optical detec-\ntion of the electron spin resonance (ESR). This method\nrelies on microwave driving of the NV spin transition\ncombined with the detection of the spin-dependent PL\nintensity of the NV defect [41]. For stronger magnetic\n\felds (>5 mT), the scanning-NV magnetometer was\nrather used in all-optical, PL quenching mode [42, 43].\nIn this case, localized regions of the sample producing\nlarge stray \felds are simply revealed by an overall re-\nduction of the PL signal induced by a mixing of the NV\ndefect spin sublevels [44]. We note that the diameter of\nthe scanning diamond tip is around 200 nm in order to\nact as an e\u000ecient waveguide for the PL emission of the\nNV defect [45, 46]. As a result, such tips cannot provide\nprecise topographic information of the sample. For the\nthinnest \rakes, the topography was thus imaged by using\nconventional, sharp AFM tips.\nFor a ferromagnetic material, stray magnetic \felds can\nbe produced by magnetization patterns presenting a non-\nzero divergencer\u0001M6= 0. Considering a CrTe 2\rake,\nthis can occur (i) at the edges of the \rake, (ii) at the\nlocation of non-collinear spin textures such as domain\n250200150100500 M (kA/m)300200100 Temperature (K)(a)\n(b)\nSubstratexyzNVaxis\u0000NV✓NVdiamond tipdNVMObjectivePLbulk CrTe2FIG. 1. (a) Temperature dependence of the magnetization M\nin a bulk crystal of CrTe 2with 1Tpolytype. The measure-\nment is performed by vibrating sample magnetometry under\na magnetic \feld of 500 mT. At room temperature, the magne-\ntization is around 120 kA/m (black dashed lines). The inset\nshows the layered crystal structure of 1 T-CrTe 2. (b) A single\nNV defect (red arrow) localized at the apex of a diamond tip\nis scanned above exfoliated CrTe 2\rakes. A microscope ob-\njective placed above the tip is used to collect the photolumi-\nnescence (PL) of the NV defect under green laser excitation.\nIn this work, the NV-to-sample distance is dNV\u001880 nm and\nthe quantization axis of the NV defect is characterized by the\nspherical angles ( \u0012NV= 58\u000e;\u001eNV= 103\u000e) in the laboratory\nframe of reference ( x;y;z ). The black arrows indicate the\nmagnetic \feld lines generated at the edges of a CrTe 2\rake\nwith uniform in-plane magnetization M.\nwalls, and (iii) at the position of thickness steps on the\n\rake. In this work, the magnetization is inferred from the\nmeasurement of the stray \feld produced at the edges of\nthe \rake [Fig. 1(b)]. Therefore, the analysis is made sim-\npler for uniformly magnetized \rakes with a homogeneous\nthickness. While uniform spin textures can be readily ob-\ntained by applying a bias magnetic \feld, obtaining thin\nCrTe 2\rakes with a uniform thickness through mechani-\ncal exfoliation is currently achieved with a low probabil-\nity. Consequently, variations in thickness must be care-\nfully taken into account when analyzing the experimental\nresults.3\nxy\n0 100 200 300\nHeight (nm)Topography(a)\nPL scan(b)\n0.6 0.8 1\nNormalized PLSimulation\nφM=20°\nM(c)\nSimulation\nφM=-80°\nM(d)\nFIG. 2. (a) AFM image of a 150 nm-thick CrTe 2\rake recorded with the scanning-NV magnetometer. (b) Corresponding PL\nmap showing a quenching contour at the edges of the \rake. The shape of the \rake is indicated by the black dashed lines.\nThe recorded PL signal is normalized with the one measured far from the \rake. The experiment is performed at zero external\nmagnetic \feld. (c,d) Simulated PL maps for two di\u000berent orientations \u001eMof the in-plane magnetization: (c) \u001eM= 20\u000eand\n(d)\u001eM=\u000080\u000e. Here, the norm of the magnetization is \fxed arbitrarily to its bulk value M= 120 kA/m. The PL quenching\ninduced by energy transfer between the NV sensor and the metallic sample is not included in the simulations.\nIII. RESULTS AND DISCUSSION\nIn a \frst experiment, we imaged a CrTe 2\rake with\na large thickness t\u0018150 nm [Fig. 2(a)]. Considering\nthe saturation magnetization of the bulk CrTe 2crystal,\nmagnetic simulations predict stray \feld amplitudes larger\nthan 10 mT at a distance dNV\u001880 nm above the edges\nof the 150-nm-thick \rake. The scanning-NV magnetome-\nter was thus operated in the PL quenching mode for such\na thick \rake. Fig. 2(b) displays the resulting PL map.\nSeveral observations can be made. First, a quenching of\nthe PL signal is observed when the NV defect is placed\nabove the \rake. This quenching e\u000bect, which is constant\nall over the \rake, does not have a magnetic origin. It is\nlinked to the metallic character of CrTe 2[47, 48]. Second,\na stronger PL quenching is obtained at the \rake edges,\nas expected for a single ferromagnetic domain. We note\nthat the additional quenching spot observed at the top-\nright edge of the \rake results from the stray \feld pro-\nduced by a large variation of the thickness [Fig. 2(a)].\nGiven the di\u000berent sources of PL quenching involved in\nthis experiment, quantitative estimates of the magneti-\nzation cannot be obtained. However, the analysis of the\nPL quenching distribution can be used to extract qual-\nitative information about the orientation of the magne-\ntization. To this end, simulations of the PL map were\ncarried out considering only the e\u000bect of magnetic \felds.\nThe geometry of the \rake used for the simulation was\ninferred from the AFM image simultaneously recorded\nwith the NV microscope [Fig. 2(a)]. Assuming a uniform\nin-plane magnetization Mwith an azimuthal angle \u001eM\nin the (x;y) plane, the stray \feld was \frst calculated\nat a distance dNVabove the \rake. The resulting PL\nquenching map was then simulated by using the model\nof magnetic-\feld-dependent photodynamics of the NVdefect described in Ref. [44]. Simulated PL maps ob-\ntained for two di\u000berent orientations of the magnetization\nare shown in Figs. 2(c,d). In micron-sized \rakes, shape\nanisotropy should favor a magnetization pointing along\nthe long axis of the \rake. Interestingly, the PL map sim-\nulated for such a magnetization direction disagrees with\nthe experimental data [Fig. 2(c)], which instead suggest\na magnetization pointing perpendicular to the long axis\nof the \rake [Fig. 2(d)]. This result is a \frst indication of\na non-negligible, in-plane magnetocrystalline anisotropy.\nThinner CrTe 2\rakes, i.e.producing less stray \feld,\nwere then studied through quantitative magnetic \feld\nimaging. To this end, a microwave excitation was ap-\nplied through a copper microwire directly spanned on the\nsample surface and the Zeeman shift of the NV defect's\nESR frequency was recorded at each point of the scan. In\nthe weak \feld regime, the ESR frequency is shifted lin-\nearly with the projection BNVof the stray magnetic \feld\nalong the NV defect quantization axis [31]. This axis\nwas precisely measured by applying a calibrated mag-\nnetic \feld [49], leading to the spherical angles \u0012NV= 58\u000e\nand\u001eNV= 103\u000ein the laboratory frame ( x;y;z ), as illus-\ntrated in Fig. 1(b). For these measurements, a bias \feld\nof 3 mT was applied along the NV defect axis in order\nto infer the sign of the stray magnetic \feld [31]. Fur-\nthermore, this bias \feld is strong enough to erase spin\ntextures such as domain walls, given the weak coercitive\n\feld of CrTe 2\rakes [28].\nA map ofBNVrecorded above a 35-nm thick CrTe 2\n\rake is shown in Fig. 3(b). A stray magnetic \feld around\n\u00061:5 mT is detected at two opposite edges of the \rake.\nIn principle, the underlying sample magnetization can\nbe retrieved from such a quantitative magnetic \feld map\nby using reverse propagation methods with well-adjusted\n\flters in Fourier-space [32]. Under several assumptions,4\nxy\n0 20 40\nHeight (nm)Topography (a)\n−1.5 0 1.5\nBNV(mT)NV image (b)\n Simulation\nM(c)\n Line profilesBNV(mT)\nLateral displacement (µm)(d)\nFIG. 3. (a) AFM image of a 35 nm-thick CrTe 2\rake. (b) Corresponding distribution of the magnetic \feld component BNV.\n(c) Simulated magnetic \feld distribution BNVfor a uniform in-plane magnetization M(black arrow) with an azimuthal angle\n\u001eM=\u0000145\u000ein the (x;y) plane and a norm M= 27 kA/m. The calculation is done at a distance dNV= 80 nm above the \rake,\nwhose shape is extracted from the topography image [black dashed lined in (a)]. (d) Fitting of an experimental line pro\fle\n(black markers) with the magnetic calculation (blue solid line). The position of the linecuts are indicated by the white dashed\nlines in (b) and (c). A magnetization M= 27\u00064 kA/m is obtained, the uncertainty being illustrated by the blue shaded area.\nThe red and green solid lines indicate the result of the calculation for M= 40 kA/m and M= 15 kA/m, respectively.\nthis method can be quite robust for magnetic materials\nwith out-of-plane magnetization [37{39]. For in plane\nmagnets, however, the reconstruction procedure ampli-\n\fes noise and is thus much less e\u000ecient [50]. As a re-\nsult, the recorded magnetic \feld distribution was rather\ndirectly compared to magnetic calculations in order to\nextract quantitative information on the sample magneti-\nzation.\nTo obtain precise information on the geometry of the\n\rake, the topography image was here measured with a\nsharp AFM tip [Fig. 3(a)]. The observed variations in\nthickness were included in the magnetic calculation (see\nAppendix A). Considering a uniformly magnetized \rake\nwith an azimuthal angle \u001eM, the stray \feld was calcu-\nlated at a distance dNVabove the \rake and then pro-\njected along the NV quantization axis in order to sim-\nulate a map of BNV. By comparing experimental data\nwith magnetic maps simulated for di\u000berent values of the\nangle\u001eM, the orientation of the in-plane magnetization\nwas \frst identi\fed, leading to \u001eM=\u0000145\u00065\u000e,i.e.\npointing along the short axis of the \rake [Fig. 3(c)].\nOnce again, this result suggests the presence of a non-\nnegligible magnetocrystalline anisotropy, in agreement\nwith recent works [28]. The norm Mof the magneti-\nzation vector was then estimated by \ftting experimental\nline pro\fles with the result of the calculation, leading to\nM= 27\u00064 kA/m [Fig. 3(d)]. We note that the stray \feld\namplitude depends on several parameters including dNV,\n\u001eM, the \rake thickness tand the NV defect quantization\naxis (\u0012NV;\u001eNV). Any imprecisions on these parameters\ndirectly translate into uncertainties on the evaluation of\nthe magnetization M. A detailed analysis of uncertain-\nties is provided in Appendix B.\nThe room-temperature magnetization measured in the\nexfoliated CrTe 2\rake is almost \fve times smaller thanthe one obtained in the bulk crystal. This observation\ncould be explained by a degradation of the sample surface\nthrough oxidation processes, leading to a thinner e\u000bective\nmagnetic thickness. However, recent experiments relying\non X-ray magnetic circular dichroism coupled to photoe-\nmission electron microscopy (XMCD-PEEM) have shown\nthat oxidation is limited to the \frst outer layers of the\nCrTe 2\rake [28]. Considering a 35-nm thick CrTe 2\rake,\nsurface oxidation can thus be safely neglected, and can-\nnot explain the observed reduction of the magnetization.\nThis e\u000bect is rather attributed to a decrease of the Curie\ntemperature in exfoliated \rakes, a phenomenon that has\nbeen observed in other vdW magnets such as Fe 3GeTe 2\nbelow 5-10 nm thicknesses [11, 51], and in more tra-\nditional ferromagnetic thin \flms below few nanometers\nthicknesses [52, 53]. The thickness marking the crossover\nfrom a thin \flm (2D-like) to a bulk magnetism (3D) is\nexpected to be strongly material-dependent and cannot\nbe predicted a priori in the case of CrTe 2[8]. However,\nfor a 35-nm thick \rake, bulk-like magnetism is a rea-\nsonable assumption and the magnetization's amplitude\nshould not be altered by the \flm thickness. In this thick-\nness regime, the main parameter altering the magnetiza-\ntion's amplitude is temperature, and how close or far it\nis fromTc. The estimation of the reduction in Curie\ntemperature was tentatively inferred by translating the\nCurie law measured for a bulk CrTe 2crystal [Fig. 1(a)]\nin order to reach M\u001827 kA/m at room temperature.\nThis is achieved for a reduction of Tcby\u001820 K. This\nvalue is in line with a recent study, which estimates a\nCurie temperature around 300 K for CrTe 2\rakes with a\nfew tens of nanometers thickness [27].\nTo support this \fnding, a similar analysis was per-\nformed for a 20-nm thick \rake [Fig. 4(a)]. Here, a stray\nmagnetic \feld is mainly detected along the bottom edge5\n0 15 30 45\nHeight (nm)Topography(a)\n−0.7 0 0.7\nBNV(mT)NV image(b)\n Simulation\nM(c)\nBNV(mT)\nLateral displacement (µm)(d)\nFIG. 4. (a) AFM image of a 20-nm thick CrTe 2\rake. (b)\nCorresponding map of the magnetic \feld component BNV. (c)\nSimulated map of BNVfor a uniform in-plane magnetization\nM(black arrow) with an azimuthal angle \u001eM=\u0000100\u000ein the\n(x;y) plane and a norm M= 26 kA/m. The black dashed line\nindicate the shape of the \rake used for the simulation. (d)\nFitting of an experimental line pro\fle (black markers) with\nthe magnetic simulation (blue solid line). The linecuts are\nindicated by the white dashed lines in (b) and (c). A mag-\nnetizationM= 26\u00064:0 kA/m is obtained, the uncertainty\nbeing illustrated by the blue shaded area. The inset shows a\nlinecut across the \rake (white dashed lines in (a)).\nof the \rake. On the opposite edge, the stray \feld distri-\nbution is quite inhomogeneous [Fig. 4(b)]. This observa-\ntion is attributed to damages of the \rake, which can be\nobserved in the AFM image. It is di\u000ecult to describe the\ncorresponding complex thickness variations, and hence to\ntake them into account in the simulations. These height\nvariations seem to correspond to several CrTe 2islands,\nwhose very small sizes could make them more prone to\noxidation than larger \rakes, and which may exhibit ran-\ndom magnetization orientations. We hence disregarded\nthe thickness variations in our structural model and the\nmagnetic calculation was performed for a \rake with uni-\nform thickness, which is a good approximation for the\nbottom and left edges of the \rake. A simulation of the\nstray \feld distribution for a magnetization with an az-\nimuthal angle \u001eM=\u0000100\u000610\u000ereproduces fairly well the\nexperimental results [Fig. 4(c)] and a quantitative anal-\nysis of line pro\fles across the bottom edge of the \rake\nleads toM= 26\u00064:0 kA/m [Fig. 4(d)], a similar value\nto that obtained for the 35-nm thick \rake. This observa-tion indicates that the magnetization is not signi\fcantly\nmodi\fed for thicknesses lying in the few tens of nanome-\nters range. A more in-depth analysis of the dependence\nof magnetization on thickness could not be carried out at\nthis stage, given the very low probability to obtain thin\nCrTe 2\rakes of homogeneous thickness through mechan-\nical exfoliation.\nIV. CONCLUSION\nTo conclude, we have used quantitative magnetic imag-\ning with a scanning-NV magnetometer to demonstrate\nthat exfoliated CrTe 2\rakes with thicknesses down to\n20 nm exhibit an in-plane ferromagnetic order at room\ntemperature with a typical magnetization in the range of\nM\u001827 kA/m. These results make CrTe 2a unique sys-\ntem in the growing family of vdW ferromagnets, because\nit is the only material platform known to date which of-\nfers an intrinsic in-plane magnetization and a Tcabove\nroom temperature in thin \rakes. These properties might\no\u000ber several opportunities for studying magnetic phase\ntransition in 2D- XY systems [54] and to design spin-\ntronic devices based on vdW magnets. The next chal-\nlenge will be to assess if the ferromagnetic order is pre-\nserved at room temperature in the few layers limit.\nACKNOWLEDGEMENTS\nThe authors warmly thank J. Vogel and M. N\u0013 u~ nez-\nRegueiro for fruitful discussions. This research has re-\nceived funding from the European Union H2020 Program\nunder Grant Agreement No. 820394 (ASTERIQs), the\nDARPA TEE program, and the Flag-ERA JTC 2017\nproject MORE-MXenes. A.F. acknowledges \fnancial\nsupport from the EU Horizon 2020 Research and Innova-\ntion program under the Marie Sklodowska-Curie Grant\nAgreement No. 846597 (DIMAF).\nAPPENDIX A: MAGNETIC FIELD SIMULATION\nAs indicated in the main text, thickness variations\nwithin the \rake can result in a magnetization pattern\nwith a non-zero divergence which produces a stray mag-\nnetic \feld. When possible, these variations were taken\ninto account in the magnetic calculation. This is illus-\ntrated in Fig. 5(a,b), where the geometry of the \rake\nused for the calculation includes a thickness step ob-\nserved in the topography image. In Fig. 5(c-e), we show\nthe magnetic \feld distributions produced at a distance\ndNV= 80 nm for three di\u000berent magnetization orien-\ntations of the \rake. First considering an out-of-plane\nmagnetization [Fig. 5(c)], the simulated magnetic \feld\ndistribution does not agree with the experimental data\nshown in Fig. 3. This con\frms that the magnetization in\nlying in the plane of the CrTe 2\rake, i.e.perpendicular6\n02040Height (nm)Topography(a)\nt=35 nmt=25 nm(b)\nMOut-of-plane\nmagnetization(c)\nIn-plane magnetization\nΦM=-30°\nM(d)\n−1.501.5\nSimulated BNV(mT)In-plane magnetization\nΦM=-145°\nM(e)\nFIG. 5. (a) AFM image of CrTe 2\rake shown in Fig. 3. (b) Geometry of the \rake used for the magnetic calculation, which\nincludes the thickness step observed in the AFM imafge. (c-e) Calculated maps of BNVfor a uniform magnetization Mpointing\nout-of plane (c), and in-plane with an azimuthal angle \u001eM=\u000030\u000e(d) or\u001eM=\u0000145\u000e(e). The norm of the magnetization is\n\fxed toM= 27 kA/m.\nto thecaxis as obtained in the bulk crystal. Simula-\ntions performed for a uniform in-plane magnetization for\ntwo di\u000berent values of the azimuthal angle \u001eMin the\n(x;y) plane are shown in Fig. 5(d,e). A comparison with\nexperimental data allows the identi\fcation of the mag-\nnetization orientation.\nAPPENDIX B: ANALYSIS OF UNCERTAINTIES\nThe norm of the magnetization Mis obtained by\n\ftting line pro\fles of the measured stray \feld dis-\ntribution with the result of the magnetic calculation\n[see Fig. 3(d)]. In this section, we analyze the uncertainty\nof this measurement by using the methodology described\nin Ref. [55]. The uncertainties result (i) from the \ftting\nprocedure itself and (ii) from those on the parameters\npi=fdNV;\u0012NV;\u001eNV;t;\u001e Mgwhich are all involved in the\nmagnetic calculation. In the following, the parameters\npiare expressed as pi= \u0016pi+\u001bpi, where \u0016pidenotes the\nnominal value of parameter piand\u001bpiits standard error.\nThese parameters are independently evaluated as follows:\n\u000fThe probe-to-sample distance dNVis inferred\nthrough a calibration measurement, following the\nprocedure described in Ref. [40], leading to dNV=\n80\u000610 nm.\n\u000fThe NV defect quantization axis is measured by\nrecording the ESR frequency as a function of theamplitude and orientation of a calibrated mag-\nnetic \feld, leading to spherical angles ( \u0012NV=\n58\u00062\u000e;\u001eNV= 103\u00062\u000e) in the laboratory frame of\nreference (x;y;z ) [see Fig. 1(b)].\n\u000fThe thickness tof the CrTe 2\rake is extracted from\nline pro\fles of the AFM image with an uncertainty\nof\u00062 nm.\n\u000fThe azimuthal angle of the in-plane magnetiza-\ntion\u001eMis obtained through the comparison be-\ntween experimental data and simulated magnetic\n\feld maps.\nWe \frst evaluate the uncertainty of the \ftting pro-\ncedure. To this end, the line pro\fle is \ftted with the\nresult of the magnetic calculation while \fxing all the\nparameters pito their nominal values \u0016 pi, leading to\nM= 26:9\u00060:7 kA/m. The relative uncertainty linked\nto the \ftting procedure is therefore given by \u000f\ft= 3%.\nWe note that the intrinsic accuracy of the magnetic \feld\nmeasurement is in the range of \u000eBNV\u00185\u0016T. The re-\nsulting uncertainty can be safely neglected.\nIn order to estimate the relative uncertainty \u000fpiintro-\nduced by each parameter pi, the \ft was performed with\none parameter pi\fxed atpi= \u0016pi\u0006\u001bpi, all the other\nparameters remaining \fxed at their nominal values. The\ncorresponding \ft outcomes are denoted M( \u0016pi+\u001bpi) and\nM( \u0016pi\u0000\u001bpi). The relative uncertainty \u000fpiintroduced by7\nthe errors on parameter piis then \fnally de\fned as\n\u000fpi=M( \u0016pi+\u001bpi)\u0000M( \u0016pi\u0000\u001bpi)\n2M( \u0016pi): (1)\nThis analysis was performed for each parameter pi. The\ncumulative uncertainty \u000fis \fnally given by\n\u000f=s\n\u000f2\n\ft+X\ni\u000f2pi; (2)\nwhere all errors are assumed to be independent.\nA summary of the uncertainties is given in Table I. We\nobtainM= 27\u00064 kA/m for the \rake shown in Fig. 3\nandM= 26\u00064 kA/m for the \rake shown in Fig. 4.TABLE I. Analysis of uncertainties in the measurement of the\nmagnetization\n(a) CrTe 2\rake shown in Figure 3\nparameterpinominal value \u0016 piuncertainty \u001bpi\u000fpi(%)\ndNV 80 nm 10 nm 8\n\u0012NV 58\u000e2\u000e7\n\u001eNV 103\u000e2\u000e2\nt 35 nm 2 nm 6\n\u001eM \u0000145\u000e5\u000e5\n\u000f=p\n\u000f2\n\ft+P\ni\u000f2pi14\n(b) CrTe 2\rake shown in Figure 4\nparameterpinominal value \u0016 piuncertainty \u001bpi\u000fpi(%)\ndNV 80 nm 10 nm 8\n\u0012NV 58\u000e2\u000e7\n\u001eNV 103\u000e2\u000e2\nt 24 nm 2 nm 8\n\u001eM \u0000100\u000e10\u000e12\n\u000f=p\n\u000f2\n\ft+P\ni\u000f2pi17\n[1] J. M. Kosterlitz and D. J. Thouless, J. Phys. C: Solid\nState Phys. 6, 1181 (1973).\n[2] H. Li, S. Ruan, and Y.-J. Zeng, Adv. Mater. 31, 1900065\n(2019).\n[3] M.-C. Wang, C.-C. Huang, C.-H. Cheung, C.-Y. Chen,\nS. G. Tan, T.-W. Huang, Y. Zhao, Y. Zhao, G. Wu, Y.-P.\nFeng, H.-C. Wu, and C.-R. Chang, Annalen der Physik\n532, 1900452 (2020).\n[4] B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein,\nR. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A.\nMcGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-\nHerrero, and X. Xu, Nature 546, 270 (2017).\n[5] C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao,\nW. Bao, C. Wang, Y. Wang, Z. Q. Qiu, R. J. Cava, S. G.\nLouie, J. Xia, and X. Zhang, Nature 546, 265 (2017).\n[6] K. S. Burch, D. Mandrus, and J.-G. Park, Nature 563,\n47 (2018).\n[7] C. Gong and X. Zang, Science 363(2019).\n[8] M. Gibertini, M. Koperski, A. F. Morpurgo, and K. S.\nNovoselov, Nature Nanotechnology 14, 408 (2019).\n[9] S. Liu, X. Yuan, Y. Zou, Y. Sheng, C. Huang, E. Zhang,\nJ. Ling, Y. Liu, W. Wang, C. Zhang, J. Zou, K. Wang,\nand F. Xiu, npj 2D Materials and Applications 1, 30\n(2017).\n[10] N. Leon-Brito, E. D. Bauer, F. Ronning, J. D. Thomp-\nson, and R. Movshovich, Journal of Applied Physics 120,\n083903 (2016).\n[11] Z. Fei, B. Huang, P. Malinowski, W. Wang, T. Song,\nJ. Sanchez, W. Yao, D. Xiao, X. Zhu, A. F. May, W. Wu,D. H. Cobden, J.-H. Chu, and X. Xu, Nature Materials\n17, 778 (2018).\n[12] Y. Deng, Y. Yu, Y. Song, J. Zhang, N. Z. Wang, Z. Sun,\nY. Yi, Y. Z. Wu, S. Wu, J. Zhu, J. Wang, X. H. Chen,\nand Y. Zhang, Nature 563, 94 (2018).\n[13] D. Weber, A. H. Trout, D. W. McComb, and J. E. Gold-\nberger, Nano Lett. 19, 5031 (2019).\n[14] H. Wang, Y. Liu, P. Wu, W. Hou, Y. Jiang, X. Li,\nC. Pandey, D. Chen, Q. Yang, H. Wang, D. Wei, N. Lei,\nW. Kang, L. Wen, T. Nie, W. Zhao, and K. L. Wang,\nACS Nano 14, 10045 (2020).\n[15] Q. Li, M. Yang, C. Gong, R. V. Chopdekar, A. T.\nN'Diaye, J. Turner, G. Chen, A. Scholl, P. Shafer,\nE. Arenholz, A. K. Schmid, S. Wang, K. Liu, N. Gao,\nA. S. Admasu, S.-W. Cheong, C. Hwang, J. Li, F. Wang,\nX. Zhang, and Z. Qiu, Nano Lett. 18, 5974 (2018).\n[16] A. F. May, D. Ovchinnikov, Q. Zheng, R. Hermann,\nS. Calder, B. Huang, Z. Fei, Y. Liu, X. Xu, and M. A.\nMcGuire, ACS Nano 13, 4436 (2019).\n[17] J. Seo, D. Y. Kim, E. S. An, K. Kim, G.-Y. Kim, S.-\nY. Hwang, D. W. Kim, B. G. Jang, H. Kim, G. Eom,\nS. Y. Seo, R. Stania, M. Muntwiler, J. Lee, K. Watanabe,\nT. Taniguchi, Y. J. Jo, J. Lee, B. I. Min, M. H. Jo, H. W.\nYeom, S.-Y. Choi, J. H. Shim, and J. S. Kim, Science\nAdvances 6(2020).\n[18] P. Mallet, F. Chiapello, H. Okuno, H. Boukari, M. Jamet,\nand J.-Y. Veuillen, Phys. Rev. Lett. 125, 036802 (2020).\n[19] S. J. Yun, D. L. Duong, D. M. Ha, K. Singh, T. L. Phan,\nW. Choi, Y.-M. Kim, and Y. H. Lee, Advanced Science8\n7, 1903076 (2020).\n[20] Z. F., B. Zheng, A. Sebastian, H. Olson, M. Liu, K. Fuji-\nsawa, Y. T. H. Pham, V. Ortiz Jimenez, V. Kalappattil,\nL. Miao, T. Zhang, R. Pendurthi, Y. Lei, A. L. Elias,\nY. Wang, N. Alem, P. E. Hopkins, S. Das, V. H. Crespi,\nM.-H. Phan, and M. Terrones, arXiv:2005.01965 (2020).\n[21] M. Bonilla, S. Kolekar, Y. Ma, H. C. Diaz, V. Kalappat-\ntil, R. Das, T. Eggers, H. R. Gutierrez, M.-H. Phan, and\nM. Batzill, Nature Nanotechnology 13, 289 (2018).\n[22] D. J. O'Hara, T. Zhu, A. H. Trout, A. S. Ahmed, Y. K.\nLuo, C. H. Lee, M. R. Brenner, S. Rajan, J. A. Gupta,\nD. W. McComb, and R. K. Kawakami, Nano Letters 18,\n3125 (2018).\n[23] J. Li, B. Zhao, P. Chen, R. Wu, B. Li, Q. Xia, G. Guo,\nJ. Luo, K. Zang, Z. Zhang, H. Ma, G. Sun, X. Duan, and\nX. Duan, Advanced Materials 30, 1801043 (2018).\n[24] J. Feng, D. Biswas, A. Rajan, M. D. Watson, F. Mazzola,\nO. J. Clark, K. Underwood, I. Markovic, M. McLaren,\nA. Hunter, D. M. Burn, L. B. Du\u000by, S. Barua, G. Bal-\nakrishnan, F. Bertran, P. Le Fevre, T. K. Kim, G. van der\nLaan, T. Hesjedal, P. Wahl, and P. D. C. King, Nano\nLett. 18, 4493 (2019).\n[25] P. K. J. Wong, W. Zhang, F. Bussolotti, X. Yin, T. S.\nHerng, L. Zhang, Y. L. Huang, G. Vinai, S. Krishna-\nmurthi, D. W. Bukhvalov, Y. J. Zheng, R. Chua, A. T.\nN'Diaye, S. A. Morton, C.-Y. Yang, K.-H. Ou Yang,\nP. Torelli, W. Chen, K. E. J. Goh, J. Ding, M.-T. Lin,\nG. Brocks, M. P. de Jong, A. H. Castro Neto, and\nA. T. S. Wee, Advanced Materials 31, 1901185 (2019).\n[26] D. C. Freitas, R. Weht, A. Sulpice, G. Remenyi, P. Stro-\nbel, F. Gay, J. Marcus, and M. N\u0013 u~ nez-Regueiro, Journal\nof Physics: Condensed Matter 27, 176002 (2015).\n[27] X. Sun, W. Li, X. Wang, Q. Sui, T. Zhang, Z. Wang,\nL. Liu, D. Li, S. Feng, S. Zhong, H. Wang, V. Bouchiat,\nM. Nunez Regueiro, N. Rougemaille, J. Coraux, A. Pur-\nbawati, A. Hadj-Azzem, Z. Wang, B. Dong, X. Wu,\nT. Yang, G. Yu, B. Wang, Z. Han, X. Han, and Z. Zhang,\nNano Research (2020).\n[28] A. Purbawati, J. Coraux, J. Vogel, A. Hadj-Azzem,\nN. Wu, N. Bendiab, D. Jegouso, J. Renard, L. Marty,\nV. Bouchiat, A. Sulpice, L. Aballe, M. Foerster,\nF. Genuzio, A. Locatelli, T. O. Mente\u0018 s, Z. V. Han,\nX. Sun, M. N\u0013 u~ nez-Regueiro, and N. Rougemaille, ACS\nAppl. Mater. Interfaces 12, 30702 (2020).\n[29] G. Balasubramanian, I. Y. Chan, R. Kolesov, M. Al-\nHmoud, J. Tisler, C. Shin, C. Kim, A. Wojcik, P. R. Hem-\nmer, A. Krueger, T. Hanke, A. Leitenstorfer, R. Brats-\nchitsch, F. Jelezko, and J. Wrachtrup, Nature 455, 648\n(2008).\n[30] J. R. Maze, P. L. Stanwix, J. S. Hodges, S. Hong, J. M.\nTaylor, P. Cappellaro, L. Jiang, M. V. G. Dutt, E. Togan,\nA. S. Zibrov, A. Yacoby, R. L. Walsworth, and M. D.\nLukin, Nature 455, 644 (2008).\n[31] L. Rondin, J.-P. Tetienne, T. Hingant, J.-F. Roch,\nP. Maletinsky, and V. Jacques, Reports on Progress in\nPhysics 77, 056503 (2014).\n[32] F. Casola, T. van der Sar, and A. Yacoby, Nature Re-\nviews Materials 3, 17088 (2018).\n[33] J.-P. Tetienne, T. Hingant, L. J. Martinez, S. Rohart,\nA. Thiaville, L. H. Diez, K. Garcia, J.-P. Adam, J.-\nV. Kim, J.-F. Roch, I. M. Miron, G. Gaudin, L. Vila,\nB. Ocker, D. Ravelosona, and V. Jacques, Nature Com-\nmunications 6, 6733 (2015).\n[34] Y. Dovzhenko, F. Casola, S. Schlotter, T. X. Zhou,F. Buttner, R. L. Walsworth, G. S. D. Beach, and A. Ya-\ncoby, Nature Communications 9, 2712 (2018).\n[35] J.-Y. Chauleau, T. Chirac, S. Fusil, V. Garcia,\nW. Akhtar, J. Tranchida, P. Thibaudeau, I. Gross,\nC. Blouzon, A. Finco, M. Bibes, B. Dkhil, D. D.\nKhalyavin, P. Manuel, V. Jacques, N. Jaouen, and\nM. Viret, Nature Materials 19, 386 (2020).\n[36] M. J. H. Ku, T. X. Zhou, Q. Li, Y. J. Shin, J. K.\nShi, C. Burch, L. E. Anderson, A. T. Pierce, Y. Xie,\nA. Hamo, U. Vool, H. Zhang, F. Casola, T. Taniguchi,\nK. Watanabe, M. M. Fogler, P. Kim, A. Yacoby, and\nR. L. Walsworth, Nature 583, 537 (2020).\n[37] L. Thiel, Z. Wang, M. A. Tschudin, D. Rohner,\nI. Guti\u0013 errez-Lezama, N. Ubrig, M. Gibertini, E. Gian-\nnini, A. F. Morpurgo, and P. Maletinsky, Science 364,\n973 (2019).\n[38] D. A. Broadway, S. C. Scholten, C. Tan, N. Dontschuk,\nS. E. Lillie, B. C. Johnson, G. Zheng, Z. Wang, A. R.\nOganov, S. Tian, C. Li, H. Lei, L. Wang, L. C. L. Hol-\nlenberg, and J.-P. Tetienne, Adv. Mater. n/a, 2003314\n(2020).\n[39] Q.-C. Sun, T. Song, E. Anderson, T. Shalomayeva,\nJ. F orster, A. Brunner, T. Taniguchi, K. Watan-\nabe, J. Gr afe, R. St ohr, X. Xu, and J. Wrachtrup,\narXiv:2009.13440 (2020).\n[40] T. Hingant, J.-P. Tetienne, L. J. Mart\u0013 \u0010nez, K. Garcia,\nD. Ravelosona, J.-F. Roch, and V. Jacques, Physical\nReview Applied 4, 014003 (2015).\n[41] A. Gruber, A. Dr abenstedt, C. Tietz, L. Fleury,\nJ. Wrachtrup, and C. V. Borczyskowski, Science 276,\n2012 (1997).\n[42] I. Gross, W. Akhtar, A. Hrabec, J. Sampaio, L. J.\nMart\u0013 \u0010nez, S. Chouaieb, B. J. Shields, P. Maletinsky,\nA. Thiaville, S. Rohart, and V. Jacques, Phys. Rev. Ma-\nterials 2, 024406 (2018).\n[43] W. Akhtar, A. Hrabec, S. Chouaieb, A. Haykal, I. Gross,\nM. Belmeguenai, M. Gabor, B. Shields, P. Maletinsky,\nA. Thiaville, S. Rohart, and V. Jacques, Phys. Rev. Ap-\nplied 11, 034066 (2019).\n[44] J.-P. Tetienne, L. Rondin, P. Spinicelli, M. Chipaux,\nT. Debuisschert, J.-F. Roch, and V. Jacques, New Jour-\nnal of Physics 14, 103033 (2012).\n[45] P. Maletinsky, S. Hong, M. S. Grinolds, B. Hausmann,\nM. D. Lukin, R. L. Walsworth, M. Loncar, and A. Ya-\ncoby, Nature Nanotechnology 7, 320 (2012).\n[46] P. Appel, E. Neu, M. Ganzhorn, A. Barfuss, M. Batzer,\nM. Gratz, A. Tsch ope, and P. Maletinsky, Review of\nScienti\fc Instruments 87, 063703 (2016).\n[47] B. C. Buchler, T. Kalkbrenner, C. Hettich, and V. San-\ndoghdar, Phys. Rev. Lett. 95, 063003 (2005).\n[48] J. Tisler, T. Oeckinghaus, R. J. St ohr, R. Kolesov,\nR. Reuter, F. Reinhard, and J. Wrachtrup, Nano Lett.\n13, 3152 (2013).\n[49] L. Rondin, J.-P. Tetienne, S. Rohart, a. Thiaville, T. Hin-\ngant, P. Spinicelli, J.-F. Roch, and V. Jacques, Nat.\nCommun. 4, 2279 (2013).\n[50] D. A. Broadway, S. E. Lillie, S. C. Scholten, D. Rohner,\nN. Dontschuk, P. Maletinsky, J.-P. Tetienne, and L. C. L.\nHollenberg, Phys. Rev. Applied 14, 024076 (2020).\n[51] C. Tan, J. Lee, S.-G. Jung, T. Park, S. Albarakati, J. Par-\ntridge, M. R. Field, D. G. McCulloch, L. Wang, and\nC. Lee, Nature Communications 9, 1554 (2018).\n[52] F. Huang, M. T. Kief, G. J. Mankey, and R. F. Willis,\nPhys. Rev. B 49, 3962 (1994).9\n[53] Y. Li and K. Baberschke, Phys. Rev. Lett. 68, 1208\n(1992).\n[54] A. Bedoya-Pinto, J.-R. Ji, A. Pandeya, P. Gargiani,\nM. Valvidares, P. Sessi, F. Radu, K. Chang, and\nS. Parkin, arXiv:2006.07605 (2020).\n[55] I. Gross, W. Akhtar, V. Garcia, L. J. Mart\u0013 \u0010nez,S. Chouaieb, K. Garcia, C. Carr\u0013 et\u0013 ero, A. Barth\u0013 el\u0013 emy,\nP. Appel, P. Maletinsky, J.-V. Kim, J. Y. Chauleau,\nN. Jaouen, M. Viret, M. Bibes, S. Fusil, and V. Jacques,\nNature 549, 252 (2017)."    },    {        "title": "2011.07813v1.Topologically_stable_bimerons_and_skyrmions_in_vanadium_dichalcogenide_Janus_monolayers.pdf",        "content": "Topologically stable bimerons and skyrmions in vanadium dichalcogenide Janus\nmonolayers\nSlimane Laref1\u0003,\u0003V. M. L. D. P. Goli1\u0003,yIdris Smaili2, Udo Schwingenschl ogl1, and Aur\u0013 elien Manchon1;2;3z\n1Physical Science and Engineering Division (PSE),\nKing Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia\n2Computer, Electrical and Mathematical Science and Engineering (CEMSE),\nKing Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia\n3Aix-Marseille Univ, CNRS, CINaM, Marseille, France\nWe investigate the magnetic phase diagram of 1T-vanadium dichalcogenide monolayers in Janus\ncon\fguration (VSeTe, VSSe, and VSTe) from \frst principles. The magnetic exchange, magne-\ntocrystalline anisotropy and Dzyaloshinskii-Moriya interaction (DMI) are computed using density\nfunctional theory calculations, while the temperature- and \feld-dependent magnetic phase diagram\nis simulated using large-scale atomistic spin modeling in the presence of thermal \ructuations. The\nboundaries between magnetic ordered phases and paramagnetic phases are determined by cross-\nanalyzing the average topological charge with the magnetic susceptibility and its derivatives. We\n\fnd that in such Janus monolayers, DMI is large enough to stabilize non-trivial chiral textures. In\nVSeTe monolayer, an asymmetrical bimeron lattice state is stabilized for in-plane \feld con\fgura-\ntion whereas skyrmion lattice is formed for out-of-plane \feld con\fguration. In VSSe monolayer,\na skyrmion lattice is stabilized for out-of-plane \feld con\fguration. This study demonstrates that\nnon-centrosymmetric van der Waals magnetic monolayers can support topological textures close to\nroom temperature.\nI. INTRODUCTION\nMagnetic skyrmions, i.e., topologically stable magnetic\ntextures, have recently attracted great interest because\nof their unique transport and topological properties1{3,\nopening avenues to novel potential applications in the\n\feld of spintronics4{10. Stable skyrmion crystals and\nmetastable isolated skyrmions are normally obtained\nfrom the competition between magnetic exchange, uniax-\nial anisotropy, antisymmetric Dzyaloshinskii-Moriya in-\nteraction (DMI)11,12, and possibly the external mag-\nnetic \feld. They have been initially reported at low\ntemperature in non-centrosymmetric magnets13,14and\nmore recently at room temperature in transition metal\nmultilayers15{19. The advent of two-dimensional van der\nWaals (2D vdW) intrinsic magnets such as Cr 2Ge2Te620,\nMnSe 221{23, VSe 224, CrI 325, and Fe 3GeTe 226opens ap-\npealing avenues for the observation of chiral magnetic\ntextures in atomically thin materials. What makes 2D\nvdW magnets particularly attractive is the possibility\nto engineer their band structure by surface chemistry.\nAn outstanding demonstration of this feature is the syn-\nthesis of transition metal Janus monolayers27{29, where\nthe transition metal element is embedded between dis-\nsimilar (chalcogen or halide) ions (see also Ref. 30).\nThis con\fguration breaks the inversion symmetry and\npromotes Rashba-type spin-orbit coupling27,31,32. In the\ncase of magnetic Janus monolayers, the inversion sym-\nmetry breaking enables spin-orbit torque33,34as well as\nDMI35{37.\nIn the present work, we investigate the onset of DMI\nand the emergence of magnetic skyrmions in vanadium-\nbased transition metal dichalcogenide Janus monolayers\nin 1T con\fguration. Speci\fcally, we focus our investiga-\ntion in VSSe, VSTe and VSeTe. In Section II, we study\nFIG. 1. (Color Online) Cartoon of the VXY Janus monolayer.\nVanadium elements are represented in grey, and the chalcogen\nelements X and Y (S, Se, and Te (X 6=Y)) are in orange and\nbrown, respectively. The red arrows indicate the direction of\nthe magnetic moment.\nthe magnetic exchange interaction and anisotropy from\n\frst principles and compute DMI using the generalized\nBloch theorem. In Section III, we investigate the mag-\nnetic phase diagram of these systems under the combined\naction of an external magnetic \feld and thermal excita-\ntions using an atomistic spin simulation method. By ex-\nploring in details the temperature-\feld magnetic phase\ndiagram of the three systems, we \fnd that chiral ground\nstates can be obtained in a reasonably large range of\ntemperatures. In particular, VSeTe displays asymmetri-\ncal bimeron lattice ground state up to room temperature,\nwhereas VSSe exhibits skyrmion lattice states.\nII. FIRST PRINCIPLES CALCULATIONS\nA. Structural and magnetic properties\nTo compute the structural and magnetic properties,\nwe perform \frst-principles calculations using the full-arXiv:2011.07813v1  [cond-mat.mtrl-sci]  16 Nov 20202\npotential linearized augmented plane-wave (FLAPW)\nmethod as implemented in FLEUR code38{40. Our calcu-\nlations are performed using the local-density approxima-\ntion exchange-correlation functional (LDA-vwn) as im-\nplemented in the FLAPW Package40. For the angular\nmomentum expansion and the reciprocal plane wave, cut-\no\u000b ofImax= 6 andkmax= 4 were applied, and we used\nradii of MT spheres around 1.7 a.u for S, 2.1 a.u for Se\nand 2.4 a.u for Te, and 2.2 a.u for V, where a.u is the\nBohrradius . \u0000-centered k-grid 32 \u000232\u00021 and 64\u000264\u00021\nhave been sampled for the whole Brillouin zone to achieve\nthe total energy without spin-orbit coupling (SOC) and\nwith SOC, respectively.\nBy de\fning ground-state geometries, crystal structures\nof the bulk VXY (X, Y= S, Se, Te) have been optimized.\nTable I lists the structural and magnetic parameters of\nVSeTe, VSSe, and VSTe materials, which are in good\nagreement with the literature41{44. All three Janus ma-\nterials exhibit C 3vsymmetry and the on-site magnetic\nmoments of V is controlled by the electron depletion due\nto the ionic bonding with the chalcogen elements. As\na phenomenological rule, the larger the electronegativ-\nity di\u000berence between X and Y elements, the larger the\ncharge depletion, and the larger the magnetic moment.\nThe magnetic exchange Jis calculated from the energy\ndi\u000berence between the ferromagnetic and antiferromag-\nnetic state. We \fnd that all three systems are ferro-\nmagnetic and Jroughly scales with the electronegativity\ndi\u000berence \u0001 \u001f, yielding the largest value (70 meV) for\nVSTe. Magnetocrystalline anisotropy is obtained by uti-\nlizing the force theorem and applying SOC within the\nsecond variation method45{47. The magnetocrystalline\nanisotropy, K=Ek\u0000E?, de\fned as the di\u000berence be-\ntween in-plane and out-of-plane magnetization energies is\nreported in Table I. Our results indicate that VSSe pos-\nsesses a weak out-of-plane uniaxial anisotropy, whereas\nboth VSeTe and VSTe exhibit easy-plane anisotropy (in\nother words, there is no variation of the magnetic energy\nwhen rotating the magnetization in the ( x;y) plane for\nthese two monolayers).\nB. Spin spiral calculations\nDMI is an antisymmetric exchange interaction that\ntends to align neighboring magnetic moments perpen-\ndicular to each other. It reads\nEDM=X\nijDij\u0001(Si\u0002Sj); (1)\nwhere Siis the magnetic moment on site i, and Dijis\nthe Dzyaloshinskii-Moriya vector that governs the DMI\nbetween sites iandj. Because DMI is controlled by in-\nversion symmetry breaking, it is oddin SOC strength and\ntherefore, a good estimation of Dijis obtained at the \frst\norder in SOC. To do so, we exploit the generalized Bloch\ntheorem48approach implemented in FLEUR code49{51.\nWe build out-of-plane N\u0013 eel spin spirals in momentum\n/s48/s50/s48/s52/s48\n/s48/s51/s54\n/s45/s48/s46/s54 /s45/s48/s46/s52 /s45/s48/s46/s50 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54/s48/s50/s48/s52/s48/s54/s48/s56/s48/s40/s97/s41\n/s40/s98/s41\n/s40/s99/s41/s32/s119/s105/s116/s104/s111/s117/s116/s32/s83/s79/s67\n/s32/s119/s105/s116/s104/s32/s83/s79/s67\n/s32/s119/s105/s116/s104/s111/s117/s116/s32/s83/s79/s67\n/s32/s119/s105/s116/s104/s32/s83/s79/s67/s32/s119/s105/s116/s104/s111/s117/s116/s32/s83/s79/s67\n/s32/s119/s105/s116/s104/s32/s83/s79/s67/s77 /s172/s71 /s174 /s77\n/s77 /s172/s71 /s174 /s77\n/s77 /s172/s71 /s174 /s77/s69\n/s113/s45/s69\n/s70/s77/s32/s40/s109/s101/s86/s32/s112/s101/s114/s32/s86/s32/s97/s116/s111/s109/s41\n/s115/s112/s105/s110/s45/s115/s112/s105/s114/s97/s108/s32/s118/s101/s99/s116/s111/s114/s32/s113/s32/s40/s50 /s112 /s47/s97/s41/s99/s111/s117/s110/s116/s101/s114/s32/s99/s108/s111/s99/s107/s119/s105/s115/s101/s32 /s172 /s32/s114/s111/s116/s97/s116/s105/s111/s110/s32/s111/s102/s32/s115/s112/s105/s110/s32/s115/s112/s105/s114/s97/s108/s115/s32 /s174 /s32/s99/s108/s111/s99/s107/s119/s105/s115/s101FIG. 2. (Color Online) Calculated energy dispersions Eqof\nrotating spin spirals without spin-orbit coupling (black sym-\nbols) and with spin-orbit coupling (red symbols) of (a) VSeTe\n(b) VSSe, and (c) VSTe.\nspace whose energy dispersion, without SOC (black) and\nat the \frst order in SOC (red), is reported on Fig. 2.\nThe corresponding di\u000berence between the spin spiral en-\nergy dispersion with and without SOC is reported on\nFig. 3 and the Dzyaloshinskii-Moriya vector is estimated\nby taking the slope of the dispersion at q= 0. This\nvalue corresponds to the DMI strength experienced by\nmagnetic textures whose length scale is much larger than\nthe crystal lattice parameter. The extracted values are\n-5.7 meV\u0001\u0017A, 1.9 meV\u0001\u0017A and 2.5 meV\u0001\u0017A for VSeTe, VSSe\nand VSTe, respectively. For the sake of comparison, the\nDMI obtained for Pt/Co(111) using the same method52is\nabout 50 meV\u0001\u0017A due to the large SOC strength of Pt. As\ndiscussed in the previous section, the DMI in vanadium-\nbased Janus monolayers is su\u000eciently strong to stabilize\nchiral magnetic textures.3\nTABLE I. Structural and magnetic parameters of T-VXY: The lattice constant (a), the distance between X and Y ( dX\u0000Y), the\nelectronegativity di\u000berence between the two chalcogen elements (\u0001 \u001f- de\fned positive), the magnetic moment of the transition\nmetal atom ( \u0016s), the Heisenberg exchange ( J), the magnetocrystalline anisotropy ( K), and DMI strength ( D). The Curie\ntemperature ( Tc) is deduced from zero-\feld susceptibility analysis as explained in Section III.\nVXY a( \u0017A)dX\u0000Y(\u0017A) \u0001\u001f \u0016s(\u0016B)J(meV)K(meV)D(meV\u0001\u0017A)Tc(K)\nVSSe 3.266 2.969 0.03 0.686 20.3 0.078 1.9 240\nVSeTe 3.673 2.887 0.45 1.391 50.82 -0.963 -5.7 630\nVSTe 3.611 2.838 0.48 1.405 71.51 -0.860 2.5 780\n/s45/s50/s48/s50\n/s45/s50/s48/s50\n/s45/s48/s46/s54 /s45/s48/s46/s52 /s45/s48/s46/s50 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54/s45/s50/s48/s50/s32/s68/s77/s73/s68 /s69\n/s83/s79/s67/s40/s113/s41/s32/s40/s109/s101/s86/s32/s112/s101/s114/s32/s86/s32/s97/s116/s111/s109/s41/s32/s68/s77/s73\n/s115/s112/s105/s110/s45/s115/s112/s105/s114/s97/s108/s32/s118/s101/s99/s116/s111/s114/s32 /s113 /s32/s40/s50 /s112 /s47/s97/s41/s32/s68/s77/s73/s77 /s172/s71 /s174 /s77\n/s77 /s172/s71 /s174 /s77\n/s77 /s172/s71 /s174 /s77/s99/s111/s117/s110/s116/s101/s114/s32/s99/s108/s111/s99/s107/s119/s105/s115/s101/s32 /s172 /s32/s114/s111/s116/s97/s116/s105/s111/s110/s32/s111/s102/s32/s115/s112/s105/s110/s32/s115/s112/s105/s114/s97/s108/s115/s32 /s174 /s32/s99/s108/s111/s99/s107/s119/s105/s115/s101\n/s40/s97/s41\n/s40/s98/s41\n/s40/s99/s41\nFIG. 3. (Color Online) Antisymmetric contribution to the\nenergy dispersion displayed on Fig. 2 as a function of the\nspin-spirals vector qfor (a) VSeTe (b) VSSe, and (c) VSTe.\nIII. MAGNETIC PHASE DIAGRAM FROM\nATOMISTIC SPIN DYNAMICS\nA. Methodology\nTo model the magnetic phases of all three VXY sys-\ntems, we consider atomic spins on each site of a triangularlattice. The Heisenberg spin Hamiltonian is given by\nH=\u0000JX\nhi;jiSi\u0001Sj\u0000X\nhi;jiDij\u0001(Si\u0002Sj)\n\u0000KX\ni(Si\u0001^z)2\u0000\u0016sX\niSi\u0001B; (2)\nwhere Siis normalized unit spin vector at site i. On the\nright side of Eq. (2), the \frst term is the Heisenberg\nexchange energy with ferromagnetic exchange strength\nJandhi;jiindicates the sum over all nearest-neighbors.\nThe second term is the DMI, Dij=D(rij\u0002^z), withDis\nthe DMI energy and rijis the unit vector between sites i\nandj. The third term represents the magnetocrystalline\nanisotropy energy that can be either easy-plane ( K < 0\nfor VSeTe and VSTe) or uniaxial out-of-plane ( K > 0 for\nVSSe). The last term is the Zeeman interaction energy\ndue to the applied \feld, B.\nTo obtain the magnetic phase diagrams, we use atom-\nistic spin dynamics technique that numerically solves the\nstochastic Landau-Lifshitz-Gilbert (LLG) equation of the\nform\n@Si\n@t=\u0000\r\n1 +\u000b2Si\u0002\u0000\nHe\u000b\ni+\u000bSi\u0002He\u000b\ni\u0001\n;(3)\nwhere\ris gyromagnetic ratio and \u000bis the intrinsic\nGilbert damping constant. We choose \u000b= 0.05. The\ne\u000bective \feld acting on the magnetic moment Siis repre-\nsented by He\u000b\ni=\u00001\n\u0016s@H\n@Si+Hth\ni, where\u0016sis the atomic\nmagnetic moment. Hth\niis the stochastic thermal \feld\narising from thermal \ructuations of the magnetic mo-\nments. By using the Langevin dynamics approach, ran-\ndom thermal \feld is added at each site. This \feld obeys\nthe Gaussian distribution in three dimensions with mean\nof zero, \u0000(t). The random thermal \feld is given by53\nHth\ni=\u0000(t)s\n2\u000bKBT\n\r\u0016s\u0001t; (4)\nwherekBis the Boltzmann constant, Tis the tempera-\nture and \u0001tis integration time step. The simulations are\ncarried out on a triangular lattice with N= 150\u0002150\nsites and periodic boundary conditions are implemented.\nThe magnetic parameters are taken from Table I. In or-\nder to characterize the magnetic state at a given point in\nthe (T;B) phase space, Eq. (3) is solved numerically to4\n0 2 4 6 8 10\nB (T)-0.6-0.4-0.200.20.40.6E - ESS (meV/atom)SS\nFM\nABX\n0 5 10 15 20 25 30\nB (T)-0.6-0.4-0.200.20.40.6E - ESS (meV/atom)SS\nFM\nSkX\n0 2 4 6 8 10\nB (T)-0.3-0.2-0.100.10.20.3E - ESS (meV/atom)SS\nFM\nSkX\n0 2 4 6 8 10\nB (T)-1-0.8-0.6-0.4-0.200.2E - ESS (meV/atom)SS\nFM(a)(b)\n(d) (c)SS\nSSSS\nSSFM FM\nFM FMSkX\nSkXABX\nFIG. 4. Zero temperature phase diagrams of VSeTe, VSSe and VSTe systems. The energies corresponding to FM, ABX and\nSkX states are shown with respect to the relative energy of SS state as a function of the applied magnetic \feld. (a) and (b)\nrepresent the phase diagrams of VSeTe with in-plane and out-of-plane applied magnetic \feld, respectively. The phase diagram\nof VSSe with out-of-plane magnetic \feld is shown in (c) while the phase diagram of VSTe is shown in (d) where in-plane\nmagnetic \feld is considered. The shaded areas represent di\u000berent ground states.\nobtain the magnetic susceptibility and average topologi-\ncal charge for VXY systems. The magnetic susceptibility\nis given by\n\u001f=\u0016s\nkBT\u0000\nhM2i\u0000hMi2\u0001\n; (5)\nwhereM=1\nNP\niSi. A simple magnetic ground state\nsuch as ferromagnetic or antiferromagnetic state and a\nskyrmion state can be di\u000berentiated by its topological\ncharge (Q). In the continuum magnetization limit, Qis\nde\fned by the following equation\nQ=1\n4\u0019Z\nS\u0001\u0012@S\n@x\u0002@S\n@y\u0013\ndxdy: (6)\nHereQdescribes the number of times magnetic moments\n(S) wrap around the unit sphere. Q= 0 denotes a triv-\nial state while Q6= 0 provides the number of skyrmions\npresent in the skyrmion state. In a magnetic state withtopologically protected magnetic texture analogous to\nskyrmion,Qis non-zero. For the discrete square lat-\ntice, Berg and L uscher proposed a method to quantify\nQ54. This method involves partitioning the lattice into\nnearest-neighbor triangles with spins at vertices of each\ntriangle. The same procedure can be applied for tri-\nangular lattice except that partitioning the lattice into\ntriangles is not required. Counter-clockwise rotation of\nvertices spins Si,SjandSkof each triangle is considered\nto calculate Qfrom the following equation55,56,\ntan (Q=2) =Si\u0001(Sj\u0002Sk)\n1 +Si\u0001Sj+Sj\u0001Sk+Sk\u0001Si:(7)\nThe spin dynamics simulations are performed with\nincreasing temperature for a \fxed magnetic \feld. At\neach point in ( T;B) phase space, the system is allowed\nto evolve for 106time steps, and the average topologi-\ncal charge and magnetic susceptibility are calculated for5\n3\u0002106averaging time steps. The considered simulation\ntime is su\u000ecient to capture all variations of temperature-\ndependent quantities.\nB. Zero-temperature phase diagram\nWe \frst compute the zero-temperature ground states\nof VSeTe, VSSe and VSTe systems under an external ap-\nplied \feld. Depending on the \feld strength, we \fnd that\nvarious types of magnetic phases can be stabilized: ferro-\nmagnetic (FM), spin spiral (SS), skyrmion crystal (SkX)\nas well as asymmetric bimeron crystal (ABX). The \feld-\ndependence of these various phases at zero temperature\nis reported on Fig. 4. In this \fgure, we report the results\nfor VSeTe (easy-plane anisotropy) with both in-plane (a)\nand out-of-plane magnetic \felds (b), VSSe (out-of-plane\nuniaxial anisotropy) with out-of-plane magnetic \feld (c),\nand VSTe (easy-plane anisotropy) with in-plane magnetic\n\feld (d).\nIn the case of VSeTe, we \fnd SS and FM states in\nthe low \feld and high \feld limits, respectively, indepen-\ndently on the \feld direction. Interestingly, at interme-\ndiate \feld we obtain two topologically non-trivial lat-\ntices, namely asymmetrical bimeron lattice [ABX - Fig.\n4(a)] for in-plane \feld con\fguration and skyrmion lat-\ntice [SkX - Fig. 4(b)] for out-of-plane \feld con\fgura-\ntion. Asymmetrical bimerons are planar counterpart of\nmagnetic skyrmions that are known to emerge under the\ncooperation of DMI with uniaxial in-plane anisotropy57.\nIn VSeTe, which possesses easy-plane rather than uni-\naxial in-plane anisotropy, the in-plane magnetic \feld is\nnecessary to stabilize the bimeron lattice. Notice that\nthe magnetization of the SS state is nearly vanishing and\nhence this state is mostly una\u000bected by the magnetic \feld\nas shown in Fig 4. In contrast, both ABX and FM states\nhave non-zero magnetization and their energy decreases\nwith increasing \feld. In our 150 \u0002150 spins system, the\nABX state with four bimerons becomes the ground state\nfor \felds above 1.2 T. The spin texture of ABX state is\nshown in Fig 5(a). It is also possible to promote lattices\nwith more bimerons with di\u000berent sizes; however, the\nenergy of these bimeron states remain higher than ABX\nstate with four bimerons and are therefore metastable.\nTherefore, the energy of ABX state depends on the num-\nber of bimerons and on their size. With increasing \feld,\nthe size of bimerons shrinks. However, ABX state main-\ntains lower energy until FM state becomes the ground\nstate. The magnetization of FM state is larger than that\nof ABX state, and hence the energy of FM state lowers\nswiftly for large magnetic \felds. Above 6.4 T, VSeTe\nacquires in-plane magnetized state. In the case of out-\nof-plane magnetic \feld con\fguration of VSeTe, SS state\nis the lowest energy state below 1.3 T as shown in Fig.\n4(b). The SkX state becomes the ground state over a\nvery large range of applied \feld, between 1.3 T and 21.4\nT. Noticeably, the skyrmions' shape is hexagonal in this\nrange, due to large skyrmion-skyrmion interaction [seeFig 5(d)]. In the SkX ground state, the skyrmion di-\nameter reduces from 21 nm to 12 nm upon increasing\n\feld. The domain wall width of skyrmion is large due to\neasy-plane anisotropy. Above 21.4T though, all spins of\nVSeTe align to produce the FM ground state.\nFIG. 5. Spin textures of VSeTe with in-plane and out-of-\nplane \felds and VSSe at di\u000berent temperature and \feld com-\nbinations. At in-plane \feld 2 T, ABX state of VSeTe with\ntemperatures (a) T = 0 K, (b) T = 60 K, and (c) T = 300\nK. At out-of-\feld 3 T, SkX state of VSeTe with temperatures\n(d) T = 0 K, (e) T = 60 K, and (f) T = 300 K. SkX state of\nVSSe with temperatures (g) T = 0 K, (h) T = 60 K and (i)\nT = 150 K at B = 3 T.\nIn the case of VSSe, shown in Fig. 4(c), a SkX state\nwith four skyrmions constitutes the ground state between\n2.5 T and 7.4 T. In this state, the skyrmion diameter\nchanges from 18 nm to 11 nm upon increasing \feld. Be-\nlow 2.5 T, the SS state is the ground state while the\nmagnetization is saturated for \felds above 7.4 T. In the\ncase of VSTe, shown in Fig. 4(d), the SS state is the\nground state in the absence of in-plane \feld. Due to the\npresence of easy-plane anisotropy and small DMI, VSTe\nhas in-plane spin texture with out-of-plane tilting. How-\never, the energy gap between SS and FM states is fairly\nsmall. VSTe acquires fully magnetized state with the\napplication of small in-plane \feld.6\nFIG. 6. (Top) ( T;B) phase diagram and (Bottom) corresponding topological charge Qfor VSeTe with (a,d) in-plane and (b,e)\nout-of-plane \felds and (c,f) VSSe systems.\nC. (T;B) magnetic phase diagram\nThe simulations are performed up to a maximum tem-\nperature of 1500 K and a maximum \feld of 25 T. The\naverage topological charge and magnetic susceptibility\nare calculated to de\fne the phase boundaries and criti-\ncal temperatures. The phase diagrams of the three sys-\ntems and their corresponding average topological charge\nare shown in Fig. 6. Along with ABX or SkX, two new\nphases emerge: \ructuation-disorder (FD) and paramag-\nnetic (PM) phases. In the FD region, both bimerons and\nskyrmions acquire a \fnite lifetime and hence the average\ntopological charge remains non-zero58. In contrast, the\nsystem goes into magnetic disordered state in the PM\nregion and exhibits zero topological charge. The details\nof these two phases are discussed below. The bound-\naries between all phases are determined by the in\rection\npoints of the temperature-dependent magnetic suscepti-\nbility and its derivatives, as discussed by Ref. 59.1. VSeTe Phase Diagram\nIn the case of VSeTe with in-plane \feld con\fguration,\ndisplayed on Fig. 6(a), the system is stabilized in cy-\ncloidal SS state at low temperature and zero magnetic\n\feld. In this state, the magnetic domains are connected\nby N\u0013 eel-type domain walls. As the in-plane \feld ap-\nproaches 1.2 T, all the domains convert into ABX state.\nIn Fig. 5(a), the magnetic texture of ABX state is shown\nwith B = 2 T and T = 0 K. As the magnetic \feld in-\ncreases to 6.4 T, the ordered ABX state breaks down into\nFM ground state. Above this \feld, bimeron states can\nbe found. However, from the zero temperature phase di-\nagram of VSeTe in Fig 4(a), these states are considered\nas metastable ABX states and they remain higher en-\nergy states. Hence, for \felds above 6.4 T, all bimerons\ndissolve and the FM state becomes the ground state.\nBecause a bimeron is a topologically protected mag-\nnetic texture analogous to skyrmion, it can be identi\fed\nby a non-zero topological charge ( Q). A bimeron with\nchargeQcan be converted into another bimeron of op-7\nposite sign,\u0000Q, merely by changing the sign of its mag-\nnetic and spatial components. Indeed, a bimeron with\nchargeQidenti\fed by its spatially dependent magnetic\ncon\fguration, [S x(x,y),Sy(x,y),Sz(x,y)], can be converted\ninto another bimeron of charge \u0000Qwith magnetic con\fg-\nuration [S x(-x,y),-Sy(-x,y),-Sz(-x,y)]57. In addition, the\ncoexistence of two bimerons with opposite signs of Qcan\noccur without loss of state energy. Using Eq. (7), the\naverage charge Qis calculated in the ( T;B) phase di-\nagram and is shown in Fig. 6(d). The \fnite value of\naverageQrepresents the ABX region and its boundary\nwith the other topologically-trivial regions. We obtain\nQ= 4 which indicates that four bimerons present in the\nABX state, a number that remains uniform throughout\nthe ABX region at low temperatures, as shown in Fig.\n6(a).\nIn order to investigate temperature dependent phases,\nwe plot the normalized magnetic susceptibility and its\n\frst and second derivatives as a function of temperature\nat speci\fc applied \felds. Figure 7 shows the suscepti-\nbility and its derivatives as a function of temperature at\nB = 2.6 T. It is di\u000ecult to determine the phase bound-\naries using the normalized susceptibility curve. However,\n@\u001f\n@Tdisplays a jump at the phase transition. The in-\n\rection points of@\u001f\n@Tseparate three regions, i.e., ABX,\nFD and PM phases. The temperature corresponding to\nthe maximum and minimum values of@\u001f\n@Tdetermine the\nboundaries between the phases. The FD region comes\nafter the ABX phase upon increasing temperature. The\nmaximum value of@\u001f\n@Tshows the lower limit of the FD\nregion at 330 K at lower \felds. This transition tempera-\nture is \feld-dependent. This region is known to display\nskyrmions with \fnite lifetime58. This can also be appli-\ncable to bimerons as well. In this region, because spins\nare excited by thermal \ructuations, creation and annihi-\nlation of bimerons of opposite signs of Qoccurs, resulting\nin a \ructuation of Qbetween -4 to +4 during the simula-\ntion. On average, the coexistence of bimerons of opposite\ntopological charge leads to Q\u00190, as shown in Fig. 6(d).\nThe minimum value of@\u001f\n@Tin Fig. 7 before the conver-\ngence represents the Curie temperature of VSeTe and it\nis 660 K at 2.6 T \feld. At zero applied \feld, the Curie\ntemperature of VSeTe is 630 K.\nIn the case of out-of-plane applied \feld, the topological\ncharge of VSeTe as a function of temperature and applied\n\feld is shown in Fig. 6(b). At low temperature and in\nSkX region, we obtain Q= -4, i.e., four skyrmions in SkX\nstate. The FD region starts at 330 K and is almost inde-\npendent of the applied \feld. In this region, the average\nQremains constant in contrast to the ABX state. This\nobservation means that the number of skyrmions remains\nconstant and no skyrmions with opposite charge \u0000Qare\ncreated in the FD region. The critical temperature of\nVSeTe is 630 K in the out-of-plane \feld con\fguration.\nTherefore, at low temperatures the critical temperature\nof VSeTe is independent of the applied \feld direction.\n0 200 400 600 800 1000\nT (K)00.20.40.60.81Normalized χχ\n-0.4-0.200.20.4\n∂χ/∂T\n∂2χ/∂T2Derivatives x 10-2\nABXFD PMFIG. 7. Temperature dependent magnetic susceptibility and\nits \frst and second order derivatives of VSeTe system with\nin-plane \feld 2.6 T. The infection points of@\u001f\n@Tseparate three\nphases, namely, ABX, FD and PM phases respectively.\n2. VSSe and VSTe Phase Diagrams\nThe magnetic phase diagram of VSSe for an external\n\feld applied out of the plane is shown in Fig 6(b). The\nshape of the skyrmions remains circular due to out-of-\nplane anisotropy [see Fig 5(g)]. From Fig. 6(e), Q= -4\nwhich indicates that four skyrmions are present in the\nground state of SkX state. From the average Qand@\u001f\n@T\ncalculations, the FD region starts at 150 K. Although\ncreation and annihilation of skyrmions occur in this re-\ngion, the average Qremains constant which suggests that\nonly skyrmions with the same sign of Qare present at\nany time. The in\rection point of@\u001f\n@Thas a minimum at\n240 K, which is the critical temperature at zero \feld.\nLet us \fnally comment on VSTe, whose phase diagram\nis reported on Fig. 6(c). This system also possesses easy-\nplane anisotropy but has a much larger exchange than\nVSeTe. For this reason, VSTe becomes ferromagnetic at\nmuch smaller in-plane \felds. In the case of VSTe, DMI\nis very small compared to the magnetic exchange Jand\nhence only SS appears without external \feld. The critical\nordering temperature of VSTe is 780 K.\nIV. CONCLUSION\nWe have shown that reasonably strong DMIs can be\nobtained in Janus VXY monolayers by using \frst prin-\nciple calculations, in spite of the relatively weak SOC of\nvanadium. This large value is directly related to the elec-\ntric dipole induced by the dissimilar chalcogen elements,\nas already observed for the Rashba spin-splitting in Ref.\n??. In addition, we \fnd that whereas VSSe possesses a8\nweak out-of-plane anisotropy, VSeTe and VSTe are char-\nacterized by strong easy-plane anisotropy. This observa-\ntion suggests that the two latter materials could be an\ninteresting platform for spin super\ruidity60, although we\nleave this aspect to future work.\nWe emphasize that the DMI we obtain is rather weak\n(about one order of magnitude smaller than in transition\nmetal multilayers52), which is partially compensated by\nthe fact that VXY is a monolayer, much thinner than tra-\nditional transition metal thin \flms. Therefore, homochi-\nral spin spirals can be stabilized down to zero external\n\feld, and metastable skyrmions and bimerons can be ob-\ntained in VSeTe and VSSe, respectively. Nonetheless,\nthe fairly large magnetic anisotropy of these monolayers\nprevents the stabilization of chiral textures without ex-\nternal magnetic \feld. We also emphasize that whereas\nour study focuses on stable ground states, the possibil-\nity to stabilize skyrmion and bimeron crystals paves the\nway to the realization of isolated metastable skyrmions\nand bimerons, which are of highest interest for potentialapplications.\nThis study, along with recent work focusing on dif-\nferent materials35,37, suggests that transition metal\ndichalcogenides monolayers in Janus con\fguration can\nhost a wealth of chiral textures in spite of their weak\nSOC. One can easily foresee that chemical surface engi-\nneering can also be favorably used to modulate the inver-\nsion symmetry breaking, which calls for further experi-\nmental exploration.\nACKNOWLEDGMENTS\nThe work was supported by King Abdullah University\nof Science and Technology (KAUST) through the award\nOSR-2018-CRG7-3717 from the O\u000ece of Sponsored Re-\nsearch (OSR). This work used the resources of the Su-\npercomputing Laboratory at King Abdullah University\nof Science and Technology (KAUST) in Thuwal, Saudi\nArabia.\n\u0003slimane.laref@kaust.edu.sa\nydurga.goli@kaust.edu.sa\nzmanchon@cinam.univ-mrs.fr\n1N. Nagaosa and Y. Tokura, Nature Nanotechnology 8, 899\n(2013).\n2K.-S. Ryu, L. Thomas, S.-H. Yang, and S. Parkin, Nature\nNanotechnology 8, 527 (2013).\n3S. Emori, U. Bauer, S.-M. Ahn, E. Martinez, and G. S.\nBeach, Nature Materials 12, 611 (2013).\n4A. Fert, V. Cros, and J. Sampaio, Nature Nanotechnology\n8, 152 (2013).\n5S. Li, W. Kang, Y. Huang, X. Zhang, and Y. Zhou, Nan-\notechnology 18, 31LT01 (2017).\n6Y. Huang, W. Kang, X. Zhang, and Y. Zhou, Nanotech-\nnology 28, 08LT02 (2017).\n7D. Prychynenko, M. Sitte, K. Litzius, B. Kr uger, G. Bouri-\nano\u000b, M. Kl aui, J. Sinova, and K. Everschor-sitte, Physi-\ncal Review Applied 9, 14034 (2018).\n8S. Luo, M. Song, X. Li, Y. Zhang, J. Hong, X. Yang,\nX. Zou, N. Xu, and L. You, Nano Letters 18, 1180 (2018).\n9J. Z\u0013 azvorka, F. Jakobs, D. Heinze, N. Keil, S. Kromin,\nS. Jaiswal, K. Litzius, G. Jakob, P. Virnau, D. Pinna,\nK. Everschor-sitte, L. R\u0013 ozsa, A. Donges, U. Nowak, and\nM. Kl aui, Nature Nanotechnology 14, 658 (2019).\n10K. M. Song, J.-s. Jeong, B. Pan, X. Zhang, J. Xia, S. Cha,\nT.-e. Park, K. Kim, S. Finizio, J. Raabe, J. Chang,\nY. Zhou, W. Zhao, W. Kang, H. Ju, and S. Woo, Na-\nture Electronics 3, 148 (2020).\n11T. Moriya, Physical Review 120, 91 (1960).\n12I. Dzyaloshinskii, Soviet Physics JETP 5, 1259 (1957).\n13S. M uhlbauer, B. Binz, F. Jonietz, C. P\reiderer, A. Rosch,\nA. Neubauer, R. Georgii, and P. B oni, Science (New York,\nN.Y.) 323, 915 (2009).\n14X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han,\nY. Matsui, N. Nagaosa, and Y. Tokura, Nature 465, 901\n(2010).15W. Jiang, P. Upadhyaya, W. Zhang, G. Yu, M. B.\nJung\reisch, F. Y. Fradin, J. E. Pearson, Y. Tserkovnyak,\nK. L. Wang, O. Heinonen, S. G. E. Velthuis, and A. Ho\u000b-\nmann, Science 349, 283 (2015).\n16G. Chen, A. Mascaraque, A. T. N'Diaye, and A. K.\nSchmid, Applied Physics Letters 106, 242404 (2015).\n17O. Boulle, J. Vogel, H. Yang, S. Pizzini, D. d. S. Chaves,\nA. Locatelli, T. O. M. A. Sala, L. D. Buda-Prejbeanu,\nO. Klein, M. Belmeguenai, Y. Roussign\u0013 e, A. Stashkevich,\nS. M. Ch\u0013 erif, L. Aballe, M. Foerster, M. Chshiev, S. Auf-\nfret, I. M. Miron, and G. Gaudin, Nature Nanotechnology\n11, 449 (2016).\n18C. Moreau-Luchaire, C. Mouta\fs, N. Reyren, J. Sampaio,\nC. A. F. Vaz, N. Van Horne, K. Bouzehouane, K. Gar-\ncia, C. Deranlot, P. Warnicke, P. Wohlh uter, J.-M. George,\nM. Weigand, J. Raabe, V. Cros, and A. Fert, Nature Nan-\notechnology 11, 444 (2016).\n19S. Woo, K. Litzius, B. Kr uger, M.-Y. Im, L. Caretta,\nK. Richter, M. Mann, A. Krone, R. M. Reeve, M. Weigand,\nP. Agrawal, I. Lemesh, M.-A. Mawass, P. Fischer,\nM. Kl aui, and G. S. D. Beach, Nature Materials 15, 501\n(2016).\n20C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao,\nW. Bao, C. Wang, Y. Wang, Z. Qiu, R. Cava, S. Louie,\nJ. Xia, and X. Zhang, Nature 546, 265 (2017).\n21D. J. O'Hara, T. Zhu, A. H. Trout, A. S. Ahmed, Y. K.\nLuo, C. H. Lee, M. R. Brenner, S. Rajan, J. A. Gupta,\nD. W. McComb, et al. , Nano Letters 18, 3125 (2018).\n22D. J. O'Hara, T. Zhu, A. H. Trout, A. S. Ahmed, Y. K.\nLuo, C. H. Lee, M. R. Brenner, S. Rajan, J. A. Gupta,\nD. W. McComb, and R. K. Kawakami, Nano Letters 18,\n3125 (2018).\n23D. J. O'Hara, T. Zhu, and R. K. Kawakami, IEEE Mag-\nnetics Letters 9, 1 (2018).\n24M. Bonilla, S. Kolekar, Y. Ma, H. C. Diaz, V. Kalappattil,\nR. Das, T. Eggers, H. R. Gutierrez, M.-H. Phan, and\nM. Batzill, Nature Nanotechnology 13, 289 (2018).9\n25B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein,\nR. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A.\nMcGuire, D. H. Cobden, et al. , Nature 546, 270 (2017).\n26Z. Fei, B. Huang, P. Malinowski, W. Wang, T. Song,\nJ. Sanchez, W. Yao, D. Xiao, X. Zhu, A. F. May, et al. ,\nNature Materials 17, 778 (2018).\n27Y. C. Cheng, Z. Y. Zhu, M. Tahir, and U. Schwingen-\nschl ogl, Europhysics Letters 102, 57001 (2013).\n28A.-Y. Lu, H. Zhu, J. Xiao, C.-P. Chuu, Y. Han, M.-H.\nChiu, C.-C. Cheng, C.-W. Yang, K.-H. Wei, Y. Yang, et al. ,\nNature Nanotechnology 12, 744 (2017).\n29J. Zhang, S. Jia, I. Kholmanov, L. Dong, D. Er, W. Chen,\nand H. Guo, ACS Nano 11, 8192 (2017).\n30L. Zhang, Z. Yang, T. Gong, R. Pan, and H. Wang, Jour-\nnal of Materials Chemistry A 8, 8813 (2020).\n31H. U. Din, M. Idrees, A. Albar, M. Sha\fq, I. Ahmad, C. V.\nNguyen, and B. Amin, Physical Review B 100, 165425\n(2019).\n32J. Chen, K. Wu, H. Ma, W. Hu, and J. Yang, RSC Ad-\nvances 10, 6388 (2020).\n33A. Manchon, J. Zelezn\u0013 y, M. Miron, T. Jungwirth,\nJ. Sinova, A. Thiaville, K. Garello, and P. Gambardella,\nReview of Modern Physics 91, 035004 (2019).\n34I. Smaili, S. Laref, J. H. Garcia, U. Schwingen-\nschlogl, S. Roche, and A. Manchon, arXiv preprint\narXiV:2007.07579v1 , 1 (2020), arXiv:2007.07579.\n35J. Liang, W. Wang, H. Du, A. Hallal, K. Garcia,\nM. Chshiev, A. Fert, and H. Yang, Physical Review B\n101, 184401 (2020).\n36F. Zhang, H. Zhang, W. Mi, and X. Wang, Phys. Chem.\nChem. Phys. 22, 8647 (2020).\n37J. Yuan, Y. Yang, Y. Cai, Y. Wu, Y. Chen, X. Yan, and\nL. Shen, Physical Review B 101, 094420 (2020).\n38D. Hamann, Physical Review Letters 42, 662 (1979).\n39E. Wimmer, H. Krakauer, M. Weinert, and A. J. Freeman,\nPhysical Review B 24, 864 (1981).\n40\\Fleur,\" .\n41Y. Ma, Y. Dai, M. Guo, C. Niu, Y. Zhu, and B. Huang,\nACS Nano 6, 1695 (2012).\n42J. Feng, L. Peng, C. Wu, X. Sun, S. Hu, C. Lin, J. Dai,\nJ. Yang, and Y. Xie, Advanced Materials 24, 1969 (2012).\n43H. Zhang, L.-M. Liu, and W.-M. Lau, Journal of Materials\nChemistry A 1, 10821 (2013).\n44H. Pan, The Journal of Physical Chemistry C 118, 13248\n(2014).\n45A. Oswald, R. Zeller, P. Braspenning, and P. Dederichs,\nJournal of Physics F: Metal Physics 15, 193 (1985).\n46A. I. Liechtenstein, M. Katsnelson, V. Antropov, and\nV. Gubanov, Journal of Magnetism and Magnetic Mate-\nrials67, 65 (1987).\n47C. Li, A. J. Freeman, H. Jansen, and C. Fu, Physical\nReview B 42, 5433 (1990).\n48L. Sandratskii, Journal of Physics: Condensed Matter 3,\n8565 (1991).\n49P. Kurz, F. F orster, L. Nordstr om, G. Bihlmayer, and\nS. Bl ugel, Physical Review B 69, 024415 (2004).\n50M. Heide, G. Bihlmayer, and S. Bl ugel, Physica B: Con-\ndensed Matter 404, 2678 (2009).\n51B. Zimmermann, M. Heide, G. Bihlmayer, and S. Bl ugel,\nPhysical Review B 90, 115427 (2014).\n52A. Belabbes, G. Bihlmayer, F. Bechstedt, S. Bl ugel, and\nA. Manchon, Physical Review Letters 117, 247202 (2016).\n53R. F. L. Evans, W. J. Fan, P. Chureemart, T. A. Ostler,\nM. O. A. Ellis, and R. W. Chantrell, Journal of Physics:Condensed Matter 26, 103202 (2014).\n54B. Berg and M. Luscher, Nuclear Physics B 190, 412\n(1981).\n55M. B ottcher, S. Heinze, S. Egorov, and J. Sinova, New\nJournal of Physics 20, 103014 (2018).\n56G. P. M uller, M. Ho\u000bmann, C. Di\u0019elkamp, D. Sch urho\u000b,\nS. Mavros, M. Sallermann, N. S. Kiselev, H. J\u0013 onsson, and\nS. Bl ugel, Physical Review B 99, 224414 (2019).\n57L. Shen, X. Li, J. Xia, and L. Qiu, arXiv preprint\narXiv:2005.11924 (2020), arXiv:arXiv:2005.11924v2.\n58L. Rozsa, E. Simon, K. Palotas, L. Udvardi, and L. Szun-\nyogh, Physical Review B 93, 024417 (2016).\n59S. V. Grigoriev, E. V. Moskvin, V. A. Dyadkin, D. Lamago,\nT. Wolf, H. Eckerlebe, and S. V. Maleyev, Physical Review\nB83, 224411 (2011).\n60S. Takei and Y. Tserkovnyak, Physical Review Letters 112,\n227201 (2014)."    },    {        "title": "2011.07826v1.Microwave_spectroscopy_of_the_low_temperature_skyrmion_state_in_Cu2OSeO3.pdf",        "content": "Microwave spectroscopy of the low-temperature skyrmion state in\nCu2OSeO 3\nAisha Aqeel,1,\u0003Jan Sahliger,1,\u0003Takuya Taniguchi,1Stefan\nM andl,1Denis Mettus,1Helmuth Berger,2Andreas Bauer,1\nMarkus Garst,3, 4Christian P\reiderer,1and Christian H. Back1, 5\n1Physik-Department, Technische Universit at M unchen, D-85748 Garching, Germany\n2\u0013Ecole Polytechnique Federale de Lausanne, CH-1015 Lausanne, Switzerland\n3Institut f ur Theoretische Festk orperphysik,\nKarlsruhe Institute of Technology, D-76131 Karlsruhe, Germany\n4Institute for quantum materials and technology,\nKarlsruhe Institute of Technology, D-76344 Eggenstein-Leopoldshafen, Germany\n5Munich Center for Quantum Science and\nTechnology (MCQST), D-80799 M unchen, Germany\n(Dated: November 17, 2020)\nAbstract\nIn the cubic chiral magnet Cu 2OSeO 3a low-temperature skyrmion state (LTS) and a concomi-\ntant tilted conical state are observed for magnetic \felds parallel to h100i. In this work, we report\non the dynamic resonances of these novel magnetic states. After promoting the nucleation of the\nLTS by means of \feld cycling, we apply broadband microwave spectroscopy in two experimental\ngeometries that provide either predominantly in-plane or out-of-plane excitation. By compar-\ning the results to linear spin-wave theory, we clearly identify resonant modes associated with the\ntilted conical state, the gyrational and breathing modes associated with the LTS, as well as the\nhybridization of the breathing mode with a dark octupole gyration mode mediated by the magne-\ntocrystalline anisotropies. Most intriguingly, our \fndings suggest that under decreasing \felds the\nhexagonal skyrmion lattice becomes unstable with respect to an oblique deformation, re\rected in\nthe formation of elongated skyrmions.\n1arXiv:2011.07826v1  [cond-mat.mtrl-sci]  16 Nov 2020Magnetic skyrmions are topologically nontrivial spin whirls that are observed in a wide\nrange of bulk materials, such as cubic chiral magnets, lacunar spinels, Heusler compounds,\nor frustrated magnets [1{6]. As an immediate consequence of their nontrivial topology,\nthe creation or annihilation of skyrmions involves rather complex winding and unwinding\nprocesses [7{9]. For instance, when undergoing the transition into the topologically trivial\nhelical state, Bloch points initiate the coalescence of neighboring skyrmions. At low tem-\nperatures, this process may give rise to magnetic textures that share similarities with both\na small-domain helical state incorporating topological disclination defects at the domain\nboundaries [10{12] and a state composed of elongated skyrmions [13].\nRecently, in addition to the well-established high-temperature skyrmion lattice state\n(HTS) [14, 15], a disconnected low-temperature skyrmion state (LTS) was identi\fed in\nCu2OSeO 3for magnetic \felds applied along h100ionly, highlighting the crucial role of\nmagnetocrystalline anisotropies for the stabilization of the LTS [16{18]. The nucleation\nof skyrmions at low temperatures is facilitated by a tilted conical state as an intermediate\nstate [19, 20]. The resulting spin texture typically exhibits a high degree of disorder and a\npronounced history dependence, most notably the volume fraction of LTS may be increased\nby magnetic \feld cycling [16, 21].\nAlthough the suppression of \fnite-temperature e\u000bects initially limits the nucleation of\nthe LTS, it also permits to address the intrinsic ground-state properties of topologically non-\ntrivial objects, complementary to studies of the metastable HTS frozen-in by means of \feld\ncooling [16, 18, 22{30]. In this context, one aspect concerns the microwave dynamics, which\nin the cubic chiral magnets are well-understood for weak spin{orbit coupling [31{33] and\nmight be even exploited for new concepts in spintronic devices [34{37]. The characteristic\nexcitations comprise +Q and {Q modes in the helimagnetic phases as well as one breathing\nand two gyrational modes in the HTS. So far, however, no information was available on the\nmicrowave dynamics in the LTS and the tilted conical phase arising for larger spin{orbit\ncoupling.\nUsing broadband ferromagnetic resonance measurements in combination with linear spin-\nwave theory, we show that the excitation spectra in the LTS are highly reminiscent of those\nin the HTS, despite distinctly di\u000berent degrees of disorder and di\u000berent stabilization mecha-\nnisms, namely thermal \ructuations for the HTS [1, 38] and magnetocrystalline anisotropies\nfor the LTS [16, 17]. All material-speci\fc parameters entering the theory were determined\n2in previous studies [16, 20, 33], allowing for a parameter-free comparison between theory\nand experiment. By reducing the symmetry, the anisotropies mediate the hybridization of\nthe breathing mode with a dark octupole gyration mode in the LTS. In addition, the ex-\ncitations observed at low \felds under decreasing \feld indicate the presence of a breathing\nmode not expected for the helimagnetic ground state, suggesting an oblique instability of\nthe hexagonal skyrmion lattice driven by the elongation of skyrmions.\nFor the present study a single-crystal cuboid of dimensions 1 :37\u00021:5\u00021:82 mm3ori-\nented along [001], [110], and [1 \u001610], was carefully polished and placed on a coplanar waveg-\nuide (CPW) with one of the (001) surfaces facing down. Two sample positions were mea-\nsured; the sample was centered either on the central signal line (width: 1 mm) or on one\nof the gaps (width: 0.3 mm). This way, the exciting ac magnetic \feld was dominated by\neither in-plane or out-of-plane components, allowing to address selectively di\u000berent resonant\nmodes. As the lateral dimensions of sample and CPW were comparable, the excitation\nalways contained a weak contribution of the nondominant component [51]. If not stated\notherwise, data shown in the following were recorded under in-plane excitation. The static\nmagnetic \feld was applied parallel to [001], i.e., perpendicular to the plane of the CPW, and\nwas changed in steps of 1 mT. Low temperatures and static magnetic \felds were provided\nby a \row cryostat with a superconducting magnet. All measurements were carried out at a\ntemperature of 5 K. Using a vector network analyzer, at each \feld value the complex trans-\nmissionS21through the CPW was measured while increasing the frequency ffrom 1 GHz\nto 6.5 GHz in steps of 0.8 MHz. Background contributions were removed by subtracting\na spectrum recorded at high magnetic \feld (500 mT), yielding the di\u000berence \u0001 S21shown\nin the following. Complementary magnetization and ac susceptibility measurements were\ncarried out using a Quantum Design physical properties measurement system.\nThe magnetic phase diagram of Cu 2OSeO 3, shown in Fig. 1(a), represents an important\npoint of reference for the discussion of the experimental data. As typical for a cubic chiral\nmagnet, it comprises paramagnetic and \feld-polarized regimes at high temperatures and\n\felds, respectively, the conical state described in terms of the superposition of a homogeneous\nmagnetization and a helix both oriented along the \feld direction, and the high-temperature\nskyrmion lattice state in \fnite \felds just below the transition temperature Tc= 58 K [14,\n15, 39{43]. In addition, for magnetic \feld parallel to h100i, in Cu 2OSeO 3a separate low-\ntemperature skyrmion state (LTS) is observed in the vicinity of the upper critical \feld\n3\u00160Hc2\u001980 mT [16]. Although the LTS represents the thermodynamic ground state, its\nnucleation is extremely slow due to the complexity of the topological winding involved. In\nfact, a two-step process is observed in which the metastable tilted conical state, characterized\nby conical helices tilting away from the \feld direction, is required as an intermediate state\nprior to the nucleation of the LTS [19, 20].\nThis complex nucleation process is also re\rected in a pronounced dependence of the\nspin texture and the associated magnetic properties on the temperature and \feld history,\nwhere Fig. 1 depicts the situation under decreasing magnetic \feld, also including metastable\nstates. For the present study, it is therefore imperative to focus on distinct measurement\nprotocols described in the following. Starting in a high magnetic \feld well above Hc2, the\n\feld is decreased until reaching the LTS, i.e., 70 mT for the given sample shape. This initial\nmeasurement from high \felds is referred to as Hinitscan. Next, the magnetic \feld is cycled\nntimes between 70 mT and 62 mT, distinctly increasing the volume fraction of LTS as will\nbe discussed below. After this cycling, the magnetic \feld is either decreased, referred to as\nHn\ndecrscan, or increased, referred to as Hn\nincrscan.\nA typical measurement consisting of a Hinitscan, 20 \feld cycles, and a H20\ndecrscan is de-\npicted in Figs. 1(b) through 1(d). The abscissa corresponds to consecutive data points/scans\nthat are separated by \feld steps of 1 mT, the actual magnetic \feld value is shown in Fig. 1(b).\nThe color bar at the bottom indicates the prevailing magnetic state using the color coding\nintroduced in Fig. 1(a), where green shading indicates data that were recorded during the\ncycling of the \feld. Solid vertical lines mark the phase boundaries as inferred conveniently\nfrom the di\u000berential susceptibility, d M=dH, and the real part of the ac susceptibility, \u001f0, both\nshown in Fig. 1(c), where we refer to Ref. [20] for a comprehensive account. Field cycling\nenhances signatures related to the LTS, such as the pronounced maximum at its low-\feld\nboundary, which is interpreted as an increasing volume fraction of LTS in agreement with\nprevious reports [16, 21].\nTypical microwave spectroscopy data are depicted in Fig. 1(d). Here and in the follow-\ning, they are shown in terms of the transmission di\u000berence \u0001 S21recorded as a function of\nfrequency at each \feld value. Dark colors indicate strong absorption due to the excitation\nof resonant modes in the sample. We start the description at high magnetic \felds (left),\nwhere a linear slope is characteristic for a Kittel resonance in the \feld-polarized regime.\nIn addition to the prominent Kittel mode, at lower frequencies several standing spin wave\n4modes are excited due to the low magnetic damping, \u000b\u001910\u00004, of Cu 2OSeO 3at low temper-\natures [44, 45]. These modes are not of central interest for the present study but nevertheless\ncomplicate the interpretation of the results at lower \felds. As a consequence, the following\ndiscussion focuses on the dominant modes in each magnetic phase, for instance refraining\nfrom an analysis of the low-intensity clockwise gyration mode in the skyrmion phase.\nWhen ignoring the \feld cycling for a moment, the microwave spectrum reminds of the\nwell-established universal spectrum of the cubic chiral magnet [32, 33, 46{48], although it is\nmore complex due to the presence of the tilted conical state and the LTS. Under decreasing\nmagnetic \feld (left to right), a change of slope in the Kittel mode marks the onset of the tilted\nconical state. Decreasing the \feld further, a broad band of absorption increases in frequency,\nreminiscent of the behavior in the conical state. The \feld cycling is re\rected in a zigzag\nshape and a continuous shift of spectral weight from high to low frequencies, in particular\nresulting in the emergence of a set of low-frequency modes. These modes essentially decrease\nin frequency under decreasing \feld until vanishing at the low-\feld boundary of the LTS.\nThe nature of the di\u000berent modes is most e\u000eciently discussed when also considering\ntheir evolution under \feld cycling, as elaborated on in Fig. 2, where initial Hinitscans are\ncombined with Hn\ndecrscans measured after di\u000berent numbers, n, of \feld cycles. For clarity,\nthe \feld cycles are omitted from the microwave spectra. In the H0\ndecrscan without cycling,\nshown in Fig. 2(a), two distinct modes may be distinguished that essentially increase in\nfrequency for decreasing \feld. These modes are marked by gray circles and remind of the\n+Q and {Q modes of the conical state. Additional absorption at reduced spectral weight\nfollows a qualitatively similar \feld dependence, suggesting standing spin wave modes as its\norigin. Changes of slope and a small discontinuity in the \u0006Q modes near Hc2are attributed\nto the tilted conical state, in which the \fnite angle between \feld and propagation direction\ninduces small deviations in the microwave response, akin to those in the helical state at\nsmall \felds, cf. Supplementary Information of Ref. [33].\nModerate cycling, such as in the H15\ndecrscan shown in Fig. 2(b), reduces the spectral weight\nof the\u0006Q modes at intermediate \felds. Instead, a set of modes emerges at distinctly lower\nfrequencies of about 2.5 GHz. Increasing the number of cycles to n= 140, as shown in\nFig. 2(c), intensi\fes this shift of spectral weight. The emerging modes exhibit frequencies\nand an evolution under decreasing \feld that are reminiscent of the counterclockwise gyration\nmode in the high-temperature skyrmion lattice. Also taking into account the redistribution\n5of spectral weight under \feld cycling and the disappearance of the modes at the low-\feld\nboundary of the LTS, these \fndings consistently suggest that the low-frequency modes are\nassociated with the LTS. From Lorentzian \fts, narrow line widths are extracted, translating\nto small damping constants \u000b\u00140:01 for these skyrmion modes. Note that the observation of\nresonant modes characteristic of skyrmions also provides direct evidence for the \fxed phase\nrelationship of the multi- Qstructure underlying the nontrivial topology of the LTS [49, 50].\nFurther information on the character of the di\u000berent modes, in particular those in the\nLTS, is obtained from measurements under di\u000berent excitation geometries as shown in Fig. 3.\nIn order to cover the entire \feld range of interest, each panel in Fig. 3 combines data from\nan H140\nincrand an H140\ndecrscan. By placing the sample either on the center of the signal line or\non one of the gaps of the CPW, the microwave excitation in the sample is dominated either\nby in-plane or out-of-plane components. In skyrmion states, in-plane excitation couples\ne\u000eciently to gyration modes, while out-of-plane excitation drives breathing modes [31]. In\nhelical or conical states, only excitation components perpendicular to the propagation vector\ncouple to the\u0006Q modes [32]. Consequently, modes observed under out-of-plane excitation\nare associated with magnetic structures that enclose \fnite angles with the magnetic \feld,\nsuch as multi-domain helical or tilted conical states.\nUnder in-plane excitation, as shown in Fig. 3(a) and all \fgures discussed so far, two groups\nof modes are prevailing below Hc2; (i) the modes in the conical and tilted conical state that\nincrease in frequency with decreasing \feld, and (ii) a lower-frequency mode that decreases\nwith decreasing \feld and is associated with the LTS (marked by cyan circles). When out-\nof-plane excitation dominates, as shown in Fig. 3(b), the overall spectral weight is reduced\nwith weak remnants of the previously discussed modes being excited due to small in-plane\nexcitation components. In addition, a prominent broad mode in the frequency range of the\n\u0006Q modes is associated with the tilted conical state. Perhaps most intriguingly, however, a\ndistinct low-frequency mode is observed across the entire \feld range below Hc2(marked by\norange circles).\nAs substantiated by the theoretical analysis presented below, the sensitivity with respect\nto the excitation geometry identi\fes the modes marked by cyan and orange circles as the\ncounterclockwise gyration mode and the breathing mode in the LTS. As a function of de-\ncreasing \feld, the resonance frequency of the breathing mode exhibits two abrupt increases.\nIn the following we will establish that the jump at \u001840 mT arises from anti-crossing due to\n6a hybridization with a dark octupole gyration mode that is also observed in the quenched\nhigh-temperature skyrmion lattice [52], while the jump at \u001820 mT is consistent with the\nformation of elongated skyrmions.\nThe presence of elongated skyrmions is supported by a comparison of microwave spectra\nrecorded after \feld cycling, i.e., a H70\ndecrscan, with corresponding spectra recorded after\ninitial zero-\feld cooling in Fig. 4. Note that the latter data are measured under increasing\nmagnetic \feld, leading to a modi\fed sequence of phase transitions, where we refer to Ref. [20]\nfor a detailed account on this hysteresis. After zero-\feld cooling, a multi-domain helical state\nforms in Cu 2OSeO 3with three equally populated domains of helices oriented along the h100i\naxes, representing the thermodynamic ground state. A magnetic \feld pointing along one\nof theh100iaxes favors the domain aligned with the \feld, resulting in an increase of its\npopulation upon increasing \feld and, eventually, in a single-domain conical state. This\nprocess is practically irreversible at low temperatures and thus a single helimagnetic domain\nis obtained after removing the \feld [11, 53]. When decreasing the \feld starting from the LTS,\nthe situation is markedly di\u000berent and the system appears to be trapped in a metastable\nstate.\nAs shown in the central part of Fig. 4, the resonance frequencies observed in zero \feld\nafter \feld cycling, cf. Fig. 4(a), di\u000ber decisively from those observed in the well-ordered\nmulti-domain helical state after zero-\feld cooling, cf. Fig. 4(b). This discrepancy suggests\nthat the dynamic properties of the complex zero-\feld magnetic texture obtained after \feld\ncycling are not captured correctly by a description in terms of helical domains. Instead,\nthe calculations presented in the following imply that, at least from the point of view of\nmicrowave excitations, a description in terms of elongated skyrmions is more accurate.\nOur theoretical treatment follows previous work [33] and uses the standard phenomeno-\nlogical model for chiral magnets supplemented by magnetocrystalline anisotropies. This\nanisotropy contribution to the free energy, \"a, proves to be key for the description of the\nproperties of the LTS in Cu 2OSeO 3. The space group P213 allows various magnetocrys-\ntalline terms but we demonstrate that already the simplest form, \"a=K(m4\nx+m4\ny+m4\nz),\ncaptures the fundamental aspects when using the anisotropy constant K=\u00002\u0001103J=m3\nas in Ref. [16], cf. Supplemental Material [] for considerations on other values of K. In\norder to address the resonances of all experimentally observed magnetic textures across the\nentire \feld range, excitation frequencies were determined for (i) non-modulated and one-\n7dimensionally modulated states, i.e, the \feld-polarized, tilted conical, and conical states,\nwithin their respective (meta-)stability range, and (ii) two-dimensionally modulated states,\ni.e., the topologically nontrivial skyrmion states. In these calculations, the \u0006Q modes and\nthe gyrational modes are driven by in-plane excitation, while the breathing mode is driven\nby out-of-plane excitation. The resonance in the tilted conical state is excited in both cases.\nForK= 0 (not shown) the calculations reproduce the characteristic modes as observed\nin the conical and the skyrmion lattice state at high temperatures where Kis small [32, 33].\nAs shown in the experimental spectra in Fig. 5(a) and the calculated spectra in Fig. 5(b), a\n\fnite anisotropy leaves the Kittel mode in the \feld-polarized regime and the \u0006Q modes in\nthe conical state highly reminiscent of their counterparts in the isotropic case. At magnetic\n\felds just below Hc2, however, the tilted conical state (gray shading) becomes energetically\nmore favorable than the regular conical state, as re\rected by a change of slope, kinks, and\nminor discontinuities in the +Q mode. Note that the irregular \feld dependence of the\nresonance frequency originates in fact in a multitude of hybridizations.\nIn both experimental and calculated spectra of the LTS, shown in Figs. 5(c) and 5(d),\nthe counterclockwise gyration mode (cyan symbols) dominates under in-plane excitation.\nFinite anisotropy leaves the character of this mode essentially unchanged, i.e., its frequency\nmonotonically decreases with decreasing \feld akin to the counterclockwise gyration mode in\nthe high-temperature skyrmion lattice. In contrast, the breathing mode (orange symbols)\nprevailing in the LTS under out-of-plane excitation is subject to fundamental changes, with\ngood agreement between experiment and theory. While for the isotropic case, K= 0, the\nfrequency of the breathing mode monotonically increases with decreasing \feld, for K6= 0\na small local minimum just below Hc2is followed by characteristic anti-crossing around\n0:6Hc2. This signature is the hallmark of the hybridization with a dark octupole gyration\nmode mediated by the magnetocrystalline anisotropies. At lower \felds, around 0 :4Hc2, the\nhexagonal skyrmion lattice becomes unstable with respect to an oblique distortion. This\ninstability is driven by the elongation of skyrmions as illustrated by the real-space images in\nFigs. 5(i) and 5(ii). The frequency of the breathing mode is enhanced for the elongated case,\ncf. Supplemental Material [] for animations, and comes close to that of \u0006Q modes of the\nhelimagnetic order but remains distinctly lower, in agreement with the experimental data\nshown in Fig. 4. On a similar note, the spectral weight of the counterclockwise gyration\nmode is distinctly reduced for the elongated skyrmions, consistent with its disappearance in\n8the experimental spectra.\nThe oblique instability of the skyrmion lattice occurs in a \feld range where skyrmions\nare only metastable. Similar to metastable isolated skyrmions [54], they experience an\nelliptical instability and become elongated, leading to an oblique distortion of the hexagonal\nlattice. As topological unwinding is energetically costly, the history employed in Fig. 5(c)\nfavors the observation of this oblique instability, notably the magnetic \feld is decreased at\nlow temperatures after the nucleation of the topologically nontrivial LTS by means of \feld\ncycling. Following this history, the observation of a low-\feld resonance that is, in contrast\nto the helical\u0006Q modes, susceptible to out-of-plane excitation (orange symbols) suggests\nthat a (metastable) oblique skyrmion state hosting elongated skyrmions was indeed realized\nin the experiment.\nIn conclusion, the cubic chiral magnet Cu 2OSeO 3was studied by means of broadband fer-\nromagnetic resonance measurements, focusing in particular on the low-temperature skyrmion\nstate. Employing \feld cycling, two excitation geometries, and comparing the experimen-\ntal results to linear spin-wave theory, resonant modes in the conical, tilted conical, and\nlow-temperature skyrmion state are clearly identi\fed, with the breathing mode in the LTS\nexhibiting a characteristic hybridization. Most intriguingly, the resonances observed in small\n\felds after \feld cycling indicate the presence of elongated skyrmions. These \fndings not\nonly highlight how the robustness of topological non-triviality may in\ruence dynamic prop-\nerties of magnetic materials, but also showcase how the study of dynamic properties may\nprovide valuable insights to static properties, such as the microscopic nature of magnetic\ntextures.\nWe wish to thank G. Benka, A. Chacon, and S. Mayr for fruitful discussions and assistance\nwith the experiments. This work has been funded by the Deutsche Forschungsgemeinschaft\n(DFG, German Research Foundation) under SPP2137 (Skyrmionics, Project No. 360506545,\nProjects 403030645, 403191981, and 403194850), TRR80 (From Electronic Correlations to\nFunctionality, Project No. 107745057, Projects E1, F7, and G9), and the excellence cluster\nMCQST under Germany's Excellence Strategy EXC-2111 (Project No. 390814868). This\nproject has received funding from the European Metrology Programme for Innovation and\nResearch (EMPIR) programme co-\fnanced by the Participating States and from the Eu-\nropean Union's Horizon 2020 research and innovation programme. A.B., D.M., and C.P.\nacknowledge \fnancial support through the European Research Council (ERC) through Ad-\n9vanced Grant No. 788031 (ExQuiSid). T.T. acknowledges funding by the JSPS Overseas\nResearch Fellowship. M.G. is supported by DFG SFB1143 (Correlated Magnetism: From\nFrustration To Topology, Project No. 247310070, Project A07), DFG Grant No. 1072/5-1\n(Project No. 270344603), and DFG Grant No. 1072/6-1 (Project No. 324327023).\n\u0003\n[1] S. M uhlbauer, B. Binz, F. Jonietz, C. P\reiderer, A. Rosch, A. Neubauer, R. Georgii, and\nP. B oni, Science 323, 915 (2009).\n[2] X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han, Y. Matsui, N. Nagaosa, and\nY. Tokura, Nature 465, 901 (2010).\n[3] Y. Tokunaga, X. Z. Yu, J. S. White, H. M. R\u001cnnow, D. Morikawa, Y. Taguchi, and Y. Tokura,\nNat. Commun. 6, 7638 (2015).\n[4] I. K\u0013 ezsm\u0013 arki, S. Bord\u0013 acs, P. Milde, E. Neuber, L. M. Eng, J. S. White, H. M. R\u001cnnow, C. D.\nDewhurst, M. Mochizuki, K. Yanai, H. Nakamura, D. Ehlers, V. Tsurkan, and A. Loidl, Nat.\nMater. 14, 1116 (2015).\n[5] A. K. Nayak, V. Kumar, T. Ma, P. Werner, E. Pippel, R. Sahoo, F. Damay, U. K. R o\u0019ler,\nC. Felser, and S. S. P. Parkin, Nature 548, 561 (2017).\n[6] T. Kurumaji, T. Nakajima, M. Hirschberger, A. Kikkawa, Y. Yamasaki, H. Sagayama,\nH. Nakao, Y. Taguchi, T. Arima, and Y. Tokura, Science 365, 914 (2019).\n[7] P. Milde, D. K ohler, J. Seidel, L. M. Eng, A. Bauer, A. Chacon, J. Kindervater, S. M uhlbauer,\nC. P\reiderer, S. Buhrandt, C. Sch utte, and A. Rosch, Science 340, 1076 (2013).\n[8] J. Wild, T. N. G. Meier, S. P ollath, M. Kronseder, A. Bauer, A. Chacon, M. Halder,\nM. Schowalter, A. Rosenauer, J. Zweck, J. M uller, A. Rosch, C. P\reiderer, and C. H. Back,\nSci. Adv. 3, e1701704 (2017).\n[9] F. Kagawa, H. Oike, W. Koshibae, A. Kikkawa, Y. Okamura, Y. Taguchi, N. Nagaosa, and\nY. Tokura, Nat. Commun. 8, 1332 (2017).\n[10] A. Dussaux, P. Schoenherr, K. Koumpouras, J. Chico, K. Chang, L. Lorenzelli, N. Kanazawa,\nY. Tokura, M. Garst, A. Bergman, C. L. Degen, and D. Meier, Nat. Commun. 7, 12430\n(2016).\n[11] A. Bauer, A. Chacon, M. Wagner, M. Halder, R. Georgii, A. Rosch, C. P\reiderer, and\n10M. Garst, Phys. Rev. B 95, 024429 (2017).\n[12] P. Schoenherr, J. M uller, L. K ohler, A. Rosch, N. Kanazawa, Y. Tokura, M. Garst, and\nD. Meier, Nat. Phys. 14, 465 (2018).\n[13] D. Morikawa, X. Yu, K. Karube, Y. Tokunaga, Y. Taguchi, T. Arima, and Y. Tokura, Nano\nLett. 17, 1637 (2017).\n[14] S. Seki, X. Z. Yu, S. Ishiwata, and Y. Tokura, Science 336, 198 (2012).\n[15] T. Adams, A. Chacon, M. Wagner, A. Bauer, G. Brandl, B. Pedersen, H. Berger, P. Lemmens,\nand C. P\reiderer, Phys. Rev. Lett. 108, 237204 (2012).\n[16] A. Chacon, L. Heinen, M. Halder, A. Bauer, W. Simeth, S. M uhlbauer, H. Berger, M. Garst,\nA. Rosch, and C. P\reiderer, Nat. Phys. 14, 936 (2018).\n[17] A. O. Leonov, ArXiv14062177 Cond-Mat (2014), arXiv:1406.2177 [cond-mat].\n[18] L. J. Bannenberg, H. Wilhelm, R. Cubitt, A. Labh, M. P. Schmidt, E. Leli\u0012 evre-Berna, C. Pap-\npas, M. Mostovoy, and A. O. Leonov, npj Quantum Mater. 4, 11 (2019).\n[19] F. Qian, L. J. Bannenberg, H. Wilhelm, G. Chaboussant, L. M. Debeer-Schmitt, M. P.\nSchmidt, A. Aqeel, T. T. M. Palstra, E. Br uck, A. J. E. Lefering, C. Pappas, M. Mostovoy,\nand A. O. Leonov, Sci. Adv. 4, eaat7323 (2018).\n[20] M. Halder, A. Chacon, A. Bauer, W. Simeth, S. M uhlbauer, H. Berger, L. Heinen, M. Garst,\nA. Rosch, and C. P\reiderer, Phys. Rev. B 98, 144429 (2018).\n[21] D. Mettus, M. Halder, A. Chacon, A. Bauer, and C. P\reiderer, unpublished (2019).\n[22] W. M unzer, A. Neubauer, T. Adams, S. M uhlbauer, C. Franz, F. Jonietz, R. Georgii, P. B oni,\nB. Pedersen, M. Schmidt, A. Rosch, and C. P\reiderer, Phys. Rev. B 81, 041203 (2010).\n[23] R. Ritz, M. Halder, C. Franz, A. Bauer, M. Wagner, R. Bamler, A. Rosch, and C. P\reiderer,\nPhys. Rev. B 87, 134424 (2013).\n[24] A. Bauer, M. Garst, and C. P\reiderer, Phys. Rev. B 93, 235144 (2016).\n[25] H. Oike, A. Kikkawa, N. Kanazawa, Y. Taguchi, M. Kawasaki, Y. Tokura, and F. Kagawa,\nNat. Phys. 12, 62 (2016).\n[26] K. Karube, J. S. White, N. Reynolds, J. L. Gavilano, H. Oike, A. Kikkawa, F. Kagawa,\nY. Tokunaga, H. M. R\u001cnnow, Y. Tokura, and Y. Taguchi, Nat. Mater. 15, 1237 (2016).\n[27] Y. Okamura, F. Kagawa, S. Seki, and Y. Tokura, Nat. Commun. 7, 12669 (2016).\n[28] A. Bauer, A. Chacon, M. Halder, and C. P\reiderer, in Topology in Magnetism , edited by\nJ. Zang, V. Cros, and A. Ho\u000bmann (Springer, 2018) p. 151.\n11[29] G. Berruto, I. Madan, Y. Murooka, G. M. Vanacore, E. Pomarico, J. Rajeswari, R. Lamb,\nP. Huang, A. J. Kruchkov, Y. Togawa, T. LaGrange, D. McGrouther, H. M. R\u001cnnow, and\nF. Carbone, Phys. Rev. Lett. 120, 117201 (2018).\n[30] M. T. Birch, R. Takagi, S. Seki, M. N. Wilson, F. Kagawa, A. \u0014Stefan\u0014 ci\u0014 c, G. Balakrishnan,\nR. Fan, P. Steadman, C. J. Ottley, M. Crisanti, R. Cubitt, T. Lancaster, Y. Tokura, and\nP. D. Hatton, Phys. Rev. B 100, 014425 (2019).\n[31] M. Mochizuki, Phys. Rev. Lett. 108, 017601 (2012).\n[32] Y. Onose, Y. Okamura, S. Seki, S. Ishiwata, and Y. Tokura, Phys. Rev. Lett. 109, 037603\n(2012).\n[33] T. Schwarze, J. Waizner, M. Garst, A. Bauer, I. Stasinopoulos, H. Berger, C. P\reiderer, and\nD. Grundler, Nat. Mater. 14, 478 (2015).\n[34] N. Nagaosa and Y. Tokura, Nat. Nanotechnol. 8, 899 (2013).\n[35] Y. Okamura, F. Kagawa, M. Mochizuki, M. Kubota, S. Seki, S. Ishiwata, M. Kawasaki,\nY. Onose, and Y. Tokura, Nat. Commun. 4, 2391 (2013).\n[36] N. Ogawa, S. Seki, and Y. Tokura, Sci. Rep. 5, 9552 (2015).\n[37] M. Garst, J. Waizner, and D. Grundler, J. Phys. D 50, 293002 (2017).\n[38] S. Buhrandt and L. Fritz, Phys. Rev. B 88, 195137 (2013).\n[39] S. Seki, J.-H. Kim, D. S. Inosov, R. Georgii, B. Keimer, S. Ishiwata, and Y. Tokura, Phys.\nRev. B 85, 220406 (2012).\n[40] S. Seki, S. Ishiwata, and Y. Tokura, Phys. Rev. B 86, 060403 (2012).\n[41] A. A. Omrani, J. S. White, K. Pr\u0014 sa, I. \u0014Zivkovi\u0013 c, H. Berger, A. Magrez, Y.-H. Liu, J. H. Han,\nand H. M. R\u001cnnow, Phys. Rev. B 89, 064406 (2014).\n[42] F. Qian, H. Wilhelm, A. Aqeel, T. T. M. Palstra, A. J. E. Lefering, E. H. Br uck, and\nC. Pappas, Phys. Rev. B 94, 064418 (2016).\n[43] A. Bauer and C. P\reiderer, in Topological Structures in Ferroic Materials: Domain Walls,\nVortices and Skyrmions , edited by J. Seidel (Springer, 2016) p. 1.\n[44] I. Stasinopoulos, S. Weichselbaumer, A. Bauer, J. Waizner, H. Berger, S. Maendl, M. Garst,\nC. P\reiderer, and D. Grundler, Appl. Phys. Lett. 111, 032408 (2017).\n[45] M. Weiler, A. Aqeel, M. Mostovoy, A. Leonov, S. Gepr ags, R. Gross, H. Huebl, T. T. M.\nPalstra, and S. T. B. Goennenwein, Phys. Rev. Lett. 119, 237204 (2017).\n[46] M. Date, K. Okuda, and K. Kadowaki, J. Phys. Soc. Jpn. 42, 1555 (1977).\n12[47] J. D. Koralek, D. Meier, J. P. Hinton, A. Bauer, S. A. Parameswaran, A. Vishwanath,\nR. Ramesh, R. W. Schoenlein, C. P\reiderer, and J. Orenstein, Phys. Rev. Lett. 109, 247204\n(2012).\n[48] S. P ollath, A. Aqeel, A. Bauer, C. Luo, H. Ryll, F. Radu, C. P\reiderer, G. Woltersdorf, and\nC. H. Back, Phys. Rev. Lett. 123, 167201 (2019).\n[49] T. Adams, S. M uhlbauer, C. P\reiderer, F. Jonietz, A. Bauer, A. Neubauer, R. Georgii, P. B oni,\nU. Keiderling, K. Everschor, M. Garst, and A. Rosch, Phys. Rev. Lett. 107, 217206 (2011).\n[50] J. Kindervater, I. Stasinopoulos, A. Bauer, F. X. Haslbeck, F. Rucker, A. Chacon,\nS. M uhlbauer, C. Franz, M. Garst, D. Grundler, and C. P\reiderer, Phys. Rev. X 9, 041059\n(2019).\n[51] I. Stasinopoulos, S. Weichselbaumer, A. Bauer, J. Waizner, H. Berger, M. Garst, C. P\reiderer,\nand D. Grundler, Sci. Rep. 7, 7037 (2017).\n[52] S. Seki, private communication (2019).\n[53] A. Bauer, A. Neubauer, C. Franz, W. M unzer, M. Garst, and C. P\reiderer, Phys. Rev. B 82,\n064404 (2010).\n[54] A. Bogdanov and A. Hubert, Phys. Status Solidi B 186, 527 (1994).\n13Frequency f (GHz) �', dM/dH �0H (mT)(b)\n(c)(a)\n(d)\nNumber of scans\n�S (arb. u.)Magnetic field���H(mT)0\nTemperature T (K)\nHigh-temp. skyrmion\nConical\n45 903060\nHinitHn\ndecrHn\nincrField cycles nskyrmionElongated\nLow-temp. skyrmionTilted \nconical\n0H || <001>\nHinit Field cycling n = 20 H20\ndecrFIG. 1. Evolution of magnetic properties under \feld cycling. (a) Magnetic phase diagram as ob-\nserved under decreasing \feld. Arrows illustrate the \feld histories used in the following. (b,c) Mag-\nnetic \feld,\u00160H, di\u000berential susceptibility, d M=dH, and real part of the ac susceptibility, \u001f0. Data\nwere recorded under decreasing magnetic \feld with n= 20 \feld cycles in the LTS. (d) Microwave\nspectra. The color encodes the transmission di\u000berence \u0001 S21. With \feld cycling spectral weight\nis redistributed. The colored bar at the bottom indicates the dominating magnetic state, vertical\nsolid lines indicate phase transitions. See text for details.\n14Frequency f(GHz)\n-Q+QKittel(a)\n(b)\n(c)\nHinit Hn\ndecr\n�S (arb. u.)\nMagnetic field �0H (mT)n = 0\nn = 15\nn = 140FIG. 2. Evolution of the microwave spectra under \feld cycling. For each panel, the transmission\ndi\u000berence \u0001 S21was recorded during an initial Hinitscan to 70 mT (dotted green line), followed by\nn\feld cycles (not shown), before \fnally decreasing the \feld to zero in a Hn\ndecrscan. (a) Without\ncycling the spectra are dominated by the \u0006Q modes of the tilted conical and conical state (yellow\ncircles). (b,c) With increasing number of cycles spectral weight is continuously redistributed from\nthe\u0006Q modes to a previously unresolved set of low-frequency modes associated with the LTS\n(cyan circles).\n150 40 80 12023456Frequency f (GHz)\n0 40 80 120\n�S (arb. u.)\n0\n-1\n-2\n-3\n-4\n-5H140\ndecr H140\nincrH140\ndecr H140\nincr\nMagnetic field �0H (mT) Magnetic field �0H (mT)G GSxy H\nG GSxy H (a) (b)FIG. 3. Microwave spectra after \feld cycling for di\u000berent excitation geometries as schematically\ndepicted in the insets. Each panel is composed of two independent scans both obtained after 140\n\feld cycles, i.e., an H140\nincrand an H140\ndecrscan. (a) Sample centered on the signal line of the CPW\nyielding predominantly in-plane ( hx) excitation. (b) Sample centered on one of the gaps of the\nCPW yielding predominantly out-of-plane ( hz) excitation. Circles mark the characteristic modes\nin the LTS, namely the counterclockwise gyration mode (cyan) and the breathing mode (orange).\n160 25 50 75 100\nMagnetic field �0H (mT) Magnetic field �0H (mT)23456\n0 25 50 75 100\n-8\n-10-6-4-20�S (arb. u.)Frequency f (GHz)Hinit H70\ndecr Zero-field cooling\n(a) (b)FIG. 4. Comparison of microwave spectra after \feld cycling and after initial zero-\feld cooling.\n(a) Data for decreasing magnetic \feld after \feld cycling composed of an Hinitand an H70\ndecrscan.\n(b) Data for increasing magnetic \feld after initial zero-\feld cooling. Note in particular the dis-\ncrepancy of the resonance frequencies in zero \feld. See text for details.\n17(d)(a)\n(b)\n(c)\nCCWBreathingElongated\nskyrmionLTS\nNormalized field H/Hc2Normalized frequency f/fc2\n(i)0.611.4 \n0.611.4 \n0 0.5 1 1.51.8(ii)+Q\n-QTC\nKittelFM \n0.611.4 \n0.611.4 Conical\nCalculatedExperiment\nCalculatedExperiment1D\n1D\n2D\n2DFIG. 5. Comparison of experimental and calculated spectra. Frequency and magnetic \feld values\nare normalized to their respective values at Hc2. (a) Experimental spectra for the \feld-polarized,\ntilted conical, and conical states. Data inferred from Fig. 2(a). (b) Calculated spectra for the non-\nmodulated and one-dimensionally modulated states. (c) Experimental spectra for the skyrmion\nstates. Data inferred from Fig. 4. (d) Calculated spectra for the two-dimensionally modulated\nstates. A counterclockwise gyration mode (cyan symbols) and a breathing mode (orange symbols)\nare identi\fed. Symbol sizes in (b) and (d) represent spectral weight. (i,ii) Calculated real-space\nimages of the magnetic texture associated with the breathing mode at high and low \felds.\n18"    },    {        "title": "2012.05076v3.Spin_lattice_model_for_cubic_crystals.pdf",        "content": "Spin-lattice model for cubic crystals\nP. Nieves1,\u0003J. Tranchida2, S. Arapan1, and D. Legut1\n1IT4Innovations, V ˇSB - Technical University of Ostrava,\n17. listopadu 2172/15, 70800 Ostrava-Poruba, Czech Republic and\n2Computational Multiscale Department, Sandia National Laboratories,\nP .O. Box 5800, MS 1322, 87185 Albuquerque, NM, United States\n(Dated: March 2, 2021)\nWe present a methodology based on the N ´eel model to build a classical spin-lattice Hamiltonian for cubic\ncrystals capable of describing magnetic properties induced by the spin-orbit coupling like magnetocrystalline\nanisotropy and anisotropic magnetostriction, as well as exchange magnetostriction. Taking advantage of the an-\nalytical solutions of the N ´eel model, we derive theoretical expressions for the parameterization of the exchange\nintegrals and N ´eel dipole and quadrupole terms that link them to the magnetic properties of the material. This\napproach allows to build accurate spin-lattice models with the desire magnetoelastic properties. We also explore\na possible way to model the volume dependence of magnetic moment based on the Landau energy. This new\nfeature can allow to consider the effects of hydrostatic pressure on the saturation magnetization. We apply this\nmethod to develop a spin-lattice model for BCC Fe and FCC Ni, and we show that it accurately reproduces the\nexperimental elastic tensor, magnetocrystalline anisotropy under pressure, anisotropic magnetostrictive coeffi-\ncients, volume magnetostriction and saturation magnetization under pressure at zero-temperature. This work\ncould constitute a step towards large-scale modeling of magnetoelastic phenomena.\nI. INTRODUCTION\nMagnetoelastic interactions couple the motion of atoms in a\nmagnetic material with atomic magnetic moments, and allow\nto transfer mechanical and thermal energies between phonon\nand magnon subsystems1. Magnetoelasticity is of great in-\nterest for applications, but also from a fundamental point of\nview. For instance, precise control of magnetization through a\nmechanical excitation of the motion of atoms in magnetic ma-\nterials, and vice versa, has enabled the development of a wide\nrange of technological applications such as sensors (torque\nsensors, motion and position sensors, force and stress sen-\nsors) and actuators (sonar transducer, linear motors, rotational\nmotors, and hybrid magnetostrictive/piezoelectric devices)2–5.\nSimilarly, the combination of magnetism and heat is exploited\nin many applications like heat-assisted magnetic recording\n(HAMR)6, thermally assisted magnetic random access mem-\nory (MRAMs)7, ultrafast all-optically induced magnetization\ndynamics8,9, magnetic refrigeration10, and biomedical mag-\nnetic hyperthermia11.\nMagnetoelastic effects can also have a strong influence on\nthe thermo-mechanical properties of materials. This is for ex-\nample the case of the phononic component of the thermal con-\nductivity. Though magnon-phonon scattering, it can abruptly\nchange through magnetic phase-transitions12,13. For metallic\noxides presenting strong magnetoelastic effects, and for which\naccurate thermal conductivity predictions can be of practical\nimportance (such as uranium dioxide14), the development of\naccurate numerical models is still an ongoing process. Sim-\nilarly, magnetoelastic effects can also play an important role\nin the thermal expansion of magnetic materials like in Invars,\nwhere is large enough to cancel the normal thermal contrac-\ntion, leading to nearly zero net thermal contraction over a\nbroad range of temperatures15.\nPresently, the theoretical and modeling techniques have\nreached a great level of development and accuracy to describe\nthe uncoupled dynamics of magnons and phonons at differentspatial and time scales. Typically, in magnetic materials this is\ndone by constraining or neglecting either the motion of atomic\nmagnetic moments or atoms. For example, in spin-polarized\nab-inito molecular dynamics (AIMD) magnetic moments are\nconstrained in certain directions and only atomic positions are\nupdated in each time step, while in classical atomistic spin\ndynamics (SD) and molecular dynamics (MD) the motion of\natoms or spins are neglected, respectively16–18. However, it is\nstill a challenge to find suitable modeling approaches to deal\nwith processes where the interaction between magnons and\nphonons is essential, like in magneto-caloric and magneto-\nelastic phenomena. The lack of such models is limiting the\nmulti-scale design of materials suitable for relevant techno-\nlogical applications based on these physical processes. Re-\ncently, novel attempts to address this problem have been pro-\nposed. Stockem et al. demonstrated that for small supercells,\na consistent interface can be designed to couple spin-polarized\nAIMD and classical SD19. Although offering an excellent\nlevel of accuracy, this approach presents the space and time\nscale limitations of first-principles approaches, and does not\nappear to be suited for running meso-scale magneto-elastic\nsimulations. Another concept, referred to as “spin-lattice dy-\nnamics”, is based on the combination of classical spin and\nmolecular dynamics (SD-MD), which includes the spatial de-\npendence of exchange integrals in the spin equation of motion,\namong other features20–26. The computational cost of this\nclassical approach scales linearly with the number of magnetic\natoms in the system26. Combined to accurate massively par-\nallel algorithms, this enables the simulation of multi-million\nmagnetic atom systems on time scales sufficient to accurately\nstudy magnon-phonon relaxation processes22,26.\nThese new ideas have opened up interesting opportuni-\nties and questions about how to model and study magneto-\ncaloric and magneto-elastic phenomena within a multi-scale\napproach. In particular, the coarse-grained modeling of spin-\norbit coupling (SOC) through magnetocrystalline anisotropy\n(MCA) in SD-MD is currently a bottle neck of this issue27.arXiv:2012.05076v3  [cond-mat.mtrl-sci]  1 Mar 20212\nThe single-ion model of MCA is widely used in SD, but\nunfortunately it does not couple atom and spin degrees of\nfreedom. This drawback can be overcome using the N ´eel\nmodel (two-ion model) that reproduces the correct symmetry\nof MCA, and couples atom and spin motion. Hence, despite\nsome limitations of the N ´eel model concerning non-magnetic\natoms and its phenomenological nature28, it seems a promis-\ning starting point to build a SD-MD model capable of simulat-\ning magneto-caloric and magneto-elastic phenomena. In this\nwork, we propose a general procedure to find the parameters\nof the N ´eel model within the Bethe-Slater curve29–31that re-\nproduces the MCA, isotropic and anisotropic magnetoelastic\nproperties, and magnetization under pressure for cubic crys-\ntals at zero-temperature accurately.\nII. METHODOLOGY\nA. Spin-Lattice Hamiltonian\nIn the following discussion, we consider the spin-lattice\nHamiltonian\nHsl(rrr;ppp;sss) =Hmag(rrr;sss)+N\nå\ni=1pppi\n2mi+N\nå\ni;j=1V(ri j); (1)\nwhere rrri,pppi,sssi, and mistand for the position, momentum,\nnormalized magnetic moment and mass for each atom iin the\nsystem, respectively, V(ri j) =V(jrrri\u0000rrrjj)is the interatomic\npotential energy and Nis the total number of atoms in the\nsystem with total volume V. Here, we include the following\ninteractions in the magnetic energy\nHmag(rrr;sss) =\u0000µ0N\nå\ni=1µi(v)HHH\u0001sssi\u00001\n2N\nå\ni;j=1;i6=jJ(ri j)sssi\u0001sssj\n+HL(v)+HN´eel(rrr;sss);(2)\nwhere µi(v)is the atomic magnetic moment that depends on\nthe volume per atom of the system v=V=N,µ0is the vac-\nuum permeability, HHHis the external magnetic field, J(ri j)\nis the exchange parameter. The quantity HLis the Landau\nenergy22,32–34\nHL(v) =N\nå\ni=1(Aiµ2\ni(v)+Biµ4\ni(v)+Ciµ6\ni(v)); (3)\nwhere Ai,BiandCiare parameters, while HN´eelis the N ´eel\ninteraction\nHN´eel=\u00001\n2N\nå\ni;j=1;i6=jfg(ri j)+l1(ri j)h\n(eeei j\u0001sssi)(eeei j\u0001sssj)\u0000sssi\u0001sssj\n3i\n+q1(ri j)h\n(eeei j\u0001sssi)2\u0000sssi\u0001sssj\n3ih\n(eeei j\u0001sssj)2\u0000sssi\u0001sssj\n3i\n+q2(ri j)\u0002\n(eeei j\u0001sssi)(eeei j\u0001sssj)3+(eeei j\u0001sssj)(eeei j\u0001sssi)3\u0003\ng;\n(4)where eeei j=rrri j=ri j, and\nl1(ri j) =l(ri j)+12\n35q(ri j);\nq1(ri j) =9\n5q(ri j);\nq2(ri j) =\u00002\n5q(ri j):(5)\nIn the case of a collinear state ( sssiksssj), the Eq. 4 is reduced to\nHN´eel=\u00001\n2N\nå\ni;j=1;i6=jfg(ri j)+l(ri j)\u0012\ncos2yi j\u00001\n3\u0013\n+q(ri j)\u0012\ncos4yi j\u00006\n7cos2yi j+3\n35\u0013\ng(6)\nwhere cos yi j=eeei j\u0001sssi. The N ´eel energy reproduces the cor-\nrect symmetry of MCA and magnetoelastic energy28. It is\nconvenient to use g(ri j)to offset the exchange and Landau\nenergy in order to allow the forces and pressure become zero\nat the ground state, as detailed in Ma et al.20. To do so, we\nwrite this function as\ng(ri j) =\u0000J(ri j)+2\nN\u00001(Aiµ2\ni(v)+Biµ4\ni(v)+Ciµ6\ni(v)):(7)\nThis offset does not affect the precession dynamics of the\nspins. However, it allows to offset the corresponding mechan-\nical forces. This particular choice of the offset also implies\nthat the spatial dependence of the exchange and Landau en-\nergy is not taken into account in the evaluation of the mag-\nnetic energy at the ground state. The exchange and Landau\nenergies determine the value of magnetic moments32. In this\nmodel, we effectively take into account this fact by parame-\nterizing the volume dependence of magnetic moment using\nfirst-principles calculations. The spatial dependence of the\nexchange and Landau energy would also contribute to the to-\ntal energy when the lattice parameter is modified, influencing\nthe energy versus volume curve from which the equation of\nstate and elastic properties are derived. However, the lack of\nthis contribution in the model should not compromise its ac-\ncuracy as long as the interatomic potential V(ri j)correctly\nreproduces the equation of state and elastic properties. Typ-\nically, interatomic potentials are developed and designed for\nthis purpose. As a result of this offset, we see that the sec-\nond term in Eq.7 cancels with the Landau energy, so that we\ncan simplify the magnetic Hamiltonian Eq.(2) by removing\nthe Landau energy\nHmag(rrr;sss) =\u0000µ0N\nå\ni=1µi(v)HHH\u0001sssi\u00001\n2N\nå\ni;j=1;i6=jJ(ri j)sssi\u0001sssj\n+HN´eel(rrr;sss);(8)\nand setting\ng(ri j) =\u0000J(ri j): (9)\nConsequently, this approach has the advantage that it avoids\nthe calculation of the parameters Ai,BiandCiin the Lan-\ndau energy (Eq.3). As shown in Section III B, the param-\neterization of the volume dependence of magnetic moment3\nmight be simpler than the calculation of the parameters in\nthe Landau energy32. According to the N ´eel model, function\ng(ri j)can be linked to the volume magnetostriction induced\nby the exchange interactions (isotropic magnetostriction)35.\nThe N ´eel dipole ( l(ri j)) and quadrupole ( q(ri j)) terms can\ndescribe the effects induced by SOC and crystal field in-\nteractions like MCA and its strain dependence (anisotropic\nmagnetostriction)35. Here, we take into account the spatial\ndependence of J(ri j),l(ri j)andq(ri j)using the Bethe-Slater\ncurve, as implemented in the SPIN package of LAMMPS26\nJ(ri j) =4aJ\u0012ri j\ndJ\u00132\"\n1\u0000gn\u0012ri j\ndJ\u00132#\ne\u0000\u0010ri j\ndJ\u00112\nQ(Rc;J\u0000ri j);\nl(ri j) =4al\u0012ri j\ndl\u00132\"\n1\u0000gl\u0012ri j\ndl\u00132#\ne\u0000\u0010ri j\ndl\u00112\nQ(Rc;l\u0000ri j);\nq(ri j) =4aq\u0012ri j\ndq\u00132\"\n1\u0000gq\u0012ri j\ndq\u00132#\ne\u0000\u0010ri j\ndq\u00112\nQ(Rc;q\u0000ri j);\n(10)\nwhere Q(Rc;n\u0000ri j)is the Heaviside step function and Rc;n\n(n=J;l;q) is the cut-off radius. The parameters an,gn,\nanddn(n=J;l;q) must be determined in order to reproduce\nthe Curie temperature ( TC) and volume magnetostriction ( ws)\nviaJ(ri j), as well as anisotropic magnetostriction and MCA\nthrough l(ri j)andq(ri j). The parameterization of J(ri j)with\nthe Bethe-Slater curve is a well established procedure. For in-\nstance, to find the values of aJ,gJ, and dJ, one can fit the\nBethe-Slater curve to exchange parameters calculated with\nDensity Functional Theory (DFT) at fixed equilibrium posi-\ntions at zero-temperature26. However, in some cases this pro-\ncedure might lead to spin-lattice models that don’t reproduce\ncorrectly either TCorws. Hence, a strategy to parameterize\nJ(ri j)using the Bethe-Slater function in order to simulate cor-\nrectly these properties is highly desirable. Similarly, the pa-\nrameterization of l(ri j)andq(ri j)with the Bethe-Slater curve\nis a quite new approach, so that it is not clear how to obtain\nthe values of these parameters yet. In Section II B, we pro-\npose a general procedure to obtain these parameters for cubic\ncrystals based on the theoretical analysis of the N ´eel model35.\nIn Section III B we explore a possible parameterization of the\nvolume dependence of magnetic moment using the Landau\nenergy. In the present work, we study this model only at zero-\ntemperature. The equations of motion of this model at finite-\ntemperature are those implemented in the SPIN package of\nLAMMPS18,26. A detailed description of these equations can\nbe found in Ref. 26.\nB. Procedure to calculate the Bethe-Slater parameters of N ´eel\ninteraction for cubic crystals\nThe basic idea to calculate the Bethe-Slater parameters for\nthe N ´eel interaction is to find the theoretical relations that link\nEq. (6) to both the MCA and magnetoelastic energies. To il-\nlustrate this method, we will apply it to simple cubic (SC),\nbody-centered cubic (BCC) and face-centered cubic (FCC)crystals. The MCA energy for cubic systems reads36\nHcub\nMCA(aaa;r) =V K1(r)(a2\nxa2\ny+a2\nxa2\nz+a2\nya2\nz); (11)\nwhere K1is the first MCA constant with units of energy per\nvolume, ris the distance to the first nearest neighbor, Vis\nthe volume of the system, and ai(i=x;y;z) are the direction\ncosines of magnetization. From this equation we have\nV K1(r) =4\u0014\nHcub\nMCA\u00121p\n2;1p\n2;0;r\u0013\n\u0000Hcub\nMCA(1;0;0;r)\u0015\n:\n(12)\nNext, we evaluate the Eq. 6 with magnetic moment directions\nsss=\u0010\n1p\n2;1p\n2;0\u0011\nandsss= (1;0;0)up to first nearest neighbors,\nand we replace it in Eq. 12 in order to ensure that the N ´eel\nenergy gives the correct MCA energy. And by doing so, we\nfind the following relations for SC, BCC and FCC37\nSC:q(r0) =V0K1(r0)\n2N=1\n2r3\n0K1(r0);\nBCC :q(r0) =\u00009V0K1(r0)\n16N=\u0000p\n3\n4r3\n0K1(r0);\nFCC :q(r0) =\u0000V0K1(r0)\nN=\u00001p\n2r3\n0K1(r0);(13)\nwhere r0is the equilibrium distance to the first nearest neigh-\nbors, and Nis the number of atoms in the equilibrium volume\nV0. Here, q(r0)has units of energy per atom. In Appendix A\nwe show that the derivative of q(r)with respect to rcan be\nwritten as\nSC:r0¶q\n¶r\f\f\f\nr=r0=3\n2r3\n0K1(r0)\u0014\n1\u0000B\nK1¶K1\n¶P\u0015\nr=r0;\nBCC :r0¶q\n¶r\f\f\f\nr=r0=\u00003p\n3\n4r3\n0K1(r0)\u0014\n1\u0000B\nK1¶K1\n¶P\u0015\nr=r0;\nFCC :r0¶q\n¶r\f\f\f\nr=r0=\u00003p\n2r3\n0K1(r0)\u0014\n1\u0000B\nK1¶K1\n¶P\u0015\nr=r0;(14)\nwhere Bis the bulk modulus and Pis pressure. Here again,\nr0¶q=¶rhas units of energy per atom. Note that the dipole\nterm in Eq. 6 cancels out after summing all first nearest neigh-\nbors, so that it does not contribute to the MCA in the cubic\ncrystal symmetry. Since we are only considering N ´eel inter-\nactions up to the first nearest neighbors, we set the cut-off\nradius Rc;qin between the first and second nearest neighbors\nin Eq. 10, that is\nq(r0) =4aq\u0012r0\ndq\u00132\"\n1\u0000gq\u0012r0\ndq\u00132#\ne\u0000\u0010r0\ndq\u00112\n: (15)\nThe derivative of this function with respect to ris\n¶q\n¶r\f\f\f\nr=r0=8aqr0e\u0000\u0010r0\ndq\u00112\nd6q\u0002\ngqr4\n0\u0000(1+2gq)d2\nqr2\n0+d4\nq\u0003\n:(16)\nHence, we have two equations with three unknown variables\naq,gq, and dq. A reasonable strategy to reduce the number4\nof unknown variables is to set dqequal to the equilibrium dis-\ntance to the first nearest neighbors r0(dq=r0) because it has\nunit of distance and can be easily estimated. Hence, solving\nEqs. 15 and 16 gives\ndq=r0;\naq=e\n8\u0014\n2q(r0)\u0000r0¶q\n¶r\f\f\f\nr=r0\u0015\n;\ngq=r0¶q\n¶r\f\f\f\nr=r0\nr0¶q\n¶r\f\f\f\nr=r0\u00002q(r0):(17)\nThese are the Bethe-Slater parameters in terms of K1and\n¶K1=¶P(via Eqs. 13 and 14) to model the physics of MCA\nwithin the N ´eel model.\nLet’s now find the values of the Bethe-Slater parameters\nthat simulate magnetostriction. The magnetoelastic energy for\ncubic systems (point groups 432, ¯43m,m¯3m) reads38,39\nHcub\nme\nV0=b0(exx+eyy+ezz)+b1(a2\nxexx+a2\nyeyy+a2\nzezz)\n+2b2(axayexy+axazexz+ayazeyz);\n(18)\nwhere b0,b1andb2are the magnetoelastic constants with\nunits of energy per volume, and ei jare the elements of the\nstrain tensor. For small deformations (infinitesimal strain the-\nory), the strain tensor can be expressed in terms of the dis-\nplacement vector uuuas40,41\nei j=1\n2\u0012¶ui\n¶rj+¶uj\n¶ri\u0013\n; i;j=x;y;z (19)\nwhere ¶ui=¶rjis called the displacement gradient. For this\ndefinition of the strain tensor, the elastic energy for cubic crys-\ntal reads40,41\nHcub\nel\nV0=c11\n2(e2\nxx+e2\nyy+e2\nzz)+c12(exxeyy+exxezz+eyyezz)\n+2c44(e2\nxy+e2\nyz+e2\nxz);\n(20)\nwhere c11,c12andc44are the elastic constants. After evalu-\nating the N ´eel energy (Eq.6) for a strained cubic crystal up to\nfirst nearest neighbors, and equalizing it to Eq.18, one finds\nfor SC, BCC and FCC35,36\nSC:l(r0) =\u0000V0b2\n2N;r0¶l\n¶r\f\f\f\nr=r0=\u0000V0b1\nN;\nBCC :l(r0) =\u00003V0b1\n8N;r0¶l\n¶r\f\f\f\nr=r0=3V0\n8N(b1\u00003b2);\nFCC :l(r0) =V0\n2N\u0012b2\n2\u0000b1\u0013\n;r0¶l\n¶r\f\f\f\nr=r0=V0\nN\u0012\nb1\u00003b2\n2\u0013\n:\n(21)\nHere, we neglected the quadrupole contribution to the mag-\nnetoelastic energy37. This approximation is reasonable when\nq(r0)\u001cl(r0). In Section III, we show that BCC Fe and FCC\nNi fulfill this condition. Next, as we did previously, insert-\ning Eq.21 into the Bethe-Slater curve and its derivative, andsetting dl=r0allow us to obtain\ndl=r0;\nal=e\n8\u0014\n2l(r0)\u0000r0¶l\n¶r\f\f\f\nr=r0\u0015\n;\ngl=r0¶l\n¶r\f\f\f\nr=r0\nr0¶l\n¶r\f\f\f\nr=r0\u00002l(r0):(22)\nThese are the Bethe-Slater parameters in terms of b1andb2\n(via Eq. 21) to model the anisotropic magnetostriction within\nthe N ´eel model.\nLastly, we show the parameterization of the exchange inter-\naction via the first term in the N ´eel model (Eq.9) to simulate\nTCandws. From the analysis of the N ´eel model up to first\nnearest-neighbours one finds35\nSC:J(r0) =kBTC\n2;r0¶J\n¶r\f\f\f\nr=r0=ws(c11+2c12)V0\n3N;\nBCC :J(r0) =3kBTC\n8;r0¶J\n¶r\f\f\f\nr=r0=ws(c11+2c12)V0\n4N;\nFCC :J(r0) =kBTC\n4;r0¶J\n¶r\f\f\f\nr=r0=ws(c11+2c12)V0\n6N;\n(23)\nwhere kBis the Boltzmann constant. The relation between\nJ(r0)andTCwas obtained using the Mean-Field Approxima-\ntion (MFA). Inserting Eq.23 into the Bethe-Slater curve and\nits derivative, and setting dJ=r0allow us to obtain\ndJ=r0;\naJ=e\n8\u0014\n2J(r0)\u0000r0¶J\n¶r\f\f\f\nr=r0\u0015\n;\ngJ=r0¶J\n¶r\f\f\f\nr=r0\nr0¶J\n¶r\f\f\f\nr=r0\u00002J(r0);(24)\nwhere the cut-off radius Rc;Jmust be in between the first and\nsecond nearest neighbors. Notice that a similar procedure to\nthis one could be used if a function with three parameters dif-\nferent to the Bethe-Slater curve is chosen to describe the func-\ntions l(r),q(r)andJ(r)in SD-MD simulations.\nC. Volume dependence of magnetic moment\nA widely used approximation in SD consists to constrain\nthe magnitude of atomic magnetic moments to a constant\nvalue16,17, which is a good approximation for many practical\napplications. However, the magnitude of the atomic magnetic\nmoments can change significantly when the volume of the sys-\ntem changes greatly. This effect can be analyzed in terms\nof the Landau expansion around the critical point where the\nmagnetization becomes zero42. The Landau expansion con-\ntains only even powers of magnetization to fulfill the time-\nreversal symmetry. For instance, the Curie temperature ( Tc) is\na critical point where the magnetization becomes zero due to5\nthe disorder of magnetic moments induced by thermal fluc-\ntuations. Another critical point is the volume per atom vc\nwhere magnetic moment collapses ( µ(vc) =0)33. Moruzzi\nidentified three types of transitions for the magnetic moment\ncollapse33,34. Here, we discuss about type I transition where\nthe behavior is continuous across vc. In particular, we ex-\nplore a possible parameterization of the magnetic moments\nbased on the Landau energy to take into account its volume\ndependence. For a system with Natoms with equal magnetic\nmoments µwe can write the Landau expansion close to vcas\nHL(v) =N\nå\ni=1(Aiµ2\ni(v)+Biµ4\ni(v))+O(µ6)\n=N(Aµ2(v)+Bµ4(v));(25)\nwhere AandBare parameters, and vis the volume per atom of\nthe system. The analysis of the Landau expansion yields33,34\nµ(v)µpv\u0000vc: (26)\nThis square root dependence describes well the magnetic mo-\nment behaviour very close to vc. However, for many practical\napplications the equilibrium volume is significantly far from\nvc, so that one needs to include additional terms in Eq.26. In\norder to do so, we perform a Taylor expansion of the square\nof magnetic moment µ2around vc, that is\nµ2(v) =µ2(vc)+¶µ2\n¶v\f\f\f\f\f\nv=vc(v\u0000vc)\n+1\n2¶2µ2\n¶v2\f\f\f\f\f\nv=vc(v\u0000vc)2\n+1\n6¶3µ2\n¶v3\f\f\f\f\f\nv=vc(v\u0000vc)3+O((v\u0000vc)4);(27)\nwhere µ2(vc) =0. Making a square root on both sides of this\nequation we have\nµ(v) =q\naµ(v\u0000vc)+bµ(v\u0000vc)2+gµ(v\u0000vc)3\n\u0001Q(v\u0000vc);(28)\nwhere\naµ=¶µ2\n¶v\f\f\f\f\f\nv=vc\nbµ=1\n2¶2µ2\n¶v2\f\f\f\f\f\nv=vc\ngµ=1\n6¶3µ2\n¶v3\f\f\f\f\f\nv=vc:(29)\nThe Heaviside step function Q(v\u0000vc)was introduced in Eq.\n28 to ensure that the magnetic moment is zero at volumes\nlower than vc. The Taylor expansion was considered up tothe third order which is enough to correctly describe the mag-\nnetic moment of BCC Fe and FCC Ni within the range of\nvolume per atom discussed in this work ( v<20˚A3/atom). For\ncases with larger volume per atom than 20 ˚A3/atom one might\nneed to include higher order terms in the Taylor expansion.\nIn the vicinity of the critical volume ( (v\u0000vc)=vc\u001c1) Eq.\n28 becomes Eq.26, so that the result derived from the Landau\nexpansion is recovered33.\nAlternatively, instead of considering the volume depen-\ndence for the parameterization of the magnetic moment µ(v),\none could consider the pressure dependence of magnetic mo-\nment µ(P). In this case, one could apply the same procedure\nbut now performing the Taylor expansion around the critical\npressure Pcwhere the magnetic moment collapses. Note that\nthe function µ(P)could only be evaluated in this way from\nPcup to the negative pressure P0at which the pressure is re-\nversed due to the large interatomic distance, see Fig. 1. In this\nmodel, longitudinal fluctuations of magnetic moments at finite\ntemperature would naturally emerge from the fluctuations of\nthe volume per atom or pressure.\nFIG. 1. Schematic of the volume and hydrostatic pressure depen-\ndence of magnetic moment. Symbols µbandµirepresent the mag-\nnetic moment for bulk at zero-pressure and isolated atom, respec-\ntively.\nIII. SPIN-LATTICE MODEL FOR BCC FE AND FCC NI\nIn this section, we build a SD-MD model for BCC Fe and\nFCC Ni based on the methodology presented in Section II.\nThe construction of the model is splitted into the following\nstages in order to systematically compute each term in Eq.1,\nwhere the magnetic Hamiltonian is given by Eq.8.\nA. Interatomic potential\nIn the model we set the modified embedded atom method\n(MEAM) potentials developed by Asadi et al.43and Lee et\nal.44for the interatomic potential V(ri j)of BCC Fe and\nFCC Ni, respectively. These potentials give an elastic ten-\nsor very close to the experimental one at zero-temperature. In\nthis first stage, it is convenient to find the equilibrium vol-\nume and bulk modulus given by the model including only\nthe MEAM potential. To do so, we compute the energy of\na set of conventional unit cells with different volume using\nthe software LAMMPS18with the SPIN package26, and we6\nfit it to the Murnaghan equation of state (EOS)45,46. We ver-\nify that the pressure in the selected equilibrium state is lower\nthan 5\u000210\u00005GPa. In Fig.2, we present the calculation of\nthe energy versus volume curve for the conventional unit cell\n(2 atoms/cell for BCC Fe and 4 atoms/cell for FCC Ni). The\nequilibrium volume and bulk modulus found with this pro-\ncedure is v0=11:586754 ˚A3/atom and B=166:73 GPa for\nBCC Fe, and v0=10:903545 ˚A3/atom and B=188:85 GPa\nfor FCC Ni. Hence, the equilibrium distance to the first near-\nest neighbor is r0=2:4690386 ˚A for BCC Fe, and 2 :4890153\n˚A for FCC Ni. At this stage, it is also convenient to compute\nthe elastic constants. To do so, we evaluate the elastic ten-\nsor with software AELAS47interfaced with LAMMPS at the\nequilibrium volume v0including the MEAM potential. The\ndeveloped interface between AELAS and LAMMPS is avail-\nable on GitHub repository48. Here, we make use of the pro-\ngram Atomsk to convert some input files49. The calculated\nvalues and experimental ones are shown in Table II. We see\nthat these interatomic potentials gives a very similar elastic\ntensor to the experiment.\nFIG. 2. Calculation of the equation of the state for (top) BCC Fe and\n(bottom) FCC Ni with the SD-MD model including only the MEAM\npotential.\nB. Magnetic moment\nNext, we find the parameterization of the volume depen-\ndence of magnetic moment µ(v). Here, we estimate it usingDFT. Namely, the parameters vc,aµ,bµandgµin Eq.28 are\nobtained by fitting this equation to the magnetic moment ver-\nsus volume curve given by DFT. The DFT calculations are\nperformed with V ASP code50–52, which is an implementa-\ntion of the projector augmented wave (PAW) method53. We\nuse the interaction potentials generated for the Perdew-Burke-\nErnzerhof (PBE) version54of the Generalized Gradient Ap-\nproximation (GGA). We set an automatic Monkhorst-Pack k-\nmesh55gamma-centered grid with length parameter Rk=60.\nThe interactions were described by a PAW potential with 14\nand 16 valence electrons for BCC Fe and FCC Ni, respec-\ntively.\nFIG. 3. Calculation of the magnetic moment versus volume under\nnormal deformations obtained with DFT (blue dots) for BCC Fe and\nFCC Ni. Red line stands for the fitting curve.\nThe results of these calculations and corresponding fitting\ncurves are shown in Fig.3. Very similar results were previ-\nously reported by Moruzzi et al. using the augmented spher-\nical wave (ASW) method56. We see that the form of Eq.28\ndescribes quite well the data obtained by DFT. In the case\nof FCC Ni the deviation between the fitted curves and DFT\ndata is slightly larger than for BCC Fe. A better fit could be\nachieved by adding higher order terms in Eq.27. The values\nof the fitting parameters vc,aµ,bµandgµare presented in\nTable I. Inserting these values into Eq.28 allows us to com-\npute the magnetic moment at the equilibrium volume given\nby the SD-MD model including only MAEM potential ob-\ntained in Section III A ( v0=11:586754 ˚A3/atom for BCC Fe,\nandv0=10:903545 ˚A3/atom for FCC Ni). This calculation\ngives µ(v0) =2:34µBfor Fe, and µ(v0) =0:67µBfor Ni, while\nthe experimental values are 2 :22µBand 0 :606µBfor Fe and\nNi, respectively36. We see that this procedure overestimates\nslightly the magnetic moment.\nThe volume dependence of magnetic moment will allow us\nto study how magnetization changes under hydrostatic pres-\nsure (normal deformation). Note that in this model the magni-\ntude of the magnetic moment will not change under volume-\nconserving deformations. To check the validity of this approx-\nimation, we run some additional DFT calculations with V ASP\nusing the same setting as before to obtain the magnetic mo-\nment under volume-conserving tetragonal deformation57for\nBCC Fe. The results are plotted in Fig.4. We observe that a\nsignificant change of the magnetic moment only takes place at\nlarge tetragonal deformations. We verify that a similar trend is\nalso observed for other types of volume-conserving deforma-\ntions like trigonal deformation57. Therefore, to some extent,\nthe model might be able to describe the behaviour of magnetic7\nFIG. 4. Magnetic moment under a volume-conserving tetragonal de-\nformation of BCC Fe ( c=a=1) calculated with DFT.\nmoment and magnetization under deformations that combines\nan arbitrarily large normal deformation (which changes the\nvolume preserving the cubic symmetry) with a small volume-\nconserving deformation that changes the crystal symmetry.\nC. Exchange interaction\nLet’s now compute the parameterization of J(r). Firstly,\nnote that the equilibrium interatomic distance, EOS and elas-\ntic constants of the ground state (collinear state) are un-\nchanged after the exchange interaction is added to the SD-MD\nmodel thanks to the offset in the exchange energy. It is inter-\nesting to analyze the influence of different types of parame-\nterization of J(r)on the volume magnetostriction ws. Hence,\nfor the parameterization of J(r)we consider the following two\nsets of parameters for aJ,gJ,dJandRc;J.\n1. Set I: Effective short range exchange\nThe set I is calculated following the procedure described\nin Section II B, so that it leads to an effective short range\nexchange interaction. As mentioned above, the equilibrium\ndistance to the first nearest neighbor at the ground state is\nnot changed by the exchange interaction due to the offset\nin the exchange energy, so that according to Eq.24 we set\nd(I)\nJ=r0=2:4690386 ˚A for BCC Fe, and d(I)\nJ=2:4890153 ˚A\nfor FCC Ni. Next, we see in Eq. 23 that we need as inputs TC,\nc11,c12andwsto compute the J(r0)and¶J=¶r. In general,\nthese inputs can be obtained by theory or experiment. For in-\nstance, here we use the experimental value of TC(1043 K for\nBCC Fe and 627 K for FCC Ni)36. For the elastic constants,\nwe will make use of the theoretical values obtained by the SD-\nMD model itself using only the MEAM potential (see Table\nII). The experimental measurement of volume magnetostric-\ntion is difficult, and one can find significant discrepancies be-tween different works15,58. Hence, we will use the theoretical\nvalue of wsat zero-temperature calculated by Shuimizu using\nthe itinerant electron model58, that is ws=1:16\u000210\u00002for\nBCC Fe and 3 :75\u000210\u00004for FCC Ni. Inserting these quan-\ntities into Eq.24 via Eq.23 gives a(I)\nJ=\u000012:5921 meV/atom\nandg(I)\nJ=2:81897 for BCC Fe, and a(I)\nJ=8:35847 meV/atom\nandg(I)\nJ=\u00000:098217 for FCC Ni.\n2. Set II: Long range exchange\nThe second set of parameters (set II) is obtained by fitting\nthe Bethe-Slater function to the exchange integrals given by\nfirst-principles calculations26. The fitted parameters ( a(II)\nJ,\ng(II)\nJandd(II)\nJ) are shown in Table I. The value of a(II)\nJtaken\nfrom Ref.26has been multiplied by 2 due to the factor 1 =2\nin the exchange energy given by Eq.8. Here, we set a large\ncut-off R(II)\nc;J=4:5˚A to take into account the exchange inter-\nactions beyond first nearest neighbors (long range exchange\ninteraction). The Bethe-Slater function with parameters from\nset I and II is plotted in Fig.5. This figure will be analyzed in\nthe context of volume magnetostriction in Section IV B 4.\nD. N ´eel energy\nNow, we are in a position to calculate the Bethe-Slater pa-\nrameters for the dipole and quadrupole terms of the N ´eel in-\nteraction given by Eqs.22 and 17, respectively. Firstly, we\nnotice that a key quantity in these equations is the equilib-\nrium distance to the first nearest neighbors r0, which obvi-\nously depends on the N ´eel interaction. Fortunately, the en-\nergy of the dipole and quadrupole terms of the N ´eel interac-\ntion for Fe and Ni are of the order of µeV/atom (see Fig.7),\nso that they are much lower than the total energy (eV/atom).\nAs a result, these terms only induce a very small change in\nr0when they are included in the SD-MD model. This fact al-\nlows us to use r0given by the SD-MD model including only\nthe MEAM potential and exchange interaction to calculate the\nBethe-Slater parameters for the dipole and quadrupole terms\nof the N ´eel interaction. Hence, according to Eqs.17 and 22,\nwe can set dl=dq=r0=2:4690386 ˚A for BCC Fe, and\ndl=dq=2:4890153 ˚A for FCC Ni.\nOnce r0is determined, we calculate aqandgqusing Eqs.\n13, 14 and 17. Here, we set the experimental values of K1and\n(1=K1)(¶K1=¶P)approximately at zero-temperature, that is,\nK1=55 KJ/m3and(1=K1)(¶K1=¶P) =\u00007:3\u000210\u00002GPa\u00001\nfor BCC Fe, and K1=\u0000126 KJ/m3and(1=K1)(¶K1=¶P) =\n\u00002:8\u000210\u00002GPa\u00001for FCC Ni59,60. As we see in Eq.17,\nwe also need the bulk modulus. In principle we could set its\nexperimental value or the one given by the EOS of this SD-\nMD model that was obtained in Section III A. In this work\nwe choose the second option in order to describe more accu-\nrately the relation between volume and pressure of the SD-\nMD model. Inserting all these quantities in Eq.17 via Eqs. 13\nand 14 leads to aq=28:5189 µeV/atom and gq=1:05331 for8\nFIG. 5. Calculation of the Bethe-Slater function J(r)andq(r)for\n(top) BCC Fe and (bottom) FCC Ni using the two set of parameters\ngiven in Table I. Vertical dash line stands for the equilibrium distance\nof the first nearest neighbors r0.\nBCC Fe, and aq=\u000049:1335 µeV/atom and gq=1:1186 for\nFCC Ni.\nLastly, we calculate the Bethe-Slater parameters for the\ndipole term ( alandgl) using Eqs. 21 and 22. In this case we\nneed the values of the anisotropic magnetoelastic constants b1\nandb2. These constants are related to the magnetostrictive\ncoefficients ( l001andl111) and elastic constants ( ci j) via38,39\nb1=\u00003\n2l001(c11\u0000c12);\nb2=\u00003l111c44:(30)\nTo calculate b1andb2we use the experimental magnetostric-\ntive coefficients l001=26\u000210\u00006andl111=\u000030\u000210\u00006for\nBCC Fe, and l001=\u000060\u000210\u00006andl111=\u000035\u000210\u00006for\nFCC Ni at zero-temperature36. For the values of the elas-\ntic constants we choose the calculated ones with the SD-MD\nmodel including only the MEAM potential (see Table II).\nDoing so, we get b1=\u00003:74166 MJ/m3andb2=10:4643MJ/m3for BCC Fe, and b1=10:0611 MJ/m3and b2=\n13:9398 MJ/m3for FCC Ni. If we insert these values in\nEq.22 via Eq.21, then we obtain al=392:747µeV/atom and\ngl=0:824409 for BCC Fe, and al=179:396µeV/atom and\ngl=1:39848 for FCC Ni.\nTABLE I. Parameters of the SD-MD model for BCC Fe and FCC Ni.\nSD-MD model\nparametersBCC Fe FCC Ni\naµ(µ2\nB\u0001atom/ ˚A3) 1.49057 0.172931\nbµ(µ2\nB\u0001atom2/˚A6) -0.0978406 -0.021997\ngµ(µ2\nB\u0001atom3/˚A9) 0.0026366 0.00096755\nvc(˚A3/atom) 6.39848 5.36535\na(I)\nJ(meV/atom) -12.5921 8.35847\ng(I)\nJ2.81897 -0.098217\nd(I)\nJ(˚A) 2.4690386 2.4890153\nR(I)\nc;J(˚A) 2.6 2.6\na(II)\nJ(meV/atom) 50.996a19.46a\ng(II)\nJ0.281a0.00011a\nd(II)\nJ(˚A) 1.999a1.233a\nR(II)\nc;J(˚A) 4.5a4.5a\nal(µeV/atom) 392.747 179.396\ngl 0.824409 1.39848\ndl(˚A) 2.4690386 2.4890153\nRc;l(˚A) 2.6 2.6\naq(µeV/atom) 28.5189 -49.1335\ngq 1.05331 1.1186\ndq(˚A) 2.4690386 2.4890153\nRc;q(˚A) 2.6 2.6\naRef.26\nThe Bethe-Slater parameters for the constructed SD-MD\nmodels are shown in Table I, while the corresponding Bethe-\nSlater functions for l(r)andq(r)using these parameters are\nplotted in Fig.6. We see that l(r0)is approximately two or-\nder of magnitude greater than q(r0). However, note that af-\nter taking into account all first nearest neighbors the N ´eel\nquadrupole and dipole energies can be of the same order of\nmagnitude close to the cubic symmetry (see Section IV A).\nFig.6 also contains interesting information about the depen-\ndence of MCA and magnetoelasticity on the distance between\nfirst nearest neighbors. For instance, we see that if we de-\ncrease the distance between first nearest neighbors from the\nequilibrium value r0(high hydrostatic pressure regime) for\nboth BCC Fe and FCC Ni, then the sign of q(r)changes,\nwhich implies a change in the sign of K1, see Eq.13. Similarly,\nif we increase the distance between first nearest neighbors for\nBCC Fe ( r0>3˚A), then the sign of l(r)changes switching the\nsign of b1, see Eq.21. In general, the physical interpretation of\nq(r)andl(r)far from the equilibrium value r0should be done\nwith caution since we only involved up to the first derivative\nof these functions evaluated at r0in their parameterization.\nIn this sense, the only meaningful region around r0may be\nwhere first order Taylor expansion at r0of the Bethe-Slater9\nfunctions of q(r)andl(r)is a good approximation. Includ-\ning up the first derivative of q(r)andl(r)in their parame-\nterization might be enough for many practical purposes since\nthe distance between first nearest neighbors oscillates close to\nthe equilibrium value at finite temperature below the melting\npoint.\nFIG. 6. Calculation of the Bethe-Slater function l(r)andq(r)for\n(top) BCC Fe and (bottom) FCC Ni using the parameters given in\nTable I. Vertical dash line stands for the equilibrium distance of the\nfirst nearest neighbors r0.\nIV . RESULTS\nA. Tests of the N ´eel interaction\nBefore evaluating the magnetoelastic properties of the SD-\nMD model, it is convenient to check that the implementation\nof the N ´eel interaction Eq.4 in the SD-MD simulation is cor-\nrect. To this end, we propose some tests by comparing the\nnumerical results of the SD-MD simulation with simple an-\nalytical solutions. For instance, if we consider a BCC struc-\nture with N ´eel interactions up to first nearest neighbor in acollinear state along sss= (0;0;1), then from Eq.6 we have\nHN´eel(0;0;1) =16Nq(r0)\n45; (31)\nwhere Nis the number of atoms in the system, r0is the\ndistance to nearest neighbor that is related to the lattice pa-\nrameter aviar0=ap\n3=2. This equation allows to verify\nthe quadrupole term. Let’s now apply to this system with\nsss= (0;0;1)a tetragonal deformation along the z-axis, where\nthe lattice parameter is cin this direction, and aalong both\nx-axis and y-axis. From Eq.6 we obtain\nHN´eel(0;0;1) =\u00004Nl(r0)\"\u0000c\na\u00012\n2+\u0000c\na\u00012\u00001\n3#\n\u000016Nq(r0)h\n2\u0000c\na\u00014\u000012\u0000c\na\u00012+3i\n35h\n2+\u0000c\na\u00012i2;(32)\nwhere\nr0=a\n2r\n2+\u0010c\na\u00112\n: (33)\nThis equation allows to check both the dipole and quadrupole\nterms. In the limit c=a\u0000 !1, the Eq.32 becomes Eq.31 ensur-\ning the continuity of the N ´eel energy under structure defor-\nmation. In Fig.7, we verify that the calculation of the N ´eel\nenergy with LAMMPS is the same to Eqs.31 and 32 using the\nBethe-Slater parameters of BCC Fe given in Table I. Similar\ntests could also be performed for other magnetic moment di-\nrections and deformations.\nFIG. 7. Calculation of the N ´eel energy with LAMMPS and (top)\nEq.32 and (bottom) Eq.31 for different values of the lattice parame-\nters.\nB. Magnetic properties at zero-temperature\nIn this section, we evaluate the magnetization and MCA\nunder pressure, anisotropic magnetostrictive coefficients, vol-10\nTABLE II. Calculated and experimental elastic constants, magnetostrictive coefficients, MCA, and MCA under hydrostatic pressure for BCC\nFe and FCC Ni at zero-temperature.\nMaterialElastic\nconstantsSD-MD\n(GPa)Expt.\n(GPa)Magnetostrictive\ncoefficientsSD-MD\n(\u000210\u00006)Expt.\n(\u000210\u00006)MCASD-MD\n(KJ/m3)Expt.\n(KJ/m3)MCA vs PSD-MD\n(GPa\u00001)Expt.\n(GPa\u00001)\nBCC Fe c11 230.0 230al001 25.9 26cK1 54.995 55d 1\nK1¶K1\n¶P-0.0727 -0.073e\nc12 134.1 135al111 -30.3 -30c\nc44 116.3 117a\nFCC Ni c11 263.9 261.2bl001 -61.9 -60cK1-125.996 -126d 1\nK1¶K1\n¶P-0.0279 -0.028e\nc12 152.1 150.8bl111 -35.4 -35c\nc44 132.8 131.7b\naRef.43,bRef.44,cRef.36,\ndRef.59,eRef.60\nume magnetostriction and saturation magnetization at zero-\ntemperature given by the developed SD-MD models for BCC\nFe and FCC Ni in Section III. We include MEAM potentials,\nexchange and N ´eel energies, and volume-dependent magnetic\nmoment in the following calculations. Magnetic collinear\nstates will be used since we are interested in properties at zero-\ntemperature. All simulations are performed with the SPIN\npackage of LAMMPS26.\n1. Ground state\nFirstly, we determine the equilibrium volume of the full\nSD-MD model (including the N ´eel interaction) for the con-\nventional unit cell of BCC Fe and FCC Ni. To this end, we\ncalculate the energy versus volume curve, and we fit it to the\nMurnaghan EOS in the same way as it was done in Fig.2\npreviously. Here, we also set the magnetic moments along\nthe easy direction ( [1;0;0]for BCC Fe and [1;1;1]for FCC\nNi) in order to get the minimum energy of the quadrupole\nterm of N ´eel interaction. The equilibrium volume found with\nthis procedure is v0=11:5867635 ˚A3/atom for BCC Fe, and\nv0=10:9035445 ˚A3/atom for FCC Ni. We verify that pres-\nsure is lower than 5 \u000210\u00005GPa in these equilibrium states.\nAs we anticipated in Section III, the dipole and quadrupole\nN´eel interactions induce a very small change in the equilib-\nrium volume when is included in the SD-MD model.\n2. Magnetocrystalline anisotropy\nNext, we compute the MCA energy at this equilibrium\nvolume by setting the magnetic moment along different di-\nrections in the XY plane. In Fig.8 we show a compari-\nson between the MCA energy calculated by SD-MD simula-\ntions with LAMMPS and Eq.11 using the experimental value\n(K1=55 KJ/m3for BCC Fe and K1=\u0000126 KJ/m3)59. The\ndirect evaluation of K1with the SD-MD model through Eq.12\ngives 54 :995 KJ/m3for BCC Fe and\u0000125:996 KJ/m3for\nFCC Ni. As we see, the SD-MD model with the Bethe-Slater\nparameters given by Table I reproduces very well the first-\norder experimental MCA.\nFIG. 8. Calculation of the MCA energy for BCC Fe and FCC Ni with\nSD-MD simulation (blue points) and Eq.11 using the experimental\nK1(red line). Magnetic moments are constrained on the XY plane.\nFIG. 9. Calculation of K1(P)=K1(0)under hydrostatic pressure using\nthe developed SD-MD model (blue dots) for BCC Fe and FCC Ni.\nThe green and red lines stand for the experimental behaviour given\nby Eq.35 and its low-pressure approximation Eq.36, respectively60.\nNow we study the effects of hydrostatic pressure on the\nMCA for this SD-MD model. To facilitate the compar-\nison between the model and experiment, we first convert\n(1=K1)(¶K1=¶P)to an integral form, that is,\n1\nK1¶K1\n¶P=z\u0000!ZK1(P)\nK1(0)dK1\nK1=ZP\n0zdP; (34)\nwhere z=\u00007:3\u000210\u00002GPa\u00001is the experimental value mea-\nsured up to P=0:5 GPa at T=77K for BCC Fe60, while for\nFCC Ni is z=\u00002:8\u000210\u00002GPa\u00001. Solving this integral we\nhave\nK1(P)\nK1(0)=ezP; (35)\nwhere in the low pressure regime ( zP\u001c1) it can be written11\nas\nK1(P)\nK1(0)\u00191+zP+O(P2): (36)\nIn Fig.9 we show the ratio K1(P)=K(0)versus pressure gener-\nated by the SD-MD model of Fe and Ni, and the experimen-\ntal behaviour given by Eq.35 and its low-pressure approxi-\nmation Eq.36. The linear fitting to the data generated by the\nSD-MD model up to P=0:5 GPa gives (1=K1)(¶K1=¶P) =\n\u00007:27\u000210\u00002GPa\u00001for BCC Fe, and\u00002:79\u000210\u00002GPa\u00001\nfor FCC Ni, which is in very good agreement with the ex-\nperimental values60. Note that Eq.35 and MCA results of\nthe model beyond the range of pressure between 0GPa and\n0:5GPa should be taken with caution due to the lack of exper-\nimental data.\nFIG. 10. Calculation of l001for BCC Fe using MAELAS inter-\nfaced with LAMMPS. (top) Quadratic curve fit to the energy ver-\nsus cell length along bbb= (0;0;1)with spin direction sss1= (0;0;1)\nunder a volume-conserving tetragonal deformation. (bottom) En-\nergy difference between states with spin directions sss2= (1;0;0)and\nsss1= (0;0;1)against the cell length along bbb= (0;0;1).\n3. Anisotropic magnetostriction\nNow, we compute the anisotropic magnetostrictive coeffi-\ncients using the SD-MD model. To this end, we apply the\nmethod proposed by Wu and Freeman61,62as implemented in\nthe program MAELAS41,57. In this method, the anisotropic\nFIG. 11. Calculation of l111for BCC Fe using MAELAS inter-\nfaced with LAMMPS. (top) Quadratic curve fit to the energy ver-\nsus cell length along bbb=\u0000\n1=p\n3;1=p\n3;1=p\n3\u0001\nwith spin direction\nsss1=\u0000\n1=p\n3;1=p\n3;1=p\n3\u0001\nunder a volume-conserving trigonal de-\nformation. (bottom) Energy difference between states with spin\ndirections sss2=\u0010\n1=p\n2;0;\u00001=p\n2\u0011\nandsss1=\u0000\n1=p\n3;1=p\n3;1=p\n3\u0001\nagainst the cell length along bbb=\u0000\n1=p\n3;1=p\n3;1=p\n3\u0001\n.\nmagnetostrictive coefficients for cubic systems (point groups\n432, ¯43m,m¯3m) are calculated as57\nl001=4(l001\n1\u0000l001\n2)\n3(l001\n1+l001\n2);l111=4(l111\n1\u0000l111\n2)\n3(l111\n1+l111\n2); (37)\nwhere l001\n1and l001\n2are the equilibrium cell lengths along\nthe length measuring direction bbb= (0;0;1)under a tetrag-\nonal deformation with collinear magnetic moment direc-\ntions sss1= (0;0;1)and sss2= (1;0;0), respectively. Simi-\nlarly, l111\n1and l111\n2are the equilibrium cell lengths along\nthe length measuring direction bbb= (1=p\n3;1=p\n3;1=p\n3)un-\nder a trigonal deformation with magnetic moment direction\nsss1=\u0000\n1=p\n3;1=p\n3;1=p\n3\u0001\nandsss2=\u0010\n1=p\n2;0;\u00001=p\n2\u0011\n, re-\nspectively. In order to obtain the equilibrium cell lengths l001\n1\nandl001\n2, one needs to evaluate the energy for a set of volume-\nconserving tetragonal distorted unit cells. Next, the energy\nversus the cell length along bbb= (0;0;1)for each magnetic\nmoment direction sss1= (0;0;1)andsss2= (1;0;0)is fitted to a12\nquadratic function\nE(x)\f\f\f\f\fsssj\nbbb=(0;0;1)=˜ajx2+˜bjx+˜cj;j=1;2 (38)\nwhere ˜ aj,˜bjand ˜ cjare fitting parameters. The mini-\nmum of this quadratic function for magnetic moment di-\nrection sss1(2)corresponds to l001\n1(2)=\u0000˜b1(2)=(2 ˜a1(2)), and it\nis the equilibrium cell length. Similarly, the equilibrium\ncell lengths l111\n1andl111\n2are obtained by applying a set of\nvolume-conserving trigonal deformations, and performing a\nquadratic fitting of the energy versus the cell length along\nbbb=\u0000\n1=p\n3;1=p\n3;1=p\n3\u0001\nwith magnetic moment directions\nsss1=\u0000\n1=p\n3;1=p\n3;1=p\n3\u0001\nandsss2=\u0010\n1=p\n2;0;\u00001=p\n2\u0011\n.\nFIG. 12. Calculation of l001for FCC Ni using MAELAS inter-\nfaced with LAMMPS. (top) Quadratic curve fit to the energy ver-\nsus cell length along bbb= (0;0;1)with spin direction sss1= (0;0;1)\nunder a volume-conserving tetragonal deformation. (bottom) En-\nergy difference between states with spin directions sss2= (1;0;0)and\nsss1= (0;0;1)against the cell length along bbb= (0;0;1).\nWe have developed an interface between the software\nMAELAS57and LAMMPS26in order to apply this method\nand extract the magnetostrictive coefficients easily. This in-\nterface is publicly available on GitHub repository63. In Fig.10\nwe show the quadratic curve fit to the energy versus cell length\nalong [0;0;1]with magnetic moment direction sss1= (0;0;1)\nto calculate l001for BCC Fe. We also plot the energy dif-\nference between states with spin directions sss1= (1;0;0)and\nsss2= (0;0;1)against the cell length along [0;0;1]. The cor-\nresponding plot for l111is presented in Fig.11. We obtainl001=25:9\u000210\u00006andl111=\u000030:3\u000210\u00006, while the ex-\nperimental values36atT=4:2K are l001=26\u000210\u00006and\nl111=\u000030\u000210\u00006. The results for FCC Ni are plotted\nin Figs. 12 and 13. Here, we get l001=\u000061:9\u000210\u00006\nandl111=\u000035:4\u000210\u00006, while the experimental values36at\nT=4:2K are l001=\u000060\u000210\u00006andl111=\u000035\u000210\u00006.\nTherefore, the developed SD-MD model for Fe and Ni also\nexhibits magnetostrictive properties very similar to the exper-\niment. Additionally, this calculation reveals that the method\nproposed by Wu and Freeman61,62is an excellent approach\nto obtain the magnetostrictive coefficients as long as both the\nelastic and magnetoelastic energies are properly described by\nthe model. This fact could not be verified before for l111of\nBCC Fe due to a possible failure of Density Function Theory\ncalculations57,64–66. In Table II we present a summary of the\nresults given by the SD-MD model for the MCA, MCA under\nhydrostatic pressure, and anisotropic magnetostrictive coeffi-\ncients.\nFIG. 13. Calculation of l111for FCC Ni using MAELAS inter-\nfaced with LAMMPS. (top) Quadratic curve fit to the energy ver-\nsus cell length along bbb=\u0000\n1=p\n3;1=p\n3;1=p\n3\u0001\nwith spin direction\nsss1=\u0000\n1=p\n3;1=p\n3;1=p\n3\u0001\nunder a volume-conserving trigonal de-\nformation. (bottom) Energy difference between states with spin\ndirections sss2=\u0010\n1=p\n2;0;\u00001=p\n2\u0011\nandsss1=\u0000\n1=p\n3;1=p\n3;1=p\n3\u0001\nagainst the cell length along bbb=\u0000\n1=p\n3;1=p\n3;1=p\n3\u0001\n.13\nTABLE III. Calculated volume magnetostriction wswith the SD-MD\nmodel for BCC Fe and FCC Ni using the set I and II of parameters in\nTable I to describe J(r). Theoretical and experimental results found\nin literature are also shown for comparison.\nSD-MD\nset I\n(\u000210\u00004)SD-MD\nset II\n(\u000210\u00004)Theory\n(\u000210\u00004)Expt.\n(\u000210\u00004)\nBCC Fe 118 -235 116a4b\n683c\nFCC Ni 3.71 -53.7 3.75a3.65e\n45.7c3.24f\n-5.1d\n-2.7b\naRef.58,bRef.67,\ncRef.68,dRef.69,\neRef.70,fRef.71\n4. Volume magnetostriction\nThe volume magnetostriction is generated by the presence\nof ferromagnetism in the magnetic material (exchange mag-\nnetostriction). It can be calculated as72\nws(T) =v0(Ms(T))\u0000v0(0)\nv0(0); (39)\nwhere v0(Ms(T))andv0(0)are the equilibrium volume per\natom in the magnetized and demagnetized (paramagnetic)\nstates, respectively. In the magnetized state, the magnetiza-\ntion is equal to the saturation magnetization Msat temperature\nT. Hence, the quantity v0(Ms(T))at zero-temperature was al-\nready calculated in Section IV B 1. To compute v0(0)we ap-\nply a similar procedure. Namely, we first calculate the energy\nof a supercell with magnetic moments oriented randomly (de-\nmagnetized state) for different values of the lattice parameter\na, preserving the cubic crystal symmetry. Next, we fit the en-\nergy versus volume curve to the Murnaghan EOS. We use a\nsupercell with 20x20x20 conventional unit cells with periodic\nboundary conditions for both BCC Fe (16000 atoms) and FCC\nNi (32000 atoms). We perform this calculation using the set I\nand II of parameters given in Table I to describe the exchange\ninteraction J(r). The results are depicted in Fig. 14. The set I\ngivesws=1:18\u000210\u00002for BCC Fe and 3 :71\u000210\u00004for FCC\nNi, reproducing fairly well the theoretical values calculated by\nShimizu58(ws=1:16\u000210\u00002for BCC Fe and 3 :75\u000210\u00004for\nFCC Ni) that we used to compute the Bethe-Slater parameters\nforJ(r)in Section III C. The set II leads to ws=\u00002:23\u000210\u00002\nfor BCC Fe and\u00005:37\u000210\u00003for FCC Ni, so they have the\nopposite sign to the results given by set I. According to Eq.\n23, these results may be understood in terms of ¶J=¶rat the\nfirst-nearest neighbors ( r=r0) since wsµ ¶J=¶r. In Fig.5,\nwe observe that set I gives ¶J=¶r>0 at r=r0for both Fe\nand Ni, while set II gives ¶J=¶r<0 atr=r0. Note that Eq.\n23 is derived assuming only exchange interactions up to first-\nnearest neighbors, and set II has a large cut-off that includes\nexchange interactions beyond first-nearest neighbors. Wang\net al. performed first-principles calculations of J(r)finding achange in the sign of ¶J=¶rclose to r=r0, and¶J=¶r>0 for\nthe second nearest neighbors73. Previous theoretical and ex-\nperimental works reported a positive volume magnetostriction\nfor BCC Fe58,67,68, while for FCC Ni one can find contradic-\ntory results with positive58,68,70,71and negative67,69values. A\nsummary of these results is presented in Table III. As seen in\nFig.5, there is a maximum of J(r)close to r0for Ni using the\nset I, so that a small increase in the lattice parameter would\nchange the sign of ¶J=¶r, and consequently the sign of ws.\nLastly, we point out that the isotropic magnetostrictive coeffi-\ncient of cubic crystals ( la) and magnetoelastic constant b0are\nrelated to the volume magnetostriction as57,74\nla=\u0000b0\u00001\n3b1\nc11+2c12=ws\n3: (40)\nHence, we see that the isotropic magnetostriction is greater\nthan the anisotropic one for both BCC Fe and FCC Ni.\nFIG. 14. Calculation of wswith the SD-MD model for ((a)-(b)) BCC\nFe and ((c)-(d)) FCC Ni using the two set of parameters given in\nTable I to describe the exchange interaction J(r).\n5. Saturation magnetization\nThe saturation magnetization at zero-temperature is com-\nputed using the following equation\nµ0Ms(v) =µ0µ(v)\nv; (41)\nwhere µ(v)is calculated using the Eq.28 with the parameters\nshown in Table I. At the equilibrium volume of the SD-MD\nmodel it gives µ0Ms(v0) =2:35T for BCC Fe, and 0 :71T for\nFCC Ni. The experimental values at zero-temperature are\nµ0Ms=2:19T for BCC Fe, and 0 :64T for FCC Ni36. We see\nthat the model slightly overestimates the saturation magne-\ntization. Next, we evaluate Msfor different volumes apply-\ning normal deformations. The results of this calculation are\nshown in Fig.15. Here, we also included the data given by\nDFT that we obtained in Section III B. We observe that the14\noverall behaviour of Msis well described by the model. As\nwe increase the volume above the equilibrium volume v0, the\npressure becomes negative and Msis decreasing. The condi-\ntion that causes Msto decrease with volume is\n¶Ms\n¶v<0\u0000!¶µ\n¶v<µ\nv: (42)\nOn the other hand, if we decrease the volume below the equi-\nlibrium volume then the pressure is positive. At high positive\npressure, Msbecomes zero when the volume per atom is lower\nthan the critical volume ( v<vc) where magnetic moment col-\nlapses ( µ=0).\nFIG. 15. Saturation magnetization against volume and pressure cal-\nculated with the SD-MD model including a volume-dependent mag-\nnetic moment (red squares) for BCC Fe and FCC Ni. Blue dots and\ngreen triangles stand for DFT and experimental data, respectively.\nV . CONCLUSIONS\nMany aspects of magnetoelastic phenomena are not fully\nunderstood yet due to the complexity of the materials at large\nscale. Advanced modeling techniques and associated numeri-\ncal tools based on a bottom-up multiscale approach could help\nto get a better understanding of magnetoelastic phenomena in\nmagnetic materials across length scales. In this sense the SD-\nMD simulations using the N ´eel model could play an important\nrole linking the atomic and macroscopic scales. Aiming at\nexploring this possibility, we showed a general methodology\nto build SD-MD models to describe MCA under hydrostatic\npressure, anisotropic magnetostriction, volume magnetostric-\ntion and saturation magnetization. To illustrate the method,\nwe successfully applied it to BCC Fe and FCC Ni at zero-\ntemperature.\nWe aim at transposing our methodology to other materi-\nals and crystal structures. For example, in magnetic oxides\nand 4-f magnets, the MCA can correspond to energies orders\nof magnitude larger than in magnetic 3-d metals. Our ap-\nproach could be used to map the subsequent interactions andbuild meso-scale models that will help reveal the influence of\nmagnetism on large-scale thermo-elastic materials properties.\nPossible extensions of these models might be also useful to\nstudy morphic effects75,76.\nAlthough this work focused on bulk magnetoelasticity, sim-\nilar effects have been shown to be important for smaller scale,\nfinite-size systems21. Previous studies have been investigat-\ning the relevance of the N ´eel model to simulate surface ef-\nfects in 3-d magnetic metals77,78. Future work could lever-\nage our framework to develop surface interaction models for\nspin-lattice simulations of magnetic nanoparticles79, as well\nas models for magnetic alloys35,80.\nThe results presented in this work also raise interesting\nquestions for future research on how these models will per-\nform at finite temperature, and under magnetic field and stress.\nIn particular, it would be interesting to study the possible cor-\nrelations between the thermal variation of the magnetoelastic\nconstants and the magnetization given by these models81–83.\nACKNOWLEDGEMENT\nThis work was supported by the ERDF in the\nIT4Innovations national supercomputing center - path to\nexascale project (CZ.02.1.01/0.0/0.0/16-013/0001791) within\nthe OPRDE. This work was supported by The Ministry of\nEducation, Youth and Sports from the Large Infrastructures\nfor Research, Experimental Development, and Innovations\nproject “e-INFRA CZ - LM2018140”. This work was also\nsupported by the Donau project No. 8X20050. P.N., D.L.,\nand S.A. acknowledge support from the H2020-FETOPEN\nno. 863155 s-NEBULA project. Sandia National Laborato-\nries is a multimission laboratory managed and operated by\nNational Technology and Engineering Solutions of Sandia,\nLLC., a wholly owned subsidiary of Honeywell International,\nInc., for the U.S. Department of Energy’s National Nuclear\nSecurity Administration under contract DE-NA-0003525.\nThis paper describes objective technical results and analysis.\nAny subjective views or opinions that might be expressed in\nthe paper do not necessarily represent the views of the U.S.\nDepartment of Energy or the United States Government.\nAPPENDIX\nAppendix A: Derivation of ¶q=¶r\nIn this appendix we show the steps to obtain the final ex-\npression for ¶q=¶rgiven by Eq. 14. Firstly, we write the\nderivative of q(r)with respect to the first nearest neighbor dis-\ntance rin Eq. 13 as\nq(r) =\u0000xV K1\nN\u0000!r0¶q\n¶r\f\f\f\nr=r0=\u0000xr0\nN¶(V K1)\n¶r\f\f\f\nr=r0;(A1)\nwhere r0is the equilibrium distance to the first nearest neigh-\nbors, Nis the number of atoms in the volume V, andxis equal15\nto\u00001=2, 9=16, and 1 for SC, BCC and FCC, respectively.\nNext, we work out this equation in the following way\nr0¶q\n¶r\f\f\f\nr=r0=\u0000xr0\nN¶(V K1)\n¶r\f\f\f\nr=r0=\u0000xr0\nN\u0014\nK1¶V\n¶r+V¶K1\n¶r\u0015\nr=r0\n=\u0000xr0\nN¶V\n¶r\f\f\f\nr=r0\u0014\nK1+V¶K1\n¶V\u0015\nr=r0\n=\u0000xr0\nN¶V\n¶r\f\f\f\nr=r0\u0014\nK1+V¶P\n¶V¶K1\n¶P\u0015\nr=r0;\n(A2)\nwhere\nSC:xr0\nN¶V\n¶r\f\f\f\nr=r0=\u00003\n2r3\n0;\nBCC :xr0\nN¶V\n¶r\f\f\f\nr=r0=3p\n3\n4r3\n0;\nFCC :xr0\nN¶V\n¶r\f\f\f\nr=r0=3p\n2r3\n0:(A3)Lastly, we make use of the definition of the bulk modulus\nB=\u0000V(¶P=¶V)in Eq.A2. Doing so, we obtain\nSC:r0¶q\n¶r\f\f\f\nr=r0=3\n2r3\n0K1(r0)\u0014\n1\u0000B\nK1¶K1\n¶P\u0015\nr=r0;\nBCC :r0¶q\n¶r\f\f\f\nr=r0=\u00003p\n3\n4r3\n0K1(r0)\u0014\n1\u0000B\nK1¶K1\n¶P\u0015\nr=r0;\nFCC :r0¶q\n¶r\f\f\f\nr=r0=\u00003p\n2r3\n0K1(r0)\u0014\n1\u0000B\nK1¶K1\n¶P\u0015\nr=r0:\n(A4)\n\u0003Corresponding author: pablo.nieves.cordones@vsb.cz\n1P.-W. Ma and S. Dudarev, Handbook of Materials Modeling:\nMethods: Theory and Modeling pp. 1017–1035 (2020).\n2F. T. Calkins, A. B. Flatau, and M. J. Dapino, Journal\nof Intelligent Material Systems and Structures 18, 1057\n(2007), https://doi.org/10.1177/1045389X06072358, URL\nhttps://doi.org/10.1177/1045389X06072358 .\n3N. Ekreem, A. Olabi, T. Prescott, A. Rafferty, and M. Hashmi,\nJournal of Materials Processing Technology 191, 96 (2007),\nISSN 0924-0136, advances in Materials and Processing\nTechnologies, July 30th - August 3rd 2006, Las Vegas,\nNevada, URL http://www.sciencedirect.com/science/\narticle/pii/S0924013607002889 .\n4V . Apicella, C. S. Clemente, D. Davino, D. Leone, and C. Visone,\nActuators 8(2019), ISSN 2076-0825, URL https://www.mdpi.\ncom/2076-0825/8/2/45 .\n5M. Dapino, Encyclopedia of Smart Materials (John Wiley and\nSons, Inc., New York, 2000).\n6M. H. Kryder, E. C. Gage, T. W. McDaniel, W. A. Challener,\nR. E. Rottmayer, G. Ju, Y . Hsia, and M. F. Erden, Proceedings of\nthe IEEE 96, 1810 (2008).\n7I. L. Prejbeanu, M. Kerekes, R. C. Sousa, H. Sibuet, O. Redon,\nB. Dieny, and J. P. Nozi `eres, Journal of Physics: Condensed\nMatter 19, 165218 (2007), URL https://doi.org/10.1088%\n2F0953-8984%2F19%2F16%2F165218 .\n8E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y . Bigot, Phys.\nRev. Lett. 76, 4250 (1996), URL https://link.aps.org/doi/\n10.1103/PhysRevLett.76.4250 .\n9T. A. Ostler, J. Barker, R. F. L. Evans, R. W. Chantrell, U. Atxi-\ntia, O. Chubykalo-Fesenko, S. El Moussaoui, L. Le Guyader,\nE. Mengotti, L. J. Heyderman, et al., Nature Communications\n3, 666 (2012), ISSN 2041-1723, URL https://doi.org/10.\n1038/ncomms1666 .\n10A. Kitanovski, J. Tu ˇsek, U. Tomc, U. Plaznik, M. O ˇzbolt, and\nA. Poredo ˇs,Magnetocaloric Energy Conversion (Springer Inter-\nnational Publishing, 2015), URL https://doi.org/10.1007%\n2F978-3-319-08741-2 .\n11E. C. Abenojar, S. Wickramasinghe, J. Bas-Concepcion, and\nA. C. S. Samia, Progress in Natural Science: Materials Interna-\ntional 26, 440 (2016), ISSN 1002-0071, special Issue for Nano\nMaterials, URL http://www.sciencedirect.com/science/article/pii/S1002007116300995 .\n12Y . Zhou, J. Tranchida, Y . Ge, J. Murthy, and T. S. Fisher, Physical\nReview B 101, 224303 (2020).\n13N. B ¨acklund, Journal of Physics and Chemistry of Solids 20, 1\n(1961).\n14M. Jaime, A. Saul, M. Salamon, V . Zapf, N. Harrison, T. Du-\nrakiewicz, J. Lashley, D. Andersson, C. Stanek, J. Smith, et al.,\nNature communications 8, 1 (2017).\n15E. Wasserman (Elsevier, 1990), vol. 5 of Handbook\nof Ferromagnetic Materials , pp. 237 – 322, URL\nhttp://www.sciencedirect.com/science/article/pii/\nS157493040580063X .\n16R. F. L. Evans, W. J. Fan, P. Chureemart, T. A. Ostler, M. O. A.\nEllis, and R. W. Chantrell, Journal of Physics: Condensed\nMatter 26, 103202 (2014), URL https://doi.org/10.1088%\n2F0953-8984%2F26%2F10%2F103202 .\n17O. Eriksson, A. Bergman, L. Bergqvist, and J. Hellsvik, Atomistic\nSpin Dynamics (Oxford University Press, 2017), URL https:\n//doi.org/10.1093%2Foso%2F9780198788669.001.0001 .\n18S. Plimpton, Journal of Computational Physics 117, 1 (1995),\nISSN 0021-9991, URL http://www.sciencedirect.com/\nscience/article/pii/S002199918571039X .\n19I. Stockem, A. Bergman, A. Glensk, T. Hickel, F. K ¨ormann,\nB. Grabowski, J. Neugebauer, and B. Alling, Phys. Rev. Lett. 121,\n125902 (2018), URL https://link.aps.org/doi/10.1103/\nPhysRevLett.121.125902 .\n20P.-W. Ma, C. H. Woo, and S. L. Dudarev, Phys. Rev. B 78,\n024434 (2008), URL https://link.aps.org/doi/10.1103/\nPhysRevB.78.024434 .\n21D. Beaujouan, P. Thibaudeau, and C. Barreteau, Phys. Rev. B 86,\n174409 (2012), URL https://link.aps.org/doi/10.1103/\nPhysRevB.86.174409 .\n22P.-W. Ma, S. Dudarev, and C. Woo, Computer Physics\nCommunications 207, 350 (2016), ISSN 0010-4655, URL\nhttp://www.sciencedirect.com/science/article/pii/\nS0010465516301412 .\n23X. Wu, Z. Liu, and T. Luo, Journal of Applied Physics 123,\n085109 (2018), https://doi.org/10.1063/1.5020611, URL https:\n//doi.org/10.1063/1.5020611 .\n24J. Fransson, D. Thonig, P. F. Bessarab, S. Bhattacharjee,\nJ. Hellsvik, and L. Nordstr ¨om, Phys. Rev. Materials 1,16\n074404 (2017), URL https://link.aps.org/doi/10.1103/\nPhysRevMaterials.1.074404 .\n25D. Perera, D. M. Nicholson, M. Eisenbach, G. M. Stocks, and\nD. P. Landau, Phys. Rev. B 95, 014431 (2017), URL https://\nlink.aps.org/doi/10.1103/PhysRevB.95.014431 .\n26J. Tranchida, S. Plimpton, P. Thibaudeau, and A. Thompson,\nJournal of Computational Physics 372, 406 (2018), ISSN\n0021-9991, URL http://www.sciencedirect.com/science/\narticle/pii/S0021999118304200 .\n27D. Perera, M. Eisenbach, D. M. Nicholson, G. M. Stocks, and\nD. P. Landau, Phys. Rev. B 93, 060402(R) (2016), URL https:\n//link.aps.org/doi/10.1103/PhysRevB.93.060402 .\n28R. Skomski, Simple Models of Magnetism (Oxford University\nPress, USA, 2008).\n29J. C. Slater, Phys. Rev. 35, 509 (1930), URL https://link.\naps.org/doi/10.1103/PhysRev.35.509 .\n30J. C. Slater, Phys. Rev. 36, 57 (1930), URL https://link.aps.\norg/doi/10.1103/PhysRev.36.57 .\n31A. Sommerfeld and H. Bethe, in Aufbau Der Zusam-\nmenh ¨angenden Materie (Springer Berlin Heidelberg,\n1933), pp. 333–622, URL https://doi.org/10.1007%\n2F978-3-642-91116-3_3 .\n32P.-W. Ma and S. L. Dudarev, Phys. Rev. B 86, 054416 (2012),\nURL https://link.aps.org/doi/10.1103/PhysRevB.86.\n054416 .\n33V . L. Moruzzi, Phys. Rev. Lett. 57, 2211 (1986), URL https:\n//link.aps.org/doi/10.1103/PhysRevLett.57.2211 .\n34J. K¨ubler, Theory of itinerant electron magnetism (Oxford Uni-\nversity Press, New York, 2009).\n35S. Chikazumi, Physics of Ferromagnetism (Oxford University\nPress, 2009).\n36R. C. O’Handley (Wiley, 2000).\n37D. S. Chuang, C. A. Ballentine, and R. C. O’Handley, Phys.\nRev. B 49, 15084 (1994), URL https://link.aps.org/doi/\n10.1103/PhysRevB.49.15084 .\n38A. Clark (Elsevier, 1980), vol. 1 of Handbook of Ferromagnetic\nMaterials , pp. 531 – 589, URL http://www.sciencedirect.\ncom/science/article/pii/S1574930405801221 .\n39J. R. Cullen, A. E. Clark, and K. B. Hathaway (VCH Publishings,\n1994), chap. 16 - Magnetostrictive Materials, pp. 529–565.\n40L. D. Landau and E. M. Lifshitz, Theory of elasticity / by L. D.\nLandau and E. M. Lifshitz ; translated from the Russian by J. B.\nSykes and W. H. Reid (Pergamon London, 1959).\n41https://github.com/pnieves2019/MAELAS .\n42L. LANDAU and E. LIFSHITZ, in Electrodynamics of Con-\ntinuous Media (Second Edition) , edited by L. LANDAU and\nE. LIFSHITZ (Pergamon, Amsterdam, 1984), vol. 8 of Course\nof Theoretical Physics , pp. 130 – 179, second edition ed., ISBN\n978-0-08-030275-1, URL http://www.sciencedirect.com/\nscience/article/pii/B9780080302751500114 .\n43E. Asadi, M. Asle Zaeem, S. Nouranian, and M. I. Baskes, Phys.\nRev. B 91, 024105 (2015), URL https://link.aps.org/doi/\n10.1103/PhysRevB.91.024105 .\n44B.-J. Lee, J.-H. Shim, and M. I. Baskes, Phys. Rev. B 68,\n144112 (2003), URL https://link.aps.org/doi/10.1103/\nPhysRevB.68.144112 .\n45F. D. Murnaghan, Proceedings of the National Academy of\nSciences of the United States of America 30, 244 (1944),\nISSN 0027-8424, 16588651[pmid], URL https://pubmed.\nncbi.nlm.nih.gov/16588651 .\n46C. L. Fu and K. M. Ho, Phys. Rev. B 28, 5480 (1983), URL\nhttps://link.aps.org/doi/10.1103/PhysRevB.28.5480 .\n47S. Zhang and R. Zhang, Computer Physics Commu-\nnications 220, 403 (2017), ISSN 0010-4655, URLhttp://www.sciencedirect.com/science/article/pii/\nS0010465517302400 .\n48https://github.com/pnieves2019/MAELAS/tree/master/\nExamples/LAMMPS/AELAS .\n49P. Hirel, Computer Physics Communications 197, 212 (2015),\nISSN 0010-4655, URL http://www.sciencedirect.com/\nscience/article/pii/S0010465515002817 .\n50G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993), URL\nhttps://link.aps.org/doi/10.1103/PhysRevB.47.558 .\n51G. Kresse and J. Furthm ¨uller, Computational Mate-\nrials Science 6, 15 (1996), ISSN 0927-0256, URL\nhttp://www.sciencedirect.com/science/article/pii/\n0927025696000080 .\n52G. Kresse and J. Furthm ¨uller, Phys. Rev. B 54, 11169 (1996),\nURL https://link.aps.org/doi/10.1103/PhysRevB.54.\n11169 .\n53G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999), URL\nhttps://link.aps.org/doi/10.1103/PhysRevB.59.1758 .\n54J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett.\n77, 3865 (1996), URL https://link.aps.org/doi/10.1103/\nPhysRevLett.77.3865 .\n55H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13,\n5188 (1976), URL https://link.aps.org/doi/10.1103/\nPhysRevB.13.5188 .\n56V . Moruzzi and P. Marcus (Elsevier, 1993), vol. 7 of\nHandbook of Magnetic Materials , pp. 97 – 137, URL\nhttp://www.sciencedirect.com/science/article/pii/\nS1567271905800435 .\n57P. Nieves, S. Arapan, S. H. Zhang, A. P. Kadzielawa, R. F. Zhang,\nand D. Legut, ArXiv e-prints (2020), 2009.01638, URL https:\n//arxiv.org/abs/2009.01638 .\n58M. Shimizu, Journal of the Physical Society of Japan 44,\n792 (1978), https://doi.org/10.1143/JPSJ.44.792, URL https:\n//doi.org/10.1143/JPSJ.44.792 .\n59M. Getzlaff (Springer, Berlin, Heidelberg, 2008).\n60A. Sawaoka, Journal of Physics and Chemistry of Solids 36, 267\n(1975), ISSN 0022-3697, URL http://www.sciencedirect.\ncom/science/article/pii/0022369775900207 .\n61R. Wu and A. J. Freeman, Journal of Applied Physics 79, 6209\n(1996), https://aip.scitation.org/doi/pdf/10.1063/1.362073, URL\nhttps://aip.scitation.org/doi/abs/10.1063/1.362073 .\n62R. Wu, L. Chen, and A. Freeman, Journal of Mag-\nnetism and Magnetic Materials 170, 103 (1997), ISSN\n0304-8853, URL http://www.sciencedirect.com/science/\narticle/pii/S0304885397000048 .\n63https://github.com/pnieves2019/MAELAS/tree/master/\nExamples/LAMMPS/MAELAS .\n64M. F ¨ahnle, M. Komelj, R. Q. Wu, and G. Y . Guo, Phys. Rev.\nB65, 144436 (2002), URL https://link.aps.org/doi/10.\n1103/PhysRevB.65.144436 .\n65N. J. Jones, G. Petculescu, M. Wun-Fogle, J. B. Restorff,\nA. E. Clark, K. B. Hathaway, D. Schlagel, and T. A. Lo-\ngrasso, Journal of Applied Physics 117, 17A913 (2015),\nhttps://doi.org/10.1063/1.4916541, URL https://doi.org/10.\n1063/1.4916541 .\n66T. Burkert, O. Eriksson, P. James, S. I. Simak, B. Johansson, and\nL. Nordstr ¨om, Phys. Rev. B 69, 104426 (2004), URL https://\nlink.aps.org/doi/10.1103/PhysRevB.69.104426 .\n67F. Richter and U. Lotter, physica status solidi (b) 34, K149 (1969),\nhttps://onlinelibrary.wiley.com/doi/pdf/10.1002/pssb.19690340261,\nURL https://onlinelibrary.wiley.com/doi/abs/10.\n1002/pssb.19690340261 .\n68J. F. Janak and A. R. Williams, Phys. Rev. B 14, 4199 (1976),\nURL https://link.aps.org/doi/10.1103/PhysRevB.14.17\n4199 .\n69Y . Tanji, Journal of the Physical Society of Japan 31, 1366 (1971),\nhttps://doi.org/10.1143/JPSJ.31.1366, URL https://doi.org/\n10.1143/JPSJ.31.1366 .\n70F. C. Nix and D. MacNair, Phys. Rev. 60, 597 (1941), URL\nhttps://link.aps.org/doi/10.1103/PhysRev.60.597 .\n71C. Williams, Phys. Rev. 46, 1011 (1934), URL https://link.\naps.org/doi/10.1103/PhysRev.46.1011 .\n72S. Khmelevskyi and P. Mohn, Phys. Rev. B 69, 140404(R) (2004),\nURL https://link.aps.org/doi/10.1103/PhysRevB.69.\n140404 .\n73H. Wang, P.-W. Ma, and C. H. Woo, Phys. Rev. B 82,\n144304 (2010), URL https://link.aps.org/doi/10.1103/\nPhysRevB.82.144304 .\n74A. Andreev (Elsevier, 1995), vol. 8 of Handbook of Magnetic Ma-\nterials , pp. 59 – 187, URL http://www.sciencedirect.com/\nscience/article/pii/S1567271905800319 .\n75J. Rouchy and E. du Tremolet de Lacheisserie, Zeitschrift f ¨ur\nPhysik B Condensed Matter 36, 67 (1979), ISSN 1431-584X,\nURL https://doi.org/10.1007/BF01333955 .\n76E. du Tremolet de Lacheisserie and J. Rouchy, Journal of Mag-\nnetism and Magnetic Materials 28, 77 (1982), ISSN 0304-8853,\nURL https://www.sciencedirect.com/science/article/pii/0304885382900312 .\n77R. Yanes, O. Chubykalo-Fesenko, H. Kachkachi, D. A.\nGaranin, R. Evans, and R. W. Chantrell, Phys. Rev. B 76,\n064416 (2007), URL https://link.aps.org/doi/10.1103/\nPhysRevB.76.064416 .\n78R. Skomski, IEEE transactions on magnetics 34, 1207 (1998).\n79G. Dos Santos, R. Aparicio, D. Linares, E. N. Miranda,\nJ. Tranchida, G. M. Pastor, and E. M. Bringa, Phys. Rev. B 102,\n184426 (2020), URL https://link.aps.org/doi/10.1103/\nPhysRevB.102.184426 .\n80N´eel, Louis, J. Phys. Radium 15, 225 (1954), URL https://\ndoi.org/10.1051/jphysrad:01954001504022500 .\n81E. du Tremolet de Lacheisserie and R. Mendia Monterroso, Jour-\nnal of Magnetism and Magnetic Materials 31-34 , 837 (1983),\nISSN 0304-8853, URL https://www.sciencedirect.com/\nscience/article/pii/0304885383907047 .\n82R. F. L. Evans, U. Atxitia, and R. W. Chantrell, Phys. Rev. B 91,\n144425 (2015), URL https://link.aps.org/doi/10.1103/\nPhysRevB.91.144425 .\n83R. F. L. Evans, L. R ´ozsa, S. Jenkins, and U. Atxitia, Phys. Rev.\nB102, 020412(R) (2020), URL https://link.aps.org/doi/\n10.1103/PhysRevB.102.020412 ."    },    {        "title": "2012.05193v2.Pressure_control_of_the_magnetic_anisotropy_of_the_quasi_two_dimensional_van_der_Waals_ferromagnet_Cr__2_Ge__2_Te__6_.pdf",        "content": "arXiv:2012.05193v2  [cond-mat.str-el]  10 Jan 2021Pressure control of the magnetic anisotropy\nof the quasi-two-dimensional van der Waals ferromagnet Cr 2Ge2Te6\nT. Sakurai,1,∗B. Rubrecht,2,3,∗L. T. Corredor,2,∗R. Takehara,4M. Yasutani,4J. Zeisner,2,3\nA. Alfonsov,2S. Selter,2,3S. Aswartham,2A. U. B. Wolter,2B. B¨ uchner,2,5H. Ohta,6and V. Kataev2\n1Research Facility Center for Science and Technology, Kobe U niversity, Kobe 657-8501, Japan\n2Leibniz IFW Dresden, D-01069 Dresden, Germany\n3Institute for Solid State and Materials Physics, TU Dresden , D-01062 Dresden, Germany\n4Graduate School of Science, Kobe University, Kobe 657-8501 , Japan\n5Institute for Solid State and Materials Physics and W¨ urzbu rg-Dresden\nCluster of Excellence ct.qmat, TU Dresden, D-01062 Dresden , Germany\n6Molecular Photoscience Research Center, Kobe University, Nada, Kobe 657-8501 Japan\n(Dated: January 12, 2021)\nWe report the results of the pressure-dependent measuremen ts of the static magnetization and\nof the ferromagnetic resonance (FMR) of Cr 2Ge2Te6to address the properties of the ferromagnetic\nphase of this quasi-two-dimensional van der Waals magnet. T he static magnetic data at hydrostatic\npressures up to 3.4GPa reveal a gradual suppression of ferro magnetism in terms of a reduction of\nthe critical transition temperature, a broadening of the tr ansition width and an increase of the field\nnecessary to fully saturate the magnetization Ms. The value of Ms≃3µB/Cr remains constant\nwithin the error bars up to a pressure of 2.8GPa. The anisotro py of the FMR signal continuously\ndiminishes in the studied hydrostatic pressure range up to 2 .39GPa suggesting a reduction of the\neasy-axis type magnetocrystalline anisotropy energy (MAE ). A quantitative analysis of the FMR\ndata gives evidence that up to this pressure the MAE constant KU, although getting significantly\nsmaller, still remains finite and positive, i.e. of the easy- axis type. Therefore, a recently discussed\npossibility of switching the sign of the magnetocrystallin e anisotropy in Cr 2Ge2Te6could only be\nexpected at still higher pressures, if possible at all due to the observed weakening of the ferromag-\nnetism under pressure. This circumstance may be of relevanc e for the design of strain-engineered\nfunctional heterostructures containing layers of Cr 2Ge2Te6.\nKeywords: van der Waals magnet, 2D material, magnetic aniso tropy, pressure effects, ferromagnetic reso-\nnance, magnetization\nI. INTRODUCTION\nThe layered van der Waals magnet Cr 2Ge2Te6[1] be-\nlongs to a class of materials which are currently the sub-\nject of intense research activities as these systems offer,\non the one hand, a materials base for experimentally ex-\nploring details of two-dimensional magnetism [ 2–4]. On\nthe other hand, the weak couplings between individual\nlayers render these materials promising as (magnetic)\ncomponents in so-called van der Waals heterostructures\n[5]. In both respects it is essential to properly charac-\nterize the magnetic anisotropies of the investigated ma-\nterials. At zero external pressure, Cr 2Ge2Te6features\na uniaxial magnetic anisotropy with the magnetic easy\naxis being oriented parallel to the crystallographic caxis,\ni.e., perpendicular to the honeycomb layers extending\nin theabplane (see, e.g., Refs. [ 1,6–8]). In a pre-\nvious study [ 9] we established the value of the uniax-\nial magnetocrystalline anisotropy energy density (MAE)\nKU= 0.48×106erg/cm3by means of ferromagnetic res-\nonance (FMR) measurements under standard conditions\nat ambient pressure. A detailed discussion of this result\nand its comparison with the band structure calculations\ncan be found in Ref. [ 9].\n∗These authors contributed equally to this work.Oneroutetomodify themagneticpropertiesofvander\nWaals magnets is the application of hydrostatic pressure\nto the crystals, which, in the case of Cr 2Ge2Te6, was al-\nready attempted in Refs. [ 10,11]. Linet al.[11] reported\nachangeintheMAEuponapplyingexternalpressuresup\nto 2GPa, in particular a change in the sign of KU. This\nimpliesthattheanisotropyofthesystemtransformsfrom\neasy-axis type (easy axis parallel to the crystallographic\ncaxis)toeasy-plane type (easydirectionperpendicularto\nthecaxis and thus in the abplane). However, details of\nthe MAE as a function of applied pressure have not been\ninvestigated so far. This motivates a systematic, com-\nbined magnetization and FMR study on Cr 2Ge2Te6at\nvarious hydrostatic pressures aiming at a quantification\nof the pressure dependence of KU. To our knowledge,\nthe FMR experiments on Cr 2Ge2Te6under pressure and\nthe magnetization studies at pressures exceeding 1GPa\nwere not reported so far.\nIn the present work we communicate a comprehensive\ninvestigation of the effect of the hydrostatic pressure on\nthe static magneticandFMR propertiesofsingle crystals\nof Cr2Ge2Te6at pressures up to P= 2.39 and 3.4GPa\napplied in the FMR and magnetization experiments, re-\nspectively. With increasing pressure, the magnetic or-\ndering temperature gradually decreases from Tc= 66K\natP= 0 down to 45K at P= 3.4GPa. Concomitantly,\nthe transition width broadens and the saturation field in-2\ncreases. Remarkably, the saturation magnetization stays\npractically constant up to P= 2.8GPa and decreases at\na higher pressure. The FMR measurements performed\natT= 4.2K≪Tcin the frequency range 50 −260GHz\nfor the two orientations of the applied magnetic field,\nH/bardblcandH⊥c, evidence that the excitation energy\ngap characteristic of an easy-axis ferromagnet closes at\nP >2GPa. The data analysis in the frame of a phe-\nnomenological theory of FMR reveals that the seemingly\nisotropic magnetism of the studied crystals manifesting\nin the gap closure is a result of the compensation of\nthe intrinsic easy-axis magnetocrystalline anisotropy of\nCr2Ge2Te6and the shape anisotropy of the sample. It\nfollows from our analysis that the MAE constant KU\nis reduced by a factor of 2 at the highest applied pres-\nsure in the FMR experiment but still remains sizable.\nThis suggests a robustness of the easy-axis type ferro-\nmagnetism of Cr 2Ge2Te6to the application of pressure\nup toP= 2.39GPa despite a reduction of Tc. Our find-\nings motivate further pressure experiments to address\nthe question of whether the sign of the MAE could be\nchanged at still high pressures before the ferromagnetic\nstate would be fully suppressed. From the technological\nperspective, the results of the present work on the pres-\nsure tunability of the magnetic anisotropy of Cr 2Ge2Te6\nmay be insightful for accessing the use of this material\nas a magnetic element of spintronic devices with strained\narchitecture.\nII. SAMPLES AND METHODS\nSingle crystals of Cr 2Ge2Te6were grown with the self\nflux technique and thoroughly characterized as described\nin detail in our previous work in Ref. [ 9]. As grown sin-\ngle crystals were thoroughly characterized by powder x-\nray diffraction and energy dispersive x-ray spectroscopy,\nboth agree well with the crystal structure in the R ¯3\nspace group as well as with the expected stoichiometry\nof Cr2Ge2Te6[12].\nThe bulk magnetization data were measured using\na custom-built pressure cell for a commercial Quan-\ntum Design superconducting quantum interference de-\nvice (SQUID) magnetometer (MPMS-XL). Inside the\nCuBecell, twoopposingcone-shapedceramicanvilscom-\npress a CuBe gasket with a cylindrical hole used as sam-\nple chamber [ 13,14]. The plateletlike-shaped single-\ncrystalline sample ( m∼3.3×10−6g) of Cr 2Ge2Te6with\nthecaxis normal to the plate was installed into the gas-\nket hole (diameter Ø = 0.6 mm and height h= 0.8 mm)\nandrestedontheflatpartoftheceramicanvil. Giventhe\nlongitudinal magnetic field direction in the SQUID mag-\nnetometer the H/bardblcorientationwaseasily achieved. The\nuniaxialforceisappliedat ambienttemperature, andit is\nconverted into hydrostatic pressure in the sample cham-\nber using Daphne oil 7575 as the pressure-transmitting\nmedium. We checked the pressure value and its homo-\ngeneity at low temperature by measuring the pressure-dependent diamagnetic response associated with the su-\nperconducting transition of a small Pb manometer in-\nserted in the sample space [ 15]. Such measurements were\nperformed in applied fields of H= 5 Oe. In order to en-\nsure a stable pressure throughout all measurements due\nto the thermal expansion of the cell upon temperature\nsweeps, athermalcyclingwasperformedoncefromambi-\nent temperature to 2 K and back to ambient temperature\nbefore the actual data acquisition.\nNotethatamagneticbackgroundisdetectedinourDC\nmagnetization measurements arising from the pressure\ncell itself. Here, we performed a detailed magnetic char-\nacterization of the empty pressure cell within the same\nconditionsastherealexperimentswiththeaimofaquan-\ntitative and reliable disentanglement of the sample sig-\nnal from the overall magnetic response. The background\ncorrection was done by subtracting the fitted signal of a\ngasketwithout a sample measured under the same condi-\ntions, given that the raw signal of the gasket in our setup\ncan be described by a magnetic dipole in our SQUID\ndetection coils. While at temperatures above ∼20 K\na temperature-independent pressure cell background en-\nsures excellent background subtraction, at low temper-\natures a strongly temperature dependent magnetization\nof the CuBe cell limits the resolution of our experiments.\nConsequently, only background-subtracted data for T >\n10 K are shown. Small kinks in the magnetization curves\naround 40 K were identified as instrumental artifacts.\nMore detailed information about the background of our\npressurecell can be found in Ref. [ 16]. Still, the combina-\ntion of the very small magnetization above the ferromag-\nnetic ordering temperature of our samples with a very\nsmall mass ( m∼3.3×10−6g) and the large uncertainty\nof absolute pressure values for T >150 K using Pb as\na manometer at low Tin our ceramic anvil pressure cell\nrestrains us from a quantitative Curie-Weiss analysis in\nthe high-temperature regime.\nFerromagneticresonanceis a resonanceresponseof the\ntotal magnetization of the ferromagnetically ordered ma-\nterial exposed to the microwaveradiation. The FMR fre-\nquency not only is determined by the strength of the ap-\nplied external magnetic field but also sensitively depends\non the magnetic anisotropy of the studied sample, as\nwill be explained in Sec. IIIB. The magnetic anisotropy\ncan be accurately quantified by measuring FMR at dif-\nferent excitation frequencies. Therefore, in the present\nworkFMR experiments were carried out using a multi-\nfrequency electron spin resonance setup equipped with\na piston-cylinder pressure cell with a maximum pres-\nsure of about 2.5 GPa. Its detailed description can be\nfound in Ref. [ 17]. Microwave radiation with frequen-\ncies in the range 50 – 260GHz was provided by a set\nof Gunn diode oscillators and detected by a4He-liquid\ncooled hot-electron InSb bolometer. Magnetic fields up\nto 10T were generated by a cryogen-free superconduct-\ning magnet. Plateletlike single-crystalline samples of\nCr2Ge2Te6with typical lateral dimensions of ∼3−4mm\nwere put inside the pressure cell in a Teflon capsule filled3\n/s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48 /s49/s52/s48 /s49/s54/s48 /s49/s56/s48 /s50/s48/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54/s49/s56\n/s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48 /s57/s48 /s49/s48/s48 /s49/s49/s48 /s49/s50/s48/s45/s48/s46 /s49/s54/s45/s48/s46 /s49/s52/s45/s48/s46 /s49/s50/s45/s48/s46 /s49/s48/s45/s48/s46 /s48/s56/s45/s48/s46 /s48/s54/s45/s48/s46 /s48/s52/s45/s48/s46 /s48/s50/s48/s46 /s48/s48/s48/s46 /s48/s50/s77 /s32/s40/s101/s109/s117/s47/s103/s41\n/s84 /s32/s40/s75/s41/s32/s48/s32/s71/s80/s97\n/s32/s49/s46/s51/s32/s71/s80/s97\n/s32/s50/s46/s50/s32/s71/s80/s97\n/s32/s50/s46/s56/s32/s71/s80/s97\n/s32/s51/s46/s52/s32/s71/s80/s97/s100/s77/s47/s100/s84\n/s84 /s32/s40/s75/s41\nFIG. 1. Temperature dependence of the magnetization M(T)\nof Cr2Ge2Te6measured in the FC mode at µ0H= 0.1T par-\nallel to the caxis for pressures up to 3.4GPa. The inset shows\nthe corresponding dM(T)/dTcurves.\nwith a pressure-transmitting fluid (Daphne 7373 oil from\nIdemitsu KosanCo., Ltd [ 18–20]). The advantageofhav-\ning a well-defined c-axis direction of the samples facili-\ntated their orientation in the FMR pressure cell. For\ntheH/bardblcfield geometry the studied sample was placed\non the flat bottom of the Teflon capsule. For H⊥cit\nwas firmly fixed with a thin Teflon tape to the side of\na small rectangular parallelepiped placed on the bottom\nof the Teflon capsule. The inner pistons of the pressure\ncell made of ZrO 2-based ceramics ensured low-loss prop-\nagation of the microwavesthrough the cell. The pressure\nwas calibrated using a superconducting tin-based pres-\nsure gauge. All FMR measurements were performed at a\ntemperature T= 4.2K for two orientations of the sample\nwith respect to the applied magnetic field, H/bardblcaxis and\nH⊥caxis.\nIII. EXPERIMENTAL RESULTS AND\nDISCUSSION\nA. Pressure-dependent magnetization study\nThe field-cooled (FC) magnetization curves M(T) as\na function of temperature are shown in Fig. 1for an ap-\nplied magnetic field µ0H= 0.1T oriented parallel to the\ncaxis under an applied pressure up to 3.4GPa. Due to\nthe small magnetization values, resulting from the very\nsmallmassofthesample,theabsolutevaluesathightem-\nperatureshow asmall artificial shift in the magnetization\naxis, coming from a highly sensitive background subtrac-\ntion.Two main characteristics are observed. First, the\nonset of the magnetic transition at approximately80K ispersistent within ±1K despite the pressure change. The\nonset was determined from the derivative dM/dTcurve\n(Fig.1, inset)whilemovingfromhightolowtemperature\nas the point of its departure from a very small, close to\nzerovalue dM/dT∼0causedbyjustasmallpolarization\nof paramagnetic moments to the negative values due the\ngradual development of the spontaneous ferromagnetic\nmoment. Second, the transition broadens continuously,\nand the ferromagnetic (FM) transition temperature Tc,\ndefined as the minimum of the dM(T)/dTcurve [21,22],\nshifts to lower temperature upon increasing pressure, as\nshown in the inset of Fig. 1. This behavior is in line with\nan earlier report [ 10] in which magnetization measure-\nments at pressures up to 1GPa showed the same ten-\ndency. Remarkably, the absolute value of the magneti-\nzation in the FM state at the given field of the measure-\nment is gradually reduced with increasing pressure up to\n3.4GPa. This can be explained by a shift of the satura-\ntion field Hsatas a function of pressure to higher values,\nwhich reduces consequently the magnetization at small\napplied fields such as 0.1T (see Fig. 2and discussion be-\nlow).\nThe magnetic field dependence of the magnetization\nM(H) is depicted in Fig. 2for the same pressure values\nas shown for the temperature dependences. To minimize\nthe influence of the nonperfect subtraction of the back-\nground signal from the pressure cell at low temperatures,\nthe curves were measured at T= 15K, which is still low\nenough to capture the characteristic behavior of the FM\nstate (see Fig. 1). TheM(H) data set in Fig. 2includes\na reference measurement performed at ambient pressure\non a more massive single crystal ( m∼4mg) and with-\nout the pressure cell. The ambient pressure data yield\na saturation moment Msof 3.05µB/Cr, which is con-\nsistent with the value found in our previous study [ 9].\nImportantly, the data in Fig. 2show that Msis pressure\nindependent up to about 2.8GPa within the experimen-\ntal error bar, the latter mostly being determined by the\ncomparably strong background signal due to the reduced\nmass of our sample. At higher pressure, however, Msre-\nduces to about 2.5 µB/Cr at 3.4GPa. Another important\nobservation is that, contrary to the saturation moment,\nthe saturationfield Hsatcontinuouslyshifts tohigher val-\nues under applied hydrostaticpressure. While saturation\ncan be achieved at ∼0.1T for ambient pressure, µ0Hsat\nincreases to ∼1.5T at 3.4GPa. A possible explanation\nfor the shift of the saturation field was proposed by Sun\net al.[10] through first principles calculations. Accord-\ning to them, the pressure-induced decrease in the Cr-Cr\nbond length favors antiferromagnetic exchange, while a\nconcomitant deviation from the 90◦Cr-Te-Cr bond angle\nleads to a suppression of the FM superexchange interac-\ntion. Increasing competition between the easy-axis MAE\nand the easy-plane shape anisotropy under pressure (see\nbelow) may also contribute to an increase of Hsat. Mag-\nnetization measurements at still higher pressures would\nbe enlightening in this regard.\nThe results of the above analysis of the M(T) and4\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53\n/s32/s48/s32/s71/s80/s97\n/s32/s49/s46/s51/s32/s71/s80/s97\n/s32/s50/s46/s50/s32/s71/s80/s97\n/s32/s50/s46/s56/s32/s71/s80/s97\n/s32/s51/s46/s52/s32/s71/s80/s97/s77 /s32/s40/s109\n/s66/s47/s67/s114/s41\n/s109\n/s48/s72/s32/s40/s84/s41\nFIG. 2. Magnetization Mas a function of applied field H/bardblc\natT= 15K for pressures up to 3.4GPa. The solid lines are\nguides to the eye. A demagnetization correction has been\napplied [ 23].\nM(H) dependences are summarized in Fig. 3, where\nwe plot the MsandTcvalues as a function of the ap-\nplied pressure with the corresponding experimental error\nbars. As already noticed above, the ordering tempera-\nture value estimated from the first derivative criterion\nis less accurate for higher pressures due to the broaden-\ning of the transition under pressure (Fig. 1, inset). Still,\nwe could clearly validate the long-range ordered ferro-\nmagnetic ground state of Cr 2Ge2Te6at 4.2K and up to\n3.4GPa, i.e., for the parameter ranges where FMR ex-\nperiments were performed (see below). In contrast to Tc,\nthe FM saturation moment is approximatelyconstant for\npressures up to 2.8GPa with a subsequent reduction for\nP >2.8GPa.\nB. Pressure-dependent FMR study\nThe FMR data sets presented in this section were col-\nlectedbymeasuringthreedifferentsingle-crystallinesam-\nples of Cr 2Ge2Te6from the same batch with similar, but\nslightlydifferent, lateraldimensions. As will be discussed\nbelow, the differences in demagnetization factors associ-\nated with different sample dimensions are, however, neg-\nligible in the analysis of the FMR data.\nSelected FMR spectra measured for the two field ge-\nometries used in this work are shown in Figs. 4(a) and\n(b). A shift of the signal to higher fields for H/bardblc\nand to smaller fields for H⊥cby applying pressure is\nclearly visible. The results of frequency-dependent FMR\nmeasurements at a temperature of 4.2K with H/bardblc\nand at different applied external pressures between 0\nand 2.39GPa are summarized in Fig. 4(c). At 0GPa,\nCr2Ge2Te6featuresaneasy-axis-typeMAEwiththeeasy\nFIG. 3. Saturation magnetization Msand the FM transition\ntemperature Tcas a function of the applied pressure (sym-\nbols).Mswas calculated as the average magnetization value\nfor applied fields µ0H≥2T, with 2T determined as the\nlower limit of the applied field at which dM/dH = 0 for all\nthe curves within the experimental error bar. The solid line s\nare guides to the eye.\naxis oriented parallel to the caxis as was investigated in\ndetail in our previous study [ 9]. This kind of anisotropy\ndirectly manifests in an FMR measurement by a shift of\nthe resonance line towards smaller magnetic fields with\nrespect to the resonance position in the paramagnetic\nstate if the external magnetic field His applied parallel\nto the magnetic easy axis ( caxis) and towards higher\nmagnetic fields if His normal to the magnetic easy axis.\nThis behavior is illustrated in Figs. 4(a) and4(b), where,\ndepending on the direction of the applied field, the FMR\nsignal at P= 0 is shifted either to the left or to the\nright side of the signal from the paramagnetic reference\nsample 2,2-diphenyl-1-picrylhydrazyl (DPPH) with a g\nfactorg≃2, which is commonly used as a magnetic\nfield marker. Remarkably, the application of an exter-\nnal pressure counteracts these tendencies by shifting the\nFMR line towards the paramagnetic position for both\nfield geometries. As a result, with increasing the applied\npressure from P= 0 to 2.01GPa the FMR signal from\nthe sample becomes isotropic; that is, the resonance field\nHresdoes not depend on the direction of the applied field\nat this value of pressure [Fig. 5(b)].\nThe frequency-field dependence of the paramagnetic\nreference sample DPPH is shown in Fig. 4as a blue\ndashed line for comparison with the ferromagnetic res-\nonance fields Hresmeasured on the Cr 2Ge2Te6samples\nforH/bardblcaxis.Hresshifts continuously to higher mag-\nnetic fields with increasing Pand thereby approaches,\nbut does not cross, the paramagnetic resonance branch\nin the frequency-field diagram. This systematic shift is\nemphasized in the inset of Fig. 4, where details of the\nfrequency-field diagram are shown for intermediate field\nstrengths. The change in the FMR line position as a5\n/s53/s46/s50 /s53/s46/s52 /s53/s46/s54 /s53/s46/s56 /s52/s46/s48 /s52/s46/s53 /s53/s46/s48\n/s32/s32\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41\n/s68/s80/s80/s72/s110 /s32/s61/s32/s49/s53/s48/s32/s71/s72/s122\n/s84 /s32/s61/s32/s52/s46/s50/s32/s75\n/s72 /s99\n/s48/s32/s71/s80/s97/s50/s46/s51/s57/s32/s71/s80/s97/s40/s98/s41/s32/s32/s65/s98/s115/s111/s114/s112/s116/s105/s111/s110/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41/s48\n/s68/s80/s80/s72/s110 /s32/s61/s32/s49/s51/s48/s32/s71/s72/s122\n/s84 /s32/s61/s32/s52/s46/s50/s32/s75\n/s72/s124/s124/s99\n/s50/s46/s51/s57/s32/s71/s80/s97/s40/s97/s41\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48\n/s32/s68/s80/s80/s72/s58/s32 /s103 /s32/s61/s32/s50\n/s32/s48/s32/s71/s80/s97\n/s32/s48/s46/s53/s32/s71/s80/s97\n/s32/s49/s46/s48/s32/s71/s80/s97\n/s32/s49/s46/s53/s49/s32/s71/s80/s97\n/s32/s50/s46/s48/s49/s32/s71/s80/s97\n/s32/s50/s46/s51/s57/s32/s71/s80/s97/s32\n/s32/s110 /s32/s40/s71/s72/s122/s41\n/s109\n/s48/s32/s72\n/s114/s101/s115/s32/s40/s84/s41/s72 /s32/s124/s124/s32/s99/s44/s32 /s84 /s32/s61/s32/s52/s46/s50/s32/s75/s48/s32/s71/s80/s97/s48/s32/s71/s80/s97\n/s50/s46/s51/s57/s32/s71/s80/s97/s50/s46/s51/s57/s32/s71/s80/s97/s40/s99/s41\n/s52/s46/s48 /s52/s46/s53 /s53/s46/s48 /s53/s46/s53/s49/s49/s48/s49/s50/s48/s49/s51/s48/s49/s52/s48/s49/s53/s48/s49/s54/s48/s110 /s32/s40/s71/s72/s122/s41\n/s109\n/s48/s32/s72\n/s114/s101/s115/s32/s40/s84/s41\nFIG. 4. Representative FMR spectra of Cr 2Ge2Te6at zero\nand the maximum applied pressure for the applied magnetic\nfield (a) parallel and (b) perpendicular to the caxis. The\nsharp spike is a signal from the paramagnetic reference DPPH\nwith agfactor of g≃2. Some distortions of the lineshape\ncan be ascribed to the interference effects of the microwaves\nin the pressure cell [ 24,25]. (c) The frequency-field diagram\nobtained from FMR measurements at various pressures up to\n2.39GPa for H/bardblc. Solid lines are fits according to Eq. ( 2)\nwhile the dashed blue line indicates the paramagnetic ν(Hres)\ndependence of the DPPH reference sample with g≃2. De-\ntails of the frequency-field diagram for intermediate fields are\nshown in the inset in order to emphasize the systematic shift\nof the resonance line with increasing pressure.\nfunction of external pressure evidences a reduction of the\nMAE of the studied system. However, the shift of the\nresonance fields of Cr 2Ge2Te6is determined not only by\nthe strength and the sign of the MAE constant KUbut\nalso by demagnetization fields resulting from the specific\nsample shapes which favor, in the case of the platelike-\nshaped Cr 2Ge2Te6crystals, an in-plane magnetization.\nAsaconsequence,theinternaldemagnetizationfieldswill\nlead to a shift of the resonance position to higher fields\nif the external magnetic field is applied perpendicular to\nthe honeycomb layers, i.e., for H/bardblcaxis. Thus, the sizeof the anisotropy constant KUderived from shifts of the\nresonance position would be underestimated, if demag-\nnetization effects were not taken into account (see the\nrelated discussion in Ref. [ 9]). These demagnetization\nfieldsHDcan be calculated using the elements of the de-\nmagnetization tensor Ni(i=x,y,z, withx,y,zbeing\nthe principal axes of the tensor) [ 23,26,27]:\nHi\nD=−4πNiM , (1)\nwhereMdenotes the magnetization of the sample. Since\nthe thickness of the plate-like samples used in this study\ncould not be quantified precisely, the real sample shape\nwas approximated using the demagnetization factors for\na flat (infinite) plate oriented perpendicular to the zaxis\n(which was chosen to coincide with the crystallographic\ncaxis of Cr 2Ge2Te6), i.e.,Nx=Ny= 0,Nz= 1. This\nassumption is supported by the fact that the lateral di-\nmensions in the abplane of the used Cr 2Ge2Te6crystals\nwere much larger than the thicknesses of the samples\nparallel to the caxis. Using the value of the satura-\ntion magnetization Ms≃3µB/Cr (which corresponds to\nMs≈201 erg/Gcm3for Cr 2Ge2Te6) yields a demagne-\ntization field µ0HDin the fully saturated ferromagnetic\nphase of -0.253T if the external field is applied parallel\nto thecaxis. The frequency-field dependences measured\nwith this field orientation were then fitted by the follow-\ning expression:\nνres=gµBµ0\nh[Hres+HD+HA], (2)\nwhereνresdenotes the resonance/microwave frequency\nandHAis the anisotropy field describing the intrinsic\nmagnetocrystalline anisotropy. Here, µB,µ0andhare\nBohr magneton, magnetic permeability and Plancks con-\nstant, respectively. Fits of Eq. ( 2) to the measured data\nare shown in Fig. 4(c) as solid lines. For the fitting, the\ngfactor and HAwere treated as free fit parameters. The\ngfactor varied very little within the error bars around\nthe mean value of 2.03 (see, Sec. IV) whereas HAsig-\nnificantly decreased with increasing pressure. Using the\nlatter parameter, it is possible to calculate the uniaxial\nMAEKUaccording to (see, for instance, Ref. [ 28])\nKfit\nU=HAMS\n2. (3)\nThe results obtained from this fitting procedure are dis-\ncussed in the following section together with results from\nan alternative approach for a quantitative analysis of the\nFMR data.\nIV. SIMULATIONS OF THE FREQUENCY\nDEPENDENCE AND DETERMINATION OF\nTHE MAGNETIC ANISOTROPIES\nInordertosimultaneouslytakeintoaccountthe results\nof our FMR measurements performed with H/bardblcand6\n/s48 /s50 /s52 /s54 /s56 /s49/s48 /s48 /s50 /s52 /s54 /s56/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48\n/s32/s72 /s32/s124/s124/s32/s99\n/s32/s72 /s32/s94 /s32/s99/s110 /s32/s40/s71/s72/s122/s41\n/s109\n/s48/s32/s72\n/s114/s101/s115/s32/s40/s84/s41/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s48/s32/s71/s80/s97/s44/s32 /s84/s32 /s61/s32/s52/s46/s50/s32/s75/s32\n/s75\n/s85/s32/s61/s32/s40/s52/s46/s56/s32 /s177 /s32/s48/s46/s51/s41 /s215/s49/s48/s53\n/s32/s101/s114/s103/s47/s99/s109/s51/s40/s97/s41 /s40/s98/s41\n/s32/s72 /s32/s124/s124/s32/s99\n/s32/s72 /s32/s94 /s32/s99\n/s109\n/s48/s32/s72\n/s114/s101/s115/s32/s40/s84/s41/s32/s32/s32/s32/s32/s32/s32/s32/s50/s46/s48/s49/s32/s71/s80/s97/s44/s32 /s84/s32 /s61/s32/s52/s46/s50/s32/s75\n/s75\n/s85/s32/s61/s32/s40/s50/s46/s55/s32 /s177 /s32/s48/s46/s52/s41 /s215/s49/s48/s53\n/s32/s101/s114/s103/s47/s99/s109/s51\nFIG. 5. Examples of simulations (solid lines) of the mea-\nsured frequency-fielddependences(symbols) at (a) 0GPa and\n(b) 2.01GPa. In these simulations experimental data sets ob -\ntained for H/bardblc(circles) and H⊥c(squares) were taken\ninto account simultaneously in order to determine the value\nofKU.\nH⊥cfield geometries, the measured frequency-field de-\npendences were numerically simulated for each pressure\nvalue. This lead to a refinement of the determination of\nKU, in particular at higher pressures. For the simula-\ntions we used a well-established phenomenological model\nof FMR (see, e.g., Refs. [ 28–30]) in which the resonance\nfrequency νresis expressed as\nν2\nres=g2µ2\nB\nh2M2ssin2θ/parenleftbigg∂2F\n∂θ2∂2F\n∂ϕ2−/parenleftBig∂2F\n∂θ∂ϕ/parenrightBig2/parenrightbigg\n.(4)\nHere,Fis the free-energy density, and θandϕare\nthe spherical coordinates of the magnetization vector\nM(Ms,ϕ,θ).\nThis phenomenological model is applicable for ferro-\nmagnets with fully saturated magnetization at tempera-\ntures sufficiently lower than Tc, i.e., the case of the fully\ndeveloped ferromagnetic phase. According to the above-\ndiscussed results of our pressure-dependent magnetiza-\ntion study, both criteria are safely fulfilled for the T,P,\nandHparameter ranges of the FMR study.\nTo obtain the resonance position of the FMR signal, F\nin Eq. (4) should be taken at the equilibrium angles ϕ0\nandθ0ofM. In the simulations, the minimum of Fwith\nrespect to θandϕfor a given set of experimental param-\neters, such as the microwave frequency and the direction\nand the strength of the magnetic field, was found numer-\nically. For Cr 2Ge2Te6, accounting for both the intrinsic\nuniaxial magnetic anisotropyand of the (extrinsic) shape\nanisotropy of the particular studied sample enabled an\naccurate description of all FMR data sets. In this casethe free energy density (in cgs units) is defined as\nF=−H·M−KUcos2(θ)+\n2πM2\ns(Nxsin2(θ)cos2(ϕ)+\nNysin2(θ)sin2(ϕ)+Nzcos2(θ))(5)\nand comprises, besides the Zeeman energy density ex-\npressed by the first term, contributions due to the uniax-\nial and shape anisotropies, the second and third terms,\nrespectively.\nRepresentative simulations of frequency-dependent\nmeasurements at 0 and 2.01GPa are shown in Figs. 5(a)\nand5(b), respectively. At 0GPa, the separation of\ntheν(Hres) curves obtained for the two different field\norientations clearly evidences the easy-axis-type mag-\nnetic anisotropy with the H/bardblccurve being shifted to\nsmaller fields and the H⊥ccurve being shifted to\nhigherfieldscomparedtotheparamagneticposition. The\n0GPa data could be simulated using the value of KU\nof 4.8×105erg/cm3obtained in our previous study [ 9].\nKeeping the value of KUfixed, the gfactor and demag-\nnetization factors were adjusted in order to consistently\ndescribe different studied samples. It turned out that\nall 0GPa data could be described successfully with an\nisotropic gfactor of 2.03, which is consistent with our\npreviousinvestigations, andthe sameset ofdemagnetiza-\ntion factors Nx=Ny= 0,Nz= 1 despite the (small) dif-\nferences in the shapes of the samples. These parameters\nwerekeptfixedin thesimulationsofthefrequencydepen-\ndences measured at several nonzero pressures. The only\nfree parameter in these simulations was the anisotropy\nconstant KUwhich allowed us to determine the pressure\ndependence of the MAE. As can be seen in Fig. 5(b),\nat a pressure of 2.01GPa, the measured frequency de-\npendences are nearly identical for both orientations of\nthe external magnetic field (a similar frequency-field di-\nagram was obtained for 2.39GPa), giving the impression\nof isotropic behavior. However, the contributions of the\nmagnetocrystalline anisotropy and the shape anisotropy\nto the free-energy density [Eq. ( 5)] have opposite signs\nwithin the pressure range of this study. Thus, the ap-\nparently isotropic frequency dependence is, in fact, a\nconsequence of the similar strengths of these two con-\ntributions. This allows us to conclude that an appli-\ncation of pressures between 2.01 and 2.39GPa reduces\nthe uniaxial MAE KUto a value that compensates the\nshape anisotropy of the crystals. With the demagnetiza-\ntion field µ0HD=−0.253T and saturation magnetiza-\ntionMs≈201 erg/Gcm3estimated above, one obtains\nthe shape anisotropy constant Kshape= (HDMs)/2≈\n−2.54×105erg/cm3. Its absolute value is indicated\nby the dashed line in Fig. 6.Moreover, the pressure-\ndependent FMR studies do not provide any evidence\nfor a pressure-induced sign change of the magnetocrys-\ntalline anisotropy or a vanishing of the intrinsic uniaxial\nanisotropy up to pressures of 2.39GPa. This is in con-\ntrastto the conclusionsdrawn in Ref. [ 11] on a spin reori-\nentation transition at pressures between 1.0 and 1.5GPa7\n/s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s48/s48/s49/s50/s51/s52/s53\n/s32/s115/s105/s109/s117/s108/s97/s116/s105/s111/s110\n/s32/s32/s102/s105/s116\n/s32/s124 /s75\n/s115/s104/s97/s112/s101/s124/s75\n/s85/s32/s40/s49/s48/s53\n/s32/s101/s114/s103/s47/s99/s109/s51\n/s41\n/s112 /s32/s40/s71/s80/s97/s41/s84 /s32/s61/s32/s52/s46/s50/s32/s75\nFIG. 6. Pressure dependence of the uniaxial magnetocrys-\ntalline anisotropy constant KUof Cr2Ge2Te6. Red circles de-\nnote the results obtained from fitting the different ν(Hres)\ncurves according to Eq. ( 2) forH/bardblc. Blue squares are\ntheKUvalues determined from simulations according to\nEq. (5) considering the experimental data sets for H/bardblc\nandH⊥csimultaneously. The horizontal dashed line shows\nthe absolute value of the negative shape anisotropy constan t\nKshape≈ −2.54×105erg/cm3. It compensates the decreas-\ning positive value of KUatp/greaterorsimilar2GPa yielding a seemingly\nisotropic behavior.\nbasedonmagneto-transportinvestigationsofCr 2Ge2Te6.\nThe pressure dependence of KUas obtained from fit-\nting ofthe H/bardblcdatato Eq.( 2) andfromthe simulations\nof all data available for both field orientations according\ntoEq.(5)isshownin Fig. 6. Qualitatively,both methods\nreveal a similar behavior, in particular a reduction of KU\nwithincreasingpressure. However,thedeviationbetween\ntheKUvalues derived from the fits and the simulation\nincreasesat higherpressures. This could be attributed to\nthe fact that the simulations simultaneously take into ac-\ncount the frequency dependences measured for both field\norientations. Therefore, the reliability of the KUvalues\nfrom simulations is higher despite the larger error bars,\nin particular at higher pressures at which differences be-\ntween both field orientations become very small. Finally,\nitisworthmentioningthatinRef.[ 31]apressure-induced\ndifference in the unit-cell volume of Cr 2Ge2Te6between\napplied pressures of 0 and 3GPa of about 4% was re-\nported. As the unit-cell volume enters into the calcu-\nlation of the saturation magnetization, such a reduction\nof volume with increasing pressure could change the ab-\nsolute value determined for KU. However, it was found\nthat the pressure-inducedcontractionofthe unit-cell vol-\nume by 4% does not affect the determination of KUbut\nis within the given error bars.V. CONCLUSIONS AND OUTLOOK\nOur combined magnetization and ferromagnetic res-\nonance study on single crystals of the quasi-two-\ndimensional ferromagnet Cr 2Ge2Te6has revealed a sig-\nnificanteffect ofthe applicationofhydrostaticpressure P\non the properties of the ferromagnetic state of this com-\npound. It manifests in a reduction of the critical temper-\natureTcand a concomitant broadening of the transition\nto the FM state, as well as in a gradual increase of the\nsaturation field Hsat. In contrast, the saturation magne-\ntization remains practically constant up to P= 2.8GPa\nand reduces by approaching the highest applied pres-\nsure of 3.4GPa. The frequency-dependent FMR mea-\nsurements performed in the fully developed FM state at\nT≪TcandH > H satat various external pressures up\nto 2.39GPa revealed a systematic shift of the resonance\nfields with increasing Pevidencing a continuous reduc-\ntion of the magnetocrystalline anisotropy constant KU.\nFrom the quantitative analysis of the data, which takes\ninto account demagnetization effects, it follows that KU,\nalthough being significantly reduced at the highest pres-\nsures applied in this study, neither vanishes nor changes\nits sign up to pressures of about 2.4GPa. Therefore,\nnoting that FMR is a very direct method for the quan-\ntitative determination of magnetic anisotropy, it can be\nconcluded with confidence that there is no evidence for\na switching of the magnetic anisotropy from the easy-\naxis to the easy-plane type within the pressure range un-\nderconsideration. Furtherpressure-dependentstudieson\nCr2Ge2Te6are desired to understand the apparent dis-\ncrepancy between our work and the magnetotransport\nstudy in Ref. [ 11] claiming a spin-reorientation transi-\ntion in Cr 2Ge2Te6at pressures 1 < P <1.5GPa and to\naddress the question of whether a pressure-induced sign\nswitching of the magnetic anisotropy could be achieved\nat pressures exceeding 2.39GPa.\nAlthough in terms of the spatial dimensionality\nwe investigated bulk, three-dimensional crystals of\nCr2Ge2Te6, we showedin our previouselectronspin reso-\nnance/FMR study that the low-temperature magnetism\nof the bulk crystals is essentially of a two-dimensional\n(2D) nature due to very weak interlayer magnetic cou-\npling in this van der Waals compound [ 9]. Consider-\ning the intrinsic magnetic two-dimensionality of the bulk\nmaterial, our pressure dependent study could be rele-\nvant for the research on Cr 2Ge2Te6in the truly 2D spa-\ntial limit. In this respect, the pressure control of the\nmagnetic anisotropy investigated in our work may pro-\nvide important hints for a targeted design of functional\nmagneto-electrical heterostructures containing layers of\nCr2Ge2Te6.\nIn particular, a recent prediction of remarkable strain\nand electric field tunability of a single layer Cr 2Ge2Te6\nis encouraging [ 32]. It remains yet an open question\nwhether the design of a heterostructure with strain ǫ\nof the order of ±(1−2)% for the Cr 2Ge2Te6layer, as\nproposed in Ref. [ 32], can be achieved. However, a siz-8\nable effect on the magnetic anisotropywhich we observed\nat hydrostatic pressures up to 2.39GPa corresponds to\nan even smaller compressive in-plane strain ǫ∼0.7%\n[31]. Such straining seems plausible to achieve since, in\ngeneral, 2D van der Waals materials are known to sus-\ntain large strain. For example, it was shown that biaxial\ncompressive and tensile strain of ∼1% can be achieved\nin single-layer molybdenum dichalcogenide deposited on\na thermally compressed or expanded polymer substrate\n[33].\nACKNOWLEDGMENTS\nWe thank Gael Bastien and Randirley Beltran Ro-\ndriguez for technical assistance with the pressure cell\nused for the magnetization studies, and the IFW work-\nshop and Juliane Scheiter for the production and treat-ment of the CuBe gaskets. The work in Kobe was par-\ntially supported by Grants-in-Aid for Scientific Research\n(C) (Grant No. 19K03746) from the Japan Society for\nthe Promotion of Science. The work in Dresden was sup-\nported by the Deutsche Forschungsgemeinschaft (DFG)\nthrough Grant No. KA1694/12-1 and within Collabora-\ntive Research Center SFB 1143“CorrelatedMagnetism –\nFrom Frustration to Topology” (Project No. 247310070)\nand the Dresden-W¨ urzburg Cluster of Excellence (EXC\n2147) “ct.qmat - Complexity and Topology in Quantum\nMatter” (Project No. 390858490). S.A. acknowledges fi-\nnancial support from the DFG through Grant No. AS\n523/4-1. A.A. acknowledges financial support form the\nDFG through Grant No. AL 1771/4-1. S.S. acknowl-\nedges financial support from the GRK-1621 Graduate\nAcademy. V.K. gratefully acknowledges the hospitality\nand financial support during his research visit to Kobe\nUniversity.\n[1]V Carteaux, D Brunet, G Ouvrard, and G Andre,\n“Crystallographic, magnetic and electronic structures\nof a new layered ferromagnetic compound Cr 2Ge2Te6,”\nJ. Phys.: Condens. Matter 7, 69 (1995) .\n[2]J.-G. Park, “Opportunities and challenges of 2D mag-\nnetic van der Waals materials: magnetic graphene?”\nJ. Phys.: Condens. Matter 28, 301001 (2016) .\n[3]Cheng Gong, Lin Li, Zhenglu Li, Huiwen Ji, Alex Stern,\nYang Xia, Ting Cao, Wei Bao, Chenzhe Wang, Yuan\nWang, Z. Q. Qiu, R. J. Cava, Steven G. Louie, Jing\nXia, and Xiang Zhang, “Discovery of intrinsic ferro-\nmagnetism in two-dimensional van der Waals crystals,”\nNature546, 265 (2017) .\n[4]Kenneth S. Burch, David Mandrus, and Je-Geun Park,\n“Magnetism in two-dimensional van der Waals materi-\nals,”Nature563, 47 (2018) .\n[5]M. Gibertini, M. Koperski, A. F. Morpurgo, and K. S.\nNovoselov, “Magnetic 2D materials and heterostruc-\ntures,”Nat. Nanotechnol. 14, 408 – 419 (2019) .\n[6]Xiao Zhang, Yuelei Zhao, Qi Song, Shuang Jia, Jing\nShi, and Wei Han, “Magnetic anisotropy of the\nsingle-crystalline ferromagnetic insulator Cr 2Ge2Te6,”\nJpn. J. Appl. Phys. 55, 033001 (2016) .\n[7]G. T. Lin, H. L. Zhuang, X. Luo, B. J. Liu, F. C.\nChen, J. Yan, Y. Sun, J. Zhou, W. J. Lu, P. Tong,\nZ. G. Sheng, Z. Qu, W. H. Song, X. B. Zhu, and\nY. P. Sun, “Tricritical behavior of the two-dimensional\nintrinsically ferromagnetic semiconductor CrGeTe 3,”\nPhys. Rev. B 95, 245212 (2017) .\n[8]Yu Liu and C. Petrovic, “Critical behavior of quasi-two-\ndimensional semiconducting ferromagnet Cr 2Ge2Te6,”\nPhys. Rev. B 96, 054406 (2017) .\n[9]J. Zeisner, A. Alfonsov, S. Selter, S. Aswartham, M. P.\nGhimire, M. Richter, J. van den Brink, B. B¨ uchner, and\nV. Kataev, “Magnetic anisotropy and spin-polarized two-\ndimensional electron gas in the van der Waals ferromag-\nnet Cr 2Ge2Te6,”Phys. Rev. B 99, 165109 (2019) .\n[10]Y. Sun, R. C. Xiao, G. T. Lin, R. R. Zhang, L. S. Ling,\nZ. W. Ma, X. Luo, W. J. Lu, Y. P. Sun, and Z. G.\nSheng, “Effects of hydrostatic pressure on spin-latticecoupling in two-dimensional ferromagnetic Cr 2Ge2Te6,”\nAppl. Phys. Lett. 112, 072409 (2018) .\n[11]Zhisheng Lin, Mark Lohmann, Zulfikhar A. Ali, Chi\nTang, Junxue Li, Wenyu Xing, Jiangnan Zhong, Shuang\nJia, Wei Han, Sinisa Coh, Ward Beyermann, and\nJing Shi, “Pressure-induced spin reorientation transi-\ntion in layered ferromagnetic insulator Cr 2Ge2Te6,”\nPhys. Rev. Mater. 2, 051004 (2018) .\n[12]S. Selter, G. Bastien, A. U. B. Wolter, S. Aswartham,\nandB.B¨ uchner,“Magnetic anisotropyandlow-fieldmag-\nnetic phase diagram of the quasi-two-dimensional ferro-\nmagnet Cr 2Ge2Te6,”Phys. Rev. B 101, 014440 (2020) .\n[13]P. L. Alireza and G. G. Lonzarich, “Miniature anvil cell\nfor high-pressure measurements in a commercial super-\nconducting quantum interference device magnetometer,”\nRev. Sci. Inst. 80, 023906 (2009) .\n[14]W. Schottenhamel, Aufbau eines hochaufl¨ osenden\nDilatometers und einer hydrostatischen SQUID-\nDruckzelle sowie Untersuchungen an korrelierten\n¨Ubergangsmetalloxiden , Ph.D. thesis, Technische Univer-\nsit¨ at Dresden (2016).\n[15]B Bireckoven and J Wittig, “A diamond anvil cell for\nthe investigation of superconductivity under pressures of\nup to 50 GPa: Pb as a low temperature manometer,”\nJ. Phys. E: Sci. Instrum. 21, 841–848 (1988) .\n[16]G. Prando, R. Dally, W. Schottenhamel, Z. Guguchia, S.-\nH. Baek, R. Aeschlimann, A. U. B. Wolter, S. D. Wilson,\nB. B¨ uchner, and M. J. Graf, “Influence of hydrostatic\npressure on the bulk magnetic properties of Eu 2Ir2O7,”\nPhys. Rev. B 93, 104422 (2016) .\n[17]T. Sakurai, K. Fujimoto, R. Matsui, K. Kawasaki,\nS. Okubo, H. Ohta, K. Matsubayashi, Y. Uwatoko, and\nH. Tanaka, “Development of multi-frequency ESR sys-\ntem for high-pressure measurements up to 2.5 GPa,”\nJ. Magn. Reson. 259, 108 – 113 (2015) .\n[18]The general pressure medium Daphne7373 ensures a hy-\ndrostatic pressure and gives good pressure homogeneity\nto the sample up to about 2.3GPa. Therefore, it is suffi-\ncientfor anFMRmeasurementwithamaximumpressure\nof 2.4GPa. The special pressure medium Daphne 7575 is9\nnecessary only for the magnetization measurement where\na pressure above 3GPa was applied in order to secure\nthe pressure homogeneity in the high-pressure range. See\nRefs. [19,20] for details.\n[19]Keizo Murata, Keiichi Yokogawa, Harukazu Yoshino,\nStefan Klotz, Pascal Munsch, Akinori Irizawa, Mo-\ntotsugu Nishiyama, Kenzo Iizuka, Takao Nanba,\nTahei Okada, Yoshitaka Shiraga, and Shoji\nAoyama, “Pressure transmitting medium Daphne\n7474 solidifying at 3.7 GPa at room temperature,”\nRev. Sci. Inst. 79, 085101 (2008) .\n[20]Keizo Murata and Shinji Aoki, “Development of high\npressure liquid medium with good hydrostatic pres-\nsure,”Rev. High. Pressure Sci. Techol. 26, 3–7 (2016) ,\n(in Japanese with abstract in English ).\n[21]We did not use the other frequently applied method, the\nArrot-plot technique [ 22], since it is not practicable for\npressure dependent studies due to the reduced resolution\nstemming from the background signal of the pressure cell\n(see Sec. II).\n[22]I. Yeung, R. M. Roshko, and G. Williams, “Arrott-\nplot criterion for ferromagnetism in disordered systems,”\nPhys. Rev. B 34, 3456–3457 (1986) .\n[23]J. A. Osborn, “Demagnetizing factors of the general el-\nlipsoid,” Phys. Rev. 67, 351–357 (1945) .\n[24]Tominimize theundesiredinterferenceeffectsinthepres-\nsure cell and, in particular, to avoid the interference\ninside the sample we followed the recipe suggested in\nRef. [25] and used very thin crystals with a thickness\nof a few tens µm which is much smaller than the short-\nest microwave wavelength of 1153 µm at the highest ap-\nplied frequency of 260GHz. Furthermore, we repeated\neach experiment several times to confirm that the reso-\nnance fields agree within the error [smaller than the sym-\nbol size in Fig. 4(c)]. This ensured that the intrinsic res-\nonance position was always correctly obtained regardless\nthe distortion.[25]G Eilers, M von Ortenberg, and R Galazka, “Origin of\nSatellite Structures of High-Field EPR in Cd 1−xMnxTe,”\nInt. J. Infrared Millimeter Waves 15, 695–722 (1994) .\n[26]D. C. Cronemeyer, “Demagnetization factors for general\nelipsoids,” J. Appl. Phys. 70, 2911 (1991) .\n[27]Stephen Blundell, Magnetism in Condensed Matter (Ox-\nford University Press, Oxford, 2001).\n[28]G. V. Skrotskii and L. V. Kurbatov, “Phenomenologi-\ncal theory of ferromagnetic resonance,” in Ferromagnetic\nResonance , edited by S. V. Vonsovskii (Pergamon Press\nLtd., 1966) Chap. Phenomenological Theory of Ferro-\nmagnetic Resonance, pp. 12–77.\n[29]J. SmitandH.G. Beljers, “Ferromagnetic ResonanceAb-\nsorption inBaFe 12O19,”Philips Res.Rep. 10,113(1955).\n[30]Michael Farle, “Ferromagnetic resonance of ultrathin\nmetallic layers,” Rep. Prog. Phys. 61, 755 (1998) .\n[31]Zhenhai Yu, Wei Xia, Kailang Xu, Ming Xu,\nHongyuan Wang, Xia Wang, Na Yu, Zhiqiang Zou,\nJinggeng Zhao, Lin Wang, Xiangshui Miao, and Yan-\nfeng Guo, “Pressure-Induced Structural Phase Tran-\nsition and a Special Amorphization Phase of Two-\nDimensional Ferromagnetic Semiconductor Cr 2Ge2Te6,”\nJ. Phys. Chem. C 123, 13885–13891 (2019) .\n[32]Kangying Wang, Tao Hu, Fanhao Jia, Guodong\nZhao, Yuyu Liu, Igor V. Solovyev, Alexander P.\nPyatakov, Anatoly K. Zvezdin, and Wei Ren,\n“Magnetic and electronic properties of Cr 2Ge2Te6\nmonolayer by strain and electric-field engineering,”\nApplied Physics Letters 114, 092405 (2019) .\n[33]Riccardo Frisenda, Matthias Drueppel, Robert Schmidt,\nSteffen Michaelis de Vasconcellos, David Perez de lara,\nRudolf Bratschitsch, Michael Rohlfing, and Andres\nCastellanos-Gomez, “Biaxial strain tuning of the opti-\ncal properties of single-layer transition metal dichalco-\ngenides,” Npj 2D Mater Appl 1, 10 (2017) ."    },    {        "title": "2012.06693v1.Electrically_Controllable_Crystal_Chirality_Magneto_Optical_Effects_in_Collinear_Antiferromagnets.pdf",        "content": "Electrically Controllable Crystal Chirality Magneto-Optical Effects in Collinear Antiferromagnets\nXiaodong Zhou,1Wanxiang Feng,1,\u0003Xiuxian Yang,1Guang-Yu Guo,2, 3and Yugui Yao1,y\n1Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurement,\nMinistry of Education, School of Physics, Beijing Institute of Technology, Beijing 100081, China\n2Department of Physics and Center for Theoretical Physics, National Taiwan University, Taipei 10617, Taiwan\n3Physics Division, National Center for Theoretical Sciences, Hsinchu 30013, Taiwan\n(Dated: December 15, 2020)\nThe spin chirality, created by magnetic atoms, has been comprehensively understood to generate and control\nthe magneto-optical effects. In comparison, the role of the crystal chirality that relates to nonmagnetic atoms\nhas received much less attention. Here, we theoretically discover the crystal chirality magneto-optical (CCMO)\neffects, which depend on the chirality of crystal structures that originates from the rearrangement of nonmagnetic\natoms. We show that the CCMO effects exist in many collinear antiferromagnets, such as RuO 2and CoNb 3S6,\nwhich has a local and global crystal chirality, respectively. The key character of the CCMO effects is the sign\nchange if the crystal chirality reverses. The magnitudes of the CCMO spectra can be effectively manipulated by\nreorienting the N ´eel vector with the help of an external electric field, and the spectral integrals are found to be\nproportional to magnetocrystalline anisotropy energy.\nAmong large family members of the magneto-optical (MO)\neffects, the Kerr [1] and Faraday [2] effects discovered in the\nmid-18th century are the representatives, which describe the\nrotations of the planes of polarization of the linearly polarized\nlight reflecting from and transmitting through magnetic me-\ndia, respectively. After they were discovered over a century,\na critical theoretical explanation of them was revealed on the\nbasis of the band theory [3]. Since then, two basic understand-\nings for the MO effects were broadly accepted [4–8]: The\nmagnitudes of MO spectra are proportional to the spontaneous\nmagnetization of magnetic media such that antiferromagnets\n(AFMs) with zero net magnetization are intuitively believed\nto present no MO signals; The simultaneous presence of band\nexchange splitting (due to finite net magnetization) and spin-\norbit coupling is the sole physical origin of the MO effects.\nNevertheless, new insights into the MO effects emerged re-\ncently. On the one hand, the MO effects were surprisingly\nfound in noncollinear [9–14] or collinear [15–18] AFMs even\nthough their net magnetization is vanishing. The MO Kerr\neffect was first predicted to exist in noncollinear (coplanar)\nAFMs Mn 3X(X=Rh, Ir, Pt) [9] by the natural lack of a\ngood symmetryTS(Tis the time-reversal symmetry; Sis a\nspatial symmetry) that is responsible for band exchange split-\nting. The collinear AFMs (e.g., MnPSe 3) were then demon-\nstrated to host the Kerr effect if an electric field is introduced\nto break theTSsymmetry [15]. On the other hand, the topo-\nlogical MO effects that originate from the scalar spin chiral-\nity was discovered in noncollinear (noncoplanar) AFMs [19].\nThe band exchange splitting and spin-orbit coupling are not\nindispensable for the topological MO effects and their quanti-\nzation [19], in sharp contrast to the conventional MO effects.\nThe above discoveries can be mainly ascribed to chiral spin\ntextures created by magnetic atoms. However, the role of non-\nmagnetic atoms that is related to crystal chirality has not been\ncompletely understood till now. In this work, using the first-\nprinciples calculations and group theory analyses, we reveal\n\u0003wxfeng@bit.edu.cn\nyygyao@bit.edu.cna new class of MO effects in collinear AFMs, termed crystal\nchirality magneto-optical (CCMO) effects. The MO Kerr and\nFaraday spectra change their signs when the crystal structure’s\nchirality alters from the left-handed state to the right-handed\nstate or vice versa. Moreover, the magnitudes of MO spectra\ncan be effectively manipulated by reorienting the N ´eel vec-\ntor with the help of an external electric field. The spectral\nintegrals are found to be proportional to magnetocrystalline\nanisotropy energy (MAE), which has not been previously re-\nported in AFMs. The CCMO effects discovered here may\nbe a promising optical means to simultaneously detect crys-\ntal structure’s chirality and N ´eel vector’s orientation.\nThe crystal structure’s chirality is generated from differ-\nent arrangements of nonmagnetic atoms by keeping the struc-\nture’s space group unchanged. The crystal chirality manifests\nin many magnetic materials with different dimensions, e.g.,\nthree-dimensional (3D) rutile RuO 2[20] andMF2(M=Mn,\nNi) [21–23], quasi-2D NNb3S6(N=V , Cr, Mn, Fe, Co,\nNi) [20, 24, 25], and 2D SrRuO 3monolayer [26]. Here, we\ntake RuO 2and CoNb 3S6as two prototypes, which has a local\nand global crystal chirality, respectively.\nRutile RuO 2has long been considered to be a Pauli param-\nagnet [27], while several recent studies revealed a collinear\nantiferromagnetic order under room temperature [28–30].\nRuO 2is a centrosymmetric system (crystallographic space\ngroupP42=mnm ) with inversion centers at magnetic Ru\natoms, therefore, it can not be regarded as a usual chiral\ncrystal [31, 32]. However, one can still define a local crys-\ntal chirality for RuO 2[20], given by \u001f=dRu1O\u0002dORu2\n(dRu1O anddORu2 are two vectors connecting the path Ru1-O-\nRu2), because nonmagnetic O atoms are not inversion centers.\nEvidently, RuO 2has two opposite chiral states, i.e., right-\nhanded chirality \u001f= +1 [Fig. 1(a)] and left-handed chiral-\nity\u001f=\u00001[Fig. 1(b)]. The two chiral states are energeti-\ncally degenerate, but their magnetic densities are redistributed\nby rotating 90 degrees. Interestingly, one chiral state is re-\nlated to the other one via a combined symmetry Tt1=2, where\nt1=2= [0:5;0:5;0:5]is the half-unit cell translation. For each\nchiral state, the N ´eel vector,n= (nRu1\u0000nRu2)=2(nRu1and\nnRu2are the spins of two Ru atoms), can be reoriented byarXiv:2012.06693v1  [cond-mat.mtrl-sci]  12 Dec 20202\nFIG. 1. (a)(b) The crystal structures of RuO 2with the right-handed ( \u001f= +1 ) and left-handed ( \u001f=\u00001) chiralities. Pink and cyan spheres\nrepresent two magnetic Ru atoms (Ru1 and Ru2), whereas yellow spheres are nonmagnetic O atoms. The dashed lines outline the octahedron\nformed by six O atoms. The magnetic density isosurfaces surrounding two Ru atoms differ by 90 degrees. Pink and cyan arrows label the\nspin orientations of two Ru atoms, and the N ´eel vector is schematically plotted along the [110] direction. Two chiral structures are related\nto each other by the symmetry Tt1=2. (c)(d) Real and imaginary parts of the diagonal element and (e)(f) real and imaginary parts of the\noff-diagonal element of optical conductivity for \u001f=\u00061states of RuO 2when the N ´eel vectornrotates within the (001) plane ( \u0012= 90\u000eand\n0\u000e6'690\u000e). The inset in (c) shows the spherical coordinates ( \u0012,') used for describing the N ´eel vectorn. (g) The momentum-resolved\n\u001b00\nyzfor the\u001f= +1 state when'= 0\u000e,45\u000e, and 90\u000e, at the incident photon energy of 2.31 eV that is marked by an arrow in (f).\nthe current-induced spin-orbit field, like in CuMnAs [33, 34].\nOur first-principles calculations [35] show that the N ´eel vec-\ntor of RuO 2points along the [001] direction with a MAE of\n2.76 meV/Ru atom, which accords well with the previous the-\noretical results [20, 28]. While the resonant X-ray scattering\ndata indicate that the N ´eel vector slightly cants from the [001]\ndirection and has nonzero [100] and [010] components [29].\nSuch a generic magnetization direction suggests that the N ´eel\nvector can be easily tuned, giving rise to the coupling between\nthe crystal chirality \u001fand N ´eel ordern.\nThe calculation of optical conductivity is the crucial step\nto obtain the MO Kerr and Faraday spectra [35]. For conve-\nnience, the vector-form notation of optical Hall conductivity,\n\u001b= [\u001byz;\u001bzx;\u001bxy] = [\u001bx;\u001by;\u001bz], is used here. Since \u001bcan\nbe regarded as a pseudovector, like the spin, its nonvanishing\ncomponents can be identified by acting on each group element\nof relevant magnetic groups [14]. And the results of symmetry\nanalyses for\u001bcan be directly applied to the Kerr and Faraday\nangles,\u001eK= [\u001ex\nK;\u001ey\nK;\u001ez\nK]and\u001eF= [\u001ex\nF;\u001ey\nF;\u001ez\nF](see\nEqs. (S6) and (S7) in Ref. [35]).\nFirst, we prejudge the nonvanishing components of \u001bby\nconsidering the N ´eel vectornlying on the (100), (010), and\n(001) planes. When npoints along the [001] direction, RuO 2\nhas a magnetic space group P420=mnm0, which contains two\nglide mirror planes M[100]t1=2andM[010]t1=2. The mir-\nrorM[100] changes the signs of \u001byand\u001bz, but preserves\n\u001bx; the half-unit cell translation t1=2plays a nothing role on\n\u001b; thus, the combined operation M[100]t1=2restricts opti-cal Hall conductivity to be a shape of \u001b= [\u001bx;0;0]. Sim-\nilarly, another glide mirror plane M[010]t1=2gives rise to\n\u001b= [0;\u001by;0]. Therefore, under the group P420=mnm0,\nthe optical Hall conductivity is zero, \u001b= [0;0;0]. When\nndeviates away from the [001] direction and rotations within\nthe (100) and (010) planes, the symmetries M[010]t1=2and\nM[100]t1=2are absent, giving rise to [\u001bx;0;0]and[0;\u001by;0],\nrespectively. Within the (001) plane, the magnetic space group\nalways contains the symmetry TM [001], which changes the\nsign of\u001bzbut preserves \u001bxand\u001by. Therefore, \u001bxand\u001by\nare potentially nonzero. In particular, if npoints along the\n[100] ([010]) direction, only \u001by(\u001bx) can be nonzero due to\nthe role ofM[010]t1=2(M[100]t1=2). To summarize, once n\ncants slightly from the [001] direction, the optical Hall con-\nductivity and the MO spectra turn to be nonzero.\nFrom the symmetry point of view, the effect of crystal chi-\nrality on\u001bdiffers from that of spin chirality. In noncollinear\nAFMs Mn 3ZN (Z=Ga, Zn, Ag, Ni) [14], the reversal of\nspin chirality changes nonvanishing components and magni-\ntudes of\u001b. While two crystal chirality states of RuO 2are\nrelated by the symmetry Tt1=2, only the sign of \u001bwill be\nchanged as\u001bis odd underT. Thus, the sign change can also\nbe achieved by reversing the N ´eel vectorn. In fact, the sign\nof\u001bfor RuO 2is determined by the sign of n\u0001\u001f.\nNext, we discuss the optical conductivity calculated from\nthe first-principles methods. In light of a recently experimen-\ntal fabrication of the [001]-orientated RuO 2thin films with\nin-plane magnetization [30], we pay our attention to the n3\nlying within the (001) plane. Figs. 1(c) and 1(d) display the\ndiagonal element of optical conductivity, \u001b0= (\u001byy+\u001bzz)=2,\nfor two chiral states of RuO 2as a function of '. The real part\n\u001b0\n0measures the average in the absorption of left- and right-\ncircularly polarized light. It has two absorptive peaks at 1.5\neV and 3.5 eV and diverges in the low-frequency region due\nto the inclusion of Drude term [35]. The imaginary part \u001b00\n0\ncan be obtained from \u001b0\n0by using Kramer-Kronig transforma-\ntion [36]. Figs. 1(c) and 1(d) shows that \u001b0is robust against\nboth the crystal chirality and N ´eel vector.\nIn contrast, the off-diagonal elements of optical conductiv-\nity are substantially affected by the crystal chirality and N ´eel\nvector. Ifnrotates within the (001) plane, two off-diagonal\nelements,\u001byzand\u001bzx, are nonzero. The evolution of \u001byz\nwith\u001fandnis plotted in Figs. 1(e) and 1(f), whereas the\nresults of\u001bzxare shown in Figs. S1(c) and S1(d) [35]. The\nsigns of\u001byzand\u001bzxare changed for \u001f= +1 and\u00001states\nrelated by the symmetry Tt1=2. Ifnpoints along the [100]\ndirection (\u0012= 90\u000eand'= 0\u000e),\u001byzis zero for both two\nchiral states due to the symmetry M[010]t1=2. By increasing\n'from 0\u000eto90\u000e, the magnitude of \u001byzgradually increases\nfor both two chiral states, which can be well accounted for the\nmomentum-resolved optical Hall conductivity [Fig. 1(g)].\nNow, we proceed to the CCMO effects for collinear AFM\nRuO 2, as depicted in Fig. 2. The Kerr and Faraday spectra\n(\u001ex\nK=#x\nK+i\"x\nKand\u001ex\nF=#x\nF+i\"x\nF) exhibit a sim-\nilar profile to the off-diagonal elements of optical conduc-\ntivity, only differing by a minus [37]. The reason is simple\nas the off-diagonal elements of optical conductivity dominate\nthe shape of MO spectra, while the diagonal elements medi-\nate the amplitude of MO spectra (see Eqs. (S6) and (S7) in\nRef. [35]). The reversal of the crystal chirality changes the\nsigns of the Kerr and Faraday spectra but retains their mag-\nnitudes. If '= 0\u000e,#x\nK;F and\"x\nK;F are zero due to the\nvanishing\u001bx(=\u001byz)[see Figs. 1(e) and 1(f)]. In the range\nof0\u000e< '690\u000e, the magnitudes of #x\nK;F and\"x\nK;F in-\ncrease monotonously with the increasing of '. An opposite\ntrend appears to #y\nK;F and\"y\nK;F (see Fig. S2). It indicates\nthat the magnitudes of CCMO effects can be effectively tuned\nby changing magnetization direction in collinear AFMs. The\nlargest Kerr and Faraday rotation angles of RuO 2are 0.62 deg\nand 2.42\u0002105deg/cm, respectively. Particularly, the Kerr ro-\ntation angle is larger than that of traditional ferromagnets, e.g.,\nbcc Fe (\u00180.6 deg) [38], hcp Co ( \u00180.48 deg) [39], and fcc Ni\n(\u00180.15 deg) [40], and is also larger than that of famous non-\ncollinear AFMs, e.g., Mn 3X(\u00180.6 deg) [9], Mn 3Y(Y=\nGe, Ga, Sn) (\u00180.02 deg) [10], and Mn 3ZN (\u00180.42 deg) [14].\nThe large CCMO effects discovered in RuO 2suggests a novel\ncollinear antiferromagnetic platform for possible applications\nin MO recording beyond transitional ferromagnets [5].\nSince the CCMO spectra are frequency-dependent quanti-\nties, additional information can be captured from their spec-\ntral integrals (SIs), defined as SI\r\nK=R1\n0+#\r\nK(!)d!and\nSI\r\nF=R1\n0+#\r\nF(!)d!with\r=fx;y;zg. Figs. 3(a) and 3(b)\nshow that when the N ´eel vectornrotates within the (100)\nplane, SIx\nKand SIx\nFare nonvanishing and their signs are oppo-\nsite for two chiral states, which can be well understood from\nthe symmetry requirements of #x\nKand#x\nF. Moreover, SIx\nK\nFIG. 2. (a)(b) Kerr rotation angle and ellipticity and (c)(d) Faraday\nrotation angle and ellipticity for the left- and right-handed chirality\nstates (\u001f=\u00061) of RuO 2when the N ´eel vectornrotates within the\n(001) plane ( \u0012= 90\u000eand0\u000e6'690\u000e). The arrows in (a) and (c)\nmark the maximums of the Kerr and Faraday rotation angles at the\nphoton energies of 2.31 and 3.53 eV , respectively.\nand SIx\nFexhibit a period of 2\u0019with\u0012, which is simply due to\n#x\nKand#x\nFare odd under time-reversal symmetry. The MAE\nof two chiral states are equivalent and have a discrete two-fold\ndegeneracy, MAE (\u0012) = MAE (\u0012+\u0019). The most interesting\nfinding is that the absolute values of SIs are proportional to\nthe MAE, i.e.,jSIx\nK;Fj/MAE, which has not been reported\nin AFMs. For the (010) plane, the nonvanishing SIy\nK;F dis-\nplay the same behaviors [Figs. 3(c) and 3(d)]. For the (001)\nplane, both SIx\nK;Fand SIy\nK;Fare nonvanishing and the MAE\nhas a period of\u0019\n2[Figs. 3(e) and 3(f)]. Nevertheless, the SIs\nare still proportional to the MAE if the SIs on the (001) plane\nare redefined as SI0\nK=R1\n0+f[#x\nK(!)]2+[#y\nK(!)]2g1=2d!and\nSI0\nF=R1\n0+f[#x\nF(!)]2+[#y\nF(!)]2g1=2d![Figs. 3(g) and 3(h)].\nFig. 3 reveals that the SIs are closely related to the MAE for\ncollinear AFMs, and the sign and size of SIs can be used to\nidentify the crystal structure’s chirality and the N ´eel vector’s\ndirection, respectively.\nFinally, we demonstrate that the CCMO effects exist also in\ncollinear AFMs with a global crystal chirality, e.g., CoNb 3S6.\nThe crystallographic space group of CoNb 3S6isP6322,\nwhich is compatible with chiral crystal structures due to the\nlacking of improper symmetry operations [31, 32]. As shown\nin Figs. 4(a) and 4(b), the left- and right-handed chiral struc-\ntures are related to each other by a mirror plane Mx, which\npreserves the positions of magnetic atoms and the orientations\nof spin magnetic moments but rearranges the nonmagnetic S\natoms. CoNb 3S6forms a collinear antiferromagnetic order\nbelow the N ´eel temperature of \u001825 K [41–43]. The mag-\nnetic space group of CoNb 3S6isC202021if two ferromagnetic\nCo layers magnetized along the [100] direction are antifer-\nromagnetically coupled along the [001] direction via a NbS 2\nlayer. The operation C2zt1=2z(t1=2z= [0;0;0:5]) forces the\ncomponents of optical Hall conductivity that are perpendic-\nular to theC2axis to be zero, and therefore gives rise to\n\u001b= [0;0;\u001bz]. As a result, only the zcomponents of Kerr4\nFIG. 3. The SIs (colored symbols and lines, left axes) and MAE (shaded regions, right axes) for the left- and right-handed chirality states\n(\u001f=\u00061) of RuO 2when the N ´eel vectornrotates within the (100), (010), and (001) planes. For the MAE on the (100)/(010) and (001)\nplanes, the total energies when npoints to the [001] and [100] directions are set to be the reference states, respectively. The MAE on the (001)\nplane are multiplied by a factor of 3. The SIs shown in (g) and (h) are subtracted by the ones when npoints to the [100] direction.\nFIG. 4. (a)(b) The crystal structures of CoNb 3S6with the right-\nhanded (\u001f= +1 ) and left-handed ( \u001f=\u00001) crystal chiralities. The\nviolet, green, and bronze spheres represent Co, Nb, and S atoms, re-\nspectively. The red arrows label the spin orientations of magnetic Co\natoms (nCo1andnCo2). The N ´eel vector,n= (nCo1\u0000nCo2)=2, is\nalong the [100] direction. (c)(d) Kerr and Faraday rotation angles for\nthe left- and right-handed chirality states of CoNb 3S6.and Faraday rotation angles, #z\nKand#z\nF, are nonvanishing, as\nshown in Figs. 4(c) and 4(d). It is obvious that the MO effects\nfor CoNb 3S6are chiral in the sense that the signs of #z\nKand\n#z\nFare reserved for the left- and right-handed crystal struc-\ntures. The reason is simple because the mirror symmetry Mx\nthat relates two chiral crystal structures changes the signs of\n#z\nKand#z\nF. The CCMO effects uncovered in CoNb 3S6are\ncharacterized by the sign change resulting from the reversal\nof crystal chirality, similarly to the scenario of RuO 2.\nIn summary, using the first-principles calculations together\nwith group theory analyses, we have demonstrated the electri-\ncally controllable CCMO effects in collinear AFMs. The sign\nand magnitude of CCMO spectra can be tuned by the crys-\ntal chirality and spin orientation, respectively. Moreover, the\nspectral integrals are found to be proportional to the magne-\ntocrystalline anisotropy energy. The electrically controllable\nCCMO effects may provide a powerful probe of crystal chi-\nrality and spin orientation and be a promising MO recording\ntechnique based on antiferromagnetic materials.\nW.F., X.Z, and Y .Y . acknowledge the support from the Na-\ntional Key R&D Program of China (Nos. 2020YFA0308800\nand 2016YFA0300600), the National Natural Science Foun-\ndation of China (Nos. 11874085 and 11734003), and the\nGraduate Technological Innovation Project of Beijing Insti-\ntute of Technology (No. 2019CX10018). G.-Y . Guo acknowl-\nedges the support from the Ministry of Science and Technol-\nogy, National Center for Theoretical Sciences, and the Far\nEastern Y . Z. Hsu Science and Technology Memorial Foun-\ndation in Taiwan.\n[1] J. Kerr, On Rotation of the Plane of Polarization by Reflection\nfrom the Pole of a Magnet, Phil. Mag. 3, 321 (1877).[2] M. Faraday, I. Experimental Researches in Electricity.–\nNineteenth Series, Phil. Trans. R. Soc. Lond. 136, 1 (1846).5\n[3] P. N. Argyres, Theory of the Faraday and Kerr Effects in Ferro-\nmagnetics, Phys. Rev. 97, 334 (1955).\n[4] W. Reim and J. Schoenes, Handbook of Magnetic Materials,\nedited by E. P. Wohlfarth and K. H. J. Buschow, V ol. 5 (Elsevier,\nNew York, 1990) Chap. 2.\n[5] M. Mansuripur, The Physical Principles of Magneto-optical\nRecording (Cambridge University, New York, 1995).\n[6] H. Ebert, Magneto-Optical Effects in Transition Metal Systems,\nRep. Prog. Phys. 59, 1665 (1996).\n[7] V . Antonov, B. Harmon, and A. Yaresko, Electronic Structure\nand Magneto-Optical Properties of Solids (Kluwer Academic\nPublishers, Dordrecht, 2004) Chap. 1.4.\n[8] W. Kuch, R. Sch ¨afer, P. Fischer, and F. U. Hillebrecht, Mag-\nnetic Microscopy of Layered Structures (Springer-Verlag Berlin\nHeidelberg, 2015) Chap. 2.\n[9] W. Feng, G.-Y . Guo, J. Zhou, Y . Yao, and Q. Niu, Large\nMagneto-Optical Kerr Effect in Noncollinear Antiferromagnets\nMn3X(X= Rh;Ir;Pt), Phys. Rev. B 92, 144426 (2015).\n[10] T. Higo, H. Man, D. B. Gopman, L. Wu, T. Koretsune, O. M. J.\nvan’t Erve, Y . P. Kabanov, D. Rees, Y . Li, M.-T. Suzuki,\nS. Patankar, M. Ikhlas, C. L. Chien, R. Arita, R. D. Shull,\nJ. Orenstein, and S. Nakatsuji, Large Magneto-Optical Kerr\nEffect and Imaging of Magnetic Octupole Domains in an Anti-\nferromagnetic Metal, Nat. Photonics 12, 73 (2018).\n[11] A. L. Balk, N. H. Sung, S. M. Thomas, P. F. S. Rosa, R. D.\nMcDonald, J. D. Thompson, E. D. Bauer, F. Ronning, and\nS. A. Crooker, Comparing the Anomalous Hall Effect and the\nMagneto-Optical Kerr Effect through Antiferromagnetic Phase\nTransitions in Mn 3Sn, Appl. Phys. Lett. 114, 032401 (2019).\n[12] M. Wu, H. Isshiki, T. Chen, T. Higo, S. Nakatsuji, and Y . Otani,\nMagneto-Optical Kerr Effect in a Non-Collinear Antiferromag-\nnet Mn 3Ge, Appl. Phys. Lett. 116, 132408 (2020).\n[13] S. Wimmer, S. Mankovsky, J. Min ´ar, A. N. Yaresko, and\nH. Ebert, Magneto-Optic and Transverse-Transport Properties\nof Noncollinear Antiferromagnets, Phys. Rev. B 100, 214429\n(2019).\n[14] X. Zhou, J.-P. Hanke, W. Feng, F. Li, G.-Y . Guo, Y . Yao,\nS. Bl ¨ugel, and Y . Mokrousov, Spin-order Dependent Anoma-\nlous Hall Effect and Magneto-Optical Effect in the Non-\ncollinear Antiferromagnets Mn3XNwithX= Ga , Zn, Ag,\nor Ni, Phys. Rev. B 99, 104428 (2019).\n[15] N. Sivadas, S. Okamoto, and D. Xiao, Gate-Controllable\nMagneto-Optic Kerr Effect in Layered Collinear Antiferromag-\nnets, Phys. Rev. Lett. 117, 267203 (2016).\n[16] F.-R. Fan, H. Wu, D. Nabok, S. Hu, W. Ren, C. Draxl, and\nA. Stroppa, Electric-Magneto-Optical Kerr Effect in a Hybrid\nOrganic–Inorganic Perovskite, J. Am. Chem. Soc. 139, 12883\n(2017).\n[17] K. Yang, W. Hu, H. Wu, M.-H. Whangbo, P. G. Radaelli,\nand A. Stroppa, Magneto-Optical Kerr Switching Properties of\n(CrI 3)2and (CrBr 3/CrI 3) Bilayers, ACS Appl. Electron. Mater.\n2, 1373 (2020).\n[18] A. De, T. K. Bhowmick, and R. Lake, Anomalous Magneto\nOptic Effects from an Antiferromagnet Topological-Insulator\nHeterostructure, arXiv:2006.02752 .\n[19] W. Feng, J.-P. Hanke, X. Zhou, G.-Y . Guo, S. Bl ¨ugel,\nY . Mokrousov, and Y . Yao, Topological Magneto-Optical Ef-\nfects and Their Quantization in Noncoplanar Antiferromagnets,\nNat. Commun. 11, 118 (2020).\n[20] L. ˇSmejkal, R. Gonz ´alez-Hern ´andez, T. Jungwirth, and\nJ. Sinova, Crystal Time-Reversal Symmetry Breaking and\nSpontaneous Hall Effect in Collinear Antiferromagnets, Sci.\nAdv. 6, eaaz8809 (2020).\n[21] S. M. Wu, W. Zhang, A. KC, P. Borisov, J. E. Pearson, J. S.Jiang, D. Lederman, A. Hoffmann, and A. Bhattacharya,\nAntiferromagnetic Spin Seebeck Effect, Phys. Rev. Lett. 116,\n097204 (2016).\n[22] L.-D. Yuan, Z. Wang, J.-W. Luo, E. I. Rashba, and A. Zunger,\nGiant Momentum-Dependent Spin Splitting in Centrosymmet-\nric Low-ZAntiferromagnets, Phys. Rev. B 102, 014422 (2020).\n[23] X. Li, A. H. MacDonald, and H. Chen, Quantum\nAnomalous Hall Effect through Canted Antiferromagnetism,\narXiv:1902.10650 .\n[24] K. Anzenhofer, J. V . D. Berg, P. Cossee, and J. Helle, The\nCrystal Structure and Magnetic Susceptibilities of MnNb 3S6,\nFeNb 3S6, CoNb 3S6and NiNb 3S6, J. Phys. Chem. Solids. 31,\n1057 (1970).\n[25] S. S. P. Parkin and R. H. Friend, 3 dTransition-Metal Interca-\nlates of the Niobium and Tantalum Dichalcogenides. I. Mag-\nnetic Properties, Philos. Mag. B 41, 65 (1980).\n[26] K. Samanta, M. Le ˇzai´c, M. Merte, F. Freimuth, S. Bl ¨ugel, and\nY . Mokrousov, Crystal Hall and Crystal Magneto-Optical Effect\nin Thin Films of SrRuO 3, J. Appl. Phys. 127, 213904 (2020).\n[27] W. D. Ryden and A. W. Lawson, Magnetic Susceptibility of\nIrO2and RuO 2, J. Chem. Phys 52, 6058 (1970).\n[28] T. Berlijn, P. C. Snijders, O. Delaire, H.-D. Zhou, T. A. Maier,\nH.-B. Cao, S.-X. Chi, M. Matsuda, Y . Wang, M. R. Koehler,\nP. R. C. Kent, and H. H. Weitering, Itinerant Antiferromag-\nnetism in RuO 2, Phys. Rev. Lett. 118, 077201 (2017).\n[29] Z. H. Zhu, J. Strempfer, R. R. Rao, C. A. Occhialini, J. Pelli-\nciari, Y . Choi, T. Kawaguchi, H. You, J. F. Mitchell, Y . Shao-\nHorn, and R. Comin, Anomalous Antiferromagnetism in\nMetallic RuO 2Determined by Resonant X-ray Scattering,\nPhys. Rev. Lett. 122, 017202 (2019).\n[30] Z. Feng, X. Zhou, L. ˇSmejkal, L. Wu, Z. Zhu, H. Guo,\nR. Gonz ´alez-Hern ´andez, X. Wang, H. Yan, P. Qin, et al. , Obser-\nvation of the Crystal Hall Effect in a Collinear Antiferromagnet,\narXiv: 2002.08712 .\n[31] H. Flack, Chiral and Achiral Crystal Structures, Helv. Chim.\nActa 86, 905 (2003).\n[32] M. Nespolo, M. I. Aroyo, and B. Souvignier, Crystallographic\nShelves: Space-Group Hierarchy Explained, J. Appl. Cryst. 51,\n1481 (2018).\n[33] P. Wadley, B. Howells, J. ˇZelezn ´y, C. Andrews, V . Hills,\nR. P. Campion, V . Nov ´ak, K. Olejn ´ık, F. Maccherozzi, S. S.\nDhesi, S. Y . Martin, T. Wagner, J. Wunderlich, F. Freimuth,\nY . Mokrousov, J. Kune ˇs, J. S. Chauhan, M. J. Grzybowski,\nA. W. Rushforth, K. W. Edmonds, B. L. Gallagher, and T. Jung-\nwirth, Electrical Switching of an Antiferromagnet, Science 351,\n587 (2016).\n[34] J. Godinho, H. Reichlov ´a, D. Kriegner, V . Nov ´ak, K. Olejn ´ık,\nZ. Ka ˇspar, Z. ˇSob´aˇn, P. Wadley, R. P. Campion, R. M. Otxoa,\nP. E. Roy, J. ˇZelezn ´y, T. Jungwirth, and J. Wunderlich, Electri-\ncally Induced and Detected N ´eel Vector Reversal in a Collinear\nAntiferromagnet, Nat. Commun. 9, 4686 (2018).\n[35] See Supplemental Material at http://link.aps.org/xxx, which in-\ncludes a detailed description of the calculation methods and\nsupplemental figures.\n[36] H. S. Bennett and E. A. Stern, Faraday Effect in Solids, Phys.\nRev.137, A448 (1965).\n[37] W. Feng, G.-Y . Guo, and Y . Yao, Tunable Magneto-Optical\nEffects in Hole-Doped Group-IIIA Metal-Monochalcogenide\nMonolayers, 2D Mater. 4, 015017 (2017).\n[38] G. Y . Guo and H. Ebert, Band-Theoretical Investigation of the\nMagneto-Optical Kerr Effect in Fe and Co Multilayers, Phys.\nRev. B 51, 12633 (1995).\n[39] G. Y . Guo and H. Ebert, Theoretical Investigation of The Orien-\ntation Dependence of The Magneto-Optical Kerr Effect in Co,6\nPhys. Rev. B 50, 10377 (1994).\n[40] P. van Engen, K. Buschow, and M. Erman, Magnetic Properties\nand Magneto-Optical Spectroscopy of Heusler Alloys Based\non Transition Metals and Sn, J. Magn. Magn. Mater. 30, 374\n(1983).\n[41] S. S. P. Parkin, E. A. Marseglia, and P. J. Brown, Magnetic\nStructure of Co 1=3NbS 2and Co 1=3TaS 2, J. Phys. C: Solid State\nPhys. 16, 2765 (1983).[42] N. J. Ghimire, A. S. Botana, J. S. Jiang, J. Zhang, Y .-S.\nChen, and J. F. Mitchell, Large anomalous Hall effect in the\nchiral-lattice antiferromagnet CoNb 3S6, Nat. Commun. 9, 3280\n(2018).\n[43] G. Tenasini, E. Martino, N. Ubrig, N. J. Ghimire, H. Berger,\nO. Zaharko, F. Wu, J. F. Mitchell, I. Martin, L. Forr ´o, and\nA. F. Morpurgo, Giant Anomalous Hall Effect in Quasi-Two-\nDimensional Layered Antiferromagnet Co1=3NbS 2, Phys. Rev.\nResearch 2, 023051 (2020)."    },    {        "title": "2012.11469v1.Magnetic_order_of_Dy___3____and_Fe___3____moments_in_antiferromagnetic_DyFeO___3___probed_by_spin_Hall_magnetoresistance_and_spin_Seebeck_effect.pdf",        "content": "Magnetic order of Dy3+and Fe3+moments in antiferromagnetic DyFeO 3probed by\nspin Hall magnetoresistance and spin Seebeck e\u000bect\nG. R. Hoogeboom1, T. Kuschel2, G.E.W. Bauer1;3, M. V. Mostovoy1, A. V. Kimel4and B. J. van Wees1\n1Physics of Nanodevices, Zernike Institute for Advanced Materials,\nUniversity of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands.\n2Center for Spinelectronic Materials and Devices, Department of Physics,\nBielefeld University, Universit atsstra \fe 25, 33615 Bielefeld, Germany.\n3AIMR & Institute for Materials Research, Tohoku University, Aoba-ku, Katahira 2-1-1, Sendai, Japan\n4Spectroscopy of Solids and Interfaces, Institute of Molecules and Materials,\nRadboud University Nijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands\n(Dated: December 22, 2020)\nWe report on spin Hall magnetoresistance (SMR) and spin Seebeck e\u000bect (SSE) in single\ncrystal of the rare-earth antiferromagnet DyFeO 3with a thin Pt \flm contact. The angular\nshape and symmetry of the SMR at elevated temperatures re\rect the antiferromagnetic order\nof the Fe3+moments as governed by the Zeeman energy, the magnetocrystalline anisotropy\nand the Dzyaloshinskii-Moriya interaction. We interpret the observed linear dependence of the\nsignal on the magnetic \feld strength as evidence for \feld-induced order of the Dy3+moments\nup to room temperature. At and below the Morin temperature of 50 K, the SMR monitors the\nspin-reorientation phase transition of Fe3+spins. Below 23 K, additional features emerge that\npersist below 4 K, the ordering temperature of the Dy3+magnetic sublattice. We conclude that\nthe combination of SMR and SSE is a simple and e\u000ecient tool to study spin reorientation phase\ntransitions and sublattice magnetizations.\nI. INTRODUCTION\nAntiferromagnets (AFMs) form an abundant class of\nmaterials that o\u000ber many advantages over ferromagnets\n(FMs) for applications in high-density magnetic logics\nand data storage devices. AFMs support high-frequency\ndynamics in the THz regime that allows faster writing of\nmagnetic bits compared to FMs. The absence of mag-\nnetic stray \felds minimizes on-chip cross-talk and allows\ndownsizing devices that are robust against magnetic per-\nturbations [1]. On the other hand, most magnetic detec-\ntion methods observe only the FM order. Recent devel-\nopments in the detection [2] and manipulation [3{5] of\nthe AFM order reveal its many opportunities.\nThe AFM DyFeO 3(DFO) belongs to a family of\nrare-earth transition metal oxides called orthoferrites\nthat display many unusual phenomena such as weak\nferromagnetism (WFM), spin-reorientation transitions,\nstrong magnetostriction, multiferroicity including a large\nlinear magnetoelectric e\u000bect [6]. Their magnetic prop-\nerties are governed by the spin and orbital momenta of\n4f rare-earth ions coupled to the magnetic moment of 3d\ntransition metal ions.\nThe magnetization of dielectrics can be detected elec-\ntrically by the spin Hall magnetoresistance (SMR) in\nheavy metal contacts with a large spin Hall angle such\nas Pt [7]. This phenomenon is sensitive to FM, but also\nAFM spin order [2, 8{10]. With a Pt contact, information\nabout AFMs can be also retrieved by the spin Seebeck\ne\u000bect (SSE) under a temperature gradient [11{13].\nHere, we track the \feld-dependence of the coupled\nDy3+and Fe3+magnetic order as a function of tem-perature by both SMR and SSE. A su\u000eciently strong\nmagnetic \feld in the abplane of DFO forces the N\u0013 eel\nvector to follow a complex path out of the abplane. A\ntheoretical spin model explains the observations in terms\nof Fe3+spin rotations that are governed by the compe-\ntition between the magnetic anisotropy, Zeeman energy,\nand Dzyaloshinskii-Moriya interaction (DMI). The Dy3+\nmoments are disordered at room temperature but nev-\nertheless a\u000bect the magnitude of the SMR. At the so-\ncalled Morin phase transition at ∼50 K the Fe3+spins\nrotate by 90○, causing a step-like anomaly in the SMR.\nAt even lower temperatures, we observe two separate fea-\ntures tentatively assigned to the re-orientation of Fe3+\nspins in an applied magnetic \feld and another related to\nthe ordering of Dy3+orbital moments. Below 23 K, the\nSMR signal is ∼1%, 1-2 orders of magnitude larger than\nreported for other materials [2, 7]. Both Fe3+and Dy3+\nmoments appear to contribute to the SSE; a magnetic\n\feld orders the Dy3+moments and suppresses the Fe3+\ncontribution. The complex SMR and SSE is evidence of\na coupling between the Fe3+and Dy3+magnetic subsys-\ntems.\nThe paper is organized as follows. In Section II we\nreview the magnetic and multiferroic properties of DFO.\nThe theory of the magnetic probing methods are dis-\ncussed in Sec. III with Subsec. III A the SMR and\nSubsec. III B the SSE. In Subsec. IV A, the fabrica-\ntion, characterization and measurement techniques are\nexplained. Further, a model including the DMI, Zeeman\nenergy and magnetic anisotropy is employed in Subsec.\nIV B. The SMR results at elevated temperatures includ-\ning the model \fts as well as SMR and SSE results at lowarXiv:2012.11469v1  [cond-mat.mtrl-sci]  21 Dec 20202\nDy3+\nFe3+\nO2-\n500/uni03BCm\nTi/Au (5/50nm)\nHy\nxa) b)\nPt (8nm)\naab b\ndevice 1 device 2\nzDyFeO/three.denominator (1mm)\nFIG. 1. (a) DFO crystal unit cell. The blue, red and\nwhite spheres represent Dy3+, Fe3+and O2−ions, respectively.\n(b) Optical image of the Pt Hall bar on top of the bulk DFO\ncrystal. The lines indicate voltage probes, AC source and an-\ngle\u000bof the external magnetic \feld H. In the two devices,\nthe crystallographic directions aandbare rotated by 45○in\nthexyplane, the reference frame of the Hall bar.\ntemperatures are described and discussed in Sec. V.\nII. MAGNETIC AND MULTIFERROIC\nPROPERTIES OF DFO\nDFO is a perovskite with an orthorhombic ( D16\n2h-\nPbnm) crystallographic structure. It consists of alter-\nnating Fe3+and Dy3+abplanes, in which the Fe ions\nare located inside O2−octahedrons (Fig. 1a)). The\nlarge Dy3+magnetic moments ( J=15/slash.left2) order at a\nlow temperature, TDy\nN=4 K. The high N\u0013 eel tempera-\ntureTFe\nN=645 K indicates strong inter- and intra-plane\nAFM Heisenberg superexchange between the Fe3+mag-\nnetic moments ( S=5/slash.left2). The AFM order of the Fe mo-\nments is of the G-type with N\u0013 eel vector G(anti)parallel\nto the crystallographic aaxis (\u0000 4symmetry [14]). The\nbroken inversion symmetry enables a DMI [15, 16] that\nin the \u0000 4-phase causes a WFM mWFM/parallel.alt1cby the small\n(∼0:5○) canting of the Fe spins [14].\nA \frst-order Morin transition from the WFM \u0000 4-phase\nto the purely AFM \u0000 1-phase occurs when lowering the\ntemperature below 50 K. At this transition, the direction\nof the magnetic easy axis abruptly changes from the a-\nto theb-direction. A magnetic \feld higher than a critical\nmagnetic \feld, Hcr, along the caxis re-orients the N\u0013 eel\nvector back to the aaxis and recovers the \u0000 4-phase.\nBelowTDy\nN, the Dy3+moments form a noncollinear\nIsing-like AFM order with Ising axes rotated by ±33○\nfrom thebaxis [17] that corresponds to a G′\naA′\nbstate in\nBertaut's notation [18]. The simultaneous presence of\nordered Fe and Dy magnetic moments breaks inversion\nsymmetry and, under an applied magnetic \feld, induces\nan electric polarization [19] by exchange striction that\ncouples the Fe and Dy magnetic sublattices [6, 20].\nHigher magnetic \felds destroy the AFM order of the\nDy3+moments and thereby the electric polarization [21].Spins in this material can be controlled by light\nthrough the inverse Faraday e\u000bect [3], as well as by tem-\nperature and magnetic \feld. Re-orientation of the Fe mo-\nments has been studied by magnetometry [22], Faraday\nrotation [23], M ossbauer spectroscopy [24] and neutron\nscattering measurements [21]. The Morin transition at\n50 K causes large changes in the speci\fc heat [25] and\nentropy [26].\nIII. PROBING METHODS\nA. Spin Hall magnetoresistance\nThe SMR is caused by the spin-charge conversion in\na thin heavy metal layer in contact with a magnet [27].\nThe spin Hall e\u000bect induces a spin current transverse to\nan applied charge current and thereby an electron spin\naccumulation at surfaces and interfaces. Upon re\rection\nat the interface to a magnetic insulator, electrons expe-\nrience an exchange interaction that depends on the an-\ngle between their spin polarization and that of the in-\nterface magnetic moments, while the latter can be con-\ntrolled by an applied magnetic \feld. The re\rected spin\ncurrent is transformed back into an observable charge\ncurrent by the inverse spin Hall e\u000bect. The interface ex-\nchange interaction is parameterized by the complex spin\nmixing conductance. The result is a modulation of the\ncharge transport that depends on the orientation of the\napplied current and the interface magnetic order. In a\nHall bar geometry, this a\u000bects the longitudinal resistance\nand causes a planar Hall e\u000bect, i.e. a Hall voltage even\nwhen the magnetic \feld lies in the transport plane.\nSMR is a powerful tool to investigate the magnetic or-\ndering at the interface of collinear [7, 27{29] and non-\ncollinear ferrimagnets [30, 31] as well as spin spirals\n[32, 33]. Recently, a \\negative\" SMR has been discovered\nfor AFMs [2, 8{10], i.e. an SMR with a 90○phase shift\nof the angular dependence as compared to FMs, which\nshows that the AFM N\u0013 eel vector Gtends to align itself\nnormal to the applied magnetic \feld. The observable in\nAFMs is therefore the N\u0013 eel vector rather than the net\nmagnetization [2].\nThe longitudinal and transverse electrical resistivities\n\u001aLand\u001aTof Pt on an AFM read [2]\n\u001aL=\u001a+\u0001\u001a0+\u0001\u001a1(1−G2\ny) (1)\n\u001aT=\u0001\u001a1GxGy+\u0001\u001a2mz+\u0001\u001aHallHz (2)\nwithGiandHiwithi∈{x;y;z}as the Cartesian com-\nponents of the (unit) N\u0013 eel and the applied magnetic \feld\nvectors, respectively. mzis the out-of-plane (OOP) com-\nponent of the unit vector in the direction of the WFM\nmagnetization. \u0001 \u001a0is an angle-independent interface\ncorrection to the bulk resistivity \u001a. \u0001\u001aHallHzis the ordi-\nnary Hall resistivity of Pt in the presence of an OOP com-\nponent of the magnetic \feld. \u0001 \u001a1(\u0001\u001a2)is proportional3\nto the real (imaginary) part of the interface spin-mixing\nconductance. \u0001 \u001a2is a resistance induced by the e\u000bective\nWFM \feld, believed to be small in most circumstances.\nThe interface Dy3+moments can contribute to the\nSMR when ordered. Below TDy\nN, the Dy3+moments are\nAFM aligned with N\u0013 eel vector GDy. AboveTDy\nNand\nin su\u000eciently large applied magnetic \felds, the Dy3+\nmoments contribute to the SMR in Eqs. (1,2) after\nreplacing the N\u0013 eel vector GDyby the (nearly perpen-\ndicular) magnetization mDy. Disregarding magnetic\nanisotropy and DMI for the moment, the spin mixing\nconductance term \u0001 \u001a1mDy\nxmDy\nyphase-shifts the SMR\nby 90○relative to the pure AFM contribution. The\nterm \u0001\u001a2mzchanges sign with mzand its contribution\n∼Hzcannot be distinguished from the ordinary Hall\ne\u000bect \u0001\u001aHallHzin Pt. We remove a linear magnetic\n\feld dependence from the OOP SMR measurements.\nResidual non-linear e\u000bects from \u0001 \u001a2mzmay persist, but\nshould be small in the \u0000 4phase. A \fnite \u0001 \u001a2mzhas\nbeen reported in conducting AFMs [34], but we do not\nobserve a signi\fcant contribution down to 60 K.\nB. Spin Seebeck e\u000bect\nA heat current in a FM excites a spin current that in\ninsulators is carried mainly by magnons, the quanta of\nthe spin wave excitations of the magnetic order. We can\ngenerate a temperature bias simply by the Joule heat-\ning of a charge current in a metal contact. A magnon\n\rowjmcan also be generated by a gradient of a magnon\naccumulation or chemical potential \u0016m[35]. Therefore\njm=−\u001bm(∇\u0016m+SS∇T) (3)\nwith\u001bmas the magnon spin conductivity and SSthe spin\nSeebeck coe\u000ecient. Thermal magnons can typically dif-\nfuse over several \u0016m [36{38], which implies that the SSE\nmainly probes bulk rather than interface magnetic prop-\nerties. The magnons in simple AFMs typically come in\ndegenerate pairs with opposite polarization that split un-\nder an applied magnetic \feld [11, 39]. The associated im-\nbalance of the magnon populations cause a non-zero spin\nSeebeck e\u000bect [13]. Paramagnets display a \feld-induced\nSSE e\u000bect [38] for the same reason, so aligned Dy3+mo-\nments can contribute to an SSE in DFO. A magnon ac-\ncumulation at the interface to Pt injects a spin current js\nthat can be observed as an inverse spin Hall e\u000bect voltage\nVISHE=\u001a\u0012SH(js×\u001b), where\u0012SHis the spin Hall angle\nand\u001bis the spin polarization. The SMR and SSE can\nbe measured simultaneously by a lock-in technique [40].\nρT\na) b)α (deg)\nGaGc\nGb αρSMR (arbitrary units)\n-45 45 0 -90 902T6TFIG. 2. N\u0013 eel vector, G=(Ga;Gb;Gc)with/divides.alt0G/divides.alt0=1, calculated\nas a function of the magnetic \feld in the abplane. The angle\n\u000b∈[−90○;90○]as de\fned in Fig. 1(b) is coded by the colored\nbar.G(\u000b)minimizes the free energy Eq. (4), for Kb=0:15 K\nper Fe ion and H=6 T (other parameters are given in the\ntext). (b) The transverse SMR (arbitrary units) due to the\nmagnetic Fe sublattice for H=6 T, i.e. the G(\u000b)from panel\n(a) (thick red line), and for H=2 T (thin blue line).\nIV. METHODS\nA. Fabrication, characterization and measurements\nWe con\frmed the crystallographic direction of our sin-\ngle crystal by X-ray di\u000braction before sawing it into slices\nalong the abplane and polishing them. Two devices\nwere fabricated on di\u000berent slices of the materials using\na three step electron beam lithography process; markers\nwere created to align the devices along two di\u000berent crys-\ntallographic directions. After fabrication of an 8 nm thick\nPt Hall bar, 50 nm Ti/Au contact pads were deposited.\nThe angular dependence of the magnetoresistance be-\nlow 50 K is complex and hysteretic. Phase changes are\nassociated by internal strains that can cause cracks in\nthe bulk crystal. We therefore carried out magnetic \feld\nsweeps at low temperatures very slowly, with a waiting\ntime of 60 seconds between each \feld step. The response\nwas measured with a 1 mA (100 \u0016A) AC current through\nthe Pt Hall bar in device 1 (device 2) with a frequency of\n7.777 Hz. The \frst and second harmonic transverse and\nlongitudinal lock-in voltages as measured with a super-\nconducting magnet in a cryostat with variable tempera-\nture insert are the SMR and SSE e\u000bects, respectively.\nBelow the transition temperature, the Morin tran-\nsition is induced by a magnetic \feld along the caxis\nthat rotates the N\u0013 eel vector from atob. For device\n1, this does not change the transverse resistance since\nGFe\nxGFe\ny=0 when the N\u0013 eel vector is in either the x- or\ny-direction. On the other hand, device 2 is optimized\nfor the observation of the Morin transition, because,\nas discussed below, the transverse resistance should be\nmaximally positive when G/parallel.alt1band maximally negative\nwhen G/parallel.alt1a.4\nB. Modelling the SMR of Pt /divides.alt0DFO\nThe orientation of the N\u0013 eel vector Gof the Fe sublat-\ntice at temperatures well above TDy\nNis governed by sev-\neral competing interactions: (a) the magnetic anisotropy,\nwhich above the Morin transition favors G/parallel.alt1a, (b) the\nZeeman energy that favors G⊥Hsince the transverse\nmagnetic susceptibility of an AFM is higher than the lon-\ngitudinal one, and (c) the coupling of the WFM moment,\nmWFM/parallel.alt1a, to the applied magnetic \feld. This compe-\ntition can be described phenomenologically by the free\nenergy density\nf=Kb\n2G2\nb+Kc\n2G2\nc+\u001f⊥\n2/bracketleft.alt1(G⋅H)2−H2/bracketright.alt−mWFMGcHa;\n(4)\nwith the \frst two terms describing the second-order mag-\nnetic anisotropy with magnetic easy, intermediate and\nhard axes along the a,bandccrystallographic direc-\ntions, respectively ( Kc>Kb>Ka=0),\u001f⊥is the trans-\nverse magnetic susceptibility, and the mWFM is the weak\nferromagnetic moment along the aaxis, induced by G/parallel.alt1c.\n/divides.alt0G/divides.alt0=1, because the longitudinal susceptibility of the Fe\nspins is very small for T/uni226ATFe\nN. The magnetic \feld His\nchosen parallel to the abplane, but Gcan have an OOP\ncomponent Gc≠0 since the third term in Eq.(4) couples\nGclinearly toHa. For the SMR at 250 K, we may dis-\nregard higher-order magnetic anisotropies that become\nimportant near the Morin transition.\nAt weak magnetic \felds, the magnetic anisotropy pins\nthe N\u0013 eel vector to the aaxis. When the Zeeman energy\nbecomes comparable with the anisotropy energy, the ro-\ntation of the magnetic \feld vector in the abplane gives\nrise to a concomitant rotation of G. In the absence of\nmagnetic anisotropy, the canting of the magnetic mo-\nments leads to G⊥Hfor any magnetic \feld orienta-\ntion due to the Zeeman energy rendering a sinusoidal\nSMR, but magnetic anisotropy can distort the angular\ndependence. This behavior is further complicated by the\nWFM: for strong magnetic \felds along the aaxis, the\nN\u0013 eel vector tilts away from the abplane towards the c\naxis, since the c-component of Ginduces a WFM mo-\nment parallel to the applied magnetic \feld [24, 41]. By\ncontrast,Gbdoes not give rise to a weak FM moment, so\nthe N\u0013 eel vector returns into the abplane when we rotate\nthe magnetic \feld away from the aaxis. The equilibrium\nN\u0013 eel vector minimizes the free energy Eq. (4) under the\nconstraint /divides.alt0G/divides.alt0=1 as a function of strength and orienta-\ntion of the magnetic \feld with in-plane (IP) angle \u000b(see\nFig. 1b)).\nWe adopt a weak magnetization parameters mWFM=\n0:133\u0016Bper Fe3+ion induced either by G/parallel.alt1calong the\naaxis [42] or by G/parallel.alt1aalong the caxis [43]. The\ntransverse magnetic susceptibility can be estimated us-\ning the Heisenberg model with an Fe-Fe exchange con-\nstantJ1=4:23 meV for Y 3Fe5O12[44] , which leads to\n\u001f⊥=\u00162\nB/slash.left(3J1), which does not depend strongly on the\nrare-earth ion. Kcgoverns the critical \feld when applied\nalong theaaxis with\u00160Hcr=9:3 T atT=270 K [24] that\nρSMR (arbitrary units)\nρSSE (arbitrary units)ρT\nρTa) b)\nα α0 0\n0 -90 90 180 -180 -45 45 0 -90 9010K\n10K250KFIG. 3. Calculated angular dependence of the transverse a)\nSMR (\u001aSMR\nT) and b) local SSE ( \u001aSSE\nT) as contributed by para-\nmagnetic Dy3+moments polarized by an applied \feld H=6 T.\nThe curve at 10 K (blue line) is calculated numerically using\nEq. 5. The 250 K curve (ampli\fed by a factor 100, red line)\nis obtained analytically from Eq. (6). Both SMR and SSE\ngrow with decreasing temperature and associated increasing\nDy3+magnetization.\nfully rotates Gfrom theato thecdirection.Kccan then\nbe estimated using Kc=mWFMHcr+\u001f⊥H2\ncr.Kbis the\nonly free temperature-dependent parameter that we \ft to\nthe \feld-dependent SMR. All other constants are taken\nto be independent of temperature. A typical calculated\ndependence of G(\u000b)and the corresponding contribution\nof the Fe spins to the SMR is shown in Fig. 2 (see below\nfor a more detailed discussion).\nOrdered rare-earth ions can also contribute to the\nSMR and SSE. The spectrum of the lowest-energy6H15/slash.left2\nmultiplet of the Dy3+ion (4f9electronic con\fguration)\nconsists of a Kramers doublet separated by \u0001 =52\ncm−1(≈75 K)from the \frst excited state [45]. At low\ntemperatures, kBT/uni226A\u0001, the Dy moments behave as\nIsing spins tilted by an angle ±\u001eDyaway from the a\naxis in the abplane (\u001eDy=57○). At high temperatures,\nkBT/uni226B\u0001, they can be described as anisotropic Heisen-\nberg spins with paramagnetic susceptibilities, \u001fDy\n∥/parenleft.alt1\u001fDy\n⊥/parenright.alt1\nfor a magnetic \feld parallel (perpendicular) to the local\nspin-quantization axis ( \u001fDy\n∥>\u001fDy\n⊥) [46].\nForkBT/uni226B\u0001, the SMR resulting from the contribu-\ntions of the four Dy sublattices (four Dy sites in the crys-\ntallographic unit cell of DFO) is\nRSMR\nT∝−A/bracketleft.alt1H2sin(2\u000b)−2Hg1Gcsin\u000b/bracketright.alt\n−2BHg 2Gcsin\u000b; (5)\nwhere the \frst term originates from the interaction of Dy\nspins with the applied magnetic \feld and the other two\nterms result from the exchange \feld induced by Fe spins\non Dy sites (for a more detailed discussion of the e\u000bective\nmagnetic \feld acting on Dy spins and the expressions\nfor A and B in terms of the magnetic susceptibilities of\nthe Dy ions see Appendix B). It can be inferred form\nFig. 2 a) that Gcis approximately proportional to cos \u000b.\nTherefore, all terms in Eq. 5 give the sin (2\u000b)dependence\nof the transverse SMR at high temperatures (thick red\nline in Fig. 3 a)). Equation (5) should be added to\nthe SMR caused by the iron sublattice with an unkown5\nweight that is governed by the mixing conductance of\nthe Dy sublattice. We may conclude however that an\nadditional sin (2\u000b)should not strongly change the shape\nof the SMR in Figure 2b).\nAt low temperatures, , T/uni226A\u0001/slash.leftkB, the Dy moments\nbehave as Ising spins. A rotation of the magnetic \feld\nin theabplane modulates the projection of the e\u000bective\nmagnetic \feld on the local spin-quantization axes of the\nfour Dy sublattices, which a\u000bects the angular dependence\nof the SMR. Since the paramagnetic model Eq. (5) can-\nnot be used anymore, we compute the Dy contribution to\nthe SMR ∼mxmynumerically for the rare-earth Hamil-\ntonian\nH(i)\nDy=gJ\u0016B(J⋅HDy)−K\n2(J⋅^zi)2; i=1;2;3;4;(6)\nwith Jas the Dy total angular momentum, gJ=4/slash.left3\nthe Land\u0013 e factor, K=\u0001/slash.left7 the anisotropy parameter,\nwhich is known to reasonably describe the low-energy\nexcited states of Dy ions and ^ziare the local easy axes\nrotated by +57○, for the Dy sublattices 1 and 3, and\n−57○, for the sublattices 2 and 4, away from the aaxis.\nThe magnetic \feld HDyacting on Dy spins is the sum\nof the applied \feld and the exchange \feld from Fe spins:\nHex=g1Gz^a±g2Gz^b, where the +/slash.left−is for the sublattices\n1;3 and 2;4, respectively. We neglect the ccomponent of\nthe exchange \feld, since the Dy magnetic moment along\nthecis small and does not a\u000bect the SMR. Using the\nHamiltonian Eq. (6), we calculate the average aandb\ncomponents of the magnetic moments of the 4 Dy sublat-\ntices at a temperature Tand the resulting contributions\nto SMR. The angular dependence of the SMR due to Dy\nspins is plotted in Fig. 3 a).\nThe calculations recover the sin (2\u000b)angular depen-\ndence of the SMR from Eq. (5) at high temperatures.\nAt 10 K (blue line) the SMR curve becomes strongly\ndeformed: The angular dependence of the SMR shows\npeaks and dips at the e\u000bective \feld directions orthog-\nonal to the quantization axis ^ziof thei-th rare-earth\nsublattice.\nFor long magnon relaxation time, the SSE generated\na spin current that is assumed to be proportional to\nthe bulk magnetization and can therefore provide addi-\ntional information. We focus here on the low tempera-\nture regime because we did not observe an SSE at ele-\nvated temperature, which is an indication that the Dy\nmagnetization plays an important role.\nA net magnetization of rare-earth moments a\u000bects the\nSSE signals in gadolinium iron [47] and gadolinium gal-\nlium [38] garnets. We assume that the SSE is dominated\nby a spin current from the bulk that is proportional to\nthe total magnetization mDy\nbof the four Dy sublattices\nthat we calculated for the Hamiltonian Eq. (6) at 10 K as\nfunction of the angle \u000bof the applied magnetic \feld. The\nmodel predicts peaks at magnetic \feld directions aligned\nwith the Ising-spin axes of the Dy moments, i.e. in be-\ntween those canted by ±33°, which enhances the mag-\nnetization. The contribution from the Fe sublattice to\nRT (mΩ)a) b)\nc) d)/gid00021 /gid00021 /gid00021 /gid00021 /gid00021 /gid00021 /gid00021 /gid00021RT (mΩ)\n∆R T/R0 (10-5)FIG. 4. (a) Transverse SMR (symbols) measured as a func-\ntion of IP magnetic \feld angle \u000band strength (indicated at\nthe top). The measurements are done on device 1 with a cur-\nrent of 1 mA at 250 K and the error bar \u0001 \u000bindicates a sys-\ntematic error due to a possible misalignment of the magnetic\n\feld direction as compared to the crystallographic axes. The\nlines are \fts obtained by adjusting Kbin the free energy model\nEq. (4). (b) The IP ( \u001e) and OOP ( \u0012) canting angles of the\nN\u0013 eel vector with respect to bas a function of the IP magnetic\n\feld direction from the \fts. (c) The maximal signal change\n\u0001RTrduring a magnetic \feld rotation depends linearily on\nthe magnetic \feld strength and (d) shows a power-law tem-\nperature dependence, \u0001 RTr/slash.leftR0∝(T)\u000f.R0is the sheet resis-\ntance obtained from the base resistance of the corresponding\nlongitudinal measurements adjusted by the geometrical fac-\ntor length/width of the Hall bar. These measurements are\ncarried out at 4 T.\nthe SSE is expected to depend as cos \u000bon the external\nmagnetic \feld direction [48]. The ratio of the Fe and Dy\ncontributions to SSE is unknown.\nV. RESULTS\nThe SMR was measured by rotating an IP magnetic\n\feld of various strengths. Temperature drift and noise\nswamped the small signal in the longitudinal resistance as\ndiscussed in Appendix A. Figure 4 a) shows the measured\nresistance of device 1 at 250 K in the transverse (planar\nHall) con\fguration using the left contacts in Fig. 1 b).\nThe results for the right Hall contacts (not shown) are\nvery similar.\nThe (negative) sign of the SMR agrees with our Fe sub-\nlattice model, suggesting that it is caused by the AFM or-\ndered Fe spins with N\u0013 eel vector Gnormal to the applied\nmagnetic \feld. However, Gcannot be strictly normal to\nthe magnetic \feld, because the SMR is not proportional\nto sin(2\u000b), as observed for example in NiO [2]. The\nstrongly non-sinusoidal angular dependence of the SMR6\na) b) c) device 2 IP device 2 OOP device 1 OOP\nFIG. 5. The relative changes in the transverse resistances RTr/slash.leftR0of (a) devices 1 and (b,c) device 2. A linear contribution\nfrom the ordinary Hall e\u000bect has been subtracted from the OOP data. O\u000bsets of the order of 10−4are removed and the curves\nare shifted with respect to each other for clarity. The magnetic \feld directions are (a,b) along zfor the OOP and (c) along\nyfor the IP con\fgurations. (a) The data for device 1 are expected to not change during the Morin transition. The observed\nSMR is symmetric with respect to current and magnetic \feld reversal and sensitive to Dy3+ordering. (b,c) Device 2 reveals the\nMorin transition by a positive step for weak magnetic \felds. Below 23 K, hysteretic resistance features emerge when sweeping\nthe \felds back and forth that vanishes at higher magnetic \felds and temperatures. The arrows indicate the magnetic \feld\nsweep directions, while the symbols highlight the critical magnetic \felds as summarized in Fig. 6.\nis evidence for a non-trivial path traced by the N\u0013 eel vec-\ntor in an applied magnetic \feld as predicted by the model\nEq. (4).\nFigure 2 a) shows the dependence of the three com-\nponentsGa,GbandGcof the N\u0013 eel vector on the IP\norientation angle \u000bof the magnetic \feld, for \u00160H=6\nT. The value of \u000b∈[−90○;90○]is indicated by the color\ncode side bar. When \u000b=0(H/parallel.alt1a), the magnetic \feld\ncauses a tilt of Gaway from the easy aaxis towards the\nhardcaxis since the N\u0013 eel vector parallel to the caxis\ninduces a magnetization along the aaxis. The excursion\nofGfrom theabplane e\u000bectively reduces the role of the\nIP magnetic anisotropy, which leads to a large rotation\nof the N\u0013 eel vector in the abplane for small \u000b(at nearly\nconstantGc). As explained above, this rotation is driven\nby the Zeeman energy of the AFM ordered Fe spins (the\nthird term in Eq.(4)), which favors G⊥Hand competes\nwith the magnetic anisotropy that favors G/parallel.alt1a(the \frst\nterm in Eq.(4)). This behavior is similar to the spin-\rop\ntransition for a magnetic \feld applied along the magnetic\neasy axis, except that Gdoes not become fully orthog-\nonal to the magnetic \feld. As the magnetic \feld vector\nrotates away from the aaxis,Gcand/divides.alt0Gb/divides.alt0decrease, and\nat\u000b=±90○,Gis parallel to the aaxis.\nThe sensitivity of Gto small\u000bgives rise to an abrupt\nchange of the transverse SMR that is proportional to\nGaGbclose to\u000b=0 (thick red line Fig. 2b). The calcu-\nlated and observed SMR scans agree well for T=250 K\nand\u00160H=6 T. Surprisingly, the shape of the experi-mental curves is practically the same at all magnetic\n\feld strengths, i.e. the SMR jumps at \u000b=0 even at\nweak \felds, while the calculation approach the geometri-\ncal sin(2\u000b)dependence (thin blue line in Fig. 2b) calcu-\nlated for\u00160H=2 T). The \fts of the observed SMR for\nall magnetic \felds require a strongly \feld-dependent IP\nanisotropy parameter Kbthat is very small in the zero\n\feld limit:Kb=(6±8)⋅10−6+(3:20±0:02)⋅10−3(H/slash.leftT)2\nK (see Fig. 4a). At present we cannot explain this be-\nhavior. The Dy3+moments should not play an important\nrole in this regime unless a Pt induced anisotropy at the\nDFO/Pt interface modi\fes their magnetism (see below).\nThe exchange coupling between the rare-earth and\ntransition-metal magnetic subsystems is re\rected by the\nsecond term in Eq.(5) of the Dy3+contribution to the\nSMR that is proportional to Gc, i.e. the AFM order of\nthe Fe spins. Since, Gcis a smooth function at \u000b=0, it\ncannot be hold responsible for the large zero-\feld magne-\ntoresistance. The angular SMR appears to be dominated\nby the N\u0013 eel vector Gof the Fe moments, in contrast to\nSmFeO 3, in which the Sm-ions determine not only the\namplitude but also the sign of the SMR [49].\nThe linear increase of the SMR with magnetic \feld\nstrength (see Fig. 4c)) can partly be explained by the\ngrowth of the maximum IP rotation angle, \u001e, of the N\u0013 eel\nvector with magnetic \feld. However, deviations from the\nlinear dependence are then expected close to the critical\nvalue,Ha∼9 T, at which the re-orientation transition\nfromG/parallel.alt1atoG/parallel.alt1cinH/parallel.alt1ais complete [24]. Nevertheless,7\nd1 OOP Dy\nd2 OOP Fe\nd2 OOP Dy\nd2 IP     Fe\n[24] M össbauer \n[21] neutron \n[21] M(T)       \nFe:GxAyFz\nDy:GxAy\nFIG. 6. Critical magnetic \felds Hcrof the observed transi-\ntions in the transverse resistance as a function of tempera-\nture. Symbols correspond to Fig. 5, where they denote the\nstep functions that trace the Morin transition in device 2. ▲\nindicates IP and ●OOP magnetic \feld directions. The lat-\nter symbol describes the peaks at lower temperatures as well.\nThe OOPHcrof the low magnetic \feld features are shown\nfor device 1 ( ▼) and device 2 ( /uni220E). The features for the IP\nmagnetic \feld directions are less pronounced and not shown.\nThe lines show a \ft by the function Hc∝(TM−T)\u000f, which\nis used to extract the ordering temperatures of 50 K and 23 K\nfor the Morin transition and a magnetic phase transition to\nan ordered Dy3+sublattice, respectively. Further data is from\nRefs. [21, 24], obtained by M ossbauer spectrometry ( /uni220E), neu-\ntron scattering ( /uni25C2) and magnetometry ( ▲).\nthe SMR signal shows no sign of saturation at \u00160Ha=6 T\nandT=250 K. The \u00160Hof Dy becomes of the order\nof kBT at a magnetic \feld strength of 37 T, indicating\ncontributions from the paramagnetic rare earth spins re-\nmains linear in the applied \feld strengths.\nFurther evidence for rare earth contributions at higher\ntemperatures is the Curie-like power-law temperature de-\npendence of the SMR (see Fig. 4d)) SMR ∼T\u000f, with\n\u000f=−1:24±0:04 at low temperatures and \u000f=−1:67±0:02\nat high temperatures.[50] For comparison, in the AFM\nNiO,\u000fis positive and the SMR signal grows quadrati-\ncally with the AFM order parameter [2]. At tempera-\ntures well below the N\u0013 eel transition TFe\nN=645 K, the Fe\nbased magnetic order is nearly temperature independent.\nThe strong magnetic \feld and temperature dependence\ntherefore suggest important contributions from polarized\nDy3+moments even at room temperature.\nThe puzzling strong magnetic \feld-dependence of Kb\nfrom the data \ft might indicate a di\u000berent coupling be-\ntween the rare earth and transition metal magnetic sub-\nsystems at the interface and in the bulk. It can be jus-\nti\fed by the following symmetry argument. The gener-\nators of the Pbnm space group of the DFO crystal are\nthree (glide) mirror planes: ~ ma, ~mbandmc, i.e. a mirror\nre\rection combined with a shift along a direction parallelto the mirror plane. mcis broken at the interface nor-\nmal to the caxis. In the absence of mc, the rare earth\norder parameters A′\naandG′\nbtransform to Gbthat de-\nscribes the AFM order of Fe spins, which allows for a\nlinear coupling between the rare earth and Fe spins at\nthe interface. Since Gbstrongly depends on \u000bat\u000b=0,\nthe same may hold for the rare earth moments at the\ninterface. The SMR is very surface sensitive and could\nbe strongly a\u000bected by this coupling.\nNext, we turn to the SMR at temperatures below the\nMorin transition at magnetic \felds around the re-entrant\n\feld,Hcr. Figure 5(a) shows the transverse SMR of\ndevice 1 in an OOP magnetic \feld, while the data for\nlongitudinal resistance are deferred to the Appendix A,\nFig. 8a). We subtracted a linear \feld dependent contri-\nbution from the OOP data that is caused by the ordinary\nHall e\u000bect in Pt.\nThe zero-\feld resistance of device 1 should not change\nunder the Morin transition when the N\u0013 eel vector direc-\ntion switches from atobnor should it be a\u000bected by weak\nmagnetic \felds H/parallel.alt1c(\u00160Hcr<0:1 T near 50 K [21]) that\nreturn the system to G/parallel.alt1a. Indeed, we do not see any\nweak-\feld anomaly of the SMR near 50 K in Fig. 5a).\nHowever, below 23 K, a negative SMR proportional to\nthe applied \feld appears. The linear \feld-dependence\nends abruptly with a positive step-like discontinuity (see\nFig. 5a)). No resistance o\u000bset has been observed between\nthe zero-\feld \u0000 1and the high-\feld \u0000 4phases. After sub-\nstraction of the strictly linear ordinary Hall e\u000bect contri-\nbution, the SMR feature is an even function of Hc. The\nmagnetic phase transition at 23 K appears to be unre-\nlated to the Morin transition and has not been reported\npreviously.\nThe Morin transition is clearly observed in the OOP\nand IP SMR of device 2, in which the crystallographic\naxes are azimuthally rotated by 45○relative to the Hall\nbar as shown in Fig. 1b). Here, an SMR signal is ex-\npected for both magnetic phases and the 90○rotation of\nthe N\u0013 eel vector from atobshould change its sign from\npositive for the AFM \u0000 1phase ( G∥b) to negative for\nthe WFM \u0000 4phase ( G∥a), for/divides.alt0Hc/divides.alt0>Hcr. The \u0000 1phase\ncan also be suppressed by an IP \feld H∥^y=^b−^athat\nrotates the N\u0013 eel vector towards ^bto lower the Zeeman\nenergy. The drop in the Hall resistance observed in de-\nvice 2 below 48 K for the OOP (Fig. 5 b)) and IP (Fig. 5\nc)) \feld directions can therefore be ascribed to the Morin\ntransition with a temperature-dependent Hcr. The SMR\nsteps are negative, as expected.\nAt even lower temperatures the model appears to break\ndown since we observe hysteretic behavior in the \feld-\ndependence of the SMR signal at low magnetic \felds for\nboth the OOP and IP directions. These features come\nup below 23 K, so appear to have the same origin as\nthe anomalies in device 1. For the OOP direction, the\nlow-\feld anomalies in device 2 are peaks while they are\nstep-like in device 1. Wang et al.[21] did not observed a\nhysteresis in the Fe3+magnetic sublattice and suggested\nthat observed hysteretic behaviour [6, 51] is an evidence8\nFIG. 7. The SSE, i.e. the detected voltage in the transverse\nHall probe divided by the squared current of device 1 at 10 K\nas a function of the magnetic \feld strength and direction \u000b.\nAt weak \felds, the SSE shows a cos \u000bdependence as expected\nfor the Fe3+magnetic sublattice. This amplitude initially\nincreases with the magnetic \feld strength but decreases again\nand \rattens for H>0:5 T.\n 0.1T 0.2T 0.3T 0.4T 0.5T 0.75T 1T 2T\n00000\n0 \n0\n0 RT (V A-2)\n-135-90-450459013510\nα\nfor long-range to short-range Dy3+magnetic order. The\nSMR might witness an ordering of Dy3+moments at the\ninterface at a higher temperature than in the bulk that\ncannot be detected by other measurements.\nAnother unexpected feature is a linear negative mag-\nnetoresistance at /divides.alt0Hy/divides.alt0>Hcrfor the IP con\fguration (see\nFig. 5c)) that might be caused by a canting of GFeto-\nwards cbyHa>1:6T[24]. A misalignment of the crys-\ntallographic axes could also a\u000bect the SMR more sig-\nni\fcantly for high magnetic \felds. However, neither of\nthese mechanisms explain the IP magnetic \feld depen-\ndence and the peaks and low magnetic \feld features in\nthe OOP measurements of both devices below 23 K (Fig.\n5a) and 5b)). Since their signs and shapes vary, we can\nexclude a paramagnetic OOP canting of the Dy3+orbital\nmoments. The Dy3+orbital moments are locked to the\nIsing axis in the abplane and the magnetization is one\norder of magnitude larger in this plane than along the c\ndirection [6]. This might explain the IP SMR features in\nterms of an IP \feld and temperature dependent order of\nthe Dy3+moments.\nThe 90○spin reorientation at the Morin transition\nmaximizes the Fe3+contribution to the SMR. The in-crease of the IP signal amplitude by one order of mag-\nnitude upon lowering the temperature, see Fig. 5(c) is\ntherefore unexpected. The signals become as large as\n1%, one order of magnitude larger than the SMR signals\nof Pt on Y 3Fe5O12[7, 27{29] and a factor four larger\nthan that of \u000b-Fe2O3[52]. Ordered Dy3+magnetic mo-\nments appear to be responsible for the anomalous signals\nbelow 23 K. They interact with the Fe sublattice by the\nexchange interaction, as observed before in the multifer-\nroic phase at temperatures exceeding TDy\nNunder a 0.5 T\nmagnetic \feld [21]. A contribution of Dy3+moments to\nthe magnetization has also been observed in terms of an\nupturn of the magnetization and hyper\fne \feld below\n23 K [53].\nThe SMR steps in device 1 around TDy\nN=4 K at which\nthe Dy moments order spontaneoulsy, are similar to those\nat higher temperature, which supports the hypothesis\nthat the latter are also related to Dy3+order. Device\n2 shows an increased H crmatching those in device 1 at\nthese temperatures. Both devices show no non-linear an-\ntisymmetric \feld dependence, indicating that the Dy3+\nordering above 4 K is \feld-induced. Li et al. [51] ob-\nserved jumps in the thermal conductivity around 4 T and\nattributed these to a spin reorientation of the Fe sublat-\ntice. However, no further transitions are observed up to\n6 T as is shown in Appendix A, so we cannot con\frm\nsuch an Fe3+transition.\nThe magnetic \feld and temperature of the occurrences\nof SMR steps at spin transitions and of SMR anomalies\nare collected in Fig. 6, including the peaks in the OOP\nmeasurements of device 2, using the same markers as in\nFig. 5. The data on the Morin transition agrees with\nprevious observations [21, 24]. The Morin point for both\nIP an OOP con\fgurations is around 50 K, whereas the\ntransitions ascribed to an ordering of the Dy3+moments\noccur around 23 K. Upon lowering the temperature, the\ntransitions associated to the Dy3+and Fe3+moments ap-\nproach each other and merge below TDy\nN, which is another\nindication of a strong inter-sublattice exchange interac-\ntion.\nFigure 7 summarizes the observed IP SSE data of de-\nvice 1 at 10 K. The angular dependence of the resistance\nat small \felds shows the cos \u000bdependence, indicating\nthat the magnon spin current jminjected into Pt is con-\nstant with angle. The amplitude initially increases lin-\nearly with \feld, but decreases again for H>0:5 T. The\nSSE signal of a uniaxial AFM has cos \u000bdependence for\nan IP rotating magnetic \feld [48]. The SSE is small at\nangles for which our model for the Dy3+contribution in\nFig. 3b) predicts a peak. However, we do not observe\nthe expected Dy3+-induced SSE contribution due to the\nDy3+magnetization shown in Fig. 3. On the contrary,\nan increase in Dy3+magnetization appears to suppress\nthe SSE signal. These results suggest that the angular\ndependence of the SSE is governed not so much by the\nordering of the Dy spins, but by their e\u000bect on the fre-\nquencies of the antiferromagnons in the Fe magnetic sub-\nsystem. The ordering of Dy spins leads to a hardening9\nof the AFM resonance modes [54]. The applied magnetic\n\feld suppresses the Dy spin ordering and results in a sub-\nstantial decrease of the spin gap [54] , which a\u000bects the\nthermal magnon \rux and, hence, the SSE. At room tem-\nperature, the SSE signal does not rise above the noise\nlevel of 0.18 V A−2.\nVI. CONCLUSION\nWe studied the rare earth ferrite DFO by measur-\ning the transverse electric resistance in Pt \flm con-\ntacts as function of temperature and applied magnetic\n\feld strength and direction. Results are interpreted in\nterms of SMR and SSE for magnetic con\fgurations that\nminimize a magnetic free energy model with magnetic\nanisotropies, Zeeman energy and DMI. The N\u0013 eel vector\nappears to slowly rotate OOP and displays jumps un-\nder IP rotating magnetic \felds. Magnetic \feld-strength\ndependences indicate that Fe3+spins are responsible for\nthe symmetry of the SMR, but that the Dy3+orbital\nmoments a\u000bect the amplitude. The \frst-order Morin\ntransition is clearly observed at temperatures below 50 K.\nAdditional sharp features emerge below 23 K at critical\n\felds below that of the Morin transition. These observed\nfeatures cannot be understood by the Fe3+N\u0013 eel vec-\ntor driven SMR. Rather, they suggest a magnetic \feld-induced ordering of Dy3+established by the competition\nbetween applied magnetic and exchange \felds with Fe3+.\nThis hypothesis is supported by the similar SMR features\nat the spontaneous Dy3+moment ordering temperature\nTDy\nN. A Dy3+order above TDy\nNalso appears to suppress\nthe SSE contributions from the Fe sublattice.\nConcluding, we report simultaneous manipulation and\nmonitoring of the ordering of both transition metal and\nrare earth magnetic sublattices and their interactions\nas a function of temperature and magnetic \feld in the\ncomplex magnetic material DFO.\nVII. ACKNOWLEDGEMENTS\nWe thank A. Wu for growing the single crystal\nDyFeO 3, J. G. Holstein, H. Adema, T. J. Schouten, H. H.\nde Vries and H. M. de Roosz for their technical assistance\nas well as R. Mikhaylovskiy and A. K. Zvezdin for discus-\nsions. This work is part of the research program Magnon\nSpintronics (MSP) No. 159 \fnanced by the Nederlandse\nOrganisatie voor Wetenschappelijk Onderzoek (NWO)\nand JSPS KAKENHI Grant Nos. 19H006450, and the\nDFG Priority Programme 1538 Spin-Caloric Transport\n(KU 3271/1-1). Further, the Spinoza Prize awarded in\n2016 to B. J. van Wees by NWO is gratefully acknowl-\nedged\n[1] S. Loth, S. Baumann, C. P. Lutz, D. M. Eigler, and A. J.\nHeinrich, Science 335, 196 (2012).\n[2] G. R. Hoogeboom, A. Aqeel, T. Kuschel, T. T. M. Pal-\nstra, and B. J. van Wees, Applied Physics Letters 111,\n052409 (2017).\n[3] D. Afanasiev, B. A. Ivanov, A. Kirilyuk, T. Rasing, R. V.\nPisarev, and A. V. Kimel, Physical Review Letters 116,\n097401 (2016).\n[4] P. Wadley, B. Howells, J. \u0014Zelezn\u0013 y, C. Andrews, V. Hills,\nR. P. Campion, V. Nov\u0013 ak, K. Olejn\u0013 \u0010k, F. Maccherozzi,\nS. S. Dhesi, S. Y. Martin, T. Wagner, J. Wunderlich,\nF. Freimuth, Y. Mokrousov, J. Kune\u0014 s, J. S. Chauhan,\nM. J. Grzybowski, A. W. Rushforth, K. Edmond, B. L.\nGallagher, and T. Jungwirth, Science 351, 587 (2016).\n[5] T. Moriyama, K. Oda, T. Ohkochi, M. Kimata, and\nT. Ono, Scienti\fc reports 8, 14167 (2018).\n[6] Y. Tokunaga, S. Iguchi, T. Arima, and Y. Tokura, Phys-\nical Review Letters 101, 097205 (2008).\n[7] H. Nakayama, M. Althammer, Y. T. Chen, K. Uchida,\nY. Kajiwara, D. Kikuchi, T. Ohtani, S. Gepr ags,\nM. Opel, S. Takahashi, R. Gross, G. E. W. Bauer, S. T. B.\nGoennenwein, and E. Saitoh, Physical Review Letters\n110, 206601 (2013).\n[8] J. Fischer, O. Gomonay, R. Schlitz, K. Ganzhorn, N. Vli-\netstra, M. Althammer, H. Huebl, M. Opel, R. Gross,\nS. T. B. Goennenwein, and S. Gepr ags, Physical Review\nB97, 014417 (2018).\n[9] Y. Ji, J. Miao, Y. M. Zhu, K. K. Meng, X. G. Xu, J. K.\nChen, Y. Wu, and Y. Jiang, Appl. Phys. Lett. 112,232404 (2018).\n[10] R. Lebrun, A. Ross, O. Gomonay, S. Bender, L. Bal-\ndrati, F. Kronast, A. Qaiumzadeh, J. Sinova, A. Brataas,\nR. Duine, and M. Kl aui, Communications Physics 2, 50\n(2019).\n[11] S. M. Wu, W. Zhang, A. Kc, P. Borisov, J. E. Pearson,\nJ. S. Jiang, D. Letterman, A. Ho\u000bmann, and A. Bhat-\ntacharya, Physical Review Letters 116, 097204 (2016).\n[12] S. M. Rezende and J. B. S. Mendes, Applied Physics\nLetters 111, 172405 (2017).\n[13] G. R. Hoogeboom and B. J. van Wees, Physical Review\nB102, 214415 (2020).\n[14] E. A. Turov, Physical Properties of Magnetically Ordered\nCrystals (Izd. Akad. Nauk SSSR, 1963).\n[15] I. Dzyaloshinsky, J. Phys. Chem. Solids 5, 253 (1958).\n[16] T. Moriya, Physical Review 120, 91 (1960).\n[17] L. M. Holmes, L. G. Van Uitert, R. R. Hecker, and G. W.\nHull, Physical Review B 5, 138 (1972).\n[18] E. F. Bertaut, \\Spin Con\fgurations of Ionic Structures:\nTheory and Practice.\" (1963).\n[19] T. Yamaguchi and K. Tsushima, Physical Review B 8,\n5187 (1973).\n[20] A. Stroppa, M. Marsman, G. Kresse, and S. Picozzi,\nNew Journal of Physics 12, 093026 (2010).\n[21] J. Wang, J. Liu, J. Sheng, W. Luo, F. Ye, Z. Zhao,\nX. Sun, S. A. Danilkin, G. Deng, and W. Bao, Phys-\nical Review B 93, 140403(R) (2016).\n[22] P. Belov, A. K. Zvezdin, M. Kadomtseva, and I. B.\nKrynetskii, Journal of Experimental and Theoretical10\nPhysics 40, 980 (1974).\n[23] A. Maziewski and R. Szymczak, Journal of Physics D:\nApplied Physics 10, 37 (1977).\n[24] L. A. Prelorendjo, C. E. Johnson, M. F. Thomas, and\nB. M. Wanklyn, Journal of Physics C: Solid State Physics\n13, 2567 (1980).\n[25] F. Zhang, S. Li, J. Song, J. Shi, and X. Sun, IEEE\nTransactions on Magnetics 51, 1000904 (2015).\n[26] Y. J. Ke, X. Q. Zhang, H. Ge, Y. Ma, and Z. H. Cheng,\nChinese Physics B 24, 037501 (2015).\n[27] Y. T. Chen, S. Takahashi, H. Nakayama, M. Altham-\nmer, S. Goennenwein, E. Saitoh, and G. Bauer, Physical\nReview B 87, 144411 (2013).\n[28] N. Vlietstra, J. Shan, V. Castel, J. Ben Youssef, G. E. W.\nBauer, and B. J. van Wees, Applied Physics Letters 103,\n032401 (2013).\n[29] M. Althammer, S. Meyer, H. Nakayama, M. Schreier,\nS. Altmannshofer, M. Weiler, H. Huebl, S. Gepr ags,\nM. Opel, R. Gross, D. Meier, C. Klewe, T. Kuschel, J. M.\nSchmalhorst, G. Reiss, L. Shen, A. Gupta, Y. T. Chen,\nG. E. W. Bauer, E. Saitoh, and S. T. B. Goennenwein,\nPhysical Review B 87, 224401 (2013).\n[30] K. Ganzhorn, J. Barker, R. Schlitz, B. A. Piot, K. Ollefs,\nF. Guillou, F. Wilhelm, A. Rogalev, M. Opel, M. Al-\nthammer, S. Gepr ags, H. Huebl, R. Gross, G. E. W.\nBauer, and S. T. B. Goennenwein, Physical Review B\n94, 094401 (2016).\n[31] B. W. Dong, J. Cramer, K. Ganzhorn, H. Y. Yuan, E. J.\nGuo, S. Goennenwein, and M. Kl aui, Journal of Physics:\nCondensed Matter 30, 035802 (2018).\n[32] A. Aqeel, N. Vlietstra, J. A. Heuver, G. E. Bauer, B. No-\nheda, B. J. Van Wees, and T. T. Palstra, Physical Review\nB92, 224410 (2015).\n[33] A. Aqeel, N. Vlietstra, A. Roy, M. Mostovoy, B. J. Van\nWees, and T. T. Palstra, Physical Review B 94, 1334418\n(2016).\n[34] K. Zhao, T. Hajiri, H. Chen, R. Miki, H. Asano, and\nP. Gegenwart, Physical Review B 100, 45109 (2019).\n[35] L. J. Cornelissen and B. J. Van Wees, Physical Review\nB93, 020403 (2016).\n[36] L. J. Cornelissen, J. Liu, R. A. Duine, J. b. Youssef, and\nB. J. Van Wees, Nature Physics 11, 1022 (2015).\n[37] R. Lebrun, A. Ross, S. A. Bender, A. Qaiumzadeh,\nL. Baldrati, J. Cramer, A. Brataas, R. A. Duine, and\nM. Kl aui, Nature 561, 222 (2018).\n[38] K. Oyanagi, S. Takahashi, L. J. Cornelissen, J. Shan,\nS. Daimon, T. Kikkawa, G. Bauer, B. J. van Wees, and\nE. Saitoh, Nature Communications 10, 4740 (2019).\n[39] R. Cheng, J. Xiao, Q. Niu, and A. Brataas, Physical\nReview Letters 113, 057601 (2014).\n[40] N. Vlietstra, J. Shan, B. J. van Wees, M. Isasa,\nF. Casanova, and J. B. Youssef, Physical Review B 90,\n174436 (2014).\n[41] V. V. Eremenko, S. L. Gnatchenko, N. F. Kharchenko,\nP. P. Lebedev, K. Piotrowski, H. Szymczak, and\nR. Szymczak, Europhysics Letters 11, 1327 (1987).\n[42] S. Cao, L. Chen, W. Zhao, K. Xu, G. Wang, Y. Yang,\nB. Kang, H. Zhao, P. Chen, A. Stroppa, R. Zheng,\nJ. Zhang, W. Ren, J. \u0013I~ niguez, and L. Bellaiche, Scienti\fc\nReports 6, 37529 (2016).\n[43] A. K. Zvezdin, V. M. Matveev, A. A. Mukhin, and A. I.\nPopov, Moscow, Izd. Nauk , 296 (1985).\n[44] C. Hahn, G. de Loubens, V. V. Naletov, J. B. Youssef,\nO. Klein, and M. Viret, European Physics Letters 108,571 (2014).\n[45] A. K. Zvezdin and V. M. Matveev, Zh. Eksp. Teor. Fiz. ,\nTech. Rep. 3 (1979).\n[46] U. V. Valiev, J. B. Gruber, S. A. Rakhimov, and O. A.\nNabelkin, physica status solidi (b) 237, 564 (2003).\n[47] S. Gepr ags, A. Kehlberger, F. Coletta, Z. Qiu, E. J.\nGuo, T. Schulz, C. Mix, S. Meyer, A. Kamra, M. Al-\nthammer, H. Huebl, G. Jakob, Y. Ohnuma, H. Adachi,\nJ. Barker, S. Maekawa, G. E. W. Bauer, E. Saitoh,\nR. Gross, S. T. B. Goennenwein, and M. Kl aui, Nature\nCommunications 7, 10452 (2016).\n[48] W. Yuan, Q. Zhu, T. Su, Y. Yao, W. Xing, Y. Chen,\nY. Ma, X. Lin, J. Shi, R. Shindou, X. C. Xie, and\nW. Han, Science Advances 4, 1098 (2018).\n[49] T. Hajiri, L. Baldrati, R. Lebrun, M. Filianina, A. Ross,\nN. Tanahashi, M. Kuroda, W. L. Gan, T. O. Mente\u0018 s,\nF. Genuzio, A. Locatelli, H. Asano, and M. Kl aui, Jour-\nnal of Physics: Condensed Matter 31, 445804 (2019).\n[50] We have not been able to identify the mechanism for the\nstep observed between 135 K and 150 K that has to our\nknowledge not been reported elsewhere either.\n[51] Z. Y. Zhao, X. Zhao, H. D. Zhou, F. B. Zhang, Q. J. Li,\nC. Fan, X. F. Sun, and X. G. Li, Physical Review B 89,\n224405 (2014).\n[52] J. Fischer, M. Althammer, N. Vlietstra, H. Huebl,\nS. Goennenwein, R. Gross, S. Gepr ags, and M. Opel,\nPhysical Review Applied 13, 014019 (2020).\n[53] S. S. K. Reddy, N. Raju, C. G. Reddy, P. Y. Reddy,\nK. R. Reddy, and V. R. Reddy, Journal of Magnetism\nand Magnetic Materials 396, 214 (2015).\n[54] T. N. Stanislavchuk, Y. Wang, Y. Janssen, G. L. Carr,\nS. W. Cheong, and A. A. Sirenko, Physical Review B\n93, 094403 (2016).\n[55] S. L. Gnatchenko, K. Piotrowski, A. Szewczyk, R. Szym-\nczak, and H. Szymczak, Journal of Magnetism and Mag-\nnetic Materials 129, 307 (1994).\n[56] J. H. Han, C. Song, F. Li, Y. Y. Wang, G. Y. Wang,\nQ. H. Yang, and F. Pan, Physical Review B 90, 144431\n(2014).\n[57] K. Oyanagi, J. M. Gomez-Perez, X. Zhang, T. Kikkawa,\nY. Chen, E. Sagasta, L. E. Hueso, V. N. Golovach, F. S.\nBergeret, F. Casanova, and E. Saitoh, arXiv:2008.02446\n(2020).\nAppendix A: Longitudinal and 2K SMR\nThe modulation of the longitudinal Pt resistance as a\nfunction of magnetic \feld are shown in Fig. 8 for com-\nparison with the transverse SMR. The longitudinal sig-\nnals are a\u000bected by a background contact resistance that\nis sensitive to temperature changes. The SMR signals\nare therefore more distorted by a small temperature drift\nthan the transverse measurements. Moreover, the back-\nground resistance su\u000ber from increased noise.\nThe OOP resistance changes of device 1 are one order\nof magnitude larger than those of device 2 and dominated\nby hysteretic e\u000bects. The signal amplitudes of OOP and\nIP con\fgurations for device 2 are similar. The measure-\nment time of one data point below 0.2 T is smaller than at\nlarger \felds, in\ruencing the shape of the graphs. Device\n2 shows hysteretic features at low magnetic \felds and be-11\nLa) b)\nLc)Ldevice 2 IP device 2 OOP device 1 OOP\nFIG. 8. Relative changes in the longitudinal SMR RL/slash.leftR0of (a) device 1 as well as (b,c) device 2 at temperatures up to 50K for\nmagnetic \feld sweeps (a,b) OOP and (c) IP. Device 1 shows large hysteretic e\u000bects at the full range of magnetic \feld strengths.\nThe amount of data points around zero magnetic \feld, and thus the waiting time per magnetic \feld change, is higher than at\nhigher \felds as to have higher resolution for the transverse Morin transition. This makes the hysteretic e\u000bects slightly distorted\ncompared to a situation with constant waiting time. Device 2 shows hysteretic e\u000bects solely at lower \feld strengths which\ncorresponds to the hysteretic features of the transverse measurements.\n1\nFIG. 9. Transverse resistance of device 2 at 2 K as a function\nof the OOP magnetic \feld up to 6 T. The resistance increases\ncontinuously with magnetic \feld strength above 2 T.\nlow 23 K, for both IP and OOP magnetic \felds that are\nsimilar to the transverse SMR features discussed in the\nmain text.\nResults of a \feld sweep up to 6 T are shown in Fig. 9.\nThe resulting continuous curve does not show transitions\non top of those discussed in the text, without evidence\nfor a phase transition at 4 T and 2 K [51, 55].Appendix B: Exchange interaction\nThe Pbnm crystal symmetry allows an exchange cou-\npling between the Dy3+moments and G-type AFM or-\ndered Fe spins. The coupling of the 4 (individual) Dy\nspins in the unit cell with the Fe spins is described as\nEDy−Fe=−g1Gc(ma\n1+ma\n2+ma\n3+ma\n4)\n−g2Gc(ma\n1−ma\n2+ma\n3−ma\n4)\n−g3Ga(mc\n1+mc\n2+mc\n3+mc\n4)\n−g4Gb(mc\n1−mc\n2+mc\n3−mc\n4); (B1)\nwhere the indices 1 ;2;3;4 label the rare-earth ions in the\nunit cell. The exchange \feld from Fe ions is estimated to\nbe∼2 T at low temperatures [45].\nForkBT/uni226B\u0001, the magnetization of the Dy sublattice\nm∥=\u001fDy\n∥H∥andm⊥=\u001fDy\n⊥H⊥for \feld components\nparallel and perpendicular to the local anisotropy axis\nandH=/radical.alt2\nH2\n∥+H2⊥. We assume that the transverse\nSMR caused by the paramagnetic Dy3+moments\npolarized by the applied \feld is proportional to mxmy\n[9, 32, 56, 57]. Adding the contributions of the four Dy\nsites in the crystallographic unit cell of DFO and the\nexchange \feld from the Fe spins acting on the Dy spins\nas described in the main text, we obtain Eq. 5 with\nA=/bracketleft.alt2(\u001fDy\n∥+\u001fDy\n⊥)2−(\u001fDy\n∥−\u001fDy\n⊥)2cos(4\u001eDy)/bracketright.alt2/slash.left2 andB=\nsin(2\u001eDy)/bracketleft.alt2(\u001fDy\n∥)2−(\u001fDy\n∥)2−(\u001fDy\n∥−\u001fDy\n∥)2cos(2\u001eDy)/bracketright.alt2.\nThe coupling constants g 3and g 4do not appear in\nthe expression for SMR since the latter does not de-\npend on the c-component of Dy spins. Moreover, the12\nc-component is very small at low temperatures, since the easy axes of Dy ions lie in the ab plane. Both g 1and g 2\nlead to (nearly) the same angular dependence of SMR."    },    {        "title": "2101.03542v1.Magnetic_anisotropy_in_van_der_Waals_ferromagnet_VI3.pdf",        "content": "1 \n Magnetic anisotropy in  van-der-Waals ferromagnet VI3 \n \nA. Ko riki1,2, M. Míšek3, J. Pospíšil1, M. Kratochvílová1, K. Carva1, J. Prokleška1, P. \nDoležal1, J. Kaštil3, S. Son4,5,6, J-G. Park4,5,6, and V. Sechovský1 \n \n1 Charles University , Faculty of Mathematics and Physics, Department of Condensed Matter Physics, Ke \nKarlovu 5, 121 16 Prague 2, Czech Republic  \n2Hokkaido University, Graduate School of Science,  Department of Condensed Matter Physics,  Kita10, Nishi \n8, Kita -ku, S apporo , 060-0810, Japan  \n3 Institute of Physics, Academy of Sciences of Czech Republic, v.v.i, Na Slovance 2, 182 21 Prague 8, Czech \nRepublic  \n4 Center for Quantum Materials, Seoul National University, Seoul 08826, Korea  \n5 Center for Correlated Electron Systems, Institute for Basic Science, Seoul 08826, Korea  \n6 Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea  \nABSTRACT  \nA comprehensive study of magnetocrystalline anisotropy of a layered  van-der-Waals ferromagnet  \nVI3 was performed. We measured  angular dependences of the torque and  magnetization with respect \nto the direction of the applied magnetic field  within  the “ac” plane  perpendicular to and within the  \nbasal  ab plane , respectively . A two-fold butterfly -like signal was detected by magnetization in the \nperpendicular “ ac” plane . This signal symmetry remains conserved throughout all magnetic regimes \nas well as through the known structural transition down to the lowest temperatures. The max imum of \nthe magnetization signal and the resulting magnetization easy axis is significantly tilted from  the \nprincipal c axis by ~40°. The c lose relation of the magnetocrystalline anisotropy to the crystal \nstructure  was documented. In contrast, a  two-fold-like angular signal was detected in the \nparamagnetic region within the ab plane in the monoclinic phase , which  transforms into a six -fold-\nlike signal below the Curie temperature TC. With further cooling, another  six-fold-like signal with an \nangular shift  of ~30° grows  approaching TFM. Below TFM, in the triclinic phase, the original six-fold-\nlike signal vanishes , being replaced by a secondary six-fold-like signal  with an angular shift of ~30° . \nI. INTRODUCTION    \nMagnetic van der Waals (vdW) materials have recently become hot subjects of interest \nbecause of their potential use in atomically thin devices for spintronic and optoelectronic \nfunction alities1-5. Exploring new chemistry paths to tune their magnetic and optical properties enable \nsignificant progress in fabricating heterostructures and ultra-compact  devices6-9 by mechanical \nexfoliation10 or well -controlled element deposition11. Despite belonging to a well -studied family of \ntransition metal trihalides, VI 3 has received significant attention only recently10, 12-14. Nevertheless, \nthe first studies pointed out that the magnetism of VI 3 is very complicated . This is probably related \nto the incomplete high spin state t2g shell in V3+ ions here, which allows for  a Jahn-Teller distortion. \nThe detailed results of 51V and 127I NMR spectroscopy complemented by temperature dependencies \nof specific -heat and magnetization15 revealed two  distinct  ferromagnetic (FM)  phases, the ground -\nstate one having the critical temperature TFM = 26 K and another existing between TFM and TC = 49.5 \nK (notation is taken from Ref.14). VI3 behaves at low temperatures as a hard ferromagnet with a high \ncoercive field 0Hc ~ 1 T at 2 K for the magnetic field applied parallel to the c-axis ( Hc)10, 12, while \nHc value is considerably lower in the perpendicular direction ( H  ab). In fields above 2 T the \nmagnetization M in H c is proportional  to ~1.2 M in H ab. This anisotropy of magnetization persists \nat least up to 9 T10, 12.  2 \n A crucial  property of the vdW materials with considerable  application potential is the strong \nmagnetocrystalline anisotropy2. In layered materials, two limits of easy‐axis are in‐plane (XY model) \nand out‐of‐plane (Ising model).  The anisotropy features of VI 3 were investigated by magnetization \nmeasurement s. These ha ve shown  a different value of saturated magnetization (higher along the c \naxis) and a significantly larger low-temperature magnetization loop hysteresis along the c axis12, 16, \n17. The Raman scattering study has also identified an acoustic magnon mode (the spin-wave gap) and \na two -magnon mode in the low-temperature FM state.  With the help of li near spin-wave theory \ncalculations, the magnetic anisotropy and intralayer exchange strength are estimated as 1.16 and 2.75 \nmeV . The sizeable  magnetic anisotropy implies the stability of the FM order in the 2D limit with a  \nhigh critical field16 . \nFirst-principles calculations have been used to investigate the magnetism and magnetic \nanisotropy of VI 3 theoretically, however, mostl y in the single -layer or bilayer limit18-21, with only a \nfew works consider ing the bulk form10, 22. Overall, magnetocrystalline anisotropy with an easy axis \nalong c has been predicted for both bulk and single layer in these works. A high orbital momentum  \non V was found in a calculation based on the linear augmented plane wave method, which led to  a \nrobust magnetic anisotropy dominated by a single -ion contribution of 15.9 meV per formula unit. \nBased on these findings , it was concluded that VI 3 behaves as a n Ising -like ferromagnet, similar to \nCrI 323. In a different calculation based on the projector augmented -wave framework t he energy \ndifference bet ween the in-plane and out -of-plane magnetization orientation was found to be strongly \ndependent on both Hubbard U as well as strain. Furthermore , this energy difference changes sign at \nU ~ 3.5 eV, close to the expected value of U24.  \nCurrently , there is a controversy concerning the VI3 crystal structure  and its conjunction with \nmagnetism . Kong et al.  have  reported a trigonal ( R-3) structure at room temperature with  a structural \ntransition below Ts = 78 K13. A contradictory result  has been reported by Son et al.12 showing  a \nstructur al phase transition at Ts = 79 K where the crystal symmetry  lowers with cooling  from the \ntrigonal P-31c structure to a monoclinic C2/c  in which  the system  become s ferromagnetic at \ntemperature TC = 50 K . Conversely , Tian et al.10 claim that the VI 3 structural phase transition is \nsimilar to the structural transition of CrI 325, i.e., they observed  a lowering of the symmetry with \nheating  through Ts from R-3 to C2/m . A detailed crystal structure study by Doležal  et al.  has \nconfirmed the  trigonal ( R-3) crystal structure at room temperature in agreement with T. Kong13 and \nfound a subsequent  symmetry lowering to a triclinic structure at TFM2 = 32 K, when the ferromagnetic \nphase FM  I transforms to a different ferromagnetic phase FM  II. The connection of the structur e with \nthe magnetic  phase  transition in VI 3 suggests a considerable role of magnetoelastic interactions in \nthis compound.  It is also suppo rted by magnetostriction -induced changes of the monoclinic -structure \nparameters at TC. Surprisingly , the temperature  of structure transition is decreased  by magnetic fields \napplied along the trigonal c-axis and is intact by  a magnetic field applied within the basal plane. These \nresults indicate that the magnetic field applied within the three -fold axis of the VI 6 octahedrons \nstabilizes the symmetry  of the  V honeycombs in the basal plane26.  \nThe direct observation of the  VI3 magnet ocrystalline anisotropy was  revealed  by the angular \ndependence of magnetization at temperature  2 K in various magnetic fields H rotated out -of-plane \nfrom the c axis to a (and b) axis and within the ab plane. The out -of-plane anisotropy exhibits a \n“butterfly -like” pattern17, different from the other 2D FM semiconductor Cr2Ge2Te627. On the other \nhand , the angular signal of  magnetization  within the basal ab plane  shows a six-fold-like symmetry . \nHowever, anisotropy studies so far performed were mostly focused on the lowest temperature \nmagnetic regime  below TFM. Further , detailed  measurements throughout the entire temperature \ninterval covering all detected magnetic regimes and structural phases  are missing and  highly desirable \nto understand complex magnetic interaction s and anisotropy in the VI3 vdW ferromagnet . The VI 3 \nmagnetic structure, exchange interactions and magnetocrystalline anisotropy are much more \ncompl icated  than earlier studied Cr tri -halides  28-32. \nThis paper focuses on  studying  magnetocrys talline anisotropy features of VI 3 in a wide \ntemperature interval by  angular -dependent magnetic torque and magnetization  measurements  for the 3 \n magnetic field applied  out-of-plane marked as “ ac” (a general perpendicular plane to the ab basal \nplane ) and within the ab basal plane.  \nII. EXPERIMEN TAL METHODS  \nThe single crystals were prepared by the chemical vapor transport m ethod , as described \nelsewhere12. The angular dependence of the magnetization was measured by MPMS (Quantum \nDesign) devices using thin single crystals with a sample area of 1.4 × 1.8 mm2. The sample was placed \non a handmade rotator with a rotation axis orthogonal to the external magnetic field. Magnetic torque \nwas measured using  a commercial Quantum Design Torque magnetometer chip attached on  a 370°  \nrotator  installed inside a 9T PPMS apparatus . To measure the angular dependence of torque within \nthe princ ipal ab plane with respect to magnetic field orientation , an L-shaped copper element had to \nbe attached to the torque  chip (Fig.  1). The sample  was fixed with Apiezon  grease . The to rque \nmagnetometer chip can typically be calibrated. However , because of installed cooper L element , \ncalibration was not possible to finalize, and so we do not focus on the absolute value of the torque \nbut scale it a ppropriately. The L-element mass contribution was tested by the rotation of the chip in \nzero magnetic  field; the observed signal was negligible compared  to the signal in magnetic field s with \nthe attached sample. The magnetic field is rotating within the ab plane of the sample for both \nmeasurements. Since the magnetic field was applied within the plane of the thin sample, the shape  of \nsamples and the demagnetization factor are  expected not to have  a significant effect on the  results. \nAll sample s used for measurement s were stored under  the inert Ar  (6N purity)  atmosphere in a \nglovebox . The contact with air was minimized only on the time of the installation of the rotators into \nthe MPMS and PPMS  instrument s. \n \nFIG. 1. The scheme of the magnetic torque chip with the installed sample attached to the copper \nL-element. The scheme was taken from the Quantum Design PPMS manual and modified.  \nIII. RESULTS  \nWithin our study , we have performed a series of angular -dependent magnetization scans \nwithin the “ac” plane, perpendicular to the ab plane (Fig.  2). Our low-temperature (2 K) and high-\nmagnetic -field (5  T) angular magnetization result s in the form  of a butterfly -like signal in polar \nprojection agree ing with the previous result in Ref.17. Both, the symmetry and positions of the  maxima \nof the angular  magnetization signal rota ted within the “ ac” plane remain conserve d throughout all \nmagnetic regimes a nd the structural transition to  the triclinic phase26 at TFM. The maximum \nmagnetization signal tilted approximately 40° out of the c-axis.  \n4 \n  \nFIG. 2. Angular -dependent magnetization isotherms in magnetic field 5  T rotated within the “ac” \nplane. The polar plot (a) and classical plot (b) shows selected curves at specific magnetic regimes; \nabove TC (60 K), below TC (50 K), in the upper vicinity of TFM (30 K) , and well below TFM (20 and \n10 K).  \n \nThe angular -dependent  magnetic torque measurements  within the basal ab plane in identical \nmagnetic field 5  T at various temperatures are shown in Fig.  3. \n \n(a) \n(b) 5 \n  \n \n \nFIG. 3. Angular -dependent magnetic torque scans in magnetic field 5  T rotated  within the ab plane. \nThe polar plot (a) shows all recorded temperatures. The classical plot (b) shows selected curves at \nspecific magnetic regime s; above TC (60 K), below T C (50 K), in the  upper  vicinity of TFM (30 K) , \nand well below TFM (10 K ). The intensity of the torque signal is on a relative scale.  \n \nAngular -dependent magnetic torque provide s information  mainly about the \nmagnetocrystalline anisotropy strength . In contrast , the angular -dependent magnetization \nmeasurement s show the size of the magnetic moment at the given orientation of the sample with \nrespect to the direction of  the external magnetic field. The isothermal  scans of the  torque have clearly \nrevealed  significant  evolution of the magnetocrystalline anisotropy  (Fig.  3). A double peak -like signal \nin the paramagnetic region  in the monoclinic phase  transforms  to a six -fold-like signal below TC. The \nsix-fold-like symmetry is present  down to the lowest temperatures.  However , a detailed analysis of \nthe curves between TC and TFM show s signature s of the gradual splitting of each maxim um when \napproaching TFM. At a temperature of 30 K in the vicinity of TFM the signal is split into two sets of \nsix-fold-like signals of similar intensity mutually rotated by  30° (see Fig.  3b – green line) . The \noriginal six -fold-like signal , which has emerged  at TC, becomes gradually reduced with decreasing \ntemperature in favor of intensity of the second set. The original  set vanishes  below TFM whereas  the \nsecond set remains down to the lowest temperature  at the fixed angular position . \n(a) \n(b) 6 \n The a ngular -dependent scans of magnetization  isotherms  with magnetic  field 5 T applied \nwithin the  basal  ab plane  are shown in  Fig. 4. \n \n \n \n \nFIG. 4. Angular -dependent magnetization isotherms  in magnetic field 5  T rotated within the ab \nplane. The polar plot ( a) shows all recorded temperatures. The classical plot (b) shows selected \n(a) \n(b) \n(c) 7 \n curves at specific magnetic regimes; above TC (60 K), below TC (50 K), in the upper vicinity of TFM \n(30 K) , and well below TFM (20 and 10 K).  Panel (c) shows the temperature evolution of the \nmagnetization averaged over  the entire angular scan. \n \nThe angular -dependent magnetization  measurements  have shown complementary result s with \nthe magnetic -torque measurements. One six-fold-like signal was detected below TC. With further \ncooling, a  clear sign ature of a second six -fold-like signal rotated by  ~30°  is observed (see green line \nin Fig.  4b). The second signal gradually grows at the expense of  the original one’s intensity . Below \nTFM, the original six-fold-like signal  vanishe s and only the second  one is detect ed. The signal in the \nparamagnetic regime  at temperatures below Ts is relatively  weak, with two-fold-like broad  maxima \naround  ~15° and ~195° (Fig.  4b) which can be distinguished  analogically  to the symmetry of \nmagnetic  torque data. The averaged angular value of the magnetic moment at given temperatures is \nplotted in Fig.  4c and saturates around value 1.7 B/V, which is lower than expected for the V3+ ion. \nIV. DISCUSSION  \nOur detailed experimental study of magnetocrystalline anisotropy by angular -dependent \nmagnetization has unambiguously proved that the magnetic moment is canted from the c-axis by ~ \n40°. A somewhat  surprising result is the robustness  of the anisotropy throughout the entire \ntemperature interval covering  both ferromagnetic phases and particularly the struct ural transition \nfrom the monoclinic to likely triclinic phase accompanying the magnetic order -to-order transition at \nTFM. \nIn contrast to the “ ac” plane, a more complex angular response of the magnetization and \ntorque were detected in the basal ab plane. In the following discussion, we will place the ab plane in \nthe plane of the VI 3 layer and take the direction of the two -fold axis as the b axis. \n \nFIG. 5. Angular dependence of torque at 60  K and fitting of the result by combining  trigonometric \nfunctions.  \n \nWhen the system reveal s three -fold rotational symmetries around the c-axis, which is the case \nof the rhombohedral R-3 space group for VI 3 above Ts, only the sin  6 term is present in the tensor z \nof magnetization torque (Eq. 1 , see Appendix ). \n𝜏𝑧=𝑀𝑥𝐻𝑦−𝑀𝑦𝐻𝑥=𝜒330𝐻6sin6𝜙     Eq. 1 \nFocusing on the paramagnetic region at 60 K below Ts (Fig.  5), the best agreement for \nexperimental torque data was found using formula (Eq.  2) \n𝜏(𝜙)=𝑘2sin2𝜙+𝑘4sin4𝜙+𝑘6sin6𝜙,      Eq. 2 \n8 \n where two -fold and four -fold signals have been observed. This is a direct indication that the symmetry \nof the crystal  below Ts is lowered at least to a monoclinic  one at this temperature because  the sin 2 \nand sin 4 contributions are forbidden for three -fold rotational symmetries in the tensor z (Eq.1) for \na rhom bohedral structure . \nSince the highest point group of a single layer in VI 3 is D3d, there are three possible \norientations of  two-fold axes when symmetry is reduced to the monoclinic structure (Fig.  6). These \nform three domains associated with each other on a three -fold axis perpendicular to the ab plane. On \nthe other hand, when a magnetic field is applied to the ab plane of a system with two -fold rotations \naround  the b (or a) axis, the torque i n the paramagnetic state can be expressed only by the sum of sin \nfunction s (Eq. 2). \n \nFIG. 6. Schematic view of three two -fold axes  and three possible domains  when the VI3 transforms  \nfrom the rhombohedral to the monoclinic structure.  \n \nIf the volume ratio of the three domains is A  : B : C (A  + B + C = 1), the torque applied to \nthe entire system is  \n𝜏tot(𝜙)=𝐴𝜏(𝜙)+𝐵𝜏(𝜙−𝜋3⁄)+𝐶𝜏(𝜙−2𝜋3⁄) \n=𝐷{𝑘2sin2(𝜙+𝛼2⁄)+𝑘4sin4(𝜙−𝛼4⁄)}+𝑘6sin6𝜙   Eq. 3 \nwhere D and α are constants determined by the volume ratio of the domains. As shown in Fig.  5, the \nphase of the sine function of the second and fourth oscillation terms is out of phase, which indicates \nthat this is the sum of multiple domains. Since the phase shift due  to the domain ratio does not occur \nin the six-fold oscillation term, the angle at which the torque becomes zero corresponds directly to \nthe main axis of the crystal.  \nAs shown in Fig.  3, at 50 K near the fer romagnetic transition at TC, a six -fold saw -tooth -like \nsignal was observed. This is due to a rapid change  in sign  due to switching of the ferromagnetic \ndomains, indicating that the magnetization is not fully polarized in the direction of the external \nmagnetic field . This implies that the magnetocrys talline energy due to the VI 6 octahedron is \nsufficiently large with respect to the Zeeman energy. The result can be understood qualitatively by \nconsidering the VI 6 octahedron to be a regular octahedron as a first approximation. The \nmagnetocrystalline aniso tropy energy  𝐸𝑎𝑐(𝜃,𝜙), represented by the symmetric  representation of the \npoint group Oh, can be written as follows  \n𝐸𝑎𝑐(𝜃,𝜙)=𝐾1{(7cos4𝜃−6cos2𝜃+3)\n12−√2\n3sin3𝜃cos𝜃cos3𝜙}…Eq. 4 \nThis equation is obtained from  the commonly use d magnetic anisotropy equation K1 \n(αx2αy2+αy2αz2+αz2αx2) where αi represents the direction cosine. We obtain this equation by making \nthe [111] axis into a new c-axis, which corresponds to the three -fold rotation axis of the VI 6 \noctahedron  by an orthogonal transformation . This formula confirms that the easy -axis direction of \nspontaneous magnetization has six values, φ = 0, π/3, 2π/3, π, 4π/3, 5π/3, independent of the sign of \nK1. Therefore, when the magnetic field is rotated within the ab plane, the moment switches to the \nmost stable d irection among these easy axes that exist every 60 ˚, as shown in Fig.  7.  \nc\n9 \n  \nFIG. 7. The angular dependence of torque at 50 K. The dashed blue line is a scaled graph of the six -\nfold oscillation term from the 60 K result (Fig. 5).  \n \nWhen the magnetic field is in the middle of the two easy axes, an abrupt change in the sign of \nthe torque takes place . Therefore, the main axis of the crystal is also considered to be in this position. \nIn Fig. 7, the six oscillations in the fitting res ults for the paramagnetic state are shown together, and \nthe two results correspond to each other.   \nAs the temperature was lowered further within the ferromagnetic  region, magnetic torque \nmeasurements indicate there was a significant change in the easy magnetization axis (Fig. 3). \nAlthough this change does not have a clear boundary temperature,  another component with an easy \naxis tilted about ~30˚ graduall y grows and finally replaces the easy axis direction at the lowest \ntemperature. This change cannot b e explained, for example, by a change in  the sign of the magnetic \nanisotropy energy 𝐸𝑎𝑐(𝜃,𝜙). Previous reports have suggested the presence of a first -order structural  \nphase transition from monoclinic to triclinic at around 30 K26, and we expect to see a qualitative \nchange reflecting this  observation . Detailed crystal structure analysis at low temperatures  below TFM \nis highly desirable . The surprising result is that the “ ac” plane signal is insensitive to signal symmetry \nchange within the ab plane. We can only speculate that the FM I and FM II magnetic structures are \nvery similar, only mutually rotated by 30 . \nThe results of the angular dependence of magnetization in Fig. 4 are in qualitative agreement \nwith the torque result; w e obtained six-fold-like signals . The folding back when the magnetization is \nreduced corresponds to ferromagnetic domain switching. The original easy axis changes with \ndecreasi ng temperature about ~30˚ below TFM, which remain conserve d down to  the lowest \ntemperature of 10 K . For simplicity, to analyze the data, we have  decompose d the model into \nspontaneous magnetization Ms, which is independent of the magnitude of the magnetic field, and \npolarized magnetization Mp, which follows the magnetic field. In this model, the result is fitted with \na function of |cosφ|+const . in Eq.  5, as shown in Fig.  8. \n \n10 \n  \nFIG. 8. Angular -dependent magnetization isotherm at 50 K and in magnetic field 5  T rotated within \nthe ab plane and the fit  at 50 K given by  the Eq. 5 . \n \n𝑀𝑠{𝐴|cos(𝜋\n𝑤(𝜙−𝜙0))|+𝐵|cos(𝜋\n𝑤(𝜙−𝜙0−60))|+𝐶|cos(𝜋\n𝑤(𝜙−𝜙0−120 ))|}+𝑀𝑝 \n(Eq.  5) \n \nFitting result Ms = 0.44(1) μB/V and Mp = 0.62(1) μB/V is roughly consistent with  previous \nmagnetization  results of previous experiments at 50  K. \n \nFIG. 9. Angular dependence of the magnetization at 10 K  rotated in a clockwise and anticlockwise \ndirection in an identical magnetic field.   \n \nFig. 9 shows the angular dependence of the magnetization at 10 K. The hysteresis  is not \nnoticeabl e at 50  K, but some non-negligible hysteresis  is observed at 10 K. Hysteresis, such as the \nintersection of clockwise and anticlockwise data that exists around 30˚ and 210˚, is typica lly expected \n11 \n to occur at the minima of magnetization where the domain switches. This suggests that the two \ncomponents, one developed at around 50 K and the other developed at a lower temperature, coexist \nas indepen dent signals.  \n \n \nFIG. 10. The projections of the magnetocrystalline anisotropy energy  𝐸𝑎𝑐(𝜃,𝜙) for anisotropy in  the \nsimplified  Oh representation . Pictures a), b), and c) for the principal axis 111 and magnetic moment \nis stable in  the 100 direction; d) p rojection of easy axes to the ac plane . \n \nThe observed anisotropy within the ab plane can be understood using projections of the \nmagnetocrystalline anisotropy energy 𝐸𝑎𝑐(𝜃,𝜙) in a simplified Oh representation  for the 111 direction \nas the principal axis  of the VI6 octahedron for magnetic moment  stable in  the 100 direction of the \ncube, i.e. the direction of Iodine  that leads to the six-fold-like signal  (see Fig.  10a, 10 c). On the other \nhand, the projection of the easy a xis in the “ac” plane (Fig.  10d) leads to a two-fold-like signal , in \nwhich the expected moment flipping position is not every 90°.  \nV. CONCLUSIONS  \nWe have performed a detailed study of  the magnetocrystalline anisotropy of VI 3 by angular -\ndependent torque magnetometry and magnetization within the ab plane  and the perpendicular “ ac” \nplane  throughout the entire  10K-60K temperature interval covering  both the two ferromagnetic \nphases and the paramagnetic phase.  Irrespective of the temperature c hange, the symmetry of the two -\nfold butterfly -like angular signal within the “ac” plane remains conserve d throughout both FM \nphases . On the other hand, s ubstantial d evelopment of the signal was detected within the ab plane. \nThe d ouble peak -like signal in the paramagnetic region in the monoclinic phase transforms to the six-\nfold-like signal below TC. The original six -fold-like signal appearing at TC disappears  below TFM and \nthe second set of six -fold-like signal rotated  by about 30° sur vives down to the lowest temperatures.  \nThe gradual emergence  of the secondary six -fold-like signal  growth  between TC and TFM and the \nangular shift of magnetization maxima  about ~30° below TFM is the subject of further research.  Since \nthe torque contains a periodic function of 2 and 4, we corroborate  that the three -fold symmetry \npresent in the rhombohedral lattice is broken  in the studied temperature range.   \nOur findings are in agreement with the presence of low symmetry monoclinic and triclinic \nphases  at low temperatures . We have shown that the easy axis is not perpendicular to the layers, but \ncanted by around 40° from the c-axis in both FM phases. VI 3 is thus not a simple Ising -like \nferromagnet. For further progress in this field , it is essential  to have  detailed knowledge about the \npredicted  triclinic crystal structure  below TFM and a neutron diffraction study to reveal the VI 3 \nmagnetic structure .  \nACKNOWLEDGMENTS  \n12 \n This work is part of the research program GACR 19 -16389J which is financed by the Czech \nScience Foundation . Work at the Center for Quantum Materials was supported by the Leading \nResearcher Program of the Nation al Research Foundation of Korea (Grant No. 2020R1A3B2079375)  \nwith partial funding  by the grant IBS -R009 -G1 provided by the Institute f or Basic Science of the \nRepublic of Korea . Experiments were performed in the Materials Growth and Measurement \nLaboratory (see: http://mgml.eu ) which is supported within the program of Czech Research \nInfrastructures (project n o. LM2018096).  We are indebted to Ross H. Colman for making language \ncorrections.  \nREFERENCES  \n1 P. Ajayan, P. Kim, and K. Banerjee, Physics Today 69, 39 (2016).  \n2 K. S. Burch, D. Mandrus, and J. G. Park, Nature 563, 47 (2018).  \n3 W. Zhang, P. K. J. Wong, R. Zhu, and A. T. S. Wee, InfoMat 1, 479 (2019).  \n4 D. Zhong, K. L. Seyler, X. Y. Linpeng, R. Cheng, N. Sivadas, B. Huang, E. Schmidgall, T. \nTaniguchi, K. Watanabe, M. A. McGuire, W. Yao, D. Xiao, K. M. C. Fu, and X. D. Xu, \nScience Advances 3, e1603113 (2017).  \n5 J. G. Park, Journal of Physics: Condensed Matter, 301001 (2016).  \n6 F. F. Li, B. S. Yang, Y. Zhu, X. F. Han, and Y. Yan, Applied Surface Science 505, 144648 \n(2020).  \n7 J. Shang, X. Tang, X. Tan, A. J. Du, T. Liao, S. C. Smith, Y. T. Gu, C. Li, and L. Z. Kou, \nAcs Applied Nano Materials 3, 1282 (2020).  \n8 F. Subhan and J. S. Hong, Journal of Physical Chemistry C 124, 7156 (2020).  \n9 K. Yang, W. T. Hu, H. Wu, M. H. Whangbo, P. G. Radaelli, and A. Stroppa, Acs Applied \nElectronic Mater ials 2, 1373 (2020).  \n10 S. Tian, J. F. Zhang, C. Li, T. Ying, S. Li, X. Zhang, K. Liu, and H. Lei, Journal of the \nAmerican Chemical Society 141, 5326−5333 (2019).  \n11 M. Gronke, B. Buschbeck, P. Schmidt, M. Valldor, S. Oswald, Q. Hao, A. Lubk, D. Wolf, \nU. Steiner, B. Buchner, and S. Hampel, Advanced Materials Interfaces 6, 1901410 (2019).  \n12 S. Son, M. J. Coak, N. Lee, J. Kim, T. Y. Kim, H. Hamidov, H. Cho, C. Liu, D. M. Jarvis, P. \nA. C. Brown, J. H. Kim, C. H. Park, D. I. Khomskii, S. S. Saxena, and J. G. P ark, Physical \nReview B 99, 041402(R) (2019).  \n13 T. Kong, K. Stolze, E. I. Timmons, J. Tao, D. R. Ni, S. Guo, Z. Yang, R. Prozorov, and R. J. \nCava, Advanced Materials 31, 1808074 (2019).  \n14 T. Hattori, Y. Ihara, K. Ishida, Y. Nakai, E. Osaki, K. Deguchi, N. K. Sato, and I. Satoh, \nJournal of the Physical Society of Japan 80, SA007 (2011).  \n15 E. Gati, Y. Inagaki, T. Kong, R. J. Cava, Y. Furukawa, P. C. Canfield, and S. L. Bud'ko, \nPhysical Revie w B 100, 094408 (2019).  \n16 B. B. Lyu, Y. F. Gao, Y. Zhang, L. Wang, X. Wu, Y. Chen, J. Zhang, G. Li, Q. Huang, N. \nZhang, Y. Chen, J. Mei, H. Yan, Y. Zhao, L. Huang, and M. Huanb, Nano Letters 20, \n6024−6031 (2020).  \n17 J. Yan, X. Luo, F. C. Chen, J. J. Gao, Z. Z. Jiang, G. C. Zhao, Y. Sun, H. Y. Lv, S. J. Tian, \nQ. W. Yin, H. C. Lei, W. J. Lu, P. Tong, W. H. Song, X. B. Zhu, and Y. P. Sun, Physical \nReview B 100, 094402 (2019).  \n18 F. Subhan and J. S. Hong, Journal of Physi cs: Condensed Matter 32, 245803 (2020).  \n19 Z. P. Guo, Q. Chen, J. N. Yuan, K. Xia, X. M. Wang, and J. Sun, Journal of Physical \nChemistry C 124, 2096 (2020).  \n20 C. Long, T. Wang, H. Jin, H. Wang, and Y. Dai, Journal of Physical Chemistry Letters 11, \n2158 (2 020).  \n21 Y. L. Ren, Q. Q. Li, W. H. Wan, Y. Liu, and Y. F. Ge, Physical Review B 101, 134421 \n(2020).  13 \n 22 M. An, Y. Zhang, J. Chen, H. M. Zhang, Y. J. Guo, and S. Dong, Journal of Physical \nChemistry C 123, 30545 (2019).  \n23 K. Yang, F. Fan, H. Wang, D. I. Kho mskii, and H. Wu, Physical Review B 101, 100402 \n(2020).  \n24 Y. P. Wang and M. Q. Long, Physical Review B 101, 024411 (2020).  \n25 G. T. Lin, X. Luo, F. C. Chen, J. Yan, J. J. Gao, Y. Sun, W. Tong, P. Tong, W. J. Lu, Z. G. \nSheng, W. H. Song, X. B. Zhu, and Y. P. Sun, Applied Physics Letters 112, 072405 (2018).  \n26 P. Doležal, M. Kratochvílová, V. Holý, P. Čermák, V. Sechovský, M. Dušek, M. Míšek, T. \nChakraborty, Y. Noda, S. Son, and J. G. Park, Physical Review Materials 3, 121401(R) \n(2019).  \n27 W. Liu, Y. H. Dai,  Y. E. Yang, J. Y. Fan, L. Pi, L. Zhang, and Y. H. Zhang, Physical \nReview B 98, 214420 (2018).  \n28 Y. Liu and C. Petrovic, Physical Review B 97, 174418 (2018).  \n29 J. L. Lado and J. F. Rossier, 2D Materials 4, 035002 (2017).  \n30 L. Webster and J. A. Yan, Phys ical Review B 98, 144411 (2018).  \n31 V. K. Gudelli and G. Y. Guo, New Journal of Physics 21, 053012 (2019).  \n32 D. Torelli and T. Olsen, 2D Materials 6, 015028 (2018).  \n \nAPPENDIX: Torque expression in the paramagnetic state  \n Considering the situation where time-reversal symmetry is preserved, we expand the \nmagnetization  as a function of the magnetic field as follows.  \n𝑀𝑖=𝜒𝑖𝑗(2)𝐻𝑗+𝜒𝑖𝑗𝑘𝑙(4)𝐻𝑗𝐻𝑘𝐻𝑙+𝜒𝑖𝑗𝑘𝑙𝑚𝑛(6)𝐻𝑗𝐻𝑘𝐻𝑙𝐻𝑚𝐻𝑛+⋯(𝑖,𝑗,𝑘=𝑥,𝑦 or 𝑧) \nEach χ(k) in this formula  represents the magnetic susceptibility tensors of the k-th rank, and \nhere, t he general sum reduction rule is used. Since these tensors are symmetrical with respect to the \nintercha nge of indices, the distinction between independent components  is determined only by the \nnumber of the subscripts x, y, and z . Therefore, for simplicity, we have  decided to denote these tensors \nby the number of its xyz , for example, as follows.  \n𝜒𝑦𝑧(2)≡𝜒011, 𝜒𝑥𝑦𝑦𝑧(4)≡𝜒121, 𝜒𝑥𝑥𝑥𝑥𝑥𝑥(6)≡𝜒600 \n We consider  the situation where a magnetic field is applied in the ab(xy) plane, so ( Hx, Hy, \nHz) = (Hcosφ, Hsinφ, 0). Then the general equation for the torque up to the 6th order can be written \nas follows.  \n𝜏𝑧={1\n2(𝜒200−𝜒020)𝐻2+1\n4(𝜒400−𝜒040)𝐻4+5\n32(𝜒600+𝜒420−𝜒240−𝜒060)𝐻6}sin2𝜙\n−{𝜒110𝐻2+1\n2(𝜒310+𝜒130)𝐻4+5\n16(𝜒510+2𝜒330+𝜒150)𝐻6}cos2𝜙\n+{1\n8(𝜒400−6𝜒220+𝜒040)𝐻4+1\n8(𝜒600−5𝜒420−5𝜒240+𝜒060)𝐻6}sin4𝜙\n−{1\n2(𝜒310−𝜒130)𝐻4+1\n2(𝜒510−𝜒150)𝐻6}cos4𝜙\n+{1\n32(𝜒600−15𝜒420+15𝜒240−𝜒060)𝐻6}sin6𝜙−{1\n16(3𝜒510−10𝜒330+3𝜒150)𝐻6}cos6𝜙 \nGenerally, t he tensor is limited in its component s by the point group symmetry of the crystal.  \nEq.1 in the discussion considers a situation in which there are three -fold rotational symmetries around \nthe c(z) axis. In this case, e ach component  of susceptibility  has the following relationship to each \nother . \n𝜒200=𝜒020, 𝜒110=0 \n𝜒400=3𝜒220=𝜒040, 𝜒310=𝜒130=0 \n𝜒600=5𝜒420=5𝜒240=𝜒060, 𝜒510=−𝜒330=𝜒150 14 \n Taking this into account, the equation of torque is expressed as follows.  \n𝜏𝑧=𝜒330𝐻6cos6𝜙 \n "    },    {        "title": "2101.10773v1.Data_driven_design_of_a_new_class_of_rare_earth_free_permanent_magnets.pdf",        "content": "Data-driven design of a new class of rare-earth free permanent magnets\nAlena Vishina,1,\u0003Daniel Hedlund,1Vitalii Shtender,2Erna K. Delczeg-Czirjak,3\nSimon R. Larsen,2Olga Yu. Vekilova,4,3Shuo Huang,3Levente Vitos,5,3\nPeter Svedlindh,1Martin Sahlberg,2Olle Eriksson,3,6and Heike C. Herper3\n1Department of Materials Science and Engineering,\nUppsala University, Box 35, 751 03 Uppsala, Sweden\n2Department of Chemistry - Ångström, Uppsala University, Box 538, 751 21, Uppsala, Sweden\n3Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120, Uppsala, Sweden\n4Department of Materials and Environmental Chemistry, Stockholm University, 10691 Stockholm, Sweden\n5Applied Materials Physics, Department of Materials Science and Engineering,\nRoyal Institute of Technology, Stockholm, SE-100 44, Sweden\n6School of Science and Technology, Örebro University, SE-701 82 Örebro, Sweden\n(Dated: January 27, 2021)\nA new class of rare-earth-free permanent magnets is proposed. The parent compound of this class\nis Co 3Mn2Ge, and its discovery is the result of first principles theory combined with experimen-\ntal synthesis and characterisation. The theory is based on a high-throughput/data-mining search\namong materials listed in the ICSD database. From ab-initio theory of the defect free material it is\npredicted that the saturation magnetization is 1.71 T, the uniaxial magnetocrystalline anisotropy\nis 1.44 MJ/m3, and the Curie temperature is 700 K. Co 3Mn2Ge samples were then synthesized and\ncharacterised with respect to structure and magnetism. The crystal structure was found to be the\nMgZn 2-type, with partial disorder of Co and Ge on the crystallographic lattice sites. From mag-\nnetization measurements a saturation polarization of 0.86 T at 10 K was detected, together with a\nuniaxial magnetocrystalline anisotropy constant of 1.18 MJ/m3, and the Curie temperature of TC\n= 359 K. These magnetic properties make Co 3Mn2Ge a very promising material as a rare-earth free\npermanent magnet, and since we can demonstrate that magnetism depends critically on the amount\nof disorder of the Co and Ge atoms, a further improvement of the magnetism is possible. From\nthe theoretical works, a substitution of Ge by neighboring elements suggest two other promising\nmaterials - Co 3Mn2Al and Co 3Mn2Ga. We demonstrate here that the class of compounds based on\nT3Mn2X (T = Co or alloys between Fe and Ni; X=Ge, Al or Ga) in the MgZn 2structure type, form\na new class of rare-earth free permanent magnets with very promising performance.\nI. INTRODUCTION\nRare earth (RE) permanent magnets dominate the\nmarket where high-performance magnetic materials are\nneeded - areas such as green-energy generation, includ-\ning wind and wave power, electric vehicle motors and\ngenerators, and many more. At the same time, the heav-\nier rare-earth elements which are necessary for obtaining\nthegoodmagneticcharacteristicsofthesematerials(such\nas Pr, Nd, Sm, Tb, or Dy) [1], can only be mined with\nmethods that leave an environmental footprint, are quite\nexpensive, and are rapidly decreasing in availability. As\na consequence many rare-earth elements are labelled crit-\nical. In order to pursue green technologies that rely on\nhigh-performance magnets, there is a growing interest in\nfinding new magnetic materials containing cheaper and\nless critical elements, while maintaining a similar high-\nperformance shown by their RE counterparts.\nIn the last decades, many attempts have been made to\ndesign both RE-lean [2, 3] and RE-free magnetic materi-\nals. Thelaterincludehighmagnetocrystallineanisotropy\nalloys (MnBi [4, 5] and MnAl [6, 7]), nanostructures [8–\n\u0003Department of Physics and Astronomy, Uppsala University, Box\n516, SE-75120, Uppsala, Sweden; alena.vishina@physics.uu.se11], thin films [12, 13], alnico permanent magnets, and\nmany more [14–17]. Techniques used for this search in-\nclude both filtering through a large number of systems\n[18–21] and investigating a single or a group of similar\ncompounds[14,15,22–29]. InRef. [18]itwasshownthat\na high-throughput density functional theory (DFT) ap-\nproachcanbeeffectiveinfilteringthroughalargenumber\nof compounds in the search for new high-performance RE\nfreepermanentmagnets, whichwouldbetime-consuming\nand expensive to synthesize experimentally. Unfortu-\nnately, a commercially relevant class of rare-earth free\ncompounds was not identified in Ref. [18]. In the cur-\nrent investigation, that data-filtering approach described\nin Ref. [18] was broadened to involve a much larger start-\ning set of compounds.\nFrom a combination of data-mining efforts, using ab-\ninitio electronic structure theory, materials synthesis,\nstructural and magnetic characterization, we can report\nonanewclassofrare-earthfreepermanentmagnets,with\nvery promising properties. In particular, we have identi-\nfied Co 3Mn2Ge, which is a new material that previously\nhas not been considered as a permanent magnet.arXiv:2101.10773v1  [cond-mat.mtrl-sci]  26 Jan 20212\nII. HIGH-THROUGHPUT AND DATA-MINING\nSEARCH\nAn initial attempt to use high-throughput searches for\nRE-free permanent magnets was published in Ref.[18],\nemploying a smaller subset of possible magnetic com-\npounds. As described in this investigation, a high-\nperformancepermanentmagnetmusthaveferromagnetic\nordering, high saturation magnetization ( MS>1T),\nhigh Curie temperature ( TC>400 K), a large uniax-\nial MAE (>1MJ/m3), and a magnetic hardness larger\nthan 1 (fo details see the Methods section). In the initial\nattempt of using high-throughput/data-mining searches,\nfocus was put on materials with a stoichiometry neces-\nsarily containing one 3 dand one 5 delement [18]. From\na pure magnetic viewpoint, this study resulted in several\npromising materials. Unfortunately, none of the identi-\nfied compounds in Ref.[18] could serve as practical re-\nplacements for the known rare-earth based compounds,\nsince they all contained expensive elements like Pt.\nIn the present study, we have made an extended search\nto involve materials with a stoichiometry that contain\ntwo types of 3d-elements, without any restriction on the\ntotal number of elements in the system. About a thou-\nsand materials from the ICSD database [30] contain two\ndifferent 3d-elements in the chemical formula. After the\nfirst high-throughput calculations of the electronic struc-\nture and magnetic moments, about 45 systems were iden-\ntified to have a magnetic saturation field larger than a\nvalue\u00180.4 T. In the next step, all the materials with\ncubic symmetry were set aside, since spin-orbit effects\nare known to influence the MAE most effectively for non-\ncubic materials [31]. We also performed an extensive lit-\nerature search, to identify if any of the compounds iden-\ntified in the intitial screening steps, had previously been\nreported as permanent magnet (e.g. the L1 0phase of\nFeNi) or if the systems that were identified, are known\nto not be ferromagnetic. These compounds were also re-\nmoved from the list of potential new rare-earth free per-\nmanent magnets. For the remaining materials, the MAE\nwas calculated. Compounds with planar or low magnetic\nanisotropy (MAE <\u00180.6 MJ/m3) were not considered\nfurther, leavingonlyfivematerialswhosesaturationmag-\nnetization and MAE are high enough to be considered as\nhigh-performance permanent magnets.\nNote that the search criteria used here are somewhat\nless strict than the criteria of magnetic properties spec-\nified from practical aspects of high performance perma-\nnent magnets. The reason for using less strict criteria,\nis to not miss new classes of compounds, that have the\npotential to become technologically relevant, after e.g al-\nloying or structural refinements. Using these criteria we\nhave identified the following compounds as potential new\npermanent magnets: ScFe 4P2, Co 3Mn2Ge, Mn 3V2Si3,\nScMnSi, and Cu 2Fe4S7and we report in Table I their cal-\nculated saturation magnetization, MAE, and magnetic\nhardness. All data were obtained from calculations at\nT= 0K and for a ferromagnetic configuration. Forthis final list of materials, we also calculated the Heisen-\nberg exchange interaction, and from Monte Carlo simu-\nlations we identified the ground state magnetic configu-\nration and, for materials with ferromagnetic configura-\ntions, the Curie temperature. Out of the five materials\nlisted above, Co 3Mn2Ge is the only one that is FM and\nhas aTCof 700 K. The rest of the materials were found\nto have a non-collinear magnetic structure and were for\nthis reason not considered further. Some of the mate-\nrials, discarded during the search, can be found in the\nAppendix , Table III. We’d like to note, that Co 3Mn2Ge\ndoes have its magnetic hardness lower than 1. However,\nthis parameter can be later on adjusted by alloying or\nother methods.\nTABLE I: Calculated MAE, saturation magnetization,\nCurie temperature, and magnetic hardness for the\nmaterials that can be considered as candidates for\nRE-free permanent magnets. NC stands for\nnon-collinear spin structure obtained from Monte Carlo\nsimulations.\nMaterial ICSD Space MAE Sat. TC\u0014\nnumber group MJ/m3magn., T K\nScFe 4P268525 136 0.71 0.65 NC 1.45\nCo3Mn2Ge 52972 194 1.44 1.71 700 0.79\nMn3V2Si3643689 193 0.85 0.73 NC 1.42\nScMnSi 86369 189 0.69 0.52 NC 1.79\nCu2Fe4S715973 51 0.90 0.37 NC 2.87\nIII. EXPERIMENTAL RESULTS\nA. Synthesis and structural characterisation\nFollowing the theoretical results presented above, fo-\ncus was then put on synthesis of Co 3Mn2Ge. Initial tri-\nals revealed the magnetic Heusler phase Co 2MnGe [32]\nas being the main competing phase in that region of the\nphase diagram. Preliminary structural refinement of the\nhexagonal Co 3Mn2Ge phase (MgZn 2-type) in a multi-\nphasesampleindicateddisorderingonthe6 hand2 asites\nimplying intermixing between Co and Ge. EDS analysis\non the multi-phase samples revealed the composition of\nthe desired hexagonal phase to be in a small homogeneity\nregion spanning Co 52Mn34Ge14and Co 53:7Mn31:7Ge14:6.\nSamples following these stoichiometries were then syn-\nthesized. The Co 52Mn34Ge14sample consisted of 95.4\nwt% of the hexagonal phase and 4.6 wt% of CoMn, a\nPauli paramagnet [33]. The diffractogram and accom-\npanying structural refinement is shown in Fig. 1. The\nstructure of the compound (R Bragg= 4.20 %) is in good\nagreement with previous studies [34] having similar unit\ncell dimensions (V = 154.60(1) \u0017A3compared to 154.61\n\u0017A3[34]). The final refinement resulted in a composition\nof Co 3:39Mn2Ge0:61with intermixing of Co and Ge on\nthe 6 hand 2 asites.3\nFIG. 1: (Color online) Refined powder diffraction data\nof the synthesized Co 52Mn34Ge14alloy. Observed\n(Yobs), calculated (Y calc), difference (Y obs-Ycalc)\ndiffraction profiles and Bragg’s peaks positions for\nCo3Mn2Ge (95.4 wt.%) and CoMn (4.6 wt.%) phases\nare shown.\nResults from XRPD refinements and microstructural\nEDS results indicate that the structure is disordered with\na tendency to contain more Co. Kuz’ma et al. [34] pro-\nposed both the ordered Mg 2Cu3Si-type and disordered\nMgZn 2-type structures from their XRPD results but a\nclear atomic distribution could not be established. To\ndetermine the atomic distribution for the theoretical cal-\nculations, precisionsinglecrystalstudieswereconducted.\nThreevariantsofthestructurewereconsideredduringre-\nfinement and are presented in Table II - the completely\nordered structure, Co-Ge intermixing, and Mn-Ge inter-\nmixing. The structure solution obtained with SHELXT-\n2014 quickly provided an ordered model for Co 3Mn2Ge.\nHowever, the wR 2value, the EDS results, and the dif-\nference Fourier map all suggest that this model should\nbe rejected. The model of Mn-Ge intermixing showed\na larger goodness-of-fit and the calculated composition\nwas too far from the measured to be accepted. The fi-\nnal model of Co-Ge intermixing overall shows the best\nparameters and is in agreement with other results. All\natoms were refined anisotropically. Detailed crystallo-\ngraphic results of the SCXRD refinements can be found\nin the Appendix , Table IV.\nB. Experimental magnetism\nThe magnetization of the Co 3Mn2Ge sample was mea-\nsured as a function of temperature and magnetic field\nto determine the magnetic ordering temperature TC,\nthe saturation magnetization, and the effective magnetic\nanisotropy constant of the material. The temperature\ndependent magnetization measurements were performedat several applied magnetic fields and in the temperature\nregion between 10 K and 900 K. The results are shown in\nFig. 2. The magnetic ordering temperature, TC= 359K,\nwas determined from the Curie-Weiss law fit to the tem-\nperature dependence of the inverse magnetic susceptibil-\nity (cf. inset in Fig. 2).\nFIG. 2: (Color online) Magnetization of Co 3Mn2Ge as a\nfunction of temperature in applied magnetic fields of\n\u00160H = 0.01 T (white open circles and white open\nrectangles) and \u00160H = 1 T (red filled rectangles). The\ninset shows the Curie-Weiss fit for the inverse magnetic\nsusceptibility with TC= 359K. The cusp in \u00160H = 1 T\nand drop in magnetization around 175 K is attributed\nto a spin–reorientation ( Tsrt).\nThe isothermal magnetization curves are shown in Fig-\nure 3. Note that Figure 3(a) shows the result for bulk\npowders whereas Figure 3(b–d) corresponds to single\ncrystal measurements at different temperatures along ( k)\nthe crystallographic c–axis and perpendicular ( ?) to it.\nIn Figure 3(a) the isothermal magnetization for the pow-\nder sample, measured at 10 K and 70 K, shows the same\napproachtosaturationandreachthesamesaturationpo-\nlarization of 0.86 T. The isothermal magnetization mea-\nsured at 170 K reaches a saturation polarization close to\nthat recorded at lower temperature; whereas the satura-\ntion polarization measured at 300 K reaches a value of\n0.60 T.\nTo analyze the behavior of the magnetic anisotropy\nand to compare to the value calculated theoretically, we\nhave used the law of approach to saturation to calcu-\nlate the effective anisotropy constant at different temper-\natures. The results are shown in Figure 4. At 10 K and\n70 K, the law of approach to saturation yields an effec-\ntiveanisotropyconstantof1.18MJ/m3andthemagnetic\nhardness parameter, \u0014, becomes 1.42. The experimental\nresults of magnetic configuration, saturation field, and\nmagnetic anisotropy hence seem consistent with the the-\noreticalpredictionsofthiscompound, whichisgratifying.\nA closer inspection of the results presented in Fig. 2 re-\nveal that below\u0018175 K the magnetization drops consid-4\nTABLE II: Results of single crystal refinements of the Co 3+xMn2Ge1\u0000xcompound. The Co-Ge disordered model\n(marked in bold) describes the measured data best.\nParameters Ordered Co-Ge disordered Mn-Ge disordered\nOccupation 6 h Co 0.84Co+0.16Ge Co\n4f Mn Mn Mn\n2a Ge 0.74Co+0.26Ge 0.4Ge+0.6Mn\nCalculated formula Co 3Mn2Ge Co3:24Mn 2Ge0:76Co3Mn2:6Ge0:4\nCalculated composition (at.%) Co 50Mn33:33Ge16:67Co54Mn 33:33Ge12:67Co50Mn43:33Ge6:67\nGoodness-of-fit on F21.214 1.170 1.353\nFinal R indices (I >2s(I)) R 1= 0.0432 R1= 0.0100 R1= 0.0149\nwR2= 0.1188 wR 2= 0.0264 wR2= 0.0360\nR indices (all data) R 1= 0.0439 R1= 0.0111 R1= 0.0160\nwR2= 0.1192 wR 2= 0.0268 wR2= 0.0364\nHighest difference peak 2.277 0.639 0.693\nDeepest hole -5.961 -0.420 -1.003\n1-\u001blevel 0.569 0.119 0.158\nFIG. 3: (Color online) (a) Magnetization of a bulk\npowder of Co 3Mn2Ge, as a function of magnetic field at\n10 K, 70 K, 170 K and 300 K. (b,c,d) Isothermal\nmagnetization of single crystals of Co 3Mn2Ge at 300 K,\n200 K and 100 K. Filled symbols show measurements\nwhere the magnetic field is perpendicular ( ?) to the\ncrystallographic c–axis whereas open symbols show\nmeasurements with the magnetic field parallell ( k)with\nthec–axis.\nerably when the applied field is small (0.01 T), whereas\na cusp–like feature is seen in an applied magnetic field\nof 1 T, both for single crystal material and bulk sam-\nples. Such features can be attributed to a change in mag-\nnetocrystalline anisotropy with temperature and we in-\nterpret this to be a spin–reorientation temperature Tsrt,\nfrom easy–axis to easy–cone anisotropy, a feature which\nis not uncommon for permanent magnets (e.g. Fe 5SiB2\n[35], MnBi [36], Cr 0:9B0:1Te [37], Gd [38] and the cel-\nebrated Nd 2Fe14B [39]). Before presenting further re-\nsults supporting a claim of a temperature induced spin–\nreorientation, we draw attention to the fact that for a\nFIG. 4: (Color online) Approach to saturation plotted\nas the magnetization normalized with the value\nmeasured at 5 T as a function of inverse applied field\nsquared for the bulk sample of Co 3Mn2Ge.\nhexagonal uniaxial material the magnetic anisotropy en-\nergy can be expressed as MAE =K1cos2\u0012+K2sin4\u0012,\nwhereVis the volume, Kiare the anisotropy constants,\nand\u0012is the angle between the magnetization vector and\nthe hexagonal c–axis [40]. Depending on the values of\nK1andK2three cases can be distinguished; the ma-\nterial has an easy–axis, an easy–plane or an easy–cone\nmagnetic anisotropy. The easy-cone state is the most fa-\nvorable one when \u00002K2\u0014K1\u00140. The temperature\ndependence of the anisotropy constants ( Ki(T)) may be\nstrong at low temperature and is usually described by\na power–law behavior, Ki(T)/M(T)\u000b[41, 42], where\nM(T)is the magnetization and \u000bis a constant. The tem-\nperaturedependenceoftheanisotropyconstantscanthen\nlead to a spin–reorientation, e.g. from an easy–axis to an\neasy–plane or an easy–axis to an easy–cone anisotropy.\nThe material under investigation here shows charac-\nteristics of an easy–axis to easy–cone spin–reorientation5\naround 175 K. We support this claim with the aid of\nisothermal magnetization curves recorded at tempera-\ntures below and above Tsrton the single crystals and\nbulk powders. The single crystal isotherms presented in\nFigure3(b–d)indicatethattheeasy–conestateispresent\nat100Kandbelow. Lookingatthemagnetizationcurves\nat 300 K (Figure 3(b)) we see that it is easier to mag-\nnetize the crystal along the c–axis than perpendicular to\nit. This shows that the material is magnetically uniax-\nial at 300 K. The same applies at 200 K (Figure 3(c)),\nwhereas at 100 K the measurements show that it is as\neasy to magnetize the material parallel or perpendicu-\nlar to the crystallographic c–axis. However, it should be\nnoted that at low field ( <0:5T) there is a deviation\nfrom a linear magnetic response in measurements per-\nformed parallel or perpendicular to the crystallographic\nc–axis. The non–linear behaviour at low fields, together\nwiththelowfieldmagnetizationversustemperaturemea-\nsurements support the easy–cone state [43].\nFigure4supportstheconclusionofamorecomplicated\nbehaviour for the magnetization of Co 3Mn2Ge. It should\nbe noted, that in order to extract an anisotropy constant\nusing the law of approach to saturation, averaging over a\nrandom distribution of crystal directions yields the con-\nstant\f= 4=15[44]. The effective anisotropy constants\nand magnetic hardness parameter should thus be seen\nas the tentative descriptions of the magnetocrystalline\nanisotropy at low temperatures.\nSummarizing the experimental results so far we con-\nclude that the theoretical prediction of Co 3Mn2Ge as a\npotential rare-earth free permanent magnet is likely. The\nsaturation magnetization, uniaxial magnetic anisotropy,\nand magnetic hardness parameter obtained from experi-\nments are consistent with the theoretical predictions. It\nmust be noted that a more detailed picture is present in\nthe experiment, with a temperature stabilized easy-cone\nstate. Thisstatewasnotspecifiedinthetheoreticalhigh-\nthroughput screening search, and could therefore not be\nexpected from the theoretical prediction. However, the\neasy axis, order of magnitude of the anisotropy, the sat-\nuration magnetization, and ordering temperature should\nagree, and here the theory has proven useful in finding a\nnew material, with a potential to be used as a permanent\nmagnet. We return to the easy cone state, below, with\na theory that takes disorder into account, and show that\ncalculations then reproduce experimental observations.\nIV. DISORDERED Co 3Mn 2Ge\nAccording to the experimental reports, there may be a\n50-50%intermixingofCoandGeatomson6 hand2 a[34]\nWyckoff positions (see Fig. 5), in addition to the excess\nof Co in comparison to Ge (see Table II), in the exper-\nimental samples. This structure and composition differ\nfrom the one reported in the ICSD [30] database, where\nCo atoms are reported to occupy 6 h, Mn atoms take 4 f,\nand Ge occupies 2 aatomic sites. It should be noted,that disorder of the type detected in our experiments is\noften observed in the related ternary Laves phases with\nMgZn 2-type [45]. In this section, we, therefore, consider,\nfrom ab-initio alloy theory, the disorder of Co 3Mn2Ge, in\norder to analyze the effect it has on the magnetic state of\nthe material and to compare the results with the theoret-\nical data obtained for the ordered structure. The effect of\nchemical and magnetic disorder on the stability and mag-\nnetization of Co 3Mn2Ge was investigated as described in\ntheMethods section (Sec. VII).\nThe results of these theoretical calculations is that the\nordered structure has a lower total energy, compared to\nthe disordered one. However, configurational entropy\ncontributions to the free energy are reported in the ap-\npendix to stabilize the disordered structure at tempera-\ntures 1700 K. The details can be found in Appendix C .\nFIG. 5: (Color online) Ordered Co 3Mn2Ge (as per\nICSD) with Co atoms occupying 6 h(light red balls),\nMn atoms taking 4 f(violet balls), and Ge (grey balls)\noccupying 2 aatomic sites (top); and the disordered\nCo3Mn2Ge structure obtained experimentally with\n50-50% intermixing between the Co and Ge sites\n(bottom).\nFollowing the experimental findings that point to an\neasy-conemagneticanisotropyatthelowertemperatures,\nwe calculated MAE for several magnetization directions\n(0\u000e< \u0012 < 90\u000e,\u001e= 0\u000e) for the ordered Co 3Mn2Ge and\ndisordered Mn 2(Co0:75Ge0:25)4phases, see Fig. 6. In-\ndeed, we can see that the disordered system shows a\npronounced deviation from the simple uniaxial behavior.\nHence, the easy-cone anisotropy observed at lower tem-\nperatures is attributed to the Co-Ge disorder observed\nin the samples. At elevated temperature, the influence\nof disorder evidently is less pronounced, and the uniax-\nial anisotropy that was obtained from theory of ordered6\nsamples, is recovered in the experiments.\nFIG. 6: (Color online) MAE for several magnetization\ndirections ( \u001e= 0\u000e) for the ordered Co 3Mn2Ge and\ndisordered Mn 2(Co0:75Ge0:25)4phases.\nV. CHEMICAL SUBSTITUTION OF Ge\nIn order to find related compounds with similar or im-\nproved magnetic properties compared to Co 3Mn2Ge, we\nperformedadditionaltheoreticalwork, wherewereplaced\nall Ge atoms in the unit cell by Al, Si, P, Ga, As, In, Sn,\nSb, Tl, and Pb. All structures were relaxed and tested\nfor stability by calculating the formation enthalpy with\nrespecttotheirelementalcomponents(fordetailsseesec-\ntionMethods). The systems found to be stable with re-\nspect to the elemental components (Tl- and Pb-based\nmaterials turned out to be unstable) were investigated\nfurther for their magnetic characteristics. Their MAE\nand the details of the magnetic state are given in Table\nVII.\nFIG. 7: (Color online) Magnetic anisotropy energy of\nCo3Mn2X, with X = Al, Si, P, Ga, Ge, As, In, Sn, Sb.\nPositive values indicate a uniaxial anisotrophy.Replacing Ge in Co 3Mn2Ge by the neighboring ele-\nments, does not change to the magnetic state much, but\nit has a large effect on the MAE, see Fig. 7. To de-\ntermine the origin of the difference in MAE values we\nanalyzed the spin-orbit coupling energy (SOC) of the\nCo and Mn atoms with their spins oriented along the\nzandxdirections as well as the partial density of states\n(DOS), similar to the analyses presented in Refs.[46–51].\nThe details of that investigation can be found in Ap-\npendixD. Summarizing the results, we find that apart\nfrom the Co 3Mn2Ge compound that is listed in the ICSD\ndatabase, Co 3Mn2Al and Co 3Mn2Ga are also expected\nto have the properties of a good permanent magnet. We\nhave shown that the main contribution to the large MAE\nof these compounds arises from Co atoms. A synthesis\nand charaterisation of these two compounds is outside\nthe scope of this investigation, but represents clearly an\ninteresting avenue forward.\nVI. DISCUSSION AND CONCLUSIONS\nUsing the high-throughput and data-mining approach\nwe filtered through the RE-free materials of the ICSD\ndatabase that contain necessarily two 3 d-elements. Only\none of those approximately thousand structures satis-\nfied our requirements for a strong permanent magnet,\nCo3Mn2Ge, with the saturation magnetization of 1.71\nT, MAE equal to 1.44 MJ/m3, andTCof 700 K. To\nreduce the cost of this material, we attempted to re-\nplace Ge by other elements. Two materials, Co 3Mn2Al\nand Co 3Mn2Ga, were found to have sufficient magneti-\nzation, MAE, and Curie temperature to be used as high-\nperformance permanent magnets. The former of the two\ncan be problematic to produce due to the competing\nHeusler compound being considerably more stable; the\nlatter doesn’t have a similar drawback.\nCo3Mn2Ge was successfully synthesized using induc-\ntion melting followed by annealing at 1073 K. The re-\nsulting crystal structure was in good agreement with the\npreviously published results. However, when the order-\ning of the system was investigated with precision single\ncrystal analysis, it indicated that the disordered struc-\nture type MgZn 2with intermixing Co and Ge on the 6 h\nand 2 asites was preferred. The structure taken from the\nICSD database and used in the high-throughput calcula-\ntion was, unlike the experimental one, ordered.\nMagnetic characteristics of the synthesized compound\nwere measured and produced the saturation magnetiza-\ntion of 0.86 T at the temperature of 10 K, uniaxial mag-\nnetic anisotropy equal to 1.18 MJ/m3above around 175\nK (with the easy-cone character of anisotropy below),\nandTC=359 K. These values, even though lower than\nthe theoretical predictions, make Co 3Mn2Ge a promis-\ning candidate for a high-performance permanent magnet.\nFurther analysis of its magnetic state as well as the ways\nto tune some of these numbers is highly desirable and\nrepresents ongoing work.7\nCalculations were also performed for the disordered\ncrystal structure of Co 3Mn2Ge obtained in the experi-\nment. We found that the crystallographic ordered struc-\nture with FM ordering is more stable than both ferro-\nmagnetically and antiferromagnetically ordered config-\nurations of the structurally disordered compound, in a\ntemperature range of 0 – 1735 K. Calculations of the\nmagnetocrystalline anisotropy were also performed for\nthe disordered Mn 2(Co0:75Ge0:25)4phase. These calcu-\nlations show a pronounced deviation from the uniaxial\ncharacter of magnetic anisotropy, in agreement with our\nexperimental low temperature data. Combining the re-\nsults of the theoretical and experimental works, leads\nto the conclusion that structural disorder influences the\nmagnetic properties, including the MAE, of Co 3Mn2Ge\nin a detrimental way, and that the best properties are\nexpected for the most ordered samples.\nIn the Appendix H we list the properties of the all\nthe previously known Co–Mn–Ge systems. Based on\nthat information we can conclude, that the disorder is\nquite common in all the Co–Mn–Ge compounds and\ndepends strongly on the sample preparation and ex-\nperimental procedures. With that in mind, we be-\nlieve that further investigation into the sample prepa-\nration or some possible doping alternatives is necessary\nfor an attempt to stabilize Co 3Mn2Ge in its ordered\nform, which is expected to have higher magnetization\nand magnetic anisotropy. Having said that, even the dis-\nordered Mn 2(Co0:75Ge0:25)4phase, with saturation po-\nlarization of 0.86 T and uniaxial magnetic anisotropy\nof 1.18 MJ/m3, possesses the characteristics of a good\npermanent magnet, although a further investigation into\nincreasing the TC= 359 is desirable.\nIt is also worth mentioning, that orientation of magne-\ntocrystallineanisotropyandthemagneticmomentofCo–\nMn–Ge systems is strongly affected by Co:Mn ratio. The\nsynthesized sample of Co 3Mn2Ge has the actual compo-\nsition of Co 52Mn34Ge14which might result in the dis-\ncrepancy between the magnetic results obtained experi-\nmentally and the ones predicted in the high-throughput\nsearch. It was proven that the easy-cone magnetocrys-\ntalline anisotropy, observed at the temperatures below\naround 175 K, is the result of the Co–Ge disorder. We\nwould like to point out, however, that one of the best\ncurrent permanent magnets, Nd 2Fe14B [39], possesses a\nsimilar feature. Nevertheless, this fact is another incen-\ntive to look for the way of stabilizing the ordered phase\nof Co 3Mn2Ge.\nThe current investigation is a promising example of\nthe material found in the theoretical data-mining search\nbeing synthesized and showing the desired characteris-\ntics of a good permanent magnet. Further experimental\ninvestigation into the magnetic state of Co 3Mn2Ge will\nbe performed as well as the computational search for im-\nproving its price-performance. With its Curie tempera-\nture close to the room temperature, we can also consider\nfine-tuning Co 3Mn2Ge for magnetocaloric applications.VII. METHODS\nA. High-throughput DFT\nThe high-throughput screening step to calculate the\nmagnetic moment of the materials was performed using\nthe full-potential linear muffin-tin orbital method (FP-\nLMTO) including spin-orbit interaction as implemented\nin the RSPt code [52, 53]. For this step the initial mag-\nnetic configuration for all the materials was ferromag-\nnetic (FM).\nThe magnetic state of materials (FM or antiferro-\nmagnetic - AFM), unless previously known, was de-\ntermined using Vienna Ab Initio Simulation Package\n(VASP) [54–57] within the Projector Augmented Wave\n(PAW) method [58], along with the Generalized Gradi-\nent Approximation (GGA) in Perdew, Burke, and Ernz-\nerhof (PBE) form [59]. VASP was also used for structure\nrelaxation at the post-high-throughput stage as well as\nto calculate spin-orbit coupling energies, for the analysis\npresented in the Appendix.\nThemagneticanisotropyenergy(MAE)wascalculated\nusing the RSPt code as \u0001E=Epl\u0000Ec; hereEcandEpl\nare the total energies with the magnetization directed\nalong and perpendicular to the c-axis. Calculations were\nperformed with the tetrahedron method with Blöchl cor-\nrection for the Brillouin zone integration [60]. A positive\nsign of the MAE corresponds to the required uniaxial\nanisotropy. For the disordered Co 3Mn2Ge, the MAE was\nevaluated for several polar angles from the force theorem\n[61, 62] as the difference of the eigenvalue sums for the\ntwo magnetization directions e\u0012andec(\u0012is a polar angle\nwhile azimuthal angle is equal to zero), while keeping the\neffective potential fixed. From the first principles calcu-\nlations, the magnetic hardness parameter was evaluated\nfrom the expression \u0014=p\n\u0001E=\u0016 0M2\nS[63], where MSis\nsaturation magnetization and \u00160is the vacuum perme-\nability.\nThe Curie temperature TCwas calculated using Monte\nCarlosimulationsimplementedwithintheUppsalaatom-\nistic spin dynamics (UppASD) software [64]. Atom-\nistic spin dynamics calculations were performed on a\n30\u000230\u000230supercell with periodic boundary conditions.\nThe required exchange parameters were calculated with\nthe RSPt code within the first nine coordination shells\n[65].\nFormation enthalpies of the materials were calculated\nwith respect to their elemental components as\n\u0001H=HCo3Mn2X\u00003HCo\u00002HMn\u0000HX;\nfor X= Al, Si, P, Ga, As, In, Sn, Sb, Tl, and Pb.\nIn this expression HCo3Mn2X,HCo,HMn, andHX\nare the enthalpies of formation for Co 3Mn2X, hexago-\nnal close-packed (hcp) cobalt, body-centered cubic (bcc)\nmanganese, and element X (such as, for example, cu-\nbic close-packed (ccp) aluminum for Co 3Mn2Al), respec-\ntively. Formation enthalpies with respect to Heusler al-\nloys Co 2MnX were obtained according to the following8\nformula\n\u0001H=HCo3Mn2X\u0000HCo2MnX\u0000HCo\u0000HMn;\nwhereHCo,HMn, andHCo2MnXare the enthalpies\nof hcp cobalt, bcc manganese, and the corresponding\nHeusler alloy.\nThe effect of chemical and magnetic disorder on the\nstability and magnetization of Co 3Mn2Ge was investi-\ngated by means of the coherent potential approximation\n[66, 67] as implemented in the Exact Muffin-Tin Or-\nbitals (EMTO) method [68, 69]. We used s,p,dand\nforbitals in the basis set. The one-electron equations\nwere solved within the soft-core and scalar-relativistic\napproximations. The Green’s function was calculated for\n16 complex energy points distributed exponentially on a\nsemi-circular contour including states within 1.1 Ry be-\nlow the Fermi level. For the one-center expansion of the\nfull charge density a lh\nmax=8 cutoff was used. The elec-\ntrostatic correction to the single-site coherent potential\napproximation was described using the screened impu-\nrity model [70] with a screening parameter of 0.6. Total\nenergies were calculated using the PBE [59] exchange-\ncorrelation functional, while local density approximation\n[71, 72] was used to calculate the magnetic moments and\nexchange interactions. The latter was calculated within\nthe magnetic force theorem [73] for the ferromagnetic\nand disordered local moment (DLM)[74, 75] configura-\ntions as implemented in the EMTO code. DLM repre-\nsentsthehightemperatureparamagnetic(PM)phase. In\nthis model, the paramagnetic phase of Co 3Mn2Ge reads\nas (Co\"\n0:5Co#\n0:5)3(Mn\"\n0:5Mn#\n0:5)2Ge. Similar formulation is\napplied to the paramagnetic phase of the alloy as well.\nB. Synthesis\nSamples of Co 3Mn2Ge were synthesized by melting Co\n(Alfa Aesar, 99.9%), Mn (Höganäs AB, 99.9%) and Ge\n(Kurt J. Lesker, 99.999%) together in an induction fur-\nnace under Ar (purity 99.999%) atmosphere. The re-\nsulting ingots were placed in Al 2O3crucibles, sealed in\nevacuated quartz glass tubes and annealed at 1073 K for\n14 days after which they were quenched in water. After\nthe final composition had been established by energy dis-\npersive X-ray spectroscopy analyses (see below), starting\nmaterials in stoichiometry of Co 52Mn34Ge14were pre-\npared using the established protocol to yield the final\nsamples. Samples were manually ground and powders\ntaken for analysis.\nC. Crystal structure analysis\nThe crystal structure was investigated using X-ray\npowder diffraction (XRPD), single crystal X-ray diffrac-\ntion (SCXRD) and scanning electron microscopy (SEM)\ncoupledwithenergydispersiveX-rayspectrocopy(EDS).The powders were mounted on single-crystal Si sample\nholders and X-ray diffraction patterns were collected us-\ningaBrukerD8AdvancewithmonochromatizedCu-K \u000b1\n(\u0015= 1.540598 Å) radiation at room temperature. Full-\nProf was used with the Rietveld refinement method to\nanalyse the data [76]. A Bruker D8 single-crystal X-ray\ndiffractometerwithMoK \u000bradiation(\u0015=0.71073Å)up-\ngraded with an Incoatec Microfocus Source (I \u0016S, beam\nsize\u0019100\u0016m at the sample position) and an APEX II\nCCD area detector (6cm \u00026cm) was utilized to collect\nSCXRD intensities at room temperature. SCXRD data\nreduction and numerical absorption corrections were per-\nformed using the APEX III software from Bruker[77].\nThe initial model of the crystal structure was first ob-\ntained with the program SHELXT-2014 and refined in\nthe program SHELXL-2014 within the APEX III soft-\nware package. The microstructure was evaluated with\na Zeiss Merlin SEM equipped with a secondary electron\n(SE) detector and an energy-dispersive X-ray spectrom-\neter. The samples for electron microscopy analysis were\nprepared by standard metallographic techniques through\ngrinding with SiC paper. For final polishing a mixture of\nSiO2and H 2O was used.\nD. Magnetic measurements\nMagnetization versus field and temperature measure-\nments were performed using a Quantum Design MPMS\nXL system. Isothermal magnetization curves were\nrecorded at several temperatures in applied magnetic\nfields up to 5 T. Magnetization measurements were per-\nformed on bulk samples as well as on single crystals. The\nsingle crystals used for these measurement were rather\nsmall hexagons, with a height of 100\u0016m and a side length\nof10\u0016m, resulting in a magnetic moment of 10\u00007–10\u00008\nAm2. Care was done to avoid common artifacts intro-\nduced when using this system [78]. The small hexagons\nwere too small to accurately measure the weight of the\nsample, and thus the hysteresis curves from single crys-\ntals were scaled to match the magnetization at the same\ntemperature for bulk samples. The temperature depen-\ndent magnetization was measured between 10 K and\n390 K in the applied magnetic fields of 0.01 T and 1\nT. The temperature dependent magnetization was also\nmeasured between 300 and 900 K and back to 300 K in\nan LakeShore VSM equipped with a furnace. The high\ntemperature measurements were performed in an applied\nmagnetic field of 0.01 T using a heating/cooling rate of\n3 K/min. The magnetization in SI units was calculated\nfromthemeasuredmagneticmomentbyusingthesample\nweight and density obtained from XRD measurements at\n298 K. The law of approach to saturation [44, 79] was\nused to calculate the effective anisotropy constant of the\nmaterial,jKeffjassuming the material to be uniaxial.9\nVIII. ACKNOWLEDGEMENT\nThe authors would like to acknowledge the support\nof the Swedish Foundation for Strategic Research, the\nSwedish Energy Agency (SweGRIDS), the Swedish Re-\nsearch Council, The Knut and Alice Wallenberg Foun-\ndation, STandUPP and the CSC IT Centre for Science,\nand the Swedish National Infrastructure for Computing\n(SNIC) for the computation resources. E. K. D.-Cz. ac-\nknowledges A. V. Ruban for valuable discussions. O.\nYu. V. acknowledges the support of Sweden’s Innovation\nAgency (Vinnova).\nIX. CONTRIBUTIONS\nO.E., H.C.H, P.S. and M.S. initiated the research.\nA.V. performed the high-throughput search and dataanalysis. V.S. and S.R.L synthesized the samples. V.S.\ncarried out the crystal structure analysis and SEM/EDS.\nD.H. performed magnetic measurements. DFT disor-\nder CPA calculations were performed and analysed by\nE.K.D.-C. and S.H. A.V., V.S., and D.H. drafted the\nmanuscript, which was reviewed and edited by all the\nauthors. All authors contributed to discussions and anal-\nysed the data.\nX. COMPETING INTERESTS\nThe authors declare no competing interests.\n[1] R. Skomski and J. M. D. Coey, Permanent magnetism\n(Institute of Physics Publishing, Bristol, UK, 1999).\n[2] H. i. d. I. Sözen, S. Ener, F. Maccari, K. P. Skokov,\nO. Gutfleisch, F. Körmann, J. Neugebauer, and\nT. Hickel, Phys. Rev. Materials 3, 084407 (2019).\n[3] T. Pandey and D. S. Parker, Phys. Rev. Applied 13,\n034039 (2020).\n[4] A.Gabay, G.Hadjipanayis, andJ.Cui,JournalofMag-\nnetism and Magnetic Materials 495, 165860 (2020).\n[5] S. Kim, H. Moon, H. Jung, S.-M. Kim, H.-S. Lee,\nH. Choi-Yim, and W. Lee, Journal of Alloys and Com-\npounds708, 1245 (2017).\n[6] H. Fang, S. Kontos, J. Ångström, J. Cedervall,\nP.Svedlindh, K.Gunnarsson, andM.Sahlberg,Journal\nof Solid State Chemistry 237, 300 (2016).\n[7] S. Shafeie, H. Fang, D. Hedlund, A. Nyberg,\nP.Svedlindh, K.Gunnarsson, andM.Sahlberg,Journal\nof Solid State Chemistry 274, 229 (2019).\n[8] B. Balasubramanian, B. Das, R. Skomski, W. Y. Zhang,\nandD.J.Sellmyer,AdvancedMaterials 13,6090(2013).\n[9] E. Lottini, A. López-Ortega, G. Bertoni, S. Turner,\nM. Meledina, G. Van Tendeloo, C. de Julián Fernán-\ndez, and C. Sangregorio, Chemistry of Materials 28,\n4214 (2016).\n[10] B. Balamurugan, B. Das, V. R. Shah, R. Skomski, X. Z.\nLi, and D. J. Sellmyer, Appl. Phys. Lett. 101, 122407\n(2012).\n[11] E. Anagnostopoulou, B. Grindi, L.-M. Lacroix, F. Ott,\nI. Panagiotopoulos, and G. Viau, Nanoscale 8, 4020\n(2016).\n[12] T. R. Gao, Y. Q. Wu, S. Fackler, I. Kierzewski,\nY. Zhang, A. Mehta, M. J. Kramer, and I. Takeuchi,\nAppl. Phys. Lett. 102, 022419 (2013).\n[13] T. J. Nummy, S. P. Bennett, T. Cardinal, and\nD. Heiman, Appl. Phys. Lett. 99, 252506 (2011).\n[14] B. Balamurugan, B. Das, W. Y. Zhang, R. Skomski,\nand D. J. Sellmyer, Journal of Physics: Condensed Mat-\nter26, 064204 (2014).[15] V. Ly, X. Wu, L. Smillie, T. Shoji, A. Kato, A. Manabe,\nand K. Suzuki, Journal of Alloys and Compounds 615,\nS285 (2014).\n[16] M. D. Kuz’min, K. P. Skokov, H. Jian, I. Radulov, and\nO. Gutfleisch, Journal of Physics: Condensed Matter\n26, 064205 (2014).\n[17] J. Cui, M. Kramer, L. Zhou, F. Liu, A. Gabay, G. Had-\njipanayis, B. Balasubramanian, and D. Sellmyer, Acta\nMaterialia 158, 118 (2018).\n[18] A. Vishina, O. Y. Vekilova, T. Björkman, A. Bergman,\nH. C. Herper, and O. Eriksson, Phys. Rev. B 101,\n094407 (2020).\n[19] A. G. Kusne, T. Gao, A. Mehta, L. Ke, M. C. Nguyen,\nK.-M. Ho, V. Antropov, C.-Z. Wang, M. J. Kramer,\nC. Long, and I. Takeuchi, Scientific Reports 4, 6367\n(2014).\n[20] Q. Gao, I. Opahle, O. Gutfleisch, and H. Zhang, Acta\nMaterialia 186, 355 (2020).\n[21] B. Fayyazi, K. P. Skokov, T. Faske, D. Y. Karpenkov,\nW. Donner, and O. Gutfleisch, Acta Materialia 141,\n434 (2017).\n[22] T. Burkert, L. Nordström, O. Eriksson, and\nO. Heinonen, Phys. Rev. Lett. 93, 027203 (2004).\n[23] G. Andersson, T. Burkert, P. Warnicke, M. Björck,\nB.Sanyal, C.Chacon,C.Zlotea, L.Nordström,P.Nord-\nblad, and O. Eriksson, Phys. Rev. Lett. 96, 037205\n(2006).\n[24] J. H. Park, Y. K. Hong, S. Bae, J. J. Lee, J. Jalli, G. S.\nAbo, N. Neveu, S. G. Kim, C. J. Choi, and J. G. Lee,\nJournal of Applied Physics 107, 09A731 (2010).\n[25] H. Fang, S. Kontos, J. Ångström, J. Cedervall,\nP.Svedlindh, K.Gunnarsson, andM.Sahlberg,Journal\nof Solid State Chemistry 237, 300 (2016).\n[26] T. Kojima, M. Ogiwara, M. Mizuguchi, M. Kotsugi,\nT. Koganezawa, T. Ohtsuki, T.-Y. Tashiro, and\nK. Takanashi, Journal of Physics: Condensed Matter\n26, 064207 (2014).\n[27] X. Zhao, C.-Z. Wang, Y. Yao, and K.-M. Ho, Phys.\nRev. B94, 224424 (2016).10\n[28] N. V. R. Rao, A. M. Gabay, and G. C. Hadjipanayis,\nJournal of Physics D: Applied Physics 46, 062001\n(2013).\n[29] S.Arapan, P.Nieves, H.C.Herper, andD.Legut,Phys.\nRev. B101, 014426 (2020).\n[30] “Inorganic crystal structure database,” http://www2.\nfiz-karlsruhe.de/icsd_home.html .\n[31] P. Bruno, Phys. Rev. B 39, 865 (1989).\n[32] K. Buschow, P. van Engen, and R. Jongebreur, Journal\nof Magnetism and magnetic materials 38, 1 (1983).\n[33] K. Adachi, K. Sato, M. Matsui, and Y. Fujio, Journal\nof the Physical Society of Japan 30, 1201 (1971).\n[34] Y. Kuz’ma and E. Gladyshevskii, Dopovidi Akademii\nNauk Ukrains’koi RSR 2, 205 (1963).\n[35] J. Cedervall, S. Kontos, T. C. Hansen, O. Balmes, F. J.\nMartinez-Casado, Z. Matej, P. Beran, P. Svedlindh,\nK.Gunnarsson, andM.Sahlberg,JournalofSolidState\nChemistry 235, 113 (2016).\n[36] M. A. McGuire, H. Cao, B. C. Chakoumakos, and B. C.\nSales, Physical Review B 90, 174425 (2014).\n[37] Y. He, G. H. Fecher, J. Kroder, H. Borrmann, X. Wang,\nL. Zhang, C.-Y. Kuo, C.-E. Liu, C.-T. Chen, K. Chen,\nF. Choueikani, P. Ohresser, A. Tanaka, Z. Hu, and\nC. Felser, Applied Physics Letters 116, 102404 (2020).\n[38] W. D. Corner and B. K. Tanner, Journal of Physics C:\nSolid State Physics 9, 627 (1975).\n[39] K. Tokuhara, Y. Ohtsu, F. Ono, O. Yamada,\nM. Sagawa, and Y. Matsuura, Solid State Communica-\ntions56, 333 (1985).\n[40] K. Krishnan, Fundamentals and Applications of Mag-\nnetic Materials (OUP Oxford, 2016).\n[41] N. Akulov, Zeitschrift für Physik 100, 197 (1936).\n[42] H. Callen and E. Callen, Journal of Physics and Chem-\nistry of Solids 27, 1271 (1966).\n[43] R. Frömter, H. Stillrich, C. Menk, and H. P. Oepen,\nPhys. Rev. Lett. 100, 207202 (2008).\n[44] J. F. Herbst and F. E. Pinkerton, Phys. Rev. B 57,\n10733 (1998).\n[45] X. Yan, X.-Q. Chen, A. Grytsiv, V. Witusiewicz,\nP. Rogl, R. Podloucky, and G. Giester, Journal of Al-\nloys and Compounds 429, 10 (2007).\n[46] S.Arapan, P.Nieves, H.C.Herper, andD.Legut,Phys.\nRev. B101, 014426 (2020).\n[47] X. Liu, D. Legut, R. Zhang, T. Wang, Y. Fan, and\nQ. Zhang, Phys. Rev. B 100, 054438 (2019).\n[48] V. Antropov, L. Ke, and D. Åberg, Solid State Com-\nmunications 194, 35 (2014).\n[49] Y. Miura, S. Ozaki, Y. Kuwahara, M. Tsujikawa,\nK. Abe, and M. Shirai, Journal of Physics: Condensed\nMatter25, 106005 (2013).\n[50] L. Ke and M. van Schilfgaarde, Phys. Rev. B 92, 014423\n(2015).\n[51] J. Zhang, B. Yang, H. Zheng, X. Han, and Y. Yan,\nPhys. Chem. Chem. Phys. 19, 24341 (2017).\n[52] J. M. Wills and B. R. Cooper, Phys. Rev. B 36, 3809\n(1987).\n[53] J. M. Wills, M. Alouani, P. Andersson, A. Delin,\nO. Eriksson, and O. Grechnyev, Full-Potential Elec-\ntronic Structure Method , Springer series in solid state\nscience (Springer, Berlin, Germany, 2010).\n[54] G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993).\n[55] G.KresseandJ.Hafner,Phys.Rev.B 49,14251(1994).\n[56] G. Kresse and J. Furthmüller, Comput. Mat. Sci. 6, 15\n(1996).[57] G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169\n(1996).\n[58] P. E. Blöchl, Phys. Rev. B 50, 17953 (1994).\n[59] J. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.\nLett.77, 3865 (1996).\n[60] P. E. Blöchl, O. Jepsen, and O. K. Andersen, Phys.\nRev. B49, 16223 (1994).\n[61] G. H. O. Daalderop, P. J. Kelly, and M. F. H. Schuur-\nmans, Phys. Rev. B 41, 11919 (1990).\n[62] M. Weinert, R. E. Watson, and J. W. Davenport, Phys.\nRev. B32, 2115 (1985).\n[63] R. Skomski and J. Coey, Scripta Materialia 112, 3\n(2016).\n[64] O. Eriksson, A. Bergman, L. Bergqvist, and J. Hellsvik,\nAtomistic Spin Dynamics: Foundations and Applica-\ntions(Oxford University Press, Oxford, 2017).\n[65] Y. O. Kvashnin, O. Grånäs, I. Di Marco, M. I. Katsnel-\nson, A. I. Lichtenstein, and O. Eriksson, Phys. Rev. B\n91, 125133 (2015).\n[66] P. Soven, Phys. Rev. 156, 809 (1967).\n[67] B. L. Gyorffy, Phys. Rev. B 5, 2382 (1972).\n[68] L. Vitos, Phys. Rev. B 64, 014107 (2001).\n[69] L. Vitos, The EMTO Method and Applications, in Com-\nputational Quantum Mechanics for Materials Engineers ,\nEngineering Materials and Processes (Springer-Verlag\nLondon, 2007).\n[70] P. A. Korzhavyi, A. V. Ruban, I. A. Abrikosov, and\nH. L. Skriver, Phys. Rev. B 51, 5773 (1995).\n[71] J. P. Perdew and Y. Wang, Phys. Rev. B 45, 13244\n(1992).\n[72] D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45,\n566 (1980).\n[73] A. I. Liechtenstein, M. I. Katsnelson, and V. A.\nGubanov, Journal of Physics F: Metal Physics 14, L125\n(1984).\n[74] A.J.Pindor, J.Staunton, G.M.Stocks, andH.Winter,\nJournal of Physics F: Metal Physics 13, 979 (1983).\n[75] B. L. Gyorffy, A. J. Pindor, J. Staunton, G. M. Stocks,\nand H. Winter, Journal of Physics F: Metal Physics 15,\n1337 (1985).\n[76] J. Rodriguez-Carvajal, IUCr Powder Diffr. Newsl. 26,\n12 (2001).\n[77] A. Saint, APEX3 software for CCD diffractometers\n(Bruker Analytical X-ray Systems Inc., Madison, WI,\n2014).\n[78] M. Sawicki, W. Stefanowicz, and A. Ney, Semiconduc-\ntor Science and Technology 26, 064006 (2011).\n[79] S. Chikazumi and C. Graham, Physics of Ferromag-\nnetism, International Series of Monographs on Physics\n(Clarendon Press, Oxford, UK, 1997).\n[80] J. H. van Vleck, Phys. Rev. 52, 1178 (1937).\n[81] P. Bruno, Phys. Rev. B 39, 865 (1989).\n[82] G. van der Laan, Journal of Physics: Condensed Matter\n10, 3239 (1998).\n[83] T. Kosugi, T. Miyake, and S. Ishibashi, Journal\nof the Physical Society of Japan 83, 044707 (2014),\nhttps://doi.org/10.7566/JPSJ.83.044707.\n[84] C. Andersson, B. Sanyal, O. Eriksson, L. Nordström,\nO. Karis, D. Arvanitis, T. Konishi, E. Holub-Krappe,\nand J. H. Dunn, Phys. Rev. Lett. 99, 177207 (2007).\n[85] A. Edström, Phys. Rev. B 96, 064422 (2017).\n[86] D. Wang, R. Wu, and A. J. Freeman, Phys. Rev. B 47,\n14932 (1993).11\n[87] E. Abate and M. Asdente, Phys. Rev. 140, A1303\n(1965).\n[88] H. Takayama, K.-P. Bohnen, and P. Fulde, Phys. Rev.\nB14, 2287 (1976).\n[89] T. Kanomata, H. Ishigaki, T. Suzuki, H. Yoshida,\nS. Abe, and T. Kaneko, Journal of Magnetism and\nMagnetic Materials 140-144 , 131 (1995), international\nConference on Magnetism.\n[90] S. Kaprzyk and S. Niziol, Journal of Magnetism and\nMagnetic Materials 87, 267 (1990).\n[91] W. Bazela, A. Szytuła, J. Todorović, Z. Tomkowicz,\nand A. Zięba, physica status solidi (a) 38, 721 (1976),\nhttps://onlinelibrary.wiley.com/doi/pdf/10.1002/pssa.2210380235.\n[92] N.T.Trung, L.Zhang,L.Caron,K.H.J.Buschow, and\nE. Brück, Applied Physics Letters 96, 172504 (2010).\n[93] T. Samanta, I. Dubenko, A. Quetz, S. Stadler, and\nN. Ali, Applied Physics Letters 101, 242405 (2012).\n[94] S. Ma, Y. Zheng, H. Xuan, L. Shen, Q. Cao, D. Wang,\nZ. Zhong, and Y. Du, Journal of Magnetism and Mag-\nnetic Materials 324, 135 (2012).[95] P. E. Markin, N. V. Mushnikov, V. I. Khrabrov, and\nM. A. Korotin, The Physics of Metals and Metallogra-\nphy106, 481 (2008).\n[96] M. Kogachi, T. Fujiwara, and S. Kikuchi, Journal of\nAlloys and Compounds 475, 723 (2009).\n[97] B. A. Collins, Y. S. Chu, L. He, D. Haskel, and F. Tsui,\nPhysical Review B 92, 224108 (2015).\n[98] A. Okubo, R. Y. Umetsu, K. Kobayashi, R. Kainuma,\nand K. Ishida, Applied Physics Letters 96, 222507\n(2010), https://doi.org/10.1063/1.3427431.\n[99] P. Webster, Journal of Physics and Chemistry of Solids\n32, 1221 (1971).\n[100] S. Nizioł, A. Wesełucha, W. Bażela, and A. Szytuła,\nSolid State Communications 39, 1081 (1981).\n[101] G. Venturini and B. Malaman, Journal of Alloys and\nCompounds 615, 886 (2014).\n[102] J.-T. Wang, D.-S. Wang, C. Chen, O. Nashima,\nT. Kanomata, H. Mizuseki, and Y. Kawazoe, Applied\nPhysics Letters 89, 262504 (2006).\n[103] Y. Fang, C. Yeh, C. Chang, W. Chang, M. Zhu, and\nW. Li, Scripta Materialia 57, 453 (2007).12\nAppendix A: Materials discarded during the final steps of high-throughput and data-mining search.\nSome of the materials with the magnetic moment above the set-up threshold of 1.0 \u0016B/f.u. were discarded later\ndue to the low or planar MAE or saturation magnetization being much lower than 1 T. These materials are listed in\nTable III.\nTABLE III: Systems with magnetic moment calculated to be higher than 1.0 \u0016B/f.u. during the high-throughput\nstep, which did not fulfill the requirements of a good permanent magnet on the further steps of data-mining.\nMaterial ICSD Space Mag. MAE (th) Sat. magn.\nnumber group state MJ/m3T\nCrNiAs 43913 189 FM 0.37 0.86\nCrNiP 43913 189 FM 0.21 0.83\nFe2Ti 103663 194 FM 0.10 0.84\nCoCrGe 409451 194 FM -0.18 0.75\nCoGeMn 623495 194 FM -0.16 1.10\nCu2Co(SnSe 4) 99296 121 FM 0.17\nCu2CoGeS 499293 121 FM 0.24\nCu2CoSiS 499292 121 FM 0.24\nCu2CoSnS 499294 121 FM 0.21\nMn2Co2C 44353 123 FM -2.21 0.43\nMnCoAs 610084 62 FM -0.88 0.90\nMnCoP 16483 62 FM -0.41 1.00\nNi5Sc 646468 191 FM 0.20\nGeMnSc 600156 189 FM 0.24 0.51\nSc2FeRh 5B251437 127 FM -0.13 0.39\nTiZn 2 106184 194 FM -0.01 0.30\nFeNiAs 610509 189 FM -1.53 0.6413\nAppendix B: Crystallographic data\nHere we list the crystallographic and structural data for the hexagonal Co 3:24Mn2Ge0:76compound along with some\nof the measurement details.\nTABLE IV: Crystallographic data and experimental details of the single crystal structure refinement for the\nhexagonal Co 3:24Mn2Ge0:76compound. Measurements were carried out at 296 K with Mo K \u000bradiation.\nEmpirical formula Co3Mn2Ge\nCalculated formula Co3:24Mn2Ge0:76\nStructure type MgZn 2\nFormula weight, Mr (g/mol) 177.92\nSpace group (No.) P63/mmc(194)\nPearson symbol, Z hP12, 4\nUnit cell dimensions:\na, Å 4.8032(2)\nb, Å 4.8032(2)\nc, Å 7.7378(4)\nV, Å 154.60(1)\nCalculated density, \u001a(g\u0001cm\u00003) 7.64\nAbsorption coefficient, \u0016(mm\u00001) 31.835\nTheta range for data collection (\u000e)4.901\u000437.119\nF(000) 324\nRange in h k l -8\u0014h\u00147,\n-8\u0014k\u00147,\n-13\u0014l\u001413,\nTotal No. of reflections 3933\nNo. of independent reflections 180(Req= 0.0264)\nNo. of reflections with I >2\u001b(I)174 (R\u001b= 0.0096)\nData/parameters 180/13\nWeighting details w=1/[\u001b2F2\no+0.5284P]\nwhere P=(F2\no+ 2F2\nC)/3\nGoodness-of-fit on F21.170\nFinalRindices [I>2\u001bI] R1= 0.0100;\nwR2= 0.0264\nRindices (all data) R1= 0.0111;\nwR2= 0.0268\nExtinction coefficient 0.00135(13)\nTABLE V: Atomic coordinates, anisotropic displacement parameters and selected interatomic distances for the\nhexagonal Co 3:24Mn2Ge0:76compound.\nAtom Site x y z U eq(Å2)\n0.837Co1+0.163Ge1 6 h0.17153(4) 0.34306(8) 1/4 0.00594(11)\nMn 4 f1/3 2/3 0.56358(7) 0.00817(16)\n0.740Co2+0.260Ge2 2 a 0 0 0 0.006667(17)\nU11 U22 U33 U12\n0.837Co1+0.163Ge1 0.00658(14) 0.00418(16) 0.00626(15) 0.00209(8)\nMn 0.00849(18) 0.00849(18) 0.0075(2) 0.00424(9)\n0.740Co2+0.260Ge2 0.0079(2) 0.0079(2) 0.0041(2) 0.00397(10)\nAtoms 1,2 d1,2(Å) Atoms 1,2 d1,2 (Å)\n2CojGe1–Co2jGe2 2.4039(2) 3Co1 jGe1–Mn1 2.7978(3)\n2Co1jGe1–Mn1 2.7815(5) 5Co2 jGe2–Mn1 2.8178(2)\nUeqis defined as one third of the trace of the orthogonalized Uijtensor. U13,U23= 0.14\nAppendix C: Ordering temperature, magnetic configuration, and crystal structure of the disordered\nCo3Mn 2Ge\nIn this section, we present the additional data for the ordered Co 3Mn2Ge and disordered Mn 2(Co0:75Ge0:25)4\nstructuresrelaxedwithEMTOcodeintheFM(Fig.8)andPM(Fig.9)phases. TableVIliststhecrystalstructureand\nmagnetic parameters obtained for these structures for both FM and PM magnetic configurations. Fig. 10 summarizes\nthe temperature effects for the FM and PM phases for the c/a-ratio fixed to the experimental value.\nTo calculate the temperature which makes the ordered and disordered state degenerate in the FM and PM phase,\nthe crystal structures of Co 3Mn2Ge and Mn 2(Co0:75Ge0:25)4were relaxed in volume and c/a ratio using the EMTO\ncode, as shown in Fig. 8 and Fig. 9. Table VI lists the crystal structure and magnetic parameters obtained for these\nstates for both FM and PM magnetic configurations. As can be seen, the chemical disorder does not have significant\neffect on the magnetic properties in the FM state. However, magnetic configuration (FM or DLM) does strongly\naffect the atomic magnetic moments, especially in the case of Co (Table VI).\nThe order-disorder transition can be estimated by the cross point of the free energies of the ordered and disordered\nstructures, i.e. \u0001F=Fdis-Ford. The temperature dependent free energy of the FM phase F(T)FMis estimated as\nF(T)FM\u0019EFM\n0+Fconf, whereEFM\n0is the internal energy for FM state at 0 K and Fconfis configurational free energy\nevaluated at different temperatures. The energy difference of Co 3Mn2Ge between the FM ordered and FM disordered\n(Mn 2(Co0:75Ge0:25)4) states is about 4.124 mRy; the difference in the configurational entropy of the alloy between\nthe ordered and disordered phases equals to 0.375 kB. SinceFconf= -TSconfwe find the transition temperature to be\naround 1735 K in the FM state.\nTemperature effects are summarized for the FM and PM phases in Figure 10 for c=a-ratio fixed to the experimental\nvalue. As we can see, there is no transition up to either the measured (359 K) or the theoretically predicted (700 K)\nmagnetic transition temperature (see left panel of Fig. 10). The ordered and disordered structures become degenerate\nin energy at\u00192300 K in the PM phase (see right panel of Fig. 10). The transition temperature is around 2200 K\nin the PM state. The stability of various antiferromagnetic configurations was investigated as well, for both ordered\nand disordered case (not shown), to account for the discrepancy between experimental and theoretical TCfor fixed\nc=a-ratio. None of the antiferromagnetic states considered in the calculations are stabilized by the configurational\nentropy up to the TC.\nThe Curie temperature for the ordered system and for Mn 2(Co0:75Ge0:25)4was estimated from Jij’s calculated for\nthe ferromagnetic reference state and performing subsequent Monte-Carlo (MC) simulations. We obtained 720 K for\nCo3Mn2Ge and 760 K for the disordered case using the theoretical lattice parameters given in Table VI. It is satisfying\nto note that the TCfor the ordered system, obtained with EMTO method, is in good agreement with the RSPt results.\nThe Curie temperature increase with disorder is due to the decrease of the antiferromagnetic Mn-Mn nearest neighbor\ninteractions as chemical disorder is applied. TCestimated for the experimental structure and composition given in\nTable II is 820 K. We canconcludethat the discrepancy between the predicted TCand the measured one does notcome\nfrom the order-disorder effect alone. To address this issue, DLM calculations were also performed. These result in the\nconsiderable drop of Co local magnetic moment in the DLM state compared to the FM moments, while Mn moment\ndoes not change (see Table VI). Magnetic moment of any atomic species, that is reduced in the DLM configuration\ncompared to a FM configuration, will lead to the reduced strength of the inter-atomic exchange interactions, and\nreduce of the ordering temperature.\nAppendix D: Origin of magnetic anisotropy in Co 3Mn 2Ge\nThe magnetocrystalline anisotropy originates from the SOC, since it is the only term in the Hamiltonian that\ncouples spin- and real-space, something that was first suggested by Van Vleck [80]. In the case of transition metals\nwhere SOC is much smaller than the bandwidth or crystal field, it can be treated as a perturbation. This lead to\nthe possibility to connect the MAE with anisotropy in orbital moment [81]. The original expression for this coupling,\nmade by Bruno, was subsequently extended, approximated, and used for various applications in [48, 50, 51, 82–86].\nIt has been shown previously [50, 51, 82, 85] that the coupling between the occupied and unoccupied states close\nto the Fermi energy \u000fFdominates the spin-orbit induced change of the total energy. Matrix elements h\u0016\u001bjL\u0001Sj\u00160\u001b0i\n[87, 88] determine the spin quantization axis direction which modifies the eigenvalues of the Kohn-Sham Hamiltonian.\nInthisexpression, \u0016representsad-orbital(withsymmetry fxy;yz;zx;x2\u0000y2;z2rg),\u001bdenotesspin, while LandSare\norbital and spin angular momentum operators. For states within the same spin channel, the couplings dxy\u0000 !dx2\u0000y2,\ndyz\u0000 !dxzpromote the uniaxial anisotropy, while dxy\u0000 !dxz,dxy\u0000 !dyz,dxz\u0000 !dz2,dyz\u0000 !dz2,dxz\u0000 !dx2\u0000y2, and\ndyz\u0000 !dx2\u0000y2favour the easy-plane magnetocrystalline anisotropy [48, 50, 51, 82, 83, 85]. The situation is reversed\nfor the couplings between opposite spin channels. Table VIII (in Appendix E ) lists the transitions that contribute\neither to easy-plane or uniaxial anisotropy along with their relative weights.15\nFIG. 8: (Color online) Volume and c=arelaxation of the ordered Co 3Mn2Ge (left) and disordered (right)\nMn2(Co0:75Ge0:25)4structures in the FM phase. rWSdenotes the Wigner-Seitz radius. Yellow lines follow the\nminimum of E( rWS) curve for a specific value of c=aand the minimum of E(c=a)curve for each rWS.\nFIG. 9: (Color online) Volume and c=arelaxation of the ordered Co 3Mn2Ge (left) and disordered (right)\nMn2(Co0:75Ge0:25)4structures in PM phase. rWSdenotes the Wigner-Seitz radius. Yellow lines follow the minimum\nof E(rWS) curve for a specific value of c=aand the minimum of E(c=a)curve for each rWS.\nTo understand the origin of the difference in MAE for Co 3Mn2X (X = Al, Si, P, Ga, As, In, Sn, Sb, Tl, and Pb), it\nis electron states close to the Fermi level ( \u000fF) one should focus on, since spin-orbit interaction that couples states just\nbelow\u000fFto states just above \u000fF, are particularly important in deciding the magnetic anisotropy [22]. To undertake\nthis analysis we inspected partial densities of states (pDOS) around \u000fF. Figure 11 shows the pDOS for Co 3Mn2Ge,\nwhich has a large uniaxial anisotropy of 1.44 MJ/m3, and Fig. 12 shows the pDOS for Co 3Mn2As, that has a large\neasy-plane anisotropy of -1.2 MJ/m3. The pDOS curves for the other Co 3Mn2X materials can be found in Appendix F ,\nFigs. 13-14. The majority spin channel of the d-states is essentially fully occupied for all the materials considered in\nthis work which shows its inertness for the magnetic anisotropy. Instead significant contributions are expected for the\nminority spin channel, that has states on either side of \u000fF. In this spin channel there are large peaks corresponding\nto thedyz,dxz, anddxystates of Co, close to the Fermi energy. These peaks are mostly empty for X = Al, Ga, Ge,\nIn, and Sn, while they lie directly on \u000fFfor X = Si and Sb, and are mostly occupied for X = P, As. All the materials\nwith uniaxial magnetic anisotropy (X = Al, Ga, Ge, In) have these large peaks of the DOS occupied, and we conclude\nthat from a microscopic point of view, these electrons are decisive for the magnetic anisotropy.\nTo analyze the MAE further we consider the difference in SOC energies with spin orientation along the zandx\naxes, \u0001Eso=Ez\nso\u0000Ex\nso, separately for all Co and Mn atoms, see Table VII (negative sign marks a contribution to\nuniaxial magnetic anisotropy). As expected, for most of X,\u0001Eso(Mn) is considerably smaller than that of Co atoms.\nThe latter can be divided into two groups (noted by the subscripts in Table VII); the contribution to \u0001Esofrom the16\nTABLE VI: Optimized crystal structure parameters ( a,c=a, and volume) and magnetic moments per unit cell\ncalculated for the ordered Co 3Mn2Ge and disordered Mn 2(Co0:75Ge0:25)4phases using the EMTO code, along with\nthe low temperature experimental lattice parameters and magnetic saturation field. Third column contains\nexperimental data. Fourth column (labeled ”Co-excess”) contains experimental data for the low temperature\nstructure, and calculated magnetic moments for this structure and for a composition given in Table II (54 % Co).\nOrdered Disordered Experiment Co-excess\nFM\na,\u0017A 4.83 4.81 4.803 4.803\nc=a 1.61 1.65 1.611 1.611\nV,\u0017A3157.6 158.5 154.6 154.5\nm6h\nCo,\u0016B1.60 1.59 1.61\nmMn,\u0016B3.30 3.28 3.27\nm2a\nCo,\u0016B 1.56 1.63\nMtot/cell, T 1.66 1.65 0.86 1.75\nDLM\na,\u0017A 4.84 4.77 4.803\nc=a 1.56 1.65 1.611\nV,\u0017A3152.7 154.9 154.5\nm6h\nCo,\u0016B0.04 0.41 0.60\nmMn,\u0016B2.92 2.99 3.05\nm2a\nCo,\u0016B 0.05 0.51\nFIG. 10: (Color online) Left panel: Normalized total energies ( E0) (full lines) and free energies ( F(T)) (dashed and\ndotted lines) calculated for the ordered (black squares) and disordered (red circles) FM state of Co 3Mn2Ge at 0 K,\n359 K (experimental TC) and 700 K (theoretical TC). Right panel: Normalized total energies ( E0) (full lines) and\nfree energies ( F(T)) calculated for the ordered (black squares) and disordered (red circles) Co 3Mn2Ge at 0 K and\n2300 K in the PM state. Dashed line stands for F(T)=E0+Fconf, dotted line denotes F(T)=E0+Fconf+Fmag.\nOn the x-axis, rWSdenotes the Wigner-Seitz radius calculated as rWS=3p\n3Vatom=4=\u0019, whereVatomis the average\nvolume of an atom in the unit cell. Fconf= -kBTP\niciln(ci) andFmag=-kBTP\niciln(1+mi).\ntwo Co atoms positioned at (0.667, 0.833, 0.75) and (0.333, 0.166, 0.25) 6 h(we denote them type 2) is different to\nthat of the remaining four Co atoms ( type 1). All Co atoms contribute to the uniaxial magnetic anisotropy for X =\nAl, Ga, and In. The values of \u0001Esoare extremely low for both Co and Mn atoms in Co 3Mn2P, which results in the\nnegligibly small value of MAE. In the case of Co 3Mn2Si and Co 3Mn2As, all Co atoms give rise to the large values\ncorresponding to easy-plane magnetocrystalline anisotropy. For the remaining materials, the two groups of Co atoms\nhave opposite signs of \u0001Eso.\nWe can also determine the d-orbitals that give the largest change to the SOC matrix element h\u0016\u001bjbHsoj\u00160\u001b0ias\nthe magnetization direction changes from ztox. Co 3Mn2X with X = Al, Ga, Ge, and In, that exhibit uniaxial\nanisotropy, all present a similar picture, see Fig. 19 ( Appendix G ) for Co 3Mn2Ge (the rest of the figures can be17\nFIG. 11: (Color online) Contribution to the density of states of Co 3Mn2Ge from the dx2\u0000y2anddz2(egset);dxy,\ndyz, anddxz(t2gset) orbitals without SOC interaction. Fermi energy is set at zero.\nFIG. 12: (Color online) Contribution to the density of states of Co 3Mn2As from the dx2\u0000y2anddz2(egset);dxy,\ndyz, anddxz(t2gset) orbitals without SOC interaction.\nfound in Appendix G , Fig. 15-21). The main contribution to the uniaxial anisotropy comes from hdx2+y2jbHsojdxyi\nfor all the Co atoms, even though there are additional contributions which are different for the Co atoms of type 1\n(Fig. 19, top) and type 2(Fig. 19, bottom). As expected (Table VIII), for these transitions to contribute to uniaxial\nanisotropy the states must be within the same spin channel. It is more difficult to distinguish any specific transitions\nfor materials with the easy-plane anisotropy, see Fig. 20 ( Appendix G ) for Co 3Mn2As (the other materials can be\nfound in Appendix G , Fig. 15-21) - there are several matrix elements promoting the negative sign in MAE. The most\nsignificant arehdx2+y2jbHsojdyzi,hdxzjbHsojdxyi, andhdz2jbHsojdyziand their size varies depending on X.\nAppendix E: d-orbitals contribution to MAE\nMatrix elements h\u0016\u001bjL\u0001Sj\u00160\u001b0i[87, 88] determine the preferable direction of spin quantization axis. Table VIII\nlists the transitions between the states below and above \u000fFwhich favour either z-axis (uniaxial) or xy-plane (planar)\nmagnetic anisotropy along with the relative weight of each transition.18\nTABLE VII: Calculated MAE (RSPt), saturation magnetization (RSPt), the average magnetic moment per Co and\nMn atoms, Curie temperature (UppASD), formation enthalpies with respect to the elemental components for the\nrelaxed Co 3Mn2X structures ( \u0001Hel), and formation enthalpies with respect to the Co 2MnX Heusler compounds (X\n= Al, Si, P, Ga, Ge, As, In, Sn, Sb, Tl, and Pb) (VASP), marked \u0001HHeus. The last three columns contain the\ndifference in SOC energy with spin orientation along zandxaxis (negative sign corresponds to uniaxial magnetic\nanisotropy) for Mn and Co atoms (VASP). Subscript 2notes two cobalt atoms positioned at (0.667, 0.833, 0.75) and\n(0.333, 0.166, 0.25) 6 hsites, subscript 1points at the remaining four Co atoms.\nMaterial MAE Sat. M (Co) M (Mn) TC\u0001Hel\u0001HHeus \u0001Eso(Mn) \u0001Eso(Co1)\u0001Eso(Co2)\nMJ/m3magn., T\u0016B\u0016BK eV/f.u. eV/f.u. meV meV meV\nCo3Mn2Al 1.38 1.77 1.45 3.37 820 -1.78 0.274 0.016 -0.42 -0.46\nCo3Mn2Si -0.64 1.63 1.14 3.32 -2.34 0.090 -0.057 0.11 1.20\nCo3Mn2P 0.047 1.50 0.76 3.35 -2.62 0.018 -0.04 0.10\nCo3Mn2Ga 0.67 1.75 1.50 3.47 800 -1.49 0.061 0.100 -0.25 -0.30\nCo3Mn2Ge 1.44 1.71 1.44 3.52 700 -1.57 0.070 0.005 -0.26 0.13\nCo3Mn2As -1.2 1.58 1.08 3.50 -1.60 -0.011 0.01 1.13\nCo3Mn2In 0.36 1.65 1.60 3.62 -0.14 0.160 -0.11 -0.25\nCo3Mn2Sn -0.42 1.63 1.53 3.59 -0.57 0.064 -0.21 0.03\nCo3Mn2Sb -0.81 1.56 1.35 3.56 -0.67 0.047 -0.19 0.68\nCo3Mn2Tl -0.21 1.63 1.63 3.67 0.87\nCo3Mn2Pb -2.7 1.58 1.58 3.66 0.93\nTABLE VIII: Transitions between the d-orbitals below and above \u000fFwhich contribute to either z-axis (uniaxial) or\nxy-plane (planar) magnetic anisotropy along with the relative weight of each transition, !\nContributes Same spin ! Opposite spin !\ntransition transition\nUniaxial dxy\u0000 !dx2\u0000y2 1dxy\u0000 !dxz,dxy\u0000 !dyz0.25\nanisotropy dyz\u0000 !dxz 0.25dxz\u0000 !dz2,dyz\u0000 !dz20.75\ndxz\u0000 !dx2\u0000y2,dyz\u0000 !dx2\u0000y20.25\nxy-plane dxy\u0000 !dxz,dxy\u0000 !dyz0.25 dxy\u0000 !dx2\u0000y2 1\nanisotropy dxz\u0000 !dz2,dyz\u0000 !dz20.75 dyz\u0000 !dxz 0.25\ndxz\u0000 !dx2\u0000y2,dyz\u0000 !dx2\u0000y20.25\nAppendix F: DOS for Co 3Mn 2X (X = Al, Si, P, Ga, In, Sn, Sb)\nHereweprovidetheDOSforthecrystalstructuresCo 3Mn2X(X=Al, Si, P,Ga, In, Sn, Sb, Tl, andPb). Thesewere\ncalculated by replacing Ge in Co 3Mn2Ge structure by the neighboring elements and relaxing their crystal structures.\nDOS of Co 3Mn2Ge and Co 3Mn2As are given above in the text.19\n(a) Density of states of Co 3Mn2Al (MAE = 1.38\nMJ/m3).\n(b) Density of states of Co 3Mn2Si (MAE = -0.64\nMJ/m3).\n(c) Density of states of Co 3Mn2P (MAE = 0.047\nMJ/m3).\n(d) Density of states of Co 3Mn2Ga (MAE = 0.67\nMJ/m3).\n(e) Density of states of Co 3Mn2In (MAE = 0.36\nMJ/m3).\n(f) Density of states of Co 3Mn2Sn (MAE = -0.42\nMJ/m3).\nFIG. 13: (Color online) Density of states of Co 3Mn2X, with X = Al, Si, P, Ga, Ge, As, In, Sn, Sb\nAppendix G: Change of the SOC matrix elements of Co when magnetization changes direction from ztoxfor\nCo3Mn 2X (X = Al, Si, P, Ga, In)\nTo determine the main orbital contribution to MAE we calculate the change in SOC matrix element h\u0016\u001bjbHsoj\u00160\u001b0i\nfor transitions between different d-orbitals as magnetization direction changes from ztox. Fig. 15-21 show contribu-\ntions from type 1Co atoms (top) and type 2Co atoms (bottom).20\nFIG. 14: (Color online) Density of states of Co 3Mn2Sb (MAE = -0.81 MJ/m3).\nFIG. 15: (Color online) Change of spin-orbit coupling matrix elements of Co, type 1(left), and Co, type 2(right) in\nCo3Mn2Al when magnetization changes direction from ztox.\nFIG. 16: (Color online) Change of spin-orbit coupling matrix elements of Co, type 1(left), and Co, type 2(right) in\nCo3Mn2Si when magnetization changes direction from ztox.\nAppendix H: The properties of the previously reported phases for Co-Mn-Ge system\nThe previously reported crystallographic phases for the Co–Mn–Ge system are presented in Table IX together with\ntheir crystal structure and some of the magnetic properties. We will briefly describe some of the key features of the\nstructural and magnetic properties of Co–Mn–Ge systems reported in the literature, which are expected to be relevant21\nFIG. 17: (Color online) Change of spin-orbit coupling matrix elements of Co, type 1(left), and Co, type 2(right) in\nCo3Mn2P when magnetization changes direction from ztox.\nFIG. 18: (Color online) Change of spin-orbit coupling matrix elements of Co, type 1(left), and Co, type 2(right) in\nCo3Mn2Ga when magnetization changes direction from ztox.\nFIG. 19: (Color online) Change of spin-orbit coupling matrix elements of Co, type 1(top), and Co, type 2(bottom)\nin Co 3Mn2Ge when magnetization changes direction from ztox.\nto the Co 3Mn2Ge. It should be noted that this outline is notintended as a review, but rather as a summary of the\nknown properties of Co–Mn–Ge phases, which we find relevant to understanding the properties of Co 3Mn2Ge.\nCoMnGe exists in two stable phases; the low-temperature orthorhombic CoMnGe (TiNiSi type) is stable below the\nmartensitic temperature ( TM) of 650 K [89] while the high-temperature hexagonal CoMnGe (BeZrSi type) is stable\nabove it [89]. The reported TC-values of the high- and low-temperature phases vary slightly, for instance, Kaprzyk and\nNiziol [90] report TC= 337 K for CoMnGe (TiNiSi) and TC= 287 K for CoMnGe (BeZrSi). As it is possible to tune22\nFIG. 20: (Color online) Change of spin-orbit coupling matrix elements of Co, type 1(top), and Co, type 2(bottom)\nin Co 3Mn2As when magnetization changes direction from ztox.\nFIG. 21: (Color online) Change of spin-orbit coupling matrix elements of Co, type 1(left), and Co, type 2(right) in\nCo3Mn2In when magnetization changes direction from ztox.\nTMto coincide with TC[89, 102, 103], a large/giant magnetocaloric effect can be achieved at the room temperature\n[92–94].\nCoGeMn is reported to exhibit a collinear [91] magnetic state. However, depending on the Co:Mn ratio,\nCoxMn1\u0000xGe samples can possess not only an easy-axis or easy-plane magnetic anisotropy but also an interme-\ndiate state [95], i.e. spin reorients with Co:Mn ratio. The substitution of Ni (Co xNi1\u0000xMnGe) produces the even\nmore diverse results; the samples can be AFM, collinear FM, and non-collinear FM, depending on the amount of Ni\nand the annealing temperature [100].\nCo2MnGe is a full Heusler compound which exhibits some common features with the prototypical full Heusler\nCu2MnAl. These include a high TCand certain peculiarities related to its structural and magnetic properties derived\nfrom the range of chemical environments, disorder, vacancies, and stacking faults. Using anomalous X-ray diffraction\ntechniques Collins et al.[97] were able to demonstrate that thin films of Co 2MnGe grown on Ge (111) show the\npreference of Mn–Ge site swapping along with the small ( <0:5%) site-interchange of Co–Mn. The authors were also\nable to show that over–stoichiometry of Ge leads to the decrease in chemical disorder, vacancies, and stacking faults.\nIt is important to note, that they were not able to grow the Co 3Mn2Ge hexagonal phases due to a too large lattice\nmismatch with the Ge (111) surface.\nKogachi et al.[96] showed that the increase in sample quenching temperature gives rise to Mn–Ge disorder and\nresults in a lower magnetization at 4.2 K. Okubo et al.[98] were able to demonstrate that the aforementioned disorder\nbrings on a considerable decrease in TC. Webster [99], on the other hand, had earlier reported that there was no\nsign of chemical disorder in Co 2MnGe and that magnetic moment originated primarily from Mn (3.58 \u0016B); Co only\ncontributes 0.75 \u0016Bonly. Some of these ”discrepancies” reported previously are thus likely due to the difference in the\npreparation of the samples, but they also show that Co 2MnGe exists in a number of states ranging from a collinear\nferromagnet with a high TCto a virtually non-magnetic material, depending on the chemical disorder.23\nTABLE IX: Co–Mn–Ge phases reported previously, with their crystal structures and transition temperatures.\nCompound Entry prototype, Remarks\nSGR Symbol and number and comments\nCoMnGe TiNiSi, Pnma(62) Low-temperature phase, stable below 650 K [89];\nTC= 337 [90].\nCoMnGe BeZrSi, P63/mmc(194) High-temperature phase, stable above 650 K [89];\ncollinear [91] with TC= 283 K [90], 334 K [89].\nTransformation with TM=TCleads to large/giant magnetocaloric\neffect [92–94].\nCoxMn1\u0000xGe Ni 2In,P63/mmc(194) Spin reorientation depending on the amount of Co; can be easy axis,\neasy plane, as well as hard/easy equally in all directions [95].\nCo2MnGe Cu 2MnAl,Fm\u00163m(225) The increase in quenching temperature leads to increase in Mn-Ge\ndisorder [96]. Disorder causes the decrease in the magnetization [96];\nhigher quenching temperature rates produce lower magnetization [96].\nHigh Mn-Ge disorder, thin films [97], enrichment of 5 at % of Ge in\nCo0:5Mn0:25Ge0:25gives the highest degree of chemical ordering [97].\nChemical disorder in Co 2MnGe decreases TC[98].\nStrong ordering in Co 2MnGe [99], almost all magnetic moment comes\nfrom Mn (3.58 \u0016Bvs. 0.75\u0016Bfor Co).\nCoxNi1\u0000xMnGe TiNiSi, Pnma(62) Samples can be either AFM or FM or non-collinear FM depending on\nxand the annealing temperature [100].\nCo3Mn2Ge Mg 2Cu3Si,P63/mmc(194) Ordered crystallographic model [34]\nCo3Mn2Ge MgZn 2,P63/mmc(194) Disordered crystallographic model [34]\nCo3Ge5Mn9Mn9Co3Ge5,R32h, (155) No magnetic properties reported. [101]\nLastly, Co 3Ge5Mn9is reported to be a rhombohedral phase with no magnetic properties reported [101]."    },    {        "title": "2102.01283v2.On_the_relationship_between_orbital_moment_anisotropy__magnetocrystalline_anisotropy__and_Dzyaloshinskii_Moriya_interaction_in_W_Co_Pt_trilayers.pdf",        "content": "arXiv:2102.01283v2  [cond-mat.mtrl-sci]  13 Aug 2022On the relationship between orbital moment anisotropy, mag netocrystalline\nanisotropy, and Dzyaloshinskii-Moriya interaction in W/C o/Pt trilayers\nZhendong Chi,1,∗Yong-Chang Lau,1,2,†Vanessa Li Zhang,3Goro Shibata,1,4Shoya\nSakamoto,1Yosuke Nonaka,1Keisuke Ikeda,1Yuxuan Wan,1,5Masahiro Suzuki,1Masashi\nKawaguchi,1Masako Suzuki-Sakamaki,6,7Kenta Amemiya,6Naomi Kawamura,8Masaichiro\nMizumaki,8Motohiro Suzuki,8Hyunsoo Yang,9Masamitsu Hayashi,1,2and Atsushi Fujimori1,10\n1Department of Physics, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan\n2National Institute for Materials Science, Tsukuba, Ibarak i 305-0047, Japan\n3School of Physics and Technology, Wuhan University, Wuhan 4 30072, China\n4Materials Sciences Research Center, Japan Atomic Energy Ag ency, Sayo, Hyogo 679-5148, Japan\n5Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba 277-8581, Japan\n6Institute of Materials Structure Science, High Energy Acce lerator Research Organization, Tsukuba, Ibaraki 305-0801 , Japan\n7Graduate School of Science and Technology, Gunma Universit y, Kiryu, Gunma 376-8515, Japan\n8Japan Synchrotron Radiation Research Institute (JASRI), S ayo 679-5198, Japan\n9Department of Electrical and Computer Engineering,\nNational University of Singapore, Singapore 117576, Singa pore\n10Department of Physics, National Tsing Hua University, Hsin chu 30013, Taiwan\n(Dated: August 16, 2022)\nWe have studied the Co layer thickness dependences of magnet ocrystalline anisotropy (MCA),\nDzyaloshinskii-Moriya interaction (DMI), and orbital mom ent anisotropy (OMA) in W/Co/Pt tri-\nlayers, in order to clarify their correlations with each oth er. We find that the MCA favors magne-\ntization along the film normal and monotonically increases w ith decreasing effective magnetic layer\nthickness ( teff). The magnitude of the Dzyaloshinskii-Moriya exchange con stant (|D|) increases with\ndecreasing teffuntilteff∼1 nm, below which |D|decreases. The MCA and |D|scale with 1 /tefffor\ntefflarger than ∼1 nm, indicating an interfacial origin. The increase of MCA w ith decreasing teff\ncontinues below teff∼1 nm, but with a slower rate. To clarify the cause of the teffdependences\nof MCA and DMI, the OMA of Co in W/Co/Pt trilayers is studied us ing x-ray magnetic circular\ndichroism (XMCD). We find non-zero OMA when teffis smaller than ∼0.8 nm. The OMA increases\nwith decreasing teffmore rapidly than what is expected from the MCA, indicating t hat factors other\nthan OMA contribute to the MCA at small teff. Theteffdependence of the OMA also suggests that\n|D|atteffsmaller than ∼1 nm is not related to the OMA at the interface. We propose that the\ngrowth of Co on W results in a strain and/or texture that reduc es the interfacial DMI, and, to some\nextent, MCA at small teff.\nI. INTRODUCTION\nUltrathin film heterostructures that consist of ferro-\nmagnetic metal (FM) layers and non-magnetic heavy\nmetal (HM) layersareattracting great interest as various\nnovel phenomena that originate from the strong spin-\norbit coupling in bulk and at interfaces have been dis-\ncovered. For example, efficient current-induced magne-\ntization reversal [1] and fast motion of magnetic domain\nwalls[2]havebeendemonstratedinheterostructureswith\nperpendicular magnetic anisotropy(PMA), which are es-\nsential in ultra-high-density magnetic memories. These\nphenomena are attributed to spin-orbit coupling-induced\neffects such as spin Hall effect [3–6], Rashba-Edelstein ef-\nfect [7, 8], and Dzyaloshinskii-Moriya interaction (DMI)\n[9, 10]. Among these effects, strong DMI is especially\nnecessary for racetrack memories because it stabilizes\n∗Present address: CIC nanoGUNE, 20018 Donostia-San Sebas-\ntian, Basque Country, Spain\n†Present address: Institute of Physics, Chinese Academy of S ci-\nences, Beijing 100190, Chinachiral N´ eel domain walls and skyrmions [11–21]. There-\nfore, solid understanding of these interfacial phenomena\nis essential to develop spintronic devices with significant\nPMA and DMI.\nRecently, the microscopic origins of PMA and interfa-\ncial DMI have been discussed in relation to the orbital\nmoment anisotropy (OMA) [22, 23] and the magnetic\ndipole moment in the FM layer [24, 25]. As OMA should\nexist both in the FM and HM elements [26], it is of high\nimportance to identify what role the OMA (of FM and\nHM elements) plays in PMA and DMI in FM/HM het-\nerostructures.\nHere, we study correlation between the magnetocrys-\ntalline anisotropy (MCA), DMI and OMA in W/Co/Pt\ntrilayers. The Co layer thickness dependences of MCA\nand DMI in W/Co/Pt trilayers are studied using vi-\nbrating sample magnetometer (VSM) and Brillouin light\nscattering spectroscopy (BLS), respectively. As for the\nOMA, we study the Co layer thickness dependences\nof the spin and orbital magnetic moments of Co in\nW/Co/Pt trilayers, where W works as a seed layer, and\nthe proximity-induced magnetization in W and Pt in\nW/Co and Pt/Co bilayers using x-ray magnetic circu-2\nlar dichroism (XMCD). We find that the MCA, DMI,\nand OMA of Co show different Co layer thickness de-\npendences in the W/Co/Pt trilayers. The origin of these\nobservations will be discussed.\nII. EXPERIMENT\nCo thin films sandwiched by W and Pt, i.e. Sub./3\nW/tCoCo/1 Pt/1 Ru (the numbers denote the nominal\nthicknesses in nm) were grown on 10 ×10 mm2thermally\noxidized Si substrates by magnetron sputtering at room\ntemperature in a base pressure better than 5 ×10−7Pa.\nThe top Ru layer is used to protect the trilayers from\noxidation. The Co layer thickness ( tCo) was varied from\n0.6 to 1.7 nm (0.6, 0.7, 0.8, 1.0, 1.1, 1.2, 1.3, 1.4, and 1.7\nnm) in different samples. The magnetic hysteresis loops\n(in-plane and out-of-plane)ofthe samplesweremeasured\nusing a VSM at room temperature. The magnetic field is\napplied up to 1.6T during the measurement. The magni-\ntude of DM exchange constant ( |D|) was investigated by\nBLS, with the same measurement setup in our previous\nstudies [27].\nX-ray absorption spectroscopy (XAS) and XMCD\nmeasurements at the Co L3,2edges were performed using\nsoft x rays at the helical undulator beamline BL-16A1\nof Photon Factory, High Energy Accelerator Research\nOrganization (KEK-PF). The spectra were measured in\nthe total electron yield (TEY) mode. The measurements\nwere performed at room temperature in a vacuum bet-\nter than 5 ×10−7Pa. The magnitude of the magnetic\nfield was set at 5 T and the field was applied parallel to\nthe incident x rays in all measurements. The XAS and\nXMCD measurements using hard x rays were conducted\nat BL39XU of SPring-8. The partial fluorescence yield\n(PFY)andx-raypolarizationswitchingmodeswereused.\nThe measurements were performed at atmospheric pres-\nsure and at room temperature. A magnetic field of up to\n2 T was applied during the measurements. In order to\nobtain the out-of-plane and in-plane components of the\nmagnetic moments, the magnetic field was applied to the\nsample along the film normal and 30◦with respect to the\nsamplesurface, referredto asout-of-planeand “in-plane”\nmagneticfieldshereafter. NotethatthepositionofthePt\nL3edge (11.563keV) is excessivelyclose to that of the W\nL2edge (11.544 keV): these two peaks will overlap with\neach other in the W/Co/Pt trilayer and complicate the\ndata analysis. Thus, two W/Co and Pt/Co bilayers were\nprepared by magnetron sputtering for measuring the W\nand PtL3,2-edge XAS and XMCD spectra. The bilayer,\ni.e. Sub./0.6 W/0.8 Co/1 Ru and Sub./0.6 Pt/0.8 Co/1\nRu, were also grown on 10 ×10 mm2thermally oxidized\nSi substrates by magnetron sputtering at room temper-\nature and the 1-nm-thick Ru served as the capping to\navoid the bilayer from surface oxidation./s32/s33/s34/s35/s36/s32/s33/s34/s35/s33/s32/s33/s34/s33/s36/s33/s33/s34/s33/s36/s33/s34/s35/s33/s33/s34/s35/s36/s37/s38/s39/s40/s41/s42/s43/s44/s35/s33/s32/s45/s43/s40/s39/s46/s47\n/s32/s48/s33 /s32/s35/s33 /s33 /s35/s33 /s48/s33\n/s32/s43/s44/s49/s50/s40/s47/s32/s33/s34/s35/s36/s37/s38/s33/s39/s40/s35/s41/s33/s42/s43\n/s43/s51/s41/s32/s52/s53/s54/s41/s40\n/s43/s38/s46/s42/s32/s38/s55/s32/s52/s53/s54/s41/s40\n/s32/s33/s34/s35/s36/s32/s33/s34/s35/s33/s32/s33/s34/s33/s36/s33/s33/s34/s33/s36/s33/s34/s35/s33/s33/s34/s35/s36/s37/s38/s39/s40/s41/s42/s43/s44/s35/s33/s32/s45/s43/s40/s39/s46/s47\n/s32/s48/s33 /s32/s35/s33 /s33 /s35/s33 /s48/s33\n/s32/s43/s44/s49/s50/s40/s47/s32/s33/s34/s35/s36/s37/s44/s33/s39/s40/s35/s41/s33/s42/s43\n/s43/s51/s41/s32/s52/s53/s54/s41/s40\n/s43/s38/s46/s42/s32/s38/s55/s32/s52/s53/s54/s41/s40\n/s32/s33/s34/s48/s32/s33/s34/s35/s33/s33/s34/s35/s33/s34/s48/s37/s38/s39/s40/s41/s42/s43/s44/s35/s33/s32/s45/s43/s40/s39/s46/s47\n/s32/s48/s33 /s32/s35/s33 /s33 /s35/s33 /s48/s33\n/s32/s43/s44/s49/s50/s40/s47/s32/s33/s34/s35/s41/s37/s36/s33/s39/s40/s35/s41/s33/s42/s43\n/s43/s51/s41/s32/s52/s53/s54/s41/s40\n/s43/s38/s46/s42/s32/s38/s55/s32/s52/s53/s54/s41/s40\n/s32/s33/s34/s48/s32/s33/s34/s35/s33/s33/s34/s35/s33/s34/s48/s37/s38/s39/s40/s41/s42/s43/s44/s35/s33/s32/s45/s43/s40/s39/s46/s47\n/s32/s48/s33 /s32/s35/s33 /s33 /s35/s33 /s48/s33\n/s32/s43/s44/s49/s50/s40/s47/s32/s33/s34/s35/s41/s37/s45/s33/s39/s40/s35/s41/s33/s42/s43\n/s43/s51/s41/s32/s52/s53/s54/s41/s40\n/s43/s38/s46/s42/s32/s38/s55/s32/s52/s53/s54/s41/s40\n/s32/s33/s34/s48/s32/s33/s34/s35/s33/s33/s34/s35/s33/s34/s48/s37/s38/s39/s40/s41/s42/s43/s44/s35/s33/s32/s45/s43/s40/s39/s46/s47\n/s32/s48/s33 /s32/s35/s33 /s33 /s35/s33 /s48/s33\n/s32/s43/s44/s49/s50/s40/s47/s32/s33/s34/s35/s41/s37/s32/s33/s39/s40/s35/s41/s33/s42/s43\n/s43/s51/s41/s32/s52/s53/s54/s41/s40\n/s43/s38/s46/s42/s32/s38/s55/s32/s52/s53/s54/s41/s40 /s32/s33/s34/s48/s32/s33/s34/s35/s33/s33/s34/s35/s33/s34/s48/s37/s38/s39/s40/s41/s42/s43/s44/s35/s33/s32/s45/s43/s40/s39/s46/s47\n/s32/s48/s33 /s32/s35/s33 /s33 /s35/s33 /s48/s33\n/s32/s43/s44/s49/s50/s40/s47/s32/s33/s34/s35/s41/s37/s46/s33/s39/s40/s35/s41/s33/s42/s43\n/s43/s51/s41/s32/s52/s53/s54/s41/s40\n/s43/s38/s46/s42/s32/s38/s55/s32/s52/s53/s54/s41/s40/s47/s48/s49 /s47/s50/s49\n/s47/s51/s49 /s47/s52/s49\n/s47/s53/s49 /s47/s54/s49\nFIG. 1. Magnetic hysteresis loops of W/Co/Pt trilayers mea-\nsured by a VSM. The magnetic field is applied perpendicular\n(blue) and parallel (red) to the film plane. The thickness of\nthe Co layer is (a) 0.6, (b) 0.8, (c) 1.0, (d) 1.2, (e) 1.3, (f) 1 .7,\nrespectively.\nIII. RESULTS AND DISCUSSION\nSelected magnetic hysteresis loops of the trilayers with\ndifferent Co thicknesses measured by using the VSM are\nshown in Fig 1. The easy axis of the trilayers are con-\nfirmed to lie in the film plane. The magnetic moment\nincreases with Co thickness. The magnetic properties of\nthe trilayers determined by the magnetic hysteresis loops\nare shown in Fig. 2. ThetCodependence of the magnetic\nmoment is shown in Fig. 2(a). A linear function is fitted\nto the data with larger weight for films with larger tCo.\nThe slope of the linear function is proportional to the\nsaturation magnetization (per unit volume) Msand the\nhorizontal axis intercept represents the magnetic dead\nlayer thickness tD. By taking into account the area of\nthe samples, we find Ms∼1350±50 emu·cm−3and\ntD∼0.14±0.05 nm. The value of Msis close to that of\nbulk Co [28]. tDis typically negative when Co faces a Pt\nlayerduetoproximity-inducedmagnetization[29,30], we\ninfer that a magnetic dead layer at the W/Co interface\nexists and compensates the negative tDat Co/Pt inter-\nface. Based on the reported values of proximity-induced\nmoment at Co/Pt interface ( tD∼ −0.06 nm) [29], the tD\nat W/Co interface is estimated as ∼0.20±0.05 nm.\nThe effective magnetic anisotropyenergy density, Keff,\nis obtained by taking the difference between the inte-3\n/s32/s33/s34/s35\n/s32/s33/s34/s32\n/s32/s33/s36/s35\n/s32/s33/s36/s32\n/s32/s33/s32/s35\n/s32/s37/s38/s39/s40/s41/s42/s43/s44/s36/s32/s45/s46/s43/s40/s39/s47/s48\n/s34/s33/s32/s36/s33/s35/s36/s33/s32/s32/s33/s35/s32\n/s32/s49/s38/s43/s44/s41/s39/s48/s44/s50/s48 /s44/s51/s48\n/s45/s34/s33/s32/s45/s36/s33/s35/s45/s36/s33/s32/s45/s32/s33/s35/s32/s32/s33/s35/s36/s33/s32/s33/s40/s52/s52/s53/s32/s40/s52/s52/s43/s44/s40/s54/s55/s53/s56/s39/s45/s34/s48\n/s34/s33/s32/s36/s33/s35/s36/s33/s32/s32/s33/s35/s32\n/s32/s40/s52/s52/s43/s44/s41/s39/s48\nFIG. 2. (a) Magnetic moment of W/Co/Pt trilayers as a\nfunction of Co layer thickness, tCo. The solid line is a linear\nfit to the data for tCo>1 nm. (b) Product of the effective\nmagnetic anisotropy energy, Keff, and effective magnetic layer\nthickness, teff, plotted as a function of teff. The solid line is a\nlinear fit to the data for teff>1 nm.\ngratedareasoftheeasy-axisandhard-axismagnetization\nhysteresis loops. With the effective magnetic layer thick-\nness defined by teff≡tCo−tD, the product of Keffand\nteff,Keffteff, is given by the following equation [30, 31]:\nKeffteff=KI+(KB−2πM2\ns)teff, (1)\nwhereKBandKIrepresent the bulk and interfacial con-\ntributions to Keff. The 2πM2\nsterm represents the shape\nanisotropy energy density. Keffteffis plotted against teff\nin Fig.2(b). Negative Keffteffcorresponds to the mag-\nnetization easy axis lying along the film plane. A lin-\near function is fitted to the data with larger weight on\nfilms with larger teff. The slope and the y-axis inter-\ncept of the linear function represent KB−2πM2\nsand\nKI, respectively. From the linear fit, we obtain KB∼\n(0.8±0.7)×106erg·cm−3andKI∼0.6±0.1 erg·cm−2.\nKIis smaller than that the values typically reported for\nstructures which include Co/Pt interfaces [29, 30]. Note\nthat the data show small but systemic deviation from the\nlinear fitting when teffis smaller than ∼1.0 nm.\nThe MCA of the trilayers is given by excluding the\n/s32/s33\n/s32/s34\n/s35\n/s36\n/s37\n/s33\n/s34/s38/s39/s40/s41/s42/s32/s34/s36/s41/s43/s44/s45/s46/s47/s48/s49/s50/s51\n/s50/s52/s34 /s33/s52/s34 /s32/s52/s34/s34\n/s32/s53/s32/s43/s54/s54/s41/s42/s55/s48/s49/s32/s51/s32/s33\n/s32/s34\n/s35\n/s36\n/s37\n/s33\n/s34/s38/s39/s40/s41/s42/s32/s34/s36/s41/s43/s44/s45/s46/s47/s48/s49/s50/s51\n/s33/s52/s34/s32/s52/s56/s32/s52/s34/s34/s52/s56\n/s32/s43/s54/s54/s41/s42/s55/s48/s51/s42/s57/s51 /s42/s58/s51\nFIG. 3. (a) teffand (b) 1 /teffdependence of magnetocrys-\ntalline anisotropoy (MCA) in W/Co/Pt trilayers. The solid\ncurves in panels (a) and (b) are calculated using Eq. (2) and\nthe values of KIandKBobtained from the fitting shown in\nFig. 2(b)./s32/s33/s34\n/s34/s33/s35\n/s34/s33/s36\n/s34/s33/s37\n/s34/s33/s38\n/s34/s39/s32/s39/s40/s40/s41/s42/s43/s44/s45/s46/s47/s48/s38/s49\n/s38/s33/s34/s32/s33/s50/s32/s33/s34/s34/s33/s50/s34\n/s33/s42/s51/s51/s40/s41/s52/s47/s49/s32/s33/s34\n/s34/s33/s35\n/s34/s33/s36\n/s34/s33/s37\n/s34/s33/s38\n/s34/s39/s32/s39/s40/s40/s41/s42/s43/s44/s45/s46/s47/s48/s38/s49\n/s38/s33/s50/s38/s33/s34/s32/s33/s50/s32/s33/s34/s34/s33/s50/s34\n/s32/s53/s33/s42/s51/s51/s40/s41/s52/s47/s48/s32/s49/s41/s54/s49 /s41/s55/s49\nFIG. 4. (a) teffand (b) 1 /teffdependence of the magni-\ntude of the Dzyaloshinskii-Moriya exchange constant ( |D|)\nin W/Co/Pt trilayers deduced by Brillouin light scattering\nspectroscopy (BLS) measurements. The solid lines in (a) and\n(b) show fit to the data from teff>1 nm.\nshape anisotropy from Keff:\nMCA≡Keff+2πM2\ns=KI\nteff+KB.(2)\nWe plot the teffand 1/teffdependences of the MCA in\nFigs.3(a) and (b), respectively. MCA increases mono-\ntonically with decreasing teff. The calculated MCA using\nthe parameters obtained from the fitting in Fig. 2(b) is\nshown by red solid lines in Fig. 3. Although the MCA is\nproportional to 1 /teffforteff>1 nm, it clearly deviates\nfrom the scaling for teff/lessorsimilar1 nm.\nTheteffand 1/teffdependences of the magnitude of\ntheDzyaloshinskii-Moriyaexchangeconstant( |D|), given\nby the BLS measurements, are plotted in Figs. 4(a) and\n(b), respectively. |D|increases with decreasing teffuntil\nteff∼1nm, belowwhichit drops. Asimilartendencyhas\nbeen observed in other HM/FM systems [32, 33], which\nis not in accordance with the simple picture of interface-\ndriven DMI. The 1 /teffdependence of |D|in Fig.4(b)\nis fitted using a linear function with a larger weight on\nthickerteff, as shown by a red solid line. The parameters\nobtained by the linear fitting areused to calculate the teff\ndependence of |D|in Fig.4(a), as shown by a red solid\nline. Figure 4displays that the experimental data devi-\nates from the linear fitting for teff<1 nm. We note that\n|D|of W/Co/Pt trilayer grown by molecular beam epi-\ntaxy (MBE) has also investigated in a recent study [34].\nThe value of |D|in a trilayer with 0.7-nm-thick Co layer,\n>2 erg/cm2, is much larger than our samples grown by\nmagnetron sputtering. We suggest the difference can be\nattributed to the different growth techniques.\nTo identify the origin of the teffdependences of MCA\nand DMI in the W/Co/Pttrilayers, the XAS and XMCD\nspectra of Co are studied. The setup of the measure-\nmentsisschematicallyillustratedinthe insetofFig. 6(a).\nFigure5(a) shows the XAS and XMCD spectra at the\nCoL3,2edges of the W/Co/Pt trilayers measured un-\nder a magnetic field of 5 T. The intensity is normalized\nto theL3peak after removing a background consisting\nof two step functions. No obvious peak shift or spectral\nline-shape change is found in both the XAS and XMCD4\n/s32/s33/s34\n/s35/s33/s36\n/s35/s33/s34\n/s34/s33/s36\n/s34/s37/s38/s39/s40/s41/s39/s42/s43/s40/s44/s37/s45/s46/s33/s47/s33/s48\n/s49/s34/s34 /s50/s51/s34 /s50/s49/s34 /s50/s50/s34\n/s52/s53/s54/s40/s54/s39/s37/s55/s39/s41/s56/s57/s44/s37/s45/s41/s58/s48/s59/s34/s33/s60/s59/s34/s33/s61/s59/s34/s33/s32/s34/s37/s62/s63/s64\n/s37/s62/s65/s66/s67/s68/s69/s32/s66/s54/s37/s66/s54/s69/s52/s40\n/s37/s32/s66/s54/s37/s70/s37/s34/s33/s60/s37/s39/s71\n/s37/s34/s33/s50/s37/s39/s71\n/s37/s34/s33/s49/s37/s39/s71\n/s37/s35/s33/s34/s37/s39/s71\n/s37/s35/s33/s35/s37/s39/s71\n/s37/s35/s33/s32/s37/s39/s71/s66/s54/s37/s33/s72\n/s66/s54/s37/s33/s32\n/s59/s35/s33/s34/s59/s34/s33/s49/s59/s34/s33/s60/s59/s34/s33/s61/s59/s34/s33/s32/s34/s38/s39/s40/s41/s39/s42/s43/s40/s44/s37/s45/s46/s33/s37/s47/s33/s48\n/s49/s34/s34/s50/s51/s36/s50/s51/s34/s50/s49/s36/s50/s49/s34/s50/s50/s36\n/s52/s53/s54/s40/s54/s39/s37/s55/s39/s41/s56/s57/s44/s37/s45/s41/s58/s48/s68/s69/s32/s66/s54/s37/s66/s54/s69/s52/s40\n/s37/s62/s65/s66/s67/s37/s43/s39/s40/s41/s57/s56/s46/s73\n/s32/s66/s54/s37/s70/s37/s34/s33/s60/s37/s39/s71\n/s37/s34/s33/s50/s37/s39/s71\n/s37/s34/s33/s49/s37/s39/s71\n/s37/s35/s33/s34/s37/s39/s71\n/s37/s35/s33/s35/s37/s39/s71\n/s37/s35/s33/s32/s37/s39/s71/s45/s46/s48\n/s45/s74/s48\nFIG. 5. XAS, and XMCD spectra of Co in W/Co/Pt tri-\nlayers. Spectra after background subtraction are shown. (a )\nXAS,XMCD and (b) the integrated XMCD spectra at the\nCoL3,2edges for samples with different Co thicknesses. The\nspectra obtained under the out-of-plane and “in-plane” mag -\nnetic fields are plotted by solid and dashed curves, respec-\ntively.\nspectra between different teff, suggesting that there is no\nsignificant changes in the chemical state of Co, i.e., the\noxidation of Co is negligibly small. The solid and dashed\ncurves in Fig. 5(a) represent the spectra measured with\nout-of-plane and “in-plane” magnetic fields, respectively.\nThe integrated XMCD spectra, as displayed by the cor-\nresponding curves in Fig. 5(b), show clear differences be-\ntween measurements under out-of-plane and “in-plane”\nmagnetic field directions. These results indicate that the\nmagnetic moment of Co is anisotropic.\nThe effective spin magnetic moment ( meff) of Co atom\nis estimated using the XMCD sum rule [35, 36]:\nmeff=mspin+7\n2mT\n=−2/integraltext\nL3∆µdν−4/integraltext\nL2∆µdν/integraltext\nL3,2µdνnh,(3)\nwhere ∆µandµare the difference and sum of the XAS\nspectra obtained using right- and left-handed circularly\npolarized light, mspinis the spin magnetic moment of theCo atom, mTis the magnetic dipole, nhis the number\nof holes in the 3 dband of the Co atom. Here, we present\nmagnetic moments in units of Bohr magneton per Co\natom using nh= 2.45[37]. The out-of-planeand in-plane\ncomponents of the magnetic moments are obtained from\nthe integrated XAS/XMCD spectra measured under the\nout-of-plane and “in-plane” magnetic fields, respectively.\nmspinis considered to be isotropic, but mTpossesses an\nangular dependence mT=−1\n2mT0/parenleftbig\n1−3sin2θ/parenrightbig\n[38, 39].\nHere,mT0represents the out-of-plane component of mT\nandθis the angle between the magentization and the\nfilm plane.\nThe estimated values of mspinandmT0are shown in\nFigs.6(a) and 6(b) as a function of teff, respectively.\nmspindeviates from its bulk value, shown by a horizontal\ndashed line [28], and tends to decrease with decreasing\nteff. Such variation of mspinwith film layer thickness\nhas also been observed in similar systems [29]. mspin\ncan be fitted against tCousing the relation mspin=\n(1−tD/tCo)mspin,active , where mspin,active is the active\nspin magnetic moment, to estimate the tDin the Co\nlayer. From the fitting, we obtain tD∼0.20±0.03 nm,\nwhich is consistent with the dead layer thickness deter-\nmined from the VSM measurements. mT0, which is con-\nsiderably smaller than mspin, represents the anisotropic\nspin-density distribution, and its strength characterizes\nthe anisotropy of the spin-density distribution of the d\norbitals. Although it has been reported that mT0is re-\nlated to the emergence of PMA [24] and DMI [25], here\nits magnitude is considerably smaller than the previous\nreports [25].\nThe orbital magnetic moment ( morb) of Co atom is\nestimated using the XMCD sum rule [35, 36]:\nmorb=−4\n3/integraltext\nL3,2∆µdν\n/integraltext\nL3,2µdνnh. (4)\nThe out-of-plane component of morb,m⊥\norb, is estimated\nfrom the XAS and XMCD spectra measured under the\nout-of-plane magnetic field. The in-plane component,\nm/bardbl\norb, is obtained using the spectra measured under the\nout-of-plane and “in-plane” fields according to the rela-\ntionship,\nmorb(θ) =m⊥\norbsin2θ+m/bardbl\norbcos2θ, (5)\nwithθ= 30◦. Theteffdependence of morbis plotted in\nFig.6(c). Both m⊥\norbandm/bardbl\norbdecrease with decreas-\ningtefffrom their bulk value. We find the decrease of\nm/bardbl\norbwithteffis stronger than that of m⊥\norb. This differ-\nence leadsto the OMAofCowhich is illustrated by black\nsquaresin Fig. 6(c), showingan increasingtrend with de-\ncreasing teff. The normalized orbital magnetic moment\nmorb/mspinis plotted against teffin Fig.6(d). The out-\nof-plane component, m⊥\norb/mspin, increases with decreas-\ningteffwhereas the in-plane component, m/bardbl\norb/mspin, de-\ncreases. The normalizedOMA of Co, illustrated by black5\nFIG. 6. teffdependence of (a) the spin moment ( mspin) and (b) the out-of-plane component of magnetic dipole ( mT0).\nteffdependence of the orbital magnetic moment ( morb) and normalized orbital magnetic moment ( morb/mspin), for different\nmagnetization directions, are shown in (c) and (d), respect ively. The spin, orbital and normalized orbital moment valu es of\nbulk hcp Co [28] are shown using dashed horizontal lines. The schematic images of the XAS and XMCD measurements setup\nare shown as the inset in panel (a).\n/s32/s33/s34\n/s32/s33/s35\n/s35/s33/s34\n/s35/s33/s35\n/s36/s35/s33/s34/s37/s38/s39/s40/s41/s37/s42/s43/s44/s41/s45/s46/s47/s48/s46/s49/s45/s47/s50/s41/s51/s52/s33/s41/s53/s33/s54/s32/s35/s33/s55/s55/s32/s35/s33/s55/s35\n/s56/s57/s58/s47/s58/s46/s41/s59/s46/s48/s60/s61/s50/s41/s51/s62/s48/s63/s54/s32/s32/s33/s34/s64/s32/s32/s33/s34/s65/s66/s41/s55/s34 /s66/s41/s55/s34/s67/s41/s32/s68 /s67/s41/s32/s55/s35/s33/s64/s41/s46/s69/s41/s67/s70/s35/s33/s71/s41/s46/s69/s41/s43/s58\n/s37/s38/s39\n/s37/s42/s43/s44/s32/s33/s34\n/s32/s33/s35\n/s35/s33/s34\n/s35/s33/s35\n/s36/s35/s33/s34/s37/s38/s39/s40/s41/s37/s42/s43/s44/s41/s45/s46/s47/s48/s46/s49/s45/s47/s50/s41/s51/s52/s33/s41/s53/s33/s54/s32/s32/s33/s64/s35/s32/s32/s33/s34/s64 /s32/s68/s33/s68/s55/s32/s68/s33/s55/s71\n/s56/s57/s58/s47/s58/s46/s41/s59/s46/s48/s60/s61/s50/s41/s51/s62/s48/s63/s54/s66/s41/s32/s35 /s66/s41/s32/s35/s56/s47/s41/s32/s68/s56/s47/s41/s32/s55/s35/s33/s64/s41/s46/s69/s41/s56/s47/s70/s35/s33/s71/s41/s46/s69/s41/s43/s58\n/s37/s38/s39\n/s37/s42/s43/s44/s51/s52/s54 /s51/s72/s54\nFIG. 7. XAS and XMCD spectra at the W (a) and Pt (b)\nL3,2edges in W/Co and Pt/Co bilayers. The intensity of the\nXMCD spectra of W (Pt) is enlarged by a factor of 25 (10).\ndiamonds in Fig. 6(d), increases with decreasing teff, es-\npecially for teff/lessorsimilar0.8 nm. These results indicate that\nthe charge redistribution at the HM/Co interfaces takes\nplace and induces OMA. Considering the density of Co\ncrystal as 8.9 g ·cm−3, we estimate the total magnetiza-\ntion of the W/Co/Pt trilayer with teff=0.9 nm as 1300\nemu·cm−3. This value is in good agreement with that\ndetermined by using the VSM.\nWe have also performed W and Pt L3,2-edge XMCD\nmeasurements on W/Co and Pt/Co bilayers to study theproximity-induced magnetization. The XAS and XMCD\nspectra are shown in Fig. 7. XMCD signals are found\nonly in the Pt/Co bilayer. These results indicate that\nproximity-induced magnetization exists at the Pt/Co in-\nterface but does not exist at the W/Co interface.\nNow, we discuss the relationship between the MCA,\nDMI and OMA of Co in W/Co/Pt trilayers. Accord-\ning to Figs. 3and4, MCA and |D|scale with 1 /teff\nforteff/greaterorsimilar1 nm, indicating the interfacial origin of the\ntwo properties. Bruno has proposed that the MCA and\nOMA are proportional to each other in FM monolayers\n[22]. Both MCA and the OMA of Co indeed increase\nwith decreasing teffsmaller than ∼0.8 nm. However,\nthe OMA tends to increase more rapidly with decreasing\nteffthan what the Bruno’s law predicts from the values\nof MCA. These results suggest additional contributions\nto MCA, such as strain or magnetic dipole mT. Previ-\nous studies have indicated that strain in the film texture\ncan weaken the MCA when the magnetic layer thickness\nis reduced to a few atomic layers [40, 41]. The teffde-\npendences of MCA and Keffteffare in accordance with\nsuch studies. van der Laan has also shown that mTaf-\nfects MCA in strongly spin-orbit coupled systems, which\nhas been associated with spin-flip virtual excitation [24].\nSuch relationship has been confirmed in recent experi-\nments [42–44]. Unfortunately, the XMCD spectra of our6\nW/Co/Pt trilayers does not have sufficient resolution to\nderivemTaccurately.\nThe DMI, on the other hand, approaches near zero as\nteffis reduced below ∼1 nm. Interfacial DMI is due to\nthe strongspin-orbitcouplingwith brokeninversionsym-\nmetry. Recent studies have shown that DMI is related\nwithmTat the HM/FM interfaces [25] and the OMA of\nthe FM atoms [23]. Comparing the results presented in\nFigs.4(a)and6(c), weconsiderthatsuchrelationsdonot\nhold in the current system because the teffdependences\nof OMA and DMI are opposite to what one expects from\nthe scaling reported in Ref. [23]. The magnitude of mT\nfound in this system is considerably smaller than that re-\nportedinRef.[25]and, therefore,itscontributiontoDMI,\nif any, is alsolikely small. Furthermore, DMI in oursput-\ntered samples is much smaller than that in MBE-grown\nW/Co/Pt trilayer [34]. Thin films grown by MBE usu-\nally show better interfacial roughness and crystal texture\ncompared to films grown by magnetron sputtering. We\nthus speculate that the DMI is more sensitive to strain\neffect or the (111) texture of Co. Theoretical studies\n[45, 46] have indicated that the crystal structure at the\nHM/FM interface influences the strength of DMI dra-\nmatically. In the present case, strain and texture of the\nCo layer near the W/Co interface may significantly de-\ngrade|D|forteff/lessorsimilar1 nm. With further increasing Co\nthickness, such effects are then mitigated by the Co/Pt\ninterface that favors the (111) texture, resulting in the\nfollowing decrease of |D|.\nIV. SUMMARY\nWe have studied the effective magnetic layer thick-\nness (teff) dependences of magnetocrystalline anisotropy\n(MCA), Dzyaloshinskii-Moriya interaction (DMI), and\norbital moment anisotropy (OMA) in W/Co/Pt trilay-\ners. For tefflarger than ∼1 nm, MCA and DMI scale\nwith 1/teff, indicating an interfacial origin. However,\nwhereas MCA continues to increase with decreasing teff,DMI tends to decrease when teffis reduced below ∼1\nnm. The OMA of Co deduced from x-ray magnetic cir-\ncular dichroism (XMCD) measurements is almost zero\n(below the detection limit) when teffis larger than ∼0.8\nnm, below which the OMA of Co increases with decreas-\ningteff. The rate at which the OMA of Co increases\nwith decreasing teffis larger than what is predicted from\nthe MCA using Bruno’s formula. The reduction of DMI\nwith decreasing tefffor films with teff/lessorsimilar1 nm, despite\nthe presence of OMA, suggests that other factors con-\ntribute to the DMI in this thickness range. We infer that\nthe strain/texture in the Co layer induced by the W un-\nderlayer significantly weakens the DMI and, to a lesser\nextent, the MCA. Further studies are necessary to clarify\nthe latter points. Our results provide a microscopic un-\nderstanding for designing viable FM/HM-interface-based\nmultifunctional spintronic devices.\nACKNOWLEDGMENTS\nWe thank H. Shimazu for samples preparation. This\nwork was supported by Grants-in-Aid for Scientific Re-\nsearch from JSPS (Grant Nos. 15H02109, 15H05702,\n16H03853, and 20K14416). The XMCD experiment was\nperformed at BL-16A of KEK-PF with the approval of\nthe Photon Factory Program Advisory Committee (pro-\nposal Nos. 2016S2-005 and 2016G066) and at BL39XU\nof SPring-8 with the approval of the Japan Synchrotron\nRadiation Research Institute (JASRI) (proposal Nos.\n2017A1048 and 2018A1058). M.H. and A.F. are ad-\njunct members of Center for Spintronics Research Net-\nwork (CSRN), the University of Tokyo, under Spintron-\nics Research Network of Japan (Spin-RNJ). Z.C. is sup-\nported by Materials Education program for the future\nleaders in Research, Industry, and Technology (MERIT)\nand JSR Fellowship, The University of Tokyo. Y.-C.L. is\nsupported by JSPS InternationalFellowship for Research\nin Japan (Grant No. JP17F17064). S.S. and Y.-X.W.\nacknowledges financial support from Advanced Leading\nGraduate Course for Photon Science (ALPS). S.S. ac-\nknowledges financial support from the JSPS Research\nFellowship for Young Scientists.\n[1]I. M. Miron, K. Garello, G. Gaudin, P. J. Zermatten,\nM. V. Costache, S. Auffret, S. Bandiera, B. Rodmacq,\nA. Schuhl, and P. Gambardella, Nature 476, 189 (2011).\n[2]I. M. Miron, T. Moore, H. Szambolics, L. D. Buda-\nPrejbeanu, S. Auffret, B. Rodmacq, S. Pizzini, J. Vogel,\nM. Bonm, A. Schuhl, and G. Gaudin, Nat. Mat. 10, 419\n(2011).\n[3]M. I. Dyakonov and V. I. Perel, Jetp Letters-Ussr 13,\n467 (1971).\n[4]J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999).\n[5]J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back,\nand T. Jungwirth, Rev. Mod. Phys. 87, 1213 (2015).\n[6]L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and\nR. A. Buhrman, Science 336, 555 (2011).[7]Y. A.BychkovandE. I. Rashba,Jetp Lett. 39, 78(1984).\n[8]V. M. Edelstein, Solid State Communications 73, 233\n(1990).\n[9]I. E. Dzyaloshinskii, JETP 5, 1259 (1957).\n[10]T. Moriya, Phys. Rev. 120, 91 (1960).\n[11]M. Bode, M. Heide, K. von Bergmann, P. Ferriani,\nS. Heinze, G. Bihlmayer, A. Kubetzka, O. Pietzsch,\nS. Bl¨ ugel, andR.Wiesendanger, Nature 447, 190(2007).\n[12]S. S. P. Parkin, M. Hayashi, and L. Thomas, Science\n320, 190 (2008).\n[13]M. Hayashi, L. Thomas, R. Moriya, C. Rettner, and\nS. S. P. Parkin, Science 320, 209 (2008).\n[14]M. Heide, G. Bihlmayer, and S. Bl¨ ugel, Phys. Rev. B\n78, 140403 (2008).7\n[15]B. B. S. M¨ uhlbauer, F. Jonietz, C. Pfleiderer, A. Rosch,\nA. Neubauer, R. Georgii, and P. B¨ oi, Science 323, 915\n(2009).\n[16]X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han,\nY. Matsui, N. Nagaosa, and Y. Tokura, Nature 465, 901\n(2010).\n[17]A. Thiaville, S. Rohart, ´E. Ju´ e, V. Cros, and A. Fert,\nEurophys. Lett. 100, 5 (2012).\n[18]S. Emori, U. Bauer, S.-M. Ahn, E. Martinez, and\nG. S. D. Beach, Nat. Mater. 12, 611 (2013).\n[19]A. Fert, V. Cros, and J. Sampaio, Nat. Nanotech. 8, 152\n(2013).\n[20]S. Tacchi, R. E. Troncoso, M. Ahlberg, G. Gubbiotti,\nM. Madami, J. ˚Akerman, and P. Landeros, Phys. Rev.\nLett.118, 147201 (2017).\n[21]X. Ma, G. Yu, S. A. Razavi, S. S. Sasaki, X. Li, K. Hao,\nS. H. Tolbert, K. L. Wang, and X. Li, Phys. Rev. Lett.\n119, 027202 (2017).\n[22]P. Bruno, Phys. Rev. B 39, 865 (1989).\n[23]K. Yamamoto, A.-M. Pradipto, K. Nawa, T. Akiyama,\nT. Ito, T. Ono, and K. Nakamura, AIP Adv. 7, 056302\n(2017).\n[24]G. van der Lann, J. Phys.: Condens. Matter 10, 3239\n(1998).\n[25]S. Kim, K. Ueda, G. Go, P.-H. Jang, K.-J. Lee, A. Be-\nlabbes, A. Manchon, M. Suzuki, Y. Kotani, T. Naka-\nmura, K. Nakamura, T. Koyama, D. Chiba, K. Yamada,\nD.-H. Kim, T. Moriyama, K.-J. Kim, and T. Ono, Nat.\nCommun. 9, 1648 (2018).\n[26]I. V. Solovyev, P. H. Dederichs, and I. Mertig, Phys.\nRev. B52, 13419 (1995).\n[27]K. Di, V. L. Zhang, H. S. Lim, S. C. Ng, M. H. Kuok,\nJ. Yu, J. Yoon, X. Qiu, and H. Yang, Phys. Rev. Lett.\n114, 047201 (2015).\n[28]J.M.D.Coey, Magnetism and Magnetic Materials (Cam-\nbridge, UK, 2010).\n[29]T. Ueno, J. Sinha, N. Inami, Y. Takeichi, S. Mitani,\nK. Ono, and M. Hayashi, Sci. Rep. 5, 14858 (2015).\n[30]Y.-C. Lau, Z. Chi, T. Taniguchi, M. Kawaguchi, G. Shi-\nbata, N. Kawamura, M. Suzuki, S. Fukami, A. Fujimori,\nH. Ohno, and M. Hayashi, Phys. Rev. Mater. 3, 104419\n(2019).\n[31]J. Sinha, M. Hayashi, A. J. Kellock, S. Fukami, M. Ya-\nmanouchi, H. Sato, S. Ikeda, S. Mitani, S.-H. Yang,\nS. S. P. Parkin, and H. Ohno, Appl. Phys. Lett. 102,\n242405 (2013).\n[32]J. Cho, N.-H. Kim, S. Lee, J.-S. Kim, R. Lavrijsen,\nA. Solignac, Y. Yin, D.-S. Han, N. J. van Hoof, H. J.\nSwagten, B. Koopmans, and C.-Y. You, Nat. Commun.\n6, 7635 (2015).\n[33]M. Belmeguenai, M. S. Gabor, Y. Roussign´ e, T. P. Jr.,\nR. B. Mos, A. Stashkevich, S. M. Ch´ erif, and C. Tiusan,\nPhys. Rev. B 97, 054425 (2018).\n[34]S. K. Jenaa, R. Islamb, E. Mili´ nskaa, M. M.\nJakubowskia, R. Minikayeva, S. Lewi´ nskaa, A. Lyn-\nnyka, A. Pietruczika, P. Aleszkiewicza, C. Autieri, and\nA. Wawro, Nanoscale 13, 7685 (2021).\n[35]B. T. Thole, P. Carra, F. Sette, and G. van der Laan,\nPhys. Rev. Lett. 68, 1943 (1992).\n[36]P. Carra, B. T. Thole, M. Altarelli, and X. Wang, Phys.\nRev. Lett. 70, 694 (1993).\n[37]N. Nakajima, T. Koide, T. Shidara, H. Miyauchi,\nH.Fukutani, A.Fujimori, K.Iio, T.Katayama, M.N´ yvlt,and Y. Suzuki, Phys. Rev. Lett. 81, 5229 (1998).\n[38]R. Wu and A. J. Freeman, Phys. Rev. Lett. 73, 1994\n(1994).\n[39]J. St¨ ohr, J. Magn. Magn. Mater. 200, 470 (1999).\n[40]M. T. Johnson, P. J. H. Bloemen, F. J. A. den Broeder,\nand J. J. de Vries, Rep. Prog. Phys. 59, 1409 (1996).\n[41]Y.-C. Lau, P. Sheng, S. Mitani, D. Chiba, and\nM. Hayashi, Appl. Phys. Lett. 110, 022405 (2017).\n[42]S. Miwa, M. Suzuki, M. Tsujikawa, K. Matsuda,\nT. Nozaki, K. Tanaka, T. Tsukahara, K. Nawaoka,\nM. Goto, Y. Kotani, T. Ohkubo, F. Bonell, E. Tamura,\nK. Hono, T. Nakamura, M. Shirai, S. Yuasa, and\nY. Suzuki, Nat. Commun. 8, 15848 (2017).\n[43]K. Ikeda, T. Seki, G. Shibata, T. Kadono, K. Ishigami,\nY. Takahashi, M. Horio, S. Sakamoto, Y. Nonaka,\nM. Sakamaki, K. Amemiya, N. Kawamura, M. Suzuki,\nK. Takanashi, and A. Fujimori, Appl. Phys. Lett. 111,\n142402 (2017).\n[44]G. Shibata, M. Kitamura, M. Minohara, K. Yoshimatsu,\nT. Kadono, K. Ishigama, T. Harano, Y. Takahashi,\nS. Sakamoto, Y. Nonaka, K. Ikeda, Z. Chi, M. Fu-\nruse, S. Fuchino, M. Okano, J. i. Fujimori, A. Uchida,\nK. Watanabe, H. Fujihira, S. Fujihira, A. Tanaka, H. Ku-\nmigashira, T. Koide, and A. Fujimori, npj Quan. Mat.\n3, 3 (2018).\n[45]H. Yang, A. Thiaville, S. Rohart, A. Fert, and\nM. Chshiev, Phys. Rev. Lett. 115, 267210 (2015).\n[46]A. Hrabec, N. A. Porter, A. Wells, M. J. Benitez, G. Bur-\nnell, S. McVitie, D. McGrouther, T. A. Moore, and C. H.\nMarrows, Phys. Rev. B 90, 020402 (2014)."    },    {        "title": "2102.05315v1.Magneto_electric_Tuning_of_Pinning_Type_Permanent_Magnets_through_Atomic_Scale_Engineering_of_Grain_Boundaries.pdf",        "content": "1 \n Magneto -electric Tuning of Pinning -Type  Permanent Magnets  \nthrough Atomic -Scale Engineering of Grain Boundaries  \n  \n \nXinglong Ye1❀, Fengkai Yan2   , Lukas Schaefer3, Di Wang1,4, Holger Ge ßwein5, Wu Wang1♪,  \nMoham med Reda Chellali1, Leigh T. Stephenson2, Konstantin Skokov3, Oliver Gutfleisch3, Dierk Raabe2, \nHorst Hahn1, Baptiste Gault2,6❀, Robert Kruk1 \n \n1 Institute of Nanotechnology, Karlsruhe Institute of Technology (KIT), Eggenstein -Leopoldshafen, Germany  \n2 Department of Microstructure Physics and Alloy Design, Max -Planck -Institut für Eisenforschung GmbH  (MPIE) , \nDüsseldorf, Germany.  \n3 Department of Material Science, Techni cal Universit y Darmstadt, Darmstadt, Germany  \n4 Karlsruhe Nano Micro Facility, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany  \n5 Institute for Applied Materials, Karlsruhe Institute of Technology, Eggenstein -Leopoldshafen,  Germany . \n6 Department of Materials, Imperial College London, London, UK  \nCorresponding authors . Email : xing-long.ye @kit.edu ; b.gault@mpie.de   \n   now at Institute of Metal Research, Chinese Academy of Science, Shenyang, China  \n♪now at Department of Physics, Southern University of Science and Technology, Shenzhen, China.  \n \nAbstract  \nPinning -type magnets maintaining high coercivity, i.e. the ability  to sustain magnetization , at high \ntemperature are at the core  of thriving  clean -energy technologies. Among the se, Sm 2Co17-based  \nmagnets are excellent candidates  owing  to their high-temperature stabilit y. However,  despite decades \nof efforts to optimize the intragranular microstructure , the coercivit y currently only reach es 20~30% \nof the theoretical limits . Here, the roles of the grain -interior nanostructure and the grain boundaries in \ncontrolling coercivity  are disentangled by an emerging  magneto -electric approach. Through hydrogen \ncharging/discharging by applying voltages of only ~ 1 V , the coercivity is reversibly tuned by an \nunprecedented value of ~ 1.3 T.  In situ magneto -structural measurements  and atomic -scale tracking  of \nhydrogen atoms reveal that the segregation of hydrogen atoms at the grain boundaries, rather than the \nchange  of the crystal structure, dominates the reversible  and substantial change  of coercivity. Hydrogen \nlowers the local magnetocrystalline anisotropy and facilitates the magnetization reversal starting from  \nthe grain boundaries . Our study  reveals the  previously neglected critical role  of grain boundaries  in the \nconventional magnetisation -switching paradigm , suggesting a critical re considerat ion of strategies to \novercome the coercivity limits  in permanent magnets, via for instance atomic -scale grain boundar y \nengineering .  \nConsequently  present work . (200 words)  \n \nKey words: Magneto -electric, Permanent magnets, H ydrogen , Grain boundary  \n 2 \n Introduction  \nPermanent magnets  with the ability to maintain the ir magnetization , i.e. a property referred to as \ncoercivity , at high temperatures are crucial  for the flourishing  clean energy technologies  such as  \nelectric vehicles  and wind powers1,2. In this regard , the pinning -type magnets, in which the coercivity \narises from  the pinning of magnetic domain walls at nano -precipitate s within the grain ,6-8 are most \npromising . As an example , the Sm 2Co17-based magnet  is the only candidate for use in electric motors \nworking above  300 ℃  owing  to its excellent temperature stability . Its coercivity  is usually believed to \nbe controlled exclusively by domain -wall pinning  because of  the nano -scale cellular microstructure \nwithin the grain9-13, while  the initial demagnetization at grain boundaries is considered  irrelevant . \nHowever, d espite  intensive  efforts to optimize  its intra-granular microstructure,  the coe rcivity currently \nonly reach es 20-30% of the theoretical anisotropy field .1,14,15 Hence , it inevitably opens the question \nabout the  possible influence of grain boundar ies during  the magnetization reversal  of the material .  \n \nThe role of grain boundar ies in pinning -type magnets may be understood if they can be modified \nseparately  from the grain interior , in conjunction with measuring  the associated coercivity change . \nTraditional  processing approaches , such as heat treatment s13, plastic deformation16and alloying9,17, can \ndramatically change the coercivity. However , these approaches often induce  irreversible modification \n(or destruction) of the microstructure both in grain boundar ies and grain interior, obscuring  the \nassessment  of their respective  impact  on coercivity . Consequently, d ecoupl ing the separate roles of the \ngrain boundary and the grain interior in magnetization reversal  becomes  technically challenging.   \n \nIt has recently been demonstrated that t he magneto -electric approach can reversibly modify the \nmagnetic properties  of materials with external voltages  without  chang ing the microstructure8,19. For \ninstance, voltage -driven proton pumping and the voltage -controlled hydrogen insertion /extraction  can \nsubstantially tune the coercivity of ferromagnetic metals.20,21 Owing  to different affinities of hydrogen \natoms to the microstructural defects , hydrogen atoms  are expected to diffuse first along the grain \nboundaries, and, then into the grain interior .22,23 This sequential diffusion, if controlled , offer s an \nopportunity  to decouple the role s of the grain boundary and the grain interior , if the associated  \ncoercivity change is monitored  at each step. Here , by employing electrochemically -controlled \nhydrogen charging/discharging, we tuned the coercivity of the Sm 2Co17-based hard magnet by ~1.3 T , \nthe largest  values ever achieved by magneto -electric approaches . The combined in situ  magn eto-\nstructural measurements  and atomic -scale mapping  of hydrogen distribution24 reveal that hydrogen \natoms strongly segregate at grain boundaries, which we akens the local magnetic anisotropy  and \naccounts for the predominant change  of coercivity . Our study  open s a way to achieve giant \nmagnetoelectric effects  by atomic -scale engineering of grain boundaries , and unveils  the critical role  \nof grain boundar ies in limiting pinning-type magnet’s performance  pointing the way  forward for future \noptimisation strategies.  \n \nResults  \nMagneto -optical observation of magnetization reversal  \nWe used commercial Sm 2Co17-based  permanent magnet s with composition s of Sm(Co 0.766 3 \n Fe0.116Cu0.088 Zr0.029)7.35 (hereafter referred to as SmCo 7.35 sample ) (Table S1 ). The hysteresis loop \nshows a coercivity of ~2.8 T ( Fig. 1A). By magneto -optical Kerr effect (MOKE) microscopy,  we \nobserved its magnetization reversal process under  demagnetiz ation  fields. Prior to  MOKE imaging , \nthe sample was  fully  magnetized at -6.8 T ( Fig. 1B), and t he domain structure imaged with the c -axis \nin the viewing plane.  \n \nAt a demagneti zation  field of 1.5 T, the sample re tained its fully-magnetized state  (Fig. 1C). When the \nfield increase d to 2 T, the reversed domains started to appear at grain boundaries  (Fig. 1D), and at 2.2 \nT expand ed into the interior of the grains ( Fig.1E). With further increasing field, the magnetic domains \nmoved massively into the grain ( Fig. 1F). At 3.0 T, only a few residual domains were un -reversed (Fig. \n1G). These observations demonstrate  the initial nucleation of magnetic domains  at grain boundaries  \nbefore their growth  into the grain interior.  \n \nReversible modification of coercivity by non-destructive hydrogen charging/discharging  \nWe employed an electrochemical three -electrode cell to charge/discharge the SmCo 7.35 sample with  \nhydrogen atoms (Fig. S1 ). In this setup, the as -prepared electrode  with the SmCo 7.35 particles was the \nworking electrode  and 1 M KOH aqueous solution the electrolyte (Fig. S2 ). During hydrogen charging , \nthe electrochemical reduction of water molecules on the metal surface provide s the hydrogen ad -atoms \nthat subsequently diffuse into the material . Convers ely, during the discharging , the hydrogen atoms on \nthe surface ( Hads) were oxidized  and removed, resulting in hydrogen desorption. Based on  the \nmeasured cyclic voltammogram  (Fig. S3 ), we used the voltages steps of -1.2 V and -0.4 V to charge \nand discharge the sample , respectively .  \nWe explored  the response  of the coercivity of the as-prepared  SmCo 7.35 sample to hydrogen \ncharging/discharging by in situ  superconducting quantum interference device (SQUID) . The coercivity \nof the as -prepared sample was ~ 2.3 T ( Fig. 2A), slightly lower than that of the bulk -form pristine  \nsample (~ 2. 8 T) (Fig. 1A). After charging at -1.2 V for one hour, the coercivity drastically  decreased \nto ~ 1.0 T. We monitored  the recovery of the coercivity during the discharg ing process  by continuously \nrecording the hysteresis loops. The coercivity increased monotonically  with the discharging time ( Fig. \n2A), and re gained  most of its initial value in the early stage of the discharging process, reaching  ~ 1.9 \nT within 10 hours ( inset in  Fig. 2A). After a prolonged time of discharging (~ 60 hours), the coercivity \nfully recovered.  \n \nIn parallel, we studied  the dynamics  of the hydrogen charging/ discharging  process by observing the \nevolution of crystal structure with in situ  X-ray diffraction ( XRD ) in transmission mode  (Fig. 2B). \nUpon hydrogen -charging , all diffraction peaks were  shifted to lower angles , and after one hour , the \npositions of the diffraction peaks stayed unchanged, indicating the complete charging  of the whole \nsample.  In the discharg ing process , two stages  were discerned . First, strikingly, only a negligible shift \nof the peaks was observed over the first 10 hours (Fig. 2C), indicating that the bulk material is still \ncharged with hydrogen atoms. Second,  only after about 80 hours of discharg ing the peak s recovered  \nto their original positions.  This observation is in strong contrast with the substantial change in \ncoercivity over the corresponding  period ( Inset in Fig. 2A). It suggests that the predominant  coercivity 4 \n change  is not ascribed to the slow hydrogen desorption from  the volume  of the material.   \nSince the predominant change of coercivity does not arise from  the volumetric slow diffusion of \nhydrogen atoms, we expect a relatively fast response of magnetization reversal to hydrogen charging \n(Fig. 2C). We first magnetized the as -prepared sample with 6.8 T (point ① in the inset). Then, the \nmagnetic field was reversed to -1.1 T  (point ②), smaller than the coerci vity of the pristine  sample (~ \n2.3 T)  and therefore , the magnetization remained positive and nearly constant . Upon hydrogen \ncharging by applying  -1.2 V (point ③), the magnetization decreased immediately and abruptly , and \nflipped fro m positive to negative in ~10 minutes. The m agneti zation started to level off after ~4 hours. \nThe immediate response of magnetization reversal to the voltage stimulus confirms the existence of \nrelatively fast diffusion path of hydrogen atoms.  \n \nAtomic -scale tracking of hydrogen atoms within the hierarchical microstructure  \nWe carried out multi -scale multi -microscopy mapping of the micro - and nano -structur al features  to \nrationalize the fast diffusion  pathways of hydrogen  atoms . Optical microscopy (Fig. 3A, Fig. 1 ) \nshowed that the sample was polycrystalline with grain s of ~26 m separated by  high-angle grain \nboundaries (HAGBs) . The grains were  further divided into sub -grains by low -angle grain boundaries  \n(LAGB s) as shown by electron back -scattered imaging (Fig. 3B). Inside  the grain transmission \nelectron microscopy show s the typical cellular structure, compos ed of the matrix cell with sizes of ~ \n40 nm , the cell boundary  and the Zr-rich lamellae crossing the cellular structure  (TEM, Fig. 3C, S4). \nAfter tilting the c-axis of the specimen  out of the viewing plane , high-resolution TEM and the \ncorresponding selected area electron diffraction (SAED) showed  that the matrix phase is Sm 2Co17 \n(rhombohedral Th 2Zn17 type) and the cell boundary phase SmCo 5 (hexagonal CaCu 5 type)  (Fig. 3D). \nThis hierarchical micro - and nanostructure matche s previous reports9-13. \n \nWe used atom probe tomography ( APT ) to locate hydrogen atoms  and deuterium  atoms  within  the \nhierarchical micro structure  of the deuterium -charged samples.  Isotopic marking by deuterium atoms  \n(D) minimize d the influence of residual hydrogen in the atom probe . We first analysed  a specimen \ncontaining  a HAGB ( Fig. S5, S6). The element -specific atom maps in Fig. 4A reveal the Cu-rich cell \nboundaries , the matrix cells and the Zr -rich platelets , match ing TEM  results . Three-dimen sional \nreconstruction shows that D mostly  segregate s in a 10 –12 nm thick layer at HAGB , reaching  a \nconcentration  of approx. 3.5 at% ( Fig. 4A). A close face-on view ( Fig. 4B) reveals  that D segregate s \nat the intersection of the cell boundaries with the HAGB . In the cells and cell boundaries, a very limited \namount of D was detected within the structure close to the detection limit , with no noticeable  \npartitioning  difference between them (Fig. S6). In addition, d euterium atoms appear slightly depleted \nfrom the Z r-phase.  \n \nWe then performed another analysis target ing LAGB s (Fig. 4C, D, S7). Again, deuterium appears \ndeplet ed in the Z-phase , and no preferential segregation within cells and the cell boundaries  (Fig. S 8). \nYet again, a strong segregation of deuterium at LAGB  was observed. The corresponding top -view \nshows a series of linear features highlighted by a set of 0.35 at.% D iso -composition surfaces, which \nare likely the dislocations that constitute the LAGB26. The D -concentration  at these dislocations can 5 \n reach up to 0.4 at% D, and H up to 4 –5 at%. Importantly, the cell edges are all connected to these \ndislocations  and the cell structure stops abruptly at LAGB ( Fig. 4D).  \n \nDiscussion  \nThe observatio n of hydrogen /deuterium  segregation at GB  regions  (Fig. 4), coupled with in situ  XRD , \ndetecting no volumetric structural change  (Fig. 2B), suggest s that the substantial change of coercivity \nin the early stage of discharging process arises from the desorption of hydrogen atoms  from  GBs. The \nability to modulate  the coercivity by only charging/discharging GBs enables the fast control of \ncoercivity, as verified by the immediate start of magnetization reversal upon hydrogen charging  (Fig. \n2B). The herein identified cr ucial  role of grain boundary in controlling the coercivity  explains  why \nreducing the volume fraction of grain boundaries27 or optimising  the cellular structure near the grain \nboundary28 can increase the coercivity  of Sm 2Co17-based magnet s.  Below we discuss how  hydrogen \nsegregation  at GBs changes the  coercivity , starting with the microstructural features  near GBs.  \n \nThree  microstructural features distinguish the GB region  from the grain interior. First, compared  with \nthe continuous  cellular structure in the grain interior ( bottom part in Fig. 4C), the cell  boundaries  are \nbroken  and terminated  near GBs (Fig. 4A, C, D). Second, the typical cell size and shape in the grain \nis approx. 40 nm and with a regular shape, but becomes larger and strongly elongated near GBs (Fig. \n4). Th ese results  agree with recent TEM reports  that the incomplete  cellular structure near GBs extend \ntowards the sub-micrometer scale29,30. Third, t he composition profiles (Fig. S 6, S8) show that the \nSmCo 5 phase contains almost twice as much Cu near the grain boundary , i.e. 30 at.% compared with \n15 at.%  in the grain interior .  \n \nThe observed  different  cellular structure and microchemistry near GBs  significantly reduce s the local \nnucleation field required for magnetization reversal. According to the micromagnetic theory, the \ncritical nucleation field, Hn, can be described by14 \n     𝐻𝑛=1\n2𝑀s∆(𝛾𝑆𝑚𝐶𝑜 5−𝛾𝐺𝐵)−𝐷𝑀s       (1) \nin which  ∆ is the width of the transition region where the domain -wall energy chan ges from 𝛾𝐺𝐵 in \nthe grain boundary to 𝛾𝑆𝑚𝐶𝑜 5 in the cell boundary , and 𝐷 is the demagnetizing factor.  The SmCo 5 \nphase has much  larger  magn etocrystalline anisotropy  than GBs, and determines  the domain -wall \nenergy  difference , (𝛾𝑆𝑚𝐶𝑜 5−𝛾𝐺𝐵) . Near GBs the Cu concentration in the SmCo 5 phase becomes  \ntwice that in the grain interior , which  substantially reduces  its magnetocrystalline anisotropy31,32 and \nthus the domain wall energy, 𝛾𝑆𝑚𝐶𝑜 5 . In addition , the dis rupted  SmCo 5 phase allows the easy \nmovement of domain  walls through the matrix phase, triggering a  macroscopic  magnetization reversal.  \nThese account for  the preferential nucleation of reversed domains near GBs ( Fig. 1 ). Moreover , when \nthe SmCo 5 phase was charged  with hydrogen atoms , its magnetocrystalline anisotropy will decrease \nby ~40%20. This further  decrease s the domain -wall energy of the SmCo 5 phase and the nucleation field. \nBesides, h ydrogen segregation  may enlarge the transition region ( ∆) between GBs and the SmCo 5 \nphase with its continuous concentration change  and reduce the nucleation field. Hence, hydrogen \nsegregation acts here as a tool to further weaken the nucleation field of the GB region and amplify  its 6 \n effect in initiating  the magnetization reversal . Next , we consider the mechanism behind the \npropagation of the initial ly-nucleated  magnetic domains near GBs into the grain interior .  \n \nIn the grain interior , the continuous  network of the SmCo 5 phase subdivides the individual grains of \nthe Sm 2Co17 matrix  into a nano scale cellular structure , rendering  them into classical pinning -type \nmagnet s. However , the models to explain their magnetization reversal  assume that grain boundar ies \nshould be non -ferromagn etic and one -to-two atomic layers  thick to reduce  the associated stray field \nnegligibly14,33. This is not the case in the current material because of  the expanded region near GBs \nwith the disintegrated  cellular structure and different microchemistry.  We can describe the \ndemagnetization process of the whole grain triggered by the initial demagnetization near GB as follows . \nThe local magnetic  field is a superposition of the external field Hext and the local demagnetization field , \nN’M, where N’ is the local or effective demagnetization factor  and M the net magnetization  of the \nsample . The latter  can be significantly inhomogeneous, and  reach values much larger  than the net \ndemagnetization field3, HD = NM, N being  the demagnetization factor of the sample  (3). As discussed \nearlier,  the nucleation field  of the GB region, Hc,GB, is much smaller  than Hc of the grain  interior , Hc,g. \nUnder a small external field , we have Hc,g>Hc,GB>Hext+NM. As the local magnetic field , Hext+NM, \napproaches Hc,GB, the initial nucleation of magnetic domains occur s near GBs  (Fig. 1 ), producing a \nlocal demagnetization field, N’M s, where Ms is spontaneous magnetization of the main Sm 2Co17 phase.  \nThen,  the adjacent inner layer with higher coercivity , Hc,g, is under  the higher magnetic field, \nHext+NM+N’M s. This additional negative field , N’M s, can be of 0.5 -1.0 T, depending on microstructural \nfeatures and Ms34. Thus, if  Hc,g-Hc,GB<N’M s, the demagnetization of GB region  will inevitably trigger  \nan avalanche -like demagnetization process in the whole  grain , driven by the local enhancement of the \ndemagnetization field.  \n \nConclusion  \nIn summary, our results  show that by electrochemically -controlled hydrogen charging/discharging the \ncoercivity of the pinning -type Sm2Co17 -based magnet can be tuned  by an unprecedented value of ~ \n1.3 T, the highest value ever reported  by a magneto -electric approach . In situ magneto -structural \ncharacterization and atomic -scale tracking of hydrogen atoms over the hierarchical microstructure \nreveals that the predominant  change of the coercivity arises from  the decoration  of the grain boundar ies \nwith hydrogen atoms. These findings  reveal , contrary to the conventional paradigm , a critical role of \nthe grain boundaries in determining  the coercivity in pinning -type magnets . Furthermore, these \ndiscoveries  are anticipated  to apply to other technologically important pinning -type magnets such as \nFePt and MnAl . Future performance -optimisation strategies should consider engineer ing of  grain \nbounda ries to enhance the coercivity of pinning -type magnets for the clean -energy  applications . The \ndemonstrated voltage -controlled and giant modification of the coercivity by hydrogen  insertion \n(extraction) into  grain boundaries also opens a way  to various applications such as magneto -electric \nactuation  or sensing  in which large magneto -electric effect are needed.   \n \n \n 7 \n  \nMaterials and methods  \nMaterials and microstructure  characterization . The Sm 2Co17-type permanent magnets with \ndimensions of Φ10 mm×6 mm were purchased from Sigma -Aldrich  (Stock No.  692832 ). The \ncomposition of the powder was analyzed by inductively -coupled plasma mass spectroscopy ( Table S1 ) \nand its microstructure characterized by optical microscopy  (KIT) , powder X -ray diffraction with a Mo \nKα, source (Philips  X’Pert Analysis , KIT ), field-emission scanning electron microscope (SEM) \nequipped with energy dispersive X -ray spectroscopy and elect ron channeling contrast imaging (ECCI)  \n(Zeiss Ultra 600 /Merlin, both at KIT and MPIE ), and transmission electron microscope ( TEM, FEI \nTitan 80 -300, KIT ). Before optical and SEM characterization , the sample surface was mechanically \npolished. The preparation of TEM samples  followed the ordinary procedure of cutting, lifting and \nmilling using FIB/SEM dual beam system (FEI Strata 400 and Zeiss Auriga 60 , KIT ). The TEM \nobservations  were taken both with the c-axis of the crystal structure out of and parallel with the viewing \nplanes.  \n \nPreparation of the Sm 2Co17 powder electrode  and the electrochemical set-up. To prepare the \nSm 2Co17 electrode, the as-received bulk sample was first charged with hydrogen  atoms. The insertion \nof hydrogen atoms caus ed the expansion  of the  sample, and, consequently , the surface of the bulk \nsample collapsed  into the very large particles. Th ese particles  were mixed with PVDF solution to form \na slurry , which was then coated onto thin copper foil s (thickness ~ 1 5 m). The slurry/Cu foil \ncomposite was then put into a homogeneous magnetic field  to magnetically align the particles.  \nAfterwards, it was  further dried at room temperature for overnight. As the last step , the composite was \ncompressed under a pressure of ~ 100 MPa  to further fix the particle s and to increase the electrical \nconductivity between Sm 2Co17 particle s and the Cu foil. We prepared the PVDF solution by dissolving  \nPVDF powder in NMP solution at a mass ratio of 5:95 with stirring  overnight.  \n \nThe charging  and d ischarging of the Sm 2Co17 electrodes were carried out under potentiostatic control \nin a three -electrode electrochem ical system (Autolab PGSTAT 302N , KIT ). The working, the counter  \nand the reference electrodes were the Sm 2Co17 powder electrode, Pt wires and a pseudo Ag/AgCl \nelectrode , respectively ; the electrolyte was an aqueous electrolyte of 1 M KOH prepared from ultrapure \nwater with a resistivity of ~ 18.2 M Ω (Fig. S2 ). The potential of the peuso Ag/AgCl electrode is \n0.300±0.002 V more positive than the standard Hg/HgO (1M KOH) electrode , and for comparison , all \nthe voltages in the paper were converted to the Hg/H gO scale . According to the cyclic voltammogram \nof the SmCo 7.35 electrode ( Fig. S3 ), the voltages steps of -1.2V and -0.4 V were used to charge and \ndischarge the sample, respectively.  \n \nIn situ XRD measurement . The crystal structure of the Sm 2Co17 electrode under the application of -\n1.2 V and -0.4 V was monitored  by in situ XRD with a parallel beam laboratory rotating anode \ndiffractometer (Mo K α radiation) in transmission geometry. The transmission geometry allow ed the \ndetection  of the entire volume of the SmCo 5 particles rather than only their surfaces. For in situ  \nmeasurement, the Sm 2Co17 electrode was attached to a glass plate (thickness ~ 0.1 mm ) and then  \nimmersed in the 1 M KOH electrolyte in plastic bag s. The counter and reference electrodes were the \nPt wire and the pseudo Ag/AgCl electrode, respectively. Diffraction patterns were collected every 10 \nminutes  with a Pilatus 300K -W area detector. NIST SRM660b LaB 6 powder was used for the detector 8 \n calibration and the determination of the instrumental resolution.  \n \nIn situ SQUID measurement . In situ magnetic measurement was carried out  with a custom -built \nminiaturized Teflon electrochemical cell in a superconducting quantum interference device (SQUID, \nMPMS3 , KIT ) at room temperature. In the electrochemical cell, the Sm 2Co17 electrode, Pt foil and \npeuso Ag/AgCl electrode were the working , counter and reference electrodes, respectively. The \nelectrolyte was 1 M KOH. The Sm 2Co17 electrode and the Pt foil were attached to the flat surface of a \nplastic rod, and the reference electrode was threading through a capillary  to determine the applied \npotential of the working  electrode . The magnetic measurements were performed at the sealed mode of \nSQUID  and with  the applied magnetic field parallel to the surface of  the Cu  foil. The magnetic \nhysteresis loop of the bulk sample w as measured with  the magnetic field along the c -axis of the \nparticles .  \n \nMOKE measurement  \nThe magnetization reversal process was monitored by characterizing the  magnetic domain structure \nunder the external magnetic field using  magneto -optical  Kerr effect (MOKE) microscopy (Zeiss Axio \nImager , D2m  evico magnetics GmbH , TU Darmstadt ). Before the MOKE observation, the sample was \nfully magnetized at a pulsed field of 6.8 T. Then, the magnetic domain structure was observed after \napplying a reversed magnetic field of 1.5 T, 2 T, 2.2 T, 2.5 T, 2.8 T, 3.0 T and 6.8 T. The magnetic field \nwas applied parallel to c -axis of the matrix Sm 2Co17 phase, and the images taken with t he c-axis in the \nviewing plane. To enhance the image contrast , the non -magnetic background image was subtracted \nfrom the  collected average image using KerrLab software.  \n \nAPT measurement  \nFor the atom probe tomography (APT , MPIE ) measurement  of hydrogen dist ribution , the Sm 2Co17 \nelectrode  was charged at -1.2 V for ~2.5 hours in 0.1 M NaOD in D 2O (instead of H 2O) using the three -\nelectrode system as described above . After the full charging, the sample was cleaned by ethanol and \nwas transferred in 10 minutes to FIB chamber for the cutting, milling and lifting at room temperature \n(FEI Helios Nanolab 600/600i).  3-4 APT tips were prepared within 3 -4 hours  at room temperature . \nAnnular milling was used to sharpen  the needle -shaped morp hology with a diameter less than ~100 \nnm. After that, a cleaning of the specimen at 5 kV was performed to remove the beam -damaged surface \nregions. The prepared tips were transferred  into the load -lock chamber of APT , waited ~ 2 hours  until \nthe vacuum reach ed ~10-8 Pa (LEAP 3000 XHR), and then transferred into analysis chamber (~10-11 \nPa) at 70 K.  We also acquired  data on Cameca LEAP 5000XR , and only waited for half an hour  before \ntransferring the sample from APT load-lock to analysis chamber . The APT experiments were \nconducted using high -voltage mode with a pulse fraction of 15% at a base temperature of 70K, a pulse \nfrequence of 200 kHz and an evaporation rate of 0.5. Atom probe data reconstruction and analysis \nwere perfor med by CAMECA IV AS 3.8.4 software.          \n \n         \n \n \n1. O. Gutfleisch, M. A. Willard, E. Brueck, C. H. Chen, S. Sankar, J. P. Liu. Magnetic materials and devices \nfor the 21st century: stronger, lighter, and more energy efficient. Adv. Mater.  23, 821 -842 (2011).  \n2. J.M.D. Coey. Hard magnetic materials: a perspective . IEEE Trans. Magn. 47, 4671 -4681 (2011).  9 \n  \n3. K. Loewe, D. Benke, C. Kübel, T. Lienig, K.P. Skokov, O. Gutfleisch. Grain boundary diffusion of different \nrare earth elements in Nd -Fe-B sintered magnets by experiment and FEM simulation . Acta Mater.  124, 421 -\n429 (2016).  \n4. A.M. Gabay, M. Marinescu, W.F. Li, J.F. Liu, G.C. Hadjipanayis . Dysprosium  saving  improvement of \ncoercivity in Nd -Fe-B sintered magnets by Dy 2S3 additions . J. Appl. Phys.  109, 083916 (2011).  \n5. Y. Dong ,  T.L. Zhang , Z.C. Xia, H. Wang , Z.H. Ma, X. Liu, W. Xia, J. M. D. Coey   and  C. B. Jiang . \nDispersible SmCo 5 nanoparticles with huge coercivity . Nanoscale  11, 16962 (2019).  \n6. J. Rial,  P. Švec,  E.M.  Palmero,  J. Camarero,  P. Švec  Sr, A. Bollero. Severe tuning of permanent magnet \nproperties in gas -atomized MnAl powder by controlled nanostructuring and phase transformation . Acta \nMater . 157, 42-52 (2018).  \n7. S. Z., Y . Y . Wu, J. M. Wang, Y . X. Jia, T. L. Zhang, T.L. Zhang, C. B. Jiang. Realization of large coercivity \nin MnAl permanent -magnet alloys by  introducing nanoprecipitates . J. Magn. Magn. Mater.  483, 164 -168 \n(2019).  \n8. O. Gutfleisch , J. Lyubina , K.H. Müller , L. Schultz . FePt hard magnets. Adv. Eng. Mater.  7, 208 -212 (2005).  \n9. M. Duerrschnabel, M. Yi, K. Uestuener, M. Liesegang, M. Katter, H. J. Kleebe, B. Xu, O. Gutfleisch, L. \nMolina -Luna. Atomic st ructure and domain wall pinning in samarium -cobalt -based permanent magnets. \nNature Commun.  8, 54 (2017).  \n10. H. F. Mildrum , M. S.Hartings , K. J.Strnat , J. G. Tront . Magnetic Properties of the In termetallic Phases \nSm 2(Co, Fe) 17. AIP Conf. Proc. 10, 618 -622 (1973) . \n11. X.Y. Xiong, T. Ohkubo, T. Koyama, K. Ohashi, Y. Tawara, K. Hono, The microstructure of sintered \nSm(Co 0.72Fe0.20Cu0.055Zr0.025)7.5 permanent magnet studied by atom probe . Acta Mater.  52, 737 -748 (2004).  \n12. O. Gutfleisch, K.H. Müller, K. Khlopkov, M. Wolf, A. Yan, R. Schaefer, T. Gemming, L. Schultz, \nEvolution of magnetic domain structures and coercivity in high -performance SmCo 2:17 -type permanent \nmagnets, Acta Mater. 54, 997 -1008 (2006).  \n13. H. Sepehri -Amin, J. Thielsch, J. Fischbacher, T. Ohkubo, T. Schrefl, O. Gutfleisch, K. Hono, Correlation \nof microchemistry of cell boundary phase and interface structure to the coercivity of \nSm(Co 0.784Fe0.100Cu0.088Zr0.028)7.19 sintered magnets. Acta Mater.  126, 1-10 (2017).  \n14. H. Kro nmüller . Theory of nucleation field in inhomogeneous ferromagnets. Phys. Stat. Sol. (b)  144, 385 -\n396 (1987).  \n15. J. M. D. Coey. Coercivity in rare-earth iron permanent magnets . Clarendon  Press (1996) . \n16. A.M.Gabaya , M.Marinescu , J.F.Liu , G.C.Hadjipanayisa . Deformation -induced texture in nanocrystalline \n2:17, 1:5 and 2:7 Sm -Co magnets. J. Magn. Magn. Mater. 321, 3318 -3323 (2009).  \n17. N.J. Yu, W.Y . Gao, M. Pan, H. F. Yang, Q. Wu, P. Y . Zhang, H.L. Ge. Influen ce mechanism  of Fe content \non the magnetic properties of Sm 2Co17-type sintered magnets: Microstructure and microchemistry. J. Alloys \nComd.  818, 152908 (2020).  \n18. A. Molinari, Hahn H., R. Kruk, V oltage -control of magnetism in all -solid -state and solid/liquid \nmagnetoelectric composites. Adv. Mater. 1806662 (2019).  \n19. A. Tan, M. Huang, C. O. Avci, F. Büttner, M. Mann, Wen Hu, C. Mazzoli, S. Wilkins, H. L. Tuller, G. S. \nD. Beach, Magneto -ionic control of magnetism using a solid -state proton pump. Nature Mater.  18, 35-41 \n(2019).  \n20. X. L. Ye, H. K. Singh, H. B. Zhang, H. Gess wein, M.  R. Chellali, R. Witte, A. Molinari, K. Skokov, O. \nGutfleisch, H. Hahn, R. Kruk. Giant  voltage -induced modification of magnetism in micron -scale  \nferromagnetic metals by hydrogen charging. Nature Commun. DOI: 10.1038/s41467 -020-18552 -z. \n21. D. A. Gilbert, A. J. Grutter . Hydrogen finds a home in ionic devices. Nature Mater. 18, 7-8 (2019).  \n22. A. Pundt and R. Kirchheim. Hydrogen in metals: microstructural aspects. Annu. Rev. Mater. Res. 36, 555 -\n608 (2006).  10 \n  \n23. M. Krautz, J. D.  Moore, K.  Skokov, J.  Liu, C. S. Teixeira, R.  Schäfer,  L. Schultz , O. Gutfleisch. Reversible  \nsolid-state hydrogen -pump driven by magnetostructural transformation in the prototype system La (Fe, \nSi)13Hy. J. Appl . Phys .112, 083918  (2012).  \n24. Y.S. Chen, H.Z. Lu, J. Liang, A. Rosenthal, H. Liu, G. Sneddon, I. McCarroll, Z. Zhao, W. Li, A. Guo, J.M. \nCairney. Observation of hydrogen trapping at dislocations,  grain boundaries, and precipitates. Science  367, \n171–175 (2020).  \n25. A. J. Breen , L. T. Stephenson , B. Sun, Y. Li , O. Kasian  , D. Raabe, M. Herbig, B. Gault. Solute hydrogen \nand deuterium observed at the near atomic scale in high -strength steel.  Acta Mater.  188, 108 -120 (2020).  \n26. V. J.  Araullo -Peters, B.  Gault, S. L.  Shrestha, L.  Yao, M. P.  Moody, S. P.  Ringer, J. M.  Cairney. Atom \nprobe crystallography: Atomic -scale 3 -D orientation mapping.  Scripta Mater . 66, 907 -910 (2012).  \n27. M. Palit, D.M. Rajkumar, S. Pandian, S.V. Kamat. Effect of grain size and magnetic properties of \nSm 2(Co,Cu,Fe,Zr) 17 permanent magnets. Mater. Chem . Phys. 179, 214 -222 (2016).  \n28. Y.Q.Wang, M. Yue, D.Wu, D.T. Zhang,  W.Q. Liu, H.G. Zhang. Microstructure modification induced giant \ncoercivity enhancement in Sm(CoFeCuZr) z permanent magnets.  Scripta Mater.  146, 231 -235 (2018).  \n29. Y. Horiuchi, M. Hagiwara, M. Endo, N. Sanada, S. Sakurada. Influence of intermediate -heat treatment on \nthe structure and magnetic properties of iron -rich Sm(CoFeCuZr) Z sintered magnets. J. Appl. Lett.  117, \n17C704 (2015).  \n30. G. Yan, W. Xia, Z. Liu, R. Chen, C. Zhang, G. Wang,  J. Liu, A. Yan. Effect of grain boundary on \nmagnetization behaviors in 2:17 type SmCo magnet. J. Magn. Magn. Mater.  489, 165459 (2019).  \n31. E. Lectard, C. H.  Allibert, R.  Ballou. Saturation magnetization and anisotropy fields in the Sm(Co 1− x Cux)5 \nphases.  J. Appl . Phys . 75, 6277 -6279  (1994).  \n32. T. Katayama, T. Shibata. Magneto crystalline anisotropy constant in Sm(Co, Cu) 5 base Alloy.  Japanese J . \nAppl . Phys . 12, 762 -764 (1973).  \n33. H. Kronmuller. Micromagnetism of hard magnetic nanocrystalline materials. Nanostructured Mater. 6, \n157-168 (1995).  \n34. W. Xia, Y.  He, H.  Huang, H.  Wang , X.  Shi, T.  Zhang, T. Zhang, J. Liu, P. Stamenov, L. Chen, J.M. Coey, \nC.B. Jiang. Initial Irreversible Losses and Enhanced High ‐Temperature Performance of Rare ‐Earth \nPermanent Magnets.  Adv. Func . Mater . 29, 1900690 (2019) . 11 \n Acknowledgements  \nThe authors appreciate financial support from Deutsche Forschungsgemeinschaft under contract \nnumber HA 1344/34 -1 (R.K, H.H.)  and CRC/TRR 270,  (L.S., K.S., O.G. , BG ) as well as Alexander \nvon Humboldt Foundation  (X.Y .) . BG and LTS acknowledges financial support from the ERC -CoG -\nSHINE -771602 . Author contributions:  X.Y., R.K., H.H. conceived the project . X.Y. designed the \nexperiments. X.Y . conducted material  preparation  and microstructural observation , built in situ \nelectrochemical charging/discharging setup and conducted magnetic measurements ; F.Y. conducted  \nAPT, part of SEM , assisted by L.T.S, and F.Y., B.G. processed APT data ; L.S., X.Y . performed MOKE ; \nH.G., X.Y . performed in-situ XRD ; M.R.C. prepared TEM samples , discussed results,  and D.W, W.W \nconducted TEM observation. K.S. and O.G contribute significantly to the interpretation  of results. X.Y . \nwrote the initial draft and B.G., K.S., R.K. revised the manuscript. All authors participate d in the \ndiscussions, contribute d to improving the manuscript and approved the submitte d manuscript. \nCompeting interests: None. Data and materials availability:  All data needed to evaluate the \nconclusions of the study are present in the paper or the supplementary materials.  \n \nSupplementary Materials:  \nFig. S1 to S 8 \nTable S112 \n  \n \n \n \n \nFig. 1 Magneto -optical Kerr effect (MOKE) microscopy  observation of the magnetization \nreversal process in Sm 2Co17-based magnet  (A) Hysteresis loops of the bulk Sm 2Co17-type magnet. \n(B-H) MOKE observation of the magnetic domain structure  under  different demagnetiz ation  field from \nthe magnetically -saturated state – the applied magnetic  field is parallel to c -axis of crystal structure  as \nindicated in (B). \n-1.5-1.0-0.50.00.51.01.5\n86 2 -6-4\n  M (10-3 A m2)\n0 \n0H (T)4 -2 -8\n-6.8 T 2 T 1.5 T\n2.2 T 2.5 T 3.0 T 6.8 TH // c-axisA B C D\nG F E H\n20 µm13 \n  \n \n \n \nFig. 2. Reversible modification of the coercivity in Sm 2Co17-based  magnet by hydrogen \ncharging /discharging  and its dynamics . (A) Hysteresis loops of the as -prepared sample and those \nafter hydrogen charging at -1.2 V for 1 hour (charged _1h) and after further discharging for 2  h (DC_2h), \n5 h (DC_5 h), 10 h (DC_10 h) and 55 h (DC_55 h). Inset  shows the continuous change of the coercivity \nagainst the discharging time. ( B) In situ  XRD monitoring the evolution of the crystal structure of the \nas-prepared sample during  charging and discharging , showing that within the first 10 hours of \ndischarging the diffraction pattern remained nearly unchanged compared with the charged sample . (C) \nTime evolution of the magnetization in the as -prepared sample with the voltages switched to -1.2 V , \nshowing the immediate  response of magnetization reversal  to voltage stimulus . Points ①, ②, ③, and \n④ indicate different magnetization states as shown in the  inset .  \nB A CFig. 2\n13 14 15 16 17 18\n DC_40 h\nDC_20 h\nDC_10 h\nDC_5 h\nDC_2 hDC_80 h\nCharged_1 hIntensity (a.u)\n2 ()\n  Sm2Co17 SmCo5\nPristine\n HC (T)\n 2.5\n2.0\n1.5\n1.0 \nDischarge time (hrs)0 20 40 60\n6.8 T \n +M 0\n+0H 4\n  -M \n0 −0H 123\n-1.1 Tcharged_1 hpristine\n0 1 2 3 4 5-0.040.000.040.080.12\n-1.2 V M (10-3 A m2)\ntime (hrs)41\n2 3\n \n \n \n \n-0.10-0.050.000.050.10\n6 2 -6-4\n  M (10-3 A m2)\n0\n0H (T)4 -2pristine\ncharged_1 h\nDC_2 h\nDC_5 h\nDC_10 h\nDC_55 h14 \n  \n \n \nFig.3. Multi -level multi -spectroscopy characterization of the hierarchical microstructure in \nSm 2Co17-based magnet . (A) An optical microscopy  image showing the polycrystalline grain s with \ngrain sizes ~ 26 m. (B) An enlarged image of the crystal grain showing low -angle grain boundaries \n(LAGB) within the grain . (C) A bright -field TEM image within the grain showing the nano -scaled \ncellular structure , composed  of the Sm 2Co17 cell, the SmCo 5 cell boundary and the Zr -rich platelets. \n(D) A close -up on the cell and cell boundary by HRTEM  and their Fourier transform ation . The c-axis \nof the matrix phase in (C) and ( D) is in and out of the viewing plane, respectively. \n15 \n  \nFig. 4 Atomic -scale tracking of hydrogen /deuterium  atoms within the microstructure in \nSm 2Co17-based  magnet . (A) 3D atom map of a deuterium -charged sample containing  a high-angle \ngrain boundary (HAGB) . The  sets of 9 at .% Cu and  6 at.% Zr iso -composition surfaces evidence the \nCu-rich cell boundaries and the Z -phase; the distribution of deuterium  reveals strong planar \nsegregation near HAGB.  (B) A face-on view on the distribution of D and Cu using 3.5 at.% D  and 9 \nat.% Cu  near the HAGB highlighting the co-location of D and Cu. Note the incomplete cellular \nstructure and its abrupt stop at HAGB . (C) 3D atom map of a deuterium -charged specimen contain ing \na low-angle grain boundary  (LAGB ). Note the disruption of the cellular structure adjacent to LAGB . \nThe distribution of D shows the strong planar  segregation  at LAGB, the depletion at Zr -rich phase and \nno partioning  between cell and cell boundary. ( D) A close -up using 0.35 at% D, 0.75 at% Hf and 4.5 \nat% H iso -surfaces show ing D segregation along a series of linear features interpreted as dislocations , \neach of which  is connected to a cell edge ; a close -up on the LAGB  showing the direction connection \nof each dislocation to a cell boundary as indicated by white arrows  (right side in D). \n \n"    },    {        "title": "2102.12788v2.Magnetic_order_and_crystalline_electric_field_excitations_of_the_quantum_critical_heavy_fermion_ferromagnet_CeRh__6_Ge__4_.pdf",        "content": "arXiv:2102.12788v2  [cond-mat.str-el]  29 Oct 2021Magnetic order and crystalline electric field excitations o f the quantum critical heavy\nfermion ferromagnet CeRh 6Ge4\nJ. W. Shu,1D. T. Adroja,2,3A. D. Hillier,2Y. J. Zhang,4Y. X. Chen,1B. Shen,1F. Orlandi,2\nH. C. Walker,2Y. Liu,1,5C. Cao,1F. Steglich,1,6H. Q. Yuan,1,5,7,8and M. Smidman1,5,∗\n1Center for Correlated Matter and Department of Physics, Zhej iang University, Hangzhou 310058, China\n2ISIS Facility, STFC Rutherford Appleton Laboratory,\nHarwell Oxford, Oxfordshire OX11 0QX, United Kingdom\n3Highly Correlated Matter Research Group, Physics Departme nt,\nUniversity of Johannesburg, P.O. Box 524, Auckland Park 200 6, South Africa\n4Institute for Advanced Materials, Hubei Normal University , Huangshi 435002, China\n5Zhejiang Province Key Laboratory of Quantum Technology and D evice,\nDepartment of Physics, Zhejiang University, Hangzhou 31005 8, China\n6Max Planck Institute for Chemical Physics of Solids, Dresde n, Germany\n7State Key Laboratory of Silicon Materials, Zhejiang Univers ity, Hangzhou 310058, China\n8Collaborative Innovation Center of Advanced Microstructu res, Nanjing 210093, China\n(Dated: November 1, 2021)\nCeRh 6Ge4is an unusual example of a stoichiometric heavy fermion ferr omagnet, which can be\ncleanly tuned by hydrostatic pressure to a quantum critical point. In order to understand the origin\nof this anomalous behavior, we have characterized the magne tic ordering and crystalline electric\nfield (CEF) scheme of this system. While magnetic Bragg peaks are not resolved in neutron powder\ndiffraction, coherent oscillations are observed in zero-fie ldµSR below TC, which are consistent with\nin-plane ferromagnetic ordering consisting of reduced Ce m oments. From analyzing the magnetic\nsusceptibilityandinelastic neutronscattering, we propo seaCEF-level schemewhichaccounts for the\neasy-planemagnetocrystalline anisotropy, wherethelow l yingfirst excitedCEF exhibitssignificantly\nstronger hybridization than the ground state. These result s suggest that the orbital anisotropy\nof the ground state and low lying excited state doublets are i mportant for realizing anisotropic\nelectroniccouplingbetweenthe f-andconductionelectrons, whichgivesrisetothehighlyan isotropic\nhybridization observed in photoemission experiments.\nIn heavy fermion materials, the competition between\nmagnetic exchange interactions which couple local mo-\nments, andtheKondointeractionbetweenlocalmoments\nand the conduction electrons, can frequently be tuned\nby non-thermal parameters such as pressure, magnetic\nfields, and/or chemical doping [ 1]. Consequently, the an-\ntiferromagnetic (AFM) ordering temperature can often\nbe continuously suppressed to zero at a quantum criti-\ncal point (QCP), where there is a breakdown of Fermi\nliquid behavior, and the large accumulation of entropy\ncan lead to emergent phases such as unconventional su-\nperconductivity [ 2,3]. Conversely, ferromagnetic (FM)\nQCP’s are not usually found [ 4], due to either a first-\norder disappearance of FM order [ 5,6], or the interjec-\ntion of different ground states [ 7–9]. Theoretically, it\nwas predicted that FM QCP’s are forbidden in clean\nitinerant FM systems [ 10,11], and while there have\nbeen reports of their occurrence in some doped mate-\nrials, including YbNi 4(P1−xAsx)2[12], CePd 0.15Rh0.85\n[13], URh 1−xRuxGe [14], and Ni 1−xRhx[15], in such\ncasesadisorderdrivensuppressionofthe first-ordertran-\nsition is difficult to exclude.\nRecently, the heavy fermion ferromagnet CeRh 6Ge4\nwithaCurietemperature TC= 2.5K[16]wasfoundtobean exception to this paradigm, from the findings that hy-\ndrostatic pressure continuously suppresses TC, yielding a\nFM QCP [ 17,18]. The QCP is accompanied by a strange\nmetal phase, with a linear temperature dependence of\nthe resisitivity and a logarithmic divergence of the spe-\ncific heat coefficient [ 17]. To account for this behavior in\nlight of the previously reported prohibition of FM QCP’s\nin itinerant systems, it was proposed that CeRh 6Ge4ex-\nhibitslocalquantum criticality, where the requisite en-\ntanglement of the local moments is generated by their\nxy-anisotropy [ 17]. Moreover, in such local FM quantum\ncriticalmodels, quasi-1Dexchangeinteractionsappearto\nbevitalforavoidingafirst-ordertransition[ 17,19],which\nis in accordance with angle-resolved photoemission spec-\ntroscopy (ARPES) measurements revealing evidence for\nhighly anisotropic c−fcoupling in CeRh 6Ge4, from the\nobservationofstronganisotropyinthehybridization[ 20].\nAlternatively, it was proposed that the pressure-induced\nfirst-order transition is avoided by the soft-modes, which\nprevent FM quantum criticality, becoming massive due\nto the antisymmetric spin-orbit coupling arising from the\nbroken inversion symmetry in the crystal lattice (space\ngroupP¯6m2) [21]. Additional studies therefore are nec-\nessary to gain insight into the origin of the FM quantum2\n/s49 /s50 /s51 /s52 /s53 /s54 /s55/s48/s51/s48/s48/s54/s48/s48/s57/s48/s48\n/s45/s48/s46/s48/s49/s48/s48/s46/s48/s49\n/s45/s48/s46/s48/s49/s48/s48/s46/s48/s49\n/s50 /s50/s46/s53 /s51 /s51/s46/s53 /s52/s45/s48/s46/s48/s49/s48/s48/s46/s48/s49\n/s48 /s50 /s52 /s54/s48/s46/s48/s56/s48/s46/s49/s50/s48/s46/s48/s56/s48/s46/s49/s50\n/s48 /s48/s46/s53 /s49 /s49/s46/s53 /s50 /s50/s46/s53 /s51 /s51/s46/s53/s48/s49/s53/s48/s51/s48/s48/s52/s53/s48/s53/s56/s46/s51/s111\n/s32/s73\n/s111/s98/s115\n/s32/s73\n/s99/s97/s108/s99\n/s32/s67/s101/s82/s104\n/s54/s71/s101\n/s52/s32\n/s32/s67/s117/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s46/s32/s117/s46/s41\n/s100 /s32/s40/s197 /s41/s32/s40/s97/s41\n/s66\n/s51\n/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s46/s32/s117/s46/s41 /s32/s32\n/s49/s50/s49/s46/s55/s111\n/s40/s48/s48/s50/s41/s40/s98/s41\n/s57/s48/s46/s48/s111\n/s100 /s32/s40/s197 /s41/s32/s53/s56/s46/s51/s111\n/s32/s50/s46/s52/s55/s32/s75/s65/s115/s121/s109/s109/s101/s116/s114/s121\n/s116/s32/s40 /s109 /s115/s41/s32/s48/s46/s50/s55/s32/s75/s40/s99/s41\n/s40/s100/s41\n/s48/s48/s46/s51/s48/s46/s54/s48/s46/s57/s49/s46/s50\n/s108 /s32/s40 /s109 /s115/s45/s49\n/s41/s66 /s32/s40/s71/s41\n/s84 /s32/s40/s75/s41/s66\n/s49\n/s66\n/s50\nFIG. 1. (Color online) (a) Neutron powder diffraction pat-\ntern of CeRh 6Ge4at 4 K, for one pair of detector banks on\nthe WISH diffractometer. The results from a Rietveld refine-\nment are also displayed. (b) Difference between the patterns\ntaken at 0.27 K, and 4 K, for three pairs of banks. The scat-\ntering angles for each of the detector banks are displayed in\nthe panels. Note that the strong features around d=1.8 and\n2.1˚A correspond to the (200) and (111) reflections of the Cu\nsample holder. (c) Zero-field µSR spectraof CeRh 6Ge4at two\ntemperatures below TC, where the solid lines show the results\nfrom the fitting described in the text. (d) Temperature de-\npendence of the internal fields (solid triangles) and Lorent zian\nrelaxation rate λ(open stars) obtained from fitting the µSR\ndata.\ncriticality, and to distinguish between the different the-\noretical scenarios. It is particularly important to both\nfurther characterize the nature of the magnetic ordering,\nand to understand the origin of the magnetic anisotropy.\nTherefore, we performed neutron diffraction, muon-spin\nrelaxation ( µSR), and inelastic neutron scattering mea-\nsurements on polycrystalline samples of CeRh 6Ge4.\nPolycrystalline samples were prepared by arc-melting\nthe constituent elements [ 16], and the as-cast sam-\nples were annealed in an evacuated quartz ampoule at\n1150◦C. Neutron powder diffraction measurements were\nperformedonthe WISHdiffractometeratthe ISISpulsed\nneutron facility [ 22,23], both at 4 K in the paramagnetic\nstate, and well below TCat 0.27 K. The data at 4 K for\none pair of detector banks is shown in Fig. 1(a), where\nthe results of a structural refinement are also displayed,\nwith the Bragg peaks from the Cu sample holder being\naccounted for using the Le-Bail method, yielding lattice\nparameters of a= 7.144(6)˚A andc= 3.846(4)˚A, in line\nwith previous reports [ 24]. In order to look for magnetic\nBragg peaks, the 4 K data were subtracted from that\nmeasured at 0.27 K, which is shown for three pairs ofdetector banks in Fig 1(b). No additional intensity is de-\ntected atd-spacings corresponding to non-integer ( hkl)\nreflections, in line with a lack of an AFM component to\nthe magnetism. ForFM order, the magnetic Braggpeaks\nare situated on the structural peaks, and therefore weak\nmagnetic peaks arising from small ordered moments will\nbe more difficult to detect. No additional intensity on\nany nuclear peaks is consistently resolved across multi-\nple pairs of detector banks. As shown in the top panel\nof Fig1(b), additional intensity is observed on the (002)\nBragg peak at 0.27 K for one pair of banks, as expected\nfor FM orderwith in-plane moments. However, this peak\nis not resolved for banks at different scattering angles,\nand no magnetic Bragg peaks at larger d-spacing are ob-\nserved, which are expected to have greater intensity [ 25].\nTherefore this increase might be an artifact arising from\nimperfect normalization to the monitor.\nEvidence for magnetic order in CeRh 6Ge4is however\nrevealed by zero-field µSR measurements, which were\nperformed on the MuSR spectrometer at the ISIS fa-\ncility [26,27]. As shown in Fig 1(c), coherent oscil-\nlations are observed below TC, demonstrating the oc-\ncurrence of long range magnetic order. The data in\nthe magnetically ordered state are best analyzed taking\ninto account three oscillation frequencies, using A(t) =\nA0+/summationtext3\ni=1Aicos(γµBit+φ)exp[−(σit)2/2]+A4exp(−λt),\nwhile in the paramagnetic state only the first and last\nterms are utilized. Here Aiare the initial asymme-\ntries corresponding to local fields Bi, with Gaussian re-\nlaxation rates σi, whileλis the Lorentzian relaxation\nrate of a non-oscillating component, and γµis the muon-\ngyromagnetic ratio. The temperature dependence of the\nthree values of Bi, as well asλ, aredisplayed in Fig. 1(d).\nB2andB3are found to increase with decreasing temper-\nature, whereas B1reaches a maximum at about 1 K, and\ndecreases slightly at lower temperatures. λexhibits a\npeak around the magnetic transition, as well as a steep\nincrease below about 0.8 K.\nThe positions of the muon stopping sites were esti-\nmated via density functional theory calculations with f-\nelectrons as core electrons [ 28,29], from the minimum\nenergy positions for a positive |e|charge. Two crys-\ntallographically inequivalent sites were identified, s1=\n(2\n3,1\n3,0), which correspondsto a global energy minimum,\nands2= (0.187,0.813,0.01)which correspondsto a local\nminimum. The latter corresponds to six equivalent crys-\ntallographic positions related by three-fold and mirror\nsymmetries, and since the three-fold symmetry is broken\nby in-plane FM order, this leads to up to three distinct\nlocalfieldsassociatedwiththe s2sites. Fromlowtemper-\nature magnetization measurements, the ordered in-plane\nmoment is estimated to be around 0 .2−0.3µB/Ce [17].\nEstimates for the local magnetic fields at the stopping\nsitess1ands2arising from in-plane FM order were cal-\nculated using the muesrpackage [ 30]. For uniform FM3\n∆\u0001= 22.1 meV|±5/2>\n|±3/2>\n|±1/2>\n\u0001+ s1(a) (b) (c)\u0000+ s2\u00027\n\u00039\n\u00048\n∆\u0002=  5.8 meV\nFIG. 2. (Color online) (a) Temperature dependence of the inv erse magnetic susceptibility of single crystal CeRh 6Ge4for two\nfield directions [ 17]. The solid lines show the results from fitting with a CEF mode l described in the text. (b) CEF level\nscheme and wavefunctions obtained from fitting with the CEF m odel, where the angular distributions of the wave functions\nare also displayed. (c) Crystal structure of CeRh 6Ge4, with Ce, Rh, and Ge shown in yellow, grey and purple, respect ively.\nFerromagnetic order with moments along the a-axis is also illustrated, as well as the proposed muon stopp ing sites s1ands2\n(orange and green spheres), and the orientation of the groun d state orbitals from the CEF model.\norder with a moment of 0 .24µB/Ce orientated along the\na-axis, local fields of 364 and 152 G are calculated for the\ns2sites, and 87 G is calculated for s1, in comparison to\nfitted values for B1,B2andB3of 405, 151, and 56 G, re-\nspectively. On the other hand, a moment of 0 .155µB/Ce\nyields 58 G for s1, in good agreement with B3, but yields\nunderestimates of 235 and 98 G for the s2sites. In mag-\nnetic metals, an accurate comparison between the calcu-\nlated and observedlocal fields requiresaccountingfor the\nmuon contact hyperfine fields, and similar discrepancies\nto calculations have been found for heavy fermion mag-\nnets [31]. Moreover, a change in the hyperfine fields with\ntemperature could lead to the non-monotonic behavior\nofB1, which together with the increase of λbelow 0.8 K,\nmay point to the low temperature evolution of the un-\nderlying correlated state. The differences may also arise\nfrom uncertainties in the positions of the muon stopping\nsites, the orientation of the moments in the basal plane,\noraspatialmodulationoftheorderedmoment, butinthe\ncase of the latter AFM Bragg peaks would be expected\nto be observed in neutron diffraction. As a result, both\nneutron diffraction and ZF- µSR are consistent with FM\norder in CeRh 6Ge4, with a small in-plane ordered mo-\nment, and indicate the absence of any significant AFM\ncomponent.\nIn order to determine the splitting of the J= 5/2 Ce\nground state multiplet of CeRh 6Ge4by crystalline elec-\ntric fields (CEF), the single crystal magnetic suscepti-\nbility [17] was analyzed using the Hamiltonian HCF=\nB0\n2O0\n2+B0\n4O0\n4, whereBm\nnandOm\nnare Stevens CEF\nparameters and operator equivalents, respectively [ 32].\nNote that since the Ce site has hexagonal point sym-\nmetry, there are only two non-zero Stevens parameters\nB0\n2andB0\n4, and therefore there is no mixing of differ-\nent|mJ/angbracketrightstates in the atomic wave functions. The re-\nsults are displayed in Fig. 2(a), with fitted values of\nB0\n2= 1.25 meV and B0\n4= 0.0056 meV, together withmolecular field parameters of λab=−52.8 mol/emu and\nλc=−111.0 mol/emu. This yields the level scheme il-\nlustrated in Fig. 2(b), with a Γ 7ground state doublet\nψ±\nGS=|±1\n2/angbracketright, a low-lying first excited state ψ±\n1=|±3\n2/angbracketright\nseparated by ∆ 1= 5.8 meV from the ground state, and\na second excited state ψ±\n2=| ±5\n2/angbracketrightat ∆2= 22.1 meV.\nNote that a |±1\n2/angbracketrightground state is anticipated, since this\nis the only eigenstate which correspondsto a non-zeroin-\nplane moment, with /angbracketleftµx/angbracketright=/angbracketleftψ∓|gJ(J++J−)/2|ψ±/angbracketright=\n1.28µB/Ce. Since /angbracketleftµx/angbracketrightis much larger than the observed\nlow temperature moment of around 0 .2−0.3µB/Ce, this\nindicates that there is a reduced ordered moment, either\nduetoKondoscreeningprocessesorsignificantzero-point\nfluctuations [ 17,33]. From the large positive value of B0\n2,\ntheab-plane is expected to correspond to the easy direc-\ntion of magnetization, in line with the observed high and\nlow temperature susceptibilities. This suggests that the\nsingle-ion anisotropy arising from the local environment\nof the Ce ions is sufficient to account for the observed\neasy-plane anisotropy, in contrast to many Kondo fer-\nromagnets which order along the hard axis [ 34]. While\nthe negative values of λabandλccould indicate the pres-\nence of coexistent antiferromagnetic correlations, as in\nCeTi1−xVxGe3[35–37], such negative values are often\nfound in Kondo ferromagnets [ 7,38–40], and an effective\nnegative molecular field can arise from the Kondo effect,\nwhich scales with TK[41,42].\nInelastic neutron scattering measurements were per-\nformed on powder samples of CeRh 6Ge4and the non-\nmagnetic analog LaRh 6Ge4, using the MERLIN spec-\ntrometer at the ISIS facility [ 43]. Figure 3displays low\nangle cuts of the data normalized to absolute units at\ntwo temperatures, for four different incident energies Ei.\nNo well defined CEF levels can be detected at energy\ntransfers up to 80 meV. On the other hand, over a large\nrange of energy transfers, the scattering from CeRh 6Ge4\nis consistently larger than the La-analog indicating the4\n/s45/s51 /s48 /s51 /s54/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48\n/s45/s53 /s48 /s53 /s49/s48 /s49/s53/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53\n/s45/s49/s48 /s48 /s49/s48 /s50/s48 /s51/s48/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53\n/s50/s48 /s52/s48 /s54/s48 /s56/s48/s48/s50/s52/s54/s56/s40/s97/s41/s32\n/s69\n/s105/s32/s32/s61/s32/s49/s50/s32/s109/s101/s86\n/s50/s48/s176 /s32/s45/s32/s54/s48/s176 /s32\n/s32/s67/s101/s82/s104\n/s54/s71/s101\n/s52/s32/s32/s55/s75\n/s32/s76/s97/s82/s104\n/s54/s71/s101\n/s52/s32/s32/s32/s55/s75\n/s32/s67/s101/s82/s104\n/s54/s71/s101\n/s52/s32/s32/s49/s48/s48/s75\n/s32/s76/s97/s82/s104\n/s54/s71/s101\n/s52/s32/s32/s32/s49/s48/s48/s75/s83 /s40/s81/s44 /s41/s32/s40/s109/s98/s32/s115/s114/s45/s49\n/s32/s109/s101/s86/s45/s49\n/s32/s102/s46/s117/s46/s45/s49\n/s41/s40/s98/s41\n/s69\n/s105/s32/s32/s61/s32/s50/s48/s32/s109/s101/s86\n/s50/s48/s176 /s32/s45/s32/s54/s48/s176 \n/s40/s99/s41/s32/s32/s32/s69\n/s105/s32/s32/s61/s32/s51/s56/s32/s109/s101/s86\n/s32/s32/s32/s49/s48/s176 /s32/s45/s32/s54/s48/s176 \n/s69/s110/s101/s114/s103/s121/s32/s116/s114/s97/s110/s115/s102/s101/s114/s32/s40/s109/s101/s86/s41/s40/s100/s41\n/s32/s32\n/s69\n/s105/s32/s32/s61/s32/s49/s48/s48/s32/s109/s101/s86\n/s49/s48/s176 /s32/s45/s32/s54/s48/s176 \nFIG. 3. (Color online) Low angle cuts of the inelastic neutro n\nscattering spectra of CeRh 6Ge4and LaRh 6Ge4at 7 K and\n100 K for incident energies of (a) 12 meV, (b) 20 meV, (c)\n38 meV, and (d) 100 meV. The integrated angular ranges are\ndisplayed in the panels.\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s48/s50/s52/s54/s56\n/s84 /s32/s61/s32/s55/s32/s75\n/s32/s32\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s69\n/s105\n/s32/s49/s50/s32/s109/s101/s86\n/s32/s50/s48/s32/s109/s101/s86\n/s32/s51/s56/s32/s109/s101/s86\n/s32/s49/s48/s48/s32/s109/s101/s86/s83\n/s109/s97/s103/s40/s81/s44 /s41/s32/s40/s109/s98/s32/s115/s114/s45/s49\n/s32/s109/s101/s86/s45/s49\n/s32/s102/s46/s117/s46/s45/s49\n/s41\n/s69/s110/s101/s114/s103/s121/s32/s116/s114/s97/s110/s115/s102/s101/s114/s32/s40/s109/s101/s86/s41/s215 /s48/s46/s49\nFIG. 4. (Color online) Magnetic contribution to the inelast ic\nneutron scattering intensity versus energy transfer at 7 K, for\nfour different incident energies Ei. The solid line shows the\ncalculated inelastic response for the CEF scheme in Fig. 2,\nwhere the FWHM of the quasielastic and inelastic peaks are\n3.3meV(2 TK), while thedashedlineshows thecasewherethe\nquasielastic FWHM is 3.3 meV but the inelastic peak corre-\nsponding to the first CEF excitation has a FWHM of 30 meV.\nNote that the slight mismatch between the data with Ei= 12\nand 20 meV is an artifact arising from slight differences in th e\nnormalization for the different incident energies.\npresence of broad magnetic scattering, extending from at\nleast the elastic line ( ∼1.5 meV for Ei= 12 meV) up\nto at least 60 meV. Note that magnetic scattering could\nnot be resolved on measurements performed at lower en-\nergies on the OSIRIS spectrometer (not displayed) [ 44],\nlikely due to the weak and broad nature of the magnetic\nresponse. The magnetic scattering of CeRh 6Ge4from\nthe MERLIN measurements was estimated by subtract-\ning low angle cuts of the LaRh 6Ge4data, taking into\naccount the different neutron scattering cross sections,\nand the results are shown in Fig. 4. It can be seen that\nthe magnetic scattering is strongest at low energies, butwith a long tail up to high energies. If this broad scat-\ntering were associated with the ground state doublet, i.e.\ncorresponding to quasielastic scattering, this would im-\nply a very large Kondo temperature TKon the order of\nhundreds of Kelvin [ 45]. This is in contrast to the mod-\nerate value of TK= 19 K deduced from comparing the\nmagneticentropytoaspin-1/2Kondomodel[ 16]. Dueto\nthe lack of well-defined excitations, the CEF parameters\nwere fixed to the values from the susceptibility analysis,\nand the solid line in Fig. 4shows the resulting calculated\ninelastic neutron spectra, with a full-width at half maxi-\nmum (FWHM) for all the excitations of 3.3 meV (2 TK).\nIt can be seen that this fails to account for the broad\nmagnetic scattering, and a well defined excitation at ∆ 1\nwould be expected to be observed. Note that no excita-\ntion at ∆ 2is expected, since the dipole matrix elements\nfor the transition from |±1\n2/angbracketrightto|±5\n2/angbracketrightare zero due to the\nneutron selection rules ∆ mJ=±1. On the other hand,\nthedashedlineshowsthe casewithaquasielasticFWHM\nof 2TK, but a much broader inelastic excitation with a\nFWHM of 30 meV, and it can be seen that this scenario\ncan well account for the broad scattering. This suggests\nthat the inelastic neutron scattering results are consis-\ntent with the CEF scheme deduced from the magnetic\nsusceptibility, but with the low-lying CEF excitation at\n5.8 meV being significantly broadened due to hybridiza-\ntion with the conduction electrons.\nThe CEF scheme displayed in Fig. 2(b) can account\nfor the in-plane orientation of the ordered moments of\nCeRh6Ge4belowTC, which was proposed to be vital\nfor generating the necessary entanglement for avoiding\na first-order transition under pressure, allowing for the\noccurrence of a ferromagnetic QCP [ 17]. The angular\ndistributions of the CEF wave functions are also dis-\nplayed. Notably, both the ground state and first ex-\ncited doublet at 5.8 meV primarily have electron density\nout of the basal plane, which may explain the strongly\nanisotropic hybridization revealed by ARPES, with sig-\nnificantly stronger hybridization along the c-axis [20].\nMoreover, the low-lying first excited doublet appears to\nhybridize much more strongly with the conduction elec-\ntrons, which may be a consequence of greater overlap\nwith the out-of-plane Rh(2) and Ge(2) atoms, while the\nground state charge density is orientated towards the\nneighboring Ce atoms [Fig. 2(c)]. Such a scenario with a\nmorestronglyhybridized excited CEF level hasbeen pre-\ndicted to give rise to metaorbital transitions [ 46], which\nwas proposed theoretically for CeCu 2Si2[47], yet has not\nbeen observed experimentally [ 48]. The influence of the\nfirst excited state on the low temperature behavior of\nCeRh6Ge4may be inferred from the Kadowaki-Woods\nratio corresponding to a ground state degeneracy N= 4,\non both sides of the QCP [ 17]. In fact, the angular distri-\nbution of the ground state 4 forbitals has been identified\nas a key parameter for tuning the hybridization of the5\nCe(Co,Ir,Rh)In 5family of heavy fermion superconduc-\ntors [49,50], where more prolate Γ 7ground states are\nassociated with stronger c−fhybridization, likely due\nto strongerhybridizationwith out-of-planeIn atoms[ 51].\nOur results suggest that the anisotropic hybridization is\nnot only driven by the quasi-one-dimensional arrange-\nment of Ce-chains, but by the angular distribution of the\nCEF orbitals arising from the localenvironment of the\nCe atoms.\nIn summary, our neutron diffraction and µSR mea-\nsurements are consistent with FM order in CeRh 6Ge4,\nwith a reduced magnetic moment compared to that ex-\npected from the CEF ground state. We propose a CEF\nscheme which can account for the easy-plane anisotropy\nof CeRh 6Ge4, which was predicted to be crucial for the\noccurrence of FM quantum criticality in this system [ 17].\nMoreover, the broad magnetic scattering observed in in-\nelastic neutron scattering suggests the presence of strong\nc−fhybridization, where the low lying first excited CEF\nlevel couples more strongly than the ground state. This\ncould potentially reconcile there being significant Kondo\nscreening processes which reduce the ordered moment,\nwiththeconclusionoflocalized4 felectronsinferredfrom\nquantum oscillation measurements [ 28]. These results\nsuggest that the anisotropy of the CEF orbitals is an im-\nportant factor in the observed anisotropic hybridization\n[20], and such anisotropic c−fcoupling may also give\nrise to quasi-1D magnetic exchange interactions, which\nhave been proposed to avoid the first-order transition\nubiquitous to isotropic systems [ 17,19]. As such, materi-\nals with similarly anisotropic ground state orbitals could\nbe good candidates for searching for additional quantum\ncritical ferromagnets. It is of particular interest to ex-\nperimentally determine if there is such a correspondingly\nlarge anisotropy in the magnetic exchange interactions of\nCeRh6Ge4, i.e. quasi-one-dimensional magnetism, which\ncould be determined from single crystal inelastic neutron\nscattering or THz spectroscopy.\nWe are very grateful to Piers Coleman for valu-\nable discussions, and to Franz Demmel for sup-\nport with measurements on OSIRIS. This work was\nsupported by the National Key R&D Program of\nChina (No. 2017YFA0303100, No. 2016YFA0300202),\nthe National Natural Science Foundation of China\n(No. 12034107, No. 11874320 and No. 11974306),\nthe Key R&D Program of Zhejiang Province, China\n(2021C01002), and the Science Challenge Project of\nChina (No. TZ2016004). DTA would like to thank the\nRoyal Society of London for Advanced Newton Fellow-\nship funding between UK and China. Experimentsat the\nISISPulsedNeutronandMuonSourceweresupportedby\na beamtime allocation from the Science and Technology\nFacilities Council (RB1820492, RB1820463, RB1820482,\nRB1820611 [ 23,27,43,44]).∗msmidman@zju.edu.cn\n[1] Z. F. Weng, M. Smidman, L. Jiao, X. Lu, and\nH. Q. Yuan, Multiple quantum phase transitions\nand superconductivity in Ce-based heavy fermions,\nReports on Progress in Physics 79, 094503 (2016) .\n[2] G. R. Stewart, Non-Fermi-liquid behavior in d- andf-\nelectron metals, Rev. Mod. Phys. 73, 797 (2001) .\n[3] P. Gegenwart, Q. Si, and F. Steglich, Quantumcriticalit y\nin heavy-fermion metals, Nature Physics 4, 186 (2008) .\n[4] M. Brando, D. Belitz, F. M. Grosche, and T. R.\nKirkpatrick, Metallic quantum ferromagnets,\nRev. Mod. Phys. 88, 025006 (2016) .\n[5] A. Huxley, I. Sheikin, E. Ressouche, N. Kerna-\nvanois, D. Braithwaite, R. Calemczuk, and J. Flou-\nquet, UGe 2: a ferromagnetic spin-triplet superconduc-\ntor,Phys. Rev. B 63, 144519 (2001) .\n[6] M. Uhlarz, C. Pfleiderer, and S. M. Hayden, Quan-\ntum phase transitions in the itinerant ferromagnet zrzn 2,\nPhys. Rev. Lett. 93, 256404 (2004) .\n[7] V. A. Sidorov, E. D. Bauer, N. A. Frederick, J. R. Jef-\nfries, S. Nakatsuji, N. O. Moreno, J. D. Thompson, M. B.\nMaple, and Z. Fisk, Magnetic phase diagram of the fer-\nromagnetic Kondo-lattice compound CeAgSb2up to 80\nkbar,Phys. Rev. B 67, 224419 (2003) .\n[8] H. Kotegawa, T. Toyama, S. Kitagawa, H. Tou,\nR. Yamauchi, E. Matsuoka, and H. Sugawara,\nPressure-temperature-magnetic field phase dia-\ngram of ferromagnetic Kondo lattice CeRuPO,\nJ. Phys. Soc. Jpn. 82, 123711 (2013) .\n[9] U. S. Kaluarachchi, V. Taufour, S. L. Bud’ko, and P. C.\nCanfield, Quantum tricritical point in the temperature-\npressure-magnetic field phase diagram of CeTiGe 3,\nPhys. Rev. B 97, 045139 (2018) .\n[10] D. Belitz, T. R. Kirkpatrick, and T. Vojta, First order\ntransitions and multicritical points in weak itinerant fer -\nromagnets, Phys. Rev. Lett. 82, 4707 (1999) .\n[11] A. V. Chubukov, C. P´ epin, and J. Rech, Instability\nof the quantum-critical point of itinerant ferromagnets,\nPhys. Rev. Lett. 92, 147003 (2004) .\n[12] A. Steppke, R. K¨ uchler, S. Lausberg, E. Lengyel,\nL. Steinke, R. Borth, T. L¨ uhmann, C. Krellner, M. Nick-\nlas, C. Geibel, F. Steglich, and M. Brando, Ferromag-\nnetic quantum critical point in the heavy-fermion metal\nYbNi4(P1−xAsx)2,Science339, 933 (2013) .\n[13] D. T. Adroja, A. D. Hillier, J.-G. Park, W. Kockel-\nmann, K. A. McEwen, B. D. Rainford, K.-H. Jang,\nC. Geibel, and T. Takabatake, Muon spin relaxation\nstudy of non-Fermi-liquid behavior near the ferro-\nmagnetic quantum critical point in CePd 0.15Rh0.85,\nPhys. Rev. B 78, 014412 (2008) .\n[14] N. T. Huy, A. Gasparini, J. C. P. Klaasse, A. de Visser,\nS. Sakarya, and N. H. van Dijk, Ferromagnetic\nquantum critical point in URhGe doped with Ru,\nPhys. Rev. B 75, 212405 (2007) .\n[15] C.-L. Huang, A. M. Hallas, K. Grube, S. Kuntz,\nB. Spieß, K. Bayliff, T. Besara, T. Siegrist, Y. Cai,\nJ. Beare, G. M. Luke, and E. Morosan, Quantum\ncritical point in the itinerant ferromagnet Ni 1−xRhx,\nPhys. Rev. Lett. 124, 117203 (2020) .\n[16] E. Matsuoka, C. Hondo, T. Fujii, A. Oshima, H. Sug-\nawara, T. Sakurai, H. Ohta, F. Kneidinger, L. Sala-6\nmakha, H. Michor, and E. Bauer, Ferromagnetic tran-\nsition at 2.5K in the hexagonal Kondo-lattice compound\nCeRh 6Ge4,J. Phys. Soc. Jpn. 84, 073704 (2015) .\n[17] B. Shen, Y. Zhang, Y. Komijani, M. Nicklas, R. Borth,\nA. Wang, Y. Chen, Z. Nie, R. Li, X. Lu, H. Lee, M. Smid-\nman, F. Steglich, P. Coleman, and H. Yuan, Strange\nmetal behavior in a pure ferromagnetic Kondo lattice,\nNature579, 51 (2020) .\n[18] H. Kotegawa, E. Matsuoka, T. Uga, M. Take-\nmura, M. Manago, N. Chikuchi, H. Sugawara,\nH. Tou, and H. Harima, Indication of ferromagnetic\nquantum critical point in Kondo lattice CeRh 6Ge4,\nJ. Phys. Soc. Jpn. 88, 093702 (2019) .\n[19] Y. Komijani and P. Coleman, Model for a ferromag-\nnetic quantum critical point in a 1D Kondo lattice,\nPhys. Rev. Lett. 120, 157206 (2018) .\n[20] Y. Wu, Y. Zhang, F. Du, B. Shen, H. Zheng, Y. Fang,\nM. Smidman, C. Cao, F. Steglich, H. Yuan, J. D. Den-\nlinger, and Y. Liu, Anisotropic c−fhybridization in\nthe ferromagnetic quantum critical metal CeRh 6Ge4,\nPhys. Rev. Lett. 126, 216406 (2021) .\n[21] T. R. Kirkpatrick and D. Belitz, Ferromagnetic quan-\ntum critical point in noncentrosymmetric systems,\nPhys. Rev. Lett. 124, 147201 (2020) .\n[22] L. C. Chapon, P. Manuel, P. G. Radaelli, C. Ben-\nson, L. Perrott, S. Ansell, N. J. Rhodes, D. Raspino,\nD. Duxbury, E. Spill, and J. Norris, Wish: The new\npowder and single crystal magnetic diffractometer on the\nsecond target station, Neutron News 22, 22 (2011) .\n[23] M. Smidman et al; (2019): STFC\nISIS Neutron and Muon Source,\nhttps://doi.org/10.5286/ISIS.E.RB1820482 .\n[24] D. Vosswinkel, O. Niehaus, U. C. Rode-\nwald, and R. Pottgen, Bismuth flux growth\nof CeRh 6Ge4and CeRh 2Ge2single crystals,\nZeitschrift f¨ ur Naturforschung B 67, 1241 (2012) .\n[25] See Supplemental Material at [] for the simulated mag-\nnetic diffraction pattern corresponding to in-plane ferro-\nmagnetic order.\n[26] P. J. C. King, R. de Renzi, S. P. Cottrell, A. D. Hillier,\nand S. F. J. Cox, ISIS muons for materials and molecular\nscience studies, Physica Scripta 88, 068502 (2013) .\n[27] M. Smidman et al; (2018): STFC\nISIS Neutron and Muon Source,\nhttps://doi.org/10.5286/ISIS.E.99690746 .\n[28] A. Wang, F. Du, Y. J. Zhang, D. Graf, B. Shen,\nY. Chen, Y. Liu, M. Smidman, C. Cao, F. Steglich,\nand H. Q. Yuan, Localized 4 f-electrons in the quan-\ntum critical heavy fermion ferromagnet CeRh 6Ge4,\nScience Bulletin 66, 1389 (2021) .\n[29] C. Cao and J.-X. Zhu, Pressure dependent electronic\nstructure in CeRh 6Ge4,arXiv:2011.14256 (2020) .\n[30] P. Bonf´ a, I. J. Onuorah, and R. D. Renzi, Intro-\nduction and a quick look at MUESR, the Magnetic\nStructure and mUon Embedding Site Refinement Suite,\nJPS Conf. Proc. 21, 011052 (2018) , Proceedings of the\n14th International Conference on Muon Spin Rotation,\nRelaxation and Resonance ( µSR2017).\n[31] F. N. Gygax, A. Schenck, and Y. ¯Onuki, Magnetic prop-\nerties of CeCu 2tested by muon-spin rotation and relax-\nation,J. Phys.: Condens. Matter 16, 2421 (2004) .\n[32] M. Hutchings, Point-charge calculations of energy lev els\nof magnetic ions in crystalline electric fields (Academic\nPress, 1964) pp. 227 – 273.[33] D. T. Adroja, W. Kockelmann, A. D. Hillier, J. Y. So,\nK. S. Knight, andB. D. Rainford, Reducedmoment mag-\nnetic ordering in aKondolattice compound: Ce 8Pd24Ga,\nPhys. Rev. B 67, 134419 (2003) .\n[34] D. Hafner, B. K. Rai, J. Banda, K. Kliemt, C. Krellner,\nJ. Sichelschmidt, E. Morosan, C. Geibel, and M. Brando,\nKondo-lattice ferromagnets and their peculiar order\nalong the magnetically hard axis determined by the crys-\ntalline electric field, Phys. Rev. B 99, 201109(R) (2019) .\n[35] W. Kittler, V. Fritsch, F. Weber, G. Fischer, D. Lam-\nago, G. Andr´ e, and H. v. L¨ ohneysen, Suppression\nof ferromagnetism of CeTiGe 3by V substitution,\nPhys. Rev. B 88, 165123 (2013) .\n[36] M. Inamdar, A. Thamizhavel, and S. K.\nDhar, Anisotropic magnetic behavior of\nsingle crystalline CeTiGe 3and CeVGe 3,\nJ. Phys. Condens. Matter 26, 326003 (2014) .\n[37] M. Majumder, W. Kittler, V. Fritsch, H. v. L¨ ohneysen,\nH. Yasuoka, and M. Baenitz, Competing magnetic corre-\nlations across the ferromagnetic quantumcritical point in\nthe Kondo system CeTi 1−xVxGe3:51V NMR as a local\nprobe,Phys. Rev. B 100, 134432 (2019) .\n[38] K. Katoh, S. Nakagawa, G. Terui, and A. Ochiai, Mag-\nnetic and transport properties of single-crystal YbPtGe,\nJ. Phys. Soc. Jpn. 78, 104721 (2009) .\n[39] C. Krellner and C. Geibel, Mag-\nnetic anisotropy of YbNi 4P2,\nJ. Phys.: Conf. Ser. 391 012032 (2012) 391, 012032 (2012) .\n[40] B. K. Rai, M. Stavinoha, J. Banda, D. Hafner, K. A.\nBenavides, D. A. Sokolov, J. Y. Chan, M. Brando,\nC.-L. Huang, and E. Morosan, Ferromagnetic ordering\nalong the hard axis in the Kondo lattice YbIr 3Ge7,\nPhys. Rev. B 99, 121109(R) (2019) .\n[41] G. Gruner and A. Zawadowski, Magnetic impurities in\nnon-magnetic metals, Rep. Prog. Phys. 37, 1497 (1974) .\n[42] H. R. Krishna-murthy, K. G. Wilson, and J. W. Wilkins,\nTemperature-dependent susceptibility of the symmet-\nric Anderson model: Connection to the Kondo model,\nPhys. Rev. Lett. 35, 1101 (1975) .\n[43] M. Smidman et al; (2018): STFC\nISIS Neutron and Muon Source,\nhttps://doi.org/10.5286/ISIS.E.RB1820492 .\n[44] M. Smidman et al; (2018): STFC\nISIS Neutron and Muon Source,\nhttps://doi.org/10.5286/ISIS.E.RB1820611 .\n[45] E. Bauer, Anomalous properties of Ce-Cu- and Yb-Cu-\nbased compounds, Advances in Physics 40, 417 (1991) .\n[46] K. Hattori, Meta-orbital transition in heavy-fermion sys-\ntems: Analysis by dynamical mean field theory and self-\nconsistent renormalization theory of orbital fluctuations ,\nJ. Phys. Soc. Jpn. 79, 114717 (2010) .\n[47] L. V. Pourovskii, P. Hansmann, M. Ferrero, and\nA. Georges, Theoretical prediction and spectroscopic\nfingerprints of an orbital transition in CeCu 2Si2,\nPhys. Rev. Lett. 112, 106407 (2014) .\n[48] A. Amorese, A. Marino, M. Sundermann, K. Chen,\nZ. Hu, T. Willers, F. Choueikani, P. Ohresser,\nJ. Herrero-Martin, S. Agrestini, C.-T. Chen, H.-\nJ. Lin, M. W. Haverkort, S. Seiro, C. Geibel,\nF. Steglich, L. H. Tjeng, G. Zwicknagl, and A. Sev-\nering, Possible multiorbital ground state in CeCu 2Si2,\nPhys. Rev. B 102, 245146 (2020) .\n[49] T. Willers, Z. Hu, N. Hollmann, P. O. K¨ orner, J. Geg-\nner, T. Burnus, H. Fujiwara, A. Tanaka, D. Schmitz,7\nH. H. Hsieh, H.-J. Lin, C. T. Chen, E. D. Bauer, J. L.\nSarrao, E. Goremychkin, M. Koza, L. H. Tjeng, and\nA. Severing, Crystal-field and Kondo-scale investigations\nof CeMIn5(M= Co, Ir, and Rh): A combined x-\nray absorption and inelastic neutron scattering study,\nPhys. Rev. B 81, 195114 (2010) .\n[50] T. Willers, F. Strigari, Z. Hu, V. Sessi, N. B.\nBrookes, E. D. Bauer, J. L. Sarrao, J. D. Thomp-son, A. Tanaka, S. Wirth, L. H. Tjeng, and\nA. Severing, Correlation between ground state\nand orbital anisotropy in heavy fermion materials,\nProc. Natl. Acad. Sci. U.S.A 112, 2384 (2015) .\n[51] J. H. Shim, K. Haule, and G. Kotliar, Modeling the\nlocalized-to-itinerant electronic transition in the heav y\nfermion system CeIrIn 5,Science318, 1615 (2007) ."    },    {        "title": "2103.03562v1.Ferromagnetic_Composite_Self_Arrangement_in_Iron_Implanted_Epitaxial_Palladium_Thin_Films.pdf",        "content": "Ferromagnetic Composite Self -Arrangement in Iron -Implanted Epitaxial \nPalladium Thin Films  \n \nA.I. Gumarov1,2,a), I.V. Yanilkin1,2, R.V. Yusupov2, A.G. Kiiamov2, \nL.R. Tagirov1,2 and R.I. Khaibullin1 \n \n1 Zavoisky  Physical -Technical Institute, FRC Kazan Scientific Centre of RAS, 420029 \nKazan, Russia  \n2 Institute of Physics, Kazan Federal University, 420008 Kazan, Russia  \na) Author to whom correspondence should be addressed: amir@gumarov.ru  \n \nABST RACT  \nWe report on the formation of the dilute Pd 1-xFex compositions with tunable magnetic \nproperties under an ion -beam implantation of epitaxial Pd thin films. Binary Pd 1-xFex \nalloys with a mean iron content x of 0.025, 0.035 or 0.075 were obtained by the \nimplantation of 40 keV Fe+ ions into the palladium films on MgO (001) substrate to the \ndoses of 0.5∙1016, 1.0∙1016 and 3.0∙1016 ions/cm2, respectively. Structural and magnetic \nstudies have shown that iron atoms occupy regular fcc -lattice Pd -sites without the \nformation of any secondary crystallographic phase. All the iron implanted Pd films reveal \nferromagnetism at low temperatures (below 2 00 K) with both the Curie temperature and \nsaturation magnetization determined by the implanted iron dose. In contrast to the \nmagnetic properties of the molecular beam epitaxy grown Pd 1-xFex alloy films with the \nsimilar iron contents, the Fe -implanted Pd fi lms possess weaker in -plane \nmagnetocrystalline anisotropy, and, accordingly, a lower coercivity. The observed \nmultiple ferromagnetic resonances in the implanted Pd 1-xFex films indicate a formation of \na magnetically inhomogeneous state due to spinodal decom position into regions, \npresumably layers, with identical crystal symmetry but different iron contents. The \nmultiphase magnetic structure is robust with respect to the vacuum annealing at 770 K, \nthough develops towards well -defined local Pd -Fe compositions.  \n \nKeywords : ion implantation, diluted palladium -iron alloy, laminar magnetic composite , \nferromagnetism, ferromagnetic resonance  \n  2 \n 1. Introduction   \nThe renaissance of an interest to Pd1-xFex alloys was triggered recently by their \npotential applications in superconducting spintronics, where diluted compositions with \nx < 0.10 serve as a weak ferromagnetic link in Josephson -junction structures for \nultrahigh -speed cryogenic devices [1–6]. Traditionally, weak ferromagnetic Pd1-xFex \nalloys have been fabricated using the magnetron sputtering [3–10] or molecular -beam \nepitaxy (MB E) [11,12] , including our recent efforts focused on epitaxial thin films of Pd1-\nxFex (0.01  x  0.1) on the MgO (001) substrate [13–16]. However, there is another \ntechnique with high potential to form dilute Fe -Pd alloys with tunable magnetic properties \n– the ion -beam implantation, widely used in modern microelectronics for silicon doping \nand microchip fabrication [17,18] . \nAlthough severa l reports on low -dose ion implantation of magnetic impurities into \nthe pure -Pd films and foils can be found in the literature [19–22], they primarily report \non the influence of irradiation -induced defects on the Curie temperature ( TC) and \nsaturation magnet ization ( Ms). The majority of the reported implantation experiments had \nbeen done at liquid -helium temperature to avoid the annealing of radiation defects during \nion bombardment. Also, the nature of these defects and the impact of thermal annealing \non the magnetic properties were studied [19,20] . \nTargeted by the potential applications of Pd1-xFex alloys in superconducting \nspintronics, we considered a high -dose (heavy) implantation of epitaxial palladium films \nwith iron at room temperature (RT) to obtain wea kly ferromagnetic layers with a \nminimum of irradiation -induced defects. Magnetometry and ferromagnetic resonance \n(FMR) studies have shown that Fe -implanted Pd films reveal ferromagnetism with the TC \nand Ms strongly dependent on the implantation dose as it was expected. In contrast to the \nMBE -grown films with a similar iron content, the implanted films have lower in -plane \nmagnetocrystalline anisotropy, and, accordingly, a lower coercive field. Depending on \ndose, revealed a presence of one to three magnetic p hases with different TC and Ms. It \nlooks that a pronounced separation into different Pd -Fe compositions takes place. These \n‘phases’ are present already in the as -implanted samples and persist (though are modified) \nafter the high -temperature vacuum annealin g indicating the tendency towards separation \ninto definite intrinsically -stable compositions. We discuss this finding in terms of \nspinodal decomposition of initially inhomogeneous distribution of Fe impurity into \nlaminar composite state across the film thi ck-ness.  \n 3 \n 2. Experimental section  \n2.1. Sample preparation and characterization  \nEpitaxial Pd thin films on MgO  (001) single -crystal substrate were used as the \nstarting materials. The films were produced from the high purity Pd (99.98%) utilizing \nultrahigh vacuum (UHV) molecular -beam epitaxy system (MBE, by SPECS GmbH, \nGermany ). The three-step synthesis of epitaxial Pd films is described  in detail in Ref.  [14]. \nThe 56Fe+ ion implantation was performed with the ILU-3 accelerator at fixed ion energy \nof 40  keV and ion currents  of 2-3 µA as measured at the sample. To obtain Pd -Fe alloys \nwith various iron concentrations, Fe+ ion doses of 0.5∙1016, 1.0∙1016 and 3.0∙1016 ions/cm2 \n(sample labels S0_5, S1_0 and S3_0, respectively ) were achieved varying the irradiation \ntime. An expected depth profile of Fe -concentration was calculated in advance for \n1.0∙1016 ions/cm2 dose following the SRIM -2013 (TRI M) algorithm  [23], and the  Fe+ \ndoses were adjusted accordingly to yield the samples with the mean iron contents of \n𝑐̅Fe ≈ 2.5, 3.5 and 7.5  at.%. Note that the original TRIM approach does not account for \nthe sputtering of the target front  surface, changes in target composition and diffusion \nduring the implantation . In order to obtain comparable thicknesses of the resulting alloy \nlayer, initial Pd films had the thicknesses of 40  nm, 60  nm and 80  nm for the three above -\nmentioned concentrations of iron, respectively. Film thickness was measured prior to and \nafter Fe+ implantation with the Bruker  DektakXT  stylus profilometer with an accuracy of \n±0.5 nm. The implanted samples had been cut into parts, and one of them was post -\nannealed in vacuum at 770 K for 20 min.  \nThe formation of the Pd 1-xFeх alloy with cubic symmetry corresponding to the crystal \nlattice of pristine Pd matrix was justified ex situ  with X-ray diffraction (XRD , Bruker D8 \nAdvance ). XRD studies were performed  utilizing the Cu-K radiation (  = 0.15418  nm) \nin the Bragg –Brentano geometry , in the  range of 2 θ angles of 40 to 72 degrees . XRD  \npatterns are presented in Fig. 1. No sign of any secondary phase ( e.g. metallic Fe \nnanoparticles) could be found at the experimental sensitivity level.  The θ-2θ scan \n(Fig.  1a) shows notable shift and broadening of the S3_0 implanted film (002) maxim um \nrelative to the pure Pd and the MBE -grown  Pd1-xFeх films due to the lattice parameter \nmodification (see detailed analysis  in Ref.  [14]) and inhomogeneous distribution of  the \niron across the film thickness , respectively . XRD φ-scan for S3_0 sample is shown in \nFig. 1b in compar ison with that for the MBE -grown Pd0.92Fe0.08 film (Fig.  1c). XRD data \nclearly  indicate the  cubic symmetry , cube -on-cube epitaxy  and single -crystalline structure \nof the implanted film . \n 4 \n  \nFIG. 1. X-ray diffraction  patterns:  2θ-scans ( a) and φ-scans ( b,c) for the S3_0 implanted film \n(red line), MBE -grown epitaxial Pd0.92Fe0.08 film (green line); pure 40  nm Pd film ( blue line) and \nMgO substrate (black line) . Eulerian cradle angle χ was set to 0 degree in 2 θ-scans and was equal \nto 45 degrees in φ-scans in order to detect <202> XRD -maxima  (angle φ is arbitrary) .  \n \n \nThe iron concentration profiles were obtained by X -ray photoelectron spectroscopy \n(XPS) in combination with depth profiling by the Ar+-ion etching technique. Overview \nXPS spectrum (Fig.  2a) for the implanted film demonstrates the set of lines corresponding \nonly to iron and palladium elements. No alien element s were  detected within the \nsensitivity  of our spectrometer  (~ 0.1%) . Figures 2b and 2c show high -resolution spectra \nof core -shell electrons Fe 2p and Pd 3d recorded with the 0.1  eV step and the analyzer \ntransmission energy of E = 25 eV. Binding energies and line shapes  are characteristic of \nPd-Fe alloy studied earlier by XPS  [14]. For comparison, we recorded also high resolution \nXPS-spectra of the pure iron and palladium films. A nnealing did not lead to any \nsignificant  modification of the shape and position of the XPS-lines. The o btained results \nsignify that Pd -Fe alloys synthetized by ion implantation or molecular -beam epitaxy  [14] \nexhibit the same dissolving  mechanism : iron atoms  substitute  for the palladium  ones in \nthe lattice  without  formation  of iron nanoparticles . The last is confirmed by the absence \nof a characteristic peak for metallic iron (Fe0) at the binding energy of  706.7 eV  [24] \n(within the sensitivity of our XPS setup).  \nDepth profiling of the samples was carried out in a step-by-step manner , combining \netching of the sample with argon ions and recording the high -resolution XPS -spectra after \nthe each etching step.  The Fe 2p and Pd 3d spectra were used to calculate the concentration \nof iron impurities. The iron impurity was detected across the en tire thickness of every \nFe+-irradiated palladium film. The film thickness after the iron implantation according to \nthe profilometry data was equal to 36  nm, 50  nm and 63  nm for doses of 0.5 ∙1016, 1.0 ∙1016, \n5 \n и 3.0∙1016 ions/cm2, respectively . Thus, a significant sputtering of the front surface of the \nfilms during the implantation occurred.  \n \n \n \nFIG. 2. a) A survey XPS spectrum recorded from the S3_0 sample ; b) and c) are the high-\nresolution XPS spectra of the inner shell Fe 2p and Pd 3d electrons, respectively, recorded for the  \nas-implanted, annealed in vacuum  iron-implanted Pd films , Pd0.92Fe0.08 alloy synthesized by \nMBE  and reference films of pure iron and palladium grown separately.  \n \n \nDistribution profiles of the iron impurity for the three doses are shown in Fig . 3. The \nshape of the distribution profile changes with the implantation dose and after the \nsubsequent thermal annealing. Obtained distributions  have essentially a stepped shape. \nPartial redistribution of the iron impurity over the film thickness takes place during the \nannealing , especially notable for the S0_5 and S3_0 samples . The most of the impurity is \nlocated up to  a depth  of 35  nm from the surface for the  S0_5 and S1_0  sample s, and to a \ndepth of 55 nm for the S3_0  sample . A local maximum  of the impurity concentration is \nobserved  near the film -substrate interface  of the as -implanted S3_0 sample . The iron \nimpurity has not been found inside the substrate for any implantation dose used in this \nwork. Thus, the MgO substrate serve s a barrier for the implanted iron impurity.  \n6 \n  \nFIG. 3. Distribution p rofiles of the iron  impurity  in S0_5 (a), S1_0 (b) and S3_0 (c) samples , \nobtained both before (black symbols and curve s) and after the thermal annealing (red symbols \nand curves ). The pink solid line in panel (b) shows the calculated profile of the iron distribution \nin the palladium matri x (see text) . Arrows indicate the position of the film/substrate interface.   \n \n \n2.2. Magnetic properties  \nMagnetic properties of the iron-implanted Pd films were measured utilizing the \nvibrating sample magnetometry  (VSM , Quantum Design  PPMS -9) and ferromagnetic \nresonance (FMR , Bruker ESP300) spectroscopy in the temperature range of 5  – 300 K \nwith the magnetic field applied either in-plane  or out-of-plane  of the films. To determine \nMs of the synthesized Pd -Fe alloy films , the combined diamagnetic contribution of the \nMgO substrate and paramagnetic one from the impurities in it were subtracted from the \nraw VSM data . Then, the saturation magnetic moment was recalculated to the number of \nBohr magnetons ( µB) per implanted Fe+-ion. \nFigure  4a shows magnetic hysteresis loops for palladium films implanted with three \ndifferent doses  of iron . For the S0_5 sample, a rather low specific saturation magnetic \nmoment of ≈ 3 µB/Fe was obtained,  which  is not far from 3.7  µB/Fe reported in [8] for the \nmagnetron sputtered film of Pd0.99Fe0.01. On the other hand, the specific moment of \n 6.5 µB/Fe was found for the MBE grown epitaxial film of 20 nm thickness with \n7 \n x = 0.019 [16], and  6.9 µB/Fe – for the bulk samples  [25–27]. Certainly, the higher value \nof the magnetic moment per iron correlates with the synthesis techniques providing more \nuniform distribution of Fe (bulk and MBE samples), while that providing less uniform \ndistribution (magnetron sputtering and ion implantation) lead to its reduc tion. \n \n \nFIG.  4. Magnetic hysteresis loops measured  with the magnetic field applied  along the hard in-\nplane magnetization axis for the S0_5, S1_0 and S3_0 samples  (a); along the in-plane easy and \nhard magnetization axes  (b); as well as out -of-plane (inset  to panel (b) ), and after the thermal \nannealing of the samples  (c). Temperature dependences of the saturation magnetization both \nbefore (open symbols) and after (solid symbols) annealing  (d). \n \n \nFigure  4b presents the magnetic hysteresis loops measured with three orientations of \nthe applied  magnetic field H with respect to the S3_0 sample plane : the in-plane  with \nH||[110] and H||[010]  crystallographic direction s of the MgO (001) substrate and  the out-\nof-plane  geometry  with H||[001] (Fig.  4b, inset) . The rectangular shape of the loop show s \nthat the < 110> direction s are the easy in-plane directions , while the <010> are the hard \nin-plane  ones in full agreement with the properties of the  MBE -grown epitaxial Pd 1-xFex \nfilms  [13–16]. Reversible character and a high er value of the saturating magnetic field in \nthe out-of-plane  geometry indicate that the obtained  material is the easy -plane magnetic \nsystem.  Qualitatively similar results were obtained for the S0_5 and S1_0 samples . \n8 \n Figure  4c shows the magnetic hysteresis loops for the implanted palladium films of \nFig. 4a measured after their thermal annealing . The annealing leads to an increase in the \nmagnetization  and coercive field of the implanted samples . Nevertheless, both the as -\nimplanted and post -annealed  Pd films reveal weaker in -plane magnetocrystalline \nanisotropy and a lower coercivity in comparison with the MBE grown Pd 1xFex films with \nsimilar iron concentration  (see Fig.  S1 of Supplementary material).  At the same time, \nfrom the temperature dependences of the magnetic moment s, Fig.  4d, it follows that TC \nfor all iron implanted Pd films increases after high-temperature annealing in vacuum . The \nmagnetometry results for as -implanted and post -annealed Pd films are summarized in \nTable  1. \n \nTable  1. Experimentally evaluated magnetic properties of as -prepared and post -annealed thin \npalladium films implanted with Fe -ions to different doses D.  \nSample label  D, \n1016 ions/cm2 𝑐̅Fe, \nat.% Ms, \nµB/Fe Bc*, \nmT TC, \nK \nAs-implanted  S0_5  0.5 2.5 3.0 0.24 40 \nS1_0  1.0 3.5 4.5 0.43 107 \nS3_0  3.0 7.5 4.4 0.56 155 \nPost-annealed  S0_5a  0.5 2.5 4.2 2.76 105 \nS1_0a  1.0 3.5 5.0 0.58 120 \nS3_0a  3.0 7.5 4.6 0.83 203 \n*The coercivity Bc is measured at  5 К \n \nThe most unusual  results for the Fe -implanted thin palladium films were obtained \nwith the ferromagnetic resonance (FMR) spectroscopy  [28]. Temperature evolution of the \nFMR  spectrum  with the magnetic field applied perpendicular to the as-implanted  S3_0 \nsample  plane is presented  in Fig.  5a. A consecutive appearance of the resonances is \nobserved as if ferromagnetic phases emerg e sequentially with  the temperature decrease . \nFMR spectra of the same sample for the magnetic field applied along the [110] (in -plane) \nand [001] (out-of-plane) directions are shown in Fig.  5b (black lines) . The modification \nof the FMR spectrum of the S3_0 sample after the thermal annealing is illustrated also by \nFig. 5b (sample S3_0a, red lines). In -plane resonance lines shift after the annealing \ntowards the lower fields, and the two out -of-plane lines – towards the higher fields being \na common feature of the spectra evolution for all the sampl es. The multiphase response \nof the magnetic system persists after the annealing.  9 \n The angular behavior  of the resonance fields  of both the as -implanted and the \nannealed samples  is typical for the easy -plane systems : in-plane resonances reside in \nlower magne tic fields with respect to the electron paramagnetic resonance lines  (a comb \ncentered around 330  mT originating from the 3d -ion impurities in the substrate ), while \nthe out-of-plane resonances are found at higher fields due to the thin-film 2D shape \nanisotr opy [28]. For comparison, the FMR spectra of the homogeneous epitaxial \nPd0.92Fe0.08 film [13] are presented in Fig.  5b. \nResonance field s for th e annealed Fe -implanted Pd -film, sample S3_0 a, in the out -\nof-plane orientation have the same values as  that for the epitaxial Pd 1-xFex films [16] with \nthe iron concentrations of x = 0.025, x = 0.04 and x = 0.063.  The S0_5 a sample with the \nlowest dose  reveals only one FMR signal at  T ≤ 40K (inset to Fig.  5b and Fig. S2a of the  \nSupplementary material ), and the S1_0 a sample – two FMR signals  (see Fig.  S2b).  \n \n \nFIG. 5. FMR spectra of the epitaxial Pd film implanted with iron to the dose of \n3.0∙1016 ions/cm2: (a) temperature evolution for the as -implanted S3_0 sample in the out-of-\nplane  geometry ; (b) in-plane  and out-of-plane  spectra of the as-implant ed (sample S3_0, black \ncurve s) and the annealed (sample S3_0a, red curve s) films , as well as the spectra of the MBE -\ngrown epitaxial Pd 0.92Fe0.08 film (green curves ). The inset shows the resonance lines of the \nannealed S0_5 a and S1_0 a samples  (ν = 9.416  GHz, T = 40 K). \n \n \n3. Discussion  \nObservation of multiple ferromagnetic resonances unambiguously indicates an \noccurrence of the corresponding set of rather well -defined , in the sense of the Ms values , \nferromagnetic components in our Fe -implanted palladium films. This, in turn, is a \nmanifestation of the Pd -Fe system separation into definite compositions. In our opinion, \nthis can take place due to a kind of a spinodal decomposition [29–32] occurring  under \nnon-equilibrium conditions in the course of the implantation, that lead, most probably, to \na formation of a laminar composite magnetic structure in the implanted Pd 1‐xFex films.  \n10 \n Indeed, because of the  doping technique, the implant distribution is in homogeneous \na priori , see Fig. 3. The distribution shape changes with increasing the dose because of \nthe front surface sputtering during the irradiation, and diffusion stimulated by the \nannealing  [33] due to a retardation  of the penetrating ions with the kinetic energy mostly \ntransferred to the heat. This could be a reason of an absence of significant  evolution  of \nthe depth distribution profiles and magnetic hysteresis loops, Fig.  4, upon post -\nimplantation thermal  annealing. Single -crystal state of the Fe -implanted Pd films (see \nFig. 1) also points at a dynamic annealing of radiation defects in the films directly during \nthe ion irradiation.  The S0_5 sample implanted with the lowest dose of 0.5∙1016 ions/ cm2 \n(the lowest time of implantation) seems to acquire  minimal self-annealing, that is why it \nshows maximal changes in the hysteresis loop shape, Ms and TC upon ex situ  annealing, \nFig. 4, c and d.  \nIn spite of inhomogeneous distribution of iron atoms, there are no indications of \nsequential reversals of different magnetic phases in hysteresis loops  (Figs.  4, a to c ). \nIndeed, if the remagnetization mechanism is a movement of domain  walls, then the most \n”soft” region (sublayer) of the film triggers  at low fields and drags the coupled \nneighboring  sublayers to complete reversal of the entire magnetic moment of a film. No \nmultiphase , laminar nature of magnetism could be recognized in the quasi -static magnetic \nmoment measurements , Fig.  4 and Fig.  S1 of the Supplementary material . The regions \nwith high iron content could lead to an increas e in coercivity  due to the domain -wall \npinning by the coupling with them (see Fig.  S1). \nOur FMR studies revealed that in the spectrum for a high-dose S3_0 sample the \nnumber of resonance lines increases from one to three with decreasing the temperature \n(Fig.  5a). This feature is quite robust, because the high -temperature anneali ng of the \nsample does not qualitatively change its FMR response (Fig.  5b). Probably, the \ninhomogeneous profile of the iron distribution in the implanted palladium  films leads to \nthe formation of a laminar  multiphase magnetic system with different temperatu res of \nferromagnetic ordering and saturation magnetizations for each individual phase  (layer) . \nIn spite of a direct contact of these phases (layers), and the resulting coupling between \nthem, they show robust individual responses with the angular behavior e xactly mimicking \nthe FMR response of thin ferromagnet ic film with the easy-plane shape anisotropy \n[13,16,28] . \nMoreover, for palladium films implanted with a minimum ( 0.5∙1016 ions/cm2, S0_5 ) \nand medium ( 1.0∙1016 ions/cm2, S1_0 ) iron doses, after bringing them to thermodynamic \nequilibrium by the annealing, we observed one FMR line for the fi rst, and two FMR lines 11 \n for the second (Fig.  S2 in the Supplementary material). The closeness of the resonance \nfields for the selected lines (see Fig . S3 of the Supplementary material) supports a \nformulation of a simple model of spinodal  decomposition into isostructural laminar \nstructure [29], each layer possessing a certain , quite  well-defined  concentration of iron in \nthe palladium . That is, the system evolves from a metastable non -equilibrium state created \nby ion implantation , to its energetically favorable state by the formation of several \nspatially -extended (macroscopic) “stable phases ”. The shift of the resonance line at \n~ 420 mT in Fig.  5b towards lower resonance fields, as well as the  appearance of \ninflections in the temperature dependence of magnetization for low and medium doses \n(Fig.  4d) after thermal annealing support this conjecture. As an ultimate hypothesis on \nthe micro scopic scale, the FMR technique reveals fine preferences for certain \ncompositio ns of Pd -Fe system  that highlights the effects of preferential local order.  \nDisclosure of this previously unknown tendency of the Pd 1-xFex binary system in the \npalladium -rich ( x < 0.10) composition range opens up new prospects for the use of these \nmaterials in superconducting spintronic heterostructures: the reported intrinsically stable \ncompositions of x = 0.025, x = 0.04, x = 0.063 (prelim inary estimates for uncoupled \nlayers) should be utilized ensuring the magnetic homogeneity of the ferromagnetic layers. \nIon-beam implantation of epitaxial palladium films with relatively low doses of iron can \nserve a route for a synthesis of near -homogeneo us low -temperature ferromagnets.  \n \n4. Conclusions  \nTo summarize, we synthesized Pd -Fe alloys with a mean iron content in the range of \n2.5-7.5 at.% using high -dose Fe-ion implantation to epitaxial Pd -films at RT . Iron-\nimplanted Pd films retain fcc cubic crystal structure typical for binary Pd -Fe alloys with \nlow content of iron.  All synthesized samples exhibit ferromagnetism at low temperatures. \nThe Curie temperature  and saturation magnetization depend on the dose o f implantation \nand are modified under subsequent thermal annealing. Magnetic hysteresis loops for the \nimplanted palladium films are narrower than those for Pd -Fe alloys with comparable \nconcentrations grown by MBE. FMR and VSM studies indicate that implantation of \nepitaxial palladium films with iron leads to the unusual formation of a laminar magnetic \ncomposite in otherwise crystallographically homogeneous film . The multiphase behavior \nis discussed within the hypothesis of spinodal deco mposition of the initially non-\nequilibrium distribution of inhomogeneously delivered dopant to isostructural , laterally \nextended laminar  sub-structures.  \n 12 \n CRediT au thorship contribution statement : \nGumarov  A.I.: Conceptualization,  Sample preparation,  Measurements, Data curation, \nWriting – original draft; Yanilkin I.V.: Measurements , Data curation;  Yusupov  R.V. : \nInvestigation, Writing – review & editing ; Kiiamov  A.G. : X-ray data acquisition & \ncuration ; Tagirov  L.R. : Formal analysis, Writing – original draft, Writing – review & \nediting ; Khaibullin R.I.: Conceptualization, Methodology, Sample preparation, \nSupervision, Funding acquisition.  \n \nDeclaration of competing interest : \nThe authors declare that they have no competing interests.  \n \nAcknowledgements :  \nAuthors thank V.I.  Nuzhdin and V.F.  Valeev for technical assistance with ion \nimplantation experiments.  \n \nFunding:  \n \nThis work was supported by the RFBR Grant No.  20-02-00981.  \n \nData available on request from the authors:  \nThe data that support the findings of this study are available from the corresponding \nauthor upon a reasonable request.  \n  13 \n References  \n[1] I.A. Golovchanskiy, V. V. Bolginov, N.N. Abramov, V.S. Stolyarov, A. Ben Hamida, V.I. \nChichkov, D. Roditchev, V.V. Ryazanov, Magnetization dynamics in dilute Pd1 -xFex thin \nfilms and patterned microstructures considered for superconducting electronics, J. Appl. \nPhys. 120 (2016) 163902. https://doi.org/10.1063/1.4965991.  \n[2] V.V. Bol’ginov, V.S. Stolyarov , D.S. Sobanin, A.L. Karpovich, V.V. Ryazanov, Magnetic \nswitches based on Nb -PdFe -Nb Josephson junctions with a magnetically soft \nferromagnetic interlayer, JETP Lett. 95 (2012) 366 –371. \nhttps://doi.org/10.1134/S0021364012070028.  \n[3] T.I. Larkin, V. V. Bol’g inov, V.S. St olyarov, V.V. Ryazanov, I. V. Vernik, S.K. Tolpygo, \nO.A. Mukhanov, Ferromagnetic Josephson switching device with high characteristic \nvoltage, Appl. Phys. Lett. 100 (2012) 222601. https://doi.org/10.1063/1.4723576.  \n[4] V.V. Ryazanov, V.V. Bol’gi nov, D.S. Sobanin, I.V. Vernik, S.K. Tolpygo, A.M. Kadin, \nO.A. Mukhanov, Magnetic Josephson Junction Technology for Digital and Memory \nApplications, Phys. Procedia. 36 (2012) 35 –41. \nhttps://doi.org/10.1016/J.PHPRO.2012.06.126.  \n[5] I.V. Vernik, V. V. Bol’Gin ov, S.V. Bakurskiy, A. A. Golubov, M.Y. Kupriyanov, \nV.V. Ryazanov, O.A. Mukhanov, Magnetic josephson junctions with superconducting \ninterlayer for cryogenic memory, IEEE Trans. Appl. Supercond. 23 (2013) 1701208 –\n1701208. https://doi.org/10.1109/TASC.2012.22 33270.  \n[6] J.A. Glick, R. Loloee, W.P. Pratt,, N.O. Birge, Critical Current Oscillations of Josephson \nJunctions Containing PdFe Nanomagnets, IEEE Trans. Appl. Supercond. 27 (2016) 1 –5. \nhttps://doi.org/10.1109/TASC.2016.2630024.  \n[7] H.Z. Arham, T.S. Khaire, R. Loloee, W.P. Pratt, N.O. Birge, Measurement of spin memory \nlengths in PdNi and PdFe ferromagnetic alloys, Phys. Rev. B - Condens. Matter Mater. \nPhys. 80 (2009) 174515. https://doi.org/10.1103/PhysRevB.80.174515.  \n[8] L.S. Uspenskaya, A.L. Rakhman ov, L.A. Dorosinskii, S.I. Bozhko, V.S. Stolyarov, \nV.V. Bolginov, Magnetism of ultrathin Pd 99 Fe 01 films grown on niobium, Mater. Res. \nExpress. 1 (2014) 036104. https://doi.org/10.1088/2053 -1591/1/3/036104.  \n[9] L.S. Uspenskaya, I.V. Shashkov, Influence o f Pd 0.99Fe0.01 film thickness on magnetic \nproperties, Phys. B Condens. Matter. (2017). https://doi.org/10.1016/j.physb.2017.09.089.  \n[10] V.V. Bol’ginov, O.A. Tikhomirov, L.S. Uspenskaya, Two -component magnetization in \nPd99Fe01 thin films, JETP Lett. 105 (2 017) 169 –173. \nhttps://doi.org/10.1134/S0021364017030055.  \n[11] M. Ewerlin, B. Pfau, C.M. Günther, S. Schaffert, S. Eisebitt, R. Abrudan, H. Zabel, \nExploration of magnetic fluctuations in PdFe films, J. Phys. Condens. Matter. 25 (2013) \n266001. https://doi.or g/10.1088/0953 -8984/25/26/266001.  \n[12] I.A. Garifullin, D.A. Tikhonov, N.N. Garif’yanov, M.Z. Fattakhov, K. Theis -Bröhl, 14 \n K. Westerholt, H.  Zabel, Possible reconstruction of the ferromagnetic state under the \ninfluence of superconductivity in epitaxial V/Pd 1−x Fex bilayers, Appl. Magn. Reson. 22 \n(2002) 439 –452. https://doi.org/10.1007/BF03166124.  \n[13] A. Esmaeili, I.R. Vakhitov, I.V. Yanilkin, A.I. Gumarov, B.M. Khaliulin, B.F.  Gabbasov, \nM.N. Aliyev, R.V. Yusupov, L.R. Tagirov, FMR Studies of Ultra -Thin Epita xial \nPd0.92Fe0.08 Film, Appl. Magn. Reson. 49 (2018) 175 –183. https://doi.org/10.1007/s00723 -\n017-0946 -1. \n[14] A. Esmaeili, I. V. Yanilkin, A.I. Gumarov, I.R. Vakhitov, B.F. Gabbasov, A.G. Kiiamov, \nA.M. Rogov, Y.N. Osin, A.E. Denisov, R.V. Yusupov, L.R. Tag irov, Epitaxial growth of \nPd1−xFex films on MgO single -crystal substrate, Thin Solid Films. 669 (2019) 338 –344. \nhttps://doi.org/10.1016/j.tsf.2018.11.015.  \n[15] W.M. Mohammed, I.V . Yanilkin, A.I. Gumarov, A.G. Kiiamov, R.V . Yusupov, \nL.R. Tagirov, Epitaxial growth and superconducting properties of thin -film PdFe/VN and \nVN/PdFe bilayers on MgO(001) substrates, Beilstein J. Nanotechnol. 11 (2020) 807 –813. \nhttps://doi.org/10.3762/bjnano .11.65.  \n[16] A. Esmaeili, I.V. Yanilkin, A.I. Gumarov, I.R. Vakhitov, B.F. Gabbasov, R.V.  Yusupov, \nD.A. Tatarsky, L.R. Tagirov, Epitaxial thin -film Pd1 -xFex alloy: a tunable ferromagnet \nfor superconducting spintronics, Sci. China Mater. (2020). \nhttps://doi .org/10.1007/s40843 -020-1479 -0. \n[17] L.A. Larson, J.M. Williams, M.I. Current, Ion implantation for semiconductor doping and \nmaterials modification, Rev. Accel. Sci. Technol. Accel. Appl. Ind. Environ. 04 (2012) \n11–40. https://doi.org/10.1142/S179362681100 0616.  \n[18] R.W. Hamm, M.E. Hamm  (eds.) , Industrial accelerators and their applications, World \nScientific , 2012. https://doi.org/10.1142/7745.  \n[19] J.D. Meyer, B. Stritzker, Reduced Susceptibility of Irradiated Palladium, Phys. Rev. Lett. \n48 (1982) 502–505. https://doi.org/10.1103/PhysRevLett.48.502.  \n[20] M. Hitzfeld, P. Ziemann, W. Buckel, H. Claus, Ferromagnetism of alloys: A new probe in \nion implantation, Solid State Commun. 47 (1983) 541 –544. https://doi.org/10.1016/0038 -\n1098(83)90495 -7. \n[21] M. Hitzfeld, P. Ziemann, W. Buckel, H. Claus, Ferromagnetism of Pd -Fe alloys produced \nby low -temperature ion implantation, Phys. Rev. B. 29 (1984) 5023 –5030. \nhttps://doi.org/10.1103/PhysRevB.29.5023.  \n[22] H. Claus, N.C. Koon, Effect of oxygen on the electroni c properties of Pd, Solid State \nCommun. 60 (1986) 481 –484. https://doi.org/10.1016/0038 -1098(86)90721 -0. \n[23] J.F. Ziegler, M.D. Ziegler, J.P. Biersack, SRIM - The stopping and range of ions in matter \n(2010), Nucl. Instruments Methods Phys. Res. Sect. B Be am Interact. with Mater. Atoms. \n268 (2010) 1818 –1823. https://doi.org/10.1016/j.nimb.2010.02.091.  \n[24] M.C. Biesinger, B.P. Payne, A.P. Grosvenor, L.W.M. Lau, A.R. Gerson, R.S.C. Smart, 15 \n Resolving surface chemical states in XPS analysis of first row transit ion metals, oxides \nand hydroxides: Cr, Mn, Fe, Co and Ni, Appl. Surf. Sci. 257 (2011) 2717 –2730. \nhttps://doi.org/10.1016/j.apsusc.2010.10.051.  \n[25] J. Crangle, W.R. Scott, Dilute ferromagnetic alloys, J. Appl. Phys. 36 (1965) 921 –928. \nhttps://doi.org/10.10 63/1.1714264.  \n[26] G.J. Nieuwenhuys, Magnetic behaviour of cobalt, iron and manganese dissolved in \npalladium, Adv. Phys. 24 (1975) 515 –591. https://doi.org/10.1080/00018737500101461.  \n[27] B. Heller, K.H. Speidel, R. Ernst, A. Gohla, U. Grabowy, V. Roth, G.  Jakob, F. Hagelberg, \nJ. Gerber, S.N. Mishra, P.N. Tandon, Transient field measurement in the giant moment \nPdFe alloy, Nucl. Instruments Methods Phys. Res. Sect. B Beam Interact. with Mater. \nAtoms. 142 (1998) 133 –138. https://doi.org/10.1016/S0168 -583X(98) 00260 -2. \n[28] M. Farle, Ferromagnetic resonance of ultrathin metallic layers, Reports Prog. Phys. 61 \n(1998) 755 –826. https://doi.org/10.1088/0034 -4885/61/7/001.  \n[29] J.W. Cahn, On spinodal decomposition, Acta Metall. 9 (1961) 795 –801. \nhttps://doi.org/10.10 16/0001 -6160(61)90182 -1. \n[30] V.P. Skripov, A.V . Skripov, Spinodal decomposition (phase transitions via unstable \nstates), Sov. Phys. - Uspekhi. 22 (1979) 389 –410. \nhttps://doi.org/10.1070/PU1979v022n06ABEH005571.  \n[31] System s Far from Equilibrium (ed. by Luis Garrido),  Springer -Verlag , Berlin -Heidelberg -\nNew York,  1980 . https://doi.org/  10.1007/BFb0025609 . \n[32] Phase Transformations in Materials  (ed. by Gernot Kostorz) , Wiley  VCH Verlag GmbH , \nWeinheim, 2001. https://doi.org/10.1002/352760264X.  \n[33] A.A. Achkeev, R.I. Khaibullin, L.R. Tagirov, A. Mackova, V. Hnatowicz, N. Cherkashin, \nSpecific features of depth distribution profiles of implanted cobalt ions in rutile TiO 2, \nPhys. Solid State. 53 (2011) 543 –553. h ttps://doi.org/10.1134/S1063783411030024.  SUPPLEMENTARY  MATERIAL  \nfor \nFerromagnetic Composite Self -Arrangement in Iron -Implanted Epitaxial \nPalladium Thin Films  \n \nA.I. Gumarov1,2,a), I.V. Yanilkin1,2, R.V. Yusupov1,2, A.G. Kiiamov2, \nL.R. Tagirov1,2 and R.I. Khaibullin1 \n \n1 Zavoisky  Physical -Technical Institute, FRC Kazan Scientific Centre of RAS, 420029 \nKazan, Russia  \n2 Institute of Physics, Kazan Federal University, 420008 Kazan, Russia  \na) Author to whom correspondence should be addressed: amir@gumarov.ru  \n \n \nComparison of Pd 1-xFex films data: Ion implantation vs. MBE  \n \n \nVSM studies  \n \n \nFIG S1.  Hysteresis loops at 5  K (a) and M(T) curves (b) measured for 80  nm epitaxial Pd film \nimplanted with 40  keV Fe+-ions with the dose of 3∙1016 ions/cm2 (○); the same sample post -\nannealed in vacuum at 770  K for 20  min ( ●); epitaxial film of Pd 0.92Fe0.08 alloy  deposited by \nMBE to (100) MgO substrate ( ●).  \n \n \n \n \n2 \n FMR studies  \n \n \nFIG S2. FMR spectra obtained in the in -plane (dotted line) and out -of-plane (solid line) \nmeasurement geometries for epitaxial Pd thin film implanted with 40  keV Fe+-ions with the \ndose of 0.5∙1016 ions/cm2 (a) and 1.0 ∙1016 ions/cm2 (b). The two samples have been measured \nafter vacuum annealing at 770 K for 20  min. Arrows  indicate  the observed resonance lines in \nthe out -of-plane orientation.  \n \n \n \n \nFIG S3.  FMR spectra for all three Fe -implanted palladium films after the annealing for \ncomparison of the resonance fields.  \n \n \n"    },    {        "title": "2103.11953v1.Field_induced_reorientation_of_helimagnetic_order_in_Cu__2_OSeO__3__probed_by_magnetic_force_microscopy.pdf",        "content": "Field-induced reorientation of helimagnetic order in Cu 2OSeO 3probed by magnetic\nforce microscopy\nPeter Milde,1,\u0003Laura K ohler,2, 3Erik Neuber,1P. Ritzinger,1Markus Garst,2, 3, 4\nAndreas Bauer,5Christian P\reiderer,5H. Berger,6and Lukas M. Eng1, 7\n1Institute of Applied Physics, Technische Universit at Dresden, D-01062 Dresden, Germany\n2Institute of Theoretical Physics, Technische Universit at Dresden, D-01062 Dresden, Germany\n3Institute for Theoretical Solid State Physics, Karlsruhe Institute of Technology, D-76131 Karlsruhe, Germany\n4Institute for Quantum Materials and Technology,\nKarlsruhe Institute of Technology, D-76344 Eggenstein-Leopoldshafen, Germany\n5Physik-Department, Technische Universit at M unchen, D-85748 Garching, Germany\n6Institut de Physique de la Mati\u0012 ere Complexe, \u0013Ecole Polytechnique F\u0013 ed\u0013 erale de Lausanne, 1015 Lausanne, Switzerland\n7Dresden-W urzburg Cluster of Excellence { Complexity and Topology\nin Quantum Matter (ct.qmat), TU Dresden, 01062 Dresden, Germany\n(Dated: March 23, 2021)\nCu2OSeO 3is an insulating skyrmion-host material with a magnetoelectric coupling giving rise to\nan electric polarization with a characteristic dependence on the magnetic \feld ~H. We report mag-\nnetic force microscopy imaging of the helical real-space spin structure on the surface of a bulk single\ncrystal of CuO 2SeO 3. In the presence of a magnetic \feld, the helimagnetic order in general reorients\nand acquires a homogeneous component of the magnetization, resulting in a conical arrangement at\nlarger \felds. We investigate this reorientation process at a temperature of 10 K for \felds close to\nthe crystallographic h110idirection that involves a phase transition at Hc1. Experimental evidence\nis presented for the formation of magnetic domains in real space as well as for the microscopic origin\nof relaxation events that accompany the reorientation process. In addition, the electric polariza-\ntion is measured by means of Kelvin-probe force microscopy. We show that the characteristic \feld\ndependency of the electric polarization originates in this helimagnetic reorientation process. Our\nexperimental results are well described by an e\u000bective Landau theory previously invoked for MnSi,\nthat captures the competition between magnetocrystalline anisotropies and Zeeman energy.\nI. INTRODUCTION\nIn the limit of weak spin-orbit coupling \u0015SOC the\ncubic chiral magnets like MnSi [1], Fe 1-xCoxSi [2, 3],\nFeGe [4, 5], and Cu 2OSeO 3[6, 7] are dominated by\nonly two coupling constants, the symmetric and anti-\nsymmetric exchange interaction, JandD, respectively\n[8]. Whereas the symmetric exchange is of zeroth order\nin\u0015SOC, the Dzyaloshinskii-Moriya interaction is of \frst\norderD\u0018O(\u0015SOC). Their ratio determines the charac-\nteristic wavevector Q=D=J of the helimagnetic order\nthat develops at zero magnetic \feld ~H= 0. For \fnite ~H,\nthe helix assumes a conical arrangement until it is fully\npolarized at the internal critical \feld \u00160Hint\nc2'D2\nJM s, with\nMsthe saturation magnetization. In addition, a small\npocket of the skyrmion lattice phase just below the or-\ndering temperature Tcis realized at intermediate values\nof~H.\nAs a result, the above-mentioned materials share a very\nsimilar magnetic phase diagram. Details of it, however,\ndepend on corrections that are parametrically smaller in\n\u0015SOC. In particular, the orientation of the helimagnetic\norder at zero \feld is determined by magnetocrystalline\nanisotropies, that are at least of fourth order in \u0015SOC,\nand generally favour the spin spiral to align either along\n\u0003peter.milde@tu-dresden.dea crystallographic h111iorh100idirection like, e.g., in\nMnSi or Cu 2OSeO 3, respectively. The Zeeman energy\ncompetes with the magnetocrystalline anisotropies re-\nsulting in a reorientation of helimagnetic order with vary-\ning magnetic \feld. Depending on the history and the\npopulation of domains, this reorientation process might\neither correspond to a crossover, or involves a \frst-order\nor second-order phase transition at the critical \feld Hc1\n[2, 9{12].\nA quantitative theory of this reorientation process that\nis valid in the limit of small \u0015SOCwas recently presented\nby Bauer et al. and veri\fed by detailed experiments on\nMnSi [12]. With the help of dc and ac susceptibilities as\nwell as neutron scattering experiments, the evolution of\nthe helix orientation, speci\fed by the unit vector ^Q(~H),\nwas carefully tracked as a function of magnetic \feld for\nvarious \feld directions. The crystallographic h100idirec-\ntion plays a special role in that two subsequent Z2tran-\nsitions could be observed con\frming a theoretical predic-\ntion of Walker [11].\nAccording to the theory of Ref. [12] the di\u000berential\nmagnetic susceptibility @HMnaturally decomposes into\ntwo parts. Whereas the \frst part derives from the helix\nwith a \fxed axis ^Q, the second part is attributed to the\n\feld dependence of ^Q(~H). The reorientation ^Q(~H) is as-\nsociated with large relaxation times \u001cbecause it requires\nthe rotation of macroscopic helimagnetic domains. As a\nconsequence, the ac susceptibility for frequencies !\u001c\u001d1\nis only sensitive to the \frst part, which was experimen-arXiv:2103.11953v1  [cond-mat.mtrl-sci]  22 Mar 20212\ntally con\frmed in Ref. [12] suggesting relaxation times\nexceeding seconds, \u001c\u00151 sec. Generally, the reorienta-\ntion depends on the history of the sample due to di\u000berent\ndomain populations, for example, realized for \fnite- or\nzero-\feld cooling. In particular, hysteresis was found at\nthe second-order phase transition at Hc1. The decrease of\nthe \feld across Hc1is accompanied with the formation of\nmultiple domains. The coexistence of di\u000berent domains\nwithin the sample might hamper the realization of the\noptimal trajectory ^Q(~H), especially, in the presence of\nlong relaxation times \u001c. As a result, distinct behavior\ncan be observed upon increasing and decreasing the \feld\nacrossHc1.\nIn Ref. [12] only bulk probes were experimentally inves-\ntigated so that the microscopic origin of the slow relax-\nation processes could not be identi\fed. However, it was\nspeculated that topological defects of the helimagnetic\norder, i.e., disclination and dislocations, might play a spe-\ncial role as they should naturally arise at the boundaries\nbetween di\u000berent domains. A slow creep-like motion of\ndislocations was indeed identi\fed by magnetic force mi-\ncroscopy (MFM) measurements on the surface of FeGe\nsamples by Dussaux et al. [13] after the system had been\nquenched from the \feld-polarized state to ~H= 0. The\nmotion of dislocations during a MFM scan results in dis-\ncontinuities of the helical pattern in the MFM image con-\nsisting of characteristic 180\u000ephase shifts. Subsequently,\nit was also demonstrated both experimentally and the-\noretically that domain walls might comprise topological\ndisclination and dislocation defects [14]. Nevertheless, a\nmicroscopic investigation of such relaxation events close\ntoHc1has not been achieved so far.\nIn the present work, we investigate the helix reorien-\ntation in the chiral magnet Cu 2OSeO 3using microscopic\nMFM measurements. This material is an insulator with a\nmagnetoelectric coupling that allows to manipulate mag-\nnetic skyrmions and helices with electric \felds, and it\ngives rise to various interesting magnetoelectric e\u000bects\n[15, 16]. This material is also promising for magnonic\napplications due to its very low Gilbert damping param-\neter [17]. In constrast to MnSi, its helix is oriented along\nah100idirection at zero \feld. The relatively large ra-\ntioHc1=Hc2\u00180:36 of Cu 2OSeO 3[18] suggests that the\nspin-orbit coupling constant \u0015SOCis larger than in MnSi.\nIndeed, additional magnetic phases stabilized by magne-\ntocrystalline anisotropies { the (metastable) canted con-\nical state as well as the low temperature skyrmion lattice\nphase { were found in Cu 2OSeO 3at low temperatures\nbut for~Honly aligned along crystallographic h100idirec-\ntions [18{20]. Recently, real space observations address-\ning these states have been reported for a thin Cu 2OSeO 3\nlamella and \feld along a h100idirection [21].\nIn previous work [22], we have already investigated\nCu2OSeO 3with MFM at higher temperatures close to\nTcand identi\fed all the magnetic phases, i.e., the helical\nand conical helimagnetic textures, the skyrmion lattice\nphase, and the \feld-polarized phase. Using Kelvin-probe\nforce microscopy (KPFM) we determined the electric po-larization and its \feld-dependence within these various\nphases. However, the reorientation process was not ad-\ndressed in Ref. [22] and it is at the focus of the present\nwork.\nDue to the restriction of our experimental setup, the\nmagnetic \feld is always aligned perpendicular to the\nplane that is scanned by MFM, and for our sample probe\nthis corresponds approximately to the crystallographic\n[110] direction, ~Hk[110]. We study the helix reorien-\ntation for this \feld direction and we determine the pe-\nriodicity of the periodic surface pattern and its in-plane\norientation. Assuming that the bulk helimagnetic order\nessentially extends towards the surface, we extract the\norientation of the helix as a function of magnetic \feld.\nIn addition, we determine the electric polarization and\nits behavior during the reorientation process. Our re-\nsults are interpreted within the e\u000bective Landau theory\nof Ref. [12] that, strictly speaking, is only controlled for\nsmall\u0015SOC, and we \fnd good agreement between the-\nory and experiment. Moreover, we present microscopic\nevidence that the motion of dislocations along domain\nboundaries contributes to the magnetic relaxation close\nto the reorientation transition.\nThe structure of the paper is as follows. In section II\nwe present the experimental methods. In section III we\nshortly review the theory of Ref. [12] and discuss its ap-\nplication to Cu 2OSeO 3. In particular, we point out the\npresence of a robust Z3transition for \felds along h111i.\nThe theoretical prediction for the current experimental\nsetup are presented and the electric polarization is eval-\nuated as a function of the applied magnetic \feld. The\nexperimental results are presented in section IV. From\nthe MFM images we extract the orientation of the heli-\nmagnetic order and the presence of various domains as a\nfunction of magnetic \feld. The relaxation processes are\nshortly analysed, and the electric polarization is deter-\nmined. Finally, we \fnish with a discussion of our results\nand a summary in section V.\nII. EXPERIMENTAL METHODS\nWe investigate the same (70 \u00065)µm thick plate sample\nwith a polished crystallographic (110) surface, as in our\nearlier work [22], where all details on the sample prepa-\nration can be found. Choosing a lower temperature at\nT= 10 K compared to the former study ensured much\nslower dynamics and accessing a broader transition re-\ngion enabled the detailed inspection of the reorientation\nof the helix axis as well as the observation of helical do-\nmains.\nFor real-space imaging, we use magnetic force mi-\ncroscopy (MFM), that proved to be a valuable tool for\nstudying complex spin textures such as magnetic bub-\nble domains [23] or helices and skyrmions in helimagnets\nand magnetic thin \flms [14, 24{28]. In the presence of\nan electric polarization, also electrostatic forces act on\nthe MFM-tip. Compensating these forces by means of3\nKelvin-probe force microscopy (KPFM) permits the de-\ntection of pristine MFM data while simultaneously re-\nvealing the contact potential di\u000berence \u0001 Ucpd[29{31].\nIn turn, this re\rects the shift of the electric potential\ninduced by the magnetoelectric coupling.\nMFM, KPFM and non-contact atomic force mi-\ncroscopy (nc-AFM) were performed in an Omicron cryo-\ngenic ultra-high vacuum STM/AFM instrument [32] us-\ning the RHK R9s electronics [33] for scanning and\ndata acquisition. For all measurements, we used PPP-\nQMFMR probes from Nanosensors [34] driven at me-\nchanical peak oscillation amplitudes of A\u001910 nm.\nMFM images were recorded in a two-step process.\nFirstly, the topography and the contact potential di\u000ber-\nence of the sample were recorded and the topographic\n2D slope was canceled. Secondly, the MFM tip was re-\ntracted 20 nm o\u000b the sample surface to record magnetic\nforces while scanning a plane above the sample surface.\nThe KPFM-controller was switched o\u000b during this second\nstep. After the \frst MFM image had been completed at\n\u0006100 mT, the magnetic \feld was changed automatically\nin constant steps of 4 mT in between consecutive images.\nIn order to ensure a correct compensation bias, we ap-\nproached the tip to the sample before every \feld step\nand switched the KPFM-controller on. Note that the\nKPFM values change for every new magnetic \feld incre-\nment. After the new \feld had been reached, we hold the\nKPFM-controller again constant and retracted the tip\nby the same lift height. After the series of images had\nbeen completed a background image in the \feld-polarized\nstate atj\u00160Hj= 250 mT was taken.\nIII. THEORY\nThe energy density for the magnetization ~Mof cubic\nchiral magnets in the limit of small spin-orbit coupling\n\u0015SOCis given byE=E0+Edip+Eaniso where\nE0=J\n2(@i~M)2+\u001bD~M(r\u0002~M)\u0000\u00160~H~M (1)\ncomprises the isotropic exchange interaction J > 0, the\nDzyaloshinskiii-Moriya interaction D > 0 and the Zee-\nman energy. Depending on the chirality of the system,\nthe sign\u001b=\u00061. The competition between the \frst\ntwo terms results in spatially modulated magnetic or-\nder with a typical wavevector given by Q=D=J. The\nsecond termEdipcontains the dipolar interaction, and\nthe last termEaniso represents the magnetocrystalline\nanisotropies that are e\u000bectively small in the limit of small\n\u0015SOC. Under certain conditions, the magnetic helix min-\nimizes the energy density Ewhere its orientation is de-\ntermined by both the Zeemann energy and the magne-\ntocrystalline anisotropies Eaniso. In general, this leads to\na helix reorientation as a function of applied magnetic\n\feld.\nAn e\u000bective theory for this helix reorientation was pre-\nsented in Ref. [12] for MnSi. In section III A and III B\n(a) tetrahedral point group T\n(b) field cooling (c) zero field cooling\n(d) (e)\n[111][100][010][001]Figure 1. (a) B20 materials possess the symmetry of the\ntetrahedral point group indicated by the di\u000berent colours on\nthe sphere. (b) and (c): trajectories for the unit vector ^Q\nfor di\u000berent \feld directions for decreasing (\feld cooling) and\nincreasing (zero \feld cooling) \feld magnitude, respectively,\nevaluated for \"(2)\nT= 0 in Eq. (2). Starting from equally pop-\nulatedh100idomains at zero \feld in (c), trajectories can be\ncontinuous or discontinuous, and can possess a kink. (d) and\n(e): The kink represents a second-order transition that occur\nfor magnetic \feld directions indicated by the blue solid lines.\nFor a \fnite \"(2)\nTthese lines are warped towards h100i.\nwe review this theory for completeness and discuss its\nvalidity for Cu 2OSeO 3. In section III C we focus on the\ntheoretical predictions for the experimental setup. In sec-\ntion III D we present a theory for the electric polarization\nin Cu 2OSeO 3and its dependence on magnetic \feld.4\nA. E\u000bective Landau potential for the helix axis\nThe helix wavevector ~Qis determined by the compe-\ntition of Dzyaloshinskii-Moriya and exchange interaction\nand, as a consequence, its magnitude is proportional to\nspin-orbit coupling j~Qj\u0018O (\u0015SOC). The orientation of\nthe magnetic helix in a certain domain is represented\nin the following by the unit vector ^Q. The compe-\ntition between magnetocrystalline anisotropies and the\nZeeman energy can be captured in the limit of small spin-\norbit coupling \u0015SOC by the Landau potential V(^Q) =\nVT(^Q) +VH(^Q) [12].\nThe \frst term represents the magnetocrystalline po-\ntential and its form is determined by the tetrahedral\npoint group T[see Fig. 1(a)] of B20 materials like MnSi\nor Cu 2OSeO 3. As indicated by the di\u000berently colored\nregions on the sphere, Texhibits a three-fold C3rota-\ntion symmetry around the h111idirections, but only a\ntwo-foldC2rotation symmetry around h100ipoints. The\npotentialVT(^Q) contains all terms consistent with Tand\nreads as\nVT(^Q) =\"(1)\nT(^Q4\nx+^Q4\ny+^Q4\nz) (2)\n+\"(2)\nT\u0000^Q2\nx^Q4\ny+^Q2\ny^Q4\nz+^Q2\nz^Q4\nx\u0001\n::::\nThe lowest order term with constant \"(1)\nT\u0018O(\u00154\nSOC) is of\nfourth order in ^Q, and it still possesses a four-fold rota-\ntion symmetry around the h100iaxes that is not present\nin the tetrahedral point group T. This symmetry is bro-\nken explicitly only at the sixth order in ^Qby the term pa-\nrameterized by \"(2)\nT\u0018O(\u00156\nSOC). Other terms of sixth and\nhigher orders are represented by the dots. In the limit of\nsmall spin-orbit coupling, the \frst term determines the\norientation of the helix at zero \feld. For \"(1)\nT>0 the\npotential is minimized for ^Qk h111ias it is the case\nfor MnSi whereas \"(1)\nT<0 favours a helix orientation\n^Qkh100ilike in Cu 2OSeO 3.\nThe second term in the Landau potential represents\nthe Zeeman energy and up to second order in the applied\nmagnetic \feld ~Hit reads\nVH(^Q) =\u0000\u00160\n2Hi\u001fijHj; (3)\nwhere\u00160is the magnetic constant. The inverse of the\nmagnetic susceptibility tensor evaluated at zero \feld is\ngiven by\n\u001f\u00001\nij=\u001f\u00001\nij;int+Nij; (4)\nwith the demagnetization tensor Nijthat is diagonal for\nan elliptical sample shape N= diagfNx;Ny;Nzgwith\ntrfNg= 1. The internal susceptibility tensor \u001fintis\nevaluated for a \fxed orientation of the helimagnetic order\nand depends on ^Q,\n\u001fij;int=\u001fint\nk^Qi^Qj+\u001fint\n?(\u000eij\u0000^Qi^Qj): (5)An explicit calculation yields \u001fint\nk= 2\u001fint\n?, i.e., only half\nof the spins respond to a transversal magnetic \feld com-\npared to a \feld applied longitudinal to ^Q.\nMinimization of the Landau potential V(^Q) =VT(^Q)+\nVH(^Q) yields the helix orientation as a function of mag-\nnetic \feld ^Q=^Q(~H). The resulting trajectories were\ndiscussed in detail for \"(1)\nT>0 in Ref. [12]. Here we fo-\ncus on\"(1)\nT<0. Next, we present a general discussion of\nthe helix reorientation before turning to the con\fguration\nof the current experimental setup.\nB. Helix reorientation transitions\nDepending on the direction of the applied magnetic\n\feld, the reorientation of the magnetic helix might in-\nvolve either a crossover, a second-order phase transition\nor a \frst-order phase transition. For the purpose of a sim-\npli\fed discussion in this section we consider a sphere-like\nsample shape with demagnetization factors Ni= 1=3.\nFirst, we will focus on the potential without sixth-order\nterms and discuss corrections due to a \fnite \"(2)\nTat the\nend of this section. \"(1)\nTin Cu 2OSeO 3is negative, i.e.,\nthe preferred directions in zero \feld are h100iindicated\nby the black colored points on the unit sphere in Fig. 1.\nFigure 1(b) presents trajectories of the helix axis ^Q(~H)\nfor di\u000berent \feld directions indicated by the coloured\ndots after \feld cooling. For high \felds ^Q(~H) =~H=j~Hj.\nWhen decreasing the \feld, the crystalline anisotropies\ngain in\ruence and the helix reorients towards h100i. De-\npending on the \feld direction, three scenarios can be\ndistinguished. The reorientation process is a crossover\nwhen the helix reorients smoothly towards the closest\nh100idirection like for the green trajectory in Fig. 1(b).\nIt involves a second-order Z2transition when the trajec-\ntory bifurcates into two at a certain critical \feld Hc1, like\nfor the blue and yellow trajectories. A special situation\narises for a magnetic \feld along h111i. Here, the trajec-\ntory can follow three paths towards one of three distinct\nh100idirections realizing a second-order Z3transition.\nThis transition is protected by the three-fold C3rotation\nsymmetry of the point group T, i.e., it is robust even in\nthe presence of a \fnite \"(2)\nT. In general, a Z3transition\ncan be \frst-order as cubic terms are allowed in the ef-\nfective Landau expansion. For the potential of Eq. (2)\nwith\"(2)\nT= 0, however, this transition turns out to be of\nsecond-order with continuous trajectories ^Q(~H).\nAfter zero \feld cooling, helimagnetic domains oriented\nalong the threeh100idirections are populated. Upon\nincreasing the magnetic \feld, the helix axis ^Qmoves to-\nwards the \feld direction [see Figure 1(c)]. In addition to\nthe reversed paths of panel (b) there exist also discontin-\nuous paths starting from domains unfavoured by the \feld\ndirection. This discontinuous reorientation correspond to\na \frst-order transition.\nThe reorientation process thus involves a second-order5\ntransition and thus a well-de\fned critical \feld Hc1only\nfor speci\fc directions of the magnetic \feld. For \"(2)\nT= 0\nthese directions are located on the great circles on the\nsphere,that connect the h111ipoints [see Fig. 1(d)]. A\n\fnite\"(2)\nTinduces a warping of these lines [see panel (e)].\nAs the ratio \"(2)\nT=\"(1)\nT\u0018\u00152\nSOCis of second order in spin-\norbit coupling this e\u000bect is expected to be small.\nIn Cu 2OSeO 3, the sixth order term only quantitatively\nin\ruences the reorientation transitions. This is di\u000berent\nto MnSi where it is crucial to take the \"(2)\nTterm into\naccount as discussed in Ref. [12]. There, \"(1)\nTis positive\nwhich yieldsh111ias preferred directions in zero \feld.\nFor a \feld along [100], four of those are equally close\nsuggesting a Z4transition. However, a \fnite \"(2)\nTsplits\nthisZ4transition into two subsequent Z2transitions.\nC. Helix orientation trajectory for the\nexperimental setup\nIn the following, we neglect the sixth-order correction\n\"(2)\nT= 0 as its in\ruence is weak and cannot be resolved\nwithin the experimental accuracy. The investigated sam-\nple [see section II] is approximately a plate so that we use\nNx=Ny= 0 andNz= 1 for the demagnetization fac-\ntors in the basis of principal axis. The normal axis of the\nplate-like sample approximately corresponds to a crys-\ntallographic [110] direction. Within the crystallographic\nbases the demagnetization tensor is then given by\nN=0\n@1=2 1=2 0\n1=2 1=2 0\n0 0 01\nA: (6)\nThe magnetic \feld is always approximately aligned along\nthe surface normal so that we restrict ourselves to the\nmagnetic \feld direction ^HT= (1;1;0)=p\n2. For the lon-\ngitudinal susceptibility of Cu 2OSeO 3we use the value\n\u001fint\nk= 1:76 given in Ref. [35]. The transition between\nthe conical and \feld-polarized phase occurs at the criti-\ncal \feld\u00160Hc2\u0019192 mT (see below), which we will use\nlater on to normalize the \feld axis.\nFor a \feld along [110] the reorientation process is ei-\nther \frst-order for the [001] domain (yellow), or second-\norder for the [100] and [010] domains (purple) [see\nFig. 2(a)]. The continuous trajectory of the helix axis\ncan be parametrized as ^QT= (cos\u001e;sin\u001e;0) with\u001e\nthe azimuthal angle that depends on the magnetic \feld,\n\u001e=\u001e(~H). Its dependence is shown in Fig. 2(b) where\nthe kink de\fnes the critical \feld, that for Eq. (6) is given\nby\n\u00160Hc1= 2(1 +\u001fint\nk)vuut\u00160\"(1)\nT\n\u001fint\nk: (7)\nAs we will see later, experimentally we \fnd \u00160Hc1\u0019\n(a) (b)\n(c) (d)\n(e)Figure 2. (a) Reorientation process for a magnetic \feld point-\ning along [110]. The helix axis ^Q= (cos\u001e;sin\u001e;0) continu-\nously moves as a function of Halong the purple paths be-\ntween [110] and either [100] or [010]. The (yellow) domain\n[001] depopulates in a discontinuous fashion upon increasing\nH. (b) Azimuthal angle \u001e(H) starting from the [100] domain\natH= 0; (c) uniform magnetization MH(H) and (d) suscep-\ntibility@MH=dH for the purple trajectories in (a). (e) Electri-\ncal polarization \u0016Pz(H) attributed to the domains located for\nH= 0 at [100] and [010] (purple) and [001] (yellow). The crit-\nical \feldHc1separates the conical state C (grey) where ^Qk~H\nand the generalized helical state H (green) where ^Q,~H.\n70 mT corresponding to a value \"(1)\nT=\u00000:0014\u0016eV/\u0017A3\nfor the magnetocrystalline potential.\nIn Fig. 3 we illustrate the Landau potential V(^Q) for\nthe above parameters for various values of the magnetic\n\feldH. At zero \feld, all h100idirections are ener-\ngetically degenerate, see panel (a). For a \fnite \feld\nalong [110], the direction [001] remains a local mini-\nmum until it disappears at a certain spinodal \feld Hsp\nofHsp=Hc2\u00190:25. At the same time, the other minima\nremain global minima and move towards the \feld direc-\ntion. They merge at the critical \feld Hc1=Hc2\u00190:36\nand a single global minimum is obtained for H\u0015Hc1.\nD. Electric polarization\nThe magnetoelectric coupling in Cu 2OSeO 3induces an\nelectric polarization that is given in terms of the magne-6\n(a) h=0 (b) h=0.1 (c) h=0.2\n(d) h=hsp≈0.25 (e) h=hc1≈0.36 (f) h=hc2=1\nVmax(h)\nVmin(h)\nFigure 3. E\u000bective Landau-potential for the helix axis,\nV(^Q), for the experimental parameters. The magnetic \feld\nis applied along [110] and various values of the reduced \feld\nh=H=H c2are shown, see text. The color coding varies\nfrom panel to panel; red and blue color correspond to the\nminimal and maximal potential value at each speci\fc \feld,\nrespectively.\ntization vector ~Mby [15]\n~P(~ r) =cME*0\n@My(~ r)Mz(~ r)\nMz(~ r)Mx(~ r)\nMx(~ r)My(~ r)1\nA+\n; (8)\nwherecMEdenotes the magnetoeletric coupling constant.\nGenerally, the expectation value in Eq. (8) can be decom-\nposed into\nhMi(~ r)Mj(~ r)i=hMi(~ r)ihMj(~ r)i+Sij(~ r) (9)\nwithSij(~ r) the correlation function. In the mean-\feld\napproximation,Sijis neglected and the polarization re-\nduces to a product of expectation values h~M(~ r)i.\nWithin the framework of the Landau theory of section\nIII A this expectation value is given in terms of a Fourier\nseries,\nh~M(~ r)i=~MH+~Mhelix(~ r) +\u000e~M(~ r): (10)\nThe second term is given by a harmonic helix\n~Mhelix(~ r) =Mhelixh\n^e1cos(~Qmin~ r) +\u001b^e2sin(~Qmin~ r)i\n(11)\nwith the orthogonal unit vectors ^ e1\u0002^e2=^Qmin=\n~Qmin=Q. Depending on the chirality of the system, see\nEq. (1), the helix can be right-handed or left-handed cor-\nresponding to \u001b= +1 or\u00001, respectively. Here, the ori-\nentation of the helix axis ^Qmin(~H) minimizes the Landau\npotentialV(^Q) at a given ~H. The \frst term in Eq. (10)\nrepresents the uniform part, ~MH=MH^H, that can be\nobtained with the help of the Landau potential:\nMH=\u00001\n\u00160@V(^Qmin(~H))\n@H: (12)The magnitude MHas well as the total susceptibility\n@HMHare shown in Fig. 2(c) and (d) respectively. The\nsusceptibility shows a pronounced mean-\feld jump at the\ncritical \feld Hc1.\nIf variations of the amplitude are negligible, the length\nof the magnetization should be locally given by the satu-\nration magnetization h~M(~ r)i2=M2\nswhich gives rise to\nanharmonicities represented by \u000e~M(~ r) in Eq. (10). Min-\nimizing the energy (1) in the presence of this constraint,\nwe \fnd in lowest order, i.e., neglecting 3 ~QFourier com-\nponents that \u000e~Malso assumes the form of a helix,\n\u000e~M(~ r)\u0019j~MH;?jh\n^e0\n1cos(2~Qmin~ r) +\u001b^e0\n2sin(2~Qmin~ r)i\n:\n(13)\nThe prefactor is determined by the magnetization pro-\njected onto the plane perpendicular to ~Qmin, i.e.,~MH;?=\n~MH\u0000(~MH^Qmin)^Qmin. The unit vectors ^ e0\n1\u0002^e0\n2=^Qmin\nare given by\n^e0\n1= ^e2(^MH;?^e2)\u0000^e1(^MH;?^e1); (14)\n^e0\n2=\u0000^e2(^MH;?^e1)\u0000^e1(^MH;?^e2); (15)\nwhere ^MH;?=~MH;?=j~MH;?j. The component \u000e~Mis\nproportional to the uniform magnetization MHand thus\nvanishes linearly with the applied magnetic \feld. More-\nover, it vanishes for the conical state where ~MHk^Qmin\nso that~MH;?= 0. The anharmonicity is thus most pro-\nnounced at intermediate \felds as observed in MnSi and\nFeGe [4, 36, 37]. Within this approximation, the ampli-\ntude of the helix is given by\nMhelix=q\nM2s\u0000~M2\nH\u0000~M2\nH;?: (16)\nIn the experiment, the polarization ~P(~ r) cannot be\nspatially resolved on the scale of the helix wavelength.\nFor this reason, we consider the polarization spatially\naveraged over a single period,\n\u0016~P=1\n2\u00192\u0019Z\n0dx~P(~ r)\f\f\fMF\nx=~Qmin~ r(17)\nwhere the upper index MF indicates that we employ the\nmean-\feld approximation. In our experimental setup,\nit turns out that only the z-component \u0016Pzis expected\nto remain \fnite. For a (110) surface this z-component\namounts to an in-plane polarization along [001], \u0016P[001]=\n\u0016Pz.\nFor the continuous trajectories, i.e., the purple paths\nof Fig. 2(a), we get\n\u0016Pz=cME\n2\u0014\nM2\nH\u00001\n2sin(2\u001e)\u0000\nM2\ns\u0000M2\nH\u0001\u0015\n; (18)\nwith the azimuthal angle \u001e=\u001e(H) of Fig. 2(b). Its mag-\nnetic \feld dependence corresponds to the purple line in7\nFig. 2(e). At the \frst critical \feld, Hc1, the polarization\nis minimal and shows a kink. For Hc2> H > H c1, the\nangle\u001e=\u0019=4 and the uniform magnetization MH=\nMsH=Hc2so that the polarization reduces to the known\nexpression [15, 22, 38] \u0016Pz=cMEM2\ns\n4(3(H=Hc2)2\u00001), and\na sign change is expected for H=Hc2=p\n3. At the second\ncritical \feld Hc2another kink re\rects the phase transi-\ntion to the \feld-polarized phase. For H\u0015Hc2, we have\nMH=Msand \u0016Pz=\u0016Pc2\u0011cME\n2M2\ns.\nFor the yellow [001] domain in Fig. 2(a), the helix\naxis is given by ^QT\nmin= (0;0;1) for \feldsjHj\u0014Hsp\u0019\n0:25Hc2up to its spinodal point where the \frst-order\ntransition must take place at the latest. Its polarization\nwithin this \feld range is then given by\n\u0016Pz=cME\n2M2\nH; (19)\nthat is shown as a yellow line in Fig. 2(e).\nIV. EXPERIMENTAL RESULTS\nIn this section, we present the experimental \fndings\nobtained via MFM and KPFM measurements that are\nboth sensitive to signals attributed to the surface of the\nmaterial. In the present setup an approximate (110)\nsurface is considered so that the surface normal ^ nT=\n(1;1;0)=p\n2. Moreover, the applied \feld is approximately\nparallel to ^n. It is important to note that helimagnetic\norder with orientation ^Qand an intrinsic wavelength\n\u0015h= 2\u0019=Q\u001960 nm [7] gives rise to periodic magnetic\nstructures appearing at the sample surface characterized\nby a projected wavevector\n~Q0=~Q\u0000^n(~Q^n) =Q(^Q\u0000^ncos \u0002); (20)\nwhere \u00022[0;\u0019=2] is the angle between the helix axis\n^Qand the surface normal ^ n. This results in a projected\nwavelength \u00150=2\u0019\nj~Q0jgiven by [25]\n\u00150=\u0015h\nsin \u0002: (21)\nFor a helix with an in-plane ^Qthe angle \u0002 = \u0019=2 and\n\u00150=\u0015h. However, for a helix oriented along the surface\nnormal \u0002 = 0 the wavelength \u00150diverges, and the surface\nshould appear homogeneous.\nFurthermore, for the later analysis we introduce the\nin-plane angle \u000b=^(^e[001];~Q0) de\fned as the angle en-\nclosed by the projected wavevector ~Q0and the in-plane\nvector ^eT\n[001]= (0;0;1).\nA. Magnetic imaging of helimagnetic order\nWe start the presentation of our experimental results\nwith typical real-space images depicted in Fig. 4. Aslideshow of the complete dataset as well as of additional\nmeasurements can be retrieved from the supplementary\nmaterials. As MFM essentially tracks the out-of-plane\ncomponent of the local magnetization, the images repre-\nsent the projection of the magnetization onto the surface\nnormal~M(~ r)^n.\nThe image series in Fig. 4 is measured on the down-\nwards branch of the hysteresis loop sweeping the exter-\nnal magnetic \feld from positive to negative values. After\napplying a saturating \feld of \u00160H= 250 mT, the \feld\nwas decreased and the series starts with \u00160H= 100 mT\nshown in Fig. 4(a) where a periodic pattern is visible.\nDecreasing the \feld further, the magnetization on the\nsurface is reconstructed at about \u00160H\u001960 mT and mul-\ntiple helical domains form and increase in size preferen-\ntially showing a stripy pattern along the [ \u0016110] direction\n[see panel (c) and (d)]. Close to zero \feld, an additional\nhelimagnetic domain oriented along the [001]-direction is\nobserved giving rise to domain walls as shown in panel\n(e). The surface wavelength \u00150associated with the var-\nious domains depends on the applied magnetic \feld in\na characteristic manner. When decreasing the \feld fur-\nther to negative values, the domains start to split and\nmagnetization reconstructs at about \u00160H\u0019 \u0000 70 mT\n[see Fig. 4(g) and (h)]. At \u00160H\u0019 \u0000 100 mT a peri-\nodic modulation oriented along [001] is again visible in\nFig. 4(i) similar to panel (a). Finally, at large \feld of\n\u00160H=\u0000250 mT, the magnetization is fully polarized\nand the corresponding image in panel (k) is featureless.\nWe observed a manifold of co-existing domains as\nshown in Fig. 4(e) after \feld cycling. In contrast, af-\nter zero \feld cooling to T= 10 K only one single domain\nwith an in-plane helix axis along [001] could be observed,\nsimilarly to our previous measurements close to the crit-\nical temperature Tc[22].\nB. Analysis of the MFM data\nAssuming that the periodic patterns observed in the\nMFM images correspond to helimagnetic ordering pro-\njected onto the sample surface, we extract the projected\nwavevector ~Q0, the projected wavelength \u00150=2\u0019\nj~Q0j, and\nthe corresponding angles \u0002 and \u000bas de\fned at the begin-\nning of section IV. Exempli\fed in Fig. 5(a), we can exper-\nimentally distinguish three types of domains depending\non the in-plane angle \u000bat zero \feld, namely type I with\n\u000b\u00180°, type II with \u000b.90°, and type III with \u000b&90°.\nIt was previously established by neutron scattering\nthat the helices at zero \feld point along a crystallo-\ngraphich100idirection [7]. Correspondingly, one expects\nindeed three di\u000berent domains for a (110) surface with\nin-plane angle \u000b= 0 for ^Qk[001] and\u000b= 90\u000efor^Q\nalong [100] and [010]. The deviations from these values\nin Fig. 6(c) indicate an uncertainty of about 4\u000edue to\na combination of systematic errors. First, the sample\nis slightly miscut so that the surface normal might be8\na +100 mT\n b +60 mT\n c +52 mT\n d +44 mT\n e 0 mT\nf -60 mT\nneg. ∆f pos.\ng -64 mT\n h -76 mT\n i -100 mT\n k -250 mT\n[001][1¯10]\n⊙[110]\n2 µm\nFigure 4. Typical scanning probe images of Cu 2OSeO 3atT= 10 K. The magnetic \feld was applied perpendicular to the\nplane of projection along the [110]-direction changing from positive to negative values. Periodic magnetic textures are observed\nwhose wavelength and orientation depend on the strength of the magnetic \feld (see text). The span of the color scale is adapted\nindividually, ranging from 1 Hz to 2.5 Hz.\nb +24 mT\n#1\n#2\n0 1.1 Hz 400 nm\na +0 mT\nα[001]\n[1¯10]/vectorQ/primeI\nIII\nII\n0 1.4 Hz 1 µm\nFigure 5. (a) Close to \feld zero three types of domains,\nI, II, and III, can be observed. The projected wavevector\n~Q0and the [001] direction enclose the in-plane angle \u000b. (b)\nRelaxation events during the scanning process give rise to\ndiscontinuity lines (red arrows) within the periodic pattern\nwith phase jumps of 180\u000e.\nslightly tilted away from [110] towards [111], while sec-\nond, the magnetic \feld might be slightly misaligned from\nthe surface normal ^ n. Third, the sample placement in the\nMFM can be slightly misaligned with a small in-plane\nrotation as well. Finally, dynamic creep of the scanning\npiezo actuator slightly a\u000bects the scanner calibration.\nThe evolution of the projected wavelength \u00150and the\ncorresponding angle \u0002 of these three types of domains is\nshown as a function of magnetic \feld for every domain\nin Fig. 6(a) and (b), respectively. This includes also do-\nmains, which where not in the image frame at zero \feldand therefore are not classi\fed as one of the three types\nin Fig. 5(a) (shown as grey dots). For the helimagnetic\ndomain oriented in-plane at zero \feld, \u00150=\u0015h(yellow\ndots, green dots after zero-\feld cooling). The other do-\nmains (blue and red dots) are characterized for a (110)\nsurface by an angle \u0002 = \u0019=4 and a projected wavelength\nof\u00150=p\n2\u0015h. A drastic change of \u00150is observed around\n70 mT that we identify with the critical \feld Hc1of the\nreorientation transition. For larger \felds, the projected\nwavelength is of order \u00150\u001810\u0015h, corresponding to an\nangle \u0002\u00185\u000e. We attribute this \fnite angle to the mis-\nalignment error mentioned above.\nC. Relaxation processes during helix reorientation\nWhenever changing the magnetic \feld, the magnetic\nstructure relaxes on relatively long time scales, especially\nclose toHc1as discussed in Ref. [12]. An example of such\na relaxation process is shown in Fig. 5(b). The MFM\nimage is scanned from top to bottom. During this scan\nthe magnetic structure might change due to relaxation\nevents. They are re\rected in discontinuity lines marked\nwith (#1) and (#2) in Fig. 5(b), where the helix pattern\nis shifted by 180\u000e. Such 180\u000eshifts were observed before\nin Ref. [13] and attributed to the motion of dislocation\ndefects in the helimagnetic background. Interestingly,\nthe discontinuity lines do not continue through the full\nimage frame but terminate. Probably, the termination\npoints coincide with a helimagnetic domain wall separat-\ning di\u000berenth100idomains [14]. This suggests that the\ndiscontinuity lines arise from motion of dislocations close9\nλ' (nm)nm))\n608010050010001500H C C\na\nsgn(nm)ΔHH)∙µ0H (nm)m)T)050100150200 -200-150-100-5095\n90\n85\n5\n0\n-5α (°)c70\n50\n30\n6\n2\n0θ (°)b90\n4\nFigure 6. Experimentally determined helix orientation char-\nacterized by (a) the projected wavelength \u00150of Eq. (21), (b)\nout-of-plane angle \u0002 = ^(^n;^Q) and (c) measured in-plane\nangle\u000b. Yellow and green dots denote [001]-domains (type\nI) observed during \feld sweeps or after zero-\feld cooling, re-\nspectively. Blue and red dots denote type II domains and\ntype III domains, respectively. Grey dots belong to domains,\nwhich where not in the image frame at zero \feld.\nto the domain wall. The creep motion of dislocations\ncontributes to the complex and slow relaxation processes,\ngiving rise to hysteretic e\u000bects even for the second-order\nphase transition at Hc1.\nD. Contact potential and polarization\nCompensating electrostatic forces at the MFM tip with\nthe help of KPFM allows to extract di\u000berences in the\ncontact potential \u0001 Ucpd. On a highly insulating sam-\nple as is Cu 2OSeO 3, this potential is measured on length\nscales of the mesoscopic MFM-tip so that it corresponds\nHC C(a)\nHC C(b)Figure 7. (a) Contact potential di\u000berence \u0001 Ucpdas a func-\ntion of increasing (yellow, orange, and red points) and de-\ncreasing (green, blue and purple points) magnetic \feld at\nT\u001910 K. (b) Same data is plotted so that all histories run\nfrom the left-hand to the right-hand side.\nto an average over entire domains. As explained in detail\nin Ref. [22], for the current experimental setup this po-\ntential \u0001Ucpdfor a single domain is proportional to the\nin-plane polarization \u0016Pzof Eqs. (18) or (19). The mea-\nsured \u0001Ucpdas a function of magnetic \feld is displayed\nin Fig. 7(a) where the background colours indicate the\nvarious phases previously identi\fed with MFM.\nSimilar to our previous measurement [22] performed\nclose to the critical temperature Tc, we \fnd a plateau-\nlike region close to the zero \feld, a minimum at the crit-\nical \feldHc1, an increase of \u0001 Ucpdwithin the conical\nphaseHc1<H <H c2, and a kink at the second critical\n\feldHc2. The main di\u000berence to the previous study in\nRef. [22] is found at intermediate \felds due to the ab-\nsence of the skyrmion phase at 10 K in the present case.\nIn addition, the hysteresis associated with the helix reori-\nentation is more pronounced at lower temperatures due\nto slow relaxation processes already mentioned in section\nIV C. Performing a closed hysteresis loop, hysteretic ef-\nfects are observed at both reorientation transitions \u0006Hc1\n[see Fig. 7(a)]. The hysteresis of both transitions is ba-\nsically the same as illustrated in panel Fig. 7(b) where\n\u0001Ucpdhas been replotted so that all histories run from\nleft to right.10\n2\n1 0.5 0 -0.51.5\n1\nH/Hc1λ'/λh\nHFP C H C FPPz/Pc21\n0.5\n-0.5-1(a)\n(b)\nHc2 Hc1 -Hc1 -Hc2\nFigure 8. Comparison between experiment (dots) and theory\n(lines) for (a) the projected wavelength \u00150of Eq. (21) and (b)\nthe electric polarization as a function of magnetic \feld. The\npurple lines correspond to the contributions from domains\nstarting in zero \feld at [100] and [010], and the yellow line\ncorresponds to the [001] domain [see Fig. 2(a)]. The experi-\nmental data were collected in a \feld sweep from negative to\npositive \felds.\nV. DISCUSSION\nIn the following, our experimental results of section IV\nare interpreted in terms of the e\u000bective theory presented\nin section III. As alluded to in section IV, the experi-\nmental system is plagued with systematic errors related\nto misalignments on the order of 5\u000e. This will be re-\n\rected in the quantitative comparison to the theoretical\npredictions that presume the magnetic \feld being strictly\naligned along [110], the direction of the surface normal.\nThe values for the critical \felds were determined from\ncharacteristic kinks in the experimental data, \u00160Hc1\u001970\nmT and\u00160Hc2\u0019192 mT. The latter value corresponds\nto an internal \feld of \u00160Hint\nc2=\u00160(Hc2\u0000NzMs)\u001960 mT\nfor our plate-like sample Nz= 0:92 and the saturation\nmagnetization Ms\u0019110 kA/m [18, 39].\nIn Fig. 8 we plot a comparison between theory and\nexperiment for the projected wavelength \u00150of Eq. (21)\nand Fig. 6(a) and the electric polarization of Fig. 7(b).\nThe experimental data (dots) were collected in an up-\nsweep from negative to positive magnetic \felds. The\nsolid lines correspond to theoretical predictions where\npurple lines are attributed to domains located at [100]\nand [010] for ~H= 0, and the yellow lines correspond to\nthe in-plane [001] domain. When the magnetic \feld isincreased from negative values beyond \u0000Hc1, the pro-\njected wavelength \u00150[see panel Fig. 8(a)], decreases in a\ncharacteristic fashion and achieves a minimum at ~H= 0\nbefore then increasing again in a similar manner on the\nother side. The theoretical curve reproduces this behav-\nior qualitatively but overestimates the projected wave-\nlength close to Hc1. Interestingly, the [001] domain ap-\npears to be spontaneously populated upon approaching\n\feld zero, undergoing a \frst-order phase transition, al-\nthough for the \feld direction [110] this domain is always\nenergetically unfavored at least within the bulk of the\nsample. As the helimagnetic axis of this domain is lo-\ncated within the surface plane a projected wavelength\n\u00150=\u0015his expected that equals the wavelength within\nthe bulk. This is indeed observed close to zero \feld, but\n\u00150slightly increases for increasing positive \felds in con-\ntrast to the theoretical prediction (yellow line).\nIn Fig. 8(b) the electric polarization is compared to\ntheory. The measured polarization is spatially averaged\nover various domains so that it presents in general the\ncombined signal from all three populated domains. Upon\nincreasing the \feld from negative values, only two of the\ndomains are populated so that the polarization closely\nfollows the purple curve. However, when the [001] do-\nmain gets spontaneously populated close to \feld zero,\nthe corresponding polarization contributes possibly ex-\nplaining the plateau-like feature of \u0016Pzfor small positive\n\felds. The behavior at the reorientation transition Hc1\nis hysteretic and strongly depends on the history, which\nwe attribute to a non-equilibrium e\u000bect similar to pre-\nvious observations in MnSi [12]. Upon increasing the\n\feld towards\u0000Hc1the single helimagnetic domain splits\ninto two, resulting in sharp signatures. However, increas-\ning the \feld towards + Hc1, two or even three populated\ndomains need to transform into a single domain which\nrequires slow relaxation processes. The \feld-sweep em-\nployed in the experiment was probably too fast for the\nsystem to equilibrate giving rise to the apparent hys-\nteretic behavior close to the continuous transition at Hc1.\nClose to the critical \feld Hc2, the measured polarization\ndeviates from the theoretical prediction as it does not\nreach the same maximal value \u0016Pc2atHc2. Furthermore,\nit decreases in the \feld-polarized phase instead of stay-\ning constant as theoretically expected. This disagree-\nment probably is due to measuring artifacts at larger\n\felds originating from a change in the distance between\nsample and tip.\nIn section IV C we provided experimental evidence for\nthe involved relaxation processes. The 180\u000ediscontinu-\nities observed in the MFM pattern hint at the motion\nof dislocations along domain boundaries [see Fig. 5(b)].\nSimilar 180\u000ediscontinuities have been previously ob-\nserved in FeGe after a \feld quench [13].\nThe assumptions employed in the theoretical model\nof section III need to be critically scrutinized. First of\nall, this model aimed at describing the reorientation of\nhelimagnetic order within the bulk of the material. Ad-\nditional contributions arising from the surface of the ma-11\nterial can easily modify our assumption that the MFM\nimages essentially re\rect the projection of bulk helimag-\nnetic order. It is known that so-called surface twists\ncan lead to modi\fcation of helimagnetic order close to\nthe surface [40], which is neglected in the present work.\nIn particular, such surface twists due to the magnetic\nboundary condition could lead to an anchoring of the\nhelix axis within the surface plane. The boundary con-\ndition is automatically ful\flled for the pristine helix in\ncase that the helix axis is aligned with the surface nor-\nmal ^Qk^n. However, in the opposite limit ^Q?^nthe\nboundary conditions will induce distortions to the heli-\ncal texture, which might lower the surface energy of the\nmagnetic structure and potentially favours surface do-\nmains with ^Q?^n. This could explain the unexpected\npopulation of the [001] domain with ^Q?^nobserved in\nthe present experiment for \feld-cooling with ~Hk[110].\nIn addition, we exclusively observed [001] helix con\fgura-\ntions at higher temperatures [22] as well as for zero-\feld\ncooling toT= 10 K. This suggests that the three h100i\ndomains, which are degenerate within the bulk at ~H= 0,\nare not evenly populated close to the surface so that the\n^Q?^nsurface domain is indeed energetically favoured.\nFurthermore, the theory presented is valid in the limit\nof small spin-orbit coupling \u0015SOC. However, this cou-\npling is su\u000eciently strong in Cu 2OSeO 3so that addi-\ntional phases are stabilized for magnetic \felds along h100i\n[19, 20]. As discussed in Ref. [18], the stabilization of a\nmetastable tilted conical phase can be phenomenologi-\ncally described in terms of a modi\fed parameter \"(1)\nTin\nEq. (2) that is \feld dependent and changes sign as a func-\ntion of~H. Whereas these e\u000bects are believed to be im-\nportant mainly for \felds along h100i, they might give rise\nto quantitative corrections for the present experimental\nsetup with ~Hk[110].\nWe note that our theory of the electric polarization, Pz,\nfor the reorientation process presented in section III D\nis also relevant for the spin-Hall magnetoresistance of\nCu2OSeO 3that was found to be proportional to Pz[38].\nIn summary, we investigated the helix reorientation in\nCu2OSeO 3for a magnetic \feld aligned close to [110] by\nmeans of magnetic force microscopy. This technique al-\nlows to probe the manifestations of the reorientation pro-cess at the sample surface with high spatial resolution.\nWe observed the formation of domains close to the transi-\ntionHc1, and we identi\fed relaxation events in real space\nthat accompany the slow reorientation process. Given\nthe experimental uncertainties, the periodicity of the ob-\nserved surface patterns are consistent with the projected\nwavelength of the bulk helimagnetic order, and its mag-\nnetic \feld dependence is in good agreement with theoret-\nical predictions. Nevertheless, we also found evidence for\na surface anchoring of the helix wavevector ^Qfavouring\ndomains with in-plane ^Q. An interesting extension of the\npresent work might be the detailed comparison between\nMFM and bulk measurements as well as the study of the\nreorientation process for \felds along h111iwhere a Z3\ntransition is expected for Cu 2OSeO 3.\nACKNOWLEDGEMENT\nP.M., E.N., and L.M.E. gratefully acknowledge \fnan-\ncial support by the German Science Foundation (DFG)\nthrough the Collaborative Research Center \\Correlated\nMagnetism: From Frustration to Topology\" (Project\nNo. 247310070, Project C05), the SPP2137 (project\nno. EN 434/40-1), and project no. EN 434/38-1 and\nno. MI 2004/3-1. L.M.E. also gratefully acknowl-\nedges \fnancial support through the Center of Excel-\nlence - Complexity and Topology in Quantum Matter\n(ct.qmat) - EXC 2147. L.K. and M.G. gratefully ac-\nknowledge \fnancial support by DFG SFB1143 (Corre-\nlated Magnetism: From Frustration To Topology, Project\nNo. 247310070, Project A07), DFG Grant No. 1072/5-1\n(Project No. 270344603), and DFG Grant No. 1072/6-\n1 (Project No. 324327023). A.B. and C.P. acknowl-\nedge \fnancial support by the Deutsche Forschungsge-\nmeinschaft (DFG, German Research Foundation) un-\nder TRR80 (From Electronic Correlations to Function-\nality, Project No. 107745057, Project E1) and the ex-\ncellence cluster MCQST under Germany's Excellence\nStrategy EXC-2111 (Project No. 390814868) as well as\nby the European Research Council (ERC) through Ad-\nvanced Grants No. 291079 (TOPFIT) and No. 788031\n(ExQuiSid) is gratefully acknowledged.\n[1] S. M uhlbauer, B. Binz, F. Jonietz, C. P\reiderer,\nA. Rosch, A. Neubauer, R. Georgii, and P. B oni,\n\\Skyrmion lattice in a chiral magnet,\" Science 13, 915\n(2009).\n[2] Sergey V Grigoriev, V A Dyadkin, D Menzel, J Schoenes,\nYu O Chetverikov, A I Okorokov, H Eckerlebe, and\nS V Maleyev, \\Magnetic structure of Fe 1\u0000xCoxSi in a\nmagnetic \feld studied via small-angle polarized neutron\ndi\u000braction,\" Physical Review B 76, 224424 (2007).\n[3] W. M unzer, A. Neubauer, T. Adams, S. M uhlbauer,\nC. Franz, F. Jonietz, R. Georgii, P. B oni, B. Pedersen,\nM. Schmidt, A. Rosch, and C. P\reiderer, \\Skyrmionlattice in the doped semiconductor Fe 1\u0000xCoxSi,\" Phys.\nRev. B 4, 041203 (2010).\n[4] B Lebech, J Bernhard, and T Freltoft, \\Magnetic struc-\ntures of cubic FeGe studied by small-angle neutron scat-\ntering,\" Journal of Physics: Condensed Matter 1, 6105{\n6122 (1989).\n[5] X.Z. Yu, N. Kanazawa, Y. Onose, K. Kimoto, W. Z.\nZhang, S. Ishiwata, Y. Matsui, and Y. Tokura, \\Near\nroom-temperature formation of a skyrmion crystal in\nthin-\flms of the helimagnet FeGe,\" Nat. Mat. 2, 106\n(2010).12\n[6] S. Seki, X.Z. Yu, S. Ishiwata, and Y. Tokura, \\Obser-\nvation of skyrmions in a multiferroic material,\" Science\n336, 198 (2012).\n[7] T. Adams, A. Chacon, M. Wagner, A. Bauer, G. Brandl,\nB. Pedersen, H. Berger, P. Lemmens, and C. P\reiderer,\n\\Long-wavelength helimagnetic order and skyrmion lat-\ntice phase in Cu 2OSeO 3,\" Phys. Rev. Lett. 108, 237204\n(2012).\n[8] Andreas Bauer and Christian P\reiderer, \\Generic as-\npects of skyrmion lattices in chiral magnets,\" in Topo-\nlogical Structures in Ferroic Materials: Domain Walls,\nVortices and Skyrmions , edited by Jan Seidel (Springer\nInternational Publishing, 2016) pp. 1{28.\n[9] Mitsuo Kataoka and Osamu Nakanishi, \\Helical spin den-\nsity wave due to antisymmetric exchange interaction,\"\nJournal of the Physical Society of Japan 50, 3888{3896\n(1981).\n[10] M L Plumer and M B Walker, \\Wavevector and spin\nreorientation in MnSi,\" Journal of Physics C: Solid State\nPhysics 14, 4689{4699 (1981).\n[11] M. B. Walker, \\Phason instabilities and successive wave-\nvector reorientation phase transitions in MnSi,\" Phys.\nRev. B 40, 9315{9317 (1989).\n[12] A. Bauer, A. Chacon, M. Wagner, M. Halder, R. Georgii,\nA. Rosch, C. P\reiderer, and M. Garst, \\Symmetry\nbreaking, slow relaxation dynamics, and topological de-\nfects at the \feld-induced helix reorientation in MnSi,\"\nPhys. Rev. B 95, 024429 (2017).\n[13] A. Dussaux, P. Schoenherr, K. Koumpouras, J. Chico,\nK. Chang, L. Lorenzelli, N. Kanazawa, Y. Tokura,\nM. Garst, A. Bergman, C. L. Degen, and D. Meier,\n\\Local dynamics of topological magnetic defects in the\nitinerant helimagnet FeGe,\" Nature Communications 7,\n12430 (2016).\n[14] P. Schoenherr, J. M uller, L. K ohler, A. Rosch,\nN. Kanazawa, Y. Tokura, M. Garst, and D. Meier,\n\\Topological domain walls in helimagnets,\" Nat. Phys.\n14, 465 (2018).\n[15] S. Seki, S. Ishiwata, and Y. Tokura, \\Magnetoelec-\ntric nature of skyrmions in a chiral magnetic insulator\nCu2OSeO 3,\" Phys. Rev. B 86, 060403(R) (2012).\n[16] M. Mochizuki and S. Seki, \\Dynamical magnetoelectric\nphenomena of multiferroic skyrmions,\" J. Phys.: Con-\ndens. Matter 27, 503001 (2015).\n[17] Markus Garst, Johannes Waizner, and Dirk Grundler,\n\\Collective spin excitations of helices and magnetic\nskyrmions: review and perspectives of magnonics in non-\ncentrosymmetric magnets,\" J. of Phys. D: Appl. Phys.\n50, 293002 (2017).\n[18] M. Halder, A. Chacon, A. Bauer, W. Simeth,\nS. M uhlbauer, H. Berger, L. Heinen, M. Garst, A. Rosch,\nand C. P\reiderer, \\Thermodynamic evidence of a sec-\nond skyrmion lattice phase and tilted conical phase in\nCu2OSeO 3,\" Phys. Rev. B 98, 144429 (2018).\n[19] A. Chacon, L Heinen, M Halder, A Bauer, W Simeth,\nS M uhlbauer, H. Berger, Markus Garst, Achim Rosch,\nand Christian P\reiderer, \\Observation of two indepen-\ndent skyrmion phases in a chiral magnetic material,\" Na-\nture Physics 31, 1 (2018).\n[20] Fengjiao Qian, Lars J. Bannenberg, Heribert Wilhelm,\nGr\u0013 egory Chaboussant, Lisa M. Debeer-Schmitt, Mar-\ncus P. Schmidt, Aisha Aqeel, Thomas T. M. Palstra,\nEkkes Br uck, Anton J. E. Lefering, Catherine Pappas,\nMaxim Mostovoy, and Andrey O. Leonov, \\New mag-netic phase of the chiral skyrmion material Cu 2OSeO 3,\"\nScience Advances 4(2018), 10.1126/sciadv.aat7323.\n[21] M.-G. Han, J. A. Garlow, Y. Kharkov, L. Camacho,\nR. Rov, J. Sauceda, G. Vats, K. Kisslinger, T. Kato,\nO. Sushkov, Y. Zhu, C. Ulrich, T. S ohnel, and J. Sei-\ndel, \\Scaling, rotation, and channeling behavior of heli-\ncal and skyrmion spin textures in thin \flms of Te-doped\nCu2OSeO 3,\" Science Advances 6(2020), 10.1126/sci-\nadv.aax2138.\n[22] P. Milde, E. Neuber, A. Bauer, C. P\reiderer, H. Berger,\nand L.M. Eng, \\Heuristic description of magnetoelectric-\nity of Cu 2OSeO 3,\" Nano Lett. 16, 5612 (2016).\n[23] A. Wadas, R. Wiesendanger, and P. Novotny, \\Bubble\ndomains in garnet \flms studied by magnetic force mi-\ncroscopy,\" J. Appl. Phys. 78, 6324 (1995).\n[24] P. Milde, D. K ohler, J. Seidel, L.M. Eng, A. Bauer,\nA. Chacon, J. Kindervater, S. M uhlbauer, C. P\reiderer,\nC. Sch utte, and A. Rosch, \\Unwinding of a skyrmion lat-\ntice by magnetic monopoles,\" Science 340, 1076 (2013).\n[25] I. K\u0013 ezsm\u0013 arki, S. Bord\u0013 acs, P. Milde, E. Neuber, L.M.\nEng, J.S. White, H.M. R\u001cnnow, C.D. Dewhurst,\nM. Mochizuki, K. Yanai, H. Nakamura, D. Ehlers,\nV. Tsurkan, and A. Loidl, \\N\u0013 el-type skyrmion lattice\nwith con\fned orientation in the polar magnetic semicon-\nductor gav 4s8,\" Nat. Mat. 14, 1116 (2015).\n[26] S. L. Zhang, A. Bauer, D. M. Burn, P. Milde, E. Neuber,\nL. M. Eng, H. Berger, C. P\reiderer, G. van der Laan,\nand T. Hesjedal, \\Multidomain skyrmion lattice state in\nCu2OSeO 3,\" Nano Letters 16, 3285 (2016).\n[27] Mirko Ba\u0013 cani, Miguel Marioni, J. Schwenk, and Hans\nHug, \\How to measure the local Dzyaloshinskii-Moriya\nInteraction in Skyrmion Thin Film Multilayers,\" Scien-\nti\fc Reports 9(2016), 10.1038/s41598-019-39501-x.\n[28] Raju Masapogu, Alon Yagil, Anjan Soumyanarayanan,\nAnthony K.C. Tan, Avior Almoalem, Fusheng Ma, Ophir\nAuslaender, and C. Panagopoulos, \\The evolution of\nskyrmions in Ir/Fe/Co/Pt multilayers and their topo-\nlogical hall signature,\" Nature Communications 10, 696\n(2019).\n[29] M.R. Weaver and D.W. Abraham, \\High resolution\natomic force microscopy potentiometry,\" J. Vac. Sci.\nTechnol. B 10598 , 1559 (1991).\n[30] M. Nonnenmacher, M.P. O'Boyle, and H.K. Wickramas-\ninghe, \\Kelvin probe force microscopy,\" Appl. Phys. Lett.\n58, 2921{2923 (1991).\n[31] U. Zerweck, Ch. Loppacher, T. Otto, S. Grafstr om, and\nL.M. Eng, \\Accuracy and resolution limits of Kelvin\nprobe force microscopy,\" Phys. Rev. B 71, 125424 (2005).\n[32] \\Omicron NanoTechnology Gmbh, Taunusstein, Ger-\nmany,\".\n[33] \\RHK Technology, Inc., 1050 East Maple Road, Troy,\nMI 48083 USA,\".\n[34] \\NANOSENSORS ™, Rue Jaquet-Droz 1, Case Postale\n216, CH-2002 Neuchatel, Switzerland,\".\n[35] T. Schwarze, J. Waizner, M. Garst, A. Bauer,\nI. Stasinopoulos, H. Berger, A. Rosch, C. P\reiderer, and\nD. Grundler, \\Universal helimagnon and skyrmion exci-\ntations in metallic, semiconducting and insulating chiral\nmagnets,\" Nature Mater. 14, 478 (2015).\n[36] S. V. Grigoriev, S. V. Maleyev, A. I. Okorokov, Yu. O.\nChetverikov, P. B oni, R. Georgii, D. Lamago, H. Eck-\nerlebe, and K. Pranzas, \\Magnetic structure of MnSi\nunder an applied \feld probed by polarized small-angle\nneutron scattering,\" Phys. Rev. B 74, 214414 (2006).13\n[37] Yusuke Kousaka, Naoki Ikeda, Takahiro Ogura, Toha\nYoshii, Jun Akimitsu, Kazuki Ohishi, Junichi Suzuki,\nHaruhiko Hiraka, Marina Miyagawa, Sadafumi Nishi-\nhara, Katsuya Inoue, and Junichiro Kishine, \\Chiral\nMagnetic Soliton Lattice in MnSi,\" JPS Conf. Proc. 2,\n010205 (2014).\n[38] A. Aqeel, N. Vlietstra, A. Roy, M. Mostovoy, B. J. van\nWees, and T. T. M. Palstra, \\Electrical detection ofspiral spin structures in Pt jCu2OSeO 3heterostructures,\"\nPhys. Rev. B 94, 134418 (2016).\n[39] J.-W.G. Bos, C.V. Colin, and T.T.M. Palstra, \\Magne-\ntoelectric coupling in the cubic ferrimagnet Cu 2OSeO 3,\"\nPhys. Rev. B 78, 094416 (2008).\n[40] Filipp N Rybakov, Aleksandr B Borisov, Stefan Bl ugel,\nand Nikolai S Kiselev, \\New spiral state and skyrmion\nlattice in 3D model of chiral magnets,\" New Journal of\nPhysics 18, 045002 (2016)."    },    {        "title": "2103.16136v1.Thermal_annealing_enhancement_of_Josephson_critical_currents_in_ferromagnetic_CoFeB.pdf",        "content": " 1 of 12Thermal annealing enhancement of Josephson critical currents in ferromagnetic CoFeB \n \nSachio Komori*, Juliet E. Thompson* \nGuang Yang, Graham Kimbell, Nadia Stelmashenko \nMark G. Blamire and Jason W. A. Robinson† \n \nDepartment of Materials Science & Metallurgy, University of Cambridge, \n 27 Charles Babbage Road, Cambridge CB3 0FS, United Kingdom \n \nThe electrical and structural properties of Co 40Fe40B20 (CoFeB) alloy are tunable with thermal annealing. \nThis is key in the optimization of CoFeB-based spintronic devices, where the advantageously low magnetic \ncoercivity, high spin polarization, and controllable magnetocrystalline anisotropy are utilised. So far, there \nhas been no report on superconducting devices based on CoFeB. Here, we report Nb/CoFeB/Nb Josephson \ndevices and demonstrate an enhancement of the critical current by up to 700% following thermal \nannealing due to increased structural ordering of the CoFeB. The results demonstrate that CoFeB is a \npromising material for the development of superconducting spintronic devices. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n*These authors contributed equally to this work. \n†Corresponding author: jjr33@cam.ac.uk   \n 2 of 12Following the discovery of giant magnetoresistance [1,2] and the development of the first spin-\nvalves [3,4], Co 40Fe40B20 (CoFeB) was identified as an alternative magnetic material to those that had \npreviously been employed, particularly due to its low magnetic anisotropy and low switching energy [5,6]. In \nCoFeB spintronic devices, the magnetoresistance can be optimized through a thermal annealing-induced \nstructural transition from amorphous to crystalline [7]. Specifically, for CoFeB/MgO/CoFeB magnetic tunnel \njunctions, the crystallization of CoFeB leads to high tunneling magnetoresistance, exceeding 600% at room \ntemperature [8]; neither CoFeB nor CoFe as-grown devices display comparable efficiencies. Studies of \ndiffusive (i.e. without a tunnel junction) CoFeB spin-valves have also demonstrated larger giant \nmagnetoresistance effects following annealing treatment [9].  \nAlthough the advantageous properties of CoFeB-based spintronic devices and their controllability \nthrough annealing have been well recognized in the field of spintronics, there has been no report on \nsuperconducting spintronic devices involving CoFeB. This may be partly due to the strong magnetic exchange \nenergy and the high resistivity of CoFeB [10], which should strongly quench the superconducting proximity \neffect, making it challenging to investigate the coupling of superconductivity and magnetism. Here, we report \nNb/CoFeB/Nb Josephson devices with thin (< 5 nm) CoFeB barriers and investigate the effect of thermal \nannealing on the critical current ( Ic). From measurement of the Josephson critical currents versus CoFeB \nbarrier thickness, we determine a superconducting coherence length in CoFeB of approximately 2 nm. \nAnnealing the devices at 400°C for 30 minutes in vacuum results in the increase in the critical current by as \nmuch as 700% for a CoFeB thickness of 4 nm. We associate this enhancement of the Josephson current with \nan increase in the electron mean free path lengths for charge and spin-flip scatter in CoFeB along with \nimproved transparency at the Nb/CoFeB interfaces. \nNb(300 nm)/CoFeB( dCoFeB)/ Nb(300 nm) trilayer stacks were fabricated on 5 mm × 5 mm quartz \nsubstrates by dc magnetron sputtering in an ultrahigh-vacuum chamber with a base pressure better than \n106 Pa. The sputtering targets (Co 40Fe40B20, Nb) were pre-sputtered for 20 minutes to clean their surfaces. \nThe films were grown in Ar at a pressure of 1.5 Pa at room temperature. Multiple quartz substrates were \nplaced on a rotating circular table that passed below a series of stationary magnetrons. A series of stacks \nwere prepared with different CoFeB thicknesses ( dCoFeB = 1.5 ─ 4.5 nm) between 300-nm-thick layers of Nb in \na single deposition. Layer thicknesses were controlled by adjusting the angular speed of the rotating table. \nCurrent-perpendicular-to-plane Nb(300 nm)/CoFeB( dCoFeB)/Nb(300 nm) nanopillar devices with \nsquare cross-sectional areas of approximately A = 500 × 500 nm2 were fabricated using a focused beam of \nGa ions as described elsewhere [11]. A pulse-tube cryogen-free measurement system (Cryogenic Ltd) was \nused to cool the devices down to 1.6 K. Resistivity and current-voltage I(V) characteristics of the nanopillars \nwere measured in a four-point configuration using the differential conductance mode of a Keithley 6221 AC-\ncurrent source and a 2182A nanovoltmeter. The Josephson critical current ( Ic) and normal state resistance \n(RN) of each device were determined by fitting the I(V) characteristics to the resistively shunted junction  3 of 12model V = RN(I2Ic2)0.5. Ic was modulated by applying a magnetic field ( H) parallel to the plane and \nperpendicular to the current direction. Electrical measurements were performed on nanopillars both before \nand after thermal annealing. Thermal annealing was performed at 400°C for 30 minutes in vacuum (10─5 Pa) \n─ the typical post-anneal condition [8,12] to promote crystallisation. Annealed CoFeB has lower resistivity \nthan amorphous CoFeB deposited at room temperature [13].  \nA typical Ic(H) Fraunhofer pattern for a Nb(300 nm)/CoFeB(3.5 nm)/ Nb(300 nm) nanopillar at 1.6 K \nbefore (solid curves) and after (dashed curves) thermal annealing is shown in Fig. 1(a) [see supplementary \nFig. S1 for all the Ic(H) data recorded in this study]. Ic(H) is hysteretic and the maximum values of Ic are \nobtained at non-zero applied fields (μ 0H = δ) due to the intrinsic barrier magnetization [14,15]. In Fig. 1(b), \nwe have plotted δ at 1.6 K versus dCoFeB, which shows a linear increase in  δ with dCoFeB. By fitting δ versus dCoFeB \nto δ = Ms(dCoFeB ─ ddead)/(2λ+dCoFeB) [14], we obtain a volume saturation magnetization \nof Ms = (643 ± 21) emu/cm3, and a magnetically dead layer thickness at each Nb/CoFeB interface of \nddead = (0.33 ± 0.09) nm, which is slightly thinner than those at Nb/Co ( ddead = 0.4 nm) and Nb/Fe ( ddead \n= 0.55 nm) interfaces [16]. Here, λ = 110 nm [17,18] is an estimate of the London penetration depth of \npolycrystalline Nb. Ms obtained here is smaller than the maximum bulk magnetization of \n1300 emu/cm3 [10,19], implying a reduced magnetization in thin (< 5 nm) CoFeB and, possibly, partial \noxidation or Ga implantation in nanopillars. For dCoFeB = 4.5 nm, the magnetization of CoFeB switches at \nμ0Hc < Ms(dCoFeB ─ ddead)/(2λ+dCoFeB) and hence the maximum in Ic occurs at δ ≈ μ0Hc, resulting in a spread in \nδ due to variations in Hc. A clear change in δ is not observed following thermal annealing, suggesting that the \nmagnetization of CoFeB in nanopillars is unaffected by annealing, consistent with our magnetization \nmeasurements of unpatterned films (see supplementary Fig. S2). In the inset of Fig. 1(b), we have plotted \nthe normalized magnetic field periodicity ( n) of Ic(H) versus dCoFeB, where n is determined from sinc  (nΦ/Φ0) \nwith Φ = μ 0HL(2λ+dCoFeB), L is the length of the junction perpendicular to the applied magnetic field, and Φ 0 \nis the magnetic flux quantum. For the dCoFeB range investigated, n ≈ 1, consistent with a dominant first \nharmonic current-phase relation. \nIn Fig. 2(a), we have plotted the total specific resistance of the nanopillars ( ARN) versus dCoFeB at \n1.6 K before and after thermal annealing. From a least square regression line fit ( ARN = ρCoFeB × dCoFeB + \n2ARNb/CoFeB), we estimate an as-grown CoFeB resistivity of ρCoFeB = (88 ± 46) μΩ·cm, which is higher than the \nresistivity of a Co 60Fe40 polycrystalline ferromagnetic alloy ( ρ ≈ 15 μΩ·cm at 10 K [20]). We also estimate the \nspecific resistances of the two Nb/CoFeB interfaces as 2 ARNb/CoFeB = (4.4 ± 1.4) fΩ·m2. The effective electron \nmean free path in as-grown CoFeB is l = 3π2ℏ/kF2e2ρCoFeB = (1.8 ± 0.9) nm, where ℏ is the Planck constant \ndivided by 2 , kF = 0.104 nm─1 [21] is the Fermi wave number in the majority band of CoFeB, and e is the \nelementary charge. We observe a decrease in ARN for all the devices following thermal annealing, suggesting  \na decrease in ρCoFeB (and increase in the electron mean free path) as a result of increased structural order.  4 of 12The increased degree of scatter in ARN versus dCoFeB for the nanopillars after thermal annealing is likely due \nto the variation of the resistance of CoFeB and Nb/CoFeB interfaces induced by annealing.  \n The decrease in RN through thermal annealing results in a notable enhancement of the Josephson \ncritical current density ( Jc) as shown in Fig. 2(b). The relative Jc change following thermal annealing [defined \nas (Jc,annealed ─ Jc,as-grown) / Jc,as-grown] is 80 ─ 700% depending on the CoFeB thickness [see inset of Fig. 2(b)]. \nWhilst an enhancement of Jc has been observed for all nanopillars investigated, the relative Jc change does \nnot show a clear dCoFeB dependence due to the relatively large variation in RN induced by thermal annealing. \nTo investigate the effect of thermal annealing upon the proximity coherence length of \nsuperconductivity in CoFeB, in Fig. 3(c) we have plotted the characteristic voltage ( IcRN) versus dCoFeB at 1.6 K \nbefore and after thermal annealing. In superconductor/ferromagnet/superconductor Josephson devices, IcRN \ntypically displays damped oscillatory behaviour as a function of ferromagnetic barrier thickness due to 0-π \nphase transitions [16,22–26]. In our devices, although IcRN exponentially decays with dCoFeB we do not observe \nevidence of 0-π oscillations. The apparent absence of these oscillations is likely due to a strong magnetic \nexchange energy of CoFeB ( Eex ≈ kBTCurie = 113 meV where kB is the Boltzmann constant and TCurie = 1313 K [27] \nis the Curie temperature of CoFeB), giving rise to a short oscillation period; π vFℏ/2Eex [28] ≈ 0.1 nm where \nvF = 1.2×104 m/s [21] is the Fermi velocity in CoFeB. Hence, the oscillation is smoothed out by the thickness \nvariation (roughness) of CoFeB and is undetectable. As shown in the inset of Fig. 2(c), the relative change of \nIcRN through annealing [defined as ( IcRN,annealed ─ IcRN,as-grown) / IcRN,as-grown] increases with increasing dCoFeB (i.e., \nthe decay slope of IcRN with dCoFeB becomes shallower as a result of annealing). By fitting the decay slope to \nIcRN ∝exp(─ξCoFeB/dCoFeB), the proximity coherence length of superconductivity in CoFeB ( ξCoFeB) is estimated \nto be (2.15 ± 0.10) nm and (2.41 ± 0.12) nm for the devices before and after annealing, respectively. \nConsidering the fact that the magnetic moment of CoFeB is unchanged following thermal annealing [see \nFig. 1(b)], the enhancement of ξCoFeB is likely due to the increase in the electron mean free path of CoFeB as \na result of improved structural ordering, consistent with the decrease in RN in Fig. 2(a).  \nIn conclusion, we have demonstrated Josephson coupling through CoFeB alloy and its optimization \nwith thermal annealing in Nb/CoFeB/Nb nanopillars. We have found a notable enhancement of the \nJosephson critical current up to a maximum of 700% following thermal annealing at 400°C, which is \nattributed to an increase in the proximity coherence length of superconductivity in CoFeB and improved \ntransparency at the Nb/CoFeB interfaces. The thermal optimization of Josephson coupling following thermal \nannealing is attractive for the development of energy efficient superconducting spintronic devices.  \n \n  5 of 12\n \nFIG. 1. (a) An Ic(H) pattern for a Nb(300 nm)/CoFeB(3.5 nm)/Nb(300 nm) nanopillar before (solid lines) and \nafter (dashed lines) thermal annealing at 400°C for 30 minutes.  The red (solid/dashed) line shows Ic with \nincreasing H and the blue (solid/dashed) line shows Ic with decreasing H. (b) In-plane magnetic hysteresis ( δ) \nfor Nb(300 nm)/CoFeB( dCoFeB)/Nb(300 nm) nanopillars before (black diamonds) and after (red circles) \nthermal annealing. The vertical error bars represent the statistical scatter of δ for multiple nanopillars \nmeasured on the same circuit. The black line is a least-squares regression line fit for the nanopillars before \nthermal annealing, giving a volume saturation magnetization of (643 ± 21) emu/cm3 and a magnetically dead \nlayer thickness at each Nb/CoFeB interface of (0.33 ± 0.09) nm. The inset shows the magnetic field periodicity \n(n) of Ic(H) vs. dCoFeB. All data at 1.6 K. \n  6 of 12\n \nFIG. 2. (a) ARN vs. dCoFeB before (black diamonds) and after thermal annealing (red circles). The black line \nshows a least-squares regression line fit for the nanopillars before annealing from which we estimate \nρCoFeB ≈ (88 ± 46) μΩ·cm and 2 ARNb/CoFeB = (4.4 ± 1.4) fΩ·m2. (b) Jc vs. dCoFeB before (black diamonds) and after \nthermal annealing (red circles) at 400°C for 30 minutes. The inset shows the relative change in Jc following \nthermal annealing [( Jc,annealed ─ Jc,as-grown) / Jc,as-grown] vs. dCoFeB. (c) IcRN vs. dCoFeB where the dashed lines are least-\nsquare regression line fits giving a proximity coherence length in CoFeB of ξCoFeB = (2.15 ± 0.10) nm and \n(2.41 ± 0.12) nm, for the nanopillars before and after thermal annealing, respectively. The inset shows the \nrelative change in IcRN following thermal annealing [( IcRN,annealed ─ IcRN,as-grown) / IcRN,as-grown] vs. dCoFeB. The error \nbars in ARN, Jc and IcRN represent the statistical scatter for multiple nanopillars. All data at 1.6 K.  7 of 12ACKNOWLEDGEMENTS \nThe authors acknowledge funding from the EPSRC Programme Grants (No. EP/N017242/1 and \nNo. EP/P026311/1). J.E.T. acknowledges funding from DTP EPSRC Grants (No.  EP/M508007/1 and \nEP/N509620/1). J.W.A.R. acknowledges funding from the Royal Society through a University Research \nFellowship. \n \nREFERENCES \n[1] M. N. Baibich, J. M. Broto, A. Fert, F. N. Van Dau, F. Petroff, P. Eitenne, G. Creuzet, A. Friederich, and \nJ. Chazelas, Giant magnetoresistance of (001)Fe/(001)Cr magnetic superlattices, Phys. Rev. Lett. 61, \n2472 (1988). \n[2] G. Binasch, P. Grünberg, F. Saurenbach, and W. Zinn, Enhanced magnetoresistance in layered \nmagnetic structures with antiferromagnetic interlayer exchange, Phys. Rev. B 39, 4828 (1989). \n[3] B. Dieny, V. S. Speriosu, S. S. P. Parkin, B. A. Gurney, D. R. Wilhoit, and D. Mauri, Giant \nmagnetoresistive in soft ferromagnetic multilayers, Phys. Rev. B 43, 1297 (1991). \n[4] B. Dieny, V. S. Speriosu, B. A. Gurney, S. S. P. Parkin, D. R. Wilhoit, K. P. Roche, S. Metin, D. T. \nPeterson, and S. Nadimi, Spin-valve effect in soft ferromagnetic sandwiches, J. Magn. Magn. Mater. \n93, 101 (1991). \n[5] S. Tsunashima, M. Jimbo, Y. Imada, and K. Komiya, Spin valves using amorphous magnetic layers, J. \nMagn. Magn. Mater. 165, 111 (1997). \n[6] M. Jimbo, K. Komiyama, and S. Tsunashima, Giant magnetoresistance effect and electric conduction \nin amorphous-CoFeB/Cu/Co sandwiches, J. Appl. Phys. 79, 6237 (1996). \n[7] A. Sharma, M. A. Hoffmann, P. Matthes, O. Hellwig, C. Kowol, S. E. Schulz, D. R. T. Zahn, and G. \nSalvan, Crystallization of optically thick films of Co xFe80−xB20: Evolution of optical, magneto-optical, \nand structural properties, Phys. Rev. B 101, 054438 (2020). \n[8] S. Ikeda, J. Hayakawa, Y. Ashizawa, Y. M. Lee, K. Miura, H. Hasegawa, M. Tsunoda, F. Matsukura, and \nH. Ohno, Tunnel magnetoresistance of 604% at 300 K by suppression of Ta diffusion in \nCoFeB/MgO/CoFeB pseudo-spin-valves annealed at high temperature, Appl. Phys. Lett. 93, 082508 \n(2008). \n[9] M. Jimbo, K. Komiyama, Y. Shirota, Y. Fujiwara, S. Tsunashima, and M. Matsuura, Thermal stability of \nspin valves using amorphous CoFeB, J. Magn. Magn. Mater. 165, 308 (1997). \n[10] S. U. Jen, Y. D. Yao, Y. T. Chen, J. M. Wu, C. C. Lee, T. L. Tsai, and Y. C. Chang, Magnetic and electrical \nproperties of amorphous CoFeB films, J. Appl. Phys. 99, 053701 (2006). \n[11] C. Bell, G. Burnell, D.-J. Kang, R. H. Hadfield, M. J. Kappers, and M. G. Blamire, Fabrication of \nnanoscale heterostructure devices with a focused ion beam, Nanotechnology 14, 630 (2003). \n[12] D. D. Djayaprawira, K. Tsunekawa, M. Nagai, H. Maehara, S. Yamagata, N. Watanabe, S. Yuasa, Y. \nSuzuki, and K. Ando, 230% room-temperature magnetoresistance in CoFeB/MgO/CoFeB magnetic \ntunnel junctions, Appl. Phys. Lett. 86, 092502 (2005). \n[13] S. X. Huang, T. Y. Chen, and C. L. Chien, Spin polarization of amorphous CoFeB determined by point-\ncontact Andreev reflection, Appl. Phys. Lett. 92, 242509 (2008).  8 of 12[14] M. G. Blamire, C. B. Smiet, N. Banerjee, and J. W. A. Robinson, Field modulation of the critical \ncurrent in magnetic Josephson junctions, Supercond. Sci. Technol. 26, 055017 (2013). \n[15] B. Börcsök, S. Komori, A. I. Buzdin, and J. W. A. Robinson, Fraunhofer patterns in magnetic \nJosephson junctions with non-uniform magnetic susceptibility, Sci. Rep. 9, 5616 (2019). \n[16] J. W. A. Robinson, S. Piano, G. Burnell, C. Bell, and M. G. Blamire, Zero to π transition in \nsuperconductor-ferromagnet-superconductor junctions, Phys. Rev. B 76, 094522 (2007). \n[17] E. Nazaretski, J. P. Thibodaux, I. Vekhter, L. Civale, J. D. Thompson, and R. Movshovich, Direct \nmeasurements of the penetration depth in a superconducting film using magnetic force microscopy, \nAppl. Phys. Lett. 95, 262502 (2009). \n[18] S. Komori, J. M. Devine-stoneman, K. Ohnishi, G. Yang, Z. Devizorova, S. Mironov, X. Montiel, L. A. B. \nO. Olthof, L. F. Cohen, H. Kurebayashi, M. G. Blamire, A. I. Buzdin, and J. W. A. Robinson, Spin-orbit \ncoupling suppression and singlet-state blocking of spin-triplet Cooper pairs, Sci. Adv. 7, eabe0128 \n(2021). \n[19] N. Heiman, R. D. Hempsted, and N. Kazama, Low-coercivity amorphous magnetic alloy films, J. Appl. \nPhys. 49, 5663 (1978). \n[20] J. Kim, J. H. Kwon, and K. Char, Quantitative analysis of the proximity effect in Nb/Co 60Fe40, Nb/Ni, \nand Nb/Cu 40Ni60 bilayers, Phys. Rev. B 72, 014518 (2005). \n[21] A. Useinov, O. Mryasov, and J. Kosel, Output voltage calculations in double barrier magnetic tunnel \njunctions with asymmetric voltage behavior, J. Magn. Magn. Mater. 324, 2844 (2012). \n[22] A. I. Buzdin, L. N. Bulaevskii, and S. V. Panyukov, Critical-current oscillations as a function of the \nexchange field and thickness of the ferromagnetic metal (F) in an S-F-S Josephson junction, JETP \nLett. 35, 178 (1982). \n[23] A. I. Buzdin and M. Y. Kupriyanov, Transition temperature of a superconductor-ferromagnet \nsuperlattice, JETP Lett. 52, 487 (1990). \n[24] H. Sellier, C. Baraduc, F. Lefloch, and R. Calemczuk, Temperature-induced crossover between 0 and \nπ states in S/F/S junctions, Phys. Rev. B 68, 054531 (2003). \n[25] T. Kontos, M. Aprili, J. Lesueur, F. Genêt, B. Stephanidis, and R. Boursier, Josephson junction through \na thin ferromagnetic layer: negative coupling, Phys. Rev. Lett. 89, 137007 (2002). \n[26] Y. Blum, A. Tsukernik, M. Karpovski, and A. Palevski, Oscillations of the superconducting critical \ncurrent in Nb-Cu-Ni-Cu-Nb junctions, Phys. Rev. Lett. 89, 187004 (2002). \n[27] K. Nagasaka, L. Varga, Y. Shimizu, S. Eguchi, and A. Tanaka, The temperature dependence of \nexchange anisotropy in ferromagnetic/PdPtMn bilayers, J. Appl. Phys. 87, 6433 (2000). \n[28] J. W. A. Robinson, S. Piano, G. Burnell, C. Bell, and M. G. Blamire, Critical current oscillations in \nstrong ferromagnetic junctions, Phys. Rev. Lett. 97, 177003 (2006). \n \n   9 of 12Supplementary information \nfor \nThermal annealing enhancement of Josephson critical currents in ferromagnetic CoFeB \n \nSachio Komori*, Juliet E. Thompson* \nGuang Yang, Graham Kimbell, Nadia Stelmashenko \nMark G. Blamire and Jason W. A. Robinson† \n \nDepartment of Materials Science & Metallurgy, University of Cambridge, \n 27 Charles Babbage Road, Cambridge CB3 0FS, United Kingdom \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n*These authors contributed equally to this work. \n†Corresponding author: jjr33@cam.ac.uk  \n 10 of 121. Fraunhofer patterns of Nb(300 nm)/CoFeB(1.5 ─ 4.5 nm)/Nb(300 nm) nanopillars \nIn Fig. S1, we have plotted the critical current ( Ic) vs. external in-plane magnetic field ( H) for \nNb(300 nm)/CoFeB(1.5 ─ 4.5 nm)/Nb(300 nm) nanopillars at 1.6 K before (solid lines) and after (dashed lines) \nthermal annealing at 400°C for 30 minutes in vacuum. All the data sets in the manuscript have been obtained \nfrom these 11 nanopillars prepared from a single sputtering deposition. The critical current density ( Jc), the \nnormal state resistance ( RN), the characteristic voltage ( IcRN), and the total specific resistance ( ARN) of the \ncorresponding nanopillars are summarized in Table S1. \nFIG. S1. Ic(H) patterns for Nb(300 nm)/CoFeB(1.5 ─ 4.5 nm)/Nb(300 nm) nanopillars before (solid lines) and \nafter (dashed lines) thermal annealing. The red lines show Ic with increasing H and the blue lines show Ic with \ndecreasing H. The inset shows the size of the nanopillars and the thickness of CoFeB. All data at 1.6 K. \n 11 of 12Table S1. Critical current density ( Jc), normal state resistance ( RN), characteristic voltage ( IcRN), and total \nspecific resistance ( ARN) of Nb(300 nm)/CoFeB(1.5 ─ 4.5 nm)/Nb(300 nm) devices at 1.6 K. The values after \nannealing are shown in brackets. \n dCoFeB (nm) Jc (A/m2) RN (mΩ) IcRN (μV) ARN (fΩm2) \na 1.5 3.01 × 1010 (5.50 × 1010) 45.0 (23.0) 170 (159) 5.66 (2.89) \nb 2.0 8.50 × 109 (1.07 × 1010) 29.4 (25.2) 57.4 (62.0) 6.75 (5.79) \nc 2.0 7.79 × 109 (1.06 × 1010) 20.7 (16.2) 51.6 (54.7) 6.62 (5.17) \nd 2.0 7.69 × 109 (1.29 × 1010) 17.3 (11.1) 48.4 (52.1) 6.30 (4.03) \ne 2.5 4.61 × 109 (2.94 × 1010) 28.8 (12.2) 30.9 (31.4) 6.70 (2.84) \nf 3.5 1.74 × 108 (4.35 × 109) 21.3 (11.0) 9.80 (12.6) 5.63 (2.89) \ng 3.5 1.01 × 109 (6.06 × 109) 13.2 (4.44) 6.43 (8.12) 3.98 (1.34) \nh 3.5 1.33 × 109 (4.28 × 109) 44.3 (21.3) 13.3 (20.3) 10.0 (4.75) \ni 4.0 5.05 × 108 (3.89 × 109) 26.7 (5.39) 4.36 (6.79) 8.64 (1.75) \nj 4.5 4.09 × 108 (6.95 × 109) 33.3 (10.8) 3.98 (5.78) 9.73 (3.16) \nk 4.5 3.36 × 108 (5.65 × 109) 25.0 (19.2) 2.76 (3.96) 8.22 (6.33) \n \n2. Effect of thermal annealing on the magnetization of CoFeB \nIn Fig. S2, we have plotted the in-plane volume magnetization ( M) versus H for an unpatterned \nNb(35 nm)/CoFeB(4 nm)/Nb(2 nm) control sample measured at 10 K before (black curve) and after (red \ncurve) thermal annealing at 400°C for 30 minutes in vacuum. Ms and Hc are unchanged following thermal \nannealing, suggesting that the exchange energy of CoFeB is unaffected by the amorphous-crystalline \ntransition. The relatively small Ms (≈ 643 ± 21 emu/cm3) estimated from δ [see Fig. 1(b) in the main \nmanuscript] compared with that obtained from the magnetization measurement of this unpatterned control \nsample (Ms ≈ 900 emu/cm3) may be due to the existence of partial oxidation or Ga implantation in nanopillars. \n \n \nFIG. S2. M (H) curve for an unpatterned Nb(35 nm)/CoFeB(4 nm)/Nb(2 nm) control sample measured at 10 K \nbefore (black curve) and after (red curve) thermal annealing. \n 12 of 123. Effect of thermal annealing on the superconductivity of Nb \nTo confirm that there is no significant effect of thermal annealing on the superconductivity of Nb, we \nprepared a Nb(30 nm) control sample and measured the superconducting transition temperature ( Tc) before \nand after thermal annealing at 400°C for 30 minutes in vacuum. As shown in Fig. S3, Tc is slightly suppressed \n(≈ 0.1 K) following thermal annealing, which is likely due to a slight oxidization of Nb as a result of a reaction \nwith SiO 2 substrate. A slight Tc-suppression might result in a slight suppression of the Josephson critical \ncurrents, rather than the increase in Josephson critical currents observed in this study. Also, we note that the \nTc (= 9.2 K) of the 300-nm-thick Nb electrodes of all the devices used in this study is unchanged after thermal \nannealing. Hence, the annealing enhancement of critical currents observed in this study is due to the change \nin the electrical properties of CoFeB and CoFeB/Nb interfaces.  \n \n \nFIG. S3. Temperature dependence of the normalized resistance for a 30-nm-thick unpatterned Nb \nmeasured before (black curve) and after (red curve) thermal annealing at 400°C for 30 minutes in vacuum. \n \n"    },    {        "title": "2103.16238v2.Thermodynamic_evidence_of_a_second_skyrmion_lattice_phase_and_tilted_conical_phase_in_Cu__2_0SeO__3_.pdf",        "content": "Thermodynamic evidence of a second skyrmion lattice phase and tilted conical phase\nin Cu 2OSeO 3\nM. Halder,1A. Chacon,1A. Bauer,1W. Simeth,1S. M uhlbauer,2\nH. Berger,3L. Heinen,4M. Garst,5A. Rosch,4and C. P\reiderer1\n1Physik Department, Technische Universit at M unchen, D-85748 Garching, Germany\n2Heinz Maier-Leibnitz Zentrum (MLZ), Technische Universit at M unchen, D-85748 Garching, Germany\n3\u0013Ecole Polytechnique Federale de Lausanne, CH-1015 Lausanne, Switzerland\n4Institut f ur Theoretische Physik, Universit at zu K oln, D-50937 K oln, Germany\n5Institut f ur Theoretische Physik, Technische Universit at Dresden, D-01062 Dresden, Germany\n(Dated: April 1, 2021)\nPrecision measurements of the magnetization and ac susceptibility of Cu 2OSeO 3are reported\nfor magnetic \felds along di\u000berent crystallographic directions, focussing on the border between the\nconical and the \feld-polarized state for a magnetic \feld along the h100iaxis, complemented by se-\nlected speci\fc heat data. Clear signatures of the emergence of a second skyrmion phase and a tilted\nconical phase are observed, as recently identi\fed by means of small-angle neutron scattering. The\nlow-temperature skyrmion phase displays strongly hysteretic phase boundaries, but no dissipative ef-\nfects. In contrast, the tilted conical phase is accompanied by strong dissipation and higher-harmonic\ncontributions, while the transition \felds are essentially nonhysteretic. The formation of the second\nskyrmion phase and tilted conical phase are found to be insensitive to a vanishing demagnetization\nfactor. A quantitative estimate of the temperature dependence of the magnetocrystalline anisotropy\nmay be consistently inferred from the magnetization and the upper critical \feld and agrees well with\na stabilization of the low-temperature skyrmion phase and tilted conical state by conventional cubic\nmagnetic anisotropies.\nI. INTRODUCTION\nIn the past decade, an abundance of magnetic mate-\nrials has been discovered featuring skyrmions, i.e., topo-\nlogically nontrivial spin textures.1An incomplete list of\nbulk systems includes cubic chiral magnets such as the\nB20 transition metal compounds2{5, rhombohedral lacu-\nnar spinels6,7, hexagonal M-type ferrites8, perovskites9,\nand tetragonal Heusler compounds.10The formation of\nskyrmions has also been reported in epitaxially thin\n\flms of these bulk systems, as well as nanowhiskers and\nnanoparticles.11,12Further, tailored magnetic bubbles\nfeaturing skyrmion characteristics have been detected in\na wide range of heterostructures13,14, as well as in care-\nfully selected atomically thin \flms.15\nWhile the microscopic interactions across this excep-\ntionally wide range of materials are inherently di\u000berent,\nall of the known systems were believed to exhibit a sin-\ngle temperature and \feld regime of the skyrmion lattice\norder. Thus, even though di\u000berent mechanisms may al-\nlow to stabilize skyrmions, there has been no example in\nwhich two di\u000berent mechanisms are su\u000eciently strong to\ncause the formation of skyrmion lattice order in the same\nmaterial in disconnected temperature and \feld regimes.\nA \frst example for two such independent skyrmion\nphases was recently reported in Cu 2OSeO 3.16Using\nsmall-angle neutron scattering, two disconnected phases\ncould be identi\fed driven by di\u000berent mechanisms. On\nthe one hand, Cu 2OSeO 3displays a well-known skyrmion\nphase near the onset of helimagnetic order in a small\napplied magnetic \feld.17,18In the following this state\nwill be referred to as high-temperature skyrmion (HTS)\nphase. As a key characteristic, the temperature versus\feld range of the HTS phase varies only very weakly un-\nder changes of \feld direction. Phenomenological con-\nsiderations as well as Monte-Carlo calculations establish\nthat the HTS phase is stabilized by entropic contribu-\ntions of thermal Gaussian \ructuations.2,19\nIn addition, a second skyrmion phase was identi\fed in\nCu2OSeO 3at the border between the conical phase and\nthe \feld polarized phase in the low-temperature limit.\nIn the following this state will be referred to as low-\ntemperature skyrmion (LTS) phase. Experimentally, the\nLTS phase di\u000bers from the HTS phase in two distinct\nways. First, as it stabilizes at low temperatures it does\nnot require the e\u000bects of thermal \ructuations. Second,\nit exists for magnetic \felds close to the crystallographic\nh100iaxis only but not for \felds along h110iandh111i.\nIn addition, the LTS phase is accompanied by another\nnew phase in which the propagation direction of the con-\nical state under increasing \feld strength increasingly tilts\naway from the \feld direction.\nThe magnetic phase diagram of Cu 2OSeO 3including\nthe two new phases may be fully accounted for in terms\nof the theoretical Ginzburg-Landau theory developed for\nthe class of cubic chiral magnets (cf. Refs.2, 3, 16, and\n20 and references therein). At the heart of this model is\nthe notion of a hierarchical set of energy scales, compris-\ning in decreasing strength of the ferromagnetic exchange,\nDzyaloshinsky-Moriya spin-orbit terms, dipolar inter-\nactions, and higher order crystal-\feld contributions21.\nEven though Cu 2OSeO 3exhibits ferri- instead of fer-\nromagnetic order with a long-wavelength modulation\ndriven by DM interactions, this Ginzburg-Landau model\nappears to be in excellent agreement with experiment.\nIndeed, taking into account conventional cubic magneticarXiv:2103.16238v2  [cond-mat.str-el]  31 Mar 20212\nanisotropies of a fairly strong h100ieasy magnetic axis,\nthe LTS and tilted conical phase may be explained in ex-\ncellent qualitative and semi-quantitative agreement with\nexperiment.16\nThe search for several topologically non-trivial phases\nin the cubic chiral magnets has a long history. For in-\nstance, data recorded in FeGe were interpreted in terms\nof complex mesophases.22{24In contrast, comprehensive\nmeasurements in MnSi unambiguously established a sin-\ngle skyrmion phase.25This motivated to revisit FeGe,\nwhere an analysis using the same set of criteria as estab-\nlished in MnSi provided a phase diagram identical to that\nobserved in MnSi.20Further, a splitting of the skyrmion\nphase reported in Cu 2OSeO 3under Zn doping could be\nattributed to chemical phase segregation.26,27\nA strong dependence on the temperature and \feld his-\ntory of the HTS phase was found in cubic chiral magnets\nsubject to disorder, notably Fe 1\u0000xCoxSi3,28and the se-\nries of Co-Zn-Mn alloys.29,30These have been exploited\nto obtain detailed information on the processes of nucle-\nation and decay of skyrmions.31,32Following the discov-\nery of two phases in Cu 2OSeO 3, a similar observation\nwas reported in Co 7Zn7Mn6with a high-temperature\nphase as observed in Cu 2OSeO 3and the B20 systems\nand a three-dimensionally disordered skyrmion phase at\nlow temperatures.33However, in contrast to Cu 2OSeO 3,\nthe stability of the low-temperature skyrmion phase is\nattributed to the interplay of DM interactions with the\ne\u000bects of frustration as opposed to magnetocrystalline\nanisotropies.\nAs the observations of the LTS and tilted conical\nphases in Cu 2OSeO 3are so far based on small-angle neu-\ntron scattering (SANS), several pressing questions exist\nconcerning the thermodynamic signatures and the sen-\nsitivity to the temperature and \feld history addressed\nin this paper. Beginning with an account of the experi-\nmental methods in section II, our paper proceeds in sec-\ntions A and B with an account of the designations of\nthe phase transitions and the magnetic phase diagrams,\nrespectively. Typical magnetization and ac susceptibil-\nity data as recorded under zero-\feld cooling (ZFC) and\nhigh-\feld cooling (HFC) are presented in sections C.1\nand C.2, respectively. This is followed by a microscopic\njusti\fcation of the transition \felds in section C.3, where\nthe intensity patterns observed in neutron scattering are\ncompared with the magnetization and susceptibility. Ev-\nidence underscoring the presence of dissipation and hys-\nteresis is presented in section D followed by information\non di\u000berent sample shapes and thus demagnetizing \felds\nin section E, which are compared to calculations in a\nGinzburg-Landau model. Finally, a comparison of the\nmagnetic \feld dependence of the speci\fc heat and mag-\nnetization at low temperatures is presented in section F.\nThe discussion of the experimental data in section IV\nbegins with an account of the anisotropies observed for\ndi\u000berent crystallographic orientations in section A. The\nsecond part of the discussion presented in section B ad-\ndresses the temperature dependence of the anisotropy en-TABLE I. Designation, \feld direction, dimensions ( a,b, and\nc), and demagnetization factor Nof the Cu 2OSeO 3single-\ncrystal samples investigated as part of this study. The mag-\nnetic \feld was applied along the direction denoted a. The\nvalue ofNcorresponds to this \feld direction. Samples are\nlisted twice when measured for di\u000berent directions of the \feld\nas stated in the table.\nsample Bka\u0002b\u0002c(mm3)N data in Figs.\nVTG1-19h100i1:87\u00021:28\u00021:53 0.28 1 to 10,12\nVTG1-20h110i1:76\u00021:27\u00021:74 0.29 1,2,3,5,8,9,10,12\nVTG1-20h111i1:74\u00021:27\u00021:76 0.30 1,2,3,8,9,10,12\nVTG1-10-1h100i3:00\u00020:50\u00020:50 0.07 6 and 7\nVTG1-18-2h100i1:85\u00020:86\u00020:78 0.18 6 and 7\nVTG1-18-2h100i0:86\u00021:85\u00020:78 0.39 6 and 7\nVTG1-18-3h100i0:15\u00020:85\u00021:88 0.77 6 and 7\nergy. The \feld dependence of the magnetic anisotropy as\nre\rected in the magnetization is \fnally discussed in sec-\ntion C. The paper closes with a brief summary in section\nV.\nII. EXPERIMENTAL METHODS\nSeveral Cu 2OSeO 3single crystals were prepared from\ningots grown by vapor transport. Numerous studies on\nsamples from the same ingots have been reported in the\nliterature.32,34{38This concerns in particular the small-\nangle neutron scattering reported in Refs.16,18as sum-\nmarized below. All single crystals investigated here dis-\nplayed the same high sample quality in terms of their\noptical appearance, the crystallinity as inferred from the\nlattice mosaic spread and sharpness of di\u000braction spots\nobserved in x-ray and neutron di\u000braction, as well as\ntheir physical properties, for instance, the value of the\nparamagnetic-to-helimagnetic transition temperature. In\nour study we focussed on \fve specimens, all of which were\nof cuboid shape. The designations, dimensions, orien-\ntations, and demagnetizing factors of these samples are\nsummarised in table I.\nEssentially all data for the \feld along h100ireported in\nthis paper were recorded in sample VTG1-19, which was\nclosest to a cubic shape with a demagnetization factor\nclose to\u00181=3. This way the sample shape and the asso-\nciated demagnetization e\u000bects were close to the spherical\nsample studied in SANS16making the di\u000berences of the\ncorrections of demagnetizing \felds tiny. For the same\nreasons data for di\u000berent crystallographic orientations\nwere recorded in sample VTG1-20, which was also close\nto cubic. All other samples served to study the e\u000bects\nof di\u000berent demagnetization factors under a \feld along\nh100i.\nThe magnetization, ac susceptibility, and speci\fc heat\nwere measured in a Quantum Design physical proper-\nties measurement system (PPMS). The magnetization\nwas determined with an extraction technique. All ac3\nsusceptibility measurements were performed at an exci-\ntation frequency f= 911 Hz and an excitation \feld of\nHac= 0:1 mT. It is important to note that the suscep-\ntibilities presented in our paper essentially represent the\ncoe\u000ecients of a Fourier expansion in response to the exci-\ntation frequency. Technically speaking, these coe\u000ecients\nare typically referred to as harmonic susceptibilities. In\ncomparison, expanding the response to a magnetic \feld\nin a Taylor series, the expansion coe\u000ecients are referred\nto as nonlinear susceptibilities. In principle, it is possible\nto convert the harmonic into the nonlinear susceptibilities\nas discussed in Sec.D and Ref. 39. For the comparison of\nthe magnetization and ac susceptibility with small-angle\nneutron scattering a data set was recorded at the beam-\nline SANS-140at MLZ following the technical procedure\ndescribed in Ref. 16.\nThe experimental data are either shown as a func-\ntion of applied \feld, without correction of demagnetiz-\ning e\u000bects, or as a function of internal \feld, where the\ne\u000bects of demagnetizing \felds were corrected by means\nof an analytical expression as reported in Ref. 41 taking\ninto account the cuboid sample shape. For those \fg-\nures in which data are shown as a function of internal\n\feld the susceptibility calculated from the magnetiza-\ntion, dM=dH, as well as the real and imaginary parts\nof the ac susceptibility, \u001f0and\u001f00, are consistently pre-\nsented with respect to the internal \feld. That is, d M=dH\nwas computed with respect to the internal \feld, whereas\nmeasured values of \u001f0and\u001f00were converted following\nconvention (for details see e.g. Ref.42).\nThe heat capacity was measured by means of a con-\nventional heat pulse method using the standard PPMS\nset up. Data were recorded in a sequence of increasing\n\feld values, where typical heat pulses generated a tem-\nperature increase of \u00181 %. All speci\fc heat data shown\nin this paper represent the average over 15 measurements\nrecorded at the same unchanged applied \feld and refer-\nence temperature.\nFor the scienti\fc questions addressed in our study, the\ntemperature and \feld history prove to be very important.\nThe protocols used in our study correspond to those used\nin the small-angle neutron scattering study reported in\nRef. 16. However, whereas data in the SANS study were\ndominantly recorded as a function of temperature, es-\nsentially all of the data reported here were recorded as a\nfunction of \feld.\nFocusing on the formation of two new phases at low\ntemperatures, we have con\frmed that our experimental\ndata in Cu 2OSeO 3may be classi\fed in terms of data\nrecorded under increasing and decreasing magnitude of\nthe magnetic \feld denoted \\up\" and \\down\", respec-\ntively. In addition, we distinguish two di\u000berent tem-\nperature versus \feld protocols, namely zero-\feld cooling\n(ZFC) and high-\feld cooling (HFC). For ZFC, the sam-\nple was cooled from a starting temperature well above Tc\nin zero magnetic \feld with a cooling rate of \u001810 K min\u00001\nto the desired temperature of the \feld sweep. It is im-\nportant to note that for the pristine helical state createdthis way the domain populations for di\u000berent h100idi-\nrections are the same. Subsequently, data were recorded\nwhile the magnitude of the magnetic \feld was increased\nin steps of 0 :5 mT.\nFor HFC, the sample was cooled from a starting tem-\nperature well above Tcin a large negative applied \feld\nof\u0000140 mT, i.e., below \u0000Hc2, with a cooling rate \u0018\n10 K min\u00001until the desired temperature of the \feld\nsweep was reached. Subsequently, data were recorded\nin a single \feld sweep in which the magnetic \feld was\nstepped from the \feld value at which the sample was\ncooled up to positive values exceeding + Hc2. Note that\nthis \feld sweep comprises data under decreasing magni-\ntude of the magnetic \feld (down) in the range \u0000Hc2<\nH < 0 and data under increasing magnitude of the mag-\nnetic \feld (up) in the range 0 <H < +Hc2, respectively.\nFor \felds exceeding Hc1, we \fnd accurately the same be-\nhavior under increasing magnitude of the magnetic \feld\nno matter whether the sweep was initiated following ZFC\nor HFC. Discrepancies in the range 0 <H\u0014+Hc1may\nbe attributed to di\u000berent domain populations in the he-\nlical state, notably for \feld along h100i, i.e., the easy\nmagnetic axis, only domains along the \feld direction are\npopulated after HFC. In the following, such di\u000berences\nbelowHc1will be pointed out where relevant.\nIII. RESULTS\nThe presentation of the experimental results begins in\nsection A with a summary of the designations and ter-\nminology used to describe the transitions of the di\u000berent\nmagnetic phases. Next, the magnetic phase diagrams are\nsummarized in section B before typical data used to infer\nthese phase diagrams are presented in section C. Higher-\nharmonic contributions in the ac susceptibility, provid-\ning \frst hand information on the presence and nature of\nhysteresis, are reported in section D. The presentation of\nthe experimental results continues in section E with data\nfor di\u000berent sample shapes and demagnetizing \felds and\ncloses in section F, where typical speci\fc heat data are\nshown.\nA. Designations and terminology\nThe temperature versus \feld diagram of Cu 2OSeO 3as\ndetermined in SANS displays \fve di\u000berent noncollinear\nmagnetic phases. In addition, the paramagnetic state at\nhigh temperatures and low \felds and the \feld-polarized\n(ferromagnetic) state at low temperatures and high \felds\nmay be distinguished. This implies considerable com-\nplexities when inferring the magnetic phase diagram from\nbulk properties such as the magnetization, ac suscepti-\nbility and speci\fc heat. In turn, it proves to be helpful\nto start with the designations and terminology used to\naddress the phase boundaries before presenting the ex-\nperimental data.4\nTABLE II. De\fnitions of the transition \felds between the\ndi\u000berent phases observed in our study. The labels \\u\" and \\d\"\ndenote increasing (up) and decreasing (down) direction of the\n\feld sweep, respectively. The subscripts \\1\" and \\2\" denote\nvalues at lower and higher absolute \feld values, respectively.\n\felds sweep dir. phase boundary\nHu\nc1,Hd\nc1up, down helical to conical\nHu\nc2,Hd\nc2up, down conical to \feld-polarized\nHu\na1,Hd\na1up, down high-temp. skyrmion, low \feld\nHu\na2,Hd\na2up, down high-temp. skyrmion, high \feld\nHu\ns1,Hd\ns1up, down low-temp. skyrmion, low \feld\nHu\ns2,Hd\ns2up, down low-temp. skyrmion, high \feld\nHu\nt1,Hd\nt1up, down tilted conical, low \feld\nHu\nt2,Hd\nt2up, down tilted conical, high \feld\nAmong the \fve noncollinear magnetic phases three are\nwell known, namely, (i) the helical order, (ii) the conical\norder, and (iii) the high-temperature skyrmion (HTS)\nphase at the border between the paramagnetic phase\nand the conical state. Recent SANS studies identi\fed\nin addition (iv) the tilted conical (TC) state and (v) a\nlow-temperature skyrmion (LTS) phase at the border be-\ntween the conical and \feld-polarized state.\nAccordingly, we distinguish in the following eight tran-\nsition \felds as summarized in Table II. The boundaries\nbetween the helical and the conical, and the conical and\n\feld-polarized phases are denoted Hc1andHc2, respec-\ntively. The boundaries of the high-temperature skyrmion\nphase are denoted Ha1andHa2. Likewise, the boundaries\nof the low-temperature skyrmion phase and the tilted\nconical phase, are denoted Hs1/Hs2andHt1/Ht2, re-\nspectively. Values determined under increasing (up) and\ndecreasing (down) magnetic \feld are labeled by the su-\nperscript \\u\" and \\d\", respectively. We note that for the\nhigh-temperature and low-temperature skyrmion phases\nas well as the tilted conical phase the subscripts 1 or 2 de-\nnote boundaries at lower and higher absolute \feld values,\nrespectively.\nB. Magnetic phase diagrams\nShown in Fig. 1 are the magnetic phase diagrams for\n\feld along the major crystallographic directions, namely\nh111i,h110i, andh100i. All diagrams are shown as a\nfunction of internal magnetic \feld, following correction of\ndemagnetizing \felds. The phase boundaries, as marked\nhere by circles, were inferred from the susceptibility data,\nwhere the de\fnitions for the transitions are given below.\nThe color coding shown in the background of the phase\ndiagrams re\rects the size of the susceptibility, d M=dH,\ncalculated from the magnetization. The temperature ver-\nsus \feld protocols used, namely zero-\feld cooled andhigh-\feld cooled, are stated in each panel, where the di-\nrection of the \feld sweeps is marked by an arrow.\nFor magnetic \felds along h111iandh110i, shown in\nFigs. 1(a) and 1(b), the behavior reported in the liter-\nature is observed.18Here the phase diagrams comprise\nhelical (H), conical (C), and HTS lattice order. With de-\ncreasing temperature both Hc1andHc2increase mono-\ntonically. Only the ZFC phase diagrams are shown, since\nHc1andHc2exhibit very little and no hysteresis between\nincreasing and decreasing \feld, respectively. It is also\nhelpful to note that the magnetization displays a clear\nsignature of the transition from the conical to the helical\nstate under decreasing \feld (not shown). Taken together,\nthe magnetic phase diagrams display all the characteris-\ntics that are well known from stoichiometric cubic chiral\nmagnets, such as MnSi and FeGe.20\nThe phase diagrams for a \feld parallel to h100i, as\nshown in Figs. 1(c) and 1(d), display the following sim-\nilarities withh111iandh110i. The HTS phase emerges\nin a temperature and \feld range, that varies somewhat\nwith crystallographic orientation.18Further, the phase\nboundary between the conical and \feld-polarized state\natHc2increases with decreasing temperature down to\nabout 40 K. Also, under ZFC, the phase boundary be-\ntween the helical and the conical state at Hc1increases,\nalbeit the absolute value is smaller than for a \feld along\nh111iandh110i.\nIn contrast to the behavior for \feld parallel to h111i\nandh110i,Hc2for a \feld parallel to h100ireaches the\nhighest value of\u001840 mT around 40 K followed by a gen-\ntle decrease. Along with this decrease in Hc2, clear sig-\nnatures in the magnetization of two new phases, iden-\nti\fed microscopically in neutron scattering16, may be\ndistinguished. Namely, below \u001830 K the tilted coni-\ncal (TC) phase emerges. This is followed by the LTS\nphase, which emerges below \u001815 K. The absence of the\ntilted conical phase and LTS phase for \feld directions\nother thanh100iprovides compelling evidence that mag-\nnetocrystalline anisotropies must play a decisive role in\ntheir formation.\nWe note that for all temperature versus \feld protocols,\nour data are consistent with the tilted conical state ap-\npearing \frst, followed by the LTS phase as described in\nthe SANS study reported in Ref. 16. Moreover, once the\ntilted conical state and LTS phase have formed, the tilted\nconical phase disappears before the LTS phase. The pre-\ncise temperature and \feld range are thereby subject to\nthe temperature versus \feld protocol. Namely, under in-\ncreasing \felds [positive \feld values in Figs. 1(c), 1(d)]\nthe LTS phase extents to larger \feld values, whereas for\ndecreasing \felds [negative \feld values in Fig. 1(d)] it per-\nsists down to lower magnetic \feld values.\nAnother di\u000berence of \feld along the h100idirection as\ncompared withh111iandh110iconcerns, \fnally, the con-\nical to helical transition at Hc1. While a clear signature\nis observed after ZFC under increasing \feld, no evidence\nsuggesting the existence of Hc1is observed under HFC.\nThis originates in di\u000berences of domain populations as5\nHTS\nHTSHTSHTSHTS\nLTS\nLTS\nLTSzero-field cooledzero-field cooledzero-field cooled\nhigh-field cooled\nFIG. 1. Magnetic phase diagrams inferred from magneti-\nzation and susceptibility data measured in \feld sweeps after\nzero-\feld cooling (a){(c) and high-\feld cooling (d) with \felds\nalongh111i(a),h110i(b), andh100i(c), (d). The following\nphases may be distinguished: helical (H), conical (C), \feld po-\nlarized (FP), tilted conical (TC), high-temperature skyrmion\n(HTS) and low-temperature skyrmion phase (LTS).\nexplained above and was previously observed in other cu-\nbic chiral magnets.28,43Namely, in this con\fguration, one\neasy axis is parallel to the magnetic \feld while the other\neasy axes are perpendicular to the \feld direction. There-\nfore, when the magnetic \feld is reduced from the conical\nto the helical state, only the domain population parallelto the \feld is populated leading to a single domain helical\nstate that is indistinguishable from the conical state.\nC. Experimental data\nThe presentation of the experimental data is organized\nin three parts. First, typical magnetization and ac sus-\nceptibility data for di\u000berent \feld orientations recorded\nunder ZFC will be reported in section C.1. This serves\nto illustrate the key features and associated de\fnitions of\nthe di\u000berent phase transitions. It is followed by related\ndata recorded under HFC in section C.2, emphasizing\nthe role of the temperature versus \feld protocol. Finally,\nselected susceptibility data are compared with neutron\nscattering data to justify the interpretation and de\fni-\ntions of the transition \felds in section C.3.\nC.1. Magnetization and susceptibility under ZFC\nShown in Figs. 2 (a){2 (d) are typical data of the mag-\nnetization, M, the susceptibility calculated from the\nmagnetization, d M=dH, as well as the real and imag-\ninary parts of the ac susceptibility, \u001f0and\u001f00, respec-\ntively. All quantities are determined with respect to the\ninternal \feld as described above. Note that the real part\nof the susceptibility is presented on a logarithmic scale,\nwhereas the imaginary part is presented on a linear scale.\nData are shown as a function of internal magnetic \feld,\nHint, across the high-temperature skyrmion phase for a\nhigh temperature of 57.5 K after ZFC, where the heli-\nmagnetic transition temperature is Tc= 58:5 K. For\n\felds alongh111i,h110i, andh100idata are presented\nin blue, green, and red, respectively. In perfect agree-\nment with the literature, the magnetization increases at\nlow \felds quasilinearly, with a distinct change of slope\natHc2. Small changes of slope at low and intermediate\n\felds are related to the helical-to-conical transition and\nthe HTS phase, respectively.\nThe detailed features associated with the transitions\nbetween the di\u000berent phases may be seen in d M=dH\nshown in Figs. 2 (b){2 (d), depicted as discrete data\npoints. Related features are also re\rected in the ac\nsusceptibility, \u001f0, shown as a line. With increasing\n\feld, three distinct maxima may be distinguished corre-\nsponding to the helical-to-conical transition at Hu\nc1, the\nconical-to-skyrmion transition at Hu\na1, and the skyrmion-\nto-conical transition at Hu\na2. The change of slope at high\n\felds, denoted as Hu\nc2, marks the transition between the\nconical and the \feld-polarized state.\nQuantitatively, the lowest value of Hu\nc2is observed\nforHkh100i, whereas the largest value is observed for\nHkh111i. In comparison with the conical state, the sus-\nceptibility is reduced both in the helical and the HTS\nphase. Further, the real part of the ac susceptibility, \u001f0,\ntracks dM=dHexcept for the transition regions around\nHu\nc1,Hu\na1, andHu\na2, suggesting the presence of reorien-6\nFIG. 2. Typical magnetization and susceptibility data as a\nfunction of internal \feld following zero-\feld cooling with \felds\naligned alongh111i(blue),h110i(green), andh100i(red).\nData were recorded at T= 57:5 K (left column) and T= 2 K\n(right column). [(a) and (e)] Magnetization data. [(b){(d)\nand (f){(h)] Susceptibility calculated from the magnetization,\ndM=dH(symbols), the real part of ac susceptibility \u001f0(black\nline) and the imaginary part of ac susceptibility \u001f00(colored\nline) for a \feld along h111i[second row: (b), (f)], \feld along\nh110i[third row: (c), (g)] and a \feld along h100i[fourth row:\n(d), (h)].\ntations of long-range textures and viscous processes oc-\ncurring on time-scales that are slow as compared with\nthe oscillation period of the ac \felds.25,28The underlying\ndissipative processes may account also for the presence of\nnonvanishing contributions to the imaginary part of the\nac susceptibility, \u001f00, atHu\na1andHu\na2.\nWith decreasing temperature, the well-known signa-\ntures of the HTS phase disappear, before the character-\nistics shown for 2 K in Figs. 2 (e){2 (h) emerge. For mag-\nnetic \feld alongh111iandh110itwo transitions may be\ndistinguished, which may be attributed to the helical to\nconical transition at Hu\nc1and the conical to \feld-polarized\ntransition at Hu\nc2. In contrast, for h100i, the \feld depen-dence is qualitatively di\u000berent and four transition \felds\nmay be discerned in the magnetization and ac suscepti-\nbility. As shown below, comparison with SANS identi\fes\nthese as the helical to conical transition at Hu\nc1, the tran-\nsitions of the tilted conical state at Hu\nt1andHu\nt2, and the\nhigh \feld transition of the LTS phase at Hu\ns2. As the sig-\nnatures of the onset of the tilted conical state are rather\npronounced, it is not possible to identify a well-de\fned\nsignature of the onset of the LTS phase at Hu\ns1.\nFor magnetic \feld between Hu\nc1andHu\nc2distinctly\nanisotropic behavior may be observed between two cross-\ning points in the \feld-dependence of the magnetization.\nNamely, at low \felds the magnetization for all three di-\nrections is at \frst essentially identical. As a side note we\nremark that the signature in Mseen forh100iatHu\nc1is\ntiny. Above Hu\nc1the magnetization for \feld along h100i\n(red curve) is slightly larger. This situation changes at\nHu\nc1forh111i, where a pronounced jump in the magne-\ntization forh111i(blue curve) de\fnes the \frst crossing\npoint. For \felds between this crossing point and the on-\nset of the tilted conical phase at Hu\nt1, the magnetization\nforh111i(blue curve) is largest. At Hu\nt1the magneti-\nzation for \feld along h100i(red curve) displays a jump\nthat de\fnes the second crossing point. For \feld values\nabove this second crossing point the magnetization for\n\feld alongh100i(red curve) is again largest.\nThus, between the two crossing points the easy mag-\nnetic axis appears to have changed from h100itoh111i.\nInterestingly, closer inspection of the crossing points re-\nveals that the magnetization is not exactly identical\nfor all three directions. However, when calculating the\nmagnetic work W(M) considered below, a sharp cross-\ning point is observed for all three directions. Thus, at\nthe crossing points, the system appears to be perfectly\nisotropic. We return to an account of the mechanism\ndriving all of this behavior in Sec. IV.\nFurther, for a \feld along along h111iandh110ithe\njump in the magnetization at the helical-to-conical tran-\nsition atHc1results in a clear peak in d M=dH. This\npeak is not seen in the real part of the ac susceptibil-\nity\u001f0, suggesting slow domain reorientations as observed\nin previous studies of MnSi and Fe 1\u0000xCoxSi25,28,43and\nviscous relaxation. Consistent with this evidence for dis-\nsipative processes, we \fnd a small amount of hysteresis\ninHc1. In contrast, at Hc2we \fnd excellent agreement\nbetween dM=dHand the real part of the ac susceptibil-\nity\u001f0. Also, there is essentially no hysteresis at Hc2(cf.\nheat capacity shown below).\nOn a related note, at low temperatures, the imaginary\npart of the ac susceptibility, \u001f00, displays a tiny contri-\nbution atHc1for \felds alongh111iandh110ionly. In\nstark contrast, we \fnd a pronounced contribution in \u001f00\nfor a \feld alongh100iin the regime of the tilted conical\nphase. Comparison with the SANS data, presented be-\nlow, reveals that Hu\nt1andHu\nt2correspond with the points\nof in\rection in \u001f00. We note that Hu\nt2is always smaller\nthanHu\ns2. Moreover, in contrast to the high-temperature\nskyrmion phase, where Ha1andHa2[cf. Figs. 2 (b){2 (d)]7\nare accompanied by strong dissipation processes causing\na large \fnite value of \u001f00, no such dissipation is observed\nfor the LTS phase in the frequency range studied.\nThe evidence for dissipation may re\rect the underly-\ning microscopic processes characteristic of the di\u000berent\nmagnetic phases and/or their creation. As concerns the\ntilted conical phase, a change of \feld strength results in a\nchange of propagation direction. In turn, it seems plau-\nsible that the strong dissipation within the tilted conical\nphases arises from changes of propagation direction with\nac \feld. In contrast, the dissipation at the boundary of\nthe HTS phase re\rects processes of creation and decay\nof skyrmions, which originate in the creation and motion\nof Bloch points.31,32The absence of such dissipation at\nthe phase boundaries of the LTS may be interpreted as\nputative evidence for a large energy barrier that inhibits\nthe creation and destruction of the skyrmions by means\nof the ac \feld. This might also hint at a di\u000berent nu-\ncleation path of the LTS phase as compared to the HTS\nphase, notably by virtue of the tilted conical phase as an\nintermediate step.\nMore generally, the absence or presence of dissipation\nmight be used to distinguish between two di\u000berent sce-\nnarios for \frst-order transitions. The presence of dissi-\npation indicates the coexistence of two phases and slow\ndynamics of phase boundaries driven by oscillating \felds.\nThe absence of dissipation is expected both for second-\nand for those \frst-order transitions, where phase coexis-\ntence is suppressed and an oscillating \feld is not able to\ninduce oscillations in the volume occupied by each phase.\nThis may occur if there is a transition from a high-energy\nmetastable state A to a state B with lower free energy.\nIn this case, only transitions from A to B but not from\nB to A are induced. Thus, an oscillating \feld will not\ninduce oscillations between the two phases. This picture\nexplains naturally the absence of dissipative e\u000bects when\nleaving the low-temperature skyrmion phase, which is ex-\npected to be metastable due to its topological protection.\nC.2. Magnetization and susceptibility under HFC\nShown in Figs. 3(a){3(d) are the magnetization and the\nsusceptibilities under HFC for di\u000berent crystallographic\ndirections as recorded at a temperature of 2 K. Again\nthe real part of the susceptibility is presented on a log-\narithmic scale, whereas the imaginary part is presented\non a linear scale. Data are presented for \feld sweeps\nfrom negative to positive \feld values as indicated by the\nblack arrows. Thus, data recorded at negative \felds (left\nhand side) were determined under decreasing absolute\n\feld strength (down) starting at H <\u0000Hc2, whereas\ndata observed at positive \felds (right hand side) were\nrecorded under increasing absolute \feld strength (up)\nstarting e\u000bectively at H= 0. As emphasized above, we\nhave con\frmed in a large number of careful tests, that\nthe data observed for positive \feld values (the right-hand\nside of Fig. 3), which start at H= 0, with the exceptionof the signature of Hc1, agrees very well with data ob-\nserved under ZFC, shown in Fig. 2.\nhigh-field cooled\nFIG. 3. Typical magnetization and susceptibility data at\nT= 2 K following high-\feld cooling. Data were recorded in\na \feld sweep from left to right. Data are shown as a func-\ntion of internal \feld along h111i(blue),h110i(green), and\nh100i(red). (a) Magnetization as a function of magnetic \feld\nunder HFC. (b){(d) Susceptibility calculated from the mag-\nnetization, d M=dH(dots), the real part of ac susceptibility\n\u001f0(black line), and the imaginary part of ac susceptibility \u001f00\n(colored line) for \felds along h111i,h110i, andh100i.\nIt is helpful again to begin with an account of the data\nfor \felds alongh111iandh110i, shown in Figs. 3(a){3(c),\nwhich correspond to the well-known properties of cubic\nchiral magnets reported in the literature. Essentially the\nsame qualitative \feld dependence is observed under ZFC\nand HFC, where the magnetization as a function of \feld\nis almost point symmetric with respect to H= 0 and\nM= 0. Starting at a large negative \feld and reducing\nthe \feld strength, a transition from the \feld-polarized\nstate to the conical state at Hd\nc2is followed by the tran-\nsition to the helical state at Hd\nc1. Increasing the \feld\nthrough zero towards positive \feld values, the system\ntransitions from the helical into the conical state at Hu\nc18\nand to the \feld-polarized state at Hu\nc2. The susceptibili-\nties associated with the di\u000berent phases display a plateau\nin the conical phase, a reduced value in the helical state,\nand a rapid decrease when entering the \feld-polarized\nregime. Concerning the evidence of magnetic anisotropy,\nwe note that Hc1is quantitatively almost identical for\nh111iandh110i, whereasHc2is highest for \feld parallel\nh111iand visibly smaller for \feld parallel h110i.\nIn comparison to \felds along h111iandh110i, the mag-\nnetization as a function of \feld for a \feld along h100i\ndisplays a sizable asymmetry with respect to H= 0 and\nM= 0. Data recorded under increasing \felds (positive\n\feld values) are equivalent to the data recorded under\nZFC data with the exception of the tiny signature of Hu\nc1\nmarked in Fig. 2(h). As explained above, this re\rects dif-\nferences in domain population arising from the di\u000berent\ntemperature versus \feld histories.\nStarting at large negative \felds as a function of de-\ncreasing \feld, the system undergoes a transition from the\n\feld-polarized to the tilted conical phase at Hd\nt2and exits\nthe tilted conical phase at Hd\nt1. The transition from the\nLTS phase to the conical state takes place at Hd\ns1much\nbelowHd\nt1. Continuing this \feld sweep through zero, the\ntransition from the conical to the tilted conical phase is\nobserved at Hu\nt1, which vanishes at Hu\nt2, followed by the\ntransition of the LTS phase to the \feld-polarized state at\nHu\ns2.\nA key result of our study concerns the detailed behav-\nior and \feld range of the tilted conical and LTS phases.\nCorrecting the e\u000bects of demagnetizing \felds, we \fnd\nno hysteresis for the transition \felds Ht1andHt2of\nthe tilted conical phase between increasing (u) and de-\ncreasing (d) \feld strength. In contrast, under decreasing\nmagnetic \felds (negative \feld values), the LTS phase ap-\npears to emerge essentially together with the tilted con-\nical phase around Hd\nt2. However, when decreasing the\n\feld further the LTS phase survives down to Hd\ns1, i.e.,\nwell below the regime of the tilted conical state vanish-\ning atHd\nt1. Strictly speaking, we infer the formation of\nthe LTS phase from the observation of Hd\ns1andHu\ns2in\ncombination with what is known from the SANS study.\nInterestingly, the small maximum of d M=dHatHd\ns1\nis related to a change of the magnetization, where the\nmagnitude of Mis larger in the presence of the low-\ntemperature skyrmion phase (cf just below Hd\ns1), while\nMcollapses onto the value of the magnetization observed\nfor the other \feld directions just below Hd\ns1. Regardless\nwhether the data are recorded under increasing or de-\ncreasing \feld, the tilted conical phase always emerges\n\frst while the low temperature skyrmion phase vanishes\nlast.\nConcerning the agreement and discrepancies between\ndM=dHand\u001f0under HFC, we \fnd essentially the same\nproperties observed under ZFC. Namely, analogous to the\nbehavior at Hu\ns2described above, where \u001f0does not track\ndM=dHand\u001f00remains vanishingly small, \u001f0does not\ntrack dM=dHatHd\ns1either, and there is also no evidence\nfor dissipation in the imaginary part of the ac susceptibil-ity. This indicates potentially important di\u000berences re-\ngarding the process of nucleation of the low-temperature\nskyrmion phase as compared with the high-temperature\nskyrmion phase that are beyond the scope of our study.\nC.3. Comparison with small-angle neutron scattering\nThe de\fnitions of the transition \felds introduced\nabove may be justi\fed by means of a direct compari-\nson with the magnetic \feld dependence of the scatter-\ning intensities observed in small-angle neutron scatter-\ning shown in Fig. 4. For the comparison a data set\nwas recorded under the same conditions reported in\nRef. 16. We note that the intensities shown here repre-\nsent peak intensities and not an integration over rocking\nscans. Therefore quantitative comparison between di\u000ber-\nent phases is not possible. For further technical details,\nwe refer to Ref. 16.\nGreat care was exercised in order to avoid systematic\nerrors. First, the neutron scattering data were recorded\nfor a spherical Cu 2OSeO 3sample prepared from the same\nbatch of material used to prepare the samples for the\nmagnetization and ac susceptibility measurements re-\nported here. Second, the magnetization and suscepti-\nbility data were recorded in a sample of almost cubic\nshape as listed in Table I. This way, the demagnetization\nfactors were similar and di\u000berences of corrections tiny.\nIn turn, data are shown as a function of internal \feld,\nwhere the demagnetization correction for the SANS data\nwas calculated from the magnetization data measured on\na di\u000berent sample. Third, the same ZFC and HFC proto-\ncols were followed when recording the neutron scattering\ndata, magnetization and susceptibility to minimize sys-\ntematic di\u000berences.\nFor what follows, it is helpful to review brie\ry the\nintensity patterns associated with the di\u000berent mag-\nnetic phases shown schematically in Figs. 4 (a1){4 (a5).\nTaken together, all magnetic phases are characterized by\nscattering intensities on the surface of a small sphere\nin reciprocal space depicted in blue shading. The ra-\ndius of this sphere re\rects the characteristic length scale\njQjof the competition of ferromagnetic exchange and\nDzyaloshinsky-Moriya interactions.\nThe illustrations shown in Figs.4(a1){4 (a5) corre-\nspond to the helical phase (green), conical phase (gray),\nhigh-temperature skyrmion phase (light red), tilted con-\nical phase (dark gray), and low-temperature skyrmion\nphase (dark red). In these depictions, the crystallo-\ngraphich100idirections are shown as black arrows, while\nthe detector plane is indicated by gray shading. It is im-\nportant to note the orientations of the applied magnetic\n\feld with respect to the detector plane and the crystal-\nlographic crystal axes.\nIn the helical state, illustrated in Fig. 4 (a1), the h100i\neasy magnetic axes result in three energetically equiv-\nalent domain populations as indicated by green-shaded\ndots. Under ZFC, these domains are populated equally.9\n(a1) helical\n(a2) conical\n(a3) HT-skyrmion\n(a4) tilted conical\n(a5) LT-skyrmion\nHHHHH (c1) (b1)\n(c2)\n(c3)\n(c4) (b4)(b3)(b2)T = 2 K, H ∥ [100]\nFIG. 4. Comparison of small-angle neutron scattering intensity with the susceptibility as a function of internal magnetic\n\feld. Data recorded under HFC and ZFC are shown on the left- and right-hand side, respectively. (a1){(a5) Typical neutron\nscattering intensity distribution in reciprocal space for the \fve di\u000berent modulated phases as denoted in each panel. The\nscattering plane is marked in gray shading. [(b1) and (c1)] Neutron scattering intensity of the helical and conical state. [(b2)\nand (c2)] Neutron scattering intensity of the tilted conical and LTS phase. [(b3) and (c3)] Susceptibility as calculated from the\nmagnetization, d M=dH(dots), and the real part of the ac susceptibility \u001f0(black line). [(b4), (c4)] Imaginary part of the ac\nsusceptibility \u001f00.\nFor su\u000eciently large \feld the conical state stabilizes. The\nassociated scattering pattern corresponds to intensities\nfor\u0006QkHas illustrated in Fig. 4 (a2). Note that for\nHkh100i, the conical intensity is indistinguishable from\nthe magnetic domain of the helical state in the \feld di-\nrection. Further, shown in Fig. 4(a3) is the scattering\npattern of the high-temperature skyrmion phase, which\nis located in a plane perpendicular to the applied mag-\nnetic \feld. Shown here is the sixfold scattering pattern\nof a single skyrmion lattice domain population, where an\nenergetically degenerate second domain population exists\nin principle under a rotation of 30\u000ewith respect to the\n\feld direction.\nIn addition to these well-known scattering patterns,\nthe tilted conical phase and the LTS phase are character-\nized by the scattering patterns illustrated in Figs. 4 (a4)\nand 4 (a5). For the tilted conical phase the di\u000braction\nspots along the magnetic \feld split into four contribu-\ntions under an angle with respect to the \feld direction.\nThis tilting angle increases under increasing magnetic\n\feld. Second, in the LTS phase, scattering intensity\nemerges in the plane perpendicular to the applied mag-netic \feld. While this pattern assumes the shape of a uni-\nform ring in most experiments, characteristic of a glassy\nappearance of the skyrmion lattice, it was demonstrated\nthat the ring represents an average of a randomly ori-\nented sixfold di\u000braction patterns.16\nThe magnetic \feld dependence of the di\u000berent scatter-\ning intensities is shown in Figs. 4(b1) and (b2) for HFC,\nas well as Figs. 4 (c1) and 4 (c2) for ZFC. Here, the color\nshading corresponds to the shading used in Figs. 4(a1){\n4 (a5). Under HFC and decreasing \feld, the tilted conical\nphase emerges at Hd\nt2, de\fned at the point of in\rection of\nthe scattering intensity, which corresponds to the point\nof in\rection of \u001f00. It is not possible to identify a clear\nsignature of the emergence of the LTS phase in the mag-\nnetization or susceptibility, which in neutron scattering\nclearly appears at a \feld value smaller than Hd\nt2. The\ntransition \feld at which the tilted conical phase vanishes,\nHd\nt1, may be de\fned at the point of in\rection in \u001f00as\nmarked. Finally, when decreasing the \feld further, the\ndisappearance of the LTS phase at Hd\ns1is clearly con-\nnected with a distinct maximum in the susceptibility in-\nferred from the magnetization, d M=dH, where neither a10\npeak in\u001f0is seen nor a contribution in \u001f00.\nIncreasing the magnetic \feld after HFC further [the\nright-hand side of panels (b1) { (b4)], the conical-to-\nhelical transition may seen at Hd\nc1(not marked for clar-\nity), which coincides with Hd\ns1, as well as Hu\nc1. Yet, no\nsignatures may be seen in the magnetization and sus-\nceptibility at Hd\nc1, which why we labelled the Hd\ns1only.\nHowever, when increasing the \feld further, the magnetic\n\feld dependence of the scattering intensities of the tilted\nconical phase displays again excellent agreement with \u001f00\nat the points of in\rection of both quantities, de\fning Hu\nt1\nandHu\nt2. Again, the LTS phase emerges at a \feld value\nHu\ns1, slightly larger than Hu\nt1, and vanishes at a \feld Hu\ns2\nwell above Hu\nt2. HereHu\ns2corresponds accurately to the\nlocation of an additional small peak in d M=dHthat is\nneither tracked in \u001f0nor visible in \u001f00, where the latter\nis vanishingly small.\nThe characteristics observed in the neutron scattering\nintensity, the magnetization, and the susceptibilities ob-\nserved under HFC are reproduced very well under ZFC as\nshown in Figs. 4(c1) through 4(c4). The only di\u000berence\nthat may be noticed here concerns the tiny signature in\ndM=dHatHu\nc1, which is not present under HFC.\nD. Higher harmonics in the susceptibility\nMotivated by the observation of clear di\u000berences be-\ntween dM=dHand\u001f0, as well as substantial contribu-\ntions in\u001f00, we explored the question of higher-harmonic\ncontributions to the ac susceptibility up to third order\nfor \feld alongh110iandh100iat selected temperatures.\nData were recorded in a single \feld sweep from nega-\ntive to positive \felds after HFC, i.e., data recorded for\npositive \feld values during these sweeps essentially cor-\nresponded to ZFC conditions.\nIn general higher harmonics re\rect the presence of\nstrong non-linearities in M(H) even for the tiny exci-\ntation amplitudes of the ac probing \feld (of the order\n0.1 mT). In the spirit of a brief introduction we note that\nthe magnetic response of a sample to an external mag-\nnetic \feld\nH(!;t) =Hdc+Haccos!t (1)\nmay be expressed in terms of the Fourier expansion\nM(!;t) =M0+1P\nn=1(M0\nncosn!t+M00\nnsinn!t) (2)\n=\u001fdcHdc+Hac1P\nn=1(\u00120\nncosn!t+\u001200\nnsinn!t) (3)\nwhere the Fourier coe\u000ecients represent unitless quanti-\nties given by\n\u00120\nn=1\n\u0019Hac2\u0019Z\n0M(!;t) cosn!td!t (4)and\n\u001200\nn=1\n\u0019Hac2\u0019Z\n0M(!;t) sinn!td!t (5)\nAs mentioned in Sec. II, in the literature, the coe\u000ecients\n\u00120\nnand\u001200\nnare also known as harmonic susceptibilities. In\ncomparison, when developing the response to an excita-\ntion in a Taylor series the expansion coe\u000ecients, denoted\n\u001f0\nnand\u001f00\nn, are referred to as nonlinear susceptibilities.\nFor mathematically well behaved properties it is possi-\nble to infer the real and imaginary parts of the nonlinear\nsusceptibilities, \u001f0\nnand\u001f00\nnfrom these Fourier coe\u000ecients\n\u00120\nnand\u001200\nn(see, e.g., Ref. 39 for details). However, since\nhigher order contributions rapidly decrease in strength,\nwe approximate the nonlinear susceptibility by the lead-\ning order contributions of the harmonic susceptibilities,\nnamely,\u001f=\u001f1\u0019\u00121,\u001f2\u00192H\u00001\nac\u00122and\u001f3\u00194H\u00002\nac\u00123.\nIn view that the magnetization curve for \feld along\nh100iis not point symmetric with respect to M= 0 and\nH= 0, the second harmonic may not be point symmetric\neither. Further, for the sharp changes of slope of the mag-\nnetization curve the second harmonic may be expected\nto be \fnite subject to the precise excitation amplitude.\nShown in Fig. 5 are typical higher order contributions\nof the harmonic susceptibility, notably \u001f,\u00122, and\u00123. In\nour measurements, we recorded the properties up to the\nthird harmonic at selected temperatures following HFC.\nData are shown as a function of internal \feld for \feld\nalongh110i(left panels) and \feld along h100i(right pan-\nels), where rows correspond to \frst, second and third\nharmonic susceptibility, while columns correspond to dif-\nferent temperatures. Data for negative \felds for h110iare\nnot shown for clarity and because they are symmetrical\nto the data at positive \feld values, thus not adding any\nadditional information.\nFor a \feld along h110iandT= 2 K, the second har-\nmonic is essentially zero except for a negative peak of\nthe real part, \u001f0\n2, atHc2. AtT= 57:5 K, tiny additional\npeaks are present at Ha2and two very weak peaks at\nHa1andHc1(hardly visible on the scale chosen here).\nMoreover, the third harmonic, \u001f3, is essentially zero at\nbothT= 2 K and 57 :5 K.\nIn comparison, for a \feld along h100i, the behavior at\nhigh temperatures is similar to that for \feld along h110i.\nNamely, at T= 57:5 K, the second harmonic, \u001f0\n2, displays\na negative peak associated with Hc2as well as additional\ntiny peaks at Ha2,Ha1, andHc1(barely resolved on the\nscale chosen here). Moreover, the third harmonic, \u001f3, is\nessentially zero. Similarly, at T= 45 K only the peak\nassociated with Hc2is observed in \u001f2, where the data\nare point symmetric with respect to H= 0 and\u001f2= 0.\nAgain the third harmonic, \u001f3, is essentially zero.\nIn contrast, at T= 2 K, a huge second harmonic is\nobserved in the \feld range of the tilted conical phase\nfor Hk h100i. Interestingly, in comparison with the\nsecond harmonic signal at high temperatures associated\nwithHc2the new contribution has the opposite sign.11\n B ∥〈110〉  B ∥〈100〉\nFIG. 5. Higher-harmonic contributions in the ac susceptibility as a function of \feld at selected temperatures for \felds along\ntheh110iandh100idirection. The set of panels on the left corresponds to Hkh110i; the set of panels on the right corresponds\ntoHkh100i. Rows display the \frst, second and third harmonic. Columns correspond to di\u000berent temperatures as marked\nin the plots. Data have been recorded following high-\feld cooling in a \feld sweep from negative to positive \feld values. The\nnegative branch of the h110idata is not shown for clarity, as it provides no further information.\nAlso, a huge third harmonic is observed in the \feld re-\ngion of the tilted conical phase. In fact, key features\nof the second and third harmonic are already present at\nT= 20 K, where they are weaker with additional \fne\nstructure (double peak).\nThe highly nonlinear response we observe in\nCu2OSeO 3represents a key characteristic of the tilted\nconical state rather than the LTS phase. We specu-\nlate that the strong nonlinear and dissipative response\narises from domain walls and other inhomogeneous tex-\ntures generated when the tilted phase is nucleated. In\nthe literature, similar properties are typically reported\nfor superconductors, and molecular and low-dimensional\nmagnets44, where speci\fc models are required for the in-\nterpretation. Unfortunately, it is at present not clear\nhow the non-linearities are generated. In the future, a\nCole-Cole plot of the interdependence of the real and\nimaginary parts of \u001ffor di\u000berent frequencies may, for\ninstance, provide information on the characteristic acti-\nvation energies-44However, this level of experimental ex-\nploration and analysis is beyond the scope of the study\npresented here.E. Demagnetizing E\u000bects\nIn view of the importance of the temperature versus\n\feld protocol and the rather subtle features in the mag-\nnetization and susceptibilities associated with the vari-\nous phase transitions we have recorded the magnetization\nand ac susceptibility for a series of samples with di\u000ber-\nent demagnetization factors. For samples with large de-\nmagnetization factors, we \fnd the reconstruction of the\nintrinsic behavior to be prohibitively di\u000ecult as small ef-\nfects are irreversibly smeared out. Also, for an irregular\nsample shape, the nucleation processes may actually be\naltered. Moreover, theoretical modeling reported in Ref.\n16 revealed that dipolar interactions play an important\nrole for the details of the magnetic phase diagram as dis-\ncussed below.\nShown in Fig. 6 are typical magnetization and suscepti-\nbility data as a function of applied \feld for various sam-\nples with di\u000berent demagnetizing factors (see also Ta-\nble I). Each column corresponds to a di\u000berent demagne-\ntization factor as stated at the top of the \fgure. Data\nwere measured in a single \feld sweep corresponding to\nHFC, i.e., analogous to the data shown in Fig. 3. For\nlack of space, data at negative \felds are shown as a func-\ntion of positive \feld values. Therefore data shown in rows12\n(a1) (a2) (a3) (a4) (a5)\n(b1) (b2) (b3) (b4) (b5)\n(c1) (c2) (c3) (c4) (c5)\n(d1) (d2) (d3) (d4) (d5)\n(e1) (e2) (e3) (e4) (e5)\nFIG. 6. Magnetization M, calculated susceptibility, d M=dH, as well as real and imaginary part of the ac susceptibility, \u001f0and\n\u001f00, respectively, for di\u000berent sample shapes with di\u000berent demagnetization factors and \feld along h100i. The demagnetization\nfactors are denoted above each column (see Table I for further details). Data were recorded following high-\feld cooling in a\nsingle \feld sweep, where data recorded at negative \felds are mirrored to positive \feld values. Data branches recorded under\ndecreasing and increasing magnitude of the \feld are shown in blue and red shading, respectively.\n(b) and (c) were recorded under increasing \feld, corre-\nsponding to the right-hand side of Fig. 3, i.e., positive\n\feld values. They are essentially equivalent to the be-\nhavior observed under ZFC. In comparison, data shown\nin rows (d) and (e) were recorded under decreasing \feld,\ncorresponding to the left-hand side of Fig. 3, i.e., negative\n\feld values.\nAssuming uniform demagnetization \felds across the\nsample volume a shearing of the data towards larger\n\feld values with increasing demagnetization factor is ex-\npected. Moreover, the bar-shaped sample geometry re-\nsults in a distribution of internal \feld directions that\ngenerates a smearing out of all features. As shown in\nthe \frst column of Fig. 6 for a tiny demagnetizing fac-\ntor,N= 0:07, all features observed under ZFC and HFC\ndescribed in detail above may be readily identi\fed. The\ntwo most important characteristics concern the presence\nof hysteresis in the magnetization due to the LTS phase,\nand the observation of dissipation in the imaginary part\nof the susceptibility \u001f00, associated with the tilted conical\nphase.For increasing demagnetization factor the steep rise\n(jump) in the magnetization associated with the onset\nof the tilted conical phase turns into a shallow rise.\nNonetheless, a non-vanishing contribution in \u001f00is ob-\nserved in a well-de\fned \feld range. This suggests that\nthe tilted conical phase continues to form in a \fnite \feld\nrange. In contrast, the signatures of the LTS phase are\nsmeared out and vanish. Without microscopic informa-\ntion it is unfortunately not possible to infer any further\ninformation on the LTS phase.\nIn order to illustrate the e\u000bects of demagnetizing \felds\non a theoretical level, we consider the Ginzburg-Landau\nmodel described in Ref. 16 taking into account the cubic\nmagnetocrystalline anisotropy, cf. Eq. (6) shown below.\nFor a given applied \feld H, the magnetic moments are\nsubject to an internal \feld Hint=H\u0000NM that depends\non the magnetization Mas well as the demagnetization\ntensor N. This results in three main physical e\u000bects: (i) a\nshift of phase boundaries due to the change of the internal\n\felds, (ii) a possible tilt of the internal \feld in the tilted\nconical phase, and (iii) the stabilization of coexistence13\nFM\nLTS\nCTC (metastable)\n0 0.20.40.60.80510152025303540\nNzµ0HFM\nLTS\nCTC (metastable)\n0 0.20.40.60.8\nNzFM LTS+FM\nLTS\nLTS+C\nC\n0 0.20.40.60.8 1\nNz0 0.20.40.60.8 1020406080100120140160\nNµ0H(mT)Hu\nt1\nHu\nt2\nHd\nt1\nHd\nt2\nHd\ns1\nHu\ns2(a) (b) (c) (d)\nFIG. 7. Magnetic phase diagrams as a function of demagnetization factor Nand applied magnetic \feld Happlied along\nh100i. Note that di\u000berent units are used for Hin the theoretical calculations as compared with experiment. (a) Theoretical\nphase diagram obtained from a Ginzburg-Landau model in the presence of the cubic anisotropy of Eq. (6), as described in\nRef. 16. Here we assumed that only single-domain states exist. We also display the metastable tilted conical state in the\nregime where its energy is higher than the LTS phase but lower than both the conical and ferromagnetic phase. It vanishes for\nsmallNz. Parameters and units correspond to those used in Ref. 16. (b) Phase diagram of (a) when including the exchange\nanisotropy cP\ni(riMi)2withc=\u00000:06, as described in the main text. The metastable tilted conical state exists even for\nNz!0. (c) Thermodynamic phase diagram [parameters as in (a)] including regions where demagnetization e\u000bects stabilize\nphase coexistence, see main text. Note that the experimental regions of phase coexistence can be much larger due to hysteresis\ne\u000bects in the region where phases are metastable. (d) Summary of the experimental critical \felds for samples with di\u000berent\ndemagnetization factor N, as detailed in Fig. 6, with data for negative \felds mirrored to positive \felds.\nregimes.\nShown in Fig. 7 (a) and (b) are phase diagrams under\nan applied \feld Halongh100ias a function of demagneti-\nzation factor Nz. The two phase diagrams only consider\nuniform phases, ignoring the possibility of phase coex-\nistence discussed below. The thermodynamically stable\nphases are the conical, the LTS and the ferromagnetic\nstate. We also show the metastable tilted conical phase\nin the regime where it has lower energy than both the\nferromagnetic and conical phase. For the chosen parame-\nters, its energy is, however, always higher than the energy\nof the LTS state. Note that we do not show a possible\nskyrmion square lattice phase. In some part of the phase\ndiagram (see Ref. 16) its energy is very close (identical\nwithin numerical errors and model uncertainties) to the\ntriangular skyrmion lattice.\nFor increasing Nzthe stability range of all phase\nboundaries shifts to larger magnetic \felds as the demag-\nnetization factor reduces internal magnetic \felds. The\nenergetics of the metastable tilted conical phase is af-\nfected in more subtle ways as the tilting can induce in-\nternal \feld components perpendicular to the external\nmagnetic \feld (the calculation assumes a single domain).\nSimilar to the pitch of the modulation, also the uniform\nmagnetization is tilted away from the applied magnetic\n\feld, notably Hkh100i. Thus, for the single domain\nstate assumed here, the internal \feld, comprising the ap-\nplied \feld, the uniform magnetization of the tilted conical\nstate and the demagnetization \feld, increasingly deviates\nfromh100i. ForNz!0 this e\u000bect is enhanced such that\nthe tilted conical phase eventually becomes energetically\nunfavorable compared to the \feld-polarized state, as thegeneral condition Nx+Ny+Nz= 1 implies that the de-\nmagnetizing \feld of the transverse \feld components gets\nboosted.\nFor the parameters used in Ref. 16 the tilted conical\nphase vanishes for Nz!0, see Fig. 7 (a). Our exper-\nimental data display, in contrast, the signatures of the\ntilted conical phase even for tiny values of N, as summa-\nrized in Fig. 7 (d). However, there are di\u000berent possibili-\nties (or combinations thereof) to account for this seeming\ndiscrepancy. First, additional anisotropies, such as ex-\nchange contributions c[(rxMx)2+(ryMy)2+(rzMz)2],\ncould stabilize the tilted conical phase further such that it\nremains metastable even for Nz!0. This is illustrated\nin Fig. 7 (b), where the additional exchange anisotropy\nterm given here has been taken into account as compared\nwith the model evaluated in Fig. 7 (a). Furthermore, the\nsuppression of the tilted conical phase by internal \feld\ncomponents perpendicular to the external \feld can be\nreduced when several domains with di\u000berent tilting di-\nrections exist [cf. Fig. 4 (a4)]. Thus the total transverse\nmagnetization might average out. We have checked that\nthis e\u000bect indeed enlarges the metastable tilted phase but\nnot up toNz= 0.\nBoth phase diagrams, Figs. 7 (a) and (b), were gener-\nated in \fnding a uniform phase with minimal energy for\na given external \feld H. At the \frst order phase transi-\ntions, however, the two states in question have di\u000berent\nmagnetization. In this case demagnetization e\u000bects sta-\nbilize regions of phase coexistence. Boundaries of regions\nof phase-coexistence (ignoring hysteresis e\u000bects and do-\nmain wall formation assuming a perfectly \\soft\" magnet\nand smooth transitions into the coexistence regimes) are14\nH H H\nFIG. 8. Magnetization (\frst row) and speci\fc heat (second row) as a function of internal \feld at 2 K for all major directions.\nData were recorded following high-\feld cooling in a single \feld sweep from negative to positive \feld values, where data recorded\nat negative \felds are mirrored to positive \feld values. Data branches recorded under decreasing and increasing are indicated\nin blue and red shading, respectively.\ndetermined from the condition that the internal magnetic\n\feld has to match the critical \feld at Nz= 0. The result-\ning phase diagram (without the metastable titled phase)\nis shown in Fig. 7 (c).\nThe experimentally observed regions of phase coexis-\ntence are much larger, see Fig. 7 (d). This can be at-\ntributed to strong hysteresis e\u000bects: after their forma-\ntion, topologically protected skyrmions remain locally\nstable over a large \feld range.45\nF. Speci\fc heat\nFor temperatures exceeding \u001810 K, the speci\fc heat\nCis dominated by contribution of phonons. In turn, a\ndetailed investigation of the temperature dependence of\nCat low temperatures was beyond the scope of the work\npresented. In order to gain some insights into possible\nsignatures of the two new magnetic phases, we focused\ninstead on the magnetic \feld dependence of Cat a single\nlow temperature.\nShown in Fig. 8 is a comparison of the magnetization,\nM, and the speci\fc heat Cas a function of magnetic \feld\nalongh111i,h110iandh100iat a constant temperature\nof 2 K. For clarity only the change of the speci\fc heat,\n\u0001C=C(B)\u0000C(H= 0) is shown. Data were recorded\nfollowing high-\feld cooling in a single \feld sweep from\nnegative to positive \feld values, where data recorded at\nnegative \felds are mirrored to positive \feld values. Data\nbranches recorded under decreasing and increasing \feldare indicated in blue and red shading, respectively. Data\nrecorded under increasing and decreasing \feld as marked\nby the arrows, corresponding essentially to ZFC and HFC\nconditions, respectively. The magnetization, shown in\nFigs. 8(a), 8(b) and 8(c) for \feld along h111i,h110iand\nh100i, respectively, displays the key characteristics asso-\nciated with the helical phase, the conical phase, the tilted\nconical phase, the low-temperature skyrmion phase and\nthe \feld polarized phase.\nThe speci\fc heat for \feld along h111iandh110ias a\nfunction of magnetic \feld, shown in Figs. 8(d) and 8(e)\nexhibits three regimes of di\u000berent slope, d C=dH. In the\nhelical state, the speci\fc heat is essentially constant, fol-\nlowed by a linear decrease in the conical state, which\nbecomes steeper in the \feld-polarized state. Apart from\na small amount of hysteresis in MatHc1, there is no\nevidence for further hysteresis.\nIn contrast, there are clear di\u000berences between increas-\ning and decreasing \feld for \feld along h100ishown in\nFig. 8 (f). Namely, under increasing magnetic \feld (red),\nthree distinct kinks corresponding to the transitions at\nHu\nt1,Hu\nt2, andHu\ns2are visible. At low magnetic \felds\nup toHu\nt1, the speci\fc heat is essentially unchanged,\n\u0001C(B)\u00190, where we note that this includes the helical\nand the conical phase. Between Hu\nt1andHu\nt2, \u0001C(B)\ndrops sharply, coinciding with the steep increase in M.\nBetweenHu\nt2andHu\ns2, the rate of change, d C=dH, de-\ncreases, followed by another decrease of d C=dHabove\nHu\ns2. In comparison, under decreasing \feld (blue), only\ntwo changes of slope corresponding to Hd\nt1andHd\nt2may15\nbe observed. Thus, there is clearly hysteresis between\nHd\nt1andHu\ns2. When further decreasing the \feld, the spe-\nci\fc heat coincides with the data recorded under increas-\ning \feld. In particular, there is very faint evidence of\nHd\ns1in the speci\fc heat, which is clearly present in the\nmagnetization.\nAs the tilted conical phase and the low-temperature\nskyrmion phase emerge as a function of magnetic \feld,\nchanges of the entropy to be expected between the dif-\nferent phases are rather tiny. Nonetheless, we observe\na suppression of the speci\fc heat with increasing mag-\nnetic \feld for all three \feld directions. The observation\nof clear signatures in the speci\fc heat at the transition\n\felds of the tilted conical phase and low-temperature\nskyrmion phase, as well as the observation of hystere-\nsis provide clear signatures of thermodynamically dis-\ntinct phases. However, some caution is necessary as the\nstrongly hysteretic behavior observed in neutron scatter-\ning, the magnetization and the speci\fc heat are charac-\nteristic of strong \frst order transitions. Both the size\nof the e\u000bects as well as the experimental method we\nuse here, notably small heat pulse relaxation calorime-\ntry, make the detection of a latent heat for a weakly \feld\ndependent phase transition line essentially impossible.\nIV. DISCUSSION\nThe changes of the magnetic phase diagram of\nCu2OSeO 3as a function of crystallographic direction in-\nferred from our magnetization, susceptibility and spe-\nci\fc heat data re\rect the presence of distinct magnetic\nanisotropies. The discussion of these anisotropies is or-\nganized in three parts. In section A a comparison of\nthe anisotropies of the magnetization of Cu 2OSeO 3is\npresented with those observed in the related compounds\nFe1\u0000xCoxSi (x= 0:2) and MnSi. This is followed\nby a quantitative estimate of the strength of the cubic\nanisotropies, presented in section B. The discussion closes\nin section C with qualitative considerations of the e\u000bec-\ntive anisotropy potential as a function of applied mag-\nnetic \feld, and how these connect with the anisotropy of\nthe magnetization and the di\u000berent magnetic phases.\nA. Anisotropy of the magnetization\nShown in Fig. 9 is a comparison of the magnetization\nof Cu 2OSeO 3, Fe 1\u0000xCoxSi (x= 0:2), and MnSi for all\nmajor crystallographic directions. Data are shown as a\nfunction of internal magnetic \feld in units of the upper\ncritical \feld Hc2, as determined in the direction exhibit-\ning the largest value of Hc2. The magnetization is pre-\nsented in units of the magnetization at high \felds.\nAs emphasized in section C, data for Cu 2OSeO 3,\nshown in Fig. 9 (a), exhibit pronounced magnetic\nanisotropy. While the magnetization appears to be es-\nsentially isotropic up to Hc1, we \fnd that the magne-\nFIG. 9. Magnetization of Cu 2OSeO 3, Fe1\u0000xCoxSi (x= 0:2),\nand MnSi, illustrating the di\u000berent strength of the cubic mag-\nnetic anisotropies. Data are shown as a function of internal\nmagnetic \feld in units of the upper critical \feld, Hc2, de-\ntermined in the direction exhibiting the largest value of Hc2.\nThe magnetization is presented in units of the magnetiza-\ntion at high \felds. (a) Magnetization of Cu 2OSeO 3, exhibit-\ning strongh100ieasy-axes anisotropy. (b) Magnetization of\nFe1\u0000xCoxSi (x= 0:2), exhibiting moderate h100ieasy-axes\nanisotropy. (c) Magnetization of MnSi, exhibiting very weak\nh111ieasy-axes anisotropy.\ntization forh111iis larger than for any other direction\nbetweenHc1forh111iandHt1forh100i. AboveHt1,\nthe magnetization for h100iis largest. The behavior at\nintermediate \felds may be interpreted in terms of a \feld\ninduced inversion of the magnetic anisotropy, where the\nh111iaxes appear to be the easy axes for a small \feld\nrange. However, as shown in further detail below, the cu-\nbic anisotropy (sign of K) does not change as a function\n\feld and theh100iaxes remain the easy axes, consistent\nwithHc2being smallest and largest for h100iandh111i,\nrespectively. Instead, the behavior observed here re\rects\nthe property of a modulated spin structure that remains\nessentially rigid in a cubic anisotropy potential.\nThe magnetization of Fe 1\u0000xCoxSi (x= 0:2) shown in\nFig. 9 (b) is reminiscent of Cu 2OSeO 3and characteristic\nof a weakh100ieasy-axes anisotropy, even though the16\nvariations in Fe 1\u0000xCoxSi are not as pronounced. In com-\nparison to Cu 2OSeO 3, the lower critical \feld Hc1seems\nessentially isotropic. It is important to emphasize that in\nparticular the helical-to-conical transition in Fe 1\u0000xCoxSi\nmay be dominated by disorder and defect-related pin-\nning, accounting for the large value of Hc1as compared\ntoHc2.28The fairly small changes of Hc2as a function of\norientation re\rect a weak magnetic anisotropy. Also, the\nmagnetization for \felds above Hc1is qualitatively di\u000ber-\nent for di\u000berent \feld directions, where the magnetization\nforh100iis steeper than for the other directions up to\nHc2. For \felds above Hc1, the slope is initially larger\nthan just below Hc2. This corresponds to the jump in M\natHc1in Cu 2OSeO 3characteristic of a broadened \frst\norder transition. Thus, in comparison to Cu 2OSeO 3, the\nmagnetic anisotropy as analyzed in terms of the magnetic\nwork presented below is by far not as pronounced. An\nexciting unresolved question for Fe 1\u0000xCoxSi concerns,\nwhether a tilted conical phase and LTS phase exist at\nleast for some x.\nThe \feld dependence of the magnetization of MnSi\nshown in Fig. 9 (c) exhibits, \fnally, the signatures of\na very weakh111ieasy-axes anisotropy. Here di\u000ber-\nences between the di\u000berent crystallographic directions\nare di\u000ecult to discern. In comparison to Cu 2OSeO 3and\nFe1\u0000xCoxSi (x= 0:2), the ratio of Hc2toHc1is largest,\nwhereas the variation of Hc2with orientation is smallest\n(see Ref. 43 for further details). Moreover, the changes of\nslope are tiny between the di\u000berent phases for all crystal-\nlographic directions. Thus, taken together, the compari-\nson of the magnetization shown here underscores that the\nanisotropy is clearly strongest in Cu 2OSeO 3withh100i\neasy axes, followed by Fe 1\u0000xCoxSi, which displays also\nh100ieasy axes, whereas MnSi exhibits the well-known,\nvery weakh111ieasy-axes anisotropy.\nB. Magnetocrystalline Anisotropy\nThe LTS and tilted conical phases arise in the pres-\nence of cubic magnetocrystalline anisotropies provided\nthat they are su\u000eciently strong.16,46The energy density\nassociated with the leading anisotropy may be expressed\nin terms of the unit vector ^Mdescribing the orientation\nof the magnetization\nFa=K(^M4\nx+^M4\ny+^M4\nz) (6)\nwhereKis the anisotropy constant. The de\fnition of\nFaused here is identical to a recent ferromagnetic res-\nonance study38and di\u000bers from the SANS study16in\nterms of the sign of Kand the units of the magneti-\nzation. As shown in Ref. 16, the cubic anisotropy is\nsu\u000ecient to give rise to a stable LTS and a metastable\ntilted conical phase for a magnetic \feld along h100i, pro-\nvided thatKis negative and exceeds a critical value Kc.\nAn estimate of the critical ansisotropy value is given by\nKc\u0019\u00000:07\u00160Hint\nc2Ms, whereHint\nc2is the internal critical\n\feld of the conical to \feld-polarized transition (evaluated\nFIG. 10. Magnetic work W, transition \felds Hc1andHc2,\nand anisotropy constant Kas inferred from the magnetiza-\ntion and the transition \felds as a function of temperature.\n(a) Temperature dependence of the magnetic work Wfor all\nmajor crystallographic orientations under magnetic \feld sat-\nurating the magnetization. (b) Di\u000berence of the magnetic\nwork \u0001Wwith respect to \feld along h100i. (c) Transition\n\feldsHc1andHc2for \feld alongh111i(blue),h110i(green),\nandh100i(red) as a function temperature. (d) Anisotropy\nconstantKas a function of temperature, as extracted from\nthe magnetic work (green, blue), and the upper critical \feld\nHc2(orange). Shown in gray shading is the regime in which\nan LTS phase becomes favorable ( K\u0014\u00000:07).\nforK= 0). (Note that we use a di\u000berent notation as\ncompared to Ref. 16.) However, the stability of the LTS\nand tilted conical phases may be reduced or enhanced\nby additional contributions to the free energy arising, for\nexample, from other anisotropy terms.\nFollowing Ref. 47 the magnetocrystalline anisotropy\nmay be estimated by considering the magnetic work\nW^H=\u00160ZMs\n0dMH (7)\nto magnetize the system to the saturated state Mswith a\n\feld pointing along the direction determined by the unit17\nvector ^H. The subscript ^Hserves to indicate the depen-\ndence on the speci\fc crystallographic direction. From\nthe experimental magnetization data, we may determine\nW^Hfor \felds along high symmetry directions, i.e., h100i,\nh110i, andh111i. Values as a function of temperature de-\ntermined this way are shown in Fig.10 (a). With decreas-\ning temperature, Wincreases monotonically below Tc.\nAt high temperatures, Wdoes not change as a function\nof \feld direction. However, towards lower temperatures a\nsmall but distinct anisotropy emerges below \u001830 K. In\nthe zero temperature limit, we \fnd W\u001920 neV /RingA\u00003,\nwhich is in quantitative agreement with density func-\ntional calculations predicting the value 23 :8 neV/RingA\u00003.48\nIn the \feld-polarized state the magnetocrystalline\nanisotropy simpli\fes for magnetic \felds along high sym-\nmetry directions\nFa=K\u00018\n><\n>:1 Mkh100i\n1=2Mkh110i\n1=3Mkh111i:(8)\nOne may, for example, estimate the anisotropy constant\nby considering the di\u000berences of the magnetic work be-\ntween di\u000berent orientations:47\nWh110i\u0000Wh100i=\u00001\n2K (9)\nWh111i\u0000Wh100i=\u00002\n3K: (10)\nThese di\u000berences are shown in Fig. 10(b). While the\ncurves are guides to the eye, it is interesting to note the\nquantitative consistency of the di\u000berent directions with\nEqs. 9 and 10. The resulting anisotropy constant Kas a\nfunction of temperature derived from these di\u000berences is\nshown by the blue and green curves in Fig.10 (d).\nIn fact, the critical \feld Hint\nc2also displays a directional\ndependence in the presence of the anisotropy Kas re-\nported in Ref. 49. The di\u000berence of critical \felds along\nh110iandh111iis proportional to Kand may be ex-\npressed as\nHint\nc2;h110i\u0000Hint\nc2;h111i=\n\u00005\n3K\n\u00160Ms\u0014\n1 +O\u0012K\u001b\n\u00160Hc2Ms\u0013\u0015; (11)\nwhere the correction, O(:::), may be neglected for small\nK. This allows to determine the value of Kwith the\nhelp of the upper critical \felds Hc2for the three crystal-\nlographic directions h100i,h110i, andh111ias shown in\nFigs. 10(c) and 10(d). At high temperatures, just below\nTcessentially no anisotropy may be observed. However,\nwith decreasing temperature, the values of Hc2for di\u000ber-\nent directions separate below \u001840 K, becoming distinctly\ndi\u000berent.\nValues of the anisotropy constant Kinferred from Hc2\nare shown as orange curve in Fig. 10(d). With decreasing\ntemperature Kdeviates from zero and decreases below\n\u001835 K. Within the accuracy of the estimates carried out\nhere it is not possible to rule out a change of sign andsmall positive value of KbetweenTcand\u001835 K. This\nmight hint at the existence of further, weaker anisotropy\nterms. The putative existence of such terms may be\nclari\fed with the help of the anisotropy of Hc1. How-\never, it is important to emphasize that despite a possible\nchange of sign of Kas a function of temperature the\neasy magnetic axes remain the h100iaxes throughout.\nIn the limit of low temperatures, the values of Kas cal-\nculated from magnetization and the upper critical \felds\nare, \fnally, quantitatively consistent with the anisotropy\nconstant determined at T= 5 K with the help of ferro-\nmagnetic resonance measurements, KFMR =\u00000:6\u0002103\nJ/m3\u0019\u00003:7 neV/ \u0017A3(see supplement of Ref. 38 for de-\ntails).\nAlso shown in Fig. 10(d) are the anisotropy con-\nstants in dimensionless units, K=(\u00160MsHint\nc2;h111i)\u0019\nK=(\u00160MsHint\nc2jK=0) (ordinate on the right-hand side),\nwhereMsandHint\nc2are temperature dependent. As\nreported in Ref. 16 the cubic anisotropy may be ex-\npected to stabilize the LTS as a metatstable state with-\nout need for further exchange anisotropies when the ratio\nK=(\u00160MsHint\nc2;h111i) falls below a threshold of \u00000:07. This\ncondition is indicated by gray shading in Fig. 12(d). In-\ndeed, below\u001815 K, where the LTS phase is observed\nforHkh100i, the ratioK=(\u00160MsHint\nc2;h111i) reaches this\nstrength suggesting that the cubic anisotropy Krepre-\nsents the main stabilization mechanism for the additional\nphases.\nC. E\u000bective energy landscape\nFor weak magnetic anisotropies it is very well estab-\nlished experimentally and theoretically, that the direction\nof the conical helix aligns with the magnetic \feld. While\nthis alignment is on the expense of magnetic anisotropy\nenergy, it allows to minimize Zeeman energy as the mag-\nnetic moments cant most e\u000bectively towards the \feld di-\nrection. In turn, this raises the question why a su\u000eciently\nstrong anisotropy stabilizes a skyrmion phase and a tilted\nconical phase in which the moments twist around propa-\ngation directions that are not aligned with the magnetic\n\feld. In the following, we present a qualitative discus-\nsion in order to provide an intuitive picture of the com-\npetition between Zeeman energy and magnetocrystalline\nanisotropy for modulated structures. A quantitative nu-\nmerical analysis has been reported in Ref. 16.\nFor what follows, it is helpful to consider the magneti-\nzation of a right-handed conical modulation,\nM(r)=Ms= cos\u000b^e3\n+ sin\u000b(^e1cos(k^e3r) + ^e2sin(k^e3r));\nwhereMsis the saturated magnetization, \u000bis the cone\nangle with cos \u000b\u0019H=H c2andkis the magnitude of the\npitch vector that is oriented along the unit vector ^k= ^e3\nwith the orthonormal right-handed basis ^ e1\u0002^e2= ^e3.18\nThe uniform magnetization component of the conical he-\nlix points into the same direction than the pitch vec-\ntor, namely ^ e3. Using the pristine conical helix as the\nstarting point of a variational calculation in the magne-\ntocrystalline potential of Eq. (6), an e\u000bective anisotropy\npotential for the orientation of the pitch vector ^kmay be\nobtained given by\nVa(^k) =Ke\u000b(\u000b)\u0010\n^k4\nx+^k4\ny+^k4\nz\u0011\n: (12)\nWe note that this approximation neglects distortions of\nthe conical helix, which are of the order of a few % as\nobserved in the SANS study (see supplement of Ref. 16;\nsee also discussion in Ref. 43). The e\u000bective anisotropy,\nKe\u000b, depends on the cone angle \u000bwhere\nKe\u000b(\u000b) =K\n64(9 + 20 cos(2 \u000b) + 35 cos(4 \u000b)): (13)\nAs a key observation, Ke\u000bchanges sign as a function of\n\u000b. Whereas the sign of KandKe\u000bare the same for small\n\u000band cone angles close to \u0019=2, they possess opposite sign\nat intermediate cone angles, 30\u000e.\u000b.70\u000e.\nThe e\u000bective anisotropy potential is illustrated in\nFig. 11. Large energies are shown in red shading, whereas\nlow energies are shown in blue shading. The \frst three\npanels represent a cubic potential (a) with K= 0, (b)\nwithK < 0 favoring the cubic h100iaxes, and (c) with\nK > 0 favouring the cubic h111iaxes. The panels in the\nnext three rows, (d){(l), indicate the energy landscape\nof the magnetocrystalline potential, Eq. (6) with K < 0,\nthat is scanned by the conical helix state of Eq. (12) for\ndi\u000berent cone angles \u000band di\u000berent orientations of the\npitch vector ^k. Finally, the last three panels represent\nthe e\u000bective anisotropy potential for the pitch orienta-\ntion ^k, i.e, the function Va(^k) for three di\u000berent cone\nangles. Whereas Ke\u000b(\u000b)<0 for\u000b= 10\u000ein panel (m)\nand for\u000b= 80\u000ein panel (o), it is positive Ke\u000b(\u000b)>0\nfor\u000b= 55\u000ein panel (n).\nWhen the magnetic \feld is applied along a h100idi-\nrection, both, the Zeeman energy as well as the e\u000bective\nanisotropy energy are minimized for Ke\u000b(\u000b)<0. How-\never, they compete for intermediate cone angles where\nKe\u000b(\u000b)>0. Whereas the Zeeman energy favors to\nalign the helix with the \feld, the e\u000bective anisotropy\nsupports a pitch vector that is oriented away from the\n\feld, since, otherwise, the magnetic moments of the he-\nlical state would dominantly point into hard h111iaxes\nresulting in a large energy penalty, see Fig. 11(e). If the\ncubic anisotropy exceeds a threshold value K <Kcwith\nKc\u0019 \u0000 0:07\u00160MsHint\nc2, the anisotropy dominates such\nthat even a LTS phase becomes favorable. A numeri-\ncal assessment16reveals, moreover, that the LTS phase\ncomprising modulations perpendicular to the \feld, even\nforms the ground state at intermediate magnetic \felds.\nThe notion of an e\u000bective magnetocrystalline\nanisotropy, Eq. (13), may also be con\frmed by consid-\nering the magnetization process. Fig. 12(a) displays\ntypical magnetization data in terms of the internal \feld\n[001]\n[010]\n[100](a) (b) (c) [001]\n[010]\n[100]\n[001]\n[010]\n[100]\n[001]\n[010]\n[100][001]\n[010]\n[100][001]\n[010]\n[100](n) (m) (o)(d)\n(g)\n(j) (k) (l)(h) (i)(e) (f) k ∥ [001] k ∥ [001] k ∥ [001]\nk ∥ [011] k ∥ [011] k ∥ [011]\nk ∥ [111] k ∥ [111] k ∥ [111]FIG. 11. Illustration of the e\u000bective anisotropy that arises\nwhen the conical helix, Eq (12), is embedded in the magne-\ntocrystalline energy landscape of Eq. (6). Large energies are\nshown in red shading, whereas low energies are shown in blue\nshading. (a){(c) explain the representation of the orienta-\ntional dependence of the energetic cost with K= 0,K < 0\nandK > 0, respectively. For K < 0 the magnetic easy axes\nareh100i, whereas they are the hard axes for K > 0. (d){(l)\nillustrate the orientations that are covered by the magnetic\nmoments of the conical helix with a certain cone angle \u000bfor\ndi\u000berent \feld orientations of the pitch vector k. (m){(o) dis-\nplay the e\u000bective anisotropy potential of Eq. (13) that changes\nsign as function of cone angle \u000b.\nHintversusMfor three crystallographic directions.\nThe data shown here were measured at a temperature\nof 2 K following ZFC with \feld along h111i(blue),\nh110i(green), andh100i(red). As pointed out \frst in\nSec. C.1 the magnetization displays two intersections\nof theh111iandh110idirections. The magnetic work\nW^H(M) =\u00160RM\n0dM0H(M0) for theh111iandh110i\ndirection as inferred from the magnetization data is\nshown in Fig. 12(b) relative to the magnetic work for the\nh100iorientation. It re\rects the behavior of the e\u000bective\nanisotropy shown in Figs. 11(m) through 11(o).\nClose to saturation as well as for small values of M\nbothh111iandh110iare energetically larger than h100i\nidentifying the latter as the easy axis. For interme-\ndiate magnetization, however, the energy of the h111i\nandh110iorientations drops below the h100iorientation,\nequivalent to a change of the e\u000bective anisotropy, i.e., a\nsign change of Ke\u000b. The magnetization interval where\nthe inversion occurs indeed corresponds to the magneti-\nzation range in which the tilted conical phase is observed.19\n02550751000Hint (mT)\n(a)111\n110\n100\n0.00 0.25 0.50 0.75 1.00 1.25\nM (B / f.u.)\n2\n1\n012W (neV/Å3)\n(b)W111 W100\nW110 W100\nFIG. 12. Magnetization and magnetic work illustrating the\n\feld dependence of the e\u000bective anisotropy. (a) Internal mag-\nnetic \feld as a function of magnetization for di\u000berent direc-\ntions. (b) Magnetic work, inferred from magnetization data\nrelative to theh100iorientation as a function of the magne-\ntization.\nV. CONCLUSIONS\nIn conclusion, we reported a comprehensive study of\nthe magnetization and ac susceptibility of the magnetic\nphase diagram of Cu 2OSeO 3. For magnetic \feld paral-\nlel to theh100iaxis in the cubic crystal structure and\nlow temperatures we found clear evidence of the forma-\ntion of two new phases, a tilted conical state and a LTS\nstate identi\fed recently in a SANS study. The mag-\nnetization and susceptibility are thereby in remarkable\nagreement with the SANS data, providing clear ther-\nmodynamic signatures of these two new phases. Com-\nplementary selected speci\fc heat data support these re-\nsults. A detailed analysis of the strength of the magnetic\nanisotropy establishes that the conventional quartic con-\ntribution to the free energy by itself is su\u000ecient to sta-bilize the LTS phase. Detailed measurements exploring\nthe role of di\u000berent sample shapes shed new insights on\nthe role of demagnetizing \felds in the stabilization of\nthe tilted conical state. Taken together, we \fnd that the\nLTS phase in Cu 2OSeO 3represents a thermodynamically\nstable ground state driven by cubic magnetocrystalline\naisotropies, whereas the tilted conical state exists as a\nmetastable phase even for tiny demagnetization factors.\nIt is \fnally instructive to speculate on the more general\nimportance of our observations in Cu 2OSeO 3. As dis-\ncussed in our paper the magnetocrystalline anisotropies\nin the B20 compounds MnSi and Fe 1\u0000xCoxSi do not\nappear to be su\u000ecient to drive the formation of a\nLTS phase. In contrast, recent reports in Co 7Zn7Mn6\nidenti\fed a second skyrmion lattice phase at low\ntemperatures interpreted as a three-dimensional order\narising from the interplay of DM interactions with the\ne\u000bects of frustration.33Judging from the literature this\nmaterial exhibits also a pronounced magnetocrystalline\nanisotropy for the h100iaxes similar to Cu 2OSeO 3.\nIt is therefore tempting to speculate, whether the low\ntemperature skyrmion phase in Co 7Zn7Mn6originates,\nin fact, in the same mechanism we identify in Cu 2OSeO 3,\nhowever in the presence large amounts of disorder. This\nspeculation \fnds further support by the observation\nthat the e\u000bects of frustration, which must be present in\nCu2OSeO 3for the sake of the same analogy, do not seem\nto appear essential for stabilizing the LTS in Cu 2OSeO 3.\nACKNOWLEDGMENTS\nWe wish to thank S. Mayr for support. AC, MH,\nand WS acknowledge \fnancial support through the TUM\nGraduate School. This project has received funding\nfrom the European Research Council (ERC) under the\nEuropean Union's Horizon 2020 research and innova-\ntion programme (grant agreement No 788031). LH\nand AR acknowledge \fnancial support through DFG\nCRC1238 (project C02). MG acknowledges \fnancial sup-\nport through DFG CRC 1143 and DFG Grant GR1072/5.\nMH, AC, AB, and CP acknowledge support through\nDFG TRR80 (projects E1, F2 and F7) and ERC-AdG\n(291079 TOPFIT). MG, AR and CP acknowledge sup-\nport through DFG SPP2137 (Skyrmionics).\n1Naoto Nagaosa and Yoshinori Tokura, \\Topological prop-\nerties and dynamics of magnetic skyrmions,\" Nature Nano.\n8, 899{911 (2013).\n2S. M uhlbauer, B. Binz, F. Jonietz, C. P\reiderer, A. Rosch,\nA. Neubauer, R. Georgii, and P. B oni, \\Skyrmion Lattice\nin a Chiral Magnet,\" Science 323, 915{919 (2009).\n3W. M unzer, A. Neubauer, T. Adams, S. M uhlbauer,\nC. Franz, F. Jonietz, R. Georgii, P. B oni, B. Pedersen,M. Schmidt, A. Rosch, and C. P\reiderer, \\Skyrmion lat-\ntice in the doped semiconductor Fe 1\u0000xCoxSi,\" Phys. Rev.\nB81, 041203(R) (2010).\n4X. Z. Yu, N. Kanazawa, Y. Onose, K. Kimoto, W. Z.\nZhang, S. Ishiwata, Y. Matsui, and Y. Tokura, \\Near\nroom-temperature formation of a skyrmion crystal in thin-\n\flms of the helimagnet FeGe,\" Nature Materials 10, 106{\n109 (2011).20\n5N Kanazawa, Y Nii, X X Zhang, A S Mishchenko, G De Fil-\nippis, F Kagawa, Y Iwasa, Naoto Nagaosa, and Y Tokura,\n\\Critical phenomena of emergent magnetic monopoles in\na chiral magnet,\" Nature Commun. 7, 11622 (2016).\n6I. Kezsmarki, S. Bordacs, P. Milde, E. Neuber, L. M.\nEng, J. S. White, H. M. Ronnow, C. D. Dewhurst,\nM. Mochizuki, K. Yanai, H. Nakamura, D. Ehlers,\nV. Tsurkan, and A. Loidl, \\Neel-type skyrmion lattice\nwith con\fned orientation in the polar magnetic semicon-\nductor GaV 4S8,\" Nature Materials 14, 1116{1122 (2015).\n7S. Bordacs, A. Butykai, B. G. Szigeti, J. S. White, R. Cu-\nbitt, A. O. Leonov, S. Widmann, D. Ehlers, H. A. Krug von\nNidda, V. Tsurkan, A. Loidl, and I. Kezsmarki, \\Equilib-\nrium Skyrmion Lattice Ground State in a Polar Easy-plane\nMagnet,\" Sci. Rep. 7, 7584 (2017).\n8Xiuzhen Yu, Maxim Mostovoy, Yusuke Tokunaga, Weizhu\nZhang, Koji Kimoto, Yoshio Matsui, Yoshio Kaneko,\nNaoto Nagaosa, and Yoshinori Tokura, \\Magnetic stripes\nand skyrmions with helicity reversals,\" Proc. Nat. Acad.\nSci.109, 8856{8860 (2012).\n9S. Ishiwata, M. Tokunaga, Y. Kaneko, D. Okuyama,\nY. Tokunaga, S. Wakimoto, K. Kakurai, T. Arima,\nY. Taguchi, and Y. Tokura, \\Versatile helimagnetic phases\nunder magnetic \felds in cubic perovskite SrFeO 3,\" Phys.\nRev. B 84, 054427 (2011).\n10Ajaya K. Nayak, Vivek Kumar, Tianping Ma, Pe-\nter Werner, Eckhard Pippel, Roshnee Sahoo, Francoise\nDamay, Ulrich K. Roessler, Claudia Felser, and Stuart\nS. P. Parkin, \\Magnetic antiskyrmions above room temper-\nature in tetragonal Heusler materials,\" Nature 548, 561{\n566 (2017).\n11X. Yu, J.P. DeGrave, Y. Hara, T. Hara, S. Jin, and\nY. Tokura, \\Observation of the Magnetic Skyrmion Lat-\ntice in a MnSi Nanowire by Lorentz TEM,\" Nano Letters\n13, 3755 (2013).\n12S. A. Meynell, M. N. Wilson, K. L. Krycka, B. J. Kirby,\nH. Fritzsche, and T. L. Monchesky, \\Neutron study of\nin-plane skyrmions in mnsi thin \flms,\" Phys. Rev. B 96,\n054402 (2017).\n13A. Fert, N. Reyren, and V. Cros, \\Magnetic skyrmions:\nadvances in physics and potential applications,\" Nature\nReviews Materials 2, 17031 (2017).\n14Wanjun Jiang, Gong Chen, Kai Liu, Jiadong Zang, S.G.E.\nte Velthuis, and A. Ho\u000bmann, \\Skyrmions in magnetic\nmultilayers,\" Physics Reports 704, 1 { 49 (2017).\n15R. Wiesendanger, \\Nanoscale magnetic skyrmions in\nmetallic \flms and multilayers: a new twist for spintron-\nics,\" Nature Reviews Materials 1, 16044 (2016).\n16A. Chacon, L. Heinen, M. Halder, A. Bauer, W. Simeth,\nS. M uhlbauer, H. Berger, M. Garst, A. Rosch, and\nC. P\reiderer, \\Observation of two independent skyrmion\nphases in a chiral magnetic material,\" Nature Physics 14,\n936 (2018), advance online publication on 22 June 2018.\n17S. Seki, X. Z. Yu, S. Ishiwata, and Y. Tokura, \\Observa-\ntion of Skyrmions in a Multiferroic Material,\" Science 336,\n198{201 (2012).\n18T. Adams, A. Chacon, M. Wagner, A. Bauer, G. Brandl,\nB. Pedersen, H. Berger, P. Lemmens, and C. P\reiderer,\n\\Long-Wavelength Helimagnetic Order and Skyrmion Lat-\ntice Phase in Cu 2OSeO 3,\" Phys. Rev. Lett. 108, 237204\n(2012).\n19Stefan Buhrandt and Lars Fritz, \\Skyrmion lattice phase\nin three-dimensional chiral magnets from monte carlo sim-\nulations,\" Phys. Rev. B 88, 195137 (2013).20A. Bauer and C. P\reiderer, \\Topological Structures in Fer-\nroic Materials: Domain Walls, Vortices and Skyrmions,\"\n(Springer International Publishing, 2016) Chap. Generic\nAspects of Skyrmion Lattices in Chiral Magnets, p. 1.\n21L. D. Landau and E. M. Lifshitz, Course of theoretical\nphysics, vol. 8 (Pergamon Press, 1980).\n22U. K. R o\u0019ler, A. N. Bogdanov, and C. P\reiderer, \\Sponta-\nneous skyrmion ground states in magnetic metals,\" Nature\n442, 797 (2006).\n23H. Wilhelm, M. Baenitz, M. Schmidt, U. K. R o\u0019ler, A. A.\nLeonov, and A. N. Bogdanov, \\Precursor Phenomena at\nthe Magnetic Ordering of the Cubic Helimagnet FeGe,\"\nPhys. Rev. Lett. 107, 127203 (2011).\n24L. Cevey, H. Wilhelm, M. Schmidt, and R. Lortz,\n\\Thermodynamic investigations in the precursor region of\nFeGe,\" Phys. Status Solidi B 250, 650 (2013).\n25A. Bauer and C. P\reiderer, \\Magnetic phase diagram of\nMnSi inferred from magnetization and ac susceptibility,\"\nPhys. Rev. B 85, 214418 (2012).\n26H. C. Wu, T. Y. Wei, K. D. Chandrasekhar, T. Y. Chen,\nH. Berger, and H. D. Yang, \\Unexpected observation of\nsplitting of skyrmion phase in Zn doped Cu 2OSeO 3,\" Sci-\nenti\fc Reports 5(2015), 10.1038/srep13579.\n27(2018), a. Stefancic, S. Moody, T.J. Hicken, T.M. Birch,\nG. Balakrishnan, S.A. Barnett, M. Crisanti, J.S.O. Evans,\nS.J.R. Holt, K.J.A. Franke, P.D. Hatton, B.M. Hud-\ndart, M.R. Lees, F.L. Pratt, C.C. Tang, M.N. Wil-\nson, F. Xiao, T. Lancaster, Origin of skyrmion lattice\nphase splitting in Zn-substituted Cu 2OSeO 3, unpublished,\narXiv/1807.04641.\n28A. Bauer, M. Garst, and C. P\reiderer, \\History de-\npendence of the magnetic properties of single-crystal\nFe1\u0000xCoxSi,\" Phys. Rev. B 93, 235144 (2016).\n29Y. Tokunaga, X. Z. Yu, J. S. White, H. M. R\u001cnnow,\nD. Morikawa, Y. Taguchi, and Y. Tokura, \\A new class of\nchiral materials hosting magnetic skyrmions beyond room\ntemperature,\" Nature Commun. 6, 7638 (2015).\n30K. Karube, J. S. White, N. Reynolds, J. L. Gavilano,\nH. Oike, A. Kikkawa, F. Kagawa, Y. Tokunaga, H. M.\nRonnow, Y. Tokura, and Y. Taguchi, \\Robust metastable\nskyrmions and their triangular{square lattice structural\ntransition in a high-temperature chiral magnet,\" Nature\nMaterials 15, 1237{1242 (2016).\n31P. Milde, D. K ohler, J. Seidel, L. M. Eng, A. Bauer,\nA. Chacon, J. Kindervater, S. M uhlbauer, C. P\reiderer,\nS. Buhrandt, C. Sch utte, and A. Rosch, \\Unwinding of a\nSkyrmion Lattice by Magnetic Monopoles,\" Science 340,\n1076{1080 (2013).\n32S. P ollath, J. Wild, L. Heinen, T. N. G. Meier, M. Kro-\nnseder, L. Tutsch, A. Bauer, H. Berger, C. P\reiderer,\nJ. Zweck, A. Rosch, and C. H. Back, \\Dynamical defects\nin rotating magnetic skyrmion lattices,\" Phys. Rev. Lett.\n118, 207205 (2017).\n33K. Karube, J. S. White, D. Morikawa, C. D. Dewhurst,\nR. Cubitt, A. Kikkawa, X. Yu, Y. Tokunaga, T. Arima,\nH. M. Ronnow, Y. Tokura, and Y. Taguchi, \\Disordered\nskymrion lattice phase stabilized by magnetic frustration\nin a chiral magnet,\" Science Advances 4, eaar7043 (2018).\n34T. Schwarze, J. Waizner, M. Garst, A. Bauer,\nI. Stasinopoulos, H. Berger, C. P\reiderer, and\nD. Grundler, \\Universal helimagnon and skyrmion exci-\ntations in metallic, semiconducting and insulating chiral\nmagnets,\" Nature Materials 14, 478{483 (2015).\n35P. Milde, E. Neuber, A. Bauer, C. P\reiderer, H. Berger,21\nand L. Eng, \\Heuristic description of magnetoelectricity of\nCu2OSeO 3,\" Nano Lett. 16, 5612 (2016).\n36S. L. Zhang, A. Bauer, D. M. Burn, P. Milde, E. Neuber,\nL. M. Eng, H. Berger, C. P\reiderer, G. van der Laan,\nand T. Hesjedal, \\Multidomain Skyrmion Lattice State in\nCu2OSeO 3,\" Nano Lett. 16, 3285 (2016).\n37S. L. Zhang, A. Bauer, H. Berger, C. P\reiderer, G. van der\nLaan, and T. Hesjedal, \\Resonant elastic x-ray scattering\nfrom the skyrmion lattice in Cu 2OSeO 3,\" Phys. Rev. B 93,\n214420 (2016).\n38I. Stasinopoulos, S. Weichselbaumer, A. Bauer, J. Waizner,\nH. Berger, S. Maendl, M. Garst, C. P\reiderer, and\nD. Grundler, \\Low spin wave damping in the insulating\nchiral magnet Cu 2OSeO 3,\" Appl. Phys. Lett. 111(2017),\n10.1063/1.4995240.\n39J. A. Mydosh, Spin glasses: an experimental introduction\n(Taylor & Francis, 1993).\n40S. M uhlbauer, A. Heinemann, A. Wilhelm, L. Karge,\nA. Ostermann, I. Defendi, A. Schreyer, W. Petry, and\nR. Gilles, \\The new small-angle neutron scattering instru-\nment SANS-1 at MLZ|characterization and \frst results,\"\nNuclear Instruments and Methods in Physics Research Sec-\ntion A: Accelerators, Spectrometers, Detectors and Asso-\nciated Equipment 832, 297{305 (2016).\n41A. Aharoni, \\Demagnetizing factors for rectangular ferro-\nmagnetic prisms,\" J. Appl. Phys. 83, 3432 (1998).\n42M. I. Youssif, Bahgat A. A., and Ali I. A., \\AC Mag-\nnetic Susceptibility Technique for the Characterization of\nHigh Temperature Superconductors,\" Egypt. J. Sol. 23,231 (2000).\n43A. Bauer, A. Chacon, M. Wagner, M. Halder, R. Georgii,\nA. Rosch, C. P\reiderer, and M. Garst, \\Symmetry break-\ning, slow relaxation dynamics, and topological defects at\nthe \feld-induced helix reorientation in mnsi,\" Phys. Rev.\nB95, 024429 (2017).\n44M. Balanda, \\AAC Susceptibility Studies of Phase Tran-\nsitions and Magnetic Relaxation: Conventional, Molecular\nand Low-Dimensional Magnets,\" Acta Physics Polonica A\n124, 964 (2013).\n45J. Wild, T.N.G. Meier, S. Poellath, M. Kronseder,\nA. Bauer, A. Chacon, M. Halder, M. Schowalter, A. Rose-\nnauer, J. Zweck, J. M uller, A. Rosch, C. P\reiderer, and\nC. H. Back, \\Entropy-limited topological protection of\nskyrmions,\" Science Advances 3, e1701704 (2017).\n46(2018), F. Qian, L.J. Bannenberg, H. Wilhelm, G.\nChaboussant, L. M. Debeer-Schmitt, M. P. Schmidt, A.\nAqeel, T.T.M. Palstra, E. Br uck, A.J.E. Lefering, C. Pap-\npas, M. Mostovoy, and A. O. Leonov, arXiv/1802.02070v2,\nunpublished.\n47N. Akulov, Z. Phys. 100, 197 (1937).\n48J. H. Yang, Z. L. Li, X. Z. Lu, M. H. Whangbo, Su-\nHuai Wei, X. G. Gong, and H. J. Xiang, \\Strong\nDzyaloshinskii-Moriya Interaction and Origin of Ferro-\nelectricity in Cu 2OSeO 3,\" Phys. Rev. Lett. 109 (2012),\n10.1103/PhysRevLett.109.107203.\n49S. V. Grigoriev, A. S. Sukhanov, and S. V. Maleyev, \\From\nspiral to ferromagnetic structure in B20 compounds: Role\nof cubic anisotropy,\" Phys. Rev. B 91, 224429 (2015)."    },    {        "title": "2104.01333v2.Predicting_synthesizable_cobalt_and_manganese_silicides__germanide_with_desirable_magnetic_anisotropy_energy.pdf",        "content": "1Predictingsynthesizablecobaltandmanganesesilicides,germanidewith\ndesirablemagneticanisotropyenergy\nZe-JinYang\nSchoolofScience,ZhejiangUniversityofTechnology,Hangzhou,310023,China\nzejinyang@zjut.edu.cn\nAbstract\nThenanoparticleCo3Si(P63/mmc)displaysremarkablemagnetism[Appl.Phys.\nLett.108,152406(2016)],wethussearchedcobaltsilicidesandseveralphasesare\nsearchedincludingaCmcmwith60meV/atomlowerthanthatofP63/mmc.A\nlower-energyComR3(-7.03eV/atom)ispredicted,whoseenergyishigherthan\nthatofknownP63/mmc(-7.04eV/atom)butislowerthanthatofmmF3(-7.02\neV/atom).Threesmall-magnetismlow-energyFe5Si3structuresaresearchedwith\nenergies30meV/atomlowerthanthatofexperimentalP63/mcm.Thestronglattice\nshapedependenceofmagnetocrystallineanisotropyenergy(MAE)isstudiedthrough\nX5Si3(X=Mn,Fe,Co).Thebuilding-blockshapeandenergyorderofcobaltsilicideis\ndominatedbyCoP63/mmc,m3mF,m3R,respectively.TheCo3CandCo3Snhave\npositiveformationofenergy,thusonlyCo3Gehassimilarstructureswiththoseof\ncounterpartsofCo3Si.Severallow-energyperfectornearly-perfecteasy-axis/plane\nMAEMn3Si,Mn5Si2,andMn5Si3structuresaresearchedandpresentimportant\napplication,asisalsothecaseinGe-containingcounterparts.AstructureI4122with\nenergy300meV/atomlowerthanthatofexperimentalMn5Si3P63/mcmissearched.2I.Introduction\nMagneticmaterialsplayanimportantroleinourlife,suchasharddiskinlaptop.\nMagnetismcomprisesofdiverseresearchfieldsinphysics,forinstancegiant\nmagnetoresistanceeffect,colossalmagnetoresistanceeffect,magnetocaloriceffect,\nmagneticexchangebiaseffect,magneticproximityeffect,spintronics,andspin\nSeebeckeffect.Recently,themostfrequentlyusedpermanentmagnetsare\nNd2Fe14B\randSmCo5-basedcompounds,bothcontainingrare-earthelement.Dueto\nlimitedrare-earthresources,thereforeitisnecessarytosynthesizematerialswithout\nrare-earthelement[1].Magnetocrystallineanisotropyenergy(MAE)playsakeyrole\ninthedevelopmentofmagneticstoragemedia.DifferentMAEmagnitudeshave\nextensiveapplicationsinthefieldsofsoftandhardmagneticmaterials.\nNanoscaleCo3Si(P63/mmc)issynthesized[2]anddisplayedexcellenteasy-plane\nMAEinitsnanoparticleform,basedonwhichtheauthorsconcludethatapath\ntowardsfabricationofnanoparticlematerialswitheasy-planeanisotropies,valuable\nenergyproducts,andinparticularcontainingearth-richsubstanceissuccessfullybuilt.\nInfact,metastablehexagonalCo3Si(P63/mmc)isobservedlongtimeago[3-5].\nHowever,thediscrepancybetweentheirtheoreticalcalculationfortheidealCo3Si\nnanoparticale(-64Merg/cm3)andexperimentalestimation(48Merg/cm3)anisotropy\nislarge,possiblyduetotheexperimentalsurplusofthe10vol.%Cointhe\nnanoparticlecomposite.Unexpectedly,theextraCofailstoformtheother\nstoichiometrycompoundssuchasCo4Si,etc.Consideringsuchimportantconclusion,\nitisverynecessarytofurtherstudythecobaltsilicidescarefully.Moreover,3nonequilibriumfabricationtechniqueusuallysynthesizesmetastablephaseswithvery\nlargeMAE.Therefore,weusetheadvancedcrystalstructurepredictionmethodto\nsearchthepossiblelow-energystructure.Moreover,theobservedCo4Sialsoisa\nmetastablephasebutitsstructureisstillunknown[4].Threedifferentphasesare\nobservedinCo2Sitillnow,itisα-Co2Si,β-Co2Si,γ-Co2Si,respectively[3-5].These\nphenomenasuggestthattherearemanymetastablephasesintheselowCo:Si\nstoichiometryratiocompounds.\nThesamegroup[2]alsoobservedthatnanoparticlesMn5Si3presentsappreciable\nmagnetocrystallineanisotropyconstants(K1=6.2Mergs/cm3at300Kandat12.8\nMergs/cm3at3K)[6].Mn-basedstructuresbecameanimportantstudyfocusfromthe\nviewpointsofdesigningrare-earth-freematerialswithusefulspin-related\napplications[7].Thus,wecarefullysearchedtheexperimentallyknownbutsimple\nMn-basedcompounds,suchasMn3Si[8],Mn5Si2[9],Mn5Si3[8],Mn6Si[10],Mn9Si2[11].\nWedidn’tsearchtheotherstoichiometrybecausethesearchandMAEcalculationare\nextremelytime-consuming.Consideringtheimportantstoichiometryof5:3,wealso\nsearchedtheFe5Si3astheprevioussearchmightmiss[1]someenergeticallysimilar\nstructures.Experimentalstructurehas53atomsperunitcell(Mn85.5Si14.5)[10]in\nMn6Si,butitis56atomsforthe8-unitcell,thesameauthorsalsofound186atoms\nfor(Mn81.5Si18.5)[11]inMn9Si2,whereasitis176atomsfor16-unitcell.Thusitis\ndifficulttocomparethesearchedperfectstructureswiththeirdata.Inthisstudy,we\nthoroughlysearchedthebinarymagneticcobaltandmanganesesilicidesaswellas\nFe5Si3inordertofindtheidealMAE,fortunately,oursearchedmanystructureshave4veryinterestingpropertiesinthediversemagneticfield.\nII.Methods\nWesearchedthestructuresbyCALYPSO(CrystalstructureAnaLYsisby\nParticleSwarmOptimization)code[12,13].Theobtainedstructuresareabinitio\noptimized[14,15]byprojector-augmentedwavemethod[16].Thespin\rpolarized\ngeneralizedgradientapproximationandthePerdew-Burke-Ernzerhoffunctionalare\nusedtomodeltheexchange-correlationenergy[17].Thestructureisoptimizedwith\ntheelectronicconvergencebelow10-7eV,thek-pointdensityisbelow2π×0.02Å−1in\ntheBrillouin-zone.Theforceontheatomisconvergedtolessthan0.01eV/Åduring\ntherelaxation.Energycutofffortheplan-wavebasisis350eV.Theseparametersare\nsufficienttoensurethereliabilityofthecalculatedresultsbasedonourcalculations\nonironsilicides[1].Theon-sitecoulombinteractiontermU(andJ)ontheenergyis\nneglectedduringallthestructurecomputation[14,15]basedontheprevious\nconclusionofironsilicides[1].\nIII.Resultsanddiscussion\nA.MAEofexperimentalnanoparticlesizeCo3Si\nWedefinethelowest-energyaxisasthereferenceaxisinthispaper.\nExperimentally[2]synthesizednanoscaleCo3Si(P63/mmc)withaneasy-plane\nanisotropywithahighK1=-64Merg/cm3(-6.4MJ/m3)andmagneticpolarizationof\n9.2kG(0.92T),whichisconfirmedbytheircalculated64-atomCo3Sinanoparticle.\nInaddition,theyalsofoundthatMn5Si3nanoparticlealsopresentsexcellent\nproperties[6].OurcalculatedCo3Sishowsa=b=4.9794Å,c=3.9755Å,substantially5differentwiththeexperimentaldataa=b=4.99Å,c=4.497Å.Ourcalculatedabsolute\ntotal-energydifferencebetweenspinorientationsinthe(100)and(001)directions\nyieldsaMAEof2.0891meVperunitcell(containing2formulaunitsor8atoms)for\nthebulkCo3Sicrystal,whichcorrespondsto|K1|=39.21Merg/cm3(or261.1412\nμeV/atom),farsmallerthantheexperimentallyreportedvalue-3.9meV(K1=-64\nMerg/cm3),whoseabsolutemagnitudeislargeandcomparabletothatoftypical\nrare-eartheasy-axisintermetallics,assaidinthatpaper.AnotherP63/mmc(Co3Si),\nwithanaverageenergyof-6.8931eV/atom,showsshortenedlatticeband\nslightly-changedaandc,asaresultwhichinducesanearly-perfecteasy-planeMAE,\nwithE001=80.28andE100=0.06μeV/atom(orenergiesalongcandaaxes),\nrespectivelyandsimilarlyhereinafter.Thus,theperfectbulkcrystalfailedtoobtain\nthecorrespondingmagnetismofnanoparticledueprobablytothesurfaceeffect.\nTorevealthepossibleoriginofthegiantaxialenergydifference,thenanoparticle\nisconstructedbyscreeningtheneighborinteractionthroughthelongvacuumlayer\nlargerthan10Å.TableS1showsthattheperfect8-atom,32-atom,64-atom\nnanoparticlecouldproduceMAEwithvaluesabout0.1,0.07,0.12meV/atom,\nrespectively,whichstillcouldn’trepeattheexperimentalMAE(-3.9meVperunitcell\nor-0.4875meV/atom).Wedidn’tcalculatethecaseof128-atomunitcellbecauseits\natomicarrangementistotallysamewiththecasesof8/32/64-atom.Onepossible\nexplanationisthattheexperimentallysynthesizedstructureexistslargedistortion\nalongthelatticeparametercorientation.DistortionusuallyinduceslargeMAEbutit\nisprobablymeaninglessinthepracticalindustrialapplication.Moreimportantly,6distortionusuallycouldn’tproduceperfectMAE.Inaddition,theirlatticeparameterc\nisalsocouldn’tbeexplainedfromtheunknownsurfacesubstanceorinternaldefect\nstructure.Thesurfaceinteractionisalmostimpossibletoelongatethelattice\nparametercwithsuchlargevalue,fromabout4.0to4.5Å,thusweignoretocalculate\ntheMAEatthecaseofc=4.5Åbecausesuchstructureishypotheticalbyuniaxial\nstrainandisalsounstable.\nTheexperimentalCo3SiP63/mmchasa~10%elongationoflatticecwhich\ninducesthegiantdistortionoflatticeshapeandbreaktheatomicequilibriumstatus.\nWetestitbytheX5Si3(X=Mn,Fe,Co),asisshowninTableS2,inwhichthestrong\nlatticeframeworkoratomicoccupancydependenceisobserved,theaverageenergy\ndifferencebetweeneasyandhardaxesofthethreelatticehasintrinsicvalue,suchas\nthestableCo5Si3hasanMAEvalueof~13μeV/atom,substitutionsofCobyMnand\nFeproduce~29(Co5Si3)and~23(Fe5Si3),respectively.ForthecaseofMn5Si3,those\ncorrespondingvaluesare~39and~39(Co5Si3)and~56(Fe5Si3).Forthecaseof\nFe5Si3,theyare~157and~88(Mn5Si3)and94(Co5Si3),respectively.\nB.Thesearchedlower-energystructuresthanthatofCo3SiP63/mmc\nWeonlysearchedthestoichiometryofCo:Si>1inordertofindthestructures\nwithlargemagnetism.Thecurrentsearchobtained5lowerenergystructuresthanthat\nofexperimentallysynthesizedCo3SiP63/mmc(-6.9eV/atom),inwhichCmcm\nstructureisabout60meV/atomlowerthanthatofP63/mmc.Theotherlower-energy\nphasesincludePmmn,(-6.9143),I4/mmm(-6.9197),I4/mcm(-6.9025),P21(-6.9071\neV/atom),asisshowninTable1.Infact,therearealsoseveralphaseshaveonly7slightlyhigherenergiesthanthatofP63/mmc.Inotherwords,themetastablephase\nwithenergyaboutseveraldozensofmeV/atomlowerthanthatofexperimentalphase\northesearchedlowest-energyphase(probablybe)stillcouldbesynthesized.\nThecurrentlycalculatedMAEofCo3SiP63/mmcis261.14μeV/atomwithan\neasy-axisenergydistribution,despitewhichisfarsmallerthanthatofpreviously\ncalculated487.5andmeasured365.625μeV/atombut~260isstillaverylargevalue.\nThustheMAEofCo3SiP63/mmc(261.14μeV/atom)isnotcomparablebutisfar\nsmallerthanthatoftypicalrare-eartheasy-axisintermetallicscompounds.\nCo3SiP63/mmc(a=b=4.9794,c=3.9754Å)isthesamelatticeframeworkwith\nthatofCoP63/mmc(a=b=2.507,c=4.069Å),thatis,the2×2×1supercellofCois\na=b=5.014Å,thusCo3SiandCohavetotallysamelatticewiththeonlyexceptionof\ntheslightdistortionduetothesubstitutionofCobySi.ProjectionsoftheCo3Si\nP63/mmctowardsaandbaxesaretotallysameowingtotheinversionsymmetry\noperationnature,thustheMAEalongaandbaxesareidentical.Moreover,the\nmixtureofnanoparticleCo3Siwithabout10vol.%ɛ-Comakestheresolutionofthis\nquestionverydifficult.ThebuildingblockofP63/mmcistetrahedronwiththe\ndistanceofCo~Si=2.4454ÅandCo~Co=2.4671Å.ThebuildingblockofCmcmisa\ncubicbodybuttheSidoesn’tdistributeatthecubiccenter,thuspresentingtwo\ndifferentkindsofdistance,itis2.4455and2.4446Åinitsfirstshellornearest\nneighbor,namely,aslightlydistortedbulkcenteredcubic(BCC)unit.\nCmcm-A(namedasA…becausemanystructureshavesamesymmetrybut\nshowobviouslydifferentenergies)isthesearchedlowest-energyphasewith8E010=0.24andE100=28.46μeV/atom,respectively.Itsbuildingblockshapeismost\nsimilarwiththatofP63/mmc.Indetail,thetwoneighboringlayer“Co3Si”hasa\ntorsionangle60ﾟinP63/mmcbutitiszerodegreeinCmcm-A.Moreover,thegeneral\nshapeofbuildingblockinCmcm-AismoresimilarwithaBCC,thedistanceof\nCo9~Co3=Co9~Co14=2.7097Å,Co11~Co14=Co11~Co3=2.547Å.Thedistancebetween\nthecentralSiandits8first-nearestneighborsareCo2,3~Si1=Co13,14~Si1=2.3227Å,\nCo5,9~Si1=2.3796Å,Co7,11~Si1=2.5162Å.ItsCo1~Si2=Si1~Co8=4.1856Å\ncorrespondstothe4.2992and4.3254ÅinP63/mmc.Thusthephasetransitionfrom\nP63/mmctoCmcm-AinCo3Siispossibleunderhighpressure.ThecentralSideviates\nthegeometricpositionevidentlyinCmcm-BrelativetothatofCmcm-A,forinstance,\nthefourpairsCo~Sidistancesare2.3837Åinfirstshellbuttheotherfouronesare\n2.482ÅinCmcm-B.Meanwhile,forthecaseofthesecondshellorsecondnearest\nneighbors,theCo~Sidistancesattheshortfirst-shellside(2.3837Å)isalsoshort\n(2.3372Å),viceversa,theotherside(2.482Å)correspondsto2.8817Å,theother\npairCo~Sidistanceswithinthesamehorizontalplaneshowsmallvariation,3.1959\nversus3.1963Å,onlythetwototally-symmetricCopresentidenticalCo~Sidistance,\nsuchasthedistancebetweenthetopandbottomofthebuildingblockis2.5666Å.\nHowever,Cmcm-B(β=90.0002ﾟ)andCmcm-A(β=90ﾟ)havelargeMAEdifference\ndueprobablytothedistortedangle.Cmcm-Cisahigher-energylayeredstructure,\nwhosetwoSilayersareseparatedbythreeColayers,projectionstowardsaandcaxes\nshowsimilarlayerseparationfeature.Itsbuildingblockisnottetragonalshapebut\ntrianglenestedshape.Co~Sidistances,whetherinthesameplaneorthenearest9neighbors,arenotexactlyanisoscelestriangle.Theβis90.0051inCmcm-D,\nmeaninganevenlargerdistortion,whosebuildingblockismorecloserwiththatof\nCom3mF,asaresultitsenergyishigherthanthestructurespackedbyCoP63/mmc.\nThesearchedCo3SiI4/mmm-A,(-6.9197eV/atom)isabout19meV/atomlower\nthanthatofP63/mmcandhasaperfecteasy-planeMAE,withavalueofE010=141.42\nμeV/atomalongthehardbaxis,whichisevidencedbytheidenticalprojections\ntowardsaandcaxesandthelatticeparametersa=c=5.0558,b=6.6642Å.\nConsideringthemultiplicityofthelatticeshapeweonlycalculatetheaxialenergy\nalongthreelatticeparametersasthisisthemostextensivelyuseddirections.\nThesearchedCo3SiI4/mmm-B(-6.9181eV/atom)isabout18meV/atomlower\nthanthatofP63/mmcandalsoshowstotally-perfecteasy-planeMAEwithavalueof\nE100=221.61μeV/atomalongthehardaaxis.ItsperfectMAEcouldbeattributed\npartiallytotheirtotally-symmetricarrangementsatthebothsidesofthemirrorthatis\nformedbythec-axisparallellinealongtheanglebisectorbetweenaandbaxis.In\nfact,weobtainseveralstructureswithsamesymmetryI4/mmmbuttheirMAEare\nevidentlydifferentowingtotheslightstrain,whichwillcauselargeaxial-energy\ndifference,suchasE001=153.08andE010=25.22inI4/mmm-CbutE001=0.03and\nE010=148.42μeV/atominI4/mmm-D,asisshowninTable1.Forsimplicity,weignore\ntocomparetheirbonddistancesastheyaregenerallysimilarwiththecasesofCmcm.\nTheotherstructuresalsohaveI4/mmmsymmetryandlargemagnetismbutpresent\nnon-perfectMAE,indictingthecomplicatedMAEfeature.10ThebuildingblockofCo3SiPmmn(-6.9143eV/atom)issimilarwiththatofCo\nP63/mmc,whoseMAEdifferseachotheralongthethreeaxesandshowslarge\ndiscrepancy.\nTheI4/mcm(-6.9025eV/atom)hasnearlysameenergywiththatofP63/mmc,\nwhoseE001=2.8andE100=211.67μeV/atom,respectively.Thisisalsoastructurewith\ntheMAElargerthan200μeV/atomandshowsnearly-perfecteasy-planeanisotropic\ndistributions.Despitethebuildingblockhaslargedistortionrelativetotheperfect\nBCC,theinter-atomicdistancestillhascertainsymmetry,forexample,\nCo1,6,7,10-Si1=2.4223,Si1-Si2,3=3.6626,Co2,5,9,12-Si1=2.7471,Co3,4,8,11-Si1=2.2918Å,\nrespectively.Thatis,theaveragedistancebetweenCo2,3,8,9~Si1andCo4,5,11,12~Si1are\nidenticalinthefirstshell,whichcontributestothelowenergyofthelattice.Moreover,\nthesecond-shellatomalsoshowshighlysymmetricarrangementsaroundtheSi1,\nwhichmightbetheoriginofthenearly-perfectMAE.Inaword,theMAEindeed\ndependsontheatomicarrangements.\nP21(-6.9071eV/atom)showsadistortedfirstandsecondshellsandthuspresents\n8/6differentCo-Sidistancesinthefirst/secondshells,respectively,whosefirst-shell\ndistancecouldgenerallybedividedintotwodifferentsortswithvaluesofabout2.38\nand2.48Å,respectively,whereasthesecond-shelldistancepresentslesssimilarities,\nincludingCo1-Si=2.5398,Co13-Si=2.4813,Co6-Si=3.048,Co8-Si=3.22,\nCo7-Si=3.1413,Co9-Si=2.2983Å,respectively.AlloftheneighborsoftheSiare\nsurroundedbytheCoatoms,whichmightcontributestoitslowerenergydespitethe\nlargedistortion.ItsaxialenergiesareE001=81.08andE100=137.16μeV/atom,11respectively,inaccordancewiththecaseofthelargely-disorderedatomic\narrangements.\nmPm3-A(-6.8891eV/atom)presentsnearly-perfecteasy-axisMAEwithvalues\nofE010=71.9andE100=72.1μeV/atom,respectively,whereasmPm3-B(-6.8876\neV/atom)showsperfecteasy-planeMAEwithavalueofE100=32.14μeV/atom.\nP4/mbm-A(-6.8976eV/atom)alsoshowsnearly-perfecteasy-planeMAEwith\nvaluesofE001=2.14andE100=144.51μeV/atom,respectively.Theothersearched\nCo3SiisshowninTableS3,inwhichmmF3-A(-6.8836eV/atom)showsgiant\nMAEwithrespectivevaluesofE001=375.13andE010=877.91μeV/atom.Mostof\nthesearchedlow-energyCo3Sihasthesimilarbuildingblockswiththatofstable\nCoP63/mmc,whereasthemajorityofthehigh-energyCo3Sihasthesimilar\nbuildingblockswiththatofmetastableCommF3,namely,thebuilding-block\nshapeoftheCodominatestheatomicarrangementandenergyorderofCo3Si.\nSinceCo3Sicouldpresentdiversestructureswithvariousmagnetismwetherefore\nalsosearchedCo3C,Co3Ge,Co3Sn,respectively.Unfortunately,onlyCo3Gehas\nnegativeformationofenergy.\nC.Thesearchedhigher-energystructuresofCo3SithanthatofP63/mmc\nThefollowingstructuresareseveralmeV/atomhigherthanthatofexperimental\nP63/mmcandmightalsobesynthesized.Asame-symmetrystructureP63/mmc-B\n(Ef=-6.8931eV/atom)withnearly-perfecteasy-planeMAEissearched,whose\nE001=80.28andE100=0.06μeV/atom,respectively.mPm3-A(Ef=-6.8891eV/atom)\npresentsnearly-perfecteasy-axisMAEwithenergiesofE010=71.9andE100=72.112μeV/atom.mPm3-Bshowsperfecteasy-planeMAE(Ef=-6.8876eV/atom)withan\nenergyofE100=32.14μeV/atom.P4/mbm-Astillpresentsnearly-perfecteasy-plane\nMAE(Ef=-6.8976eV/atom)withenergiesofE001=2.14andE100=144.51μeV/atom,\nrespectively.TheserichMAEcouldsatisfythediverseindustryrequirements.\nD.Predictedlow-energystructureofCo3Ge\nThesymmetryofthesearchedlow-energyCo3Gestructureisgenerallysame\nwiththatofCo3Si,asisshowninTable2andS4.Thesearchedtwolowest-energy\nstructuresalsoareCmcm(-6.5163)andI4/mmm(-6.4817eV/atom),withanenergy\ndifferenceof35meV/atom.Thegeneralbondlength(orsidelength)ofthemost\nstableCmcmstructureisabout2.59and2.48Åintheunit-formulaparallelogram,and\nthelongerCo-Gedistanceisabout4.4Å,asisplottedbytheredline.Weignoreto\nplotthebuildingblockasitissimilarwiththatofCo3Si.TheE010=0.24and\nE100=28.46μeV/atominCo3SiCmcmcorrespondstoE010=14.91andE100=32.19in\nCo3Ge.Similarly,aperfectandlargeeasy-axisMAE(E001=E010=261.14μeV/atom)in\nCo3Si(P63/mmc)correspondstoE001=332.58andE010=85.53μeV/atominCo3Ge\n(P63/mmc).P63/mmchassimilaratomicarrangementwiththatof1PinCo3Gebut\nthelatterpresentsnearly-perfecteasy-axisMAEwithvaluesofE010=116.38and\nE100=114μeV/atom,andtheenergyof1PislowerthanthatofP63/mmc,witha\ndifferenceofabout15meV/atom,stillrevealingthestrongatomiccoordinate\ndependenceofMAE.\nTwoCo3GeI4/mmmstructureshaveperfecteasy-planeMAE,withvaluesof\nEf=-0.0795eV/atomandE001=157.06μeV/atom,Ef=-0.0524eV/atomand13E010=9.6218μeV/atom,respectively.Infact,manyI4/mmmstructuresshow\nnearly-perfecteasy-axis/planeMAE,asisshowninTableS4,suchasonestructure\nshowsE001=156.9andE010=156.95μeV/atom,respectvely,whereasallofthese\nI4/mmmstructureshavenearlyidenticalenergies,whichmightbeagreatchallenge\nduringexperimentalsynthesization.\nPmmm(Ef=-0.0728eV/atom)hasnearly-perfecteasy-axisMAEwithvaluesof\nE001=122.41andE100=122.58μeV/atom,respectively.Theothersame-symmetry\nstructures(Pmmm)areshowninTableS4,whichfurtherdemonstratesthattheMAE\nisstronglydependentontheatomiccoordinate.Similarly,thefollowingstructures\nhaveenergiesabout70meV/atomhigherthanthatoflowest-energystructureCmcm\nandalsomightbesynthesized,suchasCmmmpresentsmediumMAEwithvaluesof\nE010=25.5andE100=26.3,P212121hasE010=24.15andE100=23.25,C2/mhas\nE001=26.18andE100=23μeV/atom,andsoon,thesenearly-perfecteasy-axisMAE\nstructureshavepotentialapplicationsinthelow-magnetismrequirementfield.Several\nstructuresshownearly-perfecteasy-planeMAEsuchasCmpresentsE001=40.92and\nE100=2.64,Ama2presentsE001=269.05andE100=0.78,C2221presentsE001=112.07,\nE010=4.75μeV/atom,andsoon.C2/mshowsrelativelylargeMAEwithvaluesof\nE010=42.56andE100=108.01μeV/atom,respectively,alloftheabovestructuresmight\nbesynthesized.Inaword,bothCo3GeandCo3Sicouldcrystallinemanylow-energy\nstructureswithdiversecoercivitymagnetism.Theselow-energyCo3Gestructuresare\nalsodominatedbythebuildingblockofCoP63/mmc,asisplottedbytheredline.\nE.PredictedCostructures14Todeterminethecommonbuildingblockshapes,wefurthersearchedtheCo\nstructure.Manylow-energyCostructuresaresearched,asisshowninTable2,in\nwhichonenewphaseispredicted,mR3(-7.0306eV/atom),whoseenergyiswell\nlocatedattheintermediaterangeofthetwoexperimentalphasesmFm3(-7.0206)and\nP63/mmc(-7.0398eV/atom).TwonewphasesincludingP42/mnm(-6.997)andCmca\n(-6.9972eV/atom)aresearchedwithenergiesof25meV/atomhigherthanthatof\nmFm3.Themagneticmomentare1.623,1.669,1.65μBforP63/mmc,m3Fm,and\nm3R,respectively.Allofthesestructureshavesimilarbuildingblockswiththe\nexceptionofcertaindistortion.SiandChavecomplicatedstructuresandalsohave\nbeensearchedformanytimesbytheresearcherinrecentyears,thusitisnot\nnecessarytosearchthemagain.\nF.PredictedcobaltsilicidesexcludingCo3Si\nFigure1showstheEfofthebinarycobaltsilicides.Forsimplicity,weconnect\nthelowest-energystructuresdirectlybecausewhethersomeofthesearched\nlowest-energystructurescouldbesynthesizedornotdependsonmanyfactors,such\nasitsownenergy,thesynthesizedtechniques,thedecomposedproducts,andsoon.As\nisshowninTableS5,CoSicrystallinesinP213atambientconditions,wesearched\nonemetastablephaseP21/c,itisabout30meV/atomhigherthanthatofP213,\nunfortunatelyitstillpresentszeromagnetism.Co2Sistabilizes[18]inPnma\n(Ef=-0.4501eV/atom)andshowsE001=32.03andE010=18.96μeV/atom,respectively.\nWepredictedonemetastableP2/m(Ef=-0.4011eV/atom)withnearly-perfect\neasy-axisenergiesofE010=23.45andE100=20.42μeV/atom,respectively.Theother15metastablestructureis50meV/atomhigherthanthatofPnma.\nThesearchedtwolowest-energyCo3Si2structureshavenegligiblemagnetism.\nAllofthefollowingstoichiometriesarelargerthan100meV/atominenergyin\ncomparisontotheirrespectivelowest-energypositionssitedwellattheconvex-hull\nline,includingCo8Si3(~160),Co5Si2(~100),Co7Si3(~100),Co9Si4(~150),Co9Si5\n(~160),Co7Si4(~160),Co8Si5(~150),Co4Si3(~100meV/atom),inaword,these\nstoichiometriesmighthavelower-energystructuresathighpressure,thusitisnot\nnecessarytostudythesestoichiometriesatambientconditionsasthecurrently\navailabletechnologymightbestillunabletosynthesizethem.Weshowthese\nlower-energystructuresandMAEforreferencepurposeonly.Allofthelow-energy\nCo8SistructuresshowsmallMAEbelow20μeV/atomandunperfecteasy-axis/plane\nMAE.WeignoretodiscusstheCo9SibecauseitsEfisabout0.Thesearched\nlow-energyCo6Sistructuresindeedhaslargemagnetism,whereasallofthemhaven’t\nperfecteasy-axis/planeMAE,onlyonestructureshowsnearly-perfecteasy-axisMAE\nwithvaluesofE001=19.89andE100=18.49μeV/atom,respectively,whichisabout30\nmeV/atomhigherthanthatofconvex-hullpoint,asisshowninTableS5.\nCo5Siexistsmanylarge-magnetismlow-energystructures,whereasnoneofthem\nshowsperfecteasy-axis/planeMAE,thelow/high-energyCo5Sistructureisalso\ndominatedmainlybythelow/high-energyCoP63/mmcandmmF3buildingblocks,\nrespectively,thatis,theenergyorderoftheCobuildingblockdeterminestheenergy\norderoftheCo5Si.Welistseverallow-energylarge-magnetismstructures,Ama2\n(Ef=-0.191,E010=336.35,E100=251.25),Cmc21(Ef=-0.1646,E001=321.31,E010=9.34),16P2(Ef=-0.1633,E010=435.12,E100=896.43),dI24(Ef=-0.1374,E001=180.58,\nE100=532.35),Cccm(Ef=-0.1171eV/atom,E001=37.85andE010=191μeV/atom),and\nsoon,someofwhichcouldbesynthesizedduetothesmallenergydifferencebetween\ntheirownenergiesandtheconvex-hullvalue,asisshowninTableS5andFigure1.\nThereareseveralnearly-perfecteasy-planestructureswithenergiesabout70\nmeV/atomhigherthanthatofthelowest-energystructureinCo5Si,suchasCc\n(Ef=-0.1646,E001=321.31,E010=9.34).Pmmm(Ef=-0.1227,E010=71.86,E100=1.78).P2\n(Ef=-0.146,E001=127.83,E010=7.5).Cccm(Ef=-0.1171,E001=37.84,E100=0.21),Fddd\n(Ef=-0.1626,E001=90.31,E010=5.5),Amm2(Ef=-0.1371eV/atom,E001=0.59,\nE100=31.4373μeV/atom),andsoon,asisshowninTableS5.Inaddition,thereare\nalsoseveralstructureswithnearly-perfecteasy-axissuchasC2,(Ef=-0.1395,\nE001=66.63,E100=71.5),P21/c(Ef=-0.1492,E010=20.81,E100=17.79),I41/amd\n(Ef=-0.1175,E001=46.8,E001=39.91).Pbcm(Ef=-0.1383eV/atom,E010=42.94,\nE100=35.17μeV/atom),andsoon,asareshowninTablesS5,6.Inaword,Co5Siisan\nimportantcompound.\nThesearchedtwolowest-energyCo9Si2Pbcm(Ef=-0.1272)andP1(Ef=-0.1275\neV/atom)areabout100meV/atomhigherthantheconvex-hullvalue,asisshownin\nTableS6,thusweignoretoanalysizethem.SeveralCo4SihavesimilarEf(-0.2\neV/atom),whichareabout50meV/atomhigherthanconvex-hullvalue,including\n3R(E001=73.12,E100=64.37),P213(E001=75.19,E010=380.39),I4/m(E010=60.73,\nE100=65.47),I4/m(E010=44.13,E100=81.44μeV/atom),inwhichoneortwostructures\nhavenearly-perfecteasy-axisMAE,asisshowninTableS5.Furthermore,P41hasa17largeMAE(E010=220.85,E100=190.99μeV/atom)butitisabout100meV/atomhigher\nthanconvex-hullvaluewithaEfof-0.1542eV/atom,asisshowninTableS6.Their\nbuildingblocksarealsodominatedbythelargelydistortedColattice.\nSomeCo7Si2mightbesynthesizedwhencomparingitsenergywiththatof\navailablemetastableCo3Si(P63/mmc)andCo4Si,asisshowninTableS5andFigure\n1,includinganearly-perfecteasy-axisMAE1P(Ef=-0.2118,E010=36.15,E100=36.96),\nP42/ncm(Ef=-0.2053,E001=43.63,E100=55.11)andP212121(Ef=-0.1911eV/atom,\nE001=2.19,E100=50.21μeV/atom),respectively.\nThesearchedtwolowest-energyCo5Si3Pbam(Ef=-0.481)andCmmm\n(Ef=-0.4737eV/atom)areabout2~3meV/atomhigherthanthatofconvex-hullvalue,\nwhereastheirMAEarenegeligible.BothFe5Si3andMn5Si3P63/mcmphasehave\nbeensynthesizedexperimentally,thecalculatedEfis-0.4212eV/atominCo5Si3\nP63/mcm,meaningametastablephase.Ourpreviouscalculationsshowthatthe\nexperimentalmetastablephaseFe5Si3(P63/mcm)exhibitsalargeeasy-planemagnetic\nanisotropywithavalueof157μeV/atom(E100=E010<E001).Theexperimentally\nsynthesizednanoscaleMn5Si3P63/mcmshowsinterestingproperties,wethus\ncalculateditsMAEandobtaintheE010=42.48andE100=37.31μeV/atom,respectively.\nThepresentCo5Si3P63/mcmphaseshowsE001=15.11andE100=12.36μeV/atom,\nrespectively,asisshowninTableS1.Featomshavetwodifferentkindsofmagnetic\nmomentinFe5Si3(P63/mcm)withvaluesof1.833,1.796,1.838,1.318,1.313B,the\ntotalmagneticmomentis15.756Binunitcell,asarealsothecasesinMn5Si3and\nCo5Si3,itis0.722,0.719,0.669,0.299,0.258BinMn5Si3withatotalvalueof5.14818B,itis0.815,0.812,0.81,0.31,0.281BinCo5Si3withatotalvalueof5.967B.\nThisphenomenonclearlydemonstratesthatthemagnetismshowsstronglattice\nframeworkortheatomiccoordinatedependence.Co5Si3haslargermagnetismthan\nMn5Si3.TheatomicradiusofMnisslightlylongerthanthatofCowithavalueof\nabout5%[19-21],thereforethelatticevolumeofMn5Si3isabout2%largerthanthat\nofCo5Si3,previousstudyfoundthattheshorteratomicdistancecorrespondstothe\nsmalleratomicmomentinironsilicides[1].Thelargeraveragemomentcontributesto\nthelargerMAEiseasilyunderstandableinFe5Si3.However,thelargeraverage\nmomentofCoinCo5Si3relativetothatofMninMn5Si3contributestofarsmaller\n(13.73μeV/atom)averageMAEthanthelatter(39.90μeV/atom)undertheindividual\nstableP63/mcmlattice,whichcouldn’tbeuniquelyillustratedbytheatomicdistance.\nTorevealthepossibleoriginweusethesameP63/mcmlatticeframeworkforX5Si3\n(X=Mn,Fe,Co)tocalculateandcomparetheirMAE,asisshowninTableS2,clearly,\nthelatticeshapeandatomiccoordinatescruciallydecidetheMAE.Forinstance,the\nMAEislessthan20μeV/atomintherelaxedCo5Si3framework,whichisalsothecase\nwhentheCoissubstitutedbyMn(Mn5Si3)andFe(Fe5Si3),respectively.Thisis\napplicabletotherelaxedMn5Si3(MnCo/Fe)andFe5Si3(FeMn/Co).Forany\ngivenframeworkandelementalspecies,theeasy-axisdirectionisunchanged,\nmeaningthatthelatticeshapedeterminetheMAE.Thelatticeframeworkalso\ndeterminestheMAEvalue,suchastheMAEintherelaxedCo5Si3isalwaysrelatively\nsmallevenforthesubstitutedMn5Si3andFe5Si3whencomparingtothecaseofFe5Si3\naswhichalwaysshowsrelativelylargeMAEwhethersubstitutionbyCo/Mnornot,19asarealsothecasefortherelaxedMn5Si3,stillindependentontheelementalspecies.\nThisconclusionhelpsustodeeplyunderstandthegiantMAEofexperimental\nnanoparticleCo3Si,whichmightoriginthelargelydistortedlatticeshapeand\nnon-equilibriumatomicarrangements.Clearly,toobtainlargeMAEinnanoscale,the\nperfectbulklatticesmusthavelargeMAE.OnlyFe5Si3P63/mcminitsstablestatus\nshowsnearly-perfecteasy-planeMAE(E001=157.25,E100=0.52μeV/atom),asis\nshowninTable4.\nThelowest-energyCo7Silocateswellattheconvexhullprofile,meaningthese\nlow-energystructurescouldbysynthesized.Forsimplicity,weonlylistthestructures\nwithenergyabout30meV/atomlowerthanthatofthelowest-energyone,infact,too\nmanystructuressiteatanenergyrangeof30~100meV/atombutthesestructureshave\nnon-perfectornearly-perfectandlargeeasy-axis/planeMAE,asisshowninTableS5.\nHowever,fortunately,twostructureshavenearly-perfecteasy-axisMAE,oneisP4132\n(Ef=-0.1188)withvaluesofE001=98.46andE010=92.28,theotheroneis1P\n(Ef=-0.1051eV/atom)withvaluesofE001=37.86andE100=35.6μeV/atom,\nrespectively.\nG.Mn3Si\nForsimplicitypurpose,weignoretoplottheconvexhullforthemanganese\nsilicides.Furthermore,wealsoignoretoplotthebuildingblockastheMnisavery\ncomplicatedcrystalcontaining58atomsinthecubiccellinitsgroundstateatambient\nconditions.Asisshownintable5,theexperimentalFm3m-A(Ef=-0.3094)Mn3Si20presentsE001=91.79andE010=61.78,respectively,theotherexperimental\nmmm/P4-A(Ef=-0.2103eV/atom)Mn3SipresentsE010=72.02andE100=29.78\nμeV/atom,respectively.TheenergydifferencebetweenFm3m-Aandmmm/P4-A\nisabout100meV/atom,indicatingthatthehigher-energymetastablephasealsocould\nbesynthesized.Astructure(Fm3m,Ef=-0.3096eV/atom)withsameMAEproportion\nwiththoseofexperimentalonesalongthethreeaxesissearchedwithvaluesof\nE100=21.39andE001=14.25μeV/atom,respectively.Aperfecteasy-axisMAEMn3Si\nP2221(Ef=-0.2103eV/atom)issearchedwithidenticalvaluesofE001=E100=51.25\nμeV/atom,averypromisingmagneticmaterial.Similarly,Ibam(Ef=-0.2207eV/atom)\nalsopresentsnearly-perfectMAEwithindividualvaluesofE001=25.98and\nE100=25.81μeV/atom,ameaningfulmagneticmaterial.Twonearly-perfecteasy-plane\nstructuresaresearched,whoseenergiesarenearlysamewiththatofP4/mmm-A,\nincludingPmma(E001=0.01,E100=51.69)andmP124(E010=1.15,E100=20.58\nμeV/atom).MostoftheMn3Silatticesaresquareshape,meaningthesimilarbuilding\nblockwiththatofmmm/P4-A.Theotherhigher-energyandsmall-magnetism\nMn3SistructuresareshowninTableS7.\nH.Mn5Si2\nAsisshownintable6,theexperimentalMn5Si2P43212presents(Ef=-0.2631\neV/atom)E001=171.25andE010=23.93μeV/atom,respectively.Twoperfecteasy-axis\nstructuresaresearched,includingIbam(Ef=-0.261,E010=E100=19.21)andI4/mmm\n(Ef=-0.2014eV/atom,E001=E100=30.13μeV/atom),bothstructuresaresynthesizeable.\nTwoperfecteasy-planeMAEstructuresPbamandC2/caresearched,whichareabout21100and80meV/atomhigherthanthatofP43212andshowE001=39.65andE100=6.25\nμeV/atom,respectively.OnestructureCmc21withgiantMAEissearched(Ef=-0.1777\neV/atom,E001=728.63andE100=696.157μeV/atom),whichmightalsohassome\nspecialapplicationsdespiteitisnottotally-perfecteasy-axisMAE.Inaword,Mn5Si2\nisanimportantcompound.TheothersearchedMn5Si2structuresareshowninTable\nS8.\nI.Mn5Si3\nAsisshownintable7,theexperimentalMn5Si3P63/mcm(Ef=-0.3306eV/atom)\nshowsnearly-perfecteasy-axisMAE,withvaluesofE010=49.11andE100=55.1\nμeV/atom,respectively.AstructureI4122(Ef=-0.6676eV/atom)withenergyabout\n300meV/atomlowerthanthatofexperimentalP63/mcmissearched,withaxial\nenergiesofE001=8.13andE100=63.56μeV/atom,respectively.Aslightdistortion\nincreasesMAEtoE001=121.64andE010=17.82μeV/atomwithachangedsymmetry\nP4122butalmostunchangedenergy(Ef=-0.6675eV/atom).\nTwoperfecteasy-planeMAEPmmaandP4122withenergiesabout30\nmeV/atomhigherthanthatofP63/mcmaresearched,withrespectivehard-axis\nenergiesofE100=48.88andE001=65.58μeV/atom.Fournearly-perfecteasy-plane\nMAEstructureswithenergiesabout10~30meV/atomhigherthanthatofP63/mcm\naresearched,includingI4/mmm(E010=0.40,E100=20.55),Pmmm(E010=12.48,\nE100=0.81),P4/mbm(E001=62.51,E010=2.65),Amm2(E001=60.55,E100=1.4μeV/atom),\nrespectively.\nPbcn(Ef=-0.3112,E010=39.30,E100=39.24)andPm(Ef=-0.2524eV/atom,22E001=82.47,E010=81.98μeV/atom)presentnearly-perfecteasy-axisMAEandcould\nbesynthesizedduetotheirlowerenergies.ThereforeMn5Si3isaninteresting\ncompound.Forsimplicity,theotherMn5Si3structuresareshowninTableS9.\nJ.Mn6Si\nExperimentalMn6Sistructurehas53atomsperunitcell(Mn85.5Si14.5),whereasit\nis56atomsin8-unitcell,thereforeweignoretocomparetheEf.Allofthesearched\nlow-energystructuresshowninFigure1haven’ttheexperimentalsymmetry3R,we\nthussystematicallyanalysizedthecellshapeanddofindplentyofcellshavethe\nhexagonalshape,whereaswedidn’tfindtheexperimentallyreportedrhombohedral\nshapecell,infactlotsofthesearchedcellaresquareshape,thereforewedidn’tdeeply\nstudytheexperimentallyreportedstructure.Thesearchednearly-perfecteasy-plane\nPbcnandPbcahavenearlysameenergies,withaEfof-0.137eV/atomandthe\nhard-axisenergyofabout54μeV/atom,asisshowninTable8,inwhichthehard-axis\norientationcouldbeswitchedduetothestructuraldistortionbetweenthem,infact,\nthisisafrequentlyoccurredphenomenon.Thenextlowest-energyP212121(Ef=-0.099\neV/atom)presentsE010=25.03andE100=76.82μeV/atom,respectively.Theeven\nhigher-energyPna21(Ef=-0.0877eV/atom)showslargerMAEwithvaluesof\nE010=241.51andE100=96.65μeV/atom,respectively.Noneperfecteasy-axisMAE\nstructureisfoundinMn6Si,severalnearly-perfecteasy-planeMAEarefoundbut\ntheirenergiesareabout50meV/atomhigherthanthatofthelowest-energystructure,\nthusweonlydiscussseverallow-energystructures.\nK.Mn9Si223WealsoignoretocomparetheexperimentalMn9Si2(Mn81.5Si18.5)asithas186\natomsintheunitcell,whereasithas176atomsin16-unitcell.AsisshowninFigure\n1,thelow-energyMn9Si2mightbesynthesizedifitsdecomposedproductsare\nmetastablephasesofMn4SiandMn5Si,thuswelistthreelow-energystructures,asis\nshowninTable8,clearly,allofthemshowmediumorsmallnon-perfect\neasy-axis/planeMAE,suchasP41212haslowestenergywithaxialenergiesof\nE001=143.17andE010=94μeV/atom,respectively.\nL.MAEofMn3Ge,Mn5Ge2,Mn5Ge3,Mn6Ge,Mn9Ge2calculatedfromtheir\nrespectiveperfectSi-containingcounterparts\nTheperfecteasy-axis/planeMAEinSi-containingcompoundsarecalculatedfor\nGe-containingcounterparts,unfortunately,onlyseveralstructureshavenegativeEf,\nthuswedidn’tcarefullydiscussthemasallofthecalculationsaredonebydirect\nsubstitutioninthecoordinatefile.TheexperimentalMn3SiFm3mpresents\nE001=91.42andE010=61.88μeV/atom,butitisE010=25.55andE100=96.15μeV/atom\ninMn3Ge(Ef=-0.0971eV/atom).Similarly,E010=69.84andE100=137.2μeV/atomin\nMn3SicorrespondstoE010=214.93andE100=207.81μeV/atominMn3Ge(Ef=-0.0542\neV/atom),displayinganearly-perfectandverylargeeasy-axisMAE.Inaddition,\nE010=39.3andE100=39.24μeV/atominMn5Si3correspondsto25.98and26.22\nμeV/atominMn5Ge3(Ef=-0.0885eV/atom),alsopresentinganearly-perfecteasy-axis\nMAE.Similarly,E001=121.64andE010=17.82μeV/atominMn5Si3correspondsto\n96.25and58.12inMn5Ge3(Ef=-0.5409eV/atom).Meanwhile,E001=143.17and\nE010=94μeV/atominMn9Si2correspondsto166.81and135.68inMn9Ge224(Ef=-0.0121eV/atom).Inaword,Ge-containingcompoundsalsohaveseveral\nstructureswithidealMAE.\nIV.Conclusions\nOurcalculatedbulklatticefailedtorepeatthepreviouslycalculatedand\nmeasuredgiantmagneticpropertiesofthenanoscaleCo3SiP63/mmcevenforthe\nlatticeparameter.Fivelower-energystructuresthanthatofmetastableCo3SiP63/mmc\naresearched,inwhichI4/mmmpresentsperfecteasy-planemagnetocrystalline\nanisotropyenergy(MAE)withahard-axisenergyof221.61μeV/atom,the\nlowest-energyCmcmshowsnearly-perfecteasy-planeMAEwithamediumenergyof\nE010=0.24andE100=28.46μeV/atomincomparisonwiththatoftheeasy-axisE001,\nanotherCmcmshowsnon-perfectlargeMAEwithaxialenergiesofE010=150.27and\nE100=6.36μeV/atom,PmmnandP21shownon-perfectlargeMAEbetween100~200\nμeV/atom,I4/mcmshowsnearly-perfecteasy-planeMAEwithvaluesofE001=2.81\nandE100=211.67μeV/atomincomparisonwiththatoftheeasy-axisE010.Astructure\nmPm3withsimilarenergywiththatofP63/mmcalsoshowsnearly-perfecteasy-axis\nMAEwithavalueofabout70μeV/atom.\nAlloftheCo3CandCo3Snpresentpositiveformationofenergy.TheMAE\nbetweenCo3SiandCo3Gearesimilarinthelowest-energyCmcm.Co3Gehassimilar\nsymmetrywiththatofCo3Si,twonearly-perfecteasy-axisMAEstructureswith\nvalueswithin110~120μeV/atomaresearchedwithenergiesabout40meV/atom\nhigherthanthatofCmcm,threenearly-perfecteasy-axisMAEstructureswithvalues\nof~25μeV/atomaresearchedwithenergiesof60~70meV/atomhigherthanthatof25Cmcm,aperfecteasy-planeMAEI4/mmmwithvaluesof~157μeV/atomissearched\nwithenergyof~30meV/atomhigherthanthatofCmcm.Fournearly-perfect\neasy-planeMAEstructureswithenergiesof60~70meV/atomhigherthanthatof\nCmcmaresearchedandshowdiverseMAEvalues.Co3Gedoesn’tpresenteasy-axis\nMAEinP63/mmc.Co3Ge1Ppresentsnearly-perfecteasy-axisMAE,whichis\ndifferentwiththatofCo3Si.\nAlow-energyComR3issearched,withalowerenergythanthe\nexperimentallymetastablemmF3,whichpresentsnearly-perfecteasy-axisMAE.\nManylow-energyCo5SihavehundredsofMAEbutnoneofthemshowsperfectMAE\nexceptforseveralnearly-perfecteasy-axis/planeMAEstructures.Theseveral\nlow-energyColatticebuildingblocksgenerallycontroltheenergyorderofthe\nsearchedindividualcobaltsilicides,namely,theCocrystalframeworkkeepsalmost\nunchangedevenaftersomesubstitutionsbytheotherelements.Ourcomparison\ncomputationconfirmedthattheMAEshowsstronglatticeframeworkortheatomic\ncoordinatedependence,asisfurtherconfirmedbythesearchedstructureswithsimilar\nenergybutwithoutsimilarMAE.ManyMn3Si,Mn5Si2,Mn5Si3existperfector\nnearly-perfectlow-energyeasy-axis/planestructures,asisalsothecaseintheseveral\nGe-containingcounterparts.\nFournearly-perfecteasy-axis/planeMAEwithmediumvaluesaresearchedin\nMn3Si.Twoperfecteasy-axisMAEwithmediumvalueissearchedinMn5Si2,two\nnearly-perfecteasy-planeMAEwithmediumvaluesaresearchedinMn5Si2,one\nMn5Si2structurewithnearly-perfecteasy-axisandgiantMAEvalues(about70026μeV/atom)issearched.Alower-energystructurewithenergyabout300meV/atom\nlowerthanthatofexperimentalMn5Si3issearched,togetherwiththreeorfour\nnearly-perfecteasy-axisMAEstructuresandtwoperfecteasy-planeandfour\nnearly-perfecteasy-planeMAEstructures.Alloftheabovementionedstructuresare\nabout100meV/atomlowerthanthatofthesearchedlowest-energystructureor\nexperimentalstructure.Inaword,mostofthecurrentlysearchedeasy-axisMAE\nstructurescouldbesynthesizedexperimentally,theseeasy-axisstructureshavevery\npromisingapplicationinthediverserequirementsfieldofspin-relatedmagnetic\ndevice.\nDataandmaterialsavailability:\nThedatathatsupportsthefindingsofthisstudyareavailablewithinthearticleandits\nsupplementarymaterial.\n[1]Z.J.Yang,S.Q.Wu,X.Zhao,M.C.Nguyen,S.Yu,T.Q.Wen,L.Tang,F.X.Li,K.M.Ho,andC.Z.\nWang,J.Appl.Phys.124(2018)073901\n[2]B.Balasubramanian,P.Manchanda,R.Skomski,P.Mukherjee,S.R.Valloppilly,B.Das,G.C.\nHadjipanayis,andD.J.Sellmyer,Appl.Phys.Lett.108(2016)152406\n[3]M.HansenandK.Anderko,ConstitutionofBinaryAlloys(McGraw-Hill,NewYork,1958).\n[4]A.Handbook,AlloyPhaseDiagrams(TheMaterialsInformationSociety,ASMInternational,\n1992).\n[5]W.B.Pearson,Pearson'sHandbookofCrystallographicDataforIntermetallicPhases(American\nSocietyforMetals,MetalsPark,OH,1985).\n[6]B.Das,B.Balasubramanian,P.Manchanda,P.Mukherjee,R.Skomski,G.C.Hadjipanayis,andD.J.\nSellmyer,NanoLett.16(2016)1132\n[7]F.Jonietz,S.Muhlbauer,C.Pfleiderer,A.Neubauer,W.Munzer,A.Bauer,T.Adams,R.Georgii,P.\nBoni,R.A.Duine,K.Everschor,M.Garst,andA.Rosch,Science330(2010)1648\n[8]J.M.Higgins,R.Ding,andS.Jin,Chem.Mater.23(2011)3843\n[9]C.B.ShoemakerandD.P.Shoemaker,ActaCryst.B32(1976)2306\n[10]C.B.ShoemakerandD.P.Shoemaker,ActaCryst.B34(1978)701\n[11]C.B.ShoemakerandD.P.Shoemaker,ActaCryst.B27(1971)227\n[12]Y.C.Wang,J.Lv,L.Zhu,andY.M.Ma,Phys.Rev.B82(2010)09411627[13]Y.C.Wang,J.Lv,L.Zhu,andY.M.Ma,Comput.Phys.Commun.183(2012)2063\n[14]G.KresseandJ.Furthmüller,Phys.Rev.B54(1996)11169\n[15]G.KresseandJ.Furthmüller,Comput.Mater.Sci.6(1996)15\n[16]G.KresseandD.Joubert,Phys.Rev.B59(1999)1758\n[17]J.P.Perdew,K.Burke,andM.Ernzerhof,Phys.Rev.Lett.77(1996)3865\n[18]S.Geller,ActaCryst.8(1955)83\n[19]J.C.Slater,J.Chem.Phys.41(1964)3199\n[20]J.C.Slater,Phys.Rev.36(1930)57\n[21]J.T.WaberandD.T.Cromer,J.Chem.Phys.42(1965)411628\nFigure1,theformationofenergy(Ef)ofthebinarycobaltsilicides,inwhichwe\nignoredtheexperimentaldataforclearpurpose,includingCo3Si(Ef=-0.2638eV/atom,\nP63/mmc),Fe5Si3(Ef=-0.3787eV/atom,P63/mcm),respectively.\nTable1,thelatticeparameter(Å),angle(ﾟ),symmetry,atomicaverageenergyE\n(eV/atom),buildingblock,MAE(μeV/atom)ofCo3Si.Allofthecolorsonlineonly.\nSymm.\nEbuilding\nblocka\nb\ncα\nβ\nγUnitcell\nCo\nSiE001\nE010\nE100\nP63/mmc\n-6.9\nExp.\n4.97\n4.97\n3.9790\n90\n120\n261.14\n261.14\n0\nI4/mmm-A\n-6.9197\n5.05\n6.66\n5.0590\n90\n90\n0\n141.42\n0\nI4/mmm-B\n-6.9181\n3.57\n3.57\n6.6690\n90\n90\n0\n0\n221.6129I4/mmm-C\n-6.9197\n5.06\n5.06\n13.2990\n90\n90\n153.08\n25.22\n0\nI4/mmm-D\n-6.9197\n5.05\n6.65\n5.0590\n90\n90\n0.03\n148.42\n0\nI4/mmm-E\n-6.8823\n5.61\n5.61\n10.9890\n90\n90\n46.9\n0\n114.09\nI4/mmm-F\n-6.8822\n7.92\n11.01\n7.9390\n90\n90\n0\n2.67\n0.06\nCmcm-A\n-6.9673\n6.26\n7.41\n3.6890\n90\n90\n0\n0.24\n28.46\nCmcm-B\n-6.9183\n6.36\n5.13\n5.2190\n90\n90\n0\n150.27\n6.36\nCmcm-C\n-6.8863\n7.21\n9.31\n2.5790\n90\n90\n76.81\n67.12\n0\nCmcm-D\n-6.8815\n5.49\n5.49\n5.4990\n90\n90\n0\n97.85\n198.27\nPmmn\n-6.9143\n4.01\n4.27\n4.9790\n90\n90\n0\n75.76\n124.73\nI4/mcm\n-6.9025\n4.84\n4.84\n7.3290\n90\n90\n2.81\n0\n211.67\nP21\n-6.9071\n6.21\n5.02\n5.4390\n90.69\n90\n81.08\n0\n137.16\nP63/mmc-B\n-6.8931\n4.96\n8.59\n8.0190\n90\n90\n80.28\n0\n0.06\nP4/mbm-A\n-6.8976\n4.83\n4.83\n7.2190\n90\n90\n2.14\n0\n144.5130mPm3-A\n-6.8891\n4.91\n9.83\n3.4790\n90\n90\n0\n71.9\n72.1\nmPm3-B\n-6.8876\n3.47\n3.47\n3.4790\n90\n90\n0\n0\n32.14\nTable2,thelatticeparameter(Å),angle(ﾟ),symmetry,formationofenergyEf\n(eV/atom),MAE(μeV/atom)ofCo3Ge.\nSymm.\nEfa\nb\ncα\nβ\nγCo\nGe\nE001\nE010\nE100\nCmcm\n-0.11377.52\n6.44\n3.7690\n90\n90\n0\n14.91\n32.19\nC2/m,\n-0.06038.62\n5.12\n6.2290\n138.35\n90\n0\n42.56\n108.01\nI4/mmm\n-0.07955.13\n5.13\n6.8790\n90\n90\n157.06\n0\n0\nI4/mmm\n-0.05245.11\n13.85\n5.1190\n90\n90\n0\n9.62\n0\nPmmm\n-0.07285.97\n5.09\n5.9789.99\n93.2\n90\n122.41\n0\n122.58\n1P\n-0.06944.09\n4.35\n5.0890\n89.98\n90.03\n0\n116.38\n114\nCmmm,\n-0.04225.06\n5.06\n3.5490\n90\n90\n0\n25.5\n26.3\nP212121,\n-0.03375.69\n5.69\n5.6690\n90\n90\n0\n24.15\n23.2531P63/mmc,\n-0.05425.07\n5.07\n4.0790\n90\n119.99\n332.58\n85.53\n0\nCm,\n-0.04315.05\n5.05\n14.3290\n100.1\n90\n40.92\n0\n2.64\nAma2,\n-0.05328.79\n5.07\n8.1490\n90\n90\n269.05\n0\n0.78\nC2221,\n-0.05138.76\n5.07\n8.1890\n90\n90\n112.07\n4.75\n0\nC2/m,\n-0.04727.16\n7.10\n5.0590\n135.01\n90\n26.18\n0\n23\nTable3,thelatticeparameter(Å),angle(ﾟ),symmetry,atomicaverageenergyE\n(eV/atom),buildingblockormotif,MAE(μeV/atom)ofCo.\nSym.\nEmotifa\nb\ncα\nβ\nγUnitcellE001\nE010\nE100\nP63/mmc\n-7.0398\n2.49\n2.49\n4.0290\n90\n120\n65.55\n115.46\n0\nm3mF\n-7.0206\n3.51\n3.51\n3.5190\n90\n90\n250.41\n55.16\n0\nm3R\n-7.0306\n4.31\n2.48\n8.2090\n100.1\n90\n0\n25.61\n23.82\nP42/mnm\n-6.9968\n4.1\n4.1\n2.5990\n90\n90\n0\n48.91\n101.7\nCmca\n-6.9972\n2.56\n4.2\n8.0890\n90\n90\n30.93\n67.03\n0\nDiamondSi\n-5.4236\n5.43\n5.4390\n90\n325.4390\nTable4,thesearchedlower-energystructureofFe5Si3.AverageatomicenergyE\n(eV/atom),formationofenergyEf(eV/atom),MAE(μeV/atom)andstructure\nparameters(Å)andangles(ﾟ)areshown.\nSym.\nE\nEfFe\nSi\na\nb\ncα\nβ\nγE001\nE010\nE100\nExp.\nP63/mcm\n-7.5687\n-0.3787\n6.69\n6.69\n4.6790\n90\n120157.25\n0\n0.52\nSearched\nCmcm\n-7.5973\n-0.41\n7.86\n5.54\n7.8290\n90\n900.87\n0\n11.82\nPnma,\n-7.5916\n-0.4044\n11.01\n5.6\n5.4590\n90\n9018.13\n0\n10.07\nP2/m,\n-7.5580\n-0.3708\n5.54\n2.78\n5.5490\n90\n900.076\n4.91\n0\nFmmm,\n-7.5941\n-0.4069\n5.53\n7.83\n7.8390\n90\n903.12\n3.18\n0\n1P,\n-7.4978\n-0.3106\n4.8\n8.27\n9.37106.16\n91.6\n101.0427.91\n5.21\n0\nCmca,\n-7.4763\n7.64\n9.6590\n9037.21\n033Table5,thesearchedlower-energystructureofMn3Si.FormationofenergyEf\n(eV/atom),MAE(μeV/atom)andstructureparameters(Å)andangles(ﾟ)areshown.\nSym.\nEfUnitcell\nMn\nSi\na\nb\ncα\nβ\nγE001\nE010\nE100\n1Exp.\nFm3m,\n-0.3094\n5.65\n5.65\n5.6590\n90\n9091.79\n61.78\n0\n2Exp.\nmmm/P4,\n-0.2103\n2.79\n2.79\n5.6890\n90\n900\n72.02\n29.78\n3searched\n1222P,\n-0.2103\n5.48\n5.7\n5.4890\n90\n9051.25\n0\n51.25\n4Ibam,\n-0.2207\n5.57\n5.55\n5.5690\n90\n9025.98\n0\n25.81\n5mP124,\n-0.2161\n3.81\n3.81\n11.8290\n90\n900\n1.15\n20.56-0.2891\n 4.78909.23\nP212121,\n-7.52683\n-0.3395\n7.66\n9.33\n4.7990\n90\n9064.4\n29.28\n034Table6,thesearchedlower-energystructureofMn5Si2.FormationofenergyEf\n(eV/atom),MAE(μeV/atom)andstructureparameters(Å)andangles(ﾟ)areshown.\nSym.\nEfUnitcell\nMn\nSi\na\nb\ncα\nβ\nγE001\nE010\nE100\n1Exp.\nP43212\n-0.2631\n8.72\n8.72\n8.5390\n90\n90171.25\n23.93\n0\n2searched\nIbam,\n-0.2615\n3.87\n7.74\n19.8890\n90\n900\n19.21\n19.21\n3I4/mmm,\n-0.2014\n3.88\n19.8\n3.8890\n90\n9030.13\n0\n30.13\n4Pbam,\n-0.1678\n8.59\n13.5\n2.6690\n90\n9039.65\n0\n0\n5C2/c\n-0.1839\n8.98\n17.22\n4.3490\n114.1\n900\n0\n6.25\n6Cmc21,\n-0.1777\n2.74\n8.59\n13.1890\n90\n90728.65\n0\n696.1535Table7,thesearchedlower-energystructureofMn5Si3.FormationofenergyEf\n(eV/atom),MAE(μeV/atom)andstructureparameters(Å)andangles(ﾟ)areshown.\nSym.\nEfUnitcell\nMn\nSi\na\nb\ncα\nβ\nγE001\nE010\nE100\n1Exp.\nP63/mcm\n-0.3306\n6.89\n6.89\n4.788990\n90\n1200\n49.11\n55.1\n2searched\nI4122,\n-0.6676\n5.64\n5.64\n11.2590\n90\n908.13\n0\n63.56\n3P4122,\n-0.6675\n3.99\n3.99\n11.2390\n90\n90121.64\n17.82\n0\n4Pmma,\n-0.2986\n2.76\n5.56\n11.2590\n90\n900\n0\n48.88\n5I4/mmm,\n-0.3194\n3.93\n3.93\n11.290\n90\n900\n0.40\n20.55\n6Pmmn,\n-0.3275\n3.99\n5.52\n7.8690\n90\n900\n12.48\n0.81\n7P4122,\n-0.2945\n5.59\n5.59\n11.0590\n90\n9065.58\n0\n0\n7P4/mbm,\n-0.2941\n7.91\n7.91\n2.7690\n90\n9062.51\n2.65\n0\n8Amm2,\n-0.2943\n11.26\n11.28\n2.7790\n90\n9060.55\n0\n1.4\n9Pbcn,\n-0.3112\n11.64\n6.72\n4.5790\n90\n900\n39.3\n39.24\n10I41/amd,\n-0.3036\n7.9\n7.9\n11.1190\n90\n907.34\n0\n8.963611Pm,\n-0.2524\n11.14\n2.82\n11.1990\n88.89\n9082.47\n81.98\n0\nTable8,thesearchedlower-energystructureofMn6SiandMn9Si2.Formationof\nenergyEf(eV/atom),MAE(μeV/atom)andstructureparameters(Å)andangles(ﾟ)\nareshown.\nSym.\nEf\nMn6SiUnitcell\nMn\nSi\na\nb\ncα\nβ\nγE001\nE010\nE100\n1Pbca\n-0.1371\n4.39\n11.63\n5.990\n90\n9054.03\n0.54\n0\n2Pbcn\n-0.137\n5.93\n4.39\n11.6790\n90\n900.67\n0\n54.28\n3P212121\n-0.0991\n3.68\n4.8\n17.0790\n90\n900\n25.03\n76.82\n4Pna21\n-0.0877\n4.4\n10.63\n6.490\n90\n900\n241.51\n96.65\nMn9Si2\n1P41212\n-0.1391\n7.76\n7.76\n8.5390\n90\n90143.17\n94\n0\nPbcn\n-0.1225\n6.21\n9.21\n8.4590\n90\n900\n9.6\n9.72\n2Pca21\n-0.1053\n8.4\n7.6\n7.5290\n90\n9030.39\n67.6\n0"    },    {        "title": "2104.01922v1.Transverse_Rashba_Effect_and_Unconventional_Magnetocrystalline_Anisotropy_in_Double_Gd_adsorbed_Zigzag_Graphene_Nanoribbon.pdf",        "content": "1  Transverse  Rashba  Effect  and Unconventional  Magnetocrystalline  \nAnisotropy  in Double -Gd-adsorbed  Zigzag  Graphene  Nanoribbon  \n \nWeifeng   Xiea,  Yu  Songb†,  Xu  Zuoa,c,d,e∗ \na College of Electronic Information and Optical Engineering, \nNankai University, Tianjin, 300350, China;  \nb College of Physics and Electronic Information Engineering, \nNeijiang Normal University, Neijiang, 641112, China;  \nc Key Laboratory of Photoelectronic Thin Film Devices and \nTechnology of Tianjin, Tianjin, 300071,  China;  \nd Engineering Research Center of Thin Film Optoelectronics Technology, Ministry \nof Education, Tianjin, 300071, China;  \ne Institute of Electronic  Engineering,  \nChina Academy of Engineering Physics, Mianyang, 621999, China.  \n† Previously with Microsystem and Terahertz Research Center, \nChina Academy of Engineering Physics, Chengdu,  \n610200, China; and Institute of Electronic Engineering, China \nAcademy of Engineering Physics, Mianyang, 621999,  China.  \n∗ Corresponding author:  xzuo@nankai.edu.cn  2  Abstract  \nThe transverse Rashba effect is proposed and investigated by the first -principle calculations based \non density functional theory in a quasi -one-dimensional antiferromagnet with a strong perpendicular \nmagnetocrystalline anisotropy, which is materialized by the Gd -adsorbed graphene nanoribbon with a \ncentric symmetry. The Rashba effect in this system is associated with the local dipole field transverse \nto and in the plane of the nanoribbon. That dipole field is induced by the off -center adsorption of the \nGd adatom above the hex -carbon ring near the nanoribbon edges. The  transverse Rashba effect at the \ntwo Gd adatoms enhances each other in the antiferromagnetic (AFM) ground state and cancels each \nother in the ferromagnetic (FM) meta -stable state, because of the centrosymmetric  atomic  structure. \nThe trans verse Rashba parameter  is 1.51 eV˚A. This  system shows a strong perpendicular \nmagnetocrystalline anisotropy (MCA), which is 1.4 meV per Gd atom in the AFM state or 2.2 meV \nper Gd atom in the FM state. The origin of the perpendicular MCA is analyzed in k−space by filter ing \nout the contribution of the transverse Rashba effect from the band structures perturbed by the spin -\norbit coupling interactions. The first -order perturbation of the orbit and spin angular momentum \ncoupling is the major source of the MCA, which is assoc iated with the one -dimensionality of the \nsystem.  The transverse Rashba effect and  the strong perpendicular magnetization hosted \nsimultaneously by the proposed AFM Gd -adsorbed graphene nanoribbon lock the up - (or down -) spin \nquantization direction to the fo rward (or backward) movement. This finding offers a magnetic \napproach to a high coherency spin propagation in one -dimensionality, and open a new door to \nmanipulating spin transportation in graphene -based spintronics.  3  I. INTRODUCTION  \n \nSpintronics, one of the most promising technologies for next -generation information \nengineering[1],  have  been  being  investigated  extensively.  Spin  in addition  to charge  degree \nof  freedom  in spintronics  is employed  for information  storage,  transmission,  and processing, \nwhich  may essentially  reshape  the electronics  with higher  density,  faster  speed,  smaller  die \nsize,  and lower  power  consumption.  The Rashba  effect,  of which  the Hamiltonian  is written \nas, \n( / )( )RR=  Η σpe\n , is prospective  for manipulating  spin transportation  in spintron - \nics, as it couples  electron  spin σ to its own momentum  p in electric  field  along  direction \ne[2–4]. The signature  of the Rashba  effect  on band  structure  is the so-called  spin-splitting, \nwhere the bands with opposite spins shift toward opposite directions in k−space. A  giant \nRashba  spin splitting  was recently  discovered  in bulk materials,  such  as BiTeI[5,  6], due to \nthe non -centrosymmetric crystal structure and the strong s pin-orbital coupling (SOC)  in- \nteraction  of  heavy  elements.  It was also discovered  in the interfaces,  such  as Bi/Ag(111)[7], \nwhere the inversion symmetry is broken by the interface.  \nAlthough it is generally recognized that breaking inversion symmetry is indispensable  for \nthe Rashba effect of nonmagnetic materials, the so -called R -2 type spin polarization that is \ngenerated by the Rashba effect was recently proposed in three -dimensional nonmagnetic \ncrystals with inversion symmetry[8]. In fact, i f a centrosymmetric crystal structure can be \ndivided into two sectors associated with the inversion symmetry and there is a local dipole  \nfield in each  sector,  the Rashba  effect  will emerge  and generate  local  spins  opposite  in \ndirection in the two sectors. It should be noted that the local spins compensate each other and \nthe total spin vanishes, as required by the inversion symmetry[9 –15]. This implies that the \nRashba effect may appear in antiferromagnets with inversion symme try as an inverse effect \nof the R -2 type spin polarization in centrosymmetric nonmagnetic crystals, in the sense that \nthe compensated spin polarization of antiferromagnet accommodates the Rashba effect.  \nThe Rashba effect  was also investiga ted in magnetic  materials[16,   17].   It’s worth noting \nthat,  in magnetic  materials,  although  the states  for opposite  magnetization  directions do not \nexist  simultaneously  in ferromagnets,  the Rashba splitting  in the SOC  band  struc tures  is \nalso equivalent to the Rashba-type spin splitting  in nonmagnetic  materials[16].  In the past \nfew decades,  the Rashba effect  in magnetic  materials  was discovered in graphene4  layer on Ni(111) surface[18], and was predicted for the ultrathin ferromagnetic film on \nnonmagnetic  heavy -metal  layer[19],  one- or two-dimensional  space  inversion  symmetry  bro- \nken antiferromagnet[20],  multi -ferromagnetic  perovskite[21],  and ferromagnetic  Gd-adsorbed \ngraphene nanoribbon[ 22, 23]. In the interfaces composed of nonmagnetic and f erromagnetic \nlayers, the electric field perpendicular to the interface plane breaks the space inversion sym - \nmetry and couples itself to the in -plane components of spin and momentum. This so -called \nperpendicular Rashba effect will vanish, when the magnetic  easy axis is perpendicular to  the \ninterface plane. On the contrary, the Rashba effect of the Gd -adsorbed zigzag graphene \nnanoribbon (ZGNR) involves the local electric field transverse to the nanoribbon, and thus \ncalled the transverse Rashba effect. As the  transverse electric field is coupled to the per - \npendicular spin component and the longitudinal crystalline momentum, it is possible to utilize \nthis effect to manipulate spin transportation even if there is perpendicular magnetic anisotropy.  \nMoreover, th e transverse electric field is local in nature, since it is associated with the off -\ncenter adsorption on the edges of the nanoribbon[ 22]. It is thus possible to decorate the \nnanoribbon with more than one magnetic heavy -metal adatoms per unit cell to achiev e the \ntransverse Rashba effect in a more complicated spin structure, such as antiferromagnetism.  \nMagnetic crystalline anisotropy (MCA) is one of the most fundamental magnetic prop - \nerties . Jointly with other parameters, it defines the magnetic behaviors of a magnet, such as the \nmagnetic easy axis, the shape of hysteresis loop, and the stability of magnetization at finite \ntemperature, which will further impact the design and implementation of magnetic devices. \nFor example, a strong perpendicular MCA is preferred for the magnetic films in magnetic \nstorage devices, such as hard disk, to boost the number of bits per unit area and enhance the \nstability against the external noises induced by thermal fluctuation, magnetic field, and electric \ncurrent. Moreover, antiferromagnetism is preferred for the integrated spintronics to minimize \nthe stray field[24 –26], although the MCA of antiferromagnets was rarely  investigated.  \nMCA is induced by the coup ling between orbit and spin angular momentums \n(LS−coupling) inside magnetic atoms[27, 28]. An accurate first -principle calculation of M - \nCA has been  pursued  from  decades.  The first-principle  MCA  is usually  derived  from  the total \nenergies calculated with th e SOC perturbation for different magnetization directions[29,  30]. \nThe force  theorem  is the essence  of the MCA  calculations,  which  requires  that the spin- 5  densities (up and down) are approximately invariant for different magnetization directions. \nHowever,  the early  calculations  of MCA  were  suffered  from  the insufficient  quality  of the cal- \nculated band structures. In fact, the LS−coupling interaction perturb s the bands near the Fermi \nenergy[28] and induces spikes to the k−space distribution of MCA for metals, which requires \na k−mesh dense enough to achieve convergence. Several approaches such as state tracking \nmethod[31] and torque method[32, 33] were propose d to solve this problem with  an \nacceptable computation cost at that time. MCA can also be calculated in a self -consistent way. \nThis approach was applied to a free -standing Rh monolayer[34].  It was shown that the \nconvergence and fluctuation of the MCA were influenced by not only the number of k−points \nsampled in the surface Brillouin zone but also the integration scheme. Electron correlation \nalso influences the first -principle MCA, as it may modify t he band structure. A LSDA+U \ncalculation predicted both the correct easy axes and the magnitudes of MCA for Fe and Ni[35].  \nThe first-principle  MCA  can be quantitatively  analyzed  by considering  the k−space  \ndistribution of MCA, that is, the difference of the summation of all occupied Kohn -Sham \neigenvalues at each k−point induced by rotating the magnetization direction. The first - \norder  perturbation  of contributes  to the MCA  at the k−points  where  the energy  difference \nbetween  the occupied  and empty  states  is comparable  to the LS−coupling  constant  (ξ) in \nthe Hamiltonian HLS = ξL · S. This contribution is called surface pair coupling (SPC). It  \nwas estimated for two -dimensional materials that the SPC contribution is proportional to  ξ3, \nwhich is an order higher than the second -order perturbation that is proportional to ξ2[31]. \nHowever, the SPC contribution can be the major one to MCA for some one -dimensional \nmaterials with a special band structure. Assume that the valence and conduction bands are \nparabolically contacted at the Γ −point in the one -dimensional Brillouin zone, in addition, \nwith a perfect electron -hole symmetry for mathematical simplicity. It follows that the length \nof SPC  region is \n(2 / )SPCLm \n , where  m∗ is the effecti ve mass  at Γ. The SPC  contribution \nto MCA is then proportional to \n3\n2  , which is a half order lower than the  second -order  \nperturbation.  \nIn addition, it was discovered by the first -principle calculations that both the Rashba ef - \nfect and the LS−coupling interaction perturb the band structure of the Gd -adsorbed ZGNR \nwhen the magnetization is perpendicular to the nanoribbon. It was proposed t hat the per - \nturbation due to the transverse Rashba effect can be extracted in k−space by simply taking  6  the  difference   of  the  SOC   perturbation   calculated   with   the  [001]   and  [00¯1]  magnetization \ndirections, respectively, and that the resulting perturbation due to the transverse Rashba ef - \nfect is an odd function in the one -dimensional Brillouin zone. In fact, the transverse  Rashba  \neffect  is antisymmetric  with respect to flipping  the spin from  the [001]  direction  to the [00\n1] \ndirection. Similarly, the LS−coupling perturbation can be extracted in k−space by taking the \nsummation  of the SOC  perturbation  calculated  with  the [001]  and [00¯1] magnetizatio n \ndirections,  respectively,  as it is symmetric  with respective  to flipping  the spin direction.  \nIn this work,  an artificial  structure  of double -Gd-adsorbed  ZGNR  with a centric  symmetry \nis proposed and investigated by the first -principle calculations based on density functional \ntheory. It is shown that the structure simultaneously accommodates the antiferromagnetic \nground state, the MCA perpendicular to the nanoribbon plane, and the Rashba effect by the \nlocal in -plane d ipole field transverse to the nanoribbon. The dipole field is associated with \nthe off -center adsorption of the Gd -adatoms on top of the hex -carbon rings near the edges. It \ncouples the spin quantization direction defined by the perpendicular MCA to the crys talline \nmomentum (\nk\n ). This enables the manipulation of spin transportation in a quasi  one-\ndimensional antiferromagnet with perpendicular MCA via the transverse Rashba effect. The \norigin of the perpendicular MCA is analyzed by consi dering the k−space distribution \nextracted from the SOC perturbed band structures. It is shown that the first -order per - \nturbation of LS−coupling between the valence band maximum and the conduction band \nminimum at the Γ −point is the major source of the perp endicular MCA. As a comparison, \nthe same atomic structure in the ferromagnetic state is investigated. The transverse Rashba \neffect of the two Gd-adatoms, however, cancels each other in the ferromagnetic state. The \nMCA is perpendicular, and the k−space distribution of MCA shows several spikes associat - \ned with the metallic band structure of the ferromagnetic state. The first -order perturbation  of \nthe LS−coupling interaction is the major source of the MCA by directly and indirectly \nchanging  the occupation  number  of the states  near the Fermi  energy.  \n \n \nII. APPROACH  \n \nFirst-principles calculations based on density functional theory are performed by using the \nprojected augmented wave (PAW) [36] method as implemented in the Vienna ab−initio \nsimulation package (VASP)[37]. SOC is included in the calculations for both antife rro-7  magnetic (AFM) and ferromagnetic (FM) states to investigate the Rashba effect and the MCA. \nExchange -correlation energy functional is treated in the Perdew -Buke -Ernzerhof pa - \nrameterization of generalized gradient approximation (GGA -PBE)[38]. The valen ce electron  \nconfiguration of Gd atom is chosen to be 4 f 75d16s2. The wave functions are expanded into  \na plane wave basis with a kinetic energy cutoff of 450 eV. It’s necessary to induce the cor - \nrection of Hubbard U term (GGA+ U )[39] to describe the strongly localized f orbits of Gd \natom,  and Uf =8 eV is adopted[ 22, 40].  The unit cell (shown in the dotted  box in FIG.  1 (a) \nand (b)) is isolated  from  its periodic images  by a vacuum  layer as thick as 15 ˚A along  the \ntrans verse direction  (x−axis,  [100]  direction)  and that as thick as 12 ˚A along  the perpendicu - \nlar direction ( z−axis, [001] direction). The nanoribbon is along the y−axis ([010] direction). \nThe carbon dangling bonds at the edges are passivated by hydrogens. The first Brillouin zone \n(FBZ) is sampled by a 1×29×1 Monkhorst -Pack grid for the structure optimization, wher e the \natomic positions are fully relaxed with the conjugate gradient algorithm until the force  on each \natom  is less than  0.01 eV ˚A−1 and the energy  error  is less than  1×10−7 eV. In the band \nstructure calculations from which the total energies are extracted, a  1×69×1 k−mesh is hired \nfor the Brillouin zone integration. According to the force theorem[29, 30], considering the SOC \ninteraction as a perturbation,  the uniaxial MCA constants (Ku) for the AFM and FM states are \ncalculated from the total ener gy difference between the y−axes ([010]) and z−axes ([001]) \nmagnetization directions, namely  Ku=E[010]−E [001]. In this work,  the convergence  of the \ncalculated  MCA  versus  the number  of k−points  has been  tested firstly and plotted in FIG. S1 \nof Supplementary Information[41]. The dipole correction[42] is applied in all the  calculations.  \n \n \nIII. RESULTS AND DISCUSSION  \n \nA. Stability of ZGNR decorated with two Gd atoms  \nThe stability of the two Gd atoms adsorbed on ZGNR with index N=4 (2Gd -4ZGNR) \nsystem (FIG. 1 (a)) is investigated in AFM and FM states.  The pure ZGNR with index  N=4 \n(4ZGNR) where the carbon dangling bonds at the edges are passivated by hydrogens contains \n32 carbons and 8 hydroge n atoms, which is fully relaxed before adsorbing two Gd atoms. The \nrelaxed results show that the 4ZGNR exhibits an AFM semiconductor property which  is in \nagreement  with the previous  reports[ 41, 43, 44]. Then  the adsorption  positions  8  \n \n \nGd 2 \n \nGd 1 \n \n(a) 1 2 3  N=4  \n(b)  \n(c) \n \n \n \nGd 1 \n \n \n \n \n \n \n \n \n \nz [001]   \n \n \n \n \n[100]   \n \n \n \nz [001]   \n \ny \n[010]  \n \n \nx [100]   \nGd 2 \n \nFIG.   1:   (a)  The  top  view  (along   z−axis  [001])   and  (b)  the  side  view  (along   x−axis  [100])   of \nthe relaxed  2Gd-4ZGNR  structure.   (c) Schematic  illustration  of the parameters  dver and dhor that \ncharacterize  the adsorption  of the Gd adatoms  horizo ntally  deviating  from  the center of hexacar bon \nring (point ‘O’).  The unit cell is indicated  by broken lines in (a) and (b).  Gray, purple  and white \nballs   represe nt  carbon,  gadolinium   and  hydrogen   atoms,   respectively.   Note  that  there   are  only \nminor  differences  between the adsorption  position  in the FM and AFM  states  (see TABLE  I). \n \nof the  two  Gd  adatoms  on  4ZGNR   are  considered   to  accomm odate  a  centric  symmetr y, \nand the unit cell constitutes  a coverage  of two Gd adatoms  per 32 carbon atoms,  shown in \nFIG.  1 (a).  Considering  two magnetic  states  of 2Gd-4ZGNR.  Firstl y, for AFM  state,  before \nfully  relaxation,  two Gd adatoms  with  opposite  magnetic  mome nt adsor bed on the relaxed \n4ZGNR  with  AFM  ground  state,  one Gd atom  places in directly  above the hex-carbon ring \nnear the edge  and the other  Gd atom  places  in direc tly below the hex-carbon ring near the \nother  edge,  and both Gd adatoms  have an identical distance  from point ‘O’ (the center of \nthe  local  edge  hexacar bon  ring,   shown  in  FIG.   1 (c)).  After  fully  relaxation,   as  shown  in \nFIG.  1 (a) and (b), the 2Gd-4ZGNR  is AFM  state.   More over, two Gd adatoms  are prefer \nmore  deviation  from  the directly  over the point ‘O’ (FIG.   1  (c)),  and  have  an  off-center \nadsorption.   It is consiste nt with   the  previous   reports  in  which  the  near-hollow  or  hollow site \nis more  favorable  than  the top or bridge  site for Gd adsor bed on graphene  nanorib bon \no dhor \ndver \n \n \ndver \n \n \ndhor \n \n[010]  \n 9  \n(c) \n (e) \nAFM  \n(b) \n[010]  \n \n(a) \n[010]  \n \nb 2Gd− 4ZGNR ,  \n \n \n \n \n \n \n \n \nz [001]   \nx [100]  AFM  \n \n \n \n \n \n \n \n \n \n \nz [001]  x [100]  \n \nFIG. 2: Spin densities of the 2Gd -4ZGNR system for (a) the AFM state and (b) the FM state \nwhere the isosurface is plotted at 5 ×10−3 e/Bohr3 with the yellow and cyan colors representing \nthe positive and negative value, respectively. The band structures without SOC perturbation for  \n(c) the AFM state and (d) the FM state. The density of states (DOS) for (e) the AFM state and \n(f) FM state. The Fermi levels are set to zero in all the band structures and DOS.  \n \nor graphene[ 22, 45]. For FM state, before fully relaxation, two Gd adatoms with the same \nmagnetic moment adsorbed on the relaxed 4ZGNR with the FM metastable state, and the \nadsorption locations of two Gd adatoms are the same as that for AFM state. After fully \nrelaxation, the whole system is FM state which is higher than the AFM state by 45.383 meV \nin energy. Similarly, two Gd atoms also have an off -center adsorption.  As a comparison, the \nstructure as well as the electronic properties of the 2Gd -4ZGNR corresponding to a coverage \nof two Gd adatoms per 16 carbon atoms has also been explored, shown  in FIG. S2 of \nSupplementary  Information[41].  \nThe binding energy of the 2Gd -4ZGNR system with (A)FM state is defined as  \n \nE(A)FM = 2EGd + E4ZGNR − E(A)FM (3.1)  \n \nwhere the EGd and  E4ZGNR  are  the  total  energies  of  an  isolated  single  Gd  atom  and  \nthe 4ZGNR  composed  of 32 carbon  atoms  and 8 hydrogen  atoms,  respectively,  and the \n(d) \n (f) \nFM \n FM 10  TABLE  I: Eb  (eV),  dver  (˚A) and dhor  (˚A) of 2Gd-4ZGNR  systems  in the AFM  and FM states.  \n Eb(eV)  dver(˚A) dhor(˚A) \nAFM  5.254  2.118  0.122  \nFM 5.267  2.123  0.192  \n \nE(A)FM \n2Gd−4ZGNR  is the total energy of 2Gd -4ZGNR system in the (A)FM state. The binding  \nenergies and structural parameters that characterize the adsorption of the Gd adatoms are \nsummarized in TABLE I for both the AFM and FM states.  The parameters dver and dhor are \nthe vertical distance from a Gd adatom to the graphene plane and the horizonta l offset to the \ncenter of the hexacarbon ring, respectively, shown in FIG. 1 (c). By comparing the adsorption \nparameters of the FM state to those of the AFM state, it is shown that the dver slightly increases \nby 0.24% and the dhor remarkably increases by 57%. The large relative difference of the dhor \nreveals that the adsorption can be impacted by the magnetic states. As shown in FIG. 2 (a) \nand (b), in the AFM state, the two Gd adatoms show opposite spin directions and so do the \ntwo edges of the ZGNR. In add ition, each Gd adatom is in a spin state opposite to that of the \nedge close to it for both the AFM and the FM states. The band structures in FIG. 2 (c) and (d) \nshow that the AFM state is a semiconductor with a band   gap of 50.52 meV and that the FM \nis a m etal. The density of states (DOS) of the AFM and FM states in FIG. 2 (e) and (f) show \nthat the Gd d-orbitals are coincident with the carbon p-orbitals in a broad energy interval, \nwhich implies a covalent bond between the Gd -adatom and the carbon atoms.  The 4 f -shell \nof each Gd adatom is half -filled and contributes about 7 µB to the atomic magnetic moment. \nIt is deeply buried in the valence band, as shown in FIG. S3 of Supplementary Information[41]. \nHence, the Gd d−orbital plays the major role in determining the magnetic  properties.  \nB. Rashba effects of AFM and FM  states  \nAs the SOC interaction is included with the spin quantization directions (SQD) along \nthe + z and −z directions, respectively, as shown in FIG. 3 (a), or along the + x and −x \ndirections,  respectively, as shown in Fig. 3 (b), the Gd d−bands near the Fermi energy \nshift in the opposite directions along the k−axis on the band diagram of the AFM state, \nindicating  a phenomenon  associated  with the Rashba  effect[16,  17]. The Rashba  Hamiltonian \ncan be written as HR = αe(σ × k) · e, where αe is the Rashba parameter regulating  the 11 \n   \n(d) \n[001]  [00\n1] \n \nk \n \n \n \n \n \nFIG.  3:  The band  structures  with  SOC -perturbation  and magnetization  along  (a) [001]  and [00¯1] \nand (b) [100]  and [¯100] directions  for the AFM  state,  where  the Rashba shifts  in energy  (ER) \nand momentum (∆ kR) are indicated in (a). (c) Band structures with SOC perturbation and \nmagnetization  along  [001]  and [00¯1] directions  for the FM state.   (d) Schematic  illustratio ns of the \ndirections  of local  spins,  screw  vector,  and wavevector  when  spin quantization  directions  is along  \n[001]  and [00¯1], respectively.  The Fermi  levels are set to zero in all the band  structures.  \n \ncoupling strength between the wave vector (k) and the spin vector (σ) associated with the \nelectric field direction ( e). Both the x− and z−components of the local built -in electric  field \nare non -zero at the two Gd adatoms due to the off -center adsorption.  By assuming that  the \ndispersion  is parabolic  and that the spin quantization  direction  (SQD)  is along  the +z \ndirection,  the resulting  parabolic  bands  with the Rashba  splitting  due to the transverse  \nelectric field component ( Ex) are given by \n2\n2()2y y x yE k k km=+\n , where \nm  and ky are the \nelectric effective mass and the longitudinal wave number, respectively, and \nx  is the Rashba \nparameter associated with the transverse electric field component Ex. This effect is coined \ntransverse Rashba effect to emphasize that it is associated with the electric field component \ntransverse to the ZGNR direction and to discriminate it from the effect associated with the  \nelectric  field perpendicular  to the ZGNR  plane.  The parabolic  split by the transverse  Rashba \neffec t can be further rewritten as \n2\n2( ) ( )2y y R RE k k k Em= + −\n , where \n2x\nRmk\n=\n  and \n2\n22x\nRmE\n=\n for electron dispersion are indicated in FIG. 3 (a). The Rashba parameter can be \nthen extracted as αx = 2ER/∆kR. For the AFM state with the SQD along the + z ([001]) direction, \nthere is an obvious Rashba splitting of the valence band maximum (VBM) or conduc tion band \nminimum (CBM) at the Γ point, from which it is extracted that the Rashba parameter  αx is \n(a) \n \n   \n \nAFM  \n(b) \nAFM  \n(c) \nFM \n-σ \n \n \n \n \n \n \nσ \n \n \n \n \n \n \n-σ 12 \n 1.51 eV˚A. This  value is comparable  to those  of the low-dimensional systems[46, 47], such as \nPt-Si nanowire, with a giant Rashba parameter[47] .\n \nWhen the SQD is along the x−axis, the Rashba splitting is associated with the \nz−component of the built -in electric field induced by the adsorption of the Gd adatoms, which \napparently induces a much weaker splitting at the highest valence band and the lowest \nconduction band than that when the SQD is along the z−axis, as shown in FIG. 3 (b). \nAnalyzing the reasons of this discrepancy, bas ed on the d−orbital −resolved diagram of Gd \natoms on band structure as shown in FIG. 6 (b), it is noticed that the zones contributed by dxy, \ndx2−y2 and dxz orbits of Gd atoms have an obvious relative large Rashba splitting for SQD along \nthe z−axis, as shown  in FIG. 3 (a). However, a much weaker Rashba splitting occurred in the \nzones where the bands dominated by dz2 and dyz orbits. On the contrary,  for SQD along the \nx−axis corresponding to the z−component of the built -in electric field (FIG. 3 (b)), the zones \nmainly contributed by dz2 , dxz and dyz orbits of Gd atoms have a large Rashba splitting, \nhowever, no apparent Rashba splitting observed in the zones where dxy and dx2−y2 orbits have  \ndominant contributions. Generally, there is no Rashba splitting in a certain band zone where \nthe components of d orbit of Gd atoms normal to the direction of electric field. Thus the \nconclusion can be obtained that the relations between the direction of th e electric field and \nthe distributions of the d-orbital components of Gd atoms on the band structure have a great \ninfluence on the magnitude of Rashba splitting at the corresponding  band.  \nIn contrast to the AFM state, there is no Rashba splitting in the FM  state with the SQD \nalong the z−axis (FIG. 3 (c)). The Rashba Hamiltonian of the 2Gd -4ZGNR system can be \nrewritten as  \nHR2 =αR,e[(σ1−σ2)×k]· e  , (3.2)  \nwhere  the inversion  symmetry  ensures  that the Rashba  parameters  are equal  and the local \nelectric  field  directions  are opposite  at the two Gd adatoms.  For the AFM  state,  σ1=−σ2, \nand thus the Rashba  splitting  at the two Gd adatoms  enhance  each  other.  For the FM state,  13  however , σ1=σ2, and thus the Rashba splitting on the two Gd adatoms perfectly cancel each \nother out. As a result, there is no Rashba splitting for the FM state with the SQD along the \nz−axis. In fact, the Rashba splitting in the AFM state is consistent with the so -calle d R−2 \ntype spin polarization proposed in the nonmagnetic 3D crystals with a centric symmetry, \nwhere local electric fields induce anti -parallel local spins on two sites associated with the \ninversion operation[8]. It should be noted that the 2Gd -4ZGNR systems in the AFM state also \nrenders a centroic symmetry that was thought to be incompatible with the Rashba effect. The \nRashba effect discovered in the AFM state can be described as the inverse effect of the R −2 \ntype spin polariz ation in the nonmagnetic crystals, as it takes place for the anti -parallel spins \nof the AFM  state.  \nIn addition, it should be noted that the Rashba effect of the AFM state is distinct from the \nRashba spin splitting of the nonmagnetic structures with or with out a centric symmetry. In \nFIG. 3 (a) and (b), the up and down spins shift toward the same direction in the one -\ndimensional k−space, and the spin symmetry of the AFM state is fully preserved. This results \nfrom the centric symmetry of the 2Gd -4ZGNR system a nd is different from the  Rashba  spin \nsplitting  of the nonmagnetic  structures,  where  the up and down  spins  are degenerate but \nseparated by shifting oppositely in k-space.  It is the SQD instead of the spin direction that is \ncoupled to the crystalline moment um (\nk\n ) in the Rashba effect of the AFM  state,  as shown  \nin FIG.  3 (a) and (b). It is more  convention  to rewrite  (3.2)  as \n \nHR2=αR,eQ· k , (3.3)  \nwhere Q≡e×(σ1−σ2) is defined to be the screw vector of the 2Gd -4ZGNR. For the transverse \nRashba effect of the AFM state when SQD is along the + z ([001]) direction, the local spins \non the two Gd adatoms are along the z−axis and antiparallel to each other, and the screw \nvector is along the + y direction and parallel to the wavevector (FIG. 3 (d)). It follows  that HR2 \n=2αR,eσzky, which  leads  to a shift of both up and down  spins  toward  the −k direction  in \nFIG.  3 (a).  As the SQD  flips from  the +z to the −z direction,  the local  spins  flip their  \ndirections,  respectively,  and the screw  vector  flips to the −y direction  (FIG. 3 (d)). It \nfollows that HR2=-2αR,eσzky, which results in the shift toward the + k direction in FIG 3  \n(a). \nC. The origin of magnetocrystalline  anisotropy  \nThe MCA  for the AFM  and FM states  are calculated  as a function  of θ and fitted  14   \n \nFIG. 4: Quadratic fitting of the calculated MCA (meV) versus cos θ (0◦ ≤ θ ≤ 180◦) for the AFM \n(black line) and FM states (red line). The inset illustrates the definition of θ.  Fitting coefficients and  \nstandard  errors  are listed  in TABLE.  S I of Supplementary  Information[41].  \nto a quadratic polynomial of cos θ, where θ is the polar an gle of the magnetization in the \nzy−plane, as shown in FIG. 4. The uniaxial MCA constant (Ku) is calculated from the total \nenergy  difference  between  the in-plane  [010]  and out-of-plane  [001]  magnetization  directions \n(Ku=E[010] −E [001]). The calculated Ku of the AFM and FM states are 2.855 meV and 4.458 \nmeV, respectively, and the out -of-plane (perpendicular) magnetization direction is preferred \nfor both the states. We notice that the Ku of the FM state is about twice that of the AFM state,  \nwhich  inspires  us to explore  the origin  of the MCA  of the two magnetic  states.  \n1. The AFM  state  \nFrom the viewpoint of electronic structure, MCA results from the variation of band \nstructure induced by rotating magnetization from one direction to another. It can thus be \nwritten as a summation in k−space[48, 49],  \nMCA ( ) ( ) { [ ( ) ( ) ( ) ( )] ( )}a a b b\ni i i i\nk FBZ i NBANDSE a E b k f k k f k w k \n= − = −  \n                        (3.4)\nwhere E(a) and E(b) represents the energy when the magnetization along a and b directions,  \n \n  \n \n 15  \n \n \n \nBand1 \nBand2  \nBand3 \nBand4  \n \n \nBand5   \n \n(a) (b) (c) \n \nFIG.  5: For the AFM  state.  (a) The distributions  of the Ku, the Rashba  part,  and the LS−coupling \npart in k−space. (b) The Rashba part of Ku distribution in k−space compared to that calculated in \nterms of (3.4) with the four or five top -most valence bands. (c) The labels of the five  top-most \nvalence  bands.  \nrespectively.  ε(k),  f (k), and w(k) are the band energy,  the filling factor, and the weight  at \na point k  in the FBZ, and the subscript i stands for the ith band and the superscripts a and b \ndenote the magnetization directions. The Ku of the AFM state is calculated to be 2.857 meV \nin terms of (3.4) with a and b set to be [010] and [001] directions respectively and i contains \nall the bands, which is almost equal to that calculated directly from  the   total energy difference. \nIn addition, the contribution to the Ku at a single k−point can be extracted from (3.4) and \nplotted as a fun ction of k, as shown by  the black curve in FIG. 5  (a), which is approximately \nantisymmetric about the Γ point like  the result of the 4ZGNR decorated with a single Gd \nadatom[ 22]. In fact, the k−space distribution of Ku can be decomposed into the odd part \nassociated to Rashba splitting and the even part associated with  LS−coupling  for the AFM  \nstate.   By setting  a and b to be [00¯1] and [001]  directions respectively, in (3.4) the \nLS−coupling part of Ku vanishes, the Rashba splitting part is calculated and plotted, as the \nred curve in FIG. 5 (a), which is rigorously an odd function of  k. As a result, the net \ncontribution to the Ku from the Rashba splitting vanishes.  In  fact, the transverse  Rashba  \neffect  of the AFM  state  causes  the opposite  shifts  of the bands  about  the  Γ−point  when   the  \nmagnetization   directions   are  set  to  be  [00¯1]  and  [001],   resulting   in that the Rashba part of \nKu is a rigorous odd function of k in the one -dimensional FBZ. By  comparing the Rashba part \nof Ku distribution in k−space calculated with all the bands of the AFM state to those \ncalculated with the four or five top -most valence bands (FIG. 5 (c)). As shown in FIG. 5 (b), \nit is shown that the five top -most valence bands approximately reproduce the Rashba  part.  \nAfter the LS−coupling part is isolated from the Ku in k−space as shown by the blue curve \n16  in FIG. 5 (a), the origin of MCA can be analyzed in terms of perturbation theory[48]. Both \nthe first and second order pe rturbations can lower down the energy of filled states, depending \non magnetization direction, and result in MCA. The first order perturbation of LS−coupling \nacts on the degenerate or nearly degenerate states at the Fermi energy, and open or enlarge the \ngap, from which the LS−coupling coefficient can be extracted. Although the second order \nperturbation can also contribute to the MCA in general, it is less important for one -\ndimensional systems, since its contribution is a half order higher than that of the fi rst order,  \nas shown  in the section  Introduction.  \nThe LS−coupling part of Ku as a function of k without the weight w(k) in (3.4)  for the \nAFM  state is plotted  in FIG.  6 (a). To isolate  the LS−coupling  effect  in the band  structure,  \na set of bands is defined by averaging the band structures calculated with the magnetization  \nalong  the [001]  and [00¯1] directions  respectively, that is, each band  of this set is defined  by \n[001] [001]( ) ( ( ) ( )) / 2i i ik k k  =+\n. It should  be noted  that there  is no Rashba effect  in the averaged  \nband  structure,  as shown  by the bands  in black  in FIG.  6 (c), because  the Rashba  splitting  \nassociated  with the [001]  and [00¯1] magnetization  directions  are opposite  to each other and \ncanceled by the averaging. In addition, the band structure calculated with the magnetization \ndirection along the [010] is plotted in FIG. 6  (c) as a comparison. In FIG. 6 (d),  it is obvious  \nthat the bandgap  is enlarged  from  49.07  meV  to 109.20  meV  by rotating  the magnetization \ndirection from the longitudinal to the perpendicular direction with respect to the ZGNR. This \nfirst-order perturbation of LS−coupling interaction lowers down the VBM of the averaged \nband structure, compared to the [010] band structure, an d results in the sharp  peak  near the Γ \npoint  (or along  the ab segment)  in the LS−coupling  part of Ku (FIG.  6 (a)). In fact, the \nLS−coupling term in Hamiltonian can be written as  \n( cos sin ) ( cos sin ),2ls z y z y l l s l l     =  = + = +H l s\n                           (3.5)  17   \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFIG. 6: All subplots are associated with the AFM state of 2Gd -4ZGNR system. (a)  LS−coupling \npart of Ku distribution  without  considering  weight  factor  w(k) in (3.4)  in k−space  which  is divided \ninto three  segments  (labeled  as ab, bc and cd) along  the Γ−M line. (b) Spin -polarized  band  structure \n(left) and projected DOS of Gd 5 d−orbitals where the markers a, b, c, d along the Γ −M line are \nthose in (a). (c) Band structures with SOC perturbation, where the bands in red are calculated \nwith the magnetization along the [010] direction and those in black are the average of the bands \ncalculated  with  the magnetization  along  the [001]  and the [00¯1] directions, respectively, that is, for \nthe ith band  at point k, \n[001] [001]( ) ( ( ) ( )) / 2i i ik k k  =+ .  The markers a, b, c, d along  the Γ−M \nline are  those in (a). (d) Zoom -in view of the gray area of (c), where the corresponding band gaps \nare marked.  \nwhere the electron spin ( s = 1/2) is in the zy−plane, λ is the LS coupling coefficient, and θ is \nthe polar angle of the spin. The band structure projected onto the d−orbitals is plotted in FIG. \n6 (b), where the VBM and CBM at the Γ point are essentially contributed by dxy and dx2−y2 \norbits,  respectively.  The matrix  representation  of (3.5)  can then be constructed  by using  dxy \nand dx2−y2 orbits as  \ncos[ ] ,coslsiHi  \n  =− +\n                                                (3.6) \nwhere ε and ε + ∆ are the VBM and CBM energy levels without the LS−coupling per - \n(c) \na  b c d \n(d) \n∆θ=109.20 meV  \n∆=49.07 meV  \n \n  \n(a) \na  b c \n  \n(b) \n dxz \ndyz \n \n \ndyz \ndxy \n \n \na b \n c \n  \nDOS  18 \n turbation at the Γ point respectively, and ∆ is the bandgap without the perturbation. By \ndiagonalizing the matrix in (3.6), two levels with the perturbation can be obtained and \nseparated by an enlarged θ-dependent  gap \n224( cos ) ,   =  +\n                                                       (3.7)  \nBy using the gaps calculated by the first -principles, ∆=49.07 meV and ∆ θ=109.20 meV at \nθ=0◦, the LS coupling coefficient λ can be calculated for the AFM state of the 2Gd -4ZGNR \nsystem to be 48.78 meV, which is close to the value of about 60 meV in Gd -adsorbed \ngraphene[45]. In fact, there is no Rashba splitting in the band structure calculated with SOC \ninteraction when the magnetization is along the [010] direction, as the spin direction is parallel \nto the wave vector. In addition, there is also no first -order LS−coupling perturbation at the Γ \npoint, as the off -diagonal elements in (3.7) at θ=90◦ vanish (TABLE. S II of Supplementary  \nInformation[41]).  \nThe contribution to the Ku from the second order perturbation of LS−coupling \ninteraction[48] is further discussed qualitatively along the bc and cd segments in FIG. 6   (a). \nIt is given by the difference between the second order perturbation energies with the \nmagnetization  along  the [010]  and [001]  directions  respectively,  that is \n2 2\n(2) (2) (2) 2\n,( ) ( ;[010]) ( ;[001]) ,zy\nu\nou uoo L u o L u\nK k E k E k −\n= − =−\n                   (3.8) \nwhere ξ  is the LS−coupling coefficient that is determined from the bandgap enlarged by  the \nfirst-order perturbation at the Γ point, and the summation is over all occupied ( o) and \nunoccupied ( u) states, both of whi ch are either spin -up or spin -down. The matrix elements of \nangular momentum operators (Lx, Ly, and Lz) are calculated with the cubic harmonics of  \nd−orbitals  and summarized  in TABLE.  S II of Supplementary  Information[41].  \nSeveral approximations can be adopted to simplify the analysis of the second -order per - \nturbation in terms of (3.8) where all transitions are considered between the occupied and \nunoccupied states. In fact, the MCA of an insulating magnet is associated with  the localized  19  d−orbitals of transition metal ions, and thus only the transitions involving the d−orbitals   of \na single Gd adatom need to be considered. In addition, the exchange splitting (the separation \nin energy) between the spin -up and  spin-down d−orbitals of a single Gd adatom is as large  \nas 1.86 eV in the AFM  state  (shown  in FIG.  S4 (a) of Supplementary  Information[41]), which \ngoes into the denominator of equation (3.8) and is two orders of magnitude stronger than the \nLS−coupling coef ficient ξ in the numerator. Thus, the transitions between the states in \nopposite spins can be neglected. More than that, the occupied d−orbitals of the two Gd \nadatoms are associated with the spin -up and spin -down channels, respectively, in the AFM \nstate (s hown in FIG. S4 (b) of Supplementary Information[41]). Thus, it is only necessary  to \nconsider  either  spin-up or spin-down  d−bands  near the bandgap.  \nThe contribution of the second -order perturbation to the Ku in k−space (FIG. 6 (a)) is \ndiscussed in terms of (3.8) with the transitions between the spin -up (or spin -down) d−orbitals \nnear the bandgap (FIG. 6 (b)).   Along the segment ab, the major contribut ion is the transition \nover the narrow bandgap near the Γ point between the highest valence band and the lowest \nconduction band that are mainly of dxy and dx2−y2 characters, respectively. This contribution \nis positive and partly canceled by the negative contribution associated with the transition \nbetween the dxy−component of the highest valence band and the dyz−component of the second \nlowest conduction band.  Along  the bc segment, there are positive and negative contributions \ncompeting with each other, which results in a weak negative Ku. The positive contribution is \nassociated with the transition between the highest valence band and the lowest conduction \nband near the b point that are both a mixture of dxy and dx2−y2 characters. The negative \ncontribution is associated with the transition between the dx2−y2−component of the highest \nvalence band and the dxz−component of the third lowest conduction band. It also comes from \nthe transition between the dxy−component of the highest valence band and the dyz−component \nof the second lowest conduction band near the b point. The magnitude of Ku is diminishing \ntoward the c point, as the denominator (Eu−Eo) in (3.8) increases due to the decrease of the \nhighest valence band. Along the cd segment,  the Ku is very weak  due to the large  Eu−Eo in \nthe denominator  of (3.8).  \n2. The FM  state  \nIn FIG. 7 (a), the LS−coupling part of Ku without the weight w(k) in (3.4) is illustrated in \nk−space for the FM state. The MCA calculated in terms of (3.4) that numerically integrates  \nthe k−space  distribution  is 4.393  meV,  almost  equal  to that determined  from  the 20  \n(a) \n (c) \na  b c d  e f g  \n \n \nFIG. 7: All subplots are associated with the FM state of 2Gd -4ZGNR system. (a) LS−coupling \npart of Ku as a function  of k without  considering  weight  factor  w(k) in (3.4),  where  the spikes  are \nmarked  out by the labels  a−g along  the k−axis.  (b) Spin -up band  structure  with the weights  of Gd \n5d−orbital  states  near the Fermi  energy  (left),  and associated  projected  DOS  (right).  The markers \na − g along  the Γ−M line are those  in (a). (c) Band  structures  calculated  with SOC  perturbations \nfor the [001]  and [010]  magnetization  directions,  respectively.  The markers  a − g along  the Γ−M \nline are those in (a). (d), (e) and (f) correspond to the zoom -in view of the gray area of (c), \nrespectively, where the corresponding band gaps in (d) are marked, the labels P 1 (P2) in (e) are \nthe crossover  points  between  the band  B1 (B2)  of the [001]  or [010]  magnetization  directions  and \nthe corresponding Fermi  energy.  \n \ndifference  between  the total energies  with the magnetization  along  [010]  and [001]  directions \nrespectively (4.394  meV).  \nThere are several spikes, either positive or negative, in FIG. 7 (a), which results from the \ndrastic change of the filling factor in (3.4) at the spikes due to the shift of Fermi energy \ninduced by rotating magnetization  direction. This effect is well -known for calculating the \nMCA of metals in terms of the force theorem[29, 30], and might lead to the numerical insta - \nbility of the result for a cubic metal, of which is the MCA usually three order of magnitude \nlower  than that calculated  here.  The so-called  state-tracking  method[31]  was proposed  to \n(e) \n \n \n  \na b c d e f g \n \nc d \n  \n(b) \ndxz \n(d) \n (f) \n \n \ndxz \n \n \n \n ∆θ=81.59 meV  \n \n \n∆=3.05 meV  \n \na   b c    d    e f \ng \n DOS  \n \n  \n  21  solve this problem, and trigged a debate over the validity of the method[34, 50]. In fact, the \nshift of Fermi energy is induced by the first -order perturbation of  the LS−coupling inter - \naction between the almost -degenerate lowest and second lowest conduction bands at the Γ \npoint, when the magnetization is along the [001] direction (FIG. 7 (c)). Those two bands are \nmainly projected onto the dxy and dx2−y2 orbits, and the first -order perturbation enlarges the  \npseudo -gap in terms  of (3.7).  From  the pseudo -gap the LS−coupling  coefficient  (λ) is \nestimated to be 40.77 meV by substituting the ab-initio  values ∆ = 3.05 meV and ∆ θ = \n81.59 meV at θ = 90 into (3.7)  (shown in FIG. 7 (d)). Note that the off -diagonal elements of \n(3.6) vanish when the magnetization along [010] direction. The enhanced pseudo -gap pushes \nthe local minimum of the lowest conduction band at the Γ point downward to the energy lower \nthan the Fermi energy.  This effect converts the lowest unoccupied states around the Γ point \nto the occupied ones when the magnetization is rotated from the [010] direction to the  [001]  \ndirection,  and thus induces  the positive  spike  at the Γ point.  \nThe lowering -down of the lowest conduction band around the Γ point not only directly \ninduces the positive spike at the Γ but also induces the negative spikes indirectly by shifting \nthe Fermi energy. The parts of the band structure (grey -background box es in FIG. 7 (c)) that \nare associated with the spikes in FIG. 7 (a) are zoomed in (FIG. 7 (d), (e), and (f)). In FIG.  7 \n(f), the Fermi energy notably shifts down and the bands near the Fermi energy slightly nudge, \nas the magnetization is rotated from [010] to [001] direction. This effect converts the occupied \nstates near the M for the [010] magnetization direction to the unoccupied states for the [001] \ndirection, and thus induces the negative spike along the fg segment in FIG. 7 (a) according to \n(3.4). This eff ect converts the occupied states near the P 1 and P 2 points (FIG.  7 (e)) for the \n[010] magnetization direction to the unoccupied states for the [001] direction, and  thus \ninduces  the negative  spikes  along  the cd and de segments  in FIG.  7 (a). \n \nIV. CONCLUSION  \n \nThe zigzag graphene nanoribbon decorated with two Gd adatoms per unit cell has been \ninvestigated by using the first -principle calculations based on the density functional theory. \nThe 2Gd -4ZGNR system feature of the anti -plane adsorption structu re and the inversion \nsymmetry is energetically stable. Its magnetic ground state is AFM, which is lower than the  \nmetastable  FM state  by 45.383  meV.  The off-center  adsorption  of the Gd adatoms  22  generates the local dipole fields in the transverse direction with respect to the nanoribbon, and \nconsequently induces the transverse Rashba effect that has been identified from the splitting \nof the SOC -included band structure for the AFM state. It is show n that the transverse Rashba \neffect at the two Gd adatoms enhances each other due to the inversion symmetry in the AFM \nstate, and cancels out each other in the FM state. The Rashba splitting in the AFM state can \nbe understood as the inverse effect of so -called R −2 type spin polarization that is AFM and \ngenerated by the Rashba effect in some centrosymmetric bulk crystals. It should be noted that \nboth up and down spins shift toward the same direction in the SOC band structure and that \nthe direction is associa ted with the screw vector of the quasi -one-dimensional AFM state.  \nThe transverse Rashba parameter ( αx) 1.51 eV˚A  from   the  splitting   in  the  SOC   band   \nstructure,   comparable   with  the  values   of  other  one-dimensional Rashba systems.  \nThe origin  of MCA  has been  explored  for the AFM  and FM states, both  of which  shows  a \nperpendicular MCA in an order of magnitude of 1 meV per Gd adatom. The uniaxial MCA \nconstant (Ku) is determined by the quadratic polynomial fitting. The k−space distribution of \nKu is calculated  for the [010],  [001],  and [00¯1] magnetization  directions,  from  which the \nRashba and LS−coupling contributions to Ku is separated in k−space.  The Rashba part is an \nodd function of k and its net contribution to total Ku vanishes. The LS−coupling part is \nhowever an even function of k and does contribute to the tota l Ku. It has been analyzed in \nterms of the SOC band structures.  For the AFM state, the positive spike at the Γ point is \nattributed to the first -order perturbation between the highest valence band and the lowest \nconduction band that pushes the former down in energy. The second -order perturbation \ncontribution is qualitatively discussed in terms of the band structure projected onto d−orbitals.  \nFor the FM state, the band structure is metallic. The first -order perturbation opens a pseudo -\ngap between the lowest  conduction band and the second lowest one, which pushes the former \ndown below the Fermi energy. This effect induces a puddle to the valence band at the Γ, and \nconsequently the positive spike to the LS−coupling contribution in k−space. More than that, \nthe puddle causes a shifting -down of Fermi energy and  converts  the occupied  states  near the \nFermi  energy to the unoccupied  ones,  which  induces several negative spikes to the \nLS−coupling contribution in k−space. This work introduces a new type of Rashba effect in a \nquasi -one-dimensional antiferromagnet based on graphene nanoribbon,  which  couples  the \nperpen dicular  magnetic  easy direction  to the wave -vector .23  This finding  implies  a new approach  to realize  and control  high coherency  spin transportation in \none dimensionality using graphene -based  spintronics.  \n \nV. ACKNOWLEDGMENTS  \n \nThis research is supported by the science Challenge Project (Grant No. TZ2016003 -1- 105); \nTianjin Natural Science F oundation (Grant No. 20JCZDJC00750); CAEP Microsystem and \nTHz Science and Technology Foundation (Grant No. CAEPMT201501); National Basic \nResearch Program of China (Grant No. 2 011CB606405).  \n \n \n \n[1] Wolf,  S. A., Awschalom,  D. D., Buhrman,  R. A., Daughton,  J. M., von Moln ar, V. S., Roukes,  \nM. L., ... and Treger, D. M. (2001). Spintronics: a spin -based electronics vision for the future. \nScience, 294(5546), 1488 -1495.  \n[2] Bychkov, Y. A., and  Rashba,  E´. I. (1984).  Oscillatory  effects  and the magnetic  susceptibili ty \nof carriers in inversion layers. Journal of physics C: Solid State Physics, 17(33), 6039.  \n[3] Bychkov, Y. A., and  Rashba,  E´. I. (1984).  Properties  of a 2D electron  gas with  lifted spectral  \ndegeneracy. JETP Lett, 39(2), 78.  \n[4] Winkler,  R. (2003).  Spin -orbit  coupling  effects  in two-dimensional  electron  and hole systems.  \nSpringer Tracts in Modern Physics, 191, 1 -8. \n[5] Ishizaka, K., Bahramy, M. S., Murakawa, H., Sakano, M., Shimojima, T., Sonobe, T., ... and \nMiyamoto,  K. (2011).  Giant  Rashba -type spin splitting  in bulk BiTeI.  Nature  Materials,  10(7), \n521-526.  \n[6] Sakano, M., Miyawaki, J., Chainani, A., Takata, Y., Sonobe, T., Shimojima,  T.,  ...  and Na - \ngaosa, N. (2012). Three -dimensional bulk band dispersion in polar BiTeI with giant Rashba -type \nspin splitting. Physical Review B, 86(8),  085204.  \n[7] Ast, C. R., Henk, J., Ernst, A., Moreschini , L., Falub, M. C.,  Pacil,  D.,  ... and Grioni,  M.  \n(2007).  Giant  spin splitting  through  surface  alloying.  Physical  Review  Letters,  98(18),  186807.  \n[8] Zhang, X., Liu, Q., Luo, J. W., Freeman, A. J., and Zunger, A. (2014). Hidden spin polarization \nin inversion -symmetric  bulk crystals.  Nature  Physics,  10(5),  387-393. \n[9] Riley,  J. M., Mazzola,  F., Dendzik,  M., Michiardi,  M., Takayama,  T., Bawden,  L., ... and 24  Kim, T. K. (2014). Direct observation of spin -polarized bulk bands i n an inversion -symmetric \nsemiconductor. Nature Physics, 10(11),  835-839. \n[10] Liu, Q., Zhang, X., Jin, H., Lam, K., Im, J., Freeman,  A. J., and Zunger, A. (2015). Search     \nand design of nonmagnetic centrosymmetric layered crystals with large local spin polarization. \nPhysical Review B, 91(23),  235204.  \n[11]S -lawinska, J., Narayan, A., and Picozzi, S. (2016). Hidden spin polarization in nonmagnetic \ncentrosymmetric BaNiS 2 crystal: Signatures from first principles. Physical Review B, 94(24), \n241114.  \n[12] Wu, S. L., Sumida,  K., Miyamoto,  K., Taguchi,  K., Yoshikawa,  T., Kimura,  A., ... and Tanaka,  \nI. (2017). Direct evidence of hidden local spin polarization in a centrosymmetric superconductor \nLaO 0.55F0.45BiS 2. Nature Communications, 8(1),  1-7. \n[13] Gotlieb , K., Lin, C. Y., Serbyn, M., Zhang, W., Smallwood, C. L., Jozwiak, C., ... and \nLanzara, A. (2018). Revealing hidden spin -momentum locking in a high -temperature   cuprate \nsuperconductor. Science, 362(6420),  1271 -1275.  \n[14] Yuan, L., Liu, Q., Zhang, X., Luo, J. W ., Li, S. S., and Zunger, A. (2019). Uncovering and \ntailoring  hidden  Rashba  spin−orbit  splitting  in centrosymmetric  crystals.  Nature  Communica - \ntions, 10(1),  1-8. \n[15] Tu, J., Chen, X. B., Ruan, X. Z., Zhao, Y. F., Xu, H. F., Chen, Z. D., ... and  Zhang,  \nY. (202 0). Direct observation of hidden spin polarization in 2H −MoTe 2. Physical Review B, \n101(3), 035102.  \n[16] Krupin,  O., Bihlmayer,  G., Starke,  K., Gorovikov,  S., Prieto,  J. E., D\nu\nbrich,  K., ... and Kaindl,  \nG. (2005). Rashba effect at magnetic metal surfaces. Physical Review B, 71(20), 201403. \n[17]Krupin, O., Bihlmayer, G., D\nu\n brich, K. M., Prieto, J. E., Starke, K., Gorovikov, S., ... and  \nKaindl,  G. (2009).  Rashba  effect  at the surfaces  of rare-earth  metals  and their  monoxides.  New \nJournal of Physics, 11(1),  013035.  \n[18] Dedkov, Y. S., Fonin, M., R\nu\n diger, U., and Laubschat, C. (2008). Rashba effect in the \ngraphene/Ni (111) system. Physical Review Letters, 100(10),  107602.  \n[19] Park, J. H ., Kim, C. H., Lee, H. W., and Han, J. H. (2013). Orbital chirality and Rashba \ninteraction in magnetic bands. Physical Review B, 87(4),  041301.  \n[20] Kawano, M., Onose, Y., and Hotta, C. (2019). Designing Rashba −Dresselhaus effect in mag - \nnetic insulators. Commu nications Physics, 2(1),  1-8. 25  [21] Yamauchi, K., Barone, P., and Picozzi, S. (2019). Bulk Rashba effect in multiferroics: A \ntheoretical  prediction  for BiCoO 3. Physical  Review  B, 100(24),  245115.  \n[22] Qin, Z., Qin, G., Shao, B., and Zuo, X. (2017). Unconventional magnetic anisotropy in one - \ndimensional Rashba system realized by adsorbing Gd atom on zigzag graphene nanoribbons. \nNanoscale, 9(32),  11657 -11666.  \n[23] Qin, Z., Qin, G., Shao, B., and Zuo, X. (2020) . Rashba spin splitting and perpendicular magnetic \nanisotropy of Gd -adsorbed zigzag graphene nanoribbon modulated by edge states under  external  \nelectric  fields.  Physical  Review  B, 101(1),  014451.  \n[24] Shick, A. B., Khmelevskyi, S., Mryasov, O. N., Wunderlich, J., and Jungwirth, T. (2010). Spin -\norbit coupling induced anisotropy effects in bimetallic antiferromagnets: A route towards \nantiferromagnetic spintronics. Physical Review B, 81(21),  212409.  \n[25] Jungwirth, T., Marti, X., Wadley, P., and Wunderlich,  J. (2016). Antiferromagnetic spintronics. \nNature Nanotechnology, 11(3),  231-241. \n[26] Baltz, V., Manchon, A., Tsoi, M., Moriyama, T., Ono, T., and Tserkovnyak, Y. (2018).  \nAntiferromagnetic spintronics. Reviews of Modern Physics, 90(1), 015005.  \n[27] Van Vleck, J. H. (1937). On the anisotropy of cubic ferromagnetic crystals. Physical Review, \n52(11),  1178.  \n[28] Lessard, A., Moos, T. H., and H\nu\n bner, W. (1997). Magnetocrystalline anisotropy energy of \ntransition -metal  thin films:  A nonperturbative  theory.  Physical  Review  B, 56(5),  2594.  \n[29] Weinert, M., Watson, R. E., and Davenport, J. W. (1985). Total -energy differences and eigen - \nvalue sums. Physical  Review B, 32(4), 2115.  \n[30] Daalderop, G. H. O., Kelly, P. J., and Schuurmans, M. F. H. (1990). First -principles calculation \nof the magnetocrystalline anisotropy energy of iron, cobalt, and nickel. Physical Review B, \n41(17),  11919.  \n[31] Wang, D. S., Wu, R., and Freeman, A. J. (1993). State -tracking first -principles det ermination of \nmagnetocrystalline anisotropy. Physical Review Letters, 70(6),  869. \n[32] Wang, X., Wu, R., Wang, D. S., and Freeman, A. J. (1996). Torque method for the theoretical \ndetermination of magnetocrystalline anisotropy. Physical Review B, 54(1),  61. \n[33] Schneider, G., Erickson, R. P., and Jansen, H. J. F. (1997). Calculation of the magnetocrys - \ntalline  anisotropy  energy  using  a torque  method.  Journal  of applied  physics,  81(8),  3869 -3871.  \n[34] Bylander, D. M., and Kleinman, L. (1995).  Effect  of k−space integration on  self-consistent  26  surface magnetic anisotropy calculations. Physical Review B, 52(3), 1437.  \n[35] Yang, I., Savrasov, S. Y., and Kotliar, G. (2001). Importance of correlation effects on magnetic \nanisotropy  in Fe and Ni. Physical  Review  Letters,  87(21),  216405.  \n[36] Bl\no\n chl, P. E. (1994). Projector augmented -wave method. Physical Review B, 50(24), 17953. \n[37]Kresse,  G., and Furthm\nu\n ller, J. (1996).  Efficiency  of ab-initio  total energy  calculations  for \nmetals and semiconductors using a plane -wave basis set. Computational Materials Science,  \n6(1),  15-50. \n[38] Perdew,  J. P., Burke,  K., and Ernzerhof,  M. (1996).  Generalized  gradient  approximation  made \nsimple. Physical Review Letters, 77(18),  3865.  \n[39] Dudarev, S. L., Botton, G. A., Savrasov, S. Y., Humphreys, C. J., and Sutton, A. P. (1998). \nElectron -energy -loss spectra and the structural stability of nickel oxide: An LSDA+U study. \nPhysical Review B, 57(3),  1505.  \n[40] Wang, B., Zhang, X., Zhang, Y., Yuan, S., Guo, Y., Dong, S., and Wang, J. (2020). Prediction   \nof a two -dimensional high -TC f -electron ferromagnetic semiconductor. Materials Horizons, 7, \n1623 -1630  \n[41] Supplementary Information is available  online  \n[42] Makov,  G., and Payne,  M. C. (1995).  Periodic  boundary  conditions  in ab initio  calculations.  \nPhysical Review B, 51(7),  4014.  \n[43] Son, Y. W., Cohen,  M. L., and Louie,  S. G. (2006).  Energy  gaps  in graphene  nanoribbons.  \nPhysical Review Letters, 97(21),  216803.  \n[44] Wu, F., Kan, E., Xiang, H., Wei, S. H., Whangbo, M. H., and Yang, J. (2009).  Magnetic states of \nzigzag graphene nanoribbons from first principles. Applied Physics Letters, 94(22), 223105.  \n[45] Lu, Y., Zhou, T. G., Shao, B., Zuo, X., and Feng, M. (2016). Carrier -dependent magnetic \nanisotropy of Gd -adsorbed graphene. AIP Advances, 6(5),  055708.  \n[46] Barke, I., Zheng, F., R\nu\n gheimer, T. K., and Himpsel, F. J. (2006). Experimental evidence for \nspin-split bands  in a one-dimensional  chain  structure.  Physical  Review  Letters,  97(22),  226405.  \n[47] Park,  J., Jung,  S. W., Jung,  M. C., Yamane,  H., Kosugi,  N., and Yeom,  H. W. (2013).  Self- \nassembled  nanowires  with giant  rashba  split bands.  Physical  Review  Letters,  110(3),  036801.  \n[48]Wang,  D. S., Wu, R., and Freeman,  A. J. (1993).  First -principles  theory  of surface  magne - \ntocrystalline  anisotropy and the diatomic -pair model. Physical Review B, 47(22), 14932.  27  [49] Van der Laan, G. (1998). Microscopic origin of magnetocrystalline anisotropy in transition \nmetal  thin films.  Journal  of Physics:  Condensed  Matter,  10(14),  3239.  \n[50] Daalderop, G. H. O., Kelly, P. J., and Schuurmans, M. F. H. (1993). Comment on State - tracking \nfirst-principles determination of magnetocrystalline anistropy. Physical Review Letters, 71(13),  \n2165.  \n \n \n \nSupplementary Information  \n \nIt is established that narrow zigzag graphene nanoribbon (ZGNR) is antiferromagnetic \n(AFM)  semiconductors,  where  each edge  is a ferromagnetic  (FM)  chain  and the two edges  are \nantiferrmagnetically coupled each other[1, 2]. The magnetic states of the ZGNR composed \nof four zigzag  carbon  chains  (4ZGNR)  are investigated  to verify  the computational  approach. \nThe edge carbon atoms are saturated by hydrogens. It is shown that the AFM state is the \nmagnetic ground state, which is lower than the FM state by 14.588 meV per edge carbon atom \nin energy and lower than the non -magnetic state by 55.436 meV, which agrees well with the \nprevious calculations[3,  4]. \nIn Fig. S1, the number of k-points can explicitly influence on the MCA. We find 1 ×69×1 k-\npoints along 1D k-space is sufficient to calculate the accurate MCA of 2Gd -4ZGNR system for \nAFM  state.  \nIn Fig. S2, the favorable  configuration  of 2Gd-4ZGNR  system  corresponding  to a coverage \nof two Gd adatoms per 16 carbon atoms and the band structures of non -SOC perturbation as \nwell as [001]  and [00¯1] magnetization  directions  are shown in Fig.  S2.  The Rashba-type \nsplitting  is shown when  the magnetization  directions  along  [001]  and [00¯1] axis for AFM  \nstate.   The calculational  binding  energ y, Rashba parameter  along  [001] ([00¯1]) magnetization \ndirection  and MCA are 5.085  eV, 0.958  eV ˚A and 2.240  meV,  respectively. \nIn Fig. S3, we can clearly see that the distributions of 4 f -orbital states of the Gd atoms in \nband structure are far away from the fermi level, which means that 4 f -orbital states have little  \ninfluence  on the properties  of whole  system.  28   \n \n \n \n[1] Son, Y. W., Cohen,  M. L., and Louie,  S. G. (2006).  Energy  gaps  in graphene  nanoribbons.  \nPhysical Review Letters, 97(21),  216803.  \n[2] Wu, F., Kan, E., Xiang, H., Wei, S. H., Whangbo, M. H., and Yang, J. (2009). Magnetic states of  \nzigzag  graphene  nanoribbons  from  first principles.  Applied  Physics  Letters,  94(22),  223105.  \n[3] Qin, Z., Qin, G., Shao, B., and Zuo, X. (2017). Unconventional magnetic anisotropy in one - \ndimensional Rashba system realized by adsorbing Gd atom on zigzag graphene nanoribbons. \nNanoscale, 9(32),  11657 -11666.  \n[4] Krychowski, D., Kaczk owski, J., and Lipinski, S. (2014). Kondo effect of a cobalt adatom on a \nzigzag  graphene  nanoribbon.  Physical  Review  B, 89(3),  035424.  29   \n \n \nFIG. S1: The influence of the k-points along the one -dimensional FBZ on MCA for AFM state of \n2Gd-4ZGNR system.  \n \nTABLE S I: The fitted values of MCA of 2Gd -4ZGNR for the AFM and FM states based on \npolynomial fitting formula ( K0 + K1 cos θ + K2 sin2θ), which are corresponding to the fitted curves  in \nFIG.  4 of text. The K0, K1, K2 and the respective  standard  errors  are presented.\nState  E − E0◦ =K0 + K1 cos θ + K2 sin2 θ Standard error  \n \nAFM  K0=6.740 ×10−3 \nK1=5.393 ×10−4 A=0.002  \nB1=0.002  \n K2=2.855  B2=0.003  \n \n \nFM K0=5.590 ×10−3 \nK1=4.266 ×10−4 A=0.009  \nB1=0.007  \n K2=4.458  B2=0.014  \n 30  \n \n(b) \n \n \n \n \n \n \n \n \n \n \nFIG. S2: (a) The top view of the equilibrium structure for 2Gd -4ZGNR structure corresponding \nto a coverage of two Gd adatoms per 16 carbon atoms.  The unit cell of structure is indicated by \nbroken lines. Gray, purple and white balls represent carbon, gadolinium and hydrogen atoms, \nrespectively. (b) The band dispersion relations of non -SOC perturbation for AFM state. (c) The \nband  structures  of SOC -pertur bed with  001 and 00¯1 magnetization  directions  for AFM  state.   Note \nthat the fermi level is s et to zero.  \n \nTABLE S II: Matrix elements of angular momentum operators Lz, Ly and Lx with the cubic \nharmonics of d−orbitals.  \n|u) Lz Ly Lx \n \n(o| dxy   dyz  dz2  dxz  dx2−y2    dxy dyz  dz2 dxz dx2−y2   dxy  dyz dz2 dxz dx2−y2 \ndxy 0 0 0 0 2i 0 i 0 0 0 0 0 0 −i 0 \ndyz 0 0 0 i 0 −i 0 0 0 0 0 0  \n3i−     0 −i \ndz2 0 0 0 0 0 0 0 0 \n3i−  0 0  \n3i  0 0 0 \ndxz 0 −i   0 0 0 0 0  \n3i  0 −i i 0 0 0 0 \ndx2−y2  −2i   0 0 0 0 0 0 0 i 0 0 i 0 0 0 \n(c) \n (a) 31  \n \n \n \nFIG. S3: The spin -up (spin -down) band structures with the distribution of Gd 4 f -orbital states.  \n \n \n(a) \n \n \n \n \n \n \n \n \n \n \n \n1.86 eV  \n \n \n \n \n \n \n \n \n \nFIG. S4: For AFM state. (a) The local projected density of states (LPDOS) about 5 d−orbital - \nresolved contributions of Gd 1 adatom. (b) The spin -up/down LPDOS of 5 d−orbital of Gd 1 and \nGd 2 adatoms.  \n "    },    {        "title": "2104.11986v2.Electric_field_driven_stability_control_of_skyrmions_in_an_ultrathin_transition_metal_film.pdf",        "content": "arXiv:2104.11986v2  [cond-mat.mtrl-sci]  26 Nov 2021Electric-field driven stability control of skyrmions in an u ltrathin transition-metal film\nSouvik Paul1,2,∗and Stefan Heinze2,†\n1Peter Gr ¨unberg Institut and Institute for Advanced Simulation,\nForschungszentrum J ¨ulich and JARA, 52425 J ¨ulich, Germany.\n2Institute of Theoretical Physics and Astrophysics,\nChristian-Albrechts-Universit ¨at zu Kiel, Leibnizstrasse 15, 24098 Kiel, Germany\n(Dated: November 29, 2021)\nTo realize future spintronic applications with magnetic sk yrmions – topologically nontrivial\nswirling spin structures – it is essential to achieve efficien t writing and deleting capabilities of these\nquasi-particles. Electric-field assisted nucleation and a nnihilation is a promising route, however, the\nunderstanding of the underlying microscopic mechanisms is still limited. Here, we show how the\nstability of individual magnetic skyrmions in an ultrathin transition-metal film can be controlled\nvia external electric fields. We demonstrate based on densit y functional theory that it is impor-\ntant to consider the changes of all interactions with electr ic field, i.e., the pair-wise exchange, the\nDzyaloshinskii-Moriya interaction, the magnetocrystall ine anisotropy energy, and the higher-order\nexchange interactions. The energy barriers for electric-fi eld assisted skyrmion writing and deleting\nobtained via atomistic spin simulations vary by up to a facto r of three more than the variations of\nthe interactions calculated from first-principles. This su rprising effect originates from the electric-\nfield dependent size of metastable skyrmions at a fixed magnet ic field. The large changes of lifetimes\nallow the possibility of electric-field assisted thermally activated writing and deleting of skyrmions.\nIntroduction\nMagneticskyrmions[1,2]showgreatpromisesasinfor-\nmation carriersin future magnetic memory, logicdevices,\nand neuromorphic computing due to their nanoscale size\nand ultralow current-driven manipulation, achieved by\nspin transfer torque (STT) [2–7] and spin orbit torque\n[8–14]. However, the main drawback of these techniques\ntorealizealow-energy-dissipationdeviceisJouleheating,\nwhich destabilizes the skyrmionic bits. The electric-field-\ninduced manipulation offers an efficient route for creat-\ning, deletingandcontrollingskyrmionsavoidingtheheat-\ning problem. The energy dissipation can be reduced by\na factor of 100 by using the electric field as compared to\nSTT [15]. Another important benefit of using an electric\nfield is that it can be applied locally and it does not dis-\nplace the skyrmionic bits, which is desirable for encoding\ninformation at a particular position.\nIn spite of recognizing the potential of electric-field-\ninduced manipulation, only a few experimental studies\nhave been reported on the electric-field-induced switch-\ning of skyrmions and skyrmion bubbles in transition-\nmetalmultilayers[16–19]andrecentlyinmultiferroichet-\nerostructures[20, 21]. Theoreticalstudies haveaddressed\neither the variation of magnetic anisotropy [22–24] or\nthe Dzyaloshinskii-Moriya interaction (DMI) [19, 25, 26]\ndirectly by the electric field or indirectly due to the\nelectric-field-induced strain [20, 21]. However, it is the\ninterplay of the exchange interaction, the DMI, and the\nanisotropywhich is responsiblefor the properties ofmag-\nnetic skyrmions [1]. Therefore, in a study on the influ-\nence of the electric field, one needs to take the varia-\ntion of all these interactions into account. Moreover,\n∗s.paul@fz-juelich.de;paul@physik.uni-kiel.de\n†heinze@physik.uni-kiel.dethe importance of higher-order exchange interactions\n(HOI) beyond the conventional Heisenberg pair-wise ex-\nchange mechanism in ultrathin films for the stability of\nskyrmions has recently been demonstrated [27].\nHere, we investigate the stability of isolated magnetic\nskyrmions in an ultrathin film from first-principles elec-\ntronic structures theory. We take an atomic Fe/Rh bi-\nlayer on the Re(0001) surface, Fe/Rh/Re(0001), as a\nmodel system, which is capable of stabilizing isolated\nskyrmions at external magnetic fields in absence of an\nelectric field [27]. From density functional theory (DFT)\ncalculations, we find that the Heisenberg pair-wise ex-\nchange interactions vary by about 15%, the DMI by\nonly 8%, while the magnetocrystalline anisotropy energy\n(MAE) variesby about30%foranelectricfield difference\nof 1 V/˚A. Among the HOI, only the four-site four spin\ninteraction shows a significant variation of about 20%\nfor an electric field difference of 1 V/ ˚A. We analyze the\nchanges of the magnetic interactions based on the spin-\ndependent screening of the electric field at the surface.\nWe study the formation and collapse of individual\nskyrmions under electric fields by atomistic spin simu-\nlations using the DFT parameters. The energy barriers\npreventing the collapse of individual skyrmions varies\nby about 60% and the barriers for skyrmion creation\nby about 30% for an electric difference of 1 V/ ˚A. The\nenhanced energy barriers with respect to the variation\nof the magnetic interactions can be explained by a shift\nof the critical magnetic field by about 0.6 T of the phase\nboundary between the skyrmion and field-polarized\nstate. Thereby, the skyrmion radius varies significantly\nwith electric field at a given magnetic field value which\nleads to the large change of the energy barriers and\nthereby of the lifetimes which makes electric-field driven\nswitching feasible.2\n020406080E(q) — EFM (meV/Fe atom)\n0.4a\n0.2 0 0.2 0.4 0.6 0.8 1\nq(2π/a)0.58uudduudd3QRight rotatingM M K \u0001Left rotating0.25\n0\n0.25\n0.02 0.05\u0000\n\u0002MCharged sheet\nRh ML\nRe(0001)\nsubstrateFe MLElectric\nfield\n-0.5 V/Åb\n+0.5 V/Å\n  0.0 V/Å\n-0.5 V/Å\nFig. 1 Effect of electric fields on the spin spiral energy dispe rsion and multi- Qstates. a Illustration of the Fe\nmonolayer (ML) on a Rh ML on the Re(0001) surface, denoted as F e/Rh/Re(0001), exposed to a perpendicular uniform\nelectric field. In the DFT calculation, the electric field is c reated by a charged sheet located at 5.3 ˚A above the Fe layer. The\ndirection of electric field lines for E=−0.5 V/˚A is shown. bEnergy dispersion E(q) of flat spin spirals along two high\nsymmetry directions ( ΓKM and ΓM) without SOC at E= +0.5 V/ ˚A (red), E= 0.0 V/ ˚A (green), and E=−0.5 V/˚A (blue).\nThe filled circles represent DFT data and the solid lines are fi ts to the Heisenberg model. The filled diamonds, highlighted by\nthe yellow border, represent the multi- Q(3Qanduudd) states without SOC, which are shown at the qpoints of the\ncorresponding single- Qstates (see Fig. 6 for spin structures). Inset shows E(q) of flat cycloidal spin spirals including DMI\nand MAE along ΓM for zero and two finite electric fields. The filled circles re present DFT data and the solid lines are fits to\nthe Heisenberg plus DMI spin model (see text for details). Gr ey (cyan) shaded region indicates right (left) rotating spi n\nspirals. Note that the MAE shifts E(q) byK/2 with respect to the FM state.\nResults\nFirst-principles calculations. Fig.1a shows the setup\nused to perform the DFT calculations including an elec-\ntric field based on the FLEURcode [28] (see methods for\ncomputational details). The electric field is introduced\nby placing a charged sheet in the vacuum region of the\nFe/Rh/Re(0001) film [29, 30]. Charge neutrality of the\nwhole system is maintained by adding or removing the\nsame amount of opposite charge to or from the film. In\nthis way, a uniform electric field perpendicular to the\nfilm plane is generated. For the electric field strength,\nwe chose values of E=±0.5 V/˚A as in the experimen-\ntal work of Ref. [16], which demonstrated switching of\nskyrmions in Fe films at these electric field strengths.\nWe first discuss the energy dispersion E(q) of ho-\nmogeneous flat spin spirals without spin-orbit coupling\n(SOC) obtained via DFT for Fe/Rh/Re(0001) along two\nhigh-symmetric directions of the two-dimensional Bril-\nlouin zone (2DBZ) (Fig. 1b). The magnetic moment Mi\nat lattice site Ri, of a spin spiral is given by Mi=\nM(cosqRi,sinqRi,0), where Mis the size of the mo-\nment (about 2 .9µBper Fe atom) and qis the spin spi-\nral vector. At E= 0, the ferromagnetic (FM) state ( Γ\npoint) is energetically lowest. The N´ eel state with an\nangle of 120◦between adjacent spins ( K point) and the\nrow-wise antiferromagnetic (AFM) state ( M point) are\nsignificantly higher in energy. The dispersion calculatedforE=±0.5 V/˚A shows the same trend as of the zero\nfield. The electric field induced modification of E(q) is\nnot visible on this scale at small q(see Supplementary\nFig. 1 for a close-up), however, it is significant at large\nqwith an energy rise (drop) at E>0 (E<0). This kind\nof modification has been observed earlier in a freestand-\ning Fe monolayer (ML) and for a Co ML on Pt(111),\nwhich can be explained by the spin spiral ( q) dependent\nscreening of the electric field [30].\nSOC adds two contributions: the MAE and the DMI\n(inset of Fig. 1b). The easy magnetization axis of the\nFe/Rh bilayer is in the film plane (easy plane) and the\nvalue of the MAE is K=−0.2 meV/Fe atom that leads\nto an energy offset of K/2 for spin spirals with respect to\nthe FM state ( Γ point). The MAE changes only slightly\nupon applying an electric field which is barely visible in\ntheinset. TheDMIarisesduetobreakingoftheinversion\nsymmetry at the surface and here it favors cycloidal spin\nspiralswithaclockwiserotationalsense(insetofFig. 1b).\nForE= +0.5 V/ ˚A, the spin spiral energy minimum is\n−0.05meV/FebelowtheFMstateandthespiralexhibits\na pitch of 18.4 nm. For E=−0.5 V/˚A, the minimum is\n−0.20meV/Fe lowerand the pitch significantlydecreases\nto 10.2 nm. Note that the field-induced change of the\nenergyminimumby0.15meV/Featomissignificantsince\nthe Zeeman energy due to an applied magnetic field of\n1 T amounts to about 0.2 meV/Fe atom.3\nElectric field J1J2 J3 J4J5J6 J7 B1Y1K1DeffK\n+0.5 V/˚A 9.37 −1.03 0.07 −0.24 0.27 −0.01−0.12−0.34 1.05 −1.23 0.87 −0.16\n0.0 V/˚A 8.85 −0.77−0.05−0.22 0.27 0.05 −0.16−0.39 1.00 −1.36 0.89 −0.20\n−0.5 V/˚A 8.08 −0.55 0.03 −0.20 0.23 0.01 −0.12−0.33 1.00 −1.53 0.94 −0.22\nTable I Variation of interaction constants with electric fie lds.Constants for the i-th nearest-neighbor exchange ( Ji),\nbiquadratic exchange ( B1), three-site four spin exchange ( Y1), four-site four spin exchange ( K1), effective DMI ( Deff), and the\nMAE (K) obtained via DFT for Fe/Rh/Re(0001) at zero and two finite el ectric fields. The positive sign of Deffindicates a\nclockwise rotational sense and negative value of Kindicates an in-plane easy magnetization axis (easy plane) . Data of E= 0.0\nV/˚A are taken from Ref. [27]. Note that the first three exchange i nteractions are modified according to Eqs. (8a-c). For more\ndetails, see Supplementary Table 1. All energies are given i n meV.\nSince higher-order interactions (HOI) beyond pair-\nwise Heisenberg type exchange can be important for\nthe magnetic ground state in transition-metal films [31–\n36] as well as for the stability of individual skyrmions\n[27] and skyrmion lattices [37], we have also calculated\ntheir variation upon applying electric fields. To cal-\nculate the biquadratic interaction as well as the three-\nsite and the four-site four spin interactions [35], we con-\nsider three multi- Qstates: the twocollinearup-up-down-\ndown (uudd) states [38] and a three-dimensional non-\ncollinear 3Q-state [31] (see methods for details). As seen\nin Fig.1b, the multi- Qstates, calculated at zero and\ntwo finite electric fields, are lower in energy compared\nto the corresponding spin spiral states (for energy differ-\nences see Supplementary Table 2). The energies of the\nmulti-Q states shift with field in a similar way as the\nenergy dispersion, i.e., their energy increases (decreases)\nfor positive (negative) electric field.\nThe total energy from DFT calculations are used to\nparametrize an atomistic spin model which is given by:\nH=−/summationdisplay\nijJij(mi·mj)−/summationdisplay\n<ij>Dij·(mi×mj)\n−/summationdisplay\niK(mz\ni)2−/summationdisplay\niµsB·mi−B1/summationdisplay\n<ij>(mi·mj)2\n−2Y1/summationdisplay\n<ijk>(mi·mj)(mj·mk)\n−K1/summationdisplay\n<ijkl>(mi·mj)(mk·ml)+(mi·ml)(mj·mk)\n−(mi·mk)(mj·ml) (1)\nwhere the magnetic moment of Fe at site iis denoted\nbyMiandmi=Mi/Mi.Jij,Dij,µsandKde-\nnote the pair-wise exchange constants, the DMI vectors,\nthe magnetic moment and the MAE constant, respec-\ntively.B1is the biquadratic constant, Y1andK1are the\nthree-site and four-site four spin constant, respectively.\nThe higher-order interactions are taken into account in\nnearest-neighbor approximation, indicated in the sum-\nmation by < .. >, since they arise from fourth-order per-\nturbation theory [35]. The DMI has been calculated in\nthe effective nearest-neighbor approximation, i.e., deter-mined from the slope of the energy contribution due to\nSOC close to q= 0 [39].\nThe DFT values of the interaction parameters are\ngiven in table Ifor zero and two finite electric fields. At\nE= +0.5 V/˚A, the ferromagnetic nearest-neighbor ex-\nchange constant, J1, is enhanced by about 6%, while at\nE=−0.5 V/˚A, it decreases by about 9% with respect to\nzero electric field. The absolute change of J1amounts to\nabout 1.3 meV for a field change by 1 V/ ˚A, which is sim-\nilar to the value of 1.2 meV, reported from a DFT study\nfor a Co ML on Pt(111) [30]. J2becomes more (less) an-\ntiferromagnetic at E>0 (E<0) with a variation almost\nlinear with electric field. The pair-wise exchange con-\nstants beyond second neighbors remain fairly constant\nwith electric fields. Among the HOI, only the four-site\nfour spin interaction, K1, varies significantly with elec-\ntric field. It shows a decrease of about 10% at E= +0.5\nV/˚A and an increase of about 13% at E=−0.5 V/˚A.\nThe effective nearest-neighbor DMI constant, Deff,\nvaries by only 0.07 meV upon changing the electric\nfield by 1 V/ ˚A, which amounts to a relative change of\n∆Deff/Deff≈0.08. The field-induced variation of the\nDMI is still twice larger than the previously reported\nvalue for a MgO/Co/Pt trilayer [25]. The MAE changes\nby 30% from E= +0.5 V/˚A toE=−0.5 V/˚A and it con-\ntributes 0.03 meV to the change of the spin spiral energy\nminimum (Fig. 1b).\nToobtaininsightintotheelectric-fieldinducedchanges\nof the magnetic interactions, we analyze the spin-\ndependent screening of the electric field in the film. The\nmodification of the charge density, ∆ ρ, at positive and\nnegative electric fields with respect to zero field is dis-\nplayed for the ferromagnetic state in Fig. 2a,b. ∆ρis\nspin-dependent and therefore, it creates surface electric\ndipoles which screen the electric field close to the sur-\nface of the film. The direction of the electric dipole at\nthe surface switches upon changing the sign of the elec-\ntric field. We observe that ∆ ρis sizable at the Fe layer,\nwhich quickly decays to small values in the film and it\ndisplays Friedel oscillations, as expected due to screening\nat a metal surface [40].\nThe largest charge differences, observed in front of the\nsurface (Fig. 2a,b), originate from the extended pzand\ndz2states, which can be understood from the orbitally4Local density of states/eV+0.5 V/Å\n  0.0 V/Å\n-0.5 V/Å\nFe dxz\nEnergy(eV)0.8 0.4 EF 0.4 0.800\nFe dz2Spin downSpin up\nSpin downSpin upc\nd\nFe sf\nSpin downSpin up\n\u0003\u0004  (arb. units)\n0.2\n00.2\n0\nFe Rh Re Re\nRh pz\nEnergy(eV)0.8 0.4 EF 0.4 0.8Spin downSpin upSpin downSpin up\n0.1\n00.2\n0Local density of states/eV\nRe pzhg 0.2\n0\nFe pzSpin downSpin upe\n00.015\n0.10.20.20.4\n0.20.2\n0.030.0150.2\n0.40.2\n0.4Spin downSpin upa\nb\n =   0.5 V/Å ε\nCharged sheet = +0.5 V/Å ε\nCharged sheet\n0.40.6\nFig. 2 Spin-dependent screening of electric fields at the sur face. a,b Electric-field induced spin-dependent charge\ndensity difference ∆ ρσ(z,E)=ρσ(z,E)−ρσ(z,0) along the z-direction and averaged over the film ( xy) plane at E=±0.5 V/˚A.\nSolid blue and red lines denote ∆ ρσ(z,E) for spin-up ( σ=↑) and spin-down ( σ=↓) electrons, respectively. The position of the\ncharged sheet, Fe, Rh and first two Re layers of Fe/Rh/Re(0001 ) film are indicated. c-fSpin-polarized local density of states\n(LDOS) of Fe dz2,dxz,pz,s, Rhpzand Repzorbitals, respectively. Each panel displays LDOS for elect ric fields of +0.5 V/ ˚A\n(red), 0 V/ ˚A (green) and −0.5 V/˚A (blue). The Fermi energy is indicated as E F.\ndecomposed spin-resolved LDOS of the Fe atom (Fig. 2c-\nf). The decrease (increase) of the spin down density at\npositive (negative) electric field in the Fe layer and the\nvacuum region is due to minority pzanddz2states, just\nbelow the Fermi energy (Fig. 2c,e). A similar field ef-\nfect arises for the spin up channel in the vacuum region,\nwhich can be attributed to the majority pzstates. On\nthe other hand, the increase of the spin up charge den-\nsity at the Fe atom for E>0 (Fig.2a) is due to an in-\ncrease of the majority dstates as can be concluded fromthe change of the integrated charge density within the\nmuffin-tin spheres (see Supplementary Table 3).\nIn contrast, the electric-field induced effect is very\nsmall for the Rh and Re atoms, for which only pzor-\nbitals are slightly affected in the LDOS (Fig. 2g-h). Note\nthat the minority pzpeak of Re and Rh just below the\nFermi energy is strongly hybridized with the Fe pzstate,\nas can be concluded from the same energy position and\npeak shape as well as from the band structures (see Sup-\nplementary Figures 2 to 5). Thereby, the pzstate of Re5\nand Rh is affected as the Fe pzstate is filled or emptied\ndue to an electric field.\nThespin-dependentimbalanceofsurfacechargecaused\nby the electric field leads to a small linear change of the\nFemagneticmomentwithelectricfield, whiletheinduced\nmoments of Rh and Re remain almost unchanged (Sup-\nplementary Figure 6). Note that the Fe moment for spin\nspirals at zero and finite electric fields remains fairly con-\nstant upon varying q.\nThe increase of the nearest-neighbor exchange interac-\ntion (cf. Table 1) in the Fe ML can be explained based\non the change of the charge density. Upon increasing the\nelectric field from E=−0.5 V/˚A toE= +0.5 V/˚A, there\nis a decrease of the total spcharge by 2.5 ×10−3electrons\nand an increase of the total dcharge by 6.8 ×10−4elec-\ntrons (see Supplementary Table 3). The Slater-Pauling\ncurve relates the Curie temperature to the number of d\nelectrons in an alloy and here, we find that such an in-\ncrease of delectrons enhances the Curie temperature of\nFe, which canbe relatedto anenhanced nearest-neighbor\nexchange interaction [30, 41].\nThe electric-field induced change of the DMI can be\nunderstood based on the electric dipoles at the inter-\nfaces of the film. The DMI constant can be related to\nthe electric dipole moment at the interfaces of magnetic\nmultilayers [42]. Since the screening charge penetrates\ninto the ultrathin film (Fig. 2a,b), it contributes to the\nelectric dipole moment. Upon changing from a positive\nto a negative electric field, switching of the dipole mo-\nment causes the observed increase and decrease of the\nDMI strength, respectively. The Re layer is quite well\nscreenedfromthe electricfield, therefore,itscontribution\nto the change of the DMI is quite small despite being a\n5dtransition metal with a large SOC constant (Supple-\nmentary Figure 7). A larger effect is expected from Rh\ndue to the non-negligible screening charge density at the\nFe/Rh interface (Fig. 2a,b). However, its contribution to\nthe total DMI is small since Rh is a 4 dtransition metal.\nAs a result, the total changeofDMI constantis relatively\nsmall compared to that of the exchange constant. Nev-\nertheless, it has a profound effect on skyrmion switching\nas discussed below.\nThe hybridization between the pzanddz2states of\nFe is significantly affected by the electric field. It has\nbeen shown previously that the electric-field dependent\np-dhybridization can explain the change of the magne-\ntocrystalline anisotropy energy [43].\nSummarizing the DFT results, we find that the electric\nfield influences the ground state through the variation\nof exchange interactions, DMI, and MAE, which leads\nto the change of the spin spiral period and the depth of\nthe energy minimum.\nAtomistic spin simulations. Next we show by atom-\nistic spin dynamics simulations based on Eq. (1) with\nthe DFT parameters that the field-dependent spin spi-\nral minimum allows to tune the onset of the FM phase,\nwhere magnetic skyrmions are metastable. From the−0.6−0.4−0.200.20.4\n−0.6−0.4−0.200.20.4Energ y relative to the homogeneous SS (meV/Fe atom)\n−0.4−0.200.20.4\n0 1 2 3 4 5\nMagnetic field (T) a\nb\nc\n−0.6SS SkX FM\n−0.5 0.0 0.5ε (V/Å) 2.32.62.9 Bc (T) ε= +0.5 V/Å\nε= 0.0 V/Å\nε= 0.5 V/Å\nFig. 3 Effect of electric fields on the phase diagram.\nZero temperature phase diagram of Fe/Rh/Re(0001) at a\nE= +0.5 V/˚A ,bE= 0 V/˚A , andcE=−0.5 V/˚A. Energies\nof the relaxed spin spiral (SS, black circles), skyrmion lat tice\n(SkX, red circles) and field-polarized ferromagnetic (FM,\ngreen circles) states are shown with reference to the\nhomogeneous spin spiral (dashed line). The SS, SkX and\nFM phases are denoted with blue, red and green background\ncolors, respectively. Inset of cshows the variation of the\ncritical field Bc(onset of FM phase) with the electric field.\nNote that the critical field is calculated where the energy of\nthe skyrmion lattice (SkX) phase becomes equal to the FM\nphase.\nzero-temperaturephasediagram(Fig. 3a-c),wefindthat,\nindependent of the electric field, a spin spiral ground\nstate occurs at zero and small magnetic fields, consistent\nwith the spin spiral minima in Fig. 1b. AtB>0.7 T,\nthe skyrmion lattice phase becomes energetically favor-\nable. With further increase of the magnetic field, the\nfield-polarized or ferromagnetic (FM) phase becomes the\nlowestenergystate. NotethattheonsetoftheFM phase,\ni.e., the critical field Bc, changes with electric field (inset6\n123456\n050100150200\n2.2 2. 7 3.2 3. \u0005 4.2 4. \u00060.5 0.0\u00070.5\nε (V/Å) 2.52.83.1 Bc (T) -0.5\n-11\n0\n4 0 4mz\ndistance from center\n(nm)-0.5\n8 8a\nb\nBcBcBc+0.5 V/Å\n 0.0 V/Å\n 0.5 V/Å\nMagnetic field (T)Energy barrier (meV)Skyrmion radius (nm)\nFig. 4 Effect of electric fields on radius and energy\nbarriers of isolated skyrmions. a Radius and benergy\nbarriers of isolated skyrmions in Fe/Rh/Re(0001) with\napplied magnetic fields at E= +0.5 V/˚A (red), E= 0 V/˚A\n(green) and E=−0.5 V/˚A (blue). In b, filled circles show\ncollapse barriers, ∆ Ecol, while filled diamonds mark creation\nbarriers, ∆ Ecrea. Inset of ashows the skyrmion profiles at\nB= 3 T for zero and two finite electric fields and bshows\nthe variation of the critical field Bc(onset of FM phase)\nwith the electric field. Note that the critical field is\ncalculated where the energy of isolated skyrmions is equal t o\nthe FM phase.\nof Fig.3c). The change of Bc(0.56 T) with the electric\nfield (1 V/ ˚A) is closeto the estimated value from the spin\nspiral energy minimum in Fig. 1b (≈0.75 T).\nNow we consider individual magnetic skyrmions in the\nfield-polarized phase. We find that the skyrmion radius\nincreases significantly for E<0, while it decreases for\nE>0 (Fig.4a). At first sight, it seems surprising that\nthe order of skyrmion radius with the electric field is\nreversed with respect to the change of the spin spiral\nperiodattheenergyminimumfoundfromDFT(Fig. 1b).\nHowever, this is due to the fact that the critical magnetic\nfield, at which the transition from the skyrmion to thefield-polarized or ferromagnetic (FM) phase occurs, also\nchanges with the electric field. Since, in an experiment,\nswitching of a skyrmion is performed at a fixed magnetic\nfield [16, 44], the variation of the skyrmion radius (see\ninset of Fig. 4a) strongly affects the energy barriers for\nskyrmion creation or annihilation as shown below [45].\nWe used the geodesic nudged elastic band (GNEB)\nmethod [46] to calculate the minimum energy path\n(MEP) between an isolated skyrmion and the FM\nbackground (Fig. 5a). We observe that the skyrmions\nannihilate by the well-known radial collapse mechanism\n[47]. From the MEP, we find the energy barrier ∆ Ecol\nprotecting the skyrmion from collapsing into the FM\nstate, i.e., the difference between the saddle point (SP),\nmaximum energy point on the path, and the initial\n(skyrmion) state (Fig. 5a). The creation barrier, ∆ Ecrea,\nis obtained from the difference between the SP and the\nfinal (FM) state.\nIt is apparent that ∆ Ecoldecreases as a function of\napplied magnetic field similar to the decrease of the\nskyrmion radius (Fig. 4b). This is due to the fact that\nthe energy terms which contribute to the barrier, in par-\nticular the DMI, scale with the number of spins in the\nskyrmion. On the other hand, ∆ Ecreadisplays only\na small nearly linear rise with applied magnetic field\n(Fig.4b). This can be understood from the fact that it\ndepends on the energy difference of the FM state and the\nsaddle point. We can relate ∆ Ecreato the energy differ-\nence between the skyrmion and the FM state, ∆ Esk−FM,\nand the collapse barrier by ∆ Ecrea= ∆Esk−FM+∆Ecol\n(Fig.5a). As discussed above, ∆ Ecoldecreases with the\nmagnetic field. However, the FM state becomes more\nfavorable with increasing magnetic field and ∆ Esk−FM\nincreases(Fig. 5a). The gradualrise of ∆ Ecreawith mag-\nnetic field is due to these two opposing contributions.\nThe magnetic field at which both barriers are equal\n(green curve in Fig. 5a) marks the transition from the\nisolated skyrmion to the FM phase. As seen in the in-\nset of Fig. 4b, this critical magnetic field shifts with the\nelectric field. At E= 0 V/˚A,Bcis about 2.9 T. If one\nchanges the electric field to +0 .5 V/˚A, at a fixed mag-\nnetic field, the creation barrier is enhanced, while the\ncollapse barrier decreases since Bcshifts to a lower value\n(Fig.3a). Therefore, the FM state becomes more favor-\nable (red curve in Fig. 5a). ForE<0, the opposite effect\noccurssince one movesinto the skyrmion phaseat a fixed\nmagnetic field of 2.9 T. The skyrmion state is therefore\nlower (blue curve in Fig. 5a) than the FM state and the\ncollapsebarrierrises. Thisexplains whycollapseand cre-\nation barriers show opposite trends upon variation of the\nelectric field. Note that the critical fields in Fig. 4b is\nslightly higher than the fields in Fig. 3c. The reason is\nthat, in the first case, the critical fields are obtained from\nthe energy equality of the isolated skyrmion to the FM\nphase, and in the second case, from the energy equality\nof the skyrmion lattice phase to the FM phase.\nNowwequantifythe energybarrierat B=3T interms7\naS \b \t \n \u000b \f \r \u000e \u000f \u0010 \u0011\nSkyrmion\nε< \u0012\nε= \u0013\nε> \u0014\nF \u0015\nC\u0016 \u0017 \u0018 \u0019 \u001a \u001b \u001c\n barrier\n\u001d\u001e \u001f  ! \" # $\n barrier% 6 & '\n(400\n)2000200400\n* + ,-200\n.1000100200300400\nTotal\n/01234589:\nA;?@BDEGHI\nJKLMNOb\ncPQ R T U V W X\nYZ [ \\ ] ^ _ `aEcrea (meV)\nbEcol (meV)\n+0.5 V/Å\n  0.0 V/Å\n 0.5 V/Å\nFig. 5 Minimum energy path, collapse and creation\nbarriers at zero and finite electric fields. a Minimum\nenergy paths (MEP) from the initial (skyrmion) to the final\n(FM) state in Fe/Rh/Re(0001) at B= 3 T for zero and two\nfinite electric fields ( E=±0.5 V/˚A). The energy difference\nbetween the saddle point and the skyrmion (FM) state\ndefines the collapse (creation) barrier. Total and individu al\nenergy contributions at B= 3 T for bthe collapse barrier\nandcfor the creation barrier. For energy decomposition of\nthe full MEP, see Supplementary Fig. 8. Combined\nexchange (C. exchange) indicates the sum of pair-wise\nexchange, biquadratic and three-site four spin interactio ns.\nThe four-site four spin interaction is denoted as 4-Spin for\nbrevity. The electric fields E= +0.5 V/˚A 0 V/˚A and−0.5\nV/˚A are shown as red, green and blue, respectively.\nof the magnetic interactions (Fig. 5b,c). At this value,\nwe find ∆ Ecol≈125 meV for zero electric field. ∆ Ecolincreases by ≈35 meV at E=−0.5 V/˚A (Fig.5b). Inter-\nestingly,thetotalenergybarrierarisesduetotheincrease\nof DMI ( ≈110 meV) and the MAE ( ≈90 meV), while\nboth the Zeeman term ( ≈ −140 meV) and the four-site\nfour spin interaction ( ≈ −20 meV) lead to opposite con-\ntributions and thus reduce the electric field effect [48].\nThe combined exchange interaction, i.e., the sum of the\ncontributions from pair-wise exchange, biquadratic and\nthree-site four spin interaction, does not contribute to\nthe change of energy barriers. For E= +0.5 V/˚A, we\nfind a decrease of ∆ Ecolby≈35 meV and the magnetic\ninteractions contribute in an analogous way.\nThe variation of ∆ Ecolby about ±30% for E=±0.5\nV/˚A (Fig.5b) cannot be directly understood from the\nelectric-field induced changes of the magnetic interac-\ntions. Since the magnetic moment does not change much\nwith the electric field, the large variation of the Zee-\nman energy must be directly linked to the change of the\nskyrmion radius with the electric field (Fig. 4a). For\nE=±0.5 V/˚A, the skyrmion radius varies by 1.3 nm\natB= 3 T. We can estimate the relative change of the\nnumber of moments in the skyrmion from ∆ Nsk/Nsk=\n(R2\nsk(E)−R2\nsk(0))/R2\nsk(0)≈0.3, which closely matches\nthe variation of the Zeeman contribution to the energy\nbarrier ∆ EZeeman/EZeeman≈0.3. The electric-field de-\npendent skyrmion radius also explains the large relative\nchange of the DMI term, ∆ EDMI/EDMI≈0.25, and the\nanisotropyterm, ∆ EMAE/EMAE≈0.45, which areabout\nfour times larger than the change of DeffandK, respec-\ntively (Table I). For the four-spin interaction, K1, this ef-\nfect is much reduced since the barriercontribution of this\nterm depends on the saddle point structure [27] which is\nless affected by the electric field.\nThe creation barrier (Fig. 5c) varies by about 15%\nforE=±0.5 V/˚A, but displays an opposite trend with\nrespect to ∆ Ecol, i.e., the barrier decreases for E<0 and\nrises for E>0. As stated above, this can be explained\nbased on the MEP (Fig. 5a). The energy decomposition\n(Fig.5c) shows that the DMI plays a minor role, while\nMAE and four-spin exchange determine the electric field\ndependence [49]. The Zeeman term shows an opposite\ntrend and again reduces the electric field effect. The\nscaling of anisotropy and Zeeman term with electric field\nis also much larger than expected from the electric-field\ninduced changes of the magnetic interactions and is due\nto the change of skyrmion radius as explained above for\n∆Ecol.\nDiscussion\nBased on our results, we can discuss the electric-field\nassisted switching of skyrmions in a scanning tunneling\nmicroscopy (STM) experiment [16]. The electric cur-\nrent injected from the STM tip results in magnon ex-\ncitations which eventually trigger the skyrmion collapse.\nIn Ref. [47], an effective temperature of the magnon bath\ndue to single hot electron events was estimated and it is\naround 50 K. The skyrmion collapse and creation rates\ncalculated based on the Arrhenius law compared well8\nwith experimental values [47]. Therefore, we use it to\nestimate the effect of the electric field on the switching.\nThe Arrhenius law is given by τ=τ0exp(∆Ecol/kBT).\nSince the lifetime τis dominated by the energy bar-\nrier ∆Ecolat low temperatures, we neglect the varia-\ntion of the prefactor τ0. Then, the ratio of lifetimes\natB= 3 T is given by τ(E= +0.5 V/˚A)/τ(E= 0)=\nexp(−35 meV/kBT). This leads to a change of the\nskyrmion lifetime by a factor of about 2 ×10−4at a tem-\nperature of T= 50 K. Therefore, a significant effect in\ndeleting individual skyrmions can be obtained by a local\nelectric field in the tunnel junction. A similar estimate\nfor the creation leads to a factor of 8 ×10−3which shows\nthat writing of skyrmions can also be greatly manipu-\nlated by an electric field.\nWe have demonstrated that the stability of isolated\nskyrmions in an ultrathin film can be changed sig-\nnificantly by external electric field. The electric-field\ninduced changes of the magnetic interactions lead to\na shift of the critical magnetic field for the onset of\nthe field-polarized phase, which exhibits metastable\nskyrmions. As a consequence, changes in the creation\nand collapse barriers are much larger than expected\nfrom the variations of the interactions which lead to\nthe possibility of writing and deleting skyrmions. Our\nstudy shows that it is indispensable to consider all\nmagnetic interactions to evaluate the electric-field effect\nfor skyrmion stability.\nMethods\nDensity functional theory calculations. All DFT\ncalculations with applied electric fields were performed\nusing the fleurcode [28]. We relaxed the top three lay-\ners of Fe/Rh/Re(0001), i.e., the Fe, the Rh and the first\nRe layer, along the z-direction by minimizing the atomic\nforce on each atom down to values of less than 0.04eV/ ˚A\nin the presence of E=±0.5 V/˚A. We chose the general-\nizedgradientapproximation(GGA)exchange-correlation\nfunctional as parametrized by Perdew, Burke and Ernz-\nerhof [50], 66 kpoints in two-dimensional Brillouin zone\n(2DBZ) and kmax= 4.0 a.u.−1for relaxation. We found\nthat there were almost no changes (below 0.01 a.u.) of\nthe top three inter-layer distances compared to the zero\nfield values given in Ref. [51]. Therefore, we take the zero\nelectric field interlayer distances of Ref. [51] for calcula-\ntions including electric fields. Similar to our result, no\nelectric-field induced relaxationwas observedin a Co ML\non the Pt(111) surface [30].\nTo check the existence of a noncollinear ground\nstate and to extract the Heisenberg pair-wise ex-\nchange parameters, we calculated the energy dispersion\nof homogeneous flat spin spirals of the form Mi=\nM(cosqRi,sinqRi,0), where Miis the magnetic mo-\nment at lattice site Riandqis the wave vector in 2DBZ\n[52]. We used the generalized Bloch theorem [53] to com-\npute spin spiralenergieswithin the chemicalunit cell. To\nstudy the surface, we chose an asymmetric film of two\natomic overlayers on nine Re substrate layers. Since weused an asymmetric film to compute the magnetic inter-\nactions and our interest is on the electric-field induced\nchanges in magnetism, we only apply electric fields per-\npendicular to the Fe surface. To be consistent with the\nzeroelectric field spinspiralcalculations[51], weused the\nlocal density approximation (LDA) exchange-correlation\nfunctional form given by Vosko, Wilk and Nusair [54],\na dense mesh of 44 ×44k-points in the full 2DBZ and\nkmax= 4.0 a.u.−1.\nThe DMI was computed within the first-order per-\nturbation theory [55] on the self-consistent spin spiral\nstate. The MAE was calculated self-consistently as\ndescribed in Ref. [56] starting from a self-consistent\nscalar-relativistic density. To obtain an accurate value\nof the MAE, we varied the substrate layers from 13 to\n17 of the asymmetric film.\nHigher-order exchange interactions from DFT.\nThe fourth-order perturbative expansion in the hopping\nparameterovertheCoulombinteractionparameterofthe\nHubbard model [57] leads to a two-site four spin (bi-\nquadratic, B1), a three-site four spin ( Y1) [58, 59] and a\nfour-site four spin ( K1) [35] term (see Eq. (1)). These\nterms arise due to hopping of the electron among two\n(two-site four spin or biquadratic), three (three-site four\nspin)andfour(four-sitefourspin)latticesites. Weevalu-\natethesethreehigher-orderinteractionconstantsdirectly\nfrom DFT within nearest-neighbor approximation.\nTo compute the constants, we consider three multi- Q\nstates (Fig. 6c-e). The multi- Qstate is constructed from\nthe superposition of spin spirals (single- Q) related to\nsymmetry equivalent qvectors of the 2DBZ (Fig. 6a,b).\nThe single- Qand multi- Qstates are energetically degen-\nerate within the Heisenberg pair-wise spin model. The\ndegeneracy is lifted when the mentioned three higher-\norder interactions are taken into account which provides\na way to compute their strength.\nWe consider two collinear uuddor 2Qstates [38] and a\nthree-dimensionalnoncollinear3 Qstate[31]todetermine\nthe threehigher-orderexchangeconstants(forspin struc-\ntures see Fig. 6c-e). The three higher-order exchange\ninteraction constants without SOC are determined from\nthe energy differences of the single- Qand multi- Qstates\nusing the following equations:\nB1=3\n32∆E3Q\nM−1\n8∆Euudd\nM/2(2)\nY1=1\n8(∆Euudd\n3K/4−∆Euudd\nM/2) (3)\nK1=3\n64∆E3Q\nM+1\n16∆Euudd\n3K/4(4)\nwhere the total energy differences between multi- Qand9\nc dM\nca b\ne+\n=\ndef g h\ni\nb1b2\nuuddM\nFig. 6 Formation and spin structures of the multi- Qstates. a Superposition of two 90◦spin spirals (1 Qstates) with\nclockwise (CW) and counterclockwise (CCW) rotational sens e results in an uuddstate (2Qstate).b2D Brillouin zone of\nhexagonal lattice and reciprocal vectors b1andb2. Two high-symmetry directions ΓK and ΓM are shown. The qvectors\ncorresponding to the uuddstate along ΓK (green circle) and along ΓM (blue circle) as well as the 3 Qstate at the M point\n(red circle) are shown. cSpin structure of the uuddstate along ΓK, which is formed by a superposition of two 90◦spin spirals\natq=±(3/4)ΓK,dspin structure of the uuddstate along ΓM, which is formed by a superposition of two 90◦spin spirals at\nq=±(1/2)ΓM and espin structure of the 3 Qstate, which is formed by a superpostion of three spin spiral s at the M point.\nNote that the spin structure of the uuddstates is collinear, whereas it is non-collinear for the 3 Qstate.\ncorresponding single- Q(spin spiral) states given by\n∆E3Q\nM=E3Q\nM−ESS\nM(5)\n∆Euudd\nM/2=Euudd\nM/2−ESS\nM/2(6)\n∆Euudd\n3K/4=Euudd\n3K/4−ESS\n3K/4(7)\nsee Fig.2and Supplementary Table 2 for energy differ-\nences.\nUpon taking the higher-order exchange constants into\naccount, we need to modify the first three Heisen-\nberg pair-wise exchange interaction parameters obtained\nfrom fitting of the homogeneous spin spirals neglecting\nhigher-order interactions as follows (for a derivation see\nRef. [27]):\nJ′\n1=J1−Y1 (8a)\nJ′\n2=J2−Y1 (8b)\nJ′\n3=J3−B1/2 (8c)\nNote that the four-site four spin interaction does not\nadjust any exchange interaction parameters, since its\ncontribution to the homogeneous spin spirals is a con-\nstant term of −12K1.Due to the large 2D unit cell, the energy of the\nmulti-Qstates were evaluated from asymmetric films\nconsisting of 8 layers in total. However, we have checked\nfor zero electric field that the energy difference between\nthe 8 and 11 layer film calculations is less than 1 meV.\nAtomistic spin dynamics simulations. To relax and\ncalculate the energy of the spin spirals, skyrmion lattice,\nFM phases as well as the isolated skyrmions, we per-\nformed spin dynamics simulations based on the Landau-\nLifshitz equation:\nℏdmi\ndt=∂H\n∂mi×mi−α/parenleftbigg∂H\n∂mi×mi/parenrightbigg\n×mi(9)\nwhereℏis the reduced Planck constant, αis the\ndamping parameter and the Hamiltonian His defined\nin Eq. (1). We used a time step of 0.1 fs, αis varied\nfrom 0.05 to 0.1 and the simulations were carried out\nover 4 to 6 million steps for relaxation. We solve Eq. ( 9)\nby semi-implicit method as proposed by Mentink et al.\n[60].\nGeodesic nudged elastic band method. We first\ncreate isolated skyrmions in the field-polarized back-10\nground, i.e., at a magnetic field above Bcfrom the\ntheoretical profile [1] and then relax the spin structure\nusing spin dynamics with the full set of DFT parameters\n(Table I). We computed the collapse and creation barri-\ners of isolated skyrmions using the GNEB method [61].\nThe method finds the minimum energy path (MEP)\nconnecting the initial state (IS), i.e., skyrmions, and final\nstate (FS), i.e., FM state, on a multidimensional energy\nsurface. Within GNEB, an initial path connecting the\nIS and FS is created by a chain of images of the system.\nThe objective of the method is to bring the initial path\nto MEP via relaxing the intermediate images. The\nrelaxation is achieved by a force projection scheme. For\nthis, the effective field is calculated at each images and\nits local tangent to the path is replaced by a spring forcewhich maintains a uniform distribution of images. The\nmaximum energy of MEP corresponds to the saddle\npoint (SP) which defines the energy barrier separating\ntwo stable states. We compute the energy of the SP\naccurately using a climbing image (CI) method on top\nof GNEB.\nData availability\nThe authors declare that the data supporting the\nfindings of this study are available within the article and\nits Supplementary Information files.\nCode availability\nThe atomistic spin dynamics code used in this work is\navailable from the authors upon a reasonable request.\n[1]Bogdanov, A. N. & Hubert, A. The properties of iso-\nlated magnetic vortices. phys. stat. sol. (b) 186, 527–543\n(1994).\n[2]Nagaosa, N. & Tokura, Y. Topological properties and dy-\nnamics of magnetic skyrmions FeRe. Nat. Nanotechnol.\n8, 899–911 (2013).\n[3]Tomasello, R. et al.A strategy for the design of skyrmion\nracetrack memories. Sci. Rep. 4, 6784 (2014).\n[4]Zhou, Y. & Ezawa, M. A reversible conversion between a\nskyrmion and a domain-wall pair in a junction geometry.\nNat. Commun. 5, 4652 (2014).\n[5]Iwasaki, J., Mochizuki, M. & Nagaosa, N. Universal\ncurrent-velocity relation of skyrmion motion in chiral\nmagnets. Nat. Commun. 4, 1463 (2013).\n[6]Sampaio, J., Cros, V., Rohart, S., Thiaville, A.&Fert, A.\nNucleation, stability and current-induced motion of iso-\nlated magnetic skyrmions in nanostructures. Nat. Nan-\notechnol. 8, 839–844 (2013).\n[7]Fert, A., Cros, V. & Sampaio, J. Skyrmions on the track.\nNat. Nanotechnol. 8, 152–156 (2013).\n[8]Chernyshov, A. et al.Evidence for reversible control of\nmagnetization in a ferromagnetic material by means of\nspin-orbit magnetic field. Nat. Phys. 5, 656 – 659 (2009).\n[9]Miron, I. M. et al.Current-driven spin torque induced\nby the rashba effect in a ferromagnetic metal layer. Nat.\nMater.9, 230 – 234 (2010).\n[10]Miron, I. M. et al.Perpendicular switching of a single\nferromagnetic layerinducedbyin-planecurrentinjection .\nNature476, 189 – 193 (2011).\n[11]Liu, L.et al.Spin-torque switching with the giant spin\nhall effect of tantalum. Science336, 555 – 558 (2012).\n[12]Woo, S. et al.Spin-orbit torque-driven skyrmion dy-\nnamics revealed by time-resolved X-ray microscopy. Nat.\nCommun. 8, 15573 (2017).\n[13]Montoya, S. A. et al.Spin-orbit torque induced dipole\nskyrmion motion at room temperature. Phys. Rev. B 98,\n104432 (2018).\n[14]MacKinnon, C. R., Lepadatu, S., Mercer, T. & Bissell,\nP.R. Roleofanadditional interfacial spin-transfertorqu e\nfor current-driven skyrmion dynamics in chiral magnetic\nlayers.Phys. Rev. B 102, 214408 (2020).\n[15]Matsukura, F., Tokura, Y. & Ohno, H. Control of mag-\nnetism by electric fields. Nat. Nanotechnol. 8, 209–220(2015).\n[16]Hsu, P.-J. et al.Electric-field-driven switching of individ-\nual magnetic skyrmions. Nat. Nanotechnol. 12, 123–126\n(2017).\n[17]Schott, M. et al.The skyrmion switch: Turningmagnetic\nskyrmion bubbles on and off with an electric field. Nano\nLett.17, 3006–3012 (2017).\n[18]Ma, C.et al.Electric field-induced creation and direc-\ntional motion of domain walls and skyrmion bubbles.\nNano Lett. 19, 353–361 (2019).\n[19]Srivastava, T. et al. Large-voltage tuning of\ndzyaloshinskii-moriya interactions: A route toward dy-\nnamic control of skyrmion chirality. Nano Lett. 18, 4871–\n4877 (2018).\n[20]Wang, Y. et al.Electric-field-driven non-volatile multi-\nstate switching of individual skyrmions in a multiferroic\nheterostructure. Nat. Commun. 11, 3577 (2020).\n[21]Ba, Y.et al.Electric-field control of skyrmions in multi-\nferroic heterostructureviamagnetoelectric coupling. Nat.\nCommun. 12, 322 (2021).\n[22]Upadhyaya, P., Yu, G., Amiri, P. K. & Wang, K. L.\nElectric-field guiding of magnetic skyrmions. Phys. Rev.\nB92, 134411 (2015).\n[23]Fook, H.T., Gan, W.L.&Lew, W.S. Gateable skyrmion\ntransport via field-induced potential barrier modulation.\nSci. Rep. 6, 21099 (2016).\n[24]Nakatani, Y., Hayashi, M., Kanai, S., Fukami, S. &\nOhno, H. Electric field control of skyrmions in magnetic\nnanodisks. Appl. Phys. Lett. 108, 1524403 (2016).\n[25]Yang, H., Boulle, O., Cros, V., Fert, A. & Chshiev, M.\nControlling Dzyaloshinskii-Moriya interaction via chira l-\nity dependent atomic-layer stacking, insulator capping\nand electric field. Sci Rep 8, 12356 (2018).\n[26]Desplat, L. et al.Mechanism for ultrafast electric-field\ndriven skyrmion nucleation. Phys. Rev. B 104, L060409\n(2021).\n[27]Paul, S., Haldar, S., von Malottki, S. & Heinze, S. Role of\nhigher-order exchange interactions for skyrmion stabilit y.\nNat. Commun. 11, 4756 (2020).\n[28]https://www.flapw.de .\n[29]Weinert, M., Schneider, G., Podloucky, R. & Redinger,\nJ. FLAPW: applications and implementations. J. Phys.:\nCondens. Matter 21, 084201 (2009).11\n[30]Oba, M. et al.Electric-field-induced modification of the\nmagnon energy, exchange interaction, and Curie temper-\nature of transition-metal thinfilms. Phys. Rev. Lett. 114,\n107202 (2015).\n[31]Kurz, P., Bihlmayer, G., Hirai, K. & Bl¨ ugel, S. Three-\ndimensional spin structure on a two-dimensional lattice:\nMn/Cu(111). Phys. Rev. Lett. 86, 1106–1109 (2001).\n[32]Romming, N. et al.Competition ofdzyaloshinskii-moriya\nand higher-order exchange interactions in Rh/Fe atomic\nbilayers on Ir(111). Phys. Rev. Lett. 120, 207201 (2018).\n[33]Kr¨ onlein, A. et al.Magnetic ground state stabilized by\nthree-site interactions: Fe /Rh(111). Phys. Rev. Lett.\n120, 207202 (2018).\n[34]Spethmann, J. et al.Discovery of magnetic single- and\ntriple-qstates in Mn /Re(0001). Phys. Rev. Lett. 124,\n227203 (2020).\n[35]Hoffmann, M. & Bl¨ ugel, S. Systematic derivation of re-\nalistic spin models for beyond-heisenberg solids. Phys.\nRev. B101, 024418 (2020).\n[36]Li, W., Paul, S., von Bergmann, K., Heinze, S. &\nWiesendanger, R. Stacking-dependent spin interactions\nin Pd/Fe bilayers on Re(0001). Phys. Rev. Lett. 125,\n227205 (2020).\n[37]S. Heinze et al. Spontaneous atomic-scale magnetic\nskyrmion lattice in two dimensions. Nat. Phys. 7, 713\n(2011).\n[38]Hardrat, B. et al.Complex magnetism of iron mono-\nlayers on hexagonal transition metal surfaces from first\nprinciples. Phys. Rev. B 79, 094411 (2009).\n[39]von Malottki, S., Dup´ e, B., F. Bessarab, P., Delin, A. &\nHeinze, S. Enhanced skyrmion stability due to exchange\nfrustration. Sci. Rep. 7, 12299 (2017).\n[40]Mitsui, T. et al.Magnetic friedel oscillation at the\nFe(001) surface: Direct observation by atomic-layer-\nresolved synchrotron radiation57Fe M¨ ossbauer spec-\ntroscopy. Phys. Rev. Lett. 125, 236806 (2020).\n[41]Takahashi, C., Ogura, M. & Akai, H. First-principles cal-\nculation of the Curie temperature Slater-Pauling curve.\nJ. Phys.: Condens. Matter 19, 365233 (2007).\n[42]Jia, H., Zimmermann, B., Michalicek, G., Bihlmayer, G.\n& Bl¨ ugel, S. Electric dipole moment as descriptor for\ninterfacial Dzyaloshinskii-Moriya interaction. Phys. Rev.\nMaterials 4, 024405 (2020).\n[43]Nakamura, K. et al.Giant modification of the magne-\ntocrystalline anisotropy in transition-metal monolayers\nbyan external electric field. Phys. Rev. Lett. 102, 187201\n(2009).\n[44]N. Romming et al.Writing and Deleting Single Magnetic\nSkyrmions. Science341, 636 (2013).\n[45]Note that the spin spiral periods obtained at zero mag-\nnetic field in the atomistic spin dynamics simulations are\nconsistent with those from the DFT calculations.\n[46]Bessarab, P. F., Uzdin, V. M. & J´ onsson, H. Method\nfor finding mechanism and activation energy of magnetic\ntransitions, applied to skyrmion and antivortex annihila-\ntion.Comput. Phys. Commun. 196, 335 – 347 (2015).\n[47]Muckel, F. et al.Experimental identification of two dis-\ntinct skyrmion collapse mechanisms. Nat. Phys. (2021).\n[48]Note that for K1<0, the four-site four spin interaction\ndestabilizes skyrmions [27].\n[49]Note that the four-spin term exhibits the same field de-\npendence for creation and collapse barriers since it has a\nsimilar contribution to the fm state as to the skyrmion\nstate.[50]Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized\ngradient approximation made simple. Phys. Rev. Lett.\n77, 3865–3868 (1996).\n[51]Paul, S. & Heinze, S. Tailoring magnetic interactions in\natomic bilayers of Rh and Fe on Re(0001). Phys. Rev. B\n101, 104408 (2020).\n[52]Kurz, P., F¨ orster, F., Nordstr¨ om, L., Bihlmayer, G. &\nBl¨ ugel, S. Ab initio treatment of noncollinear magnets\nwith the full-potential linearized augmented plane wave\nmethod. Phys. Rev. B 69, 024415 (2004).\n[53]Sandratskii, L. M. Energy Band Structure Calculations\nfor Crystals with Spiral Magnetic Structure. Phys. Status\nSolidi B 136, 167 (1986).\n[54]Vosko, S. H., Wilk, L. & Nusair, M. Accurate spin-\ndependent electron liquid correlation energies for local\nspin density calculations: a critical analysis. Can. J.\nPhys.58, 1200–1211 (1980).\n[55]Heide, M., , Bihlmayer, G. & Bl¨ ugel, S. Describing\nDzyaloshinskii-Moriya spirals from first principles. Phys.\nB: Condens. Matter 404, 2678 (2009).\n[56]Li, C., Freeman, A. J., Jansen, H. J. F. & Fu, C. L. Mag-\nnetic anisotropy in low-dimensional ferromagnetic sys-\ntems: Fe monolayers on Ag(001), Au(001), and Pd(001)\nsubstrates. Phys. Rev. B 42, 5433–5442 (1990).\n[57]Hubbard, J. Electron correlations in narrow energy\nbands.Proc. R Soc. London. Ser. A Mathematical and\nPhysical Sciences 276, 238–257 (1963).\n[58]MacDonald, A. H., Girvin, S. H. & Yoshioka, D.t\nUex-\npansion for the Hubbard model. Phys. Rev. B 37, 9753–\n9756 (1988).\n[59]Takahashi, M. Half-filled Hubbard model at low tem-\nperature. J. Phys. C Solid State Phys. 10, 1289–7301\n(1977).\n[60]H. Mentink, J., V. Tretyakov, M., Fasolino, A., I. Kat-\nsnelson, M. & Rasing, T. Stable and fast semi-implicit\nintegration of the stochastic Landau–Lifshitz equation.\nJ. Phys.: Condens. Matter 22, 176001 (2010).\n[61]F. Bessarab, P., M. Uzdin, V. & J´ onsson, H. Method\nfor finding mechanism and activation energy of magnetic\ntransitions, applied to skyrmion and antivortex annihila-\ntion.Comput. Phys. Commun. 196, 335–347 (2015).\nAcknowledgements\nWe gratefully acknowledge computing time at the\nsupercomputer of the North-German Supercomput-\ning Alliance (HLRN) and financial support from\nthe Deutsche Forschungsgemeinschaft (DFG, Ger-\nman Research Foundation) via project no. 414321830\n(HE3292/11-1). S.P. graciously acknowledge financial\nsupport from the European Research Council (ERC)\nunder the European Union’s Horizon 2020 research and\ninnovation program (Grant No. 856538, project “3D\nMAGiC”. S.P. and S.H. thank G. Bihlmayer and M.\nHoffmann for useful discussion. S.H. thanks M. Goerzen\nfor insightful discussions.\nAuthor contributions\nS.H. devised the project. S.P. performed the calcula-\ntions. S.P. and S.H. analyzed the results and wrote the\npaper.\nCompeting interests12\nThe authors declare no competing financial interests. Additional information\nSupplementary information is available for this paper\nat"    },    {        "title": "2104.13381v2.Observation_of_a_phase_transition_within_the_domain_walls_of_ferromagnetic_Co3Sn2S2.pdf",        "content": "1 \n Observation of a phase  transition  \nwithin the domain walls of ferromagnet ic Co 3Sn2S2 \n \nChangmin Lee1, Praveen Vir2, Kaustuv Manna2,3, Chandra Shekhar2, J. E. Moore1,4, M. A. \nKastner5,6,7, Claudia Felser2, and Joseph Orenstein1,4* \n \n1 Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, California , USA  \n2 Max Planck Institute for Chemical Physics of Solids, Dresden, Germany  \n3 Department of Physics, Indian Institute of  Technology Delhi, New Delhi, India  \n4 Department of Physics, University of California at Berkeley, Berkeley, California , USA  \n5 Department of Physics, Stanford University, Palo Alto, California , USA  \n6 Stanford Institute for Materials and Energy Science, SLAC National Accelerator Laboratory, Menlo Park, \nCalifornia , USA  \n7 Department of Physics, Massachusetts Institute of Technology, Cambridge, MA , USA  \n* Corresponding author: jworenstein@lbl.gov  \nAbstract  \n \nThe ferromagnet ic phase of  Co3Sn2S2 is widely considered to be a topological Weyl semimetal , \nwith evidence  for momentum -space monopoles of Berry curvature from transport  and \nspectroscopic probes.  As the bandstructure is highly sensitive to the magnetic order , \nattention has focused on anomalies in magn etization, susceptibility and transport \nmeasurements that are seen  well below the Curie temperature , leading to specula tion that a \n“hidden” phase coexists with ferromagnetism. Here we report spatially -resolved \nmeasurements  by Kerr effect microscopy  that identify this phase.  We find that the anomal ies \ncoincide with a  deep minimum in domain wall  (DW)  mobility, indicating a crossover between \ntwo regimes of DW propagation. We demonstrate that this crossover is a manifestation of a \n2D phase  transition that o ccurs with in the DW , in which  the magnetization texture changes \nfrom continuous rotation to unidirectional variation . We propose that the existence of this \n2D transition deep within the ferromagnetic state of the bulk  is a consequence of  a giant  \nquality factor for  magnetocrystalline anisotropy  unique to this compound.  This work \nbroadens the horizon of the conventional binary classification of DWs into Bloch and Néel 2 \n walls, and  suggests  new strategies for manipulation of domain walls and their rol e in electron \nand spin transport.  \n \nIntroduction  \nThe crystal structure of Co 3Sn2S2 comprises quasi -2D Co 3Sn layers separated by sulfur atoms, \nwith the magnetic Co atoms arranged on a kagom é lattice1.  The ferromagnetic state is \ncharacterized by strong easy -axis anisotropy favoring magnetization in the c direction, that is, \nperpendicular to the layers. Recent research  has focused on evidence for Weyl semimetal \ntopology2-4 in Co 3Sn2S2 based o n a giant anomalous Hall effect5-9, and observation of Fermi arcs \nby photoemission10 and scanning tunneling spectros copies11.  However, a s the bandstructure  is \nexpected to be highly sensitive to the magnetization texture, attention has been drawn to \nunexpected structure s in the temperature dependence near 130 K (well below the Curie \ntemperature of 175 K) that is seen by static  and ac susceptibility12, muon spin rotation13, Hall \neffect14, magneto -optic spectroscopy8, and magnetic force microscopy15 measurements.   \nAlthough these structures are relatively small in some cases,  their ubiquity across laboratories \nand probes has led to proposals  that a phase  transition to a coexisting antiferromagnet13 or spin \nglass14 takes place deep within the dominant ferromagnetic phase.  Here the scanning Kerr \nmicroscopy measurements described below show that such a phase transition does indeed occur, \nbut within  the 2D manifold of the ma gnetic domain walls (DWs) rather than the bulk of the 3D \ncrystal.    \n \nResults  \nA schematic of the experimental setup is shown in Fig . 1(a), which illustrates local probing of \nmagnetization dynamics  using the polar Kerr effect  (see Methods for details) . Scanning the \nsample under the laser focus yields a map (Fig. 1( b)) of the Kerr ellipticity , Φ(𝒓), revealing \ndomains of magnetization directed parallel and anti -parallel to the surface normal, or c-axis \ndirection . The change  in Φ(𝒓) across a  DW is proport ional to the  zero-field magnetization \n(shown as  a function of temperature in Fig. S1 of Supplementary Information ). For local \nmeasurements of the DW dynamics, a  coil surrounding the sample generates an oscillating \nmagnetic field parallel to the easy axis, 𝑯(𝑡)=𝒄̂𝐻𝑎𝑐𝑠𝑖𝑛𝜔𝑡 . This ac magnetic field tilts the \nenergy landscape, inducing DW displacement in the direction that increases the volume of 3 \n magnetization aligned parallel to 𝑯(𝑡). The oscillating displacement in turn generates a  \nsynchronous modula tion in ellipticity, 𝛿Φac(𝒓), that p eaks at the domain boundaries. (Fig. S 2(b) \nof Supplementary Information is a map  of 𝛿Φac(𝒓) that shows  peak contrast at the domain walls \nimaged in Fig. 1(b)).  \n \nFigure 1(c) -(j) shows a series of maps that illustrate the temperature dependence of 𝛿Φac(𝒓)/Φ𝑑𝑐 \nin the range from 120 K to the Curie temperature. Two features are immediately clear from the \nsequence of images. First, as the temperature is increased to  the Cur ie temperature, each domain \nboundary becomes increasingly convoluted, suggesting a progressi ve decrease in surface tension. \nA second, and less expected feature is that the DWs are nearly invisible in the 140 K map, \nindicating a deep minimum in  𝛿Φac(𝒓)/Φ𝑑𝑐, and therefore the oscillating displacement Δ𝑥, as a \nfunction of 𝑇.  \n \nThe key advantage of scanned local probe over global ac susceptibility  (𝜒𝑎𝑐) measurements is the \nability to determine the displacement, Δ𝑥, of individual  segments of  DWs.  The displ acement is \nobtained with ≈1 nm sensitivity from  the ratio  of 𝛿Φ𝑎𝑐 at the domain wall center , 𝛿Φ𝑎𝑐(0), to \nthe step in ellipticity across the DW , Φ𝑑𝑐, through the relation 𝛿Φ𝑎𝑐(0)/Φ𝑑𝑐=erf(Δ𝑥/√2𝜎), \nwhere 𝜎 is the probe focal radius   (see Supplementary Information Section III).   Panels (a) -(d) of \nFigure 2 illustrate the local dynamics of a representative segment of a DW  (see Supplementary  \nInformation  Section IV for details on the spatial variation  of DW dynamics ). Fig. 2(a) present s an \noverview of 𝛿Φ𝑎𝑐(0)/Φ𝑑𝑐 in the 𝜔−𝑇 plane using a color scale; Fig. 2(b) shows several line cuts \nthrough the plane at constant 𝜔.  A deep minimum in 𝛿Φ𝑎𝑐(0)/Φ𝑑𝑐 is evident in both plots. Fig. \n2(c) shows the  corresponding  DW displacement  as a function of 𝐻𝑎𝑐 for several temperatures . The \ndisplacement  displays a threshold , indicating collective pinning  behavior; Δ𝑥(𝐻𝑎𝑐) is near ly zero \nbelow a frequency dependent field, 𝐻𝑡ℎ, and increases  linearly with 𝐻−𝐻𝑡ℎ for 𝐻>𝐻𝑡ℎ.  Finall y, \nthe wall displacement measured with 𝐻𝑎𝑐= 28 Oe  is plotted in Fig. 2d  on an expanded \ntemperature scale that highlights the step -like change  centered on 𝑇=135 K . \n \nThe step change in displacement near 135 K  cannot be understood within the standard  theory of \nDW dynamics , which assumes a Bloch wall texture  in which  𝑴 rotates with constant magnitude \nin the plane of the DW.  In this theory Bloch DW mobility is proportional to the product of the 4 \n Gilbert parameter , which  describ es the damping rate of transve rse magnetization fluctuations , and \nthe DW width16.  Both of the se parameters are expected t o vary smoothly and monotonically at \ntemperatures well below 𝑇𝑐.  While the disorder that gives rise to collective pinning could account \nfor a monotonic decrease in mobility as 𝑇 is lowered, it cannot  explain the subsequent increase of \nDW displacement  below≈135 K.  \n \nStudies  of the ac susceptibility , 𝜒𝜔(𝑇), of insulating hexaferrite magnets performed in the early \n1990’s provide a clue as to the origin of the anomalous DW displacement in Co3Sn2S2.  These \nmeasurements revealed  nonmonotonicity in 𝜒𝜔(𝑇), resembling the minimum in DW displacement  \nin Co3Sn2S2, but with the qualitative difference that  it was found  in the  critical  regime , 𝑇=\n0.995𝑇𝑐, rather that deep with in the ferromagnetic phase17-19.  This phenomenon was traced to the \nprediction by Bulaevskii and Ginzburg (BG)20 that a “linear” wall, in which 𝑴 is aligned along \nthe easy axis an d passes through zero at the DW center, has lower energy than the Bloch wall , \nabove a transition temperature,  𝑇𝐵𝐺. Later th e same underlying physics was rediscovered in the \ncontext of  a transition from  sine-Gordon  to 𝜙4solitons21,22 as a function of the amplitu de of \nrotational symmetry breaking.    \n \nThe origin o f the BG phase transition  and the associated minimum in 𝜒𝜔(𝑇) is a crossover between \nthe energy  required to twist the magnetization and the energy required to change  its amplitude.  \nThis competiti on can be quantified by a free energ y of the form,  \n𝐹∝𝜌𝑠(∇𝑀)2+𝐾𝑀𝑇2+1\n8𝜒𝑐(𝑀2−𝑀𝑠2)2/𝑀𝑠2, (1) \nwhere 𝜌𝑠 is the spin stiffness, 𝐾≡1/2𝜒𝑎𝑏 is the uniaxial  (easy axis)  anisotropy factor, and 𝜒𝑐 is \na susceptibility that parameterizes the energy cost of changing the amplitude of the magnetization \nfrom its equilibrium value . 𝑀𝑇 and 𝑀𝑠 are the in -plane (or transverse) and saturation \nmagnetization s, respectively.  As 𝑇→𝑇𝑐, 𝜒𝑐(𝑇) diverges, whereas 𝐾(𝑇) does no t, and therefore \nvariations in the amplitude of 𝑴 cost less  energ y than rotations.  Consequently, in a range of \ntemperature below 𝑇𝑐 that depends on parameters,  the energy to create a linear wall is less than the \nenergy of a Bloch wall.  Minimizing the free energy in Eq. 1 yields a continuous phase transition \nfrom linear to elliptical to circular (Bloch) walls.  The order parameter of the BG transition is \n𝑀𝑇(0), the amplitude of the magnetization transverse to the propa gation direction evaluated at the 5 \n center of the DW.  This order parameter  manifests in  DW dynamics because propagation of a \nlinear wall  in which 𝑀𝑇(0)=0 involves purely longitudinal changes in magnetization, whereas \nmotion of a Bloch wall, where 𝑀𝑇(0)=𝑀𝑠, requires rotati on of 𝑴. \n \nThe signature of such a continuous phase transition  is enhanced susceptibility of  the order \nparamete r to its conjugate field , in this case a  static magnetic field  𝐻𝑇 applied perpendicular to the \neasy axis .  Figure 3(a)  shows the DW displacement vs. temperature for 𝐻𝑇=0,500Oe and \n1000Oe; Fig. 3(b) is the corresponding change in DW displacement with respect to 𝐻𝑇=0.  \nClearly, the susceptibility  to in -plane field is confined to a temperature regime close to the \ntransition.  Figs. 3(c) and 3(d) provide a more detailed view of the field dependence of the wall \ndisplacement.  Fig. 3(c) shows Δ𝑥𝑣𝑠.𝐻𝑎𝑐 for several values of 𝐻𝑇 at 𝑇=139𝐾; Fig. 3(d) is a \nplot of Δ𝑥𝑣𝑠.𝐻𝑇 at 𝐻𝑎𝑐=28Oe for temperatures near the transition.  Near  the transition  \ntemperature  a transverse  field of 1000 Oe  is sufficient  to switch  the DW from its low  to high \nmobility state , that is, changing 𝑀𝑇(0) from zero to 𝑀𝑠.  This indicates a remarkable enhancement \nof transverse susceptibility , as the field required to overcome the bulk anisotropy and rotate the \nspins to the basal plane is ~ 20 T.  \n \nThe free energy in Eq. 1  predicts  a mean -field transition occurring when the control parameter \n𝜏(𝑇)≡4𝜒𝑐/𝜒𝑎𝑏 reaches unity and an order parameter critical exponent of one -half, such that . \n𝜌≡𝑀𝑇(0)/𝑀𝑠=[1−𝜏(𝑇)]1/2.  However, given th at the transition occurs in a  two-dimensional \n(2D) manifold, one might anticipate strong fluctuation effects.  Indeed, Lawrie and Lowe have \nshown th eoretically that the BG transition is in the universality class of 2D Ising models  in which \nfluctuations reduce the transition temperature  and lead to an order parameter  critical exponent of \n1/823.  The Ising nature of the transition reflects the two discrete choices for the helicity of the \nBloch wall.  \n \nTo test the prediction of a 2D Ising transition within the DW , we determine d 𝜏(𝑇) through \nmeasurements of 𝑀𝑣𝑠.𝐻 for both in - and out -of-plane directions  at intervals of 1 K in the \ntemperature range from 110 to 160 K. An example of the isothermal  magnetizat ion data at 𝑇=\n120K,  characteristic of a ferromagnet with strong uniaxial anisotropy, is shown in Fig. 4 (a). The \nin-plane susceptibility  𝜒𝑎𝑏 is extracted from the slope of 𝑀𝑎 vs. 𝐻𝑎and is related to the anisotropy 6 \n parameter through the relation 𝐾=1/2𝜒𝑎𝑏. The longitudinal susceptibility  𝜒𝑐 is obtained from \nthe slope of 𝑀𝑐 vs. 𝐻𝑐 just above the saturation point (between 0.2 and 1 T). The temperature \ndependence of 𝜒𝑐 and 𝜒𝑎𝑏 (divided by 8 for clarity) is plotted in Fig. 4b . \n \nFig. 4(c)  shows the local DW dynamics over the same expanded temperature range as in Fig. 2(d), \nbut now plotted as a function of 𝜏(𝑇) rather than 𝑇.  To compare with a theory for the dynamics \nto be discussed below, we have converted the vertical axis scale from displacement, Δ𝑥, to an \neffective mobility, 𝜇eff, through the relation, 𝜇eff≡𝜔Δ𝑥/(𝐻𝑎𝑐−𝐻𝑡ℎ).  The observation  that the \ntransition takes place when  𝜏(𝑇) is of order unity gives additional strong support for the \nidentifying the step in mobility as the BG transition.  Moreover, the fact that the transition occurs \nat 𝜏(𝑇)≈0.45 rather than precisely unity  suggests a reduction  critical temperature typical of 2D \nIsing models, where for example  the 𝑇𝑐 for the Ising transition on a square lattice is reduced from \nthe mean field value by a factor 0.5724.   \n \nTo further pursue the mean field vs. 2D Ising comparison, we consider  the rate at which the  order \nparameter  grows below the transition , using the mobility as a proxy. To do so requires a \nphenomenological model for the mobility as a function the BG order parameter22. We introduce \nthe coefficients Γ𝐿 and Γ𝑇, the relaxation  rate of longitudinal  and transverse fluctuations of  𝑴, \nrespectively , which quantify the rate at which energy is dissipated as the DW propagates.   The DW \nmobility is obtained from the steady state condition that the dissipation rate  equal s the rate at which \nenergy is gained  as the magnetization aligns with the applied field (see Supplementary Section V I \nfor details) . The resulting expression for 𝜇eff[𝜌(𝜏)] interpolates between the mobilities of perfectly \ncircular walls where 𝜇eff=𝑤Γ𝑇/𝑀𝑠 and linear walls for which 𝜇eff=3𝑤Γ𝐿/2𝑀𝑠. The dashed  \nblue line in  Fig. 4(c), which is a plot of  𝜇eff(𝜏) assuming  𝜌=(1−𝜏)1/2, shows that a mean -field \ndescription fails not only for the critical value of 𝜏, but the order parameter exponent as well.  The \nmean field or 3D Ising  fits with lower critical values of 𝜏 near 0.5 also fail to describe the data \naccurately near the phase transition (see Supplementary Section VII for details).  \nThe red dashed line, which assumes the 2D Ising critical exponent  of 1/8  and a reduced  critical  \nvalue of 𝜏, such that  𝜌=(1−𝜏/0.45)1/8, provides a better description of the mobility data.    \n \n 7 \n Discussion  \nFinally, a natural question is why a DW phase transition deep in the FM state is unique ly observed \nin Co3Sn2S2 and not in other metallic uniaxial ferromagnets.   We suggest that what sets this \ncompound apart is the giant value of its dimensionless anisotropy factor, 𝐾.  Table  1 summarizes \nthe magnetocrystalline anisotropy of several uniaxial ferromagnets  that are receiv ing attention for \npotential applications . Notice that while the dimensionful anisotropy energy, 𝜇0𝐾𝑀𝑠2, is similar \nfor each compound, the saturation magnetization of  Co3Sn2S2 is much smaller; consequently,  the \nratio of anisotropy to magnetostatic energy is anomalously large. This leads to the absence of  the \ndomain branching that complicates the structure of domain walls at the surface and disrupts the \nBG transition  in other FM systems .  The large value of 𝐾 likely originates from  the extreme \nsensitivity of the low -energy electronic states  to the magnetization25.  The giant  anisotrop y in \nCo3Sn2S2, and potentially other systems in which the electron dispersion  is highly sensitive to \nmagnetization texture , may lead to new strategies  for manipulation of domain walls and their role \nin electron and spin transport.  \n \n \nMethods  \nScanning MOKE microscopy  \nA linearly polarized 633 nm HeNe laser beam was focused at normal incidence onto a 1 μm spot \non the sample surface with an objective lens (Olympus LMPFLN 50x, NA = 0.5). The reflected \nbeam was then collected by a 50:50 non -polarizing beam splitt er, which directs the beam to  a \nquarter wave p late and a Wollaston prism for  balanced photodetection of Kerr ellipticity . The \nsample position was raster scanned with xy piezoelectric scanners (Attocube ANPx101) and the \nfocusing was fine -adjusted with a z piezoelectric scanner (Attocube ANPz102). For local \nmeasurements of the DW dynamics, a n out -of-plane AC magnetic field was applied through a coil \n(Woodruff Scientific, 156 turns, inner diameter: 5mm, height: 1.5 mm)  that surrounds the sample. \nThe modulation  in Kerr ellipticity δΦac(𝒓) was measured using  a standard lock-in detection  at the \nfrequency of the coil. The sensitivity of DC Kerr measurements is typically ~1 mrad, due to various \nsources of 1/𝑓 fluctuations.   By shifting the observation frequency using modulation of 𝐻𝑎𝑐, we \nare able to measure Kerr effect amplitudes of less than 1 µrad.  8 \n Data availability Statement  \nThe datasets generated during and/or analysed during the current study are avai lable from the \ncorresponding author on reasonable request.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 9 \n References  \n1 Weihrich, R. & Anusca, I. Half Antiperovskites. III. Crystallographic and Electronic \nStructure Effects in Sn2−xInxCo3S2. Zeitschrift für anorganische und allgemeine \nChemie  632, 1531 -1537, doi:10.1002/zaac.200500524 (2006).  \n2 Wan, X., Turner, A. M., Vishwanath, A. & Savrasov, S. Y. Topological semimetal and \nFermi -arc surface states in the electronic structure of pyrochlore iridates. Physical  Review \nB 83, doi:10.1103/PhysRevB.83.205101 (2011).  \n3 Burkov, A. A. & Balents, L. Weyl semimetal in a topological insulator multilayer. Phys \nRev Lett  107, 127205, doi:10.1103/PhysRevLett.107.127205 (2011).  \n4 Armitage, N. P., Mele, E. J. & Vishwanath, A. W eyl and Dirac semimetals in three -\ndimensional solids. Reviews of Modern Physics  90, doi:10.1103/RevModPhys.90.015001 \n(2018).  \n5 Xu, Q.  et al.  Topological surface Fermi arcs in the magnetic Weyl semimetal Co3Sn2S2. \nPhysical Review B  97, doi:10.1103/PhysRevB. 97.235416 (2018).  \n6 Liu, E.  et al.  Giant anomalous Hall effect in a ferromagnetic Kagome -lattice semimetal. \nNat Phys  14, 1125 -1131, doi:10.1038/s41567 -018-0234 -5 (2018).  \n7 Wang, Q.  et al.  Large intrinsic anomalous Hall effect in half -metallic ferromagnet \nCo3Sn2S2 with magnetic Weyl fermions. Nat Commun  9, 3681, doi:10.1038/s41467 -\n018-06088 -2 (2018).  \n8 Okamura, Y.  et al.  Giant magneto -optical responses in magnetic Weyl semimetal \nCo3Sn2S2. Nat Commun  11, 4619, doi:10.1038/s41467 -020-18470 -0 (2020).  \n9 Yang, H.  et al.  Giant anomalous Nernst effect in the magnetic Weyl semimetal Co 3Sn2S2. \nPhysical Review Materials  4, doi:10.1103/PhysRevMaterials.4.024202 (2020).  \n10 Liu, D. F.  et al.  Magnetic Weyl semimetal phase in a Kagome crystal. Science  365, 1282 -\n1285, doi:10 .1126/science.aav2873 (2019).  \n11 Morali, N.  et al.  Fermi -arc diversity on surface terminations of the magnetic Weyl \nsemimetal Co3Sn2S2. Science  365, 1286 -1291, doi:10.1126/science.aav2334 (2019).  \n12 Kassem, M. A., Tabata, Y., Waki, T. & Nakamura, H. Low -field anomalous magnetic \nphase in the kagome -lattice shandite Co3Sn2S2. Physical Review B  96, \ndoi:10.1103/PhysRevB.96.014429 (2017).  10 \n 13 Guguchia, Z.  et al.  Tunable anomalous Hall conductivity through volume -wise magnetic \ncompetition in a topological kagome magnet. Nat Commun  11, 559, doi:10.1038/s41467 -\n020-14325 -w (2020).  \n14 Lachman, E.  et al.  Exchange biased anomalous Hall effect driven by frustration in a \nmagnetic kagome lattice. Nat Commun  11, 560, doi:10.1038/s41467 -020-14326 -9 (2020).  \n15 Howlader, S., Ramachandran, R., Monga, S., Singh, Y. & Sheet, G. Domain structure \nevolution in the ferromagnetic Kagome -lattice Weyl semimetal Co3Sn2S2. J Phys \nCondens Matter , doi:10.1088/1361 -648X/abc4d1 (2020).  \n16 Schryer, N. L. & Walker, L. R. The motion of 180° domain walls in uniform dc magnetic \nfields. Journal of Applied Physics  45, 5406 -5421, doi:10.1063/1.1663252 (1974).  \n17 Kotzler, J., Garanin, D. A., Hartl, M. & Jahn, L. Evidence for critical fluctuations in \nBloch walls near their disordering temperature. Phys Rev Lett  71, 177 -180, \ndoi:10.1103/PhysRevLett.71.177 (1993).  \n18 Kötzler, J., Hartl, M. & Jahn, L. Signature of linear (Bulaevskii –Ginzburg) domain walls \nnear Tc of Ba‐hexaferrite. Journal of Applied Physics  73, 6263 -6265, \ndoi:10.1063/1.352664 (1993).  \n19 Hartl -Malang, M., Kotzler, J. & Garanin, D. A. Domain -wall relaxation near the disorder \ntransition of Bloch walls in Sr hexaferrite. Phys Rev B Condens Matt er 51, 8974 -8983, \ndoi:10.1103/physrevb.51.8974 (1995).  \n20 Bulaevskii, L. N. & Ginzburg, V. L. Temperature Dependence of the Shape of the \nDomain Wall in Ferromagnetics and Ferroelectrics. Sov. Phys. JETP  18 (1964).  \n21 Sarker, S., Trullinger, S. E. & Bishop,  A. R. Solitary -wave solution for a complex one -\ndimensional field. Physics Letters A  59, 255 -258, doi:10.1016/0375 -9601(76)90784 -2 \n(1976).  \n22 Garanin, D. A. Dynamics of elliptic domain walls. Physica A: Statistical Mechanics and \nits Applications  178, 467 -492, doi:10.1016/0378 -4371(91)90033 -9 (1991).  \n23 Lawrie, I. D. & Lowe, M. J. Domain wall singularity in a Landau -Ginzburg model. \nJournal of Physics A: Mathematical and General  14, 981 -992, doi:10.1088/0305 -\n4470/14/4/025 (1981).  \n24 Onsager, L. Crystal Statis tics. I. A Two -Dimensional Model with an Order -Disorder \nTransition. Physical Review  65, 117 -149, doi:10.1103/PhysRev.65.117 (1944).  11 \n 25 Ghimire, M. P.  et al.  Creating Weyl nodes and controlling their energy by magnetization \nrotation. Physical Review Researc h 1, doi:10.1103/PhysRevResearch.1.032044 (2019).  \n26 Chen, B.  et al.  Magnetic Properties of Layered Itinerant Electron Ferromagnet \nFe3GeTe2. Journal of the Physical Society of Japan  82, 124711, \ndoi:10.7566/jpsj.82.124711 (2013).  \n27 Dillon, J. F. & Olson, C . E. Magnetization, Resonance, and Optical Properties of the \nFerromagnet CrI3. Journal of Applied Physics  36, 1259 -1260, doi:10.1063/1.1714194 \n(1965).  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 12 \n Acknowledgement s \nWe would like to thank Jeffrey Orenstein and Simon Woodruff for design of the magnetic coils \nused in the measurements, Veronika Sunko and Dylan Rees for assistance in installing the DC \nmagnetic field module, and Jonathan Denlinger for assistance in sample characterization. This \nwork was primarily supported by the Spin Physics  program under the Director, Office of \nScience, Office of Basic Energy Sciences, Materials Sciences and Engineering Division, of the \nU.S. Department of Energy, Contract No. DE -AC02 -76SF 00515 .  C. L. and J.O. acknowledge \nsupport from the Quantum Materials program under the Director, Office of Science, Office of \nBasic Energy Sciences, Materials Sciences and Engineering Division, of the U.S. Department of \nEnergy, Contract No.  DE-AC02 -05CH11 231. J.O. received support for optical measurements \nfrom the Gordon and Betty Moore Foundation’s EPiQS Initiative through Grant GBMF4537 to \nJ.O. at UC Berkeley. P.V., K.M., C.S., and C.F. acknowledge financial support from the \nEuropean Research Council (ER C) Advanced Grant No. 742068 “TOP -MAT”; European \nUnion’s Horizon 2020 research and innovation program (Grant Nos. 824123 and 766566) and \nDeutsche Forschungsgemeinschaft (DFG) through SFB 1143. K. M. acknowledges the Max \nPlanck Society for the funding suppo rt under Max Planck –India partner group project.  \n \nAuthor Contributions  \nC.L. performed the optical measurements. C.L. and J.O. analyzed the data and wrote the \nmanuscript. P.V., K.M., C.S., and C.F. grew and characterized the crystals. C.S. performed the \nmagnetic susceptibility measurements. J.O. and J.E.M. developed the theoretical models. All \nauthors commented on the manuscript.  \n \nCompeting interests Statement  \nThe authors declare no competing interests.  \n \n 13 \n Compound  Magnetization \n(𝑴𝒔) \n×105𝐴/𝑚 Magnetostatic energy  \n(𝜇0𝑀𝑠2)×105𝐽/𝑚3 Anisotropy energy  \n(𝜇0𝐾𝑀𝑠2)×105𝐽/𝑚3 Anisotropy \nfactor (𝑲) \nCo3Sn2S210 0.56 0.042  3.8 90 \nFe3Ge2Te226 4.0 2.0 12.1 6.0 \nCrI 327 2.1 0.58 3.1 5.3 \n \nTable 1 | Summary of various magnetic constants in uniaxial ferromagnets  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 14 \n  \n \nFigure 1 | Scanning Kerr microscopy. a  Overview of ac MOKE microscopy setup. The sample \nis surrounded by a coil that generates an out -of-plane ac magnetic field.  b Unmodulated Kerr \nellipticity map taken at T = 120 K reveals stripe -like magnetic domains.  c-j ac MOKE maps \nmeasured at temperatur es ranging from 120 to 174 K. A 2 8 Oe ac magnetic field was applied at a \nfrequency of 1kHz. The normalized ac Kerr ellipticity amplitude ΦAC/ΦDC is significantly \nreduced at 140 K.  \n  \n \n \n15 \n  \n \nFigure 2 | DW displacement and linear wall phase transition.  a Normalized ac Kerr response \nmeasured at a segment of a domain wall over a range of frequency (5 Hz – 10 kHz) and temperature \n(40 – 175 K). A sharp decrease in the Kerr response is noticeable near T=140 K. b Selected plots \nof the temperature dependence at constant frequency (horizontal cuts through the frequency -\ntemperature map shown in a). c Measurements of the domain wall displacement plotted against ac \nmagnetic field amplitude at temperatures in the range of 120-140 K ( 𝑓=1kHz , See \nSupplementary Fig. 4 for data at higher temperatures) . d DW displacement versus temperature. \nThe non -monotonic temperature dependence of the displacement can be understood as a phase \ntransition from a e, circular or f, elliptical wall at low temperatures to a g, linear wall at  high \ntemperatures in which the magnetization vanishes at the wall center.  \n \n \n \n \n \n16 \n  \n \nFigure 3 | Magnetic field dependence of the DW ellipticity. a . DW displacement plotted against \ntemperature for various in -plane dc magnetic field 𝐻𝑦 applied to the sample.  b. The changes in \nDW displacement Δ𝑥 induced by dc field values of 𝐻𝑦=500Oe and 1000Oe show sharp peaks \nat 𝑇=135𝐾 and 136𝐾, indicating a phase transition into the linear wall state, c. DW \ndisplacement versus 𝐻ac for different  values of in -plane dc fields 𝐻𝑦 at 𝑇=139K. A clear change \nin the DW displacement and mobility is observed by tuning the wall ellipticity with an in -planed \nfield 𝐻𝑦 applied to the sample near 𝑇BG. d. DW displacement plotted versus in -plane field 𝐻𝑦 for \nvarious temperatures. A saturation behavior is observed at lower temperatures toward the Bloch \nwall phase.  \n \n \n  \n17 \n  \n \nFigure 4 | Magnetic susceptibility and 2D Ising transition. a Magnetization versus magnetic \nfield along out -of-plane ( z) and in -plane ( x) directions, respectively.  b Magnetic susceptibility \nplotted against temperature. The in -plane susceptibility data divided by 8 for clarity. c Effective \nmobility, 𝜇eff=Δ𝑥\nΔ𝐻, of the DW is plotted against 𝜏≡4𝜒𝑐/𝜒𝑎𝑏, which is the control parameter for \nthe linear wall phase transition. The 𝜇eff data fits is consistent with the DW mobility predicted by \nthe 2D Ising model (red dotted line)24, but not with the mean field theory (blue dotted line), which \npredict a phase transition at 𝜏∗~1. Error bars represent the uncertainties of least square fitting of \ndata.   \n \n \n \n \n \n \n \n \n \n \n \n18 \n  \nSupplementary Information  \n“Observation of a phase transition  \nwithin the domain walls of ferromagnetic Co 3Sn2S2” \n \nI. DC Kerr ellipticity on traversing a domain wall  \nThe unmodulated, dc Kerr ellipticity was measured as a function of position transverse to the \ndomain walls. Typically, three consecutive scans through a wall were recorded to improve signal \nto noise ratio (Fig. S1(a)).  The results were then fitted to an e rror function whose amplitude is \nproportional to the 𝑧 component of the magnetization, 𝑀𝑧, within a domain.  Φ𝑑𝑐 is therefore a \nmeasure of the equilibrium 𝑀𝑧 in zero applied field.   \n \nThe temperature dependence of Φ𝑑𝑐 is shown in Fig. S1(b), with two fits (red curves) that span \nthe low temperature spin wave regime and the high temperature critical regime. In the high \ntemperature regime, the best fit was obtained with Curie temperature 𝑇𝑐=175.2±0.2K and \ncritical exponent 𝛽=0.33±0.03, which is in  good agreement with 𝛽=0.326 predicted by the \n3D Ising model.  \n \n \nFigure S1 | a dc Kerr ellipticity plotted as a function of position relative to the center of the domain \nwall for various temperatures. The ellipticity data (dots) are subsequently fit to er ror functions \n19 \n (lines). b The temperature dependence of dc Kerr ellipticity compared to predictions for the spin \nwave (low temperature) and critical (high temperature) regimes, respectively.  \n \nII. Comparison of unmodulated and ac Kerr ellipticity maps  \nIn order to verify that the ac Kerr response tracks the DW displacement, we show maps of \nunmodulated and ac Kerr ellipticity maps taken across the same region of the sample in Figs. S2(a) \nand (b), respectively.  \n \nFigure S2 | a  Unmodulated Kerr ellipticity map taken at T = 120 K reveals stripe -like magnetic \ndomains.  b Modulated (f = 1 kHz) ac Kerr map measured over the same region of sample exhibits \na strong signal at the domain boundaries. The signal is normalized by the dc Ker r ellipticity \nobtained at 120 K.   \n \nIII. Analysis of domain wall displacement  \nDomain wall (DW) displacement was determined by measuring both Φ𝑑𝑐 and the ac modulation \nsynchronous with the applied magnetic field, 𝛿Φ𝑎𝑐, as a function of position along a direc tion \nnormal to the DW. Scanning the laser probe transverse to the domain wall yields a peak in \n𝛿Φ𝑎𝑐/Φ𝑑𝑐 at the DW center, as discussed in the main text.  There are three length scales that \ndetermine the peak amplitude: DW displacement, Δ𝑥, DW width, 𝑤, and laser focus radius, 𝜎.  We \nobtain 𝜎≈1𝜇𝑚 using standard optical methods for determining the focal diameter.  We estimate \nthat 𝑤 is on the order of several nanometers based on neutron scattering experiments, where a \nferromagnetic spin wave disp ersion was observed, 𝐸(𝑞)=𝐸𝑔𝑎𝑝+𝐷𝑞2, with 𝐸𝑔𝑎𝑝≈2𝑚𝑒𝑉  and \n20 \n 𝐷≈8𝑚𝑒𝑉−𝑛𝑚2.  Thus the our measurements are performed in the regime where 𝜎≫𝑤, \nwhere the peak amplitude is independent of 𝑤 and related to the displacement through the formula,  \n𝛿Φ𝑎𝑐(0)/Φ𝑑𝑐=erf(Δ𝑥/√2𝜎).\n \nIV. Spatial variation of the domain wall displacement  \nThe ac Kerr ellipticity exhibits a certain amount of spatial variation. Depending on the specific \nlocation of the measurement spot (Fig. S5a), the DW displacement can vary by a factor of two, \nwhile the threshold field 𝐻𝑡ℎ exhibits a much smaller variatio n (Fig. S5b). However, the non -\nmonotonic temperature dependence of the DW displacement is observed for all DWs, as shown in \nthe ac maps plotted in Fig. 1 of the main t ext.  \n \nFigure S3 | a  ac MOKE map measured at 130 K. b Plot of DW displacement at the fiv e different \nlocations indicated panel a, showing variations in DW mobility.  \n \n \n \n \n21 \n V. Domain wall displacement at high temperatures (T > 140 K)  \n \nFigure S4 | DW displacement above the BG phase transition. Measurements of the domain \nwall displacement plotted against ac magnetic field amplitude at temperatures ranging from \nT=140 K to 150 K ( 𝑓=1kHz). Larger wall displacements and mobilities can be observed \ntoward higher temperatures.  \n \nVI. Relation of domain wall mobility to Bulaevskii -Ginzburg order parameter  \nWe consider the propagation of an elliptical domain wall described by magnetization,  \n𝑴(𝑥,𝑡)=𝑀𝑠[𝒛̂tanh(𝑥−𝑣𝑡\n𝑤)+𝒚̂𝜌\ncosh(𝑥−𝑣𝑡\n𝑤)], (1) \nwhere 𝑀𝑠 is the saturation magnetization, 𝑤 is the wall width, and 𝜌=𝑀𝑦/𝑀𝑠 is the normalized \nin-plane magnetization at the DW center. The power dissipated by the wall moving with velocity \n𝑣 is given by,  \n𝑃𝑑𝑖𝑠𝑠=1\nΓ𝐿∫𝑑𝑥𝑀2̇+1\nΓ𝜃∫𝑑𝑥𝑀𝑦2̇, (2) \n \nwhere Γ𝐿 and Γ𝜃 are relaxation rates of longitudinal and transverse fluctuations of magnetization.  \nUsing change of variables 𝑢=(𝑥−𝑣𝑡)/𝑤, \n22 \n 𝑃𝑑𝑖𝑠𝑠=𝑣2\n𝑤[1\nΓ𝐿∫𝑑𝑢(𝑑𝑀\n𝑑𝑢)2\n+1\nΓ𝜃∫𝑑𝑢(𝑑𝑀𝑦\n𝑑𝑢)2\n]. (3) \nThe integral ∫𝑑𝑢(𝑑𝑀\n𝑑𝑢)2\nwas calculated numerically. For convenienc e in fitting we used the \nempirical form (4/3)𝑀𝑠2(1−𝜌)2.2 which approximates the numerical result to within 2%. The \nintegral ∫𝑑𝑢(𝑑𝑀𝑦\n𝑑𝑢)2\n gives 𝑀𝑠2𝜌2, so that the dissipated power can be written,  \n𝑃𝑑𝑖𝑠𝑠=𝑣2𝑀𝑠2\n𝑤[1\nΓ𝐿2\n3(1−𝜌)2.2+𝜌2\nΓ𝜃]. (4) \nFor steady state motion we equate the dissipated power with the rate at which stored magnetostatic \nenergy is lowered,  𝑈=𝑣𝑀𝑠𝐻, where 𝐻 is the magnetic field applied parallel to the easy axis.  \nFinally solving for the ratio of the velocity to the applied f ield yields the relation between mobility \nand BG order parameter plotted in Fig. 4(c) of the main text,  \n𝜇=𝑤\n𝑀𝑠1\n2\n3Γ𝐿(1−𝜌)2.2+𝜌2\nΓ𝜃. \n \n \n \n \n \n \n \n \n \n \n 23 \n VII. Various fits to the effective DW mobility  \n \nFigure S5 | Effective DW mobility and fits to the data. Effective mobility, 𝜇eff=Δ𝑥\nΔ𝐻, of the DW \nis plotted against 𝜏≡4𝜒𝑐/𝜒𝑎𝑏, which is the control parameter for the linear wall phase transition. \nThe 𝜇eff data fits much better with the 2D Ising model (red dotted line, 𝛽=0.125), compared to \nthe 3D Isi ng model (blue dotted line,  𝛽=0.326) or the mean field theory (green dotted line, 𝛽=\n0.5) with critical values of 𝜏 near 0.5. In all cases, 𝜏∗ was adjusted give the best fit.  \n \n \nReferences  \n1 Garanin, D. A. Dynamics of elliptic domain w alls. Physica A: Statistical Mechanics and \nits Applications  178, 467 -492, doi:10.1016/0378 -4371(91)90033 -9 (1991).  \n \n"    },    {        "title": "2105.04893v2.On_the_connection_between_magnetic_interactions_and_the_spin_wave_gap_of_the_insulating_phase_of_NaOsO___3__.pdf",        "content": "On the connection between magnetic interactions and the spin-wave gap of the\ninsulating phase of NaOsO 3\nNikolaos Ntallis,1Vladislav Borisov,1Yaroslav O. Kvashnin,1Danny Thonig,2Erik\nSj oqvist,1Anders Bergman,1Anna Delin,3, 4Olle Eriksson,1, 2and Manuel Pereiro1,\u0003\n1Department of Physics and Astronomy, Uppsala University, Uppsala 751 20, Sweden\n2School of Science and Technology, Orebro University, SE-701 82, Orebro, Sweden\n3Department of Applied Physics, School of Engineering Sciences,\nKTH Royal Institute of Technology, AlbaNova University Center, SE-10691 Stockholm, Sweden\n4Swedish e-Science Research Center (SeRC), KTH Royal Institute of Technology, SE-10044 Stockholm, Sweden\n(Dated: May 13, 2021)\nThe scenario of a metal-insulator transition driven by the onset of antiferromagnetic order in\nNaOsO 3calls for a trustworthy derivation of the underlying e\u000bective spin Hamiltonian. To deter-\nmine the latter we rely on ab initio electronic-structure calculations, linear spin-wave theory, and\ncomparison to experimental data of the corresponding magnon spectrum. We arrive this way to\nHeisenberg couplings that are .45% to .63% smaller than values presently proposed in the literature\nand Dzyaloshinskii-Moriya interactions in the region of 15% of the Heisenberg exchange J. These\ncouplings together with the symmetric anisotropic exchange interaction and single-ion magnetocrys-\ntalline anisotropy successfully reproduce the magnon dispersion obtained by resonant inelastic X-ray\nscattering measurements. In particular, the spin-wave gap fully agrees with the measured one. We\n\fnd that the spin-wave gap is de\fned from a subtle interplay between the single-ion anisotropy,\nthe Dzyaloshinskii-Moriya exchange and the symmetric anisotropic exchange interactions. The re-\nsults reported here underpin the local-moment description of NaOsO 3, when it comes to analyzing\nthe magnetic excitation spectra. Interestingly, this comes about from a microscopic theory that\ndescribes the electron system as Bloch states, adjusted to a mean-\feld solution to Hubbard-like\ninteractions.\nIn the context of correlated electronic systems, 5 dox-\nides hold a distinct position, realizing the peculiar regime\nwhere Coulomb interactions (the Hubbard Uin partic-\nular), thed{dhopping matrix elements (referred to as\nt's), and the spin-orbit (SO) coupling ( \u0015) have similar\nmagnitudes ( \u0015\u0018t\u0018U). In contrast, in 3 dand 4f\nbased magnets, it is possible to identify a smallest energy\nscale, since the SO coupling de\fnes a much smaller en-\nergy scale in 3 dtransition-metal compounds ( \u0015\u001ct<U ),\nand the 4f{4fhopping is minute in rare-earth systems\n(t\u001c\u0015<U ).\nThe SO coupling in iridates and osmates, for exam-\nple, is of order 0 :5\u00000:6 eV. In addition, the 5 dorbitals,\nbeing much more extended than, e.g., the rare-earth 4 f\nstates, causes the 5 delectrons on neighboring ions to\ninteract much more e\u000bectively. In other words, reason-\nably well localized magnetic moments can still be in place\nbut ligand-mediated superexchange is greatly enhanced\nas compared to in, e.g., 4 finsulators. Combined with the\nstrong SO coupling, it may generate highly anisotropic\nintersite magnetic interactions, with the remarkable sit-\nuation encountered in t5\n2giridates of having either an-\ntisymmetric Dzyaloshinskii-Moriya (DM) [1, 2] or sym-\nmetric Kitaev [1, 3, 4] e\u000bective coupling parameters that\nare even larger than the isotropic Heisenberg constant.\nSincet\u0018U, metal-insulator transitions (MIT's) may\nalso occur. Extensively discussed in this respect is the\nosmium oxide perovskite compound NaOsO 3. However,\nrather than a Mott MIT, the scenario of a Slater MIT has\nbeen proposed for NaOsO 3[5{9], where the formation ofantiferromagnetic (AF) order is opening the insulating\ngap.\nIn this frame of reference, detailed knowledge of the un-\nderlying magnetic interactions is crucial. Here we shed\nlight on this matter by means of ab initio calculations\nand subsequent atomistic spin-dynamics simulations em-\nploying the derived e\u000bective magnetic interactions. The\ncomputed magnon spectra are compared with experimen-\ntal data as obtained by resonant inelastic X-ray spec-\ntroscopy (RIXS) [10]. We calculate from ab initio den-\nsity functional theory (DFT) a Heisenberg exchange of\n\u00184\u00006 meV, that can be easily reconciled with the band-\nwidth of the magnon spectrum; \u001880 meV, reported on\nthe basis of RIXS measurements[10]. However, our anal-\nysis shows that a single-ion anisotropy (SIA), the DM\ninteraction, Heisenberg exchange as well as the symmet-\nric anisotropic exchange interaction, are large, and nec-\nessary for theory to reproduce the magnon spectrum of\nNaOsO 3.\nFrom the structural point of view, NaOsO 3has the oc-\ntahedral environment of Os5+O6, so that the electronic\ncon\fguration is 5d3, suggesting that the t2g-band is\nhalf \flled. Moreover, this compound shows Curie-Weiss\nmetallic nature and abruptly but continuously turns to\nan AF insulator at 410 K [5]. The structural, electronic\nand magnetic properties of bulk NaOsO 3are studied the-\noretically using density functional theory (DFT)[11, 12]\nwithin the generalized-gradient approximation in the\nPBE parameterization[13]. Electronic properties are cal-\nculated within the all-electron full-potential fully rela-arXiv:2105.04893v2  [cond-mat.str-el]  12 May 20212\ntivistic approach, with linear mu\u000en-tin orbitals as ba-\nsis functions in both LDA+U and LSDA+U approxima-\ntions, as implemented in the RSPt electronic structure\ncode [14{16], with the U values ranged from 0 to 4 eV.\nFigure 1(a) shows the experimental crystal struc-\nture of NaOsO 3. The cell dimensions are a=5 :3842 \u0017A,\nb=5:3282 \u0017A and c=7:5804 \u0017A. The magnetic ground state\nfor all used values of U has a G-type antiferromagnetic\n(AF) ordering, as shown in Fig. 1(a). In Fig. 2, the\nspin-, orbital- and total magnetic moment per Os atom\nis plotted with respect to the U value for the di\u000berent\napproximations, i.e. LDA+U and LSDA+U. The two\napproaches produce di\u000berent values of the magnetic mo-\nments as function of U. Notably, the LDA+U approx-\nimation would produce a zero magnetic moment for Os\natoms by decreasing U down to 2 eV, according to Fig. 2.\nThis result is natural, since all exchange splitting in this\nlevel of approximation is in the static Hubbard type in-\nteraction, while the spin-density functional by construc-\ntion provides no exchange splitting. In the limit of \fnite\nU, where both of the approximations produce almost the\nsame orbital moment, the di\u000berence on the spin part is\nclose to 0:35\u0016Bper atom. Also, one can reproduce the\nexperimental values of the Os moment ( \u00181\u0016B) for both\napproximations, albeit for di\u000berent values of U.\nDi\u000berent types of magnetic interactions, such as the\nHeisenberg and DM interaction, were evaluated for the\nG-type antiferromagnetic reference state based on the\nfully-relativistic generalization [17] of the Lichtenstein-\nKatsnelson-Antropov-Gubanov (LKAG) formula [18],\nwhich we have recently applied to di\u000berent correlated sys-\ntems [19]. The purpose of this is to map the electronic\nsystem onto a generalized classical Heisenberg model:\nH=\u0000X\ni6=je\u000b\ni^J\u000b\f\nije\f\nj; \u000b;\f =x;y;z; (1)\nwhere the unit vectors eiindicate the direction of local\nspins and the fully-relativistic exchange tensor J\u000b\f\nijcon-\na)\n b)\nFIG. 1. a) Pnma crystal structure of NaOsO 3. Red and\nyellow spheres indicate O and Na ions, respectively. b) G-type\nAF order within the Os sublattice. The NN exchange paths\n(within the acplane and along the baxis) are highlighted.\nFIG. 2. Calculated total (solid symbols) and spin (un\flled\nsymbols) magnetic moments in a) together with orbital mo-\nments (opposite to the spin moments) per atom of NaOsO 3in\nb). The moments are plotted for di\u000berent values of U and for\nLSDA+U and LDA+U implementations with \fxed J H= 0:6\neV. The dotted line represents the magnetic moment mea-\nsured in Ref. [6].\ntains contributions from the Heisenberg exchange as well\nas the DM interaction ^Dand the symmetric anisotropic\nexchange ^\u0000 de\fned by\nDz\nij=1\n2(Jxy\nij\u0000Jyx\nij) (2)\n\u0000z\nij=1\n2(Jxy\nij+Jyx\nij) (3)\nand similar expressions for the x- andy-components.\nAccording to Fig. 2, the values of U which produce to-\ntal magnetic moments closest to the experimental ones\nare U\u00190:8 eV for LSDA+U, similarly to [20{22], and\nU\u00193:5 eV for LDA+U. Figure 3 shows the calcu-\nlated magnetic interaction strengths with respect to the\natomic distance. All interaction strengths decay quite\nfast with respect to distance indicating that the major\ninteractions to be considered are the \frst and second or-\nder ones. The LDA+U approximation (with U = 3 :5 eV)\nproduces Heisenberg exchange interaction that is roughly\ntwice the magnitude compared to the LSDA+U data\n(with U = 0 :8 eV). Similarly the LDA+U calculation re-\nsults in a value of Dthat is approximately three or four\ntimes larger than the one for LSDA+U approximation.\nIn order to assess the quality of the exchange interac-\ntions we used them to evaluate adiabatic magnon spec-\ntra, that can be compared to experimental results (see\ndiscussion below). We note, however, already here that\nsuch an analysis reproduces experimental results quite\npoorly from a theory based on LDA+U (see supplemen-\ntary information (SM) for further details on the LDA+U\nresults)[23], while LSDA+U calculations give reasonable\nresults. For this reason we focus in the rest of this paper\non results from LSDA+U calculations.\nDespite the same magnetic pattern along the xandz\ndirections and very similar distances ( dx= 3:787 A and\ndz= 3:790 A), the Heisenberg exchange interactions J1x\nandJ1zare markedly di\u000berent. This could be related to\nthe large e\u000bect of magnetic anisotropy which aligns the\nmagnetic moments and the N\u0013 eel vector preferably along\ntheydirection. It is interesting that the J1z=J1xratio3\nFIG. 3. Calculated interaction strengths J+D+ \u0000 with\nrespect to the atomic distance for U=0.8 eV, J H= 0:6 eV\n(top panel) and U=1.0 eV, J H= 0:6 eV (bottom panel). All\ninteraction are marked following the prescription 1 !3 given\nin Fig. 1.\nFIG. 4. a) Calculated DM vectors Dfor the nearest neighbor\nbonds in NaOsO 3shown by the orange ( D1), green ( Da\n3) and\nmagenta ( Db\n3) arrows. The Da;b\n3vectors are magni\fed by the\nfactor of 2.5 for clarity. The G-type antiferromagnetic order-\ning of Os magnetic moments moriented along the y-direction\nis depicted by the red and blue arrows. The magnetic inter-\nactions were obtained using LSDA+ UwithU= 0:8 eV and\nJH= 0:6 eV. b) The nearest-neighbor Heisenberg exchange\nJ1along the [100] and [001] directions and the y-component\nof the DM interaction D1in the abplane as functions of the\ncorrelation strength U.\nincreases as a function of the correlation strength U from\n0.62 at U = 0 :8 eV to 0.85 at U = 4 eV. This may suggest\nthat the system becomes magnetically more isotropic for\nstronger correlations where the system is more insulating,\nbased on the calculated value of the electronic band gap.\nWe also notice that the scaled Heisenberg parameters\nJ=m2\nsare far from the 1=Uscaling expected for insula-\ntors. This can be explained by the fact that correlations\nin the narrow-gap insulator NaOsO 3are not strong com-\npared to the electron hopping amplitude t, so that t=Uis\nnot a small parameter and the perturbation theory, which\ncould lead to the 1=Uscaling, is not applicable. Interest-\ningly, the nearest-neighbor parameters J1=m2\nsincrease as\nfunctions of U up to U = 3 :0\u00003:5 eV which deserves a\nmore detailed study in the future. On the other hand,\nthe DM interaction shows a non-monotonous variation of\nitsy-component, which is dominating for the bonds alongthexy-directions, with a maximum around U = 1 eV fol-\nlowed by a decreasing trend for larger U values.\nThe numerical values of the exchange interactions are\ncollected in Table I for LSDA+U calculations, while the\nDM vectors for the nearest-neighbor bonds are shown\nin Fig. 4. These interactions were used to calculate\nthe magnon spectra, which was evaluated by rewriting\nEq. (1), in the form of a bilinear, e\u000bective Hamiltonian\nHi;jcontaining the isotropic Heisenberg, the DM as well\nas symmetric anisotropic exchange. Adding single-site\nmagnetic anisotropy we end up with the following ex-\npression for the full Hamiltonian of the spin system:\nHi;j\nmod=\u0000JijSi\u0001Sj\u0000Dij\u0001Si\u0002Sj\u0000Si\u0000ijSj(4)\n\u0000KX\nk=i;j\u0000\nSk\u0001eY\nk\u00012;\nwhere Si,Sjare normalized atomic spin moments. The\nisotropic Heisenberg coupling is given by JijandDij\nstands for the DM vector, while the tensor \u0000 ijrepresents\nthe symmetric anisotropic exchange interaction between\natomic sites iandj. The Heisenberg exchange interac-\ntionJij, the norm of Dijand mean value of \u0000 ijtensor\ntake the values J1,jD1j, \u00001andJ2,jD2j, \u00002for bonds\n1 and 2 (cf. Fig. 1(b)), respectively. The orientation of\nthe DM vectors depend on the position in the unit cell,\nsee Fig. 4. The DM vectors Da\n3andDb\n3for the third-\nneighbours atoms are slightly di\u000berent (cf. Table I and\nFig. 4) because the cell dimensions in the ab plane are\nalso slightly di\u000berent (a 6=b). As regards the magnetic\nanisotropy, each site possesses an easy-axis orientation\ndirected along the local magnetic y-axis eY\nk. The mag-\nnetic anisotropy parameter ( Kin Eq. (4)) was calculated\nbased on the sum of eigenvalues in the one-shot relativis-\ntic calculations using the RSPt code [14{16] where the\nmagnetization axis is oriented along the x-,y- andz-\ndirections. The obtained estimates are K= 2:05 meV\nfor U=0:8 eV andJH= 0:6 eV while the calculation with\nU=1:0 eV andJH= 0:6 eV resulted in K= 2:44 meV.\nIn order to validate the parameters calculated by ab\ninitio electronic structure theory, we performed calcu-\nlations of magnon dispersions, based on a collinear AF\nground state, in the framework of linear spin-wave the-\nory (LSWT) [2] by using the UppASD software [24]. Ef-\nfective coupling parameters as collected in Table I were\nutilized, along with a moment as illustrated in Fig. 2a),\nfrom LSDA+U approximation using U=0.8-1.0 eV and\nJH= 0.6 eV.\nIn order to understand which interactions dominate\nthe spin-wave gap of NaOsO 3, we calculated adiabatic\nmagnon spectra by progressively adding di\u000berent energy\nterms as present in Eq. (4), see Fig. 7. It is evident\nfrom the \fgure that the uniaxial single ion anisotropy\nis the term responsible for the spin-wave gap and the\nvalue of U that best \ft the experimental spin-wave gap4\nTABLE I. Intersite interaction vectors, e\u000bective couplings,\nand DM vectors for di\u000berent Os-O-Os bond angles. The units\nof the coupling parameters are given in meV. The reference\nsystem is speci\fed in Fig. 1(b).\n\u00121= 155:2\u000eU= 0:8 eV U= 1:0 eV\nJ1 -6.64 -7.58\n\u00001 0.62 0.57\nrij (\u00060:5;\u00060:5;0) (\u00060:5;\u00060:5;0)\nD1\u0006(0:04;\u00000:92;\u00000:37)\u0006(0:08;\u00000:92;\u00000:40)\n\u00122= 153:9\u000e\nJ2 -4.12 -5.11\n\u00002 0.07 0.05\nrij (0:0;0:0;\u00060:5) (0 :0;0:0;\u00060:5)\nD2 (\u00000:50;0:15;0:00) (\u00000:41;0:07;0:00)\n\u00123=;\nJ3 0.67 0.68\n\u00003 0.18 0.18\nrij (\u00060:5;\u00060:5;\u00060:5) (\u00060:5;\u00060:5;\u00060:5)\nDa\n3\u0006(0:20;\u00000:01;\u00000:18)\u0006(0:20;\u00000:03;\u00000:18)\nDb\n3\u0006(\u00000:20;0:01;\u00000:11)\u0006(0:20;0:00;\u00000:10)\nFIG. 5. Adiabatic magnon spectrum of NaOsO 3by progres-\nsively adding energy contributions to the e\u000bective spin Hamil-\ntonian. We illustrate the cases with contributions given by J,\nJ+D+\u0000,J+\u0000+K,J+D+\u0000+K, for U=0.8 eV, J H= 0:6 eV\nfrom LSDA+U theory (top panel) and U=1.0 eV, J H= 0:6\neV from LSDA+U theory (bottom panel). Experimental data\nprovided by RIXS measurements are shown in black dots [10].\nis achieved by U=1.0 eV while for U=0.8 eV, the gap\nis slightly too low (the magnon excitations are evaluated\nfor a wider range of values of U in the SM[23]). Further-\nmore, the contribution of the DM interaction together\nwith the symmetric anisotropic exchange interaction even\nthough they are small, they are necessary to correctly de-\nscribe the full experimental spin-wave dispersion. Inter-\nestingly, the calculated pro\fle of the magnon spectra for\nU=1 eV agrees quite well with the experimental spectra,even though the exchange coupling parameters together\nwith the anisotropy constant di\u000ber substantially with re-\nspect to the ones suggested from the experimental data\nin Ref. [10]. A direct \ftting procedure of the magnon\nspectra, with a relative large number of parameters in a\nspin-Hamiltonian, can agree with experimental data but\nthe parameters of the Hamiltonian obtained from such\n\fttings, might be completely o\u000b compared to interactions\ncalculated from ab initio electronic structure theory. This\nillustrates that ab initio theory is crucial for extracting\nproper interaction types, strengths and ranges.\nThe calculated adiabatic magnon excitation energies\nare in good agreement with the experimental data re-\nported in Ref. [10], as shown in Fig. 7. However, AMS\nis formally calculated at T= 0K, whereas the experi-\nmental measurements are done at room temperature, i.e.\nT= 300K. Therefore theory and experiments refer to data\nobtained at di\u000berent temperatures. In order to shed more\nlight on this point, we show in Fig. 6 the dynamical struc-\nture factor calculated in the framework of LSWT, which\nis proportional to the di\u000berential cross section measured\nin scattering experiments. It depends on the crystal mo-\nmentum ( q) and energy transfer ( !) and is de\fned as a\n3\u00023 matrix:\nS(q;!) =1\n2\u0019NX\ni;jeiq(ri\u0000rj)Z1\n\u00001e\u0000i!thSi(t)ST\nj(t)idt;\n(5)\nwhere indices i;jlabel the atomic positions and riis the\nposition vector of atom i.Nis the number of magnetic\natoms in the unit cell and Sirepresents a column vector\nFIG. 6. Dynamical structure factor of NaOsO 3for a) U=0.8\neV, J H= 0:6 eV and at T=5K, b) for U=0.8 eV, J H=\n0:6 eV and at T=300K, c) for U=1.0 eV, J H= 0:6 eV and\nat T=5K and d) U=1.0 eV, J H= 0:6 eV and at T=300K.\nAll the aforementioned parameters are calculated at the level\nof LSDA+U theory. Energy is convoluted with a 0.7 meV\nfull width at half maximum Gaussian. Experimental data\nprovided by RIXS measurements are shown by green dots\n[10].5\nof thex,y, andzspin operator components. The sym-\nbolh:::idenotes the ensemble average in thermal equilib-\nrium with the environment at temperature T. Since we\nare dealing with magnons, i. e., bosons, Tis introduced\nin the thermodynamic average by using Bose-Einstein\nstatistics, with a Bose factor\nn(!) =1\ne~!=kBT\u00001: (6)\nIn Fig. 6(a),(c), we show S(q;!) for NaOsO 3using\nmagnetic interactions according to Eq. (4) and Table I\natT= 5 K and in Fig. 6(b),(d) the dynamical structure\nfactor is calculated for T= 300 K, the value of the tem-\nperature at which the experimental measurements have\nbeen reported in Ref. [10]. The dispersion of the curve\nin Fig. 6 is naturally the same as in Fig. 7 but additional\ninformation is obtained from the intensity, which reveals\nhow likely a scattering event is. Notably, for T=5 K only\nin the vicinity of the \u0000 point does the cross section show\na signi\fcant intensity indicating that at very low tem-\nperatures only these speci\fc modes would be captured.\nOn the other hand, at a temperature of T= 300 K, the\ntwo lowest energy branches show a considerable intensity\nalong the q-path in agreement with experimental data.\nNote here that according to Fig. 7, the \frst acoustic and\n\frst optical branches are very close in energy and thus\nboth of them are activated by increasing temperature.\nFigure 6 also reveals that at 300 K, the experimental\nconditions of the RIXS measurements, the experimental\napparatus only can capture the acoustic and \frst opti-\ncal branches while the additional third and fourth optical\nbranches remain hidden to the experiment.\nIn summary, the very rare situation in which ab initio\ncalculations of Heisenberg exchange results in .45% to\n.63% smaller values compared to those obtained through\n\ftting of experimental magnon dispersions by LSWT, has\nled us to a careful analysis of the spin excitation spectrum\nof the osmate perovskite NaOsO 3. To this end, we carried\nout atomistic spin-dynamics simulations using as input\nthe e\u000bective magnetic interactions determined from ab\ninitio calculations. These interactions involved not only\nHeisenberg but also DM and symmetric anisotropic inter-\nsite couplings, plus single-ion spin anisotropies. The ef-\nfective DM interaction parameters turn out to be sizable,\n\u001814% of the Heisenberg J1's for U=1 eV, while the single-\nion anisotropies are also robust. Our theoretical results\nreproduce with good accuracy the observations, albeit\nwith a distinct di\u000berence in that the spectra along the\nX!\u0000!Z path are slightly steeper than the measured\nspectra. Moreover, the third and fourth optical branches\ndemonstrated in the theory, are not found in the exper-\nimental measurements. We propose a mechanism based\non thermal \ructuations for why these two modes are hid-\nden from experimental detection. The analysis put forth\nhere di\u000bers from that of Ref. [10] in that our calculations\nprovide somewhat smaller magnetic anisotropy, but no-tably, a sizable DM exchange which was not suggested in\nRef. [10]. We note that the DM interaction can on gen-\neral grounds not be neglected in NaOsO 3since the O ion\nmediating superexchange between two Os NN's does not\nsit at an inversion center and SO interactions are large\nfor the 5dshell. Hence, one should anticipate that the\nDM interaction is large, which our theoretical calcula-\ntions con\frm. We have demonstrated that the single ion\nanisotropy along with DM interaction and to a less extent\nthe symmetric anisotropic exchange interaction, clearly\nhas a role in the spin-wave gap of NaOsO 3. The contri-\nbution of the DM and \u0000 interaction as the microscopic\nmechanism behind a spin-wave gap is seldom discussed,\nand NaOsO 3is a unique material in this sense.\nWhile our results for the magnon modes faithfully re-\nproduce the dispersion reported on the basis of RIXS\nexperiments [10], we also point out that in the regime of\nobserved magnons, the optical and acoustic modes are\nsometimes close in energy and in order to identify which\nmode is observed, it is important to theoretically anal-\nyse the scattering amplitudes. Furthermore, we \fnd that\na full account of all interaction parameters are needed\nin order to reproduce experimental observations of this\ncomplex compound.\nFinally, we note that it was argued in Ref. [10] that\nNaOsO 3is a system on the boundary between local-\nmoment and itinerant magnetism, based on the fact\nthat the SOC is very strong with a spin gap of 58 meV\naround the \u0000 point. Our results can explain experi-\nmental magnon spectra by assuming an e\u000bective on-site\nCoulomb repulsion of Ue\u000b=U\u0000JH= 0:4 eV which\ndrives the system out of the purely itinerant regime.\nAcknowledgements. The authors acknowledge \fnan-\ncial support from Knut and Alice Wallenberg Founda-\ntion through Grant No. 2018.0060. A.D. acknowledges\n\fnancial support from the Swedish Research Council\n(VR) through Grants No. 2015-04608, No. 2016-05980,\nand No. 2019-05304. O.E. also acknowledges support\nfrom eSSENCE, SNIC, the Swedish Research Council\n(VR) and the ERC (synergy grant FASTCORR, project\n854843). D.T. acknowledges support from the Swedish\nResearch Council (VR) through Grant No. 2019-03666.\nE.S. acknowledges \fnancial support from the Swedish\nResearch Council (VR) through Grant No. 2017-03832.\nThe work of Y.O.K. is supported by VR under the project\nNo. 2019-03569. Some of the computations were per-\nformed on resources provided by the Swedish National\nInfrastructure for Computing (SNIC) at the National Su-\npercomputer Center (NSC), Link oping University, the\nPDC Centre for High Performance Computing (PDC-\nHPC), KTH, partially funded by the Swedish Research\nCouncil through grant agreement no. 2016-07213 and the\nHigh Performance Computing Center North (HPC2N),\nUme\u0017 a University.6\nSupplementary Information\nTABLE II. Calculated values of J1,D1and \u0000 1, for LDA+U\napproximation. Only values of UandJH(in eV) are shown\nthat produce a magnetic moment per Os atom close to the\nexperimental value. All values of couplings are in meV.\nU J H J1 D1 \u00001\n3:5 0 :6\u000014:42\u00001:67 0 :54\n4:0 0 :6\u000016:87\u00001:67 0 :44\nFIG. 7. Adiabatic magnon spectrum of NaOsO 3for LDA+U\napproximation. Only U=3 :5 eV,JH= 0:6 eV and U=4 :0 eV,\nJH= 0:6 eV are shown as they produce a magnetic moment\nper atom close to the experimental one. Experimental data\nprovided by RIXS measurements are shown in black dots [10].\n\u0003Corresponding author: manuel.pereiro@physics.uu.se\n[1] G. Jackeli and G. Khaliullin, Mott Insulators in the\nStrong Spin-Orbit Coupling Limit: From Heisenberg to a\nQuantum Compass and Kitaev Models, Phys. Rev. Lett.\n102, 017205 (2009).\n[2] R. Yadav, M. Pereiro, N. A. Bogdanov, S. Nishimoto,\nA. Bergman, O. Eriksson, J. van den Brink, and L. Hozoi,\nHeavy-mass magnetic modes in pyrochlore iridates due to\ndominant Dzyaloshinskii-Moriya interaction, Phys. Rev.\nMaterials 2, 074408 (2018).\n[3] Y. Yamaji, Y. Nomura, M. Kurita, R. Arita, and\nM. Imada, First-Principles Study of the Honeycomb-\nLattice Iridates Na 2IrO 3in the Presence of Strong Spin-\nOrbit Interaction and Electron Correlations, Phys. Rev.\nLett. 113, 107201 (2014).\n[4] V. M. Katukuri, S. Nishimoto, V. Yushankhai, A. Stoy-\nanova, H. Kandpal, S. Choi, R. Coldea, I. Rousochatza-\nkis, L. Hozoi, and J. van den Brink, Kitaev interac-tions between j=1/2 moments in honeycomb Na 2IrO 3are\nlarge and ferromagnetic: insights from ab initio quantum\nchemistry calculations, New J. Phys. 16, 013056 (2014).\n[5] Y. G. Shi, Y. F. Guo, S. Yu, M. Arai, A. A. Belik,\nA. Sato, K. Yamaura, E. Takayama-Muromachi, H. F.\nTian, H. X. Yang, J. Q. Li, T. Varga, J. F. Mitchell, and\nS. Okamoto, Continuous metal-insulator transition of the\nantiferromagnetic perovskite NaOsO 3, Phys. Rev. B 80,\n161104 (2009).\n[6] S. Calder, V. O. Garlea, D. F. McMorrow, M. D. Lums-\nden, M. B. Stone, J. C. Lang, J.-W. Kim, J. A. Schlueter,\nY. G. Shi, K. Yamaura, Y. S. Sun, Y. Tsujimoto, and\nA. D. Christianson, Magnetically Driven Metal-Insulator\nTransition in NaOsO 3, Phys. Rev. Lett. 108, 257209\n(2012).\n[7] Y. Du, X. Wan, L. Sheng, J. Dong, and S. Y. Savrasov,\nElectronic structure and magnetic properties of NaOsO 3,\nPhys. Rev. B 85, 174424 (2012).\n[8] M.-C. Jung, Y.-J. Song, K.-W. Lee, and W. E. Pickett,\nStructural and correlation e\u000bects in the itinerant insulat-\ning antiferromagnetic perovskite NaOsO 3, Phys. Rev. B\n87, 115119 (2013).\n[9] I. L. Vecchio, A. Perucchi, P. Di Pietro, O. Limaj,\nU. Schade, Y. Sun, M. Arai, K. Yamaura, and S. Lupi,\nInfrared evidence of a Slater metal-insulator transition in\nNaOsO 3, Sci. Rep. 3(2013).\n[10] S. Calder, J. G. Vale, N. Bogdanov, C. Donnerer,\nD. Pincini, M. Moretti Sala, X. Liu, M. H. Upton,\nD. Casa, Y. G. Shi, Y. Tsujimoto, K. Yamaura, J. P. Hill,\nJ. van den Brink, D. F. McMorrow, and A. D. Christian-\nson, Strongly gapped spin-wave excitation in the insulat-\ning phase of NaOsO 3, Phys. Rev. B 95, 020413 (2017).\n[11] P. Hohenberg and W. Kohn, Inhomogeneous Electron\nGas, Phys. Rev. 136, B864 (1964).\n[12] W. Kohn and L. J. Sham, Self-Consistent Equations In-\ncluding Exchange and Correlation E\u000bects, Phys. Rev.\n140, A1133 (1965).\n[13] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized\nGradient Approximation Made Simple, Phys. Rev. Lett.\n77, 3865 (1996).\n[14] J. M. Wills and B. R. Cooper, Synthesis of band and\nmodel Hamiltonian theory for hybridizing cerium sys-\ntems, Phys. Rev. B 36, 3809 (1987).\n[15] J. Wills, O. Eriksson, M. Alouani, and D. Price, Full-\nPotential LMTO Total Energy and Force Calculations\nin Electronic structure and physical properties of solids ,\nSpringer-Verlag Berlin Heidelberg (2000).\n[16] J. Wills, M. Alouani, P. Andersson, A. Delin, O. Eriks-\nson, and O. Grechnyev, Full-Potential Electronic Struc-\nture Method , volume 167, Springer-Verlag Berlin Heidel-\nberg (2010).\n[17] Y. O. Kvashnin, A. Bergman, A. I. Lichtenstein, and\nM. I. Katsnelson, Relativistic exchange interactions in\nCrX3(X= Cl, Br, I) monolayers, Phys. Rev. B 102,\n115162 (2020).\n[18] A. Liechtenstein, M. Katsnelson, V. Antropov, and\nV. Gubanov, Local spin density functional approach to\nthe theory of exchange interactions in ferromagnetic met-\nals and alloys, Journal of Magnetism and Magnetic Ma-\nterials 67, 65 (1987).\n[19] V. Borisov, Y. O. Kvashnin, N. Ntallis, D. Thonig,\nP. Thunstr om, M. Pereiro, A. Bergman, E. Sj oqvist,\nA. Delin, L. Nordstr om, and O. Eriksson, Theory of\nHeisenberg and anisotropic exchange interactions in mag-7\nnetic materials with correlated electronic structure and\nsigni\fcant spin-orbit coupling, arXiv:2011.08209 (2020).\n[20] S. Middey, S. Debnath, P. Mahadevan, and D. D. Sarma,\nNaOsO 3: A high Neel temperature 5 doxide, Phys. Rev.\nB89, 134416 (2014).\n[21] S. Mohapatra, C. Bhandari, S. Satpathy, and A. Singh,\nE\u000bect of structural distortion on the electronic band\nstructure of NaOsO 3studied within density functional\ntheory and a three-orbital model, Phys. Rev. B 97,\n155154 (2018).\n[22] P. Liu, J. He, B. Kim, S. Khmelevskyi, A. Toschi,\nG. Kresse, and C. Franchini, Comparative ab initio study\nof the structural, electronic, magnetic, and dynamical\nproperties of LiOsO 3and NaOsO 3, Phys. Rev. Materials\n4, 045001 (2020).\n[23] See Supplemental Material at [URL] for further details.\n[24] B. Skubic, J. Hellsvik, L. Nordstr om, and O. Eriksson, A\nmethod for atomistic spin dynamics simulations: imple-\nmentation and examples, J. Phys.: Condens. Matter 20,\n315203 (2008)."    },    {        "title": "2105.09273v1.Experimental_Evidence_of_a_change_of_Exchange_Anisotropy_Sign_with_Temperature_in_Zn_Substituted_Cu2OSeO3.pdf",        "content": "Experimental Evidence of a change of Exchange Anisotropy Sign with Temperature in\nZn-Substituted Cu 2OSeO 3\nS. H. Moody,1,\u0003P. Nielsen,1M. N. Wilson,1D. Alba Venero,2A.\u0014Stefan\u0014 ci\u0014 c,3G. Balakrishnan,3and P. D. Hatton1\n1Durham University, Department of Physics, South Road, Durham, DH1 3LE, United Kingdom\n2ISIS Neutron and Muon Source, Rutherford Appleton Laboratory, Didcot OX11 0QX, UK.\n3University of Warwick, Department of Physics, Coventry, CV4 7AL, United Kingdom\n(Dated: April 20, 2022)\nWe report small-angle neutron scattering from the conical state in a single crystal of Zn-substituted\nCu2OSeO 3. Using a 3D vector-\feld magnet to reorient the conical wavevector, our measurements\nshow that the magnitude of the conical wavevector changes as a function of crystallographic direc-\ntion. These changes are caused by the anisotropic exchange interaction (AEI), whose magnitude\ntransitions from a maxima to a minima along the h1 1 1iandh1 0 0icrystallographic directions respec-\ntively. We further \fnd that the AEI constant undergoes a change of sign from positive to negative\nwith decreasing temperature. Unlike in the related compound FeGe, where similar behaviour of the\nAEI induces a reorientation of the helical wavevector, we show that the zero \feld helical wavevec-\ntor in (Cu 0.98Zn0.02)2OSeO 3remains along the h1 0 0idirections at all temperatures due to the\ncompeting fourth-order magnetocrystalline anisotropy becoming dominant at lower temperatures.\nThe lack of inversion symmetry in the family of\nB20 chiral cubic materials allows a non-vanishing\nDzyaloshinskii-Moriya interaction (DMI), which com-\npetes with numerous other magnetic interactions to sta-\nbilize a wide variety of incommensurate magnetic states\n[1{8]. At zero \feld and at temperatures below TC,\nthe isotropic exchange interaction and DMI (which pre-\nfer spin-aligned with constant Aand spin-perpendicular\nwith constant Drespectively), twist the magnetic texture\ninto a helix with a periodicity determined by the ratio of\nthe strengths of the two interactions, q\u0019D=A. Typ-\nically,qis close to 0.1 nm\u00001, suitable for measurement\nwith magnetic small angle neutron scattering (SANS) ex-\nperiments. The non-centrosymmetric crystal structure\nallows for non-zero o\u000b-diagonal terms in the exchange\ntensor [9], which gives rise to the so-called anisotropic\nexchange interaction (AEI) with a strength \r, where\nj\rj<jAj. In MnSi, the helical wavevector was shown\nto align along the h1 1 1idirections [10] due to a negative\nAEI constant [11{14]. \ris also known to be temperature\ndependent, with a change of sign with temperature being\nresponsible for the helical wavevector reorientation in the\nrelated material, FeGe [15, 16].\nThe orientation of these incommensurate textures can\nalso be manipulated by the application of a magnetic\n\feld, which induces a transition from the multi-domain\nmagnetic helix into a single-domain magnetic cone state,\nwhose wavevector typically follows the direction of the\napplied magnetic \feld [10]. As shown in Fig. 1a, in\nthe cone state the spins cant towards the \feld direction\nto minimize the Zeeman interaction. Applying a higher\n\feld reduces the conical angle \u0012, but maintains a constant\nwavevector, q, until the cone angle reaches zero to form\na forced ferromagnetic state [17].\nThe behaviour of chiral magnets adopting a B20 struc-\nture materials near TCis further enriched by thermal\n\ructuations, which induce a hexagonal lattice of mag-netic skyrmions within small region at non-zero \feld.\nMagnetic skyrmions are vortex-like whirls of magnetiza-\ntion that are the 2D solitonic solutions to systems host-\ning a number of competing magnetic interactions [1, 18{\n22]. Their particle-like nature and topological proper-\nties make magnetic skyrmions appealing for applications\nwithin a variety of spintronic devices, ranging from race-\ntrack memory schemes and logic devices, to stochastic\nand neuromorphic computing [23{29].\nThe e\u000bects of anisotropic interactions such as magne-\ntocrystalline anisotropy (MCA) and the AEI on these\nmagnetic textures are typically overlooked, as the in-\nherent cubic symmetry of the B20 materials limits the\nstrength of these energy terms [30]. Despite this,\nanisotropy-stabilised magnetic textures within the mul-\ntiferroic B20 material Cu 2OSeO 3have been found to ex-\nist at low temperatures, such as the low-temperature\nskyrmion (LTS) [31{34] and tilted conical (TC) phase,\nwhere the conical wavevector deviates from the direction\nof the magnetic \feld applied along a [1 0 0], with qcant-\ning towards theh1 1 1i[35]. The forms of the MCA and\nAEI for a magnetic cone with angle \u0012and wavevector-\nangle\u001eare shown in Fig. 1b. In (ii), the local minima\nalong theh1 1 1idirections for K > 0 explain the ori-\ngin of the TC, but an additional AEI shown in (i) is\nessential to prevent the texture collapsing into a \feld-\npolarized state. Currently, the e\u000bects of the AEI have\nnot been investigated on the LTS, and decoupling the\ne\u000bects of MCA and the AEI remains a challenge as the\nconventional method of torque magnetometry only inves-\ntigates the \feld-polarized state where the AEI vanishes\n[36]. Distinguishing and quantifying these anisotropic\ninteractions is therefore important to better understand\nthe low-temperature behaviour of Cu 2OSeO 3.\nIn this letter, we utilize a 3D vector-\feld magnet to-\ngether with small-angle neutron scattering (SANS) to\nshow that the AEI changes sign from positive to negativearXiv:2105.09273v1  [cond-mat.str-el]  19 May 20212\n                                 \n−90−45045 90\nφ(◦)04590\nθ(◦)\n−90−45045 90\nφ(◦)04590θ(◦)\n0.00 0.17 0.33\n 0.33 0.66 1.00\n(a)[1 1 0]\nq\n(b)\n(i) (ii)\n(c)E=\r E=K\nn0H\n[0 01][11 1][1 1 0]\nFIG. 1: a) One period of the magnetic conical state with\nwavevector, q, where localized moments cant towards qby\nthe angle\u0012. The direction of qis given by the polar\ncoordinate \u001e, with\u001e= 0;55;90\u000ebeing the [1 1 0], [1 1 1] and\n[0 01] directions respectively. b) AEI(i) and MCA(ii) energy\nlandscapes of a conical state normalised by the constants \r\nandK. c) SANS scattering geometry with 3D vector\nmagnet. Incoming neutrons ( n0) parallel to the [1 1 0]\ndirection, with the applied magnetic \feld perpendicular to\nn0within the scattering plane (Blue Circle).\nwith decreasing temperature in (Cu 0.98Zn0.02)2OSeO 3.\nUnlike in the related compound FeGe, this change of\nsign does not induce a reorientation of the zero-\feld mag-\nnetic helix. This demonstrates that the 4thorder MCA\nalso increases in magnitude with decreasing temperature,\ngreater that the AEI in order to stabilize the zero-\feld\nmagnetic helices along the h1 0 0idirections at all tem-\nperatures [37]. Our \fndings provide insight into the low\ntemperature behavior of Cu 2OSeO 3that will be essential\nfor complete understanding of the creation and stabiliza-\ntion mechanisms of the recently discovered LTS and TC\nphases within pristine Cu 2OSeO 3.\nA 20.5 mg single crystal of (Cu 0.98Zn0.02)2OSeO 3was\ngrown at the University of Warwick utilizing the chemi-\ncal vapour transport technique, see [38] for details. For\nthe SANS experiment, the LARMOR instrument at the\nISIS Pulsed Neutron and Muon Source was used. The\nsample was aligned with an x-ray Laue camera (Multi-wire Laboratories) such that the [1 1 0] direction was ver-\ntical in the laboratory frame and the [1 1 0] direction was\nparallel with the incident neutron beam. This orienta-\ntion allows the magnetic textures whose periodic compo-\nnents lie within a plane spanned by the all 3 high cubic-\nsymmetry directions ( h1 0 0i,h1 1 0i, andh1 1 1i) to simul-\ntaneously satisfy the Bragg condition [39], see Fig. 1c.\nA zinc-substituted sample was chosen for our study due\nto the reduction of the exchange interaction compared\nto pristine Cu 2OSeO 3, resulting in a greater value of q\n[37]. Previously, we have shown strong similarities of\nthe magnetic behavior between Zn-substituted and pris-\ntine samples [37], and hence we expect our results here\nwill extend to pristine samples. The time-of-\right neu-\ntron di\u000braction data was reduced with Mantid [40], using\nscattered neutron wavelengths between 0.9 and 13.5 /RingA.\nTo observe the e\u000bects of the anisotropic interactions on\nthe magnetic textures within Cu 2OSeO 3, we performed\n\feld scans by rotating the direction of the \feld at a con-\nstant magnitude. Initially, the \feld was applied vertically\nin the laboratory frame after zero-\feld cooling. The \feld\norientation was then rotated 90 degrees in 46 steps about\nan axis parallel with the neutron beam, such that the\n[11 0], [1 1 1], [0 0 1] directions were parallel with the \feld\nat angles of 0, 35.3 and 90 degrees respectively. This\nprocedure was performed at 5, 12 and 50 K at magnetic\n\feld magnitudes of 70, 60 and 40 mT respectively. These\n\felds were chosen to avoid phase coexistence between the\nmagnetic conical and helical states, see Supplementary\nMaterial [URL will be inserted by publisher]. A selection\nof frames from the \feld scans are shown in Fig. 2a.\nIn the \frst frame of the 50 K dataset, Fig. 2a(i),\nthe vertically applied magnetic \feld induces a magnetic\nconical state with wavevector along the [1 1 0] direction,\nwhich we detect as a single pair of vertical peaks with\nq\u00190:011/RingA\u00001, corresponding to a real space length of\n57 nm. Upon rotating the \feld, Fig. 2a(ii-v), the conical\nwavevector rotates in an attempt to follow the direction\nof the magnetic \feld. This behaviour is expected in the\nlimit in which the AEI and MCA are much smaller than\nthe isotropic Dzyaloshinskii-Moriya and exchange inter-\nactions (\r;K << A;D ), allowing the conical wavevec-\ntor to rotate freely to minimize the Zeeman interaction.\nHowever, this behaviour is not replicated within the data\ntaken at lower temperatures in Fig. 2b(i-v), where an-\ngle of the conical wavevector ( \u001e) lags behind the mag-\nnetic \feld angle ( H\u001e), particularly after the \feld passes\nthrough the [1 1 1] direction. This o\u000bset between the \feld\ndirection and qincreases in magnitude with further rota-\ntion of the magnetic \feld, up to a maxima just before the\n\feld is applied along the [0 0 1] direction. This magnetic\nstate is characteristic of the tilted conical state seen pre-\nviously in pristine Cu 2OSeO 3[35], and shows that while\nisotropic interactions dominate at temperatures near TC,\nthe anisotropic interactions become increasingly impor-\ntant at low temperatures and prevent free rotation of the3\n−0.01 0.01qx−0.010.000.01qy\n0 200 400 600 800\n0 350 700 1050 1400\n(a)\n(b)I50K(a.u)\nI5K(a.u)50 K\n5 K40 mT\n70 mT(i) (ii) (iii) (iv) (v)\n(i) (ii) (iii) (iv) (v)[1 1 0] [1 1 1]\n[0 01]\nFIG. 2: Selection of reduced SANS patterns for \feld rotations at 40 mT, 50 K(a) and 70 mT, 5 K(b). The \feld angles for\neach frame are 0, 22, 44, 68 and 90\u000efor frames (i-v) and are shown by the thick dashed lines. The thin-dashed circle gives the\nlocation of equidistant q= 0:011/RingA\u00001. Crystallographic directions are shown by the dotted lines in a(i).\nmagnetic conical state.\nIn order to quantify the e\u000bects of the anisotropic inter-\nactions, the conical peaks collected during the \feld scans\nshown in Fig. 2 and an additional set at 12 K were \ft\nusing two dimensional Gaussian functions in polar coor-\ndinates, allowing the extraction of the magnetic wavevec-\ntor angle,\u001e, and magnitude, q. The results of this \ftting\nfor \feld angle scans at the three di\u000berent temperatures\nare shown in Fig. 3, with qnormalized to the value at\nH\u001e= 90 degrees ( q0) for each dataset.\nThe behaviour of qas a function of \feld angle is shown\n0.951.001.05q/q0(a.u)\n0 15 30 45 60 75 90\nHφ(◦)04590φ(◦)\n(a)\n(b)\n[1 1 0] [1 1 1][0 01]\nFIG. 3: a) Normalised wavevector magnitude, and\nwavevector angle (b), \u001ein Fig. 1a, of the conical state as a\nfunction of applied \feld angle, H\u001e. Dark blue, light blue and\norange indicate temperatures of 5, 12, and 50 K respectively.\nDashed line shows [1 1 1] direction. Dotted line shows H\u001e.in Fig. 3a. A clear di\u000berence between the high tempera-\nture and low temperature regimes is observed. In the 50\nK datasetqinitially decreases slightly as the applied \feld\nis rotated from the [1 1 0] to [1 1 1] direction and then in-\ncreases as the \feld is rotated further, up to a maximum\nwhen H is along [0 0 1]. This behaviour is reversed at low\ntemperatures (both at 12 K and 5 K), where the magni-\ntude of the conical wavevector instead increases to a max-\nimum when the \feld is rotated from [1 1 0] to [1 1 1], be-\nfore quickly decreasing as the \feld angle approaches the\n[0 01] direction. In Fig. 3b, the e\u000bects of the anisotropic\ninteractions can also clearly be seen to be more signi\f-\ncant in the low-temperature datasets, as in them \u001edevi-\nates substantially from linearity after the magnetic \feld\npasses through [1 1 1], as compared to the 50 K dataset\nwhere\u001edeviates only slightly from the expected linear\ntrend in the absence of anisotropic interactions.\nThe dependence of qonH\u001eallows us to decouple the\ntwo dominant anisotropic interactions. We start by using\nthe non-constant terms within the free energy expansion\nderived by Bak and Jensen[11], which is valid for sys-\ntems ofP213 crystal symmetry that host slowly-varying\nmagnetization densities m(r)[41]:\nF(r) =Dm(r)\u0001(r\u0002 m(r))\n+1\n2A[(rmx)2+ (rmy)2+ (rmz)2]\n+1\n2\r[(@mx\n@x)2+ (@my\n@y)2+ (@mz\n@z)2]\n+K(m4\nx+m4\ny+m4\nz)(1)\nWhereD;A are the familiar Dzyaloshinskii-Moriya\nand exchange sti\u000bness, and K;\r are the 4thorder MCA4\nand AEI constants respectively. In the case of D= 0,\nandA;\r > 0, the spin-texture reduces to a simple ferro-\nmagnet, with the spin direction given by the sign of the\nMCA constant K. We approximate a model spin-texture\nusing a conical ansatz:\nm(r)=Ms= sin\u0012(cos( q\u0001r)^ e1+sin( q\u0001r)^ e2)+cos\u0012^ e3(2)\nwhere we follow the convention that \u0012is the conical\nangle, andf^ engde\fne three mutually orthogonal ba-\nsis vectors, with ^ e3kq. To compare with our ex-\nperimental observations, we consider a conical texture\nwith the wavevector restricted to the plane spanned\nby the three high cubic-symmetry directions, such that\nq= (qp\n2sin\u001e;qp\n2sin\u001e;qcos\u001e). Using this form for the\nmagnetic wavevector and substituting (2) into (1), in-\ntegrating over one conical period, \u0015, and di\u000berentiating\nwith respect to qwe \fnd a single stable solution:\n@\n@qF(q;\u001e) = sin2\u0012[D+Aq+1\n2\rqsin2\u001e(3 cos2\u001e+ 1)]\nq=jDj\nA+1\n2\rsin2\u001e(3 cos2\u001e+ 1)\n(3)\nThe implications of this result can be seen in Fig. 4(a-\nc), which show the experimentally measured values of\nthe conical wavevector qfor di\u000berent conical wavevec-\ntor angles, \u001e, at 50, 12 and 5 K respectively. At 50\nK, the increased wavevector at [0 0 1] compared to [1 1 1]\nis a clear indication of a positive exchange anisotropy\nconstant\r, where energy costs for directions other than\ntheh1 0 0i, are compensated by a shrinking of the coni-\ncal wavevector. This behaviour reverses with decreasing\ntemperature, shown in Fig. 4b and c, where the wavevec-\ntor maxima are located at the [1 1 1] and minima towards\nthe [0 0 1], suggesting a change of sign of \r.\nUsing the experimentally determined value for the ex-\nchange sti\u000bness, A= 4:4\u000210\u000013J=m fromTC= 57\nK [38, 42], and \ftting (3) to the data in Fig. 4, we \fnd\n\r= 2:1(2)\u000210\u000014J=m at 50 K,\u00003:4(4)\u000210\u000014at 12 K,\nand\u00006:7(3)\u000210\u000014J=m at 5 K. The low-temperature\nvalues are consistent with theoretical values of \rrequired\nfor TC formation [35], in agreement with the TC states\nobserved in this study.\nDuring the re\fnement of the low temperature datasets,\nonly angles above 55 degrees were used. This is because\nat low temperatures, the direction of the conical wavevec-\ntor deviates signi\fcantly from the magnetic \feld direc-\ntion for\u001e < 55 degrees, inducing a component of the\nmagnetic \feld perpendicular to the conical wavevector.\nApplying transverse \felds to a helical state with a pinned\nqdirection is known to deform the helix [43], introducing\nhigher-order components to the Fourier transform of the\nmagnetic state. The presence of higher order peaks in\n0 20 40 60 80\nφ(◦)0.01120.01140.0116q(˚A−1)\n40 50 60 70 80 90\nφ(◦)0.01040.01060.01080.0110q(˚A−1)\n40 50 60 70 80 90\nφ(◦)0.01040.01060.01080.0110q(˚A−1)\n−0.02 0.00 0.02\nqx(˚A−1)−0.020.000.02qy(˚A−1)\n101102103(a)\n(b)\n(c)\n(d)[1 1 0]\n[1 11][0 01]\n\r= 2:1(2)\u000210\u000014J=m\n\r=\u00003:4(4)\u000210\u000014J=m\n\r=\u00006:7(3)\u000210\u000014J=m\nFIG. 4: Values of the extracted wavevector magnitude ( q)\nagainst wavevector angle ( \u001e) with (a-c) being the 5, 12 and\n50 K datasets respectively. Fits to Eq. 4 are shown by the\nbold lines, with dashed lines showing data ignored in the\n\ftting procedure due to the appearance of higher-order\npeaks, shown in (d). Values of \rshown in panels using a\nvalue ofA= 4:4\u000210\u000013J=m.5\nour SANS pattern shown in Fig. 4d suggest that similar\ndeformations are occurring in our tilted conical state. For\nhelices, these deformations are known to reduce the value\nofqas the perpendicular \feld strength increases, which\nwould be consistent with our observations of a large q\ndiscernibly where the higher order peaks are prominent.\nFully accounting for these deformations within our model\nspin-texture remains an avenue intended for future study.\nAs shown in Fig. 1b(i), in the absence of cubic\nanisotropy, a change of sign of the AEI from positive to\nnegative with decreasing temperature would induce a he-\nlical reorientation from the h1 0 0itoh1 1 1idirections. In\norder for the orientation of the helical wavevector to re-\nmain aligned along the h1 0 0idirections(consistent with\nexperimental observations), we \fnd that Kmust be neg-\native, with the condition jKj>4\n3j\rjq2. This requires that\nthe magnitude of Kincreases with decreasing tempera-\ntures. This increase in jKjat low temperatures is also\nin agreement with previous work discussing the origin of\nthe low temperature skyrmion state [33].\nIn conclusion, we performed SANS on a single crystal\nof (Cu 0.98Zn0.02)2OSeO 3, using a novel 3D vector mag-\nnet to decouple the anisotropic interactions within the\nmaterial by investigating the behaviour of the magnetic\nconical state with a rotating magnetic \feld of constant\nmagnitude. We observed a change of behaviour of the\nmagnitude of the conical wavevector, q, as a function of\nwavevector angle, \u001e, whereby the crystal directions cor-\nresponding to a maximum or minimum in qreversed with\ncooling from 50 K to 12 K. We have explain this using\na mean-\feld model and \fnd that the AEI changes sign\nbetween these temperatures, with \ftted values consistent\nwith those required for tilted conical state formation. Un-\nlike in the related compound FeGe, helical reorientation\ndoes not occur within Cu 2OSeO 3with the change in sign\nof the AEI. This is due to an increasingly negative cu-\nbic anisotropy, which increases in magnitude faster than\nthe AEI, causing the helical wavevectors to remain along\ntheh1 0 0idirections. We believe our \fnding that the\nAEI changes sign at similar temperatures to the occur-\nrence of the LTS and TC magnetic phases will be highly\nuseful for understanding the formation and stabilisation\nof these newly discovered magnetic textures within this\nmaterial.\nACKNOWLEDGEMENTS\nThis work was supported by the UK Skyrmion Project\nEPSRC Programme Grant (No. EP/N032128/1). The\nSANS experiment at the ISIS Pulsed Neutron and Muon\nSource was supported by a beamtime allocation from the\nScience and Technology Facilities Council, Proposal No.\n1920501. M.N.W. acknowledges the support of the Natu-\nral Sciences and Engineering Research Council of Canada\n(NSERC).\u0003samuel.h.moody@durham.ac.uk\n[1] T. Lancaster, Skyrmions in magnetic materials, Contem-\nporary Physics 60, 246 (2019).\n[2] I. E. Dzyaloshinskii, Theory of helicoidal structures in an-\ntiferromagnets, J. Exptl. Theoret. Phys. 46, 1420 (1964).\n[3] A. Neubauer, C. P\reiderer, B. Binz, A. Rosch, R. Ritz,\nP. G. Niklowitz, and P. B oni, Topological Hall e\u000bect\nin the A phase of MnSi, Phys. Rev. Lett. 102, 186602\n(2009).\n[4] T. Tanigaki, K. Shibata, N. Kanazawa, X. Yu, Y. Onose,\nH. S. Park, D. Shindo, and Y. Tokura, Real-space obser-\nvation of short-period cubic lattice of skyrmions in mnge,\nNano Letters 15, 5438 (2015).\n[5] H. Wilhelm, M. Baenitz, M. Schmidt, U. K. R o\u0019ler, A. A.\nLeonov, and A. N. Bogdanov, Precursor phenomena at\nthe magnetic ordering of the cubic helimagnet FeGe,\nPhys. Rev. Lett. 107, 127203 (2011).\n[6] X. Z. Yu, N. Kanazawa, Y. Onose, K. Kimoto, W. Z.\nZhang, S. Ishiwata, Y. Matsui, and Y. Tokura, Near\nroom-temperature formation of a skyrmion crystal in\nthin-\flms of the helimagnet FeGe, Nature Materials 10,\n106 (2011).\n[7] S. Seki, X. Z. Yu, S. Ishiwata, and Y. Tokura, Observa-\ntion of skyrmions in a multiferroic material, Science 336,\n198 (2012).\n[8] Y. Tokunaga, X. Z. Yu, J. S. White, H. M. R\u001cnnow,\nD. Morikawa, Y. Taguchi, and Y. Tokura, A new class\nof chiral materials hosting magnetic skyrmions beyond\nroom temperature, Nature Communications 6, 7638\n(2015).\n[9] S. Blundell, Magnetism in Condensed Matter (Oxford\nUniversity Press, Oxford, 2001).\n[10] Y. Ishikawa, K. Tajima, D. Bloch, and M. Roth, Helical\nspin structure in manganese silicide MnSi, Solid State\nCommunications 19, 525 (1976).\n[11] P. Bak and M. H. Jensen, Theory of helical magnetic\nstructures and phase transitions in MnSi and FeGe, Jour-\nnal of Physics C Solid State Physics 13, L881 (1980).\n[12] O. Nakanishi, A. Yanase, A. Hasegawa, and M. Kataoka,\nThe origin of the helical spin density wave in MnSi, Solid\nState Communications 35, 995 (1980).\n[13] S. V. Maleyev, Cubic magnets with Dzyaloshinskii-\nMoriya interaction at low temperature, Phys. Rev. B 73,\n174402 (2006).\n[14] V. Ukleev, O. Utesov, L. Yu, C. Luo, K. Chen, F. Radu,\nY. Yamasaki, N. Kanazawa, Y. Tokura, T.-h. Arima, and\nJ. S. White, Signature of anisotropic exchange interaction\nrevealed by vector-\feld control of the helical order in a\nFeGe thin plate, Phys. Rev. Research 3, 013094 (2021).\n[15] B. Lebech, J. Bernhard, and T. Freltoft, Magnetic struc-\ntures of cubic FeGe studied by small-angle neutron scat-\ntering, Journal of Physics: Condensed Matter 1, 6105\n(1989).\n[16] M. L. Plumer, Wavevector and spin-\rop transitions in\ncubic FeGe, Journal of Physics: Condensed Matter 2,\n7503 (1990).\n[17] Y. Ishikawa, T. Komatsubara, and D. Bloch, Magnetic\nphase diagram of MnSi, Physica B 86-88 , 401 (1977).\n[18] K. Ishimoto, H. Yamauchi, Y. Yamaguchi, J. Suzuki,\nM. Arai, M. Furusaka, and Y. Endoh, Anomalous region\nin the magnetic phase diagram of (Fe,Co)Si, Journal of6\nMagnetism and Magnetic Materials 90-91 , 163 (1990).\n[19] S. M uhlbauer, B. Binz, F. Jonietz, C. P\reiderer,\nA. Rosch, A. Neubauer, R. Georgii, and P. B oni,\nSkyrmion lattice in a chiral magnet, Science 323, 915\n(2009).\n[20] U. K. R o\u0019ler, A. N. Bogdanov, and C. P\reiderer, Sponta-\nneous skyrmion ground states in magnetic metals, Nature\n442, 797 (2006).\n[21] M. Vousden, M. Albert, M. Beg, M.-A. Bisotti, R. Carey,\nD. Chernyshenko, D. Cort\u0013 es-Ortu~ no, W. Wang, O. Hov-\norka, C. H. Marrows, and H. Fangohr, Skyrmions in thin\n\flms with easy-plane magnetocrystalline anisotropy, Ap-\nplied Physics Letters 108, 132406 (2016).\n[22] A. N. Bogdanov and C. Panagopoulos, Physical founda-\ntions and basic properties of magnetic skyrmions, Nature\nReviews Physics 2, 492 (2020).\n[23] X. Zhang, M. Ezawa, and Y. Zhou, Magnetic skyrmion\nlogic gates: conversion, duplication and merging of\nskyrmions, Scienti\fc Reports 5, 9400 (2015).\n[24] A. Fert, N. Reyren, and V. Cros, Magnetic skyrmions:\nadvances in physics and potential applications, Nature\nReviews Materials 2, 17031 (2017).\n[25] S. L. Zhang, W. W. Wang, D. M. Burn, H. Peng,\nH. Berger, A. Bauer, C. P\reiderer, G. van der Laan,\nand T. Hesjedal, Manipulation of skyrmion motion by\nmagnetic \feld gradients, Nature Communications 9, 2115\n(2018).\n[26] A. F. Sch a\u000ber, L. R\u0013 ozsa, J. Berakdar, E. Y. Vedmedenko,\nand R. Wiesendanger, Stochastic dynamics and pattern\nformation of geometrically con\fned skyrmions, Commu-\nnications Physics 2, 72 (2019).\n[27] H. Zhang, D. Zhu, W. Kang, Y. Zhang, and W. Zhao,\nStochastic computing implemented by skyrmionic logic\ndevices, Phys. Rev. Applied 13, 054049 (2020).\n[28] S. Li, W. Kang, Y. Huang, X. Zhang, Y. Zhou, and\nW. Zhao, Magnetic skyrmion-based arti\fcial neuron de-\nvice, Nanotechnology 28, 31LT01 (2017).\n[29] K. M. Song, J.-S. Jeong, B. Pan, X. Zhang, J. Xia,\nS. Cha, T.-E. Park, K. Kim, S. Finizio, J. Raabe,\nJ. Chang, Y. Zhou, W. Zhao, W. Kang, H. Ju, and\nS. Woo, Skyrmion-based arti\fcial synapses for neuromor-\nphic computing, Nature Electronics 3, 148 (2020).\n[30] W. Sucksmith and J. E. Thompson, The magnetic\nanisotropy of cobalt, Proceedings of the Royal Society of\nLondon. Series A. Mathematical and Physical Sciences\n225, 362 (1954).\n[31] A. Chacon, L. Heinen, M. Halder, A. Bauer, W. Simeth,\nS. M uhlbauer, H. Berger, M. Garst, A. Rosch, and\nC. P\reiderer, Observation of two independent skyrmion\nphases in a chiral magnetic material, Nature Physics 14,\n936 (2018).\n[32] L. J. Bannenberg, R. Sadykov, R. M. Dalgliesh, C. Good-\nway, D. L. Schlagel, T. A. Lograsso, P. Falus, E. Leli\u0012 evre-\nBerna, A. O. Leonov, and C. Pappas, Skyrmions and\nspirals in MnSi under hydrostatic pressure, Phys. Rev. B\n100, 054447 (2019).[33] M. Halder, A. Chacon, A. Bauer, W. Simeth,\nS. M uhlbauer, H. Berger, L. Heinen, M. Garst, A. Rosch,\nand C. P\reiderer, Thermodynamic evidence of a sec-\nond skyrmion lattice phase and tilted conical phase in\nCu2OSeO 3, Phys. Rev. B 98, 144429 (2018).\n[34] A. Aqeel, J. Sahliger, T. Taniguchi, S. M andl, D. Mettus,\nH. Berger, A. Bauer, M. Garst, C. P\reiderer, and C. H.\nBack, Microwave spectroscopy of the low-temperature\nskyrmion state in Cu 2OSeO 3, Phys. Rev. Lett. 126,\n017202 (2021).\n[35] F. Qian, L. J. Bannenberg, H. Wilhelm, G. Chabous-\nsant, L. M. Debeer-Schmitt, M. P. Schmidt, A. Aqeel,\nT. T. M. Palstra, E. Br uck, A. J. E. Lefering, C. Pappas,\nM. Mostovoy, and A. O. Leonov, New magnetic phase\nof the chiral skyrmion material Cu 2OSeO 3, Science Ad-\nvances 4(2018).\n[36] M. Prei\u0019inger, K. Karube, D. Ehlers, B. Szigeti, H. A. K.\nvon Nidda, J. S. White, V. Ukleev, H. M. R\u001cnnow,\nY. Tokunaga, A. Kikkawa, Y. Tokura, Y. Taguchi, and\nI. K\u0013 ezsm\u0013 arki, Vital role of anisotropy in cubic chiral\nskyrmion hosts, (2020), arXiv:2011.05967 [cond-mat.str-\nel].\n[37] M. T. Birch, S. H. Moody, M. N. Wilson, M. Crisanti,\nO. Bewley, A. \u0014Stefan\u0014 ci\u0014 c, G. Balakrishnan, R. Fan,\nP. Steadman, D. Alba Venero, R. Cubitt, and P. D. Hat-\nton, Anisotropy-induced depinning in the Zn-substituted\nskyrmion host Cu 2OSeO 3, Phys. Rev. B 102, 104424\n(2020).\n[38] A. \u0014Stefan\u0014 ci\u0014 c, S. H. Moody, T. J. Hicken, M. T. Birch,\nG. Balakrishnan, S. A. Barnett, M. Crisanti, J. S. O.\nEvans, S. J. R. Holt, K. J. A. Franke, P. D. Hatton, B. M.\nHuddart, M. R. Lees, F. L. Pratt, C. C. Tang, M. N. Wil-\nson, F. Xiao, and T. Lancaster, Origin of skyrmion lattice\nphase splitting in Zn-substituted Cu 2OSeO 3, Phys. Rev.\nMaterials 2, 111402 (2018).\n[39] S. M uhlbauer, D. Honecker, E. A. P\u0013 erigo, F. Bergner,\nS. Disch, A. Heinemann, S. Erokhin, D. Berkov,\nC. Leighton, M. R. Eskildsen, and A. Michels, Mag-\nnetic small-angle neutron scattering, Rev. Mod. Phys.\n91, 015004 (2019).\n[40] O. Arnold, J. Bilheux, J. Borreguero, A. Buts, S. Camp-\nbell, L. Chapon, M. Doucet, N. Draper, R. Ferraz\nLeal, M. Gigg, V. Lynch, A. Markvardsen, D. Mikkel-\nson, R. Mikkelson, R. Miller, K. Palmen, P. Parker,\nG. Passos, T. Perring, P. Peterson, S. Ren, M. Reuter,\nA. Savici, J. Taylor, R. Taylor, R. Tolchenov, W. Zhou,\nand J. Zikovsky, Mantid|data analysis and visualization\npackage for neutron scattering and \u0016SR experiments, Nu-\nclear Instruments and Methods in Physics Research Sec-\ntion A: Accelerators, Spectrometers, Detectors and As-\nsociated Equipment 764, 156 (2014).\n[41] L. D. Landau and E. M. Lifshitz, Statistical Physics , 2nd\ned., 1, Vol. 9 (Pergamon Press, Pergamon Press Ltd.,\nHeadington Hill Hall, Oxford OX3 0BW, England, 1980).\n[42] R. M. White, Quantum Theory of Magnetism (Springer,\nBerlin, 1983).\n[43] Y. A. Izyumov, Modulated, or long-periodic, magnetic\nstructures of crystals, Soviet Physics Uspekhi 27, 845\n(1984)."    },    {        "title": "2106.01844v1.Macrospin_model_of_an_assembly_of_magnetically_coupled_core_shell_nanoparticles.pdf",        "content": "Macrospin model of an assembly of magnetically coupled core-shell nanoparticles\nNikolaos Ntallis,1Corisa Kons,2Hariharan Srikanth,2Manh-Huong Phan,2D.A. Arena,2and Manuel Pereiro1,\u0003\n1Department of Physics and Astronomy, Uppsala University, Uppsala 751 20, Sweden\n2Department of Physics, University of South Florida, Tampa, Florida 33620, USA\nHighly sophisticated synthesis methods and experimental techniques allow for precise measure-\nments of magnetic properties of nanoparticles that can be reliably reproduced using theoretical\nmodels. Here, we investigate the magnetic properties of ferrite nanoparticles by using theoretical\ntechniques based on Monte Carlo methods. We introduce three stages of sophistication in the macro-\nmagnetic model. First, by using tailor-made hamiltonians we study single nanoparticles. In a second\nstage, the internal structure of the nanoparticle is taken into consideration by de\fning an internal\n(core) and external (shell) region, respectively. In the last stage, an assembly of core/shell NPs are\nconsidered. All internal magnetic couplings such as inter and intra-atomic exchange interactions or\nmagnetocrystalline anisotropies have been estimated. Moreover, the hysteresis loops of the afore-\nmentioned three cases have been calculated and compared with recent experimental measurements.\nIn the case of the assembly of nanoparticles, the hysteresis loops together with the zero-\feld cooling\nand \feld cooling curves are shown to be in a very good agreement with the experimental data. The\ncurrent model provides an important tool to understand the internal structure of the nanoparticles\ntogether with the complex internal spin interactions of the core-shell ferrite nanoparticles.\nI. Introduction\nVarious technological \felds have motivated the design\nand fabrication of nanostructures with the ability of tai-\nloring the magnetic properties. The category of magnetic\nnanoparticles (NPs) are interesting from both fundamen-\ntal and technological points of view1{6. Thus, for ex-\nample, ferrite nanoparticles with the general formula of\nMFe 2O4(M = Fe, Co, Ni and Mn) have attracted great\nattention of researchers due to their potential applica-\ntions in biomedicine and industry2. In particular, spinel\nferrites are an attractive class of ferrimagnets o\u000bering\nboth soft and hard phases as well as a common crystal\nstructure that allow for a high quality crystal interface\nbetween core and shell layers7. The metallic ions are lo-\ncated at either octahedral coordinated sites|also called\nB-sites|or forming a tetrahedral geometry|denoted as\nA-sites. The ratio of the occupation of these sites pro-\nduces di\u000berent spinel structures de\fned by the inversion\ndegree number X. The normal and inverse spinel struc-\ntures represent the two limiting cases with X= 0 and\nX= 1, respectively. Between these two limits, a mixed\nspinel exists where the divalent transition metal M is dis-\ntributed between sites A and B8,10. A value of X= 2=3\nrepresents a random distribution of metallic ions between\nsites A and B. The degree of inversion can a\u000bect the\nmagnetic properties of ferrites as, for example, satura-\ntion magnetization and coercive \feld11,12. Ferrites are\nfound to have a ferrimagnetic magnetic ordering, which is\ndue to the dominant antiferromagnetic exchange between\nsites A and B8,10.\nThe inverse spinel structure is of paramount impor-\ntance for the current work where Fe3+cations are equally\ndistributed at both A and B sites while the divalent M\nions are found only at octahedral sites. In this work,\nCoFe 2O4(CFO) was selected due to its high anisotropy\nthat should limit the degree of canting as spins will be\nmore tightly bound to the crystal lattice compared toFe3O4(FO), which has a relatively high moment but\nmuch lower anisotropy. A common crystal structure and\nnegligible di\u000berences in lattice constant between the two\nmaterials (8.40 \u0017A for FO, 8.39 \u0017A for CFO8) enables syn-\nthesis of high quality core/shell NPs and, consequently,\na bi-magnetic structure of these two compounds can be\nconstructed without introducing large lattice mismatch\ndistortions. Moreover, as CFO is in a hard magnetic\nphase while FO is in a soft magnetic phase at normal\nconditions, the variants of these compounds can possess\ninteresting exchanged coupled related properties.\nThus, two variants of core/shell NP assemblies have al-\nready been synthesized and magnetically characterized9.\nMore speci\fcally, a conventional assembly of NPs is ex-\nperimentally synthesised by adding CFO to the core\nof the NP and capping the core with a shell of FO\n(CFO@FO). The inverted assembly is achieved by plac-\ning FO in the core and adding CFO in the shell\n(FO@CFO). In order to give a better understanding of\ntheir underlying reversal mechanisms, we employ Monte\nCarlo (MC) atomistic simulations for both type of as-\nsemblies based on tailor-made hamiltonians. Thus, we\nintroduce progressively di\u000berent degrees of sophistication\nof the model. Initially, we performed MC simulations\nfor single nanoparticles by varying several of their inter-\nnal degrees of freedom in order to examine how interface\nand surface e\u000bects a\u000bect their magnetic response. In a\nsecond stage, the hamiltonian is improved to describe a\nsingle nanoparticle composed of a central region denoted\nas core and an external shell region. The hamiltonian\nalso considers the interface e\u000bects between the core and\nshell regions. Finally, the hamiltonian, and consequently,\nthe macrospin model is recasted in a more sophisticated\nshape so that the model now is capable to describe not\nonly single nanoparticles with internal structure but also\nan assembly of core-shell nanoparticles. A depiction of\nthe three NP systems modeled is shown in Fig. 1. More-\nover, we also indicate in several colors the regions ofarXiv:2106.01844v1  [cond-mat.mes-hall]  3 Jun 20212\nthe nanoparticles that are essential to understand the\ndi\u000berent terms considered in the hamiltonians given by\nEqs. (1-3).\nII. Model and Results\nInitally, we employed atomistic Monte Carlo simula-\ntions with the implementation of the Metropolis algo-\nrithm of isolated nanoparticles composed of CoFe 2O4and\nFe3O4. The Hamiltonian used for the calculations is\nH=\u0000X\ni;jJijSiSj\u0000X\niKicos2\u0012i\u0000g\u0016BX\niBSi(1)\nwherei;jdenote \frst-neighbour atomic positions and\nSiis the atomic magnetic moment. The \frst term de-\nscribes the Heisenberg exchange interaction, Jij, between\natomic sites iandj. The ansatz given in Eq. (1) implies\nthat a ferromagnetic (antiferromagnetic) interaction is\ndescribed by a positive (negative) exchange coupling. In\nthe second term, a uniaxial atomic magnetic anisotropy\nof strength Kiis introduced to collect the spin-orbit ef-\nfects. The angle \u0012ide\fnes the deviation of the atomic\nmagnetic moment ( Si) with respect to the atomic easy\naxis anisotropy. The last term represents the Zeeman\nterm under the in\ruence of a global external magnetic\n\feld. For the very small sizes in diameter of the nanopar-\nticles considered here ( \u00186 nm for CFO and \u00187 nm for\nFO), the surface e\u000bects become important, and thus, for\nall the simulated systems, unless stated otherwise, we de-\n\fne a surface shell thickness of width d(cf. Fig. 1 a)).\nThe widthdis chosen to be equal to the unit cell length\nof the associated material. For the core spins, the easy\naxis anisotropy is set along zdirection whereas for the\nsurface atoms the anisotropy axis direction is randomly\nselected for each spin site in agreement with experimen-\ntal evidence9. BulkJijvalues are collected in Table I for\nCFO and FO nanoparticles13. Likewise, the bulk values\nof the magnetic anisotropy for the core region (red color\nin Fig. 1 a)) of the nanoparticles are ki= 0:0036 mRy\nandki= 0:000112 mRy per atom for CFO and FO, re-\nspectively8,10. Both CFO and FO have an inverse spinel\nstructure, i.e., the divalent ion (Co+2for CFO and Fe+2\nfor FO) is found only in B sites. The Co+2ion has a\nmoment of 3 :0\u0016B=atom, whereas Fe+2presents a mo-\nment of 4:0\u0016B=atom. In both samples, Fe+3ions have a\nmoment of 5.0 \u0016B=atom. The spherical particles with ra-\ndiusrare constructed by replication of the bulk inverse\nspinel structure. The spherical shape is reproduced by\nremoving the unit cells that are a distance bigger than r\naway from center of the nanoparticle.\nFigure 2 shows hysteresis loops for CFO and FO single\nnanoparticles at T= 5Kby varying the strength of the\nsurface anisotropy ( Ks\ni) with respect to the one of the\ncore (Kc\ni). As surface atoms interact with a number of\nneighbouring oxygen atoms smaller than the number of\noxygen atoms interacting with bulk atoms, the exchange\ncouplings of the surface atoms are expected to be smaller\nFIG. 1. Since the studied nanoparticles are very small in size,\nthe surface e\u000bects become very important. Thus, we sketch\nin a) an spherical single nanoparticle with the region in gold\nrepresenting the portion of the nanoparticle that belongs to\nthe surface. The width of the surface is denoted by d. In b),\na core-shell nanoparticle is shown with the di\u000berent regions\nindicated by using di\u000berent colors. Finally, in c) is represented\nan assembly of core-shell nanoparticles, while on the right is\ndrawn the internal structure of a single nanoparticle taken\nfrom the assembly. The arrows represent the macrospins used\nin the model.\nTABLE I. Bulk exchange coupling constants for CFO and FO\nwith di\u000berent interaction transition-metal (TM) atomic sites.\nAll values are given in energy units of mRy.\nCFO TMi TMj i j J ij\nFe Fe A A \u00000:094\nFe Co A B \u00000:143\nFe Fe A B \u00000:164\nCo Co B B 0 :296\nFe Co B B \u00000:117\nFe Fe B B \u00000:047\nFO Fe Fe A A \u00000:132\nFe Fe A B \u00000:150\nFe Fe A B \u00000:177\nFe Fe B B 0 :308\nFe Fe B B \u00000:08\nFe Fe B B \u00000:06\nthan the exchange couplings for bulk atoms. Thus, we set\nJs\nij=Jc\nij= 0:5. This value is chosen on a perfect spherical\nsurface formed by a cubic unit cell with the mean value\nof the coordination number drooping to half in the sur-\nface. Thus, by setting the aforementioned ratio to 0 :5 we\nassume that, in mean average, the coordination number3\nFIG. 2. Calculated hysteresis loops for CFO (left panel) and\nFO (right panel) single nanoparticles for di\u000berent ratios of the\nsurface anisotropy with respect to the core anisotropy\u0010\nKs\ni\nKc\ni\u0011\nat a temperature of 5 K.\nin the surface is half of the one for the atoms in the bulk.\nFO single nanoparticles clearly shows a larger saturation\nmagnetization with a value of 1 :31\u0016B=atom with respect\nto CFO which has a value of 0 :97\u0016B=atom. On the con-\ntrary, CFO possesses a larger coercive \feld reaching a\nvalue\u00182:5 T. It is worthwhile to mention here that the\nvariation of the coercive \feld with respect to the ratio\nKs\ni=Kc\niis almost negligible, in particular, for FO single\nnanoparticles. This is a direct consequence of the the\nrandomly easy axis distribution of the surface atoms.\nBy increasing the level of sophistication of the model,\nwe proceed to the calculation of single core-shell nanopar-\nticles for the two aforementioned variants, i.e., CFO@FO\nand FO@CFO. In order to study the core-shell morphol-\nogy, we introduce the following hamiltonian:\nH=\u0000bulkX\ni;jJijSiSj\u0000interf :X\ni;jaiJijSiSj\u0000surf:X\ni;jasJijSiSj\n\u0000X\niKicos2\u0012i\u0000g\u0016BX\niBSi (2)\nIn Eq. (2), we have introduced two additional terms\nwhich separate the interface and surface e\u000bects from the\nbulk of the nanoparticle. We assume that both the width\nof the surface (d s) and interface (d i) are equal to the\nCFO unit cell constant, i.e. d s=di=0:835 nm. As both\nCFO and FO possess the same inverse spinel structure\nand very similar lattice constant, we would expect an al-\nmost perfect match in the interface. However, due to the\nspherical shape of the nanoparticle, and consequenly, the\ncurvature of the interface, there is still a slight mismatch\nbetween the boundaries in the interface which produces a\nnonzero surface tension. For the aforementioned reasons,\nin the current model we assume that the spinel structure\nis preserved but it is slightly distorted on the interface\nand surface. The distortion is introduced by rescaling\nthe exchange constants Jijon the interface by the factorTABLE II. Structural parameters of CFO@FO and FO@CFO\nnanoparticles. All values are given in distance units of nm.\nCore Radius Shell Thickness Diameter\nCFO@FO 2.9 1.7 9.2\nFO@CFO 3.5 1.3 9.6\naiand on the surface by as. The initial value of Jijon\nthe interface is assumed to be the mean average of the\nsum ofJijof the associated bulk values of the di\u000berent\nstructures. Unless stated elsewhere, ashas a value of 0 :5,\ni.e., on the surface we assume that interaction strengths\ndrop to the half value than the one found in the bulk.\nAs a consequence, we assume the anisotropy becomes\nrandomized on the surface but uniaxial on the interface.\nThe structural parameters of the core-shell nanoparticles\nstudied here are given in Table II.\nIt has to be noted here that the size of both types of\ncore-shell nanoparticles are far below the coherent radius\nlimit of 3:6lex, wherelexis the exchange length14. Typ-\nical values of the exchange length are 4.9 nm and 5.2\nnm for Fe 3O4and CoFe 2O4bulk systems, respectively8.\nThus, domain formation is very unlikely. In consequence,\nthe magnetization reversal process can only be attributed\nto incoherent states on the interface or surface. Figures\n3 and 6 show hysteresis loops at 5K for CFO@FO and\nFO@CFO core-shell nanoparticles, respectively. The hys-\ntersis loops are plotted for di\u000berent scaling factors, i.e.,\nai,asandak. The anisotropy scaling parameter per atom\nis represented by ak. Both types of nanoparticles possess\nsimilar properties as single particles. Thus, under the\nabsence of magnetocrystalline anisotropy (black dotted\ncurves,ak= 0) both types of core-shell nanoparticles\nshow a knee-like behaviour close to saturation indicat-\ning the frustration produced by Jijin ferrites. Also in\nthe range from -3 T to 3 T in the applied external mag-\nnetic \feld, none of the CFO@FO and FO@CFO core-\nshell nanoparticles possess a full closed loop. However,\nby scaling (reducing) the exchange constants on the in-\nterface without magnetocrystalline anisotropy, the afore-\nmentioned e\u000bect is highly reduced although it is worth-\nwhile to mention here that the core-shell nanoparticles\nstill have a non-zero coercive \feld (cf. red curves in\nFigs. 3 and 6). The latter arises from the fact that the\nmismatch between the core and shell introduces canting\nin the interface spins and thus induces an increase of the\nexchange anisotropy of the system. Introduction of mag-\nnetocrystalline anisotropy into the nanoparticles led to\nan increase in the coercive \feld. For example, for a scal-\ning factorak= 2:2, the coercive \feld is calculated to\nbe 2:3 T and 1:9 T for FO@CFO and CFO@FO, respec-\ntively.\nFor exchange coupled systems, the exchange interac-\ntion on the interface is a critical parameter regardless\nof the soft or hard phase of the nanoparticle. Figure\n4 shows the variation of the coercive \feld with respect4\nto the interface scaling factor for CFO@FO nanopar-\nticle at 5K for two di\u000berent values of the anisotropy\nscaling factor. In the limit of no anisotropy coupling\n(ai!0), the coercive \feld approximates to that of the\nhard phase. For both anisotropy scaling parameters, the\ncoercive \feld possess a non-monotonic behaviour. When\nthe anisotropy constant approaches the bulk values, i.e.,\nak!1, the coercive \feld continuously decreases but the\ndecay is broken in the interval [0 :3;0:5]. In this region,\nthe coercive \feld curve shows a plateau with a very tiny\nincrease. After this value the decrease rate reduces, i.e.,\nthe coercive \feld decreases but with a smaller slope. On\nthe other hand, for ak= 2:2 the coercive \feld shows a\npeak forai\u00180:2, so that, the interface e\u000bects induce\nmagnetic \ructuations for such coupled systems. Notably\nthe hardness of the bi-magnetic systems is heavily depen-\ndent on the interplay between the anisotropy barrier and\nthe exchange interactions on the interface. As already\nmentioned above, the disorder induced by the interface\ne\u000bects, even under the in\ruence of strong anisotropy, re-\nsults in a non-zero coercive \feld. By increasing of the\nanisotropic scaling factor, a high coercive \feld can be\nreproduced with a small interface coupling factor. On\nthe other hand, for the anisotropy bulk value case, a\nvery strong interface coupling must be considered in or-\nder to achieve the maximum coercive \feld. Therefore, a\nnanoparticle system requires an almost perfect interface\nwith negligible imperfections or dislocations. From these\nfacts, we can conclude that for bi-magnetic nanoparticle\nsystems, the interface shows an enhanced anisotropy.\nApart form the coercive \feld, the remanent magneti-\nzation is a critical parameter for magnetic applications.\nFigure 5 shows the evolution of the normalized rema-\nnent magnetization with respect to the interface scaling\nfactor. For both ak= 1:0 andak= 2:2, the remanent\nmagnetization shows an increasing trend up to a critical\nvalue of the interface scaling factor ai= 0:3 and then\nclearly reduces down to a value of Mr=Ms= 0:2. For\nboth anistropy scaling factors, the Mr=Msratio show a\nmaximum close to 0 :8.\nIndependently from the interface coupling, the surface\nis free to start a reversal process due to reduced coordina-\ntion of the atomic spins. From our simulations it is found\nthat, in order to reach the maximum Mr=Msa larger\nvalue of the interface coupling constant is needed, with\nrespect to the one for the maximum coercive \feld. As\nthe surface and interface are already disordered an even\nstronger coupling between the phases is needed in order\nfor the soft phase to overrule and rise the remament state.\nInterestingly though when the Mr=Msis at its maximum\nvalue still the nanoparticle possess a considerable coer-\ncive \feld as the disorder of the interface does not allow\nfor a full reversal process to be completed. Despite that,\nit is clear that in ferrites the exchange interactions are\nextremely strong and play a major role in the magnetic\nbehaviour. As seen in Figs. 4 and 5, even with a scaling\nofak= 2:2 for a value of ai= 0:3, a clearly exchanged\ncoupled behaviour can been achieved. Moreover, the size\nFIG. 3. Hysteresis loops for CFO@FO core-shell nanoparticle\nat 5 K. Loops are plotted for di\u000berent scaling factors ai,as\nandak.\nFIG. 4. Coercive \feld as a function of the interface scaling\nfactor aifor CFO@FO nanoparticles.\nof the particle is also extremely important as it does not\nallow for any kind of domain formation14.\nExperimental measurements have been performed on\ndense assemblies of CFO@FO and FO@CFO core-shell\nnanoparticles9. Under these experimental conditions, the\ndipolar interaction can a\u000bect the magnetic response, es-\npecially for NPs composed of CFO and FO due to their\nconsiderable magnetic moments. As the size of the parti-\ncles is below the coherent radius limit, in order to study\nthe assembly in a computationally e\u000ecient form, we de-\nvelop a coarse-grain macrospin model for each nanopar-\nticle. To be more speci\fc, we used 3 to 6 macrospins\nper particle. Under this level of theory, we recast the\nHamiltonian in the following form:5\nFIG. 5. Variance of the normalized remanent magnetization\nfor CFO@FO nanoparticle with respect to interface scaling\nfactor ai.\nFIG. 6. Hysteresis loops for FO@CFO single nanoparticle at\n5 K. Loops are plotted for di\u000berent scaling factors ai,asand\nak.\nH=\u0000X\ni;jJijSiSj\u0000X\nm;nJinterMmMn+X\nm;nMm[D]Mn\n\u0000X\nik(cos)2\u0012i\u0000g\u0016BX\niSiB (3)\nwherefSigrepresent the moment of macro spins within\na nanoparticle while fMngis the net moment of\nparticlen, so that, indices fi;jgdenote summation\nwithin a nanoparticle (intra-nanoparticle interaction)\nwhilefm;ngdenote summation between nanoparticles\n(inter-nanoparticles interaction). The parameter Jijis\nthe Heisenberg interaction coupling between macrospins\niandjwithin the nanoparticle while Jinter describes the\ninterparticle interactions of the Heisenberg form. The\nthird term in Eq. (3) represents dipolar interactions be-\ntween nanoparticles with [ D] being the dipolar tensor andit is calculated as an interaction between the net moment\nof each particle Mn. The magnetic anisotropy constant is\nrepresented by kand\u0012iis the angle between Siand easy\naxis direction while the last term represent the Zeeman\ncontribution with Bbeing the applied external magnetic\n\feld.\nFor each nanoparticle, the total magnetic moment is\nsubdivided into Nmacrospins, N1for the core and N2\nfor shell, and the sum of the macrospins for the core and\nshell equals the total moment per nanoparticle, as deter-\nmined from bulk magnetometry. Thus, the total moment\nper nanoparticle is divided between the core and shell\ncontribution, separately. For the CFO@FO core-shell\nnanoparticle, the core and shell moments are estimated to\nbe 4:14\u000210\u000020Am2and 12:38\u000210\u000020Am2, respectively;\nwhile for FO@CFO nanoparticle, the core and shell mo-\nments are 6 :80\u000210\u000020Am2and 10:74\u000210\u000020Am2,\nrespectively. The multiple macrospins per core and shell\ncan simulate the e\u000bects of spin canting at the core-shell\nand shell-vacuum interface. The ensemble of nanopar-\nticles is simulated by placing the macrospins for each\nnanoparticle on a 12 \u000212\u000212 grid ( i.e.,123nanoparti-\ncles) with a mean spacing of 10 nm between nanoparticles\nand periodic boundary conditions.\nIn order to de\fne the interaction parameters, we\nperform a \ftting procedure of the calculated single-\nnanoparticles energy. The atomistic energy of the sys-\ntem is calculated for di\u000berent magnetic con\fgurations\nand mapped back into the macrospin model. The toler-\nance of the \ftting procedure ensures that all \ftted pa-\nrameters have an error smaller than 10\u00004meV/atom. In\nTables III and IV, we present the parameters used for\nthe macrospin model. For both compounds, the indices\nf1\u00003grefer to the core and f4\u00006gto the shell. In\nall cases, the anisotropy was assumed to be uniaxial for\neach nanoparticle, and its axis was randomly selected per\nparticle in the assembly of nanoparticles. The interpar-\nticle interaction, Jinter, was set to 0 :1 mRy. With the\naim to reduce the number of free \ftting parameters, the\nmacrospins labelled as f1gfor the core and f4gfor the\nshell, are imposed to interact with the same exchange\ninteraction strength (see Table III).\nIn Table IV, the anisotropy contributions are distin-\nguished between core and shell. FO region shows a con-\nsiderable enhancement when it is interfaced with CFO,\nthus reducing the volume of the soft phase in each\nnanoparticle. The CFO region, being the harder mag-\nnetic phase, is the most probable nucleation region which\nsparks the magnetization reversal process. According to\nthis model, the nucleation process is initiated from the\nsurface of the CFO@FO or the core of the FO@CFO.\nThe application of at least three macrospins per material\nallows us to implicity take into account by a mean mag-\nnetization state the incoherent states on the interface or\nsurface.\nFigure 7 shows the calculated hysteresis loops from\nthe current macro spin model. The agreement with ex-\nperimental data9is remarkably good, especially in low6\nTABLE III. Heisenberg interaction constant Jijfor the\nmacrospin model. All Jijare given in units of mRy .\nCompound J12J13J23J24J34J25J35J45J46J56\nCFO@FO 3.4 3.4 -2.1 2.3 1.9 2.2 1.8 2.4 2.4 -1.8\nFO@CFO 3.1 3.1 -1.8 2.1 2.2 2.3 1.7 2.8 2.8 -1.9\ntemperature regime. The small value used for Jinter in-\ndicates that even with a dense ensemble of particles, the\nmagnetic behaviour is dominated by the intra-particle\ncharacteristics. For low temperatures, both samples are\ncharacterized by a knee behaviour appearing right after\nthe remanent state. Even though we have assumed a non-\nperfect interface, this behaviour is not so pronounced for\nthe single nanoparticles. Thus, for the case where the\nnanoparticles are grouped forming an assembly, it is pos-\nsible to attribute a random anisotropy axis distribution\ncreating now a range of energy barriers to be bypassed in\nthe cycle to complete the magnetization reversal. The in-\ntroduction of dipolar interactions has an e\u000bect, that the\ncoercive \feld drops down to values of about 1 :1T\u00001:3T.\nThis value is clearly lower than the one calculated for\nsingle nanoparticles. Even though the assembly is dense,\ndipolar interactions do not dramatically decrease the co-\nercive \feld due to the moderate magnetic moment of the\nnanoparticles. It has to be noted here that even though\nthe assembly is quite dense, we managed to reproduce the\nexperimental data by using a macrospin model. This fact\nmeans that we do not need to take into account the dis-\ntribution of the dipolar \felds because the intra-particle\ninteractions are quite strong and, as already explained\nabove, there cannot be any kind of magnetic domain for-\nmation in the nanoparticles. The magnetization reversal\nprocess depends strongly on the incoherent modes on the\ninterface and surface arising from the exchange strength\nvariations.\nThe accuracy of the current model decreases as the\ntemperature of the system is increased (cf. Fig. 7). For\nthis reason, we also simulate the the ZFC/FC curves of\nthe assemblies as shown in Fig. 8. Notably the pro\fle of\nthe ZFC/FC curves is reproduced quite well with the cur-\nrent macrospin model however, in both cases, the block-\ning temperature TBis overestimated. The latter e\u000bect\nis more pronounced in the FO@CFO variant where the\nTBis overestimated by 30 K. This overestimation is also\nfound in the loop simulations as the coercive \feld is over-\nestimated in all cases for temperatures larger than 5 K.\nThe latter is a clear indication that there is a tempera-\nture dependence of the magnetic parameters, disregarded\nin the current model.\nIII. Summary and Conclusions\nIn conclusion, using progressively more complex mod-\nels beginning with the case of isolated single nanoparti-TABLE IV. Anisotropy constant kfor the macrospin model.\nThe units of the uniaxial magnetocrystalline anisotropy are\ngiven in mRy .\nCore Shell\nCompound k1 k2 k3 k4 k5 k6\nCFO@FO 7.1 13.1 13.1 8.1 7.6 0.1\nFO@CFO 0.8 6.4 6.4 9.2 8.1 8.0\nFIG. 7. Hysteresis loops for CFO@FO (left column) and\nFO@CFO (right column) assembly nanoparticles at several\ntemperatures (5 K, 75 K and 100 K). In blue is shown the ex-\nperimental hysteresis loop taken from Ref. [9] for comparison\nwith the prediction of the theoretical model.\ncles, we have investigated the magnetic properties of fer-\nrite nanoparticles. The proposed macrospin model is able\nto describe substantially well the experimental measure-\nments of an assembly of such core-shell nanoparticle sys-\ntems. By taking the bulk values of the exchange interac-\ntions and magnetocrytalline anisotropy, we discussed the\ncalculated hysteresis loops in terms of the ratio between\nthe surface anisotropy with respect to the core anistropy\nfor CFO and FO nanoparticles (cf. Fig. 1 a)). By gaining\nsome information about the internal parameters of iso-\nlated nanoparticles, we improved the model by consider-\ning regions with di\u000berent magnetic texture, so that, the\nparticles now have a core, interface, shell and surface (cf.\nFig. 1 b)). This improvement requires consideration of\nmore types of exchange coupling between the di\u000berent re-\ngions as well as magnetocrystalline anistropies per region.\nBy scaling exchange interactions and anisotropies in the\ninterface and surface, we can get information about the\ndetails of the hysteresis loops so that we have full control\nover features seen in the experimental hysteresis loops,\nsuch as, the knee present in the vicinity of the remanent\n\feld that are measured in experiments for an assembly of\nnanoparticles (cf. Fig. 7). In the last stage of the macro-7\nFIG. 8. Zero-\feld cooling (ZFC) and \feld cooling (FC) curves\nfor the assembly nanoparticles CFO@FO (left panel) and\nFO@CFO (right panel). The predictions of the theoretical\nmodel (circle symbol) are compared with the experimental\nmeasurements taken from Ref.[9] (solid line).\nmagnetic model, we brought the core-shell nanoparticles,\nwith their internal structure, all together forming a cubic\ncrystal structure (see Fig. 1 c)). This system is describe\nby the hamiltonian of Eq. (3). In doing so, the model is\ncapable of describing with high \fdelity the experimental\nhysteresis loops, ZFC and FC curves. Overall, experi-\nmental and theoretical results are in close agreement al-\nthough discrepancies in the hysteresis loops increase at\nhigher temperatures. It is well-known in literature that\nthe Heisenberg exchange interaction is a function of the\ntemperature15. We speculate here that the small devi-\nation of the calculated hysteresis loops with respect to\nthe experimental ones at 100 K is due to the fact we donot include a temperature dependence in the estimated\nHeisenberg exchange interactions.\nOverall, and based on the good description of the ex-\nperimental results, the current model underpins the pro-\nposed internal structure of the nanoparticles, not only\nwith the core and shell, but also with two thin layers\nof interface and surface, so that, the interface plays an\nimportant role in describing the observed knee in the ex-\nperimental hysteresis loops.\nAcknowledgments\nThe computations were enabled by resources provided\nby the Swedish National Infrastructure for Computing\n(SNIC) at Chalmers Center for Computational Science\nand Engineering (C3SE), High Performance Comput-\ning Center North (HPCN), and the National Supercom-\nputer Center (NSC) partially funded by the Swedish Re-\nsearch Council through grant agreement no. 2016-07213.\nNN and MP acknowledge \fnancial support from the\nKnut and Alice Wallenberg Foundation through grant\nno. 2018.0060. HS and MHP acknowledge support (syn-\nthesis and magnetic measurements) from the US Depart-\nment of Energy, O\u000ece of Basic Energy Sciences, Divi-\nsion of Materials Sciences and Engineering under Award\nNo. DE-FG02-07ER46438. This material is based upon\nwork supported by the National Science Foundation un-\nder Grant No. ECCS-1952957. DAA acknowledges the\nsupport of the USF Nexus Initiative and the Swedish\nFulbright Commission. We also thank Joshua Robles for\nassistance with the nanoparticle synthesis.\n\u0003Corresponding author: manuel.pereiro@physics.uu.se\n1I. Shari\f, H. Shokrollahi and S. Amiri, Ferrite-based mag-\nnetic nano\ruids used in hyperthermia applications, J.\nMagn. Magn. Mater. 324, 903 (2012).\n2S. Y. Srinivasan, K. M. Paknikar, D. Bodas and V.\nGajbhiye, Applications of cobalt ferrite nanoparticles\nin biomedical nanotechnology, Nanomedicine 13, 1221\n(2018).\n3M. W. Mushtaq, F. Kanwal, M. Imran, N. Ameen, M.\nBatool, A. Batool, S. Bashir, S. M. Abbas, A. Ur Rehman,\nS. Riaz, S. Naseem and Z. Ullah, Synthesis of surfactant-\ncoated cobalt ferrite nanoparticles for adsorptive removal\nof acid blue 45 dye, Mater. Res. Express. 5, (2018).\n4H. Zhu, S. Zhang, Y.-X. Huang, L. Wu and S. Sun,\nMonodisperse MxFe3\u0000xO4(M = Fe, Cu, Co, Mn)\nNanoparticles and their electrocatalysis for oxygen reduc-\ntion reaction, Nano Lett. 13, 2947 (2013).\n5A. Quesada, C. M. Granados, O. Lopez, A. Erokhin, S.\nLottini, E. Pedrosa, J. Bollero, A. Aragon, Ana M., F.\nStingaciu, M. Bertoni, C. de Juli\u0013 an Fern\u0013 andez, C. Sangre-\ngorio, J. F. Fernandez, D. Berkov, and M. Christensen,\nEnergy Product Enhancement in Imperfectly Exchange-\nCoupled Nanocomposite Magnets, Adv. Elect. Mater. 2,\n1500365 (2016).6Ferrite nanoparticles: Synthesis, characterisation and ap-\nplications in electronic device\", Materials Science and En-\ngineering: B 215, 37 (2017).\n7M. A. G. Soler and L. G. Paterno Nanostructures (William\nAndrew Publishing, 2017)\n8J. M. D. Coey, Magnetism and Magnetic Materials (Cam-\nbridge University Press, 2012).\n9C. Kons, K.L. Krycka, J. Robles, N. Ntallis, M. Pereiro,\nM.-H. Phan, H. Srikanth, J.A. Borchers and D.A. Arena,\nSpin Canting in Exchange Coupled Bi-Magnetic Nanopar-\nticles: Interfacial E\u000bects and Hard/Soft Layer Ordering,\narXiv:2105.11501 [cond-mat.mes-hall].\n10B. D. Cullity and C. D. Graham Introduction to magnetic\nMaterials (Willey, 2008)\n11Y.H. Hou, Y.J. Zhao, Z.W. Liu, H.Y. Yu, X.C. Zhong,\nW.Q. Qiu, D.C. Zeng and L.S. Wen, Structural, elec-\ntronic and magnetic properties of partially inverse spinel\nCoFe2O4: A \frst-principles study, J. Phys. D.: Appl.\nPhys. 43, 445003 (2010).\n12N. Da\u000be, F. Choueikani, S. Neveu, M. Arrio, A. Juhin, P.\nOhresser, V. Dupuis, P. Sainctavit, N. Da\u000be,F. Choueikani,\nS. Neveu, M. Arrio, A. Juhin, V. Dupuis, P. Sainc-\ntavit, Magnetic anisotropies and cationic distribution in\nCoFe2O4 nanoparticles prepared by co-precipitation route:8\nIn\ruence of particle size and stoichiometry, J. Magn.\nMagn. Mater. 460, 243 (2018).\n13C.M. Srivastava, G. Srinivasan and N.G. Nanadikar, Ex-\nchange constants in spinel ferrites, Phys. Rev. B 19, 1\n(1979).\n14N. Ntallis and K.G. Efthimiadis, Size dependence of\nthe magnetization reversal in a ferromagnetic particle, J.\nMagn. Magn. Mater. 99, 373 (2015).\n15A. Szilva, M. Costa, A. Bergman, L. Szunyogh, L. Nord-\nstr om and O. Eriksson, Interatomic Exchange Interac-\ntions for Finite-Temperature Magnetism and Nonequilib-\nrium Spin Dynamics, Phys. Rev. Lett. 111, 127204 (2013)."    },    {        "title": "2106.03624v3.MAELAS_2_0__A_new_version_of_a_computer_program_for_the_calculation_of_magneto_elastic_properties.pdf",        "content": "MAELAS 2.0: A new version of a computer program\nfor the calculation of magneto-elastic properties\nP. Nievesa,\u0003, S. Arapana, S. H. Zhangb,c, A. P. K ˛ adzielawaa, R. F. Zhangb,c,\nD. Leguta\naIT4Innovations, VŠB - Technical University of Ostrava, 17. listopadu 2172/15, 70800\nOstrava-Poruba, Czech Republic\nbSchool of Materials Science and Engineering, Beihang University, Beijing 100191, PR China\ncCenter for Integrated Computational Materials Engineering, International Research Institute\nfor Multidisciplinary Science, Beihang University, Beijing 100191, PR China\nAbstract\nMAELAS is a computer program for the calculation of magnetocrystalline\nanisotropy energy, anisotropic magnetostrictive coefficients and magnetoelastic\nconstants in an automated way. The method originally implemented in version\n1.0 of MAELAS was based on the length optimization of the unit cell, proposed\nby Wu and Freeman, to calculate the anisotropic magnetostrictive coefficients. We\npresent here a revised and updated version (v2.0) of MAELAS, where we added a\nnew methodology to compute anisotropic magnetoelastic constants from a linear\nfitting of the energy versus applied strain. We analyze and compare the accuracy\nof both methods showing that the new approach is more reliable and robust than\nthe one implemented in version 1.0, especially for non-cubic crystal symmetries.\nThis analysis also help us to find that the accuracy of the method implemented\nin version 1.0 could be improved by using deformation gradients derived from\nthe equilibrium magnetoelastic strain tensor, as well as potential future alterna-\ntive methods like the strain optimization method. Additionally, we clarify the role\nof the demagnetized state in the fractional change in length, and derive the ex-\npression for saturation magnetostriction for polycrystals with trigonal, tetragonal\nand orthorhombic crystal symmetry. In this new version, we also fix some issues\nrelated to trigonal crystal symmetry found in version 1.0.\nKeywords: Magnetostriction, Magnetoelasticity, Magnetocrystalline anisotropy,\n\u0003Corresponding author.\nE-mail address: pablo.nieves.cordones@vsb.cz\nPreprint submitted to Computer Physics Communications September 16, 2021arXiv:2106.03624v3  [cond-mat.mtrl-sci]  15 Sep 2021High-throughput computation, First-principles calculations\nPROGRAM SUMMARY\nProgram Title: MAELAS\nDeveloper’s respository link: https://github.com/pnieves2019/MAELAS\nLicensing provisions: BSD 3-clause\nProgramming language: Python3\nJournal reference of previous version: P. Nieves, S. Arapan, S.H. Zhang, A.P. K ˛ adzielawa,\nR.F. Zhang and D. Legut, Comput. Phys. Commun. 264, 107964 (2021)\nDoes the new version supersede the previous version?: Yes\nReasons for the new version: To implement a more accurate methodology to compute\nmagnetoelastic constants and magnetostrictive coefficients, and fix some issues related to\ntrigonal crystal symmetry.\nSummary of revisions:\n• New method to calculate magnetoelastic constants and magnetostrictive coeffi-\ncients derived from the magnetoelastic energy.\n• Correction of the trigonal crystal symmetry.\n• Implementation of the saturation magnetostriction for polycrystals with trigonal,\ntetragonal and orthorhombic crystal symmetry.\n• In the visualization tool MAELASviewer, we included the possibility to choose the\ntype of reference demagnetized state in the calculation of the fractional change in\nlength\nNature of problem: To calculate anisotropic magnetostrictive coefficients and magnetoe-\nlastic constants in an automated way based on Density Functional Theory methods.\nSolution method: In the first stage, the unit cell is relaxed through a spin-polarized cal-\nculation without spin-orbit coupling (SOC). Next, after a crystal symmetry analysis, a\nset of deformed lattice and spin configurations are generated using the pymatgen library\n[1]. The energy of these states is calculated by the Vienna Ab-initio Simulation Package\n(V ASP) [2], including SOC. The anisotropic magnetoelastic constants are derived from\nthe fitting of these energies to a linear polynomial. Finally, if the elastic tensor is provided\n[3], then the magnetostrictive coefficients are also calculated from the theoretical relations\nbetween elastic and magnetoelastic constants.\nAdditional comments including restrictions and unusual features: This version supports\nthe following crystal systems: Cubic (point groups 432, ¯43m,m¯3m), Hexagonal (6 mm,\n622, ¯62m, 6=mmm ), Trigonal (32, 3 m,¯3m), Tetragonal (4 mm, 422, ¯42m, 4=mmm ) and\nOrthorhombic (222, 2 mm,mmm ).\n2References\n[1] S. P. Ong, W. D. Richards, A. Jain, G. Hautier, M. Kocher, S. Cholia, D. Gunter, V .\nL. Chevrier, K. A. Persson, and G. Ceder, Comput. Mater. Sci. 68, 314 (2013).\n[2] G. Kresse, J. Furthmüller, Phys. Rev. B 54 (1996) 11169.\n[3] S. Zhang and R. Zhang, Comput. Phys. Commun. 220, 403 (2017).\n1. Introduction\nMAELAS aims to provide an efficient and reliable computational program for\nthe study of magnetostriction by automated first-principles calculations. The ver-\nsion 1.0 of MAELAS was originally published in Ref. [1], where we implemented\nand generalized the length optimization method proposed by Wu and Freeman to\ncalculate the magnetostrictive coefficients ( l) [2]. In version 1.0, we also pro-\nposed to compute the magnetoelastic constants ( b) indirectly from the theoretical\nrelations between the elastic constants ( Ci j) and magnetostrictive coefficients us-\ning the calculated values of these quantities. The present release of the MAELAS\nprogram (version 2.0) aims to add a more robust, rigorous and accurate alternative\nmethodology to calculate the magnetoelastic constants with respect to the previ-\nous version. This is accomplished by implementing cell deformations and spin\ndirections theoretically derived from the magnetoelastic energy for each crystal\nsymmetry. In this new approach, the magnetoelastic constants are directly ob-\ntained from a linear fitting of the energy versus strain data.\nA quantitative analysis of the accuracy of these methods can be very useful to\nknow their reliability and to optimize them. In every calculation with MAELAS\nwe can identify two main sources of errors. The first source of errors comes from\nthe methodology used to compute the magnetostrictive coefficients and magne-\ntoelastic constants, while the second source is the evaluation of the total energies\nperformed with Density Functional Theory (DFT) itself. For instance, in some\ncases DFT gives energy values that can not fitted well to the polynomial used\nin these methods, due to numerical or physical reasons. Frequently, the lack of\navailable experimental data makes it difficult to estimate the reliability and pre-\ncision of these calculations. Aiming to overcome these limitations, we propose\nhere to analyze the systematic error coming from the methodologies implemented\nin MAELAS by evaluating the exact energy from the theory of magnetostriction.\nThis strategy allows us to compare the accuracy between the length optimization\n3method (originally implemented in version 1.0) and the new approach (added in\nversion 2.0). It also helps us to identify some issues with the trigonal crystal sym-\nmetry in the previous version of MAELAS, which is fixed in the present version\nand discussed here.\nTypically, in studies of magnetostriction the change in length is referred to an\ninitial demagnetized state. However, this fact can lead to difficulties in the in-\nterpretation of the results since the demagnetized state may not correspond to a\nunique reference state [3]. Aiming to clarify this point, we also study the role\nof the demagnetized state in the fractional change in length. The paper is orga-\nnized as follows. In Section 2, we describe the main updates in the new version,\nwhile the accuracy of MAELAS is analyzed in Section 3. The new implemented\nmethod is benchmarked in Section 4. The paper ends with a summary of the main\nconclusions and future perspectives (Section 5).\n2. Main updates in new version\nIn this section, we explain in detail three major updates implemented in the\nnew version of MAELAS. An introduction to the theory of magnetostriction and\noverview of the theoretical background were already provided in the publication\nof version 1.0 of MAELAS [1]. Here, we use the same notation, definitions and\nconventions as in Ref. [1].\n2.1. New method for direct calculation of anisotropic magnetoelastic constants\nWe have added an alternative method to compute anisotropic magnetoelas-\ntic constants in version 2.0. This approach is derived from the total energy ( E)\nincluding elastic ( Eel), magnetoelastic ( Eme) and unstrained magnetocrystalline\nanisotropy ( E0\nK) terms\nE(eee;aaa) =Eel(eee)+Eme(eee;aaa)+E0\nK(aaa); (1)\nwhere eeeis the strain tensor and aaais the normalized magnetization ( jaaaj=1). The\nbasic idea of this method is to subtract the total energy of two magnetization\ndirections aaa1andaaa2for a deformed unit cell in such a way that we can get the i-\nth anisotropic magnetoelastic constant bifrom a linear fitting of the energy versus\nstrain data\n1\nV0\u0002\nE(eeei(s);aaai\n1)\u0000E(eeei(s);aaai\n2)\u0003\n=Gibis+Fi(K1;K2); (2)\n4where V0is the equilibrium volume, Giis a real number, Fican depend on the\nmagnetocrystalline anisotropy constants K1andK2, and sis the parameter used\nto parameterize the strain tensor ei(s). In practice, Eq. 2 is fitted to a linear\npolynomial\nf(s) =As+B; (3)\nwhere AandBare fitting parameters, so that the i-th anisotropic magnetoelastic\nconstant biis given by\nbi=A\nGi: (4)\nAdditionally, we note that in some cases from the fitting parameter Bis possible\nto estimate the magnetocrystalline anisotropy constants since B=Fi(K1;K2). In\nAppendix A we describe the procedure to generate the deformations for each\nmagnetoelastic constant, while in Table 1 we show the selected set of aaai\n1andaaai\n2,\nand the corresponding value of Githat fulfils Eq. 2. We implemented this method\nfor all supported crystal symmetries in version 1.0 [1].\nTo illustrate this method, let’s apply it for the calculation of b1of cubic sym-\nmetry since it is easy to handle. For cubic (I) systems (point groups 432, ¯43m,\nm¯3m) the energy terms in Eq.1 read [1]\n1\nV0Ecub\nel(eee) =cxxxx\n2(e2\nxx+e2\nyy+e2\nzz)+cxxyy(exxeyy+exxezz+eyyezz)\n+2cyzyz(e2\nxy+e2\nyz+e2\nxz);\n1\nV0Ecub(I)\nme(eee;aaa) =b0(exx+eyy+ezz)+b1(a2\nxexx+a2\nyeyy+a2\nzezz)\n+2b2(axayexy+axazexz+ayazeyz);\n1\nV0E0;cub\nK(aaa) =K0+K1(a2\nxa2\ny+a2\nxa2\nz+a2\nya2\nz)+K2a2\nxa2\nya2\nz;(5)\nwhere cxxxx=C11,cxxyy=C12andcyzyz=C44are the elastic constants, b0,b1and\nb2are the magnetoelastic constants, and K0,K1andK2are the magnetocrystalline\nanisotropy constants. In Table 1, we see that the deformation gradient FFFto cal-\nculate b1is given by Eq. A.4, which leads to the following strain tensor through\nEq.A.3\neeeb1(s) =0\n@exxexyexz\neyxeyyeyz\nezxezyezz1\nA=0\n@s0 0\n0 0 0\n0 0 01\nA: (6)\n5Table 1: Selected magnetization directions ( aaa1,aaa2) in the new method implemented in MAELAS\nversion 2.0 to calculate the anisotropic magnetoelastic constants. The first column shows the\ncrystal system and the corresponding lattice convention set in MAELAS based on the IEEE format\n[4]. In the fifth and sixth columns we show the values of the parameters GandFthat are defined\nin Eq.2. Last column presents the equation of the deformation gradient Fi jthat we used in Eq.2\nfor the calculation of each magnetoelastic constant. The symbols a;b;ccorrespond to the lattice\nparameters of the relaxed (not distorted) unit cell.\nCrystal systemMagnetoelastic\nconstantaaa1 aaa2 G F FFF\nCubic (I) b1 (1;0;0)\u0010\n1p\n2;1p\n2;0\u0011\n1\n2\u0000K1\n4Eq.A.4\naaakˆxxx,bbbkˆyyy,ccckˆzzz b 2\u0010\n1p\n2;1p\n2;0\u0011 \u0010\n\u00001p\n2;1p\n2;0\u0011\n2 0 Eq.A.5\nHexagonal (I) b21 (0;0;1)\u0010\n1p\n2;1p\n2;0\u0011\n1\u0000K1\u0000K2Eq.A.6\naaakˆxxx,ccckˆzzz b 22 (0;0;1)\u0010\n1p\n2;1p\n2;0\u0011\n1\u0000K1\u0000K2Eq.A.7\nbbb=\u0010\n\u0000a\n2;p\n3a\n2;0\u0011\nb3 (1;0;0) ( 0;1;0) 1 0 Eq.A.8\na=b6=c b 4\u0010\n1p\n2;0;1p\n2\u0011 \u0010\n\u00001p\n2;0;1p\n2\u0011\n2 0 Eq.A.9\nTrigonal (I) b21 (0;0;1)\u0010\n1p\n2;1p\n2;0\u0011\n1\u0000K1\u0000K2Eq.A.6\naaakˆxxx,ccckˆzzz b 22 (0;0;1)\u0010\n1p\n2;1p\n2;0\u0011\n1\u0000K1\u0000K2Eq.A.7\nbbb=\u0010\n\u0000a\n2;p\n3a\n2;0\u0011\nb3 (1;0;0) ( 0;1;0) 1 0 Eq.A.8\na=b6=c b 4\u0010\n1p\n2;0;1p\n2\u0011 \u0010\n\u00001p\n2;0;1p\n2\u0011\n2 0 Eq.A.9\nb14\u0010\n1p\n2;1p\n2;0\u0011 \u0010\n\u00001p\n2;1p\n2;0\u0011\n2 0 Eq.A.10\nb34\u0010\n1p\n2;0;1p\n2\u0011 \u0010\n\u00001p\n2;0;1p\n2\u0011\n2 0 Eq.A.11\nTetragonal (I) b21 (0;0;1)\u0010\n1p\n2;1p\n2;0\u0011\n1\u0000K1\u0000K2Eq.A.6\naaakˆxxx,bbbkˆyyy,ccckˆzzz b 22 (0;0;1)\u0010\n1p\n2;1p\n2;0\u0011\n1\u0000K1\u0000K2Eq.A.7\na=b6=c b 3 (1;0;0) ( 0;1;0) 1 0 Eq.A.8\nb4\u0010\n1p\n2;0;1p\n2\u0011 \u0010\n\u00001p\n2;0;1p\n2\u0011\n2 0 Eq.A.9\nb0\n3\u0010\n1p\n2;1p\n2;0\u0011 \u0010\n\u00001p\n2;1p\n2;0\u0011\n2 0 Eq.A.12\nOrthorhombic b1 (1;0;0) ( 0;0;1) 1 K1 Eq.A.13\nc<a<b b 2 (0;1;0) ( 0;0;1) 1 K2 Eq.A.13\nb3 (1;0;0) ( 0;0;1) 1 K1 Eq.A.14\nb4 (0;1;0) ( 0;0;1) 1 K2 Eq.A.14\nb5 (1;0;0) ( 0;0;1) 1 K1 Eq.A.15\nb6 (0;1;0) ( 0;0;1) 1 K2 Eq.A.15\nb7\u0010\n1p\n2;1p\n2;0\u0011\n(0;0;1) 1K1\n2+K2\n2Eq.A.16\nb8\u0010\n1p\n2;0;1p\n2\u0011\n(0;0;1) 1K1\n2Eq.A.17\nb9\u0010\n0;1p\n2;1p\n2\u0011\n(0;0;1) 1K2\n2Eq.A.18\n6We also see in Table 1 that the two magnetization directions to calculate b1are\naaa1= (1;0;0)andaaa2= (1=p\n2;1=p\n2;0). Hence, Eq. 2 becomes\n1\nV0\u0014\nE(eeeb1(s);1;0;0)\u0000E\u0012\neeeb1(s);1p\n2;1p\n2;0\u0013\u0015\n=1\n2b1s\u00001\n4K1; (7)\nwhere we recover G=1=2, as shown in Table 1. Therefore, b1can be estimated\nfrom the fitting parameter Aof the linear polynomial (Eq. 4) as b1=A=G=\n2A. We also observe that F=\u0000K1=4, so the first magnetocrystalline anisotropy\nconstant K1for the unstrained cubic state can be estimated as well from the fitting\nparameter Bof the linear polynomial ( K1=\u00004B), see Eq.3.\nThe corresponding workflow for this new methodology is depicted in Fig.1. It\ncontains the same steps as the workflow for the method implemented in version\n1.0 for direct calculation of anisotropic magnetostrictive coefficients [1]. How-\never, here the generated deformations and spin configurations for the Vienna Ab\ninitio Simulation Package (V ASP) [5–7] in step 3 corresponds to the new method.\nIn step 5, after processing V ASP output data, MAELAS v2.0 will calculate di-\nrectly the magnetoelastic constants. Additionally, if the elastic constants are also\nprovided in AELAS format [4], then it will also calculate the anisotropic magne-\ntostrictive coefficients ( l) from the theoretical equations l(bk;Ci j)given in Ref.\n[1]. The calculated magnetostrictive coefficients can be analyzed and visualized\nwith the online tool MAELASviewer [8]. This new method is executed by using\ntag -mode 2 in the command line. The length optimization method [2] originally\nimplemented in version 1.0 for direct calculation of anisotropic magnetostrictive\ncoefficients [1] can also be executed in version 2.0 by using tag -mode 1, see Fig.2.\n2.2. Revision of the trigonal (I) symmetry\nThe analysis of the accuracy of MAELAS that is presented in Section 3 helped\nus to identify some issues in version 1.0 [1] related to the implementation of the\nlength optimization method (originally proposed by Wu and Freeman [2]) for the\ntrigonal (I) symmetry. First of all, we point out a misprint in the publication of\nversion 1.0 [1] in the elastic energy that should read\nEtrig(I)\nel\u0000E0\nV0=1\n2cxxxx(e2\nxx+e2\nyy)+cxxyyexxeyy+cxxzz(exx+eyy)ezz+1\n2czzzze2\nzz\n+2cyzyz(e2\nxz+e2\nyz)+(cxxxx\u0000cxxyy)e2\nxy\n+cxxyz(4exyexz+2exxeyz\u00002eyyeyz):\n(8)\n7Figure 1: Workflow of the new methodology for direct calculation of anisotropic magnetoelastic\nconstants implemented in MAELAS v2.0. This new method is executed by tag -mode 2. The\nmethod originally implemented in version 1.0 for direct calculation of anisotropic magnetostrictive\ncoefficients [1] based on the length optimization [2] can also be executed in version 2.0 by using\ntag -mode 1.\nwhere cxxxx=C11,cxxyy=C12,cxxzz=C13,cxxyz=C14,czzzz=C33, and cyzyz=C44.\nSecondly, the correct theoretical relations between the magnetostrictive coeffi-\n8Figure 2: Diagram showing the two methods available in MAELAS v2.0 to calculate anisotropic\nmagnetostrictive coefficients ( l) and magnetoelastic constants ( b). The method -mode 1 corre-\nsponds to the approach originally implemented in version 1.0 [1] based on the length optimization\n[2], while -mode 2 is the new method added in version 2.0 (Section 2.1).\ncients lg;1,lg;2andl21, and the elastic and magnetoelastic constants are\nlg;1=1\n2C14b14\u00001\n2C44b3\n1\n2C44(C11\u0000C12)\u0000C2\n14;\nlg;2=\u00001\n2b4(C11\u0000C12)+b34C14\n1\n2C44(C11\u0000C12)\u0000C2\n14;\nl21=\u00001\n2b14(C11\u0000C12)+b3C14\n1\n2C44(C11\u0000C12)\u0000C2\n14:(9)\nLastly, based on the equation of the relative length change Dl=l0, for the calcu-\nlation l12in version 1.0 we proposed to compute the cell length in the direction\nbbb= (a=p\na2+c2;0;c=p\na2+c2)using the deformation gradient [1]\nFFF\f\f\fl12(version 1:0)\nbbb=(a;0;c)p\na2+c2(s) =W0\n@1 0sc\n2a\n0 1 0\nsa\n2c0 11\nA: (10)\nwhere W=3p\n4=(4\u0000s2),aandcare the lattice parameters of the relaxed (not\ndeformed) unit cell. However, this choice leads to the same theoretical total en-\nergy for the selected magnetization directions aaa1= (0;1=p\n2;1=p\n2)andaaa2=\n(0;1=p\n2;\u00001=p\n2), that is, E(eeel12;aaa1) =E(eeel12;aaa2). Consequently, in practice\nthis would always give very similar equilibrium lengths l1andl2for the cases with\n9aaa1andaaa2, respectively. Thus, a negligible magnetostrictive coefficient l12(l12\u0018=\n0) will be obtained always. To fix this problem in version 2.0, we use the same\nmagnetization directions aaa1= (0;1=p\n2;1=p\n2)andaaa2= (0;1=p\n2;\u00001=p\n2), and\nreplace the measuring length direction by bbb= (1;0;0), and deformation gradient\nby\nFFF\f\f\fl12(version 2:0)\nbbb=(1;0;0)(s) =0\nB@1+s 0 0\n01p1+s0\n0 01p1+s1\nCA: (11)\nIn this case, we have\nl12=2(l1\u0000l2)\nr(l1+l2)=4(l1\u0000l2)\n(l1+l2); (12)\nwhere r=1=2, and l1andl2are the equilibrium cell length along bbb= (1;0;0)\nwhen the magnetization points to aaa1= (0;1=p\n2;1=p\n2)andaaa2= (0;1=p\n2;\u00001=p\n2),\nrespectively.\n2.3. The demagnetized state and its role in the fractional change in length\nThe magnetostrictive coefficients calculated with MAELAS correspond to the\nvalues at the magnetic saturated state. In experiment, these coefficients can be\nobtained by measuring the change in length of the material under saturation with\nrespect to an initial length at a reference demagnetized state. Thus, the measured\nchange in length, as well as the form of the equation describing the fractional\nchange in length, depend on the initial demagnetized state that is used as a ref-\nerence state. Unfortunately, this state may not correspond to a unique reference\nstate since it could have different distribution of magnetic domains [3]. This fact\nis important to take into account in order to compare experiment and theory prop-\nerly, see the analysis of L1 0FePd in Section 4.5. In this section, based on Birss’s\nwork [3], we analyze the influence of the demagnetized state on the fractional\nchange in length for the crystals supported by MAELAS. To facilitate the com-\nparison between theory and experiment we included the possibility to choose the\ntype of reference demagnetized state in the calculation of the fractional change in\nlength in the visualization tool MAELASviewer [8]. On the other hand, in ver-\nsion 1.0 we already implemented the expressions of fractional change in length\nfor polycrystals with cubic (I) and hexagonal (I) symmetries [1, 3]. Here, we\nderive the corresponding equation to compute the saturated magnetostriction for\npolycrystals with trigonal (I), tetragonal (I) and orthorhombic symmetries, which\nare implemented in version 2.0.\n102.3.1. Cubic (I)\nLet’s consider a single crystal with Cubic (I) symmetry with length l0at an ar-\nbitrary direction bbbin an initial demagnetized state with randomly oriented atomic\nmagnetic moments. After the magnetization of the crystal is saturated in the di-\nrection aaa, the length in the direction bbbisl, see Fig.3. In this case we have the\nfollowing fractional change in length [1]\nl\u0000l0\nl0\f\f\f\f\faaa\nbbb=la+3\n2l001\u0012\na2\nxb2\nx+a2\nyb2\ny+a2\nzb2\nz\u00001\n3\u0013\n+3l111(axaybxby+ayazbybz+axazbxbz):(13)\nThe ideal demagnetized state with randomly oriented atomic magnetic moments\ncan not be achieved in experiment below the Curie temperature ( TC) due to the\nferromagnetic order. The lattice parameters for single crystal at this hypothet-\nical demagnetized state below TCcan be experimentally estimated by extrapo-\nlating the lattice parameters above TCvia the Debye theory and the Grüneisen\nrelation [9]. In practice, the real experimental demagnetized states below TC\nexhibit different distribution of magnetic domains that depend on the demagne-\ntization method [3]. Thus, it is convenient to define a standard reference state\nas a demagnetized state with many ferromagnetic domains, where there are the\nsame number of domains oriented in each of the crystallographically equiva-\nlent directions of easy magnetization [3]. For instance, if the easy directions\nare the quaternary axes <100>, as in BCC Fe, then we have six easy direc-\ntionsaaa= (\u00061;0;0),aaa= (0;\u00061;0)andaaa= (0;0;\u00061). On the other hand, if the\neasy directions are the ternary axes <111>, as in FCC Ni, then we have eight\neasy directions aaa= (\u00061=p\n3;\u00061=p\n3;\u00061=p\n3),aaa= (\u00071=p\n3;\u00061=p\n3;\u00061=p\n3),\naaa= (\u00061=p\n3;\u00071=p\n3;\u00061=p\n3)andaaa= (\u00061=p\n3;\u00061=p\n3;\u00071=p\n3). These de-\nmagnetized states are schematically represented in Fig. 3, where the length in the\ndirection bbbis labeled with l0\n0andl00\n0for easy ternary and quaternary axes, respec-\ntively.\nIf the initial state is the ideal demagnetized state with randomly oriented atomic\nmagnetic moments and the final state is the demagnetized state with the same\nnumber of domains oriented in each of the easy ternary directions, then the frac-\n11Figure 3: (a) Ideal demagnetized state of a cubic single crystal with randomly oriented atomic\nmagnetic moments. Demagnetized state with magnetic domains along all possible easy directions\n(b)<100>and (c) <111>. (d) Saturated magnetic state.\ntional change in length can be written as\nl0\n0\u0000l0\nl0\f\f\f\f\f\nbbb=N\nå\ni=1l0\n0;i\u0000N\nå\ni=1l0;i\nN\nå\ni=1l0;i\f\f\f\f\f\nbbb=N\nå\ni=1l0;i\nl00\n@l0\n0;i\u0000l0;i\nl0;i\f\f\f\f\faaai\nbbb1\nA\n=6\nå\nj=1N(aaaj)\nå\ni=1l0;i\nl00\n@l0\n0;i\u0000l0;i\nl0;i\f\f\f\f\faaaj\nbbb1\nA;(14)\nwhere Nis the total number of magnetic domains within a cross section along bbb,\nas shown in Fig.3, and N(aaaj)is the number of magnetic domains with the same\neasy magnetization direction aaajjj. In the last step, we grouped the domains with\nequal easy magnetization direction together. Each term in the last summation can\nbe computed by applying Eq.13 to each magnetic domain individually\n6\nå\nj=1N(aaaj)\nå\ni=1l0;i\nl00\n@l0\n0;i\u0000l0;i\nl0;i\f\f\f\f\faaaj\nbbb1\nA=la+3\n2l001(b2\nx\"N(aaa1)\nå\ni=1l0;i\nl0#\n+b2\nx\"N(aaa2)\nå\ni=1l0;i\nl0#\n+b2\ny\"N(aaa3)\nå\ni=1l0;i\nl0#\n+b2\ny\"N(aaa4)\nå\ni=1l0;i\nl0#\n+b2\nz\"N(aaa5)\nå\ni=1l0;i\nl0#\n+b2\nz\"N(aaa6)\nå\ni=1l0;i\nl0#\n\u00001\n3);(15)\n12where aaa1= (1;0;0),aaa2= (\u00001;0;0),aaa3= (0;1;0),aaa4= (0;\u00001;0),aaa5= (0;0;1)\nandaaa6= (0;0;\u00001). Next, we assume the same contribution of the sum of all mag-\nnetic domains with equal easy magnetization direction to the total length (equally\nweighted domain distribution [3]), that is,\nN(aaaj)\nå\ni=1l0;i\nl0=1\nNe;j=1;::;Ne (16)\nwhere Neis the number of easy directions for magnetization. For example, for\nquaternary axes we have Ne=6. Using Eq.16 in Eq.15 gives\n6\nå\nj=1N(aaaj)\nå\ni=1l0;i\nl00\n@l0\n0;i\u0000l0;i\nl0;i\f\f\f\f\faaaj\nbbb1\nA=la: (17)\nAnd, replacing Eq. 17 in Eq.14 yields\nl0\n0\u0000l0\nl0\f\f\f\f\f\nbbb=la: (18)\nHence, only the isotropic magnetostriction related to the exchange interaction\n(volume magnetostriction ws'3la[9, 10]) takes place effectively. We may write\nthis calculation in the following general and compact form\nl0\n0\u0000l0\nl0\f\f\f\f\f\nbbb=1\nNeNe\nå\nj=1l\u0000l0\nl0\f\f\f\f\faaaj\nbbb; (19)\nwhere each term inside the summation corresponds to the Eq.13 evaluated at the\neasy direction aaaj. On the other hand, if the initial state is the demagnetized state\nwith the same number of domains oriented in each of the easy ternary directions\nand the final state is the saturated state with magnetization along aaa, then the frac-\ntional change in length is\nl\u0000l0\n0\nl0\n0\f\f\f\f\faaa\nbbb=l\u0000l0\n0\nl0\u00011\n1\u0000\u0010l0\u0000l0\n0\nl0\u0011\f\f\f\f\faaa\nbbb=l\u0000l0\n0\nl0\u0014\n1+\u0012l0\u0000l0\n0\nl0\u0013\n+:::\u0015\f\f\f\f\faaa\nbbb\n\u0018=l\u0000l0\n0\nl0\n0\f\f\f\f\faaa\nbbb=l\u0000l0\nl0\f\f\f\f\faaa\nbbb\u0000l0\n0\u0000l0\nl0\f\f\f\f\f\nbbb\n=3\n2l001\u0012\na2\nxb2\nx+a2\nyb2\ny+a2\nzb2\nz\u00001\n3\u0013\n+3l111(axaybxby+ayazbybz+axazbxbz):(20)\n13Thus, in this case only the anisotropic magnetostriction related to the SOC and\ncrystal field takes place effectively. The same result is found for the case with\neasy quaternary directions\nl00\n0\u0000l0\nl0\f\f\f\f\f\nbbb=la;\nl\u0000l00\n0\nl00\n0\f\f\f\f\faaa\nbbb=3\n2l001\u0012\na2\nxb2\nx+a2\nyb2\ny+a2\nzb2\nz\u00001\n3\u0013\n+3l111(axaybxby+ayazbybz+axazbxbz):(21)\nHowever, we point out that including higher order terms in Eq.13 leads to different\nresults between easy ternary and quaternary directions [3]. We see that using\nthe initial demagnetized state with domains pointing to all easy directions as a\nreference state allows to characterize the fractional change in length without the\nneed to measure or calculate the isotropic magnetostrictive coefficient la.\nTo compute the fractional change in length for a polycrystal, one needs to\nperform the following average of the corresponding fractional change in length\nfor a single crystal (uniform stress approximation)[3]\n<f(aaa;bbb)>=1\n8pZp\n0Z2p\n0Z2p\n0f(aaa0;bbb0)sinqdqdfdy; (22)\nwhere the magnetization and measuring length directions inside the integral are\nparameterized as\na0\nx=cosQsinqcosf+sinQ(cosqcosfcosy+sinfsiny);\na0\ny=cosQsinqsinf+sinQ(cosqsinfcosy\u0000cosfsiny);\na0\nz=cosQcosq\u0000sinQsinqcosy;\nb0\nx=sinqcosf;\nb0\ny=sinqsinf;\nb0\nz=cosq;(23)\nwhere Qis the angle between aaaandbbb. For instance, inserting the Eq.13 in Eq.22\nyields\n<l\u0000l0\nl0\f\f\f\f\faaa\nbbb>=x+h(aaa\u0001bbb)2=la+3\n2lS\u0014\n(aaa\u0001bbb)2\u00001\n3\u0015\n; (24)\n14where\nx=la\u00001\n2lS;\nh=3\n2lS;\nlS=2\n5l001+3\n5l111:(25)\nIf we consider that the initial state of the polycrystal is the demagnetized state with\nthe same number of domains oriented in each of the easy ternary or quaternary\ndirections, then using Eq.22 we find\n<l\u0000l0\n0\nl0\n0\f\f\f\f\faaa\nbbb>=<l\u0000l00\n0\nl00\n0\f\f\f\f\faaa\nbbb>=x+h(aaa\u0001bbb)2=3\n2lS\u0014\n(aaa\u0001bbb)2\u00001\n3\u0015\n; (26)\nwhere\nx=\u00001\n2lS;\nh=3\n2lS;\nlS=2\n5l001+3\n5l111:(27)\nRecently, we verified the Eq.24 by means of finite element methods [11].\n2.3.2. Hexagonal (I)\nLet’s now consider a single crystal with Hexagonal (I) symmetry with length l0\nin an arbitrary direction bbbat an initial demagnetized state with randomly oriented\natomic magnetic moments. After the magnetization of the crystal is saturated in\nthe direction aaa, the length in the direction bbbisl, see Fig.5. In this case we have\nthe following fractional change in length using Clark’s notation [1, 12]\nl\u0000l0\nl0\f\f\f\f\faaa\nbbb=la1;0(b2\nx+b2\ny)+la2;0b2\nz+la1;2\u0012\na2\nz\u00001\n3\u0013\n(b2\nx+b2\ny)\n+la2;2\u0012\na2\nz\u00001\n3\u0013\nb2\nz+lg;2\u00141\n2(a2\nx\u0000a2\ny)(b2\nx\u0000b2\ny)+2axaybxby\u0015\n+2le;2(axazbxbz+ayazbybz);(28)\nAs in the cubic case, we define a standard reference state as a demagnetized state\nwith many ferromagnetic domains, where there are the same number of domains\n15oriented in each of the crystallographically equivalent directions of easy magne-\ntization [3]. For uniaxial crystals (hexagonal, trigonal and tetragonal symmetries)\nthere are three types of easy directions: easy axis, easy plane and easy cone [13],\nsee Fig.4.\nFigure 4: Phase diagram of magnetocrystalline anisotropy energy for uniaxial crystals. A\nmetastable anisotropy occurs when K1>0 and K2<\u0000K1=2. In this case, easy axis and easy\nplane are in coexistence, one is an absolute minimum and the other one is a local minimum.\nThe fractional change in length for hexagonal (I) crystals assuming an initial\ndemagnetized state with domains pointing to all possible easy directions for easy\naxis and easy plane was previously derived by Birss [3]. Here, we reproduce\nBirss’s results using Clark’s notation, and include the case with easy cone. The\nconsidered demagnetized states are schematically represented in Fig. 5, where the\nlength in the direction bbbis labeled with l0\n0,l00\n0andl000\n0for the cases with easy axis,\neasy plane and easy cone, respectively.\nFor the case with easy axis, there are two easy directions ( Ne=2) along the\nlattice vector ccc, parallel to the z-axis, aaa1= (0;0;1)andaaa2= (0;0;\u00001). Hence,\napplying the general formula given by Eq.21 we obtain\nl0\n0\u0000l0\nl0\f\f\f\f\f\nbbb=1\n22\nå\nj=1l\u0000l0\nl0\f\f\f\f\faaaj\nbbb\n=la1;0(b2\nx+b2\ny)+la2;0b2\nz+2\n3la1;2(b2\nx+b2\ny)+2\n3la2;2b2\nz:(29)\nHence, if the initial state is the demagnetized state with the same number of do-\nmains oriented in each of the easy directions aaa1= (0;0;1)andaaa2= (0;0;\u00001),\n16Figure 5: (a) Ideal demagnetized state of a hexagonal single crystal with randomly oriented atomic\nmagnetic moments. Demagnetized state with magnetic domains along all possible easy directions:\n(b) easy axis, (c) easy plane and (d) easy cone. (e) Saturated magnetic state.\nand the final state is the saturated state with magnetization along aaa, then the frac-\ntional change in length is\nl\u0000l0\n0\nl0\n0\f\f\f\f\faaa\nbbb\u0018=l\u0000l0\nl0\f\f\f\f\faaa\nbbb\u0000l0\n0\u0000l0\nl0\f\f\f\f\f\nbbb=la1;2\u0000\na2\nz\u00001\u0001\n(b2\nx+b2\ny)\n+la2;2\u0000\na2\nz\u00001\u0001\nb2\nz+lg;2\u00141\n2(a2\nx\u0000a2\ny)(b2\nx\u0000b2\ny)+2axaybxby\u0015\n+2le;2(axazbxbz+ayazbybz):(30)\nIn the case of easy plane, the number of easy directions is Ne=2p. Thus, we\nshould formally replace the summation in Eq.21 by an integral as\nl00\n0\u0000l0\nl0\f\f\f\f\f\nbbb=1\n2pZ2p\n0l\u0000l0\nl0\f\f\f\f\faaa=(cosq;sinq;0)\nbbbdq\n=la1;0(b2\nx+b2\ny)+la2;0b2\nz\u00001\n3la1;2(b2\nx+b2\ny)\u00001\n3la2;2b2\nz:(31)\nTherefore, if the initial state is the demagnetized state with the same number of\ndomains oriented in each of the easy plane directions, and the final state is the\n17saturated state with magnetization along aaa, then the fractional change in length is\nl\u0000l00\n0\nl00\n0\f\f\f\f\faaa\nbbb\u0018=l\u0000l0\nl0\f\f\f\f\faaa\nbbb\u0000l00\n0\u0000l0\nl0\f\f\f\f\f\nbbb=la1;2a2\nz(b2\nx+b2\ny)+la2;2a2\nzb2\nz\n+lg;2\u00141\n2(a2\nx\u0000a2\ny)(b2\nx\u0000b2\ny)+2axaybxby\u0015\n+2le;2(axazbxbz+ayazbybz):(32)\nThe results obtained by Birss[3] up to second order are recovered by using in the\nabove equations the conversion formulas between Clark and Birss definition of\nthe magnetostrictive coefficients [1, 12]\nQ0=la1;0+2\n3la1;2\nQ1=la2;0+2\n3la2;2\u0000la1;0\u00002\n3la1;2\nQ2=\u0000la1;2\u00001\n2lg;2\nQ4=la1;2+1\n2lg;2\u0000la2;2\nQ6=2le;2\nQ8=lg;2:(33)\nThe magnetocrystalline anisotropy with easy cone is obtained when K1<0\nandK2>\u0000K1=2 [13]. The cone angle Wwith respect to the z-axis is given by\n[13]\nW=arcsins\njK1j\n2K2(34)\nIn this case, the total number of easy directions is Ne=4psinW. Formally, we\n18should replace the summation in Eq.21 by an integral as\nl000\n0\u0000l0\nl0\f\f\f\f\f\nbbb=1\n4psinWZ2p\n0l\u0000l0\nl0\f\f\f\f\faaa=(sinWcosq;sinWsinq;cosW)\nbbbsinWdq\n+1\n4psinWZ2p\n0l\u0000l0\nl0\f\f\f\f\faaa=(sinWcosq;sinWsinq;\u0000cosW)\nbbbsinWdq\n=la1;0(b2\nx+b2\ny)+la2;0b2\nz\n+\u0012\ncos2W\u00001\n3\u0013\nla1;2(b2\nx+b2\ny)+\u0012\ncos2W\u00001\n3\u0013\nla2;2b2\nz:(35)\nWe see that setting the cone angle to zero ( W=0) in Eq.35 leads to the same result\nas in Eq.29 for case with the easy axis. Similarly, if we set the cone angle W=p=2,\nthen we recover the result for the case with easy plane given by Eq.31. Next, if the\ninitial state is the demagnetized state with the same number of domains oriented\nin each of the easy cone directions, and the final state is the saturated state with\nmagnetization along aaa, then the fractional change in length is\nl\u0000l000\n0\nl000\n0\f\f\f\f\faaa\nbbb\u0018=l\u0000l0\nl0\f\f\f\f\faaa\nbbb\u0000l000\n0\u0000l0\nl0\f\f\f\f\f\nbbb\n=\u0000\na2\nz\u0000cos2W\u0001\nla1;2(b2\nx+b2\ny)+\u0000\na2\nz\u0000cos2W\u0001\nla2;2b2\nz\n+lg;2\u00141\n2(a2\nx\u0000a2\ny)(b2\nx\u0000b2\ny)+2axaybxby\u0015\n+2le;2(axazbxbz+ayazbybz):(36)\nWe see that assuming easy cone directions in the initial demagnetized state allows\nto derive general and compact formulas of the fractional change in length for\nuniaxial crystals that include the easy axis and easy plane as particular cases when\nthe cone angle is W=0 and W=p=2, respectively. For instance, Eqs.30 and 32\nare recovered by setting W=0 and W=p=2 in Eq.36, respectively. Similarly,\n19using Mason’s definitions of the magnetostrictive coefficients[1, 14], we find\nl\u0000l000\n0\nl000\n0\f\f\f\f\faaa\nbbb=lA[(axbx+ayby)2\u0000(axbx+ayby)azbz\u00001\n2(b2\nx+b2\ny)sin2W]\n+lB[(cos2W\u0000a2\nz)(1\u0000b2\nz)\u0000(axbx+ayby)2+1\n2(b2\nx+b2\ny)sin2W]\n+lC[(cos2W\u0000a2\nz)b2\nz\u0000(axbx+ayby)azbz]+4lD(axbx+ayby)azbz;\n(37)\nwhere the original result derived by Mason[14], assuming a demagnetized con-\ndition with equal numbers of domains directed along the z-axis (easy axis), is\nrecovered by setting the cone angle W=0.\nLastly, let’s compute the fractional change in length for polycrystals with\nhexagonal (I) crystal symmetry. If we consider a reference initial demagnetized\nstate with randomly oriented atomic magnetic moments and final saturated state,\nthen we should replace Eq.28 in Eq.22\n<l\u0000l0\nl0\f\f\f\f\faaa\nbbb>=x+h(aaa\u0001bbb)2; (38)\nwhere\nx=10\n15la1;0+1\n3la2;0+2\n45la1;2\u00002\n45la2;2\u00002\n15le;2\u00002\n15lg;2;(39)\nh=\u00002\n15la1;2+2\n15la2;2+2\n5le;2+2\n5lg;2: (40)\nFor the case with a reference initial demagnetized state made by domains aligned\nto all possible easy directions and final saturated state, we only need to compute\nthe crystal with easy cone because it includes the easy axis and easy plane as\nparticular cases of the cone angle. Thus, inserting Eq.36 in Eq.22 gives\n<l\u0000l000\n0\nl000\n0\f\f\f\f\faaa\nbbb>=x(W)+h(aaa\u0001bbb)2; (41)\nwhere\nx(W) =4\n15la1;2+1\n15la2;2\u00002\n15le;2\u00002\n15lg;2\u00001\n3(2la1;2+la2;2)cos2W;\nh=\u00002\n15la1;2+2\n15la2;2+2\n5le;2+2\n5lg;2: (42)\n20The cases with easy axis and easy plane are obtained by setting the cone angle\nW=0 and W=p=2 in Eq.42, respectively. By doing so, one recovers the results\nderived by Birss [1, 3] with the help of the conversion formulas between Clark\nand Birss definitions given by Eq.33.\n2.3.3. Trigonal (I)\nThe case with trigonal (I) symmetry is very similar to the hexagonal one since\nboth are uniaxial crystals. Thus, we use the same notation as shown in Fig.5.\nLet’s first consider a single crystal with Trigonal (I) symmetry with length l0in\nan arbitrary direction bbbin an initial demagnetized state with randomly oriented\natomic magnetic moments. After the magnetization of the crystal is saturated in\nthe direction aaa, the length in the direction bbbisl. In this case we have the following\nfractional change in length [1, 15]\nl\u0000l0\nl0\f\f\f\f\faaa\nbbb=la1;0(b2\nx+b2\ny)+la2;0b2\nz+la1;2\u0012\na2\nz\u00001\n3\u0013\n(b2\nx+b2\ny)\n+la2;2\u0012\na2\nz\u00001\n3\u0013\nb2\nz+lg;1\u00141\n2(a2\nx\u0000a2\ny)(b2\nx\u0000b2\ny)+2axaybxby\u0015\n+lg;2(axazbxbz+ayazbybz)+l12\u00141\n2ayaz(b2\nx\u0000b2\ny)+axazbxby\u0015\n+l21\u00141\n2(a2\nx\u0000a2\ny)bybz+axaybxbz\u0015\n:\n(43)\nFollowing the same analysis as in the case of hexagonal (I) symmetry, we find\nthat assuming a reference initial demagnetized state with domains pointing to all\npossible easy cone directions and a final saturated state leads to the fractional\n21change in length\nl\u0000l000\n0\nl000\n0\f\f\f\f\faaa\nbbb=la1;2\u0000\na2\nz\u0000cos2W\u0001\n(b2\nx+b2\ny)+la2;2\u0000\na2\nz\u0000cos2W\u0001\nb2\nz\n+lg;1\u00141\n2(a2\nx\u0000a2\ny)(b2\nx\u0000b2\ny)+2axaybxby\u0015\n+lg;2(axazbxbz+ayazbybz)+l12\u00141\n2ayaz(b2\nx\u0000b2\ny)+axazbxby\u0015\n+l21\u00141\n2(a2\nx\u0000a2\ny)bybz+axaybxbz\u0015\n;\n(44)\nwhere the cases with easy axis and easy plane are obtained by setting the cone\nangle W=0 and W=p=2, respectively.\nIn the case of polycrystals, if we consider an initial demagnetized state with\nrandomly oriented atomic magnetic moments and final saturated state, then we\nfind\n<l\u0000l0\nl0\f\f\f\f\faaa\nbbb>=x+h(aaa\u0001bbb)2; (45)\nwhere\nx=10\n15la1;0+1\n3la2;0+2\n45la1;2\u00002\n45la2;2\u00002\n15lg;1\u00001\n15lg;2;(46)\nh=\u00002\n15la1;2+2\n15la2;2+2\n5lg;1+1\n5lg;2: (47)\nOn the other hand, if the reference initial state of the polycrystal is a demagnetized\nstate with domains pointing to all easy cone directions and the final state is the\nsaturated state, then we obtain\n<l\u0000l000\n0\nl000\n0\f\f\f\f\faaa\nbbb>=x(W)+h(aaa\u0001bbb)2; (48)\nwhere\nx(W) =4\n15la1;2+1\n15la2;2\u00002\n15lg;1\u00001\n15lg;2\u00001\n3(2la1;2+la2;2)cos2W;\nh=\u00002\n15la1;2+2\n15la2;2+2\n5lg;1+1\n5lg;2: (49)\n22Again, the corresponding cases with easy axis and easy plane are obtained by\nsetting the cone angle W=0 and W=p=2, respectively. Interestingly, we see that\nxandhdo not depend on l12andl21since their angular dependence gives zero\nin the integration of Eq.22. Further studies are needed to clarify the validity of\nthis result.\n2.3.4. Tetragonal (I)\nThe case with tetragonal (I) symmetry is also very similar to the hexagonal\nand trigonal ones. Again, we use the same notation as shown in Fig.5. Let’s\nstart by considering a single crystal with Tetragonal (I) symmetry with length l0\nin an arbitrary direction bbbin an initial demagnetized state with randomly oriented\natomic magnetic moments. After the magnetization of the crystal is saturated in\nthe direction aaa, the length in the direction bbbisl. In this case we have the following\nfractional change in length [1, 15]\nl\u0000l0\nl0\f\f\f\f\faaa\nbbb=la1;0(b2\nx+b2\ny)+la2;0b2\nz+la1;2\u0012\na2\nz\u00001\n3\u0013\n(b2\nx+b2\ny)\n+la2;2\u0012\na2\nz\u00001\n3\u0013\nb2\nz+1\n2lg;2(a2\nx\u0000a2\ny)(b2\nx\u0000b2\ny)+2ld;2axaybxby\n+2le;2(axazbxbz+ayazbybz);(50)\nFollowing the same analysis as in the case of hexagonal (I) crystal, we find that\nassuming a reference initial demagnetized state with domains pointing to all pos-\nsible easy cone directions and a final saturated state leads to the fractional change\nin length\nl\u0000l000\n0\nl000\n0\f\f\f\f\faaa\nbbb=la1;2\u0000\na2\nz\u0000cos2W\u0001\n(b2\nx+b2\ny)+la2;2\u0000\na2\nz\u0000cos2W\u0001\nb2\nz\n+1\n2lg;2(a2\nx\u0000a2\ny)(b2\nx\u0000b2\ny)+2ld;2axaybxby\n+2le;2(axazbxbz+ayazbybz);(51)\nwhere the cases with easy axis and easy plane are obtained by setting the cone\nangle W=0 and W=p=2, respectively.\nIn the case of polycrystals, if we consider an initial demagnetized state with\nrandomly oriented atomic magnetic moments and final saturated state, then we\n23find\n<l\u0000l0\nl0\f\f\f\f\faaa\nbbb>=x+h(aaa\u0001bbb)2; (52)\nwhere\nx=10\n15la1;0+1\n3la2;0+2\n45la1;2\u00002\n45la2;2\u00002\n15le;2\u00001\n15lg;2\u00001\n15ld;2;(53)\nh=\u00002\n15la1;2+2\n15la2;2+2\n5le;2+1\n5lg;2+1\n5ld;2: (54)\nOn the other hand, if the reference initial state of the polycrystal is a demagnetized\nstate with domains pointing to all easy cone directions and the final state is the\nsaturated state, then we obtain\n<l\u0000l000\n0\nl000\n0\f\f\f\f\faaa\nbbb>=x(W)+h(aaa\u0001bbb)2; (55)\nwhere\nx(W) =4\n15la1;2+1\n15la2;2\u00002\n15le;2\u00001\n15lg;2\u00001\n15ld;2\n\u00001\n3(2la1;2+la2;2)cos2W;\nh=\u00002\n15la1;2+2\n15la2;2+2\n5le;2+1\n5lg;2+1\n5ld;2: (56)\nAgain, the corresponding cases with easy axis and easy plane are obtained by\nsetting the cone angle W=0 and W=p=2, respectively.\n2.3.5. Orthorhombic\nNow, we analyze the case of Orthorhombic crystals. Let’s consider a single\ncrystal with Orthorhombic symmetry and length l0in an arbitrary direction bbbin\nan initial demagnetized state with randomly oriented atomic magnetic moments.\nAfter the magnetization of the crystal is saturated in the direction aaa, the length\nin the direction bbbisl, see Fig.7. In this case, we have the following fractional\n24change in length [1, 14]\nl\u0000l0\nl0\f\f\f\f\faaa\nbbb=la1;0b2\nx+la2;0b2\ny+la3;0b2\nz+l1(a2\nxb2\nx\u0000axaybxby\u0000axazbxbz)\n+l2(a2\nyb2\nx\u0000axaybxby)+l3(a2\nxb2\ny\u0000axaybxby)\n+l4(a2\nyb2\ny\u0000axaybxby\u0000ayazbybz)+l5(a2\nxb2\nz\u0000axazbxbz)\n+l6(a2\nyb2\nz\u0000ayazbybz)+4l7axaybxby+4l8axazbxbz+4l9ayazbybz:\n(57)\nAs in the previous crystal symmetries, we define a standard reference state as a\ndemagnetized state with many ferromagnetic domains, where there are the same\nnumber of domains oriented in each of the crystallographically equivalent direc-\ntions of easy magnetization [3].\nFigure 6: Phase diagram of magnetocrystalline anisotropy energy for orthorhombic crystals.\nThe magnetocrystalline anisotropy energy in an unstrained orthorhombic crys-\ntal up to second-order in ais[1, 14]\nE0\nK\nV0=K0+K1a2\nx+K2a2\ny: (58)\nThe analysis of this equation shows that there are two types of easy directions:\neasy axis and easy plane. However, the case with easy axis is more relevant since\nthe easy plane is a singular transition state between two different easy axis direc-\ntions, see Fig.6. Thus, here we only study the three possible easy axis directions:\neasy axis along the lattice vector cccparallel to z-axis, easy axis along the lattice\n25vector aaaparallel to x-axis, and easy axis along the lattice vector bbbparallel to y-\naxis. Hence, in each case there are two easy directions Ne=2. The considered\ndemagnetized states are schematically represented in Fig. 7, where the length in\nthe direction bbbis labeled with l0\n0,l00\n0andl000\n0for the cases with easy axis along the\nlattice vector ccc,aaaandbbb, respectively. We assume the same IEEE lattice conven-\ntion ( c<a<b) that is used in MAELAS and AELAS [4].\nFigure 7: (a) Ideal demagnetized state of a orthorhombic single crystal with randomly oriented\natomic magnetic moments. Demagnetized state with magnetic domains along all possible easy\ndirections: (b) easy axis along the lattice vector cccparallel to z-axis, (c) easy axis along the lat-\ntice vector aaaparallel to x-axis and (d) easy axis along the lattice vector bbbparallel to y-axis. (e)\nSaturated magnetic state. MAELAS uses the IEEE lattice convention c<a<b.\nApplying the same procedure used for cubic and hexagonal crystals, we obtain\nthe following fractional change in length assuming an initial demagnetized state\nwith domains pointing to all easy directions parallel to the lattice vector cccand\nfinal saturated state\nl\u0000l0\n0\nl0\n0\f\f\f\f\faaa\nbbb\u0018=l\u0000l0\nl0\f\f\f\f\faaa\nbbb\u0000l0\n0\u0000l0\nl0\f\f\f\f\f\nbbb\n=l1(a2\nxb2\nx\u0000axaybxby\u0000axazbxbz)\n+l2(a2\nyb2\nx\u0000axaybxby)+l3(a2\nxb2\ny\u0000axaybxby)\n+l4(a2\nyb2\ny\u0000axaybxby\u0000ayazbybz)+l5(a2\nxb2\nz\u0000axazbxbz)\n+l6(a2\nyb2\nz\u0000ayazbybz)+4l7axaybxby+4l8axazbxbz+4l9ayazbybz:\n(59)\nThus, we recover the result originally derived by Mason [14]. In the case with\n26easy axis parallel to the lattice vector aaawe have\nl\u0000l00\n0\nl00\n0\f\f\f\f\faaa\nbbb\u0018=l\u0000l0\nl0\f\f\f\f\faaa\nbbb\u0000l00\n0\u0000l0\nl0\f\f\f\f\f\nbbb\n=l1([a2\nx\u00001]b2\nx\u0000axaybxby\u0000axazbxbz)\n+l2(a2\nyb2\nx\u0000axaybxby)+l3([a2\nx\u00001]b2\ny\u0000axaybxby)\n+l4(a2\nyb2\ny\u0000axaybxby\u0000ayazbybz)+l5([a2\nx\u00001]b2\nz\u0000axazbxbz)\n+l6(a2\nyb2\nz\u0000ayazbybz)+4l7axaybxby+4l8axazbxbz+4l9ayazbybz;\n(60)\nwhile for the case of easy axis parallel to the lattice vector bbbwe find\nl\u0000l000\n0\nl000\n0\f\f\f\f\faaa\nbbb\u0018=l\u0000l0\nl0\f\f\f\f\faaa\nbbb\u0000l000\n0\u0000l0\nl0\f\f\f\f\f\nbbb\n=l1(a2\nxb2\nx\u0000axaybxby\u0000axazbxbz)\n+l2([a2\ny\u00001]b2\nx\u0000axaybxby)+l3(a2\nxb2\ny\u0000axaybxby)\n+l4([a2\ny\u00001]b2\ny\u0000axaybxby\u0000ayazbybz)+l5(a2\nxb2\nz\u0000axazbxbz)\n+l6([a2\ny\u00001]b2\nz\u0000ayazbybz)+4l7axaybxby+4l8axazbxbz\n+4l9ayazbybz:(61)\nIn the case of polycrystals, if we consider an initial demagnetized state with\nrandomly oriented atomic magnetic moments and final saturated state, then we\nfind\n<l\u0000l0\nl0\f\f\f\f\faaa\nbbb>=x+h(aaa\u0001bbb)2; (62)\nwhere\nx=1\n3(la1;0+la2;0+la3;0)+2\n15(l1+l4\u0000l7\u0000l8\u0000l9)\n+1\n6(l2+l3+l5+l6); (63)\nh=\u00001\n15l1\u00001\n15l4\u00001\n6(l2+l3+l5+l6)+2\n5(l7+l8+l9):(64)\nOn the other hand, if the reference initial state of the polycrystal is a demagnetized\nstate with domains pointing to all easy directions along the lattice vector cccand the\n27final state is the saturated state, then we obtain\n<l\u0000l0\n0\nl0\n0\f\f\f\f\faaa\nbbb>=x+h(aaa\u0001bbb)2; (65)\nwhere\nx=2\n15(l1+l4\u0000l7\u0000l8\u0000l9)+1\n6(l2+l3+l5+l6); (66)\nh=\u00001\n15l1\u00001\n15l4\u00001\n6(l2+l3+l5+l6)+2\n5(l7+l8+l9):(67)\nIn the case with easy axis parallel to the lattice vector aaawe have\n<l\u0000l00\n0\nl00\n0\f\f\f\f\faaa\nbbb>=x+h(aaa\u0001bbb)2; (68)\nwhere\nx=2\n15\u0012\n\u00003\n2l1+l4\u0000l7\u0000l8\u0000l9\u0013\n+1\n6(l2\u0000l3\u0000l5+l6); (69)\nh=\u00001\n15l1\u00001\n15l4\u00001\n6(l2+l3+l5+l6)+2\n5(l7+l8+l9);(70)\nwhile for the case of easy axis parallel to the lattice vector bbbwe obtain\n<l\u0000l000\n0\nl000\n0\f\f\f\f\faaa\nbbb>=x+h(aaa\u0001bbb)2; (71)\nwhere\nx=2\n15\u0012\nl1\u00003\n2l4\u0000l7\u0000l8\u0000l9\u0013\n\u00001\n6(l2\u0000l3\u0000l5+l6); (72)\nh=\u00001\n15l1\u00001\n15l4\u00001\n6(l2+l3+l5+l6)+2\n5(l7+l8+l9):(73)\n3. Analysis of the accuracy of the methods implemented in MAELAS\nThere are two main sources of errors in any calculation with MAELAS, (i) one\ncoming from the used methodology to compute the magnetostrictive coefficients\n28and magnetoelastic constants and other from (ii) the DFT calculation of the total\nenergy for each deformed state. Unfortunately, the lack of available experimental\ndata for some crystal symmetries makes it difficult to estimate the reliability and\nprecision of these calculations. In this Section, we try to estimate the systematic\nerror coming from the methodologies implemented in MAELAS by evaluating\nthe exact energy of each deformed state from the theory of magnetostriction. In\nthis way, we can get rid of possible errors from DFT, and reveal the accuracy of\nthe implemented methods, as well as possible issues. To do so, we first consider\na theoretical unit cell for each supported crystal symmetry in MAELAS that is\ncharacterized by an imposed set of lattice parameters, elastic ( Cexact\ni j) and magne-\ntoelastic ( bexact\nk) constants. The imposed values for these quantities are shown in\nthe first, third and fifth columns of Table 2. Next, using these elastic and magne-\ntoelastic constants, we calculate the exact magnetostrictive coefficients (shown in\nthe seventh column of Table 2) from the theoretical relations between these quan-\ntitieslexact(bexact\nk;Cexact\ni j) [1]. Finally, we also use these elastic and magnetoelastic\nconstants to evaluate the exact elastic and magnetoelastic energies [1]\nEexact(eee;aaa) =Eexact\nel(eee)+Eexact\nme(eee;aaa); (74)\nfor each deformed state and magnetization direction generated by the two imple-\nmented methods in MAELAS. Then these exact energies are used as inputs in\nthe step 5 of MAELAS workflow (see Fig.1) to compute the magnetoelastic con-\nstants and magnetostrictive coefficients. The exact expression of the elastic and\nmagnetoelastic energies for cubic (I) symmetry are given in Eq. 5, while the cor-\nresponding equations for the other supported symmetries can be found in Ref. [1].\nBy comparing the exact values and MAELAS results for bandlwe can estimate\nthe relative error of the calculation of MAELAS as\nbRel:Error\ni (%) =bexact\ni\u0000bMAELAS\ni\nbexact\ni\u0002100; (75)\nlRel:Error\ni (%) =lexact\ni\u0000lMAELAS\ni\nlexact\ni\u0002100: (76)\nThe estimated relative errors are shown in Table 2 for -mode 1 (method originally\nimplemented in version 1.0 based on the length optimization [1, 2], including the\ncorrections of the trigonal (I) symmetry given in Section 2.2) and -mode 2 (new\nmethod implemented in version 2.0 that is described in Section 2.1). We observe\nthat the new methodology (-mode 2) improves significantly the accuracy of the\n29calculations for all magnetoelastic constants and magnetostrictive coefficients ex-\nhibiting relative errors lower than 10\u00004%. The -mode 1 gives good results for\ncubic symmetry, but for lower symmetries the relative error is significantly large\nin some cases ( >10%), due to the deformation gradients implemented in version\n1.0 for the cell length optimization [1]. In Appendix B, we analyze this problem\nin more detail and discuss possible ways to improve the accuracy of -mode 1 . In\nthis appendix, we also describe an alternative potential approach called strain opti-\nmization method, which is still under evaluation and is not currently implemented\nin version 2.0. Based on these results, we recommend to use -mode 2 instead\nof -mode 1, especially for non-cubic crystals. The -mode 2 is set as the default\nmethod in version 2.0. We note that performing the calculations with these two\nindependent methods and comparing their results can also be useful to confirm the\ncorrect execution of the program.\n4. Tests for the new method\nIn this section, we perform some calculations with the new method described\nin Section 2.1 (-mode 2). Here, MAELAS is interfaced with V ASP [5–7] to com-\npute the energies of each deformed state. For a fair comparison with the method\nimplemented in version 1.0 (-mode 1), we consider the same materials (BCC Fe,\nFCC Ni, HCP Co, Fe 2Si space group (SG) 164, L1 0FePd and YCo SG 63) as\nin the publication of version 1.0 [1], and use the same relaxed unit cells, V ASP\nsettings (energy cut-off, pseudopotentials, exchange correlation,...) and elastic\nconstants calculated with AELAS [1, 4]. A summary of the results is presented in\nTable 3.\n4.1. FCC Ni\nThe new method implemented in MAELAS v2.0 (-mode 2) gives very similar\nresults to -mode 1 for FCC Ni (cubic (I) crystal symmetry), see Table 3. The\nanalysis of the calculated magnetoelastic constants with -mode 2 as a function of\nthe k-points in the Brillouin zone is shown in Fig. 8. We see that both b1and\nb2converge well above 4 \u0002104k-points and are in fairly good agreement with\nexperiment. In Fig.8 we also plot the linear fitting of the energy versus strain ( exx)\ndata to compute b1.\n4.2. BCC Fe\nThe results given by -mode 2 for BCC Fe (cubic (I) crystal symmetry) are also\nvery similar to -mode 1, see Table 3. The analysis of the calculated magnetoelastic\n30Table 2: Summary of the analysis of the accuracy of the methods implemented in MAELAS.\nThe relative error for the magnetoelastic constants ( bRel:Error) and magnetostrictive coefficients\n(lRel:Error) is computed using Eqs. 75 and 76, respectively.\nCrystal\nsystemCi jCexact\ni j\n(GPa)bbexact\n(MPa)llexact\n(\u000210\u00006)-mode 1\nbRel:Error\n(%)-mode 1\nlRel:Error\n(%)-mode 2\nbRel:Error\n(%)-mode 2\nlRel:Error\n(%)\nCubic (I) C11 243 b1 -4.1 l001 26.0317 0.015 0.013 7 \u000110\u000068\u000110\u00006\na=2:8293Å C12 138 b2 10.9 l111 -29.7814 0.005 0.004 6 \u000110\u000063\u000110\u00006\nC44 122\nHexagonal (I) C11 307 b21 -31.9 la1;295.0656 -17.6 -17.6 6 \u000110\u000061\u000110\u00005\na=2:4561Å C12 165 b22 25.5 la2;2-125.9316 -17.4 -17.5 8 \u000110\u000062\u000110\u00005\nc=3:9821Å C13 103 b3 -8.1 lg;257.0423 17.0 17.0 6 \u000110\u000066\u000110\u00006\nC33 358 b4 42.9 le;2-286.0 0.002 0.002 5 \u000110\u000067\u000110\u00006\nC44 75\nTrigonal (I) C11 428 b21 43.1 la1;2-104.9605 22.9 10.8 1 \u000110\u000051\u000110\u00005\na=3:9249Å C12 164 b22 -34.2 la2;2143.1326 -38.6 -16.4 1 \u000110\u000058\u000110\u00005\nc=4:8311Å C13 133 b3 60:7lg;1-202.6605 -11.9 -8.8 1 \u000110\u000051\u000110\u00005\nC14 -27 b4-34.3 lg;2204.2029 -15.8 -42.3 1 \u000110\u000051\u000110\u00005\nC33 434 b14 -42.4 l12 -377.9282 -28.1 46.8 1 \u000110\u000051\u000110\u00005\nC44 118 b34 55.4 l21 266.5791 37.9 -34.8 1 \u000110\u000051\u000110\u00005\nTetragonal (I) C11 324 b21 -2.4 la1;2-20.4581 42.4 7.3 8 \u000110\u00006\u00002\u000110\u00005\na=2:6973Å C12 67 b22 -15.2 la2;278.1888 18.3 15.4 6 \u000110\u00006\u00004\u000110\u00005\nc=3:7593Å C13 133 b3 -7.9 lg;230.7393 -14.2 -14.2 6 \u000110\u00006\u00002\u000110\u00005\nC33 264 b4 -5.6 le;227.7228 0.002 0.002 5 \u000110\u000067\u000110\u00006\nC44 101 b0\n3 -7.9 ld;2106.7568 0.007 0.007 5 \u000110\u000066\u000110\u00006\nC66 37\nOrthorhombic C11 76 b1 43:1 l1 -632.9614 37.4 24.7 7 \u000110\u000065\u000110\u00006\na=4:0686Å C12 45 b2-34.2 l2 681.5787 6.9 7.9 6 \u000110\u000066\u000110\u00006\nb=10:3157Å C13 48 b3 60:7 l3 -752.4503 19.6 0.03 7 \u000110\u000067\u000110\u00006\nc=3:8956Å C23 55 b4-34.3 l4 471.7839 -11.6 -18.5 6 \u000110\u000066\u000110\u00006\nC22 102 b5-42.4 l5 809.6944 -46.8 -10.8 7 \u000110\u000065\u000110\u00006\nC33 141 b6 55.4 l6 -808.9638 -7.6 -5.6 6 \u000110\u000066\u000110\u00006\nC44 40 b7 35.4 l7 -284.9353 47.3 54.3 6 \u000110\u000067\u000110\u00006\nC55 27 b8-22.6 l8 253.4425 43.2 11.6 9 \u000110\u000068\u000110\u00006\nC66 39 b9 38.7 l9 -326.1699 -119.8 -85.6 5 \u000110\u000062\u000110\u00005\n31Table 3: Anisotropic magnetostrictive coefficients and magnetoelastic constants calculated with\nthe two methods (-mode 1 and -mode 2) available in MAELAS v2.0. In parenthesis we show\nthe magnetostrictive coefficients with Mason’s definitions [1, 14]. These data correspond to the\nsimulations with the same V ASP settings, relaxed unit cell and elastic constants as in Ref. [1].\nFor Fe 2Si we repeated the calculations with -mode 1 as implemented in version 2.0 [v2.0], that is,\nincluding the corrections described in Section 2.2.\nMaterial Crystal systemDFT\nExchange\nCorrelationMagnetostrictive\ncoefficientMAELAS\n-mode 1\n(\u000210\u00006)MAELAS\n-mode 2\n(\u000210\u00006)Expt.\n(\u000210\u00006)Magnetoelastic\nconstantMAELAS\n-mode 1\n(MPa)MAELAS\n-mode 2\n(MPa)Expt.\n(MPa)\nFCC Ni Cubic (I) GGA l001 -78.4b-72.7 -60ab1 15.5b14.4 9.9b\nSG 225 l111 -46.1b-44.0 -35ab2 19.4b18.5 13.9b\nBCC Fe Cubic (I) GGA l001 25.7b29.1 26ab1 -5.2b-5.9 -4.1b\nSG 229 l111 17.2b15.7 -30ab2 -5.3b-4.9 10.9b\nHCP Co Hexagonal (I) LSDA+U la1;2(lA) 111 (-109)b75 (-74) 95 (-66)cb21 -21.3b-16.2 -31.9b\nSG 194 J=0:8eV la2;2(lB) -251 (-114)b-156 (-77) -126 (-123)cb22 48.3b28.4 25.5b\nU=3eV lg;2(lC) 4 (251)b2 (156) 57 (126)cb3 -0.7b-0.4 -8.1b\nle;2(lD) -51 (10)b-57 (-8) -286 (-128)cb4 7.1b7.9 42.9b\nFe2Si Trigonal (I) GGA la1;2-9b[v1.0] -5 b21 3.1b[v1.0] 0.7\nSG 164 la2;215b[v1.0] 17 b22 -4.2b[v1.0] -6.2\nlg;18b[v1.0] 6 b3 -0.7b[v1.0] -1.9\nlg;229b[v1.0] 29 b4 3.3b[v1.0] -3.5\nl12 -3b[v1.0] -5 b14 -1.4b[v1.0] 1.9\nl21 -13b[v1.0] -14 b34 -0.4b[v1.0] 1.4\nGGA la1;2-9 [v2.0] b21 3.1 [v2.0]\nla2;215 [v2.0] b22 -4.2 [v2.0]\nlg;18 [v2.0] b3 -2.5 [v2.0]\nlg;229 [v2.0] b4 -3.7 [v2.0]\nl12 -11 [v2.0] b14 2.0 [v2.0]\nl21 -13 [v2.0] b34 2.2 [v2.0]\nL10FePd Tetragonal (I) GGA la1;2-21b-41 b21 -2.4b5.1\nSG 123 la2;279b82 b22 -15.2b-10.7\nlg;231b27 b3 -7.9b-6.9\nle;228b26 b4 -5.6b-5.3\nld;2106b106 b0\n3 -7.9b-7.9\nYCo Orthorhombic LSDA+U l1 -11b-36 b1 -1.7b0.8\nSG 63 J=0:8eV l2 32b57 b2 1.2b-2.9\nU=1:9eV l3 70b75 b3 -3.8b-2.2\nl4 -74b-64 b4 4.3b0.6\nl5 -30b-35 b5 -0.1b1.7\nl6 7b24 b6 2.3b-2.2\nl7 36b38 b7 -4.4b-4.2\nl8 -20b-34 b8 1.1b1.8\nl9 35b-7 b9 -8.7b-0.5\naRef.[16],bRef.[1],cRef.[17]\nconstants with -mode 2 as a function of the k-points in the Brillouin zone is shown\nin Fig. 9. We see that both b1andb2converge well above 5 \u0002104k-points. While\nb1is in good agreement with experiment, b2has the opposite sign to experiment\ndue to a possible failure of DFT [18, 19]. In Fig.9 we also plot the linear fitting\nof the energy versus strain ( exy) data to compute b2. Since the slope of the fitted\nlinear polynomial is negative, then the calculation gives b2<0.\n32Figure 8: (Left) Calculated magnetoelastic constants for FCC Ni with the new method imple-\nmented in MAELAS v2.0 (-mode 2). (Right) Calculation of b1for FCC Ni through a lin-\near fitting of the energy difference between magnetization directions aaa1= (1;0;0)andaaa2=\n(1=p\n2;1=p\n2;0)versus strain ( exx) data.\nFigure 9: (Left) Calculated magnetoelastic constants for BCC Fe with the new method imple-\nmented in MAELAS v2.0 (-mode 2). (Right) Calculation of b2for BCC Fe through a lin-\near fitting of the energy difference between magnetization directions aaa1= (1=p\n2;1=p\n2;0)and\naaa2= (\u00001=p\n2;1=p\n2;0)versus strain ( exy) data.\n4.3. HCP Co\nThe results of both methods for HCP Co (hexagonal (I) crystal symmetry)\nfollow the experiment, except for b4and its corresponding magnetostrictive co-\nefficient le;2which are clearly underestimated. Hence, the new method confirms\nthe issue with b4andle;2that was already identified with -mode 1 in the publi-\ncation of version 1.0 [1]. Calculated magnetoelastic constants for HCP Co with\nthe new method implemented in MAELAS v2.0 (-mode 2) for different values of\nthe Hubbard U parameter is shown in Fig.10. In this figure we also plot the linear\n33fitting of the energy versus strain data to compute b4. We see that the linear fitting\nis quite good (R-squared=0.9989), so that the low value of b4is not related to a\npossible failure in the fitting procedure.\nFigure 10: (Left) Calculated magnetoelastic constants for HCP Co with the new method imple-\nmented in MAELAS v2.0 (-mode 2) for different values of the Hubbard U parameter. (Right)\nCalculation of b4for HCP Co through a linear fitting of the energy difference between magnetiza-\ntion directions aaa1= (1=p\n2;0;1=p\n2)andaaa2= (\u00001=p\n2;0;1=p\n2)versus strain ( exz) data.\n4.4. Fe 2Si\nFor Fe 2Si (SG 164, trigonal (I) symmetry) we first recalculated the magne-\ntostrictive coefficients and magnetoelastic constants with -mode 1 including the\ncorrections implemented in version 2.0 for trigonal (I) symmetry that were de-\nscribed in Section 2.2. The corrected deformation gradient for l12(Eq. 11)\nin -mode 1 gives l12=\u000011\u000210\u00006, while the old deformation gradient (Eq.\n10) implemented in version 1.0 provides a negligible magnetostriction ( l12=\n\u00003\u000210\u00006), as expected from the discussion about this issue in Section 2.2. Ad-\nditionally, the corrections of the theoretical relations for lg;1,lg;2andl2;1given\nby Eq.9 fix the issue with the sign of b4,b14andb34found in the calculations with\nversion 1.0. In Fig. 11 we show the calculation of b22andb4through a linear\nfitting of the energy versus strain data generated with -mode 2.\n4.5. L1 0FePd\nIn the case of L1 0FePd (tetragonal (I) symmetry), -mode 2 gives magnetoe-\nlastic constants similar to -mode 1, except for b21that has opposite sign due to the\ndeviation obtained in la1;2(b21=\u0000(C11+C12)la1;2\u0000C13la2;2). Both methods\nalso lead to similar magnetostrictive coefficients, where the largest deviation is\n34Figure 11: Calculation of (left) b22and (right) b4for Fe 2Si through a linear fitting of the energy\nversus strain data.\nFigure 12: Calculation of (left) b3and (right) b0\n3for L1 0FePd through a linear fitting of the energy\nversus strain data.\nfound in la1;2. In Fig.12 we show the linear fitting of the energy versus strain\n(exy) data to compute b3andb0\n3using -mode 2.\nIn the publication of version 1.0 [1], we discussed the measurements of the\nlongitudinal magnetostriction (measuring length direction is parallel to the ap-\nplied field HHHkbbb) for L1 0FePd along the lattice vector aaa(lka, where aaa= (1;0;0)\nandbbb(1;0;0)) and lattice vector ccc(lkcwhere ( aaa= (1;0;0)andbbb(1;0;0)) per-\nformed by Shima et al.[20]. In such analysis, we applied Eq.50 to obtain the rela-\ntion between measured longitudinal magnetostriction ( lkaandlkc) and calculated\n35magnetostrictive coefficient with MAELAS finding\nlka\u0011l\u0000l0\nl0\f\f\f\f\faaa===(((111;;;000;;;000)))\nbbb===(((111;;;000;;;000)))=la1;0\u0000la1;2\n3+lg;2\n2; (77)\nlkc\u0011l\u0000l0\nl0\f\f\f\f\faaa===(((000;;;000;;;111)))\nbbb===(((000;;;000;;;111)))=la2;0+2la2;2\n3: (78)\nThe application of Eq.50 implies the assumption that the initial reference demag-\nnetized state in the measurement of the fractional change in length corresponds\nto a hypothetic demagnetized state with randomly oriented atomic magnetic mo-\nments. However, an initial reference demagnetized state with magnetic domains\nseems a more reasonable assumption to interpret the results given by Shima et al.,\njudging by the experimental setup [20]. For instance, if we assume our standard\nreference demagnetized state with magnetic domains along all possible easy di-\nrections, then we should applied Eq.51 with cone angle W=0 since L1 0FePd has\nan easy axis. In this case, we find\nlka\u0011l\u0000l0\n0\nl0\n0\f\f\f\f\faaa===(((111;;;000;;;000)))\nbbb===(((111;;;000;;;000)))=\u0000la1;2+lg;2\n2; (79)\nlkc\u0011l\u0000l0\n0\nl0\n0\f\f\f\f\faaa===(((000;;;000;;;111)))\nbbb===(((000;;;000;;;111)))=0: (80)\nInserting our calculated values of la1;2andlg;2(shown in Table 3) in Eq.79\ngives lka=36:5\u000210\u00006andlka=54:5\u000210\u00006with -mode 1 and -mode 2,\nrespectively. Here, the result obtained by -mode 2 is likely to be more accu-\nrate than the one with -mode 1, based on the analysis performed in Section 3.\nShima et al. measured[20] lka=100\u000210\u00006which is about two times larger\nthan our calculation with -mode 2. This deviation could be related to the dif-\nferent strength of SOC in the experiment and theory. This fact could be esti-\nmated from the magnetocrystalline anisotropy energy at the unstrained state. In\nour calculation we obtained an uniaxial magnetocrystalline anisotropy constant\nKU= [E(1;0;0)\u0000E(0;0;1)]=(a2c) =1:24 MJ/m3[1], while in the experimental\nsample (L1 0Fe48Pd52) Shima et al. measured KU=2:5 MJ/m3[20] at 4 :2 K,\nlarger than other reported values [21, 22], and approximately two times larger\nthan in our calculation, as in the case of lkawith -mode 2. Concerning lkc, the\n36theoretical result given by Eq.80 is in fairly good agreement to the small values\nmeasured by Shima et al. lkc'0 at applied field H=5 kOe and lkc'20\u000210\u00006\natH=50 kOe [20]. This kind of analysis can be done easily with the visualization\ntool MAELASviewer[8] that now includes the possibility to select the type of de-\nmagnetized state to be used as a reference state in the calculation of the fractional\nchange in length, see Fig.13.\nFigure 13: Calculation of the fractional change in length of L1 0FePd when the magnetization is\nsaturated along the lattice vector aaaparallel to the x-axis aaa= (1;0;0). As reference state, we used\na demagnetized state with domains along all easy directions. Each point on the surface represent\nthe fractional change in length along this particular crystallographic direction (measuring length\ndirection bbb). We used the magnetostrictive coefficients calculated with -mode 2 of MAELAS\nwhich are shown in Table 3, as well as the calculated magnetocrystalline anisotropy constants K1=\n1:24 MJ/m3andK2=0. This figure was generated with the new update of MAELASviewer[8]\nthat allows to select the type of reference demagnetized state.\n4.6. YCo\nIn YCo (SG 63, orthorhombic), we observe larger discrepancies between -\nmode 1 and -mode 2 than in the previous examples, especially in the results for\nthe magnetoelastic constants. This fact is partially due to the lower accuracy of -\nmode 1 with respect -mode 2, as discussed in Section 3. Fig.14 presents the linear\nfitting of the energy versus strain ( exy) data to compute b3andb8using -mode 2.\n5. Conclusions\nIn summary, version 2.0 of MAELAS includes an alternative method (-mode\n2) to compute directly the anisotropic magnetoelastic constants from a linear fit-\n37Figure 14: Calculation of (left) b3and (right) b8for YCo through a linear fitting of the energy\nversus strain data.\nting of the energy versus strain data, which is derived from the magnetoelastic en-\nergy of each crystal symmetry. The analysis of the accuracy of MAELAS reveals\nthat this new method is more precise and convenient than the method originally\nimplemented in version 1.0, especially for non-cubic symmetries. The fact that\nthese two independent methods give similar results in many cases can also be a\nuseful way of confirming the correct execution of the calculations. For instance,\nthis analysis helped us to identify some issues in version 1.0 for the trigonal (I)\nsymmetry, which has been fixed in version 2.0. Additionally, it also allows us to\nfind that the accuracy of -mode 1 could be improved by using deformation gradi-\nents derived from the equilibrium magnetoelastic strain tensor, as well as potential\nalternative approaches like the strain optimization method.\nTo facilitate the comparison between theory and experiment, we implemented\na new feature in the visualization tool MAELASviewer that allows to choose the\ntype of reference demagnetized state in the calculation of the fractional change in\nlength. This new option helped us to have better understanding of the experimental\nmeasurements of magnetostriction for L1 0FePd.\nLastly, we also derived and implemented the fractional change in length for\npolycrystals with trigonal, tetragonal and orthorhombic crystal symmetries, which\nhas the same general form as the cubic and hexagonal crystals.\nAcknowledgement\nThis work was supported by the ERDF in the IT4Innovations national super-\ncomputing center - path to exascale project (CZ.02.1.01/0.0/0.0/16-013/0001791)\nwithin the OPRDE. This work was also supported by The Ministry of Educa-\ntion, Youth and Sports from the Large Infrastructures for Research, Experimental\n38Development, and Innovations project “e-INFRA CZ (ID:90140)\", by the Donau\nproject No. 8X20050, and the computational resources provided by the Open\nAccess Grant Competition of IT4Innovations National Supercomputing Center\nwithin the projects OPEN-18-5, OPEN-18-33, and OPEN-19-14. In addition,\nDL, SA, and APK acknowledge the Czech Science Foundations grant No. 20-\n18392S and P.N., D.L., and S.A. acknowledge support from the H2020-FETOPEN\nno. 863155 s-NEBULA project.\nAppendix A. Generation of the deformed unit cells in the new method to\ncalculate magnetoelastic constants\nIn this appendix we present the procedure to generate the deformed unit cells\nin the new methodology (-mode 2 ) of MAELAS 2.0 to calculate anisotropic mag-\nnetoelastic constants that is described in Section 2.1. The deformed unit cells are\ngenerated by multiplying the lattice vectors of the initial unit cell aaa= (ax;ay;az),\nbbb= (bx;by;bz),ccc= (cx;cy;cz)by the deformation gradient Fi j[23]\n0\n@a0\nxb0\nxc0\nx\na0\nyb0\nyc0\ny\na0\nzb0\nzc0\nz1\nA=0\n@FxxFxyFxz\nFyxFyyFyz\nFzxFzyFzz1\nA\u00010\n@axbxcx\naybycy\nazbzcz1\nA (A.1)\nwhere a0\ni,b0\niandc0\ni(i=x;y;z) are the components of the lattice vectors of the\ndeformed cell. In the infinitesimal strain theory, the deformation gradient is re-\nlated to the displacement gradient ( ¶ui=¶rj) asFi j=di j+¶ui=¶rj, where di jis\nthe Kronecker delta. Since the strain tensor ( ei j) can be expressed in terms of the\ndisplacement vector uuuas [24]\nei j=1\n2\u0012¶ui\n¶rj+¶uj\n¶ri\u0013\n; i;j=x;y;z (A.2)\nthen the strain tensor ei jcan be written in terms of the deformation gradient as\neee=0\n@exxexyexz\neyxeyyeyz\nezxezyezz1\nA=1\n20\n@2(Fxx\u00001)Fxy+Fyx Fxz+Fzx\nFxy+Fyx2(Fyy\u00001)Fyz+Fzy\nFxz+Fzx Fyz+Fzy2(Fzz\u00001)1\nA: (A.3)\nWe see, that the strain tensor is symmetric ( ei j=eji,i6=j), but the deformation\ngradient is not necessarily symmetric. For the new methodology implemented in\nMAELAS 2.0 we consider deformation gradients that fulfill Eq.2 with the mag-\nnetization directions aaa1andaaa2given by Table 1. For the sake of simplicity in\nimplementing this new approach, we use deformations that don’t preserve the\nvolume of the unit cells (non-isochoric deformation).\n39Appendix A.1. Cubic (I) system\nFor cubic (I) systems the new method (-mode 2) generates two set of deformed\nunit cells using the following deformation gradients\nFFFb1(s) =0\n@1+s0 0\n0 1 0\n0 0 11\nA; (A.4)\nFFFb2(s) =0\n@1s0\ns1 0\n0 0 11\nA: (A.5)\nThe parameter scontrols the applied deformation, and its maximum value can be\nspecified through the command line of the program MAELAS using tag \u0000s. The\ntotal number of deformed cells can be chosen with tag \u0000n.\nAppendix A.2. Hexagonal (I) system\nIn the case of hexagonal (I), the new method (-mode 2) generates 4 sets of\ndeformed cells using the following deformation gradients\nFFFb21(s) =0\n@1+s\n20 0\n0 1 +s\n20\n0 0 11\nA; (A.6)\nFFFb22(s) =0\n@1 0 0\n0 1 0\n0 0 1 +s1\nA; (A.7)\nFFFb3(s) =0\n@1+s\n20 0\n0 1\u0000s\n20\n0 0 11\nA; (A.8)\nFFFb4(s) =0\n@1 0 s\n0 1 0\ns0 11\nA: (A.9)\nAppendix A.3. Trigonal (I) system\nIn the case of trigonal (I), the new method (-mode 2) generates 6 sets of de-\nformed unit cells. The deformation gradients for b12,b22,b3andb4are the same\n40as in the hexagonal (I) case, (Eqs.A.6, A.7, A.8 and A.9), while for b14andb34\nthe deformation gradients are\nFFFb14(s) =0\n@1 0 s\n0 1 0\ns0 11\nA; (A.10)\nFFFb34(s) =0\n@1s0\ns1 0\n0 0 11\nA: (A.11)\nAppendix A.4. Tetragonal (I) system\nIn the case of tetragonal (I), the new method (-mode 2) generates 5 sets of\ndeformed unit cells. The deformation gradients for b12,b22,b3andb4are the\nsame as in the hexagonal (I) case, (Eqs.A.6, A.7, A.8 and A.9), while for b0\n3the\ndeformation gradient is\nFFFb0\n3(s) =0\n@1s0\ns1 0\n0 0 11\nA: (A.12)\n41Appendix A.5. Orthorhombic system\nFor orthorhombic crystals the new method (-mode 2) generates 9 sets of de-\nformed cells using the following deformation gradients\nFFFb1(s) = FFFb2(s) =0\n@1+s0 0\n0 1 0\n0 0 11\nA; (A.13)\nFFFb3(s) = FFFb4(s) =0\n@1 0 0\n0 1+s0\n0 0 11\nA; (A.14)\nFFFb5(s) = FFFb6(s) =0\n@1 0 0\n0 1 0\n0 0 1 +s1\nA; (A.15)\nFFFb7(s) =0\n@1s0\ns1 0\n0 0 11\nA; (A.16)\nFFFb8(s) =0\n@1 0 s\n0 1 0\ns0 11\nA; (A.17)\nFFFb9(s) =0\n@1 0 0\n0 1 s\n0s11\nA: (A.18)\nAppendix B. Analysis of the accuracy of the length optimization method and\npotential ways to improve it\nIn Section 3 we found that the length optimization method (-mode 1) leads\nto significantly large relative errors in the calculation of some magnetostrictive\ncoefficients. In this appendix we analyze this issue in more detail.\nThe implementation of the length optimization method in MAELAS contains\ntwo approximations. The first approximation is used in the formula to compute\nthe magnetostrictive coefficients from the equilibrium lengths l1andl2along the\ndirection bbbwhen magnetization points to direction aaa1andaaa2, respectively [1]\nl\u0018=2(l1\u0000l2)\nr(l1+l2): (B.1)\n42This is a fairly good approximation for all known magnetostrictive materials ( l<\n10\u00002), and gives small relative error ( <10\u00002). The second approximation is re-\nlated to the estimation of the equilibrium lengths l1andl2in Eq.B.1 through a\nparameterized deformation gradient FFF(s). Let’s analyze in more detail the influ-\nence of this approximation on the calculation of l001for Cubic (I) crystal and\nla2;2for Hexagonal (I) crystal.\nAppendix B.1. Cubic (I)\nIdeally, one needs to make use of the equilibrium magnetoelastic strain tensor\neeeeq(aaa)in order to construct the parameterized deformation gradient that contains\nthe exact equilibrium state as a particular case of the parameter. The anisotropic\npart of the equilibrium strain tensor for Cubic (I) systems reads [1, 25]\neeeeq(aaa) =0\n@3\n2l001\u0000\na2\nx\u00001\n3\u00013\n2l111axay3\n2l111axaz\n3\n2l111axay3\n2l001\u0000\na2\ny\u00001\n3\u00013\n2l111ayaz\n3\n2l111axaz3\n2l111ayaz3\n2l001\u0000\na2\nz\u00001\n3\u00011\nA: (B.2)\nIn -mode 1, for the calculation of l001we selected the magnetization directions\naaa1= (0;0;1)andaaa2= (1;0;0)[1]. Hence, the equilibrium strain tensor becomes\neeeeq(0;0;1) =0\n@\u00001\n2l001 0 0\n0\u00001\n2l001 0\n0 0 l0011\nA;\neeeeq(1;0;0) =0\n@l001 0 0\n0\u00001\n2l001 0\n0 0\u00001\n2l0011\nA:(B.3)\nThe corresponding deformation gradients that give these equilibrium strain tensors\nare obtained using Eq. A.3\nFFF\f\f\fl001\naaa1=(0;0;1)=0\n@1\u00001\n2l001 0 0\n0 1\u00001\n2l001 0\n0 0 1 +l0011\nA;\nFFF\f\f\fl001\naaa2=(1;0;0)=0\n@1+l001 0 0\n0 1\u00001\n2l001 0\n0 0 1 \u00001\n2l0011\nA:(B.4)\nTherefore, if we parameterize these deformation gradients by setting the magne-\ntostrictive coefficients as free parameters ( li\u0000 !si), then they generate a set of\n43deformed cells that contains the exact equilibrium state that corresponds to the\nparticular case when the free parameters are equal to the magnetostrictive coeffi-\ncients (minimum energy). Hence, these deformation gradients should be parame-\nterized as\nFFF\f\f\fl001\naaa1=(0;0;1)(s) =0\n@1\u00001\n2s 0 0\n0 1\u00001\n2s 0\n0 0 1 +s1\nA; (B.5)\nFFF\f\f\fl001\naaa2=(1;0;0)(s) =0\n@1+s 0 0\n0 1\u00001\n2s 0\n0 0 1\u00001\n2s1\nA; (B.6)\nwhere sis a free parameter. Let’s try to estimate the relative error in the calcu-\nlation of l001using these parameterized deformation gradients. At the minimum\nenergy (equilibrium state), the free parameter in Eqs. B.5 and B.6 is equal to\nthe magnetostrictive coefficient ( s=l100). Hence, these parameterized defor-\nmation gradients give the exact equilibrium lengths l1andl2along the direction\nbbb= (0;0;1)(via Eq. A.1)\nlexact\n1= (1+l001)cz;\nlexact\n2=\u0012\n1\u00001\n2l001\u0013\ncz;(B.7)\nwhere czis the z-component of the relaxed (not deformed) lattice vector ccc=\n(0;0;cz). Therefore, the Eq.B.1 used in -mode 1 to compute the magnetostric-\ntive coefficient gives [1]\nlapprox\n001\u0018=2(lexact\n1\u0000lexact\n2)\nr(lexact\n1+lexact\n2)=4l001\n4+l001; (B.8)\nwhere r=3=2. The relative error using the exact value in Table 2 ( l001=\n26:0317\u000210\u00006) is\nlRel:Error\n001 =l001\u0000lapprox\n001\nl001\u0002100=0:0007% ; (B.9)\nwhich is smaller than in the example shown in Section 3 using the -mode 1 of\nMAELAS ( lRel:Error\n001=0:013%). The relative error is larger in MAELAS because\n44we implemented the following deformation gradient for both magnetization direc-\ntionsaaa1= (0;0;1)andaaa2= (1;0;0)[1]\nFFF\f\f\fl001\naaa1=(0;0;1)(s) =FFF\f\f\fl001\naaa2=(1;0;0)(s) =0\nB@1p1+s0 0\n01p1+s0\n0 0 1 +s1\nCA; (B.10)\nwhich is not completely consistent with the parameterized deformation gradients\ngiven by Eqs.B.5 and B.6, derived from the theoretical equilibrium strain tensor\neeeeq(aaa). As a result, the deformation gradient implemented in -mode 1 of MAE-\nLAS generates a set of deformed cells that do not contain the exact equilibrium\nstate when the magnetization is constrained along the direction aaa. Consequently,\nit gives approximated values of the equilibrium lengths lapprox\n1andlapprox\n2along\nthe direction bbbwhen magnetization points to direction aaa1andaaa2, respectively.\nInterestingly, this analysis reveals a potential alternative approach to the length\noptimization method that requires only one magnetization direction to compute\nl001, and without using any approximation. Namely, after a cell relaxation, a set of\ndeformed unit cells are generated with the deformation gradient given by Eq.B.5.\nNext, the energy of these unit cells are evaluated constraining the direction of\nmagnetization to aaa= (0;0;1). The calculated energies versus the parameter s\nused in the deformation gradient are fitted to a cubic polynomial (according to our\npreliminary tests using the exact energy Eq.74, a quadratic polynomial may not\nbe sufficient for s\u00190:01)\nE(s) =As3+Bs2+Cs+D; (B.11)\nwhere A,B,CandDare fitting parameters. The value of sat the local minimum\nof this function ( Emin=E(smin)) corresponds to the magnetostrictive coefficient\nl001\n¶E\n¶s=0\u0000 !l001=smin=\u0000B\u0006p\nB2\u00003AC\n3A: (B.12)\nSince the local minimum of the cubic polynomial is the critical point where the\nsecond derivative is positive, then l001is equal to the solution in Eq.B.12 that\nmeets the condition\n¶2E\n¶s2>0\u0000 !6Asmin+2B>0: (B.13)\n45In Fig.B.15 we show the calculation of l001with this approach. Here, the energy\nis evaluated using the theoretical formula for the cubic (I) symmetry (Eq.74) with\nthe elastic and magnetoelastic constants in Table 2. The relative error in the cal-\nculation of l001islRel:Error\n001=\u00000:00009% which comes purely from the fitting\nprocedure since no approximations are used in this method. Since the calculation\nof the magnetostrictive coefficient is done through an optimization of the strain\n(the magnetostrictive coefficient is equal to the strain that minimizes the energy),\nthen we can call this approach as the strain optimization method. Currently, we\nare verifying the reliability of this method with first principle calculations. Thus,\nwe have not implemented it in version 2.0 of MAELAS.\nFigure B.15: Blue points correspond to the energy for the deformed unit cells with magnetization\ndirection aaa= (0;0;1)generated with the deformation gradient given by Eq.B.5. The energy is\nevaluated using the theoretical formula for the cubic (I) symmetry (Eq.74) with the elastic and\nmagnetoelastic constants in Table 2. Red line is the fitting curve to a cubic polynomial. The\nminimum of this function corresponds to l001.\nAppendix B.2. Hexagonal (I)\nThe calculation of la2;2in -mode 1 is performed using the following deforma-\ntion gradient for the length optimization along bbb= (0;0;1)of both magnetization\ndirections aaa1= (0;0;1)andaaa2= (1;0;0)[1]\nFFF\f\f\fla2;2\naaa1=(0;0;1)(s) =FFF\f\f\fla2;2\naaa2=(1;0;0)(s) =0\nB@1p1+s0 0\n01p1+s0\n0 0 1 +s1\nCA: (B.14)\n46As in the cubic case, this deformation gradient generates a set of deformed cells\nthat do not contain the exact equilibrium state when the magnetization is con-\nstrained along the direction aaa. Consequently, it gives approximated values of the\nequilibrium lengths l1andl2along the direction bbbwhen magnetization points to\ndirection aaa1andaaa2, respectively.\nAgain, one needs to make use of the equilibrium magnetoelastic strain tensor\neeeeq(aaa)in order to find the parameterized deformation gradient that contains the\nexact equilibrium state when the magnetization is constrained along the direction\naaa. The anisotropic part of the equilibrium strain tensor for Hexagonal (I) systems\nreads [1, 25]\neeeeq(aaa) =0\nB@la1;2\u0000\na2\nz\u00001\n3\u0001\n+lg;2\n2\u0000\na2\nx\u0000a2\ny\u0001\nlg;2axay le;2axaz\nlg;2ayax la1;2\u0000\na2\nz\u00001\n3\u0001\n\u0000lg;2\n2\u0000\na2\nx\u0000a2\ny\u0001\nle;2ayaz\nle;2azax le;2azay la2;2\u0000\na2\nz\u00001\n3\u00011\nCA:\n(B.15)\nIn -mode 1, for the calculation of la2;2we selected the magnetization directions\naaa1= (0;0;1)andaaa2= (1;0;0)[1]. Hence, the equilibrium strain tensor becomes\neeeeq(0;0;1) =0\n@2\n3la1;20 0\n02\n3la1;20\n0 02\n3la2;21\nA;\neeeeq(1;0;0) =0\n@\u00001\n3la1;2+1\n2lg;20 0\n0\u00001\n3la1;2\u00001\n2lg;20\n0 0 \u00001\n3la2;21\nA:(B.16)\nThe corresponding deformation gradients that give these equilibrium strain tensors\nare obtained using Eq. A.3\nFFF\f\f\fla2;2\naaa1=(0;0;1)=0\n@2\n3la1;2+1 0 0\n02\n3la1;2+1 0\n0 02\n3la2;2+11\nA;\nFFF\f\f\fla2;2\naaa2=(1;0;0)=0\n@\u00001\n3la1;2+1\n2lg;2+1 0 0\n0\u00001\n3la1;2\u00001\n2lg;2+1 0\n0 0 \u00001\n3la2;2+11\nA:\n(B.17)\nTherefore, if we parameterize these deformation gradients by setting the magne-\ntostrictive coefficients as free parameters ( li\u0000 !si), then they generate a set of\n47deformed cells that contains the exact equilibrium state that corresponds to the\nparticular case when the free parameters are equal to the magnetostrictive coeffi-\ncients (minimum energy). Hence, these deformation gradients should be parame-\nterized as\nFFF\f\f\fla2;2\naaa1=(0;0;1)(s1;s2) =0\n@2\n3s1+1 0 0\n02\n3s1+1 0\n0 02\n3s2+11\nA; (B.18)\nFFF\f\f\fla2;2\naaa2=(1;0;0)(s1;s2;s3) =0\n@\u00001\n3s1+1\n2s2+1 0 0\n0\u00001\n3s1\u00001\n2s2+1 0\n0 0 \u00001\n3s3+11\nA;(B.19)\nwhere s1,s2ands3are free parameters. We see that these deformation gradients\nare also diagonal matrices like the one implemented in -mode 1 given by Eq.B.14\nbut with different diagonal elements. For instance, Eq.B.14 is a particular case of\nEq.B.18 when the free parameters s1ands2are constrained to\ns1=3\n2\u00121p1+s\u00001\u0013\n; (B.20)\ns2=3\n2s; (B.21)\nwhere sis the single parameter used in Eq.B.14. In Fig.B.16 we show the en-\nergy of the deformed unit cells with magnetization direction aaa1= (0;0;1)gen-\nerated with the deformation gradients given by Eq.B.14 (red line) and Eq.B.18\n(2D surface). In this figure we evaluated the energy using the theoretical formu-\nlas for the hexagonal (I) symmetry (Eq.74) with the elastic and magnetoelastic\nconstants in Table 2. The blue point in Fig.B.16 stands for the equilibrium en-\nergy for the deformed cells generated with Eq.B.18, where the free parameters s1\nands2are equal to the magnetostrictive coefficients s1=la1;2=95:0656\u000210\u00006\nands2=la2;2=\u0000125:9316\u000210\u00006, see Table 2. We observe that the minimum\nenergy given by Eq.B.14 (red line) is an approximation of the exact equilibrium\nstate. Consequently, it gives an approximated value of the equilibrium length l1in\nthe direction bbb= (0;0;1)when magnetization points to direction aaa1= (0;0;1).\nLet’s try to estimate the relative error in the calculation of la2;2induced by the\ndeformation gradients Eqs. B.18 and B.19. At the minimum energy (equilibrium\nstate), the free parameters in Eqs. B.18 and B.19 are equal to the magnetostrictive\ncoefficients ( si=li). Hence, these parameterized deformation gradients give the\n48Figure B.16: Energy of the deformed unit cells with magnetization direction aaa1= (0;0;1)gen-\nerated with the deformation gradients given by Eq.B.14 (red line) and Eq.B.18 (2D surface). The\nenergy is evaluated using the theoretical formulas for the hexagonal (I) symmetry (Eq.74) with the\nelastic and magnetoelastic constants in Table 2. Black lines are the projected isoenergy contours of\nthe 2D surface. The blue point stands for the equilibrium energy for the deformed cells generated\nwith Eq.B.18, where the free parameters s1ands2are equal to the magnetostrictive coefficients\ns1=la1;2=95:0656\u000210\u00006ands2=la2;2=\u0000125:9316\u000210\u00006, see Table 2.\nequilibrium lengths l1andl2along the direction bbb= (0;0;1)(via Eq. A.1)\nlexact\n1=\u00122\n3la2;2+1\u0013\ncz;\nlexact\n2=\u0012\n\u00001\n3la2;2+1\u0013\ncz;(B.22)\nwhere czis the z-component of the relaxed (not deformed) lattice vector ccc=\n(0;0;cz). Therefore, the formula used in -mode 1 to compute the magnetostrictive\ncoefficient gives [1]\nla2;2\napprox =2(lexact\n1\u0000lexact\n2)\nr(lexact\n1+lexact\n2)=2la2;2\n1\n3la2;2+2; (B.23)\nwhere r=1. The relative error using the exact value in Table 2 ( la2;2=\u0000125:9316\u0002\n10\u00006) is\nla2;2\nRel:Error=la2;2\u0000la2;2\napprox\nla2;2\u0002100=\u00000:002% ; (B.24)\n49which is clearly much smaller than in the example shown in Section 3 using the\ndeformation gradient with a single parameter implemented in -mode 1, Eq.B.14\n(\u001817%). Hence, for hexagonal crystals deformation gradients with more than\none parameter based on the equilibrium strain tensor could improve the accuracy\nof -mode 1 since they give more accurate values for the equilibrium lengths l1and\nl2. Unfortunately, they will also increase the number of deformed cells needed\nto calculate the magnetostrictive coefficients through the sampling of the param-\neters in the deformation gradients, making this approach more computationally\ndemanding. In Section 4 we see that the implemented deformation gradients with\na single parameter in -mode 1 still provide reasonable results (similar to -mode 2)\nin many cases. Although, in general, we recommend to use -mode 2.\nReferences\n[1] P. Nieves, S. Arapan, S. Zhang, A. K ˛ adzielawa, R. Zhang, D. Legut,\nMaelas: Magneto-elastic properties calculation via computational high-\nthroughput approach, Computer Physics Communications 264 (2021)\n107964. doi:https://doi.org/10.1016/j.cpc.2021.107964 .\nURL https://www.sciencedirect.com/science/article/pii/\nS0010465521000801\n[2] R. Wu, A. J. Freeman, First principles determinations of magnetostriction\nin transition metals (invited), Journal of Applied Physics 79 (8) (1996)\n6209–6212. arXiv:https://aip.scitation.org/doi/pdf/10.1063/\n1.362073 ,doi:10.1063/1.362073 .\nURL https://aip.scitation.org/doi/abs/10.1063/1.362073\n[3] R. Birss, The saturation magnetostriction of ferromagnetics, Advances in\nPhysics 8 (31) (1959) 252–291. arXiv:https://doi.org/10.1080/\n00018735900101198 ,doi:10.1080/00018735900101198 .\nURL https://doi.org/10.1080/00018735900101198\n[4] S. Zhang, R. Zhang, AELAS: Automatic ELAStic property derivations via\nhigh-throughput first-principles computation, Computer Physics Communi-\ncations 220 (2017) 403 – 416. doi:10.1016/j.cpc.2017.07.020 .\nURL http://www.sciencedirect.com/science/article/pii/\nS0010465517302400\n50[5] G. Kresse, J. Hafner, Ab initio molecular dynamics for liquid metals, Phys.\nRev. B 47 (1993) 558–561. doi:10.1103/PhysRevB.47.558 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.47.558\n[6] G. Kresse, J. Furthmüller, Efficiency of ab-initio total energy cal-\nculations for metals and semiconductors using a plane-wave ba-\nsis set, Computational Materials Science 6 (1) (1996) 15 – 50.\ndoi:10.1016/0927-0256(96)00008-0 .\nURL http://www.sciencedirect.com/science/article/pii/\n0927025696000080\n[7] G. Kresse, J. Furthmüller, Efficient iterative schemes for ab initio total-\nenergy calculations using a plane-wave basis set, Phys. Rev. B 54 (1996)\n11169–11186. doi:10.1103/PhysRevB.54.11169 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.54.11169\n[8] P. Nieves, S. Arapan, A. P. K ˛ adzielawa, D. Legut, MAELASviewer: An\nonline tool to visualize magnetostriction, Sensors 20 (22) (2020). doi:10.\n3390/s20226436 .\nURL https://www.mdpi.com/1424-8220/20/22/6436\n[9] A. Andreev, Chapter 2 Thermal expansion anomalies and spontaneous\nmagnetostriction in rare-earth intermetallics with cobalt and iron, V ol. 8\nof Handbook of Magnetic Materials, Elsevier, 1995, pp. 59 – 187.\ndoi:10.1016/S1567-2719(05)80031-9 .\nURL http://www.sciencedirect.com/science/article/pii/\nS1567271905800319\n[10] P. Nieves, J. Tranchida, S. Arapan, D. Legut, Spin-lattice model for cubic\ncrystals, Phys. Rev. B 103 (2021) 094437. doi:10.1103/PhysRevB.103.\n094437 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.103.094437\n[11] P. Nieves, D. Legut, Influence of grain morphology and orientation on\nsaturation magnetostriction of polycrystalline terfenol-d (2021). arXiv:\n2105.04315 .\n[12] A. E. Clark, B. F. DeSavage, R. Bozorth, Anomalous Thermal Expansion\nand Magnetostriction of Single-Crystal Dysprosium, Phys. Rev. 138 (1965)\n51A216–A224. doi:10.1103/PhysRev.138.A216 .\nURL https://link.aps.org/doi/10.1103/PhysRev.138.A216\n[13] J. M. D. Coey, Magnetism and Magnetic Materials, Cambridge University\nPress, 2010.\n[14] W. P. Mason, Derivation of magnetostriction and anisotropic energies for\nhexagonal, tetragonal, and orthorhombic crystals, Phys. Rev. 96 (1954) 302–\n310. doi:10.1103/PhysRev.96.302 .\nURL https://link.aps.org/doi/10.1103/PhysRev.96.302\n[15] J. R. Cullen, A. E. Clark, K. B. Hathaway, Materials, science and technology,\nVCH Publishings, 1994, Ch. 16 - Magnetostrictive Materials, pp. 529–565.\n[16] R. C. O’Handley, Modern magnetic materials, Wiley, 2000.\n[17] A. Hubert, W. Unger, J. Kranz, Messung der magnetostriktionskonstanten\ndes kobalts als funktion der temperatur, Z. Physik 224 (1) (1969) 148–155.\ndoi:10.1007/BF01392243 .\nURL https://doi.org/10.1007/BF01392243\n[18] N. J. Jones, G. Petculescu, M. Wun-Fogle, J. B. Restorff, A. E. Clark,\nK. B. Hathaway, D. Schlagel, T. A. Lograsso, Rhombohedral magnetostric-\ntion in dilute iron (Co) alloys, Journal of Applied Physics 117 (17) (2015)\n17A913. arXiv:https://doi.org/10.1063/1.4916541 ,doi:10.1063/\n1.4916541 .\nURL https://doi.org/10.1063/1.4916541\n[19] M. Fähnle, M. Komelj, R. Q. Wu, G. Y . Guo, Magnetoelasticity of fe: Pos-\nsible failure of ab initio electron theory with the local-spin-density approx-\nimation and with the generalized-gradient approximation, Phys. Rev. B 65\n(2002) 144436. doi:10.1103/PhysRevB.65.144436 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.65.144436\n[20] H. Shima, K. Oikawa, A. Fujita, K. Fukamichi, K. Ishida, Magnetic\nanisotropy and mangnetostriction in L1 0FePd alloy, Journal of Mag-\nnetism and Magnetic Materials 272-276 (2004) 2173 – 2174, pro-\nceedings of the International Conference on Magnetism (ICM 2003).\ndoi:10.1016/j.jmmm.2003.12.898 .\nURL http://www.sciencedirect.com/science/article/pii/\nS0304885303018316\n52[21] V . MAYKOV , A. YERMAKOV , G. IV ANOV A, V . KHRABROV ,\nL. MAGAT, INFLUENCE OF LONG-RANGE ORDER DEGREE ON\nMAGNETIC-PROPERTIES OF COPT AND FEPD SINGLE-CRYSTALS,\nFIZIKA METALLOV I METALLOVEDENIE 67 (1) (1989) 79–84.\n[22] H. Shima, K. Oikawa, A. Fujita, K. Fukamichi, K. Ishida, A. Sakuma, Lat-\ntice axial ratio and large uniaxial magnetocrystalline anisotropy in L1 0-type\nFePd single crystals prepared under compressive stress, Phys. Rev. B 70\n(2004) 224408. doi:10.1103/PhysRevB.70.224408 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.70.224408\n[23] E. B. Tadmor, R. E. Miller, Modeling Materials, Cambridge University\nPress, 2009. doi:10.1017/cbo9781139003582 .\nURL https://doi.org/10.1017/cbo9781139003582\n[24] L. D. Landau, E. M. Lifshitz, Theory of elasticity / by L. D. Landau and\nE. M. Lifshitz ; translated from the Russian by J. B. Sykes and W. H. Reid,\nPergamon London, 1959.\n[25] A. Clark, Chapter 7 magnetostrictive rare earth-Fe 2compounds, V ol. 1\nof Handbook of Ferromagnetic Materials, Elsevier, 1980, pp. 531 – 589.\ndoi:10.1016/S1574-9304(05)80122-1 .\nURL http://www.sciencedirect.com/science/article/pii/\nS1574930405801221\n53"    },    {        "title": "2106.05063v1.Unraveling_the_role_of_the_magnetic_anisotropy_on_thermoelectric_response__a_theoretical_and_experimental_approach.pdf",        "content": " Unraveling the role of the magnetic anisotropy on thermoelectric response: a theoretical and experimental approach M. A. Corrêa1,2,a , M. Gamino1, A. S. de Melo1, M. V. P. Lopes1, J. G. S. Santos1, A. L. R. Souza1, S. A. N. França Junior1, A. Ferreira2, S. Lanceros-Méndez2,3,4 , F. Vaz2 and F. Bohn1  1)Departamento de Física, Universidade Federal do Rio Grande do Norte, 59078-900 Natal, RN, Brazil 2)Centro de Física, Universidade do Minho, 4710-057 Braga, Portugal 3)BCMaterials, Basque Center for Materials, Applications and Nanostructures, UPV/EHU Science Park, 48940 Leioa, Spain 4)IKERBASQUE, Basque Foundation for Science, E-48009 Bilbao, Spain  \n a marciocorrea@fisica.ufrn.br  Abstract Magnetic anisotropies have key role to taylor magnetic behavior in ferromagnetic systems. Further, they are also essential elements to manipulate the thermoelectric response in Anomalous Nernst (ANE) and Longitudinal Spin Seebeck systems (LSSE).  We propose here a theoretical approach and explore the role of magnetic anisotropies on the magnetization and thermoelectric response of noninteracting multidomain ferromagnetic systems. The magnetic behavior and the thermoelectric curves are calculated from a modified Stoner Wohlfarth model for an isotropic system, a uniaxial magnetic one, as well as for a system having a mixture of uniaxial and cubic magnetocrystalline magnetic anisotropies. It is verified remarkable modifications of the magnetic behavior with the anisotropy and it is shown that the thermoelectric response is strongly affected by these changes. Further, the fingerprints of the energy contributions to the thermoelectric response are disclosed. To test the robustness of our theoretical approach, we engineer films having the specific magnetic properties and compare directly experimental data with theoretical results. Thus, experimental evidence is provided to confirm the validity of our theoretical approach. The results go beyond the traditional reports focusing on magnetically saturated films and show how the thermoelectric effect behaves during the whole magnetization curve. Our findings reveal a promising way to explore the ANE and LSSE effects as a powerful tool to study magnetic anisotropies, as well as to employ systems with magnetic anisotropy as sensing or elements in technological applications. Keywords: Longitudinal Spin Seebeck Effect, Nernst Effect, Magnetic anisotropy, Spintronics      I. INTRODUCTION  Thermoelectric effects driven by spin currents thermally activated have attracted increasing interest not just due to their relevance in the context of fundamental physics of phenomena associated with spin caloritronics [1–5], but also due to their potential relevance in a wide variety of technological applications [6–8]. Remarkably, the Anomalous Nernst Effect (ANE) [9–12] and the Longitudinal Spin Seebeck Effect (LSSE) [7,13–17] arise as the most studied effects in the field. ANE and LSSE are observed when a magnetic material is simultaneously submitted to a temperature gradient ∇𝑇 and a magnetic field 𝐻$$⃗ perpendicular to each other. However, while ANE appears exclusively in ferromagnetic (FM) metals, LSSE is often disclosed for bilayers consisting of an insulating ferrimagnetic (FMI) or FM layer capped by non-magnetic metal (NM) with high spin-orbit coupling [7,13–17]. It is worth highlighting that, in ferromagnetic metallic bilayer heterostructures, the thermoelectric response of the whole system is a result of the combination of both effects, ANE and LSSE. Hence, to circumvent this fact and suppress any contribution associated with ANE or thermomagnetic effects and to , thus obtain a pure LSSE signal, systems are commonly engineered, taking into consideration a ferrimagnetic insulator, for instance the well-known Y3Fe5O12 (YIG) material, to compose the bilayers heterostructure [18–21]. Indeed, the magnetic and electrical nature of YIG suggests the YIG/NM bilayers as natural candidates for investigations addressing the LSSE.  Despite the suitability of the YIG/NM bilayers for such goal, the experimental procedures needed to induce the ferrimagnetic phase in YIG thin films often involve several steps, including  reactive magnetron sputtering and/or annealing, the latter typically performed in a controlled atmosphere [22]. In addition, the magnetic behavior of the YIG/NM systems is commonly found to be isotropic in the film-plane. These features make the achievement of YIG/NM bilayers combining specific structural and magnetic properties a complex task. Given these issues raised by YIG/NM bilayers, a different path would be highly convenient. Thus, we bring to light that ferromagnetic films, such as Co-rich alloys, may reveal magnetic field dependent thermoelectric responses with interesting characteristics [23]. Through the appropriate selection of substrates and/or experimental deposition parameters, it is possible to induce/control the magnetic anisotropies of the FM system. Consequently, the thermoelectric response may be modified by both, the magnetic anisotropy and the orientation of the external magnetic field [23]. While the thermoelectric effects have been widely investigated from the theoretical perspective, the role of the magnetic anisotropy on thermoelectric effects has rarely been addressed, despite of its scientific and technological potential relevance [24]. Recently, the modifications of the magnetic properties and thermoelectric effects, as ANE, have been explored in flexible magnetostrictive Co40Fe40B20 thin films with induced uniaxial magnetic anisotropy and under external stress [23]. Moreover, the influence of the cubic magnetocrystalline anisotropy of Co58.73Fe27.83Al13.44 Heusler alloys, grown onto substrates with distinct orientations, on thermoelectric effects has been studied by the LSSE [25]. For both situations, the results corroborate that the thermoelectric curves are strongly dependent on the magnitude and orientation of the magnetic field with respect to the anisotropies [23–25]. In addition, the thermoelectric measurements may be yet used to probe the magnetic anisotropy, as well as to identify the field ranges in which the anisotropic and Zeeman energy terms represent the major contribution to the magnetic free energy density governing the magnetization process [25].  In this work, we propose a theoretical approach from a modified Stoner Wohlfarth model [26] and explore the role of magnetic anisotropies on the magnetic and thermoelectric responses of noninteracting multidomain ferromagnetic systems. The magnetic behavior has been calculated together with the thermoelectric response for an isotropic system, a uniaxial magnetic one, as well as for a system having a mixture of uniaxial and cubic magnetocrystalline magnetic anisotropies. Modifications of the magnetic behavior with the anisotropy have been verified and it is shown that the thermoelectric response is strongly affected by these changes. To test the robustness of the developed theoretical approach, films have been engineered having the specific magnetic properties and a direct comparison between experimental data and theoretical results has been carried out. The results go beyond the traditional reports focusing on magnetically saturated films and show how the behavior of the thermoelectric effect during the whole magnetization curve.   II. THEORETICAL APPROACH A. Thermoelectric effects The Anomalous Nernst Effect corresponds to the generation of a voltage by applying a temperature gradient to a ferromagnetic metal or a semiconductor [9,10]. The electrical field associated with the ANE can be expressed as  E\"\"⃗!\"#=−λ\"µ∘(m\"\"\"⃗×∇T),\t(1) in which λ! is the anomalous Nernst coefficient, µ∘\tis the vacuum magnetic permeability, 𝑚$$⃗ is the magnetization vector, and ∇𝑇 is the temperature gradient.  The Spin Seebeck Effect (SSE), in turn, consists in the generation of a voltage by applying a temperature gradient to FM/NM or FIM/NM heterostructures [7,27]. The electrical field due to the SSE generated in the non-magnetic metal is described by  E\"\"⃗%%#=−λ%%#(σ\"\"⃗×∇T).\t(2) Here, λ##$ is the spin Seebeck coefficient, σ\"\"⃗ is the spin polarization whose orientation is given by the magnetization direction, and ∇𝑇 represents the temperature gradient.  From Eqs. (1) and (2), it can be noticed the remarkable similarity of the effects coming also from the experimental configuration, where ∇𝑇 and 𝐻$$⃗ are perpendicular to each other, as represented in Fig. 1. In the present case, it is assumed that ∇𝑇 is perpendicular to the plane of the film, while 𝐻$$⃗ remains in the plane of the film, as depicted in Fig. 1. For the calculations, the thermoelectric voltage V detected at the electrical contacts at the ends of the main axis of the film is given by:  V=−∫E\"\"⃗⋅dl⃗&',\t(3) the limits of integration being related to the distance between the electrical contacts used in the experiment. Here we assume y as the direction for the detection of V, as shown in Fig. 1.    FIG. 1. Schematic representation of (Left) ANE and (Right) LSSE experiments. For the LSSE system, the non-magnetic metal shall have high-spin orbit coupling constant, hence yielding the conversion of the spin current to charge current through the Inverse Spin Hall Effect. In our calculation, it is assumed y as the direction to the V detection. The SSE voltage is measured by means of the Inverse Spin Hall Effect (ISHE) [28]. The relation between the spin current Js, generated from the temperature gradient, and the charge current Jc, produced in the NM metal layer, is given by: J⃗(=θ%)7*+ℏ89J⃗-×σ:;,\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t(4)  in which σ* is the spin-polarization direction and θ#% is the spin Hall angle that measures the efficiency of the spin-charge conversion. B. Magnetic free energy density  From the magnetic free energy density ξ describing the system with a given anisotropy configuration [29], a routine for the energy minimization provides the values of the equilibrium angles of the magnetization at each magnetic field 𝐻$$⃗. Then, once the magnetization response is obtained for a specific field orientation, the thermoelectric voltage V can be obtained. This \n numerical procedure enables to achieve the dependence of the thermoelectric voltage for systems with distinct anisotropy configurations. To proceed with the magnetic and thermoelectrical calculations, the film defining the magnetic system is first described. Here, a theoretical system is considered that is divided into 50 noninteracting magnetic domains, what allows to represent a sample having magnetic anisotropy dispersion. The magnetization of the whole system is then obtained averaging the magnetization of the domains. The magnetic free energy density for each domain is based on a modified Stoner-Wohlfarth model [30,31], in which the magnetic anisotropy is considered. Here, we have focus on systems having uniaxial and cubic magnetocrystalline anisotropies, as well as on an isotropic one. For the sake of simplicity, the magnetization of each domain is considered in the plane of the film, and consequently, the free energy density can be written by  ξ.=−m-.\tHcos(φ)−φ/.)−k0.\tsin*(φ1.−φ/.)−23ξ(2..\t(5) The first term is the Zeeman interaction, the second one describes the uniaxial magnetic anisotropy, and the last one is related to the cubic magnetocrystalline anisotropy. For the numerical calculations, θ&'=90∘\tand φ&' are defined as the equilibrium angles of the magnetization for each domain at a given magnetic field 𝐻$$⃗\t. At the same time,  θ&' is the angle of the magnetization vector with respect to the z axis, while φ&' is the angle between the projection of the 𝑚$$⃗' in the xy plane with respect to the x axis, as indicated in Fig. 2.  Similarly, θ% and φ% are defined as the angles describing the orientation of the magnetic field 𝐻$$⃗. In addition, 𝑚*' is the unit vector of the magnetization, msi is the saturation magnetization, and k0.=/!\"4#\"*, where ℎ(' is the anisotropy field related to the uniaxial magnetic anisotropy in the 𝑢3(' direction for each domain, defined by θ('  and φ('. For all quantities, the subscript i indicate that they are associated to the i-th magnetic domain composing the system. Finally, ξ(2.  describes the in-plane component of cubic magnetocrystalline anisotropy, which can be written as ξ(2.=k(2.9α2.*α*.*+α2.*α5.*+α*.*α5.*;,                                  (6) with α2.=cosφ/.sinθ/.,  α*.=sinφ/.sinθ/.,\t(7) α5.=cosθ/.. \n FIG. 2. Schematic diagram of the theoretical system and definitions of the vectors employed for the numerical calculation. Here we consider m as the magnetization vector, whose orientation for each given magnetic field value is set by θ! and φ!, the equilibrium angles with respect to the z and x axes, respectively. The 𝐻99⃗ corresponds to the external magnetic field vector, described by θ\" e φ\". In particular, the uniaxial anisotropy vector (not shown here) is in the film-plane, i. e. 𝜃#$=90∘.  \n Looking at Eq. (5), a system with uniaxial magnetic anisotropy is represented taking 𝑘)'≠0 and 𝑘*+'=0. From our concern here, we may mention that taking 𝑘)'≠0 and 𝑘*+'≠0, a system is described having both uniaxial and cubic magnetocrystalline anisotropies, while by considering 𝑘)'≠0, φ&=φ% and\t𝑘*+'=0, a system is described in which the magnetization is aligned to the magnetic field, i.e. having isotropy in the film-plane.\t 1. Magnetic system with isotropy in the film-plane  Figure 3 presents the magnetic and thermoelectric responses for a magnetic system having magnetic isotropy in the plane of the film. As previously mentioned, it is assumed 𝑘)'≠0, φ('=φ%  and 𝑘*+'=0  to describe the magnetic free energy density of this case of study. To our numerical calculations, a system is considered consisting of a single magnetic domain with the following general parameters: 𝑚,'=143 emu/cm3 and ℎ('=30 Oe, with θ('=90∘, φ('=φ%, and θ%=90∘. Notice that the use of the multidomain structure is not required for this case. For the thermoelectric calculation, it is assumed ∆T =\t30 K and an effective thermoelectric coefficient of λ-..=1.3×10/0 V/K [23]. Here λ-..\tbrings the contributions of λ!\tand 𝜆##$ of the system. For a system in which the LSSE is also present, such as YIG/NM, it is assumed λ-..=λ##$. For sake of simplicity, the results are addressed in terms of the normalized magnetization and thermoelectric voltage values. To this end, the quantities 𝑚/𝑚, and 𝑉/𝑉&12 are defined, where 𝑚, and 𝑉&12 are the maximum values obtained for the magnetization and the thermoelectric voltage, respectively, when φ%=0∘.  Figure 3(a) shows the magnetization behavior as a function of the field for distinct φ% values. As expected, all the curves have squared shapes, with coercive field of 30 Oe, being similar irrespective of the field orientation given by φ%, a fact due to the constant alignment between the 𝑚,' and 𝐻$$⃗ for isotropic systems.  Figure 3(b) in turn shows the thermoelectric response for such isotropic magnetic system. The shape of the curves mirrors the magnetization loops, even with varying φ%. However, it is observed a significant reduction in the amplitude of the curve as φ% increases. This feature is directly related to the decrease of the component of the induced electric field along the V detection direction defined by the electrical contacts.  Figure 3(c) presents the angular dependence of the 𝑉/𝑉&12 for constant magnetic field value. Here we show the curves for two magnetic field values, 300 Oe and 2000 Oe. For both situations, the sample is magnetically saturated, and the signal response has a cosine-shaped form, being proportional to 𝑐𝑜𝑠φ%.  Finally, it is worth emphasizing that the results for the isotropic magnetic system are in agreement with the ones found in previous reports for ferrimagnetic materials, such as the YIG/Pt bilayers, samples that are widely explored in investigations addressing LSSE effects [7,32].    \nFIG. 3. (a) Normalized magnetization curves and (b) thermoelectric voltage curves at distinct φ\" for a system having isotropic magnetic features in the film-plane. For the calculations, the following parameters are considered: 𝑚&$=143 emu/cm3, ℎ#$=30 Oe, θ#$=90∘, φ#$=φ\", θ\"=90∘, φ\" ranging from 0∘ up to 90∘, λ'((=1.3×10)*  V/K and ∆T =\t30 K. (c) 𝑉/𝑉!+,, at 𝐻=300 Oe and 2000 Oe and with ∆T =\t30 K, as a function of φ\". 2. Magnetic system with uniaxial magnetic anisotropy  For a magnetic system with uniaxial magnetic anisotropy, it is assumed 𝑘)'≠0 and 𝑘*+'=0. Specifically, a system is considered with the following parameters: 𝑚,'=1330 emu/cm3, ℎ('=100 Oe, θ('=90∘, θ%=90∘, and λ-..=1.3×10/0  V/K. In addition, it is now taken into account the multidomain structure in order to mimic the magnetic anisotropy dispersion. The uniaxial anisotropy direction is first set at 𝜑('=90∘, with linear dispersion of 5∘. This procedure enables to represent a ferromagnetic system in which the uniaxial magnetic anisotropy is affected/dispersed by, for instance, the stress stored in a film during deposition.  Figure 4 shows the evolution of the magnetic and thermoelectric responses with 𝜑% for a magnetic system with uniaxial magnetic anisotropy, in which the easy magnetization axis lies in the plane of the film. For the thermoelectric calculations, it is also assumed ∆T =\t30 K. \n Figure 4(a) shows the magnetization curves for distinct values of φ%. From a general point of view, it is verified the well-known behavior with a near-squared shape for the curves calculated at φ%=80∘, revealing an easy magnetization axis located nearby this orientation. For the φ%=0∘, the form of the magnetization curve directly allows to infer a hard magnetization axis, corroborating the behavior for an uniaxial magnetic system when the field is perpendicular to the easy axis. Remarkably, it is observed a small coercive field for φ%=0∘, a feature arising from the magnetic domains having anisotropy dispersion. In this sense, the presented theoretical approach considering multidomains appears as an appropriate tool to describe details of the magnetic response of the systems. Considering the thermoelectric behavior, presented in Fig. 4(b), the shape of the curves is not strongly modified as the φ% increases. For φ%=0∘, the thermoelectric curve mirrors the one observed for the magnetization. This feature is expected, since the induced electrical field is along the V detection direction given by the electrical contacts for this configuration. However, although the shape of the curves remains unchanged even with increasing φ%(from 0 to 90°), it is observed a decrease in the 𝑉/𝑉&12 value at maximum field.  Figure 4(c) depicts the dependence of 𝑉/𝑉&12 with φ% for a constant magnetic field. Here, an interesting feature is observed with respect to the shape of the curves.  For an external magnetic field of 2000 Oe, it is observed that the response of 𝑉/𝑉&12 as a function of φ% is perfectly described by a cosine, as expected according to theory for a magnetically saturated sample. For an external field of 300 Oe in turn, it is verified a change in the profile of the curve and the sample is magnetically saturated at this smaller field value, an intriguing fact at a first glance.   At 2000 Oe, the field value is much above the saturation and anisotropy fields. As a consequence, the Zeeman energy is, at least, one order of magnitude higher than the uniaxial anisotropy energy, representing the major contribution to the magnetization dynamics. Hence, the Zeeman energy defines the magnetization direction, which follows the orientation of the field, leading to a dependence of 𝑉/𝑉&12\twith\tcos φ%.  The most striking finding here resides in the fact that the deviation from the cosine shape is found when the field is 300 Oe. This distortion can be interpreted in terms of the competition of two energy contributions. Specifically, at a field value right above the saturation, the Zeeman energy and the uniaxial magnetic anisotropy energy are of the same order of magnitude, in a sense that the response of the whole system is a result of the balance of both of terms. Given that, small deviations of the magnetization are identified through the thermoelectric experiment. Remarkably, although the magnetometric techniques are not efficient enough to probe small deviation of the magnetization at saturate states, in which the field is right above the saturation field, the ANE or LSSE experiments provide an increased sensitivity for such task.  \nFIG. 4. (a) Normalized magnetization curves and (b) thermoelectric voltage curves for a system with uniaxial magnetic anisotropy with φ#$=90∘. For the calculations, the following parameters are considered: 𝑚&$=1330 emu/cm3, ℎ#$=100 Oe, θ#$=90∘, θ\"=90∘, φ\" ranging from 0∘ up to\t90∘,λ'((=1.3×10)*\t\tV/K\tand\t∆𝑇\t=\t30\tK. (c) 𝑉/𝑉!+,, at 𝐻=300 Oe and 2000 Oe and with ∆T =\t30 K, as a function of φ\". \n To explore the modification on the thermoelectric curves when the uniaxial anisotropy direction is modified, now it is considered φ('=0∘, with a linear dispersion of 5∘\t, as used in the previous case. Fig. 5(a) shows the magnetization curves, allowing to observe the quasi-static magnetic behavior for a uniaxial system with in-plane magnetic anisotropy. Instead of the previous situation (φ('=90∘), the magnetization curves start from a squared shape for φ%=0∘. Besides, the modification of φ% shows similar curves to the ones observed in the previous case, in which φ('=90∘. On the other hand, remarkable modifications are observed in the thermoelectric curves, as shown in Fig. 5(b). While the shape of the curves is unchanged in the previous situation with varyingφ%, here the thermoelectric curves present a strong dependence on φ% values. For φ%=0∘\tthe thermoelectric curves mirror the magnetization curves, with similar coercive field values, as expected, since the electric field is aligned with the electrical contacts. However, as φ% increases, a decrease in the thermoelectrical intensity at high fields is observed, keeping unchanged the maximum values reached by the thermoelectric voltage when 𝐻$$⃗ is close to ℎ('. The shape of the curves is a result of the competition between the Zeeman and the uniaxial anisotropy, what is expected given that the sample is not saturated. For the curves at higher φ% values and high 𝐻$$⃗ intensities, the magnetization is aligned with the magnetic field direction. However, as the magnetic field intensity decreases, the uniaxial magnetic anisotropy leads to a rotation of the magnetization and, consequently, of the electric field in the LSSE or ANE experiments. This feature leads to an increase in the thermoelectric voltage for fields near to ℎ(' values.  Once again, for the 𝑉/𝑉&12 as a function of the φ% curves (Fig. 5 (c)), it is observed a slight distortion on the cosine curve for an external field of 300 Oe, a feature associated to the competition of the Zeeman and uniaxial anisotropy. However, the cosine shape is observed for the curves in which the field is 2000 Oe, in agreement with the previous discussions.  Considering the remarkable modification in the thermoelectric curves observed when the φ(' values are changed (Figs. 4 and 5), the magnetic and thermoelectric curves have been calculated for a fixed φ%, and the φ(' varying from 0∘\tup to 180∘, these curves being depicted and discussed in Fig. 1S of the Supplementary Material. \n FIG. 5. (a) Normalized magnetization curves and (b) Normalized thermoelectric voltage curves for a system with uniaxial magnetic anisotropy with φ#$=0∘. For the calculations, the following parameters have been considered: 𝑚&$=1330 emu/cm3, ℎ#$=100 Oe, θ#$=90∘, θ\"=90∘, φ\" ranging from 0∘ up to 90∘, λ'((=1.3×10)*  V/K and ∆T =\t30 K. (c) 𝑉/𝑉!+,, at 𝐻=300 Oe and 2000 Oe and with ∆T =\t30 K, as a function of φ\". 3. Magnetic system with uniaxial and cubic magnetocrystalline anisotropies  A more complex magnetic anisotropy configuration was further explored. In the following, a magnetic system having both uniaxial and cubic magnetocrystalline anisotropies is addressed. Such system discloses very interesting magnetization curves and thermoelectric responses, with potential for technological applications, such as in magnetic logic keys.  Specifically, a system is considered with the parameters previously employed for the uniaxial one, 𝑚,'=1330 emu/cm3, ℎ('=100 Oe, θ('=90∘, θ%=90∘, and λ-..=1.3×10/0  V/K, with the additional quantity of  𝑘*+'=2.8𝑘)' associated with the cubic magnetocrystalline anisotropy. \n The multidomain structure is also kept to represent the magnetic anisotropy dispersion, by setting the uniaxial anisotropy direction at 𝜑('=90∘, with a linear dispersion of 5∘.  Figure 6 presents the numerical calculations for the magnetization curves and thermoelectric responses for φ('=0∘, φ('=45∘, and φ('=90∘.  Considering φ('=0∘, Figs. 6 (a.I) and (a.II) depict the magnetization and thermoelectric behavior as a function of the external magnetic field for distinct φ% values. The addition of the cubic magnetocrystalline anisotropy yields, besides the easy and hard magnetization axes, the intermediate one at φ=45°. The emergence of such configuration of magnetic anisotropy and the competition of anisotropy axes lead to magnetization curves with interesting behavior, in particular for high φ% values. Given that the uniaxial anisotropy axis is along the V detection direction, the richness of details found in the magnetization curves is not observed in the thermoelectric ones. However, when φ('≠0∘, the refined structure of the magnetization curves, full of details associated to the magnetization process, is also identified in the thermoelectric curves. For φ('=45∘,\tchanges in both, magnetization and thermoelectric curves are observed as the φ% is modified (see Figs. 6 (b.I) and (b.II)). For φ% values smaller than 45∘,\tthe thermoelectric curves present similar shape to the one observed on the magnetization curves. On the other hand, for φ% values higher than φ('=45∘, the shapes of the thermoelectric and magnetic curves disclose fundamental differences, with a decrease in 𝑉/𝑉&12. Taking into account φ('=90∘ \t(Fig. 6 (c.I) and (c.II)), the mirror of the magnetization curves and thermoelectric one is verified, for φ%=0∘\tand φ%=20∘\tfield directions. On the other hand, a remarkable modification on the thermoelectric curves is verified for φ% above 45◦. This behavior reflects the modifications observed in the previous situations. From a general point of view, important modifications are observed in the shape of the  thermoelectric curves when φ% is higher than 45∘, irrespectively of the φ(' direction. This behavior corroborates previous results observed in magnetostrictive systems, in which the effective magnetic anisotropy has been modified by stress applications [23]. It is worth remarking that the addition of the cubic magnetocrystalline to an uniaxial anisotropy axis (keeping constant the anisotropy parameters considered in the previous sections and assuming  𝑘*+'=2.8𝑘)') describes a system with harder magnetic properties, if compared with the ones found in Fig. 3(a), 4(a) and 5(a). This fact becomes evident when the coercive fields are observed in Fig. 6(a.I, b.I and c.I), as well as through the fact that most of the curves in Fig. 6 do not achieve the magnetic saturation at 300 Oe.  Such behavior is reflected in the dependence of 𝑉/𝑉&12 with φ%, as presented in Fig. 6 (d.I) for a system with  φ('=0∘. Specifically, distortions in the curves are evidenced when the external magnetic field is 300 Oe. As previously discussed, deviations in the expected cosine form are attributed to a competition between the contributions of the Zeeman energy and anisotropy terms. For a field of 2000 Oe in turn, the Zeeman energy becomes at least one order of magnitude greater than the uniaxial and cubic anisotropies ones, having the major contribution to the magnetization dynamics. In this case, the system is magnetically saturated, and the cosine shape is recovered.  To highlight this feature, Fig. S2 in the Supplementary material presents calculations of the magnetization and thermoelectric response for a field of ±2000 Oe. To highlight the influence of the magnetic state on the thermoelectric response, Fig. 6(d.II) brings the evolution of the 𝑉/𝑉&12, at 300 Oe, with φ% for distinct φ(' values. From this, the competition of the anisotropies and Zeeman energies in this complex magnetic system can be disclosed.    FIG. 6. Numerical calculation for a system with uniaxial and cubic magnetocrystalline anisotropies. Normalized magnetization, at selected field orientations, for a system with (a.I) φ#$=0∘, (b.I) φ#$=45∘\tand (c.I) φ#$=90∘. Thermoelectric curves for a system with (a.II) φ#$=0∘, (b.II) φ#$=45∘\tand (c.II)) φ#$=90∘. (d.I)  𝑉/𝑉!+, as a function of φ\" for  φ#$ fixed at 0∘\tand magnetic fields of 300 Oe and 2000 Oe. (d.II) 𝑉/𝑉!+, as a function of φ\" for several φ#$ values at a magnetic field of 300 Oe. \n III. COMPARISON WITH EXPERIMENTAL RESULTS Three magnetic systems with distinct anisotropy configurations were produced to test the robustness of the theoretical approach.  The first sample is a YIG (3 m)/Pt (6 nm) bilayer grown onto a [111]-oriented Gd3Ga5O12 substrate, a sample with isotropic magnetic properties. The sample geometry and the employed materials make this sample suitable to the study of pure LSSE.  The second sample is a Co40Fe40B20 (300 nm) film grown onto a glass substrate, corresponding to a uniaxial magnetic system. In this case, given the sample structure, the thermoelectric voltage is associate with the pure ANE response. The third sample consists in a Co58.73Fe27.83Al13.44 (53 nm)/W (2 nm) bilayer deposited onto [100]-oriented GaAs substrate. Such system has cubic magnetocrystalline and uniaxial magnetic anisotropies. Here, the thermoelectric response is a result of the mixing of LSSE and ANE. Although the LSSE and ANE have different physical origins, their correspondence with respect to hysteretic behavior is similar. The experimental procedures used for sample preparation and characterization are presented in the Supplementary Material. The investigated samples allow verifying not just the dependence of the magnetic and thermoelectric features with the magnetic anisotropy, but they also provide tools to show that the ANE and LSSE experiments present very similar behavior and can be described by the developed theoretical approach.   Figure 7 shows the experimental and theoretical results for the YIG (3 m)/Pt (6 nm) bilayer. The measurements are obtained with Δ𝑇\t\t=\t27 K. For the calculations, it is considered 𝑚,=143 emu/cm3, λ-..=5.2×10/0 V/K, and ℎ('=4.8 Oe. Further, it is assumed φ('=30∘+𝜑%. This value allows to reproduce the experimental results, for which it is observed a deviation between 𝜑(\tand 𝜑%\t. Anyway, the isotropic behavior of the YIG (3 m)/Pt (6 nm) bilayer as a function of the  φ%is evident and corroborates the decrease of 𝑉/𝑉&12 with increasing 𝜑% and the absence of modifications in the shape of the curve with varying field orientation. Considering the curves obtained at φ%=90∘\t(not show), no thermoelectrical signal is observed. This feature is associated with the perpendicular alignment of the electric field and the electric contact direction (see Fig. 1).    \nFIG. 7. Comparison between experimental and theoretical results. Normalized magnetization curves obtained for (a) φ\"=0∘,  (b) φ\"=30∘. (c) φ\"=60∘.  The experimental results are obtained for the YIG (300 nm)/Pt (6 nm) bilayer grown onto [111]-oriented Gd3Ga5O12 substrate. The open circles depict the experimental results, while the red line brings the numerical calculations obtained with the theoretical approach for an isotropic magnetic system. To perform the calculation, a system with 𝑚&=143 emu/cm3, ℎ#$=4.8 Oe, φ#$=30∘+φ\", and λ'((=5.2×10)*V/K is considered. Besides, ∆T =\t27 K has been used during the LSSE measurement. Considering the uniaxial magnetic system, the experimental results for a Co40Fe40B20 film deposited onto a glass substrate are presented.  During the deposition a magnetic field of 1.0 kOe has been applied to induce the uniaxial magnetic behavior. Figure 8 presents thermoelectric curves for the selected φ% values of 0∘, 30∘, and 60∘. For the numerical calculations, 𝑚,=1026 em/cm3, 𝑘)'=7.072×103 ergs/cm3, 𝑘*+=0, Δ𝑇\t=\t27 K, λ-..=1.3×10/0 V/K, and φ('=85∘ have been used From figure 8, a very interesting evolution in the shape of the thermoelectric response with φ% is observed. In particular, this feature is a consequence of the change in the orientation of the field, \n which is rotating from the easy magnetization axis direction to the hard magnetization one. For φ%=0∘, i.e. along the easy axis, it is verified a mirror between the thermoelectric and magnetization curves (the latter not shown). This behavior is due to the fact that the detection direction of the thermoelectric voltage, defined by the electrical contacts, is perpendicular to the easy magnetization axis. By rotating the magnetic field to φ%=30∘ and φ%=60∘, at a high magnetic field regime, the component of the induced electric field along the direction of the electrical contacts decreases, leading a reduction of the thermoelectrical voltage. On the other hand, at a low magnetic field regime, the uniaxial magnetic anisotropy leads the magnetization to the easy axis, rotating the electric field and, consequently, increasing the thermoelectrical voltage, as confirmed from the results. This mechanism is observed in both, experimental results and numerical calculations. Here, it is worth pointing out the alignment between the electric field and contact direction is essential to describe the experimental curves. In addition, although a multidomain system is simulated, the complex domain structure of a real system leads to an increase in the coercive field for a real sample. This feature is responsible for the small discrepancy between the coercivity values of the experimental and theoretical results observed as φ% increases.   FIG. 8. Comparison between experimental and theoretical results. Normalized magnetization curves obtained for (a) φ\"=0∘,  (b) φ\"=30∘. (c) φ\"=60∘.  The experimental results are obtained for the Co40Fe40B20 film grown onto the glass substrate, presenting uniaxial magnetic anisotropy behavior. The open circles depict the experimental results, while the red line brings the numerical calculations obtained with the theoretical approach for an isotropic magnetic system. To perform the calculation, it has been considered 𝑚&=1026 em/cm3, 𝑘-$=7.072×10. ergs/cm3, 𝑘/0=0, Δ𝑇\t=\t27 K, λ'((=1.3×10)* V/K and 𝜑#$=85∘. Figure 9 shows the experimental and theoretical results obtained for the thermoelectric curves of Co58.73Fe27.83Al13.44/W bilayers grown onto [100]-oriented GaAs substrates and submitted to a 1.0 kOe during the deposition (details on the experimental procedure can be found in Supplementary Material). The employed substrates and the annealing process allows to obtain a magnetic system with uniaxial and cubic magnetocrystalline anisotropies. For the numerical calculations, it has been considered 𝑚,=1330 emu/cm3, 𝑘)'=1.66×100 ergs/cm3, 𝑘*+'=2.8\t𝑘)', Δ𝑇\t=\t27 K, an φ('=90∘  , and\tλ-..=3.79×10/4 V/K.  From a general point of view, the evolution of the curve’s shape as φ% increases has been observed. The curves present a remarkable modification on the shape when φ% is higher than 45∘. This behavior is similar to the one observed in the theoretical results presented in Fig. 6. As discussed before, at φ(=45∘, the studied system shows an intermediate magnetization axis. For this reason, the φ% value is an important angle.  For φ% values smaller than this critical value, the thermoelectric curves seem to present similar behavior to the \n one presented by the magnetization. On the other hand, for φ% values larger than 45∘, the shape changes drastically. Hence, the cross of φ% through this anisotropy axis leads to modifications in the shape of the thermoelectric response, an issue associated with the anisotropy configuration, discussed in detail in Ref. [23].   \nFIG. 9. Comparison between experimental and theoretical results. Normalized magnetization curves obtained for (a) φ\"=0∘,  (b) φ\"=30∘. (c) φ\"=60∘.  The experimental results are obtained for the Co58.73Fe27.83Al13.44/W bilayer grown onto [100]-oriented GaAs substrate, a system having cubic and uniaxial magnetic anisotropies. The open circles depict the experimental results, while the red line brings the numerical calculations obtained with the theoretical approach for an isotropic magnetic system. To perform the calculation, it has been applied 𝑚&=1330 emu/cm3, 𝑘-$=1.66×10* ergs/cm3, 𝑘/0=2.8𝑘-$, Δ𝑇\t=\t27 K, and 𝜑#$=90∘, λ'((=3.79×10)1\tV/K. IV. CONCLUSION In summary, the role of magnetic anisotropies on the magnetic and thermoelectric responses of noninteracting multidomain ferromagnetic systems has been addressed, based on a theoretical approach based in a modified Stoner Wohlfarth model. The approach has been tested for an isotropic system, a uniaxial magnetic one, as well as for a system having a mixture of uniaxial and cubic magnetocrystalline magnetic anisotropies. It has been verified remarkable modifications of \n the magnetic behavior with the anisotropy and it has been shown that the thermoelectric response is strongly affected by these changes. Further, the fingerprints of the energy contributions to the thermoelectric response have been disclosed, revealing that the anisotropy energy terms represent a fundamental contribution to the energy balance even in saturated states when the field is right above the saturation field. To test the robustness of the developed theoretical approach, three films have been engineered with specific magnetic properties and have been compared to the corresponding magnetic and thermoelectric experimental data. From the comparison and the remarkable agreement between experiment and theory, experimental evidence has been provided to confirm the validity of the theoretical approach. The results go beyond the traditional reports focusing on magnetically saturated films and show how the thermoelectric effect behaves during the whole magnetization curve. The results emphasize the importance of selecting specific field alignments and field amplitudes to provide reliable interpretation of the ANE and LSSE experiments. Our findings reveal a suitable and robust way to explore the ANE and LSSE effects as a powerful tool to study magnetic anisotropies, as well as to employ systems with magnetic anisotropy as sensing elements for technological applications.   ACKNOWLEDGMENTS This work was partially supported by the Brazilian agencies CNPq and CAPES. M. A. Correa acknowledges CNPq through the project 304943/2020-7, and CAPES through process 88887.573100/2020-00. Further, this work was also supported by the Portuguese Foundation for Science and Technology (FCT) in the framework of the Strategic Funding UID/FIS/04650/2019 and project PTDC/BTM-MAT/28237/2017. A. Ferreira acknowledges the FCT for the Junior Research  Contract (CTTI-31/18). Financial support from the Basque Government Industry Department under the ELKARTEK, and PIBA (PIBA-2018-06) programs is also acknowledged.      REFERENCES [1] G.E.W. Bauer, E. Saitoh, B.J. van Wees, Spin caloritronics, Nat. Mater. 11 (2012) 391–399. doi:10.1038/nmat3301. [2] R.C.M. S. R. Boona, J.P. Heremans, Spin caloritronics, Energy Environ. Sci. 7 (2014) 885–910. doi:10.1039/C3EE43299H. [3] J. Holanda, O. Alves Santos, R.O. Cunha, J.B.S. Mendes, R.L. Rodríguez-Suárez, A. Azevedo, S.M. Rezende, Longitudinal spin Seebeck effect in permalloy separated from the anomalous Nernst effect: Theory and experiment, Phys. Rev. B. 95 (2017) 214421. doi:10.1103/PhysRevB.95.214421. [4] M. Johnson, Spin caloritronics and the thermomagnetoelectric system, Solid State Commun. 150 (2010) 543–547. doi:10.1016/j.ssc.2009.10.027. [5] M. Zeng, Y. Feng, G. Liang, Graphene-based Spin Caloritronics, Nano Lett. 11 (2011) 1369–1373. doi:10.1021/nl2000049. [6] A. Kirihara, K. Kondo, M. Ishida, K. Ihara, Y. Iwasaki, H. Someya, A. Matsuba, K. Uchida, E. Saitoh, N. Yamamoto, S. Kohmoto, T. Murakami, Flexible heat-flow sensing sheets based on the longitudinal spin Seebeck effect using one-dimensional spin-current conducting films, Sci. Rep. 6 (2016) 23114. doi:10.1038/srep23114. [7] K. Uchida, M. Ishida, T. Kikkawa, A. Kirihara, T. Murakami, E. Saitoh, Longitudinal spin Seebeck effect: from fundamentals to applications, J. Phys. Condens. Matter. 26 (2014) 343202. doi:10.1088/0953-8984/26/34/343202. [8] P. Sengupta, Y. Wen, J. Shi, Spin-dependent magneto-thermopower of narrow-gap lead chalcogenide quantum wells, Sci. Rep. 8 (2018) 5972. doi:10.1038/s41598-018-23511-2.  [9] A.W. Smith, The Transverse Thermomagnetic Effect in Nickel and Cobalt, Phys. Rev. (Series I). 33 (1911) 295–306. doi:10.1103/PhysRevSeriesI.33.295. [10] N. Nagaosa, J. Sinova, S. Onoda, A.H. MacDonald, N.P. Ong, Anomalous Hall effect, Rev. Mod. Phys. 82 (2010) 1539–1592. doi:10.1103/RevModPhys.82.1539. [11] Y. Pu, D. Chiba, F. Matsukura, H. Ohno, J. Shi, Mott Relation for Anomalous Hall and Nernst Effects in Ga1-x Mnx As Ferromagnetic Semiconductors, Phys. Rev. Lett. 101 (2008) 117208. doi:10.1103/PhysRevLett.101.117208. [12] D. Xiao, Y. Yao, Z. Fang, Q. Niu, Berry-Phase Effect in Anomalous Thermoelectric Transport, Phys. Rev. Lett. 97 (2006) 026603. doi:10.1103/PhysRevLett.97.026603. [13] D. Tian, Y. Li, D. Qu, X. Jin, C.L. Chien, Separation of spin Seebeck effect and anomalous Nernst effect in Co/Cu/YIG, App. Phys. Lett. 106 (2015) 212407. doi:10.1063/1.4921927. [14] K.T. Shinji Isogami, M. Mizuguchi, Dependence of anomalous Nernst effect on crystal orientation in highly ordered γʹ-Fe 4 N films with anti-perovskite structure, Appl. Phys. Express. 10 (2017) 73005. http://stacks.iop.org/1882-0786/10/i=7/a=073005. [15] T.C. Chuang, P.L. Su, P.H. Wu, S.Y. Huang, Enhancement of the anomalous Nernst effect in ferromagnetic thin films, Phys. Rev. B. 96 (2017) 174406. doi:10.1103/PhysRevB.96.174406. [16] H. Kannan, X. Fan, H. Celik, X. Han, J.Q. Xiao, Thickness dependence of anomalous Nernst coefficient and longitudinal spin Seebeck effect in ferromagnetic NixFe100−x films, Sci. Rep. 7 (2017) 6175. doi:10.1038/s41598-017-05946-1. [17] K.I. Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, E. Saitoh, Observation of longitudinal spin-Seebeck effect in magnetic insulators, Appl. Phys. Lett. 97 (2010) 172505. doi:10.1063/1.3507386.  [18] S. Meyer, R. Schlitz, S. Geprägs, M. Opel, H. Huebl, R. Gross, S.T.B. Goennenwein, Anomalous Hall effect in YIG|Pt bilayers, Appl. Phys. Lett. 106 (2015) 132402. doi:10.1063/1.4916342. [19] C. Hahn, G. De Loubens, O. Klein, M. Viret, V. V. Naletov, J. Ben Youssef, Comparative measurements of inverse spin Hall effects and magnetoresistance in YIG/Pt and YIG/Ta, Phys. Rev. B. 87 (2013) 174417. doi:10.1103/PhysRevB.87.174417. [20] R. Iguchi, K. Uchida, S. Daimon, E. Saitoh, Concomitant enhancement of the longitudinal spin Seebeck effect and the thermal conductivity in a Pt/YIG/Pt system at low temperatures, Phys. Rev. B. 95 (2017) 174401. doi:10.1103/PhysRevB.95.174401. [21] D. Meier, D. Reinhardt, M. van Straaten, C. Klewe, M. Althammer, M. Schreier, S.T.B. Goennenwein, A. Gupta, M. Schmid, C.H. Back, J.-M. Schmalhorst, T. Kuschel, G. Reiss, Longitudinal spin Seebeck effect contribution in transverse spin Seebeck effect experiments in Pt/YIG and Pt/NFO, Nat. Commun. 6 (2015) 8211. doi:10.1038/ncomms9211. [22] Y.-M. Kang, S.-H. Wee, S.-I. Baik, S.-G. Min, S.-C. Yu, S.-H. Moon, Y.-W. Kim, S.-I. Yoo, Magnetic properties of YIG (Y3Fe5O12) thin films prepared by the post annealing of amorphous films deposited by rf-magnetron sputtering, J. Appl. Phys. 97 (2005) 10A319. doi:10.1063/1.1855460. [23] A.S. Melo, A.B. de Oliveira, C. Chesman, R.D. Della Pace, F. Bohn, M.A. Correa, Anomalous Nernst effect in stressed magnetostrictive film grown onto flexible substrate, Sci. Rep. 9 (2019) 15338. doi:10.1038/s41598-019-51971-7. [24] Z. Li, J. Krieft, A.V. Singh, S. Regmi, A. Rastogi, A. Srivastava, Z. Galazka, T. Mewes, A. Gupta, T. Kuschel, Vectorial observation of the spin Seebeck effect in epitaxial NiFe 2 O 4 thin  films with various magnetic anisotropy contributions, Appl. Phys. Lett. 114 (2019) 232404. doi:10.1063/1.5092774. [25] M.V. Lopes, E.C. de Souza, J.G. Santos, J.M. de Araujo, L. Lima, A.B. de Oliveira, F. Bohn, M.A. Correa, Modulating the Spin Seebeck Effect in Co2FeAl Heusler Alloy for Sensor Applications, Sensors. 20 (2020) 1387. doi:10.3390/s20051387. [26] C. Tannous, J. Gieraltowski, The Stoner-Wohlfarth model of ferromagnetism, Eur. J. Phys. (2008). doi:10.1088/0143-0807/29/3/008. [27] S. Hoffman, K. Sato, Y. Tserkovnyak, Landau-Lifshitz theory of the longitudinal spin Seebeck effect, Phys. Rev. B. 88 (2013) 064408. doi:10.1103/PhysRevB.88.064408. [28] H. Adachi, K. Uchida, E. Saitoh, S. Maekawa, Theory of the spin Seebeck effect., Rep. Prog. Phys. 76 (2013) 036501. doi:10.1088/0034-4885/76/3/036501. [29] M.A. Correa, F. Bohn, Manipulating the magnetic anisotropy and magnetization dynamics by stress: Numerical calculation and experiment, J. Magn. Magn. Mater. 453 (2018) 30–35. doi:10.1016/J.JMMM.2017.12.087. [30] E.C. Stoner, E.P. Wohlfarth, A mechanism of magnetic hysteresis in heteresis in heterogeneous alloys, Phil. Trans. Roy.Soc. 240 (1948) 599--642. doi:10.1098/rsta.1948.0007. [31] E.P. Wohlfarth, Relations between Different Modes of Acquisition of the Remanent Magnetization of Ferromagnetic Particles, J. Appl. Phys. 29 (1958) 595–596. doi:10.1063/1.1723232. [32] H. Bai, X.Z. Zhan, G. Li, J. Su, Z.Z. Zhu, Y. Zhang, T. Zhu, J.W. Cai, Characterization of YIG thin films and vacuum annealing effect by polarized neutron reflectometry and magnetotransport measurements, Appl. Phys. Lett. 115 (2019) 182401.  doi:10.1063/1.5124832. [33] M. Gamino, J.G. S. Santos, A.L. R. Souza, A.S. Melo, R.D. Della Pace, E.F. Silva, A.B. Oliveira, R.L. Rodríguez-Suárez, F. Bohn, M.A. Correa, Longitudinal spin Seebeck effect and anomalous Nernst effect in CoFeB/non-magnetic metal bilayers, J. Magn. Magn. Mater. 527 (2021) 167778. doi:10.1016/j.jmmm.2021.167778.    "    },    {        "title": "2106.08791v1.Magnetic_and_geometrical_control_of_spin_textures_in_the_itinerant_kagome_magnet_Fe__3_Sn__2_.pdf",        "content": "Magnetic and geometrical control of spin textures in the itinerant kagome magnet\nFe3Sn2\nMarkus Altthaler,1, 2, 3,\u0003Erik Lysne,2, 3,\u0003Erik Roede,2Lilian Prodan,1Vladimir Tsurkan,1, 4\nMohamed A. Kassem,5Stephan Krohns,1Istv\u0013 an K\u0013 ezsm\u0013 arki,1,yand Dennis Meier2, 3,z\n1Experimental Physics V, University of Augsburg, 86135 Augsburg, Germany\n2Department of Materials Science and Engineering,\nNorwegian University of Science and Technology (NTNU), 7043 Trondheim, Norway\n3Center for Quantum Spintronics, Department of Physics,\nNorwegian University of Science and Technology (NTNU), 7491 Trondheim, Norway\n4Institute for Applied Physics, MD-2028, Chisinau, Moldova\n5Department of Physics, Assiut University, 171516 Assiut, Egypt\n(Dated: June 17, 2021)\nMagnetic materials with competing magnetocrystalline anisotropy and dipolar energies can de-\nvelop a wide range of domain patterns, including classical stripe domains, domain branching, as\nwell as topologically trivial and non-trivial (skyrmionic) bubbles. We image the magnetic domain\npattern of Fe 3Sn2by magnetic force microscopy (MFM) and study its evolution due to geometric\ncon\fnement, magnetic \felds, and their combination. In Fe 3Sn2lamellae thinner than 3 µm, we\nobserve stripe domains whose size scales with the square root of the lamella thickness, exhibiting\nclassical Kittel scaling. Magnetic \felds turn these stripes into a highly disordered bubble lattice,\nwhere the bubble size also obeys Kittel scaling. Complementary micromagnetic simulations quanti-\ntatively capture the magnetic \feld and geometry dependence of the magnetic patterns, reveal strong\nreconstructions of the patterns between the surface and the core of the lamellae, and identify the\nobserved bubbles as skyrmionic bubbles. Our results imply that geometrical con\fnement together\nwith competing magnetic interactions can provide a path to \fne-tune and stabilize di\u000berent types\nof topologically trivial and non-trivial spin structures in centrosymmetric magnets.\nI. INTRODUCTION\nProgress in information and communication technol-\nogy is driven by the discovery of new materials and phe-\nnomena that enable components with improved func-\ntionality. In a recent development, topologically pro-\ntected spin textures, such as magnetic skyrmions and\nmerons, are explored as functional nanoscale objects\nthat can be utilized to process and store information\n[1{3]. Initially linked to the Dzyaloshinskii{Moriya in-\nteraction [4] (DMI), skyrmions were mostly studied in\nnon-centrosymmetric crystals, such as cubic chiral [5{\n7] and axially symmetric polar magnets [8{10]. It was\ndemonstrated that the emergence of skyrmions often is\naccompanied by peculiar electromagnetic properties like\nthe topological Hall e\u000bect [11], a zoo of collective exci-\ntations [12{14], pinning-free motion at ultra low current\ndensities and polar dressing via the magnetoelectric e\u000bect\n[15].\nThe formation of skyrmions and topologically equiv-\nalent skyrmionic bubbles [16] is not restricted to non-\ncentrosymmetric materials. In centrosymmetric mag-\nnets, such topologically non-trivial spin textures can be\nstabilized by frustrated exchange interactions [17], orig-\ninating from competing dipolar interactions and mag-\nnetic anisotropy [16, 18{23]. In general, the spin texture\n\u0003equal contribution\nyistvan.kezsmarki@physik.uni-augsburg.de\nzdennis.meier@ntnu.noof both skyrmions and skyrmionic bubbles is described\nby three parameters: the polarity ( p=\u00061) correspond-\ning to the direction of magnetiztion at the core, the he-\nlicity (\r) determined by the o\u000bset to the polar phase\n(\b(\u001e) =m\u001e+\r) and the vorticity\nm=1\n2\u0019Z2\u0019\n0d\b(\u001e); (1)\nwhich counts the number of revolutions of the polar an-\ngle [24]. Here, \u001eand \b are the azimuthal angle in real\nspace and the azimuthal angle of the local magnetiza-\ntion, respectively. In centrosymmetric magnets, bubbles\nwithm= 0 are topologically trivial, whereas m= 1 in-\ndicates a topologically protected state, corresponding to\nskyrmions/skyrmionic bubbles. Depending on the rel-\native magnitudes of the involved interactions, topolog-\nically trivial and non-trivial bubbles can even coexist,\nleading to complex mixed states with solitons of topolog-\nical charge [18, 25].\nFor achieving such exotic magnetic states and, hence,\nnovel functional responses, materials with perpendicu-\nlar magnetocrystalline anisotropy (PMA) and compara-\nble values of uniaxial and shape anisotropy are particu-\nlarly promising [18, 26]. The delicate balance of the two\nanisotropy terms creates substantial frustration, which\nenables the formation of unconventional magnetic order,\nincluding modulated magnetic structures, Bloch points\nand lines, as well as skyrmionic bubbles with opposite\nhelicity [27].\nHere, we study Fe 3Sn2, a kagome layered-based itiner-\nant magnet, where uniaxial ( Ku) and shape anisotropiesarXiv:2106.08791v1  [cond-mat.str-el]  16 Jun 20212\nare comparable [28]. The latter is described by the qual-\nity factorQ= 2Ku=\u00160M2\nsat, which has a value of about\n0.15. The material has a centrosymmetric rhombohe-\ndral structure formed of Fe-Sn bilayers alternating with\nSn layers along the crystallographic caxis. The anoma-\nlous Hall e\u000bect [29{31], the exceptionally high Curie tem-\nperature and massive Dirac fermions [31] in the vicinity\nof the Fermi energy make Fe 3Sn2highly attractive for\ntechnological applications [32, 33]. Geometric con\fne-\nment has been demonstrated to have a strong impact on\nthe nucleation of bubbles. More precisely, bubbles with\nm= 0 andm= 1 (type II and type I, respectively) are\npermitted to coexist or topologically trivial bubbles are\ncompletely expelled [28]. Futhermore, by con\fning the\ngeometry, Hou et al. have demonstrated that the criti-\ncal \felds required to stabilize type I bubbles are signi\f-\ncantly reduced, reaching realistic values for technological\napplications. Early studies on current-driven dynamics\nhave also demonstrated that the helicity can be electri-\ncally tuned via spin transfer torque [34]. The presented\nexperiments by Lorenz transmission electron microscopy\n(LTEM), however, were limited to electron transparent\nsamples with thicknesses <\u0018200 nm. In order to ex-\npand the research and systemically study the impact of\nthe demagnetization energy for thicknesses varying over\nlarger length scales, we apply magnetic force microscopy\n(MFM). We \fnd that micrometer thick lamella-shaped\nsamples exhibit bulk-like behavior, developing the pecu-\nliar dendrite pattern reported for Fe 3Sn2bulk samples\n[26, 35]. For smaller thickness ( <\u00181µm), we observe the\nformation of stripe and bubble domains which decrease\nin size with decreasing lamella thickness, following Kit-\ntel's law [36]. In addition, we study how variable mag-\nnetic \felds transform the magnetic stripe domains into\nskyrmionic bubbles while keeping the sample thickness\n\fxed and, vice versa, how the magnetic structure evolves\nas the sample thickness varies under constant magnetic\n\feld. The MFM measurements are complimented by mi-\ncromagnetic simulation of the 3D magnetic order and\nits response to the applied manipulation, providing new\ninsight into the physics of topological spin textures in\nmagnetic materials with competing uniaxial and shape\nanisotropies.\nII. METHODS\nA. Crystal growth\nFe3Sn2single crystals are grown by the chemical trans-\nport reactions method. As starting material for growth,\nthe preliminary synthesized polycrystalline powder is\nused. It has been prepared by solid state reactions from\nthe high-purity elements: Fe (99.99%) and Sn (99.995%).\nIodine is utilized as the transport agent in the single crys-\ntal growth. The growth is performed at temperatures be-\ntween 730 and 680 °C. Plate-like samples of thickness 20\n- 40µm along the c-axis and 3-5 mm within the ab-plane\nFIG. 1. (a) MFM imaging of a polished ab-plane surface of a\nFe3Sn2single crystal reveals a peculiar dendrite pattern. (b)\nRepresentative SEM image showing a plan-view lamella cut\nfrom an ab-plane surface as shown in (a). (c) MFM image of a\n2:7µm thick lamella revealing a pattern of out-of-plane mag-\nnetized alternating ferromagnetic stripe-like domains. Each\nindividual stripe exhibits an undulated pattern characteris-\ntic for the onset of domain branching. (d) MFM scan of a\n200 nm thick lamella, showing a stripe-like domain pattern.\nAll scale-bars (red) have a width of 4 µm.\nare obtained.\nB. Lamella preparation\nFe3Sn2plan-view lamellae are prepared using a FIB-\nSEM at 30 kV ion beam acceleration voltage. We em-\nploy a Thermo Scienti\fc G4UX DualBeam and a Zeiss\nCrossbeam 550 system equipped with an EasyLift EX\nNanoManipulator and a Kleindiek MM3A-EM microma-\nnipulator, respectively. A triangular prism-shaped vol-\nume is isolated by milling undercuts at 45 °to the surface\nfrom a Fe 3Sn2single crystal. The specimen is then ex-\ntracted using an micromanipulator and transferred to a\ncopper half grid. By milling parallel to the original crys-\ntal surface, the samples are thinned to the desired thick-\nness. The lamellae are then lifted o\u000b again and trans-\nferred to a substrate covered with 200 nm of gold. In\norder to attach the lamella, the edges are welded to the\nsurface using carbon or platinum as seen in the SEM im-\nage in Figure 1(b). Lamellae with controlled thicknesses\nbetween 200 nm and several microns are prepared and\n\fxed \rat on a substrate.\nFollowing the same procedure, a prism-shaped volume\nis prepared for the wedge-like lamella. After polishing\nthe original surface, the sample is transferred to a silicon\nsubstrate with the surface facing down. Mounted on a 45 °\nstub, the prism is then cut from the apex at a low angle\nversus the base plane resulting in a wedge-like lamella\nwith a thickness varying from 400 nm to 900 nm along3\nthe long axis.\nC. Magnetic imaging\nTo investigate the magnetic domain textures Fe 3Sn2,\nwe perform AC-MFM on an Oxford Instruments Cypher\nES Environmental AFM at room temperature, map-\nping the phase shift of the read-out signal. For low-\ntemperature and \feld-dependent studies, an attocube at-\ntoAFM I, equipped with a superconducting magnet is\nused, recording the frequency shift with a phase sensi-\ntive feedback loop. In both setups, the obtained image\ncontrast is proportional to the projection of the gradi-\nent of the stray magnetic \feld to the tip magnetization\n[37]. In this work, we use Nanosensors TM PPP-MFMR\nprobes magnetized prior to the scans.\nThe periodicity of the magnetic patterns in the MFM\nimages is quanti\fed using a MATLAB script. At each\npoint of the MFM images the local periodicity is deter-\nmined by fast Fourier transformation (FFT) over an area\nthat is su\u000eciently large not to a\u000bect the obtained period-\nicity by a spatial cuto\u000b. For wedged samples, the period-\nicity is averaged along lines of equal specimen thickness.\nD. Micromagnetic simulations\nMicromagnetic simulations are performed using Mu-\nMax3 [38]. The energy terms included in the model\nrepresent the Heisenberg exchange, 1st order uniaxial\nanisotropy, Zeeman, and demagnetization energy terms,\n\"=Z\nVs\u0002\nAex(rm)2\u0000Kum2\nz\n+MsatBext\u0001m\u00001\n2MsatBdem\u0001m\u0003\ndr:(2)\nHere, m(x;y;z )\u0011M(x;y;z )=Msatis the reduced magne-\ntization. The orientation of the magnetization M(x;y;z )\nwithin the sample volume Vsis represented as a con-\ntinuous unit vector \feld. Msatis the saturation mag-\nnetization, Aexis the exchange constant, Kuis the 1st\norder uniaxial anisotropy constant, Bextis the external\nmagnetic \feld, and Bdemis the demagnetization \feld.\nBased on former studies at room temperature [28, 32{\n34], the magnetic easy axis is chosen to point along the\nzaxis. In order to avoid edge-induced e\u000bects, periodic\nboundary conditions are applied along the in-plane di-\nrections ( xand y). Open boundary condition is used\nalong the out-of-plane direction ( z), re\recting the con-\nstraints of ab-plane cuts of varying thickness. The mesh\nis constructed by 8 nm \u00028 nm\u00028 nm cells. The geom-\netry is set to a square cuboid with length and width of\n4096 nm. We investigate the thickness dependence of the\nmagnetic patterns for a series of cuboids with di\u000berent\nthicknesses. The magnetization is relaxed from an initialuniform out-of-plane magnetized state, where nucleation\nsites are included in the form of cylinders magnetized in\nthe opposite direction to the surrounding matrix. The\nlatter allows for stabilizing the maze-like ground state\nobserved experimentally against uniform saturation.\nIII. RESULTS\nFIG. 2. (a) The inset displays the 3D visualization of the to-\npography of the wedge-like lamella colored with the magnetic\ntexture, where the bright and dark contrast corresponds to do-\nmains with magnetization up and down. The local periodicity\n(blue data points) is obtained via fast Fourier transformation\n(FFT) based analysis of the MFM signal and plotted versus\nthe sample thickness. The analysis shows a square root de-\npendence of the domain width, i.e., Kittel scaling (red line).\nA selection of simulated structures, representing a 4x4 µm2\narea with periodic boundary conditions, is displayed in (b)-\n(e) where the sample thickness is 192 nm, 384 nm, 576 nm,\nand 768 nm. In the simulations brightness corresponds to the\nz-component of the magnetization, whereas colors encode the\nin-plane components in accordance with the color wheel in\npanel (e). The simulated trend in periodicity is in agreement\nwith the experimental observations shown in (a), where the\ngreen data points show the periodicity derived from simula-\ntions scaled by a factor of 1.12 to match the scaling law of the\nexperimental data. In (f) and (g) representative cross sections\nfor the thin and thick end of the wedge lamella taken along\nthe lines marked in (b) and (e) are shown. The color code is\nthe same as in with the simulations presented in top view.4\nMFM scans of the polished ab-plane surface of a\nFe3Sn2crystal reveal a peculiar pattern of alternating\nbright and dark phase contrast, showing branched ferro-\nmagnetic domains, Fig. 1(a). Such domain branching is\ncharacteristic for Fe 3Sn2[35], and materials with PMA in\ngeneral [26]. The order of branching is expected to scale\nwith the width of the underlying domains, vanishing with\ndecreasing sample thickness [26]. Experimentally, we ob-\nserve that the domain branching in Fe 3Sn2disappears at\na sample thickness of about 2 :7µm. The corresponding\nMFM image, Fig. 1(c), reveals predominantly stripe-like\ndomains with undulated domain walls characteristic for\nthe onset of domain branching. In contrast, a maze-like\npattern of stripe-like domains with straight domain walls\nis observed in thinner lamellae as presented in Fig. 1(d),\nconsistent with literature [32]. The MFM data con\frms\nthat the applied FIB-based extraction procedure does not\nsuppress or hinder the magnetic domain formation, con-\n\frming that the applied approach is suitable for prepa-\nration of high-quality Fe 3Sn2lamellae.\nFigure 2 presents the e\u000bect of geometrical con\fnement.\nThe possibility to utilize FIB milling to shape lamellae\nenables many opportunities to alter the geometry sys-\ntematically and with nanometer precision. We seize this\noption to prepare a wedged lamella with thickness rang-\ning from 400 nm to 900 nm, to gain insight into the con-\ntinuous variation of the magnetic pattern with the sam-\nple thickness. The MFM image recorded on the wedged\nlamella is displayed as an inset to Fig. 2(a). Qualitatively,\nthe alternating ferromagnetic stripe-like pattern resem-\nbles the magnetic domain state observed in the lamella\nin Fig. 1(d). Along the thickness gradient a clear widen-\ning of the stripes can be observed towards the thicker\nend of the lamella. Perpendicular to the gradient, no sig-\nni\fcant variation of the stripe width is found, pointing\nto a direct connection between the local thickness of the\nlamella and the width of the stripe domains. Our quan-\ntitative analysis shows a square root dependence of the\ndomain width on the lamella thickness, i.e., the stripe\nwidth follows classical Kittel scaling [36].\nThe variation of the stripe width versus sample thick-\nness, revealed within the wedged lamella, provides a\nbasis to re\fne the magnetic interaction parameters in\nFe3Sn2for more accurate micromagnetic simulations.\nFigures 2(b)-(e) show the results of our simulations for\nsquare cuboid lamellae with various thicknesses (192 nm,\n384 nm, 576 nm and 768 nm). With the parameter val-\nuesMsat= 566 kA m\u00001,Aex= 14:0 pJ m\u00001andKu=\n30:0 kJ m\u00003, the simulations quantitatively reproduce the\nthickness dependence of the stripe width (c.f. Fig. 2(a)),\nas observed in the surface region of the lamella. Impor-\ntantly, the re\fned micromagnetic simulations provide an\nin-depth view on the 3D magnetization pattern. Fig-\nures 2(f) and (g) show cross-section images of the mag-\nnetization representative for the thin and thick end of\nthe wedged lamella, respectively. In both cases alternat-\ning ferromagnetic domains, coaligned with the uniaxial\nanisotropy, constitute the core of the cuboids. In thesurface region N\u0013 eel capping occurs, explaining the grad-\nual transition between domains in the MFM response.\nWhile the rotation of the magnetization is rather smooth\nin the surface region, the walls between up and down do-\nmains become considerably sharper in the internal region.\nThis e\u000bect is strongest for the largest lamella thickness\nin Fig. 2(g).\nAfter clarifying the thickness dependence of the mag-\nnetic texture, we investigate its magnetic \feld-driven evo-\nlution. Fig. 3(a) shows the \feld dependence of the mag-\nnetization for a macroscopic single crystalline sample at\n300 K with the magnetic \feld applied out of plane, i.e.,\nparallel to the caxis. In zero magnetic \feld, the sample\nhas no remanent magnetization in agreement with the\nstripe-domain pattern observed by MFM and in the mi-\nFIG. 3. (a) The magnetization data for a macroscopic sample\n(presented by the black data points) indicates that saturation\nmagnetization is reached at about 750 mT and a hysteresis\nfree cycle. Magnetic-\feld dependent MFM scans recorded on\na 450 nm thick lamella with stripe domains are presented as\ninsets, revealing the local response to the applied \feld ( T=\n200 K). All images displayed show a 4x4 µm2area with a\n25 Hz color gradient. As the magnetic \feld increases, stripe\ndomains (region marked by the underlying blue color) grad-\nually transform to bubbles (green) before reaching the satu-\nrated state (purple). The residual contrast observed in the\nMFM scans obtained after Fe 3Sn2has been \feld polarized\nis attributed to the surface topography shown in the inset\nof (a). Upon decreasing the \feld, this evolution is reversed\nwith spontaneously forming bubbles that subsequently elon-\ngate. The initially fragmented stripes grow and merge, re-\nlaxing back into a maze-like domain pattern. Panels (b)-(d)\nshow three distinct \feld steps illustrating the \feld evolution\nin more detail.5\ncromagnetic simulations. Following a linear increase, the\nmagnetization saturates at 750 mT to the value of 2 \u0016B\nper Fe atom. This value is in good agreement with ear-\nlier reports [39, 40] and converts to Msat= 571 kA m\u00001,\nthat agrees within 1% with the value re\fned in the micro-\nmagnetic simulations. The detailed \feld evolution of the\ndomain pattern is documented by a series of MFM im-\nages recorded on a 450 nm thick Fe 3Sn2lamella at 200 K.\nMFM images recorded over the same area in 100 mT\nsteps are shown both for increasing the \feld up to 900 mT\nand decreasing it back to zero. With increasing magnetic\n\feld, the stripe domains gradually transform to bubble-\nlike domains in between 300 mT and 600 mT before being\nexpunged at the transition to the uniform ferromagnetic\nphase at approximately 700 mT. Analogous results are\nobserved at room temperature as discussed later. At\nhigh magnetic \felds, a residual contrast is observed in\nthe MFM scans independent of further variations in the\n\feld strength, which we attribute to topographic fea-\ntures shown in the inset to Fig. 3(a). With decreasing\n\feld, bubble-like domains form spontaneously just be-\nlow 700 mT and subsequently elongate. Below 500 mT\nthe fragmented stripes gradually grow and merge at low\n\felds. The evolution of the magnetic pattern can be fol-\nlowed in more detail in Figs. 3(b)-(d), which show three\nrepresentative images recorded in zero \feld, 300 mT and\n600 mT. In 300 mT, the area magnetized opposite to the\n\feld shrinks and some of the stripes break up into bub-\nbles. In 600 mT, i.e., just below the saturation \feld, a\npure bubble phase is realized. Thus, the behavior ob-\nserved in the lamella is quantitatively similar to the do-\nmain evolution observed in classical ferromagnets with\nPMA, such as CoCr and TbGdFeCo, under application\nof a magnetic \feld [26]. The emergence of completely\ndi\u000berent domain wall positions at zero \feld before and\nafter ramping the magnetic \feld further re\rects that no\nsubstantial pinning occurs in the Fe 3Sn2lamella. Thus,\npronounced e\u000bects from defects, e.g., Ga implantation\nfrom the FIB preparation, can be excluded.\nNext, we investigate the combined e\u000bect of thickness\nvariation and magnetic \feld on the domain patterns in\nFe3Sn2by studying the domain structure in a wedged\nlamella as function of magnetic \feld. The experimental\nresults and complementary micromagnetic simulations,\ndisplayed in Fig. 4, show the magnetic textures in zero\n\feld (a) and 450 mT (b). In zero \feld, stripes form a\nmaze-like pattern, where the stripe width scales with\nthe thickness according to Kittel's law, as presented in\nFig. 2(a). In 450 mT, this pattern transforms to a state\nwhere shorter stripes coexist with bubbles. The nucle-\nation of bubbles seems to start at the edges of the lamella,\nwhereas the remaining stripes are concentrated within\nthe internal part of the lamella. The phase coexistence\nof stripes and bubbles in 450 mT is consistent with the\nobservations at 200 K. Interestingly, we observe that not\nonly the remaining fragmented stripe-like domains follow\nKittel scaling, but also the diameter of the newly formed\nbubbles scales accordingly.\nFIG. 4. (a) The MFM image in zero \feld shows the gradual\nwidening of the magnetic domains in a wedge-shaped lamella\nfollowing Kittel's scaling law analogous to Fig. 2(a). (b) By\napplying a magnetic \feld of 450 mT along the out-of-plane di-\nrection, the micromagnetic texture can be driven toward the\nbubble domain phase. Both the remaining stripes as well as\nthe induced bubbles follow Kittel scaling. (c), (d) Calculated\nmagnetic texture in a region of 4x4 µm2, considering a sample\nthickness of 384 nm and 768 nm, respectively, which represents\nthe thin and thick end of the lamella in zero \feld. (e), (f) Cal-\nculated magnetic textures in external \feld of 450 mT, showing\nthe coexistence of residual stripe domains and bubbles, repro-\nducing the experimentally observed behavior. The insets to\npanel (e) show to coexistence of skyrmionic bubbles of oppo-\nsite helicity (red, green), as well as trivial bubbles (black).\n(g), (h) show cross-sections illustrating the \feld evolution at\nthe lines marked in (c) and (e). Both bubble domains and\nstripe domains are being clipped towards the surface by in-\nplane closure domains. All scale-bars (red) have a width of\n4µm.\nMicromagnetic simulations representing the domain\nstructure in the thin and the thick end of the lamella,\nrespectively, are shown in Figure 4 (c) and (d). The sim-\nulations reproduce the domain pattern observed in zero\n\feld, including the thickness dependence of the stripe\nwidth. In 450 mT, similar to the experimental data,\na mixed state of residual stripe-like domains and bub-\nble domains is realized in the simulation, as shown in\nFigs. 4(e) and (f). The insets to panel (e) emphasize the\ncoexistence of skyrmionic and topologically trivial bub-\nbles, which cannot be concluded from the MFM scans\nalone. Furthermore, the simulated 3D magnetization re-\nveals bubbles of opposite helicity. A cross-section view of\nthe thin end of the sample in zero \feld and 450 mT, Figs.\n4 (g) and (h), show the magnetic texture in the core of the\nsample. In zero \feld, the pattern is similar to that found\nin the thick sample displayed in Fig. 2(g). In 450 mT, the\narea magnetized parallel to the \feld is extended and the\nremaining bubble domains of opposite magnetization are\ncon\fned by sharp domain walls in the core and clipped at6\nthe surface by the in-plane closure domain pattern. As a\nconsequence, the diameter of the bubble domains varies\nsubstantially between core and surface.\nIV. SUMMARY\nIn this work, we studied the response of the micro-\nmagnetic spin texture in the frustrated kagome magnet\nFe3Sn2under geometrical con\fnement and in magnetic\n\feld. Bulk samples reveal a branched [26] pattern of al-\nternating ferromagnetic domains, which is characteristic\nfor materials with PMA. In contrast, lamellae with thick-\nnesses in the order of a few 100 nm develop stripe patterns\nof alternating ferromagnetic up and down domains. The\nthickness limit for the onset of domain branching is about\n2:7µm. Below this limit, the size of the stripe domains\nfollows classical Kittel scaling. The experimental results\nare corroborated by micromagnetic simulations, implying\nthat the competition between dipolar interactions and\nuniaxial anisotropy is the driving force behind the for-\nmation and scaling of the domains. Furthermore, the\nmicromagnetic simulations reveal N\u0013 eel capping, i.e., the\nformation of closure domains at the surface. Analogous\nto the response to geometrical con\fnement, classical do-\nmain poling is also achieved by the application of external\nmagnetic \felds. We observe that stripe domains contract\nand break up into bubbles before a \feld-polarized single-\ndomain state is reached. Intermediate Q-materials, likeFe3Sn2, exhibit classical domain responses, thus enabling\ndomain engineering following established strategies: in-\ntroduction of geometrical con\fnement and application of\nexternal magnetic \feld. Importantly, this behavior also\napplies to both topologically trivial and non-trivial bub-\nble domains. This yields the opportunity to tailor the\nhost material to develop speci\fc skyrmionic bubbles and\nmixed states, by optimizing domain size and densities for\npotential future applications.\nV. ACKNOWLEDGMENTS\nThis work was supported by the DFG via the\nTransregional Research Collaboration TRR 80 From\nElectronic Correlations to Functionality (Augs-\nburg/Munich/Stuttgart) and via the DFG Priority\nProgram SPP2137, Skyrmionics, under Grant No. KE\n2370/1-1 and by the project ANCD 20.80009.5007.19\n(Moldova). E.L., E.R., and D.M. thank the Research\nCouncil of Norway for funding (project number 263228)\nand support through the Norwegian Micro- and Nano-\nFabrication Facility, NorFab (project number 295864).\nM.A., E.L., and D.M. acknowledge support by the\nResearch Council of Norway through its Centres of\nExcellence funding scheme, Project No. 262633, \\QuS-\npin\". D.M. thanks NTNU for support via the Onsager\nFellowship Program and the Outstanding Academic\nFellows Program.\n[1] T. H. R. Skyrme, A uni\fed \feld theory of mesons and\nbaryons, Nuclear Physics 31, 556 (1962).\n[2] R. Tomasello, E. Martinez, R. Zivieri, L. Torres, M. Car-\npentieri, and G. Finocchio, A strategy for the design of\nskyrmion racetrack memories, Scienti\fc Reports 4, 6784\n(2014).\n[3] X. Zhang, G. P. Zhao, H. Fangohr, J. P. Liu, W. X.\nXia, J. Xia, and F. J. Morvan, Skyrmion-skyrmion and\nskyrmion-edge repulsions in skyrmion-based racetrack\nmemory, Scienti\fc Reports 5, 7643 (2015).\n[4] I. Dzyaloshinsky, A thermodynamic theory of \\weak\" fer-\nromagnetism of antiferromagnetics, Journal of Physics\nand Chemistry of Solids 4, 241 (1958).\n[5] S. M uhlbauer, B. Binz, F. Jonietz, C. P\reiderer,\nA. Rosch, A. Neubauer, R. Georgii, and P. B oni,\nSkyrmion Lattice in a Chiral Magnet, Science 323, 915\n(2009).\n[6] W. M unzer, A. Neubauer, T. Adams, S. M uhlbauer,\nC. Franz, F. Jonietz, R. Georgii, P. B oni, B. Pedersen,\nM. Schmidt, A. Rosch, and C. P\reiderer, Skyrmion lat-\ntice in the doped semiconductor Fe 1{xCoxSi, Phys. Rev.\nB81, 041203(R) (2010).\n[7] X. Z. Yu, N. Kanazawa, Y. Onose, K. Kimoto, W. Z.\nZhang, S. Ishiwata, Y. Matsui, and Y. Tokura, Near\nroom-temperature formation of a skyrmion crystal in\nthin-\flms of the helimagnet FeGe, Nature Mater 10, 106\n(2011).[8] I. K\u0013 ezsm\u0013 arki, S. Bord\u0013 acs, P. Milde, E. Neuber, L. M.\nEng, J. S. White, H. M. R\u001cnnow, C. D. Dewhurst,\nM. Mochizuki, K. Yanai, H. Nakamura, D. Ehlers,\nV. Tsurkan, and A. Loidl, N\u0013 eel-type skyrmion lattice\nwith con\fned orientation in the polar magnetic semicon-\nductor GaV 4S8, Nature Materials 14, 1116 (2015).\n[9]\u0013A. Butykai, S. Bord\u0013 acs, I. K\u0013 ezsm\u0013 arki, V. Tsurkan,\nA. Loidl, J. D oring, E. Neuber, P. Milde, S. C. Kehr, and\nL. M. Eng, Characteristics of ferroelectric-ferroelastic do-\nmains in n\u0013 eel-type skyrmion host GaV 4S8, Scienti\fc Re-\nports 7, 44663 (2017).\n[10] S. Bord\u0013 acs, A. Butykai, B. G. Szigeti, J. S. White, R. Cu-\nbitt, A. O. Leonov, S. Widmann, D. Ehlers, H.-A. K. von\nNidda, V. Tsurkan, A. Loidl, and I. K\u0013 ezsm\u0013 arki, Equilib-\nrium skyrmion lattice ground state in a polar easy-plane\nmagnet, Scienti\fc Reports 7, 7584 (2017).\n[11] A. Neubauer, C. P\reiderer, B. Binz, A. Rosch, R. Ritz,\nP. G. Niklowitz, and P. B oni, Topological hall e\u000bect in the\naphase of MnSi, Phys. Rev. Lett. 102, 186602 (2009).\n[12] M. Garst, J. Waizner, and D. Grundler, Collective spin\nexcitations of helices and magnetic skyrmions: review\nand perspectives of magnonics in non-centrosymmetric\nmagnets, Journal of Physics D: Applied Physics 50,\n293002 (2017).\n[13] T. Schwarze, J. Waizner, M. Garst, A. Bauer,\nI. Stasinopoulos, H. Berger, C. P\reiderer, and\nD. Grundler, Universal helimagnon and skyrmion exci-\ntations in metallic, semiconducting and insulating chiral7\nmagnets, Nature Materials 14, 478 (2015).\n[14] D. Ehlers, I. Stasinopoulos, V. Tsurkan, H.-A. Krug von\nNidda, T. Feh\u0013 er, A. Leonov, I. K\u0013 ezsm\u0013 arki, D. Grundler,\nand A. Loidl, Skyrmion dynamics under uniaxial\nanisotropy, Phys. Rev. B 94, 014406 (2016).\n[15] E. Ru\u000b, S. Widmann, P. Lunkenheimer, V. Tsurkan,\nS. Bord\u0013 acs, I. K\u0013 ezsm\u0013 arki, and A. Loidl, Multiferroicity\nand skyrmions carrying electric polarization in GaV 4S8,\nScience Advances 1(2015).\n[16] X. Yu, M. Mostovoy, Y. Tokunaga, W. Zhang, K. Ki-\nmoto, Y. Matsui, Y. Kaneko, N. Nagaosa, and Y. Tokura,\nMagnetic stripes and skyrmions with helicity reversals,\nPNAS 109, 6 (2011).\n[17] T. Okubo, S. Chung, and H. Kawamura, Multiple-q\nStates and the Skyrmion Lattice of the Triangular-\nLattice Heisenberg Antiferromagnet under Magnetic\nFields, Phys. Rev. Lett. 108, 017206 (2012).\n[18] X. Yu, Y. Tokunaga, Y. Taguchi, and Y. Tokura, Vari-\nation of Topology in Magnetic Bubbles in a Colos-\nsal Magnetoresistive Manganite, Advanced Materials 29,\n1603958 (2017).\n[19] W. Wang, Y. Zhang, G. Xu, L. Peng, B. Ding, Y. Wang,\nZ. Hou, X. Zhang, X. Li, E. Liu, S. Wang, J. Cai,\nF. Wang, J. Li, F. Hu, G. Wu, B. Shen, and X.-X. Zhang,\nA Centrosymmetric Hexagonal Magnet with Superstable\nBiskyrmion Magnetic Nanodomains in a Wide Tempera-\nture Range of 100{340 K, Advanced Materials 28, 6887\n(2016).\n[20] J. Tang, L. Kong, Y. Wu, W. Wang, Y. Chen, Y. Wang,\nJ. Li, Y. Soh, Y. Xiong, M. Tian, and H. Du, Target Bub-\nbles in Fe 3Sn2Nanodisks at Zero Magnetic Field, ACS\nNano 14, 10986 (2020).\n[21] S. A. Montoya, S. Couture, J. J. Chess, J. C. T. Lee,\nN. Kent, D. Henze, S. K. Sinha, M.-Y. Im, S. D. Ke-\nvan, P. Fischer, B. J. McMorran, V. Lomakin, S. Roy,\nand E. E. Fullerton, Tailoring magnetic energies to form\ndipole skyrmions and skyrmion lattices, Phys. Rev. B 95,\n024415 (2017).\n[22] B. Ding, J. Cui, G. Xu, Z. Hou, H. Li, E. Liu, G. Wu,\nY. Yao, and W. Wang, Manipulating Spin Chirality of\nMagnetic Skyrmion Bubbles by In-Plane Reversed Mag-\nnetic Fields in (Mn 1{xNix)65Ga35(x= 0.45) Magnet,\nPhys. Rev. Applied 12, 054060 (2019).\n[23] B. Ding, Z. Li, G. Xu, H. Li, Z. Hou, E. Liu, X. Xi,\nF. Xu, Y. Yao, and W. Wang, Observation of Mag-\nnetic Skyrmion Bubbles in a van der Waals Ferromagnet\nFe3GeTe2, Nano Lett. 20, 868 (2020).\n[24] B. G obel, I. Mertig, and O. A. Tretiakov, Beyond\nskyrmions: Review and perspectives of alternative mag-\nnetic quasiparticles, Physics Reports 895, 1 (2021), be-\nyond skyrmions: Review and perspectives of alternative\nmagnetic quasiparticles.\n[25] J. C. Loudon, A. C. Twitchett-Harrison, D. Cort\u0013 es-\nOrtu~ no, M. T. Birch, L. A. Turnbull, A. \u0014Stefan\u0014 ci\u0014 c, F. Y.\nOgrin, E. O. Burgos-Parra, N. Bukin, A. Laurenson,\nH. Popescu, M. Beg, O. Hovorka, H. Fangohr, P. A.\nMidgley, G. Balakrishnan, and P. D. Hatton, Do Images\nof Biskyrmions Show Type-II Bubbles?, Advanced Mate-\nrials31, 1806598 (2019).\n[26] A. Hubert, Magnetic Domains: The Analysis of Magnetic\nMicrostructures (Springer, 1998) pp. XXIII, 696.\n[27] A. O. Leonov and M. Mostovoy, Multiply periodic states\nand isolated skyrmions in an anisotropic frustrated mag-net, Nature Communications 6, 8275 (2015).\n[28] Z. Hou, Q. Zhang, G. Xu, S. Zhang, C. Gong, B. Ding,\nH. Li, F. Xu, Y. Yao, E. Liu, G. Wu, X.-x. Zhang,\nand W. Wang, Manipulating the Topology of Nanoscale\nSkyrmion Bubbles by Spatially Geometric Con\fnement,\nACS Nano 13, 922 (2019).\n[29] T. Kida, L. A. Fenner, A. A. Dee, I. Terasaki, M. Hagi-\nwara, and A. S. Wills, The giant anomalous hall e\u000bect\nin the ferromagnet Fe 3Sn2|a frustrated kagome metal,\nJournal of Physics: Condensed Matter 23, 112205 (2011).\n[30] Q. Wang, S. Sun, X. Zhang, F. Pang, and H. Lei, Anoma-\nlous hall e\u000bect in a ferromagnetic Fe 3Sn2single crystal\nwith a geometrically frustrated Fe bilayer kagome lattice,\nPhys. Rev. B 94, 075135 (2016).\n[31] L. Ye, M. Kang, J. Liu, F. von Cube, C. R. Wicker,\nT. Suzuki, C. Jozwiak, A. Bostwick, E. Rotenberg, D. C.\nBell, L. Fu, R. Comin, and J. G. Checkelsky, Massive\ndirac fermions in a ferromagnetic kagome metal, Nature\n555, 638 (2018).\n[32] Z. Hou, W. Ren, B. Ding, G. Xu, Y. Wang, B. Yang,\nQ. Zhang, Y. Zhang, E. Liu, F. Xu, W. Wang, G. Wu,\nX. Zhang, B. Shen, and Z. Zhang, Observation of Var-\nious and Spontaneous Magnetic Skyrmionic Bubbles at\nRoom Temperature in a Frustrated Kagome Magnet with\nUniaxial Magnetic Anisotropy, Advanced Materials 29,\n1701144 (2017).\n[33] Z. Hou, Q. Zhang, G. Xu, C. Gong, B. Ding, Y. Wang,\nH. Li, E. Liu, F. Xu, H. Zhang, Y. Yao, G. Wu, X.-\nx. Zhang, and W. Wang, Creation of Single Chain of\nNanoscale Skyrmion Bubbles with Record-High Temper-\nature Stability in a Geometrically Con\fned Nanostripe,\nNano Lett. 18, 1274 (2018).\n[34] Z. Hou, Q. Zhang, X. Zhang, G. Xu, J. Xia, B. Ding,\nH. Li, S. Zhang, N. M. Batra, P. M. F. J. Costa, E. Liu,\nG. Wu, M. Ezawa, X. Liu, Y. Zhou, X. Zhang, and\nW. Wang, Current-Induced Helicity Reversal of a Sin-\ngle Skyrmionic Bubble Chain in a Nanostructured Frus-\ntrated Magnet, Advanced Materials 32, 1904815 (2020).\n[35] K. Heritage, B. Bryant, L. A. Fenner, A. S. Wills,\nG. Aeppli, and Y.-A. Soh, Images of a \frst-order spin-\nreorientation phase transition in a metallic kagome fer-\nromagnet, Advanced Functional Materials 30, 1909163\n(2020).\n[36] C. Kittel, Theory of the structure of ferromagnetic do-\nmains in \flms and small particles, Phys. Rev. 70, 965\n(1946).\n[37] O. Kazakova, R. Puttock, C. Barton, H. Corte-Le\u0013 on,\nM. Jaafar, V. Neu, and A. Asenjo, Frontiers of magnetic\nforce microscopy, Journal of Applied Physics 125, 060901\n(2019).\n[38] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen,\nF. Garcia-Sanchez, and B. Van Waeyenberge, The design\nand veri\fcation of MuMax3, AIP Advances 4, 107133\n(2014).\n[39] G. L. Caer, B. Malaman, and B. Roques, Mossbauer ef-\nfect study of Fe 3Sn2, Journal of Physics F: Metal Physics\n8, 323 (1978).\n[40] B. Malaman, D. Fruchart, and G. L. Caer, Magnetic\nproperties of Fe 3Sn2. ii. neutron di\u000braction study (and\nmossbauer e\u000bect), Journal of Physics F: Metal Physics\n8, 2389 (1978)."    },    {        "title": "2106.09631v1.Computational_Optimization_of_MnBi_to_Enhance_Energy_Product.pdf",        "content": "Computational Optimization of MnBi to Enhance  Energy Product  \nTula R Paudel1a), Bhubnesh Lama1, Parashu Kharel2 \n1 Department of Physics, South Dakota School of Mines and Technology, Rapid City , SD 5770 1 \n2Department of Physics, South  Dakota State University, Brookings, SD  57707  \nABSTRACT  \nHigh energy  density magnets are preferred over induction magnets for many application s, \nincluding electric motors used in flying rovers, electric ve hicles,  and wind turbines. However , \nseveral issues re lated to cost and supply with state-of-the-art rare -earth -based ma gnet necessiti es \ndevelop ment of  high-flux ma gnets containing low cost , earth -abundant materia ls. He re, we \ndemonstrate the possibility of tuning magn etizatio n and magnetocryst alline anisotropy of one of \nthe candidate materials , MnBi , by alloying  it with foreign elements .  By using the density \nfunctional theory in the high -throughput fashi on, we consider the possibil ity of alloying MnBi \nwith all possible metal and non -metal elements in the periodic table and found that MnBi -based \nalloys  with Pd, Pt, Rh, Li, and O are stable against decompo sition to constituent elements and have \nlarger magnetization , energy product  compared  and magnetic anisotropy compared  to MnBi   We \nconsider the possibility of these elements occupying half and all of the available empty sites. \nCombined with other favorable properties of MnBi, such as high Curie temperature and earth \nabundancy of constituents elements, we envision the possibility o f  MnBi -based high-energy -\ndensity ma gnets. \n \na)  Author to whom correspondence should be addressed:tula.paudel@sdsmt.edu  \n MnBi, a member of Mn -A alloys,  where A can  be any element  with 3+ oxidation state  as \nAl, Ga,  In, has potential  to be a good magnet  [1–4].  The compound posses considerabl e \nmagnetocrystalline anisotropy energy  of  ( ~0.163  meV /uc) [5] that increases with temperature up \nto 553  K  [6] and a high coercive field   17kOe  [7]  at room temperature ; such a  combination leads \nto high energy product  (BH)max o f 17 MGOe  [5] interesting for magnetic applications. \nAdditionally , the compound ha s a large  Curie temperature of 628K  [1] and well suited for high -\ntemperature applications.  Even with such favorable properties, the  MnBi and MnBi -based  \ncompound s suffer from low saturation magnetization  [8]. Various ideas , including defect \nengineering  [2,9–14], exchange coupling with t he soft magnets  [8,11,15 –21], and  microstructural \nengineering  [22–26], have been tested to  improve the energy products in the se compound s. \nHoweve r, they have so far yielded mixed results. While most of the doping  with various foreign \nelements and microstructure refinement have shown an increase of coercivity  at the expense of \nmagnetization   [2,3,9,27] , Sn doping  [3] is found to increase  magnetization and  magnetic \nanisotropy , leading to an increase in  energy product s.   \nIn this manuscript, we test an alternative approach  of alloying   MnBi  with foreign elements  \nto enhance magnetization  and magnetocrystalline anisotropy  leading to en hance d energy product  \n(BH) max, of the materials. We use the high throughput density functional theory  for screening \nsuitable elements to form an alloy  with MnBi.  The structure of MnBi offers clues on why alloying \nmay be  possible in MnBi. The l ow-temperature phase of the MnBi crystallizes in hexagonal NiAs \ntype structure (Space Group No. 194) , as shown in Fig. 1 . The lattice vectors  of such a crystal are \n𝐴1=1/2 𝑎𝑥̂+√3/2 𝑎𝑦̂ ;𝐴2=1/2 𝑎𝑥̂−√3/2 𝑎𝑦̂   and 𝐴3=𝑐𝑧̂, where 𝑎  and 𝑐  are lattice \nconstants. Mn occupies 2a Wyckoff’s positions  with D3d  site symmetry : 0 and (1/2  𝐴3);  Bi \noccupies 2c Wyckoff’s positions with D3h symmetry  (1/3 𝐴1+2/3 𝐴2+1/4 𝐴3)  and \n(2/3 𝐴1+1/3 𝐴2+3/4 𝐴3) while the other high symmetry  Wyckoff’s  2d position s :  \n(2/3 𝐴1+1/3 𝐴2+1/4 𝐴3)  and (1/3 𝐴1+2/3 𝐴2+3/4 𝐴3) are empty.  We incorporat e \nforeign elements to these \nsites and search for the \nelements that increase \nmagnetization and  coercivity.   \nWe consider two cases for \neach elemental ( el) insertion . \ni) only one of two vacant sites \nis filled with foreign elements \n(half -filled case) ;  the other \nhalf-filled configuration \nobtained by filling the rest \nempty  position  is expected to \nbehave similarly because of the similarity of lattice symmetry and the local environment \nsurrounding the elements when placed in that position s, and ii)  both of two vacant sites are filled \n(full-filled case). Our strategy to find the suitable element for alloying is to first evaluate the \nformation energy of each  alloy to find the stable one ; second,  determine saturation  magnetization  \nof stable alloys ; and finally , calculate magnetic anisotropy energy of alloys that are stable and have \nmagnetization larger than undoped MnBi.   \n  \nFigure 1: NiAs type low -temperature structure of MnBi side \nview in the left and top view on the right.  \nFor the stable alloy , the formation energy , Δ𝐻, must  be negative , which ascertain s energy \ngain by alloying . The formation energy of the MnBi:el  is evaluated as Δ𝐻=𝐸( MnBi :el)−\n 𝐸(𝑀𝑛𝐵𝑖 )−𝐸(𝑒𝑙). Here 𝐸(𝑀𝑛𝐵𝑖 ) and 𝐸(𝑒𝑙) are energ ies of MnBi and ground state energy of \nelement , el. Similarly,  E(MnBi:el) is the energy of the ground state configuration of the alloy .  We \ndetermine the ground state energy of the alloy  by comparing the energies of two configurati ons: \nfirst, ferromagnetic  (FM)  configuration  in which magnetic moment of alloying elements  aligns \nparallel to the magnetic moments of Mn;  and second  antiferromagnetic (AFM) configura tions in \nwhich the magnetic moment of alloying element is aligned anti -parallel to the magnetic moments \nof Mn , determine their energ ies. The calculated formation energy from these calculations is likely \nto be in the order of ~0.1 eV /formula unit , corresponding to ~10 KJ/mol  [28]. \nFigure 2 shows the formation energy of the alloy with the various element s in the periodic \ntable.   Out of all those calculations, the attractive  alloy s are the one s with the negative formation \nenergy.  In the half -filled case, only F and Pd doped  alloy s have negative formation energ ies. These \nare marked with light color in Figure 1.  Similarly,  in the full-filled  case, F, Pd, Li, O, Y, Rh doped \nalloy s also ha ve negative formation energ ies, indicating the alloy s to be stable with respect to \ndecomposition to the elemental phase.  Other half-filled alloys with Rh, Pt, and Sc have formation \nenerg ies less than 0.2  eV; these dopants only partially occupy these vacant sites.    We calculate \nsite occupancy 𝑁𝑜𝑐𝑐/𝑁=exp(−Δ𝐻/𝐾𝐵𝑇) at 500 K,  where N is the number of available site s, KB \nis the Boltzmann constant , and T is the temperature .  We found that Rh, Pt , and Sc occupy 40 %, \n25%, and 2% of  empty sites , respectively . Similarly,  full-filled MnBi alloys with  Pt, Ca , and La  \nhas slightly positive formation energy (<0. 2 eV) and only partia lly occupy the empty sites : Pt \noccupies 23%, Ca occupies 9% , and La occupies 6% of the empty sites. When Δ𝐻 > 0.2 eV \ninterstitial sites  of MnBi remains empty.    \nFigure 2. The f ormation energy of the MnBi  alloy with the various elements. The two number s \nabove and below each element in the box represent the formation energy, Δ𝐻, of half -filled  \n(MnBi:el 1/2) and full -filled (MnBi :el) alloy with the element  (el). The isolated box shows the \nlegend.  The highli ghted elements form a stable alloy with MnBi .    \nAll the alloys with negative formation energy are found to have a larger volume compared \nto the MnBi.  The lattice volume increases , but modestly , according to the increase in the atomic \nradius  [29]. The smallest volume increase is observed in F (1%) , and the largest  volume increase \nis observed in the case of Y (42%). Volume increa ses more in the full -filled than half-filled case; \nhowever, the increase is not linear ; for example, the volume  increase  of full-filled MnBiRh is ~ \n19%, whereas that of half -filled MnBiRh 1/2 is ~ 14%. This nonlinear incre ase in volume indicate s \nhybridization between the alloying elements  and MnBi while both empty sites are occupied . \nDespite the increase in volume, the volume formation, which is defined in a similar way as the \nformation energy,  𝛥VF = V(MnBi:nel) – (VMnBi – nVel), where 𝑛 is number of   volume 𝑉𝑒𝑙  and \n𝑉𝑀𝑛𝐵𝑖 is the volume of MnBi,  is generally negative  (except for MnBi:  Pt) as shown in Table I. The \nnegative  𝛥VF also indicate s hybridization between the valence orbitals of alloying  elements with \nvalence orbitals of Mn  and Bi in MnBi.    \nHaving found the stable alloy s, we next analyze their magnetic properties .  Table I. shows  \nchange s in magnetization with respect to the  3.54 𝜇𝐵/Mn of bulk MnBi .  Out of the alloys with \nnegative formation energy alloying with F, Sc, Y reduces the net moment, where alloying with Li, \nO, Rh, Pd , and Pt increases net moment by up to 14% compared to the undoped case. In the case \nof F doping, F -p orbit al hybridize s strongly with Bi -p orbitals  (Fig. 2c) , which couples to Mn -d \norbitals antiferromagnetically like that in bulk MnBi , thereby reducing the net moment. In the case \nof Sc and Y, their 3 d-states , as shown in Fig 2d and Fig 2e, mainly lie in the conduction band ; \nthus,  they hybridize  with Mn minority d -states , inducing small moments in themselves and \nreduc ing moments in Mn, and the moments in Y and Sc couple antiferromagnetically with Mn \nmoments  further reducing  the overall m agnetizations .   Table I: Change in saturation mag netization, anisotropy, volume formation, 𝑐/𝑎, estimated \nCurie Temperature ( 𝑇𝑐) and change in energy product, Δ(𝐵𝐻)𝑚𝑎𝑥(%) as a function of alloying \nelements (el) in half -filled MnBi: el 1/2 and full-filled MnBi:el alloys.  \nCase  𝚫M(%)  ΔAE(MJ/m3) Δ𝑉𝐹(Å3) c/a Tc (K) Δ(𝐵𝐻)𝑚𝑎𝑥(%) \nMnBi  0 0  1.33 580 0 \nMnBi:Li  9 4 -1.27 1.24 190 18 \nMnBI:O  9 5 -0.72 1.05 555 18 \nMnBi:F 1/2 -2 -     \nMnBi:F  -7 -     \nMnBi:Sc 1/2 -9 -     \nMnBi:Y  -7 -     \nMnBi:Rh 1/2 8 3.0 -1.53 1.33 522 16 \nMnBi:Rh  14 0.92 -0.02 1.21 312 32 \nMnBi:Pd 1/2 9 20.7 -0.46 1.33 457 18 \nMnBi:Pd  14 0.33 -0.46 1.22 327 30 \nMnBi:Pt  10 4.79 0.06 1.20 417 21 \n Alloying with Rh, Pd, Pt increases the net magnetization of the system. 4 d orbitals of the \nRh (Fig. 2f) and Pd (Fig 2g) and 5d orbitals of the Pt (Fig 2h) lies m ainly in the valence band and \nhybridize strongly with the majority spin -channel of Mn , which induces small magnetic moments \non themselves and enhances magnetic moments on Mn. These moments  on Rh, Pd, and Pt,  \nhowever , couples ferromagnetically with Mn moments. This is opposite to interaction between Mn \nat the regular sites and interstitial sites , which couples antiferromagnetically  [30,31]  or interaction   \nFigure 2: Atom, spin and orbital resolved density of states (DOS) of MnBi  (a) and (b) and its \nalloying elements (el) in stable full -filled MnBi:el and half -filled MnBi:el 1/2 as indicated in left of \npanels (c) to (j). In the cases (F, Rh and Pd), where both half -filled and full -filled alloys are stable, \nthe DOS of element in full -filled alloy is shown with dotted line in the background of solid line \nthat corresponds to DOS of the el ement in half -filled case. Colors red, blue, and green corresponds \nto 𝑑, 𝑝 and 𝑠 orbitals respectively. ×𝑥 in  c-f and g -j  corresponds to the enlargement factor for \nDOS for better visualization.  \nbetween Sc, Y in interstitial sites and Mn at regular sites, as we discussed previously.  The \ndifference comes from the large exchange splitting of Mn -d bands leading to almost no interaction \nbetween  elect rons wi th different spins  located at the different sites. However, in Rh, Pd, and Pt, \neven though intra -atomic exchange interaction leads to d-orbitals splitting according to spin like  \nthat in Mn, the splitting is much smaller. As a result, bands corresponding to both spin channel s \nare occupied  though there is a s light shift between them , which  result s in a small magnetic  moment. \nAdditionally, they have a significant overlap  with Mn d-orbit als occupied by the majority spin \nchannel, resulting in ferromagnetic direct exchange . \n Alloying group II elements like Li and O also increases overall magnetization.  The s-\nstates of Li and 𝑝−states of O lies on the opposite side of the Fermi energy structure: the O -p \nstates lie primari ly in the valence band as shown in Fig. 2j, while  Li-s states lie in the conduction \nband  as shown in Fig. 2i . As a result , the hybridization between Li-s states and  Mn-d states , and \nO-p and Mn -d states  is spin -dependent ; the oxygen sta tes hybridize mostly with the Mn -d majority \nstates, while  Li-s states hybridize mostly with Mn -d minority states leading to  the enhancement of \nMn moments and development of small moments in Li and O them selves. The magnetic moment s \nof Li and O also couple ferr omagnetically because of direct exchange with Mn moments and help \nincreasing overall magnetization . We verify this hypothesis by  explicitly compar ing the energy of \nthe system with the moments of Mn and moments of Li and O coupling ferromagnetically and \nantiferromagnetically and found that the ferromagnetically coupled system has lower energy . \nUsing the energy difference Δ𝐸=𝐸𝐹𝑀−𝐸𝐴𝐹𝑀 we estimate the Curie temperature in the mean -\nfield approximations  by using 𝑇𝑐=Δ𝐸/3 [31].  For the bulk MnBi , we calculate Δ𝐸=150 \nmeV/Mn  and 𝑇𝑐 ≈ 580 K , which is reso nably close to extrapolated experimental value of 775K [6]; \nThe structural transition at 628K hinder measur ing real T c of  MnBi.    Using the same approach, \nwe eavaluate the  𝑇𝑐 of stable MnBi  alloys with enhanced magnetizations. The second to last \ncolumn  of the table I shows the calculated results, which shows that  𝑇𝑐 reduces in general upon \ndoping. This is consistent with the fact that upon doping the volume increases ,  c/a decreases  and \nhence the first exchange parameter 𝐽1 , which contribute the most to total exchange 𝐽,  decreases \ndue to reduced overlap between the orbital along out -of-plane directions.  This estimate ignore the \ncontribution to the Tc second and higher other exchange a nd should represent the lower bound.   \nNext, we calculate change ( 𝛥) in magnetic anisotropy energy MAE, 𝛥MAE  = MAE  \n(MnBi:el /el1/2)  - MAE(MnBi) , in stable alloy s with magnetization larger than  that of bulk MnBi.  \nMAE is computed  by taking the energy  difference between the system with  magnetization point ing \nalong [100] direction and the system with magnetization point ing along  [001] direction , 𝑀𝐴𝐸 =\n𝐸(𝑀||[100 ])−𝐸(𝑀||[001 ]). The results in Table I show that anisotropy energy increases upon \nalloying com pared to MnBi ( ~1MJ/m3) in all cases.   Since the anisotropy energy 𝐸=𝐾1sin2𝜃+\n𝐾2sin4𝜃 , where 𝜃 is the angle between easy axis [001] and the direction of magnetization, and \n𝐾2 << 𝐾1  ΔMAE  roughly corresponds to change  in 𝐾1.  We note that magnetization of MnBi alloy \nwith Rd and Pd1/2 points along [001] direction, similar to the bulk MnBi, the  magnetization of \nMnBi alloy with Li, O, Pd , and Pt changes to in -plane [100] direction.  Qualitatively, s uch change s \nin MAE and easy axis are related to subtle changes in band distribution near the Fermi level and \nmay require separate detail ed investigations . The magnetic anisotropy is  a relativistic effect driven \nby spin -orbit coupling . As the spin -orbit coupling ener gy is much smaller than the bandwidth of \nMn-3d states , perturbative expansion of energy can be used to explain the spin-orbit coupling \neffect. The lowest non -zero correction to the energy in such expansion includes the term proportional to 1/(E unocc – Eocc) where Eocc ( unocc) is the energy of occupied (unoccupied) state \naround Fermi energy . The non-zero DOS of alloying elements near the Fermi level indicates the \npossibility of such changes near Fermi levels . \nFinally, we estimate the theoretical  value of maximum energy product (BH) max assuming \napplied field 𝐻 much smaller than ma gnetization , M. In this appr oximation ,  magnetic induction \nfield 𝐵=4𝜋𝑀  in CGS units , and 𝑀𝑟=𝑀𝑠  when magnetic hysteresis loop is  almost  rectangular. \nThe  (𝐵𝐻)𝑚𝑎𝑥=𝐵𝑟2/4, where the remnant field 𝐵𝑟=4𝜋𝑀𝑟=4𝜋𝑀𝑠 [32].  The formula allows us \nto estimate the energy -produc t using only the intrinsic properties such as actual magnetization \n(emu/cc)  and homogeneous sample without using  coercivity that relies on the extrinsic facto rs \nsuch as shape and size of the samp le.  The last column of Table I shows the calculated percentage \nchange in energy product to calculated energy product of 20M GOe , which is similar to the value \nof 17 MGOe  [33,34]  measured experimentally.  The energy product  enhances when MnBi is \nalloyed with Pd, Pt, Rh, Li, and O per the change in magnetizations.   \nIn summary, we used the first-principles density functional theory to find an alloy with the \nincrease d magnetization and anisotropy . We predict MnBi alloy with Pd, Pt, Rh, Li , and O are  \nstable against decomposition to constituent elements and have larger magnetization and anisotropy \ncompared to MnBi  and have a high energy product .  Magnetic easy axis of MnBi alloy with Pd 1/2 \nand Rh remains the same  as bulk MnBi and lies out -of-plane direction, while the  magnetic -easy-\naxis of MnBi alloy with Li, O, Pd and Pt lie in -plane. Pd can rotate the magnet ic-easy-axis [35] \nfrom in-plane to out -of-plane depending upon the percentage of Pd that would be incorporated in \nMnBi .   We anticipate this comprehensive study of MnBi alloy spurs more theoretical and \nexperimental study .  We note that  the alloying of the se elements with MnBi may require using out-\nof-equilibrium growth methods as their formation energ ies are relatively smal l.  \nComputational Methods  \nWe employ the projected augmented wave (PAW)  [36] method for the electron -ion \npotential and the gradient density approximation (GGA) for exchange -correlation potential, as \nimplemented in the Vienna ab-initio  simulation package (VASP)  [37,38]  with the recommended \nset of pseudopotentials . The calculations are carried out using the kinetic energy cutoff of 340 eV \nand 661 k-point mesh for Brillouin zone integration of pseudocubic  unit cells. The k -points are \nscaled according to size. We fully relax ionic coordinates with the force convergence limit of 0.001 \neV/atom. For the anisotropy calculations , we include 0.1an addi tional onsite Coulomb interaction \nHubbard (U -J) parameter of 3eV to Mn -3d states  so that  the sign of  magnetocrystalline of bulk \nMnBi is consistent with the experiments  and includ es spin-orbit coupling term s explicitly.  \nAcknowledgments  \nThis work was supported by South Dakota -NASA EPSCoR Research Initiation Grant  no \n80NSSC19M0063.  Computations were performed utilizing the Holland Computing Center at the \nUniversity of Nebraska, Lincoln , and the Gamow Computing Cluster at the Department of  Physics, \nSouth Dakota School of Mines and Technology. The authors acknowledge discussion with Alex \nLeary, Research Materials Engineer at the NASA Glenn Research Center, Cleveland OH.  \nAvailibily of Data  \nThe data that support the findings of this study are available from the corresponding author upon \nreasonable request .  \n[1] P. Kharel, P. Thapa, P. Lukashev, R. F. Sabirianov, E. Y. Tsymbal, D. J. Sellmyer, and B. \nNadgorny, Transport Spin Polarization of High Curie  Temperature MnBi Films , Phys. Rev. \nB - Condens. Matter Mater. Phys. 83, 1 (2011).  \n[2] P. Kharel, X. Z. Li, V. R. Shah, N. Al -Aqtash, K. Tarawneh, R. F. Sabirianov, R. Skomski, \nand D. J. Sellmyer, Structural, Magnetic, and Electron Transport Properties of MnBi:Fe \nThin Films , J. Appl. Phys. 111, 07E326 (2012).  \n[3] W. Zhang, B. Balasubramanian , P. Kharel, R. Pahari, S. R. Valloppilly, X. Li, L. Yue, R. \nSkomski, and D. J. Sellmyer, High Energy Product of MnBi by Field Annealing and Sn \nAlloying , APL Mater. 7, (2019).  \n[4] V. N. Antonov and V. P. Antropov, Low-Temperature MnBi Alloys: Electronic an d \nMagnetic Properties, Constitution, Morphology and Fabrication (Review Article) , Low \nTemp. Phys. 46, 3 (2020).  \n[5] J. Park, Y. -K. Hong, J. Lee, W. Lee, S. -G. Kim, and C. -J. Choi, Electronic Structure and \nMaximum Energy Product of MnBi , Metals (Basel). 4, 455 (2014).  \n[6] X. Guo, X. Chen, Z. Altounian, and J. O. Ström -Olsen, Temperature Dependence of \nCoercivity in MnBi , J. Appl. Phys. 73, 6275 (1993).  \n[7] S. Das, B. Wang, T. R. Paudel, S. M. Park, E. Y. Tsymbal, L. -Q. Chen, D. Lee, and T. W. \nNoh, Enhanced Fl exoelectricity at Reduced Dimensions Revealed by Mechanically Tunable \nQuantum Tunnelling , arXiv.  \n[8] Y. L. Ma, X. B. Liu, K. Gandha, N. V. Vuong, Y. B. Yang, J. B. Yang, N. Poudyal, J. Cui, \nand J. P. Liu, Preparation and Magnetic Properties of MnBi -Based H ard/Soft Composite \nMagnets , J. Appl. Phys. 115, 19 (2014).  \n[9] R. F. Sabiryanov and S. S. Jaswal, Magneto -Optical Properties of MnBi Doped with Cr , J. \nAppl. Phys. 85, 5109 (1999).  \n[10] A. Sakuma, Y. Manabe, and Y. Kota, First Principles Calculation of Magn etocrystalline \nAnisotropy Energy of MnBi and MnBi 1 - x Sn X , J. Phys. Soc. Japan 82, 073704 (2013).  \n[11] V. V. Ramakrishna, S. Kavita, R. Gautam, T. Ramesh, and R. Gopalan, Investigation of \nStructural and Magnetic Properties of Al and Cu Doped MnBi Alloy , J. Magn. Magn. Mater. \n458, 23 (2018).  \n[12] P. Rani, A. Taya, and M. K. Kashyap, Enhancement of Magnetocrystalline Anisotropy of \nMnBi with Co Interstitial Impurities , AIP Conf. Proc. 1942 , (2018).  \n[13] Y. Yang, J. W. Kim, P. Z. Si, H. D. Qian, Y. Shin, X. W ang, J. Park, O. L. Li, Q. Wu, H. \nGe, and C. J. Choi, Effects of Ga -Doping on the Microstructure and Magnetic Properties \nof MnBi Alloys , J. Alloys Compd. 769, 813 (2018).  \n[14] K. Anand, N. Christopher, and N. Singh, Evaluation of Structural and Magnetic Property \nof Cr -Doped MnBi Permanent Magnet Material , Appl. Phys. A 125, 870 (2019).  \n[15] R. Skomski and J. M. D. Coey, Giant Energy Product in Nanostructured Two -Phase \nMagnets , Phys. Rev. B 48, 15812 (1993).  \n[16] R. Skomski and J. M. D. Coey, Exchange Coupling and Energy Product in Random Two -\nPhase Aligned Magnets , IEEE Trans. Magn. 30, 607 (1994).  \n[17] R. Skomski, Aligned Two -Phase Magnets: Permanent Magnetism of the Future? (Invited) , \nJ. Appl. Phys. 76, 7059 (199 4). \n[18] H. Zeng, J. Li, J. P. Liu, Z. L. Wang, and S. Sun, Exchange -Coupled Nanocomposite \nMagnets by Nanoparticle Self -Assembly , Nature 420, 395 (2002).  \n[19] X. Xu, Y. K. Hong, J. Park, W. Lee, A. M. Lane, and J. Cui, Magnetic Self -Assembly for the Synthe sis of Magnetically Exchange Coupled MnBi/Fe -Co Composites , J. Solid State \nChem. 231, 108 (2015).  \n[20] Q. Dai, M. Asif warsi, J. Q. Xiao, and S. Ren, Solution Processed MnBi -FeCo Magnetic \nNanocomposites , Nano Res. 9, 3222 (2016).  \n[21] T. R. Gao, L. Fang, S . Fackler, S. Maruyama, X. H. Zhang, L. L. Wang, T. Rana, P. \nManchanda, A. Kashyap, K. Janicka, A. L. Wysocki, A. T. N’Diaye, E. Arenholz, J. A. \nBorchers, B. J. Kirby, B. B. Maranville, K. W. Sun, M. J. Kramer, V. P. Antropov, D. D. \nJohnson, R. Skomski, J.  Cui, and I. Takeuchi, Large Energy Product Enhancement in \nPerpendicularly Coupled MnBi/CoFe Magnetic Bilayers , Phys. Rev. B 94, 2 (2016).  \n[22] F. Yin, N. Gu, T. Shigematsu, and N. Nakanishi, Sintering Formation of Low Temperature \nPhase MnBi and Its Disord ering in Mechanical Milling , Journal of Materials Science and \nTechnology.  \n[23] J. B. Yang, Y. B. Yang, X. G. Chen, X. B. Ma, J. Z. Han, Y. C. Yang, S. Guo, A. R. Yan, \nQ. Z. Huang, M. M. Wu, and D. F. Chen, Anisotropic Nanocrystalline MnBi with High \nCoerciv ity at High Temperature , Appl. Phys. Lett. 99, 2011 (2011).  \n[24] H. Kronmüller, J. B. Yang, and D. Goll, Micromagnetic Analysis of the Hardening \nMechanisms of Nanocrystalline MnBi and Nanopatterned FePt Intermetallic Compounds , \nJ. Phys. Condens. Matter 26, 064210 (2014).  \n[25] E. Céspedes, M. Villanueva, C. Navío, F. J. Mompeán, M. García -Hernández, A. Inchausti, \nP. Pedraz, M. R. Osorio, J. Camarero, and A. Bollero, High Coercive LTP -MnBi for High \nTemperature Applications: From Isolated Particles to Film -like Structures , J. Alloys \nCompd. 729, 1156 (2017).  \n[26] Y. L. Huang, Z. Q. Shi, Y. H. Hou, and J. Cao, Microstructure, Improved Magnetic \nProperties, and Recoil Loops Characteristics for MnBi Alloys , J. Magn. Magn. Mater. 485, \n157 (2019).  \n[27] P. Kharel, R. S komski, R. D. Kirby, and D. J. Sellmyer, Structural, Magnetic and Magneto -\nTransport Properties of Pt -Alloyed MnBi Thin Films , J. Appl. Phys. 107, 66 (2010).  \n[28] V. Stevanović, S. Lany, X. Zhang, and A. Zunger, Correcting Density Functional Theory \nfor Accu rate Predictions of Compound Enthalpies of Formation: Fitted Elemental -Phase \nReference Energies , Phys. Rev. B 85, 115104 (2012).  \n[29] E. Clementi, D. L. Raimondi, and W. P. Reinhardt, Atomic Screening Constants from SCF \nFunctions. II. Atoms with 37 to 86 E lectrons , J. Chem. Phys. 47, 1300 (1967).  \n[30] A. E. Taylor, T. Berlijn, S. E. Hahn, A. F. May, T. J. Williams, L. Poudel, S. Calder, R. S. \nFishman, M. B. Stone, A. A. Aczel, H. B. Cao, M. D. Lumsden, and A. D. Christianson, \nInfluence of Interstitial Mn on Magnetism in the Room -Temperature Ferromagnet \nMn1+δSb , Phys. Rev. B - Condens. Matter Mater. Phys. 91, 1 (2015).  \n[31] T. J. Williams, A. E. Taylor, A. D. Christianson, S. E. Hahn, R. S. Fishman, D. S. Parker, \nM. A. McGuire, B. C. Sales, and M. D. Lumsden, Extended Magnetic Exchange \nInteractions in the  High -Temperature Ferromagnet MnBi , Appl. Phys. Lett. 108, (2016).  \n[32] J. M. D. Coey, Magnetism and Magnetic Materials  (Cambridge University Press, 2001).  \n[33] R. G. Pirich and J. David J. Larson, United States Patent ( 19 ) , 4784703 (1988).  \n[34] W. Zhang , P. Kharel, S. Valloppilly, L. Yue, and D. J. Sellmyer, High -Energy -Product \nMnBi Films with Controllable Anisotropy , Phys. Status Solidi Basic Res. 252, 1934 (2015).  \n[35] V. G. Myagkov, L. E. Bykova, V. Y. Yakovchuk, A. A. Matsynin, D. A. Velikanov, G. S.  \nPatrin, G. Y. Yurkin, and G. N. Bondarenko, High Rotatable Magnetic Anisotropy in MnBi \nThin Films , JETP Lett. 105, 651 (2017).  [36] P.E.Blöchl, Projector Augmented -Wave Method , Phys. Rev. B 50, 17953 (1994).  \n[37] G. Kresse and J. Furthmüller, Efficient It erative Schemes for Ab Initio Total -Energy \nCalculations Using a Plane -Wave Basis Set , Phys. Rev. B 54, 11169 (1996).  \n[38] G. Kresse, From Ultrasoft Pseudopotentials to the Projector Augmented -Wave Method , \nPhys. Rev. B 59, 1758 (1999).  \n "    },    {        "title": "2106.10249v2.Towards_hexagonal__C__2v___systems_with_anisotropic_DMI__Characterization_of_epitaxial_Co__10_bar_1_0___Pt__110___multilayer_films.pdf",        "content": "Towards hexagonal C2vsystems with anisotropic DMI:\nCharacterization of epitaxial Co (1 0 1 0)/Pt (1 1 0) multilayer \flms\nYukun Liu, Michael D. Kitcher, Marc De Graef, and Vincent Sokalski\nDepartment of Materials Science & Engineering,\nCarnegie Mellon University, Pittsburgh, PA, USA 15213\n(Dated: October 14, 2021)\nFerromagnet/heavy metal (FM/HM) multilayer thin \flms with C2vsymmetry have the poten-\ntial to host antiskyrmions and other chiral spin textures via an anisotropic Dzyaloshinkii-Moriya\ninteraction (DMI). Here, we present a candidate material system that also has a strong uniaxial\nmagnetocrystalline anisotropy aligned in the plane of the \flm. This system is based on a new\nCo/Pt epitaxial relationship, which is the central focus of this work: hexagonal closed-packed\nCo(1 0 1 0)[0 0 0 1] kface-centered cubic Pt(1 1 0)[0 0 1]. We characterized the crystal structure and\nmagnetic properties of our \flms using X-ray di\u000braction techniques and magnetometry respectively,\nincluding q-scans to determine stacking fault densities and their correlation with the measured\nmagnetocrystalline anisotropy constant and thickness of Co. In future ultrathin multilayer \flms, we\nexpect this epitaxial relationship to further enable an anisotropic DMI while supporting interfacial\nperpendicular magnetic anisotropy. The anticipated con\ruence of these properties, along with the\ntunability of multilayer \flms, make this material system a promising testbed for unveiling new spin\ncon\fgurations in FM/HM \flms.\nI. INTRODUCTION\nMagnetic skyrmions are promising candidates for fu-\nture spintronic data storage and logic devices because\nof their small size, heightened stability, and high mo-\nbility [1{5]. These structures are typically stabilized\nvia the Dzyaloshinskii-Moriya interaction (DMI) which\nemerges in magnetic systems with high spin-orbit cou-\npling and no inversion symmetry [6{8]. In polycrys-\ntalline ferromagnet/heavy metal (FM/HM) thin \flms,\nonly N\u0013 eel skyrmions have been stabilized, due to the high\nsymmetry of these systems [2, 3, 5, 7, 9{14]. However,\nrecent research has shown that low-symmetry FM/HM\nthin \flm interfaces exhibit anisotropic DMI which, in\nturn, could stabilize antiskyrmions|the topological op-\nposites of skyrmions [12{16]. Speci\fcally, antiskyrmions\nare preferred over N\u0013 eel skyrmions when the DMI has\nopposing signs along orthogonal crystallographic direc-\ntions. To date, research on anisotropic DMI in FM/HM\nsystems has focused on \flms based on a body-centered\ncubic (BCC) heavy metal [13, 15{19]. Moreover, across\nthe limited experimental studies carried out thus far, a\nsuitable antiskyrmion host has not been found.\nTo expand this search, we present a system that\nexhibits the growth of hexagonal close-packed (HCP)\nCo on a face-centered cubic (FCC) Pt underlayer, re-\nsulting in a singular magnetocrystalline easy axis that\nlies in the plane of the \flm. When coupled with an\n(an)isotropic DMI and/or an interface-induced perpen-\ndicular easy axis, such in-plane uniaxial anisotropy is\nlikely to broaden the range of spin con\fgurations that can\nexist in FM/HM \flms. In this work, we characterized the\nepitaxial relationship that enables such a system; inves-\ntigated the prevalence of structural defects|namely, in-\ntrinsic stacking faults (SFs)|during the growth of \flms;\nand measured the magnetocrystalline anisotropy (MCA)\nby probing the resulting in-plane hysteresis.\nFIG. 1. Orientation and interatomic relationship between\nFCC Ag(1 1 0), FCC Pt(1 1 0), FCC Co(1 1 0), and HCP\nCo(1 0 1 0).\nII. METHODS\nAll \flms were grown via DC magnetron sputtering\nin an Ar atmosphere at a \fxed working pressure of 2.5\nmTorr. Prior to deposition, single-crystal Si(1 1 0) sub-\nstrates were HF-etched for 1 minute to remove any native\noxide that would otherwise impede epitaxial growth. Fol-\nlowing Yang et al., we deposit a Ag bu\u000ber layer which is\nknown to grow with the FCC (1 1 0) texture on Si(1 1 0)\n[20]. This growth seed facilitates the deposition of our\n\flms at room temperature, thus avoiding the severe inter-\nlayer di\u000busion that occurs at elevated temperatures [20].\nMoreover, the degree of lattice mismatch between Ag and\nPt is rather moderate ( \u00184.3%). Pt[10 nm]/Co[20 nm,arXiv:2106.10249v2  [cond-mat.mtrl-sci]  13 Oct 20212\n50 nm]/Pt[3 nm] thin \flms were subsequently deposited.\nTo identify the phases and textures of the di\u000berent lay-\ners, 2\u0012-!coupled scans and azimuthal ( ') scans were\nperformed on the deposited thin \flms using a Phillips\nX'Pert Pro X-ray di\u000bractometer with un\fltered Cu K\u000b\nradiation. Furthermore, the density of SFs in HCP Co for\neach sample was quantitatively determined from X-ray\nreciprocal space q-scans. Using measurements of the full\nwidth at half maximum ( wh) of di\u000berent Co peak pro\fles\nalong di\u000berent directions, we extracted the broadening ef-\nfects caused by di\u000berent intrinsic SFs. The corresponding\nSF densities were subsequently calculated.\nThe magnetic anisotropy of the deposited thin \flms\nwas examined using a Princeton Measurements alternat-\ning gradient force magnetometer (AGFM). M{Hhys-\nteresis curves were obtained for orthogonal in-plane di-\nrections and their corresponding magnetic anisotropy en-\nergy densities were calculated from the area between one\nhalf of the idealized hard axis loop and the M-axis.\nIII. RESULTS AND DISCUSSION\nThe representative 2 \u0012-!coupled scan pro\fles pre-\nsented in Figure 2 show dominant HCP Co(1 0 1 0), FCC\nPt(2 2 0), and FCC Ag(2 2 0) di\u000braction peaks, indicat-\ning that while Pt grows with the same preferred FCC\n(1 1 0) orientation of the Ag underlayer, Co adopts the\nHCP (1 0 1 0) texture. Here, we use Bravais-Miller in-\ndexing notation ( HKIL ) to represent hexagonal struc-\ntures where I=\u0000(H+K). These results are corrob-\norated by the azimuthal scan pro\fles in Figure 3 which\nreveal HCP Co(1 0 1 1) planes at the same azimuth as\nFCC (1 1 1) planes in Pt and Ag. Correspondingly, the\nepitaxial relationship between layers is Co(1 0 1 0)[0 0 0 1]\nkPt(1 1 0)[0 0 1]kAg(1 1 0)[0 0 1]kSi(1 1 0)[0 0 1]. The\nstrong (1 1 0) texture of the Ag growth seed in these \flms\nagrees with the work of Yang et al., who noted that a 4 \u00024\nmesh of Ag unit cells on a 3 \u00023 mesh of Si unit cells has\na lattice mismatch of less than 0.45% [20].\nGiven the prevalence of \\all-FCC\" Co/Pt multilay-\ners in magnetics research, it may seem counterintuitive\nthat a cubic Pt(1 1 0) underlayer would give rise to a\nhexagonal Co layer. However, as shown in Figure 1,\nthe HCP Co(1 0 1 0) surface has signi\fcantly lower lattice\nmismatch with the Pt(110) surface along [001] as com-\npared to the FCC Co equivalent, while the mismatches\nalong the [1 1 0] of Pt are nearly identical. Together with\ncobalt's thermodynamic preference for the HCP phase at\nroom temperature, the potential for a lower strain drives\nthe formation of the observed texture. However, the the-\noretical lattice mismatch along Pt [1 1 0] at the observed\nCo/Pt interface is considerable. We therefore explored\nthe prevalence of intrinsic SFs in these \flms and their im-\npact on the magnetocrystalline uniaxial anisotropy along\nthe c-axis of Co.\nIt has been established that intrinsic SFs in HCP\ncrystals broaden di\u000braction peaks along [0 0 0 1] when\nIntensity (Counts/s)Co (10!10)Ag (220)Pt (220)Si (220) KβSi (220) Kα\nPt (111)Ag (111)050010001500200025003000\n352θ(deg)35404550556065707580FIG. 2. Di\u000braction pro\fles of 2 \u0012-!coupled scans performed\non the Co(20 nm)/Pt \flm.\nIntensityCounts/sPt(111)2θ=39.67°ψ=34.03°Ag(111)2θ=38.22°ψ=33.97°Co(10\"11)2θ=47.54°ψ=29.26°0.511.5\n10002000300040005000×104\n100200300400500-100-50φ(deg)050100150200250\nFIG. 3. Di\u000braction pro\fles of azimuthal ( ') scans performed\non the Co(20 nm)/Pt \flm.\nH\u0000K6= 3N, whereN2Z. Furthermore, the degree\nof broadening depends on the parity of L. In particu-\nlar, the normalized broadening, \u0001 wh;sf, of these peaks\nin reciprocal space and the density of SFs are related by:\n\u0001wh;sf=3\u000b+ 3\f\n\u0019; Lmod 2 = 0; (1a)\n\u0001wh;sf=3\u000b+\f\n\u0019; Lmod 2 = 1; (1b)\nwhere\u000band\frepresent the densities of deformation\nand growth SFs, respectively [21]. However, the di\u000brac-\ntion pro\fle of a reciprocal space q-scan is also a\u000bected3\nby the broadening e\u000bect caused by mosaicity; the ob-\nserved broadening must therefore be deconvolved during\nthis analysis. A summary of the possible broadening ef-\nfects for di\u000berent peaks is shown in Table I.\nTo characterize the broadening e\u000bects of SFs and mo-\nsaicity, X-ray reciprocal space q-scans were performed\non the Co(2 0 2 0), (1 0 1 0), and (2 0 2 1) di\u000braction peaks\nalong multiple directions for each sample. As a refer-\nence, the reciprocal space map of thin \flms is shown in\nFigure 4. Figure 5 presents measured reciprocal space\nq-scan di\u000braction pro\fles, where the dash lines in the\ninsets indicate the scanning directions. The whof each\npeak was estimated by \ftting peaks with Gaussian func-\ntions. The results were then normalized with respect to\nthe experimentally measured spacing between Co(2 0 21)\nand (2 0 2 1) peaks in reciprocal space; this spacing is\nmarked in Figure 4 as the \\Reference Length\". Table I\nshows the normalized whvalues. The normalized whof\nthe cobalt (1 0 1 0) peak along the (1 0 1 0) plane nor-\nmal was used as a baseline for the other peak widths,\nas it is not a\u000bected by the broadening e\u000bects from either\nSFs or mosaicity. The broadening e\u000bects caused by SFs\ncan be determined by comparing the di\u000braction pro\fle\nof the Co(1 0 1 0) di\u000braction peak along the Co(1 0 1 0),\n(1 21 0), and (0 0 0 1) plane normals. As shown in Fig-\nure 5 and Table I, the whvalues of the Co(1 0 1 0) and\n(2 01 0) di\u000braction peaks along Co(0 0 0 1) plane normal\nare considerably larger than those measured along the\n(1 21 0) and (1 0 1 0) plane normals, indicating the pres-\nence of stacking faults.\nFIG. 4. Reciprocal space map for the deposited thin \flm.\nThe orientation of the \flm is indicated by the coordinate axes.\nThe reciprocal space distance between Co(2 0 21) and (2 0 2 1)\npeaks is marked as the Reference Length which is used for\npeak width normalization for whvalues shown in Table I. The\nlimitations of the instrument prohibit the probing of regions\nwithin the shaded blue areas and beyond the blue arc.\nFor each sample, \u0001 wh;sfwas calculated as the di\u000ber-\nence in normalized whbetween Co(1 0 1 0) peak widths\nmeasured along the (0 0 0 1) and ( 1 21 0) plane normals.\nBy solving Equation (1a), the sum of deformation and\ngrowth SF densities in each sample was calculated, as\nshown in Table II. Our calculations reveal considerable\nSF densities of 7.6% and 9.1% for our 20 nm and 50 nm\n\flms respectively, suggesting that SF density increases\nIntensity (Counts/s) FIG. 5. Di\u000braction pro\fle of reciprocal space q-scans per-\nformed on the Co(1 0 1 0) peak of Co(20 nm)/Pt thin \flm\nalong the Co(1 0 1 0), Co( 1 21 0), and Co(0 0 0 1) plane nor-\nmal directions. The inset of each \fgure is the reciprocal\nspace map which shows the identity of the measured peaks\nand the dashed line indicates the scanning direction of X-\nray. For clarity, the rightmost section of the scan along the\nCo(1 0 1 0) normal direction was not shown, as it is obscured\nby the Si(2 2 0) substrate peak (shown in Figure 1).\nTABLE I. Broadening e\u000bects present and whvalues for cobalt\npeaks in the Co (20 nm)/Pt and Co (50 nm)/Pt thin \flms\nmeasured by reciprocal space q-scans along primary direc-\ntions. S denotes the broadening e\u000bect caused by stacking\nfaults; M refers to the broadening e\u000bect caused by mosaic-\nity; and N signi\fes the absence of both e\u000bects. The values\nwere normalized with respect to the reciprocal space distance\nbetween Co(2 0 21) and Co(2 0 2 1) peaks.\nCo Scan Broadening whwh\nPeak Direction E\u000bects (20 nm) (50 nm)\n(1 01 0) (0 0 0 1) S+M 0 :1071 0:1360\n(1 01 0) ( 1 21 0) M 0 :0341 0:0487\n(1 01 0) (1 0 1 0) N 0 :0149 0:0099\n(2 02 0) (0 0 0 1) S+M 0 :1553 0:0683\n(2 02 0) ( 1 21 0) M 0 :0550 0:0578\n(2 02 0) (1 0 1 0) N 0 :0188 0:0177\n(2 02 1) (0 0 0 1) S+M 0 :0671 0:0675\nwith thickness within this thickness regime. In princi-\nple, the individual densities of deformation and growth\nSFs can be calculated by comparing the wh;sfvalues for\nthe (2 0 2 0) and (2 0 2 1) peaks. (The (1 0 1 1) peaks are\ninaccessible, as indicated in Figure 4.) However, mea-\nsuring the (2 0 2 1) peak width along the ( 1 21 0) normal\ndirection requires a change in \flm orientation that would\nintroduce additional broadening, thereby compromising\nour analysis. Moreover, the unusual thickness trends for4\nthe (2 0 2 0) peak widths warrant further investigation.\nBased on the characterized M{Hhysteresis curves, the\ndeposited thin \flms exhibit a strong in-plane anisotropy,\nas shown in Figure 6. Seeing as the uniaxial symme-\ntry axis (the c-axis, i.e., h0 0 0 1i) of HCP Co lies within\nthe \flm plane for Co(1 0 1 0) \flms, these results suggest\nthat the MCA is dominant over any interface-induced\nanisotropy present. The anisotropy energy density, Ku,\nis shown in Table II for the two \flms. In both cases,\nKuis considerably lower than the bulk K1value of 450\nkJ=m3[22], supporting the hypothesis that SFs lower the\nMCA energy of HCP Co. Moreover|as expected| Ku\ndecreases with increasing intrinsic SF density. Finally,\nwe note that lattice strain along the Co[1 1 2 0] direction\nmay also in\ruence the values of Kumagnetoelastically.\nAsymmetric ultrathin \flms based on all-FCC (1 1 1)\nCo/Pt layers possess a strong interfacial DMI stem-\nming primarily from Co/Pt interface, as well as a\nstrong perpendicular magnetic anisotropy (PMA) which\nis enhanced in the presence of Co/Ni layers. Be-\nyoxpecting similar properties in ultrathin asymmetric\nCo(1 0 1 0)/Pt(1 1 0){based \flms, we anticipate the emer-\ngence of DMI anisotropy, as well as the additional pres-\nence of a signi\fcant uniaxial in-plane MCA that is modu-\nlated by the degree to which the HCP phase of Co is pre-\nserved. By capitalizing on the tunability of a multilayer\nC2vCo/Pt system, one could explore the interplay be-\ntween all three e\u000bects in FM/HM \flms|and potentially\nstabilize elusive chiral structures such as antiskyrmions\nand bimerons, the latter being the in-plane equivalents\nof skyrmions [23].\nTABLE II. The combined deformation ( \u000b) & growth ( \f)\nstacking fault densities for Co and the magnetocrystalline\nanisotropy energy density ( Ku) in Co/Pt \flms di\u000bering in\nthe thickness of Co ( tco).\ntCo \u000b+\f(%) Ku(kJ=m3)\n20 nm 7.644 64.8\n50 nm 9.142 52.4\nIV. CONCLUSION\nIn this work, we characterized Co(20nm)/Pt and\nCo(50nm)/Pt C2v\flms sputtered at room tempera-\nture on Si(1 1 0) substrates with a Ag growth seed.\nThe epitaxial relationship between the Pt and Co lay-\ners was identi\fed as HCP Co(1 0 1 0)[0 0 0 1]kFCC\nPt(1 1 0)[0 0 1]; to our knowledge, this is the \frst re-\nport of HCPkFCC epitaxy in an FM/HM \flm system.\nThese \flms exhibit a de\fnitive uniaxial in-plane mag-\nnetic anisotropy along Co[0 0 0 1], in agreement with the\nobserved HCP (1 0 1 0) texture of Co. An analysis of the\nbroadening of Co(1 0 1 0) peaks along principal directions\nrevealed stacking fault densities of 7.6% and 9.1% in the\n20 nm and 50 nm \flms, respectively. Both \flms have\nSubstrate OrientationAlong Co [0001]Along Co [1!210]Pt/Co (20 nm)m/ms\n-1.5-1-0.500.51.01.5\n-2-1.5-1-0.500.511.52H (kOe)\nSubstrate OrientationAlong Co [0001]Along Co [1!210]Pt/Co (50 nm)m/msm/ms\n-1.5-1-0.500.51.01.5\n-2-1.5-1-0.500.511.52H (kOe)Substrate OrientationAlong Co [0001]Along Co [1!210]Pt/Co (50 nm)FIG. 6. Measured M{Hhysteresis curves for Co/Pt \flms.\nThe thickness of Co in each \flm in given in parentheses. The\nmeasurement directions are indicated with respect to Co.\nsigni\fcantly lower Kuvalues compared to bulk cobalt,\nwhich we attribute|at least in part|to the apprecia-\nble intrinsic stacking fault densities observed. We an-\nticipate that the observed epitaxial relationship and the\nsingular in-plane easy axis will be preserved in thinner\n\flms|where the collective impact of interface-induced\nPMA, anisotropic DMI and in-plane uniaxial anisotropy\non the stability of various chiral spin structures in mul-\ntilayer FM/HM \flms could be explored.\nACKNOWLEDGMENTS\nAll authors acknowledge the support of the De-\nfense Advanced Research Agency Program (DARPA)\non Topological Excitations in Electronics (Grant No.\nD18AP00011) and the use of the Materials Characteriza-\ntion Facility at CMU, supported by grant MCF-677785.\nM.D.K. is also grateful for the \fnancial support of the\nNational GEM Consortium as well as the Neil & Jo Bush-5\nnell and Bradford & Diane Smith Engineering Fellow- ships, all awarded by the College of Engineering at CMU.\n[1] A. Fert, V. Cros, and J. Sampaio, Nature Nanotechnol-\nogy8, 152{156 (2013).\n[2] N. Nagaosa and Y. Tokura, Nature Nanotechnology vol-\nume8, 899{911 (2013).\n[3] K. Everschor-Sitte, J. Masell, R. M. Reeve, and M. Kl aui,\nJournal of Applied Physics 124, 240901 (2018).\n[4] D. Prychynenko, M. Sitte, K. Litzius, B. Kr uger,\nG. Bouriano\u000b, M. Kl aui, J. Sinova, and K. Everschor-\nSitte, Physical Review Applied 9, 014034 (2018).\n[5] C. Back, V. Cros, H. Ebert, K. Everschor-Sitte, A. Fert,\nM. Garst, T. Ma, S. Mankovsky, T. L. Monchesky,\nM. Mostovoy, N. Nagaosa, S. S. P. Parkin, C. P\reiderer,\nN. Reyren, A. Rosch, Y. Taguchi, Y. Tokura, K. von\nBergmann, and J. Zang, Nature Physics 7, 713{718\n(2011).\n[6] I. E. Dzyaloshinskii, Journal of Physics and Chemistry of\nSolids 4, 241{255 (1958).\n[7] T. Moriya, Physical Review 120, 91{98 (1960).\n[8] U. K. R o\u0019ler, A. N. Bogdanov, and C. P\reiderer, Nature\n442, 797{801 (2006).\n[9] S. M uhlbauer, B. Binz, F. Jonietz, C. P\reiderer,\nA. Rosch, A. Neubauer, R. Georgii, and P. B oni, Science\n323, 915{919 (2009).\n[10] X. Yu, Y. Onose, N. Kanazawa, J.-H. Park, J. H. Han,\nY. Matsui, N. Nagaosa, and Y. Tokura, Nature 465,\n901{904 (2010).\n[11] S. Heinze, K. V. Bergmann, M. Menzel, J. Brede, A. Ku-\nbetzka, R. Wiesendanger, G. Bihlmayer, and S. Bl ugel,\nNature Physics 7, 713{718 (2011).\n[12] U. G ung ord u, R. Nepal, O. A. Tretiakov, K. Be-\nlashchenko, and A. A. Kovalev, Physical Review B 93,064428 (2016).\n[13] M. Ho\u000bmann, B. Zimmermann, and G. P. M uller, Nature\nCommunications 8, 308 (2017).\n[14] A. Raeliarijaona, R. Nepal, and A. A. Kovalev, Physical\nReview Materials 2, 124401 (2018).\n[15] L. Camosi, S. Rohart, O. Fruchart, S. Pizzini,\nM. Belmeguenai, Y. Roussign\u0013 e, A. Stashkevich, S. M.\nCherif, L. Ranno, M. de Santis, and J. Vogel, Physical\nReview B 95, 214422 (2017).\n[16] S. Huang, C. Zhou, G. Chen, H. Shen, A. K. Schmid,\nK. Liu, and Y. Wu, Physical Review B 96, 144412 (2017).\n[17] L. Camosi, N. Rougemaille, O. Fruchart, J. Vogel, and\nS. Rohart, Physical Review B 97, 134404 (2018).\n[18] L. Camosi, J. Pe~ na-Garcia, O. Fruchart, S. Pizzini,\nA. Locatelli, T. O. Mente\u0018 s, F. Genuzio, H. N.\nJustin Shaw, and J. Vogel, New Journal of Physics 23,\n013020 (2021).\n[19] S. Tsurkan and K. Zakeri, Physical Review B 102,\n060406(R) (2020).\n[20] W. Yang, D. N. Lambeth, and D. E. Laughlin, Journal\nof Applied Physics 85, 8 (1999).\n[21] M. T. Sebastian and P. Krishna, Random, Non-Random\nand Periodic Faulting in Crystals (Gordon and Breach\nScience Publishers, 1994).\n[22] B. D. Cullity and J. Chad D. Graham, Introduction to\nMagnetic Materials (Wiley-IEEE Press, 2008) p. 227.\n[23] B. G obel, A. Mook, J. Henk, I. Mertig, and O. A. Tre-\ntiakov, Magnetic bimerons as skyrmion analogues in in-\nplane magnets, Phys. Rev. B 99, 060407 (2019)."    },    {        "title": "2106.11947v1.Quantum_confined_charge_transfer_that_enhances_magnetic_anisotropy_in_lanthanum_M_type_hexaferrites.pdf",        "content": "Quantum-con\fned charge transfer that enhances magnetic anisotropy in lanthanum\nM-type hexaferrites\nChurna Bhandari,1Michael E. Flatt\u0013 e,2, 3and Durga Paudyal1, 4\n1The Ames Laboratory, U.S. Department of Energy, Iowa State University, Ames, IA 50011, USA\n2Department of Physics and Astronomy, University of Iowa, Iowa City, IA 52242, USA\n3Department of Applied Physics, Eindhoven University of Technology, Eindhoven, The Netherlands\n4Departments of Electrical and Computer Engineering,\nand Computer Science, Iowa State University, Ames, Iowa 50011, USA\nIron-based hexaferrites are critical-element-free permanent magnet components of magnetic de-\nvices. Of particular interest is electron-doped M-type hexaferrite i.e., LaFe 12O19(LaM) in which\nextra electrons introduced by lanthanum substitution of barium/strontium play a key role in up-\nlifting the magnetocrystalline anisotropy. We investigate the electronic structure of lanthanum\nhexaferrite using a localized density functional theory which reproduces semiconducting behavior\nand identi\fes the origin of the very large magnetocrystalline anisotropy. Localized charge transfer\nfrom lanthanum to the iron at the crystal's 2 asite produces a narrow 3 dz2valence band strongly\nlocking the magnetization along the caxis. The calculated uniaxial magnetic anisotropy energies\nfrom fully self-consistent calculations are nearly double the single-shot values, and agree well with\navailable experiments. The chemical similarity of lanthanum to other rare earths suggests that LaM\ncan host for other rare earths possessing non-trivial 4 felectronic states for, e.g., microwave-optical\nquantum transduction.\nINTRODUCTION\nSince the discovery of M-type hexaferrites in 1950,\nthese complex oxides have been of continuing research\ninterest for permanent magnets and magnetic memory\ndevices1{4, however the unique interplay between charge,\nspin, and orbital degrees of freedom suggest broader ap-\nplications, including to quantum information science. M-\ntype hexaferrites have a chemically and thermally sta-\nble crystal structure composed of easily accessible con-\nstituent elements in nature, especially barium hexafer-\nrite and strontium hexaferrite2. The Gorter's type5hex-\naferrite has alternately aligned parallel and anti-parallel\nmagnetic moments of ferric Fe3+ions with respect to\nthe hexagonal axis, aided by superexchange interac-\ntion via oxygen resulting in a large magnetic moment\n(20\u0016B/f.u.). The unique crystal structure leads to\na huge uniaxial magnetocrystalline anisotropy (MCA)\nalong the hexagonal axis, which assists their usage for\nhigh-frequency microwave elements ( i.e.at low applied\nmagnetic \felds). An increase in the magnetic anisotropy\nconstant (K1) leads directly to higher-frequency mi-\ncrowave operation, assuming the microwave loss remains\nlow.\nA known approach to enhance the MCA is through\nreplacement of the divalent barium with trivalent lan-\nthanum in M-type hexaferrites6{8. The increase in the\nmagnetic anisotropy was attributed to the extra electron\nadded6, with the suggestion that it leads to the forma-\ntion of one Fe2+per formula unit and thus enhances\nthe orbital angular momentum. The increase of the or-\nbital angular momentum provides, through the spin-orbit\ninteraction (SOC), a stronger magnetic anisotropy. Al-\nthough the crystal structure consists of \fve Fe-sublattices\nat inequivalent crystal site locations (labelled 2a, 2b,\n12k, 4f 1, and 4f 2), it is not clear where the added elec-tron resides. Ref. 6 suggested the Fe (2a) site as a pre-\nferred site of electron localization, but there was no di-\nrect experimental measurement to con\frm the formation\nof Fe2+. Ref. 8 studied the charged states of Fe ions\nand inferred a partially quenched orbital moment of Fe\n(2a) from the anomalous behavior of the hyper\fne \feld\nsplitting in their M ossbauer spectroscopy measurement.\nA similar charge state is discussed in other M ossbauer\nexperiments9,10, but they did not conclusively determine\nthe magnitude of orbital moment which is required to\ncon\frm the Fe2+character.\nStandard density functional theory (DFT) calculations\npredict a delocalized electronic state resulting from the\nadditional electron introduced by the lanthanum10, and\nthus predicts metallic behavior in disagreement with the\nexperimentally observed semiconducting behavior. Thus\nit is not surprising that these calculations also yield an\nMCA much smaller than experimentally measured. In\nLaM, although Refs. 11 and 12 employed the same full-\npotential linearized augmented plane wave (FP-LAPW)\nmethods, they obtained very di\u000berent results, including\nfor the spin magnetic moments of Fe. Ref. 12 could not\nconclusively \fnd localization of extra electrons, which is\nexpected in the delocalized electron state calculation. So,\nthe discussion of MCA is irrelevant there. Ref. 11 found\nelectrons localized at Fe (2a), however their calculated\nK1is smaller by a factor of two compared to the ex-\nperiment for both SrM and LaM. A clear understanding\nof the proper electronic structure of lanthanum M-type\nhexaferrite (LaM) would thus provide a clear pathway\ntowards describing the properties of 4 f-spin containing\nrare-earth-doped materials, such as are now used in other\nsemiconducting rare-earth hosts like yttrium orthovana-\ndate and yttrium oxide.13{15\nHere we investigate the electronic structure, optical be-\nhavior and magnetic properties of Ba/SrM and LaM us-arXiv:2106.11947v1  [cond-mat.mtrl-sci]  22 Jun 20212\nFIG. 1. ( Left) Crystal structure of M-type hexaferrite\n(LaFe 12O19) with Gorter's-type spin con\fgurations of the dif-\nferent Fe sublattices. Violet, brown, and purple balls are Fe\n(2a), Fe (12k), and Fe (2b) with spin- \"(green) and dark green\nand magenta balls are Fe (4f 1) and Fe (4f 2) with spin-#(red),\nrespectively. Yellow and black balls are La and O atoms.\n(Right ) The polyhedra of each Fe-sublattice based on their\nnearest O atoms are shown with the site symmetries.\ninglocalized density functional theory (LDFT). The key\nresults are: (i) replacement by La modi\fes the electronic\nband structure of BaM (SrM) signi\fcantly, producing a\nstrongly localized band in the gap region, (ii) The elec-\ntronic bandgap occurs between states of the same spin\nand is reduced, (iii) La-substituted electrons are strongly\nlocalized around the Fe (2a) site and occupy a localized\n(3dz2) band, (iv) The spin magnetic moment of Fe (2a)\nis reduced accordingly due to the formation of an Fe2+\ncharge state, and (v) The strongly localized extra elec-\ntron enhances the K1by approximately two compared to\nBaM/SrM, which agrees with experiment.\nMETHODS AND CRYSTAL STRUCTURE\nThe \frst-principles calculations were performed by us-\ning the Vienna ab-initio Simulation Package (VASP) in\nthe all-electron (AE) projector augmented wave (PAW)\nform16,17with the generalized gradient approximation\n(GGA) for the exchange-correlation functional including\nthe Hubbard U corrections for Fe atoms. Following pre-\nvious work11, we used a Hubbard Ueff= 4:5 eV in our\nGGA +Ucalculations within Ref. 18's simpli\fed rota-\ntionally invariant formalism in which only the e\u000bective\nvalue ofUeff=U\u0000Jis relevant. We used a total en-\nergy plane wave cut o\u000b of 500 eV and 7 \u00027\u00021k-mesh for\nthe Brillouin zone sampling. In VASP, the spin-orbit cou-pling (SOC) correction can be obtained by fully relativis-\ntic noncollinear methods19. To compute the magnetic\nanisotropy, we used both single-shot and self-consistent\nschemes for the total energy calculations. We did two set\nof DFT calculations: i) standard and ii) localized. In the\nlocalized DFT calculations a really high value of Ueffis\nused and it is subsequently reduced to a realistic value\nto avoid a higher-energy delocalized state solution.\nWe used experimental lattice parameters measured\nby the x-ray di\u000braction method10,20in our calculations.\nWe note that LaM undergoes a structural transition\nfrom hexagonal to orthorhombic at low temperature, but\nwe neglect the low temperature structure in this work.\nThe room temperature crystal structure has a hexagonal\nprimitive unit cell with space group 194-P63/mmc (D4\n6h)\nas shown in Fig. 1, which consists of two formula units\nof LaFe 12O19and a total of 11 sublattices viz., \fve Fe at\n2a, 2b, 12k and 4f 1, 4f2, \fve O at 4f 1, 4e, 12k, 12k, 6h,\nand 1 La at 2d Wycko\u000b positions. Fe (2a), Fe (12k), and\nFe (4f 2) atoms form octahedral networks with O atoms\nwithD3d(-3m),Cs(m), andC3v(3m) point group symme-\ntry, while Fe (4f 1) forms tetrahedra with C3v(3m) and Fe\n(2b) forms a bipyramidal structure with D3h(-6m2) point\ngroup symmetry as shown in Fig. 1 ( Right ). We used the\nexperimental structure parameters in our calculations to\navoid the systematic lattice parameter overestimate from\nthe GGA functional. We also did test calculations using\nthe fully relaxed ab initio structure and found no signif-\nicant changes in the properties.\nRESULTS AND DISCUSSION\nEvolution of electronic structure with\nLa-substitution\nWe begin our discussion with the electronic band struc-\nture of the Gorter's type BaM ferrimagnet calculated\nalong high symmetry kpoints as shown in Fig. 2. The\nelectronic band gap occurs between spin majority valence\nbands of Fe (3 d\") and spin minority conduction bands of\nFe (3d#). The valence bands consist of a strong admix-\nture of Fe (3 d\") with oxygen (2 p), whereas conduction\nbands are mostly Fe (3 d#). As is shown in Fig. 2(a),\nBaM has a direct band gap of 1.82 eV at \u0000, computed\nwith GGA + U methods, in excellent agreement with\nrecent experiment21(\u00181:82\u00001:97 eV). The lowest indi-\nrect band gap of 1.84 eV between the valence band max-\nimum (VBM) (at A) and the conduction band minimum\n(CBM) (at \u0000) is also not di\u000berent from the direct gap\nas observed in the experiment. Although the experimen-\ntally observed indirect gap ( \u00181:72\u00001:77 eV) is slightly\nsmaller than the direct gap, one cannot really distinguish\nthem since they are very close. Also, the high symmetry\nk-point A is pretty much the same as \u0000 because the lat-\ntice constant cis much larger ( c\u00184a) as compared to\na.\nThe optical band gap related to the transitions be-3\n(a) (b) (c)\nFIG. 2. Electronic band structure calculated using GGA + U of (a) BaM and (b) LaM and (c) GGA + U + SOC of LaM\nmethods along high symmetry kpoints. The high symmetry points are: A =1\n2(0;0;1), \u0000 = (0;0;0), M =1\n2(1;0;0), and\nK =1\n3(1;1;0) in units of primitive reciprocal lattice vectors, where ~ a=a(1;0;0),~b=a(\u00001=2;p\n3=2;0), and~ c= (0;0;c) are\nprimitive translation lattice vectors.\ntween the valence and conduction bands with the same\nspins is larger by \u00180:5 eV than the lowest direct gap.\nWe found quite a bit increase of gap (2.34 eV) in the op-\ntimized structure, but the other properties such as mag-\nnetic moment and magnetic anisotropy constant remain\nthe same as discussed elsewhere11. We also note that\nthe magnitudes of calculated band gaps depend on the\nchoice of Hubbard parameters, UandJ. The sensible\nvalue ofUeff= 4:5 eV used in the calculations is based\non its similarity to that obtained for Fe3+in\u000b-Fe2O3\nwith spectroscopic measurements22.\nNext, we discuss the electronic band structure of LaM.\nOrdinary DFT predicts a delocalized state of LaM lead-\ning to a half-metal (this band structure is shown in the\nAppendix Fig. 8). In contrast, experimentally LaM is\na semiconductor7. The calculated electron from La-\nsubstitution is spread over all the Fe-sublattices, even\nthough the occupancy is larger on Fe (2a) than the other\niron sites. In essence, the small amount of the extra elec-\ntron, when shared among the Fe sites, will not modify\nthe Fe3+charge state to Fe2+. Indeed, the calculated\nvalues of the spin magnetic moments con\frm these re-\nsults, which are about the same (4 :10\u0016B) for all the, Fe\natoms as found in undoped BaM.\nThe true ground state is obtained by the localized DFT\ncalculation which predicts correctly LaM as a semicon-\nductor agreeing with the experiment. The calculated to-\ntal energy per unit cell is also about 1 eV smaller than\nthat of the delocalized solution, con\frming the global\nground state property of the localized result. Since the\ndelocalized solution is metastable, hereafter, we only dis-\ncuss the results from the localized calculations.\nPrevious calculations in Ref. 11 and 12 did not study\nthe band structure apart from structural change andcharge state. In Ref. 12 authors calculated the density of\nstates but it shows an incorrect metallic state. Therefore,\nto the best of our knowledge, the correct band structure\nof LaM has not yet been studied in the literature.\nLa substitution dramatically changes the band struc-\nture of BaM near the Fermi level. One can see the fully\noccupied two distinct subbands in the gap region in Fig.\n2(b). The lower subbands consist of four and the upper\nsubbands consist of a doubly degenerate band. A small\nindirect band gap of 0.77 eV opens up between the VBM\nat K and the CBM at \u0000, which is about half of the BaM\nband gap. The reduction of the band gap is expected\nsince the new bands show up in the middle of the gap.\nWhereas our calculated band gap agrees with experi-\nmental measurements, its magnitude is larger than that\nobtained by the resistivity measurement (0.18 eV) in the\nceramic LaM7. The band gap measured with this type\nof experiment would also be smaller for BaM (See the\nRef. 7 for SrM). So, it would be interesting to see if one\ncould perform a spectroscopic measurement of the gap in\na LaM single crystal. The other interesting feature is the\nappearance of a pseudo-Dirac-like cone at the K-point in\nthe lower subbands, which may lead to anomalous behav-\nior in transport measurements in hole-doped LaM, which\nis worth future investigation. The GGA + U + SOC com-\nputed band structure in Fig. 2(c) does not show much\ndi\u000berence compared to a calculation without SOC, ex-\ncept for a few split-o\u000b bands, which is expected because\nthe SOC is weaker than the crystal \feld splittings in 3 d\nelements.4\nFIG. 3. Computed optical absorption coe\u000ecient ( \u000b) in units\nof cm\u00001as a function of photon energy ( in eV) for linearly\npolarized light in two di\u000berent directions EkaandEkc\nusing the GGA + U method in (a) BaM and (b) LaM. In-\nset \fgures show the real part of the dielectric constant. The\ncomputed optical spectra show anisotropy between light po-\nlarization parallel to plane ( ab) and perpendicular to plane\n(c) directions.\nOptical anisotropy\nTo further scrutinize the band gap, we computed the\noptical absorption coe\u000ecient ( \u000b) using GGA + U. The\nresults are shown in Fig. 3. We \fnd \u000bxxfor light po-\nlarization along Ekais stronger than \u000bzzfor the light\npolarization Ekcre\recting an anisotropy in the optical\nabsorption in BaM. We note that x,y, andzcorrespond\nto the crystalline a,b, andcdirections, respectively. The\noptical absorption is much weaker in the energy range\n(1.8 - 2.0 eV) which is expected since it corresponds to a\n3d-3dtransition. The optically disallowed transitions be-\ncome allowed due to an admixture of Fe-3 dwith 4sand\n3porbitals as is also seen in other 4 d=5doxides23,24. The\nreal part of the dielectric constants \"xx=\"yy= 5:32 and\n\"zz= 4:79 are consistent with the \u000b, which also con\frms\nthe optical anisotropy.\nIn LaM, the optical absorption peak is red shifted by\nabout 1 eV as expected due to the reduced band gap.\nThe real part of the dielectric constants, \"xx= 5:5 and\n\"zz= 4:95, are similar to those in BaM. However the \u000bin LaM shows a higher peak in the low energy region,\ndi\u000bering that from BaM.\nQuantum-con\fned charge transfer\nThere are \fve di\u000berent Fe sublattices in LaM, but\nit is not clear which one is the most favorable for La-\nsubstituted electron localization. We address this issue\nby analyzing i) localized vs. delocalized solutions, ii)\nsymmetry, and iii) partial density of states (PDOS). As\ndescribed above, it is now clear that the delocalized solu-\ntion is not correct as it predicts LaM is a metal. This has\ntwo main consequences: \frst, an electron cannot occupy\nthe Fe atoms at 12k and 4f 1sites because one electron\nhas to be shared among multiple Fe atom that are equiv-\nalent by symmetry. This scenario would lead LaM to be\na metal. Second, if an electron would occupy the 4f 1(4f2)\nsites, then it would further increase the net magnetic mo-\nment by 2\u0016Bper cell, which is also not correct because\nthe opposite magnetic moments would decrease. This\ncontradicts with the experimentally observed magnetic\nmoment (38 \u0016Bper unit cell).\nTo gain a better insight into the electron localization,\nwe employ the local symmetry of the Fe-O networks to\nanalyze the single particle energy levels. According to\nC3vsite symmetry, the 3 dorbitals of Fe tetrahedron at\n4f1site split into A1,A2, and doubly degenerate Eir-\nreducible representations (IRReps). In the Fe (4f 2) the\nhighly distorted octahedra with C 3valso lead to similar\nsplittings of orbitals. The Fe (12k) octahedra has the\nlowest symmetry in which orbitals split into AandA0\nIRReps. In either case it is inappropriate for an electron\nto occupy any speci\fc empty levels because of the con-\ntinuum states present due to orbital overlaps between\nthe nearest Fe atoms. Then we are left with the two\nsublattices at the 2a and 2b sites. In this case, one can\nanticipate a semiconducting state if one electron occupies\neither one of these two sites.\nThe local site symmetry of Fe (2b) is D 3hforming a\nbipyramidal (trigonal) FeO 5unit. The Fe (3 d) orbitals\ntransform as e00(dyz;dzx ),a0(dz2), ande0(dxy;dx2\u0000y2)\nIRReps25as given in Fig. 4 for the ground state con\fgu-\nrationA0\n1with S=5/2 and L=0. The next electron would\noccupy the orbitally degenerate state e00which might lead\nto the well-known Jahn Teller distortion26. It is inappro-\npriate for electron to occupy this site as it would cost\nextra energy.\nOn the other hand, the local site symmetry of Fe (2a)\nis D 3d, in which Fe (3d) orbitals transform as a1g,e00\ng, and\negIRReps. These two orbitally degenerate states with eg\nIRReps are two dimensional and can mix with each other\nwhilea1gcannot. Because of the trigonally distorted oc-\ntahedral structure of Fe (2a), the ground state con\fgu-\nration consists of singly occupied \fve 3 d-orbitals in BaM\nand the corresponding levels are shown in Fig. 5(a). The\nlowest level is a dz2orbital which transforms as a1g. IR-\nReps with respect to the hexagonal axis indicating that5\nFIG. 4. Schematic diagram showing the crystal \feld splitting\nof single particle levels of Fe (2b) (FeO 5) with 3d5valence\nelectrons. Here a0andeare irreducible representations of the\nsingle particle 3 dlevels belonging to the D3hpoint group of\nFe (2b).\nFIG. 5. Schematic diagrams showing the crystal \feld split-\nting of single particle levels for Fe (2a) (FeO 6octahedra) with\n(a) 3d5and (b) 3d6valence electrons. Here a1gandegare\nirreducible representations of the single particle 3 dlevels be-\nlonging to the D3dpoint group of Fe (2a).\nany extra electron would occupy it (the opposite spin\nchannel) \frst as shown in Fig. 5(b). We note that the\nsymmetry of the single particle wavefunctions does not\nchange due to the SOC, except the splitting of Fe (12k)\ninto two sublattices.\nTo further con\frm the electron localization as dictated\nby symmetry, we computed the PDOS for each atom with\nand without La substitution as shown in Fig. 6. As ex-\npected, the 5 dstates of La are located far above the Fermi\nlevel which is re\rected in PDOS (shown in Appendix\nFig. 9) indicating that it simply acts as an electron donor\n(similar to Ba except it donates an extra electron). Next,\nwe discuss the PDOS of Fe atoms with and without La\nsubstitution. It is evident from Fig. 6(a) that the PDOS\nof all Fe atoms except Fe (2a) are unchanged with La sub-\nstitution. No occupied states of Fe atoms are seen near\nthe Fermi level except for Fe (2a). But the entire 3 d-band\nof Fe (2a) is shifted upwards showing a sharp peak below\nthe Fermi level. To the best of our knowledge, this is the\n\frst prediction of such a quantum-con\fned charge trans-\nfer state in the La-substituted hexaferrites. Remarkably,\nit lies in the gap region and it is sharply localized atTABLE I. Net amount of electron transfer to di\u000berent Fe\natoms in LaFe 12O19. The tabulated values are per Fe atom\nand are expressed in units of electrons.\nAtom Fe (2a) Fe (2a) F e(12k) F e(4f 1) Fe (4f 2) La/O\nelectron 0.8127 0.0005 0.0038 0.0002 0.0065 0.1505\nthe Fe (2a) site consistent with the appearance of two\nnarrow bands (each coming from one Fe (2a)). These\nresults suggest that the La-substituted electron occupies\na single orbital of the Fe (2a) site. As shown in Fig. 5,\nthe transferred charge will occupy the a1g(dz2) orbital\nwhich is parallel to the hexagonal axis (not the octahe-\ndralz-axis). To further con\frm this, we computed the\norbitally resolved partial density of states (Fig. 7), which\nshows that the band is derived from the dz2orbital of\nFe (2a) consistent with the symmetry analysis. Electron\ncharge distribution associated with this band is shown in\nFig. 6 (b) by a charge density contour computed in the\nenergy range from 0.5 eV below to the Fermi level. The\ncontour shows a perfect dz2like shape pointing along the\nhexagonal axis at Fe (2a).\nTo quantify the charge localization, we computed the\nnet charge transfer to each Fe atom by integrating the\noccupied region of the PDOS near the Fermi level as\ngiven in Table I. We \fnd that 81% of the added electrons\noccupies Fe (2a), and the remaining amount is mostly\nshared between O and La atoms. This in turn suggests\nthat the extra charge added to Fe (2a) would modify\nits Fe3+charge state to Fe2+consistent with the anoma-\nlous behavior of the hyper\fne \feld splitting in M ossbauer\nspectroscopy9,10.\nMagnetic moments and Curie temperature\nThe spin magnetic moments of individual atoms in\nBaM/SrM and LaM are given in Table II. The net mag-\nnetic moment of BaM/SrM is 39 :98\u0016Bper unit cell which\nis consistent with the Gorter's type magnetic order as\ndiscussed above. The sum of the magnetic moments of\nthe individual atoms is slightly smaller than the total\nvalue as expected since these are calculated within the\natomic spheres, which exclude the interstitial contribu-\ntions. Our calculated magnetic moments for individual\nFe atoms are slightly larger than experimental values due\nto the following factors: (1) theoretical values depend on\nthe choice of Hubbard parameters and the size of the\natomic radii used in the calculations, (2) the experimen-\ntal values measured at \fnite temperature are about 10%\nsmaller than ideal values (5 \u0016B) at zero K, and (3) there\nis strong hybridization of Fe (3 d) with O(p). Indeed, the\nO atoms carry a spin magnetic moment as large as 0.33\n\u0016Bas given in Table II.\nAll experiments see a smaller value for Fe (2a) than the\nother Fe sites. We again emphasize that the discrepancies\nbetween theoretical and experimental values appear to6\n(a) (b)\nFIG. 6. (a) Partial density of states (PDOS) of Fe atoms computed using GGA + U for BaM and LaM. The shared red\nregion below the Fermi level indicates the electron transfer to Fe (2a) from La. The solid (dashed) lines represent the PDOS\nin LaM(BaM) respectively. (b) Electron charge density contours computed in occupied region of valence bands (spin-down\nchannel) with isosurface level 0.002 e/ \u0017A3in LaM, where Fe atoms are labeled according to their site symmetries. The contours\nfurther con\frm that La-substituted extra electron charge is localized to Fe (2a) and nearby O atoms. The shape of contours\naround Fe atom is dz2-like and directed along the hexagonal axis.\nFIG. 7. Orbital resolved partial density of sates of Fe (2a)\ncomputed with GGA + U method in LaFe 12O19. The sharply\nlocalized state below the Fermi level indicates the transfer of\nthe La substituted electron into the dz2-derived orbital.\nbe similar for La substituted cases or not, as discussed\nabove.\nAs the calculated spin magnetic moment suggests the\nformation of a Fe2+state at the 2a site, the orbital mag-\nnetic moment expected be large following Hund's rule.\nHowever, its magnitude is much smaller ( \u00180:1\u0016B) dueTABLE II. Spin magnetic moment of individual atoms in\nSr/Ba/LaFe 12O19(Sr/Ba/LaM), the total magnetic moment\nper unit cell (which includes the interstitial contribution), and\ncomparison with experiment. Also shown are experimental\nmagnetic moments (average values for LaM measured at room\ntemperature).\nAtom SrM BaM LaM LaM\n(site) This work This work This work Expt.\nFe (2a) 4.11 4.23 3.69 3.35\nFe (2b) 4.08 4.14 4.15 3.83\nFe (12k) 4.16 4.24 4.24 3.73\nFe (4f1) -3.97 -4.12 -4.19 -3.83\nFe (4f2) -4.11 -4.18 -4.11 -3.90\nO (4e) 0.34 0.33 0.33\nO (4f) 0.09 0.08 0.11\nO (6h) 0.05 0.04 0.04\nO (12k) 0.17 0.17 0.12\nO (12k) 0.09 0.03 0.05\nSr/La (2d) 0.00 0.00 -0.01\nTotal 39.98 39.97 38.00\nto the quenching of the orbital moment by the crystal\n\feld. The PDOS shows that the extra charge occupies\nthedz2orbital with L\u00180. Although the orbital mo-\nment is small, it is still increased by about 10% in LaM\nas compared to SrM for Fe (2a) and it qualitatively sup-7\nTABLE III. Calculated Curie temperature T Cand its\ncomparison with previous theory and experiment for\nSr/Ba/LaFe 12O19(Sr/Ba/LaM).\nMethods SrM BaM LaM\nExpt. 73729723 695\nThis worka893 880 851\nTheoryb- 1514 1419\nTheoryc- 1009 946\nTheoryd-\u00181500 -\nTheorye-\u00181000 -\naHubbard, Ueff=4.5 eV\nbMean \feld approximation (MFA), Ueff=3.4 eV27\ncRandom phase approximation (RPA), Ueff=3.4 eV27\ndMFA, Ueff=4 eV28\neRPA, Ueff=4 eV28\nports the hyper\fne splitting data in which the Fe at 2a\nsite shows a transformation from Fe3+to Fe2+8.\nWe now discuss the Curie temperature (T C) and com-\npare with previous results27,28. We consider an isotropic\nexchange interaction ( Jij) between the nearest neigh-\nbor Fe atoms within the mean \feld approximation. Al-\nthough more involved approaches such as the random\nphase approximation (RPA) and mean-\feld approxima-\ntion (MFA) have been employed previously27,28to esti-\nmate T C, its magnitude is overestimated relative to ex-\nperiments when reasonable values of the Hubbard U are\nused. Our straightforward calculations, taking the the\ntotal energy di\u000berence between ferrimagnetic and ferro-\nmagnetic con\fgurations, show that the energy required\nto break a bond between the nearest neighbor Fe atoms\nis:Jij= 12:32, 12.13, and 11.73 meV in SrM, BaM, and\nLaM, respectively. The total energy di\u000berence (also the\nenergy required to break a bond) reduces with an increas-\ning value of U thereby reducing the T C. Remarkably, by\nusing T C\u0018S2Jij=kB, withS= 5=2 for each Fe atom,\nwe get 893, 880, and 851 K for respective compounds ob-\ntained by using a realistic value of U= 4.5 eV; these agree\nvery well with experimental values as shown in Table III.\nAlthough the magnitudes are slightly overestimated, the\ntrend of the relative di\u000berences agrees with the experi-\nments.\nMagnetic anisotropy\nThe magnetic energy cost to rotate the spontaneous\nmagnetization M=Ms(sin\u0012cos\u001e^x+sin\u0012sin\u001e^y+cos\u0012^z)\nwith respect to the crystalline axis is the magnetic\nanisotropy energy ( Ea). Here,\u0012and\u001eare magnetization\nangles and ^ x, ^y, and ^zare unit vectors. Phenomenologi-\ncally, the magnetic anisotropy energy density for a given\nmagnet is given byEa\nV=K1sin2\u0012+K2sin4\u0012+K3sin6\u0012,\nwhereKiis the ith order magnetic anisotropy constant\nandVis the volume of the magnet30.\nFor hard magnets the lowest order magnetic anisotropyTABLE IV. Calculated magnetic anisotropy constant K1in\nSr/Ba/LaFe 12O19(Sr/Ba/LaM) and its comparison with ex-\nperiment and previous DFT. The values are presented in units\nof MJ/m3. The values outside(inside) the parenthesis corre-\nspond to results obtained by fully self-consistent (single-shot)\nspin-orbit coupling via noncolliiear relativistic methods.\nK1 SrM BaM LaM\nThis worka0.18 0.18 0.32\nThis workb0.34 0.33 0.66\nExpt.31,320.35-0.36 0.32-0.33 0.5-0.8\nTheoryc0.18 0.36\naSingle-shot calculations\nbFully self-consistent calculations\ncFLAPW methods with U eff=4.5 eV11\nconstantK1dominates and its magnitude depends on\nthe single ion and the dipolar moment (pair contribu-\ntion) interaction energies. The single ion energy depends\non the strength of the spin-orbit coupling ( \u0015) and crystal\n\feld (\u0001) and is much larger than the pair contribution30.\nRef. 25 also showed from a model based on quantum\nmany-body wavefunctions that the leading contribution\nis from a single ion for Fe (2b). The zero-\feld splitting\n(uniaxial splitting) parameter D\u00182cm\u00001is two orders\nof magnitude larger than the dipolar spin-spin contribu-\ntion. Therefore, here, we focus only on the single ion\ncontribution to K1.\nFirst we computed the total energy along the two\ndi\u000berent crystalline axes viz., E100along the aand\nE001along the c-axis respectively. Then, the mag-\nnetic anisotropy energy is obtained as: Ea=E100\u0000\nE001. The computed values of K1=Ea=Vare posi-\ntive in all cases (Table IV) suggesting magnetocrystalline\nanisotropy with an easy axis along the crystalline c-axis.\nThe values outside (inside) the parentheses are obtained\nwith and without full self-consistent total energy calcu-\nlations using the GGA + U + SOC methods. The mag-\nnitudes ofK1are sensitive to the methods and depend\non whether it is a single shot or fully self-consistent cal-\nculation, despite \u0015's relative weakness for Fe (3 d) atoms.\nFor instance, the computed value of K1is 0.34 MJ/m3\nfor BaM with the fully self-consistent calculations, twice\nthe single-shot value of 0.18 MJ/m3. In principle the self-\nconsistent method is a better approach because the e\u000bect\nof SOC is included properly in the charge density and\ntherefore it is expected to predict better results. Indeed,\nthe value of K1obtained with the fully self-consistent cal-\nculations is in excellent agreement with the experimen-\ntal value (0.35 MJ/m3). Similar values are predicted for\nK1in SrM in excellent agreement with the experiment,\nwhich is expected since in both cases we have similar\nchemical and structural environments e.g., sp-electron el-\nements (Sr or Ba).\nAgain, in LaM the self-consistent value of K1is en-\nhanced by about a factor of two compared to the single-\nshot value, as was found for the parent compounds8\n(Ba/SrM). The magnitude of K1is increased in LaM\nby a factor\u00182 compared to Ba/SrM which is in very\ngood agreement with experiment (0.5-0.8 MJ/m3)6. The\nincrease in magnetic anisotropy can be understood in\nterms of charge transfer, spin, and orbital moment. As\ndiscussed above, the charge from La substitution occu-\npies thedz2orbital of Fe (2a) resulting in the net charge\npolarization along the c-axis along with the formation\nof a Fe2+charge state. Although the orbital moment is\nexpected to be quenched for Fe2+, e\u000bectively it is still in-\ncreased by\u001810% in LaM. Interestingly, it is about three\ntimes larger along the c-axis than in the plane, which in-\ndicates that it would cost more magnetic energy to rotate\nstrongly polarized charge away from the c-axis leading to\nthe enhancement of K1in LaM.\nCONCLUSION\nWe used ab-initio calculations to study the evolution of\nthe electronic band structure and optical spectra of the\nhexaferrites after chemical substitution of a divalent sp-\nelement (Ba/Sr) with a trivalent 5 d-element (La). Our\nlocalized DFT calculations predict a very sharply local-\nized band in the gap in LaM which is revealed by the\nelectronic band structure and PDOS. The origin of this\nlocalized band is due to quantum-con\fned charge trans-\nfer from the substituted La to a speci\fc site [Fe (2a)], oc-\ncupying the symmetry-allowed 3 dz2-derived orbital. The\nreduced value of the spin magnetic moment of Fe (2a)\nis consistent with localization of the additional electron\nthere, leading to the formation of a Fe2+charge state\nat the 2a site. The increase in electronic charge, which\nleads to a reduction in the spin moment and an increase\nin the orbital moment of Fe (2a) driven by this La substi-\ntution, results in an increase of the magnetic anisotropy\nby a factor of two in LaM as compared to Sr/BaM and\nagrees very well with available experiment. Our study\nopens up the possibility of exploring rare-earth-based\nelectron-substituted hexaferrites as quantum spin sys-\ntems because of the formation of this spatially localized\nquantum state in the gap. It also provides the descrip-\ntion of a potential host for other 4 felements that would\nprovide sharp optical transitions within the optical gap\nof the host, with these rare earth elements substituting\nfor the lanthanum.\nACKNOWLEDGMENTS\nThis work is supported by the Critical Materials In-\nstitute, an Energy Innovation Hub funded by the U.S.\nDepartment of Energy, O\u000ece of Energy E\u000eciency and\nRenewable Energy, Advanced Manufacturing O\u000ece. The\nAmes Laboratory is operated for the U.S. Department of\nEnergy by Iowa State University of Science and Tech-\nnology under Contract No. DE-AC02-07CH11358. MEF\nis supported by NSF DMR-1921877. We would like to\nFIG. 8. Electronic band structure of metalstable ferrimag-\nnet LaM calculated using the GGA + U method along high\nsymmetry kpoints.\nthank C. S \u0018ahin for his useful remarks on the paper. We\nwould also like to acknowledge Ed Moxley for maintain-\ning and updating computational facilities and software.\nThe U.S. government retains and the publisher, by\naccepting the article for publication, acknowledges that\nthe U.S. government retains a nonexclusive, paid-up,\nirrevocable, worldwide license to publish or reproduce\nthe published form of this manuscript or allow others\nto do so, for the U.S. government purposes. U.S.\nDepartment of Energy (DOE) will provide public\naccess to these results of federally sponsored research\nin accordance with the DOE Public Access Plan\n(http://energy.gov/downloads/doe-public-access-plan).\nAppendix A: Band structure of LaM with a\ndelocalized electron\nHere we present the band structure of the ferrimag-\nnetic metallic metastable state of LaM shown in Fig. 8,\ncomputed using delocalized (standard) DFT. In this case,\nthe electrons from La substitution partially occupy all the\nFe sublattices, resulting in a half metal (as visible in the\n\fgure).\nAppendix B: Density of states of LaM\nWe compare the PDOS of individual atoms in LaM in\nFig. 9. The oxygen PDOS shows the average value of all\noxygen atoms in unit cell as denoted by O (ave.). The La\nPDOS shows that 3 dstates are located farther away from\nthe Fermi level, suggesting that it will donate an extra9\nFIG. 9. PDOS of individual atoms in LaM where O (ave.)\ndenotes the average PDOS of oxygens present in the unit cell.electron to the system. Indeed, the electron transfers\nto Fe (2a), because it has an extra localized band (with\nopposite spin) just below the Fermi level as discussed in\nthe main text.\n1Kimura Tsuyoshi. Magnetoelectric hexaferrites. Annual\nReview of Condensed Matter Physics , 3(1):93{110, 2012.\n2J. M. D. Coey. Hard magnetic materials: A perspective.\nIEEE Transactions on Magnetics , 47(12):4671{4681, 2011.\n3Robert C. Pullar. Hexagonal ferrites: A review of the syn-\nthesis, properties and applications of hexaferrite ceramics.\nProgress in Materials Science , 57(7):1191{1334, 2012.\n4Global permanent magnets market share report, 2020-\n2027.\n5E. W. Gorter. Saturation magnetization of some ferrimag-\nnetic oxides with hexagonal crystal structures. Proceedings\nof the IEE - Part B: Radio and Electronic Engineering ,\n104(5):255{260, 1957.\n6F.K. Lotgering. Magnetic anisotropy and saturation of\nLaFe 12O19and some related compounds. Journal of\nPhysics and Chemistry of Solids , 35(12):1633 { 1639, 1974.\n7M. K upferling, P. Nov\u0013 ak, K. Kn\u0012 \u0010\u0014 zek, M. W. Pieper,\nR. Gr ossinger, G. Wiesinger, and M. Reissner. Mag-\nnetism in La substituted Sr hexaferrite. Journal of Applied\nPhysics , 97(10):10F309, 2005.\n8R. Gr ossinger, M. K upferling, M. Haas, H. M uller,\nG. Wiesinger, and C. Ritter. Magnetic anisotropy and\nmagnetostriction of LaFe 12O19.Journal of Magnetism and\nMagnetic Materials , 310(2, Part 3):2587 { 2589, 2007. Pro-\nceedings of the 17th International Conference on Mag-\nnetism.\n9D. Seifert, J. T\u0013 opfer, F. Langenhorst, J.-M. Le Breton,\nH. Chiron, and L. Lechevallier. Synthesis and magnetic\nproperties of La-substituted m-type Sr hexaferrites. Jour-\nnal of Magnetism and Magnetic Materials , 321(24):4045 {\n4051, 2009.10Michaela K upferling, Roland Gr ossinger, Martin W.\nPieper, G unter Wiesinger, Herwig Michor, Clemens Ritter,\nand Frank Kubel. Structural phase transition and mag-\nnetic anisotropy of la-substituted m-type Sr hexaferrite.\nPhys. Rev. B , 73:144408, Apr 2006.\n11Vojt \u0014 ech Chlan, Karel Kou\u0014 ril, Kate \u0014 rina Uli\u0014 cn\u0013 a, Helena\n\u0014St\u0014 ep\u0013 ankov\u0013 a, J org T opfer, and Daniela Seifert. Charge lo-\ncalization and magnetocrystalline anisotropy in La, Pr,\nand Nd substituted Sr hexaferrites. Phys. Rev. B ,\n92:125125, Sep 2015.\n12P. Nov\u0013 ak, K. Kn\u0013 \u0010\u0014 zek, M. K upferling, R. Gr ossinger, and\nM. W. Pieper. Magnetism of mixed valence (LaSr) hex-\naferrites. The European Physical Journal B - Condensed\nMatter and Complex Systems , 43(4):509{515, Feb 2005.\n13Jonathan M. Kindem, John G. Bartholomew, Philip J. T.\nWoodburn, Tian Zhong, Ioana Craiciu, Rufus L. Cone,\nCharles W. Thiel, and Andrei Faraon. Characterization of\n171Yb3+: YVO 4for photonic quantum technologies. Phys.\nRev. B , 98:024404, Jul 2018.\n14John G. Bartholomew, Jake Rochman, Tian Xie,\nJonathan M. Kindem, Andrei Ruskuc, Ioana Craiciu,\nMi Lei, and Andrei Faraon. On-chip coherent microwave-\nto-optical transduction mediated by ytterbium in YVO 4.\nNature Communications , 11(1):3266, Jun 2020.\n15D. Serrano, C. Deshmukh, S. Liu, A. Tallaire, A. Ferrier,\nH. de Riedmatten, and P. Goldner. Coherent optical and\nspin spectroscopy of nanoscale Pr3+: Y2O3.Phys. Rev.\nB, 100:144304, Oct 2019.\n16G. Kresse and J. Hafner. Ab initio molecular dynamics for\nliquid metals. Phys. Rev. B , 47:558{561, Jan 1993.\n17G. Kresse and D. Joubert. From ultrasoft pseudopotentials\nto the projector augmented-wave method. Phys. Rev. B ,10\n59:1758{1775, Jan 1999.\n18S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J.\nHumphreys, and A. P. Sutton. Electron-energy-loss spec-\ntra and the structural stability of nickel oxide: An\nLSDA+U study. Phys. Rev. B , 57:1505{1509, Jan 1998.\n19D. Hobbs, G. Kresse, and J. Hafner. Fully unconstrained\nnoncollinear magnetism within the projector augmented-\nwave method. Phys. Rev. B , 62:11556{11570, Nov 2000.\n20J. A. Kohn and D. W. Eckart. New hexagonal ferrite,\nestablishing a second structural series. Journal of Applied\nPhysics , 35(3):968{969, 1964.\n21Alan Ba~ nuelos-Fr\u0013 \u0010as, Gerardo Mart\u0013 \u0010nez-Guajardo, Leo\nAlvarado-Perea, L\u0013 azaro Canizalez-D\u0013 avalos, Facundo Ruiz,\nand Claudia Valero-Luna. Light absorption properties of\nmesoporous barium hexaferrite, BaFe 12O19.Materials Let-\nters, 252:239{243, 2019.\n22R Zimmermann, P Steiner, R Claessen, F Reinert,\nS H ufner, P Blaha, and P Dufek. Electronic structure\nof 3d-transition-metal oxides: on-site coulomb repulsion\nversus covalency. Journal of Physics: Condensed Matter ,\n11(7):1657{1682, jan 1999.\n23Churna Bhandari, Zoran S Popovi\u0013 c, and S Satpathy. Elec-\ntronic structure and optical properties of Sr 2IrO4under\nepitaxial strain. New Journal of Physics , 21(1):013036,\nJan 2019.\n24Churna Bhandari and S. Satpathy. E\u000bect of epitaxial strain\non the optical properties of NaOsO 3.Journal of Physics\nand Chemistry of Solids , 128:265{269, 2019. Spin-Orbit\nCoupled Materials.\n25Nobuko Fuchikami. Magnetic anisotropy of magneto-\nplumbite BaFe 12O19.Journal of the Physical Society ofJapan , 20(5):760{769, 1965.\n26H. A. Jahn, E. Teller, and Frederick George Donnan.\nStability of polyatomic molecules in degenerate electronic\nstates - i - orbital degeneracy. Proceedings of the Royal\nSociety of London. Series A - Mathematical and Physical\nSciences , 161(905):220{235, 1937.\n27Chuanjian Wu, Zhong Yu, Ke Sun, Jinlan Nie, Rongdi\nGuo, Hai Liu, Xiaona Jiang, and Zhongwen Lan. Cal-\nculation of exchange integrals and curie temperature for\nLa-substituted barium hexaferrites. Scienti\fc Reports ,\n6(1):36200, Oct 2016.\n28P. Nov\u0013 ak and J. Rusz. Exchange interactions in barium\nhexaferrite. Phys. Rev. B , 71:184433, May 2005.\n29Z.F. Zi, Y.P. Sun, X.B. Zhu, Z.R. Yang, J.M. dai,\nand W.H. Song. Structural and magnetic properties of\nsrfe12o19hexaferrite synthesized by a modi\fed chemical\nco-precipitation method. Journal of Magnetism and Mag-\nnetic Materials , 320(21):2746 { 2751, 2008.\n30Ralph Skomski. Simple Models of Magnetism . Oxford\nUniversity Press, Oxford University Press Inc., New York,\n2008.\n31L. Jahn and H. G. M uller. The coercivity of hard ferrite\nsingle crystals. physica status solidi (b) , 35(2):723{730,\n1969.\n32B. T. Shirk and W. R. Buessem. Temperature dependence\nof ms and k1 of BaFe 12O19and SrFe 12O19single crystals.\nJournal of Applied Physics , 40(3):1294{1296, 1969."    },    {        "title": "2106.12808v1.Large_scale_epitaxy_of_two_dimensional_van_der_Waals_room_temperature_ferromagnet_Fe5GeTe2.pdf",        "content": "1 \n Large -scale epitaxy of two -dimensional van der Waals \nroom -temperature ferromagnet Fe 5GeTe 2  \nAuthors  \nMário Ribeiro1*, Giulio Gentile1, Alain Marty1, Djordje  Dosenovic2, Hanako Okuno2, \nCéline Vergnaud1, Jean -François Jacquot3, Denis Jalabert2, Danilo Longo4, Philippe \nOhresser4, Ali Hallal1, Mairbek Chshiev1,5, Olivier Boulle1, Frédéric Bonell1*, Matthieu \nJamet1 \n1 University Grenoble Alpes, CEA, CNRS, IRIG -Spintec, F -38000 Grenoble, France  \n2 University Grenoble Alpes, CEA, IRIG -MEM, F -38000 Grenoble, France  \n3 University Grenoble Alpes, CEA, IRIG -SYMMES, F -38000 Grenoble, France  \n4 Synchrotron SOLEIL, L’Orme des Merisiers, Saint -Aubin, 91192 Gif -sur-Yvette, France  \n5 Institut Universitaire de France (IUF), 75231, Paris, France  \n* Corresponding authors: mario.oliv eiraribeiro@cea.fr ; frederic.bonell@cea.fr  \n  2 \n Abstract  \nIn recent years, two -dimensional van der Waals materials have emerged  as an important  \nplatform for the observation of long -range ferromagnetic order in atomically thin layers. \nAlthough heterostructures of such materials can be conceived to harness and couple a wide \nrange of magneto -optical and magneto -electrical properties, technologically  relevant \napplications require Curie temperatures at or above room -temperature and the ability to grow \nfilms over large areas.  Here we demonstrate the large -area growth of single -crystal ultrathin \nfilms of stoichiometric Fe 5GeTe 2 on an insulating substrat e using molecular beam epitaxy. \nMagnetic measuremen ts show the persistence of soft  ferromagnetism up to room temperature,  \nwith a Curie temperature of 293  K, and a weak out-of-plane magnetocrystalline anisotropy. \nSurface, chemical, and structural characteri zations confirm the layer -by-layer growth, 5:1:2 \nFe:Ge:Te stoichiometric elementary composition, and single crystalline character of the films.  \nMAIN  TEXT  \nIntroduction  \nUntil recently, long -range magnetic ordering was not thought possible in two -dimensional  \n(2D) systems. The Mermin -Wagner theorem states  that thermal fluctuations in a low -\ndimensionality isotropic Heisenberg model block the emergence of ferromagnetism  (FM)1. In \n2017, ferromagnetism was observed in  exfoliated flakes of the 2D van der Waals (vdW) \nmateria l Cr 2Ge2Te6 2, and later in  CrI 33. These materials are electrical insulators , exhibit  \nperpendi cular magnetic anisotro py, and have Curie temperature ( TC) well below room \ntemperature (RT). The prevailing understanding is  that long -range ferromagnetic order  in 2D \nvdW systems  survives  due to the spin -wave excitation gap opened by the strong intrinsic \nmagnetocrystalline aniso tropy (MCA). 2D vdW materials became a rich pla tform to explore \nlow-dimensionality magnetism, and in a few years  the list of 2D vdW FMs increased with the 3 \n addition of FePS 34, Fe 3GeTe 25, Fe 4GeTe 26, and Fe 5-δGeTe 27. Fe-Ge-Te (FGT) ternary \ncompounds in particular stood out due to their itinerant fer romagneti sm, which is of great \ninterest for  spintronic applications.  \nAmong the family of 2D vdW FMs, FGT  compounds host a variety of noticeable magnetic \neffects, namely  large anomalous - and planar topological Hall effect8,9, magnetic \nskyrmions10,11, Kondo lattices12, and giant tunneling magnetoresistance13. To the best of our \nknowledge, t he highest TC in pristine, bulk 2D vdW layered materials has been observed in \nFe5-δGeTe 2, with TC~310 K7. Fe 3GeTe 2 exhibits a  TC of 230 K14,15 in bulk form and 130 K5 for \na single layer. In these systems, the TC can be tuned via the use of extrinsic mechanisms such \nas ionic gating16, doping17, ion implantation18, proximity to topological insulators19, and \npattern ing methods20. Metastable states can be stabilized via temperature quenching and \ntemperature -cycles21. From a technological standpoint, high TC is among the magnetic \nproperties of highest interest for next -generation spintronic applications integrating 2D vdW \nFMs. Fe 5-δGeTe 2 stands out for its near RT TC and itinerant ferromagnetism. However, the \nmechanisms governing its magnetic properties are still unclear , with variable magnetic \nproperties ( TC and anisotropy)  being reported. M ost experimental realizations have been done \nusing Fe -deficient  bulk crystals  or micrometer -sized flakes exfoliated from them 7,21–23. The \nproperties of  Fe5GeTe 2 films with exact 5:1:2 stoichiometry remain unexplored.  Recent \nstudies pointed out  the role played by Fe vacancies and  the different Fe sublattices  on the \ncomplex range of magnetic behaviors observed6,7,21. Additionally, m agnetic critical exponents \nof bulk Fe 5-δGeTe 2 are close to those of a 3D Heisenberg  and 3D XY  system s23, remaining \nunclear if these  model s still hold in  ultrathin regimes.  \nWhile many of the demonstrations of vdW FMs materials relied on bulk crystals and on \nexfoliated flakes  in the 2D limit , growth by molecular  beam epitaxy (MBE) has  been pursued \nto expand the complexity of the designed heterostructures and large -area scalability19. MBE is 4 \n a well -established deposition technique for the realization of atomically thin single -crystal \nvdW materials and heteros tructures with sharp interfaces24. Any outlook on future \nheterostructures and device concepts based on 2D vdW FMs requires the large -area \nfabrication of high -quality ultrathin films24–28, for which MBE is well suited.  \nHere, we demonstrate the MBE growth of single -crystal thin films of stoichiometric \nFe5GeTe 2 on an insulating substrate, Al 2O3(001). In -situ reflection high -energy electron \ndiffraction (RHEED) measurements demonstrate  the layer-by-layer growth of single -\ncrystalline  films. Chemical and structural characterization using Rutherford backscattering \nspectroscopy (RBS), scanning transmission electron microscopy (STEM), and X -ray \ndiffraction (XRD) , confirm t he 5:1:2 Fe:Ge:Te  stoichiometry  and rhombohedral crystal \nstructure. Superconducting quantum interference device (SQUID)  and X -ray magnetic \ncircular dichroism  measurements (XMCD) show a large magnetization saturation  (MS) of 644 \nkA/m at 10 K (1.95  µB/Fe), TC of 29 3 K, and a  magnetic critical exponent β of 0.2 93, which  \napproaches the universal window for critical -exponents in 2D systems.  \n \nResults   \n \nThe MBE growth of Fe 5GeTe 2 was carried out in a home -built UHV deposition system \nwith a base pressure in the low 10−10mbar (see  Methods for further information). The \nAl2O3(001) substrate s were first annealed in air  at 1000 ºC in a tube furnace to promote the \nformation of a terrace -and-step smooth surface, and again annealed in the UHV chamber at \n800 ºC for 20 minutes prior to the deposition. Ge and Te were co -deposited using Knudsen \neffusion and thermal cracker cell s, respectively, and Fe using e -beam evaporation. The \nsubstrate temperature for the growth of Fe5GeTe 2 was set to 350 ºC, t he flux ratio of Fe:Ge \nwas kept to the stoichiometric 5:1 , and the Te flux on overpressure  (approximately twice the 5 \n stoichiometric fl ux). After the deposition, the films were annealed for 20 minutes at 550 ºC \nunder Te flux , and then capped with a 3 nm -thick Al film at RT. The growth of Fe 5GeTe 2 on \nAl2O3(001) was monitored in-situ using RHEED . We report mainly on the characterization of \n12-nm-thick Fe 5GeTe 2 films  (1 monolayer  ~ 0.977 nm) . In Figure 1A, the top left side panel  \nshow s well-defined RHEED  patterns of the Al 2O3(001) surface with the electron -beam  (e-\nbeam) aligned along  [110], while the bottom left panel shows the diffraction pattern for the \nsame direction after the growth of 12 nm of Fe5GeTe 2.  Similarly, t he top and bottom  right -\nside panels show the diffraction patterns of the Al 2O3(001) surface before and after the growth \nof 12 Fe5GeTe 2 layers, respectively, with the e-beam aligned along [210] (see Supplementary  \nFigure  1 for RHEED snapshots during the  deposition and annealing of 12 -nm and 2 -nm-thick \nsamples).  \nDespite the la rge lattice misfit with Al 2O3(001)(~ -15%), t he sharp  streaks and anisotropic \nRHEED patterns demonstrate  the single crystalline character  of Fe 5GeTe 2, with c-axis \nperpendicular to the surface. The two az imuths are aligned with those of the Al 2O3(001) \nsurface, showing the Al2O3(001)[100]/ Fe5GeTe 2(001)[100]  epitaxial relationship .  The real -\ntime monitoring of the RHEED intensity  of the 1st order diffraction streaks  allowed for the \nobservation of periodical intensity oscillations characteristic of a layer -by-layer growth mode \n(Figure 1B). By indexing the successive os cillations as the formation of  a single  layer, the rat e \nof growth was estimated at 1.69  monolayers per minute of deposition. Atomic force \nmicroscopy  (AFM)  scans of the surface of 12-nm-thick films  not capped with Al, taken within \nminutes of exposing the surface to the environment, show a smooth surface with root -mean -\nsquare rou ghness  (RMS)  of 0.5 nm , without clearly defined terraces , but with atomic steps \ncorresponding to those of one or multiple monolayers  (Figure 1C) . \nTo gather further information on the crystallographic structure, domain size, mosaic \nspread , and lattice parameter distribution of the films dep osited, we performed in -plane 6 \n (grazing incidence) and out -of-plane XRD measurements (see Methods for further details).  \nFe5GeTe 2 is a highly anisotropic layered vdW ferromagnet with trigonal crystal system. \nTypical lattice cons tants reported in literature for  bulk Fe 5-δGeTe 2 are a ≈ 4.04 Å and c ≈ \n29.19 Å7,21,29. Unit cell s are typically represented with  three stacked Fe5GeTe 2 layers, with \nvdW gaps between adjacent Te planes. The complex atomic arrangement, the common \npresence of stacking faults and vacancy -induced disorder makes the precise determination of \nthe crystallographic structure an open issue7,21,22,29. In Figure 2A, we depict the unit cell and \nthree Fe 5GeTe 2 layers  viewed along the [ 110] direction , according to the crystallographic \nmodel  of ref.  22 .The model considers Fe -deficiency via fractional  occupation probabil ities of \nthe Fe sites, and includes a split  Ge and Fe 1 site with half -occupation probability . Figure 2B \nshows the out -of-plane θ/2θ diffraction scan of a 12 -nm-thick film, where we index the \nAl2O3(00l) and Fe 5GeTe 2(00l) Bragg peaks. The determined unit cell lattice constant  c ≈ 29.3 \nÅ is in goo d agreement with previous reports  (thickness of 1 ML = c/3)7,21,29. The fring es \naround the Bragg peaks of  Fe5GeTe 2(009) indicate  low surface and interface roughness  (also \nconfirmed by the RMS of 0.5 nm measured by AFM in Figure 1C), and allows for the \nestimation of a film thickness of  12.5 ± 0.3 nm, in strong agreement with the expected growth \nof 12 layers determi ned from the RHEED oscillations (See Sup plementary  Figure 2  for the \nXRD analysis of a 2 -nm-thick film).  In Figure 2C, we show in -plane radial scans θ///2θ// that \nwere acquired  along the ( hh0) and (0 h0) reciprocal directions, 30º apart from each other. The \nfull width at half maxi mum (FWHM) is also displayed above the peaks. From these scans, we \nobtain an in -plane lattice parameter a = 4.068  Å, expanded by roughly 0.7% compared to the \nreference value of 4.04  Å. The mutually exclusive presence of the ( hh0) and (0 h0) peaks on \neach radial scan indicate the absence of domains rota ted by at least 30º. This is further \nconfirmed in Figure 2D, which  shows the azimuthal dependence  of the intensity of the  Bragg \npeaks  identified in the radial scans  (grey curves) . The baseline of the azimuthal scans  is the 7 \n sum of the instrumental background  and of  the signal from misaligned crystallites  with \nisotropic distribution  of orientations . The observed correspondence between  the azimuthal  and \nradia l base lines impl ies that there is no isotropic component in the crystallites angular \ndistribution . The film is single crystalline with the expected 60º periodicity of the basal plane \nof the crystal lattice . Noticeably, the FWHM for the Bragg peaks is substantially larger. From \nthe study of the momentum transfer for each radial and azimuthal Bragg diffraction peak and \nits dispersion , one can estimate the domain size D, lattice parameter distribution Δa/a, and in-\nplane m osaic spread  Δξ of the film31,32 (see Supplementary Note 1  and Supplementary Figure \n3 for details on  the calculations) . For the 12 -nm-thick f ilm, D was determined to be  33 nm, \nwhich must be considered as a lower bound for the domain size due to the instrumental \nlimited resolution,  Δa/a ≈0.66 %, and Δξ ≈ 3.34 º, values  that corroborate the single -crystalline \ncharacter  of the deposited films.  \nIn order to take a closer look at disorder and stacking of the Fe 5GeTe 2 ultrathin films, we \nrecorded atomic -resolution real -space images by STEM (see Methods for further \ninformation). Figure 3A shows a high -angle annular  dark-field (HAADF) cross -section image \nalong  [110]. The atomically flat Al 2O3 promoted the epitaxy of well -oriented layers that are \nuniform over large areas, parallel to each other, and with no perceivable  rotated domains . At \nthe interface between the  two materials, there is a thin  amorphized gap of 6 Å, after which the \nfirst Fe 5GeTe 2 layer develops.  \nChemical analysis using RBS spectroscopy confirms the nominal 5:1:2 stoichiometry of \nthe Fe:Ge:Te elementary composition of the film (inset of Figure 3A). Energy dispersive X -\nray spectroscopy (EDX) shows a homogenous distribution of each element in the film ( Figure \n3B). In Figure 3C, Z contrast enhanced HAADF over a smaller section of the film yields a site \ndistribution with clear indication of the presence of heavier atoms at the edges of the layers. \nElement resolved EDX over two layers shows the clear presence of Fe and Ge in between Te 8 \n sites. Fe shows a broader dependence on position than Ge, as expected from  its distribution in \nthe cell .  The presence of the Fe1 atom is not clearly resolved in our me asurements, but the \nsymmetry of the images points to a 50% occupation probability in the two possible sites, as in \nthe model of ref. 22. In order to confirm it, we performed simulations of Z-contrast HAADF \nimages for several occupation probabilities  (Figure 3D). For a 100%  occupation probability, \nthe simulation indicates that an STEM scan  should be able to  clearly  resolve the site  occupied \nby the Fe1 atom, while at 0% the background strongly contrasts with the atomic sites . At \n50%,  we observe a tail connecting the  inner Fe2 atoms to Te. This reproduces well the \nexperimental result,  and strongly suggest s that a crystallographic model with a Fe1 half-split \nsite applies to our epitaxial films . \nThe magnetic properties of the films were characterized using SQUID and XMC D (see \nMaterials and Methods for further information). In Figure 4A and Figure 4B we plot \nisothermal magnetic hysteresis loops  at different temperatures  with the magnetic field applied \nin the  ab basal plane , and  along  the c-axis of the layers, respectively.  The da ta have  been \ncorrected by subtracting the  magnetic background observed at temperatures well above TC, \nand a residual slope due to a paramagnetic contribution from the substrate below 50 K.  The \nsame slope was subtracted from the  in-plane and out -of-plane curves. This treatment assumes \nthat the hig h temperature magnetic signal originates from the substrate, which is justified by \nour XMCD and SQUID measurements, as explained in the following . The insets show the \nraw magnetization curves  and the considered background in grey . The in -plane  hysteresis  \nloops show low coercivity  (~11 mT)  and saturation fields  (~200 mT) . The saturation \nmagnetization  averaged over in -plane and out -of-plane measurements amounts to Ms = (644 ± \n54) kA/m  at 10 K , corresponding to a magnetic moment m = (1.95 ± 0.12) µB/Fe.  These \nvalues are in-line with those  previously reported for  Fe-deficient Fe 5-δGeTe 2 (δ = 0.1  – 0.3) 9 \n films  exfoliated  from bulk crystals 21,33. Perpendicular to the surface, the saturation field  \nlargely  increases, indicating  an easy -plane magnetic anisotropy.  \nIn Figure 4C, the temperature dependence of the magnetization remane nce ( Mr) closely  \nfollows  the scaling behavior of an ordered ferromagnetic system, 𝑀r∝(𝑇C−𝑇\n𝑇𝐶)𝛽\n, yielding TC \n≈ 293  K and β ≈ 0.294 . This behavior is in striking contrast to the complex temperature \ndependences observed in bulk and Fe -deficient Fe5-δGeTe 2 films7,21, suggesting that MBE -\ngrown films present a higher structural and chemical homogeneity. The Mr can be reasonably \nfitted with 3D models (see Supplementary Figure 4). However , our determined β significantly \ndeviates from the  corresponding  magnetic critical exponent and from the one observed in  \nbulk-crystals β ≈ 0.34623. With a value of β ≈ 0.294 , our results approaches the window for \ncritical exponents of two -dimensional systems34, and possibly indica tes a characteristic 2D \nbehavior in ultrathin films. Figure 4D shows  the normalize d in-plane and out -of-plane \nisoth ermal magnetic loops  at 50  K, 100  K, 200 K, and 300 K. W e determine the  effect ive \nanisotropy  Keff from the  area enclosed  between the curves. The uniaxial anisotropy Ku is \ndeduced from  Keff and the measured MS, with the  relation  𝐾𝑒𝑓𝑓=𝐾𝑢+𝐾𝑑=𝐾𝑢−\n(µ02⁄)𝑀𝑆2, where Kd is the shape anisotropy and µ0 the vacuum permeability.  The \ntemperature  dependence of Keff, Ku, and Kd is displayed on Figure 4E. We estimate a small \npositive uniaxial  anisotropy Ku ≈ 0.09 J/cm3 that favor s an easy c-axis, but remains weaker \nthan shape anisotropy at all temperatures . Figure 4F shows  the magnetic moment s of this \nsample (S1)  determined  by SQUID and  compares it to the magnetic moment of a second \nsample (S2) measured by both SQUID and XMCD.  Sample S2 was capped with  5 nm of  Te \ninstead of Al . For sample S2, t he excellent agreem ent between SQUID and XMCD confirms \nthe validity of the background  correction of SQUID measurements . The smaller magnetic \nmoment of sample S2 compared to  that of  sample S1 (whose 5:1:2 Fe:Ge:Te com position was \nascertained by RBS , in Figure 3A) might originate from Fe deficiency.  In Figure 4G, H we 10 \n detail the XMCD characterization of sample S2, and show the XMCD spectra at  the Fe and \nGe L 2,3 edges  at 4 K  (see Methods for further det ails). The measurements were performed in \ntotal electron yield mode , with X -ray incidence normal to the surface ( α = 90°) , and the ±5 T \nmagnetic field parallel to the beam direction. Additional measurements were carried out at α = \n30° (not shown). We applie d the conventional sum rules analysis35,36 to the Fe absorption \nsignal in order to determine the effective spin ( 𝑚𝑆𝑒𝑓𝑓,𝛼=𝑚𝑆+𝑚𝐷𝛼) and orbital ( 𝑚𝐿𝛼) \nmoments  (see Supplementary Note 2  for further details on the determination of the Fe \nmagnetic moments with XMCD sum rules) . From those, we derive the isotropic spin ( mS) and \norbital ( mL) moments, as well as the intra -atomic dipole moment anisotropy Δ𝑚𝐷=𝑚𝐷⊥−\n𝑚𝐷∥ and the orbital moment anisotropy Δ𝑚𝐿=𝑚𝐿⊥−𝑚𝐿∥ (⊥ and ∥ denote the out -of-plane and \nin-plane components , respectively ). We obtained mS = 1.54  µB, mL = 0.08  µB, ΔmD = 0.21 µB \nand ΔmL = -0.01 µB. The total magnetic moment amounts to mS + mL = 1.62  µB, in excellent \nagreement with the theoretical value  determined from first principles  reported in Table 1 (see \nMethods for furth er details) . \nThe small mL and ΔmL are co nsistent with the weak out -of-plane uniaxial magnetic \nanisotropy deduced by SQUID measurements. We also observed a small polarization of Ge \natoms, in agreement with ref.  33 and with our ab initio  results (Table 1).  As depicted in Figure \n4H, both 2 p→4d (ΔL=+1) and 2 p→4s (ΔL=-1) transitions present a finite XMCD signal. The \nsign of the XMCD for 4 d (4s) final states is equal (opposite) to the one of Fe. Given the \nopposite  sign of ΔL for these transitions, we deduce that t he magnetic moment of both 4 d and \n4s bands is  opposite to the Fe one. In ref. 33, a sizeable magnetic moment was also observed \non Te atoms, but we could not confirm it in our film due to the use of a Te cappi ng layer.  \nNote that we neglected th e Ge and Te contributions to determine the Fe magnetic mom ent by \nSQUID. According to Table 1, this assumptions leads to an underestimation of the Fe \nmagnetic moment by only 2%.  11 \n Discussion  \nIn summary,  we demonstrate d the large -scale growth of few -layer, single -crystal Fe 5GeTe 2 \non Al2O3(001)  using  molecular beam epitaxy , and provide d a thorough structural and \nmagnetic characterization . We show ed that the MBE technique yiel ds Fe5GeTe2 layers with \nhigh structural quality, layer -by-layer growth, vi rtually no F e deficiency, and room \ntemperature ferromagnetism. The  ultrathin films exhibit soft ferromagnetism,  a large \nmagnetization of about 644 kA/m at 10 K, easy -plane magnetizat ion, with  a weak  c-axis \nmagnetocrystalline anisotropy  Ku ~ 0.09 J/cm3. The deviation of the magnetic critical \nbehavior from a 3D Heisenberg /3D XY  model  to one close to that of  2D systems , with β = \n0.294 might be  a fingerprint of  the effects of reduced dimensionality on the physical \nproperties of the films.   STEM images and simulations clearly indicate  a crystallographic \nstructure with half -split o ccupation probability of the Fe1  atomic site.  \nOur work highlights the role of  MBE a s a key tool for the fine tailoring of 2D vdW FMs \nsingle crystals. In the particular case of Fe 5GeTe 2, we have grown a benchmark material with \nexact stoichiometry and well -defined crystalline structure to well -establish the magnetic \nproperties of this nove l vdW ferromagnetic material. It will serve as a platform to further \nunveil the complex dependence of the magnetic properties on the film composition and crystal \nstructure, and as a stepping stone towards the integration in large -scale vdW multilayers for \nspintronics .   \n \nMethods  \nMBE  \n1×1 cm2 Al2O3(001) substrates were cleaned prior to the tube furnace annealing with a \nstandard  isopropanol/acetone ultrasonic bath. For the growth of Fe 5GeTe 2, high purity Ge 12 \n (99.999%) was evaporated at 1060 ºC, while high -purity Te (99.999%) was evaporated at 330 \nºC and cracked at 1000 º C. Fe (99.99%) was evaporated following standard e -beam \nprocedures. Fe and Te deposition rates were  calibrated by AFM  on pre-patterned sample s. \nThe Ge deposition rate was calibrated with homoepitaxial R HEED oscillations on Ge(111). \nThe Fe flux rate was  monitored during the deposition using a quartz microbalance, while the \nGe and Te flux rates were monitored before and after the depositions with  a UHV pressure \ngauge.  \nXRD  \nIn-plane XRD measurements were performed with a SmartLab Rigaku diffractometer \noperating at 45 kV and 200 mA.  Collimators with a resolution of 0.5º were used both in the \nsource and the detector. The measurements were done wi th a copper rotating anode beam tube \n(Kα = 1.54 Å)  at a grazing incident angle of 0.4º. The out -of-plane XRD measurements were  \nperformed using a Panalytical Empyrean diffractometer  operated at 35 kV and 50 mA, with a \ncobalt source, ( Kα = 1.79 Å)   A PIXcel -3D detector allowed  a resolution of 0.02° per pixel , in \ncombination with a divergence slit of 0.125º .  \nAFM  \nThe film to pography was characterized with  a Bruker D imension -Icon microscope using  a \nscanasyst -air cantilever, operated in ScanAsyst mode , and under ambient conditions . The \nroot-mean -square roughness was evaluated on 1×1 µm2 areas.  \nSTEM  \nSTEM measurements were performed using a Cs -corrected FEI Themis at 200 keV. \nHAADF -STEM images were acquired using a convergence semi -angle of 20 mrad and \ncollecting scattering from >60 mrad. EDX was performed for elemental mapping using a \nBruker EDX system consisting of four silicon drift detectors in the Themis microscope. 13 \n STEM specimens were prepared by the focused ion beam lift -out technique using a  Zeiss \nDualBeam Cross Beam uSTE750 at 30 kV.  \nHAADF -STEM simulations were carried out using mu ltislice simulation software , namely  \nJEMS37,  and µSTEM38  including the experimental conditions use d for the image acquisition. \nThe model structures were  constructed based on the crystal  information given in refs.22,29. \nRBS  \nRutherford backscattering was done using  an ion -beam of  4He+ ions with 2MeV , at a \nchamber pressure  of 10-6 mbar, with a 160º angle between the incident beam and the detec tor, \nand an angle of incidence of 75º.   \nSQUID   \nSQUID measurements were performed using a Quantum Design magnetic property \nmeasurement system MPMS®3  following standard procedures . Magnetic field sweeps were \nmade in no -overshoot, persistent mode. The temperature sweeps were made at a rate of 5 K \nper minute. The estimated magnetic moments were determined from an average of two scans. \nFor magnetization remanence me asurements , the films were field -cooled from RT to 10 K \nwith an external field of 2 T a pplied in -plane , and the superconducting magnet quenched \nbefore starting the temperature dependence.  \nXMCD  \nThe XMCD measurements were performed on the DEIMOS beamline39 of synchr otron \nSOLEIL. A XAS spectrum i s obtained by continuously moving the energy of the \nmonochromator over the energy range of interest while synchronizing the energy and \npolarization of the undulator to the en ergy of the monochromator40. The XMCD sp ectrum is \nobtained from four measurements, where  both the circular helicity and the direction  of the \napplied magnetic field are flipped.  14 \n Ab initio  calculations  \nFirst-principle calculations were performed using the Vienna Ab -initio Simulation Package \n(VASP)41,42, employing projector augmented wave (PAW)43,44 for potentials with the plane \nwave energy cutoff in all the calculations set to 550 eV, and the generalized gradient \napproximation (GGA)45 for the exchange -correlation energy. As a first step, ionic relaxation \nof Fe 5GeTe 2 bulk stru cture was performed with k-point grids of 7×7×1 and 0.01 eV/Å force \ntolerance during optimization of the atomic positions. The in -plane (out -of-plane) lattice \nparameter was fixed to 4.07 (29.3) Å. Next, a self -consistent calculation with 21×21×3 k-\npoint gr ids was realized to obtain the charge density wave and the atomic spin moments. \nFinally, the orbital moments were obtained by performing a non -self-consistent calculatio n \nwith spin -orbit coupling  included with the quantization axis set along [001].  \n \nAcknowledgments  \nThe MBE reactors were funded by the project Minatec LABS 2018 N°2018 AURA P3 and \nthe project EPI2D of the University Grenoble Alpes IDEX . The authors acknowledge the \nfinancial support from the UGA IDEX IRS/EVASPIN, the ANR projects ELMAX (A NR-20-\nCE24 -0015) and MAGICVALLEY (ANR -18-CE24 -0007), and the King Abdullah University \nof Science and Technology under grant number OSR -2018 -CRG7 -3717 , the DARPA TEE  \nprogram through Grant MIPR# HR0011831554 from the DOI, and the LANEF framework \n(ANR -10-LABX -51-01) for its support with mutualized infrastructure.  \n \nAuthor contributions:   \nThe s ample preparation , growth,  and the SQUID measurements were done by MR, G G, and  \nFB. XRD was performed by A M. The AFM scans were made by GG.  The STEM cross -\nsections and simu lations  were done by HO and DD . XMCD was done by FB, GG, D L, and 15 \n PO. RBS was perform by CV, J -FJ and DJ.  AH and MC performed the ab initio simulations. \nThe study was supervised by FB, O B, and M J. MR and FB  wrote the paper, and all authors \ncontributed to the discussion of the results . \n \nCompeting interests:  \n Authors declare that they have no competing interests.   \n  16 \n References  \n1. Mermin, N. D. & Wagner, H. Absence of Ferromagnetism or Antiferromagnetism in One - \nor Two -Dimensional Isotropic Heisenberg Models. Phys. Rev. Lett.  17, 1133 –1136 \n(1966).  \n2. Gong, C. et al.  Discovery of intrinsic ferromagnetism in two -dimensional van der Wa als \ncrystals. Nature  546, 265 –269 (2017).  \n3. Huang, B. et al.  Layer -dependent ferromagnetism in a van der Waals crystal down to the \nmonolayer limit. Nature  546, 270 –273 (2017).  \n4. Wang, X. et al.  Raman spectroscopy of atomically thin two -dimensional magnet ic iron \nphosphorus trisulfide (FePS3) crystals. 2D Mater.  3, 031009 (2016).  \n5. Fei, Z. et al.  Two-dimensional itinerant ferromagnetism in atomically thin Fe3GeTe2. \nNature Materials  17, 778 –782 (2018).  \n6. Seo, J. et al.  Nearly room temperature ferromagnetis m in a magnetic metal -rich van der \nWaals metal. Science Advances  6, eaay8912 (2020).  \n7. May, A. F. et al.  Ferromagnetism Near Room Temperature in the Cleavable van der \nWaals Crystal Fe5GeTe2. ACS Nano  13, 4436 –4442 (2019).  \n8. Kim, K. et al.  Large anomalous Hall current induced by topological nodal lines in a \nferromagnetic van der Waals semimetal. Nature Materials  17, 794 –799 (2018).  \n9. You, Y. et al.  Angular dependence of the topological Hall effect in the uniaxial van der \nWaals ferromagnet Fe3GeTe2. Phys. Rev. B  100, 134441 (2019).  \n10. Ding, B. et al.  Observation of Magnetic Skyrmion Bubbles in a van der Waals \nFerromagnet Fe3GeTe2. Nano Lett.  20, 868 –873 (2020).  \n11. Park, T. -E. et al.  Neel -type skyrmions and their current -induced motion in v an der Waals \nferromagnet -based heterostructures. Phys. Rev. B  103, 104410 (2021).  17 \n 12. Zhang, Y. et al.  Emergence of Kondo lattice behavior in a van der Waals itinerant \nferromagnet, Fe3GeTe2. Science Advances  4, eaao6791 (2018).  \n13. Wang, Z. et al.  Tunnelin g Spin Valves Based on Fe3GeTe2/hBN/Fe3GeTe2 van der \nWaals Heterostructures. Nano Lett.  18, 4303 –4308 (2018).  \n14. Deiseroth, H. -J., Aleksandrov, K., Reiner, C., Kienle, L. & Kremer, R. K. Fe3GeTe2 and \nNi3GeTe2 – Two New Layered Transition -Metal Compounds: Crystal Structures, \nHRTEM Investigations, and Magnetic and Electrical Properties. European Journal of \nInorganic Chemistry  2006 , 1561 –1567 (2006).  \n15. Chen, B. et al.  Magnetic Properties of Layered Itinerant Electron Ferromagnet Fe3GeTe2. \nJ. Phys. Soc. Jpn.  82, 124711 (2013).  \n16. Deng, Y. et al.  Gate -tunable room -temperature ferromagnetism in two -dimensional \nFe3GeTe2. Nature  563, 94–99 (2018).  \n17. Tian, C. et al.  Tunable magnetic properties in van der Waals crystals (Fe1−xCox)5GeTe2. \nAppl. Phys. Lett.  116, 202402 (2020).  \n18. Yang, M. et al.  Highly Enhanced Curie Temperature in Ga -Implanted Fe3GeTe2 van der \nWaals Material. Advanced Quantum Technologies  3, 2000017 (2020).  \n19. Wang, H. et al.  Above Room -Temperature Ferromagnetism in Wafer -Scale Two -\nDimensional v an der Waals Fe3GeTe2 Tailored by a Topological Insulator. ACS Nano  \n14, 10045 –10053 (2020).  \n20. Li, Q. et al.  Patterning -Induced Ferromagnetism of Fe3GeTe2 van der Waals Materials \nbeyond Room Temperature. Nano Lett.  18, 5974 –5980 (2018).  \n21. May, A. F., Br idges, C. A. & McGuire, M. A. Physical properties and thermal stability of \nFe5GeTe2 single crystals. Phys. Rev. Materials  3, 104401 (2019).  18 \n 22. May, A. F., Du, M. -H., Cooper, V. R. & McGuire, M. A. Tuning magnetic order in the \nvan der Waals metal Fe5GeTe2 by cobalt substitution. Phys. Rev. Materials  4, 074008 \n(2020).  \n23. Li, Z. et al.  Magnetic critical behavior of the van der Waals Fe5GeTe2 crystal with near \nroom temperature ferromagnetism. Scientific Reports  10, 15345 (2020).  \n24. Liu, S. et al.  Wafer -scale two -dimensional ferromagnetic Fe 3 GeTe 2 thin films grown \nby molecular beam epitaxy. npj 2D Materials and Applications  1, 1–7 (2017).  \n25. Burch, K. S., Mandrus, D. & Park, J. -G. Magnetism in two -dimensional van der Waals \nmaterials. Nature  563, 47–52 (2018).  \n26. Li, P. et al.  Single -layer CrI3 grown by molecular beam epitaxy. Science Bulletin  65, \n1064 –1071 (2020).  \n27. Bedoya -Pinto, A. et al.  Intrinsic 2D -XY ferromagnetism in a van der Waals monolayer. \narXiv:2006.07605 [cond -mat]  (2021).  \n28. Li, B. et al.  Van der Waals epitaxial growth of air -stable CrSe2 nanosheets with \nthickness -tunable magnetic order. Nature Materials  1–8 (2021) doi:10.1038/s41563 -021-\n00927 -2. \n29. Stahl, J., Shlaen, E. & Johrendt, D. The van der Waals Ferromagnets Fe5 –δGeTe2 a nd \nFe5–δ–xNixGeTe2 – Crystal Structure, Stacking Faults, and Magnetic Properties. \nZeitschrift für anorganische und allgemeine Chemie  644, 1923 –1929 (2018).  \n30. Momma, K. & Izumi, F. VESTA 3 for three -dimensional visualization of crystal, \nvolumetric and mor phology data. J Appl Cryst  44, 1272 –1276 (2011).  \n31. Renaud, G., Barbier, A. & Robach, O. Growth, structure, and morphology of the \nPd/MgO(001) interface: Epitaxial site and interfacial distance. Phys. Rev. B  60, 5872 –\n5882 (1999).  19 \n 32. Guillet, T. et al.  Magnetotransport in Bi2Se3 thin films epitaxially grown on Ge(111). \nAIP Advances  8, 115125 (2018).  \n33. Yamagami, K. et al.  Itinerant ferromagnetism mediated by giant spin polarization of the \nmetallic ligand band in the van der Waals magnet Fe5GeTe2. Phys. Rev . B 103, L060403 \n(2021).  \n34. Taroni, A., Bramwell, S. T. & Holdsworth, P. C. W. Universal window for two -\ndimensional critical exponents. J. Phys.: Condens. Matter  20, 275233 (2008).  \n35. Chen, C. T. et al.  Experimental Confirmation of the X -Ray Magnetic Cir cular Dichroism \nSum Rules for Iron and Cobalt. Phys. Rev. Lett.  75, 152 –155 (1995).  \n36. Stöhr, J. Exploring the microscopic origin of magnetic anisotropies with X -ray magnetic \ncircular dichroism (XMCD) spectroscopy. Journal of Magnetism and Magnetic Materi als \n200, 470 –497 (1999).  \n37. JEMS. https://www.jems -swiss.ch/.  \n38. Allen, L. J., D ׳Alfonso, A. J. & Findlay, S. D. Modelling the inelastic scattering of fast \nelectrons. Ultramicroscopy  151, 11–22 (2015).  \n39. Ohresser, P. et al.  DEIMOS: A beamline dedicated  to dichroism measurements in the \n350–2500 eV energy range. Review of Scientific Instruments  85, 013106 (2014).  \n40. Joly, L. et al.  Fast continuous energy scan with dynamic coupling of the monochromator \nand undulator at the DEIMOS beamline. J Synchrotron R ad 21, 502 –506 (2014).  \n41. Kresse, G. & Furthmüller, J. Efficiency of ab -initio total energy calculations for metals \nand semiconductors using a plane -wave basis set. Computational Materials Science  6, \n15–50 (1996).  \n42. Kresse, G. & Furthmüller, J. Efficien t iterative schemes for ab initio total -energy \ncalculations using a plane -wave basis set. Phys. Rev. B  54, 11169 –11186 (1996).  \n43. Blöchl, P. E. Projector augmented -wave method. Phys. Rev. B  50, 17953 –17979 (1994).  20 \n 44. Kresse, G. & Joubert, D. From ultraso ft pseudopotentials to the projector augmented -\nwave method. Phys. Rev. B  59, 1758 –1775 (1999).  \n45. Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized Gradient Approximation Made \nSimple. Phys. Rev. Lett.  77, 3865 –3868 (1996).  \n \nData and materials availability:   \nAll data needed to evaluate t he conclusions in the paper are present in the paper and/or the \nSupplementary Materials. Additional data related to this paper may be requested from the \nauthors.  \n  21 \n FIGURES  \n \nFigure 1 Surface characterization of Fe 5GeTe 2. (A) RHEED patterns of the surface \nbefore and after the deposition of a 12 -nm thick film of Fe 5GeTe 2 along the [110] and [210] \nazimuths. ( B) RHEED intensity oscillations of the (010) streak. The solid line is a moving \naverage of the raw scatter data. Monolayer number identified on each oscilation. ( C) Atomic \nforce microscopy image of a 12 -nm-thick film. Inset: Height profile showing resolved \nmonolayer steps (1 ML ~ 0.977 nm)  \n  \n22 \n  \nFigure 2 Structur al characterization of Fe 5GeTe 2 by X-ray diffraction.  (A) Unit cell \nprojected along [110], with two -halfs of an FGT layer separated by the vdW gap between \nadjacent Te sublayers. The right -side image shows the same projection for several repetitions \nof the unit cell. The occupation probability is represented by half -filled solid balls. Images \ngenerated using VESTA software30 for the crystallographic model of ref. 22 .  (B) Out -of-plane \nθ/2θ XRD scan of the films, showing Al 2O3(00l) and Fe 5GeTe 2 (00l) peaks. Inset: zoom on the \n(009) Bragg peak to show the fringes used to estimate the film thickness. ( C) In-plane radial \nXRD scans along the (110) reciprocal direction (R), and (010), (R+30º). Each diffraction \npeak is labelled with the full widh at h alf maximum.( D) In-plane azimuthal XRD scan over the \nfamilies {110} and {300} showing the 6 -fold symmetry of the crystal. Grey lines show the \ncorresponding radial peak.  \n  \n23 \n  \nFigure 3 Transmission electron microscopy of Fe 5GeTe 2. (A) High -angle annular dark -\nfield (HAADF) cross section image of the 12 nm -thick film along [110]. Left inset: High -pass \nfiltered image of the layers. Top inset: Rutherford backscattering spectroscopy showing \nstoichiometric 5:1:2 Fe:Ge:Te composition. Bottom inset: Zoomed view of the interface \nbetween Al 2O3 and Fe 5GeTe 2. (B) Element distribution across the layers, determined using \nenergy dispersive X -ray spectroscopy (EDX). (C) Atomic number contrast enhanced HAADF \nimage of the Fe 5GeTe 2 layers. The left plot s hows an EDX scan over two layers, selective to \nTe, Ge, and Fe. (D) Comparison between experimental (top left panel) and simulated (top \nright panel) Z -constrast enhanced HAADF images. Bottom panels are zooms on the Fe1 site. \nThe simulations consider differe nt occupation probabilities of the Fe1 site.  \n  \n24 \n  \nFigure 4 Magnetic characterization of 12 -nm thick Fe 5GeTe 2 using SQUID and XMCD . \n(A) Isothermal magnetic hysteresis loops with the external magnetic field H in the ab basal \nplane for  different temperatures. The magnetic background at temperatures well above T C has \nbeen subtracted for all measurements.Inset: Raw data highlighting the magnetic background \nconsidered. (B) Same as (A), with H parallel to the c -axis. (C) Magnetization reman ence as a \nfunction of temperature. Fitting to the scaling behavior of a magnetic system in an ordered \nstate yields T C ≈ 293 K and β≈0.294. The inset shows the scaling behavior of the remanence \nfor different 2D and 3D magnetic critical exponents (see Supple mentary Figure 3). (D) \n25 \n Normalized in -plane and out -of-plane magnetic hysteresis loops  for different temperatures. \n(E) Effective magnetic anisotropy, K eff, shape anisotropy,K d, and uniaxial anisotropy, K u, as a \nfunction of temperature.(F) Magnetic moment p er Fe atom as a function of temperature \ndetermined from SQUID and XMCD for two 12 -nm thick samples (S1 and S2) (G) Normalized \nFe L 2,3-edge X -ray absorption spectroscopy (XAS) (µ++µ-) and XMCD (µ+-µ-) spectra. µ+(-)is \nthe absorption for parallel (antiparallel) alignment of the light helicity and the sample \nmagnetization.  (H) Normalized Ge L 2,3-edge XAS and XMCD.  \n  26 \n TABLES  \nTable 1 Spin (m S), orbital (m L) and total (m S+m L) magnetic moment per atom in Fe 5GeTe 2, \nestimated from first principles calculations. A single Fe1 site was considered to have 100% \noccupancy (see text).The last two lines give the calculated averaged Fe magnetic moments \nand the experimental values measured by XMCD, respectively.  \nAtom 𝑚𝑆 \n(µB/at.) 𝑚𝐿 \n(µB/at.) 𝑚𝑆+𝑚𝐿 \n(µB/at.) \nTe1  0.00  0.00  0.00 \nFe1 -0.16 -0.03 -0.19 \nFe2  2.35  0.05  2.40 \nFe3  2.10  0.04  2.14 \nGe -0.06  0.00 -0.06 \nFe4  1.50  0.03  1.53 \nFe5  2.60  0.08  2.68 \nTe2 -0.10 -0.02 -0.12 \nFe \n(averaged)   1.68  0.03  1.71 \nFe (exp.)   1.54  0.08  1.62 \n \n \n "    },    {        "title": "2107.03768v1.Magnetic_properties_of_3d__4d__and_5d_transition_metal_atomic_monolayers_in_Fe_TM_Fe_sandwiches__Systematic_first_principles_study.pdf",        "content": "arXiv:2107.03768v1  [cond-mat.mtrl-sci]  8 Jul 2021Magnetic properties of 3 d, 4d, and 5 dtransition-metal atomic monolayers in\nFe/TM/Fe sandwiches: Systematic first-principles study\nJustyn Snarski-Adamski∗, Justyna Rychły, Mirosław Werwiński\nInstitute of Molecular Physics, Polish Academy of Sciences ,\nM. Smoluchowskiego 17, 60-179 Poznań, Poland\nAbstract\nPrevious studies have accurately determined the effect of tr ansition metal point defects on the properties of bcc iron.\nThe magnetic properties of transition metal monolayers on t he iron surfaces have been studied equally intensively. In\nthis work, we investigated the magnetic properties of the 3 d, 4d, and 5 dtransition-metal (TM) atomic monolayers in\nFe/TM/Fe sandwiches using the full-potential local-orbit al (FPLO) scheme of density functional theory. We prepared\nmodels of Fe/TM/Fe structures using the supercell method. W e selected the total thickness of our system so that the Fe\natomic layers furthest from the TM layer exhibit bulk iron-b cc properties. Along the direction perpendicular to the TM\nlayer, we observe oscillations of spin and charge density. F or Pt and W we obtained the largest values of perpendicular\nmagnetocrystalline anisotropy and for Lu and Ir the largest values of in-plane magnetocrystalline anisotropy. All TM\nlayers, except Co and Ni, reduce the total spin magnetic mome nt in the generated models, which is in good agreement\nwith the Slater-Pauling curve. Density of states calculati ons showed that for Ag, Pd, Ir, and Au monolayers, a distinct\nvan Hove singularity associated with TM/Fe interface can be observed at the Fermi level.\n1. Introduction\nOver the past decades, the magnetic thin films and\nlayered structures have attracted considerable attention\nin theoretical and applied physics [ 1]. These systems ex-\nhibit novel physical phenomena such as enhanced mag-\nnetic moments, magnetocrystalline anisotropy (MAE), os-\ncillatory interlayer coupling, and spin and charge-densit y\nwaves [ 2,3]. These quantum phenomena related to mag-\nnetism and spin-orbit coupling are also the subject of inter -\nest in spintronics. An important example of a spintronic\ndevice is the spin-transfer torque memory, which uses a\nspin-polarized tunneling current to switch magnetization\n[4]. Although there is huge potential in creating mag-\n∗Corresponding author\nEmail address: justyn.snarski-adamski@ifmpan.poznan.pl\n(Justyn Snarski-Adamski )netic multilayer structures, their control is difficult. Mor e-\nover, controlling and growing clean and sharp interfaces\nrequires a large amount of labor and knowledge. Previous\nstudies have predicted strong effects of 3 dTM monolay-\ners on ferromagnetic moments on metallic overlayers [ 5,6]\nor sandwich bilayers [ 7]. For the systems considering the\nsandwiches model with Mn bilayers, oscillations of spin\nmagnetic moments were observed [ 7]. The magnetic prop-\nerties of 3 dTM on metallic substrates such as Pd(001)\nand Ag(001) show great similarity to the interactions of 3 d\nmagnetic impurities in bulk alloys [ 8]. For 4 dand 5 dmono-\nlayer on Ag(001) and Au(001) substrates, the magnetism\nhas been explained as real effect of two-dimensional band\nstructure [ 9]. An exceptionally large perpendicular MAE\nwas found for Ir monolayer capped on Fe(001) surface [ 10].\nThere has also been interest in forcing ferromagnetic cou-\nPreprint submitted to Elsevier July 9, 2021Figure 1: A model of the crystal structure of the Fe/TM/Fe-\nsandwich system which crystallizes in a tetragonal structu re [space\ngroup P4/mmm (No. 123), a=b= 2.83 Å and c= 31.13 Å]. (a)\nModel showing the periodic multiplication of the unit cell, which is\nthe subject of our consideration. (b) Unit cell of the Fe/TM/ Fe-\nsandwich system.\npling in rare-earth-metal/Fe sandwiches systems using 3 d\nelements, which allow to obtaine huge magnetic moments\nof the order of 10 µB/rare-earth atom [ 11,12]. Further-\nmore, the ferromagnetic coupling in the Fe/TM/Gd sand-\nwich system with 4 dand 5 dmetal spacers was found to be\nstronger than that with 3 dspacers [ 13]. In this work, we\nfocused on the systems consisting of a Fe-bcc matrix and\na transition metal monolayer, see Fig. 1.\n2. Computational details\nIn this paper, the results of computations performed\nin the framework of density functional theory (DFT) are\npresented. The models of 3 d, 4d, and 5 dtransition-\nmetal atomic monolayers in Fe/TM/Fe sandwiches were\ninvestigated using the full-potential local-orbital sche me\n(FPLO18.00-52) [ 14,15]. To model the systems, we used\nthe supercell method [ 16]. The strucutral parameters of\nour models are a=b= 2.83 Å and c= 31.13 Å. The\ngeneralized gradient approximation (GGA) in the Perdew-\nBurke-Ernzerhof (PBE) form [ 17] was used. The lattice\nparameters were set as for bulk Fe-bcc and multiply 11\ntimes in the cdirection, whereas the Wyckoff positions\nwere optimized for each considered system using a spin--100.0-50.00.050.0100.0\n3dMAE (µeV·atom-1)\nLu   Hf   Ta    W   Re   Os    Ir    Pt   Au   HgSc   Ti    V     Cr   Mn  Fe   Co   Ni   Cu   Zn\nY     Zr   Nb   Mo  Tc   Ru   Rh   Pd  Ag   Cd4d\n5d\nFigure 2: Magnetocrystalline anisotropy energy (MAE) of\nFe/TM/Fe-sandwiches for various 3 d, 4d, and 5 dtransition metal\n(TM) elements. Calculations were performed with the FPLO18 us-\ning the PBE functional and supercell model. Considered unit cell\ninclude 21 atoms of Fe and 1 atom of TM.\npolarized scalar-relativistic approach, see Fig. 1. The sub-\nstitution of TM element was intended to create a single\ndopant monolayer in the center of this structure. Our cal-\nculation did not include optimization of lattice parameter s.\nA 12 ×12×1k-mesh was used for geometrical optimiza-\ntion and accuracy of forces was set as 10−3eV Å−1. An\n80×80×10k-mesh was found to lead to well converged re-\nsults of the magnetocrystalline anisotropy energies (MAE) .\nThe convergence criterion for the charge density was set\nas 10−6. MAE was evaluated as a difference between the\nfully-relativistic total energies calculated for quantiz ation\naxes [100] and [001]. The applied full-potential approach\nplays an important role in the correct determination of the\nvalues of magnetic moments and MAE [ 18]. For Fe 22, the\nobtained result of the MAE value determines the accuracy\nof our calculations, as we would expect zero for such a sys-\ntem. Thus, the accuracy of our calculations is estimated\nat 1 µeV/atom. The VESTA code was used to visualize\nthe crystal structure [ 19].\n3. Results and discussion\nSince we considered all 3 d, 4d, and 5 dtransition met-\nals in Fe/TM/Fe-sandwich models, we obtained a complete\n242.044.046.048.0\n3dTotal spin magnetic moment ( µB )\nLu   Hf   Ta    W   Re   Os    Ir    Pt   Au   HgSc   Ti    V     Cr   Mn  Fe   Co   Ni   Cu   Zn\nY     Zr   Nb   Mo  Tc   Ru   Rh   Pd  Ag   Cd4d\n5d\n(a)m- Fe/TM/Fe-1.00.01.02.0\n3dSpin magnetic moment ( µB )\nLu   Hf   Ta    W   Re   Os    Ir    Pt   Au   HgSc   Ti    V     Cr   Mn  Fe   Co   Ni   Cu   Zn\nY     Zr   Nb   Mo  Tc   Ru   Rh   Pd  Ag   Cd4d\n5d\n(b)ms- Fe/TM/Fe-0.10-0.050.000.050.10\n3dOrbital magnetic moment ( µB )\nLu   Hf   Ta    W   Re   Os    Ir    Pt   Au   HgSc   Ti    V     Cr   Mn  Fe   Co   Ni   Cu   Zn\nY     Zr   Nb   Mo  Tc   Ru   Rh   Pd  Ag   Cd4d\n5d\n(c)ml- Fe/TM/Fe\nFigure 3: ( a) Total spin magnetic moment per formula unit, ( b) spin magnetic moment as calculated for spin quantization a xis along the\ndistinguished axis [001] (easy axis), ( c) orbital magnetic moment as calculated for spin quantizati on axis along the [001] (easy axis) for various\n3d, 4d, and 5 dTM elements in the Fe/TM/Fe-sandwiches. Calculations were performed with FPLO18, using the PBE functional and superce ll\nmethod.\npicture of the variations of MAE, total spin magnetic mo-\nment, and magnetic moments on TM, see Figs. 3band3c.\nWe can see that 5 delements can lead to a more extreme\nMAE than the 3 dand 4 delements due to the stronger\nspin-orbit coupling. Positive and negative MAE values in-\ndicate alignment of magnetic moments perpendicular and\nin-plane of the TM layer, respectively. For Pt and W we\nobtained the largest values of perpendicular magnetocrys-\ntalline anisotropy and for Lu and Ir the largest values of in-\nplane magnetocrystalline anisotropy. The largest value of\nMAE for Pt interlayer is not surprising considering the ex-\ntremely high MAE values observed for the L1 0FePt phase\n[20]. The previous calculations also showed an increase in\nthe value of MAE for systems with W substitution [ 18,21].\nAlmost all TM elements, except Co and Ni, decrease\nthe total spin magnetic moment relative to the Fe atom in\nthe structures under our consideration, see Fig. 3a, which\nis in good agreement with the Slater-Pauling curve [ 22].\nWe also see that almost all TMs, except Co, Ni, and Pt,\ncontribute a lower orbital magnetic moment than Fe, see\nFig.3c. Against this background, Pt stands out again, as it\nhas the highest orbital moment among all the TMs consid-\nered. The orbital magnetic moment for bcc Fe calculated\nin this paper, which is 0.045 µB, is underestimated rela-\ntive to the experimental value of 0.085 µB, which suggests\nthat the calculated values of orbital magnetic moment areabout twice underestimated. [ 23]. Calculations of the spin\nand orbital magnetic moment on the atomic monolayer of\nthe transition metal show explicit trends with increasing\natomic numbers. Analogous trends have been found for\npoint substitutions in bulk-iron, both theoretically [ 24,25]\nand experimentally [ 26].\nTherefore, we conducted a detailed case study for a\nsystem with Pt for which we obtained the maximum value\nof MAE. In Fig. 4a, we have shown the oscillation of charge\nand spin magnetic moment in Fe/Pt/Fe-sandwich system.\nFor Pt we observe spin magnetic moment of 0.42 µBand\nfor three Fe layers nearest to Pt the values are as follows:\n2.76, 2.32, and 2.25 µB. However, the Fe layers farthest\nfrom the Pt layer exhibit bulk Fe-bcc magnetic properties.\nWe present the calculation of density of states to\nshow what happens near the interface in the Fe/Pt/Fe-\nsandwich, see Fig. 5. In Fig. 5b we present the total\nDOS for the Fe/Pt/Fe system with the comparisons in\nFig.5a for Fe 22-bcc in the same model as here discussed\n(Fe/Fe/Fe-sandwich system). The comparison of Fe 22\nDOS with Fe/Pt/Fe DOS shows great similarity and few\ndifferences resulting from about two additional electrons\non the Pt 5 dshell compared to the Fe 3 dshell. The c-g\npanels show the contributions from the dorbitals the two\nclosest (Fe1, Fe2) and farthest (F10, Fe11) Fe layers in re-\nlation to the Pt layer. For the Pt 5 dband, one spin channel\n3-15 -10 -5 0 5 10 150.51.01.52.02.53.0\n-15 -10 -5 0 5 10 15-0.020.000.020.040.060.08Magnetic moment ( µB )\nExcess electron (e)\nDistance from Pt layer (Å)\n(a)\n(b)\nFigure 4: ( a) Calculation of spin magnetic moment and charge os-\ncillations in Fe/Pt/Fe-sandwich system. ( b) A model of the crystal\nstructure of Fe/TM/Fe-sandwich.\nis partially empty and the other is completely filled. The\nadjacent Fe1 layer shows an increased spin polarization\nvalue ( ms= 2.76 µB). The DOS for Fe10 and Fe11, far-\nthest from Pt layer, resemble the spectra observed for Fe 22.\nThe van Hove singularity on the Fermi level observed for\nFe/Pt/Fe-sandwich system comes from the Pt 5 dstates.\nThis narrow maximum at the Fermi level can have a sig-\nnificant impact on the electrochemical properties of the\nconsidered system. Unlike the band-types typical of va-\nlence band states, it can be interpreted as a characteristic\nof an atomic-like state of electrons.\n4. Summary and conclusions\nWe have presented the first-principles results for the\nFe/TM/Fe-sandwich system. Almost all TM layers, ex-\ncept Co and Ni layers, reduce the total spin magnetic\nmoment in the considered models, which agrees well\nwith the Slater-Pauling curve. We observe that W and\nPt monolayers induce strong perpendicular magnetocrys-050100\n050100\n024\n024\n024\n024\n-8 -6 -4 -2 0 2 4024(a) Fe/Fe/Fe Total DOS\n(b) Fe/Pt/Fe Total DOS\n(c) Pt 5d\n(d) Fe1 3d\n(e) Fe2 3d\n(f) Fe10 3 d\n(g) Fe11 3 dDOS [states·(eV·spin·atom or u.c.)-1]\nE - EF (eV)\nFigure 5: Calculated densities of states (DOS) for the consi dered\nFe/TM/Fe structure. (a) Fe 22total DOS, (b) Fe/Pt/Fe total DOS,\n(c) Pt 5 dDOS, (d, e) Fe1 and Fe2 3 dDOS for the two Fe layers\nclosest to the Pt layer, (f, g) Fe10 and Fe11 3 dDOS for the two Fe\nlayers furthest from the Pt layer.\ntalline anisotropy, while Lu and Ir most strongly prefer in-\nplane anisotropy. Our calculations indicate oscillation o f\nspin magnetic moment and charge. We have also observed\nthe van Hove singularity in the formed Fe/TM interface\nfor TM = Ag, Pd, Pt, Ir, and Au.\nAcknowledgements\nWe acknowledge the financial support of the Na-\ntional Science Center Poland under the decision DEC-\n2019/35/O/ST5/02980 (PRELUDIUM-BIS 1) and DEC-\n2018/30/E/ST3/00267 (SONATA-BIS 8). Part of the\ncomputations were performed on resources provided by the\nPoznan Supercomputing and Networking Center (PSNC).\nWe thank Paweł Leśniak and Daniel Depcik for compiling\nthe scientific software and administration of the comput-\ning cluster at the Institute of Molecular Physics, Polish\nAcademy of Sciences.\n4Appendix\nTable 1provides the values shown in Figs. 2,3a,3b,\nand3c.\nReferences\n[1] T. Thersleff, S. Muto, M. Werwinski, J. Spiegelberg, Y. Kv ash-\nnin, B. Hjorvarsson, O. Eriksson, J. Rusz, K. Leifer, Toward s\nsub-nanometer real-space observation of spin and orbital m ag-\nnetism at the Fe/MgO interface, Sci. Rep. 7 (1) (2017) 44802.\ndoi:10.1038/srep44802 .\n[2] R. S. Fishman, Spin-density waves in Fe/Cr trilayers and mul-\ntilayers, J. Phys. Condens. Matter 13 (13) (2001) R235–R269 .\ndoi:10.1088/0953-8984/13/13/201 .\n[3] C. Turtur, G. Bayreuther, Magnetic moments in ultrathin Cr\nfilms on Fe(100), Phys. Rev. Lett. 72 (10) (1994) 1557–1560.\ndoi:10.1103/PhysRevLett.72.1557 .\n[4] J. C. Slonczewski, Current-driven excitation of magnet ic mul-\ntilayers, J. Magn. Magn. Mater. 159 (1) (1996) L1–L7.\ndoi:10.1016/0304-8853(96)00062-5 .\n[5] C. L. Fu, A. J. Freeman, T. Oguchi, Prediction of Strongly\nEnhanced Two-Dimensional Ferromagnetic Moments on Metal-\nlic Overlayers, Interfaces, and Superlattices, Phys. Rev. Lett.\n54 (25) (1985) 2700–2703. doi:10.1103/PhysRevLett.54.2700 .\n[6] C. Li, A. J. Freeman, Giant monolayer magnetization of Fe on\nMgO: A nearly ideal two-dimensional magnetic system, Phys.\nRev. B 43 (1) (1991) 780–787. doi:10.1103/PhysRevB.43.780 .\n[7] S. T. Purcell, M. T. Johnson, N. W. E. McGee, R. Co-\nehoorn, W. Hoving, Two-monolayer oscillations in the antif er-\nromagnetic exchange coupling through Mn in Fe/Mn/Fe sand-\nwich structures, Phys. Rev. B 45 (22) (1992) 13064–13067.\ndoi:10.1103/PhysRevB.45.13064 .\n[8] S. Blügel, B. Drittler, R. Zeller, P. H. Dederichs, Magne tic prop-\nerties of 3 dtransition metal monolayers on metal substrates,\nAppl. Phys. A 49 (6) (1989) 547–562. doi:10.1007/BF00616980 .\n[9] S. Blügel, Two-dimensional ferromagnetism of 3 d, 4d\n, and 5 dtransition metal monolayers on noble metal\n(001) substrates, Phys. Rev. Lett. 68 (6) (1992) 851–854.\ndoi:10.1103/PhysRevLett.68.851 .\n[10] D. Odkhuu, S. H. Rhim, N. Park, S. C. Hong, Extremely\nlarge perpendicular magnetic anisotropy of an Fe(001) sur-\nface capped by 5 dtransition metal monolayers: A den-\nsity functional study, Phys. Rev. B 88 (18) (2013) 184405.\ndoi:10.1103/PhysRevB.88.184405 .[11] B. Sanyal, C. Antoniak, T. Burkert, B. Krumme, A. Warlan d,\nF. Stromberg, C. Praetorius, K. Fauth, H. Wende, O. Eriksson ,\nForcing Ferromagnetic Coupling Between Rare-Earth-Metal\nand 3 dFerromagnetic Films, Phys. Rev. Lett. 104 (15) (2010)\n156402. doi:10.1103/PhysRevLett.104.156402 .\n[12] C. Autieri, P. A. Kumar, D. Walecki, S. Webers, M. A. Gubb ins,\nH. Wende, B. Sanyal, Recipe for High Moment Materials with\nRare-Earth and 3d Transition Metal Composites, Sci. Rep. 6 ( 1)\n(2016) 29307. doi:10.1038/srep29307 .\n[13] C. Autieri, B. Sanyal, A systematic study of 4 dand 5 dtran-\nsition metal mediated exchange coupling between Fe and Gd\nnanolaminates, J. Phys. Condens. Matter 29 (46) (2017) 4658 02.\ndoi:10.1088/1361-648X/aa8f1e .\n[14] K. Koepernik, H. Eschrig, Full-potential nonorthogon al local-\norbital minimum-basis band-structure scheme, Phys. Rev. B\n59 (3) (1999) 1743–1757. doi:10.1103/PhysRevB.59.1743 .\n[15] H. Eschrig, 2. The Essentials of Density Functional The ory\nand the Full-Potential Local-Orbital Approach, in: W. Herg -\nert, M. Däne, A. Ernst (Eds.), Computational Materials Sci-\nence: From Basic Principles to Material Properties, Lectur e\nNotes in Physics, Springer, Berlin, Heidelberg, 2004, pp. 7 –21.\ndoi:10.1007/978-3-540-39915-5_2 .\n[16] A. Edström, Magnetocrystalline anisotropy of Laves\nphase Fe 2Ta1−xWxfrom first principles: Effect of 3 d-\n5dhybridization, Phys. Rev. B 96 (6) (2017) 064422.\ndoi:10.1103/PhysRevB.96.064422 .\n[17] J. P. Perdew, K. Burke, M. Ernzerhof, Generalized Gradi ent\nApproximation Made Simple, Phys. Rev. Lett. 77 (18) (1996)\n3865–3868. doi:10.1103/PhysRevLett.77.3865 .\n[18] M. Werwiński, A. Edström, J. Rusz, D. Hedlund, K. Gun-\nnarsson, P. Svedlindh, J. Cedervall, M. Sahlberg, Mag-\nnetocrystalline anisotropy of Fe 5PB2and its alloys with\nCo and 5 delements: A combined first-principles and ex-\nperimental study, Phys. Rev. B 98 (21) (2018) 214431.\ndoi:10.1103/PhysRevB.98.214431 .\n[19] K. Momma, F. Izumi, VESTA : A three-dimensional\nvisualization system for electronic and structural anal-\nysis, J. Appl. Crystallogr. 41 (3) (2008) 653–658.\ndoi:10.1107/S0021889808012016 .\n[20] R. A. Ristau, K. Barmak, L. H. Lewis, K. R. Coffey, J. K.\nHoward, On the relationship of high coercivity and L10 order ed\nphase in CoPt and FePt thin films, J. Appl. Phys. 86 (8) (1999)\n4527–4533. doi:10.1063/1.371397 .\n[21] A. Edström, M. Werwiński, D. Iuşan, J. Rusz, O. Eriksson ,\nK. P. Skokov, I. A. Radulov, S. Ener, M. D. Kuz’min, J. Hong,\nM. Fries, D. Y. Karpenkov, O. Gutfleisch, P. Toson, J. Fidler,\nMagnetic properties of (Fe 1-xCox)2B alloys and the effect of\ndoping by 5 delements, Phys. Rev. B 92 (17) (2015) 174413.\n5Table 1: Values of magnetocrystalline anisotropy energy [M AE (µeV/atom)], total spin magnetic moment [ m(µBu.c.−1)], spin magnetic\nmoment [ ms(µBu.c.−1)] on transition metal (TM), and orbital magnetic moment [ ml(µBu.c.−1)] on TM for various 3 d, 4d, and 5 dTM\nelements in the Fe/TM/Fe-sandwiches.\n3d 4d 5d\nTM MAE m m s ml TM MAE m m s ml TM MAE m m s ml\nSc -20.46 42.93 -0.50 0.0039 Y -52.96 42.77 -0.43 0.0024 Lu -69.73 42.98 -0.32 0.0102\nTi -1.06 43.53 -0.81 0.0123 Zr -32.93 43.54 -0.48 0.0121 Hf -7.88 43.46 -0.45 0.0292\nV -8.47 44.32 -1.03 0.0186 Nb -18.24 44.44 -0.38 0.0141 Ta 1.27 44.29 -0.43 0.0207\nCr -2.25 45.05 -0.74 0.0068 Mo 6.38 45.01 -0.30 0.0018 W 63.41 44.91 -0.28 -0.0232\nMn -3.69 46.50 1.07 0.0070 Tc -10.62 45.53 -0.16 -0.0075 Re 3.65 45.35 -0.17 -0.0484\nFe -0.63 47.62 2.16 0.0451 Ru -6.52 46.17 0.23 -0.0140 Os -18.48 45.61 -0.01 -0.0692\nCo -2.67 48.27 1.61 0.0771 Rh -3.06 47.55 0.74 0.0427 Ir -97.39 46.41 0.37 -0.0027\nNi -2.76 47.73 0.69 0.0601 Pd 14.28 46.83 0.29 0.0351 Pt 124.80 46.99 0.41 0.1008\nCu 15.08 46.39 -0.04 0.0101 Ag 23.30 45.60 -0.05 0.0094 Au 25.82 45.81 0.02 0.0425\nZn 24.08 45.77 -0.22 0.0008 Cd 12.22 44.54 -0.19 0.0017 Hg -13.64 44.68 -0.15 0.0092\ndoi:10.1103/PhysRevB.92.174413 .\n[22] A. Williams, V. Moruzzi, A. Malozemoff, K. Terakura,\nGeneralized Slater-Pauling curve for transition-metal\nmagnets, IEEE Trans. Magn. 19 (5) (1983) 1983–1988.\ndoi:10.1109/TMAG.1983.1062706 .\n[23] C. T. Chen, Y. U. Idzerda, H.-J. Lin, N. V. Smith, G. Meigs ,\nE. Chaban, G. H. Ho, E. Pellegrin, F. Sette, Experimental Con -\nfirmation of the X-Ray Magnetic Circular Dichroism Sum Rules\nfor Iron and Cobalt, Phys. Rev. Lett. 75 (1) (1995) 152–155.\ndoi:10.1103/PhysRevLett.75.152 .\n[24] H. Akai, Nuclear spin-lattice relaxation of impuritie s in fer-\nromagnetic iron, Hyperfine Interact. 43 (1) (1988) 253–270.\ndoi:10.1007/BF02398306 .\n[25] P. H. Dederichs, R. Zeller, H. Akai, H. Ebert, Ab-initio cal-\nculations of the electronic structure of impurities and all oys of\nferromagnetic transition metals, J. Magn. Magn. Mater. 100 (1)\n(1991) 241–260. doi:10.1016/0304-8853(91)90823-S .\n[26] R. Wienke, G. Schütz, H. Ebert, Determination of local\nmagnetic moments of 5d impurities in Fe detected via spin-\ndependent absorption, J. Appl. Phys. 69 (8) (1991) 6147–614 9.\ndoi:10.1063/1.348786 .\n6"    },    {        "title": "2107.11204v1.Reduced_magnetocrystalline_anisotropy_of_CoFe__2_O__4__thin_films_studied_by_angle_dependent_x_ray_magnetic_circular_dichroism.pdf",        "content": "Reduced magnetocrystalline anisotropy of CoFe 2O4thin \flms studied by\nangle-dependent x-ray magnetic circular dichroism\nYosuke Nonaka,1,\u0003Yuki K. Wakabayashi,2Goro Shibata,1, 3Shoya Sakamoto,1Keisuke Ikeda,1Zhendong Chi,1\nYuxuan Wan,1Masahiro Suzuki,1Tsuneharu Koide,4Masaaki Tanaka,2, 5Ryosho Nakane,2, 6and Atsushi Fujimori1, 7\n1Department of Physics, the University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan\n2Department of Electrical Engineering and Information Systems,\nThe University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan\n3Department of Applied Physics, Tokyo University of Science, Katsushika-ku, Tokyo 125-8585, Japan\n4Photon Factory, Institute of Materials Structure Science,\nHigh Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan\n5Center for Spintronics Research Network, Graduate School of Engineering,\nThe University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan\n6Institute for Innovation in International Engineering Education,\nThe University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan\n7Department of Applied Physics, Waseda University, Shinjuku-ku, Tokyo 169-8555, Japan\n(Dated: July 26, 2021)\nSpinel-type CoFe 2O4is a ferrimagnetic insulator with the N\u0013 eel temperature exceeding 790 K, and\nshows a strong cubic magnetocrystalline anisotropy (MCA) in bulk materials. However, when a\nCoFe 2O4\flm is grown on other materials, its magnetic properties are degraded so that so-called\nmagnetically dead layers are expected to be formed in the interfacial region. We investigate how the\nmagnetic anisotropy of CoFe 2O4is modi\fed at the interface of CoFe 2O4/Al2O3bilayers grown on\nSi(111) using x-ray magnetic circular dichroism (XMCD). We \fnd that the thinner CoFe 2O4\flms\nhave signi\fcantly smaller MCA values than bulk materials. The reduction of MCA is explained by\nthe reduced number of Co2+ions at the Ohsite reported by a previous study [Y. K. Wakabayashi\net al. , Phys. Rev. B 96, 104410 (2017)].\nINTRODUCTION\nSpinel-type cobalt ferrite CoFe 2O4is a classical fer-\nrimagnetic insulator having the N\u0013 eel temperature ex-\nceeding 790 K and exhibits a strong cubic magnetocrys-\ntalline anisotropy (MCA). The cubic MCA has been suc-\ncessfully explained by the single-ion anisotropy of the\nCo2+ions at the inequivalent Ohsites [1]. Recently,\nheterostructures incorporating thin CoFe 2O4layers have\nattracted much attention as spintronics devices [2] be-\ncause of the spin-dependent band gap [3] and high N\u0013 eel\ntemperature of CoFe 2O4[4]. For example, a CoFe 2O4-\nbased tunnel barrier acts as a spin \flter because electrons\nhave spin-dependent tunneling probabilities [5]. How-\never, the experimentally obtained spin-\fltering e\u000eciency\nof CoFe 2O4-based tunnel barriers still remains lower than\ntheoretical values of 100% [2, 6, 7]. As a possible cause of\nthe low spin-\fltering e\u000eciency, it has been proposed that\nstructural and/or chemical disorder lead to the formation\nof impurity states in the spin-dependent gap [3, 8, 9]. In\norder to improve the spin \fltering e\u000eciency, it is thus\nessential to understand the electronic and magnetic phe-\nnomena at the interfaces.\nIn a recent work [10], the magnetic properties of\nCoFe 2O4(111)/Al 2O3(111)/Si(111) structures [11] were\nstudied using the element-speci\fc probe of x-ray ab-\nsorption spectroscopy (XAS) and x-ray magnetic circu-\nlar dichroism (XMCD). A schematic illustration of the\nstacking structure of the sample and an example of crosssectional transmission-electron-microscope (TEM) image\nare shown in Fig. 1(a). Because they employed the total-\nelectron yield (TEY) method to detect the absorption\nsignals, several nanometers from the surface were pref-\nerentially probed [12, 13]. By reducing the \flm thick-\nnesses below the probing depth of a few nm, they ob-\ntained XAS and XMCD spectra re\recting the magnet-\nically dead layers of \u00181.4 nm thickness and the cation\nredistribution near the CoFe 2O4/Al2O3interface as il-\nlustrated in Fig. 1(b). They revealed a high density\nof Co-Fe antisite defects and the reduced concentra-\ntion of the Co2+ions at the Ohsites in the magneti-\ncally dead layers. Since the Co2+ions at the Ohsites\nare considered to be the origin of the cubic MCA in\nbulk CoFe 2O4, a signi\fcant modi\fcation of the magnetic\nanisotropy is expected in the interfacial region. In the\npresent study, we investigate how the strong cubic MCA\nof CoFe 2O4is a\u000bected at the CoFe 2O4/Al2O3interface\nby angle-dependent XMCD that is a powerful local probe\nto study element-speci\fc magnetic anisotropies [14, 15].\nThe surface-sensitive XMCD method allowed us to in-\nvestigate the magnetic properties in the interfacial region\nwith high sensitivity. Furthermore, by using XMCD, dia-\nmagnetic signals from substrates, which prevent us from\nthe observation of intrinsic paramagnetic signals, can be\nautomatically excluded.arXiv:2107.11204v1  [cond-mat.mtrl-sci]  23 Jul 20212\nd\nAl2O3CoFe2O4Probi ng depth\n(several nm)X-ray\ne-\nMagnetically \ndead l ayer(a)\n(b)Si substrateCoFe2O4\n(d=11nm)\nAl2O3\nSiOx\nFIG. 1. Schematic pictures of the sample structure\nand the method to probe the interfacial region. (a)\nSample structure and cross sectional TEM image of the\nCoFe 2O4(111)/Al 2O3(111)/Si(111) sample with d= 11 nm,\nadapted from Ref. [10]. (b) Illustration of probing the mag-\nnetically dead interfacial layers with XMCD by changing the\nthickness of the CoFe 2O4layer.\nEXPERIMENTAL METHODS\nEpitaxial CoFe 2O4(111) thin \flms with the thicknesses\nofd= 1.4, 2.3, 4, and 11 nm were grown on a 2.4 nm-\nthick\r-Al2O3(111) bu\u000ber layer/ n+-Si(111) substrate us-\ning the pulsed laser deposition method. In order to avoid\ncharging of the samples during the XAS and XMCD mea-\nsurements, we used heavily phosphorus-doped Si(111)\nwafers with low resistivities of 2 m\n cm. For the epi-\ntaxial growth of the \r-Al2O3bu\u000ber layers on the Si sub-\nstrates, we used solid-phase reaction of Al and SiO 2. A\nmore detailed description of the sample preparation and\ncharacterization is given in Ref. [10].\nMagnetic \feld-angle-dependent XAS and XMCD mea-\nsurements were performed at room temperature using\na superconducting vector-magnet XMCD apparatus [16]\ninstalled at the undulator beamline BL-16A of Photon\nFactory, High Energy Accelerator Research Organization\n(KEK-PF). The magnetization Mof CoFe 2O4thin \flm\nis rotated by the magnetic \feld H. The magnetic \feld\nangle\u0012Hand magnetization angle \u0012Mare de\fned rela-\ntive to the surface normal. The absorption signals were\n/s32/s33/s34/s35/s33/s32/s33/s34/s36/s37/s32/s33/s34/s36/s33/s32/s33/s34/s33/s37/s33/s33/s34/s33/s37/s38/s39/s40/s41/s39/s42/s43/s40/s44/s45/s46/s47/s34/s45/s48/s34/s49\n/s50/s51/s33 /s50/s35/s37 /s50/s35/s33 /s50/s36/s37 /s50/s36/s33 /s50/s33/s37\n/s52/s53/s54/s40/s54/s39/s45/s55/s39/s41/s56/s57/s44/s45/s46/s41/s58/s49/s32/s33\n/s32/s59/s33/s60\n/s50/s33/s60/s61/s41/s45/s32/s35/s62/s51/s63/s41/s64/s57/s41/s45\n/s65/s66/s67/s68\n/s45/s63/s35/s33/s60/s45/s36/s34/s33\n/s33/s34/s37\n/s33/s38/s39/s40/s41/s39/s42/s43/s40/s44/s45/s46/s47/s34/s45/s48/s34/s49/s67/s54/s61/s41/s35/s69/s70/s45/s46/s36/s36/s36/s49\n/s61/s41/s45/s32/s35/s62/s51/s63/s41/s64/s57/s41/s45/s65/s71/s72\n/s33/s45/s73/s45/s36/s36/s45/s39/s74\n/s34/s33/s35/s45/s73/s45/s33/s34/s50/s45/s75\n/s36/s35/s45/s73/s45/s33/s60\n/s35/s34/s33\n/s36/s34/s37\n/s36/s34/s33\n/s33/s34/s37\n/s33/s34/s33\n/s32/s33/s34/s37\n/s32/s36/s34/s33\n/s32/s36/s34/s37/s34/s76/s56/s54/s77/s45/s46/s34/s78/s79/s80/s34/s48/s34/s49\n/s32/s81/s33 /s32/s70/s37 /s33 /s70/s37 /s81/s33\n/s66/s47/s57/s39/s41/s40/s43/s82/s45/s61/s43/s41/s83/s64/s45/s71/s39/s57/s83/s41/s45 /s36/s35/s37/s45/s46/s64/s41/s57/s34/s49/s55/s84/s76/s40/s34\n/s85/s54/s45/s47/s39/s43/s42/s54/s40/s56/s54/s76/s44/s46/s86/s49\n/s46/s82/s49/s46/s47/s49\n/s46/s64/s49/s55/s84/s76/s41/s56/s43/s74/s41/s39/s40/s47/s83/s45/s57/s41/s54/s74/s41/s40/s56/s44\n/s32/s33/s36/s38\n/s34/s33/s38/s76/s56/s54/s77/s72/s47/s74/s76/s83/s41/s67/s61/s69/s87/s36/s36/s36/s88\n/s70/s37/s60\n/s72/s43/s87/s36/s36/s35/s88/s84/s45/s56/s47/s44FIG. 2. Angle-dependent XMCD of the 11 nm-thick\nCoFe 2O4\flm at the Fe L2;3edges. (a) Experimental ge-\nometry of the angle-dependent XMCD measurements. (b)\nXAS spectrum at \u0012H= 0\u000eafter background subtraction. (c)\nXMCD spectra for various \u0012H's. (d)\u0012Hdependence of the\nrelative magnetic moment projected onto the x-ray incident\nvector, deduced from the L3-edge XMCD intensity (see main\ntext). The gray dashed curve (sine curve) shows the simula-\ntion for the magnetically isotropic situation.\ndetected in the TEY mode. The XMCD signals were col-\nlected by switching the photon helicity at the rate of 10\nHz [17]. In order to eliminate the saturation e\u000bect [18],\nwhich induces an extrinsic angle dependence to spectral\nline shapes, we \fxed the x-ray incident angle at 45\u000eand\napplied magnetic \felds of 0.7 T to the sample along var-\nious directions. The geometry of the present measure-\nments is illustrated in Fig. 2(a).\nRESULTS AND DISCUSSION\nFigures 2(b) and (c) show the Fe L2;3-edge XAS\n(\u0016++\u0016\u0000) and XMCD ( \u0016+\u0000\u0016\u0000) spectra of the 11 nm-\nthick CoFe 2O4thin \flm. Here, \u0016+(\u0016\u0000) denotes the\nabsorption coe\u000ecient for photons with positive (nega-\ntive) helicity. Since the spectral line shape of XAS did\nnot show any appreciable \u0012Hdependence, only the spec-\ntrum at\u0012H= 0\u000eis shown. Figure 2(c) shows that\nthe XMCD spectrum systematically changes with \u0012H.\nSince the XMCD intensity is proportional to the mag-\nnetic moment projected onto the x-ray incident direc-\ntion (Mproj=Mcos(\u0012M\u000045\u000e)), its\u0012Hdependence re-\n\rects the change of the magnetization direction \u0012Mun-\nder varying \u0012H. Since the total magnetic moment and\nthe FeL2;3-edge XMCD spectra of these samples were\nalready obtained in Ref.[10], we deduced Mprojfrom the3\nintensity of Fe L2;3-edge XMCD under the assumption\nthat the Fe L2;3-edge XMCD intensity is proportional\ntoM. Figure 2(d) shows the \u0012Hdependence of Mproj.\nIf this \flm has neither magnetocrystalline nor magnetic\nshape anisotropy, Mwould be fully aligned to the mag-\nnetic \feld direction ( \u0012H=\u0012M) and thusMprojshould be\nMcos(\u0012H\u000045\u000e), as shown by a gray dashed curve. The\ndeviation of the experimental data from Mcos(\u0012H\u000045\u000e)\nthus shows the magnetic anisotropy.\nIn order to analyze the obtained \u0012Hdependence of\nMproj, we use the Stoner-Wohlfarth model [14, 19]. Ac-\ncording to the model, the magnetic energy density Eof\na thin \flm is given by:\nE=\u0000\u00160MH cos (\u0012M\u0000\u0012H) +\u00160\n2M2cos2\u0012M+EMCA;\n(1)\nwhere\u00160is the vacuum permeability, and the other vari-\nables are de\fned in Fig. 2(a). In Eq. (1), Eis the sum\nof the Zeeman energy [ \u0000\u00160MH cos (\u0012M\u0000\u0012H)] and the\nmagnetic anisotropy energy. The magnetic anisotropy\nenergy consists of two contributions. One is the shape\nanisotropy (SA) energy, which originates from the de-\nmagnetizing \feld of the \flm, and the other is the MCA\nenergy, which has a microscopic origin. Note that the\nStoner-Wohlfarth model is applicable not only to the fer-\nromagnetic state but also to the paramagnetic state if\nthe actual magnetization Mlower than the saturation\nmagnetization is used. Since CoFe 2O4has a cubic MCA\nin addition to the uniaxial anisotropy and has crystal\ndomains in the present samples [10], the MCA term can-\nnot be fully calculated with the Stoner-Wohlfarth model.\nTherefore, we have calculated Mprojby incorporating\nonly the Zeeman and SA terms in Eq. (1), and attribute\nthe di\u000berence between the measured Mprojand the cal-\nculatedMprojto the MCA.\nFigures 3(a){(d) show the measured Mprojand calcu-\nlatedMprojfor all the samples, where colored dots are\nmeasurements and black solid lines are calculations. One\ncan see a clear di\u000berence between the calculation and\nthe experiment for the 11 nm-thick \flm as shown in\nFig. 3(a), and the di\u000berence decreases with decreasing\nthickness as shown in Figs. 3(b), 3(c), and 3(d). (The\n11 nm-thick \flm is ferrimagnetic with hysteresis but the\nmagnetic \feld of 0.7 T was not su\u000ecient to saturate the\nmagnetization. The ferrimagnetic behavior is gradually\nlost with decreasing thickness and the 1.4 nm-thick \flm\nis almost paramagnetic [10].) For each \flm thickness,\nthe di\u000berence is the largest at around \u0012H= 0\u000e, as shown\nin Fig. 3(e),which is consistent with the cubic magnetic\nanisotropy of bulk CoFe 2O4with the [111] hard axis [20].\nConsidering that the relative contribution of the inter-\nface to the XAS and XMCD increases with decreasing\n\flm thickness, the present observation indicates a weak-\nening of the MCA of CoFe 2O4in the interfacial region\nincluding the magnetically dead layer. The magnitude of\nthe di\u000berence between the calculation and measurementis plotted as a function of thickness din Fig. 3(f).\nNow we discuss the microscopic origin of the reduc-\ntion of the MCA near the interface. The MCA of\nbulk CoFe 2O4is well explained by a single-ion model\n[1], according to which the exceptionally high single-\nion anisotropy of Co2+(Oh) governs the MCA. There-\nfore, the distribution of Co2+(Oh) is expected have dom-\ninant e\u000bects on the MCA. Such modi\fcations of mag-\nnetic anisotropy induced by the distribution of Co2+(Oh)\nare also reported for the CoFe 2O4nanoparticles [21{23].\nThe cation distribution in these \flms has been studied\nusing XMCD by Wakabayashi et al. [10], according to\nwhich the inversion parameter y, de\fned by the chem-\nical formula [Co 1\u0000yFey]Td[Fe2\u0000yCoy]OhO4, suddenly de-\ncreases in the d\u00142:3 nm region as shown in Fig. 3(f).\nFrom the comparison of yand the normalized MCA (the\nmagnitude of MCA divided by M) as functions of the\n\flm thickness dplotted in Fig. 3(f), one can see that the\nreduction of MCA with decreasing \flm thickness qualita-\ntively follows the reduction of y. This result implies that\nthe reduction of the normalized MCA originates from the\nreduction of the Co2+(Oh) ions in the interfacial region.\nCONCLUSION\nWe have investigated the magnetic anisotropy of the\nepitaxial CoFe 2O4(111) thin \flms (thicknesses d=11, 4,\n2.3, and 1.4 nm) grown on the 2.4 nm-thick \r-Al2O3(111)\nbu\u000ber layer/ n+-Si(111) substrates using magnetic \feld-\nangle-dependent XAS and XMCD. We have found that\nthe MCA of the interfacial region including the magneti-\ncally dead layer is reduced. We attribute the reduction of\nMCA to the thickness-dependent cation redistribution in\nthe interfacial region, that is, to the abrupt reduction of\nthe Co2+(Oh)-ion concentration in the interfacial region.\nWe would like to thank Kenta Amemiya and Masako\nSuzuki-Sakamaki for technical support at KEK-PF BL-\n16. This work was supported by a Grant-in-Aid for\nScienti\fc Research from JSPS (15H02109, 26289086,\n15K17696, and 19K03741). The experiment was done\nunder the approval of the Photon Factory Program Ad-\nvisory Committee (Proposal No. 2016S2-005). Y. K.\nW. and Z. C. acknowledges \fnancial support from Ma-\nterials Education Program for the Futures leaders in Re-\nsearch, Industry and Technology (MERIT). S. S. and Y.\nW. acknowledges \fnancial support from Advanced Lead-\ning Graduate Course for Photon Science (ALPS). Y. K.\nW. and S. S. also acknowledge support from the JSPS\nResearch Fellowship Program for Young Scientists. A.F.\nis an adjunct member of Center for Spintronics Research\nNetwork (CSRN), the University of Tokyo, under Spin-\ntronics Research Network of Japan (Spin-RNJ).4\n/s32/s33/s34\n/s32/s33/s35\n/s32/s33/s32\n/s36/s32/s33/s35/s37/s38/s39/s40/s41/s42/s43/s44/s45/s37/s46/s47/s41/s40/s42/s45/s48 /s32/s49/s50/s51/s33/s52/s33/s53\n/s54/s32/s32/s55/s32/s32/s36/s55/s32\n/s37/s38/s39/s40/s41/s42/s43/s44/s45/s56/s43/s41/s57/s58/s45/s59/s40/s39/s57/s41/s45 /s33/s34/s35/s45/s60/s58/s41/s39/s33/s61/s32/s32/s33/s32/s34/s32/s35/s36\n/s45/s62/s63/s64/s41/s65/s43/s47/s41/s40/s42/s45\n/s45/s66/s67/s38/s64/s41/s45/s38/s40/s43/s68/s46/s33\n/s45/s45/s45/s45/s45/s45/s45/s48/s44/s38/s57/s44/s52/s57/s38/s42/s43/s46/s40/s53/s69/s33/s32\n/s54/s33/s32\n/s32/s33/s32\n/s36/s54/s33/s32/s37/s38/s39/s40/s41/s42/s43/s44/s45/s37/s46/s47/s41/s40/s42/s45/s48 /s32/s49/s50/s51/s33/s52/s33/s53\n/s54/s32/s32/s55/s32/s32/s36/s55/s32\n/s37/s38/s39/s40/s41/s42/s43/s44/s45/s56/s43/s41/s57/s58/s45/s59/s40/s39/s57/s41/s45 /s33/s34/s35/s45/s60/s58/s41/s39/s33/s61/s45/s62/s63/s64/s41/s65/s43/s47/s41/s40/s42/s45\n/s45/s66/s67/s38/s64/s41/s45/s38/s40/s43/s68/s46/s33\n/s45/s45/s45/s45/s45/s45/s45/s48/s44/s38/s57/s44/s52/s57/s38/s42/s43/s46/s40/s53/s32/s32/s33/s32/s37/s37/s32/s35/s36/s48/s38/s53\n/s48/s70/s53/s32/s33/s71\n/s32/s33/s69\n/s32/s33/s54\n/s32/s33/s32\n/s36/s32/s33/s54\n/s36/s32/s33/s69/s37/s38/s39/s40/s41/s42/s43/s44/s45/s37/s46/s47/s41/s40/s42/s45/s48 /s32/s49/s50/s51/s33/s52/s33/s53\n/s54/s32/s32/s55/s32/s32/s36/s55/s32\n/s37/s38/s39/s40/s41/s42/s43/s44/s45/s56/s43/s41/s57/s58/s45/s59/s40/s39/s57/s41/s45 /s33/s34/s35/s45/s60/s58/s41/s39/s33/s61/s32/s32/s33/s32/s38/s39/s40/s32/s35/s36\n/s45/s62/s63/s64/s41/s65/s43/s47/s41/s40/s42/s45\n/s45/s66/s67/s38/s64/s41/s45/s38/s40/s43/s68/s46/s33\n/s45/s45/s45/s45/s45/s45/s45/s48/s44/s38/s57/s44/s52/s57/s38/s42/s43/s46/s40/s53\n/s32/s33/s54/s55\n/s32/s33/s54/s32\n/s32/s33/s32/s55\n/s32/s33/s32/s32\n/s36/s32/s33/s32/s55\n/s36/s32/s33/s54/s32/s37/s38/s39/s40/s41/s42/s43/s44/s45/s37/s46/s47/s41/s40/s42/s45/s48 /s32/s49/s50/s51/s33/s52/s33/s53\n/s54/s32/s32/s55/s32/s32/s36/s55/s32\n/s37/s38/s39/s40/s41/s42/s43/s44/s45/s56/s43/s41/s57/s58/s45/s59/s40/s39/s57/s41/s45 /s33/s34/s35/s45/s60/s58/s41/s39/s33/s61/s32/s32/s33/s32/s37/s39/s34/s32/s35/s36\n/s45/s62/s63/s64/s41/s65/s43/s47/s41/s40/s42/s45\n/s45/s66/s67/s38/s64/s41/s45/s38/s40/s43/s68/s46/s33\n/s45/s45/s45/s45/s45/s45/s45/s48/s44/s38/s57/s44/s52/s57/s38/s42/s43/s46/s40/s53/s48/s44/s53\n/s48/s58/s53/s36/s32/s33/s35/s36/s32/s33/s69/s32/s33/s32/s32/s33/s69/s37/s38/s39/s40/s41/s42/s43/s44/s45/s37/s46/s47/s41/s40/s42/s45/s48 /s32/s49/s50/s51/s33/s52/s33/s53\n/s54/s32/s32/s55/s32/s32/s36/s55/s32\n/s37/s38/s39/s40/s41/s42/s43/s44/s45/s56/s43/s41/s57/s58/s45/s59/s40/s39/s57/s41/s45 /s33/s34/s35/s45/s60/s58/s41/s39/s33/s61/s72/s38/s57/s44/s33/s45/s73/s45/s62/s63/s64/s42/s33\n/s54/s33/s35/s45/s40/s47\n/s69/s33/s71/s45/s40/s47\n/s45/s45/s45/s35/s45/s40/s47\n/s45/s54/s54/s45/s40/s47/s48/s41/s53\n/s48/s51/s53/s32/s33/s34\n/s32/s33/s74\n/s32/s33/s75\n/s32/s33/s55/s76/s40/s77/s41/s65/s68/s43/s46/s40/s45/s64/s38/s65/s38/s47/s41/s42/s41/s65/s45 /s36\n/s54/s69/s34/s35/s32\n/s78/s67/s43/s44/s79/s40/s41/s68/s68/s45 /s37/s45/s48/s40/s47/s53/s54/s33/s32\n/s32/s33/s55\n/s32/s33/s32/s80/s46/s65/s47/s38/s57/s43/s81/s41/s58/s45/s38/s65/s41/s38/s45/s43/s40/s45/s64/s38/s40/s41/s57\n/s45/s48/s41/s53/s45/s58/s43/s77/s43/s58/s41/s58/s45/s70/s82/s45 /s38/s45/s48/s38/s33/s45/s52/s33/s53/s69/s33/s32\n/s54/s33/s55\n/s54/s33/s32\n/s32/s33/s55\n/s32/s33/s32/s38/s45/s52/s68/s41/s58/s45/s43/s40/s45/s42/s67/s41/s45/s44/s38/s57/s44/s52/s57/s38/s42/s43/s46/s40/s68\n/s46/s51/s45/s64/s38/s40/s41/s57/s68/s45/s48/s38/s53/s36/s48/s58/s53/s45/s45/s48 /s32/s49/s50/s51/s33/s52/s33/s53\nFIG. 3. Experimental and calculated Mproj[Mprojde\fned in Fig. 2(a)] of CoFe 2O4thin \flms. (a){(d) Experimentally\nobtainedMprojand calculations for the \flms with various thicknesses ( d= 11, 4, 2.3, and 1.4 nm). (e) Di\u000berence curves\nbetween the experimental and calculated Mprojin panels (a){(d). Note that the vertical scale gradually changes from (a) to\n(d). (f) Comparison between the di\u000berence curves shown in panels (a){(d) and the inversion parameter yreported in Ref. [10].\nNote that the di\u000berence curves [shaded area in panel (e)] have been divided by the magnetization Min each \flm.\nDATA AVAILABILITY STATEMENT\nThe data that support the \fndings of this study are\navailable from the corresponding author upon reasonable\nrequest.\n\u0003Current a\u000eliation: Nippon Steel Corporation, Steel Re-\nsearch Laboratories.\n[1] M. Tachiki, Prog. Theor. Phys. 23, 1055 (1960).\n[2] J.-B. Moussy, J. Phys. D. Appl. Phys. 46, 143001 (2013).\n[3] Z. Szotek, W. M. Temmerman, D. K odderitzsch,\nA. Svane, L. Petit, and H. Winter, Phys. Rev. B 74,\n174431 (2006), arXiv:0608168 [cond-mat].\n[4] G. A. Sawatzky, F. van der Woude, and A. H. Morrish,\nJ. Appl. Phys. 39, 1204 (1968).\n[5] M. G. Chapline and S. X. Wang, Phys. Rev. B 74, 14418\n(2006).\n[6] S. Matzen, J. B. Moussy, R. Mattana, K. Bouzehouane,\nC. Deranlot, and F. Petro\u000b, Appl. Phys. Lett. 101,\n042409 (2012).\n[7] J. S. Moodera, T. S. Santos, and T. Nagahama, J. Phys.:\nCondens. Matter 19, 165202 (2007).\n[8] A. V. Ramos, M.-J. Guittet, J.-B. Moussy, R. Mattana,\nC. Deranlot, F. Petro\u000b, and C. Gatel, Appl. Phys. Lett.\n91, 122107 (2007), arXiv:0707.3823.\n[9] Y. K. Takahashi, S. Kasai, T. Furubayashi, S. Mitani,\nK. Inomata, and K. Hono, Appl. Phys. Lett. 96, 072512(2010).\n[10] Y. K. Wakabayashi, Y. Nonaka, Y. Takeda, S. Sakamoto,\nK. Ikeda, Z. Chi, G. Shibata, A. Tanaka, Y. Saitoh,\nH. Yamagami, M. Tanaka, A. Fujimori, and R. Nakane,\nPhys. Rev. B 96, 104410 (2017).\n[11] R. Bachelet, P. de Coux, B. Warot-Fonrose, V. Skum-\nryev, G. Niu, B. Vilquin, G. Saint-Girons, and\nF. S\u0013 anchez, CrystEngComm 16, 10741 (2014).\n[12] B. T. Thole, G. van der Laan, J. C. Fuggle, G. A.\nSawatzky, R. C. Karnatak, and J.-M. Esteva, Phys. Rev.\nB32, 5107 (1985).\n[13] B. H. Frazer, B. Gilbert, B. R. Sonderegger, and G. De\nStasio, Surf. Sci. 537, 161 (2003).\n[14] G. Shibata, M. Kitamura, M. Minohara, K. Yoshi-\nmatsu, T. Kadono, K. Ishigami, T. Harano, Y. Taka-\nhashi, S. Sakamoto, Y. Nonaka, K. Ikeda, Z. Chi, M. Fu-\nruse, S. Fuchino, M. Okano, J.-i. Fujihira, A. Uchida,\nK. Watanabe, H. Fujihira, S. Fujihira, A. Tanaka, H. Ku-\nmigashira, T. Koide, and A. Fujimori, npj Quantum\nMater. 3, 3 (2018), arXiv:arXiv:1706.05183v3.\n[15] S. Sakamoto, G. Zhao, G. Shibata, Z. Deng, K. Zhao,\nX. Wang, Y. Nonaka, K. Ikeda, Z. Chi, Y. Wan,\nM. Suzuki, T. Koide, A. Tanaka, S. Maekawa, Y. J. Ue-\nmura, C. Jin, and A. Fujimori, ACS Appl. Electron.\nMater. 3, 789 (2021).\n[16] M. Furuse, M. Okano, S. Fuchino, A. Uchida, J. Fuji-\nhira, and S. Fujihira, IEEE Trans. Appl. Supercond. 23,\n4100704 (2013).\n[17] K. Amemiya, M. Sakamaki, T. Koide, K. Ito,\nK. Tsuchiya, K. Harada, T. Aoto, T. Shioya, T. Obina,\nS. Yamamoto, and Y. Kobayashi, J. Phys. Conf. Ser.5\n425, 152015 (2013).\n[18] R. Nakajima, J. St ohr, and Y. Idzerda, Phys. Rev. B 59,\n6421 (1999).\n[19] E. C. Stoner and E. P. Wohlfarth, Philos. Trans. R. Soc.\nLondon. Ser. A 240, 599 (1948).\n[20] H. Shenker, Phys. Rev. 107, 1246 (1957).\n[21] N. Da\u000b\u0013 e, F. Choueikani, S. Neveu, M.-A. Arrio, A. Juhin,\nP. Ohresser, V. Dupuis, and P. Sainctavit, J. Magn.Magn. Mater. 460, 243 (2018).\n[22] C. Moya, A. Fraile Rodr\u0013 \u0010guez, M. Escoda-Torroella,\nM. Garc\u0013 \u0010a del Muro, S. R. V. Avula, C. Piamonteze,\nX. Batlle, and A. Labarta, J. Phys. Chem. C 125, 691\n(2021).\n[23] M. Fantauzzi, F. Secci, M. Sanna Angotzi, C. Passiu,\nC. Cannas, and A. Rossi, RSC Adv. 9, 19171 (2019)."    },    {        "title": "2107.13201v1.Field_tunable_three_dimensional_magnetic_nanotextures_in_cobalt_nickel_nanowires.pdf",        "content": " \n Field tunable three -dimensional  \nmagnetic nanotextures in cobalt -nickel \nnanowires  \n \nI.M Andersen†, D. Wolf‡, L.A. Rodriguez§◊, A. Lubk‡, D. Oliveros†, C. Bran¶, T. \nNiermann♠, U.K. Rößler‡, M. Vazquez¶, C. Gatel†, E. Snoeck†  \n†Centre d’Élaboration de Matériaux et d’Etudes Structurales -CNRS, 29 rue Jeanne Marvig, 31055 Toulouse, France \n‡Leibniz I nstitute for Solid State  and Materials  Research, IFW Dresden, Helmholtzstra ße 20, 01069 Dresden, \nGermany \n§Departamento de Física  – Universidad del Valle, A.A. 25360 Cali, Colombia  \n◊Centro de Excelencia en Nuevos Materiales – Universidad del Valle, A.A. 25360 Cali, Colombia  \n¶Instituto de Ciencia de Materiales de Madrid – CSIC, Sor Juana I nés de la Cruz, 3, Madrid 28049 , Spain \n♠ Institut für Optik und Atomare Physik, Technische Universität Berlin, Straße des 17. Juni 135, 10623 Berlin, \nGermany  \n \n \n \n \nABSTRACT Cylindrical magnetic nanowires with large transversal magneto crystalline anisotropy have \nbeen shown to sustain non-trivial magnetic configurations resulting from the interplay of spatial \nconfinement, exchange , and anisotropies. Exploiting these peculiar 3D spin configurations  and their \nsolitonic inhomogeneities  are prospected to improve magnetization switching in future spintronics , such \nas power -saving magnetic memory and logic applications. Here we employ  holographic vector -field \nelectron tomography to reconstruct the  remanent magnetic states in CoNi nanowires with  10 nm \nresolution in 3D , with a particular focus on domain walls between remanent states  and ubiquitous  real-\nstructure effects stemming from irregular morphology and anisotropy variations . By tuning the applied \nmagnetic field direction , both longitudinal and transverse multi -vorte x states of different chiralities and \npeculiar 3D features such as shifted vortex cores are stabilized. The chiral domain wall between the \nlongitudinal vortices of opposite  chiralities exhibit s a complex 3D shape characterized by a push  out of the \ncentral vortex line  and a gain in exchange and anisotropy energy . A similar complex 3D texture , including \nbent vortex lines , forms  at the domain boundary between transverse -vortex states and longitudinal \nconfigurations . Micromagnetic simulations allow an understan ding of the origin of the observed complex \nmagnetic states . \n \n  \n 1. INTRODUCTION  \nFerromagnetic nanostructures have been vastly studied over the past two decades, motivated by the \nprospect of developing improved spintronic devices based on the physics of magnetic soliton  (e.g.,  domain \nwalls, bubbles, skyrmions) dynamics.1–3 Advances in nanofabrication4–7 and measurement tools and \ntechniques8–10 have not only enabled additional miniaturization  to improve the power and efficiency of \ndevices  but further  lead to the explor ation  of more sophisticated three -dimensional (3D) magnetic \nnanostructures as candidates for new types of data storage and logic device s.11–13 In particular , \ngeometrical confinement on length scales comparable to the characteristic magnetic length scales has \nproven to be a powerful concept to stabilize 3D magnetic textures with unique properties that would \notherwise not form  14,15. The complexity of the se magnetic configuration s relies on the competition \nbetween  the local exchange, anisotropies, and long -range dipole -dipole couplings enforcing magnetic flux  \nclosure.16 Moreover, localize d solitonic configurations of the magnetization in a nanoscale specimen, such \nas multifarious magnetic domain walls (DWs), generically break spatial parities present in the material. \nThis magnetic chirality has important implications for the possibility to  exert torques and to drive such \nconfigurations, e.g.,  by electrical currents or by external fields – one of the basic operations of spintronics. \nDetailed control of these specific magnetization configurations is therefore crucial for technological \nadvance s in nanomagnetism. That particularly also includes consideration of “real -structure” effects from , \ne.g.,  granular materials, misalignment of effective anisotropy axes, inhomogeneous surface anisotropies, \nmorphology variations , or local  imperfections and d efects , which  are known to exert a large influence on \nmagnetic configurations  and their properties, e.g.,  DW velocities ,17,18 in practically all realistic materials \nsystems . \nConsequently , even geometrically simple cylindrical magnetic nanowires  (NW s) of magnetic materials \nwith strong perpendicular magnetocrystalline anisotropy  have been shown to exhibit a rich phase diagram \nof m agnetic configurations ranging from uniform magnetization to longitudinal and transver se (multi -\n)domain states, depending on diameter, length, aspect ratio, externally applied field , and magnetic history  \nof the NW .19–24 Cylindrical magnetic  CoNi NWs  with a hexagonal close -packed ( hcp) crystal structure \noriented with their c-axis close to perpendicular to the NW axis are a prominent member of this class, \nwhich may be additionally tuned such to als o display face -centered cubic ( fcc) crystal structure  with \nweaker crystalline anisotropy .25–29 Previous experimental studies have revealed the existence of (A) a \nremanent longitudinal vortex state stabilized through the application of a longitudinal external field \nand/or moderate Ni -to-Co content  ratio , and, more recently, (B) a multi -vortex/multi -transverse domain \nstate  stabilized by the application of a transverse field and/or low Ni-to-Co content  ratio .30,31 It has been \nrevealed that the longitudinal vortex core is displaced off -center in response to crystalline anisotropy and \nthat particular boundary  regions  emerge between lo ngitudinal vortex states of opposite chirality and \nlongitudinal and transverse -vortex states.  \nNotwithstanding the identification of these general states and their fundamental properties, a detailed \nunderstanding of the important transition regions between  the different domains, referred to as domain \nwalls in a generalized sense in the following , is largely missing.32,33 A reason  for this  is a lack of \nexperimental techniques allowing to probe these  complex 3D nanomagnetic configurations with strong \nhysteretic character down to nanometer resolution and the elusive exploration of solitonic states with  \n solely theoretic tools (micromagneti c simulations ) in the presence of  unknown parameters such as \neffective anisotropies, real -structure effects and unknown details of the magnetic history.  Consequently, \nquestions concerning the precise magnetic configuration and energetics (e.g., the balance between \nreduced anisotropy and increased demagnetizing field) of  the DW between longitudinal vortices of \nopposite chirality  and between  regions with different crystalline texture , but also the general impact of \nmorphology, effective magnetocrystalline  anisotropies,  or defec ts remain largely unresolved in these \nsystems to date. Similarly, 3D effect s in the transverse multiple vortex state  (e.g., induced by the curvature \nof the NW) have not been addressed previously for the same reasons.  \nIn the following , we will address these  points by combining the first 3D reconstruction of pertaining \nmagnetic configurations in Co/Ni NWs at nanometers resolution with detailed micromagnetic simulations \nconsidering the experimental geometry. The step from 2D to 3D magnetic imaging represents a  huge \nmethodological challenge for the pertaining electron microscopy and x -ray imaging techniques, which \nhave been overcome only recently .34–40 Of those, holographic vector -field electron tomography (VFET)  in \nthe transmission electron microscope (TEM) currently provides for the largest spatial resolution slightly \nbelow 10 nm , which has been demonstrated  in a recent reconstruction of  all three components of the \nmagnetic induction in a cylindrical NW.41 As a high spatial resolution is paramount in this study , we \nadapted and employed VFET despite  rather strict sample criteria, such as  restrictions on the sample \nthickness  imposed by TEM . Once the sample is prepared,  the TEM instrumentation offers a large range of \ncharacterization modes exploiting the manifold electron -specimen interaction, allowing different kinds of \nmeasurements to be performed on the same sample region, including  local structural ( e.g., by nano \ndiffract ion) and compositional analysis ( e.g., by electron energy loss spectroscopy ). \nIn the following, we use the above -mentioned  methodology  to image the nanoscale material basis and \nthe magnetic configuration in a Co -based ferromagnetic cylindrical NW  with a diameter of 70 nm  \ndisplaying  a dominant transverse magnetocrystalline anisotropy , with the  hcp c-axis oriented almost \nperpendicular to the NW axis, and a definite granular crystalline microstructure. We employ holographic \nVFET to explore the 3D mag netic remanent state ( RS) of the CoNi NW, where  two characteristic different \nRSs are distinguished after applying  an external saturation field of 2 T either (i) perpendicular or (ii) parallel \nto the NW axis, referred to as RS -1 and RS -2, in the following. The experimental results are combined with \nnumerical simulations of co -existing magnetic states by micromagnetic continuum theory to resolve the ir \ncomplex nature. By microscopically resolving the magnetic configurations for two RS in such wires, we \nshow th at these states resemble earlier suggestions regarding possible ground - and metastable states but \nare considerably more complex, particular ly concerning the DWs separating the domains.  \n2. RESULTS AND DISCUSSI ON \nFigure 1 shows the morphology and structure of a 1.6 µm long segment of a representative cylindrical \nCo85Ni15 NW as determined by a combination of several TEM -based techniques. The 3D electrostatic \npotential reconstructed from holographic VFET  (see experim ental details) , displayed in Fig ure 1(a), reveals \na small deformation of the NW with respect to an ideal cylindrical geometry , where the NW  is fairly \nstraight, but with a  slightly oval -shaped and irregular  cross -section  in different regions of the wire , and a \nsplit in the NW tip. The bright -field (BF) TEM image in Fig ure 1(b) reveals a projected NW width of 70  ± 5 \nnm in the imaged region and the existence of differently oriented crystallographic grains visible due to  \n change in diffraction contrast. A detail ed structural analysis  (Figs. 1(c)-1(f)) was performed  using the \ncommercial  ASTAR  software environment , a scanning precession electron diffraction  (SPED)  assisted \nautomated crystal orientation mapping (ACOM -TEM)  (see experimental details) . The SPED results show \nthe coexistence of hcp (red) and fcc (blue) crystal phases within the studied NW segment : a central long \nhcp grain surrounded by regions with a polycrystalline texture of a mix between fcc and hcp grains . The \nhcp grain extending ~1 µm alo ng the NW axis is oriented with its c-axis (magnetocrystalline easy axis) \ntilted at a  mean  angle of 75° to the NW axis , consistent with  a previous report .22 This hcp grain is the \nregion of interest for  the following experiments.  The above variations in morphology and effective \nmagnetocrystalline anisotropy arising from different grain phases and orientations are common  for such  \nnanowires synthesized by template -assisted electrodeposition.23,42 In combination with inhomogeneous \nsurface anisotropies and crystal defects, they impose important res trictions on the magnetic configuration \nand their dynamics, which we refer to as real -structure effects in the following.   \nFigs. 1(g) and 1( h) show magnetic phase images of two principal remanent magnetic configurations, where \ntwo different RSs in the hcp grain  can be stabilized  by varying the external saturation field direction:  a \ntransverse -vortex chain22 with a perpendicular Hsat (RS-1) and longitudinal states when a parallel Hsat is \napplied (RS -2). Moreover, we observe domain boundaries between different longitudinal states and at \nboundaries between fcc and hcp grains. Note, however, that the details of the magnetic configuration, \nparticular ly at the domain boundaries, cannot be recovered from such 2D projections. To recover their  \nfull 3D texture, a tomographic reconstruction from mul tiple projections is required, as detailed below.  \n \nFigure 1:  Morphology, crystallographic structure, and magnetic phase shift of a typical CoNi N W sample after  \nappl ying  an external magnetic field in the direction perpendicular (H1) and parallel (H2) to the NW axis. ( a) Iso-surface \n \n rendering of the CoNi  NW extracted from the 3D electrostatic potential reconstructed by electron holographic \ntomography , where the scale bar marks 100 nm.  (b) Bright -field (BF) TEM image of the region of interest. ( c) \nStructura l map obtained by SPED ACOM -TEM , where blue and red color marks  the highest match for respectively fcc \nand hcp crystal phase s. (d)-(f) Crystal grain orientation maps represented by the color bar indicating  the degrees of  \ndeviation of 0001 hcp and 111 fcc crystal orientatio n with respect to the x-, y-, and z -axes. (g)-(h) Magnetic phase shift \nalong the electron propagation direction as indicated in (a) reconstructed by electron holography for the two \nremanent state s: RS-1 (H1) and RS -2 (H2) with the color scale from -4 to +4 radians . The scale bar in ( a) is common to \n(b) and equal to 200 nm.  Images from (b) to (h) have the same scale.    \n \nA. Micromagnetic simulations of idealized case  \nTo facilitate a comprehensive discussion of observed 3D RSs, particularly including domain boundaries , \nwe begin with micromagnetic simulations  using the OOMMF code package .43 These preliminary  \nmicromagnetic simulations  were performed  for an ideal ized, perfectly cylindrical CoNi NW  consisting of \nthree segments ( fcc-hcp-fcc grains)  with dimension s and morphology schematically described in Fig ure \n2(a). Although the  simulations were performed for  an idealized version of the studied CoNi NWs , they \nrepresent the experimentally observed magnetic configurations  in their basic features . Consequently, \nthey a re well suited as a starting reference to explain the  origin and for discussing the following \nexperimental results , which are additionally affected by a host of real -structure and hysteretic effec ts. \nFigs. 2(b) and 2( c) depict  3D magnetization maps of the RS s after applying a  saturation field  of 2 T  \nperpendicular ( along positive z-axis) and Figs. 2(d)-2(g) parallel (along the positive x-axis)  to the NW axis  \nto mimic the magnetic history of the experimentally observed  RSs in Fig s. 1(g) and 1( h). In the fcc grains, \na parallel magnetization alignment is created in both cases, with a magnetization flux closure at the NW \nends to minimize the magnetostatic stray fields energy.44,45 As in the experiment s, the direction of the \napplied field determines  the rem anent magnetization configuration of the hcp grain, where again two \ntypes of magnetic states are found: (RS -1) a transverse -vortex chain when a perpendicular Hsat (see Fig ure \n2(b)), and (RS -2) two longitudinal vortex states separated by a DW when a parallel Hsat is applied (see \nFigure 2(d)). These two 3D magnetic states have been observed previously  and predicted to have similar \nenergies46 in cylindrical NWs with  a diameter of around 80 nm.   \nIn RS -1, the Hsat applied perpendicular to the NW axis induced the formation of a chain of transverse -\nvortex states  with alternat ing helicity .22,29,46 The vortex cores are aligned close to the middle of the NW \nand have the same polarity  with  cores pointing in the positive z-direction, i.e., Hsat direction (see Fig ure \n2(c)). The vortices have a core -to-core distance of about 75 nm. While these characteristics are true for \nthe transverse -vortex states in the middle of the chain , the ones near the fcc grains have tilted vortex \ncores and  even  opposite polarity  (Fig. 2 (c)). The same figure also shows that the vortices approaching the \nfcc|hcp domain boundary  do not have their cores aligned at the center of the NW axis, so their \nmagnetization rotation is not symmetrical as compared to the rest of the vortices in the middle of the  \nchain. This correspond s with a similar transverse -vortex chain configuration in a study comparing \nmicromagnetic simulations with 2D magnetic imaging.22  \nRS-2 (Figure 2(e)) consists of two longitudinal vortex states32,47 with opposite chirality but  the same \npolarity, where the cores are aligned towards the Hsat direction (positive x-axis) (see Fig ure 2(e) and 2 (g)). \nThe presence of longitudinal vortices at remanence (also called vortex structure) ha s previously been  \n observed in single -crystal and textured cylindrical  Co-based  NWs with a uniaxial anisotropy.28,32 The \nmagnetization vectors  general ly have an x-component predominantly pointing to the  previously applied  \nsaturation field direction , revealing a slight canting of the vortex states . This canting  effect  of the \nlongitudinal vortex states is caused by the tilted magnetocrystalline anisotropy , which favors an alignment \nof magnetization vectors with the nearest easy axis direction (here, at 77° relative  negative x-axis, see \nSuppl. Material note 1 ). For the same reason , the vortex cores are slightly shifted off -center.  \nThe longitudinal vortex st ates are separated by a generalized  DW, where individual magnetization vectors \nrotate through a transverse state  while going from left to right vortex state with opposite chirality. \nHowever, t here is no switch of the magnetization direction of the longitud inal vortex domains  relative to \nthe NW axis , i.e., no head -to-head or tail -to-tail polarity domains like in a transverse  vortex DW. We , \ntherefore , call this a chiral DW (CDW), referencing the change in the magnetization rotation around the \nwire axis  without a switch in the polarization of the domains . At the middle of the CDW, the magnetization \nvectors are oriented parallel to each other (Figure 2 (g)) at an average angle of 50° with respect to the NW \naxis, which approach es the direction of the magneto crystalline easy axis. These  DWs were  predicted  by \nmicromagnetic simulations over a decade ago48,49 and experimentally observed  in a publication by Y . P. \nIvanov  et al.,33  however, a thorough discussion of the detailed structure or energetics (CDW energy) has \nnot been conducted yet .  \nThese idealized simulations serve as a reference  for the configuration in a real NW as obtained by VFET . \nThe expe rimental results contain distinct structural and morphological features of the NW and are \naffected by persisting limitations of the holographic VFET in terms of spatial resolution, incomplete tilt \nrange , and noise. To perform a simulation closely resemblin g the experimental case, we  conducted an \nadditional  set of micromagnetic simulations  for a pure hcp crystal structure, using the same magnetic \nparameters but dimension s and morphology of  the real NW volume  as found experimentally from the 3D \nelectrostatic potential reconstructed by VFET.  These  simulations are in astonishing agreement with the \nremanent magnetic states observed experimentally, despite  not considering local changes in crystal \norientations and grain boundaries, which demonstrates the crucial im pact of the NW morphology details .   \n  \nFigure 2:  3D magnetization of a simplified/idealized CoNi NW computed by micromagnetic simulations. (a) Scheme \nof the geometry and crystal structure  of the simulated CoNi NW. ( b,c) 3D vector map s of the RS after the application \nof a saturation field ( Hsat) perpendicular to the NW axis  with  color -coding of (b) m x and (c) m z, the reduced \ncomponents (normalized with the saturation magnetization) . (d) 3D vector map of RS after the application of Hsat \nparallel to the NW axis  with color -coded m x. (e-g) Cross -section view s of longitudinal vortex states with ( e) positive \nand ( g) negative chirality, and ( f) the chiral domain wall (CDW) in between.  \n \n \n  \n \n B. Perpendicular saturation: transverse -vortex chain  \nWe begin the discussion of the tomographically reconstruct ed magnetic induction vector -field with the \nRS-1 configuration  after applying a 2  T saturati ng magnetic field perpendicular  to the NW axis along the \nz-axis (i.e., also almost perpendicular to the  crystallographic  c-axis) . Fig ure 3 shows the main results \ncompar ed to realistic  micromagnetic simulations  based on the real morphology obtained from the \nexperimental tomogram  (see Figure 3(c)  as well as 3D iso -surface rendering in Suppl. Material Movie 1) . \nThe reconstructed  magnetic stray field  around the NW has been removed because it is influenced  by \nexperimental art ifacts due to its weaker signal -to-noise ratio compared to the magnetic flux density inside \nthe NW . In the xy-plane  (Fig. 3 (a)), a series of tran sversal vortices with alternating direction of rotation  \n(chirality)  is observed . This coincides with previous findings,22 where the magnetic configuration of the \nsame type of NWs was stu died by 2D electron holography. The xz-plane  (Fig. 3b) , which contains the \ndirection parallel to the electron beam , reveals homogeneous alternating segments of positive (blue) and \nnegative (yellow) y-components of the B-field , suggesting this transverse magnetic texture to be  virtually \ninvariant in the z-direction  (see also 3D volume rendering in Suppl. Material Movies 2,3) . Accordingly, in \nFigure 3(b), the B-field obtained from the micromagnetic simulation (corresponding magnetization  in \nFigure 2(b)) is visualized  using the same color scale and positions  as the experimental 3D data in \nFigure 3(a). The similarities between experiment and simulation are striking regarding their overall \nappearance in terms of size , e.g., 75 nm average distance between vortic es, and shape. A detailed view \ninside a single vortex , corresponding to the red and blue rectangles on experimental and simulated results , \nrespectively, is shown in Fig s. 3(d)-3(f) for all three planes xy, xz, and yz. While  the experimental data \nallows  for resolving the magnetic z-component in a few vortex cores  only , they suggest  an overall core \nalignment  in the same direction due to the applied saturation field. Based on micromagnetic simulations, \nsuch a spin configuration is characteristic for all th e vortex cores along the entire vortex chain, with the \ncore pointing in the same (positive) z-direction but with a slight continuous transition in the axial direction \ntoward the  end of the chain.  Consequently, all the vortices in the middle of the chain favor an in -plane \n(xy-plane) rotation of the magnetic vectors around a core oriented along the z-axis (out -of-plane), and \ntherefore perpendicular to both the NW axis and the alternating transversal magnetic domains.  \n  \n  \nFigure 3:  3D reconstruction of the magnetic B-field inside a CoNi NW by holographic VFET and comparison with \nmicromagnetic simulation after appl ying  a saturation field ( H) in the z-direction, i.e. , perpendicular to the NW axis  \n(indicated by black arrow) . (a) Central slices in axial (xy) and (xz) direction through the 3D B-field inside the NW \nvisualized by arrow -plots and color -coded volume rendering of  Bx and By in Tesla  (T) reveal ing a transverse vortex \nchain in the hcp region. ( b) B-field from  the micromagnetic simulation of the NW incorporating the real NW \nmorphology and assuming  hcp-single -crystal wire.  (c) Iso -surface rendering of the electrostatic potential  showing  the \nNW morphology. The positions of the axial slices  in (c) are d epicted as red and blue rectangles  in (a) and (b) . (d,e) \nZoom -in of the regions indicated in (a) and ( b). (f) Cross -sectional slice (yz) through the vortex -core marked as  a black \nvertical line in (d,e) superimposed by the color -coded 𝐵𝑧 component.  The sc ale bar in (a) is common to all images in \n(a) and (c) and is equal to 100 nm.   \nThe overall  transverse  vortex chain configuration can be compared to a double Halbach configuration,50 \nwhere the driving effect for the chain formation is the minimization of stray fields outside the structure \nby concentrating the fields within the NW. Accordingly, the configur ation can be consid ered a string of \nantiparallel perpendicular domains ( aligned with the y-axis), separated by transverse DWs , i.e., the \nvortices  pointing in the z -direction .  Furthermore, there is an almost complete absence of variations of \nthe observed t exture along the z-axis in the middle of the vortex chain.  \nApart from the transverse vortex configuration within the chain, there is also a  transition into an axial \nconfiguration at the ends of the chain . Focusing on the region at the left side of the tran sverse -vortex \n \n chain , several particular features can be distinguished from the 3D data. First of all, the magnetic \nconfiguration starts to evolve  from a transverse vortex chain in to a longitudinal vortex. This transition, \nwhich may also be considered a domain wall between transverse and longitudinal vortex state , is fully \nestablished  in the simulation , whereas the experiment shows  are more complex behavior (see further \nbelow) . When approaching the domain boundary interface , the center of th e transverse vortices  (marked \nin Figs.  3(a) and 3( b) as black circles  and blue/red vertical lines ) deviate from the NW axis in a characteristic \nintensifying zig-zag pattern . Additionally, the vortex lines are tilted with respect to the z-axis. A possible \nexplanation of this peculiar phenomenon is the accommodation of a continuous transiti on to the adjacent \nlongitudinal configuration . In the simulations this change is believed to come from changes in the NW \nshape, and the transition is from transverse -vortex  states into a longitudinal vortex domain.  The \ntopologically protected longitudinal vortex core (indicated by a red line in Fig. 3 (b) and also discussed in \nFig. 5 (f) and 5(g) ) must be expulsed from the NW , thereby distorting and shifting neighboring  vortex cores \nforming the vortex chain.  This mechanism  breaks the symmetry of the vortex chain arrangement and \npromote s a shift of the vortex cores away from the wire’s xz-mid-plane. A similar mechanism \ncharacterized by a bending  of vortex lines is also ob served in the longitudinal vortex configuration , as \ndiscussed further below.  This is believed to be the origin of the transition between longitudinal and \ntransverse vortex states, where the vortex line in a longitudinal vortex bends  out towards  the surface  of \nthe nanowire, transitioning into transverse vortex states.  An example of such a transition  observed by 2D \nelectron holography  is presented in Suppl . Material  note 5. While a similar tilting of the vortex lines is also \nobserved in the  3D experimental results, the resulting longitudinal domain is more turbulent, i.e., neither \na homogeneous parallel domain, nor a clear longitudinal vortex. This is believed to be due to the multi -\ngranular structure containing an increasing amount of fcc struct ure that proceeds the magnetic transition \nregion. This is in agreement with previous results showing variations in the curling component of \nlongitudinal vortex domains in NWs with a mixture of fcc and hcp crystal phase as discussed in a previously \nmentione d paper (Andersen et al.22).  \nBoth the simulated and the experimental vortex chain also show a slight variation in the core -to-core \ndistance, where the last couple of vortices before the  right -hand NW end are slightly more spread out. \nMoreover , the vortex chain in the experimental results contains one extra vortex core (15) compared to \nthe simulated  (14), and its effective chain region stretches over a longer part of the NW. While the gen eral \ninter -core distance is similar for the two cases, there is a difference when analyzing the region before the \nNW tip. The simulated vortex chain only display s a slight increase in distance between two vortex cores \ntowards the end of the NW (around 10% between smallest and largest spac ing), whereas the experimental \nconfiguration shows an inter -core distance of the last two vortices before the NW tip (rightmost in Fig ure \n3(a)) being  more than 4 0% of the mean spacing.  One possibl e explanation for this discrepancy is the fact \nthat the simulation does not consider local variations in crystal grains and orientations  of the real NW . By \ncorrelating the magnetic configuration (Figure 3(a)) with the crystal structure (Figures 1(c)-1(f)), the last \nvortex is loc ated in a polycrystalline region consisting of several crystal grains. In addition, the proximity \nto the NW tip and the vortex chain makes it energetically more favorable for the chain to continue past \nthe large hcp grain. Also , the relative +77° orientati on of the  c-axis with respect to  the positive or negative \nx-axis (two degrees higher tha n the average measured value)  is no longer trivial for the resulting simulated  \nRS (Suppl . Material note 1,2).  \n  \nC. Parallel saturation: longitudinal vortex domains  \nFigure 4 presents the main results of holographic VFET and micromagnetic simulations performed at \nremanence after applying the magnetic saturation field  along  the NW axis (RS -2). According  to \nmicromagnetic simulations  performed in the idealized cylindrical NW , a similar configuration is expected \nto be established even if the appl ied saturation field is oriented with only a few degrees of variations from \nthe perpendicular direction (see Suppl . Material note 3 ). This is important , as the experimental procedure \ndoes not allow for applying the external magnetic field perfectly parallel to the NW axis ( see Experimental \ndetails ). In Fig ure 4(a), the resulting magnetic B-field is visualized for both the xy- and xz-plane  (see also \n3D volume rendering in Suppl. Material  Movies 2,3) . As in the case of RS -1, the magnetic stray fields have \nbeen removed from the tomogram. In both experiment  (Figure 4 (a)) and micromagnetic  simulation  \n(Figure 4 (b)), the CDWs  are present in crossover regions  indicated by  a blue -white -yellow color map. The \nadjacent longitudinal vortex domains on both sides of the CDW are well-represented  in the  corresponding  \nxz-plane s. All vortex domains have a positive x-component of the B-field , indicating  equal polarity but \ndifferent chira lity. This is particular ly visible in the detailed view of the CDW s (Figures 4(c)-4(f)), which also \nclearly exhibit the left - and right -handed vortex lines being expulsed to the NW  surfaces because they \ncannot be transformed into each other  in a continuo us way  (see also Fig. 5) . The different rotation of the \nvortex domains can be clearly seen in the cross -sectional views in the yz-plane for both  experimental \n(Figure 4(g)) and simulat ed (Figure 4(h)) results . The longitudinal vortices are slightly canted with respect \nto the NW axis, producing  a small number of magnetic charges at the NW surface and stray fields  (Figure \n1h) outside the structure , not only at the CDW s. The canting is also linked to the magnetoc rystalline \nanisotropy easy axis of the hcp grain.   \n  \nFigure 4 . 3D reconstruction of the magnetic B-field inside a CoNi NW by holographic VFET and comparison with \nmicromagnetic simulation after appl ying  a saturation field ( H) in the x-direction  (indicated by black arrow) , i.e. , \nparallel to the NW axis. (a) Central slices in axial (xy) and (xz) direction through the 3D B-field inside the NW  (axial \nslices indicated in Fig ure 3c) visualized by arrow -plots and color -coded volume rendering of  Bx and By in Tesla (T). (b) \nB-field from micromagnetic simulation incorporating the real NW morphology and assuming hcp crystalline \nanisotropy  (parameter  khcp). The simulation results are of the same cross -sectional slices and visualized in the same \nway as the experimental data.  (c-f) Zoom -in of the regions indicated in (a) and ( b). (g,h) Cross -sectional slices (yz) at \npositions marked as black vertical lines in ( c-f) superimposed by the color -coded 𝐵𝑧 component show the transition \nbetween two vortex states with opposite chirality separated by a CDW. The scale bar in (a) is common to all images \nin (a) and ( b) and is equal to 100 nm.   \nAgain,  there are certain differences between the experiment and the real-shape simulation  in terms of \nthe number of CDWs and their positions, e.g., the obvious existence of an additional CDW in the \nexperiment  (Figure 4 (a) and 4(b)). Furthermore, the orientation s of the CDWs are more perpendicular to \nthe NW axis in the experiment compared to the simulation. Since t he observed NW is not a \nmonocrystalline homogeneous cylinder (Figure 1a -b), the DW positioning  might be affected by pinning \nsites originating from modulation s in the NW thickness or crystal phase. Conclusively, it is more \nenergetically favorable for the system to form multiple  domain s, where the DW position is influenced by \n \n a combination of the NW mor phology, as well as the position and orientations of crystal grains , thus \nexplaining  the differen ce in location and number of DWs  between the  simulated and experimental RS. \n \nD. Domain boundary / domain wall energetics  \nHaving completed the detailed description of the magnetization textures in the observed remanent states \nRS-1 (transverse -vortex chain) and RS -2 (longitudinal vortex) focusing  on the domain boundaries/walls \noccurring in these systems, we finally investigate  the formation energies of the latter. Fig. 5 (a) displays \nthe vortex line skeleton (determined by tr acing the B-field vorticity maxima ) at the boundary  between \nlongitudinal and transverse -vortex domain  (see also Fig. 3 (b), 4(b), and Suppl. Material Movies 4 and 5 ) \ntogether with the corresponding 1D micromagnetic energy densities, obtained by integrating the volume \nenergy densities obtained from the micromagnetic simulations over the yz cross  sections. While this \nparticular domain wall is not present in the 3D reconstructed NW region  because of the magnetization \nhistory and the mixed fcc and hcp phase at the left NW tail, there is plenty of experimental evidence of \nsuch a transition from other CoNi NW (regions) . An exemplary  combined holography and micromagnetic \nstudy  of such a pure hcp region , which was not deliberately polarized by an external field,  is presented in \nSuppl . Material  note 5.  \nIn Fig. 5(a), one clearly observes the reduction of the demagnetizing field energy in the tra nsverse vortex \ncores , which is compensated by an increase in exchange and anisotropy energy. The comparison of \naverage total energies between longitudinal vortex state region and the transverse -vortex chain state \nshows a slightly higher stability of the former, which explains why the transverse -vortex chain is only \nformed when the external field is perpendicular to the NW and the  crystallographic c-axis. The domain \nwall separating both regions  is comparable in total energy to the transverse vortex chain state.  Comparing \nthe domain wall (transition) between longitudinal and transverse vortex state and the CDW in Fig. 5(b) we \nfurthermore note that both a related, i.e., exhibit a similar bending of the vortex line towards the \nlongitudinal vortex.  \nThe CDW formation relies on a decrease of both exchange and anisotropy energy while increasing the \ndemagnetizing field energy compared to a single longitudinal vortex ( Fig. 5b). The reducti on of exchange \nenergy occurs due to the suppression of the vortex texture in the CDW. This promotes a parallel alignment \nof the magnetization vector , as shown in Fig. 2(f)) . The anisotropy energy is then reduced by the alignment \nof the CDW magnetization with the crystallographic c-axis. Note that the increase of the demagnetizing \nfield energy surmounts the combined exchange and anisotropy energy drop, rendering the single CDW \nconfiguration metastable. However, the RS of both real -shape simulation (Figure 4b) and experimental \nresults (Figure 4a) contains more than two longitudinal vortex domains separated by multiple CDW.  This \nconfiguration decreases the demagnetizing field energy, which helps to  stabiliz e the observed multi -CDW \nstate during creation (see also Ref. 33 for other multi -CDW states). Th e observed energetic s of the CDW \nsets it apart from conventional DW between homogeneously magnetized domains as a rather unusual \nkink or exchange and anisotropy energy gaining texture, with unique static (i.e., stray field generating) \nand dynamical properties .32  \n  \n  \nFigure 5. Vortex line skeleton and 1D energy densities of ( a) longitudinal and transverse -vortex  domain boundary and \n(b) CDW. The blue and red lines indicate the position of the clockwise  and counterclockwise rotating vortex lines as \nobtained from micromagnetic simulations.  \n \n3. SUMMARY  \nWe have conducted a  comprehensive 3D investigation of the magnet ic remanent states in cylindrical NW \nincluding pertaining domain walls  and boundaries , tunable through the orientation of an external  \nmagnetic saturation field . To this end, we  have performed 3D quantitative nanoscale magnetic imaging in \nCo-rich CoNi NWs  with a nearly perpendicular magnetocrystalline anisotropy , revealing two distinctly \ndifferent remanent configurations depending on  the direction of an applied saturation field . A transverse -\nvortex chain is formed f or a saturation field simultaneously orie nted perpendicular to  both  the NW axis \nand the magneto -crystalline c-axis, whereas for a field applied parallel to the wire axis , a longitudinal \nvortex configuration develops. The transverse -vortex chain mainly displays in -plane magnetic \ncomponents, with  a small 3D modulation  along the vortices . Characteristic  zig-zag modulations of the \nvortex lines are observed for the transverse -vortex states located close to the  chain’s extremities , \neventually rotating the lines into a longitudinal vortex . The second  remanent magnetic configuration \nconsist s of longitudinal vortex states separated by chiral DWs , characterized by an expulsion of the \nopposite vortex lines from the nanowire . The chiral domain walls are furthermore distinguished by a \nreduction of both exchange and anisotropy energy from conventional DWs. By combin ing the \nexperimental results and micromagnetic simulations using both an ideal cylindrical shape and the real -\nshape retrieved by VFET, we observed a large impact of real -structure effects on the resulti ng RSs of the \nsystem. This shows the importance of carefully control ling the system morphology for accurate \nmanipulation of its magnetic configuration.  Both magnetic conf igurations and their solitonic domain \nboundaries/walls are interesting candidates for applications such as magnetic memories. For instance , the \ndeliberate manipulation of the topologically protected CDWs , e.g., through currents or external fields , \nmay be e xploited for racetrack memories. Similarly, deliberate manipulation of the polarity in the vortex \nchain may hold potentials for extremely compact magnetic memory devices.  \n \n  \n4. EXPERIMENTAL DETAILS  \nA. Nanowire fabrication and sample preparation  \nThe Co85Ni15 nanowires were grown by electrodeposition into self -assembled pores of anodic aluminum  \noxide templates in a three -electrode cell using a Watts -type bath electrolyte. The alumina templates were \nmade by a two -step anodization process. An electrolyte composit ion of 0.124 M CoSO 4·7H 2O + 0.085 M \nCoCl 2·6H 2O + 0.064 M NiSO 4·6H 2O + 0.064 M NiCl 2·6H2O + 0.32 M H3BO 3 was used with electroplate voltage \nof -1.2 V versus the Ag/AgCl reference electrode, and a pH value of 3.0. To enable TEM investigation of \nindividual wires, the electrodeposited NWs were then released from the template by chemically dissolving \nthe alumina  in a chromic oxide and phosphoric acid  solution . Finally, they were transferred to a copper \nTEM -grid (100 -mesh hexagonal) with Quantifoil S7/2 carbon support and 2 nm carbon film (Quantifoil \nMicro Tools, Jena, Germany).  \nB. Application of the external magn etic field s \nA magnetic field of 2 T into the image plane direction was applied using the objective lens of the TEM \ninstrument prior to the measurements.  The effective magnetic component relative the sample axis can \nthen be controlled by tilting the sample plane. Where H OL is the amplitude of the magnetic field generated \nby the objective lens, we will then get that H 1 = H, while H 2 = H sin , where  is the sample tilt.  Specifically, \nthe external magnetic field was applied at zero tilt (I), and at 70° tilt angle with tilt axis perpendicular to \nNW axis such that the NW tip goes into or out of the image plane. This corresponds to the application of \na magnetic saturation field (I) perpendicular and (II) parallel to the nanowire axis, respectively. The \nmagnetocr ystalline easy axis has an orientation close to perpendicular to both the electron beam direction \nand the NW axis (Figure 1 (c)-1(f)), meaning that both the applied fields were close to perpendicular to the \nc-axis.  \nC. Quantitative 3D magnetic imaging  \nHolographic VFET was performed on the studied sample to obtain the magnetic induction of single \nnanowires. The holographic tilt series were recorded on a n FEI Titan 80 -300 Holography Special Berlin \nTEM instrument equipped with a 2K slow -scan CCD camera (Ga tan Inc .’s Ultrascan 1000P) operated at \n300 kV in image -corrected Lorentz -mode using a double biprism setup.51 The holograms were acquired \nwith a pixel size of 0.67 nm and an interference fringe spacing of 4.95 pixels per fringe. A dedicated dual -\naxis tomography sample holder (Model 2040 from E.A. Fischione Instruments, Inc.) was used for \nestablishing high tilt angles and manual in -plane rotation as required for recording a dual -tilt-axis \ntomography tilt series.41  For each remanent sta te of the NW (RS -1 and RS -2), a hologram tilt series were \nrecorded around two different axes rotated by 90° in -plane with respect to each other. Then, in the case \nof RS -1, the sample was flipped upside down, and another two tilt series were acquired around  exactly \nthe same tilt axes as in the non -flipped case. All six series covering a tilt range from -66° to +66° with a \nstep size of 3° were obtained by an in -house built semi -automatic software (THOMAS) that efficiently \nperforms and monitors the tilting pro cess and acquisition of object holograms and reference holograms.52 \nFlipped and non -flipped phase image series were corrected for image distortions ( see Suppl . Material  note \n4) and displacement, subtracted and added in order to separate the magnetic and the electric \ncontributions to the recorded phase images retrieved from the holograms by Fourier reconstruction.53  \n From the magnetic phase tilt series, the Bx (para llel component to the tilt axis) and By (parallel component \nto the tilt axis after 90° in -plane rotation) components were reconstructed in 3D using weighted \nsimultaneous iterative reconstruction technique (W -SIRT).54 From the electric phase tilt serie s, the \nelectrostatic potential  was reconstructed in 3D using W -SIRT.  Because the experimental conditions of \nthe VFET experiment (tilt range, tilt steps, resolution of the projection data , object size ) were very similar \nto those reported in Ref. 41, we ca n assume the same spatial resolution  as determined in Ref. 41, i.e., \n10nm in x- and y-direction  and slightly reduced resolution in z-direction due to the limited tilt range.  \nD. Mapping of nanowire texture  \nThe crystal grains and orientations of the studied nanowire were mapped using  scanning  precession \nelectron diffraction ( SPED)55,56 with the commercial software package NanoMegas ASTAR ( the product \nname) , an automated crystal orientat ion mapping (ACOM -TEM ) system .57 The ACOM technique  consist s \nof scanning the defined sample area and  acquiring  PED pattern s of each defined pixel in the sample  scan . \nThese patterns contain the crystallographic information of the area, allowing the use r to obtain the crystal \nstructure and orientation information, by an automatic comparison  of the experimental and theoretical \ndiffraction patterns  for any given material . The mapping was done  on a Philips CM20 -FEG TEM operated \nat 200 kV with a condenser aperture of 50 µm. SPED patterns were acquired at a camera length of 235 \nmm and a magnification  of 58 kx , scanned with an  electron beam of ~1 nm spot size with a step size of 4 \nnm (pixel of 4 nm2) over  the sample and a precession angle of 0.7°.  \nE. Micromagnetic simulations  \nThe micromagnetic simulations were performed by using the OOMMF code.58 We simulate the rem anent \nstate of an idealized 70 -nm-diameter cylindrical CoNi nanowire formed by two crystal phases , and a real -\nshape and hcp-single -crystal CoNi NW. The real -shape NW morphology was obtained from iso -surface \nrendering extracted from the 3D electrostatic pot ential retrieved by holographic VFET. Imitating what was \ndone experimentally, simulated remanent states were obtained by appl ying a saturation field of 2 T in \ntwo different directions: perpendicular and parallel to the NW axis. Considering that the deposit ion \nprocess produces CoNi NWs with possibly coexisting crystal phases ( fcc and hcp),22 the idealized  NW is a \ntextured structure formed by two 200 nm long cylindrical fcc grains located a t each end of the wire, with \na 1200 nm long cylindrical hcp grain in between them (Figure 2 (a)). For the magnetic parameters, we \nconsider both crystal phases to have the saturation magnetization ( MS) and exchange constant ( A) of the \nCo85Ni15 alloy reported by Samardak et al.23 (MS = 1273 kA/m; A = 26 pJ/m); a cubic anisotropy with an \nanisotropy constant Kfcc = 100 kJ/m3 for the fcc grains;59 and a uniaxial anisotropy with an anisotropy \nconstant Khcp = 350 kJ/m3 for the hcp grain. An easy axis orientation at 81° relative to the NW axis was \nchosen for the fcc phase considering the nanodiffraction (ASTAR) measurements performed on the \nstudied NW and another NW containing a long fcc grain of a NW from the same batch as the presented \nNW (see Supporting  Material note  1), while an easy axis orientation at 77° relative to NW axis was tuned \nfor the hcp phase, for both idealized and real -shape NWs, considering ASTAR measurements and \nsimulations at different e asy axis orientations. The cylindrical and real shape of the NW s were generated \nby stacking magnetic unit cells of sizes 5  5  5 nm3 and 2.7  2.7  2.7 nm3, respectively.   \n  \nASSOCIATED CONTENT  \nSupplemental Material  accompanies this paper  and contains additional details about the \nexperimental methods, simulations , and data analysis process.  \n Note 1: Easy axis direction of CoNi nanowires for micromagnetic simulations . \n Note 2: Shape effects and magnetocrystalline easy axis orientation study by \nmicromagne tic simulations.  \n Note 3:  Simulated remanent states of the system with different saturation field \ndirection . \n Note 4: Correction of anisotropic magnification distortion in phase images . \n Movie 1: CoNi nanowire tomogram of  mean inner potential . \n Movie 2: Co Ni nanowire tomogram of magnetic induction x -component . \n Movie 3: CoNi nanowire tomogram of magnetic induction y -component . \nMovies 4,  5: CoNi nanowire simulation of magnetic induction y -component with \nextracted vortex lines.  \n \nConflict of interest:  The authors declare no financial/commercial Conflict of Interest.  \n \nACKNOWLEDGEMENTS  \nThe research leading to these results has received funding from the European Union Horizon 2020 \nresearch and innovation program under grant agreement No. 823717 – ESTEEM3. We e xpress our \ngratitude to M. Lehmann (TU Berlin) for providing access to the FEI Titan 80 –300 Holography Special \nBerlin. D.W.  and A.L. acknowledge funding from the European Research Council (ERC) under the \nHorizon 2020 research and innovation program of the  European Union (grant agreement number \n715620 ). C.B and M. V. acknowledge funding from Spanish MINECO under Project MAT2016 -76824 -C3-\n1-R and PID2019 -108075RB -C31, and the Regional Government of Madrid under Project S2018/NMT -\n4321 NANOMAGCOSTCM.  \n \nAUTHOR CO NTRIBUTIONS  \nI.M.A., D.W., L.A.R., and A.L. have contributed equally to this work. I.M.A., D.W and T.N. performed the \nholographic VFET experiments, and D.W aligned, reconstructed, and evaluated the 3D data. L.A.R. \nperformed the micromagnetic simulations. D. O. and I.M.A. performed the ASTAR© measurements and \nthe associated data treatment. C.B. fabricated the NWs. A.L. contributed with  data analysis  and \nmagnetic calculations. I.M.A., A.L., D.W , and L.A.R. collaborated on the main discussions of the results  \n and wrote the preliminary manuscript. U.K.R., M.V., C.G. , and E.S. contributed to discussions and the \ndevelopment of the manuscript.  \n \nREFERENCES  \n(1)  Forster, H.; Schrefl, T.; Suess, D.; Scholz, W.; Tsiantos, V.; Dittrich, R.; Fidler, J. Domain Wall \nMotion in Nanowires Using Moving Grids (Invited). J. Appl. Phys.  2002 , 91 (10), 6914. \nhttps://doi.org/10.1063/1.1452189.  \n(2)  Burrowes, C.; Mihai, A. P.; Ra velosona, D.; Kim, J. -V.; Chappert, C.; Vila, L.; Marty, A.; Samson, Y.; \nGarcia -Sanchez, F.; Buda -Prejbeanu, L. D.; Tudosa, I.; Fullerton, E. E.; Attané, J. -P. Non -Adiabatic \nSpin -Torques in Narrow Magnetic Domain Walls. Nat. Phys.  2010 , 6 (1), 17 –21. \nhttps ://doi.org/10.1038/nphys1436.  \n(3)  Sander, D.; Valenzuela, S. O.; Makarov, D.; Marrows, C. H.; Fullerton, E. E.; Fischer, P.; McCord, J.; \nVavassori, P.; Mangin, S.; Pirro, P.; al.,  et. The 2017 Magnetism Roadmap. J. Phys. Appl. Phys.  \n2017 , 50, 363001. htt ps://doi.org/10.1088/1361 -6463/aa81a1.  \n(4)  Phatak, C.; Liu, Y.; Gulsoy, E. B.; Schmidt, D.; Franke -Schubert, E.; Petford -Long, A. Visualization of \nthe Magnetic Structure of Sculpted Three -Dimensional Cobalt Nanospirals. Nano Lett.  2014 , 14 \n(2), 759 –764. h ttps://doi.org/10.1021/nl404071u.  \n(5)  Ivanov, Y. P.; Chuvilin, A.; Lopatin, S.; Kosel, J. Modulated Magnetic Nanowires for Controlling \nDomain Wall Motion: Toward 3D Magnetic Memories. ACS Nano  2016 , 10 (5), 5326 –5332. \nhttps://doi.org/10.1021/acsnano.6b01337.  \n(6)  Pablo -Navarro, J.; Sangiao, S.; Magén, C.; de Teresa, J. M. Diameter Modulation of 3D \nNanostructures in Focused Electron Beam Induced Deposition Using Local Electric Fields and \nBeam Defocus. Nanotechnology  2019 , 30 (50), 505302. https://doi.org/10.1088/1361 -\n6528/ab423c.  \n(7)  Fernández -Pacheco, A.; Serrano -Ramón, L.; Michalik, J. M.; Ibarra, M. R.; Teresa, J. M. D.; O’Brien, \nL.; Petit, D.; Lee, J.; Cowburn, R. P. Three Dimensional Magnetic Nanowi res Grown by Focused \nElectron -Beam Induced Deposition. Sci. Rep.  2013 , 3, 1492. https://doi.org/10.1038/srep01492.  \n(8)  Lichte, H. Electron Holography: Optimum Position of the Biprism in the Electron Microscope. \nUltramicroscopy  1996 , 64 (1), 79 –86. https:/ /doi.org/10.1016/0304 -3991(96)00017 -4. \n(9)  McCartney, M. R.; Smith, D. J. Electron Holography: Phase Imaging with Nanometer Resolution. \nAnnu. Rev. Mater. Res.  2007 , 37 (1), 729 –767. \nhttps://doi.org/10.1146/annurev.matsci.37.052506.084219.  \n(10)  Donnelly, C.; Finizio, S.; Gliga, S.; Holler, M.; Hrabec, A.; Odstrčil, M.; Mayr, S.; Scagnoli, V.; \nHeyderman, L. J.; Guizar -Sicairos, M.; Raabe, J. Time -Resolved Imaging of Three -Dimensional \nNanoscale Magnetization Dynamics. Nat. Nanotechnol.  2020 , 15 (5), 356 –360.  \nhttps://doi.org/10.1038/s41565 -020-0649 -x. \n(11)  Parkin, S. S. P.; Hayashi, M.; Thomas, L. Magnetic Domain -Wall Racetrack Memory. Science  2008 , \n320 (5873), 190 –194. https://doi.org/10.1126/science.1145799.  \n(12)  Mattheis, R.; Glathe, S.; Diegel, M.; Hübne r, U. Concepts and Steps for the Realization of a New \nDomain Wall Based Giant Magnetoresistance Nanowire Device: From the Available 24 Multiturn \nCounter to a 212 Turn Counter. J. Appl. Phys.  2012 , 111 (11), 113920. \nhttps://doi.org/10.1063/1.4728991.  \n(13)  Hoffmann, A.; Bader, S. D. Opportunities at the Frontiers of Spintronics. Phys. Rev. Appl.  2015 , 4 \n(4), 047001. https://doi.org/10.1103/PhysRevApplied.4.047001.   \n (14)  Fischer, P.; Sanz -Hernández, D.; Streubel, R.; Fernández -Pacheco, A. Launching a New Dime nsion \nwith 3D Magnetic Nanostructures. APL Mater.  2020 , 8 (1), 010701. \nhttps://doi.org/10.1063/1.5134474.  \n(15)  Stano, M.; Fruchart, O. Magnetic Nanowires and Nanotubes. ArXiv180804656 Cond -Mat  2018 , \n27, 155 –267. https://doi.org/10.1016/bs.hmm.2018.08.002.  \n(16)  Arrott, A. S. Micromagnetism as a Prototype for Complexity. In Spintronics Handbook: \nSpintransport and Magnetism ; CRC Press, 2019; Vol. 1, p 95. \nhttps://doi.org/10.1201/9780429423079 -3. \n(17)  Garcia -Sanchez, F.; Szambolics, H.; Mihai, A. P.; Vila, L .; Marty, A.; Attané, J. -P.; Toussaint, J. -Ch.; \nBuda -Prejbeanu, L. D. Effect of Crystalline Defects on Domain Wall Motion under Field and \nCurrent in Nanowires with Perpendicular Magnetization. Phys. Rev. B  2010 , 81 (13), 134408. \nhttps://doi.org/10.1103/Phy sRevB.81.134408.  \n(18)  Thiaville, A.; Nakatani, Y.; Miltat, J.; Suzuki, Y. Micromagnetic Understanding of Current -Driven \nDomain Wall Motion in Patterned Nanowires. Europhys. Lett. EPL  2005 , 69 (6), 990 –996. \nhttps://doi.org/10.1209/epl/i2004 -10452 -6. \n(19)  Ivanov, Y. P.; Trabada, D. G.; Chuvilin, A.; Kosel, J.; Chubykalo -Fesenko, O.; Vázquez, M. \nCrystallographically Driven Magnetic Behaviour of Arrays of Monocrystalline Co Nanowires. \nNanotechnology  2014 , 25 (47), 475702. https://doi.org/10.1088/0957 -4484/25/ 47/475702.  \n(20)  Cantu -Valle, J.; Betancourt, I.; Sanchez, J. E.; Ruiz -Zepeda, F.; Maqableh, M. M.; Mendoza -\nSantoyo, F.; Stadler, B. J. H.; Ponce, A. Mapping the Magnetic and Crystal Structure in Cobalt \nNanowires. J. Appl. Phys.  2015 , 118 (2), 024302. http s://doi.org/10.1063/1.4923745.  \n(21)  Arshad, M. S.; Proenca, M. P.; Trafela, S.; Neu, V.; Wolff, U.; Stienen, S.; Vazquez, M.; Kobe, S.; \nRožman, K. Ž. The Role of the Crystal Orientation ( c -Axis) on Switching Field Distribution and the \nMagnetic Domain Co nfiguration in Electrodeposited Hcp Co –Pt Nanowires. J. Phys. Appl. Phys.  \n2016 , 49 (18), 185006. https://doi.org/10.1088/0022 -3727/49/18/185006.  \n(22)  Andersen, I. M.; Rodríguez, L. A.; Bran, C.; Marcelot, C.; Joulie, S.; Hungria, T.; Vazquez, M.; Gatel, \nC.; Snoeck, E. Exotic Transverse -Vortex Magnetic Configurations in CoNi Nanowires. ACS Nano  \n2020 , 14 (2), 1399 –1405. https://doi.org/10.1021/acsnano.9b07448.  \n(23)  Samardak, A. S.; Ognev, A. V.; Samardak, A. Yu.; Stebliy, E. V.; Modin, E. B.; Chebotkevich, L. A.; \nKomogortsev, S. V.; Stancu, A.; Panahi -Danaei, E.; Fardi -Ilkhichy, A.; Nasirpouri, F. Variation of \nMagnetic Anisotropy and Temperature -Dependent FORC Probing of Compositionally Tuned Co -Ni \nAlloy Nanowires. J. Alloys Compd.  2018 , 732, 683 –693. \nhttps: //doi.org/10.1016/j.jallcom.2017.10.258.  \n(24)  Liu, Z.; Chang, P. -C.; Chang, C. -C.; Galaktionov, E.; Bergmann, G.; Lu, J. G. Shape Anisotropy and \nMagnetization Modulation in Hexagonal Cobalt Nanowires. Adv. Funct. Mater.  2008 , 18 (10), \n1573 –1578. https://d oi.org/10.1002/adfm.200701010.  \n(25)  García, J.; Vega, V.; Iglesias, L.; Prida, V. M.; Hernando, B.; Barriga‐Castro, E. D.; Mendoza‐\nReséndez, R.; Luna, C.; Görlitz, D.; Nielsch, K. Template -Assisted Co –Ni Alloys and \nMultisegmented Nanowires with Tuned Magn etic Anisotropy. Phys. Status Solidi A  2014 , 211 (5), \n1041 –1047. https://doi.org/10.1002/pssa.201300731.  \n(26)  Vega, V.; Böhnert, T.; Martens, S.; Waleczek, M.; Montero -Moreno, J. M.; Görlitz, D.; Prida, V. M.; \nNielsch, K. Tuning the Magnetic Anisotropy of  Co–Ni Nanowires: Comparison between Single \nNanowires and Nanowire Arrays in Hard -Anodic Aluminum Oxide Membranes. Nanotechnology  \n2012 , 23 (46), 465709. https://doi.org/10.1088/0957 -4484/23/46/465709.  \n(27)  Cho, J. U.; Wu, J. -H.; Min, J. H.; Ko, S. P.; Soh , J. Y.; Liu, Q. X.; Kim, Y. K. Control of Magnetic \nAnisotropy of Co Nanowires. J. Magn. Magn. Mater.  2006 , 303 (2), e281 –e285. \nhttps://doi.org/10.1016/j.jmmm.2006.01.082.   \n (28)  Bran, C.; Fernandez -Roldan, J. A.; Palmero, E. M.; Berganza, E.; Guzman, J.; d el Real, R. P.; Asenjo, \nA.; Fraile Rodríguez, A.; Foerster, M.; Aballe, L.; Chubykalo -Fesenko, O.; Vazquez, M. Direct \nObservation of Transverse and Vortex Metastable Magnetic Domains in Cylindrical Nanowires. \nPhys. Rev. B  2017 , 96 (12), 125415. https://doi.org/10.1103/PhysRevB.96.125415.  \n(29)  Bran, C.; Fernandez -Roldan, J. A.; P. Del Real, R.; Asenjo, A.; Chen, Y. -S.; Zhang, J.; Zhang, X.; Fraile \nRodríguez, A.; Foerster, M.; Aballe, L.; Chubykalo -Fesenko, O.; Vazquez, M. Unveiling  the Origin of \nMultidomain Structures in Compositionally Modulated Cylindrical Magnetic Nanowires. ACS Nano  \n2020 , 14 (10), 12819 –12827. https://doi.org/10.1021/acsnano.0c03579.  \n(30)  Berganza Eguiarte, E.; Jaafar, M.; Bran, C.; Fernandez -Roldan, J.; Chubyk alo-Fesenko, O.; Vázquez, \nM.; Asenjo, A. Multisegmented Nanowires: A Step towards the Control of the Domain Wall \nConfiguration. Sci. Rep.  2017 , 7, 11576. https://doi.org/10.1038/s41598 -017-11902 -w. \n(31)  Vivas, L. G.; Vazquez, M.; Escrig, J.; Allende, S.; Altbir, D.; Leitao, D. C.; Araujo, J. P. Magnetic \nAnisotropy in CoNi Nanowire Arrays: Analytical Calculations and Experiments. Phys. Rev. B  2012 , \n85 (3), 035439. https://doi.org/10.1103/PhysRevB.85.035439.  \n(32)  Ivanov, Y. P.; Vázquez, M.; Chubykalo -Fesenk o, O. Magnetic Reversal Modes in Cylindrical \nNanowires. J. Phys. Appl. Phys.  2013 , 46 (48), 485001. https://doi.org/10.1088/0022 -\n3727/46/48/485001.  \n(33)  Ivanov, Y. P.; Chuvilin, A.; Vivas, L. G.; Kosel, J.; Chubykalo -Fesenko, O.; Vázquez, M. Single \nCrysta lline Cylindrical Nanowires – toward Dense 3D Arrays of Magnetic Vortices. Sci. Rep.  2016 , \n6, 23844. https://doi.org/10.1038/srep23844.  \n(34)  Chapman, J. N.; Scheinfein, M. R. Transmission Electron Microscopies of Magnetic \nMicrostructures. J. Magn. Magn. Mater.  1999 , 200 (1), 729 –740. https://doi.org/10.1016/S0304 -\n8853(99)00317 -0. \n(35)  Biziere, N.; Gatel, C.; Lassalle -Balier, R.; Clochard, M. C.; Wegrowe, J. E.; Snoeck, E. Imaging the \nFine Structure of a Magnetic Domain Wall in a Ni Nanocylinder. Nano Let t. 2013 , 13 (5), 2053 –\n2057. https://doi.org/10.1021/nl400317j.  \n(36)  Wartelle, A.; Pablo -Navarro, J.; Staňo, M.; Bochmann, S.; Pairis, S.; Rioult, M.; Thirion, C.; Belkhou, \nR.; Teresa, J. M. de; Magén, C.; Fruchart, O. Transmission XMCD -PEEM Imaging of an Engineered \nVertical FEBID Cobalt Nanowire with a Domain Wall. Nanotechnology  2018 , 29 (4), 045704. \nhttps://doi.org/10.1088/1361 -6528/aa9eff.  \n(37)  Jaafar, M.; Serrano -Ramón, L.; Iglesias -Freire, O.; Fernández -Pacheco, A.; Ibarra, M. R.; De Teresa, \nJ. M.; A senjo, A. Hysteresis Loops of Individual Co Nanostripes Measured by Magnetic Force \nMicroscopy. Nanoscale Res. Lett.  2011 , 6 (1), 407. https://doi.org/10.1186/1556 -276X -6-407.  \n(38)  Rodríguez, L. A.; Magén, C.; Snoeck, E.; Serrano -Ramón, L.; Gatel, C.; Córd oba, R.; Martínez -\nVecino, E.; Torres, L.; De Teresa, J. M.; Ibarra, M. R. Optimized Cobalt Nanowires for Domain Wall \nManipulation Imaged by in Situ Lorentz Microscopy. Appl. Phys. Lett.  2013 , 102 (2), 022418. \nhttps://doi.org/10.1063/1.4776709.  \n(39)  McViti e, S.; McGrouther, D.; McFadzean, S.; MacLaren, D. A.; O’Shea, K. J.; Benitez, M. J. \nAberration Corrected Lorentz Scanning Transmission Electron Microscopy. Ultramicroscopy  2015 , \n152, 57–62. https://doi.org/10.1016/j.ultramic.2015.01.003.  \n(40)  Donnelly, C.; Gliga, S.; Scagnoli, V.; Holler, M.; Raabe, J.; Heyderman, L. J.; Guizar -Sicairos, M. \nTomographic Reconstruction of a Three -Dimensional Magnetization Vector Field. New J. Phys.  \n2018 , 20 (8), 083009. https://doi.org/10.1088/1367 -2630/aad35a.  \n(41)  Wolf, D.; Biziere, N.; Sturm, S.; Reyes, D.; Wade, T.; Niermann, T.; Krehl, J.; Warot -Fonrose, B.; \nBüchner, B.; Snoeck, E.; Gatel, C.; Lubk, A. Holographic Vector Field Electron Tomography of \nThree -Dimensional Nanomagnets. Commun. Phys.  2019 , 2 (1), 1 –9. \nhttps://doi.org/10.1038/s42005 -019-0187 -8.  \n (42)  Barriga -Castro, E. D.; García, J.; Mendoza -Reséndez, R.; Prida, V. M.; Luna, C. Pseudo -\nMonocrystalline Properties of Cylindrical Nanowires Confinedly Grown by Electrodeposition in \nNanoporous Alumina Temp lates. RSC Adv.  2017 , 7, 13817 –13826. \nhttps://doi.org/10.1039/C7RA00691H.  \n(43)  Donahue, M. J.; Porter, D. G. OOMMF User’s Guide, Version 1.0 ; Interagency Report; National \nInstitute of Standards and Technology: Gaithersburg, MD, 1999.  \n(44)  Wolf, D.; Rodri guez, L. A.; Béché, A.; Javon, E.; Serrano, L.; Magen, C.; Gatel, C.; Lubk, A.; Lichte, \nH.; Bals, S.; Van Tendeloo, G.; Fernández -Pacheco, A.; De Teresa, J. M.; Snoeck, E. 3D Magnetic \nInduction Maps of Nanoscale Materials Revealed by Electron Holographic T omography. Chem. \nMater.  2015 , 27 (19), 6771 –6778. https://doi.org/10.1021/acs.chemmater.5b02723.  \n(45)  Usov, N. A.; Zhukov, A.; Gonzalez, J. Remanent Magnetization States in Soft Magnetic Nanowires. \nIEEE Trans. Magn.  2006 , 42 (10), 3063 –3065. https://doi.o rg/10.1109/TMAG.2006.880104.  \n(46)  Lebecki, K. M.; Donahue, M. J. Comment on ``Frustrated Magnetization in Co Nanowires: \nCompetition between Crystal Anisotropy and Demagnetization Energy’’. Phys. Rev. B  2010 , 82 (9), \n096401. https://doi.org/10.1103/PhysRev B.82.096401.  \n(47)  Fernandez -Roldan, J. A.; P. del Real, R.; Bran, C.; Vazquez, M.; Chubykalo -Fesenko, O. Electric \nCurrent and Field Control of Vortex Structures in Cylindrical Magnetic Nanowires. Phys. Rev. B  \n2020 , 102 (2), 024421. https://doi.org/10.1103 /PhysRevB.102.024421.  \n(48)  Vila, L.; Darques, M.; Encinas, A.; Ebels, U.; George, J. -M.; Faini, G.; Thiaville, A.; Piraux, L. \nMagnetic Vortices in Nanowires with Transverse Easy Axis. Phys. Rev. B  2009 , 79 (17), 172410. \nhttps://doi.org/10.1103/PhysRevB.79 .172410.  \n(49)  Chandra Sekhar, M.; Liew, H. F.; Purnama, I.; Lew, W. S.; Tran, M.; Han, G. C. Helical Domain Walls \nin Constricted Cylindrical NiFe Nanowires. Appl. Phys. Lett.  2012 , 101 (15), 152406. \nhttps://doi.org/10.1063/1.4758469.  \n(50)  Majernik, N.; R osenzweig, J. B. Halbach Undulators Using Right Triangular Magnets. Phys. Rev. \nAccel. Beams  2019 , 22 (9), 092401. https://doi.org/10.1103/PhysRevAccelBeams.22.092401.  \n(51)  Harada, K.; Akashi, T.; Togawa, Y.; Matsuda, T.; Tonomura, A. Optical System for Do uble -Biprism \nElectron Holography. J. Electron Microsc. (Tokyo)  2005 , 54 (1), 19 –27. \nhttps://doi.org/10.1093/jmicro/dfh098.  \n(52)  Wolf, D.; Lubk, A.; Lichte, H.; Friedrich, H. Towards Automated Electron Holographic Tomography \nfor 3D Mapping of Electrostatic  Potentials. Ultramicroscopy  2010 , 110 (5), 390 –399. \nhttps://doi.org/10.1016/j.ultramic.2009.12.015.  \n(53)  Lehmann, M.; Lichte, H. Tutorial on Off -Axis Electron Holography. Microsc. Microanal.  2002 , 8 (6), \n447–466. https://doi.org/10.1017/S1431927602020147 . \n(54)  Wolf, D.; Lubk, A.; Lichte, H. Weighted Simultaneous Iterative Reconstruction Technique for \nSingle -Axis Tomography. Ultramicroscopy  2014 , 136, 15–25. \nhttps://doi.org/10.1016/j.ultramic.2013.07.016.  \n(55)  Rauch, E. F.; Portillo, J.; Nicolopoulos, S. ; Bultreys, D.; Rouvimov, S.; Moeck, P. Automated \nNanocrystal Orientation and Phase Mapping in the Transmission Electron Microscope on the \nBasis of Precession Electron Diffraction. Z. Für Krist.  2010 , 225, 103 –109. \nhttps://doi.org/10.1524/zkri.2010.1205.  \n(56)  Viladot, D.; Véron, M.; Gemmi, M.; Peiró, F.; Portillo, J.; Estradé, S.; Mendoza, J.; Llorca -Isern, N.; \nNicolopoulos, S. Orientation and Phase Mapping in the Transmission Electron Microscope Using \nPrecession -Assisted Diffraction Spot Recognition: Sta te-of-the-Art Results: REVIEW OF PACOM \n(ASTAR) APPLICATION. J. Microsc.  2013 , 252, 23–34. https://doi.org/10.1111/jmi.12065.  \n(57)  Moeck, P.; Rouvimov, S.; Rauch, E. F.; Véron, M.; Kirmse, H.; Häusler, I.; Neumann, W.; Bultreys, \nD.; Maniette, Y.; Nicolopou los, S. High Spatial Resolution Semi‐automatic Crystallite Orientation  \n and Phase Mapping of Nanocrystals in Transmission Electron Microscopes. Cryst. Res. Technol.  \n2011 , 46, 589 –606. https://doi.org/10.1002/crat.201000676.  \n(58)  Donahue, M. J. OOMMF User’s  Guide, Version 1.0. - 6376  1999 . \nhttp://dx.doi.org/10.1002/https://dx.doi.org/10.6028/NIST.IR.6376.  \n(59)  Carl, A.; Weller, D.; Savoy, R.; Hillebrands, B. HCP -FCC Phase Transition in Co -Ni Alloys Studied \nwith Magneto -Optical Kerr Spectroscopy. MRS Online Proc. Libr. Arch.  1994 , 343. \nhttps://doi.org/10.1557/PROC -343-351.  \n "    },    {        "title": "2108.02926v2.Strong_magneto_optical_and_anomalous_transport_manifestations_in_two_dimensional_van_der_Waals_magnets_Fe__n_GeTe__2____n____3__4__5_.pdf",        "content": "Strong magneto-optical and anomalous transport manifestations in two-dimensional\nvan der Waals magnets Fe nGeTe 2(n= 3, 4, 5)\nXiuxian Yang,1, 2, 3Xiaodong Zhou,1, 2Wanxiang Feng,1, 2,\u0003and Yugui Yao1, 2\n1Centre for Quantum Physics, Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurement (MOE),\nSchool of Physics, Beijing Institute of Technology, Beijing, 100081, China\n2Beijing Key Lab of Nanophotonics & Ultra\fne Optoelectronic Systems,\nSchool of Physics, Beijing Institute of Technology, Beijing, 100081, China\n3Kunming Institute of Physics, Kunming 650223, China\n(Dated: October 6, 2021)\nTwo-dimensional ferromagnetic materials have recently been attracted much attention mainly due\nto their promising applications towards multi-functional devices, e.g., microelectronics, spintronics,\nand thermoelectric devices. Utilizing the \frst-principles calculations together with the group theory\nanalysis, we systematically investigate the magnetocrystalline anisotropy energy, magneto-optical\ne\u000bect, and anomalous transport properties (including anomalous Hall, Nernst, and thermal Hall\ne\u000bects) of monolayer and bilayer Fe nGeTe 2(n= 3, 4, 5). The monolayer Fe nGeTe 2(n= 3, 4,\n5) exhibits the out-of-plane, in-plane, and in-plane ferromagnetic orders with considerable mag-\nnetocrystalline anisotropy energies of -3.17, 4.42, and 0.58 meV/f.u., respectively. Ferromagnetic\norder is predicted in bilayer Fe 4GeTe 2while antiferromagnetic order prefers in bilayer Fe 3GeTe 2and\nFe5GeTe 2. The group theory analysis reveals that in addition to monolayer ferromagnetic Fe nGeTe 2\n(n= 3, 4, 5), the magneto-optical and anomalous transport phenomena surprisingly exist in bilayer\nantiferromagnetic Fe 5GeTe 2, which is much rare in realistic collinear antiferromagnets. If spin mag-\nnetic moments of monolayer and bilayer Fe nGeTe 2are reoriented from the in-plane to out-of-plane\ndirection, the magneto-optical and anomalous transport properties enhance signi\fcantly, presenting\nstrong magnetic anisotropy. We also demonstrate that the anomalous Hall e\u000bect decreases with\nthe temperature increases. The gigantic anomalous Nernst and thermal Hall e\u000bects are found in\nmonolayer and bilayer ferromagnetic Fe nGeTe 2, and the largest anomalous Nernst and thermal Hall\nconductivities, respectively, of -3.31 A/Km and 0.22 W/Km at 130 K are observed in bilayer ferro-\nmagnetic Fe 4GeTe 2. Especially, bilayer antiferromagnetic Fe 5GeTe 2exhibits large zero-temperature\nanomalous Hall conductivity of 2.63 e2/h as well as anomalous Nernst and thermal Hall conductivi-\nties of 2.76 A/Km and 0.10 W/Km at 130 K, respectively. Our results suggest that two-dimensional\nvan der Waals magnets Fe nGeTe 2(n= 3, 4, 5) have great potential applications in magneto-optical\ndevices, spintronics, and spin caloritronics.\nI. INTRODUCTION\nAccording to the Mermin-Wagner theorem, ferromag-\nnetism is hardly survived in two-dimensional (2D) sys-\ntem due to the enhanced thermal \ructuations [1]. Re-\ncently, exfoliating 2D magnetic materials from van der\nWaals (vdW) layered magnets has been con\frmed to\nbe an e\u000ecient method. For example, the 2D materi-\nals, CrI 3[2] and Cr 2Ge2Te6[3], were \frst experimentally\nreported to have the long-range magnetic order at low\ntemperature in 2017. The existence of atomically thin\nmagnetic materials reveals that the magnetic anisotropy\ncould counter thermal \ructuations and thus stabilized\nthe long-range magnetic order. Therefore, the 2D mag-\nnetic materials have attracted huge interest and are ex-\npected to be used in various multi-functional devices,\nsuch as microelectronics, spintronics, and thermoelec-\ntric devices. However, due to the low Curie tempera-\nture (TC) of CrI 3(\u001845 K) [2] and Cr 2Ge2Te6(\u001865\nK) [3], it poses a big challenge for using 2D vdW mag-\nnetic materials in any realistic applications. In 2016,\n\u0003wxfeng@bit.edu.cnthrough the density-functional theory calculations, the\nmonolayer (ML) Fe 3GeTe 2(F3GT) was predicted to be\na stable 2D magnetic system with large magnetocrys-\ntalline anisotropy energy (MAE) [4], which provides a\ntheory basis for experimental exfoliation. Subsequently,\nthe 2D vdW layered material, F3GT, was successfully\nexfoliated from the bulk materials and the TCof ex-\nfoliated ML is 130 K [5]. Additionally, the TCof tri-\nlayer F3GT can be imporved to the room temperature\nby ionic-liquid gating technology [6]. As the \frst 2D fer-\nromagnetic metal, F3GT has been extensively studied,\nsuch as the electronic-, optical-, transport-, and spin-\nrelated properties [4, 7{14].\nThe exploration of 2D vdW magnetic materials with\nhigherTChas never stopped. Following F3GT, ferromag-\nnetic Fe 5\u0000xGeTe 2(sometimes referred to as Fe 5GeTe 2,\nF5GT) and Fe 4GeTe 2(F4GT) have also been synthe-\nsized [15{17]. By re\rective magnetic circular dichroism\nand Hall e\u000bect measurements, F5GT exhibits intrinsic\nferromagnetic order at room temperature in bulk ma-\nterial (TC= 310 K) and nano\rakes ( TC= 270\u0018300\nK), respectively [16]. Similarly, F4GT also shows high\nTC(270 K) in the thin \rakes [17]. Compared with\nF3GT, F4GT and F5GT have higher TC, and therefore\nthey may be more suitable for realistic device applica-arXiv:2108.02926v2  [cond-mat.mtrl-sci]  5 Oct 20212\ntions. However, with the increase of iron occupancy,\nthe magnetic properties of F4GT and F5GT turn to\nbe more complex. For example, in nano\rakes F4GT\n(thick than seven-layer), the spin changes from the out-\nof-plane to in-plane direction around 110 K, while the\nspin-reorientation temperature is signi\fcantly enhanced\nup to 200 K if the thickness of nano\rakes reduces down\nto seven-layer [17]. The out-of-plane magnetization also\nemerges in the nano\rakes F5GT and magnetic anisotropy\nis enhanced with decreasing thickness down to \fve unit-\ncell layer [18]. Besides the temperature- and thickness-\ndependent magnetic anisotropy, other fundamental phys-\nical manifestations, such as magneto-optical response as\nwell as anomalous electronic, thermal, and thermoelec-\ntric transports, are still unclear in the atomically thin\nlimit of Fe nGeTe 2(n= 3, 4, 5), especially for F4GT and\nF5GT. It essentially hinders the practical applications\nof FenGeTe 2, and therefore it is time to uncover these\nphysical properties of Fe nGeTe 2.\nIn this paper, utilizing the \frst-principles calculations,\nwe \frst determine the magnetic ground states of ML\nand bilayer (BL) Fe nGeTe 2(n= 3, 4, 5). The cal-\nculated MAE indicates that ML F3GT exhibit strong\nout-of-plane ferromagnetic order, while ML F4GT and\nF5GT show in-plane ferromagnetic order. The out-of-\nplane magnetization of Fe nGeTe 2(n= 3, 4, 5) increases\nwith increasing the number of layers. For BL struc-\ntures, F3GT and F5GT exhibit interlayer antiferromag-\nnetic coupling, while F4GT shows strong interlayer fer-\nromagnetic coupling. Subsequently, we use magnetic\ngroup theory to analyze the shape of optical conductiv-\nity tensor, and hence reveal the symmetry requirements\nfor magneto-optical e\u000bect (MOE) and anomalous magne-\ntotransport phenomena, including anomalous Hall e\u000bect\n(AHE), anomalous Nernst e\u000bect (ANE), and anomalous\nthermal Hall e\u000bect (ATHE). These e\u000bects are symmet-\nrically allowed in ML ferromagnetic F3GT with out-of-\nplane magnetization, and in ML ferromagnetic Fe nGeTe 2\n(n= 4, 5) and BL ferromagnetic Fe nGeTe 2(n= 3, 4, 5)\nwith both in-plane and out-of-plane magnetization. More\nintriguing, we \fnd that BL antiferromagnetic F5GT sat-\nis\fes the symmetry requirements and can appear MOE,\nAHE, ANE, and ATHE, whereas these e\u000bects are pro-\nhibited in BL antiferromagnetic F3GT and F4GT. For\nthe magnetic structures that satisfy the symmetry re-\nquirements, we calculate the magneto-optical Kerr and\nFaraday spectra, zero- and \fnite-temperature anoma-\nlous Hall conductivity (AHC), anomalous Nernst con-\nductivity (ANC), and anomalous thermal Hall conductiv-\nity (ATHC). The dependence of MOE, AHE, ANE, and\nATHE on magnetization direction and thin-\flm thick-\nness are clearly observed in Fe nGeTe 2. The calculated\nmagneto-optical and anomalous transport properties are\nimpressively large, which are comparable to or even\nlarger than that of many famous ferromagnets and anti-\nferromagnets. Our results suggest that 2D vdW magnets\nFenGeTe 2have potential applications in magneto-optical\ndevices, spintronics, and spin caloritronics.II. THEORY AND COMPUTATIONAL DETAILS\nThe MOE is one of the fundamental physical e\u000bects\nin condensed-matter physics, which can be described as\nthat when a linearly polarized light hits on the sur-\nface of a magnetic material, the polarization planes\nof re\rected and transmitted lights shall rotate, called\nmagneto-optical Kerr [19] and Faraday [20] e\u000bects, re-\nspectively. Nowadays, the MOE has become to be a pow-\nerful and nondestructive probe of magnetism in 2D mate-\nrials [2, 3]. To quantitatively characterize the magneto-\noptical performance of a given magnetic material, the\nKerr (\u0012K) and Faraday ( \u0012F) rotation angles, the de\rec-\ntion of the polarization planes with respect to the inci-\ndent light, are used. Moreover, the \u0012K(F)(rotation angle)\nand\"K(F) (ellipticity) are usually combined into the so-\ncalled complex Kerr (Faraday) angle, written as [21{23]\n\u001eK=\u0012K+i\"K=i2!d\nc\u001bxy\n\u001bsxx; (1)\nand\n\u001eF=\u0012F+i\"F=!d\n2c(n+\u0000n\u0000); (2)\nn2\n\u0006= 1 +4\u0019i\n!(\u001bxx\u0006i\u001bxy); (3)\nwherec,!, anddare the speed of light in vacuum, fre-\nquency of incident light, and thin-\flm thickness, respec-\ntively.\u001bxyis the o\u000b-diagonal element of the optical con-\nductivity tensor for a magnetic thin-\flm and \u001bs\nxxis the\ndiagonal element of the optical conductivity tensor for a\nnonmagnetic substrate (e.g., SiO 2). SiO 2is a large band\ngap insulator and \u001bs\nxx=i(1\u0000n2)!=4\u0019with the refractive\nindex ofn= 1.546.n2\n\u0006are the eigenvalues of the dielec-\ntric tensor for a magnetic thin-\flm. From the expres-\nsions of complex Kerr and Faraday angles [Eq. (1){(3)],\none can see that the crucial quantity is the optical con-\nductivity, which can be calculated by Kubo-Greenwood\nformula [24, 25],\n\u001b\u000b\f=\u001b1\n\u000b\f(!) +i\u001b2\n\u000b\f(!)\n=ie2~\nNk\ncX\nkX\nn;mfmk\u0000fnk\n\"mk\u0000\"nk\n\u0002h nkj^v\u000bj mkih mkj^v\fj nki\n\"mk\u0000\"nk\u0000(~!+i\u0011); (4)\nwhere the subscripts \u000b;\f2fx;yg. The superscripts 1\nand 2 represent the real and imaginary parts of \u001bxy, \nc\nis the volume of unit cell and Nkis the total number of\nk-points for sampling the Brillouin zone (here a k-mesh\nof 300\u0002300\u00021 is used),  nkand\"nkare the Wannier\nfunction and interpolated energy at the band index nand\nmomentum k, respectively, ^ vx;yare the velocity operators\nalong thexorydirection, ~!is the photon energy, \u0011is an\nadjustable parameter for energy smearing (here \u0011= 0.15\neV), andfnkis the Fermi-Dirac distribution function.3\nIn condensed-matter physics, the AHE is another im-\nportant physical phenomenon, which is characterized by\na transverse voltage drop generated by a longitudinal\ncharge current in the absence of an external magnetic\n\feld [26]. Physically speaking, the dc limit of the real\npart of the o\u000b-diagonal element of optical conductivity,\n\u001b1\nxy(!!0), is nothing but the intrinsic AHC. There-\nfore, the AHE and MOE are closely related to each other\nand have similar physical origins. Based on the linear\nresponse theory, the intrinsic AHC at zero-temperature\n(\u001bT=0\nxy) can be calculated [27],\n\u001bT=0\nxy=\u0000e2\n~VX\nn;kfnk\nn\nxy(k); (5)\nwhere \nn\nxy(k) is the band- and momentum-resolved Berry\ncurvature, given by,\n\nn\nxy(k) =\u0000X\nn06=n2Im[h nkj^vxjh n0kih n0kj^vyjh nki]\n(\"nk\u0000\"n0k)2:\n(6)\nDue to the band crossings, the Berry curvature can be\nsensitive to the k-mesh. After carefully examined the\nconvergence of intrinsic AHC, a k-mesh of 350\u0002350\u00021\nis ensured to obtain reliable results (see Fig. S1 in Sup-\nplementary Material).\nThe transverse charge current can be induced by a lon-\ngitudinal temperature gradient \feld (instead of a longi-\ntudinal electric \feld), which is referred as the ANE [28].\nIn general, the transverse thermal current concomitantly\narise under the longitudinal temperature gradient \feld,\ncalled ATHE or anomalous Righi-Leduc e\u000bect [29]. The\nANE and ATHE are usually regarded as the thermoelec-\ntric counterpart and the thermal analogue of the AHE,\nrespectively. The AHE, ANE, and ATHE can be physi-\ncally related to each other via the anomalous transport\ncoe\u000ecients in the generalized Landauer-B uttiker formal-\nism [30{32],\nR(n)\nxy=Z1\n\u00001d\"(\"\u0000\u0016)n(\u0000@f\n@\")\u001bT=0\nxy(\"); (7)\nwhere\",\u0016, andfare energy, chemical potential, and\nFermi-Dirac distribution function, respectively. Then,\nthe temperature-dependent AHC ( \u001bA\nxy), ANC (\u000bA\nxy) and\nATHC (\u0014A\nxy) read\n\u001bA\nxy=R(0)\nxy; (8)\n\u000bA\nxy=\u0000R(1)\nxy=eT; (9)\n\u0014A\nxy=R(2)\nxy=e2T: (10)\nThe \frst-principles calculations were performed using\nthe projector augmented wave method [33], implemented\nin Vienna ab initio simulation package ( vasp ) [34,35]. The generalized gradient approximation with the\nPerdew-Burke-Ernzerhof parameterization [36] was uti-\nlized to treat the exchange-correlation functional. The\nplane-wave cuto\u000b energy was set to be 400 eV. For the\nBrillouin zone integration, we used an 18 \u000218\u00021k-\nmesh. The convergence of total energy with respect\nto the cuto\u000b energy and k-mesh has been well exam-\nined, and the results of MAE are reliable (see Tabs. S1\nand S2 in Supplementary Material). The vacuum layer\nwith the thickness of at least 25 \u0017A for all structures was\nused to avoid interactions between adjacent layers and\nthe DFT-D3 method [37, 38] was adopted to treat the\nvdW correction of BL systems. The spin-orbit coupling\nwas included in the calculations of MAE, band struc-\ntures, MOE, AHE, ANE, and ATHE. The maximally\nlocalized Wannier function s were obtained by WAN-\nNIER90 package [24], where the p-orbitals of Ge and\nTe atoms as well as the d-orbitals of Fe atoms were pro-\njected onto the Wannier functions. The band structures\nobtained from Wannier-interpolation are identical to the\nDFT ones around the Fermi level, as shown in supple-\nmentary Figs. S2{S4. To obtain the magnetic point\nand space groups of various magnetic structures, the\nISOTROPY software [39] is used, in which the lattice\nconstants, atomic positions, and the direction and size of\nspin magnetic moment on each atom are necessary input\nparameters.\nIII. RESULTS AND DISCUSSION\nIn this section, we shall present our results as follows.\nFirst, the crystal structures of ML and BL Fe nGeTe 2(n\n= 3, 4, 5) are discussed and the magnetic ground states\nare con\frmed by the calculations of MAE. Then, utilizing\nmagnetic group theory, the shape (nonzero o\u000b-diagonal\nelements) of optical conductivity tensor is determined\nand hence the symmetry requirements for the MOE and\nanomalous transport properties (AHE, ANE, and ATHE)\nare obtained. Subsequently, the symmetrically-allowed\nMOE, AHE, ANE, and ATHE for ML and BL ferro-\nmagnetic Fe nGeTe 2are calculated by the \frst-principles\nmethods. Finally, the BL antiferromagnetic Fe 5GeTe 2\nthat surprisingly generates the MOE, AHE, ANE, and\nATHE is detailedly discussed as a special case.\nA. Crystal, magnetic and electronic structures\nThe top and side views of the vdW layered magnets\nFenGeTe 2(n= 3, 4, 5) with hexagonal structure ex-\nfoliating from bulk crystals [5, 6, 16, 17] are shown in\nFig. 1. The optimized lattice constants and atomic posi-\ntions of ML and BL F3GT, F4GT, and F5GT are listed\nin Tab. I and supplementary Tabs. S3{S8. Each F3GT\n(F4GT, F5GT) ML is consisted of three (four, \fve) Fe\natoms, one Ge atom, and two Te atoms. From Figs. 1(a)\nand 1(c), one can see that for ML F3GT and F4GT, ev-4\nFe3GeTe2F e4GeTe2F e5GeTe2y\nx\nz\ny\n(\nb)( d)( f)3.14 Å3 .03 Å2 .96 ÅGeFeT\ne(\na)( c)( e)\nFIG. 1. (Color online) (a,c,e) Top views of monolayer Fe nGeTe 2(n= 3, 4, 5). The silver, green, and orange spheres represent\nFe, Ge, and Te atoms, respectively. The pink dashed lines draw up the 2D primitive cell. (b,d,f) Side views of bilayer Fe nGeTe 2\n(n= 3, 4, 5). The interlayer vdW gaps of Fe 3GeTe 2, Fe4GeTe 2, and Fe 5GeTe 2are 2.96 \u0017A, 3.03 \u0017A, and 3.14 \u0017A, respectively.\nery three identical atoms form a triangle and other atoms\nare located in the center of the triangle. However, for ML\nF5GT, six Fe atoms form a hexagon. The calculated ef-\nfective thickness of ML F3GT, F4GT, and F5GT are 8.13\n\u0017A, 9.42 \u0017A, and 9.83 \u0017A, respectively, which are consistent\nwith the experimental data [5, 6, 16, 17]. To investigate\nbilayer structures, the stacking order should be primar-\nily determined. As shown in supplementary Fig. S5 and\nTab. S9, we constructed di\u000berent stacking orders of BL\nFenGeTe 2and calculated the total energies of ferromag-\nnetic and antiferromagnetic structures. Our results show\nthat the stacking orders taken from experimental bulk\ncrystal structures, as shown in Figs. 1(b), 1(d), and 1(f),\nhave the lowest energies. The relaxed interlayer vdW\ngaps of BL F3GT, F4GT, and F5GT are 2.96 \u0017A, 3.03\n\u0017A, and 3.14 \u0017A, respectively. The vdW gap of F3GT is\nconsistent with the experimental data (2.95 \u0017A) [6]. The\nlarge vdW gaps of F4GT and F5GT indicate their weak\ninterlayer coupling, and it is likely to obtain atomicallythin structures (ML or BL) by mechanically exfoliating\nfrom bulk crystals, similarly to F3GT [5, 6].\nAs one knows, the magnetic anisotropy, which is a pre-\nferred direction for spin aligning, is bene\fcial to stabi-\nlize the long-range magnetic order in 2D systems. The\nmagnetic anisotropy has various sources, such as shape\nanisotropy, stress anisotropy, and the foremost magne-\ntocrystalline anisotropy [3]. The MAE is usually de\fned\nas the di\u000berence of total energy between two di\u000berent\nspin directions, e.g., \u0001 EMAE(\u0012;') =E(\u0012;')\u0000E(\u0012=\n90\u000e;'= 0\u000e), hereE(\u0012;') is the total energy when the\nspin points to the direction labeled by the polar ( \u0012) and\nazimuthal ( ') angles. Positive \u0001 EMAE indicates an fa-\nvored in-plane magnetization along the x-axis (\u0012= 90\u000e,\n'= 0\u000e), and negative \u0001 EMAE indicates the out-of-plane\nmagnetization. The MAE of ML and BL F3GT, F4GT,\nand F5GT are shown in Fig. 2 and Tab. I. Fig. 2 illus-\ntrates that ML F3GT is an Ising ferromagnet since there\nexists an easy axis along the z-direction, while ML F4GT5\n045901\n351\n802\n252\n70315045901\n351\n802\n252\n70315045901\n351\n802\n252\n70315\n0 meV(b)Fe4GeTe2/s113\n or /s106 (degree)Z(Y)X\n4.420 4\n.42 meV-3.170 Z\n(Y)X\nunit: meV\n0\n meV(c)0\n0 .580\n.58 meVFe5GeTe2/s113\n or /s106 (degree)XZ(Y)unit: meV\n0\n meV/s113\n or /s106 (degree)Fe3GeTe2-\n3.17 meVunit: meV(a)\nFIG. 2. (Color online) The calculated magnetocrystalline anisotropy energy of monolayer Fe nGeTe 2(n= 3, 4, 5). Top patterns\nare the three-dimensional mapping of MAE, whereas bottom patterns are the extracted data from the equator and longitude\nlines (black dashed lines) of the MAE map. Apparently, F3GT favors the out-of-plane magnetization, while F4GT and F4GT\npresent the in-plane magnetization.\nand F5GT belong to the category of XY ferromagnets\ndue to the magnetic isotropy on the xyplane (i.e., the\neasy plane) [40]. Moreover, in the zxplane the angular\ndependence of MAE can be \ftted well by the equation,\nMAE(\u0012) =K1cos2\u0012+K2cos4\u0012, whereK1andK2are\nthe magnetocrystalline ansiotropy coe\u000ecients. The \ft-\nted coe\u000ecients (in unit of meV) for ML F3GT, F4GT,\nand F5GT are K1=\u00002:59 andK2=\u00000:58,K1= 4:48\nandK2=\u00000:06, andK1=\u00000:67 andK2=\u00000:09, re-\nspectively. The maximum values of MAE in ML F3GT,\nF4GT, and F5GT between the out-of-plane ( z-axis) and\nin-plane (x-axis) magnetization reach to -3.17, 4.42 and\n0.58 meV/f.u., respectively, which are larger than that of\nfamous 2D ferromagnets CrI 3(0.5 meV/f.u. [41], 0.305\nmeV/f.u. [42]) and Cr 2Ge2Te6(0.4 meV/f.u.) [43], sug-\ngesting that ML F3GT and F4GT have appreciable ap-\nplications in magnetic data storage. Additionally, our\ncalculated MAE value of ML F3GT is close to the ex-\nperimental measurement (2.0 meV/f.u) [6] and is con-\nsistent with the previously theoretical calculation (3.0\nmeV/f.u.) [11]. In contrast to the out-of-plane magneti-\nzation of ML F3GT, ML F4GT and F5GT exhibit thein-plane magnetization, as shown in Figs. 2(a-c). How-\never, in recent experimental reports the nanometer-thick\nF4GT and F5GT samples show the out-of-plane mag-\nnetic anisotropy [16{18]. The di\u000berence between com-\nputational and experimental results can be explained by\nthe thickness of F4GT and F5GT thin-\flms. In our cal-\nculations, F3GT, F4GT, and F5GT are ML structures,\nwhereas the e xperimental samples are multilayer struc-\ntures (the thinnest sample is seven-layer for F4GT [17]\nand \fve-layer for F5GT [18]). Additionally, the com-\nposition ratio of Fe, Ge, and Te atoms in experimental\nsamples is not strictly to be 4:1:2 [17] or 5:1:2 [16, 18],\nwhich may also a\u000bect the magnetic ground states. From\nTab. I, one can observe that the MAE of F3GT decreases\nwith increasing number of layers, which is in good agree-\nment with other theoretical work [11]. This trend also\nappears in F4GT and F5GT, suggesting the out-of-plane\nmagnetic anisotropy enhances when the number of lay-\ners increases. Therefore, a reasonable guess is that with\nincreasing number of layers, multilayer F4GT and F5GT\neventually exhibit the out-of-plane magnetization. To\nverify, we further calculate the MAE of bulk F4GT and6\nF5GT, listed in Tab. I, from which one can indeed \fnd\nthat the bulk F4GT and F5GT prefer to out-of-plane\nmagnetization.\nThe crystal structures of BL Fe nGeTe 2are plotted in\nbottom row of Figs. 1. We further consider two magnetic\nstructures of BL Fe nGeTe 2which have ferromagnetic or\nantiferromagnetic interlayer coupling. The calculated en-\nergy di\u000berence and magnetic ground states are summa-\nrized in Tab I, from which one can see that BL F4GT\nprefers to be ferromagnetism, while BL F3GT and F5GT\nare favorable to be antiferromagnetism. However, BL an-\ntiferromagnetic F3GT is extremely fragile to dopants or\ndefects [44], and the ferromagnetic phase of BL F3GT\nwas observed by experiments [5, 6]. The BL ferromag-\nnetic F4GT has been con\frmed and the intralayer ferro-\nmagnetic coupling of F4GT is stronger than F3GT due\nto the two Fe-Fe dumbbells [17]. Because of tunable Fe\ncontents and Fe vacancies, F5GT exhibits more complex\nmagnetic behaviors than F3GT [45]. Additionally, the\ncrystal quality of F5GT strongly depends on how the\nsamples are cooled down [18]. From Tab. I, one can see\nthat the perfect BL F5GT shows the interlayer antiferro-\nmagnetic coupling. Compared with ferromagnets, anti-\nferromagnets are more advantageous for spintronics and\nspin caloritronics because of robustness against magnetic\nperturbation, negligible stray \feld, and ultrafast spin dy-\nnamics [46]. Therefore, BL antiferromagnetic F5GT may\nbe more applicable to spintronics and spin caloritronics.\nTABLE I. The calculated lattice constants ( a), MAE\n(\u0001EMAE), and magnetic ground states (MGS) of monolayer\nand bilayer Fe nGeTe 2(n= 3, 4, 5). Here, \u0001 EMAE =E(\u0012=\n0\u000e;'= 0\u000e)\u0000E(\u0012= 90\u000e;'= 0\u000e) and positive (negative)\nvalue indicates a favored in-plane (out-of-plane) magnetiza-\ntion. The energy di\u000berence between ferromagnetic and anti-\nferromagnetic bilayer Fe nGeTe 2, \u0001EFM-AFM , are given in the\nparentheses, where positive (negative) value indicates that\nAFM (FM) structure is energetically stable. The units of\n\u0001EMAE and \u0001EFM-AFM are meV/f.u. and the unit of ais\u0017A.\na \u0001EMAE MGS (\u0001 EFM-AFM )\nF3GT ML 4.02 -3.17 FM\nBL 4.01 -3.23 AFM (2.89)\nBulk 4.04 -3.25\nF4GT ML 3.92 4.42 FM\nBL 3.93 1.98 FM (-13.48)\nBulk 3.97 -1.33\nF5GT ML 3.96 0.58 FM\nBL 3.97 0.32 AFM (0.83)\nBulk 4.00 -0.55\nWe then discuss the electronic structures of Fe nGeTe 2\n(n= 3, 4, 5) by taking the ML ferromagnetic states with\nthe out-of-plane magnetization as examples, as plotted\nin Fig. 3. Around the Fermi level, the d-,p-, andp-\norbitals of Fe, Ge, and Te atoms, respectively, are domi-nant components. Moreover, the p-orbitals of Ge and Te\natoms have nearly equal contributions. One can see that\nML ferromagnetic Fe nGeTe 2are metals with a complex\nmulti-band character at the Fermi level. ML ferromag-\nnetic FenGeTe 2with the in-plane magnetization as well\nas BL ferromagnetic and antiferromagnetic Fe nGeTe 2\nwith both out-of-plane and in-plane magnetization are\nalso multi-band metals, see supplementary Figs. S2{S4.\nB. Magnetic group theory\nBefore practically calculating the MOE, AHE, ANE,\nand ATHE, a preceding step is to \fnd their symmetry\nrequirements. According to Eqs. (7)-(10), the \fnite-\ntemperature AHC, ANC, and ATHC can be obtained\nfrom the zero-temperature AHC. Additionally, the dc\nlimit of the real part of the o\u000b-diagonal element of\noptical conductivity is just the zero-temperature AHC\n[Eqs. (4)-(5)]. Therefore, the AHE, ANE, and ATHE as\nwell as MOE should share the same symmetry require-\nments. And we only need to study the shape (nonzero\no\u000b-diagonal elements) of optical conductivity tensor ( \u001b).\nMagnetic group theory is a powerful tool to identify non-\nvanishing elements of \u001b. Because of the translation-\nally invariance of \u001b, it is su\u000ecient to restrict our anal-\nysis to magnetic point group. Here, the magnetic point\ngroups of various magnetic structures are calculated by\nISOTROPY software [39] and the results are summa-\nrized in Tab. II and supplementary Tab. S10. Addition-\nally, the o\u000b-diagonal elements of \u001bcan be regarded as\na pseudovector, like spin, and therefore its vector-form\nnotation,\u001b= [\u001bx;\u001by;\u001bz] = [\u001byz;\u001bzx;\u001bxy], is used for\nconvenience. It should be mentioned that in 2D systems\n\u001bxand\u001byare always equal to zero, and only \u001bz(\u0011\u001bxy)\nis potentially nonzero.\nThe magnetic point groups of ML Fe nGeTe 2(n= 3,\n4, 5) as a function of polar ( \u0012) and azimuthal ( ') angles\nwhen the spin rotates within the xzandxyplanes are\nsummarized in supplementary Tab. S10. In the main\ntext, we only discuss three special directions, namely,\nwhen the spin points to the x-axis (\u0012= 90\u000e;'= 0\u000e),\ny-axis (\u0012= 90\u000e;'= 90\u000e), andz-axis (\u0012= 0\u000e;'= 0\u000e), as\nlisted in Tab II. Let us start with analyzing the magnetic\npoint groups of ML F3GT. The magnetic point groups of\nML F3GT are m0m20,m0m02, and \u00166m020when the spin\npoints tox-,y-, andz-axes, respectively. First, group\nm0m20has one mirror plane Mx, which is perpendicular\nto the spin orientation ( x-axis) and is parallel to the z-\naxis. Such a mirror operation reverses the sign of \u001bz, thus\nsuggesting\u001bz= 0. Group m0m02 contains one combined\nsymmetryTMz, whereTis the time-reversal symmetry\nandMzis parallel to the spin orientation ( y-axis) and\nis perpendicular to z-axis. Note that TMzoperation\nreverses the sign of \u001bz, and hence \u001bz= 0. When the spin\nis along the z-axis, ML F3GT belongs to magnetic point\ngroup of \u00166m020. This group has a mirror plane Mz, which\nis perpendicular to the spin orientation ( z-axis) and does7\n-101-\n101-\n101E (eV) Fe d orbits( a)F\n3GTF 3GT Ge p orbits  Te p orbits(\nb)F\n4GTE (eV)F\n4GT(\nc)F\n5GTE (eV)Γ\nK M Γ F5GTΓ\nK M Γ \nFIG. 3. (Color online) Orbital-projected band structures of monolayer ferromagnetic Fe nGeTe 2(n= 3, 4, 5) with the out-of-\nplane magnetization. The blue, orange, and green represent the d-,p-, andp-orbitals of Fe, Ge, and Te atoms, respectively.\nTABLE II. Magnetic point groups of monolayer and bilayer Fe nGeTe 2(n= 3, 4, 5) when the spin points to the x-,y-, and\nz-axes. We consider monolayer structures with only ferromagnetic order, but bilayer structures with both ferromagnetic and\nantiferromagnetic orders. Y (yes) and N (no) indicate that there exists nonzero \u001bzor not.\nML (FM) BL (FM) BL (AFM)\nx y z x y z x y z\nF3GT m0m20(N) m0m02 (N) \u00166m020(Y) 2 =m(N) 20=m0(Y) \u001631m0(Y) 20=m(N) 2 =m0(N) \u0016301m0(N)\nF4GT 2 =m(N) 20=m0(Y) \u001631m0(Y) 2 =m(N) 20=m0(Y) \u001631m0(Y) 20=m(N) 2 =m0(N) \u0016301m0(N)\nF5GT m(N) m0(Y) 3 m01 (Y) m(N) m0(Y) 3 m01 (Y) m(N) m0(Y) 3 m01 (Y)8\nnot change the sign of \u001bz. On the other hand, this group\ncontains three combined symmetries ( TM), one of which\nisTMx. The other two mirror planes can be obtained by\ntheC3symmetry. Both the TandMoperations reverse\nthe sign of \u001bz, indicating that \u001bzis even under the TM\nsymmetry. Overall, \u001bzcan exist in the magnetic point\ngroup of \u00166m020, that is,\u001b= [0;0;\u001bz].\nThe magnetic point groups of ML F4GT are 2 =m,\n20=m0, and \u001631m0when the spin points to the x-,y-, and\nz-axes, respectively. The group 2 =mhas a mirror plane\nMx, indicating \u001bz= 0. For group 20=m0, there is a com-\nbined symmetryTMx. Both theTandMxoperations\nreverse the sign of \u001bz, indicating \u001bzis even underTMx\nsymmetry, and hence \u001bz6= 0. The group \u001631m0contains\nthree combined symmetries TM, which are the same as\ngroup \u00166m020. The combined symmetries do not change\nthe sign of \u001bz, giving rise to the nonzero \u001bz. Next, we\nanalyze the magnetic point groups of ML F5GT, which\narem,m0, and 3m01 when the spin is along the x-,y-,\nandz-axes, respectively. There is a mirror plane Mxin\ngroupm, reversing the sign of \u001bz, and thus\u001bzshould be\nzero. Group m0includes a combined symmetry TMx,\nwhich keeps \u001bz6= 0. Group 3 m01 has three combined\nsymmetriesTM, like groups \u00166m020and\u001631m0, giving rise\nto the nonzero \u001bz.\nFinally, we discuss the magnetic point groups of BL\nferromagnetic and antiferromagnetic Fe nGeTe 2(n= 3,\n4, 5). It should be noted that BL F3GT and F4GT share\nthe same magnetic point groups, as listed in Tab. II. The\nmagnetic point groups of BL ferromagnetic F3GT and\nF4GT are 2 =m, 20=m0, and \u001631m0when the spin points\nto thex-,y-, andz-axes, respectively. As previously\nmentioned, we conclude that the nonzero \u001bzcan exist\nwhen the spin points to the y- andz-axes, while \u001bz= 0 if\nthe spin is along x-axis. The magnetic point groups of BL\nantiferromagnetic F3GT and F4GT are 20=m, 2=m0and\n\u0016301m0, all of which have a combined symmetry TP(Pis\nthe spatial inversion symmetry), which does not allow \u001bz.\nAdditionally, ML and BL F5GT share the same magnetic\npoint groups when the spin is along the x-,y-, andz-axes.\nTherefore, from our group theory analysis, \u001bzexists well\nin antiferromagnetic BL, just like ferromagnetic ML and\nBL. Recently, there has been a consensus that \u001bzcan\nexist in some noncollinear antiferromagnets even though\nthe net magnetization is zero, presenting attractive MOE\nand anomalous transport phenomena [47{62]. However,\nit is relatively rare in collinear antiferromagnets [63, 64].\nTherefore, BL antiferromagnetic F5GT may be of great\ninterest to experimental attention.\nIn general, \u001bzis nonzero in ML ferromagnetic F3GT\nwhen the spin is along the z-axis and is also nonzero in\nML ferromagnetic F4GT and F5GT, BL ferromagnetic\nF3GT, F4GT, and F5GT, as well as BL antiferromag-\nnetic F5GT when the spin points to the y- andz-axes,\nwhich are all summarized in Tab. II. In the following,\nwe restrict our discussion on the MOE, AHE, ANE, and\nATHE of the above-mentioned magnetic structures.C. Magneto-optical e\u000bects\nThe band structures of ML and BL Fe nGeTe 2(n=\n3, 4, 5) are plotted in supplementary Figs. S2{S4, from\nwhich the multi-band metals can be easily con\frmed. Ac-\ncording to Eqs. (1)-(3), the magneto-optical Kerr and\nFaraday angles are closely related to the optical conduc-\ntivity, which are shown in supplementary Figs. S6-S9.\nThe diagonal elements of optical conductivity are nearly\nisotropy regardless of the spin's direction, while the o\u000b-\ndiagonal elements of optical conductivity exhibit strong\nanisotropy between the in-plane and out-of-plane mag-\nnetization. Consequently, the magneto-optical Kerr and\nFaraday angles also show strong anisotropy, as shown in\nsupplementary Figs. S10 and S11. Although the MAE\nresults show that ML and BL F4GT and F5GT prefer to\nthe in-plane magnetization, the magnetized direction of\n2D materials can be feasibly modulated by applying an\nexternal magnetic \feld [4]. In the main text, we take the\nout-of-plane magnetization as an example to discuss the\nMOE as well as the following AHE, ANE, and ATHE.\nFigure 4 displays the magneto-optical Kerr and Fara-\nday angles of ML and BL ferromagnetic Fe nGeTe 2with\nthe out-of-plane magnetization (i.e., z-axis) as a func-\ntion of photon energy. In the energy range of 0 \u00184\neV, the Kerr ( \u0012K) and Faraday ( \u0012F) angles oscillate fre-\nquently. For ML F3GT, F4GT, and F5GT, the promi-\nnent peaks of Kerr angle locate at f0.39, 2.25, 2.80g\neV,f0.69, 1.90, 2.42, 2.88 geV, andf0.45, 1.96geV, re-\nspectively. For BL structures, the locations of promi-\nnent peaks appear close to that of ML structures. Re-\nmarkably, the amplitudes of the \u0012Kof BL structures are\nlarger than that of ML structures. The maximum val-\nues of\u0012Kfor ML (BL) F3GT, F4GT, and F5GT are\n1.19\u000e(2.07\u000e), 1.14\u000e(1.90\u000e), 0.74\u000e(1.44\u000e), respectively.\nOur calculated Kerr angles of ML and BL F3GT are in\nagreement with the previously theoretical report (1.65\u000e\nand 2.76\u000efor ML and BL, respectively) [11]. The cal-\nculated largest values of \u0012Ffor ML (BL) F3GT, F4GT,\nand F5GT are 222.66\u000e=\u0016m (147.73\u000e=\u0016m), 109.12\u000e=\u0016m\n(82.95\u000e=\u0016m), 50.28\u000e=\u0016m (54.07\u000e=\u0016m), respectively.\nThe\u0012Kof ML and BL ferromagnetic Fe nGeTe 2are\nlarger than that of the most traditional ferromagnets.\nFor example, the \u0012Kof bcc Fe, hcp Co, and fcc Ni are\nonly -0.5\u000e, -0.42\u000e, and -0.25\u000e[65{67], respectively. More-\nover, the\u0012Kof ML and BL ferromagnetic Fe nGeTe 2are\nlarger than or comparable to that of many famous 2D\nmagnetic materials, such as CrI 3[0.29\u000e(ML), 2.86\u000e(tri-\nlayer)] [2], Cr 2Ge2Te6[0.9\u000e(ML), 1.5\u000e(BL), 2.2\u000e(tri-\nlayer)] [43], CrTe 2[0.24\u000e(ML), -1.76\u000e(trilayer)] [68],\nblue phosphorene [0.12\u000e(ML), 0.03\u000e(BL)] [69], gray\narsenene [0.81\u000e(ML), 0.14\u000e(BL)] [69], and InS [0.34\u000e\n(ML)] [70]. Additionally, the \u0012Fof ML and BL ferro-\nmagnetic Fe nGeTe 2are larger than or comparable to\nthat of MnBi thin-\flms [80\u000e=\u0016m] [23, 71], Cr 2Ge2Te6\n[\u0018120\u000e=\u0016m (ML)] [43], CrI 3[108\u000e=\u0016m (TL)] [41], and\nCrTe 2[-173\u000e=\u0016m (ML)] [68]. The large magneto-optical\nKerr and Faraday e\u000bects suggest that ML and BL ferro-9\n-202-\n2020\n2 4 -202-2000200-\n20002000\n2 4 -2000200/s113K (deg) ML \nBLF\n3GT(a)(\nb)F\n4GT/s113K (deg)F\n5GT(c)/s113 K (deg)P\nhoton energy (eV)(d)F\n3GT/s113F (deg/µm) ML \nBL(\ne)F\n4GT/s113F (deg/µm)(\nf)F\n5GT/s113F (deg/µm)P\nhoton energy (eV)\nFIG. 4. (Color online) The calculated magneto-optical Kerr (a-c) and Faraday (d-f) rotation angles for monolayer (olive lines)\nand bilayer (pink lines) ferromagnetic Fe nGeTe 2(n= 3, 4, 5) with the out-of-plane magnetization (along the z-axis).\nmagnetic Fe nGeTe 2may have potential applications in\nvarious magneto-optical devices.\nD. Anomalous Hall, anomalous Nernst, and\nanomalous thermal Hall e\u000bects\nNow, let us focus on the anomalous transport prop-\nerties of ML and BL ferromagnetic Fe nGeTe 2(n=\n3, 4, 5). The zero-temperature AHC calculated via\nEq. (5) are plotted in Figs. 5(a-c) and supplementary\nFigs. S12(a-c) and S13(a-c). Similarly to the MOE,\nthe zero-temperature AHC also exhibits large magnetic\nanisotropy, i.e., the results of out-of-plane magnetiza-\ntion are much larger than that of in-plane magnetiza-tion. Hence, we only discuss the case of out-of-plane\nmagnetization in the main text. The calculated zero-\ntemperature AHC for ML F3GT, F4GT, and F5GT are\n0.73e2=h, 1.70e2=h, and 0.46e2=hat the Fermi energy\n(EF), respectively. With the increasing number of lay-\ners, AHC can be greatly enhanced. For example, the\nzero-temperature AHC for BL F3GT, F4GT, and F5GT\nincrease up to 1.57 e2=h, 2.54e2=h, and 1.50 e2=hat\ntheEF, respectively. Our calculated zero-temperature\nAHC of ML and BL F3GT agree well with the previously\ntheoretical report [0.37 e2=h(ML), 1.50 e2=h(BL)] [10].\nBy appropriate doping electrons or holes, not only the\nCurie temperature can be increased, but also the mag-\nnetotransport properties can be modulated. Intuitively\nspeaking, doping electrons or holes can shift the posi-10\n-2024-\n4-2024-2024-\n4-2024-2024-\n0.20 .00 .20.00.2-\n0.20 .00 .20.00.2-4-2024-\n0.20 .00 .20.00.2-2024-\n2024-\n2024/s115T=0x\ny (e2/h) ML \nBLF\n3GT(a)(\ng)F\n3GT/s97Ax\ny (A/Km)(b)F\n4GT(\nh)F\n4GT(c)F\n5GT(\nj)F\n3GTκAx\ny (W/Km)E\n (eV)(k)F\n4GTE\n (eV)(i)F\n5GT(\nl)F\n5GTE\n (eV)(d)F\n3GT/s115Ax\ny (e2/h)(e)F\n4GT(f)F\n5GT\nFIG. 5. (Color online) The calculated anomalous Hall conductivities at 0 K (a,b,c) and 130 K (d,e,f), anomalous Nernst\nconductivity at 130 K (g,h,i), and anomalous thermal Hall conductivity at 130 K (j,k,l) for monolayer and bilayer ferromagnetic\nFenGeTe 2(n= 3, 4, 5) with the out-of-plane magnetization (along the z-axis) as a function of the Fermi energy. The black\nsquares in (a) indicate the positions where the Berry curvatures shown in Fig. 6 are calculated. The black circles in (a) and (g)\nindicate the large slopes of anomalous Hall conductivity and the sharp peaks of anomalous Nernst conductivity, respectively.\ntion of the Fermi energy. From Figs. 5(a-c), one can \fnd\nthat by doping electrons or holes, the AHC of ML F3GT\nand F4GT do not improve too much, while the AHC of\nML F5GT as well as of BL F3GT, F4GT, and F5GT en-\nhance signi\fcantly. For example, the AHC of BL F3GT,\nF4GT, and F5GT increase up to -2.16 e2=h, 4.48e2=h,\nand 3.46e2=hwhen the Fermi energy shifts to 0.09 eV, -\n0.07 eV, and -0.04 eV, respectively. Among Fe nGeTe 2(n\n= 3, 4, 5), BL F4GT shows the largest AHC of 4.48 e2=h\n(\u0019920 S/cm), which is larger than bulk F3GT (140{540S/cm) [72], thin-\flms F3GT (360{400 S/cm) [12], and\nthin-\flms F4GT ( \u0018180 S/cm) [17], and is even com-\npared with bcc Fe (751 S/cm [27], 1032 S/cm [73]).\nTaking F3GT as an example, we interpret the varia-\ntion of the AHC for ML and BL structures as well as\nfor the doping e\u000bect in BL structures. Figure 6 plots\nthe momentum-resolved Berry curvature of ML and BL\nF3GT. At the true Fermi energy ( EF), the negative spots\nof Berry curvature are obviously larger than the positive\nones [Fig. 6(a)], resulting in the positive value of AHC11\n2500\n500-\n250\n-\n500(a)( b)( c)\nFIG. 6. (Color online) The calculated momentum-resolved Berry curvature (in arbitrary units) of monolayer (a) and bilayer\n(b,c) Fe 3GeTe 2on thekx\u0000kyplane. The Berry curvatures in (a) and (b) are calculated at the true Fermi energy ( EF), while\nin (c) is calculated at EF+ 0:09 eV. The black dashed lines draw up the \frst Brillouin zone.\n[refer to Eqs. (5)and (6) and see Fig. 5(a)]. Additionally,\nthe darker red and blue colors indicate that the di\u000berence\nbetween positive and negative spots of Berry curvature\nbecomes larger [Fig. 6(b)], giving rise to the larger AHC\nin BL F3GT than that in ML F3GT [see Fig. 5(a)]. In\nFig. 6(c), when the Fermi energy is shifted to EF+ 0:09\neV for BL F3GT, the positive spots of Berry curvature\nturn to dominate in the \frst Brillouin zone, and thus lead\nto a large peak of negative AHC [see Fig. 5(a)].\nAccording to Eqs. (7)-(10), the temperature-dependent\nAHC, ANC, and ATHC can be computed from zero-\ntemperature AHC. The metallicity and high TCof 2D\nFenGeTe 2(n= 3, 4, 5) have been experimentally con-\n\frmed [5, 6, 16, 17], such as the TCof\u0018200 K in four-\nlayer F3GT, of 270 K in thin-\flms F4GT, and of 280 K\nin thin-\fms F5GT. Additionally, the ML F3GT has been\nprepared by mechanically exfoliating from bulk materi-\nals and the TCis reported to be 130 K [5]. In our cal-\nculations, the temperature-dependent AHC, ANC, and\nATHC are uniformly calculated at 130 K, as plotted in\nthe second, third, and fourth rows of Fig. 5, respectively.\nA trend is that the AHC decreases with increasing of\nthe temperature, especially for BL Fe nGeTe 2[compare\nthe \frst and second rows of Fig. 5]. At the tempera-\nture of 130 K, the largest AHC of BL F3GT, F4GT, and\nF5GT reach to 1.08 e2=h, 3.94e2=h, and 2.51 e2=hat\nthe Fermi energies of 0.0 eV, -0.08 eV, and -0.03 eV, re-\nspectively. Although the AHC at 130 K is smaller than\nthe zero-temperature AHC, the evolution of AHC as a\nfunction of the Fermi energy is similar to the case of zero-\ntemperature, and for instance, the AHC peaks roughly\nat the same energy.\nThe ANE is usually regarded as the thermoelectric\ncounterpart of the AHE, which is a celebrated e\u000bect from\nthe realm of spin caloritronics [74, 75]. Equation (9) can\nbe derived to the Mott relation at low temperature, which\nrelates the ANC to the energy derivative of AHC [76],\n\u000bA\nxy=\u0000\u00192\n3k2\nBT\ne\u001bA\nxy(\u0016)0: (11)\nFor example, there appears two large peaks of ANC neartheEF[see black arrows in Fig. 5(g)], where the slopes\nof AHC are rather step [see black arrows in Fig. 5(a)].\nThe ANC of ML and BL F3GT are almost zero at the\nEF. However, the ANC of ML F3GT reaches up to 1.85\nA/Km at 0.12 eV by electron doping. Moreover, the\nANC of BL F3GT reaches up to 2.14 A/Km and -2.52\nA/Km at 0.07 eV and 0.13 eV by electron doping, and\nreaches up to -1.40 A/Km at -0.03 eV by hole doping,\nrespectively. The calculated ANC of BL F3GT is higher\nthan thin-\flms F3GT ( \u00180.3 A/Km) [12]. The ANC\nof ML and BL F4GT are also almost zero at the EF,\nsimilarly to F3GT. After doping appropriate electrons or\nholes, the pronounced peaks of ANC arise. For example,\nthe ANC of ML F4GT can be -1.85 A/Km at -0.02 eV,\nand the ANC of BL F4GT appears to be -3.31 A/mK at\n-0.13 eV, see Fig. 5(h). Figure 5(i) plots the ANC curves\nas a function of the Fermi energy for ML and BL F5GT.\nThe ANC of ML F5GT is almost zero at the EFbut can\nexceed to\u00062:0 A/Km in the energy range of -0.15 \u0018-\n0.2 eV. In contrast to ML F5GT, BL F5GT has a large\nANC of 1.68 A/Km at the EF. Thus, to improve the\nthermoelectric performance of Fe nGeTe 2, doping appro-\npriate electrons or holes is an e\u000bective way. The calcu-\nlated ANC of ML and BL Fe nGeTe 2are larger than that\nof many popular magnetic materials, e.g., Heusler com-\npounds (1\u00188 A/Km) [77], Co 3Sn2S2(\u00182 A/Km) [78],\nTi2MnAl (\u00181.31 A/Km) [79], suggesting that ML and\nBL FenGeTe 2may have potential applications in ther-\nmoelectric devices.\nFurthermore, we present the ATHE (or called anoma-\nlous Righi-Leduc e\u000bect) [29], that is, a transverse ther-\nmal current induced by a longitudinal temperature gra-\ndient \feld, which is normally thought to be the ther-\nmal analogue of the AHE. The ATHC of ML and BL\nferromagnetic Fe nGeTe 2(n= 3, 4, 5) are plotted in\nlast row of Fig. 5. At the EF, the calculated ATHC\nof ML (BL) F3GT, F4GT, and F5GT are 0.11 (0.04)\nW/Km, 0.12 (0.18) W/Km, and 0.07 (0.11) W/Km, re-\nspectively. These values of ATHC can be further en-\nhanced by doping electrons or holes. For example, the\nlargest ATHC in ML (BL) F3GT, F4GT, and F5GT can12\nreach up to 0.16 (0.05) W/Km, 0.14 (0.22) W/Km, and\n0.19 (0.16) W/Km, respectively. The calculated ATHC\nof ML and BL ferromagnetic Fe nGeTe 2are larger than\nthat of Fe 3Sn2(\u00180.09 W/Km at 300 K) [80] and is\nslightly smaller than that of Co 2MnGa (\u00180.6 W/Km at\nroom temperature) [81], suggesting that 2D Fe nGeTe 2\nare excellent material platform to study the anomalous\nthermal transports.\nE. Antiferromagnetism of bilayer Fe 5GeTe 2\nThe understanding of MOE and AHE is gradually de-\nveloping. In early stage, the MOE and AHE are usually\nassumed to be linearly proportional to net magnetization,\nand hence most of host materials are ferromagnetic or\nferrimagnetic that have \fnite net magnetization. In con-\ntrast, the antiferromagnets with zero net magnetization\nare expected to have neither MOE nor AHE. Until 2014,\nChen et al. [82] predicted that Mn 3Ir with noncollinear\nantiferromagnetic order has large AHE. Subsequent stud-\nies have identi\fed that the MOE and AHE can occur in\nsimilar noncollinear antiferromagnets [47{62]. Despite\nthe comprehensive understanding of MOE and AHE in\nthe above noncollinear magnetic systems, it has not been\nunderstood well in more popular collinear antiferromag-\nnets. A common wisdom is that the MOE and AHE are\nnot easily activated in collinear antiferromagnets. The\nreason is simple that (1) although the time-reversal sym-\nmetryTis broken, however, the symmetries combined\nTand some spatial symmetries S(translation or inver-\nsion) exist more generally in collinear antiferromagnets\nrather than noncollinear antiferromagnets; (2) Such com-\nbined symmetries TSforce the MOE and AHE to be\nzero. Sivadas et al. [83] theoretically predicted that the\nmagneto-optical Kerr e\u000bect can occur in collinear anti-\nferromagnetic MnPSe 3[83], but an external electric \feld\nis needed to break all combined symmetries. On the ex-\nperimental aspect, the magneto-optical Kerr e\u000bect was\nobserved in bilayer collinear antiferromagnetic CrI 3us-\ning an external electric \feld [84, 85]. Very recently, the\nAHE and MOE are predicted in two collinear antifer-\nromagnets RuO 2and CoNb 3S6with the special crystal\nchirality [63, 64], which are naturally absent of any com-\nbined symmetries TS.\nIn this subsection, we shall show that the MOE, AHE,\nANE, and ATHE surprisingly exist in BL F5GT with\ncollinear antiferromagnetic order, which is needless of\nany external conditions, such as an electric \feld. This\nis much rare in realistic collinear antiferromagnets, espe-\ncially for 2D materials. The MAE and magnetic group\ntheory of BL F5G have been discussed in the preced-\ning subsections. The nonzero Berry curvature are ex-\npected in BL antiferromagnetic F5GT (both out-of-plane\nand in-plane magnetization) because of the symmetry re-\nquirements, and the anomalous transport properties that\noriginate from nonzero Berry curvature are also symmet-\nrically allowed. The band structure plotted in supple-\n-0.4-0.20.00.20\n2 4 -20020/s113K (deg) z-axis \ny-axis (×20)(a)(\nb)/s113 F (deg/µm)P\nhoton energy (eV)FIG. 7. (Color online) The calculated magneto-optical Kerr\n(a) and Faraday (b) rotation angles for BL antiferromagnetic\nF5GT with the out-of-plane (along the z-axis) and in-plane\n(along they-axis) magnetization. The data of \u0012Kand\u0012Ffor\nthe in-plane magnetization are multiplied by a factor of 20.\nmentary Figs. S4(e) and S4(f) clearly show the metallic\ncharacter of BL antiferromagnetic F5GT.\nFigure 7 plots the magneto-optical Kerr ( \u0012K) and\nFaraday (\u0012F) rotation angles of BL antiferromagnetic\nF5GT with the out-of-plane (along the z-axis) and in-\nplane (along the y-axis) magnetization. It can be found\nthat BL antiferromagnetic F5GT also exhibits strong\nmagneto-optical anisotropy as the results of out-of-plane\nmagnetization are much larger than that of in-plane mag-\nnetization, similarly to BL ferromagnetic F5GT. The\nlargest\u0012Kand\u0012Fare -0.23\u000eand 15.94\u000e=\u0016m at 0.87 eV\nand 2.23 eV, respectively. The \u0012Kof BL antiferromag-\nnetic F5GT is comparable to that of some famous non-\ncollinear and collinear antiferromagnets, such as Mn 3X\n(X= Rh, Ir, Pt) (0.2 \u00180.6\u000e) [47], Mn 3X0N (X0= Ga,\nZn, Ag, or Ni) (0.3 \u00180.4\u000e) [59], RuO 2(\u00180.62\u000e) [63], and\nCoNb 3S6(\u00180.2\u000e) [63], and is larger than that of non-\ncollinear antiferromagnets Mn 3Y(Y= Ge, Sn) (0.008 \u0018\n0.02\u000e) [60, 86]. The strong MOE in BL antiferromag-\nnetic F5GT suggests a new material platform for antifer-\nromagnetic magneto-optical devices.\nFinally, let us turn our attention to the transverse elec-\ntronic, thermoelectric, and thermal transport properties\nof BL antiferromagnetic F5GT, as shown in Fig. 8. Like13\n-202-\n0.20 .00 .2-202-\n0.20 .00 .20.000.050.100\n1 002 003 00-1.0-0.50.0-202(a)/s115T=0x\ny (e2/h) z-axis \ny-axisE\n (eV)(c)/s97Ax\ny (A/Km)(d)/s107Ax\ny (W/Km)E\n (eV)(\ne)/s97Ax\ny (A/Km)T\n (K)01 002 003 000.000.050.10(f)/s107Ax\ny (W/Km)T\n (K)(b)/s115Ax\ny (e2/h)\nFIG. 8. (Color online) The calculated anomalous Hall conductivities at 0 K (a) and 130 K (b), anomalous Nernst conductivity\nat 130 K (c), and anomalous thermal Hall conductivity at 130 K (d) for BL antiferromagnetic F5GT with the out-of-plane\n(along thez-axis) and in-plane (along the y-axis) magnetization as a function of the Fermi energy. (e) and (f) are the anomalous\nNernst and thermal Hall conductivities as a function of temperature calculated at the true Fermi energy.\nthe magneto-optical Kerr and Faraday spectra, the AHC,\nANC, and ATHC appeared in the out-of-plane magneti-\nzation are much stronger than that in the in-plane mag-\nnetization. For the out-of-plane magnetization, the zero-\ntemperature AHC is -0.28 e2=hat theEFand can be\nsigni\fcantly enhanced by electron doping, such as 2.63\ne2=hatEF+ 0:10 eV [Fig. 8(a)]. When the temperature\nincreases, the AHC tends to be weaker [Fig. 8(b)]. For\nexample, the AHC calculated at 130 K is -0.35 e2=hat\ntheEFand the largest value is 2.19 e2=hatEF+ 0:10\neV. The largest zero-temperature AHC (2.63 e2=h\u0019518\nS/cm) is larger than that of some famous noncollinearantiferromagnets, e.g., Mn 3Y(Y= Ge, Ga, Sn) (100 \u0018\n300 S/cm) [49], and Mn 3X0N (X0= Ga, Zn, Ag, or Ni)\n(-359\u0018344 S/cm) [59].\nFigures 8(c) and 8(d) depict the ANC and ATHC cal-\nculated at 130 K as a function of energy, respectively.\nThe ANC and ATHC are -1.05 A/Km and -0.004 W/Km\nat theEF, respectively. Through doping electrons, the\nANC can reach up to -2.14 A/Km and 2.76 A/Km at 0.06\neV and 0.15 eV, respectively, which are larger than that\nof Mn 3X(X= Sn, Ge) (-0.89 \u0018-0.54 A/Km) [49] and\nbcc Fe (1.8 A/Km) [51]. The ATHC reaches up to about\n0.10 W/Km at 0.10 eV, which is comparable to that of14\nBL ferromagnetic F5GT and is slightly larger than some\nfamous noncollinear antiferromagnets, such as Mn 3Sn\n(0.04 W/Km) [51] and Mn 3Ge (0.015 W/Km) [56]. Since\nthe N\u0013 eel temperature of BL antiferromagnetic F5GT is\nunknown yet, we calculate the ANC and ATHC at the\nEFas a function of temperature, as shown in Figs. 8(e)\nand 8(f). The dependence of the ANC and ATHC on\ntemperature in the range of 10 \u0018300 K is not much ob-\nvious for the in-plane magnetization. On the other hand,\nfor the out-of-plane magnetization, the temperature has\na great in\ruence on the ANC and ATHC. For example,\nthe ANC and ATHC increase from almost zero at 10 K\nto -1.28 A/Km and 0.10 W/Km at 300 K, respectively.\nOur results suggest that BL antiferromagnetic F5GT is\nan ideal material platform for the applications of antifer-\nromagnetic spin caloritronics and thermal devices instead\nof usual ferromagnets.\nIV. SUMMARY\nIn conclusion, utilizing the \frst-principles calculations\ntogether with group theory analysis, we systematically\ninvestigated the MAE, MOE, AHE, ANE, and ATHE in\nML and BL Fe nGeTe 2(n= 3, 4, 5). The calculated\nMAE indicates that ML and BL F3GT prefer to the\nout-of-plane magnetization, while the ML and BL F4GT\nand F5GT are in favor of the in-plane magnetization.\nMoreover, the MAE shows the magnetic isotropy in the\nxyplane and as the number of layers increases the out-\nof-plane magnetization enhances. Additionally, we also\ncon\frmed the magnetic ground states of BL Fe nGeTe 2.\nExcept that BL F4GT is energetically stable in its ferro-\nmagnetic state, both BL F3GT and BL F5GT have lower\nenergies in the antiferromagnetic states. We have deter-\nmined the symmetry requirements of the MOE, AHE,\nANE, and ATHE via magnetic group theory analysis.\nIn short, these closely related physical phenomena can\narise in ML and BL ferromagnetic F3GT, F4GT, and\nF5GT with the z-axis magnetization, in BL ferromag-\nnetic F3GT with the y-axis magnetization, in ML and BL\nferromagnetic F4GT and F5GT with the y-axis magne-\ntization, and more intriguingly in BL antiferromagnetic\nF5GT with the y- orz-axis magnetization. The results of\ngroup theory analysis are fully consistent with our \frst-\nprinciples calculations.\nThe MOE, AHE, ANE, and ATHE show the strong\nanisotropy between the in-plane and out-of-plane mag-\nnetization. The largest magneto-optical Kerr and Fara-\nday rotation angles are found in the ferromagnetic states\nwith the out-of-plane magnetization. The Kerr angles of\n1.19\u000e(2.07\u000e), 1.14\u000e(1.90\u000e), and 0.74\u000e(1.44\u000e) are cal-\nculated in ML (BL) F3GT, F4GT, and F5GT, respec-\ntively; the Faraday angles of 222.66\u000e=\u0016m (147.73\u000e=\u0016m),\n109.12\u000e=\u0016m (82.95\u000e=\u0016m), and 50.28\u000e=\u0016m (54.07\u000e=\u0016m)\nare calculated in ML (BL) F3GT, F4GT, and F5GT, re-\nspectively. Furthermore, the calculated zero-temperature\nAHC at the EFare 0.73 (1.57) e2=h, 1.70 (2.54) e2=h, and0.46 (1.50)e2=hin ML (BL) ferromagnetic F3GT, F4GT,\nand F5GT with the out-of-plane magnetization. By dop-\ning electrons or holes, the zero-temperature AHC of BL\nferromagnetic F3GT, F4GT, and F5GT can be signi\f-\ncantly enhanced to -2.16 e2=h, 4.48e2=h, and 3.46e2=h,\nrespectively. According to the generalized Landauer-\nB uttiker formalism, the temperature-dependent AHC,\nANC, and ATHC can be obtained from zero-temperature\nAHC. The overall trend of \fnite-dependent AHC as a\nfunction of the Fermi energy is very similar to that of\nzero-temperature AHC, but the magnitude of AHC re-\nduces markedly with the increasing of temperature. At\nthe temperature of 130 K, the largest ANC can reach\nup to 1.85 (-2.52) A/Km, -1.85 (-3.31) A/Km, and 2.37\n(1.68) A/Km in ML (BL) F3GT, F4GT, and F5GT by\nappropriate electron or hole doping, respectively. The\nlargest ATHC in ML (BL) F3GT, F4GT, and F5GT are\ncalculated to be 0.16 (0.05) W/Km, 0.14 (0.22) W/Km,\nand 0.19 (0.16) W/Km at 130 K, respectively. The note-\nworthy MOE, AHE, ANE and ATHE discovered in ML\nand BL ferromagnetic Fe nGeTe 2(n= 3, 4, 5) are com-\nparable to or even larger than that of many familiar fer-\nromagnets, suggesting that 2D Fe nGeTe 2have potential\napplications in magneto-optical devices, spintronics, spin\ncaloritronics, and etc.\nAnother interesting \fnding in our work is that the\nMOE, AHE, ANE, and ATHE surprisingly appear in BL\ncollinear antiferromagnetic F5GT, which is much rare in\n2D magnets. The \frst-principles results show that the\nMOE, AHE, ANE and ATHE in BL collinear antifer-\nromagnetic F5GT are considerably large. For the out-\nof-plane magnetization, the largest Kerr angle, Faraday\nangle, zero-temperature AHC, ANC, and ATHC are -\n0.23\u000e, 15.94\u000e=\u0016m, -0.28e2=h, -1.05 A/Km (130 K), and\n-0.004 W/Km (130 K), respectively. By doping electrons\nor holes, the AHC, ANC, and ATHC can increase up\nto 2.63e2=h, 2.76 A/Km (130 K) and 0.1 W/Km (130\nK), respectively. On the other hand, by increasing tem-\nperature to 300 K, the ANC and ATHC can reach up\nto -1.28 A/Km and 0.10 W/Km at the EF. These re-\nsults are comparable to or even larger than that of some\nfamous noncollinear and collinear antiferromagnets, indi-\ncating the BL collinear antiferromagnetic F5GT is a good\nmaterial for multi-functional device applications based on\nantiferromagnetism instead of usual ferromagnetism.\nACKNOWLEDGMENTS\nThe authors thank Xiaoping Li, Jin Cao, Chaoxi Cui,\nShifeng Qian, Jun Sung Kim, Xiaobo Lu, and Li Yang\nfor their helpful discussion. This work is supported by\nthe National Natural Science Foundation of China (Grant\nNos. 11874085, 11734003, and 12061131002), the Sino-\nGerman Mobility Programme (Grant No. M-0142), and\nthe National Key R&D Program of China (Grant No.\n2020YFA0308800).15\n[1] N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133\n(1966).\n[2] B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein,\nR. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A.\nMcGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-\nHerrero, and X. Xu, Nature 546, 270 (2017).\n[3] C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao,\nW. Bao, C. Wang, Y. Wang, Z. Q. Qiu, R. J. Cava, S. G.\nLouie, J. Xia, and X. Zhang, Nature 546, 265 (2017).\n[4] H. L. Zhuang, P. R. C. Kent, and R. G. Hennig, Phys.\nRev. B 93, 134407 (2016).\n[5] Z. Fei, B. Huang, P. Malinowski, W. Wang, T. Song,\nJ. Sanchez, W. Yao, D. Xiao, X. Zhu, A. F. May, W. Wu,\nD. H. Cobden, J.-H. Chu, and X. Xu, Nat. Mater. 17,\n778 (2018).\n[6] Y. Deng, Y. Yu, Y. Song, J. Zhang, N. Z. Wang, Z. Sun,\nY. Yi, Y. Z. Wu, S. Wu, J. Zhu, J. Wang, X. H. Chen,\nand Y. Zhang, Nature 563, 94 (2018).\n[7] \u001f. Johansen, V. Risingg\u0017 ard, A. Sudb\u001c, J. Linder, and\nA. Brataas, Phys. Rev. Lett. 122, 217203 (2019).\n[8] X. Wang, J. Tang, X. Xia, C. He, J. Zhang, Y. Liu,\nC. Wan, C. Fang, C. Guo, W. Yang, Y. Guang, X. Zhang,\nH. Xu, J. Wei, M. Liao, X. Lu, J. Feng, X. Li, Y. Peng,\nH. Wei, R. Yang, D. Shi, X. Zhang, Z. Han, Z. Zhang,\nG. Zhang, G. Yu, and X. Han, Sci. Adv 5, eaaw8904\n(2019).\n[9] S. Albarakati, C. Tan, Z.-J. Chen, J. G. Partridge,\nG. Zheng, L. Farrar, E. L. H. Mayes, M. R. Field, C. Lee,\nY. Wang, Y. Xiong, M. Tian, F. Xiang, A. R. Hamilton,\nO. A. Tretiakov, D. Culcer, Y.-J. Zhao, and L. Wang,\nSci. Adv 5, eaaw0409 (2019).\n[10] X. Lin and J. Ni, Phys. Rev. B 100, 085403 (2019).\n[11] M.-C. Jiang and G.-Y. Guo, arXiv:2012.04285.\n[12] J. Xu, W. A. Phelan, and C.-L. Chien, Nano Lett. 19,\n8250 (2019).\n[13] L. Yin and D. S. Parker, Phys. Rev. B 102, 054441\n(2020).\n[14] G. Zheng, W.-Q. Xie, S. Albarakati, M. Algarni, C. Tan,\nY. Wang, J. Peng, J. Partridge, L. Farrar, J. Yi,\nY. Xiong, M. Tian, Y.-J. Zhao, and L. Wang, Phys.\nRev. Lett. 125, 047202 (2020).\n[15] J. Stahl, E. Shlaen, and D. Johrendt, Z. Anorg. Allg.\nChem. 644, 1923 (2018).\n[16] A. F. May, D. Ovchinnikov, Q. Zheng, R. Hermann,\nS. Calder, B. Huang, Z. Fei, Y. Liu, X. Xu, and M. A.\nMcGuire, ACS Nano 13, 4436 (2019).\n[17] J. Seo, D. Y. Kim, E. S. An, K. Kim, G.-Y. Kim, S.-\nY. Hwang, D. W. Kim, B. G. Jang, H. Kim, G. Eom,\nS. Y. Seo, R. Stania, M. Muntwiler, J. Lee, K. Watanabe,\nT. Taniguchi, Y. J. Jo, J. Lee, B. I. Min, M. H. Jo, H. W.\nYeom, S.-Y. Choi, J. H. Shim, and J. S. Kim, Sci. Adv\n6, eaay8912 (2020).\n[18] T. Ohta, K. Sakai, H. Taniguchi, B. Driesen, Y. Okada,\nK. Kobayashi, and Y. Niimi, Appl. Phys. Express 13,\n043005 (2020).\n[19] J. Kerr, Phil. Mag. 3, 321 (1877).\n[20] M. Faraday, Phil. Trans. R. Soc. 136, 1 (1846).\n[21] Y. Suzuki, T. Katayama, S. Yoshida, K. Tanaka, and\nK. Sato, Phys. Rev. Lett. 68, 3355 (1992).\n[22] G. Y. Guo and H. Ebert, Phys. Rev. B 51, 12633 (1995).\n[23] P. Ravindran, A. Delin, P. James, B. Johansson, J. M.Wills, R. Ahuja, and O. Eriksson, Phys. Rev. B 59,\n15680 (1999).\n[24] A. A. Mosto\f, J. R. Yates, Y.-S. Lee, I. Souza, D. Van-\nderbilt, and N. Marzari, Computer Physics Communi-\ncations 178, 685 (2008).\n[25] J. R. Yates, X. Wang, D. Vanderbilt, and I. Souza, Phys.\nRev. B 75, 195121 (2007).\n[26] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and\nN. P. Ong, Rev. Mod. Phys. 82, 1539 (2010).\n[27] Y. Yao, L. Kleinman, A. H. MacDonald, J. Sinova,\nT. Jungwirth, D.-s. Wang, E. Wang, and Q. Niu, Phys.\nRev. Lett. 92, 037204 (2004).\n[28] W. Nernst, Ann. Phys. 267, 760 (1887).\n[29] T. Qin, Q. Niu, and J. Shi, Phys. Rev. Lett. 107, 236601\n(2011).\n[30] N. W. Ashcroft and N. D. Mermin, Solid state physics\n(SaundersCollege Publishing, Philadelphia, 1976).\n[31] H. van Houten, L. W. Molenkamp, C. W. J. Beenakker,\nand C. T. Foxon, Semicond. Sci. Technol. 7, B215 (1992).\n[32] K. Behnia, Fundamentals of thermoelectricity (Oxford\nUniversity Press, 2015).\n[33] P. E. Bl ochl, Phys. Rev. B 50, 17953 (1994).\n[34] G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993).\n[35] G. Kresse and J. Furthm uller, Phys. Rev. B 54, 11169\n(1996).\n[36] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.\nLett. 77, 3865 (1996).\n[37] S. Grimme, J. Antony, S. Ehrlich, and H. Krieg, The\nJournal of Chemical Physics 132, 154104 (2010).\n[38] S. Grimme, S. Ehrlich, and L. Goerigk, J Comput Chem\n32, 1456 (2011).\n[39] H. T. Stokes, D. M. Hatch, and B. J. Campbell,\nhttps://stokes.byu.edu/iso/isotropy.php .\n[40] H. L. Zhuang and R. G. Hennig, Phys. Rev. B 93, 054429\n(2016).\n[41] V. K. Gudelli and G.-Y. Guo, New J. Phys. 21, 053012\n(2019).\n[42] X. Lu, R. Fei, L. Zhu, and L. Yang, Nat. Commun. 11,\n4724 (2020).\n[43] Y. Fang, S. Wu, Z.-Z. Zhu, and G.-Y. Guo, Phys. Rev.\nB98, 125416 (2018).\n[44] S. W. Jang, M. Y. Jeong, H. Yoon, S. Ryee, and M. J.\nHan, arXiv:1904.04510.\n[45] H. Zhang, R. Chen, K. Zhai, X. Chen, L. Caretta,\nX. Huang, R. V. Chopdekar, J. Cao, J. Sun, J. Yao,\nR. Birgeneau, and R. Ramesh, Phys. Rev. B 102, 064417\n(2020).\n[46] V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono,\nand Y. Tserkovnyak, Rev. Mod. Phys. 90, 015005 (2018).\n[47] W. Feng, G.-Y. Guo, J. Zhou, Y. Yao, and Q. Niu, Phys.\nRev. B 92, 144426 (2015).\n[48] A. K. Nayak, J. E. Fischer, Y. Sun, B. Yan, J. Karel,\nA. C. Komarek, C. Shekhar, N. Kumar, W. Schnelle,\nJ. K ubler, C. Felser, and S. S. P. Parkin, Sci. Adv. 2,\ne1501870 (2016).\n[49] G.-Y. Guo and T.-C. Wang, Phys. Rev. B 96, 224415\n(2017).\n[50] M. Ikhlas, T. Tomita, T. Koretsune, M.-T. Suzuki,\nD. Nishio-Hamane, R. Arita, Y. Otani, and S. Nakat-\nsuji, Nat. Phys 13, 1085 (2017).\n[51] X. Li, L. Xu, L. Ding, J. Wang, M. Shen, X. Lu, Z. Zhu,16\nand K. Behnia, Phys. Rev. Lett. 119, 056601 (2017).\n[52] T. Jungwirth, J. Sinova, A. Manchon, X. Marti, J. Wun-\nderlich, and C. Felser, Nat. Phys. 14, 200 (2018).\n[53] L. \u0014Smejkal, Y. Mokrousov, B. Yan, and A. H. MacDon-\nald, Nat. Phys. 14, 242 (2018).\n[54] J. \u0014Zelezn\u0013 y, P. Wadley, K. Olejn\u0013 \u0010k, A. Ho\u000bmann, and\nH. Ohno, Nat. Phys. 14, 220 (2018).\n[55] K. Sugii, Y. Imai, M. Shimozawa, M. Ikhlas, N. Kiy-\nohara, T. Tomita, M.-T. Suzuki, T. Koretsune,\nR. Arita, S. Nakatsuji, and M. Yamashita, (2019),\narXiv:1902.06601.\n[56] L. Xu, X. Li, X. Lu, C. Collignon, H. Fu, J. Koo,\nB. Fauqu\u0013 e, B. Yan, Z. Zhu, and K. Behnia, Sci. Adv\n6, eaaz3522 (2020).\n[57] X. Zhou, J.-P. Hanke, W. Feng, S. Bl ugel, Y. Mokrousov,\nand Y. Yao, Phys. Rev. Materials 4, 024408 (2020).\n[58] Y. Zhang, Y. Sun, H. Yang, J. \u0014Zelezn\u0013 y, S. P. P. Parkin,\nC. Felser, and B. Yan, Phys. Rev. B 95, 075128 (2017).\n[59] X. Zhou, J.-P. Hanke, W. Feng, F. Li, G.-Y. Guo, Y. Yao,\nS. Bl ugel, and Y. Mokrousov, Phys. Rev. B 99, 104428\n(2019).\n[60] M. Wu, H. Isshiki, T. Chen, T. Higo, S. Nakatsuji, and\nY. Otani, Applied Physics Letters 116, 132408 (2020).\n[61] W. Feng, J.-P. Hanke, X. Zhou, G.-Y. Guo, S. Bl ugel,\nY. Mokrousov, and Y. Yao, Nat Commun 11, 118 (2020).\n[62]\u0014S. Libor, A. H. MacDonald, J. Sinova, S. Nakatsuji, and\nT. Jungwirth, (2021), arXiv:2107.03321.\n[63] X. Zhou, W. Feng, X. Yang, G.-Y. Guo, and Y. Yao,\nPhys. Rev. B 104, 024401 (2021).\n[64] L. \u0014Smejkal, R. Gonz\u0013 alez-Hern\u0013 andez, T. Jungwirth, and\nJ. Sinova, Sci. Adv. 6, eaaz8809 (2020).\n[65] V. Antonov, B. Harmon, and A. Yaresko, Elec-\ntronic structure and magneto-optical properties of solids\n(Springer Science & Business Media, 2004).\n[66] G. Y. Guo and H. Ebert, Phys. Rev. B 51, 12633 (1995).\n[67] P. M. Oppeneer, T. Maurer, J. Sticht, and J. K ubler,\nPhys. Rev. B 45, 10924 (1992).\n[68] X. Yang, X. Zhou, W. Feng, and Y. Yao, Phys. Rev. B\n103, 024436 (2021).\n[69] X. Zhou, W. Feng, F. Li, and Y. Yao, Nanoscale 9, 17405\n(2017).\n[70] W. Feng, G.-Y. Guo, and Y. Yao, 2D Mater. 4, 015017(2017).\n[71] G. Q. Di and S. Uchiyama, Phys. Rev. B 53, 3327 (1996).\n[72] K. Kim, J. Seo, E. Lee, K.-T. Ko, B. Kim, B. G. Jang,\nJ. M. Ok, J. Lee, Y. J. Jo, W. Kang, et al. , Nat. Mater.\n17, 794 (2018).\n[73] P. N. Dheer, Phys. Rev. 156, 637 (1967).\n[74] G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nat.\nMater. 11, 391 (2012).\n[75] S. R. Boona, R. C. Myers, and J. P. Heremans, Energy\nEnviron. Sci. 7, 885 (2014).\n[76] D. Xiao, Y. Yao, Z. Fang, and Q. Niu, Phys. Rev. Lett.\n97, 026603 (2006).\n[77] J. Noky, Y. Zhang, J. Gooth, C. Felser, and Y. Sun, npj\nComput Mater 6, 77 (2020).\n[78] S. N. Guin, P. Vir, Y. Zhang, N. Kumar, S. J. Watz-\nman, C. Fu, E. Liu, K. Manna, W. Schnelle, J. Gooth,\nC. Shekhar, Y. Sun, and C. Felser, Advanced Materials\n31, 1806622 (2019).\n[79] J. Noky, J. Gayles, C. Felser, and Y. Sun, Phys. Rev. B\n97, 220405 (2018).\n[80] H. Zhang, C. Q. Xu, and X. Ke, Phys. Rev. B 103,\nL201101 (2021).\n[81] L. Xu, X. Li, L. Ding, T. Chen, A. Sakai, B. Fauqu\u0013 e,\nS. Nakatsuji, Z. Zhu, and K. Behnia, Phys. Rev. B 101,\n180404 (2020).\n[82] H. Chen, Q. Niu, and A. H. MacDonald, Phys. Rev.\nLett. 112, 017205 (2014).\n[83] N. Sivadas, S. Okamoto, and D. Xiao, Phys. Rev. Lett.\n117, 267203 (2016).\n[84] S. Jiang, J. Shan, and K. F. Mak, Nat. Mater. 17, 406\n(2018).\n[85] B. Huang, G. Clark, D. R. Klein, D. MacNeill,\nE. Navarro-Moratalla, K. L. Seyler, N. Wilson, M. A.\nMcGuire, D. H. Cobden, D. Xiao, W. Yao, P. Jarillo-\nHerrero, and X. Xu, Nat. Nanotech. 13, 544 (2018).\n[86] T. Higo, H. Man, D. B. Gopman, L. Wu, T. Koretsune,\nO. M. J. van 't Erve, Y. P. Kabanov, D. Rees, Y. Li, M.-\nT. Suzuki, S. Patankar, M. Ikhlas, C. L. Chien, R. Arita,\nR. D. Shull, J. Orenstein, and S. Nakatsuji, Nat. Pho-\ntonics 12, 73 (2018)."    },    {        "title": "2108.05528v1.Magnetic_interactions_and_spin_excitations_in_van_der_Waals_ferromagnet_VI__3_.pdf",        "content": "Magnetic interactions and spin excitations in van der Waals ferromagnet VI 3\nElijah Gordon,1V. V. Mkhitaryan,1Haijun Zhao,1, 2Y. Lee,1and Liqin Ke1\n1Ames Laboratory, U.S. Department of Energy, Ames, Iowa 50011\n2School of Physics, Southeast University, Nanjing 211189, China.\n(Dated: August 13, 2021)\nUsing a combination of density functional theory (DFT) and spin-wave theory methods, we inves-\ntigate the magnetic interactions and spin excitations in semiconducting VI 3. Exchange parameters\nof monolayer, bilayer, and bulk forms are evaluated by mapping the magnetic energies of various\nspin con\fgurations, calculated using DFT+ U, onto the Heisenberg model. The intralayer cou-\nplings remain largely unchanged in three forms of VI 3, while the interlayer couplings show stronger\ndependence on the dimensionality of the materials. We calculate the spin-wave spectra within a\nlinear spin-wave theory and discuss how various exchange parameters a\u000bect the magnon bands. The\nmagnon-magnon interaction is further incorporated, and the Curie temperature is estimated using\na self-consistently renormalized spin-wave theory. To understand the roles of constituent atoms\non magnetocrystalline anisotropy energy (MAE), we resolve MAE into sublattices and \fnd that\na strong negative V-I inter-sublattice contribution is responsible for the relatively small easy-axis\nMAE in VI 3.\nI. INTRODUCTION\nMagnetic 2D van der Waals materials (m2Dv) have\ngreat potential for future energy-e\u000ecient spin-based de-\nvices. Despite the tremendous advancement in the\nresearch \feld of 2D materials since the discovery of\nGraphene, \fnding and developing robust magnetic 2D\nmaterials remain a signi\fcant challenge. In practice,\nit is nontrivial to induce magnetism into non-magnetic\n2D materials by the magnetic proximity e\u000bects or de-\nfect engineering and control them systematically and re-\nliably. On the other hand, the development of intrinsic\nmagnetic 2D materials have signi\fcantly been motivated\nby the recent experimental breakthrough, demonstrat-\ning that magnetism could sustain down to mono-layer\nCrI3and bilayer CrGeTe 3at a lower temperature (tens\nof Kelvins), [1{11]. Broader applications require m2Dv\nwith a higher TC, which is essentially determined by their\nintrinsic magnetic properties, particularly, exchange cou-\nplings and magnetocrystalline anisotropy. The latter is\nof enhanced importance in 2D as it lifts the constraint\nof the Mermin-Wagner theorem by inducing a spin-wave\n(SW) gap and stabilizing the long-range magnetic order-\ning at \fnite temperatures.\nBesides CrI 3and CrGeTe 3, other intrinsic 2D magnetic\nmaterials of bulk or layer form have been explored ever\nsince. Among them, VI 3is a semiconductor with an en-\nergy gap of\u00180:32{0:67 eV [12, 13] in its bulk form. It\nis ferromagnetic (FM) below 50 K in a relatively small\nmagnetic \feld of about 0 :1 T [12{16]. VI 3experiences\ntwo FM transitions at TC\u001936 K and 50 K at ambient\npressure; the two transitions merge into one as hydro-\nstatic pressure above 0 :6 GPa is applied [17, 18]. Bulk\nVI3has easy-axis magnetic anisotropy ( Ku= 37 kJ=m3at\n10 K) [19]. Remarkably, VI 3is a hard ferromagnet with\na 9:1 kOe coercive \feld at 2 K, signi\fcantly larger than in\nCr-based m2Dv [12, 13]. In general, coercivity depends\nnot only on magnetic anisotropy but also on extrinsic fac-\ntors such as microstructures. The mechanism behind thevery di\u000berent coercivity between VI 3and CrI 3is not well\nunderstood yet. It is also unclear how di\u000berent are the\nexchange couplings and the nature of magnetocrystalline\nanisotropy (MA) in VI 3compared to those in Cr-based\nm2Dv.\nUnlike CrI 3, no exfoliated monolayer VI 3has been re-\nported yet. However, cleavage energies are theoretically\nfound to be between 0 :18 and 0:29 J=m2, which is com-\nparable to those for Cr-based m2Dv [15, 16, 20, 21].\nThus, the exfoliation of VI 3in the near future might\nbe expected, and it is desirable to understand how mag-\nnetic interactions evolve with the material con\fguration\nchanging from bulk to multiple layers, as this understand-\ning is relevant to the real applications, where multiple\nlayers or heterostructures are used instead of bulk mate-\nrials.\nTheoretical studies have also been carried out to un-\nderstand the intrinsic magnetic properties of monolayer\nVI3and con\ricting results regarding the exchange cou-\nplings and electronic structures have been reported. For\nexample, large exchange coupling and higher Curie tem-\nperature,TC, resulting from a half-metallic ground state,\nhave been reported in Ref. [22]. On the other hand, Yang\nand coworkers have found a semiconducting ground state\nand estimated the energy di\u000berence between FM and\nN\u0013 eel-AFM states, suggesting smaller exchange couplings\nandTCin monolayer VI 3[21]. Similarly, di\u000berent depen-\ndences of magnetism on layer stacking in bilayer VI 3have\nbeen reported [23, 24]. These discrepancies demonstrate\nthe sensitivity of calculated intrinsic magnetic properties\non the underlying electronic structure.\nInelastic neutron scattering (INS) is the tool of choice\nto characterize the spin-wave spectrum and underlying\nmagnetic interactions. Unfortunately, unlike CrI 3[25],\nno INS study on VI 3has been reported so far. A com-\nprehensive theoretical analysis of exchange couplings and\nthe resulting spin-wave spectra may provide a helpful in-\nsight to understand magnetic properties in VI 3and make\npredictions upon the prospective INS experiments.arXiv:2108.05528v1  [cond-mat.mtrl-sci]  12 Aug 20212\nIt is believed that MAE in m2Dv results from the in-\nterplay between the spin polarization of 3 datoms and the\nlarge spin-orbit coupling (SOC) of the heavier pelements.\nBy scaling the SOC strength on Cr and I sublattices,\nLado et al. have shown that the heavier I atom plays a\nsigni\fcant role in inducing the magnetic anisotropy in\nCrI3[26]. On the other hand, Chen et al. have argued\nthat a large single-ion anisotropy is expected in VI 3[21]\nbut it is absent in CrI 3. A numerical resolution of MAE\ninto atomic sites or pairs may help to better understand\nMAE in VI 3.\nIn this work, we investigate the exchange couplings\nin monolayer, bilayer, and bulk VI 3within the density\nfunctional theory (DFT). Using the calculated exchange\nparameters, we employ a spin-wave theory to investigate\nthe spin-wave spectra and estimate the Curie tempera-\ntures. By scaling the SOC strength, we decompose the\nMAE contribution into sublattices and compare it with\nCrI3. We also discuss the validity of the second-order\nperturbation theory (SOPT) [27] in calculating MAE in\nthese systems.\nII. METHODS\nThe DFT calculations were performed within the\nframework of the plane-wave projector-augmented wave\nformalism [28], as implemented in the Vienna ab initio\nsimulation package ( vasp ) [29, 30]. The Perdew-Burke-\nErnzerhof generalized gradient approximation [31] was\nused for the exchange-correlation functional. All calcula-\ntions were performed with a plane wave cuto\u000b energy of\n400 eV. To properly describe the additional Coulomb re-\npulsion between electrons beyond DFT, the on-site Hub-\nbardUwas applied on V-3 delectrons using the double-\ncounting scheme introduced by Dudarev et al. [32].\nThe experimental [15] lattice parameters and atomic\npositions of bulk VI 3were employed for all calculations\nunless explicitly stated otherwise. For bulk VI 3, we con-\n\frmed that the volume of the optimized structure is\nwithin\u00184% of the experimental value. The reported\noptimized lattice constants for monolayer VI 3are di\u000ber-\nent only by\u00183% from the experimental values of bulk\nVI3[16, 33]. Thus, employing experimental structural\nparameters is a reasonable approximation for this study.\nA more comprehensive investigation of theoretical struc-\ntural optimization, which often depends on the function-\nals used, is beyond the scope of the current study. In\nthe bilayer and monolayer models, a vacuum region of\n16\u0017Awas included to avoid the interaction between peri-\nodic images.\nA. Exchange parameters\nTo estimate the exchange couplings, we map the total\nenergies of various collinear spin con\fgurations onto a\nHeisenberg model de\fned as\nNéel\nZigzag\nStripy\n(a)\n (b)\nFIG. 1. (a) Schematic representation of the V sublattices\nin rhombohedral VI 3. The hexagonal conventional unit cell\nis used to depict pair exchange parameters for the \frst few\nneighbors of V atoms. Two V sublattices, V 1and V 2, are\nindicated with green and purple spheres, respectively. (b)\nN\u0013 eel-, zigzag-, and stripy-AFM intralayer spin con\fgurations.\nA four-atom unit cell, denoted by the dashed blue lines, is\nused to describe the three spin con\fgurations. Green and\nblue spheres denote the spin-up and spin-down orientations,\nrespectively.\nH=\u0000X\ni6=jJij^ei\u0001^ej; (1)\nwhere ^eiis the unit vector of magnetic moment on site\ni. The \frst six nearest-neighbor exchange couplings Jij,\nincluding three intralyer ( J1,J2, andJ3) and three inter-\nlayer ( ~J0,~J1\u0000, and ~J1+) exchange couplings indicated in\nFig. 1(a), are included to describe the bilayer and bulk\ncases.\nFor monolayer VI 3, we calculate the energies of four\ncollinear spin con\fgurations, namely, FM, N\u0013 eel-, zigzag-\n, and stripy-AF [34] to derive the three intralayer ex-\nchanges as follows:\n0\n@J1\nJ2\nJ31\nA=0\n@6 0 6\n2 8 6\n4 8 01\nA\u000010\n@\u0001EN\n\u0001EZ\n\u0001ES1\nA: (2)\nHere, \u0001E\u0016=E\u0016\u0000EFM,\u0016=N, Z, and S, are the energies\n(per V atom) of N\u0013 eel, Zigzag, and Stripy con\fgurations\n(illustrated in Fig. 1(b)) with respect to the reference FM\ncon\fguration.\nTo obtain the interlayer couplings for bilayer and bulk\nVI3, we double each of these four intralayer con\fgu-\nrations along the out-of-plane direction in a larger su-\npercell. For each intralayer con\fguration, there are\ntwo (analog to FM/AFM) possible interlayer orderings,\nwhose energy di\u000berence is used to extract the interlayer\ncouplings.3\nB. Linear spin-wave theory\nUsing the obtained exchange parameters we calculate\nthe SW spectra within a linear SW theory. First, we\nrewrite Eq. (1) in terms of spin Sinstead of unit vector\n^e, asH=\u00001\n2P\ni6=jJS\nijSi\u0001Sj, whereJS\nij= 2Jij=S2,\nandS=jSj=m=2 is the on-site spin of V atoms.\nNext, we bosonize the spin operator via the Holstein-\nPrimako\u000b transformation [35] and truncate the Hamilto-\nnian at quadratic order in bosons to diagonalize it. There\nare two V sublattices in the primitive unit cell of bulk R-\nVI3, resulting in two SW branches with energies written\nas [36]:\n!\u0006(q) =S\u0000\nJIntra\nq=0+JInter\nq=0\u0000JIntra\nq\u0006jJInter\nqj\u0001\n;(3)\nwhereJIntra\nq andJInter\nq sum over the intra-sublattice and\ninter-sublattice exchange couplings, respectively. With\nthe considered range of exchange couplings, we have\nJIntra\nq =J2(q) +~J1\u0000(q);\nJInter\nq =J1(q) +J3(q) +~J0(q) +~J1+(q): (4)\nHere,\nJi(q) =JS\ni\ri(q) =JS\niX\n\u000eie\u0000i2\u0019q\u0001\u000ei; (5)\nand\ri(q) is the structure factor, while \u000eistanding for\nthe connecting vectors of corresponding exchange JS\ni.\nC. Non-linear spin-wave theory\nThe linear SW theory brie\ry discussed above neglects\nthe interactions between magnons. Therefore it is appli-\ncable only at the lower temperatures, where INS experi-\nments are often used to measure SW spectra. In the crit-\nical region close to the Curie temperature, interactions\nbetween the spin waves become progressively more rele-\nvant. To the leading order, these interactions are given by\nthe quartic terms of the Holstein-Primako\u000b transformed\nHamiltonian. To assess the SW interaction e\u000bects, we\ninvolve quartic terms and treat them within a Hartree-\nFock-like decoupling approximation [37{40]. This pro-\ncedure leads to an e\u000bective quadratic Hamiltonian con-\ntaining renormalization factors that encapsulate SW in-\nteraction e\u000bects. The e\u000bective quadratic Hamiltonian is\nthen solved self-consistently, e.g., for the magnetization\nversus temperature behavior, M(T). In such a way, we\nset up a self-consistently renormalized spin-wave theory\n(SRSWT) [41], which is the analogue of previous self-\nconsistently renormalized theories [37, 42{44] for layered\nsystems with hexagonal intralayer structure.\nWithin our SRSWT, renormalization factors de\fning\nthe e\u000bective Hamiltonian are combinations of the average\nspin polarization, \u0016S, and the average short-range two-\nboson correlations between the exchange-coupled sites.Thus, the e\u000bective Hamiltonian is a function of four pa-\nrameters in monolayer VI 3or of seven parameters in\nbulk VI 3. The self-consistency equations are found by\nderiving relations for \u0016Sand the average two-boson cor-\nrelations from the e\u000bective Hamiltonian. Subsequently,\nthese equations are solved numerically by successive iter-\nations. The magnetization is de\fned as the normalized\naverage spin polarization, M(T) =\u0016S(T)=S, and the crit-\nical temperature is taken as the temperature at which\nM(T) turns to zero.\nUsing the calculated values of exchange couplings, we\n\fndM(T) and the corresponding critical temperatures\nfor monolayer, bilayer, and bulk VI 3, by means of SR-\nSWT. The analysis of monolayer and bulk systems is fa-\ncilitated by the fact that these systems consist of two\nequivalent sublattices of magnetic atoms. In contrast,\nwe regard the few-layer systems with Lnumber of lay-\ners as consisting of 2 Lsublattices { two per hexagonal\nlayer. This approach allows us to take into account the\nphysical di\u000berence between the surface and bulk layers.\nAt the same time, it requires a substantially larger pa-\nrameter space for the self-consistency equations, as the\naverage spin polarization and two-boson correlation val-\nues at di\u000berent sublattices are generally di\u000berent. The\ndetails of our SRSWT method and implementation can\nbe found elsewhere [41].\nD. Sublattice-resolved MAE\nMAE is calculated as the total-energy di\u000berence be-\ntween two FM states, in which the magnetization is\naligned along the [100] or [001] directions, respectively.\nSOC was included using the second-variation method [45,\n46]. We resolve MAE contributions into sublattices via\nscaling the SOC strength \u0018ion V and I sites with param-\neters\u0015i; the SOC Hamiltonian becomes\nHsoc=X\ni=V,I\u0015i\u0018i(Li\u0001Si): (6)\nAssuming that MAE can be well described by SOPT,\ntotal MAE can be written as [47]\nK(\u0015V;\u0015I) =KV-V\u00152\nV+KV-I\u0015V\u0015I+KI-I\u00152\nI;(7)\nwhereKV-V,KV-I, andKI-Idescribe the response of\nMAE to the SOC strength of V, V and I simultaneously,\nand I, respectively. They can be determined by \ftting\ntheK(\u0015V;\u0015I) curve or directly evaluated within SOPT,\nand the inter-sublattice term KV-Iarise from the cou-\npling between the two sublattices [47].\nIII. RESULTS AND DISCUSSION\nA. Electronic properties\nFigure 2 shows the calculated non-SOC band struc-\ntures, the partial density of states (PDOS) projected4\n M K  A−3−2−10123Energy [eV]1a2ex 2ey\n1ex 1ey(c) (a)\n2ex = α(d xy)  + β(d xz) \n2ey = α(d x2 - y2 ) + β(d yz)\n1ex = β(d xy) - α(d xz) \n1ey = β(d x2 - y2 ) - α(d yz)\n1a = d z2\n-2 -1 1 2-4-2024PDOS [States/eV]\n0(xy,x2-y2) \n(xz,yz)  \nz2\nEnergy [eV](b)\nFIG. 2. (a) Band structure of VI 3forU= 2 eV. Solid and dotted lines indicate the majority and minority spin channels,\nrespectively. Fermi level EFis set at zero energy. This \fgure was constructed using sumo [48]. (b) Partial density of states\nprojected on the V- dstates. (c) The splitting and occupation of V- dorbitals in VI 3, where\u000b=p\n1=3 and\f=p\n2=3; thet2g\norbital set splits into doubly-degenerate (1 ex, 1ey) and singly-degenerate (1 a) orbital sets.\non V-dorbitals, and their corresponding orbital charac-\nters near the Fermi level EF, calculated in DFT+ Uwith\nU= 2 eV. The obtained bandgap of 0 :42 eV is within the\nrange of reported experimental values [12, 13].\nThe oxidation state of V in VI 3is +3, with electronic\ncon\fguration [Ar]4 s03d2. V atoms arrange in a honey-\ncomb structure within the layer with edge-sharing octa-\nhedral coordination by I ligands. The octahedral ligand\nand crystal \felds split the V- dstates into the doublet eg\nand triplet t2gsubsets, with the latter states lying lower\nin energy. Therefore, the two unpaired electrons of V3+\noccupy the t2gorbitals, both with spins up according to\nHund's \frst rule, forming the spin state S= 1, consistent\nwith the calculated on-site magnetic moment of V atom\nm\u00192\u0016B/V.\nCorrelation e\u000bects beyond DFT are required to de-\nscribe the semiconducting ground state of VI 3cor-\nrectly [49]. Figure 2(c) shows that the t2gorbital set fur-\nther splits into doubly degenerate (1 ex, 1ey) and singly\ndegenerate (1 a) orbital sets. While DFT gives a half-\nmetallic state, the on-site Coulomb correction shifts the\n1astate upward relative to the (1 ex, 1ey) states, result-\ning in a bandgap in between [23, 49]. We note that\none may need to control the initial orbital occupancy in\nthe DFT+Ucalculation to converge to the experimen-\ntal semiconducting ground state for VI 3, which may ex-\nplain the discrepancy of ground states found in previous\nDFT+Ucalculations [21, 23, 50].\nWe also qualitatively explore the e\u000bects of compressive\nstrain on the electronic structure by gradually decreas-\ning the lattice parameters (volume) of bulk VI 3while\nkeeping the aspect ratio \fxed. We found that the (1 ex,\n1ey) states shift upward relative to the 1 astate, and a\nsemiconductor-to-metal transition occurs at a strain of\n5{6% when the gap closes, and all three t2gstates be-\ncome partially occupied. A more systematic and compre-\nhensive theoretical investigation on structural propertiesmay help compare with or guide the future experiments\nin this direction.\nB. Exchange coupling in monolayer, bilayer, and\nbulk VI 3\nTABLE I. Pairwise intralayer exchange parameters Ji,i=\n1, 2, 3 and interlayer exchange parameters ~Ji,i= 0, 1+,\n1\u0000for the Heisenberg Hamiltonian H=\u0000P\ni6=jJij^ei\u0001^ejin\nmonolayer, bilayer, and bulk R-VI 3calculated within DFT+ U\nwith various Uvalues (in the unit of eV). Positive (negative)\nJijvalues correspond to FM (AFM) couplings. Exchange\nparameters are illustrated in Fig. 1 for the bulk structure. The\ndegeneracy (No.) and distance ( Rij) ofJijare also provided.\nR-VI 3Lbl. No.Rij Jij(meV)\nForm ( \u0017A)U=2U=3U=3.7\nMonolayerJ1 3 3.946 2.26 2.07 1.99\nJ2a6a6.835 0.06 0.08 0.09\nJ3 3 7.893 -0.23 -0.18 -0.16\nBilayerJ1 3 3.946 2.26 2.07 2.03\nJ2 6 6.835 0.04 0.06 0.06\nJ3 3 7.893 -0.16 -0.14 -0.12\n~J0 1/0b6.552 -0.14 -0.07 -0.08\n~J1\u0000 3 7.660 0.08 0.06 0.04\n~J1+ 0/3b7.672 0.25 0.17 0.18\nBulkJ1 3 3.946 2.18 2.02 1.98\nJ2a6 6.835 0.10 0.09 0.08\nJ3 3 7.893 -0.20 -0.16 -0.15\n~J0 1 6.552 0.27 0.17 0.14\n~J1\u0000a6 7.660 0.18 0.13 0.11\n~J1+ 3 7.672 -0.05 -0.03 -0.02\naIntra-sublattice couplings for monolayer and bulk cases.\nbDi\u000berent V sites have di\u000berent numbers of neighbors coupled\nthrough ~J0or~J1+.5\nU (eV)\nHexagonal Stacking\nMonoclinic Stacking\nFIG. 3. Interlayer magnetic oupling energy, \u0001 E=EL-L\nAFM\u0000\nEL-L\nFM, as a function of on-site Coulomb interaction parameter\nUin bilayer VI 3with hexagonal and monoclinic stackings.\nWe next discuss the dependence of exchange couplings\non dimensionality, additional on-site Coulomb interac-\ntionU, and stacking order. Table I summarizes the ex-\nchange parameters calculated in monolayer, bilayer, and\nbulk VI 3with di\u000berent Uvalues. Although increasing U\nvalues generally decrease the exchange coupling, suggest-\ning a more localized-moment picture, qualitative details\nremain the same. In particular, there is no change in\ntheir signs for the considered Uvalues.\nIn all the monolayer, bilayer, and bulk forms, the near-\nest neighbor intralayer exchange coupling J1, arises from\na superexchange via the near-90\u000eV-I-V bonds. It is\nfound to be dominant and FM, while J2andJ3are\nweakly FM and AFM, respectively. In bulk VI 3, the\n\frst two nearest neighbor interlayer parameters ~J0and\n~J1\u0000are found to be positive, leading to a favorable FM\ninterlayer ordering. Note that although ~J1+is negative,\nits magnitude is small. The overall interlayer coupling\nis dominated by ~J0and ~J1\u0000, especially the latter, which\nhas a larger coordination number of six.\nInterestingly, the interlayer couplings show a much\nstronger dependence on dimensionality than the in-\ntralayer ones. As shown in Table I, the intralayer cou-\nplings only change slightly in three di\u000berent forms. In\ncontrast, the overall interlayer coupling in bilayer remains\nFM as in bulk, the individual interlayer couplings ~J0and\n~J1+change signs when we compare the bilayer and bulk\ncases. Note that the coordination number of interlayer\ncouplings decreases by a factor of two in the bilayer form.\nWe also investigate the stacking-dependent magnetism\nin VI 3and found a similar magnetostructural coupling\nas previously found in CrI 3[51{53]. Figure 3 shows\ntheUdependence of interlayer magnetic coupling en-\nergy in bilayer VI 3with hexagonal stacking and mon-\noclinic stacking. The energy di\u000berence is calculated as\n\u0001E=EL-L\nAFM\u0000EL-L\nFM, withEL-L\nAFM andEL-L\nFMbeing the en-\nergies of AFM-ordered layer-layer and FM-ordered layer-layer con\fgurations, respectively. For the hexagonal\nstacking, FM interlayer ordering between the layers is\nfound to be stable over a wide range of U. For the mon-\noclinic stacking, in contrast, the energy di\u000berence \u0001 E\nis very small at U\u00192 eV, indicating competing inter-\nlayer AFM and FM states. Moreover, a crossover from\nthe interlayer-FM to interlayer-AFM magnetic ground\nstate occurs at U\u00192:25 eV. These results point to the\nstrong magnetostructural coupling in bilayer VI 3. Con-\nsequently, control of the magnetic con\fguration, for ex-\nample, via strain or electric \feld, is feasible, facilitating\ndevice applications.\nC. Spin-wave dispersion\nM K Γ Z\nM K K Γ(a)\n(b)\nl\nl\nFIG. 4. (a) Spin-wave spectra calculated with only intralayer\nexchange couplings Ji,i= 1, 2, 3 involved (red), intralayer Ji,\ni= 1, 2, 3, and interlayer ~J0involved (blue dashed), and all\ncalculatedJiand ~Jiexchange couplings involved (black). For\nall three plots, the exchange coupling values used are those\ncalculated for the bulk VI 3. (b) Spin-wave spectra calculated\nalong [h,k,l] with various l(kz) values; [h,k,0] is along the\n\u0000{K{Mpath (i.e., it is equivalent to the red line in panel a).\nUtilizing the exchange parameters for the bulk VI 3, we\ncalculated the linear SW spectra and investigated how\ndi\u000berent exchanges a\u000bect the spectra features. Such un-\nderstanding may be useful to compare with future INS\nexperiments on VI 3. Figure 4(a) shows the SW spectra\ncalculated along the in-plane \u0000{ K{Mpath and out-of-\nplane \u0000{Zpath. To better illustrate the e\u000bects of di\u000ber-\nent exchanges on SW excitation, we consider three cases6\nwith di\u000berent numbers of interlayer couplings included in\nthe Hamiltonian: 1) only intralayer couplings ( J1-J2-J3;\nsolid red lines), 2) intralayer and the 1 stnearest-neighbor\ninterlayer couplings ( J1-J2-J3-~J0; blue dashed lines), 3)\nall of the considered intralayer and interlayer couplings\n(J1-J2-J3-~J0-~J1\u0000-~J1+; solid black lines).\nWith only intralayer coupling considered, two SW\nbranches cross exactly at K, which is caused by the van-\nishing of the inter-sublattice coupling JInter\nq, and thus\nthe energy di\u000berence between in-phase and out-of-phase\ninter-sublattice precessing. Including the \frst nearest in-\nterlayer coupling ~J0leaves the acoustic branch unchanged\nbut lifts the optical branch by a constant and correspond-\ningly shifts the Dirac crossing toward Mpoint along the\n\u0000{K{Mpath. The constant shifting of in-plane optical\nSW spectra is because ~J0is along the zdirection. On\nthe other hand, the SW spectra along the \u0000{ Zdirection\nbecome dispersive with ~J0. Unlike ~J0, interlayer cou-\nplings ~J1+and ~J1\u0000have nonzero in-plane components in\ntheir connecting vectors (similar to those of J1). They\ninduce additional dispersion on in-plane SW spectra but\npreserve the Dirac crossing at K(not shown) in the ab-\nsence of ~J0. We note that comparing the dispersions of\nthe acoustic and optical modes along the \u0000{ Zdirection\nmay tell whether the dominant FM interlayer coupling\nis intra-sublattice or inter-sublattice. For example, as\nshown in Fig. 4(a), FM ~J1\u0000increases both branches' en-\nergies while the FM ~J0increases the acoustic mode en-\nergy but lowers the optical mode energy when we move\nalong \u0000{Z. It would be interesting if future INS can be\nused to unveil the nature of interlayer couplings.\nFigure 4(b) shows the SW spectra in bulk VI 3calcu-\nlated along paths that are parallel to \u0000{ K{Mbut at var-\niouskzplanes (denoted by di\u000berent colors in Fig. 4(b)).\nAll of the six exchange couplings are included. At \fnite\nkz, the interlayer couplings rotate the Dirac crossing o\u000b\nthis path. Even at kz= 0 plane, including further in-\nterlayer couplings can also rotate the Dirac crossing o\u000b\nthe high-symmetry line and opens up a gap along the\n\u0000{K{Mhigh-symmetry path [36].\nRelativistic exchanges and MAE are not included in\nthe model spin Hamiltonian to calculate SW spectra.\nEasy-axis MAE, in either single-ion or anisotropic ex-\nchange (equivalently two-ion MAE) form, introduces a\ngap of \u0001 = (2 S\u00001)Kat \u0000. On the other hand,\nDzyaloshinkii-Moriya interactions (DMI) and Kitaev in-\nteraction can open up a global gap [25] between two\nmagnon branches.\nD. Magnetization vs Temperature from non-linear\nspin-wave theory\nWe further use SRSWT to explore the temperature\ndependence of magnetization, M(T), of the monolayer,\nbilayer, and bulk VI 3, by utilizing the calculated MAE\nand exchange parameters. For the bulk system, exchange\nparameters calculated for on-site Coulomb correction of\n01 02 03 04 05 00.81.0MT\n(K)bilayermonolayerbulk010203040500.81.0 \nU = 2 eV \nU = 3 eV \nU = 3.7 eVMT\n(K)bulkFIG. 5. (Color online) Magnetization dependence on temper-\nature in zero magnetic \feld calculated using self-consistently\nrenormalized spin-wave theory [41]. Plots for the monolayer,\nbilayer, and bulk VI 3are obtained by using exchange param-\neters calculated for U= 2 eV, as listed in Table I, and the\ncalculated MAE value of 71 \u0016eV. Inset: the same dependence\nfor the bulk system, resulting from the three di\u000berent sets of\nexchange couplings calculated for U= 2, 3, and 3 :7 eV.\n2, 3, and 3.7 eV yield magnetic ordering temperature val-\nues of 48.9, 42.3, and 39.5 K, respectively (see Fig. 5 in-\nset). By noting that the ordering temperature found for\nU= 2 eV is in excellent agreement with the experimen-\ntally reported value of 49 K [13], for the analysis of mono-\nlayer and bilayer systems, we use the exchange couplings\ncalculated with this value of U. For the momentum-space\nintegrals involved in the SRSWT self-consistency equa-\ntions, we use a very \fne k-point mesh of 100 \u0002100 for 2D\nand 100\u0002100\u0002100 for 3D.\nThe plots in Fig. 5 show a typical mean-\feld-like\nbehavior, implying a \frst-order phase transition with\nthe magnetization abruptly vanishing at a critical tem-\nperature. As expected, the magnetization is strongly\nquenched in lower dimensions; The ordering tempera-\nture decreases to about 13 K and 24.3 K in monolayer\nand bilayer VI 3, respectively. Overall, these results attest\nto the feasibility of intrinsically ferromagnetic monolayer\nand bilayer VI 3.\nE. Constituents of MAE and validity of\nperturbation treatment\nThe calculated MAE in VI 3is much smaller than in\nCrI3. To understand the role of constituent atoms on\nthe MA in two compounds we vary the SOC strength\non V/Cr and I sites independently or simultaneously, to\nstudy how MA evolves in monolayer VI 3and compared it\nwith monolayer CrI 3. Figure 6 shows the calculated MAE\nas functions of scaling parameter \u0015with three di\u000berent7\n0 0.5 1 1.5\nCr,I (SOC scale)00.511.5MAE (meV/formula unit)SOC Scaling I with Cr = 0\nSOC Scaling Cr with I = 0\nSOC Scaling using Cr = I\n0 0.5 1.5\nV,I1\n (SOC scale)00.511.5I with V = 0 SOC Scaling \nSOC Scaling V with I = 0\nSOC Scaling using V = I(a)\n(b)\nDeviation from SOPT at λ = 1: \nI - 18.60 %\nV - 0.08 %\nVI3 - 42.06 %Deviation from SOPT at λ = 1: \nI - 11.03  %\nCr - 0.93 %\nCrI3 - 9.24 %MAE (meV/formula unit)\nFIG. 6. MAE as a function of the SOC scaling factor \u0015M;I(M\n= V or Cr) in monolayer CrI 3(a) and VI 3(b). The calculated\nvalues are presented by symbols, and the lines are \fts using\nthe data calculated with \u00152[0;0:4]. The red dashed, blue\nstraight, and black dotted lines correspond to scaling with \u0015I\n(at\u0015M= 0),\u0015M(at\u0015I= 0), and\u0015I=\u0015M, respectively. The\ncalculated MAE at \u0015I=\u0015M= 1 corresponds to normal DFT\nresult.\nscaling schemes, as denoted by red cross, blue triangle,\nand blue circles. We extract corresponding sublattice\ncontributions by \ftting Eq. (7) data calculated with \u00152\n[0;0:4].\nThe large SOC on I sites is essential in providing the\nuniaxial anisotropy in VI 3and CrI 3. Without includ-\ning SOC on I site, CrI 3has a negligibly small easy-plane\nanisotropy from KCr-Cr while VI 3exhibits an even larger\neasy-plane MAE from KV-V. A small positive Cr-I con-\ntributionKCr-Iin CrI 3; a 9% decrease in MAE is ob-\nserved when one removes SOC from Cr in CrI 3. Our\nDFT results for CrI 3are consistent with previous cal-\nculations [26, 54]. Remarkably, the much smaller MAE\nin VI 3than in CrI 3results mostly from a large negative\ninter-sublattice contribution KV\u0000I; which had also been\nfound inL10FeNi and FePd [27, 47]. Turning o\u000b SOC\non V increases the MAE in VI 3by 335%.\nSOPT can be very useful to describe and understandMAE in various systems [55, 56]. Figure 6 also shows\nthe quadratic \fts of MAE values, on the basis of Eq. (7),\nusing data calculated for \u00152[0;0:4], and corresponding\nextrapolation to \u0015= 1:5. The deviation of the \ft from\nthe calculated data indicates the validity of the SOPT\ntreatment of the SOC and MAE. For small perturbations\n(small 3dSOC comparing to bandwidth) one expects a\nperfect quadratic dependence of MAE on the scaling fac-\ntors with the SOPT, as shown in the plots in Fig. 6 for\nthe scaling scheme that \u0015V/Cr is varied while \u0015I= 0.\nHowever, the large SOC constant of I- porbitals causes\ndeviations of MAE from SOPT in CrI 3and VI 3, espe-\ncially the latter. Comparing the \ftted K(\u0015V=\u0015I) with\nthe calculated ones, VI 3shows a dramatically larger de-\nviation of 42% than 9% in CrI 3. This large deviation\nleads to the conclusion that SOPT describes MA e\u000bects\nbetter in CrI 3than in VI 3. To con\frm this conclusion,\nwe have also calculated MAE of CrI 3and VI 3using force\ntheorem and SOPT with full-potential linear augmented\nplane wave (FP-LAPW) method [57]. The calculations\nproduced about 75% (18%) di\u000berent MAE values be-\ntween the two di\u000berent methods for VI 3(CrI 3). Note\nthat VI 3has a much smaller bandgap than CrI 3, which\nlikely makes SOPT less valid in describing the MAE in\nVI3.\nIV. CONCLUSIONS\nIn summary, we have theoretically studied the elec-\ntronic and magnetic properties of the two-dimensional\n(2D) van der Waals magnet VI 3in various forms. The in-\nterlayer exchange couplings show a stronger dependence\non the dimensionality of materials than the intralayer\ncouplings. Moreover, VI 3also shows a strong stacking-\ndependent interlayer coupling, similar to CrI 3, render-\ning the bilayer ground state interlayer ordering read-\nily switchable between the FM and AFM con\fgurations\nupon external stimuli, such as strain and electric \feld.\nConsidering the relative positions of and the hybridiza-\ntions between cation-3 dand anion-pstates can be es-\nsential to determine the superexchange coupling in VI 3,\nfuture studies going beyond DFT+ Uand including non-\nlocal exchange correlations will be useful to investigate\nthe magnetic interactions in VI 3. Within SRSWT, the\ncalculatedTCin bulk agrees well with experiments while\nTCin the monolayer and bilayer are quenched down to\n14 and 24 K, respectively. The sublattice-resolved MAE\ncalculated in the monolayer VI 3reveals that a strong V-I\ninter-ion easy-plane contribution to MAE lowers the over-\nall easy-axis MAE in VI 3. Finally, we demonstrate that\nthe second-order perturbation treatment of the MAE is\nless valid in VI 3than in CrI 3, due to the combination of\nlarge SOC on I sublattice and a smaller bandgap in VI 3.8\nACKNOWLEDGMENTS\nThe authors thank B. Harmon, P. Ong and G. Miller\nfor valuable discussions. This work was supported by\nthe U.S. Department of Energy, O\u000ece of Science, O\u000ece\nof Basic Energy Sciences, Materials Sciences and Engi-\nneering Division, and Early Career Research Program.\nAmes Laboratory is operated for the U.S. Department of\nEnergy by Iowa State University under Contract No. DE-\nAC02-07CH11358.\nAppendix A: Intralayer coupling\nFor monolayer, the energies (per magnetic atom) of\nFM, N\u0013 eel-, zigzag-, and stripy-AF con\fgurations, EFM,\nEN,EZ, andES, respectively, are mapped onto Eq. (1)as follows:\nE\u0016=E0+3X\nk=1C\u0016\nkJk; \u0016 = FM;N;Z;S; (A1)\nwhereE0is the non-magnetic energy, and the coe\u000ecients\nC\u0016\nkare listed in Table II.\nTABLE II. Coe\u000ecients of mapping of di\u000berent spin con\fgura-\ntions onto the spin Hamiltonian de\fned as H=\u0000PJij^ei\u0001^ej.\nThe right columns represent the di\u000berence, \u0001 C\u0016=C\u0016\u0000CFM.\nCon\fgurationsC \u0001C\nC\u0016\n1C\u0016\n2C\u0016\n3 \u0001C\u0016\n1\u0001C\u0016\n2\u0001C\u0016\n3\nFM -3 -6 -3 0 0 0\nN\u0013 eel 3 -6 3 6 0 6\nZigzag -1 2 3 2 8 6\nStripy 1 2 -3 4 8 0\n[1] M. Gibertini, M. Koperski, A. F. Morpurgo, and K. S.\nNovoselov, Magnetic 2d materials and heterostructures,\nNature Nanotechnology 14, 408 (2019).\n[2] Y. Liu, Y. Huang, and X. Duan, Van der waals integra-\ntion before and beyond two-dimensional materials, Na-\nture567, 323 (2019).\n[3] K. S. Burch, D. Mandrus, and J.-G. Park, Magnetism in\ntwo-dimensional van der Waals materials, Nature 563,\n47 (2018).\n[4] C. Lin, Y. Li, Q. Wei, Q. Shen, Y. Cheng, and W. Huang,\nEnhanced valley splitting of transition-metal dichalco-\ngenide by vacancies in robust ferromagnetic insulating\nchromium trihalides, ACS Applied Materials & Interfaces\n11, 18858 (2019).\n[5] B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein,\nR. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A.\nMcGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-\nHerrero, and X. Xu, Layer-dependent ferromagnetism in\na van der waals crystal down to the monolayer limit,\nNature 546, 270 (2017).\n[6] S. Jiang, L. Li, Z. Wang, K. F. Mak, and J. Shan, Con-\ntrolling magnetism in 2d CrI 3by electrostatic doping,\nNature Nanotechnology 13, 549 (2018).\n[7] P. M. Coelho, K. Nguyen Cong, M. Bonilla, S. Kolekar,\nM.-H. Phan, J. Avila, M. C. Asensio, I. I. Oleynik, and\nM. Batzill, Charge density wave state suppresses ferro-\nmagnetic ordering in VSe 2monolayers, The Journal of\nPhysical Chemistry C 123, 14089 (2019).\n[8] P. Huang, P. Zhang, S. Xu, H. Wang, X. Zhang, and\nH. Zhang, Recent advances in two-dimensional ferromag-\nnetism: materials synthesis, physical properties and de-\nvice applications, Nanoscale 12, 2309 (2020).\n[9] X. Li and J. Yang, First-principles design of spintronics\nmaterials, National Science Review 3, 365 (2016).\n[10] M. M. Otrokov, I. I. Klimovskikh, H. Bentmann, D. Es-\ntyunin, A. Zeugner, Z. S. Aliev, S. Ga\u0019, A. U. B. Wolter,\nA. V. Koroleva, A. M. Shikin, M. Blanco-Rey, M. Ho\u000b-\nmann, I. P. Rusinov, A. Y. Vyazovskaya, S. V. Ere-meev, Y. M. Koroteev, V. M. Kuznetsov, F. Freyse,\nJ. S\u0013 anchez-Barriga, I. R. Amiraslanov, M. B. Babanly,\nN. T. Mamedov, N. A. Abdullayev, V. N. Zverev, A. Al-\nfonsov, V. Kataev, B. B uchner, E. F. Schwier, S. Ku-\nmar, A. Kimura, L. Petaccia, G. Di Santo, R. C. Vi-\ndal, S. Schatz, K. Ki\u0019ner, M. Unzelmann, C. H. Min,\nS. Moser, T. R. F. Peixoto, F. Reinert, A. Ernst, P. M.\nEchenique, A. Isaeva, and E. V. Chulkov, Prediction and\nobservation of an antiferromagnetic topological insulator,\nNature 576, 416 (2019).\n[11] D. Soriano, M. I. Katsnelson, and J. Fern\u0013 andez-Rossier,\nMagnetic two-dimensional chromium trihalides: A theo-\nretical perspective, Nano Letters 20, 6225 (2020), pMID:\n32787171, https://doi.org/10.1021/acs.nanolett.0c02381.\n[12] S. Son, M. J. Coak, N. Lee, J. Kim, T. Y. Kim, H. Hami-\ndov, H. Cho, C. Liu, D. M. Jarvis, P. A. C. Brown, J. H.\nKim, C.-H. Park, D. I. Khomskii, S. S. Saxena, and J.-G.\nPark, Bulk properties of the van der waals hard ferromag-\nnet VI 3, Phys. Rev. B 99, 041402 (2019).\n[13] T. Kong, K. Stolze, E. I. Timmons, J. Tao, D. Ni, S. Guo,\nZ. Yang, R. Prozorov, and R. J. Cava, VI 3{ a new lay-\nered ferromagnetic semiconductor, Advanced Materials\n31, 1808074 (2019).\n[14] J. A. Wilson, C. Maule, P. Strange, and J. N. Tothill,\nAnomalous behaviour in the layer halides and oxyhalides\nof titanium and vanadium: a study of materials close to\ndelocalisation, Journal of Physics C: Solid State Physics\n20, 4159 (1987).\n[15] S. Tian, J.-F. Zhang, C. Li, T. Ying, S. Li, X. Zhang,\nK. Liu, and H. Lei, Ferromagnetic van der waals crystal\nVI3, Journal of the American Chemical Society 141, 5326\n(2019).\n[16] M. An, Y. Zhang, J. Chen, H.-M. Zhang, Y. Guo,\nand S. Dong, Tuning magnetism in layered\nmagnet VI 3: A theoretical study, The Jour-\nnal of Physical Chemistry C 123, 30545 (2019),\nhttps://doi.org/10.1021/acs.jpcc.9b08706.9\n[17] E. Gati, Y. Inagaki, T. Kong, R. J. Cava, Y. Furukawa,\nP. C. Can\feld, and S. L. Bud'ko, Multiple ferromagnetic\ntransitions and structural distortion in the van der waals\nferromagnet VI 3at ambient and \fnite pressures, Phys.\nRev. B 100, 094408 (2019).\n[18] J. Valenta, M. Kratochv\u0013 \u0010lov\u0013 a, M. M\u0013 \u0010\u0014 sek, K. Carva,\nJ. Ka\u0014 stil, P. Dole\u0014 zal, P. Opletal, P. \u0014Cerm\u0013 ak, P. Proschek,\nK. Uhl\u0013 \u0010\u0014 rov\u0013 a, J. Prchal, M. J. Coak, S. Son, J.-G. Park,\nand V. Sechovsk\u0013 y, Pressure-induced huge increase of\ncurie temperature of the van der waals ferromagnet vi 3\n(2020), arXiv:2010.10319 [cond-mat.mtrl-sci].\n[19] J. Yan, X. Luo, F. C. Chen, J. J. Gao, Z. Z. Jiang, G. C.\nZhao, Y. Sun, H. Y. Lv, S. J. Tian, Q. W. Yin, H. C. Lei,\nW. J. Lu, P. Tong, W. H. Song, X. B. Zhu, and Y. P. Sun,\nAnisotropic magnetic entropy change in the hard ferro-\nmagnetic semiconductor VI 3, Phys. Rev. B 100, 094402\n(2019).\n[20] J. He, S. Ma, P. Lyu, and P. Nachtigall, Unusual dirac\nhalf-metallicity with intrinsic ferromagnetism in vana-\ndium trihalide monolayers, J. Mater. Chem. C 4, 2518\n(2016).\n[21] K. Yang, F. Fan, H. Wang, D. I. Khomskii, and H. Wu,\nVI3: A two-dimensional ising ferromagnet, Phys. Rev. B\n101, 100402 (2020).\n[22] J. Yang, J. Wang, Q. Liu, R. Xu, Y. Li, M. Xia, Z. Li, and\nF. Gao, Enhancement of ferromagnetism for VI 3mono-\nlayer, Applied Surface Science 524, 146490 (2020).\n[23] Y.-P. Wang and M.-Q. Long, Electronic and magnetic\nproperties of van der waals ferromagnetic semiconductor\nVI3, Phys. Rev. B 101, 024411 (2020).\n[24] C. Long, T. Wang, H. Jin, H. Wang, and Y. Dai,\nStacking-independent ferromagnetism in bilayer VI 3with\nhalf-metallic characteristic, The Journal of Physical\nChemistry Letters 11, 2158 (2020).\n[25] L. Chen, J.-H. Chung, B. Gao, T. Chen, M. B. Stone,\nA. I. Kolesnikov, Q. Huang, and P. Dai, Topological spin\nexcitations in honeycomb ferromagnet CrI 3, Phys. Rev.\nX8, 041028 (2018).\n[26] J. L. Lado and J. Fern\u0013 andez-Rossier, On the origin of\nmagnetic anisotropy in two dimensional CrI 3, 2D Mate-\nrials4, 035002 (2017).\n[27] M. Blanco-Rey, J. I. Cerd\u0013 a, and A. Arnau, Validity of\nperturbative methods to treat the spin{orbit interaction:\napplication to magnetocrystalline anisotropy, New Jour-\nnal of Physics 21, 073054 (2019).\n[28] P. E. Bl ochl, O. Jepsen, and O. K. Andersen, Improved\ntetrahedron method for brillouin-zone integrations, Phys.\nRev. B 49, 16223 (1994).\n[29] G. Kresse and D. Joubert, From ultrasoft pseudopoten-\ntials to the projector augmented-wave method, Phys.\nRev. B 59, 1758 (1999).\n[30] G. Kresse and J. Furthm uller, E\u000eciency of ab-initio total\nenergy calculations for metals and semiconductors using\na plane-wave basis set, Computational Materials Science\n6, 15 (1996).\n[31] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized\ngradient approximation made simple, Phys. Rev. Lett.\n77, 3865 (1996).\n[32] S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J.\nHumphreys, and A. P. Sutton, Electron-energy-loss spec-\ntra and the structural stability of nickel oxide: An\nLSDA+U study, Phys. Rev. B 57, 1505 (1998).\n[33] F. Subhan and J. Hong, Magnetic anisotropy and curie\ntemperature of two-dimensional VI 3monolayer, Journalof Physics: Condensed Matter 32, 245803 (2020).\n[34] N. Sivadas, M. W. Daniels, R. H. Swendsen, S. Okamoto,\nand D. Xiao, Magnetic ground state of semiconducting\ntransition-metal trichalcogenide monolayers, Phys. Rev.\nB91, 235425 (2015).\n[35] T. Holstein and H. Primako\u000b, Field dependence of the\nintrinsic domain magnetization of a ferromagnet, Phys.\nRev.58, 1098 (1940).\n[36] L. Ke and M. I. Katsnelson, Electron correlation e\u000bects\non exchange interactions and spin excitations in 2D van\nder Waals materials, npj Computational Materials 7, 1\n(2021).\n[37] M. Bloch, Magnon renormalization in ferromagnets near\nthe curie point, Physical Review Letters 9, 286 (1962).\n[38] S. Sarker, C. Jayaprakash, H. R. Krishnamurthy, and\nM. Ma, Bosonic mean-\feld theory of quantum heisen-\nberg spin systems: Bose condensation and magnetic or-\nder, Phys. Rev. B 40, 5028 (1989).\n[39] B.-G. Liu, A nonlinear spin-wave theory of quasi-\n2d quantum heisenberg antiferromagnets, Journal of\nPhysics: Condensed Matter 4, 8339 (1992).\n[40] Z. Li, T. Cao, and S. G. Louie, Two-dimensional ferro-\nmagnetism in few-layer van der waals crystals: Renor-\nmalized spin-wave theory and calculations, Journal of\nMagnetism and Magnetic Materials 463, 28 (2018).\n[41] V. Mkhitaryan and L. Ke, Self-consistently renormailzed\nspin-wave theory of layered ferromagnets on honeycomb\nlattice (2021), arXiv:2107.04085 [cond-mat.mtrl-sci].\n[42] P. D. Loly, The Heisenberg ferromagnet in the selfconsis-\ntently renormalized spin wave approximation, Journal of\nPhysics C: Solid State Physics 4, 1365 (1971).\n[43] E. Rastelli, A. Tassi, and L. Reatto, Selfconsistently\nrenormalized spin-wave approximation for some two-\ndimensional magnetic systems, Journal of Physics C:\nSolid State Physics 7, 1735 (1974).\n[44] M. G. Pini, E. Rastelli, A. Tassi, and V. Tognetti, The\nin\ruence of the anisotropy on the temperature renormal-\nisation of the magnetic excitations in FeCl 2, Journal of\nPhysics C: Solid State Physics 14, 3041 (1981).\n[45] C. Li, A. J. Freeman, H. J. F. Jansen, and C. L. Fu, Mag-\nnetic anisotropy in low-dimensional ferromagnetic sys-\ntems: Fe monolayers on Ag(001), Au(001), and Pd(001)\nsubstrates, Phys. Rev. B 42, 5433 (1990).\n[46] A. B. Shick, D. L. Novikov, and A. J. Freeman, Rela-\ntivistic spin-polarized theory of magnetoelastic coupling\nand magnetic anisotropy strain dependence: Application\nto Co/Cu(001), Phys. Rev. B 56, R14259 (1997).\n[47] L. Ke, Intersublattice magnetocrystalline anisotropy us-\ning a realistic tight-binding method based on maximally\nlocalized Wannier functions, Phys. Rev. B 99, 054418\n(2019).\n[48] A. M. Ganose, A. J. Jackson, and D. O. Scanlon, sumo:\nCommand-line tools for plotting and analysis of periodic\nab initio calculations, Journal of Open Source Software\n3, 717 (2018).\n[49] Y. Lee, T. Kotani, and L. Ke, Role of nonlocality in ex-\nchange correlation for magnetic two-dimensional van der\nWaals materials, Phys. Rev. B 101, 241409 (2020).\n[50] C. Huang, F. Wu, S. Yu, P. Jena, and E. Kan, Discovery\nof twin orbital-order phases in ferromagnetic semicon-\nducting VI 3monolayer, Phys. Chem. Chem. Phys. 22,\n512 (2020).\n[51] N. Sivadas, S. Okamoto, X. Xu, C. J. Fennie, and\nD. Xiao, Stacking-dependent magnetism in bilayer CrI 3,10\nNano Letters 18, 7658 (2018).\n[52] D. Soriano, C. Cardoso, and J. Fern\u0013 andez-Rossier, Inter-\nplay between interlayer exchange and stacking in cri3 bi-\nlayers, Solid State Communications 299, 113662 (2019).\n[53] P. Jiang, C. Wang, D. Chen, Z. Zhong, Z. Yuan, Z.-Y.\nLu, and W. Ji, Stacking tunable interlayer magnetism in\nbilayer CrI 3, Phys. Rev. B 99, 144401 (2019).\n[54] C. Xu, J. Feng, H. Xiang, and L. Bellaiche, Interplay\nbetween Kitaev interaction and single ion anisotropy in\nferromagnetic CrI 3and CrGeTe 3monolayers, npj Com-\nputational Materials 4, 57 (2018).[55] L. Ke and M. van Schilfgaarde, Band-\flling e\u000bect on\nmagnetic anisotropy using a Green's function method,\nPhys. Rev. B 92, 014423 (2015).\n[56] L. Ke, D. A. Kukusta, and D. D. Johnson, Origin of\nmagnetic anisotropy in doped Ce 2Co17alloys, Phys. Rev.\nB94, 144429 (2016).\n[57] P. Blaha, K. Schwarz, F. Tran, R. Laskowski,\nG. K. H. Madsen, and L. D. Marks, WIEN2k: An\nAPW+lo program for calculating the properties of\nsolids, Journal of Chemical Physics 152, 074101 (2020),\nhttps://doi.org/10.1063/1.5143061."    },    {        "title": "2108.10965v2.Shape_anisotropy_effect_on_magnetization_reversal_induced_by_linear_down_chirp_pulse.pdf",        "content": "Shape anisotropy effect on magnetization reversal induced by linear\ndown chirp pulse\nZ. K. Juthya, M. A. J. Pikula, M. A. S. Akandaaand M. T. Islama,<\naPhysics Discipline, Khulna University, Khulna 9208, Bangladesh\nARTICLE INFO\nKeywords :\nMagnetization reversal\nShape anisotropy\nEnergy barrier\nStimulatedenergyabsorption _emissionABSTRACT\nWe investigate the influence of shape anisotropy on the magnetization reversal of a single-domain\nmagnetic nanoparticle driven by a circularly polarized linear down-chirp microwave field pulse\n(DCMP). Based on the Landau-Lifshitz-Gilbert equation, numerical results show that the three con-\ntrollingparametersofDCMP,namely,microwaveamplitude,initialfrequencyandchirprate,decrease\nwith the increase of shape anisotropy. For certain shape anisotropy, the reversal time significantly\nreduces. Thesefindingsarerelatedtothecompetitionofshapeanisotropyanduniaxialmagnetocrys-\ntalline anisotropy and thus to the height of energy barrier which separates the two stable states. The\nresultofdampingdependenceofmagnetizationreversalindicatesthatforacertainsampleshape,there\nexistsanoptimaldampingsituationatwhichmagnetizationisfastest. Moreover,itisalsoshownthat\ntherequiredmicrowavefieldamplitudecanbeloweredbyapplyingthespin-polarizedcurrentsimulta-\nneously. Theusageofanoptimumcombinationofbothmicrowavefieldpulseandcurrentissuggested\ntoachievecostefficiencyandfasterswitching. Sothesefindingsmayprovidetheknowledgetofabri-\ncate the shape of a single domain nanoparticle for the fast and power-efficient magnetic data storage\ndevice.\n1. Introduction\nObtaining fast and energy-efficient magnetization rever-\nsalofsingle-domainoftheperpendicularlymagnetizednanopar-\nticle is an interesting issue owing to its potential applica-\ntion in high-density data storage devices [1, 2, 3] and rapid\ndata access [4]. For high density, high thermal stability and\nlow error rate in device application, high anisotropy mate-\nrials [5] are needed. But it is a challenge to find out a way\nwith low energy to achieve the fastest magnetization rever-\nsalforhigh-anisotropymagneticnanoparticle. Overthepast\ntwo decades, several controlling parameters _driving forces\narediscoveredtoachievefastestmagnetizationreversalwith\nlow cost. Namely, magnetization reversal using a constant\nmagnetic field [6, 7], reversal by microwave field of con-\nstant frequency or time dependent frequency, either with or\nwithout a polarized electric current [8, 9, 10, 11, 12, 13,\n14, 15, 16, 17, 18, 19] and by spin transfer torque (STT)\nor spin orbit torque (SOT) [20, 21, 22, 23, 24, 25, 26, 27,\n28, 29, 30, 31, 32, 33, 34]. All methods mentioned above\naresufferingfromspecificdrawbacks[6,35,36,37,38,39,\n40, 41]. For example, magnetic field or microwave field\ndriven magnetization is not energy efficient and fast as ex-\npected. Incaseofcurrent(bySTT)orSOTdriven,thehigher\nthreshold current requirement is a bottleneck. Later on, re-\nsearchers are digressed to employ the microwave chirped\npulses (microwave with time-dependent frequency) which\ninduce fast and energy-efficient magnetization reversal [13,\n14, 15, 16, 17, 19] but still the required field amplitude, ini-\ntial frequency and chirp rate are not small as desired prac-\n<Corresponding authors:\ntorikul@phy.ku.ac.bd (M.T. Islam)\nORCID(s): 0000-0002-4168-8695 (Z.K. Juthy); 0000-0002-7500-1572\n(M.A.J. Pikul); 0000-0002-6742-2158 (M.A.S. Akanda);\n0000-0002-3846-4009 (M.T. Islam)tically. Recent studies [42, 43] have demonstrated that the\nfastmagnetizationreversalofacubicnanoparticle(i.e.,with\nzerodemagnetization _shapeanisotropy)obtainedbythelin-\neardownchirpmicrowavepulse(DCMP).Thephysicalmech-\nanismofthismodelisthattheDCMP(whosefrequencylin-\nearlydecreasesfrominitial +f0tofinal*f0)stimulatesmi-\ncrowave energy absorption (emission) by the magnetic mo-\nment before (after) crossing the energy barrier. The effi-\nciencyofthemicrowaveenergyabsorption _emissiondepends\nonhowcloselythemicrowavefrequencyandmagnetization\nintrinsic frequency match during reversal. This frequency-\nmatching is related to the energy barrier [44, 45]. However,\ninsmall-scalenanoparticle,demagnetizationfield _shapeanisotropy\n(whichwasexcludedpreviously)hasagreatimportance[46,\n47,44]because,itinteracts _opposesthedominateduniaxial\nmagnetocrystalline anisotropy and thus reduces the energy\nbarrier between two stable states. So, the magnetization re-\nversal of a nanoparticle with finite shape anisotropy might\nbefasterandenergyefficient. Inaddition,fabricatingperfect\ncubicnanoparticle(i.e.,withzerodemagnetizationfield)isa\nchallengingissuefrompracticalpointofview. Forthemost\ndevice applications, the desired microwave pulse (DCMP)\nshould be withsmaller amplitude, initial frequency and low\nchirp rate.\nTherefore,thisstudyfocusesonhowtheshapeanisotropy\nHshapeaffects the energy consumption i.e., the amplitude\nHmw, initial frequency foand pulse chirp rate \u0011of DCMP\nand the magnetization switching time ts. Interestingly, we\nfoundthattheparametersofDCMPi.e., Hmw,fo,and\u0011de-\ncrease with the increase of shape anisotropy Hshape. In ad-\ndition,thereisacritical Hshapeorcross-sectionalarea A,for\nwhichorlarger, tssignificantlyreducesto0.346nswhichis\nclose to the theoretical limit [40]. For instance, in case of\na sample of cross-sectional area 3232nm2, orHshape=\n0:598T, we estimated ts= 0:346ns,Hmw= 0:0187T,\nZ. K. Juthy et al.: Preprint submitted to Elsevier Page 1 of 7arXiv:2108.10965v2  [cond-mat.mes-hall]  2 Oct 2021Shape anisotropy effect on magnetization reversal induced by linear down chirp pulse\nfo=5:2GHz,andoptimal \u0011=24:96ns*2whicharesignif-\nicantly smaller than ts=0:56ns,Hmw=0:045T,fo=21\nGHz,andoptimal \u0011=67:2ns*2forthecubicshapedorzero\nHshape. Thephysicalreasonofthesefindingsisrelatedtothe\ncompetitionbetweenshapeanisotropyandtheuniaxialmag-\nnetocrystalline anisotropy. The findings of damping depen-\ndence of magnetization reversal indicate that, for a specific\nAorHshape,thereexistsaoptimaldampingsituation. More-\nover,thisstudyshowsthattherequiredmicrowavefieldam-\nplitude of DCMP can be lowered by applying direct current\nsimultaneously along with DCMP. Therefore, an optimum\ncombination of microwave field pulse and current could be\napplied to achieve faster and power efficient magnetization\nreversal. So these findings might be significant to design\ntheshapeofsingledomainnanoparticletoproducethelow-\nenergy cost magnetic storage device with high speed data\nprocessing.\n2. Analytical Model and Method\nWe consider a spin valve system that consists of a free\nand fixed ferromagnetic layers by sandwiching a nonmag-\nneticspacer,asshownschematicallyinFigure1(a). Thefree\nlayerisasquarenanoparticleofarea Aandthickness d. The\nmagnetization direction of the fixed and free layer is along\nzdirection, i.e., both are perpendicularly magnetized. The\nmagnetizationofthefreelayercanbetreatedasamacrospin\nrepresented by the unit moment mwith total magnetization\nAdMs, whereMsrepresents the saturation magnetization.\nThe demagnetization field, which generated by the magne-\ntization of nanoparticle, can be approximated by an easy-\nplane shape anisotropy. The demagnetization field _shape\nanisotropy field is expressed as Hshape=Hshapemzzsuch\nthatHshape=*\u00160.Nz*Nx/Ms,istheshapeanisotropyco-\nefficient,where NzandNxaredemagnetizationfactors[45]\nand\u00160=4\u001910*7N_A2is the vacuum magnetic perme-\nability. Because of the strong uniaxial magnetocrystalline\nanisotropy Hani=Hanimzz, the magnetization of the free\nlayer prefers to stay in two ground states, i.e., mparallel to\nzand*z.\nUnderthecircularlypolarizedDCMPandspinpolarized\ncurrent, the magnetization dynamics mis governed by the\nLandau-Lifshitz-Gilbert (LLG) equation [7, 9, 37, 48]\ndm\ndt=*\rmHeff+\u000bmdm\ndt*\rhsm.pm/(1)\nwhere\u000bisthedimensionlessGilbertdampingconstantand,\n\risgyromagneticratio. Heffisthetotaleffectivefieldwhich\ncontains the microwave magnetic field Hmw, the exchange\nfield2A\nMs(2mwhereAistheexchangestiffnessconstant,and\ntheeffectiveanisotropyfield Hkalongzdirection,i.e., Hk=\nHkmzz.hsis the strength of spin transfer torque(STT),\nhs=`PJ\n2e\u0016oMsd(2)\nwhereJ,e,`,P,\u00160andddenote the current density, elec-\ntron charge, the Planck’s constant, spin polarization of cur-\nmf0\n0-f(b) (a)\nτ\n-\n+\npFigure 1: (a) Schematic diagram of the system in where mand\npdenote the unit vectors of the magnetization of free layer\nand fixed layer, respectively. A linear down-chirp microwave\nfield and an electric current are applied on a single domain\nnanoparticle. (b) The frequency profile illustrates (sweeping\nfrom+f0to*f0) the linear down-chirp microwave field.\nrent,thevacuumpermeability,andthicknessofthefreelayer,\nrespectively.\nTheHkhas two components: an uniaxial magnetocrys-\ntalline anisotropy Haniand a shape anisotropy Hshapefield\nwhich can be expressed as Hk=Hani+Hshape= [Hani*\n\u00160.Nz*Nx/Ms]mzz. Thus, the resonant frequency of the\nnanoparticlecanbefoundfromtheKittelformula f0=\r\n2\u0019[Hani*\n\u00160.Nz*Nx/Ms]:\nWefirstapplyonlythemicrowavefield-drivenmagneti-\nzationreversal,i.e.,withoutelectriccurrent(SST)term,the\nrate of energy change is expressed as\n \u000f=*\u000b\róómHeffóó2*m\fHmw (3)\nSince damping is positive, so, the first term is always nega-\ntive. For time-dependent external microwave field, depend-\ning on the angle between the instantaneous magnetization\nmandHmw, the second term can be either positive or neg-\native, in other words, the microwave field pulse can trigger\nstimulated energy absorption or emission.\nIt has been shown that the fast magnetization reversal is\nachieved by DCMP [42] with the physical picture: the fre-\nquency of DCMP matches the frequency change of magne-\ntizationprecession. Hence,ittriggersstimulatedmicrowave\nabsorption (emission) by (from) the magnetization before\n(after) it crosses over the energy barrier.\nThemicrowavefieldthatwehaveappliedtakestheform,\nHmw=Hmw\u0004cos\u001e.t/x+sin\u001e.t/y\u0005(4)\nwhereHmwistheamplitudeofthemicrowavefieldand \u001e.t/\nis the phase. In this paper, we use a DCMP whose instanta-\nneous frequency is defined as f.t/ =1\n2\u0019d\u001e\ndt, which linearly\ndecreases with time at constant rate \u0011(in units of ns*2) as\nshown in Figure 1(b). The phase \u001e.t/and the instantaneous\nfrequencyf.t/are given by,\n\u001e.t/=2\u0019.f0t*\u0011\n2t2/;f.t/=f0*\u0011t (5)\nZ. K. Juthy et al.: Preprint submitted to Elsevier Page 2 of 7Shape anisotropy effect on magnetization reversal induced by linear down chirp pulse\n/s40/s98/s41/s40/s99/s41\n/s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48/s48/s49/s50/s116\n/s115/s32/s40/s110/s115/s41\n/s104 /s32/s40/s110/s115/s45/s50\n/s41/s32/s32 /s65\n/s48/s32\n/s32/s32 /s65\n/s49/s32/s32/s32\n/s32/s32 /s65\n/s50/s32\n/s32/s32 /s65\n/s51/s32\n/s32/s32 /s65\n/s52\n/s48/s46/s48 /s48/s46/s51 /s48/s46/s54/s49/s53/s51/s48/s52/s53/s54/s48/s104 /s32/s40/s110/s115/s45/s50\n/s41\n/s72\n/s115/s104/s97/s112/s101/s40/s84/s41/s48/s46/s48 /s48/s46/s51 /s48/s46/s54 /s48/s46/s57/s45/s49/s48/s49/s109\n/s122\n/s116/s32 /s40/s110/s115/s41/s32/s32 /s65\n/s48\n/s32/s32 /s65\n/s49/s32/s32\n/s32/s32 /s65\n/s50/s32\n/s32/s32 /s65\n/s51/s32\n/s32/s32 /s65\n/s52\n/s32/s32 /s65\n/s53\n/s32/s32 /s65\n/s54\n/s32/s32 /s65\n/s55\n/s32/s32 /s65\n/s56/s40/s97/s41\nFigure 2: Model parameters of the free layer, Ms=106A/m,Aex=1310*12J_m,Hk=0:75T,\r=1:761011rad/(T \fs),\nand\u000b= 0:01. (a) Temporal evolution of mzinduced by DCMP with Hmw=0.045 T for the samples of different A. (b) The\ndependence of switching time tson the chirp rate \u0011for different cross-sectional areas while keeping the microwave field amplitudes\nHmwconstant. The vertical dashed lines indicate the lower and upper limits of \u0011for magnetization switching. (c) The dependence\nof chirp rate \u0011on shape anisotropy Hshapefor switching time within 1 ns.\n/s40/s99/s41/s40/s98/s41\n/s45/s49\n/s48\n/s49/s45/s49/s48/s49\n/s45/s49/s48/s49\n/s109\n/s122\n/s109/s121\n/s109\n/s120/s45/s49\n/s48\n/s49/s45/s49/s48/s49\n/s45/s49/s48/s49\n/s109\n/s122\n/s109/s121\n/s109\n/s120\n/s45/s57/s48 /s48 /s57/s48/s45/s48/s46/s54/s45/s48/s46/s51/s48/s46/s48/s48/s46/s51\n/s115/s115/s115/s115/s115\n/s105/s105/s105/s105/s105/s105/s101 /s47/s77\n/s115/s72\n/s107\n/s113 /s32/s40/s100/s101/s103/s41/s32/s32 /s65\n/s48\n/s32/s32 /s65\n/s50/s32/s32\n/s32/s32 /s65\n/s52/s32\n/s32/s32 /s65\n/s53/s32\n/s32/s32 /s65\n/s54\n/s32/s32 /s65\n/s55\n/s32/s32 /s65\n/s56/s105/s115/s40/s97/s41\nFigure 3: (a) The potential \"along the line \u001e= 0 for the\nsamples of different A. Angles\u0012and\u001eare defined as cos*1mz\nandtan*1.mx_my/. The symbols i and s represent the initial\nstateandthesaddlepoint,respectively. Magnetizationreversal\ntrajectories of (b) A0=88nm2and (c)A8=3232 nm2.\nwherefoistheinitialfrequencyat t=0. Thedurationofthe\nmicrowavepulseis \u001c=2fo\n\u0011whichmeansthefinalfrequency\nis*fo. AccordingtotheappliedDCMP,thesecondtermof\nEq. (3) can be expressed as\n \"=*Hmwsin\u0012.t/sin\b.t/d\u001e\ndt(6)\nwhere\b.t/istheanglebetween mt(thein-planecomponent\nofmandHmw). Therefore, the microwave field pulse can\ntrigger the stimulated energy absorption (for *\b.t/) which\nis occurred before crossing the energy barrier and emission\n(for\b.t/) after crossing the energy barrier.\nFor this study, the parameters of Permalloy are chosen\nfrom typical experiments on microwave-driven magnetiza-\ntion reversal as Ms=106A_m,Hani=0:75T,\r=1:76\n1011rad_.T\fs/,Aex= 1310*12J_m and\u000b= 0:01. Al-\nthough, the demonstrated strategy and findings would still\nbe valid for other materials. The cell size used throughout\nthis study is .222/ nm3. We solved the LLG equa-tionnumericallyusingtheMUMAX3package[49]fortime-\ndependent circularly polarized microwave field. We con-\nsider the switching time window of 1ns and least magne-\ntization reversal mz=*0:9.\n3. Numerical Results\nIn a small-scale nanoparticle, demagnetization field or\nshapeanisotropyfield Hshapeisanimportantpropertytotake\ninto account in the investigation of the magnetization rever-\nsal. In order to approximate demagnetization field _shape\nanisotropyHshape,wechoosecuboidshapedsample _nanoparticle\nofvolumeV=Ad,whereAisthecross-sectionalareaand d\nistheheight. Toincreasethedemagnetizationfieldstrength\norHshape,Aisgraduallyincreasedwhilethethickness d=8\nnm is kept fixed. Specifically, we intend to investigate the\nmagnetization reversal of the samples, A1=1010 ,A2=\n1212,A3=1414 ,A4=1616 ,A5=2020 ,A6=\n2424,A7=2828 ,A8=3232 nm2. Foreachsample,\nby finding the demagnetization factors [45] NzandNx, the\nshape anisotropy coefficient Hshape=\u00160.Nz*Nx/Msare\ncalculated analytically. Hshapeactually acts along *zi.e.,\nit opposes the intrinsic easy axis anisotropy field Haniand\nhence reduces the effective anisotropy. Therefore, for each\nA,theresonance _theoreticalfrequencyisestimatedbywell-\nknownKittel-formula f0=\r\n2\u0019\u0004Hani*\u00160.Nz*Nx/Ms\u0005as\nshown in the Table 1.\nToinvestigatetheDCMP-drivenmagnetizationreversals\nof nanoparticles of different A, by keeping Hmw=0:045T\nandf0(close to resonant frequency corresponding to Aor\nHshape) fixed, we estimate the optimal \u0011for each sample or\nHshape. Figure2(b)showsthe \u0011dependenceof tsforseveral\nsamples orHshape. It is noted that there exists a window of\n\u0011denoted by vertical dashed line. For a small range of \u0011in\nthe right edge of window, the tsis minimal. It means that\nthereisaflexiblespacetochoose \u0011. Inaddition,theoptimal\n\u0011.Hshape/decreaseswith HshapeasshownintheFigure2(c).\nThese findings are useful in device application since it was\nreported that generating DCMP with higher chirp rate is a\nZ. K. Juthy et al.: Preprint submitted to Elsevier Page 3 of 7Shape anisotropy effect on magnetization reversal induced by linear down chirp pulse\nTable 1\nSimulated values of the controlling parameters for different samples of cross-sectional area A\nCross*sectional area Shape anisotropy field Natural frequency Simulated minimal frequency\nA(nm2) hshape(T) f0(GHz) f0(GHz)\nA1 0.09606 18.3 18.3\nA2 0.17718 16 17\nA3 0.2465 14.1 14.\nA4 0.3064 12.4 12.4\nA5 0.4049 9.6 10.2\nA6 0.4827 7.49 9\nA7 0.5457 5.72 7.5\nA8 0.5980 4.26 5.2\nchallenge [42]. Afterward, DCMP (with Hmw= 0:045,f0\ncorrespondingto Hshapeandoptimal\u0011(Hshape))drivenmag-\nnetization reversals are investigate and thus the time evolu-\ntions ofmzfor different HshapeorAare shown in the Fig-\nure 2(a). It is found that with the increase of Hshape, the\nmagnetization switching time tsremains almost same until\nA7= 2828 nm2but forA8g3232nm2, the mag-\nnetizationreversesverysmoothlyi.e., tsdropsto=0.34 ns\nwhichisclosetothetheoreticallimit(0.39 ns)[40]. So,itis\nmentionedthatthereisacritical Aandforlargervaluesthan\nit,theswitchingtimebecomessignificantlysmallerwhichis\nan ultimate demand in device application.\nTo understand the reason why fast reversal is obtained,\none can recall that the energy barrier is proportional to the\neffectiveanisotropyfield[39,40]. Sincethe Hshapeopposes\nthe anisotropy field Haniand thus reduces the height of the\nenergy barrier (energy difference between the initial state\nand saddle point). This reduction of energy barrier for dif-\nferentAispresentedintheFigure3(a). Therefore,themag-\nnetizationreversalbecomeseasierandfasterbecausethemi-\ncrowave frequency and magnetization precession frequency\nclosely matches as expected. To be more explicit, one may\nlookatthetrajectoriesofmagnetizationreversalfortwocases\nofAorHshapein Figure 3(b) and 3(c). For A0= 88\nnm2orHshape= 0, the magnetization takes a longer time\nbecause of some additional processions while crossing the\nhigher energy barrier. However, for A8= 3232 nm2\norHshape= 0:598T, the magnetization reversal is fairly\nsmoothi.e.,withoutanyadditionalprocessionssincetheen-\nergy barrier is reduced by Hshape. For more justification,\nFig4(a)showsthechangetheangle \b(blue)andmz(black\nline)withtimefor A8=3232 nm2. Notethattheangle \b\nremains almost 90oduring magnetization reversal which is\ndesiredfortriggeringanefficientmicrowaveenergyabsorp-\ntion (emission) by magnetization. Thus the corresponding\nenergychangingrate  \u000f(red),inFig4(c)isobtainedwithab-\nsorption and emission peaks (no distortion) which lead fast\nreversal. However, Fig 4(b), for A0= 88nm2i.e., for\nzeroHshape, shows the  \u000f(red line) absorption and emission\npeaks (distorted) indicating inefficient i.e., slower magneti-\nzation reversal (black line).\nThen,fortheDCMPwithfixed Hmw=0:045Tandop-\ntimal chirp rate \u0011.Hshape/, we obtain the fast magnetization\n/s40/s99/s41/s40/s98/s41/s40/s97/s41\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51/s45/s49/s48/s49\n/s116/s32/s40/s110/s115/s41/s109\n/s122\n/s45/s50/s55/s48/s45/s49/s56/s48/s45/s57/s48/s48/s57/s48/s49/s56/s48/s50/s55/s48/s70 /s32/s40/s100/s101/s103/s41\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52/s45/s49/s48/s49\n/s116/s32/s40/s110/s115/s41/s109\n/s122\n/s45/s49/s48/s49/s101 /s32/s40/s49/s48/s49/s53\n/s74/s47/s115/s32/s109/s51\n/s41/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54/s45/s49/s48/s49\n/s116/s32/s40/s110/s115/s41/s109\n/s122\n/s45/s51/s45/s50/s45/s49/s48/s49/s50/s51/s101 /s32/s40/s49/s48/s49/s53\n/s74/s47/s115/s32/s109/s51\n/s41/s40/s97/s41Figure 4: (a) ForA8= 3232 nm2, the plot of the relative\nangle\bagainst time (blue line) and the time dependence of\nmz(black line). Plots of the energy changing rate  \u000fof mag-\nnetization against time (red line) and the time dependence of\nmz(black line) for the cross-sectional area (b) A0=88nm2\nand (c)A8=3232 nm2\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54/s48/s53/s49/s48/s49/s53/s50/s48/s32/s102\n/s111/s32/s40/s84/s104/s101/s111/s114/s101/s116/s105/s99/s97/s108/s41\n/s32/s102\n/s111/s32/s40/s83/s105/s109/s117/s108/s97/s116/s101/s100/s41\n/s32/s72\n/s109/s119\n/s72\n/s115/s104/s97/s112/s101/s32/s40/s84/s41/s102\n/s111/s32/s40/s71/s72/s122/s41\n/s48/s46/s48/s50/s48/s48/s46/s48/s50/s52/s48/s46/s48/s50/s56/s48/s46/s48/s51/s50/s72\n/s109/s119/s32/s40/s84/s41\nFigure 5: The dependence of initial frequency foand min-\nimum required field amplitude Hmwas a function of shape\nanisotropyHshapefor switching time within 1 ns.\nreversal of different Afor a certain range of f0around each\nresonance_naturalfrequency. Thereexistaminimal f0,i.e.,\nminimalf0correspondingtoeachsample,forwhichswitch-\ning time is lowest. Figure 5 shows the simulated minimal\nfrequencyf0(bluecircles)asafunctionof Hshapeandfound\nthatthesimulated f0(bluecircles)ofDCMPdecreaseswith\nincreasingHshape. These can be explained as the Hshapere-\nduces the effective anisotropy Hk. For further justification,\nZ. K. Juthy et al.: Preprint submitted to Elsevier Page 4 of 7Shape anisotropy effect on magnetization reversal induced by linear down chirp pulse\n/s48/s46/s48/s48 /s48/s46/s48/s51 /s48/s46/s48/s54/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s116\n/s115/s40/s110/s115/s41\n/s97/s32/s32 /s65\n/s48/s32\n/s32/s32 /s65\n/s49/s32/s32\n/s32/s32 /s65\n/s50/s32/s32\n/s32/s32 /s65\n/s51/s32\n/s32/s32 /s65\n/s52/s32\n/s45/s49\n/s48\n/s49/s45/s49/s48/s49\n/s45/s49/s48/s49\n/s109\n/s122\n/s109/s121\n/s109\n/s120/s40/s99/s41/s40/s98/s41\n/s45/s49\n/s48\n/s49/s45/s49/s48/s49\n/s45/s49/s48/s49\n/s109\n/s122\n/s109/s121\n/s109\n/s120/s40/s97/s41\nFigure 6: (a) Minimal switching time for the samples of dif-\nferentAas a function of Gilbert damping \u000b. Magnetization\nreversal trajectories of A3=1414 nm2for damping (b) \u000b=\n0.010 and (c) \u000b=0:017.\nthetheoreticallycalculatedfrequencies(blackline)areplot-\nted in same Figure 5 and noted that the decreasing trends\nare consistent as expected. Later on, by keeping the min-\nimalf0.Hshape/and optimal \u0011.Hshape/(i.e., corresponding\ntodifferentAorHshape)fixed,weinvestigatetheminimally\nrequiredamplitude Hmw(redsquare)asafunctionof Hshape.\nFigure5showsthat Hmwalsodecreaseswiththeincreaseof\nHshape. This is explained by the same reason of reducing\nthe height of the energy barrier with Hshapeas shown in the\nFigure 2(b). So even the smaller amplitude Hmwcan drive\nfaster magnetization reversal.\nTheGilbertdampingcoefficient \u000bisanimportantmate-\nrial parameter which has a significant effect on the magne-\ntization reversal process as well as switching time. In gen-\neral, damping has two influences. First, it hinders energy\nabsorption (cost more energy) during climbing the energy\nbarrier for magnetization reversal [50]. Second, after cross-\ning the energy barrier, it dissipates the energy of magne-\ntization promptly and thus it accelerates the magnetization\nto reach the ground _stable state [51, 52]. However, in the\nmagnetizationswitchingprocess,themagnetizationrequires\nto climb an energy barrier (originated from the anisotropy\nfield) and then goes to a stable state. Ideally, for the fast\nandenergy-efficientmagnetizationreversal,smaller(larger)\ndamping (\u000b) before (after) cross over the energy barrier is\nbetter. Therefore, this study emphasizes finding the optimal\n\u000bofthesampleofdifferent Aatwhichthefastestreversalis\nvalid. Accordingly, DCMP-driven magnetization as a func-\ntion of\u000bfor different Ais studied. Figure 6(a) shows the\nGilbert damping dependence of switching time for different\nA. It is found that for A0= 88nm2(uniaxial sample),\nthe range of \u000bis small but for finite shape anisotropy (biax-\nial sample), the range of \u000bis large which is useful in device\napplication. Moreover, for the sample of 1414nm2, the\nswitchingtimeislowest( ts=0:482ns)at\u000b=0:017. Tobe\nmore explicit, Figure 6(b) and 6(c) show the trajectories of\nmagnetization reversal for \u000b= 0:01and\u000b= 0:017respec-\ntively and observed that for \u000b= 0:017, the reversal path is\nshorter. This is happened because, after cross over the en-\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s48/s46/s48/s49/s48/s46/s48/s50/s48/s46/s48/s51/s72\n/s109/s119/s32/s40/s84/s41\n/s74 /s32/s40/s49/s48/s49/s50\n/s32/s65/s47/s109/s50\n/s41/s32/s32 /s65\n/s48/s32\n/s32/s32 /s65\n/s49/s32/s32\n/s32/s32 /s65\n/s50/s32/s32\n/s32/s32 /s65\n/s51/s32\n/s32/s32 /s65\n/s52/s32Figure 7: Phase diagram, for magnetization switching within\n1 ns, in terms of DCMP field amplitudes Hmwand current\ndensityJfor the samples of different A.\nergy barrier, the larger \u000breduces the magnetization preces-\nsion by dissipating the magnetization energy promptly and\nthus,itleadstofastreversal. Itisindicatedthatthereisaspe-\ncific sample that has an optimal damping situation at which\nthe magnetization reversal might be fastest.\nFormostdeviceapplication,thedesiredmicrowavepulse\n(DCMP) should be of smaller amplitude Hmw, smaller ini-\ntialfrequency foandlowchirprate \u0011,butwehaveobtained,\nforasampleof A=1616 nm2,minimalHmw=0:0278T,\noptimumfo=12:4GHz, and\u0011=40:672ns*2. In order to\nfurther reduce Hmw, we simultaneously apply a direct cur-\nrent such that it assists the magnetization reversal induced\nbyDCMP.ForDCMPcontribution,wetune Hmwwhilethe\nminimalfoandoptimal \u0011correspondingtodifferent Akept\nfixed. Figure 7 shows the Hmw*Jphase diagram for the\nsample of different Aunder the condition of magnetization\nreversal time 1 ns. As an example, for A= 1616 nm2,\nwithout DCMP, the required current to obtain fast switch-\ning gets very high i.e., J= 2:31012A_m2while without\ncurrenti.e.,J=0,theHmw=0:0278Tislargefrompracti-\ncal point of view. Therefore, with simultaneous applying of\nDCMPandcurrent,therequiredvaluesof HmwandJcould\nbe smaller than the above mentioned values (for individual\ncase)whichgivesalargescopeofpracticalrealizationofthis\nDCMP-driven magnetization reversal strategy.\n4. Discussion and Conclusions\nThisstudyinvestigatestheinfluenceofshapeanisotropy\nontheDCMP-drivenmagnetizationreversaltime tsandthe\nthreecontrollingparameters( Hmw,foand\u0011)ofDCMP,and\ndamping dependence of magnetization reversal. The shape\nanisotropy interestingly assists magnetization reversal since\nthe parameters of DCMP Hmw,foand\u0011decrease with the\nthe increase of shape anisotropy Hshape. For the sample of\nA8g3232nm2,tssignificantly reduces to 0.346 ns.\nThe minimal parameters of DCMP are estimated as Hmw=\n0:0187T,fo=5:2GHz,andoptimal \u0011=24:96ns*2which\nZ. K. Juthy et al.: Preprint submitted to Elsevier Page 5 of 7Shape anisotropy effect on magnetization reversal induced by linear down chirp pulse\nare useful in device application. These findings can be at-\ntributed to the shape anisotropy Hshapewhich reduces the\neffective anisotropy and thus reduces the height of the en-\nergybarrier. TheDCMPstimulatesefficientenergyabsorp-\ntion and emission during magnetization reversal. A recent\nstudy [43] reported that the temperature effect also assists\nthe magnetization reversal, i.e., it reduces the parameters of\nmicrowave field pulse. Therefore, it is desired that, at finite\ntemperature,theestimatedvaluesof Hmw,foand\u0011wouldbe\ndecreased further (which is beyond the scope of this study).\nThe result of damping dependence of magnetization rever-\nsal indicates that for a specific AorHshape, there exists an\noptimal damping situation. Moreover, it is also shown that\nthe required microwave field amplitude can be lowered by\napplying the direct current simultaneously. The usage of an\noptimumcombinationofbothmicrowavefieldpulseandcur-\nrent is suggested to achieve cost efficient and faster switch-\ning. So these findings may provide the knowledge to fabri-\ncate the shape of a single domain nanoparticle to generate\nthe magnetic data storage device.\n5. Acknowledgements\nThisworkwassupportedbytheMinistryofScienceand\nTechnology (Grant No. 440 EAS) and the Ministry of Edu-\ncation (BANBEIS, Grant No. SD2019972).\nReferences\n[1] S. Sun, C. B. Murray, D. Weller, L. Folks, A. Moser, Monodisperse\nFePt nanoparticles and ferromagnetic FePt nanocrystal superlattices,\nScience 287 (5460) (2000) 1989–1992.\n[2] S. Woods, J. Kirtley, S. Sun, R. Koch, Direct investigation of su-\nperparamagnetism in Co nanoparticle films, Physical Review Letters\n87 (13) (2001) 137205.\n[3] D. Zitoun, M. Respaud, M.-C. Fromen, M. J. Casanove, P. Lecante,\nC. Amiens, B. Chaudret, Magnetic enhancement in nanoscale CoRh\nparticles, Physical Review Letters 89 (3) (2002) 037203.\n[4] B. Hillebrands, K. Ounadjela, Spin dynamics in confined magnetic\nstructuresI&II,Vol.83,SpringerScience&BusinessMedia,2003.\n[5] S.Mangin,D.Ravelosona,J.Katine,M.Carey,B.Terris,E.E.Fuller-\nton, Current-induced magnetization reversal in nanopillars with per-\npendicular anisotropy, Nature materials 5 (3) (2006) 210–215.\n[6] A. Hubert, R. Schäfer, Magnetic domains: the analysis of magnetic\nmicrostructures (1998).\n[7] Z. Sun, X. Wang, Fast magnetization switching of stoner particles:\nA nonlinear dynamics picture, Physical Review B 71 (17) (2005)\n174430.\n[8] G. Bertotti, C. Serpico, I. D. Mayergoyz, Nonlinear magnetization\ndynamics under circularly polarized field, Physical Review Letters\n86 (4) (2001) 724.\n[9] Z.Sun,X.Wang,Strategytoreduceminimalmagnetizationswitching\nfield for stoner particles, Physical Review B 73 (9) (2006) 092416.\n[10] S.I.Denisov,T.V.Lyutyy,P.Hänggi,K.N.Trohidou,Dynamicaland\nthermaleffectsinnanoparticlesystemsdrivenbyarotatingmagnetic\nfield, Physical Review B 74 (10) (2006) 104406.\n[11] S. Okamoto, N. Kikuchi, O. Kitakami, Magnetization switching be-\nhavior with microwave assistance, Applied Physics Letters 93 (10)\n(2008) 102506.\n[12] J.-G. Zhu, Y. Wang, Microwave assisted magnetic recording utiliz-\ningperpendicularspintorqueoscillatorwithswitchableperpendicular\nelectrodes, IEEE Transactions on Magnetics 46 (3) (2010) 751–757.\n[13] C.Thirion,W.Wernsdorfer,D.Mailly,Switchingofmagnetizationbynonlinearresonancestudiedinsinglenanoparticles,Naturematerials\n2 (8) (2003) 524–527.\n[14] K. Rivkin, J. B. Ketterson, Magnetization reversal in the anisotropy-\ndominated regime using time-dependent magnetic fields, Applied\nphysics letters 89 (25) (2006) 252507.\n[15] Z.Wang,M.Wu,Chirped-microwaveassistedmagnetizationreversal,\nJournal of Applied Physics 105 (9) (2009) 093903.\n[16] N. Barros, M. Rassam, H. Jirari, H. Kachkachi, Optimal switching\nof a nanomagnet assisted by microwaves, Physical Review B 83 (14)\n(2011) 144418.\n[17] N. Barros, H. Rassam, H. Kachkachi, Microwave-assisted switching\nofananomagnet: Analyticaldeterminationoftheoptimalmicrowave\nfield, Physical Review B 88 (1) (2013) 014421.\n[18] T. Tanaka, Y. Otsuka, Y. Furomoto, K. Matsuyama, Y. Nozaki, Se-\nlectivemagnetizationswitchingwithmicrowaveassistanceforthree-\ndimensionalmagneticrecording,JournalofAppliedPhysics113(14)\n(2013) 143908.\n[19] G. Klughertz, P.-A. Hervieux, G. Manfredi, Autoresonant control of\nthe magnetization switching in single-domain nanoparticles, Journal\nof Physics D: Applied Physics 47 (34) (2014) 345004.\n[20] J.C.Slonczewski,etal.,Current-drivenexcitationofmagneticmulti-\nlayers, Journal of Magnetism and Magnetic Materials 159 (1) (1996)\nL1.\n[21] L.Berger,Emissionofspinwavesbyamagneticmultilayertraversed\nby a current, Physical Review B 54 (13) (1996) 9353.\n[22] M.Tsoi,A.Jansen,J.Bass,W.-C.Chiang,M.Seck,V.Tsoi,P.Wyder,\nExcitation of a magnetic multilayer by an electric current, Physical\nReview Letters 80 (19) (1998) 4281.\n[23] J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, D. C. Ralph,\nCurrent-driven magnetization reversal and spin-wave excitations in\nCo_Cu_Co pillars, Phys. Rev. Lett. 84 (2000) 3149–3152.\n[24] X. Waintal, E. B. Myers, P. W. Brouwer, D. C. Ralph, Role of spin-\ndependent interface scattering in generating current-induced torques\nin magnetic multilayers, Phys. Rev. B 62 (2000) 12317–12327.\n[25] J. Z. Sun, Spin-current interaction with a monodomain magnetic\nbody: A model study, Physical Review B 62 (1) (2000) 570.\n[26] J. Sun, Spintronics gets a magnetic flute, Nature 425 (6956) (2003)\n359–360.\n[27] M. D. Stiles, A. Zangwill, Anatomy of spin-transfer torque, Physical\nReview B 66 (1) (2002) 014407.\n[28] Y.B.Bazaliy,B.Jones,S.-C.Zhang,Current-inducedmagnetization\nswitchinginsmalldomainsofdifferentanisotropies,PhysicalReview\nB 69 (9) (2004) 094421.\n[29] R. Koch, J. Katine, J. Sun, Time-resolved reversal of spin-transfer\nswitching in a nanomagnet, Physical Review Letters 92 (8) (2004)\n088302.\n[30] W. Wetzels, G. E. Bauer, O. N. Jouravlev, Efficient magnetization\nreversal with noisy currents, Physical review letters 96 (12) (2006)\n127203.\n[31] A. Manchon, S. Zhang, Theory of nonequilibrium intrinsic spin\ntorque in a single nanomagnet, Physical Review B 78 (21) (2008)\n212405.\n[32] I.M.Miron,G.Gaudin,S.Auffret,B.Rodmacq,A.Schuhl,S.Pizzini,\nJ. Vogel, P. Gambardella, Current-driven spin torque induced by the\nrashba effect in a ferromagnetic metal layer, Nature materials 9 (3)\n(2010) 230–234.\n[33] I.M.Miron,K.Garello,G.Gaudin,P.-J.Zermatten,M.V.Costache,\nS.Auffret,S.Bandiera,B.Rodmacq,A.Schuhl,P.Gambardella,Per-\npendicular switching of a single ferromagnetic layer induced by in-\nplane current injection, Nature 476 (7359) (2011) 189–193.\n[34] L. Liu, C.-F. Pai, Y. Li, H. Tseng, D. Ralph, R. Buhrman, Spin-\ntorque switching with the giant spin hall effect of tantalum, Science\n336 (6081) (2012) 555–558.\n[35] J. Grollier, V. Cros, H. Jaffres, A. Hamzic, J.-M. George, G. Faini,\nJ.B.Youssef,H.LeGall,A.Fert,Fielddependenceofmagnetization\nreversal by spin transfer, Physical Review B 67 (17) (2003) 174402.\n[36] H. Morise, S. Nakamura, Stable magnetization states under a spin-\npolarized current and a magnetic field, Physical Review B 71 (1)\nZ. K. Juthy et al.: Preprint submitted to Elsevier Page 6 of 7Shape anisotropy effect on magnetization reversal induced by linear down chirp pulse\n(2005) 014439.\n[37] T. Taniguchi, H. Imamura, Critical current of spin-transfer-torque-\ndrivenmagnetizationdynamicsinmagneticmultilayers,PhysicalRe-\nview B 78 (22) (2008) 224421.\n[38] Y.Suzuki,A.A.Tulapurkar,C.Chappert,Spin-injectionphenomena\nandapplications,in: NanomagnetismandSpintronics,Elsevier,2009,\npp. 93–153.\n[39] Z. Sun, X. Wang, Theoretical limit of the minimal magnetization\nswitchingfieldandtheoptimalfieldpulseforstonerparticles,Physi-\ncal review letters 97 (7) (2006) 077205.\n[40] X. Wang, Z. Sun, Theoretical limit in the magnetization reversal of\nstoner particles, Physical Review Letters 98 (7) (2007) 077201.\n[41] X. Wang, P. Yan, J. Lu, C. He, Euler equation of the optimal tra-\njectoryforthefastestmagnetizationreversalofnano-magneticstruc-\ntures, EPL (Europhysics Letters) 84 (2) (2008) 27008.\n[42] M. T. Islam, X. Wang, Y. Zhang, X. Wang, Subnanosecond mag-\nnetization reversal of a magnetic nanoparticle driven by a chirp mi-\ncrowave field pulse, Physical Review B 97 (22) (2018) 224412.\n[43] M. Islam, M. Pikul, X. Wang, Thermally assisted magnetization re-\nversal of a magnetic nanoparticle driven by a down-chirp microwave\nfieldpulse,JournalofMagnetismandMagneticMaterials537(2021)\n168174.\n[44] T. Yoo, D. Shin, J. Kim, H. Kim, S. Lee, X. Liu, J. Furdyna, Effect\nofshapeanisotropyonthemagnetizationreversalprocessof(ga,mn)\nasferromagneticsemiconductors,JournalofAppliedPhysics103(7)\n(2008) 07D119.\n[45] J. Dubowik, Shape anisotropy of magnetic heterostructures, Physical\nReview B 54 (2) (1996) 1088.\n[46] R. Skomski, Nanomagnetics, Journal of Physics: Condensed Matter\n15 (20) (2003) R841.\n[47] M.Jamet,W.Wernsdorfer,C.Thirion,D.Mailly,V.Dupuis,P.Méli-\nnon, A. Pérez, Magnetic anisotropy of a single cobalt nanocluster,\nPhysical Review Letters 86 (20) (2001) 4676.\n[48] T. L. Gilbert, A phenomenological theory of damping in ferromag-\nnetic materials, IEEE transactions on magnetics 40 (6) (2004) 3443–\n3449.\n[49] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-\nSanchez, B. Van Waeyenberge, The design and verification of mu-\nmax3, AIP Advances 4 (10) (2014) 107133.\n[50] Z.Wang,M.Wu,Chirped-microwaveassistedmagnetizationreversal,\nJournal of Applied Physics 105 (9) (2009) 093903.\n[51] Y. Uesaka, Y. Suzuki, O. Kitakami, Y. Nakatani, N. Hayashi,\nH. Fukushima, Switching time of a single spin in linearly varying\nfield, Journal of Applied Physics 111 (12) (2012) 123907.\n[52] C.Y.Mao,J.-G.Zhu,R.M.White,T.Min,Effectofdampingconstant\nontheswitchinglimitofathin-filmrecordinghead,Journalofapplied\nphysics 85 (8) (1999) 5870–5872.\nZ. K. Juthy et al.: Preprint submitted to Elsevier Page 7 of 7"    },    {        "title": "2108.11277v1.__textit_Ab_initio___theory_of_magnetism_in_two_dimensional__1T__TaS__2_.pdf",        "content": "Ab initio theory of magnetism in two-dimensional 1T-TaS 2\nDiego Pasquier\u0003and Oleg V. Yazyevy\nInstitute of Physics, Ecole Polytechnique F\u0013 ed\u0013 erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland\nWe investigate, using a \frst-principles density-functional methodology, the nature of magnetism\nin monolayer 1 T-phase of tantalum disul\fde (1 T-TaS 2). Magnetism in the insulating phase of\nTaS 2is a longstanding puzzle and has led to a variety of theoretical proposals including notably\nthe realization of a two-dimensional quantum-spin-liquid phase. By means of non-collinear spin\ncalculations, we derive ab initio spin Hamiltonians including two-spin bilinear Heisenberg exchange,\nas well as biquadratic and four-spin ring-exchange couplings. We \fnd that both quadratic and\nquartic interactions are consistently ferromagnetic, for all the functionals considered. Relativistic\ncalculations predict substantial magnetocrystalline anisotropy. Altogether, our results suggest that\nthis material may realize an easy-plane XXZ quantum ferromagnet with large anisotropy.\nIntroduction. Materials with \rat bands near the\nFermi level are an ideal platform to study correlated-\nelectrons phenomena. There has recently been a renewal\nof interest in \rat-band materials because of the discovery\nof correlated-insulating phases and superconductivty in\ntwisted bilayer graphene near the so-called magic angle\n[1, 2]. Full characterization and understanding of mag-\nnetism in such systems remains a theoretical challenge.\nIn certain transition metal dichalcogenides (TMDs), \rat\nbands emerge near the Fermi level due to the occurence of\na commensurate charge-density-wave (CDW) phase with\nstar-of-David (SOD) reconstruction [3{6]. Among those\nmaterials, the most studied one is undoubtedly octahe-\ndral tantalum disul\fde, which will be the focus of the\npresent work.\nThe polymorph of tantalum disul\fde with distorted\noctahedral coordination of metal atoms (1 T-TaS 2) is a\nspecial member of the family of transition metal dichalco-\ngenides family of materials. Like many other metallic\nTMDs [7{11], it undergoes charge-density-wave (CDW)\ninstabilities as the temperature is lowered. At low tem-\nperature, it crystallizes in the so-called SOD phase (illus-\ntrated in Fig.1), withp\n13\u0002p\n13 periodicity and 13 Ta\natoms per supercell. In this SOD phase, insulating be-\nhaviour is observed, which has been commonly attributed\nto electron localization due to strong correlations [3, 12].\nThe reason is the following: in the SOD phase, the num-\nber of electrons per unit cell is odd, since each Ta atom\ncontributes one conduction t2gelectron [13], so conven-\ntional band theory would predict a metal. This argument\nis clear for the monolayer, but there is actually a caveat\nfor the multi-layer and bulk cases: the stacking is not\nexactly known, so the three-dimensional character could\nbecome important and the exact role of interlayer inter-\nactions is unclear [14].\nA longstanding question regarding TaS 2concerns mag-\nnetism or, to be more precise, the apparent absence of it\n[14], as even the signature of local moment formation\nis not observed. Several scenarios have been proposed\n\u0003diego.pasquier@ep\r.ch\nyoleg.yazyev@ep\r.chin the past decades. The peculiar properties of TaS 2\nwere part of the inspiring data behind Anderson's theory\nof resonating valence bonds [15]. Fazekas and Tosatti\nsuggested that the strong spin-orbit coupling (SOC) of\nTa atoms could suppress the magnetic moments [16]. A\n\ructuating N\u0013 eel order was proposed in Ref. [17] and re-\ncently, the idea of a spin liquid in TaS 2has gained mo-\nmentum after Law and Lee proposed that TaS 2might\nbe a rare realization of a quasi-2D quantum-spin-liquid\n(QSL) phase [14]. In follow-up work [18], the theory was\nworked out, yielding the interesting proposal of the pos-\nsible realization of a spinon Fermi surface in TaS 2. This\ninteresting proposal has triggered experimental e\u000bort ob-\nserve these unconventional states, not only in TaS 2but\nalso in similar materials such as TaSe 2[19, 20]. A key\ningredient in the theory is the presence, in the e\u000bective\nspin Hamiltonian, of a sizeable antiferromagnetic four-\nspin ring-exchange term beyond the conventional Heisen-\nberg model.\nThe purpose of the present work is to propose a real-\nistic picture of the nature of magnetism in 2D 1 T-TaS 2\nderived from \frst principles, using a density-functional-\ntheory (DFT) framework. Using non-collinear density-\nfunctional calculations, we aim to provide estimates for\nthe magnetic interaction parameters. In particular, we\naddress whether magnetism in this material is expected\nto deviate strongly from Heisenberg behavior indeed, as\ndiscussed above. While estimating Heisenberg exchange\ncouplings using DFT is a relatively simple (although del-\nicate) and well-established procedure, the calculation of\nquartic terms is less common and more challenging. The\napproach that we shall adopt here is that of non-collinear\nspin calculations (see e.g. Refs. [21, 22]). The method\nconsists in calculating the total energy for a series of an-\ngles between the spins on di\u000berent sites of the super-\nlattice. While two-spin bilinear terms lead to a linear\ndependency on the angle cosine only, quartic terms lead\nto a quadratic dependency, allowing one to extract the\nbiquadratic and ring-exchange couplings.\nModel Hamiltonian. Following He et al. [18] and ne-\nglecting anisotropy for the moment, we write H, the low-\nenergy Hamiltonian describing spin physics in the 2DarXiv:2108.11277v1  [cond-mat.mtrl-sci]  25 Aug 20212\nFIG. 1. Ball-and-stick representation of TaS 2in the com-\nmensurate CDW phase. Grey balls represent Ta atoms, while\npurple balls stand for S atoms. Only the shortest Ta-Ta bonds\nwere drawn in order to facilitate visualization of the star-of-\nDavid pattern.\nMott-insulating phase of 1 T-TaS 2:\nH=HHeis+Hplaq+Hb; (1)\nwhereHHeisis the usual Heisenberg Hamiltonian, de-\nscribing bilinear spin interactions, and Hplaqdescribes\nquartic four-spin plaquette terms. In Eq. 1, we also ex-\nplicitly include biquadratic interaction Hb.\nThe Heisenberg Hamiltonian is given by\nHHeis=JX\n<ij>~Si\u0001~Sj; (2)\nwhere<ij > denotes nearest-neighbor (NN) sites of the\ntriangular lattice, Jis the nearest-neighbor Heisenberg\ncoupling, and ~Sistands for the S= 1=2 spin opera-\ntor at site i. Next-to-nearest-neighbor interactions can\nbe safely neglected, because they are much smaller com-\npared to NN ones. The biquadratic Hamiltonian is given\nbyHb=BP\n<ij>(~Si\u0001~Sj)2, and the four-spin Hamilto-\nnianHplaqdescribes higher-order spin-spin interactions\nand reads:\nHplaq=KX\n<ijkl>(~Si\u0001~Sj)(~Sk\u0001~Sl) + (~Si\u0001~Sl)(~Sj\u0001~Sk)\n\u0000(~Si\u0001~Sk)(~Sj\u0001~Sl);\n(3)\nwhere<ijkl> denotes a plaquette (see Fig. 2), and K\nis the ring-exchange coupling.\nMethodology. All the \frst-principles calculations pre-\nsented in this paper were carried out using the Quan-\ntum ESPRESSO package [23] . Lattice parameters and\natomic positions were obtained by minimizing forces and\nstress in thep\n13\u0002p\n13 supercell, using the generalized-\ngradient approximation (GGA) [24, 25]. A grid of 8 \u00028\nk-points was used, with a Marzari-Vanderbilt (MV) [26]\nxy\njk l\ni⃗ a1⃗ a2FIG. 2. Schematic representation of the spin lattice em-\nployed for the spin-polarized calculations. Each black dot\nrepresents a David star with 39 atoms. Thep\n3p\n13\u0002p\n3p\n13\nsupercell contains three spin sublattices (one of each color).\nThe red diamond illustrates an example of plaquette on the\ntriangular lattice.\nsmearing of 0 :01 Ry. Projector-augmented-wave (PAW)\n[27] pseudo-potentials from Ref. [28], including explic-\nitlysandpsemicore states for Ta atoms, were used to\ndescribe interaction between core and valence electrons.\nPlane-wave cuto\u000bs were set to 60 Ry and 300 Ry for wave\nfunctions and charge density, respectively.\nHeisenberg interactions were calculated by compar-\ning total energies with di\u000berent spin con\fgurations in\nap\n3p\n13\u0002p\n3p\n13 supercell (2p\n13\u0002p\n13 for the eval-\nuation of the biquadratic term), using both the local-\ndensity approximation (LDA) and the GGA, with and\nwithout on-site Hubbard Uparameter for Ta 5 dorbitals.\nFor DFT+Ucalculations, we have set U= 2:5 eV. For\nspin-polarized calculations, we have used a smaller MV\nsmearing of 0 :001 Ry, together with a grid of 3 \u00023k-\npoints in the supercell. These parameters were chosen to\nensure convergence of the exchange couplings.\nStructure. We begin by optimizing the lattice param-\neters and atomic positions, in ap\n13\u0002p\n13 supercell con-\ntaining 13 Ta and 26 S atoms. The resulting structure\nminimizing forces and stress was found to be the well-\nknown SOD phase, illustrated in Fig.1. The calculated\nsuperlattice parameter was 12.18 \u0017A and the shortest Ta-\nTa bonds inside the SOD was 3.18 \u0017A (compared to 3.38\n\u0017A in the high-symmetry phase), in line with previous re-\nports [4]. Fig. 3 shows the noninteracting DFT bands,\nexhibiting a half-\flled narrow band with a bandwidth\nof\u001830 meV. In accordance with the existing literature\n[4, 29, 30], we also \fnd the corresponding electronic struc-\nture insulating when spin polarization is allowed. Param-\neters of an e\u000bective Hubbard model can be roughly esti-\nmated from the DFT results. Through a Wannier trans-\nformation we \fnd a NN hopping of t\u00192:5 meV, and the3\n−1−0.500.51\nΓMK Γ−30−20−1001020\nΓMK Γ(a) (b)\n Energy (eV)\nEnergy (meV)\nFIG. 3. (a) DFT non-polarized band dispersion of\nsingle-layer TaS 2in the star-of-David phase, along the high-\nsymmetry directions of the mini-Brillouin zone. The Fermi\nenergy is set to zero. (b) Dispersion of the half-\flled \rat\nband.\ne\u000bective Hubbard interaction can be estimated by exam-\nining the correlation gap, which gives Ue\u000b\u0019125 meV\nwith DFT and Ue\u000b\u0019330 meV with DFT+ U, mean-\ning theUe\u000b=t\u0018132 ratio sits well in the Mott-Hubbard\nregime.\nAb initio parameters: isotropic case. We turn to the\ninvestigation of magnetic properties and examine \frst the\ncollinear case, neglecting SOC and higher-order interac-\ntions. This will allow us to compare with the values ob-\ntained from non-collinear calculations, in order to check\nthe robustness and consistency of the methodology. We\nadopt ap\n3p\n13\u0002p\n3p\n13 supercell (see Fig. 2 ), contain-\ning three David stars. In the calculations, we can force\nthe solution to converge to di\u000berent spin con\fgurations\nby appropriately choosing the initial spin orientation in\nthe self-consistent cycle. The exchange coupling is then\ngiven by:\nJcollinear =E\"\"\"\u0000E#\"\"\n12S2; (4)\nwithS= 1=2 for TaS 2. In Eq. 4, E\"\"\"stands for the\nenergy of the con\fguration with all the spin parallel to\neach other, while E#\"\"is the energy of the con\fguration\nin which the spin of one of the three stars points antipar-\nallel to the other two. Results for di\u000berent functionals are\nshown in Table I. While the exact values of the parame-\nter are somewhat sensitive to the choice of functional and\nHubbard parameter, the qualitative picture remains ro-\nbust, i.e. coupling is predicted to be ferromagnetic (with\na rather small value of \u00181 meV), in line with the cousin\nmaterials NbSe 2and NbS 2for which similar values of the\nparameters were reported [5, 31].\nIn a one-band Hubbard model, the antiferromagnetic\nexchange would be JAFM = 4t2=Ue\u000b, which gives\u0019\n0:08 meV with our estimated DFT+ Uparameters. This\nis an order of magnitude smaller than the estimated J,\nindicating that other mechanisms prevail. Multi-band\n0123\n0 π(a)\n0 π(b)\nEnergy (meV)\nα αDFT\nlinear fit\nquadratic fitFIG. 4. Total energy as a function of the angle between spins,\ncalculated with GGA+ U, in the (a)p\n3p\n13\u0002p\n3p\n13 and (b)\n2p\n13\u0002p\n13 supercells. Fits wit linear and quadratic functions\nare also presented.\nHubbard models can have ferromagnetic kinetic exchange\n(see Refs. [32, 33]) due to third-order (e.g. cyclic) pro-\ncesses, which tend to become dominant in the limit of\na very \rat band. Magic-angle twisted bilayer graphene\nwas recently also proposed to be a ferromagnetic Mott\ninsulator [34], driven by a large intersite direct Coulomb\nexchange that dominates the kinetic JAFM, the latter be-\ning small due to the bands' \ratness.\nWe then turn out attention the non-collinear case. Let\nus de\fne\u000bas the angle between the spin direction of dif-\nferent stars in the supercell, as shown in Fig. 2. We can\nwrite the total energy as a function of \u000bin the following\nway:\nE(\u000b) =E0+E1cos(\u000b) +E2cos2\u000b : (5)\nThe two-spin and four-spin exchange can then be ex-\ntracted from the E1andE2parameters as:\nJ=2E1\u0000E2+ 6BS4\n12S2;K=E2\u00006BS4\n6S4:(6)\nThe system of equations 6 is underdetermined, so we need\nto \frst determine Busing a di\u000berent supercell. This\ncan be done by considering the 2p\n13\u0002p\n13 supercell for\nwhich the quadratic term E2\u00021\n2depends only on B, with\nB=E2\u00021\n2=4S4.\nFig. 4 shows the calculated total energies in the\nGGA+Ucase for a set of selected angles in bothp\n3p\n13\u0002p\n3p\n13 and 2p\n13\u0002p\n13 supercells. Fig. 4 also shows\nthe least-square \fts with linear and quadratic functions,\nshowing that the quadratic form allows one to almost\nperfectly \ft the DFT results, whilst the linear \ft, which\ncorresponds to a mapping to a simple Heisenberg model,4\nTABLE I. Ab initio Heisenberg, biquadratic and ring-exchange parameters (in meV), calculated with di\u000berent methods and\napproximations.\nJ(collinear) J B K J0K0\nLDA \u00001:11 \u00001:15\u00000:22\u00000:02\u00001:12\u00000:24\nLDA+U\u00001:19 \u00001:13\u00000:07\u00000:18\u00001:12\u00000:24\nGGA \u00001:62 \u00001:66\u00000:73\u00000:04\u00001:57\u00000:77\nGGA+U\u00001:11 \u00001:11\u00000:55\u00000:03\u00001:04\u00000:57\nshows deviations. However, it is evident from Fig. 4 that\nthe deviations from the Heisenberg model are only mod-\nest. This is re\rected by the ratio E2=E1\u00180:1 that\nshows that contributions beyond bilinear two-spin terms\nare subdominant. Using LDA, the ratio found is even\nsmaller (E2=E1\u00180:05). Although GGA is in general\nan improvement compared to LDA, the formulation of\nnon-collinear magnetism with gradient corrections is am-\nbiguous, so we would expect the LDA results to be more\nrigorous. We have also checked that higher-order terms\n(i.e. higher power in the expansion) are negligible. This\nis one of the central results of the present work, that \frst-\nprinciples calculations predict no substantial deviations\nfrom Heisenberg physics, i.e. from a Hamiltonian trun-\ncated to quadratic terms in the spin operators. It was\nhowever not obvious a priori since it had been hypoth-\nesized otherwise in theoretical work, and was therefore\nworth checking. Our \fnding can be understood in light\nof the large Ue\u000b=t\u001d1 ratio found, which is di\u000berent than\nwhat was often taken in model Hamiltonians [17, 18, 35].\nIn Table I, values for J B, andKobtained with\ndi\u000berent functionals are reported. We also show as\na sensitivity analysis J0andK0, which were obtained\nby mapping the DFT results to a model without bi-\nquadratic exchange. Bilinear Heisenberg terms obtained\nfrom collinear and non-collinear calculations are consis-\ntent, demonstrating the soundness of our methodology.\nBiquadratic and ring exchange are also found to be fer-\nromagnetic for both LDA and GGA, with or without the\nHubbard correction U. Table I shows the extracted value\nofKis very sensitive to the choice of model (i.e. with\nor without biquadratic coupling) and also quite sensitive\nto the choice of functional.. Nevertheless, the sign of\nthe coupling is robust and indicates a QSL phase is not\ncompatible with the DFT results because unlike other\ncombinations [36], the scenario with both negative Jand\nKis not predicted to host such state.\nSpin anisotropy. Our results suggest that a ferro-\nmagnetic state might be observed at low temperature\nin monolayer TaS 2. In two-dimensional systems, a \f-\nnite Curie temperature requires spin anisotropy, driven\nby the SOC. We calculated, using the LDA(+ U)+SOC\nfunctional, the in-plane and out-of-plane bilinear cou-\nplings denoted JxJyandJz, withzthe out-of-plane\ndirection. We consider here the quadratic Hamiltonian\nH=P\n<ij>(JxSx\niSx\nj+JySy\niSy\nj+JzSz\niSz\nj), neglecting pos-\nsible anisotropy of the quartic terms. The methodologyused is the same as for the evaluation of isotropic Jus-\ning collinear calculations, except that we consider three\ndi\u000berent spin orientations. For these non-collinear calcu-\nlations including the SOC, we have not used the GGA\nbecause of possible convergence issues in that case.\nExtracted parameters are shown in Table II. In-\nplane couplings JxandJyare equal, indicating in-plane\nisotropy. We observe smaller couplings in the relativistic\ncase compared to isotropic J, that we understand as due\nto the reduced magnetic moment induced by the SOC\n(\u00190.82\u0016B/SOD with LDA+ U). The total energy di\u000ber-\nence between the ferromagnetic state with xandypolar-\nization is negligibly small, EFM\nx\u0000EFM\ny<0:1\u0016eV/SOD,\nwhich we interpret as evidence for almost if not perfect\nisotropy. On the other hand, the out-of-plane coupling\nJzis signi\fcantly smaller, giving a negative anisotropy\nterm\r=Jz=Jx\u00001 =\u00000:3 in DFT+ U. The ex-\nchange anistropy \rwas found to be insu\u000ecient to ac-\ncount for the calculated magnetocrystalline anistropy\nEFM\nz\u0000EFM\nx\u00190:27 meV/SOD in LDA+ U. The simplest\nway to account for the remaining di\u000berence is through an\non-site anisotropy term DP\ni(Sz\ni)2in the Hamiltonian.\nThe corresponding extracted values for Dare shown in\nTable II.\nOur \frst-principles results thus indicate that, at\nthe DFT(+ U) level of theory, 2D TaS 2in the SOD\nphase maps into an easy-plane quantum ferromagnet.\nFor the XXZ model with easy-plane anisotropy, the\nMermin-Wagner theorem applies meaning that no \fnite-\ntemperature second-order phase transition is expected\n[37]. This is unlike the easy-axis case for which a \f-\nnite Curie temperature is predicted due to the opening\nof a gap in the spin-wave dispersion (see e.g. Ref. [38]\nin the case of CrI 3). Classical Monte-Carlo simulations\nof the easy-axis XXZ model on a two-dimensional lat-\ntice show a Berenzinskii-Kosterlitz-Thouless transition\nwithTBKT\u0018JS2[39]. Results for the quantum model\nwere less conclusive, since large \ructuation in the ex-\ntreme quantum limit ( S= 1=2) may reduce the e\u000bective\nanisotropy and exchange coupling magnitude [40]. We\nhave found that the four-spin terms are also predicted\nto be ferromagnetic (and rather small), and in that case\nthe expected e\u000bect of the ring-exchange term is to merely\nrenormalize the spin wave dispersion [41].\nDiscussion. Magnetism in the SOD phase of TaS 2is\na longstanding question, which as led to several interest-\ning proposals. Theoretical understanding has been so far5\nTABLE II. Ab initio relativistic Heisenberg and on-site\nanisotropy parameters (in meV), and corresponding exchange\nanisotropy \r.\nJxJyJzD \r\nLDA+SOC \u00000:94\u00000:94\u00000:75 0:19\u00000:20\nLDA+U+SOC\u00000:78\u00000:78\u00000:54 0:37\u00000:30\nhindered by two factors. First, experimental work prob-\ning magnetism was only done on layered form of TaS 2.\nUncertainty on the exact stacking remains and the ap-\nproximation of a quasi-2D material is questionable. Sec-\nondly, an ab initio model Hamiltonian for spins was not\nknown, and parameters were mostly derived from simpli-\n\fed one-band Hubbard models. Our results di\u000ber signi\f-\ncantly from these models, as we have found opposite sign\nfor both two-spin and four-spin couplings (i.e we have\nfound both terms to be ferromagnetic). In theory, our\nresults are derived from \frst principles so they should\nbe more robust compared to a Hubbard model studied\nperturbatively to derive a spin Hamiltonian. However,\nin the presence of strong correlations, we cannot exludethat the approximations of the DFT+ Umethod break\ndown and that it fails qualitatively, even though it usu-\nally performs well for magnetism in various correlated\nsystems.\nThe present work suggests that single-layer TaS 2could\nrealize a 2D easy-plane ferromagnet. This could help in-\nterpret future experiments, not only on TaS 2but also on\nsimilar materials such as TaSe 2, although the generaliz-\nability of the present results to other materials remains to\nbe established. If con\frmed, that would also have bear-\ning on the understanding of bulk TaS 2, as it would mean\nthat a quasi-2D spin-liquid cannot explain the absence of\nmagnetic ordering observed. There is growing evidence\nthat interlayer interactions are more important than ini-\ntially thought and that this material might actually be\nquasi-one-dimensional rather than quasi-2D [4, 42, 43].\nThe origin of the observed insulating behaviour is still\nunclear and might therefore be explained from a di\u000ber-\nent perspective than a Mott insulator [44]. Recently, it\nwas proposed that bulk TaS 2could actually be a simple\nband insulator, with gap formation driven by the stack-\ning pattern [45, 46].\n[1] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi,\nE. Kaxiras, and P. Jarillo-Herrero, Nature 556, 43\n(2018).\n[2] Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken,\nJ. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe,\nT. Taniguchi, E. Kaxiras, et al. , Nature 556, 80 (2018).\n[3] B. Sipos, A. F. Kusmartseva, A. Akrap, H. Berger,\nL. Forr\u0013 o, and E. Tuti\u0014 s, Nature Materials 7, 960 (2008).\n[4] P. Darancet, A. J. Millis, and C. A. Marianetti, Physical\nReview B 90, 045134 (2014).\n[5] D. Pasquier and O. V. Yazyev, Physical Review B 98,\n045114 (2018).\n[6] Y. Chen, W. Ruan, M. Wu, S. Tang, H. Ryu, H.-Z. Tsai,\nR. Lee, S. Kahn, F. Liou, C. Jia, et al. , Nature Physics\n16, 218 (2020).\n[7] J. A. Wilson, F. Di Salvo, and S. Mahajan, Advances in\nPhysics 24, 117 (1975).\n[8] A. C. Neto, Physical Review Letters 86, 4382 (2001).\n[9] K. Rossnagel, Journal of Physics: Condensed Matter 23,\n213001 (2011).\n[10] S. Manzeli, D. Ovchinnikov, D. Pasquier, O. V. Yazyev,\nand A. Kis, Nature Reviews Materials 2, 17033 (2017).\n[11] D. Pasquier and O. V. Yazyev, Physical Review B 100,\n201103 (2019).\n[12] P. Fazekas and E. Tosatti, Physica B+ C 99, 183 (1980).\n[13] D. Pasquier and O. V. Yazyev, 2D Materials 6, 025015\n(2019).\n[14] K. T. Law and P. A. Lee, Proceedings of the National\nAcademy of Sciences 114, 6996 (2017).\n[15] P. W. Anderson, Materials Research Bulletin 8, 153\n(1973).\n[16] P. Fazekas and E. Tosatti, Philosophical Magazine B 39,\n229 (1979).[17] L. Perfetti, T. Gloor, F. Mila, H. Berger, and M. Grioni,\nPhysical Review B 71, 153101 (2005).\n[18] W.-Y. He, X. Y. Xu, G. Chen, K. T. Law, and P. A.\nLee, Physical Review Letters 121, 046401 (2018).\n[19] S. Ma~ nas-Valero, B. M. Huddart, T. Lancaster, E. Coron-\nado, and F. L. Pratt, npj Quantum Materials 6, 1 (2021).\n[20] W. Ruan, Y. Chen, S. Tang, J. Hwang, H.-Z. Tsai,\nR. Lee, M. Wu, H. Ryu, S. Kahn, F. Liou, et al. , arXiv\npreprint arXiv:2009.07379 (2020).\n[21] N. S. Fedorova, C. Ederer, N. A. Spaldin, and A. Scara-\nmucci, Physical Review B 91, 165122 (2015).\n[22] N. S. Fedorova, A. Bortis, C. Findler, and N. A. Spaldin,\nPhysical Review B 98, 235113 (2018).\n[23] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car,\nC. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococ-\ncioni, I. Dabo, et al. , Journal of physics: Condensed mat-\nter21, 395502 (2009).\n[24] J. P. Perdew and W. Yue, Physical Review B 33, 8800\n(1986).\n[25] J. P. Perdew, K. Burke, and M. Ernzerhof, Physical\nReview Letters 77, 3865 (1996).\n[26] N. Marzari, D. Vanderbilt, A. De Vita, and M. Payne,\nPhysical Review Letters 82, 3296 (1999).\n[27] P. E. Bl ochl, Physical Review B 50, 17953 (1994).\n[28] A. Dal Corso, Computational Materials Science 95, 337\n(2014).\n[29] D. C. Miller, S. D. Mahanti, and P. M. Duxbury, Physical\nReview B 97, 045133 (2018).\n[30] S. Yi, Z. Zhang, and J.-H. Cho, Physical Review B 97,\n041413 (2018).\n[31] C. Tresca and M. Calandra, 2D Materials 6, 035041\n(2019).\n[32] K. Penc, H. Shiba, F. Mila, and T. Tsukagoshi, Physical6\nReview B 54, 4056 (1996).\n[33] Z. Huang, D. Liu, A. Mansikkam aki, V. Vieru, N. Iwa-\nhara, and L. F. Chibotaru, Physical Review Research 2,\n033430 (2020).\n[34] K. Seo, V. N. Kotov, and B. Uchoa, Physical review\nletters 122, 246402 (2019).\n[35] L. Perfetti, P. A. Loukakos, M. Lisowski, U. Bovensiepen,\nM. Wolf, H. Berger, S. Biermann, and A. Georges, New\nJournal of Physics 10, 053019 (2008).\n[36] T. Grover, N. Trivedi, T. Senthil, and P. A. Lee, Physical\nReview B 81, 245121 (2010).\n[37] N. D. Mermin and H. Wagner, Physical Review Letters\n17, 1133 (1966).\n[38] J. L. Lado and J. Fern\u0013 andez-Rossier, 2D Materials 4,\n035002 (2017).\n[39] A. Cuccoli, V. Tognetti, and R. Vaia, Physical ReviewB52, 10221 (1995).\n[40] A. Cuccoli, V. Tognetti, P. Verrucchi, and R. Vaia, Phys-\nical Review B 51, 12840 (1995).\n[41] S. A. Owerre, Canadian Journal of Physics 91, 542\n(2013).\n[42] A. S. Ngankeu, S. K. Mahatha, K. Guilloy, M. Bianchi,\nC. E. Sanders, K. Han\u000b, K. Rossnagel, J. A. Miwa,\nC. Breth Nielsen, M. Bremholm, and P. Hofmann, Phys-\nical Review B 96, 195147 (2017).\n[43] C. Butler, M. Yoshida, T. Hanaguri, and Y. Iwasa, Na-\nture Communications 11, 2477 (2020).\n[44] J. Lee, K.-H. Jin, and H. W. Yeom, Physical Review\nLetters 126, 196405 (2021).\n[45] T. Ritschel, H. Berger, and J. Geck, Physical Review B\n98, 195134 (2018).\n[46] S.-H. Lee, J. S. Goh, and D. Cho, Physical Review Let-\nters122, 106404 (2019)."    },    {        "title": "2108.11547v2.Magnetoelastic_anisotropy_in_Heusler_type_Mn___2_δ__CoGa___1_δ___films.pdf",        "content": "arXiv:2108.11547v2  [cond-mat.mtrl-sci]  6 Mar 2022Magnetoelastic anisotropy in Heusler-type Mn 2−δCoGa 1+δfilms\nTakahide Kubota,1,2,∗Daichi Takano,1Yohei Kota,3Shaktiranjan Mohanty,4\nKeita Ito,1,2Mitsuhiro Matsuki,1Masahiro Hayashida,1Mingling Sun,1Yukiharu\nTakeda,5Yuji Saitoh,5Subhankar Bedanta,4Akio Kimura,6,7and Koki Takanashi1,2,8\n1Institute for Materials Research, Tohoku University, Send ai 980-8577, Japan\n2Center for Spintronics Research Network,\nTohoku University, Sendai 980-8577, Japan\n3National Institute of Technology, Fukushima College, Iwak i, Fukushima 970-8034, Japan\n4Laboratory for Nanomagnetism and Magnetic Materials (LNMM ),\nSchool of Physical Sciences, National Institute of Science Education and Research,\nAn OCC of Homi Bhabha National Institute (HBNI), Jatni 75205 0, Odisha, India\n5Materials Sciences Research Center,\nJapan Atomic Energy Agency, Hyogo 679–5148, Japan\n6Graduate School of Science, Hiroshima University,\nHigashi-hiroshima 739–8526, Japan\n7Graduate School of Advanced Science and Engineering,\nHiroshima University, Higashi-hiroshima 739-8526, Japan\n8Center for Science and Innovation in Spintronics (Core Rese arch Cluster),\nTohoku University, Sendai 980-8577, Japan\n(Dated: March 8, 2022)\n1Abstract\nPerpendicular magnetization is essential for high-densit y memory application using magnetic\nmaterials. High-spin polarization of conduction electron s is also required for realizing large elec-\ntric signals from spin-dependent transport phenomena. Heu sler alloy is a well-known material\nclass showing the half-metallic electronic structure. How ever, its cubic lattice nature favors in-\nplane magnetization and thus minimizes the perpendicular m agnetic anisotropy (PMA), in gen-\neral. This study focuses on an inverse-type Heusler alloy, M n2−δCoGa1+δ(MCG) with a small\noff-stoichiometry ( δ) , which is expected to be a half-metallic material. We obser ved relatively\nlarge uniaxial magnetocrystalline anisotropy constant ( Ku) of the order of 105J/m3at room tem-\nperature in MCG films with a small tetragonal distortion of a f ew percent. A positive correlation\nwas confirmed between the c/aratio of lattice constants and Ku. Imaging of magnetic domains\nusing Kerr microscopy clearly demonstrated a change in the d omain patterns along with Ku. X-ray\nmagnetic circular dichroism (XMCD) was employed using sync hrotron radiation soft x-ray beam\nto get insight into the origin for PMA. Negligible angular va riation of orbital magnetic moment\n(∆morb) evaluated usingtheXMCD spectra suggested a minor role of t he so-called Bruno’s term to\nKu. Our first principles calculation reasonably explained the small ∆morband the positive correla-\ntion between the c/aratio and Ku. The origin of the magnetocrystalline anisotropy was discu ssed\nbased on the second-order perturbation theory in terms of th e spin–orbit coupling, claiming that\nthe mixing of the occupied ↑- and the unoccupied ↓-spin states is responsible for the PMA of the\nMCG films.\nI. INTRODUCTION\nHeusler alloys have been attracting interest because of their vario us physical proper-\nties with selection of a plenty of compositions [ 1]. Especially, in spintronics research field,\nhalf-metallic electronic structure, providing fully spin-polarized con duction electrons, has\nmotivated spin-dependent transport studies such as magnetore sistance effects [ 2,3]. Among\nseveral kinds of Heusler alloys, Co-based Heusler alloys, e.g.Co2YZ,Y: a transition metal,\nZ: a 12 or 13 group element, are representative materials that have been predicted to show\n∗takahide.kubota@tohoku.ac.jp ; author to whom correspondenceshould be addressed; present ly at Depart-\nment of Advanced Spintronics Medical Engineering, Graduate Scho ol of Engineering, Tohoku University,\nSendai 980-0845, Japan.\n2the half-metallicity [ 4–8]. The half-metallicity was experimentally deduced from the large\nmagnetoresistance effects, [ 9–16] the X-ray absorption spectroscopy [ 17–20] and photoelec-\ntron spectroscopy [ 21,22]. Recent bulk-sensitive angle-resolved photoelectron spectrosc opy\nexperiments of Co 2MnGe with soft X-ray synchrotron radiation beam uncovered thre e-\ndimensional energydispersions, whichareingoodagreementwithth etheoreticalbandstruc-\ntures [20]. In contrast to Co-based Heusler alloys, Mn-based Heusler alloys, Mn2YZgained\nrenewed interests as a new class of half-metallic Heusler alloys [ 6,23]. The major advantage\nof the Mn-based Heusler alloy is relatively small saturation magnetiza tion (Ms) which re-\nduces critical current density for current induced magnetization switching phenomena based\non spin-transfer-torque (STT) [ 24]. In addition, some of the Mn-based Heusler alloys with\nthe inverse-Heusler structure (X Aphase) were predicted to possess novel electronic struc-\ntures,e.g.half-metallic spin-gapless semiconductor [ 25–27]. Among the Mn-based Heusler\nalloys, Mn 2CoGa is highlighted in this paper. Mn 2CoGa is a theoretically predicted half-\nmetallic inverse-type Heusler alloy [ 28–30]. The composition dependent magnetocrystalline\nanisotropy is an interesting feature in the Mn 2CoGa and the related compositions: Tetrago-\nnal distortion of the lattice occurs due to the band Jahn-Teller effe ct in Mn 3−xCoxGa alloys\nwithxbelow 0.5, and relatively large magnetocrystalline anisotropy of the o rder of 106J/m3\nhas been observed [ 28,31,32]. The tetragonal distortion destroys the half-metallic energy\ngap, while the conduction electron maintains relatively high spin polariz ation due to the\ndifference in the electron mobilities between the majority and minority spin channels [ 30].\nRecent studies also reported another interesting property of a M n2CoGa film showing per-\npendicular magnetic anisotropy (PMA) originating from the magneto crystalline anisotropy\nin a nearly cubic lattice [ 33]. In the previous study, Mn 2−δCoGa1+δ(MCG) films ( δ∼0.1)\ngrown onto MgO (001) single crystal substrates exhibited perpen dicular magnetization and\nin-plane magnetization with and without a Cr buffer layer underneath , respectively. Such\na drastic change can be ascribed to a small difference in the lattice st rain,c/aratios which\nwere 1.00 and 1.01 for the films without and with the Cr buffer layer, re spectively. The pre-\nvious results suggested that the small lattice strain induced magne tocrystalline anisotropy\nin MCG in the cubic phase, i.e., the magnetoelastic anisotropy was induced, which is totally\ndifferent from the interface PMA in Co-based full-Heusler alloys cont aining Fe, for which\nPMA energy of the order of 105J/m3was reported in ultra-thin, ∼1 nm, films layered with\nMgO [34–37]. The PMA in highly spin polarized materials is essential for spintronic de vices,\nsuch as STT-type magnetoresistive random access memories. Con cerning the MCG, how-\n3ever, thehalf-metallicity inthestrained latticeaswell astheoriginof themagnetocrystalline\nanisotropy remains unsolved thus far.\nIn this study, a set of MCG epitaxially grown films showing different PMA andc/aratios\nwere prepared, and correlations between the magnetic propertie s, magnetic domain images,\nand electronic structures were discussed based on experimental results and theoretically\ncalculated electronic structures. To figure out the chemical orde r and electronic structures\nof film samples, synchrotron radiation soft x-ray absorption spec tra (XAS) and soft x-ray\nmagnetic circular dichroism (XMCD) were also investigated. There we re many reports\nwith XAS and XMCD results on Heusler alloys containing Mn atoms [ 17,18,38] including\nMn2CoGa [29,39,40]. In the previous studies, the degree of the electron localization wa s\ndiscussed using the XAS and XMCD spectra, which are also addresse d in this study.\nII. EXPERIMENTAL PROCEDURES\nAllsamplesweredepositedontoMgO(001)single-crystalsubstra tesbyusinganultrahigh-\nvacuum magnetron sputtering machine with a base pressure below 3 ×10−7Pa. The MCG\nlayers were prepared by co-sputtering technique using a Mn 55Ga45target and a Co tar-\nget. The film composition of the MCG layer was Mn 42.6Co24.3Ga33.1(at.%) corresponding\nto Mn 1.7CoGa1.3in the atomic ratio, which was characterized by an electron probe mic ro\nanalyzer with a standard sample for the quantitative analysis. We em ployed two materials\nfor the buffer layer dependence study: Ag and Cr. The stacking st ructures of samples and\nthe name for the sample series were as follows: Series A (Ag buffer): MgO sub. /Cr (20\nnm) /Ag (40 nm) / MCG ( t) /Ta (3 nm), and series B (Cr buffer): MgO sub. /Cr (20 nm)\n/ MCG ( t) /Ta (3 nm). The thicknesses ( t) of the MCG layer were 5, 10, 20 and 30 nm.\nBefore installing the samples inside the vacuum chamber, the substr ates were cleaned using\nacetone and ethyl alcohol, and after introducing them inside the ch amber, the substrates\nwere annealed at 700 °C to clean the surface. All layers were deposited at room tempera-\nture, and in situpost-annealing was carried out at 700 °C (500 °C) for the Cr (MCG) layer.\nThe morphology of the sample surfaces was observed by using an at omic force microscope\n(AFM) after depositing a Ta capping layer. The crystal structure s and magnetic properties\nwere characterized by x-ray diffraction (XRD), and using a vibratin g sample magnetometer\n(VSM), respectively, at room temperature.\nThe hysteresis loop and domain images were recorded simultaneously by magneto-optical\n4(c) 5 nm\n (d) 30 nm\n(a) 5 nm\n (b) 30 nm\nSeries A Series A\nSeries B Series B\n17 \n(nm)\n1.7\n(nm)\n2.0\n(nm)\n2.0\n(nm)\nFIG. 1. Atomic force microscope images of the sample surface s. Series A: (a) t: 5, and (b) 30 nm.\nSeries B: (c) t: 5, and (d) 30 nm. Scan-area is 1 ×1µm2for all the images.\nKerr effect (MOKE) microscopy manufactured by Evico Magnetics L td. Germany. All the\nsamples were measured in polar MOKE (P-MOKE) geometries in which th e sample surface\nwas perpendicular to the applied magnetic field.\nThe XAS and XMCD measurements were carried out on L2,3-edges of Mn and Co atoms\nby a total electron-yield method at the twin helical undulator beamlin e BL23SU of SPring–\n8 [41]. External magnetic fields ( H) of±3 T and ±8 T were applied for measurements at\nout-of-planefield ( θ: 0°) and pseudo in-plane field ( θ: 54.7°), respectively. The measurement\ntemperature was set at 300 K.\nIII. RESULTS AND DISCUSSION\nA. Surface Morphology and Crystal Structure\nSurface morphologies were observed using AFM and the results are shown in figures 1(a)–\n1(d)fortheseries AandB with tof5and30nm. Forthe5-nm-thicksamples, several islands\nwith a height of several nanometers are observed in series A (Fig. 1(a)), while relatively flat\nsurface is observed in series B (Fig. 1(c)). Regarding the 30-nm-thick samples, the surfaces\nare relatively flat for both series A and B. The thickness dependenc e of the average surface\nroughness ( Ra) and peak-to-valley height ( P-V) are summarized in figures. 2(a) and2(b),\nrespectively. The values of RaandP-Vdrastically drop at t= 10 nm for series A, while,\nthose values show no dependence on tand relatively small values for series B. These results\nsuggest a difference in the growth mode of the MCG layers depending onthe buffer material:\n51.8\n28 \n26 \n5\n0P-V (nm) \n30 20 10 02.0\n0.5Ra (nm) \n0(a)\n(b)\nt (nm)Series A Series A\nSeries B Series B\nFIG. 2. Summaryof (a) average surface roughness ( Ra) and (b) peak-to-valley height ( P-V) values\nas a function of t.\nFor the series A using Ag buffer layer, the Volmer-Weber (VW) mode, which describes a\nthree-dimensional growth, is considered for the initial growth of t he MCG layer. It then\npossibly changes to the Frank-van-der-Merwe (FM) mode for a lay er-by-layer growth mode\nwith increasing t. For the series B using Cr buffer layer, the film was grown possibly in th e\nFM mode throughout the whole thicknesses. The lattice spacings of both buffer materials\nare smaller than that of a bulk Mn 2CoGa [28], and the lattice mismatches are approximately\n1.6% and 1.4% to the Cr and Ag buffer layers, respectively, which are o f the same order.\nA reason for the difference in the growth modes between the two sa mple series can be\nqualitatively understood by considering surface energies for the b uffer/MCG layer system in\nthe following way: According to the quasi-equilibrium description, a su m of surface energies\n(σsum) can be written as σsum=σf−σb+σi, whereσf,σb, andσiare surface energies\nof a film, a buffer layer, and the interface energy between the film an d the buffer layer,\nrespectively [ 42,43]. Here, strain energies for the film and the buffer layer are neglecte d\nbecause the lattice strain is small ( ∼a few %) in the present samples. We may also neglect\nthe difference between σivalues for the cases of series A (Ag /MCG interface) and series B\n(Cr/MCG interface), because σidepends on the lattice misfits between the layers which are\ncomparable to each other for the present sample series. In additio n, contribution of σito\nthe value of σsumcould be very small because the potential energy for the interfac e energy\nexhibits exponential decrease when the misfit is close to zero [ 44–46]. Based on the points\nstated above, σfandσbcan be dominant parameters affecting the sign of σsum∼σf−σb,\ni.e.VW (FM) mode is expected for σsum>0 (<0) for which σfis larger (smaller) than σb.\n6TABLE I. Reported surface energies in literature.\nSurface energy (J/m2)\nRefs. 47a48b49c\nCr 2.056 2.300 3.979\nAg 1.302 1.250 1.200\naThermodynamic calculation for poly-crystalline system.\nbExperimental value for poly-crystalline samples.\ncCalculated values for (001) surface based on full-charge density lin ear muffing-tin orbitals method.\nReported surface energies for poly-crystalline and (001) surfac e of the cubic systems are\nsummarized in Table I[47–49]. Although the surface energy of MCG is unknown here, we\nmay qualitatively understand the different growth modes considerin g the surface energies of\nCr and Ag as σb,i.e., the surface energies of Cr are larger than those of Ag for all the cases\nshown inTable I. Thus, the σsumcanberelatively small fortheCr buffer samplecompared to\nthat using the Ag buffer sample, which possibly resulted inbetter cov erage of the MCG layer\ndeposited on the Cr buffer than that on the Ag buffer. In addition to the possibility of the\ndifferent growth modes explained above, different situations for th e surface reconfiguration\ncaused by the post-annealing can be another factor: i.e., the MCG layer forms islands (the\nlayer structure) on the Ag (Cr) buffer after the annealing becaus e of the relatively small\n(large)σb, especially for the relatively thin 5-nm-thick samples. For t≥10 nm, the good\ncoverage was likely realized because of the thick (relatively large volu me) of the MCG layer\nfor the as-deposited stage and surface reconfiguration caused by the post-annealing.\nThe out-of-plane and in-plane XRD measurements were carried out to investigate the\ncrystal structure. The XRD profiles of 5-nm-thick samples are sh own in figure 3for series\nA (Figs. 3(c) and3(d)) and B (Figs. 3(e) and3(f)). From the MCG layer, a series of h0 0\n(h= 2 or 4) diffractions are observed for the diffraction profile with /vectorQ||MgO[110], and only\n2 2 0 diffractions are observed for /vectorQ||MgO[100] where /vectorQis a diffraction vector. Features of\nthese diffraction profiles were similar for all other samples indicating t he epitaxial growth\nwith a relationship of MgO(001)[100] |MCG(001)[110]. Figures 4(a),4(b), and 4(c) show\nthetdependence of the out-of-plane lattice constant ( c), the in-plane lattice constants ( a),\nand thec/aratio, respectively. For the series A, the c/aratio shows nearly no dependence\n7Q || MgO<110>\n70 60 50 40 30 20 (f) Series B(d) Series A\n70 60 50 40 30 20 \n2θχ /φ (degree)(e) Series BIntensity (arb. unit) (c) Series AQ || MgO<100>\n2θχ /φ (degree)(a) (b)\nMgO 200 \nAg 200 MCG 220 \nMgO 220 \nAg220 MCG 400 \nCr 200 Cr 110 \n*\n*\nMCG 200 \n▼▼\nFIG. 3. In-plane XRD profiles of 5-nm-thick samples. (a, b) Di ffraction positions for MCG, Ag,\nCr layers, and MgO substrates. (c, d) Diffraction profiles of se ries A, and (e, f) those of series B.\nDiffraction vectors( /vectorQ)areparalleltoMgO <100>andMgO <110>for(c, e)and(d, f), respectively.\nWeak peaks around 67 °– 69°in (c) and (e) are probably from the Ta capping layers.\nont, on the other hand for the series B, although the c/aratio is 1.04 at t= 5 nm and\nrelaxes with increasing t, the tetragonal distortion is maintained at a relatively thick tof\n30 nm for which the c/aratio is 1.02. The different lattice strain possibly originates from\nthe difference in the growth modes of the MCG layers: For the series A, as it was found in\nthe observation of morphology using the AFM images, the strain insid e the lattice could be\nrelaxed because of the three-dimensional growth mode in the initial stage. For the series B,\non the other hand, the layer-by-layer growth in the initial stage re sulted in the restriction of\nthe MCG lattice to the in-plane direction, which remained unrelaxed ev en at the relatively\nlarge thickness.\nB. Magnetic Properties\nFigures5and6show magnetization curves of all samples for the series A and B, res pec-\ntively. In-plane magnetization is observed for the series A, and per pendicular magnetization\nis observed for the series B. The perpendicular magnetization in ser ies B is consistent with\na previous study on a Mn 1.8Co1.2Ga1.0film [33]. These features for the easy magnetization\ndirections were the same for all other layer thicknesses down to 5 n m in each sample se-\nries. Figure 7(a) shows the tdependence of saturation magnetic moments ( µs) which were\ncalculated using Msvalues obtained from the magnetization curves (Figs. 5and6) and\n8c-axis\na-axis\nc-axis\na-axis\n1.06\n1.04\n1.02\n1.00\n0.98c/a\n30 20 10 0\nt (nm)0.62\n0.60\n0.58\n0.56Lattice constants (nm) 0.62\n0.60\n0.58\n0.56(a) Series A\n(b) Series B\nSeries B\nSeries A(c)\nFIG. 4. Thickness ( t) dependence of lattice constants of the samples: (a) series A on Ag buffers,\n(b) series B on Cr buffers, and (c) the tdependence of c/a ratios for series A and B. The error\nbars range within the size of data points\n-400-2000200400\n-1 -0.5 00.5 1-400-2000200400\n-1 -0.5 00.5 1t: 30 nm┴|| \nt: 20 nm┴|| t: 10 nm┴|| \nt: 5 nm┴|| M (10 3 A/m) \nµ0H (T) µ0H (T)M (10 3 A/m) (a) (b)\n(c) (d)Series A: Mn2- δCoGa1+ δ on Ag buffer\nFIG. 5. Magnetization curves for series A samples. Marks ⊥and||represent the out-of-plane and\nin-plane directions, respectively, for applied magnetic fi elds. The layer thicknesses ( t) are (a) 5 (b)\n10, (c) 20, and (d) 30nm. The measurements were carried out at room temperature.\nthe lattice constants (Fig. 4). Theµsshows nearly no dependence on t, which suggests\na similar degree of chemical order among the samples even for the re latively low thickness\nregion. The magnitudes of µsrange from 1.1 to 1.4 µBper a formula unit (f.u.), which\nare relatively small compared to other ferromagnetic Heusler alloys showing the moment\n9-400-2000200400\n-400-2000200400\n-2 -1 01 2-2 -1 0 12t: 30 nm t: 20 nmt: 10 nm t: 5 nmM (10 3 A/m) \nµ0H (T) µ0H (T)M (10 3 A/m) (a) (b)\n(c) (d)\n┴\n|| ┴\n|| ┴\n|| ┴|| Series B: Mn2- δCoGa1+ δ on Cr buffer\nFIG. 6. Magnetization curves for series B samples. Marks ⊥and||represent the out-of-plane and\nin-plane directions, respectively, for applied magnetic fi elds. The layer thicknesses ( t) are (a) 5 (b)\n10, (c) 20, and (d) 30nm. The measurements were carried out at room temperature.\nSeries BSeries A(a)\n(b)Series B\nSeries A3\n2\n1\n0µ0Hk┴ or ||  (T) \n30 20 10 0\nt (nm)µ0Hk|| \nµ0Hk┴2\n1\n0µs ( µB/f.u.) \nFIG. 7. (a) Saturation magnetic moment ( µs) per a formula unit (f.u.) and (b) saturation field\n(H⊥(or||)\nk) as a function of t, whereH⊥(or||)\nkrepresents anisotropy field for out-of-plane (in-plane)\ndirection. The measurements were carried out at room temper ature.\nof the order of 4 – 6 µBbecause of the ferrimagnetic alignment of the atomic moments.\nAlthough the experimental values are smaller than theoretical valu es of approximately 2\nµBfor Mn 2CoGa in literature [ 50] as well as in our calculation (see Sec. IV), those are of\nthe same order of other calculated values of about 1 µBin which the off-stoichiometry and\nthe disorder effects were considered [ 51]. Comparing the two sample series, the samples in\nseries A exhibit slightly larger µsthan those in series B. The difference is possibly because\nof a small difference of chemical order for each sample series[ 52]. As another possibility, the\n102.0 \n1.5 \n1.0 \n0.5 \n0\n-0.5 Ku (10 5 J/m 3)\n1.06 1.04 1.02 1.00 0.98 \nc/aSeries B\nSeries A\nFIG. 8. The c/aratio dependence of uniaxial magnetocrystalline anisotro py constant ( Ku).\ndifference in composition distribution probably caused by a difference in a small amount of\ninterdiffusion around the Ag (or Cr) / MCG interface may be consider ed[53]. Figure 7(b)\nshows the tdependence of saturation field for hard magnetization directions w hich are the\nout-of-plane field ( H⊥\nk) and the in-plane field ( H||\nk) for series A and B, respectively. The H⊥\nk\nmonotonically decreases as tincreases for the series A, while H||\nkshows a drop around the t\nof greater than 10 nm for the series B. The uniaxial magnetocryst alline anisotropy constant\n(Ku) of MCG films was evaluated from the magnetization curves using the following for-\nmula;Ku=Keff\nu+(µ0/2)M2\ns, where ( µ0/2)M2\nsis the shape anisotropy energy, and Keff\nuwas\nderived from the area enclosed by the out-of-plane and in-plane ma gnetization curves. A\npositive value of Keff\nurepresents the perpendicular magnetization in this study. The valu es\nofKuare summarized as a function of the c/aratios in figure 8. Here, the Kuexhibits\npositive correlation with the c/aratio indicating that it is caused by the magnetoelastic ef-\nfect. The maximum value of Kuwas 1.6×105J/m3at thec/aof 1.04. The c/adependent\nKuis theoretically discussed in Sec. IV. Although the strain could be maximum around the\ninterface region and be relaxed around the surface region in the re al samples, the following\ndiscussion is based on the uniform strain for the simplicity.\nC. Domain Observations\nThe values of Kuexhibit clear dependence on t, especially in the samples of series B,\nfor which the difference in magnetic domain structures was also obse rved using Kerr mi-\ncroscopy [ 54,55]. Figure 9shows hysteresis loops for the samples of series B with t= 10\n(Fig.9(a)), 20 (Fig. 9(b)) and 30 nm (Fig. 9(c)) measured by MOKE in polar mode. The\nsquare like shapes of the hysteresis indicate that the magnetizatio n reversal occurs via do-\nmain nucleation and domain wall motion for all the samples [ 54,56]. To clarify the change of\n111.0\n0\n-1.0(a) t: 10 nm\n1.0\n0\n-1.0t: 20 nm\n1.0\n0\n-1.0Normalized Intensity (arb. unit) \n-200 -100 0100 200\nµ0H (mT)(c)\nt: 30 nm(b)i\nii \niii\niv \nv\niii \niii\niv \nv\ni\nii \niii\niv \nv\nFIG. 9. The hysteresis loops for t= (a) 10, (b) 20, and (c) 30 nm in series B measured with Kerr\nmicroscopy in polar mode. The red dots and numbers i – v on the h ysteresis loops corresponds to\nthe points for which domain images are shown in the next figure i.e., Fig.10.\ndomain pattern, five points were chosen for MOKE microscopy obse rvation: (i) a saturation\npoint at a positive field (+ Hsat), (ii) a point around nucleation at a negative field ( −Hnucl),\n(iii) a point around coercivity at a negative field ( −Hc), (iv) a point near saturation in a\nnegative field ( ∼ −Hsat), and (v) a saturation point for the negative field ( −Hsat). The\nnumbers, i – v are also annotated in each panel of Fig. 9, and the respective field values are\nshown in each panel of MOKE images in Fig. 10. The domain images observed by MOKE\nmicroscopy in polar mode of three samples are shown in Fig. 10. Note that the domain\nimage was not studied for t= 5 nm because of large Hc(µ0Hc∼210 mT) which was out\nof the range of the MOKE setup. For the 10-nm-thick sample, relat ively large size of do-\nmains appears, which contain a 180 °domain wall and are visible in Figs. 10(b) to10(d).\nThe domain pattern changes for t= 20 nm, in which large patch-like domains are observed\n(Figs.10(g) –10(i)). With increasing tto 30 nm, the domain size becomes small (Figs. 10(l)\n–10(n)). It is observed here that the domain size is observed to be com paratively large for\nthe sample with large anisotropy energy. The surface roughness lik ely causes the magnetic\ndomain pinning[ 57].\n12500 µm(i) + Hsat (ii) - HNucl(iii) ~ - Hc(iv) ~ -Hsat (v) -Hsat +H\n(a) 198.6 mT\n(f) 59.6 mT\n(k) 59.5 mT(b) -146.0 mT\n(g) -38.7 mT\n(l) -25.5 mT(c) -149.6 mT\n(h) -40.0 mT\n(m) -27.0 mT(d) -153.6 mT\n(i) -41.5 mT(e) -198.4 mT\n(j) -59.5 mT\n(o) -60.0 mTt: 10 nm\n20 nm\n(n) -28.5 mT\n30 nmH\nFIG. 10. The domain images for (a – e) t= 10, (f – j) 20, and (k – o) 30 nm in series B, whose\nhysteresis loops are shown in Figs. 9(a),9(b), and 9(c), respectively. The field points (i – v) which\nare shown above the panels are annotated in Fig. 9. The exact field values are shown in each panel.\nThe scale bar and field direction are shown in (a), which is val id for all the images.\nD. X-ray Magnetic Circular Dichroism\nElement specific magnetic moment of the 30-nm-thick sample showing perpendicular\nmagnetization(series B)wascharacterized using synchrotron ra diationsoftx-rayatBL23SU\nof SPring-8 [ 41]. The XAS (= µ−+µ+) and XMCD (= µ−−µ+) spectra are shown in\nfigure11for MnL2,3and CoL2,3absorption edges measured at θof 0°and 54.7 °, where\nµ+(−)represents the soft x-ray absorption for the positive (negative ) helicity. The XAS and\nXMCD spectra of previously studied bulk Mn 2VAl [38] and Co 2MnGa [19] samples are also\nshown in Fig. 11to discuss the origin of structures observed in the spectra for th e series B\nsample, which is to be described in the latter part of this section. Elem ent specific hysteresis\nloopsmeasured at the L3absorptionedges ofMnandCoatomsareshown infigure 12. MCD\nsignals for Mn and Co atoms simultaneously switch around the coerciv e field. For θ= 0, the\ncoercive field is consistent with that observed in VSM measurements shown in Figs. 5and6.\nThese results indicate strong ferromagnetic coupling between the Co moments and the net\nmoment for Mn atoms. To quantitatively discuss the angular depend ence of XMCD, orbital-\n(mθ\norb) andeffective spin-moments ( meff\nspin) per anatomwere evaluated using magneto-optical\nsum rules [ 58–63] which state the moments using the following equations;\nmθ\norb=−4q\n3rnh, (1)\n130.02\n0.003θ: 0°\n54.7°\nθ: 0°\n54.7°\n-(1/150)×MVA*θ: 0°\n54.7°\n(1/150)×CMG**(d)\n(1/150)×CMG**(b)Mn L3L2 Co L3L2\n(a)\nθ: 0°\n54.7°XAS (arb. unit) MCD (arb. unit) \n660 650 640 630 800 790 780 770\nPhoton Energy (eV)(c)\nFIG. 11. (a, b) Normalized soft x-ray absorption spectra (XA S) and (c, d) soft x-ray magnetic\ncircular dichroism (XMCD) spectra for the series B sample ar ound the (a, c) Mn L2,3and (b, d)\nCoL2,3absorption edges. Red lines and blue lines represent spectr a taken at θ= 0°and 54.7 °,\nrespectively, where θ= 0°is the perpendicular to the film plane direction. MVA and CMG a re\nreference spectra of bulk Mn 2VAl [38] and Co 2MnGa [19] samples, respectively. Positions for the\nshoulders in XMCD spectra are marked by dotted lines and soli d lines for those in MVA and CMG,\nrespectively, and the corresponding structures are also ma rked in the spectra of series B sample\nusing the same markers.\nMCD (arb. unit) \n-1.0 -0.5 00.5 1.0\nµ0H (T)at Mn L3\n-1.0 -0.5 00.5 1.0at Co L3MCD (arb. unit) \nµ0H (T)θ: 0°\n54.7°θ: 0°\n54.7°(a) (b)\n0.002\n0.002\nFIG. 12. Element specific hysteresis loops measured at L3absorption edges of (a) Mn and (b)\nCo atoms. Angles, θshown in each pannel represent magnetic fied direction respe ct to the sample\nnormal.\nmeff\nspin=mspin+7mθ\nT=−6p−4q\nrCnh, (2)\nwheremspinandmθ\nTare the spin-moment and magnetic dipole moment, respectively. The\nquantities, p,q, andrare the XMCD integrations for the L3-edge,L3+L2-edges, and\n14the integral of XAS which are defined as the following equations: p=/integraltext\nL3(µ−−µ+)dE,\nq=/integraltext\nL3+L2(µ−−µ+)dE, andr=/integraltext\nL3+L2(µ−+µ+−b.g)dE(b.g is a background spectrum\nshowing a step-function shape). A correction factor ( C) is included in equation ( 2) to take\ninto account the so-called jjmixing effect [ 64–66]. Formeff\nspinof Mn and Co, the Cof 1.5\nand 1.1 were assumed, respectively [ 38,64,65]. The numbers of 3d holes ( nh) were assumed\nto be 4.44 ±0.08 and 2.29 ±0.02 for the Mn and Co atoms, respectively, based on the\ntheoretical values for Mn 2CoGa obtained from the first principles calculations in section\nIV. Here, the values of nhare averaged over four cases of Mn 2CoGa with X Aand L2 1b\nphases and c/aratios of 1.00 and 1.04. The values of error are the standard deviat ion of\nall cases including possible inequivalent atomic sites for Mn atoms which are described in\ndetail later. In a strict manner, the inequivalent sites should be dec onvoluted from the\nXAS/XMCD spectra, and the equations ( 1) and (2) are applied to each site, however, the\npresentnhfor the Mn sites showing similar values enable us to evaluate average v alues using\nthe equations ( 1) and (2) as an approximation [ 67]. The values of meff\nspin,mθ\norbandmorb/meff\nspin\nare summarized in figure 13. Corresponding total magnetic moment was calculated to be\n1.18±0.03µB/f.u.usingmeff\nspinandmθ\norbatθ= 54.7 °, and the film composition. The\ntotal magnetic moment by the sum rule is of the same order as the µsof 1.27µB/f.u.\nmeasured by VSM. Concerning the angular dependence, no θdependence was found either\nformeff\nspinormθ\norb. Here, the θof 54.7 °is a magic angle for which mθ\nT= 0 by applying the\napproximate symmetry relation of 2 m90°\nT+m0°\nT= 0, where mθ\nT=m90°\nTsin2θ+m0°\nTcos2θfor\n3d metals [ 60,68]. Thus, the angular independent meff\nspinvalues suggest that the anisotropy\nof the magnetic dipole moment (∆ mT=m0°\nT−m90°\nT=3\n2m0°\nT) is less than 0.036 µB/atom\nwhich is in an error-bar range in the present sample. The angular inde pendent mθ\norbvalues\nprovide information for the discussion of origin of PMA in series B: The magnetocrystalline\nanisotropy energy ( EMCA) has been theoretically evaluated using expressions based on the\nsecond-order perturbationtheoryintermsofspin-orbitinterac tions[69–71]. Notethat EMCA\nis defined as the energy difference when the magnetization points to the perpendicular and\nin-plane direction: EMCA=Eθ=90−Eθ=0,i.e., the positive sign of EMCAindicates PMA in\nthis definition. Within a formulation EMCAcan be expressed as follows [ 69,70]:\nEMCA≃ξ\n4∆morb−21\n2ξ2\nEex∆mT, (3)\nwhereξ,Eex, and ∆morbare the spin–orbit coupling constant, exchange splitting of 3d\nbands, and the anisotropy of orbital magnetic moment defined as ∆ morb=mθ=0\norb−mθ=90°\norb,\n1554.7° 0° \nPhoton angle, θ (°)54.7° 0° Co Mn morb θ (µB/atom) morb θ/mspin eff mspin eff  (µB/atom) (a)\n(b)\n(c)0.15\n0.10\n0.05\n0\n-0.050.08\n0.06\n0.04\n0.02\n0\n-0.020.8\n0.4\n0Series B\nCo Mn \nCo Mn \nFIG. 13. Summary of (a) the effective spin-moment ( meff\nspin) and (b) orbital moment ( mθ\norb) per a\natom evaluated using magneto-optical sum rule. (c) mθ\norb/meff\nspinvalues. Photon angles, θ= 0°and\n54.7°are defined as the out-of-plane and pseudo-in-plane directi ons, respectively. All data points\nare for the series B sample.\nrespectively, for which the first term on the right side correspond s to the contribution to\nthe energy difference from the spin-conservation terms in virtual excitations, and the second\nterm corresponds to that from the spin-flip term. The experiment ally observed angular\nindependent mθ\norbimplies small contribution of the spin-conservation terms to Ku, and the\nother contribution of the spin-flip terms discussed in refs. 70and71is more important.\nTheoretical values of Kuandmorbas well as the origin of PMA with respect to the spin-flip\nterm are to be discussed further in section IV.\nThe sub-peaks and shoulder structures in XAS and XMCD spectra p rovide additional\ninformation on the chemical order of the atoms in Heusler structur es [17–19,29,38–40,52,\n72]. Notethatapossibility ofoxidizationisexcluded, becausetheshape sofXASandXMCD\nspectra arecompletely different fromthosereportedformateria lscontaining Mn-and/orCo-\noxides [73,74]. In addition, shoulders and sub-peaks are clearly observed at 300 K for MCD\nspectra shown in Figs. 11(c) and11(d), for which the measurement temperature is higher\nthan the magnetic transition temperatures for most of Mn-/Co-o xide materials. Figure 14\nshows a schematic illustration of a crystal structure for Heusler a lloys and the notations of\nthe atomic sites for the inverse-Heusler structure (X Aphase), the full-Heusler structure (L2 1\nphase), and an L2 1bphase. In addition to these three phases, fully-disordered D0 3phase\n16A: \nB: \nC: \nD: Site XA L2 1b L2 1\nGa Ga Ga \nMn Mn \nMn \nMn Mn Co Co \nMn   or Co \nMn   or Co \nabc\nFIG. 14. Schematic illustration of the Heusler structure, a nd possible atomic sites for Mn 2CoGa.\nRelative directions of magnetic moments for each atom theor etically reported in ref. 50are de-\nscribed using black and white arrows beside the name of atoms .\nwas considered as possible chemical phases in previous studies on bu lk samples [ 50,75].\nAccording to total energy calculations in ref. 50, the X Aphase was the most stable at the\nground state, and relatively small energy difference of 0.136 eV/f.u . was reported for the\nL21bphase. The energy differences with respect to the X Aphase were much higher for the\nL21and D0 3phases than that for the L2 1bphase. Concerning the magnitudes of magnetic\nmoment for each atom, the values were similar in X Aand L2 1bphases to each other and\nferrimagnetic coupling was reported as schematically shown in Fig. 14, which resulted in\na total moment of about 2.0 µB/f.u. for both. On the other hand, those were relatively\nlarge (small) for the L2 1(D03) phase showing ferromagnetic (ferrimagnetic) coupling with\na total moment of about 7.7 (1.0) µB/f.u. As it was discussed on Fig. 7in Sec.IIIB,\nthe total magnetic moments of the present samples are of the ord er of the reported values\nfor the X Aand L2 1bphases, and considering the calculated energy difference mentione d\nabove, the L2 1and D0 3phases are excluded in the following discussion. Note that the L2 1b\nphase was experimentally proposed for the previous bulk Mn 2CoGa samples based on high-\nangle annular dark field scanning electron transmission microscope im ages [75] and neutron\ndiffraction experiments [ 50]. On the other hand, the X Aphase was proposed in another\npreviousstudyonanepitaxiallygrownMn 2CoGafilmbasedonXASandXMCDresults [ 39].\nThe shapes of XAS and XMCD spectra of Mn 2CoGa films were previously discussed based\non the first-principles calculation for the X Aphase. In the previous studies [ 39,40], several\nshoulders, sub-peaks, and dip structures were mentioned. In th e present results, similar\nstructures are observed, i.e., broad shoulders in the XAS, and sub-peaks and dip structures\nin the XMCD spectra are observed, which are marked by solid and bro ken lines in Fig.\n11. The marked structures in the XMCD spectra can also be understo od as a sum of two\nspectra [29,39,40,72,76] which are, e.g., MVA [38] and CMG [ 19] for the markers of broken\n17and solid lines, respectively. Here, MVA is a typical example of a mater ial with Mn atom(s)\nat D(C)-site, and CMG is another example with a Mn atom at B-site in Fig .14. The\nbroad feature is mainly from the Mn atom(s) D(C)-site showing relat ively itinerant nature\nof electrons [ 38], on the other hand, the Mn atom at the B-site exhibits relatively loca lized\nnature showing remarkable shoulders in XAS [ 17–19]. Concerning the XMCD spectra for\nCo, shoulders are also observed around the post-absorption edg es with the solid markers in\nFig.11(d) which also includes a reference spectrum of CMG. The origin of th e shoulders is\nattributed to the 2p →3deg(oreu) transition [ 19].\nIV. FIRST-PRINCIPLES CALCULATION\nFirst-principles calculation of electronic structure and magnetic pr operties of bulk\nMn2CoGa with the X Aand L2 1bphases were performed by employing the tight-binding\nlinear muffin-tin orbital method under the local spin-density functio nal approximation [ 77].\nThe coherent potential approximation was adopted to treat the p artial disorder between\nMn and Co in C- and D-sublattices of the L2 1bphase (see Fig. 14). The lattice constant\na0= 0.5879 nm [ 50] was adopted. We have confirmed that the electronic structure is\nhalf-metallic in both phases and the calculated magnetization is close t o integer 2 µB/f.u.,\nwhich is consistent with a previous calculation by Umetsu, Tsujikawa et al[50]. For the\ncalculation of Kuandmorb, spin–orbit interaction was taken into account [ 78]. TheKu\nvalue was evaluated from the energy difference between when the m agnetization aligned\nalong the c- anda-axis directions, i.e.,Ku= (Ea−Ec)/V[79,80], within the magnetic\nforce theorem ( V: volume of a unit cell). The number of k-point sampling was set to 55,44 0\nin the full-Brillouin zone.\nFigure15shows the calculated result of Kuof Mn 2CoGa in the X Aand L2 1bphases\nas a function of the axial ratio c/a, wherec/awas changed under the assumption of con-\nstant volume deformation. PMA appears for c/a >1 in both phases. The Kuvalues at\nc/a= 1.04intheX AandL2 1bphases are1 .5×105J/m3(0.049meV/f.u.) and2 .6×105J/m3\n(0.081 meV/f.u.), respectively, which are quantitatively consistent values with the obtained\nexperimental result shown in Fig. 8. Figure16shows the morbof Mn andCo atoms when the\nrelative angle between magnetization and crystal c-axis is 0 °, 54.7°, and 90 °forc/a= 1.04.\nNote that the positive (negative) sign of morbmeans that the direction of orbital magnetic\nmoment and overall magnetization are the same (opposite). Thus, the directions of spin\n18FIG. 15. Axial ratio c/adependence of uniaxial magnetic anisotropy constant Kuof Mn2CoGa\nwith the X Aand L2 1bphases obtained from first-principles calculation.\nFIG. 16. Calculated orbital magnetic moment morbof Mn and Co atom in Mn 2CoGa with the X A\nand L2 1bphases. The angle of 0◦, 54.7◦, and 90◦means the relative angle between magnetization\nand crystal c-axis.\nand orbital magnetic moments are the same each other in Mn at D-su blattice with X A\nphase and in Mn at C/D-sublattice with the L2 1bphase, because the spin magnetic moment\nis arranged in the opposite direction with the magnetization. This res ult is in agreement\nwith a previous first-principles calculation [ 39]. In Fig. 16, we find that the ∆ morbis of\nthe order of 0.001 µB/atom in Co and even smaller in Mn. Based on Eq. ( 3), therefore,\nthe orbital moment anisotropy cannot explain the magnetocrysta lline anisotropy observed\nin our experiment. This fact implies that the PMA in Mn 2CoGa is mainly induced by the\nhybridization of ↑-spin and ↓-spin states via spin–orbit coupling, which is similar to the\ndiscussion of magnetocrystalline anisotropy in previous studies [ 81–83].\nFor a detailed analysis of magnetocrystalline anisotropy, we introdu ce the expression of\n19FIG. 17. (a), (b) Total and local DOS of Mn and Co at C- and/or D- sublattice in Mn 2CoGa\nwith the X Aand L2 1bphases for c/a= 1.00. (c)–(f) Projected DOS into t2g-baseddxy,dyz, and\ndzxorbitals, and into eg-baseddx2−y2andd3z2−r2orbitals for c/a= 1.00 and 1.04. The values\nin parenthesis denote c/a. The upper and lower panels of each figure present the DOS of ↑- and\n↓-spin states, respectively.\nKubased on the second-order perturbation theory in terms of the s pin–orbit coupling,\nKu=ξ2\n4V/summationdisplay\noσ/summationdisplay\nuσ′2δσ,σ′−1\nεuσ′−εoσ/bracketleftbig\n|/angbracketleftuσ′|ℓz|oσ/angbracketright|2−|/angbracketleftuσ′|ℓx|oσ/angbracketright|2/bracketrightbig\n, (4)\nwhereεoσand|oσ/angbracketright(εuσand|uσ/angbracketright)denotetheeigenvalueandeigenvector ofthenonpertubative\noccupied (unoccupied) states, respectively ( σ=↑or↓). In addition, ℓz(ℓx) is thez(x)\n20component of the orbital angular momentum operator. From the s ublattice decomposition\nanalysis of magnetocrystalline anisotropy energy [ 84,85], we confirmed that Mn and Co\nin C- and/or D-sublattice mainly yields the PMA in tetragonally distorte d Mn2CoGa for\nc/a >1 in both of the X Aand L2 1bphases [86]. In figure 17, we depict the density of states\n(DOS) of X A- and L2 1b-Mn2CoGa. Figs 17(a) and17(b) show the total DOS and the local\nDOS of Mn and Co at C- and/or D-sublattice for the cubic ( c/a= 1.00) case. Furthermore,\nFigs.17(c)–17(f) show the DOS projected into the d-orbitals ( dxy,dyz,dzx,dx2−y2,d3z2−r2)\nnear the Fermi energy ( EF) for the cubic and tetragonal ( c/a= 1.04) cases. The degeneracy\nint2g(dxy,dyz,dzx) based states and that in eg(dx2−y2,d3z2−r2) based states are removed\nby the tetragonal distortion. In particular, we can see a significan t DOS splitting in dx2−y2\nandd3z2−r2states, because d3z2−r2(dx2−y2) orbital spreads to the parallel (perpendicular)\ndirection to the crystal c-axis.\nLet us discuss a possible origin for PMA in tetragonally distorted Mn 2CoGa qualitatively,\nusing the expression of Eq. ( 4) and the projected DOS shown in Figs. 17(c)–17(f). With\nreference to previous studies [ 71,83,87], we assumed εuσ−εoσ′in Eq. (4) as a constant\nvalue which is independent of wave-number kfor simplicity [ 88]. In Figs. 17(c)–17(f), the\npeaks of d↓\nx2−y2andd↓\n3z2−r2states are located just above EF, and also the d↑\nyzstate lies\nbelowEF, in Mn and Co at C- and/or D-sublattice with both of the X Aand L2 1bphases.\nThese states hybridize each other via spin–orbit coupling, which res ults in non-zero matrix\nelement of /angbracketleftdyz|ℓx|dx2−y2/angbracketrightand/angbracketleftdyz|ℓx|d3z2−r2/angbracketright. In Eq. ( 4), the mixing of occupied d↑\nyzstate\nand unoccupied d↓\nx2−y2state, and the mixing of occupied d↑\nyzstate and unoccupied d↓\n3z2−r2\nstate make for the positive Kuvalue,i.e., promoting PMA. Note here that δ↑↓= 0 in\neq. (4). In Figs. 17(c)–17(f), the peak of the unoccupied d↓\n3z2−r2state of Mn and Co\nshifts to lower energy side with increasing c/a; this enhances the positive contribution of\nKu, because the energy difference from the occupied d↑\nyzstate,εu↓−εo↑, becomes smaller,\nin Eq. (4). In contrast, the peak of d↓\nx2−y2shifts to the higher energy side, reducing the\npositive contribution. We notice that there is a tradeoff relation enh ancing and reducing the\npositiveKuwith increasing c/a; however, the former contribution is three times as large as\nthe latter, because of |/angbracketleftdyz|ℓx|d3z2−r2/angbracketright|2= 3 and |/angbracketleftdyz|ℓx|dx2−y2/angbracketright|2= 1. Therefore, the PMA\nis induced by increasing the positive Kuvalue in the tetragonally distorted MCG films for\nc/a >1 through the spin-flip term in Eq. ( 4), and a qualitative mechanism for the PMA is\nalmost the same regardless of whether crystal structure is the X Aor L21bphases. On the\nother hand, the in-plane magnetic anisotropy arises for c/a <1, as shown in Fig. 15, based\n21on the opposite scenario of the above.\nV. SUMMARY\nCrystal structure and magnetic properties of MCG films on Cr or Ag buffer were in-\nvestigated experimentally and theoretically. Epitaxial growth of all samples was confirmed\nusing XRD profiles, and the different growth modes were suggested depending on the buffer\nmaterials, which possibly caused the different c/aratios among the samples. Tetragonal\ndistortion of the MCG layers was found for the Cr buffer samples up t o the layer thickness\nof 30 nm, whereas Ag buffer samples exhibited cubic structure which was stable in bulk\nMn2CoGa samples in literature. From magnetization curves measuremen ts, perpendicular\nmagnetization was found in all the Cr buffer samples; on the other ha nd, all Ag buffer\nsamples exhibited in-plane magnetization. The c/aratio dependence of Kuclearly exhibited\npositive correlation, and the maximum value of Kuwas 1.6×105J/m3at roomtemperature\nfor the 5-nm-thick MCG film on the Cr buffer. Magnetic domain patter n observations using\nthe magneto-optical Kerr microscopy revealed changes of the do main structures depending\non the magnitude of Ku. XAS and XMCD spectra were also acquired to elucidate the origin\nof magnetocrystalline anisotropy in MCG, and unraveled the chemica l order of L2 1bor XA\nphase. In addition, the angular dependence of XMCD spectra reve aled that the change of\nthemorbwas negligibly small suggesting a negligible contribution of the spin cons erved term,\nso called Bruno’s term, to Ku. Theoretical values of Kuwere evaluated by means of the\nfirst principles electronic structure calculation. The positive corre lation was also confirmed\nbetween the c/aratios and the values of Kuin the calculation. In addition, the calculated\nangular dependence of morbwas negligibly small. Those results are consistent with the\nexperimental results. Based on the analysis of the second-order perturbation theory, the\norigin of the magnetocrystalline anisotropy is proposed that the hy bridization between two\ncrystal-field split states with opposite spins, i.e., the spin-flip term near the Fermi level,\npromotes positive contribution to Ku.\nACKNOWLEDGMENTS\nThe XAS and XMCD experiments were performed under the Shared U se Program of\nJAEA Facilities (2018B-E24, and 2019B-E17) with the approval of t he Nanotechnology\n22Platform project supported by the Ministry of Education, Culture , Sports, Science and\nTechnology (A-18-AE-0042, and A-19-AE-0037). The synchrot ron radiation experiments\nwere performed at the JAEA beam line BL23SU in SPring-8 (2018B384 2, and 2019B3844).\nTK, DT and MM would like to thank Mr. Issei Narita for technical supp ort. SB, SM thank\ndepartment of atomic energy (DAE), Government of India for fina ncial support for the Kerr\nmicroscopy facility. This work was partially supported by KAKENHI (J P20K05296) from\nJSPS, and by a cooperative research program (No.20G0414) of th e CRDAM-IMR, Tohoku\nUniv.\nCONFLICT OF INTEREST\nThe authors have no conflicts to disclose.\nDATA AVAILABILITY STATEMENTS\nThe data that support the findings of this study are available within it s supplementary\nmaterial, and from the corresponding author upon reasonable req uest.\n[1] C. Felser and A. Hirohata, eds., Heusler Alloys (Springer International Publishing, 2016).\n[2] C. T. Tanaka, J. Nowak, and J. S. Moodera, J. Appl. Phys. 81, 5515 (1997) .\n[3] J. A. Caballero, Y. D. Park, J. R. Childress, J. Bass, W.-C . Chiang, A. C. Reilly, W. P. Pratt,\nand F. Petroff, Journal of Vacuum Science & Technology A: Vacuum, Surfaces, and Films 16,\n1801 (1998) .\n[4] J. K¨ ubler, A. R. William, and C. B. Sommers, Phys. Rev. B 28, 1745 (1983) .\n[5] S. Ishida, S. Fujii, S. Kashiwagi, and S. Asano, J. Phys. Soc. Japan 64, 2152 (1995) .\n[6] I. Galanakis, P. H. Dederichs, and N. Papanikolaou, Phys. Rev. B 66, 174429 (2002) .\n[7] Y. Miura, K. Nagao, and M. Shirai, Phys. Rev. B 69, 144413 (2004) .\n[8] H. C. Kandpal, G. H. Fecher, C. Felser, and G. Sch¨ onhense ,Phys. Rev. B 73, 094422 (2006) .\n[9] K. Inomata, S. Okamura, R. Goto, and N. Tezuka, Jpn. J. Appl. Phys. 42, L419 (2003) .\n[10] Y. Sakuraba, J. Nakata, M. Oogane, H. Kubota, Y. Ando, A. Sakuma, and T. Miyazaki, Jpn.\nJ. Appl. Phys. 44, L1100 (2005) .\n23[11] J. Schmalhorst, D. Ebke, A. Weddemann, A. H¨ utten, A. Th omas, G. Reiss, A. Turchanin,\nA. G¨ olzh¨ auser, B. Balke, and C. Felser, J. Appl. Phys. 104, 043918 (2008) .\n[12] T. Kubota, S. Tsunegi, M. Oogane, S. Mizukami, T. Miyaza ki, H. Naganuma, and Y. Ando,\nAppl. Phys. Lett. 94, 122504 (2009) .\n[13] M. Yamamoto, T. Ishikawa, T. Taira, G.-F. Li, K.-I. Mats uda, and T. Uemura, J. Phys.:\nCondens. Matter 22, 164212 (2010) .\n[14] H.-X. Liu, T. Kawami, K. Moges, T. Uemura, M. Yamamoto, F . Shi, and P. M. Voyles, J.\nPhys. D: Appl. Phys. 48, 164001 (2015) .\n[15] Y. Sakuraba, M. Ueda, Y. Miura, K. Sato, S. Bosu, K. Saito , M. Shirai, T. J. Konno, and\nK. Takanashi, Appl. Phys. Lett. 101, 252408 (2012) .\n[16] T. Kubota, Z. Wen, and K. Takanashi, J. Magn. Magn. Mater. 492, 165667 (2019) .\n[17] K. Miyamoto, K. Iori, A. Kimura, T. Xie, M. Taniguchi, S. Qiao, and K. Tsuchiya, Solid State\nCommun. 128, 163 (2003) .\n[18] N. D. Telling, P. S. Keatley, G. van der Laan, R. J. Hicken , E. Arenholz, Y. Sakuraba,\nM. Oogane, Y. Ando, and T. Miyazaki, Phys. Rev. B 74, 224439 (2006) .\n[19] T. Yoshikawa, V. N. Antonov, T. Kono, M. Kakoki, K. Sumid a, K. Miyamoto, Y. Takeda,\nY. Saitoh, K. Goto, Y. Sakuraba, K. Hono, A. Ernst, and A. Kimu ra,Phys. Rev. B 102,\n064428 (2020) .\n[20] T. Kono, M. Kakoki, T. Yoshikawa, X. Wang, K. Goto, T. Mur o, R. Y. Umetsu, and\nA. Kimura, Phys. Rev. Lett. 125, 216403 (2020) .\n[21] M. Kolbe, S. Chadov, E. A. Jorge, G. Sch¨ onhense, C. Fels er, H.-J. Elmers, M. Kl¨ aui, and\nM. Jourdan, Phys. Rev. B 86, 024422 (2012) .\n[22] M. Jourdan, J. Min´ ar, J. Braun, A. Kronenberg, S. Chado v, B. Balke, A. Gloskovskii,\nM. Kolbe, H. Elmers, G. Sch¨ onhense, H. Ebert, C. Felser, and M. Kl¨ aui, Nat Commun 5\n(2014), 10.1038/ncomms4974 .\n[23] S. Ishida, S. Asano, and J. Ishida, J. Phys. Soc. Japan 53, 2718 (1984) .\n[24] J. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996) .\n[25] S. Ouardi, G. H. Fecher, C. Felser, and J. K¨ ubler, Phys. Rev. Lett. 110, 100401 (2013) .\n[26] I. Galanakis, K. ¨Ozdo˘ gan, E. S ¸a¸ sıo˘ glu, and S. Bl¨ ugel, J. Appl. Phys. 115, 093908 (2014) .\n[27] Y. Xin, H. Hao, Y. Ma, H. Luo, F. Meng, H. Liu, E. Liu, and G. Wu,Intermetallics 80, 10\n(2017).\n[28] V. Alijani, J. Winterlik, G. H. Fecher, and C. Felser, Appl. Phys. Lett. 99, 222510 (2011) .\n24[29] P. Klaer, B. Balke, V. Alijani, J. Winterlik, G. H. Feche r, C. Felser, and H. J. Elmers, Phys.\nRev. B84, 144413 (2011) .\n[30] S. Chadov, J. Kiss, and C. Felser, Adv. Funct. Mater. 23, 832 (2012) .\n[31] S. Ouardi, T. Kubota, G. H. Fecher, R. Stinshoff, S. Mizuka mi, T. Miyazaki, E. Ikenaga, and\nC. Felser, Appl. Phys. Lett. 101, 242406 (2012) .\n[32] T. Kubota, S. Ouardi, S. Mizukami, G. H. Fecher, C. Felse r, Y. Ando, and T. Miyazaki, J.\nAppl. Phys. 113, 17C723 (2013) .\n[33] T. Kubota, S. Mizukami, Q. L. Ma, H. Naganuma, M. Oogane, Y. Ando, and T. Miyazaki, J.\nAppl. Phys. 115, 17C704 (2014) .\n[34] Y. Wang, J. Zhang, X.-G. Zhang, H.-P. Cheng, and X. F. Han ,Phys. Rev. B 82, 054405\n(2010).\n[35] Z. Wen, H. Sukegawa, S. Mitani, and K. Inomata, Appl. Phys. Lett. 98, 242507 (2011) .\n[36] T. Kamada, T. Kubota, S. Takahashi, Y. Sonobe, and K. Tak anashi,IEEE Trans. Magn. 50,\n2600304 (2014) .\n[37] T. Kubota, Y. Ina, M. Tsujikawa, S. Morikawa, H. Narisaw a, Z. Wen, M. Shirai, and\nK. Takanashi, J. Phys. D: Appl. Phys. 50, 014004 (2016) .\n[38] K. Nagai, H. Fujiwara, H. Aratani, S. Fujioka, H. Yomosa , Y. Nakatani, T. Kiss, A. Sekiyama,\nF. Kuroda, H. Fujii, T. Oguchi, A. Tanaka, J. Miyawaki, Y. Har ada, Y. Takeda, Y. Saitoh,\nS. Suga, and R. Y. Umetsu, Phys. Rev. B 97, 035143 (2018) .\n[39] M. Meinert, J.-M. Schmalhorst, C. Klewe, G. Reiss, E. Ar enholz, T. B¨ ohnert, and K. Nielsch,\nPhys. Rev. B 84, 132405 (2011) .\n[40] S. Ouardi, G. H. Fecher, T. Kubota, S. Mizukami, E. Ikena ga, T. Nakamura, and C. Felser,\nJ. Phys. D: Appl. Phys. 48, 164007 (2015) .\n[41] Y. Saitoh, Y. Fukuda, Y. Takeda, H. Yamagami, S. Takahas hi, Y. Asano, T. Hara, K. Shira-\nsawa, M. Takeuchi, T. Tanaka, and H. Kitamura, J. Synchrotron Radiat. 19, 388 (2012) .\n[42] E. Bauer, Appl. Surf. Sci. 11-12, 479 (1982) .\n[43] M. T. Kief and W. F. Egelhoff, Phys. Rev. B 47, 10785 (1993) .\n[44] J. H. V. D. Merwe, J. Appl. Phys. 34, 117 (1963) .\n[45] J. H. V. D. Merwe, J. Appl. Phys. 34, 123 (1963) .\n[46] N. H. Fletcher, J. Appl. Phys. 35, 234 (1964) .\n[47] L. Z. Mezey and J. Giber, Jpn. J. Appl. Phys. 21, 1569 (1982) .\n25[48] G. Guisbiers and M. Jos´ e-Yacaman, in Encyclopedia of Interfacial Chemistry (Elsevier, 2018)\npp. 875–885.\n[49] L. Vitos, A. Ruban, H. Skriver, and J. Koll´ ar, Surf. Sci. 411, 186 (1998) .\n[50] R. Y. Umetsu, M. Tsujikawa, K. Saito, K. Ono, T. Ishigaki , R. Kainuma, and M. Shirai, J.\nPhys.: Condens. Matter 31, 065801 (2018) .\n[51] The calculation details for the magnetic moments with t he offstoichiometry and the disorder\nare described in Supplementary Material.\n[52] T. Kubota, K. Kodama, T. Nakamura, Y. Sakuraba, M. Oogan e, K. Takanashi, and Y. Ando,\nAppl. Phys. Lett. 95, 222503 (2009) .\n[53] X. Xu, Z. Chen, Y. Sakuraba, A. Perumal, K. Masuda, L. Kum ara, H. Tajiri, T. Nakatani,\nJ. Wang, W. Zhou, Y. Miura, T. Ohkubo, and K. Hono, Acta Mater. 176, 33 (2019) .\n[54] R. Belhi, A. Fassatoui, A. A. Adjanoh,andK.Abdelmoula ,J.Appl. Phys. 116, 013908 (2014) .\n[55] S. Mallick, S. S. Mishra, and S. Bedanta, Sci. Rep. 8, 11648 (2018) .\n[56] A. Hubert and R. Sch¨ afer, “Magnetic domains,” (Spring er-Verlag GmbH, 1998) pp. 107–354.\n[57] P. Bruno, G. Bayreuther, P. Beauvillain, C. Chappert, G . Lugert, D. Renard, J. P. Renard,\nand J. Seiden, J. Appl. Phys. 68, 5759 (1990) .\n[58] B. T. Thole, P. Carra, F. Sette, and G. van der Laan, Phys. Rev. Lett. 68, 1943 (1992) .\n[59] P. Carra, B. T. Thole, M. Altarelli, and X. Wang, Phys. Rev. Lett. 70, 694 (1993) .\n[60] J. St¨ ohr and H. K¨ onig, Phys. Rev. Lett. 75, 3748 (1995) .\n[61] G. van der Laan and A. I. Figueroa, Coord. Chem. Rev. 277-278 , 95 (2014) .\n[62] T. Koide, H. Miyauchi, J. Okamoto, T. Shidara, A. Fujimo ri, H. Fukutani, K. Amemiya,\nH. Takeshita, S. Yuasa, T. Katayama, and Y. Suzuki, Phys. Rev. Lett. 87, 257201 (2001) .\n[63] K. Edmonds, G. van der Laan, and G. Panaccione, Semicond Sci. Technol. 30, 043001 (2015) .\n[64] Y. Teramura, A. Tanaka, and T. Jo, J. Phys. Soc. Japan 65, 1053 (1996) .\n[65] H. A. D¨ urr, G. van der Laan, D. Spanke, F. U. Hillebrecht , and N. B. Brookes, Phys. Rev. B\n56, 8156 (1997) .\n[66] E. Goering, Philos. Mag. 85, 2895 (2005) .\n[67] F. Takata, K. Ito, Y. Takeda, Y. Saitoh, K. Takanashi, A. Kimura, and T. Suemasu, Phys.\nRev. Materials 2, 024407 (2018) .\n[68] D. Weller, J. St¨ ohr, R. Nakajima, A. Carl, M. G. Samant, C. Chappert, R. M´ egy, P. Beauvil-\nlain, P. Veillet, and G. A. Held, Phys. Rev. Lett. 75, 3752 (1995) .\n[69] P. Bruno, Phys. Rev. B 39, 865 (1989) .\n26[70] G. van der Laan, J. Phys.: Condens. Matter 10, 3239 (1998) .\n[71] Y. Kota and A. Sakuma, J. Phys. Soc. Jpn. 83, 034715 (2014) .\n[72] J. Karel, F. Bernardi, C. Wang, R. Stinshoff, N.-O. Born, S . Ouardi, U. Burkhardt, G. H.\nFecher, and C. Felser, Phys. Chem. Chem. Phys. 17, 31707 (2015) .\n[73] T. Burnus, Z. Hu, H. H. Hsieh, V. L. J. Joly, P. A. Joy, M. W. Haverkort, H. Wu, A. Tanaka,\nH.-J. Lin, C. T. Chen, and L. H. Tjeng, Phys. Rev. B 77, 125124 (2008) .\n[74] H. Jung, S.-J. Lee, M. Song, S. Lee, H. J. Lee, D. H. Kim, J. -S. Kang, C. L. Zhang, and S.-W.\nCheong, New J. Phys. 11, 043008 (2009) .\n[75] K. Minakuchi, R. Umetsu, K. Kobayashi, M. Nagasako, and R. Kainuma, J. Alloys 645, 577\n(2015).\n[76] P. Klaer, C. A. Jenkins, V. Alijani, J. Winterlik, B. Bal ke, C. Felser, and H. J. Elmers, Appl.\nPhys. Lett. 98, 212510 (2011) .\n[77] I. Turek, V. Drchal, J. Kudrnovsk´ y, M. Sob, and P. Weinb erger,Electronic Structure of\nDisordered Alloys, Surfaces and Interfaces (Springer US, 1996).\n[78] I. Turek, V. Drchal, and J. Kudrnovsk´ y, Philos. Mag. 88, 2787 (2008) .\n[79] Y. Kota and A. Sakuma, J. Appl. Phys. 111, 07A310 (2012) .\n[80] Y. Kota and A. Sakuma, J. Phys. Soc. Japan 81, 084705 (2012) .\n[81] S. Miwa, M. Suzuki, M. Tsujikawa, K. Matsuda, T. Nozaki, K. Tanaka, T. Tsukahara,\nK. Nawaoka, M. Goto, Y. Kotani, T. Ohkubo, F. Bonell, E. Tamur a, K. Hono, T. Naka-\nmura, M. Shirai, S. Yuasa, and Y. Suzuki, Nat. Commun. 8, 15848 (2017) .\n[82] K. Ikeda, T. Seki, G. Shibata, T. Kadono, K. Ishigami, Y. Takahashi, M. Horio, S. Sakamoto,\nY. Nonaka, M. Sakamaki, K. Amemiya, N. Kawamura, M. Suzuki, K . Takanashi, and A. Fu-\njimori,Appl. Phys. Lett. 111, 142402 (2017) .\n[83] J. Okabayashi, Y. Miura, Y. Kota, K. Z. Suzuki, A. Sakuma , and S. Mizukami, Sci. Rep. 10,\n9744 (2020) .\n[84] I. V. Solovyev, P. H. Dederichs, and I. Mertig, Phys. Rev. B 52, 13419 (1995) .\n[85] L. Ke, Phys. Rev. B 99, 054418 (2019) .\n[86] TheestimatedanisotropyenergyofMnandCoinC-and/or D-sublatticeusingtheformulation\nin Ref. [84] is on the order of 0.01 meV with PMA, and further that of B-Mn i s one order of\nmagnitude smaller. Therefore, the contribution of B-Mn is r elatively small compared to the\ncontribution of the other Mn and Co.\n27[87] K. Masuda, H. Itoh, Y. Sonobe, H. Sukegawa, S. Mitani, an d Y. Miura, Phys. Rev. B 103,\n064427 (2021) .\n[88] The k-resolved analysis is desirable for more detailed investig ation ofKubased on the second\norder perturbation theory. However, this treatment is cons idered difficult to apply to bulk\nsystems including several species of atoms and randomness, compared to slab geometry sys-\ntems such as ferromagnetic/nonmagnetic layer junctions. [ 89–92] Thus, we give a simplified\nanalysis for discussion.\n[89] K. Masuda, S. Kasai, Y. Miura, and K. Hono, Phys Rev B 96, 174401 (2017) .\n[90] K. Masuda and Y. Miura, Phys Rev B 98, 224421 (2018) .\n[91] S. Kwon, P.-V. Ong, Q. Sun, F. Mahfouzi, X. Li, K. L. Wang, Y. Kato, H. Yoda, P. K. Amiri,\nand N. Kioussis, Phys Rev B 99, 064434 (2019) .\n[92] S. Jiang, S. Nazir, and K. Yang, Phys Rev B 101, 134405 (2020) .\n28"    },    {        "title": "2109.00093v2.Antiferromagnetic_Hysteresis_above_the_Spin_Flop_Field.pdf",        "content": "Antiferromagnetic Hysteresis above the Spin Flop Field\nM. J. Grzybowski,1, 2,\u0003C. F. Schippers,1O. Gomonay,3K. Rubi,4M. E. Bal,4U. Zeitler,4A. Kozioł-Rachwał,5\nM. Szpytma,5W. Janus,5B. Kurowska,6S. Kret,6M. Gryglas-Borysiewicz,2B. Koopmans,1and H. J. M. Swagten1\n1Department of Applied Physics, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands\n2Faculty of Physics, University of Warsaw, Pasteura 5, Warsaw, Poland\n3Institute of Physics, Johannes Gutenberg-University Mainz, 55128 Mainz, Germany\n4High Field Magnet Laboratory (HFML-EMFL), Radboud University, 6525 ED Nijmegen, The Netherlands\n5Faculty of Physics and Applied Computer Science, AGH University of Science and Technology, 30-059 Kraków, Poland\n6Institute of Physics, Polish Academy of Sciences, Lotników 32/46, Warsaw, Poland\n(Dated: November 22, 2022)\nMagnetocrystalline anisotropy is essential in the physics of antiferromagnets and commonly treated as a\nconstant, not depending on an external magnetic field. However, we demonstrate that in CoO the anisotropy\nshould necessarily depend on the magnetic field, which is shown by the spin Hall magnetoresistance of the\nCoO | Pt device. Below the Néel temperature CoO reveals a spin-flop transition at 240 K at 7.0 T, above which\na hysteresis in the angular dependence of magnetoresistance unexpectedly persists up to 30 T. This behavior is\nshown to agree with the presence of the unquenched orbital momentum, which can play an important role in\nantiferromagnetic spintronics.\nAntiferromagnetic thin-film materials have attracted a lot\nof attention recently due to their unique properties that cre-\nate potential applications in spintronics such as data storage\n[1, 2] or long-distance spin transport [3, 4]. Their robustness\nagainst external magnetic fields is an important promise for\nusing these antiferromagnetic materials. Therefore, they de-\nmand strong magnetic fields for reorientation of the spins, and\nthe understanding of the impact of such fields is essential. The\nbehavior of an antiferromagnet (AF) in high magnetic fields\nis governed by the competition between magneto-crystalline\nanisotropy and antiferromagnetic exchange interactions. De-\ntailed knowledge of this anisotropy is key to determine possi-\nble magnetic configurations of an antiferromagnet as well as\nproper device design and experimental geometry for potential\napplications. It is common and well-documented to treat the\nanisotropy as a physical parameter that is strictly constant for\na given temperature [5–7].\nIn this letter, we demonstrate that the magnetic anisotropy\nof a thin-film CoO can be modified by strong magnetic fields.\nIt manifests itself in the angular dependent magnetoresistance\nof the CoOjPt as clear, reproducible and abrupt changes of\nresitivity coexisting with the hysteresis around the hard axis,\nwhich are present up to the highest tested magnetic fields of\nB=30T. Such behavior cannot be reproduced by a simple\nmacrospin model nor a domain model. We explain it by the\ncontribution of the unquenched orbital momentum into the\nanisotropy.\nWe start with an experimental study of spin reorientation in-\nduced by the strong magnetic fields in CoO with an adjacent Pt\nlayer. When the magnetic field Bis applied to an AF along an\neasy axis, the Néel vector n, which is the difference of the two\nsublattice magnetizations, may reorient perpendicularly to the\nfield. It is a well-known spin-flop transition and occurs at the\nspin-flop field Bsf, which depends on the magnetic anisotropy.\nThis reorientation of ncan be detected by the spin Hall mag-\nnetoresistance (SMR) [5–9]. The angular dependence of the\nSMR allows extracting the spin-flop field values for different\nFIG. 1. The scheme of the structure used for the experiments. Jrep-\nresents the current direction. Magnetic easy (hard) axes are marked\nby orange (violet) arrows. Blue and red arrows represent the spin\naccumulation in Pt due to the spin Hall effect.\ngeometrical configurations of the magnetic field and thus, the\nintriguing behavior of the anisotropy in CoO can be observed.\nHere, we show that the spin-flop transition is clearly detected\ntogether with a remarkable presence of the hysteretic behav-\nior of the AF that cannot be explained by a constant anisotropy\nfield commonly applied for antiferromagnets. This can be un-\nderstood as a magnetic anisotropy that should necessarily de-\npend on the external magnetic field magnitude and orientation.\nThe electrical measurements were performed on a Hall bar\ndevices made of a CoO thin film grown on MgO (001) sin-\ngle crystals and capped with 5 nm Pt layer. In the main text\nwe present data for the 2 nm CoO prepared by reactive DC\nmagnetron sputtering (Fig. 1). For validation purposes, other\nMBE grown CoO layers, which yielded quantitatively similar\nresults [10], were investigated. Similarly, a reference poly-\ncrystalline 5 nm Pt layer was sputtered directly on MgO and\nexamined which is presented in the supplementary part [10].\nWe focus on the analysis of the transverse resistance Rxy\nvariations as it can be measured with better accuracy due to\nthe smaller zero-field offset and smaller temperature sensi-\ntivity than the longitudinal resistance. The longitudinal data\nwas also analyzed and can be consistently described within\nthe same model [10]. Moreover, a reference device made ofarXiv:2109.00093v2  [cond-mat.mtrl-sci]  21 Nov 20222\nMgOjPt grown and fabricated in an analogous manner has\nbeen studied to rule out effects originating from the mere Pt\nlayer [10] such as the Hanle magnetoresistance [11]. All data\npresented here were collected below the Néel temperature at\nT=240K. The background signal coming likely from small\nsample misalignment with respect to the magnetic field was\nsubtracted [10].\nInitially, the expected in-plane biaxial anisotropy of the\nCoO thin film with easy axes along [110]and[110](orange\narrows in Fig. 1) is verified [9, 12]. To do this, the in-plane\nmagnetic field of 30T is applied along the expected easy axis\n[110]to set a well-defined AF state of nk[110]. Afterwards,\nthe actual field sweep from 0 to 30T is performed with the\nfixed orientation of Bk[110]and the resistance is measured\n(Fig. 2). Upon increasing the magnitude of Bthe signal is ini-\ntially constant and then an abrupt change of Rxycan be seen at\nB=7T. This significant change appears only when the mag-\nnitude of Bis increasing. Similar behavior is observed with\nthe magnetic field pointing along another easy axis (see [10]).\nThe abrupt change of Rxydoes not appear above the Néel tem-\nperature [10]. Moreover, it does not correlate with any ran-\ndom fluctuations of temperature. Finally, the probing current\ndensity does not change during the experiments, so thermal\neffects can be ruled out. Therefore, the electrical signal can\nbe attributed to the transverse SMR. The abrupt change of re-\nsistance at 7T reflects the spin-flop transition, where the Néel\nvector nchanges the orientation from BkntoB?nwhen Bis\nincreasing. The resistance change at the spin flop is consistent\nwith the negative sign of SMR [5, 9, 13]. Nonlinear behav-\nior of SMR in high fields can originate from progressive spin\ncanting and the occurrence of small net magnetic moment m\nas depicted symbolically by double-color arrows in Fig. 2.\nTo verify the qualitative understanding of the obtained re-\nsults, we construct a macrospin model. Two magnetic mo-\nments coupled antiferromagnetically are considered. The en-\nergy expression [14] can be formulated as follows:\nE(n) =2\nB2\nsf(B\u0001n)2\u0000\u0000\nn4\nx+n4\ny\u0001\n+Bank\nBan?n2\nz: (1)\nIn-plane biaxial anisotropy field and out-of-plane uniaxial\nanisotropy field are represented by Ban?andBank, respec-\ntively. The two orthogonal easy directions in-plane are de-\nnoted as x;yand correspond to [110]and[110], respectively\n(Fig. 1). The spin-flop field is denoted by Bsfand equals\nBsf=2pBexBan?[14], where Bexis the exchange field. The\nvalue of Bsfis determined from the experimental data (Fig. 2).\nBremains in the plane of the sample in all experiments. The\nspins are expected to remain in-plane due to thin-film charac-\nter of the sample [9, 12] and the out-of-plane anisotropy field\nis set Bank\u001dBan?. We perform the energy minimization with\nBsfand coordinates of Bas the only parameters, from which\nwe obtain the expected orientation of n. With the assumption\nofB\u001cBex, the magnitude of finite mperpendicular to ndue\nto the tilt of the spin sublattices can be estimated [10, 14]. The\nSMR signal can be modeled by Rxyµrnnkn?+rmmkm?. For\nFIG. 2. Transverse magnetoresistance for the CoO structure col-\nlected for Balong the [110]easy axis. Orange (black) data points\ncorrespond to the experiment with increasing (decreasing) magnetic\nfield. The order of the measurements is indicated by the digits. The\nexpected shape of the SMR signal calculated within the macrospin\nmodel for Bsf=7:0T is presented by the red curve. The thick arrows\nrepresent the directions of the spins (spin canting, angle not to scale)\nwith respect to the magnetic field (thin grey arrow). The electrical\nsignal is expressed as DRxy=R, where DRxyis the difference in resis-\ntance between a data point taken in a magnetic field and the initial\nvalue Rxy(B=0). Sheet resistance averaged over different magnetic\nfields is denoted by R.\nqualitative analysis, it is enough to approximate the ratio be-\ntween rnandrm[10]. Even when allowing for the possibility\nthat SMR probes not only the spin part but rather the total the\nangular momentum this would not influence the position of\nthe sharp resistance transition and the hysteresis. Therefore,\nwhatever component contributes to the signal, the hysteresis\nwidth would be the same for both the spin and the orbital com-\nponent.\nComparison of the model to the field sweep along [110]pre-\nsented in Fig. 2 reveals good qualitative agreement. The mod-\neled SMR marked as a red line exhibits the similar behavior\nto the measurements. The model (red solid line in Fig. 2)\nalso reproduces the monotonic increase of the SMR that can\nbe due to spin canting in high fields. It should be noted that\nthe quantitative difference between the model and data can\nbe explained by the fact that we deal with a domain struc-\nture and only part of the domains are initially oriented parallel\nto the field. Therefore, the magnitudes of the abrupt resis-\ntance change at 7T and the progressive resistance increase for\nB>7T cannot be easily captured by the simple macrospin\nmodel.\nDespite a good correlation between the model and the ex-\nperimental data for the field along an easy direction, we\nfind that the model cannot describe the measurements for\nthe magnetic field along a hard axis [10]. To clarify it, we\nperform an angular dependent magnetoresistance (ADMR)3\nmeasurements at different magnetic fields and show the data\nforB=10T and B=25T in Fig. 3 and in Fig. 4, respec-\ntively. These angle-dependent measurements determine how\nnevolves upon gradually changing the magnetic field direc-\ntion from an easy to a hard axis. In ADMR, the magnetic field\nvector has a fixed magnitude but it rotates with respect to the\ncrystalline axes. The resistance is recorded as a function of an\nangle abetween the magnetic field and [010], which is a hard\naxis. Above the Bsf=7T,nis expected to follow the direc-\ntion perpendicular to B, which is high enough to overcome the\nin-plane anisotropy [7]. This should be reflected in the SMR\nsignal that is shown as the green dashed lines in Fig. 3 and in\nFig. 4, which is the result of the macrospin model calculation\nwith experimentally determined Bsf.\nRemarkably, the experiments show a completely different\nSMR dependence. When the magnetic field changes the ori-\nentation from a=90\u000etoa=\u000090\u000eand back, the electrical\nsignal follows the same trend except for the region close to\na=0\u000e. In the vicinity of a=0\u000e, the point where the mag-\nnetic field points along the hard axis, the SMR shows a hys-\nteresis loop (Fig. 3 and Fig. 4). At higher magnetic fields, the\nhysteresis becomes narrower. The hysteresis can be charac-\nterized by its width and plotted as a function of B(Fig. 5).\nThen, it can be clearly seen that hysteresis loops appear in the\nwhole range of tested magnetic fields between Bsfup to 30T.\nThe hysteresis loops are absent above TNand thus they can\nbe attributed to the antiferromagnetic behavior of CoO. Be-\nlowBsf, no abrupt resistance changes are detectable and only\nslight residual hysteretic behavior around a=0 can be ob-\nserved that can be due to possible domain structure in the sys-\ntem [10]. The ADMR measurements reveal that for Bk[010]\nandB>Bsfdifferent states of the AF are possible depending\non the history and the system behaves as if it is always below\nthe spin-flop field. The experimental results can be understood\nas the Néel vector lagging behind the magnetic field due to the\nstrong in-plane anisotropy. However, such effect is expected\nonly below Bsf, but should be absent above the spin-flop field\nin our current understanding of the interplay between Zeeman\nenergy, antiferromagnetic exchange, and anisotropy energy,\nwhere nshould follow the direction perpendicular to the mag-\nnetic field [7].\nFirst, we check whether the existence of a domain struc-\nture can explain the experimental observations. We consider a\nset of Néel vectors that experience different anisotropy fields\nwhich reflects possible inhomogeneity of the layer that can\nresult in antiferromagnetic domains. We simulate an ADMR\nmeasurement for each case separately using equation (1) and\naverage obtained results assuming a Gaussian distribution of\nthe spin-flop fields centered around 7T [10]. We refer to this\nas a domain model [10]. As can be seen in the blue line in\nFig. 3 such a procedure can indeed result in appearance of a\nsmall hysteresis in ADMR above the spin-flop field. However,\nit completely fails to reproduce the abrupt transitions visible\nin strong magnetic fields (Fig. 4) and is therefore not appro-\npriate to explain this observation.\nTherefore, we consider another approach to interpret the ex-\nFIG. 3. The angular dependence of magnetoresistance for the CoO\nstructure collected at B=10T is depicted by the black dots. It is\nexpressed as relative changes of the resistance as a function of an an-\ngleathatBforms with the [010] crystalline direction. Green (gray)\narrows represent the spin (magnetic field) orientation with respect to\ncrystalline axes (coordinates in bottom right corner). Black arrows\ndepict the angle sweeping direction. The green dashed line displays\nmodeled behavior of the ADMR with Bsf=7T. Introduction of the\nnew energy term into the model with x=\u00004:02 can reproduce the\nhysteresis as illustrated by the red continuous curve. The blue line\nrepresents the domain model.\nFIG. 4. The angular dependence of the magnetoresistance for\nB=25T. Experimental results (black points) are compared to the\nmacrospin models with x=0 (green dashed line) and x=\u00004:02\n(red continuous curve) and the domain model (blue line). The corre-\nsponding spins (magnetic field) orientation with respect to the crys-\ntalline axes with the coordinates in bottom right corner are symbol-\nized by the thick green (gray) arrows.4\nperimental data. We assume that the magnetic field modifies\nthe magnetic anisotropy of the antiferromagnet. We describe\nthis contribution by introducing an additional, phenomenolog-\nical energy term, relevant above Bsf:\nEa=x\nB2\nsfBxBynxny (2)\ninto the equation (1) with a coefficient x. It appears only if\nthe magnetic field has nonzero components for both easy di-\nrections. Thus, it follows the symmetry of the experimental\nobservations and can describe the unexpected hysteretic be-\nhavior of the AF when a strong magnetic field is parallel to a\nhard direction. For negative x, such contribution enhances the\nenergy barrier between the states with different orientations\nof the Néel vector seen as an enhancement of the effective\nanisotropy field. We believe that this contribution originates\nfrom the field-induced tilting of the orbital momentum from\nthe easy axis; and hence we call it L-model.\nThe existence of the hysteresis loop in SMR can be repro-\nduced using the L-model with x=\u00004:02 as demonstrated by\nthe solid red lines in Fig. 3 and Fig. 4. Good agreement of\nthe hysteresis width between the model and experiments is\nobtained for all magnetic fields tested experimentally (Fig. 5).\nSlight deviations from the model that can be seen in the exper-\nimental data in Fig. 3 as a rounded shape of the hysteresis loop\nand a difference in the curvature near a=\u000645\u000ecan be inter-\npreted either as a signature of a domain structure and magnetic\ninhomogeneity of the system (these are also the features of the\nblue, domain model curve) or the contribution of the angular\nmomentum into the SMR signal. For the model with x=0,\nwhich corresponds to the original equation (1) and constant\nmagnetic anisotropy, no hysteresis above the Bsfis expected\nas indicated by green dashed lines in Fig. 3 and 4 as well as\ngreen circles in Fig. 5.\nA likely explanation for the observed hysteretic behavior\nand the additional energy term Ea(equation 2) is the un-\nquenched orbital momentum that is known to be present in\nCoO and has a strong contribution to the magnetocrystalline\nanisotropy in this material [15, 16]. It has also been observed\nto manifest itself in the magnon spectrum [17]. Furthermore,\nthe orbital part of the angular momentum Lhas already been\nshown to determine the angular magnetoresistance of CeSb,\nwhich is a 4f monopnictide with unquenched L[18]. In gen-\neral, it is observed that the orbital part can induce a wide va-\nriety of transport phenomena [19], which belong to the most\nrecent discipline of orbitronics.\nIn our study, we restrict our discussion to the simplest case\nto qualitatively justify the orbital momentum dependence on\nthe magnetic field and its influence on the anisotropy. First,\nwe note that spin–orbit coupling induces an antiparallel align-\nment of the effective L=1 orbital angular momenta L1and\nL2to the corresponding magnetic spins for each sublattices\n[17]. The external magnetic field B\u0014Bsfaligned parallel to\nthe magnetic spins does not change this configuration; the or-\nbital angular momenta are aligned antiparallel to spins with\nLx j=\u00061 (where the quantization axis xis parallel to the\nFIG. 5. Summary of the hysteresis width observed in ADMR as\na function of the magnetic field (yellow squares and circles). The\nmacrospin model with x=0 (green circles) predicts the occurrence\nof a hysteresis only slightly below Bsf, whereas setting x=\u00004:02\n(red diamonds) yields good agreement with the measurements. Lines\nare guides for the eye.\neasy axis). However, the magnetic field, oriented generically\nin the xy- plane, induces mixing of the states with the pro-\njections Ljx=\u00061, and 0. This mixing, in turn, results in a\nnonzero expectation values of quadrupolar variables, hˆLjxˆLjyi\n(j=1;2) and additional spin anisotropy Kha\nanAxy(B)nxnywith\nAxy(B) =åj=1;2hˆLjxˆLjyi.\nTo calculate the quantum states of the orbital momenta we\nintroduce the hamiltonian [17]:\nˆH=å\nj=1;2h\n\u0000KL\u0000ˆLjej\u00012+lSjˆLj+2mBBˆLji\n;(3)\nwhere KLis anisotropy of the angular momentum, the con-\nstant lparametrizes spin-orbit coupling, mBis the Bohr mag-\nneton. We assume that the orientation of the Néel vectors\nis fixed along xand the local quantization axes ejare paral-\nlel/antiparallel to the Néel vector ( e1\"#e2\"\"ˆx).\nFigure 6 shows the field dependence of Axy(B)(see supple-\nmentary material for more details). Below the critical field\n(red line), Axy(B)is close to zero, as the quadrupoles hˆL1xˆL1yi\nandhˆL2xˆL2yihave opposite signs and almost compensate each\nother. However, close to the critical field value, the magnetic\nfield is large enough to align both orbital momenta parallel\nto the magnetic field, and both quadrupoles have the same\nsign. This induces a rapid increase of Axy(B)above the criti-\ncal field with the maximum anisotropy value achieved for the\nfield parallel to the hard axis. There is a nonlinear depen-\ndence of Axy(B)on the magnetic field [10] above the critical\nfield, which we substitute with the B2relation for simplicity in\nour phenomenological energy term (equation 2). The value of\nthe critical field scales with spin–orbit coupling land corre-\nsponds to the spin-flop field BsfifBis parallel to an easy axis.\nField-induced rotation of orbital states can induce additional5\nFIG. 6. The value of the field-induced anisotropy Axy(B)\u0011\nåj=1;2hˆLjxˆLjyi(color code, dimensionless) as a function of the ori-\nentation and the magnitude of the magnetic field. The star line shows\nthe value of the threshold field Bth(a)below which the field-induced\nanisotropy is negligible. The HA and EA denote a hard and an easy\naxis, respectively.\nstrain uxyµAxythat through a magnetoelastic mechanism con-\ntributes into effective magnetic anisotropy. We conjecture that\na direct experimental confirmation of such behavior of the an-\ngular momentum could be performed, for example, by study-\ning the magnetic field induced frequency shift of the magnon\nmodes with optical excitation. We also notice a similarity of\nthe considered case to the Pashen–Back effect, in which the\nmagnetic field is much stronger than the spin-orbit coupling\nof a system [20].\nTo summarize, we report the observation of the hysteresis\nloops in the angular dependence of magnetoresistance of a\nthin film of CoO with adjacent Pt layer. The hysteretic behav-\nior is unexpectedly present above the spin-flop field and per-\nsists up to the highest tested magnetic fields (30T). It can be\ninterpreted by the dependence of the magnetic anisotropy on\nthe external magnetic field. The unquenched orbital angular\nmomentum is a likely reason for the observed effects. These\nfindings highlight the role of the anisotropy variations induced\nby the magnetic field and the role of the unquenched orbital\nmomentum in the physics of antiferromagnets and their poten-\ntial applications. The anisotropy dependence on the magnetic\nfield may modify the expected equilibrium spin state of an-\ntiferromagnets in electrical experiments that strongly rely on\nthe generation of the effective magnetic field.\nWe would like to thank Rembert Duine, Reinoud Lavri-\njsen, Michał Baj and Tomasz Dietl for helpful discussions.\nWe acknowledge Pim Lueb for VSM-SQUID measurements.\nWe acknowledge support from HFML-RU, member of the\nEuropean Magnetic Field Laboratory (EMFL). Sample fab-\nrication was performed using NanoLabNL facilities. The re-\nsearch was funded by the Dutch Research Council (NWO)\nunder Grant 680-91-113. O.G. acknowledges support from\nthe Alexander von Humboldt Foundation, the ERC SynergyGrant SC2 (No. 610115), and the Deutsche Forschungsge-\nmeinschaft (DFG, German Research Foundation) - TRR 173 –\n268565370 (project B12). This research was funded in part by\nNational Science Centre, Poland 2021/40/C/ST3/00168. For\nthe purpose of Open Access, the author has applied a CC-BY\npublic copyright licence to any Author Accepted Manuscript\n(AAM) version arising from this submission.\n\u0003Michal.Grzybowski@fuw.edu.pl\n[1] P. Wadley, B. Howells, J. Zelezny, C. Andrews, V . Hills,\nR. P. Campion, V . Novak, K. Olejnik, F. Maccherozzi, S. S.\nDhesi, S. Y . Martin, T. Wagner, J. Wunderlich, F. Freimuth,\nY . Mokrousov, J. Kune, J. S. Chauhan, M. J. Grzybowski, A. W.\nRushforth, K. W. Edmonds, B. L. Gallagher, and T. Jungwirth,\nElectrical switching of an antiferromagnet, Science 351, 587\n(2016).\n[2] T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, Antiferro-\nmagnetic spintronics, Nature Nanotechnology 11, 231 (2016).\n[3] R. Lebrun, A. Ross, S. A. Bender, A. Qaiumzadeh, L. Baldrati,\nJ. Cramer, A. Brataas, R. A. Duine, and M. Kläui, Tunable long-\ndistance spin transport in a crystalline antiferromagnetic iron\noxide, Nature 561, 222 (2018).\n[4] J. Han, P. Zhang, Z. Bi, Y . Fan, T. S. Safi, J. Xiang, J. Finley,\nL. Fu, R. Cheng, and L. Liu, Birefringence-like spin transport\nvia linearly polarized antiferromagnetic magnons, Nature Nan-\notechnology 15, 563 (2020).\n[5] G. R. Hoogeboom, A. Aqeel, T. Kuschel, T. T. M. Palstra, and\nB. J. van Wees, Negative spin Hall magnetoresistance of Pt on\nthe bulk easy-plane antiferromagnet NiO, Applied Physics Let-\nters111, 052409 (2017).\n[6] L. Baldrati, A. Ross, T. Niizeki, C. Schneider, R. Ramos,\nJ. Cramer, O. Gomonay, M. Filianina, T. Savchenko, D. Heinze,\nA. Kleibert, E. Saitoh, J. Sinova, and M. Kläui, Full angular\ndependence of the spin Hall and ordinary magnetoresistance\nin epitaxial antiferromagnetic NiO(001)/Pt thin films, Physical\nReview B 98, 024422 (2018).\n[7] S. Geprägs, M. Opel, J. Fischer, O. Gomonay, P. Schwenke,\nM. Althammer, H. Huebl, and R. Gross, Spin Hall magne-\ntoresistance in antiferromagnetic insulators, Journal of Applied\nPhysics 127, 243902 (2020).\n[8] J. Fischer, O. Gomonay, R. Schlitz, K. Ganzhorn, N. Vliet-\nstra, M. Althammer, H. Huebl, M. Opel, R. Gross, S. T. B.\nGoennenwein, and S. Geprägs, Spin Hall magnetoresistance in\nantiferromagnet/heavy-metal heterostructures, Physical Review\nB97, 014417 (2018).\n[9] L. Baldrati, C. Schmitt, O. Gomonay, R. Lebrun, R. Ramos,\nE. Saitoh, J. Sinova, and M. Kläui, Efficient Spin Torques\nin Antiferromagnetic CoO/Pt Quantified by Comparing Field-\nAnd Current-Induced Switching, Physical Review Letters 125,\n77201 (2020).\n[10] Suplementary Material.\n[11] S. Vélez, V . N. Golovach, A. Bedoya-Pinto, M. Isasa,\nE. Sagasta, M. Abadia, C. Rogero, L. E. Hueso, F. S. Bergeret,\nand F. Casanova, Hanle Magnetoresistance in Thin Metal Films\nwith Strong Spin-Orbit Coupling, Physical Review Letters 116,\n016603 (2016).\n[12] W. N. Cao, J. Li, G. Chen, J. Zhu, C. R. Hu, and Y . Z.\nWu, Temperature-dependent magnetic anisotropies in epitax-\nial Fe/CoO/MgO(001) system studied by the planar Hall effect,6\nApplied Physics Letters 98, 262506 (2011).\n[13] H. Nakayama, M. Althammer, Y .-T. Chen, K. Uchida, Y . Ka-\njiwara, D. Kikuchi, T. Ohtani, S. Geprägs, M. Opel, S. Taka-\nhashi, R. Gross, G. E. W. Bauer, S. T. B. Goennenwein, and\nE. Saitoh, Spin Hall Magnetoresistance Induced by a Nonequi-\nlibrium Proximity Effect, Physical Review Letters 110, 206601\n(2013).\n[14] H. V . Gomonay and V . M. Loktev, Spin transfer and current-\ninduced switching in antiferromagnets, Physical Review B 81,\n144427 (2010).\n[15] J. Kanamori, Theory of the Magnetic Properties of Ferrous and\nCobaltous Oxides, I, Progress of Theoretical Physics 17, 177\n(1957).\n[16] J. Kanamori, Theory of the Magnetic Properties of Ferrous and\nCobaltous Oxides, II, Progress of Theoretical Physics 17, 197(1957).\n[17] T. Satoh, R. Iida, T. Higuchi, Y . Fujii, A. Koreeda, H. Ueda,\nT. Shimura, K. Kuroda, V . I. Butrim, and B. A. Ivanov, Exci-\ntation of coupled spin–orbit dynamics in cobalt oxide by fem-\ntosecond laser pulses, Nature Communications 8, 638 (2017).\n[18] J. Xu, F. Wu, J.-K. Bao, F. Han, Z.-L. Xiao, I. Martin, Y .-Y .\nLyu, Y .-L. Wang, D. Y . Chung, M. Li, W. Zhang, J. E. Pearson,\nJ. S. Jiang, M. G. Kanatzidis, and W.-K. Kwok, Orbital-flop\nInduced Magnetoresistance Anisotropy in Rare Earth Monop-\nnictide CeSb, Nature Communications 10, 2875 (2019).\n[19] D. Go, D. Jo, H.-W. Lee, M. Kläui, and Y . Mokrousov, Orbi-\ntronics: Orbital currents in solids, EPL (Europhysics Letters)\n135, 37001 (2021).\n[20] L. D. Landau and E. M. Lifshitz, Quantum Mechanics Non-\nRelativistic Theory (Pergamon Press Ltd., 1958)."    },    {        "title": "2109.06769v2.Valence_state_determines_the_band_magnetocrystalline_anisotropy_in_2D_rare_earth_noble_metal_compounds.pdf",        "content": "arXiv:2109.06769v2  [cond-mat.mtrl-sci]  25 Jan 2022Valence state determines the band magnetocrystalline anis otropy in 2D\nrare-earth/noble-metal compounds\nM. Blanco-Rey,1,2R. Castrillo-Bodero,3K. Ali,2,3P. Gargiani,4F. Bertran,5\nP.M. Sheverdyaeva,6J.E. Ortega,7,3,2L. Fernandez,7,3and F. Schiller3,2\n1Departamento de Pol´ ımeros y Materiales Avanzados: F´ ısic a, Qu´ ımica y\nTecnolog´ ıa, Universidad del Pa´ ıs Vasco UPV-EHU, 20018 Sa n Sebasti´ an, Spain\n2Donostia International Physics Center, E-20018 Donostia- San Sebasti´ an, Spain\n3Centro de F´ ısica de Materiales CSIC/UPV-EHU-Materials Ph ysics Center, E-20018 San Sebasti´ an, Spain\n4ALBA Synchrotron Light Source, Carretera BP 1413 km 3.3, 082 90 Cerdanyola del Vall` es, Spain\n5Synchrotron SOLEIL, CNRS-CEA, L’Orme des Merisiers, Saint -Aubin-BP48, 91192 Gif-sur-Yvette, France\n6Instituto di Struttura della Materia, Consiglio Nazionale delle Ricerche, 34149 Trieste, Italy\n7Departamento de F´ ısica Aplicada I, Universidad del Pa´ ıs V asco UPV/EHU, E-20018 San Sebasti´ an, Spain\n(Dated: January 26, 2022)\nIn intermetallic compounds with zero-orbital momentum ( L= 0) the magnetic anisotropy and\nthe electronic band structure are interconnected. Here, we investigate this connection on divalent\nEu and trivalent Gd intermetallic compounds. We find by X-ray magnetic circular dichroism an\nout-of-plane easy magetization axis in 2D atom-thick EuAu 2. Angle-resolved photoemission and\ndensity-functional theory prove that this is due to strong f−dband hybridization and Eu2+valence.\nIn contrast, the easy in-plane magnetization of the structu rally-equivalent GdAu 2is ruled by spin-\norbit-split d-bands, notably Weyl nodal lines, occupied in the Gd3+state. Regardless of the Lvalue,\nwe predict a similar itinerant electron contribution to the anisotropy of analogous compounds.\nI. INTRODUCTION\nThe spin-orbit coupling (SOC) is responsible for the\nsplitting of a 4 fnorbital into multiplet states |JJz/angbracketrightde-\nfined by the total angular momentum Jand its projection\nJzon the magnetization direction. If these levels are ac-\ncessible by the crystal field energies, quenching of /angbracketleftJz/angbracketright\ncan occur. The orientation dependence of this mixing of\nstates is at the origin of the strong magnetocrystalline\nanisotropy of systems based in rare earths (RE) [1, 2]. In\nrecent years, magnets in the zero-dimensional (0D) limit\nhave been successfully realized using REs, either as iso-\nlated ad-atoms [3–7] or as centers in organic molecules [8–\n14]. Prior to these achievements it was known that\nthe 4felectron shell confers to bulk intermetallic RE\ncompounds uniaxial anisotropy and high Curie temper-\naturesTC(and consequently large coercive fields) [15],\nwhere the magnetic coupling follows a RKKY mecha-\nnism [16]. In two-dimensional (2D) rare-earth/noble-\nmetal (RE/NM) surface compounds large TCvalues are\nretained [17–22]. Anisotropy is a requisite for stable long-\nrange magnetic ordering in 2D [23–25]. Magneto-optic\nKerr effect (MOKE) and X-ray magnetic circular dichro-\nism (XMCD) studies reveal that the magnetocrystalline\nanisotropy in these compounds does not follow a com-\nmon trend. For example, the HoAu 2monolayer (ML)\ngrown on Au(111) is strongly anisotropic with out-of-\nplane (OOP) easy axis, whereas the GdAu 2one is more\neasily magnetized in-plane (IP). The HoAu 2case can be\nunderstood in terms of the5H8state given by Hund’s\nrule for 4 f10(Ho3+), since the oblate shape of the orbital\nand the surface charge distribution favour an OOP easy\naxis [9, 22]. However, explaining the GdAu 2anisotropy\ndemands an extension of the model. In fact, the 4 fshellof Gd3+is half-filled, i.e. the total orbital quantum num-\nber isL= 0 (8S7/2ground state), and the hybridization\nwith the surface bands is negligible, and therefore the\nanisotropy must have a different origin.\nAt half-filling 4 f7, classical dipolar interactions tend\nto dominate the anisotropy in bulk Gd compounds [26,\n27], yet the magnetocrystalline contribution is not fully\nquenched, as observed in metallic hcpGd [28]. The lat-\nter results from the spin polarization of the 5 dconduc-\ntion electrons, which are also subject to SOC [29, 30].\nThis leads to sizable magnetocrystalline anisotropy en-\nergy (MAE) values and non-zero orbital momenta [31].\nIn this work, we show that the magnetocrystalline\nanisotropy of RE-NM 2MLs is not defined by the 4 f\nsingle-orbital anisotropy only, but there exists an addi-\ntional term originated at the itinerant electrons. Since\nall the compounds of the RE-NM 2family display similar\nband dispersion features, the calculations presented here\nforL= 0 systems, namely EuAu 2and GdAu 2, predict\nthat the itinerant electrons contribute to the MAE with\n≈1 meV in general and that the RE valence state de-\ntermines whether this contribution favours an OOP or\nIP easy axis of magnetization. This means that, for RE\nmaterials with a large 4 fsingle orbital anisotropy, the\ntotal anisotropy will not depend on the band dispersion.\nFor other RE metals, however, the band dispersion may\ndefine the magnetic anisotropy. We show that Eu in the\nEuAu2ML on Au(111) behaves as a divalent species and\nthus it is nominally in a8S7/2ground state, as Gd3+\nin the GdAu 2ML. However, an EuAu 2ML presents an\nOOP easy axis of magnetization and MAE = 1 .6 meV, in\ncontrast to the IP easy axis observed in GdAu 2. The IP\neasy axis of GdAu 2is explained in terms of the SOC lift-\ning degeneracies in dispersive valence bands with Gd( d)\ncharacter. On the other hand, in divalent EuAu 2the2\nEu(d) states are unoccupied and do not contribute. In-\nstead, the anisotropy is caused by the strong f−dband\nhybridization.\nII. METHODS\nA. Experimental Methods\nSample preparation of EuAu 2has been carried out by\nthermal deposition of Eu onto a clean Au(111) single\ncrystalline surface. The formation of the monolayer is\nachieved when a complete layer with moir´ e is observed\nin Scanning Tunneling Microscopy (STM), or the surface\nstate emission of the Au(111) Shockley state in angle-\nresolved photoemission spectroscopy (ARPES) [32, 33]\nhas completely vanished. The substrate temperature is\naround 675 K for GdAu 2, whereas for EuAu 2a precise\ntemperature of 575 K is required. The prepared EuAu 2\nand GdAu 2ML systems reveal a√\n3×√\n3R30◦atomic\narrangement and a long range order moir´ e lattice with\nrespect to the underlaying Au(111) substrate. This su-\nperstructure is easily distinguishable in low-energy elec-\ntron diffraction (LEED) images and serves as a quality\nindicator in XMCD synchrotron preparations.\nThe EuAu 2XMCD measurements were performed at\nBoreas beamline of the Spanish synchrotron radiation fa-\ncility ALBA using a 90% circularly polarized light from a\nhelical undulator. The measurements were undertaken at\n2-20 K with a variable magnetic field up to ±6 T. The ap-\nplied magnetic field /vectorHwas aligned with the photon prop-\nagation vector. The XMCD spectrum is the difference\nbetween the two X-ray absorption spectroscopy (XAS)\nspectra recorded with opposite orientation of the mag-\nnetic field and/or the circular helicity of the light, which\nwe callµ+andµ−for simplicity. The XMCD signal is\nproportional to the projection of the magnetization in the\ndirection of the applied magnetic field. At normal light\nincidence, the field is also normal to the sample surface\n(out-of-plane geometry) while at grazing incidence (here\n70◦), the magnetic field is nearly parallel to the surface\n(in-plane geometry). The magnetization curves are taken\nby varying continuously the applied field with the sample\nkept in one of the mentioned geometries for both circular\nhelicities. For normalization issues, at each field value\ntwo XAS absorption values were taken, the first one cor-\nresponding to the photon energy of the maximum of the\nXMCD signal, the second one at a slightly lower photon\nenergy prior to the XAS absorption edge. In order to de-\ntermine the Curie temperature TCof the material such\nmagnetization curves were taken for several temperatures\nbelow and above TC.\nARPES experiments were taken at CASSIOPEE\nbeamline of SOLEIL synchrotron, France, and at our\nhome laboratory in San Sebasti´ an (Spain) together with\nthe STM and LEED analysis. Photoemission data in\nSan Sebastian were acquired using Helium I α(hν=\n21.2 eV) light. In San Sebasti´ an and at SOLEIL achannelplate-based display type hemispherical analyzer\nwas used (Specs 150 and Scienta R4000 electron analyz-\ners) with angular and energy resolution set to 0.1◦and\n40 meV, respectively. At the synchrotron, p-polarized\nlight was used and the sample temperature during mea-\nsurements was 70 K, while sample temperature during\nHelium I αARPES measurements were 120 K. Resonant\nphotoemission spectroscopy (ResPES) is achieved at the\nEu 4d→4fabsorption edge. In such measurements the\nphotoemission signal is resonantly enhanced due to a su-\nperposition/interference of the direct photoemission pro-\ncess and an Auger decay and leads to a broad resonant\nmaximum above the 4 dabsorption threshold accompa-\nnied by a number of narrow peaks caused by several decay\nprocesses that was initially explained for the 4 f7config-\nuration of Gd [34] and later for the same configuration of\nEu [35]. In mixed-valent Eu compounds slightly different\nresonant energies [36] can be used to differentiate di- and\ntri-valent contributions. Here, we used h ν= 141 eV and\n146 eV corresponding to two close photon energies in or-\nder to minimize photoemission cross-section changes but\nstill resonantly enhance di- or tri-valent signals. The off-\nresonant energy h ν= 130 eV corresponds to a simple\n(4f7)8S7/2→(4f7)7FJphotoemission transition. The\nindividual Jcomponents cannot be resolved easily at\nsuch high photon energies due to worse energy resolu-\ntion, but have been observed for pure Eu metal at low\nphoton energies [37].\nB. Theoretical Methods\nDensity-functional theory (DFT) calculations were\nperformed in the full-potential linearized augmented\nplane wave (FLAPW) formalism [38–40] at the GGA+U\nlevel to describe strong correlation [41, 42] in the full-\nlocalized limit approximation, since the Eu(4 f) orbital\nis half filled [43]. The Perdew-Burke-Ernzerhof (PBE)\nexchange and correlation functional [44] was used. The\nparameter U= 5.5 eV is found to match the 4 fbands\nbinding energies observed in ARPES. In the case of the\ncalculation parameters for GdAu 2, we refer the reader to\nprevious work in Ref. [21], where U= 7.5 eV was em-\nployed. The lattice constant of the model geometries\nwas fixed to the EuAu 2experimental value. For the sup-\nported models a√\n3×√\n3R30◦supercell geometry was\nused with fccstacking of atomic planes and an interlayer\ndistance of 2.25 ˚A between the flat EuAu 2monolayer and\nthe Au(111) substrate. For the local FLAPW basis, as\nin previous works with REAu 2, RE-6s,4f,5dand Au-\n6s,5delectrons were included as valence electrons, and\nRE-5s,5pand Au-5 pas linear orbitals. Partial wave ex-\npansions up to lmax= 8 and 10 were set inside the Eu\nand Au muffin-tin spheres of radii 1.43 and 1.48 ˚A, re-\nspectively.\nSOC was included in the calculation both self-\nconsistently and in the force theorem perturbative ap-\nproximation [45–48], i.e. without carrying out further3\nself-consistent optimization of the charge density, for\nspins oriented in-plane ( X) and out-of-plane ( Z) [see\nFig. 2(a)]. Using the force theorem, the MAE is com-\nputed as the energy diference between the band energies,\ni.e. it is a rigid-band approximation. The working prin-\nciples of the method are described in Section III E. The\nMAE accuracy is highly dependent on the fine details of\nthe band structure, specially the gap openings at band\ncrossings and the Fermi level. Therefore, a fine sampling\nof the first Brillouin zone (BZ) and the sharpest possi-\nble Fermi-Dirac function are required. In this work, the\nMAE was converged with a tolerance of ∼0.1 meV using\na smearing width σ= 5 meV for the Fermi level (first-\norder Methfessel-Paxton method [49]) and a mesh of at\nleast 25×25k-points, with plane wave expansion cutoffs\nof 4 and 12 bohr−1for the wavefunctions and potential,\nrespectively.\n5.0nmPhotoemission Intensity (arb. un.)\n-10 -8 -6 -4 -2 0Eu 4f\nh (eV)/c110\n130 (off-res.)\n136\n141 (on-res. 2 )+\n146 (on-res. 3 )+\n7FAu 5d\nE-E (eV)F(a) (b)\n/c109+/c109--/c109 /c109- +2.2\n2.0\n1.8\n1.6\n1.4\n1.2\n1.0\n1160 1140 1120-0.40.0XAS (arb. units)\nPhoton energy (eV)/c113= 0° Eu M4,5\nM5\nM4T= 3K(d)\n/c1090H (T)-1.0-0.50.00.51.0\n-6 -4 -2 0 2 4 6Eu M XMCD signal (arb. un.)5Eu M5\n/c113= 0°T= 3K\n/c113= 70°\n- (70°)/c109 /c109- +-0.50.00.520 10 0\nM (arb. un.)02Tc=13 KT (K)/c1090H= 6T) (c\nEuAu LEED2\nAu(0,0)\n1180hypothetical\nEu 4f3+Eu 4f2+\nFIG. 1. Structure, Eu valence and magnetic proper-\nties of EuAu 2.(a) STM image (U = -2V, I = 200 pA).\nInset: LEED pattern at E= 40eV energy, the red circles\nmark the Au(111) substrate spots prior to EuAu 2formation.\n(b) Resonant photoemission across the 4 d→4fabsorption\nedge determining the pure monolayer formation of divalent\nEu atoms. The final state multiplets are taken from [50, 51].\n(c) X-ray absorption at the Eu M 4,5edge with circularly left\n(µ−) and right ( µ+) polarized light at an applied field of µ0H\n= 6T and at T= 3K in normal incidence geometry ( θ= 0).\nThe difference spectrum (XMCD) is shown for normal and\ngrazing ( θ= 70◦) incidences. (d) Magnetization curves taken\nat the Eu M 5edge in these two geometries and in the inset\nthe corresponding Arrot plot analysis to determine TC.III. RESULTS AND DISCUSSION\nA. Structure Characterization\nThe EuAu 2ML is characterized by a hexagonal moir´ e\nsuperstructure visible in the scanning tunneling mi-\ncroscopy (STM) micrograph and the LEED pattern of\nFig. 1(a). Similar moir´ e superstructures are found in\nother RE/NM metal surfaces [18, 21, 22, 52]. The STM\nanalysis of EuAu 2yields a moir´ e superlattice constant\nofam= (3.3±0.1) nm with a coincidence lattice of\n(11.5±0.3)×(11.5±0.3) with respect to the Au(111)\nsurface (aAu= 0.289 nm), in agreement with LEED. The\nEuAu2ML reveals a (√\n3×√\n3)R30◦reconstruction on\ntop of the Au(111) surface and a lattice parameter of\n0.55 nm. The amof EuAu 2is 10% smaller than the\none found in GdAu 2, HoAu 2, or YbAu 2, which show\nam≈3.6 nm [18, 22].\nB. Valency Analysis by ResPES\nThe ResPES spectrum [see Fig. 1(b)] displays unequiv-\nocally the presence of a single Eu2+multiplet peak and\nno further peaks related to Eu3+revealing the presence of\nonly divalent Eu atoms exclusively located at the EuAu 2\nML. Eu atoms below the surface (either di- or trivalent)\nwould give rise to an additional shifted multiplet [36] that\ndoes not appear in the spectra. Therefore, the existence\nof a single EuAu 2ML with divalent character is probed.\nC. Magnetic Properties by XAS and XMCD\nFig. 1(c) shows the XAS and XMCD spectra of an\nEuAu2ML at the Eu M 4,5absorption edges ( T= 3 K;\nµ0H= 6 T and normal incidence). The XAS line shape\nconfirms the divalent character of the Eu atom in EuAu 2\nML deduced from the ResPES spectra of Fig. 1(b) [see\nalso Supplemental Material (SM) [53] Fig. S4 for compar-\nison to XAS spectrum of trivalent Eu 2O3]. Magnetiza-\ntion curves of Fig. 1(d) are taken at different geometries\nand at the photon energy of the maximum M 5XMCD\npeak while changing µ0H. The out-of-plane magnetiza-\ntion curves with the field applied perpendicular to the\nsurface saturates at approx. 2 T, while the in-plane mag-\nnetization curve reaches saturation at much higher ap-\nplied fields, close to 6 T. This clearly reveals that the easy\naxis of magnetization is perpendicular to the plane. Mag-\nnetization loops were recorded at various temperatures,\nallowing an Arrot plot estimation [21, 54] of the Curie\ntemperature of the surface compound of TC= 13 K [see\ninset of Fig. 1(d)]. Further details are found in Fig. S5.\nThe OOP easy axis of magnetization in EuAu 2MLs\ndiffers from the IP one of GdAu 2[20, 21], despite of the\nidentical 4 felectronic configuration (8S7/2) and atomic\nstructure. In the following, we explain this behavior in4\nterms of their respective band structure measured by\nARPES, which exhibits a clear dependence on the va-\nlence character of the REs, namely, divalent for Eu and\ntrivalent for Gd. Further analysis by DFT provides in-\nsight in the spin-orbit effects on the individual bands.\nD. Electronic Structure by ARPES and DFT\nResults of ARPES measurements performed on EuAu 2\nMLs show the characteristic surface dispersive localized\nbands with Eu( d)-Au(s,p) character that are commonly\nfound in all REAu 2surface compounds [19, 21, 22], see\nFig. 2(d,e) and SM Figs. S1 and S2. Following the nota-\ntion of Ref. [22], these valence bands are labelled A,B,C,\nandC′. As already seen in other REAu 2materials, some\nof these bands are better detected in the first BZ while\nother bands gain intensity in higher order BZ’s. The\n4femissions of Eu are seen in the binding energy range\nbetween 1.1 and 0.4 eV, in agreement with ResPES of\nFig.1(b). To better visualize the valence band structure,\na low photon energy hν= 21.2 eV was choosen, where the\n4fcross-section is relatively low. Signs of hybridization\nbetween the localized Eu 4 flevel and the EuAu 2valence\nbands are observed as kinks in the linear dispersion of the\nCbands in the second BZ and are marked with arrows\nin Fig. 2(e) (see also Fig. S2). Such 4 f-valence band\nhybridizations have been observed for other Eu-based\nbulk [55] and in YbAu 2surface compounds [22]. The C\nband in EuAu 2is characterized by a nearly conical shape\nat¯Γ, with the apex above the Fermi level EF. DFT band\nstructure calculations for isolated and supported EuAu 2\nMLs disclose the intersection above EFof theCband, of\ndxycharacter, with bands of dx2−y2anddxz,yz character,\nas shown in Fig. 2(c). These band crossings occur be-\nlow (above) the Fermi energy for trivalent (divalent) RE\nions [21, 22]. Adding the 3 ML Au(111) substrate slab\nprovide additional dispersive bands [21, 22], which cause\nan upward shift of ≈0.1 eV in the RE( d)-Au(s) hybrid\nband manifold, and a strong renormalization of 4 flev-\nels. Indeed, in the calculated free-standing EuAu 2band\nstructure the 4 fstates lie ≈0.7 eV lower. The 4 fband\nof EuAu 2is half-filled but, as the individual bands are\nheavily split by hybridization with the AandCbands,\nit shows some degree of dispersion across the BZ. In con-\ntrast, Gd(4 f) orbital in GdAu 2barely interacts with the\nsurrounding metal, showing a small crystal field splitting\nand no significant dispersion (see SM Fig. S3).\nE. Force Theorem Analysis\nSOC at the crossings of the two-dimensional RE( d)-\nAu(s) bands are sources of anisotropy that we character-\nize by DFT in the following. The SOC matrix element\nfor two electrons belonging to orbitals of the same shell\nis\n/angbracketleftlmσ|ξll·s|lm′σ′/angbracketright (1)wherelandsare the one-electron orbital and spin mo-\nmentum operators; l,mandσindices label the eigen-\nstates of l,lz, ands, respectively; and ξlis the SOC\nstrength, which is approximately constant for all the elec-\ntrons in each l-shell. Intershell SOC is negligible. For hy-\ndrogenic atomic orbitals, these matrix elements are ana-\nlytical and have been tabulated [56, 57]. Selected combi-\nnations of mvalues and spin orientations give non-zero\nmatrix elements. A second-order perturbative treatment\nof the SOC hamiltonian term shows that this behaviour\nis inherited by hybrid bands of an extended system [58–\n64]. In such case, degeneracy lifting at band crossings\nobeys the rules imposed by the crystallographic symme-\ntry of the system and the orbital symmetry of the bands\nfor a given orientation of the spins.\nIn the so-called force theorem approach, where the\nSOC correction enters non-self-consistently in the DFT\ncalculation, the MAE is obtained from the spin-\norientation-dependent band energy contribution to the\ntotal energy of the system [45, 47]:\nMAE =/summationdisplay\nknǫX\nknfFD(ǫX\nkn−EX\nF)−/summationdisplay\nknǫZ\nknfFD(ǫZ\nkn−EZ\nF)\n(2)\nwhere the sum runs over the eigenenergies ǫˆsa\nkn(kand\nnare thek-point and band indices, respectively), fFD\nis the Fermi-Dirac function, and Eˆsa\nFare the Fermi en-\nergies for the spin orientations ˆ sa=Z(out-of-plane)\nandX[the in-plane direction along nearest RE atoms,\nshown in Fig. 2(a)]. Azimuthal dependence of the MAE\nis small. According to this equation, non-zero contribu-\ntions to the MAE are generated at band crossings that\nbecome gapped for one spin orientation, but not for the\nother. The final easy-axis direction results from the inte-\ngration of all individual gap contributions, of positive and\nnegative small values, over occupied states. At crossings\nof fully occupied bands, the energy dispersion around the\ndegeneracy point has to be asymmetrical in order to have\na sizable contribution to MAE, otherwise band energies\naround the gapped feature will cancel out each other. If\nthe degeneracy lifting occurs exactly at the Fermi level,\nthe contribution to the MAE will be large [47, 65].\nA spectral analysis of Eq. 2is not possible, as the Fermi\nenergy depends on the spin orientation. Instead, the\ncommon practice is to plot MAE vs.Ne, i.e. MAE( Ne),\nwhereNeis the number of electrons with respect to the\nneutral case ( Ne= 0) calculated as an integral over the\ndensity of states of the system for each spin orientation,\ni.e. each Nevalue provides EX,Z\nF(Ne) values [64–68].\nTherefore, the MAE( Ne) curve allows to asign a positive\nor a negative MAE contribution to the region of the en-\nergy range close to EX,Z\nF(Ne). As the Fermi energy fluc-\ntuation with the spin direction is small (a few meV), we\nuse theEF(Ne) value calculated without SOC to guide\nthe eye in the graphs. The value obtained at neutrality,\nMAE(Ne= 0), corresponds to the expected anisotropy\nof the system. This type of MAE analysis relies on the\nvalidity of the force theorem approximation, namely, it5\n-2.0-1.00.01.02.0\nΚ /c71 ΜEuAu2\n0100\n4f weight (%)\nΚ /c71 ΜEuAu2/ 3ML Au\nΚ /c71/c50/c71/c50 Μ\nΚ/c71(a) )(c\nΜ-2.0-1.00.0(d)\n/c71dxydx -y2 2\nd ,\ndxz\nyz\n4fA\nBCC’\nΚ /c71/c50 ΜE-E (eV)F\n-2.0-1.00.0(e)\n/c71A\nBC C’(b)\nGdAu2 EuAu2\nxy\nzRE\nAu\n4f\nFIG. 2. Band structures of GdAu 2and EuAu 2.(a) Model of the REAu 2ML unit cell. (b) Path in the reciprocal space\nof the band structures measured by ARPES at photon energy h ν= 21.2eV for GdAu 2(d) and EuAu 2(e). (c) Band structure\nfrom DFT calculations along the indicated high-symmetry di rection for a EuAu 2monolayer, both free-standing (left) and on\n3ML Au(111) slab-supported (right). The Eu 4 fcontribution to the states is indicated in a colored scale.\nassumes that there are only minor changes in the electron\ndensity (essentially, loss of collinearity) caused by the ef-\nfect of SOC [46, 47]. Despite the two REAu 2compounds\nunder study being heavy-atom systems, by comparing to\nfully self-consistent calculations of the band structures\nwith SOC, we find that the force theorem properly ac-\ncounts for the spin-orbit induced gaps in an energy range\nof several eV around the Fermi energy (see SM Figs. S6\nand S7 for GdAu 2and EuAu 2, respectively).\n1. GdAu 2\nWe apply this analysis to the band structure of free-\nstanding and supported GdAu 2MLs, shown in Fig. 3(b-\nc). The band crossings labelled αandα′in the figures are\ndegenerate for in-plane magnetization, but split by OOP\nmagnetization. Fig. 3(a) panels shows, as a guide for the\neye, the orbital characters and spin polarities in the ab-\nsence of SOC for free-standing GdAu 2. Theαandα′\nfeatures are crossings between dxyanddx2−y2(m=±2)\nwithequal spin polarization, a situation where the matrix\nelement Eq. 1foresees splitting by OOP magnetization.\nThe Weyl nodal line β[69], which is a ring-shaped cross-\ning around the Γ point between dxyanddx2−y2bands\nwithopposite spin polarization [see Fig. 3(a)], is split by\nin-plane magnetization according to Eq. 1.\nFig. 3(d) shows the obtained MAE( Ne). The curves\nEF(Ne)−EF(0) without SOC, shown in the SM Fig. S9\nestablish an approximate mapping between band fillingsand binding energies [a few discrete values are given also\nin the right-hand side axes of Fig. 3(b,c)]. For GdAu 2\nclose to neutrality ( Ne= 0) the essential contributions\nto the MAE are those of α,α′andβfeatures. In the\nfree-standing case, the βnodal ring results in the net\nsharp MAE peak of negative values in the dashed red\nline atNe=−0.9 (i.e. contributing to IP easy axis of\nmagnetization), whereas features αandα′contribute to\nthe OOP easy magnetization axis with peaks of positive\nMAE at Ne=−1 and just above Ne= 0 in the red\ndashed curve of Fig. 3(d). At neutrality we obtain an\nOOP easy axis with MAE(0) = 0 .17 meV.\nThe same features α,α′andβappear for GdAu 2/3 ML\nAu(111), albeit at binding energies higher by ∼0.1 eV\n[see Figs. 3(b,c)], which also result in positive and neg-\native contributions to the MAE around the neutrality\npoint. The βline causes the in-plane peak at Ne=−0.5\n(solid blue line), and αandα′contribute to the OOP\nmagnetization easy axis with positive-valued MAE peaks\natNe=−1.3 and 0.8, respectively. At the neutrality\npoint we find MAE(0) = −0.9 meV, i.e. IP easy axis of\nmagnetization, same as observed experimentally. This\nis due to the contribution of the βgapped line, which\ndominates over αandα′single-point gaps at Γ, aided\nby the fact that the βgapped line lies closer to EFin\nthe supported case. The contribution of the GdAu 24f7\nelectrons to the anisotropy lies within the force theorem\nerror, since the departure from sphericity of this orbital\nis negligible when embedded in the alloy (in other words,\nthe Gd(4 f) spin density cloud would be almost insen-6\nΜ Γ Κ-1.0-0.9-0.700.51.5Ne (b) GdAu2\nX\nZα\nα'β β\n-0.8-0.40.00.4\nΜ Γ Κ-3.1-1.3-0.500.82.3Ne (c) GdAu2/ 3ML Au\nX\nZα\nα'\n-0.8-0.40.00.4\nΜ Γ Κ(a) GdAu2(no SOC)\nα\nα'β βd↑\nx2-y2d↓\nx2-y2\nd↑\nxyd↓\nxy\n-4.0-2.00.02.0\n-4-3-2-1012MAE (meV)\nNe(d)\nGdAu2/ 3ML Au\nGdAu2\nα'\nβα\nE-EF(eV)\nE-EF(eV)\nβ β\nFIG. 3. Band structure and MAE of GdAu 2as a function of band filling. (a) Orbital and spin characters of free-\nstanding GdAu 2bands in the absence of SOC. Band structure of a free-standin g (b) and supported (c) GdAu 2ML for spins\naligned to the X (violet) and Z (green) directions in the forc e theorem approximation, each calculation referred to its o wn\nFermi level. The SOC-induced gaps at the band crossings rele vant to the MAE are indicated with circles labelled α,α′andβ.\nThe right-hand side axes show the corresponding band filling , i.e. the number of electrons Nereferred to charge neutrality. (d)\nMAE as a function of Nefor free-standing (solid) and supported (dashed) GdAu 2. Positive (negative) MAE values mean OOP\n(IP) easy magnetization axis.\nsitive to the applied field orientation, as shown in SM\nFig. S3). Therefore, we attribute the experimental easy\nmagnetization plane essentially to the MAE contribution\nof the dispersive Gd( d)-Au(s) bands.\n2. EuAu 2\nNext, we apply the force theorem methodology to the\nEuAu2bands. Bare inspection of the band structures\nof both free-standing and supported EuAu 2MLs rules\nout the possibility of the anisotropy being dominated by\nthe SOC effects on the Eu( d)-Au(s) hybrid bands for two\nreasons: (i) due to the divalent character of Eu, the dxy\nanddx2−y2band crossings relevant for MAE are unoc-\ncupied [see Fig. 2(c)] and thus they cannot contribute\nto the observed anisotropy, and (ii) the f-band is heav-\nily broadened by hybridization, with individual bands\nshowing significant dispersion and spin orientation de-\npendence [see SM Fig. S3(d)].\nThe force theorem Eq. 2applied to the U= 5.5 eV\nband structures yields an OOP easy magnetization\naxis both for free-standing and supported EuAu 2, with\nMAE(0) = 1 .0 and 1.54 meV, respectively [these are, re-\nspectively, the values taken by the thick red dashed and\nblue solid curves, at neutrality in Fig. 4(a)]. The MAE\npeaks due to α′,βandαband crossing features are visible\nin the free-standing EuAu 2bands above the charge neu-\ntrality level (see also Fig. S8). For EuAu 2/3 ML Au(111),\nFig.4(b) shows two large broad peaks at each MAE( Ne)\ncurve, which take values ≃ ±0.1 eV at the electron fill-\ning ranges that correspond to the binding energy ranges\nwhere the hybridized 4 fbands lie. The peak pairs ac-\ncount for the filling of the SOC-split band manifolds 4 f7/2\nand 4f5/2. In the calculation with U= 5.5 eV, these\npeaks occur at Ne=−10 and−5, respectively, corre-sponding to binding energies close to -1 eV [see also SM\nFig. S9(b,c)]. These bands yield a non-negligible pos-\nitive residual contribution to MAE(0) by virtue of the\nf−dhybridization. In a single-ion picture, this would\nbe interpreted as a loss of sphericity of the half-filled 4 f\nshell, which acquires a net non-zero /angbracketleftL/angbracketright[9]. The SM\nFig. S3 shows the magnetization distribution projected\non ˆsa[mˆsa(r)] of the Eu(4 f) shell embedded in the free-\nstanding alloy monolayer, obtained by integration of the\nKohn-Sham states in the binding energy range between\n-2.25 and -1.25 eV. The magnetization anisotropy distri-\nbutionmX(r)−mZ(r) takes positive and negative values\nin-plane and out-of-plane, respectively, which can be in-\nterpreted as the Eu( f) orbital spin density tendency to\nbecome deformed along the applied field direction. Note\nthat the same quantity integrated for Gd( f) between -10\nand -8.5 eV is an order of magnitude smaller.\nThe 4fcontribution to the MAE of EuAu 2can be fur-\nther probed by modifying the Ucorrelation parameter.\nWe have carried out force theorem analyses for supported\nEuAu2withU= 3.5 and 7.5 eV. When Uis decreased,\nthe MAE energy is increased by ≃1 meV and when Uis\nincreased, the easy axis of magnetization remains out-of-\nplane, but the MAE(0) value is reduced (see Fig. 4). The\ndensities of states [SM Fig. S9(d)] show that a larger U\nvalue implies less hybridization between the 4 felectrons\nand the hybrid bands of dcharacter crossing the Fermi\nlevel. Indeed, for U= 7.5 eV the 4 fband lies below\nEF−1.5 eV, i.e. close to the Au substrate states and be-\nlow thedxyband. For U= 3.5 eV, however, the 4 fband\nis centred at EF−0.5 eV, producing hybrid f−dstates\nclose to the Fermi level. This shows that the degree of\nf−dhybridization drives the MAE behaviour of EuAu 2.7\n-2024\n-4 -3 -2 -1 0 1 2(a)\nMAE (meV)\nNeEuAu2/ 3ML Au:\nU = 3.5 eV\nU = 5.5 eV\nU = 7.5 eV\nEuAu2: U = 5.5 eVα'\nβα\n-2002040\n-20 -15 -10 -5 0 5(b)\nMAE (meV)\nNe4f7/24f5/2\nFIG. 4.MAE of EuAu 2as a function of band filling (a)\nMAE as a function of Nefor free-standing (solid blue) and\nsupported (dashed red) EuAu 2calculated with U= 5.5eV\nclose to the neutrality condition Ne= 0. Positive (negative)\nMAE values contribute to the OOP (IP) easy magnetization\naxis. Additionalsolid curvesshowtheresultsforEuAu 2/3ML\nAu(111) with other Uvalues. (b) Same curves in a wider Ne\nto show the large contributions of 4 felectrons, switching the\neasy axis of magnetization from IP (negative MAE) to OOP\n(positive MAE). The rectangle corresponds to the close-up\nshown in panel (a).\nIV. CONCLUSIONS\nTo summarize, our XMCD experiments have deter-\nmined that a EuAu 2ML on Au(111) shows out-of-plane\neasy magnetization axis. Using photoemission experi-\nments combined with DFT calculations, we show that\nthis is a consequence of the Eu2+valence state and the\nf−dband hybridization. In contrast, the easy magneti-\nzation plane of GdAu 2ML is explained by Gd3+valence\nand spin-orbit splitting of the point and line degenera-\ncies of bands with dxy,x2−y2character. The role of the RE\nvalence is to determine whether these specific states are\noccupied or empty. This model of the contribution of the\nitinerant electrons to the magnetocrystalline anisotropyis a general result for the REAu 2family of monolayer\ncompounds. For RE=Eu,Gd this is the only mechanisms\nat work, since the 4 forbital is half-filled, i.e. L= 0. For\nRE atoms with non-zero Lquantum numbers, there is a\nsingle-ion anisotropy that results from the interplay be-\ntween spin-orbit and crystal-field splittings of the 4 fmul-\ntiplet. For example, this multiplet mechanism dominates\nthe OOP easy magnetization axis behaviour in HoAu 2,\nwhere the valence is Ho3+[22]. We predict the itiner-\nant electrons to yield MAE values of ≈1 meV in REAu 2\nmonolayers, irrespective of the Lvalue. Therefore, the\nband and multiplet mechanisms may eventually compete.\nACKNOWLEDGMENTS\nDiscussions with the late J.I. Cerd´ a are\nwarmly thanked. Financial support from\nprojects MAT-2017-88374-P, PID2020-116093RB-\nC44 and PID2019-103910GB-I00, funded by\nMCIN/AEI/10.13039/501100011033/, the Basque\nGovernment (grants IT-1255-19, IT1260-19), and the\nUniversity of the Basque Country (UPV/EHU) (grant\nGIU18/138) is acknowledged. Computational resources\nwere provided by DIPC. The SOLEIL based research\nleading to the results has been supported by the project\nCALIPSOplus under Grant Agreement 730872 from the\nEU Framework Programme for Research and Innovation\nHORIZON 2020. L.F. acknowledges financial support\nfrom the EU’s Horizon 2020 research and innovation\nprogramme under the Marie Sk/suppress lodowska-Curie grant\nagreement MagicFACE No 797109.8\n[1]K. H. J. Buschow and F. de Boer, Physics of Magnetism\nand Magnetic Materials (Kluwer Academic / Plenum\nPublishers, New York, 2003).\n[2]R. Skomski and D. Sellmyer,\nJournal of Rare Earths 27, 675 (2009) .\n[3]F. Donati, A. Singha, S. Stepanow, C. W¨ ackerlin,\nJ. Dreiser, P. Gambardella, S. Rusponi, and H. Brune,\nPhys. Rev. Lett. 113, 237201 (2014) .\n[4]F. Donati, S. Rusponi, S. Stepanow, C. W¨ ackerlin,\nA. Singha, L. Persichetti, R. Baltic, K. Diller, F. Patthey,\nE. Fernandes, J. Dreiser, ˇZ.ˇSljivanˇ canin, K. Kum-\nmer, C. Nistor, P. Gambardella, and H. Brune,\nScience352, 318 (2016) .\n[5]A. Singha, R. Baltic, F. Donati, C. W¨ ackerlin, J. Dreiser,\nL. Persichetti, S. Stepanow, P. Gambardella, S. Rusponi,\nand H. Brune, Phys. Rev. B 96, 224418 (2017) .\n[6]F. D. Natterer, F. Donati, F. m. c. Patthey, and\nH. Brune, Phys. Rev. Lett. 121, 027201 (2018) .\n[7]F. Donati, S. Rusponi, S. Stepanow, L. Persichetti,\nA. Singha, D. M. Juraschek, C. W¨ ackerlin, R. Baltic,\nM. Pivetta, K. Diller, C. Nistor, J. Dreiser, K. Kummer,\nE. Velez-Fort, N. A. Spaldin, H. Brune, and P. Gam-\nbardella, Phys. Rev. Lett. 124, 077204 (2020) .\n[8]N. Ishikawa, M. Sugita, and W. Wernsdorfer,\nAngewandte Chemie International Edition 44, 2931 (2005) .\n[9]J. D. Rinehart and J. R. Long,\nChem. Sci. 2, 2078 (2011) .\n[10]M. J. Mart´ ınez-P´ erez, S. Cardona-Serra, C. Schlegel,\nF. Moro, P. J. Alonso, H. Prima-Garc´ ıa, J. M.\nClemente-Juan, M. Evangelisti, A. Gaita-Ari˜ no,\nJ. Ses´ e, J. van Slageren, E. Coronado, and F. Luis,\nPhys. Rev. Lett. 108, 247213 (2012) .\n[11]M. Mannini, F. Bertani, C. Tudisco, L. Mala-\nvolti, L. Poggini, K. Misztal, D. Menozzi,\nA. Motta, E. Otero, P. Ohresser, P. Sainctavit,\nG. G. Condorelli, E. Dalcanale, and R. Sessoli,\nNature Communications 5, 4582 (2014) .\n[12]B. W. Heinrich, L. Braun, J. I. Pascual, and K. J. Franke,\nNano Letters 15, 4024 (2015) .\n[13]A.Chiesa, F. Cugini, R. Hussain, E. Macaluso, G. Allodi,\nE. Garlatti, M. Giansiracusa, C. A. P. Goodwin, F. Ortu,\nD. Reta, J. M. Skelton, T. Guidi, P. Santini, M. Solzi,\nR. De Renzi, D. P. Mills, N. F. Chilton, and S. Carretta,\nPhys. Rev. B 101, 174402 (2020) .\n[14]M. Briganti, E. Lucaccini, L. Chelazzi, S. Ciat-\ntini, L. Sorace, R. Sessoli, F. Totti, and M. Perfetti,\nJournal of the American Chemical Society 143, 8108 (2021) .\n[15]Givord, D., Laforest, J., Li, H. S., Li´ enard,\nA., de la Bˆ athie, R. Perrier, and Tenaud, P.,\nJ. Phys. Colloques 46, C6 (1985) .\n[16]K.Buschow, Reports on Progress in Physics 42, 1373 (1979) .\n[17]M. Corso, M. J. Verstraete, F. Schiller, M. Ormaza,\nL. Fern´ andez, T. Greber, M. Torrent, A. Rubio, and J. E.\nOrtega,Phys. Rev. Lett. 105, 016101 (2010) .\n[18]M. Corso, L. Fern´ andez, F. Schiller, and J. E. Ortega,\nACS Nano 4, 1603 (2010) .\n[19]M. Ormaza, L. Fern´ andez, S. Lafuente, M. Corso,\nF. Schiller, B. Xu, M. Diakhate, M. J. Verstraete, and\nJ. E. Ortega, Phys. Rev. B 88, 125405 (2013) .\n[20]L. Fern´ andez, M. Blanco-Rey, M. Ilyn, L. Vi-\ntali, A. Maga˜ na, A. Correa, P. Ohresser,J. E. Ortega, A. Ayuela, and F. Schiller,\nNano Letters 14, 2977 (2014) .\n[21]M. Ormaza, L. Fern´ andez, M. Ilyn, A. Maga˜ na,\nB. Xu, M. J. Verstraete, M. Gastaldo, M. A. Val-\nbuena, P. Gargiani, A. Mugarza, A. Ayuela, L. Vi-\ntali, M. Blanco-Rey, F. Schiller, and J. E. Ortega,\nNano Letters 16, 4230 (2016) .\n[22]L. Fernandez, M. Blanco-Rey, R. Castrillo-Bodero,\nM. Ilyn, K. Ali, E. Turco, M. Corso, M. Or-\nmaza, P. Gargiani, M. A. Valbuena, A. Mugarza,\nP. Moras, P. M. Sheverdyaeva, A. K. Kundu, M. Ju-\ngovac, C. Laubschat, J. E. Ortega, and F. Schiller,\nNanoscale 12, 22258 (2020) .\n[23]N. D. Mermin and H. Wagner,\nPhys. Rev. Lett. 17, 1133 (1966) .\n[24]V. Berezinskii, Soviet Physics JETP-USSR 32, 493+\n(1971).\n[25]J. M. Kosterlitz and D. J. Thouless,\nJournal of Physics C: Solid State Physics 6, 1181 (1973) .\n[26]D. J. W. Geldart, P. Hargraves, N. M. Fujiki, and R. A.\nDunlap, Phys. Rev. Lett. 62, 2728 (1989) .\n[27]M. Rotter, M. Loewenhaupt, M. Doerr, A. Lind-\nbaum, H. Sassik, K. Ziebeck, and B. Beuneu,\nPhys. Rev. B 68, 144418 (2003) .\n[28]J. J. M. Franse and R. Gersdorf,\nPhys. Rev. Lett. 45, 50 (1980) .\n[29]M. Wietstruk, A. Melnikov, C. Stamm,\nT. Kachel, N. Pontius, M. Sultan, C. Gahl,\nM. Weinelt, H. A. D¨ urr, and U. Bovensiepen,\nPhys. Rev. Lett. 106, 127401 (2011) .\n[30]M. Colarieti-Tosti, S. I. Simak, R. Ahuja, L. Nordstr¨ om,\nO. Eriksson, D. ˚Aberg, S. Edvardsson, and M. S. S.\nBrooks, Phys. Rev. Lett. 91, 157201 (2003) .\n[31]H. Jang, B. Y. Kang, B. K. Cho, M. Hashimoto,\nD. Lu, C. A. Burns, C.-C. Kao, and J.-S. Lee,\nPhys. Rev. Lett. 117, 216404 (2016) .\n[32]F. Reinert, G. Nicolay, S. Schmidt, D. Ehm, and\nS. H¨ ufner, Phys. Rev. B 63, 115415 (2001) .\n[33]G. Nicolay, F. Reinert, S. H¨ ufner, and P. Blaha,\nPhys. Rev. B 65, 033407 (2001) .\n[34]F. Gerken, J. Barth, and C. Kunz,\nPhys. Rev. Lett. 47, 993 (1981) .\n[35]O.-P. Sairanen and S. Aksela,\nJournal of Physics: Condensed Matter 4, 3337 (1992) .\n[36]W.-D. Schneider, C. Laubschat, and B. Reihl,\nPhys. Rev. B 27, 6538 (1983) .\n[37]G. Kaindl, A. H¨ ohr, E. Weschke, S. Vandr´ e,\nC. Sch¨ ußler-Langeheine, and C. Laubschat,\nPhys. Rev. B 51, 7920 (1995) .\n[38]H. Krakauer, M. Posternak, and A. J. Freeman,\nPhys. Rev. B 19, 1706 (1979) .\n[39]E. Wimmer, H. Krakauer, M. Weinert, and A. J. Free-\nman,Phys. Rev. B 24, 864 (1981) .\n[40]Fleursite:http://www.flapw.de .\n[41]V. I. Anisimov, F. Aryasetiawan, and A. I. Lichtenstein,\nJournal of Physics: Condensed Matter 9, 767 (1997) .\n[42]A. B. Shick, A. I. Liechtenstein, and W. E. Pickett,\nPhys. Rev. B 60, 10763 (1999) .\n[43]V. I. Anisimov, I. V. Solovyev, M. A. Ko-\nrotin, M. T. Czy˙ zyk, and G. A. Sawatzky,\nPhys. Rev. B 48, 16929 (1993) .9\n[44]J. P. Perdew, K. Burke, and M. Ernzerhof,\nPhys. Rev. Lett. 77, 3865 (1996) .\n[45]M. Weinert, R. E. Watson, and J. W. Davenport,\nPhys. Rev. B 32, 2115 (1985) .\n[46]C. Li, A. J. Freeman, H. J. F. Jansen, and C. L. Fu,\nPhys. Rev. B 42, 5433 (1990) .\n[47]G. H. O. Daalderop, P. J. Kelly, and M. F. H. Schuur-\nmans,Phys. Rev. B 41, 11919 (1990) .\n[48]D.-s. Wang, R. Wu, and A. J. Freeman,\nPhys. Rev. B 47, 14932 (1993) .\n[49]M. Methfessel and A. T. Paxton,\nPhys. Rev. B 40, 3616 (1989) .\n[50]F.Gerken, Journal of Physics F: Metal Physics 13, 703 (1983) .\n[51]W. D. Schneider, C. Laubschat, G. Kalkowski, J. Haase,\nand A. Puschmann, Phys. Rev. B 28, 2017 (1983) .\n[52]Y. Que, Y. Zhuang, Z. Liu, C. Xu,\nB. Liu, K. Wang, S. Du, and X. Xiao,\nThe Journal of Physical Chemistry Letters 11, 4107 (2020) .\n[53]See Supplemental Material at (URL to be provided) ,\nwhich includes additional Refs. [70–72], for further de-\ntails.\n[54]A. Arrott, Phys. Rev. 108, 1394 (1957) .\n[55]S. Danzenb¨ acher, D. V. Vyalikh, Y. Kucherenko,\nA. Kade, C. Laubschat, N. Caroca-Canales, C. Krell-\nner, C. Geibel, A. V. Fedorov, D. S. Dessau,\nR. Follath, W. Eberhardt, and S. L. Molodtsov,\nPhys. Rev. Lett. 102, 026403 (2009) .\n[56]E. Abate and M. Asdente,\nPhys. Rev. 140, A1303 (1965) .\n[57]C. Elsasser, M. Fahnle, E. H. Brandt, and M. C. Bohm,\nJournal of Physics F: Metal Physics 18, 2463 (1988) .\n[58]H. Takayama, K.-P. Bohnen, and P. Fulde,\nPhys. Rev. B 14, 2287 (1976) .\n[59]P. Bruno, Phys. Rev. B 39, 865 (1989) .\n[60]M. Cinal, D. M. Edwards, and J. Mathon,\nPhys. Rev. B 50, 3754 (1994) .[61]G.vanderLaan, Journal of Physics: Condensed Matter 10, 3239 (1998) .\n[62]L. Ke and M. van Schilfgaarde,\nPhys. Rev. B 92, 014423 (2015) .\n[63]O.ˇSipr, S. Mankovsky, S. Polesya, S. Bornemann,\nJ. Min´ ar, and H. Ebert, Phys. Rev. B 93, 174409 (2016) .\n[64]M. Blanco-Rey, J. I. Cerd´ a, and A. Arnau,\nNew Journal of Physics 21, 073054 (2019) .\n[65]G. H. O. Daalderop, P. J. Kelly, and M. F. H. Schuur-\nmans,Phys. Rev. B 50, 9989 (1994) .\n[66]T. Moos, W. H¨ ubner, and K. Bennemann,\nSolid State Communications 98, 639 (1996) .\n[67]A. Lessard, T. H. Moos, and W. H¨ ubner,\nPhys. Rev. B 56, 2594 (1997) .\n[68]P. Ravindran, A. Kjekshus, H. Fjellv˚ ag, P. James,\nL. Nordstr¨ om, B. Johansson, and O. Eriksson,\nPhys. Rev. B 63, 144409 (2001) .\n[69]B. Feng, R.-W. Zhang, Y. Feng, B. Fu,\nS. Wu, K. Miyamoto, S. He, L. Chen,\nK. Wu, K. Shimada, T. Okuda, and Y. Yao,\nPhys. Rev. Lett. 123, 116401 (2019) .\n[70]B. T. Thole, G. van der Laan, J. C. Fuggle,\nG. A. Sawatzky, R. C. Karnatak, and J.-M. Esteva,\nPhys. Rev. B 32, 5107 (1985) .\n[71]T. Kinoshita, H. Gunasekara, Y. Takata, S.-\ni. Kimura, M. Okuno, Y. Haruyama, N. Ko-\nsugi, K. Nath, H. Wada, A. Mitsuda, M. Shiga,\nT. Okuda, A. Harasawa, H. Ogasawara, and A. Kotani,\nJournal of the Physical Society of Japan 71, 148 (2002) .\n[72]A. Tcakaev, V. B. Zabolotnyy, C. I. Fornari,\nP. R¨ ußmann, T. R. F. Peixoto, F. Stier, M. Dettbarn,\nP. Kagerer, E. Weschke, E. Schierle, P. Bencok, P. H. O.\nRappl, E. Abramof, H. Bentmann, E. Goering, F. Rein-\nert, and V. Hinkov, Phys. Rev. B 102, 184401 (2020) .arXiv:2109.06769v2  [cond-mat.mtrl-sci]  25 Jan 2022Supplemental Material for: “Valence state determines the b and magnetocrystalline\nanisotropy in 2D rare-earth/noble-metal compounds”\nM. Blanco-Rey,1,2R. Castrillo-Bodero,3K. Ali,2,3P. Gargiani,4F. Bertran,5\nP. M. Sheverdyaeva,6J.E. Ortega,2,3,7L. Fernandez,3,7and F. Schiller2,3\n1Departamento de Pol´ ımeros y Materiales Avanzados: F´ ısic a, Qu´ ımica y\nTecnolog´ ıa, Universidad del Pa´ ıs Vasco UPV-EHU, 20018 Sa n Sebasti´ an, Spain\n2Donostia International Physics Center, E-20018 Donostia- San Sebasti´ an, Spain\n3Centro de F´ ısica de Materiales CSIC/UPV-EHU-Materials Ph ysics Center, E-20018 San Sebasti´ an, Spain\n4ALBA Synchrotron Light Source, Carretera BP 1413 km 3.3, 082 90 Cerdanyola del Vall` es, Spain\n5Synchrotron SOLEIL, CNRS-CEA, L’Orme des Merisiers, Saint -Aubin-BP48, 91192 Gif-sur-Yvette, France\n6Instituto di Struttura della Materia, Consiglio Nazionale delle Ricerche, 34149 Trieste, Italy\n7Departamento de F´ ısica Aplicada I, Universidad del Pa´ ıs V asco UPV/EHU, E-20018 San Sebasti´ an, Spain\n(Dated: January 26, 2022)\nS1. ADDITIONAL PHOTOEMISSION RESULTS\n-2.0-1.5-1.0-0.50.0\n1.6 1.2 0.8 0.4 0.0 -0.4\n-0.4-0.20.00.20.4\n1.6 1.2 0.8 0.4 0.0 -0.4E = EF\nk (Å )||,x-1k (Å )||,y-1\nE-E (eV)F\n/c71 /c71/c50\n-2.0-1.5-1.0-0.50.0\n-0.4 -0.2 0.0 0.2 0.4/c71/c50\nk (Å )||,y-1/c77/c75\n/c75/c71 /c77/c75/c75/c50/c77/c50\n/c71/c50EuAu2\nFIG. S1. ARPES intensity mappings of EuAu 2at hν= 21.2eV. Fermi energy intensity plot (Fermi surface mapping)\nof the EuAu 2surface and band structure cuts along the ¯Γ¯M¯Γ2and¯K2¯Γ2¯K2surface Brillouin zone directions.S2\n-2.0-1.5-1.0-0.50.0\n-0.4 0.0 0.4\n-2.0-1.5-1.0-0.50.0\n-0.4 0.0 0.4\n-2.0-1.5-1.0-0.50.0\n-0.8 -0.4 0.0 0.4 0.8\n-2.0-1.5-1.0-0.50.0\n-0.8 -0.4 0.0 0.4 0.8\n-2.0-1.5-1.0-0.50.0\n-0.8 -0.4 0.0 0.4 0.8E-E (eV)F/c71\n/c77 /c77\nk||(Å )-1h =\n26eV/c110\nh =\n31eV/c110\nh =\n41eV/c110\nh =\n44eV/c110\nh =\n50eV/c110\nFIG. S2. ARPES intensity plots along the ¯M¯Γ¯Mhigh symmetry direction in EuAu 2.Photoemission intensity\nmappings along the ¯Γ2¯M¯Γ¯M¯Γ2high symmetry line. The yellow circle at -0.27 ˚A and -1.2eV reveal that the observed kink in\nthe C band does not disperse for different photon energies.S3\nFIG. S3. Electronic structure of the 4fstates in free-standing GdAu 2and EuAu 2.(a,b) Off- and on-resonant\nphotoemission of GdAu 2and EuAu 2, respectively, obtained at the indicated photon energies. The indicated multiplets are\ntaken from Refs. [1, 2]. (c,d) Detail of the self-consistent ly calculated spin-orbit-split 4 fbands of free-standing GdAu 2(c) and\nEuAu 2(d) for spins oriented along the X(purple) or Z(green) directions. Note the contrast between the weak disp ersion\nand strong hybridization of the GdAu 2and EuAu 2cases, respectively. (e,g) Real-space representations of the RE(4 f) charge\ndensityρ(r), obtained from the integration of the Kohn-Sham states wit h spins along the Zdirection shown in panels (c) and\n(d), respectively. The depicted isoelectronic surface is 0 .1ea−3\n0. In both atoms, the deviation from sphericity are negligibl e\nby bare eye. The next quantity can, in contrast, probe the asp hericity of the orbital with more precision. (f,h) Real-spa ce\nrepresentations of the anisotropy of the 4 fmagnetization density, calculated as ∆ m(r) =mX(r)−mZ(r), with red (blue)\nindicating ∆ m(r)>0 (<0). The depicted isosurfaces in this case are 0.0001 ea−3\n0for Gd(4 f) (g) and 0.001 ea−3\n0for Eu(4 f)\n(h). The mGd(r) sensitivity to the magnetization axis is smaller than that ofmEu(r) by an order of magnitude, which means\nthat Eu(4 f) is more prone to be deformed by a magnetic field, i.e. it can pr ovide non-negligible contributions to the MAE by\ndeparture of its spherical shape, as shown in the main paper F ig. 4.S4\nS2. ADDITIONAL XAS RESULTS\n1.8\n1.6\n1.4\n1.2\n1.0XAS (arb. units)\n1180 1170 1160 1150 1140 1130 1120\nPhoton energy (eV) EuAu2\n Eu2O3\nFIG. S4. X-ray absorption spectra taken with linear horizontal pola rized light at normal incidence geometry\nfor divalent EuAu 2and trivalent Eu 2O3.X-ray absorption spectroscopy results can be used to easily distinguish di- and\ntri-valent Eu atoms in compounds. Here we show the spectra of completely divalent Eu in EuAu 2and trivalent Eu in Eu 2O3.\nFor comparison, theoretical and experimental spectra of di valent Eu atoms in different compounds can be found in Refs. [3 –5].S5\n/c109+/c109--/c109 /c109- +2.2\n2.0\n1.8\n1.6\n1.4\n1.2\n1.0\n1160 1140 1120-0.40.0XAS (arb. units)\nPhoton energy (eV)/c113= 0° Eu M4,5\nM5\nM4T= 3K(a) (b)\n/c1090H (T)-1.0-0.50.00.51.0\n-6 -4 -2 0 2 4 6Eu M XMCD signal (arb. un.)5Eu M5\n/c113= 0°T= 3K\n/c113= 70°\n- (70°)/c109 /c109- +-0.50.00.520 10 0\nM (arb. un.)02Tc=13 KT (K)/c1090H= 6T\n1.6\n1.4\n1.2\n1.0\n1240 1220 1200 1180-0.4-0.3-0.2-0.10.00.1\n/c109+/c109--/c109 /c109- +/c113= 0° Gd M4,5\nM5\nM4T= 2K\n- (60°)/c109 /c109- +/c1090H= 6T\n-6 -4 -2 0 2 4 62\n1\n0\n-130 20 10\nM (arb. un.)02T (K)Gd M5\n/c113= 0°T= 2K\n/c113= 60°\nTc=19 K\nPhoton energy (eV) /c1090H (T)-1.0-0.50.00.51.0\nGd M XMCD signal (arb. un.)5) (d) (c\nFIG. S5. Comparison of the magnetic properties of EuAu 2and GdAu 2.(a),(c) X-ray absorption spectra taken with\nleft and right polarized light at T= 3K and with an applied field of µ0H= 6T with the field applied normal to the surface for\nEuAu 2and GdAu 2, respectively. The resulting difference, the XMCD signals, are shown in the bottom part for fields normal\nto the surface θ= 0◦, atθ= 70◦and 60◦, respectively. (b),(d) The magnetization curves are obtai ned by acquisition of the\nXAS signal at the maximum of the XMCD signal (and a pre-edge va lue for normalization) varying continually the applied fiel d\nfrom +6T to -6T and then from -6T to +6T for both circular light polarizations. The Curie temperature TCcan be extracted\nby the Arrott plot analysis [6, 7] of magnetization curves ta ken at several temperatures around TC. As a result we obtain TC=\n13K and 19K for the EuAu 2and GdAu 2surface compounds, respectively. Most important here is th e change in the anisotropy,\nresulting in an out-of-plane easy axis of magnetuzation for Eu atoms in EuAu 2and an in-plane easy axis for Gd in GdAu 2.S6\nS3. TESTS ON THE VALIDITY OF THE FORCE THEOREM APPROACH\n-3-2-1 0 1\nΜ Γ ΚE-EF (eV)GdAu2 / 3ML Au; SOC X\n FT\n SCF\n-3-2-1 0 1\nΜ Γ ΚE-EF (eV)GdAu2 / 3ML Au; SOC Z\n FT\n SCF-3-2-1 0 1\nΜ Γ ΚE-EF (eV)GdAu2; SOC X\n FT\n SCF\n-3-2-1 0 1\nΜ Γ ΚE-EF (eV)GdAu2; SOC Z\n FT\n SCF\nFIG. S6. Self-consistent and force-theorem bands for free-standin g and supported GdAu 2structures. Band\nstructures of free-standing and supported GdAu 2are shwon for spins aligned to in-plane (panels labelled as S OC X) and\nout-of-plane (SOC Z) directions. The eigenenergies obtain ed for SOC included in the force theorem approximation (labe lled\nFT, red) are compared with the self-consistent ones (SCF, bl ue). Each individual calculation is referred to its own Ferm i level\n(note that the Fermi energy changes if the spin orientation c hanges). Agreement between FT and SCF calculations in the ca se\nof a 3ML Au substrate is reduced with respect to that of free-s tanding GdAu 2due to the intense SOC at the substrate bands,\nwhich introduces small overall shifts in the band structure . Despite this, the overall band structure, and in particula r the gap\nopenings, for the two spin directions are satisfactorily re produced by the force theorem method.S7\n-3-2-1 0 1\nΜ Γ ΚE-EF (eV)EuAu2 / 3ML Au; SOC X\n FT\n SCF\n-3-2-1 0 1\nΜ Γ ΚE-EF (eV)EuAu2 / 3ML Au; SOC Z\n FT\n SCF-3-2-1 0 1\nΜ Γ ΚE-EF (eV)EuAu2; SOC X\n FT\n SCF\n-3-2-1 0 1\nΜ Γ ΚE-EF (eV)EuAu2; SOC Z\n FT\n SCF\nFIG. S7. Self-consistent and force-theorem bands for free-standin g and supported EuAu 2structures. Same\ninformation as in Fig. S6 for the EuAu 2case.S8\nΜ Γ Κ-3.1-0.300.20.72.03.5Ne(b)    EuAu2\n X\n Zα\nα'β β\n-2.0-1.00.01.02.0\nΜ Γ ΚE-EF (eV)(a)   EuAu2 (no SOC)\nα\nα'β βd↑\nx2-y2d↓\nx2-y2\nd↑\nxyd↓\nxy\nFIG. S8. Band structure for free-standing EuAu 2.(a) To guide the eye, orbital and spin characters of free-sta nding\nEuAu 2bands in the absence of SOC. (b) Band structure of free-stand ing EuAu 2with SOC (force theorem) for spins aligned\nto the X (violet) and Z (green) directions. The right-hand si de axis shows the corresponding band filling Ne(see also Fig. S9).\nNote that the gap openings that were relevant in the anisotro py of trivalent GdAu 2(labelled α,α′andβ) lie above the Fermi\nlevel in the EuAu 2case and therefore do not contribute to the observed MAE.S9\nS4. VISUAL GUIDE TO INTERPRET THE MAE( Ne) CURVES\n-1-0.5 0 0.5\n-4-3-2-1 0 1 2(a)E-EF (eV)\nNeGdAu2 / 3ML Au\nGdAu2\n-2-1 0 1\n-4-3-2-1 0 1 2(b)E-EF (eV)\nNeEuAu2, U = 5.5 eV\n-2-1 0 1 2\n-20 -10  0  10(c)EuAu2 / 3ML AuE-EF (eV)\nNe 0 10 20 30\n-2-1 0 1 2(d)DOS (arb.u.)\nE-EF (eV)U = 3.5 eV\n U = 5.5 eV\n U = 7.5 eV\nFIG. S9. Correspondence between Neand Fermi energies. Dependence of EuAu 2/ 3ML Au electronic structure\nand MAE with the correlation parameter. (a-c) Guides to the eye to interpret the MAE( Ne) curves in the main paper.\nThe curves represent the Fermi energy that corresponds to a g iven filling with respect to neutrality ( Ne= 0), as in the\nright-hand axis tics of main paper Figs. 3(b,c). These fillin gs are calculated as an integral of the density of states for e ach\none of the four considered systems without including SOC. (a ) free-standing and supported GdAu 2to interpret the MAE( Ne)\ncurves of Fig. 3(d). (b) Same for EuAu 2[Fig. 4(a)]. (c) A wider range of fillings allows to identify t he Eu(4f) contributions\nin Fig. 4(b), located at different binding energies (indicat ed by horizontal dashed lines) or Neinterval (plateaus) depending\non theUvalues. The 4 fbroadened band is displaced toward the Fermi level for lower Uvalues. (d) Total density of states\ncalculated for EuAu 2/ 3ML Au with spins perpendicular to the surface in the force t heorem approximation for three values\nof the correlation parameter U. ForU= 5.5eV (blue), the broad peak centered at energies E−EF≈ −1eV corresponds to\nthe hybridized 4 fbands, in agreement with the experimental binding energies . ForU= 7.5eV (purple) the 4 fband lies at the\ntop of Au substrate dband.S10\n[1] F. Gerken, Journal of Physics F: Metal Physics 13, 703 (1983).\n[2] W.-D. Schneider, C. Laubschat, and B. Reihl, Phys. Rev. B 27, 6538 (1983).\n[3] B.T.Thole, G.vanderLaan, J.C.Fuggle, G.A.Sawatzky,R .C. Karnatak,andJ.-M.Esteva,Phys. Rev. B 32, 5107 (1985).\n[4] T. Kinoshita, H. Gunasekara, Y. Takata, S.-i. Kimura, M. Okuno, Y. Haruyama, N. Kosugi, K. Nath, H. Wada, A. Mitsuda,\nM. Shiga, T. Okuda, A. Harasawa, H. Ogasawara, and A. Kotani, Journal of the Physical Society of Japan 71, 148 (2002).\n[5] A. Tcakaev, V. B. Zabolotnyy, C. I. Fornari, P. R¨ ußmann, T. R. F. Peixoto, F. Stier, M. Dettbarn, P. Kagerer,\nE. Weschke, E. Schierle, P. Bencok, P. H. O. Rappl, E. Abramof , H. Bentmann, E. Goering, F. Reinert, and V. Hinkov,\nPhys. Rev. B 102, 184401 (2020).\n[6] A. Arrott, Phys. Rev. 108, 1394 (1957).\n[7] M. Ormaza, L. Fern´ andez, M. Ilyn, A. Maga˜ na, B. Xu, M. J. Verstraete, M. Gastaldo, M. A. Valbuena, P. Gargiani,\nA. Mugarza, A. Ayuela, L. Vitali, M. Blanco-Rey, F. Schiller , and J. E. Ortega, Nano Letters 16, 4230 (2016)."    },    {        "title": "2109.07240v1.Nonlinear_One_Dimensional_Constitutive_Model_for_Magnetostrictive_Materials.pdf",        "content": "Nonlinear One-Dimensional Constitutive Model for\nMagnetostrictive Materials\nAlecsander N. Imhof\nE-mail: alecsanderi@vt.edu\nDepartment of Biomedical Engineering and Mechanics, Virginia Tech.\nJohn P. Domann\nE-mail: jpdomann@vt.edu\nDepartment of Biomedical Engineering and Mechanics, Virginia Tech.\nAbstract. This paper presents an analytic model of one dimensional magnetostric-\ntion. We show how speci\fc assumptions regarding the symmetry of key micromagnetic\nenergies (magnetocrystalline, magnetoelastic, and Zeeman) reduce a general three-\ndimensional statistical mechanics model to a one-dimensional form with an exact so-\nlution. We additionally provide a useful form of the analytic equations to help ensure\nnumerical accuracy. Numerical results show that the model maintains accuracy over\na large range of applied magnetic \felds and stress conditions extending well outside\nthose produced in standard laboratory conditions. A comparison to experimental data\nis performed for several magnetostrictive materials. The model is shown to accurately\npredict the behavior of Terfenol-D, while two compositions of Galfenol are modeled\nwith varying accuracy. To conclude we discuss what conditions facilitate the descrip-\ntion of materials with cubic crystalline anisotropy as transversely isotropic, to achieve\npeak model performance.arXiv:2109.07240v1  [physics.app-ph]  15 Sep 2021Nonlinear One-Dimensional Constitutive Model for Magnetostrictive Materials 2\n1. Introduction\nMagnetostrictive materials enable the use of numerous technologies including energy\nharvesters, ultrasonic transducers, vibration dampers, and even novel antennas [1{10].\nThese materials intrinsically couple a material's magnetic and mechanical degrees of\nfreedom, allowing the magnetization M(H;\u001b) and magnetostriction \"m(H;\u001b) to be\ndescribed as a function of the magnetic \feld Hand stress\u001b(or total strain \"). However,\nit is di\u000ecult to accurately model the strongly coupled nonlinear magneto-mechanical\nconstitutive response of these materials at the macroscale. The lack of an accurate\nnon-linear magnetostrictive constitutive model has likely inhibited the design of novel\nmagnetostrictive technologies.\nFigure 1 schematically illustrates two potential magnetization curves M(H;\u001b0)\nat \fxed stress \u001b0. Two commonly encountered behaviors are highlighted. Type I\ncurves are concave down until saturation, while Type II curves transition from concave\nup to concave down between demagnetization and saturation. Type I curves are\ncommonly seen in materials like Terfenol-D (Tb 0:3Dy0:7Fe19:2) [11, 12], while Type\nII curves are encountered in some compositions of Galfenol (e.g., Fe 81:6Ga18:4) when\nplaced in compression [12]. The speci\fc MH curve displayed by a given material is\nstrongly dependent on its crystalline structure in addition to any externally controlled\nanisotropies, like an applied stress. Numerous modeling approaches have been utilized\nin an attempt to capture the behaviors seen in Figure 1. Three common approaches\nare to either 1) construct polynomial series expansions, 2) assume a modi\fed Langevin\nbehavior, or 3) utilize statistical mechanics [13{21].\nFigure 1. Schematic of two representative MH curves for magnetostrictive materials\nat \fxed stress. Depending on the crystal structure of the material the responses can\ngenerally be described by curves that are always concave down (Type I) or curves that\ntransition from concave up to down. (Type II)\nPolynomial series expansions start by constructing phenomenological Taylor series\nexpansion of either the Helmholtz f(H;\") or Gibbsg(H;\u001b) free energy density. OnceNonlinear One-Dimensional Constitutive Model for Magnetostrictive Materials 3\na series is constructed g\u0019\u0000\u00160\u001fijHiHj=2 +:::, classical thermodynamics is utilized\nto obtain conjugate \feld variables like the magnetization \u00160M=\u0000(@g=@H) and\nmagnetostriction \"m=\u0000(@g=@\u001b). These models are straightforward to utilize once\nthe expansion coe\u000ecients (i.e., material properties) are identi\fed [13, 14]. However,\nconducting the requisite number of experiments to measure the expansion coe\u000ecients\ncan be technically challenging and costly. Routinely only \frst order expansions\nare utilized, resulting in a preponderance of 'piezomagnetic' models, even though\npiezomagnetism is a fairly rare phenomenon most prevalent in antiferromagnets [22{25].\nOne important restriction on polynomial approaches is immediately evident from the\ncurves in Figure 1. Notably, \fnite expansions are incapable of capturing the saturating\nbehavior inherent in magnetic phenomena. As a result the use of polynomial models\nmust be accompanied by knowledge of their bias conditions and limited range of validity.\nWhile the use of higher-order expansions can increase the range of validity, their use\ncomes at the large cost of rapidly increasing the number of expansion coe\u000ecients. In\npractice unknowns or variations in test setups can result in di\u000eculties using these\nlinearized material properties when modeling an actual nonlinear device.\nA common alternative to polynomial models is to a priori assume a magnetic\nconstitutive response that saturates and also assume micromagnetic expressions for\nmagnetostriction are valid at the macroscale. We refer to this below as `Langevin\nmagnetostriction'. An example of this approach is to assume \"m/M\nM, withM=\nMsL(h)^hwhereMsis the saturation magnetization and Lis the Langevin function\ndepending on the reduced magnetic \feld h[26, 27]. While the reduced \feld h(H;T)\nis conventionally obtained from a ratio of the magnetic \feld energy divided by the\nthermal energy, several authors have made ad-hoc assumptions that extend h(H;\u001b;T)\nto depend on stress \u001bas well [15{18,26{28]. Langevin magnetostriction can qualitatively\nsimulate a family of the Type I magnetization curves shown in Figure 1, where stress\nmodulates the susceptibility. While Langevin magnetostriction provides a path to\nincorporating saturating behavior and magneto-mechanical coupling, it also introduces\ninconsistencies compared to experimental observations, and more advanced models (e.g.,\nmicromagnetics). One large inconsistency introduced by Langevin magnetostriction\nconcerns the \u0001E e\u000bect.\nWhile the assumption \"m/M\nMis often justi\fed by appealing to the\nmicromagnetic response \"m/M\nM=M2\ns=m\nm, it is not generally possible to\nsimply extend micromagnetic behavior to the macroscale without a suitable averaging\nprocedure (e.g., it's more accurate to say \"m/hm\nmi, whereh:iis a relevant thermal\naverage). Consider that when H= 0 an anhysteretic constitutive model predicts\nM= 0, and therefore the assumptions above identically produce zero macroscopic\nmagnetostriction \"m(H= 0;\u001b) = 0. However, magnetic domains are easily controlled\nwith stress at zero magnetic \feld, which is exempli\fed in numerous experimental\nstudies that observe the peak \u0001 Ee\u000bect occurs when H= 0 [29{31]. This observed\nresponse is explicitly due to zero-\feld magnetostriction. Attempting to use Langevin\nMagnetostriction models while designing a device reliant on the \u0001 Ee\u000bect, like aNonlinear One-Dimensional Constitutive Model for Magnetostrictive Materials 4\nmagnetometer [32{34], is therefore expected to be a challenging endeavour. Instead\nofa priori assumingM=MsL(h)^h, we highlight that the Langevin model for classical\nparamagnetism is derived using statistical mechanics, indicating these assumptions likely\ndo not need to be made.\nModels constructed with statistical mechanics are capable of capturing the\nfull nonlinear saturating behavior and strong magneto-mechanical coupling of\nmagnetostrictive materials. The Gibbs Canonical Ensemble can be used to model a\nmagnetostrictive material in thermal equilibrium with a heat bath (the environment) at\nconstant temperature, capable of having work done on it. As shown in Equations (1) -\n(2) this can be used to calculate the partition function Zand the expected macroscale\nfree energy Gfor a given (microscopic) Landau free energy GL[35].\nZ=Z\nMexp\u0012\n\u0000GL\nkbT\u0013\ndm (1)\nG=\u0000kbTlnZ (2)\nIn these equations kbTis the thermal energy and integration is over all possible\nstates of the system (i.e. Mis the set of all admissible magnetization distributions).\nThe Landau free energy contains any of the relevant energy densities used in\nmicromagnetics. This allows the impact of external magnetic \felds, magnetostatic \felds\n(demag), magnetocrystalline anisotropy (MCA), exchange energy, and magnetostriction\nto be incorporated [35{40]. Once the expected energy Gis constructed, classical\nthermodynamics can be utilized and a series of partial derivatives leads to the average\nproperties of the system. For a uniform system with volume V, the average magnetic\nmomenth\u0016i=hMiV=\u0000(@G=@H)=\u00160, and magnetostriction h\"mi=\u0000(@G=@\u001b)=V.\nWhile statistical models can potentially model paramagnetic, ferromagnetic,\nferrimagnetic, and antiferromagnetic materials with arbitrary anisotropy energies, and\neven account for polycrystalline materials [39{41], the requisite integral equations\ngenerally lack closed form solutions and therefore necessitate numerical approximations\n[19{21]. This has resulted in common simplifying assumptions including zero\nexchange coupling (i.e., paramagnetic behavior) [41{44], potentially with simpli\fed\nmagnetocrystalline anisotropies that treat polycrystalline cubic materials as transversely\nisotropic [43{46]. Improving the computational e\u000eciency of these models has been\nthe focus of recent research that has shown an excellent ability to \ft these models to\nexperimental data [20,41,43,45{47].\nWhile generating a general 3D model is certainly a goal for this line of research,\nsimplifying the model to one dimension has several bene\fts. Most notably, as we\nwill show below there is a closed form analytical solution for a suitably simpli\fed\n1D model. Additionally, we note that most macroscale experimental studies have\nutilized conditions where the applied magnetic \feld and surface tractions are parallel\nHkT[11,12,16,47{51]. Each of the cited experimental studies has therefore provided\ne\u000bectively one-dimensional data, and not tested general 3D loading conditions. In\naddition to explaining existing experimental data, a valid 1D model is also expectedNonlinear One-Dimensional Constitutive Model for Magnetostrictive Materials 5\nto \fnd use in reduced order models, including magnetostrictive rod and beam theories\n[52{54].\nThe work in this paper presents an analytic constitutive model for magnetostrictive\nmaterials that is derived using statistical mechanics. This is made possible by restricting\nthe allowed orientations of the applied magnetic \feld, stress, and MCA, in addition to\nrestricting the allowable type of MCA. In the following sections we present the necessary\nassumptions, derive a 1D magnetostrictive constitutive model, provide a convenient\nnumerical implementation, and use the model to simulate experimental data from the\nliterature. Results show this model is capable of accurately simulating the response\nof materials with isotropic Joulian magnetostriction when the magnetocrystalline\nanisotropy of the material produces Type I magnetization curves as depicted in Figure\n1).\n2. Model Development\n2.1. Boltzmann Statistics\nIn this section we use the Gibbs Canonical Ensemble to describe a paramagnetic\nmagnetostrictive material in thermal equilibrium with an environment at constant\ntemperature, capable of having work done on it. For a paramagnetic material we\nfocus on the average response of an isolated magnetic moment, allowing us to write\nthe partition function and expected free energy density as\nz=Z\nSexp (\u0000\fgL)dm (3)\ng=\u0000\f\u00001lnz (4)\nwheregLis the Landau free energy density, \f\u00001the thermal energy density, and\nintegration is now restricted to all orientations of m=M=Ms(i.e., over the unit\nsphereS). Shown in Equation (5), we consider a free energy density composed of\nZeemanfz, magnetoelastic anisotropy fme, and magnetocrystalline anisotropy fmca\nenergy densities [35{38].\ngL=fz(m;H) +fme(m;\u001b) +fmca(m;K) (5)\nwhere the notation f(m;:::) indicatesmis a prescribed parameter and Kis the MCA\ntensor. Once the partition function is constructed the equilibrium magnetization and\nmagnetostriction are calculated by taking partial derivatives of the expected free energyNonlinear One-Dimensional Constitutive Model for Magnetostrictive Materials 6\nin Equation (4), leading to\nhMi=Ms1\nzZ\nSmexp (\u0000\fgL)dm\n=Mshmi (6)\nh\"mi=3\u0015s\n21\nzZ\nSm\nmexp (\u0000\fgL)dm\n=3\n2\u0015shm\nmi (7)\nwhere\u0015sis the saturation magnetostriction. Based on Equations (6) and (7) for the\naverage magnetization and magnetostriction, we can identify the probability density\nfunctionP(m;H;\u001b;\f) = exp (\u0000\fgL)=z. This allows equations (6) and (7) to be\ninterpreted as thermal averages of the micromagnetic equations for magnetization and\nisotropic Joulian magnetostriction.\nIn addition to obtaining the magnetization and magnetostriction, the nonlinear\nmaterial properties can be calculated by taking derivatives of the previous expressions.\nThe properties are readily shown to be\nh\u001fi=\f\u00160(hM\nMi\u0000hMi\nhMi) (8)\nhSmi=\f(h\"m\n\"mi\u0000h\"mi\nh\"mi) (9)\nhqi=\f(hM\n\"mi\u0000hMi\nh\"mi) (10)\nwhereh\u001fiis the magnetic susceptibility at constant stress, hSmithe magnetostrictive\ncompliance at constant magnetic \feld, and hqithe piezomagnetic coupling tensor de\fned\nbyhqi=@hMi=@\u001b=\u0016\u00001\n0@h\"mi=@H. We brie\ry note that the total compliance\nS=Sel+hSmi, whereSelis the elastic compliance (e.g., inverse Young's modulus\nfor 1D loading). The form of Equations (8)-(10) reveals that the macroscopic material\nproperties are proportional to the statistical variance / \ructuation of the underlying\nmicroscopic \felds (i.e., var( x) =hx2i\u0000hxi2).\nWe brie\ry note that equations (3) - (10) can commonly be simpli\fed by shifting\nthe free energy with functions that are independent of m. The magnetization h~Miand\nmagnetostriction h~\"miobtained from the free energy ~ gL=gL+f(H;\u001b) is related to\nthe average magnetization hMiand magnetostriction h\"miobtained from gLby\n~z=Z\nSexp (\u0000\f~gL)dm=zexp(\u0000\ff) (11)\nD\n~ME\n=\u00001\n\u00160@f\n@H+hMi (12)\nh~\"mi=\u0000@f\n@\u001b+h\"mi: (13)\nIn practice, a function independent of mcan be added to gLto ensure the Boltzmann\nterm exp (\u0000\f~gL)\u00141 which aids in numerical calculations (i.e., so the exponential\ndoesn't lead to numerical inaccuracies when gL<<\u00001). In what follows, terms withNonlinear One-Dimensional Constitutive Model for Magnetostrictive Materials 7\ntilde overbars are understood to include an energy o\u000bset as described in Equations\n(11)-(13).\nAs previously stated the integrals used in Boltzmann statistics do not generally\npossess closed form solutions. However, we now show that exact closed form solutions\nexist for a speci\fc type of MCA and restrictions on the orientations of the magnetic\n\feld and stress.\n2.2. Quadratic Anisotropy\nA closed form solution to Equations (3) - (10) can be obtained under the following\nassumptions: 1) the MCA can be represented as a quadratic form fmca=m\u0001Km , 2)\nthe material displays isotropic magnetostriction (i.e., \u0015100=\u0015111=\u0015s), 3) the combined\nmagnetoelastic anisotropy and MCA is transversely isotropic, and 4) the magnetic \feld is\nperpendicular to the isotropic anisotropy plane. To satisfy these requirements we assume\nthe material has transversely isotropic MCA, with only two unique eigenvalues Ki. We\nassume the unique direction to be the 1-direction, while the 23-plane is isotropic (i.e.,\nK16=K2=K3). Additionally, we assume the stress state has only 2 unique principal\nstresses\u001b16=\u001b2=\u001b3, where the stress and MCA eigenvectors are parallel. These\nstress assumptions are commonly satis\fed for long rod or beam-like materials with one\nunique axis, and are consistent with numerous experimental studies [11, 12, 16, 47{51].\nThe assumption of transversely isotropic MCA is a key mathematical assumption in this\nmodel that restricts the possible materials this model is suitable for. This point will be\nexamined further in the results section.\nTo see how these assumptions lead to a closed form solution we \frst construct the\ntotal anisotropy energy density fA\nfA=fmca+fme (14)\nfmca=m\u0001Km =Kijmimj (15)\nfme=m\u0001\u0006m=\u00003\n2\u0015s\u001bijmimj; (16)\nwhere the isotropic magnetostriction is described by \u0006=\u00003\u0015s\u001b=2 and\u001bis the Cauchy\nstress. Both Kand\u0006are symmetric rank 2 tensors. We combine them to de\fne\nthe non-dimensionalized anisotropy tensor A=\u0000\f(K+\u0006). The non-dimensionalized\nmagnetic \feld is h=\f\u00160MsH.\nExpressing these energies with components parallel to the eigenvectors of A,\n\u0000\fgL=Aim2\ni+himi, whereAiare the eigenvalues of A. Additionally, the transversely\nisotropicAhas only two unique eigenvalues A16=A2=A3, and following assumption 4)\nh=h^e1is parallel to the 1-axis. Accounting for the fact that jmj= 1 is a unit vector,\nand discarding the resultant term independent of m, we have\u0000\fgL=Am2\n1+hm1,\nwhereA=A1\u0000A2is the change in anisotropy energy from the isotropic plane to\nthe transverse axis. Summarizing, based on the assumptions above we can simplifyNonlinear One-Dimensional Constitutive Model for Magnetostrictive Materials 8\nEquation (3) to\nz= 2\u0019Z1\n\u00001exp\u0000\nAm2\n1+hm1\u0001\ndm1 (17)\nThe solution to the integral in (17) is a summation of Dawson functions D().\nHowever, when A< 0 the Dawson function produces complex numbers and calculations\nusing this function can rapidly accumulate large numerical errors. In that case the\nintegral can be simpli\fed and written in terms of the error function Erf() as summarized\nin Equation(18)\nz\n2\u0019=8\n<\n:z+=exp (A+h) D(\u000b+)+exp (A\u0000h) D(\u000b\u0000)p\nAA> 0\nz\u0000=\u0000exp\u0010\n\u0000h2\n4A\u0011p\u0019(Erf(\r+)\u0000Erf(\r\u0000))\n2p\u0000AA< 0(18)\nwhere\u000b\u0006= (2A\u0006h)=(2p\nA) and\r\u0006= (h\u00062A)=(2p\n\u0000A). Following equations (6) to\n(10) above, derivatives of these expressions can be used to obtain the magnetization,\nmagnetostriction, and nonlinear material properties. However, we \frst make several\nre\fnements to these equations as there are numerous points where, although they are\nanalytically correct, they can accrue large numerical errors when evaluated.\nThe \frst point at which the current solutions can become inaccurate is when\nh>>jAj. Notice that in (18) as A!0 the equations become indeterminate. A simple\n\fx for this problem is to utilize a series expansion of the partition function (Equation\n(17)) for small anisotropy values about A= 0. The general form of the series expansion\nis provided in the Appendix in section 5.1. Additionally this approximation can be used\nto solve for the magnetization, magnetostriction, and material properties. The series\nexpansion should be used when jAj=h<\u000f , where\u000fwill depend on the speci\fc numerical\nimplementation. When testing second order expansions using Matlab we found a cuto\u000b\nratio ofjAj=h< 10\u00007preserved accuracy in the magnetization and magnetostriction.\nThe use of the error function Erf( x) is susceptible to numerical errors when x>> 1.\nThe numerical accuracy can be improved by introducing the scaled complimentary error\nfunction Erfcx( x), where Erf( x) = 1\u0000exp(\u0000x2) Erfcx(x). This has the advantage of\nsimplifying several exponential terms, and helping to avoid arithmetic under\row [55].\nAs previously stated these solutions can be further simpli\fed by shifting the free energy.\nThe speci\fc energy o\u000bsets used below were all chosen to ensures that exponential\nterms in the solutions remain bounded as the magnetic \feld and stress become large.\nCombined these changes simplify Equation (18) to\n~z\n2\u0019=8\n>><\n>>:~z+=D(\u000b+)+exp (\u00002h) D(\u000b\u0000)p\nAA> 0\n~z\u0000=\u0000p\u0019(\u0000Erfcx(\r+)+exp (\u00002h) Erfcx(\r\u0000))\n2p\u0000AA< 0; \r+>0\n~z\u0000=\u0000p\u0019(\u00002+exp (\u0000\r2\n+)Erfcx(\r+)+exp(\u0000\r2\n\u0000)Erfcx(\r\u0000))\n2p\u0000AA< 0; \r+<0;(19)Nonlinear One-Dimensional Constitutive Model for Magnetostrictive Materials 9\nwhere the energy o\u000bsets applied to Equation (19) are\nf=8\n>><\n>>:\u0000A\u0000h A> 0\n\u0000A\u0000h A< 0; \r+>0\nh2=4A A< 0; \r+<0:(20)\nThese o\u000bsets were selected to ensure the exponential terms in (18) converge to zero as\nfh; Ag!1 . The derivatives of equations (19) and (20) with respect to hresults in\nthe average magnetization,\nhMi\nMs=8\n>><\n>>:\u0000h\n2A\u0000\u00001+exp(\u00002h)\n2A~z+A> 0\n\u0000h\n2A\u00001\u0000exp(\u00002h)\n2A~z\u0000A< 0; \r+>0\n\u0000h\n2A+exp(\u0000\r2\n+)\u0000exp(\u0000\r2\n\u0000)\n2A~z\u0000A< 0;\r+<0:(21)\nAdditionally, the derivative of (18) with respect to Aresults in an average\nmagnetostriction of\n2\n3h\"mi\n\u0015s=8\n>>>><\n>>>>:h2\u00002A\n4A2+h\n~z+\u0010\n1+exp (\u00002h)\n2Ah\u00001\u0000exp (\u00002h)\n4A2\u0011\nA> 0\nh2\u00002A\n4A2+h\n~z\u0000\u0010\n1+exp (\u00002h)\n2Ah\u00001\u0000exp (\u00002h)\n4A2\u0011\nA< 0; \r+>0\nh2\u00002A\n4A2+h\n~z\u0000\u0012\nexp(\r2\n+)+exp(\u0000\r2\n\u0000)\n2Ah\u0000exp(\r2\n+)+exp(\u0000\r2\n\u0000)\n4A2\u0013\nA< 0; \r+<0(22)\nExpressions for the nonlinear material properties in Equations (8) - (10) can be\nobtained through additional derivatives of the equations above. As these expressions\nare reasonably lengthy, we provide them in section 5.2 of the Appendix.\nAs a \fnal note, the equations above are written assuming that h\u00150. Values for\nh < 0 can readily be obtained by noting the magnetization is an odd function with\nrespect toh, while the magnetostriction is even (i.e., for h <0M(h) =\u0000M(jhj). We\nalso note that in contrast to the Langevin Magnetostriction models discussed in the\nintroduction, when h= 0 the magnetostriction h\"mi6= 0 in this model. Instead, when\nh= 0 a net magnetostriction is induced due to the competing MCA and magnetoelastic\nanisotropy energies. Finally, similar equations for hMiandh\"mihave previously\nappeared in the literature, most closely matching the results in Equation (21) and (22)\nwhenA > 0 [56, 57]. However the authors believe this is the \frst time a numerically\naccurate solution for both A< 0 andA> 0 has been presented.\n3. Results and Discussion\nThis section provides 1) a numerical comparison of the closed form solutions presented\nin equations (19), (21), and (22) to conventional numerical integration, 2) a qualitative\nassessment of material response these solutions produce, and \fnally 3) assesses the\nmodel's ability to simulate the experimentally measured response of several common\nmagnetostrictive materials.Nonlinear One-Dimensional Constitutive Model for Magnetostrictive Materials 10\nTable 1. Accuracy compared to numerical integration\nFunction Avg. Error Max Error\n~z 2:4\u000210\u000036:8\u000210\u00001\nhMi 1:9\u000210\u0000112:3\u000210\u00009\nh\"mi 2:4\u000210\u000053:9\u000210\u00003\n3.1. Numerical Accuracy\nTo evaluate the numerical accuracy of these solutions they were compared to standard\nnumerical integration. A grid of N= 100 logarithmically spaced \feld points 10\u00002\u0014h\u0014\n106andN= 200 logarithmically spaced anisotropies 10\u00002\u0014j\u0006Aj\u0014106were generated\nforNtotal= 20;000 points. At each grid point numerical integration was performed using\nMatlab's integral() function with relative and absolute errors of 10\u000012. The relative\nerrors for each equation were calculated as jfnum\u0000feqnj=fnum, wherefis the parameter\nof interest. While we utilized Matlab's built-in scaled complimentary error function\nerfcx() , the built-in Dawson function was quite slow, and instead we approximated\nD() using McCabe's continued fraction. This achieves a precision to 10\u000015, and has a\nsimple and fast numerical implementation [58].\nThe accuracy results are summarized in Table 1. Additionally, detailed error\nsurfaces are shown in section 5.3 of the Appendix. The data in Table 1 shows 1)\nthe average relative error per grid point (i.e., sum of individual errors divided by Ntotal)\nand 2) the maximum relative error at a single grid point. The relative errors of all three\nfunctions were approximately 10\u000014for the majority of the tested grid points. The only\nregion where the solutions became inaccurate was when the series expansion was used for\njAj=h< 10\u00007. The maximum errors were all measured in the expansion region. It should\nbe noted that without the series expansion the error can quickly climb above 100% in\nthat region (i.e., the expansion worked). While the partition function had a maximum\nerror of over 50% for the expansion, the resulting magnetization and magnetostriction\nstill remained accurate as they depend on derivatives of znot its absolute value. The\nmax relative error in the magnetostriction only reached 0 :4%.\nTo provide more context for this information, if we consider the material properties\nfor Terfenol-D that are presented in Section 3.3 below, we can show that accuracy\nanalysis above is valid for applied \felds 0 \u0014H\u0014200 T and stresses 0 \u0014\u001b\u001410\nTPa, respectively. This range is clearly far outside those used in standard laboratory\nsettings, leaving the authors to conclude the presented equations remain numerically\naccurate for \felds and stress conditions typically applied to these materials. Material\nfailure and additional phenomena would need to be considered before these equations\nbecome numerically inaccurate.Nonlinear One-Dimensional Constitutive Model for Magnetostrictive Materials 11\nFigure 2. Model predictions of (a) magnetization and (b) magnetostriction and (c)\nthe \u0001Ee\u000bect for possible handAvalues.\n3.2. Qualitative Assessment\nHaving shown the equations above accurately solve the integral expressions used in\nstatistical mechanics, we now turn our attention to analyzing the qualitative response of\nthe resulting model. Figure 2 illustrates the (a) magnetization and (b) magnetostriction\ncurves at \fxed anisotropy A. Figure 2 (c) shows the \u0001 Ee\u000bect this model predicts at\n\fxed values of h. Starting with Figure 2 (a) we note that in addition to properly\nsaturating, the model has a lower susceptibility when A is negative (compression) and a\nhigher susceptibility when A is positive (tension). Furthermore, when A= 0 the model\nreduces to the Langevin function as required. Finally, while only small values of Aare\npresented in Figure 2(a), over the entire range of tested values 10\u00002\u0014j\u0006Aj\u0014106only\nType I behavior was observed. This model does not qualitatively describe materials\nwith Type II MH curves.\nFigure 2 (b) shows the predicted magnetostriction curves as a function of hwith\ndi\u000berent lines at \fxed anisotropy A. Note that the zero \feld magnetostriction has been\nsubtracted from each curve to clearly display all curves on the same graph. Therefore\nthe plot showsh\"m(h;A)i\u0000h\"m(0;A)i. For a positive magnetostrictive material in\ntensionA > 0, the model correctly predicts a small change in magnetostriction as h\nincreases. For this case tension produces an initial magnetization con\fguration where\nthe microscale magnetization starts either parallel or anti-parallel to the magnetic \feld\ndirection. Upon increasing hthe anti-parallel domains transition to parallel, however\nthere is no resulting magnetostriction as \"m/m2\n1is even in the magnetization.\nConversely, compression causes the magnetization to initially align perpendicular to\nthe applied \feld direction. Increasing the applied magnetic \feld then forces the\ndomains to become parallel. For a large enough initial compression this results in\nh\"m(h;A)i\u0000h\"m(0;A)i= 3\u0015s=2 ashincreases.\nConcluding with Figure 2 (c) the total Young's modulus of the material EtotwasNonlinear One-Dimensional Constitutive Model for Magnetostrictive Materials 12\ncalculated using the using the magnetostrictive compliance Smand elastic Young's\nModulusEel,\nEtot= (1=Eel+hSmi)\u00001: (23)\nAs the intention of Figure 2 is qualitative assessment of the predicted behavior, a Young's\nModulus of 1 Pa was chosen. Changing this value changes the amplitude of Etotand\n\u0001E, but not the locations of the peaks. The trends in this graph are consistent with\nexperimental data in which the largest \u0001 Ee\u000bect is observed at zero magnetic \feld [29].\nFurthermore, recalling that A is composed of MCA and magnetoelastic anisotropy, this\nmodel shows that in order to maximize the \u0001 Ee\u000bect an applied stress that cancels out\nMCA is required. The maximum value of hSmiobtained from Equation (9) is equal to\n\f\u00152\ns=5, leading to the maximum \u0001 E=Eel\u0000Etot.\nmax \u0001E=\u00121\nEel+5\n\f\u00152\nsE2\nel\u0013\u00001\n(24)\n3.3. Experimental Comparison\nIn addition to con\frming the numerical accuracy of the magnetization and\nmagnetostriction in Equations (21) and (22), the solutions were compared to\nmagnetization and magnetostriction curves found in the literature for three di\u000berent\nmaterials. We digitized data for Terfenol-D Tb 0:3Dy0:7Fe19:2[11], and two compositions\nof Galfenol Fe 79:1Ga20:9, and Fe 81:6Ga18:4[12]. The saturation magnetization Msand\nsaturation magnetostriction \u0015swere directly obtained from the experimental data, with\nMs= max(jMj), and\u0015s= 2=3 max(j\"mj). The magnetostriction calculation came from\ncurves where we assume the initial bias stress was large enough to exclusively produce\n180\u000edomain walls in the material when H= 0. AfterMsand\u0015swere obtained, the only\nremaining unknowns in the model are the magnetocrystalline anisotropy coe\u000ecient K,\nand the thermal energy density term \f. For each material Kand\fwere identi\fed by\nminimizing the relative error between the constitutive model and the experimental data.\nThis calculation produces error surfaces that only depend on two variables. Therefore\nwe did not use a nonlinear optimization routine, but instead calculated the error over a\ngrid ofKand\fvalues. These error surfaces are presented in the results below.\nThe values of Kand\fobtained from this procedure, along with the average and\nmaximum relative error between the modeled and experimental data are presented\nin Table 2. As we calculate errors when comparing the 1) magnetization and 2)\nmagnetostriction data, we present three sets of values for fK;\fg, and their errors for\neach material. The three cases are for 1) \ftting the combined data set, 2) \ftting only\nthe magnetization data, and 3) \ftting only the magnetostriction data. The last two \fts\ncan be utilized for models requiring only hMiorh\"mi(i.e., in one-way-coupled models).\nFigure 3 parts (a) and (b) compare the model with experimental data for Terfenol-\nD using the combined \ft parameters in Table 2. The solid lines are the digitized\nexperimental data, while the circular markers are modeled data points. Figures 3 (c)-(e)Nonlinear One-Dimensional Constitutive Model for Magnetostrictive Materials 13\nTable 2. Model Parameters and Results*\nMaterial Fit K \fAvg. Error Max. Error\nhMi h\"mi hMi h\"mi\nTb0:3Dy0:7Fe19:2Combined -4697 6 :5\u000210\u000047.8 % 3.0 % 33 % 20 %\nMs= 7:8\u0002105OnlyhMi6515 1:6\u000210\u000045.0 % | 55 % |\n\u0015s= 1:4\u0002103Onlyh\"mi-4697 6:6\u000210\u00004| 3.0 % | 19 %\nFe79:1Ga20:9 Combined 363.6 1 :1\u000210\u000036.4 % 12 % 26 % 38 %\nMs= 1:2\u0002106OnlyhMi-767.7 7:4\u000210\u000044.0 % | 13 % |\n\u0015s= 1:3\u0002102Onlyh\"mi1010 2:8\u000210\u00003| 6.7 % | 37 %\nFe81:6Ga18:4 Combined 909.2 1 :7\u000210\u000036.8 % 7.0 % 23 % 31 %\nMs= 1:2\u0002106OnlyhMi1333 1:6\u000210\u000036.5 % | 22 % |\n\u0015s= 1:7\u0002102Onlyh\"mi1636 4:6\u000210\u00003| 5.7 % | 31 %\n*Units:Ms(A/M),\u0015s(ppm),K(J/m3),\f(m3/J)\nFigure 3. (a) Magnetization and (b) magnetostriction of Tb 0:3Dy0:7Fe19:2at constant\nstress values compared to the 1D constitutive model with fK;\fgminimizing the\ncombined error. Relative errors of (c) the combined data, (d) only the magnetization,\nand (e) only the magnetostriction for a parametric sweep of Kand\f. In parts (c)-\n(e) the red triangular markers are the locations of minimum combined error, while the\nyellow circular markers are locations of of minimum magnetization or magnetostriction\nerror. Data digitized from [11].\nshow the average relative error per data point for the data in 3(a) and 3(b). The red\ntriangular markers show the location of the optimal values for the combined error, while\nthe yellow circles are the locations of minimum error when \ftting just the magnetization\n/ magnetostriction. When using the fK;\fgthat minimize the combined error, there is\n7:8% and 3:0% relative error per data point in the magnetization and magnetostrictionNonlinear One-Dimensional Constitutive Model for Magnetostrictive Materials 14\ncomparison, respectively. It is worth noting that the best \ft parameters for the\nmagnetization 3(d) and magnetostriction 3(e) occur for di\u000berent fK;\fg. The combined\nresult in 3(c) is skewed towards the optimal magnetostriction location as a wide range\noffK;\fgvalues produce errors close to the global magnetization minima of 5 :0% (as\nseen in 3(d)). It is possible to reduce the average error to 5 :0% if only the magnetization\nis compared, or 3 :0% if only the magnetostriction is compared.\nFigure 4 parts (a) and (b) show a comparison of the experimental data for\nFe79:1Ga20:9using the combined \ft parameters, with formatting identical to 3. Figures\n4(c)-(e) show the average relative error for the data in 4(a) and 4(b) is 6 :4% and 12% per\ndata point respectively. For this material the magnetization data is accurately described,\nwhile the model struggles to capture the magnetostriction data in compression. It is\npossible to reduce the average error to 4 :0% if only the magnetization is compared, or\n6:7% if only the magnetostriction is compared (i.e., the presented magnetization is close\nto its best \ft, while the magnetostriction can be improved).\nFigure 4. (a) Magnetization and (b) magnetostriction of Fe 79:1Ga20:9at constant\nstress values compared to constitutive model. Relative errors of (c) the magnetization\n(d) the magnetostriction and (e) the combination of the two over a parametric\nsweep ofKand\f. In parts (c)-(e) the red triangular markers are the locations of\nminimum combined error, while the yellow circular markers are locations of of minimum\nmagnetization or magnetostriction error. Data digitized from [12].\nFigure 5 parts (a) and (b) show a comparison of the experimental data for\nFe81:6Ga18:4using the combined \ft parameters. Figures 5 (c) shows the average relative\nerror for the data in 5(a) and (b) is 6 :8% and 7% per data point respectively. Once\nagain the combined result in 5(c) is skewed towards the optimal magnetization. Despite\nthis low relative error, a close look at 5(a) shows that the constitutive model doesn't\nqualitatively capture the nonlinear (Type II) behavior that occurs when the material isNonlinear One-Dimensional Constitutive Model for Magnetostrictive Materials 15\nsubjected to large compressive stresses. As a result, we do not recommend using the\nanalytical model for this composition of Galfenol.\nFigure 5. (a) Magnetization and (b) magnetostriction of Fe 81:6Ga18:4at constant\nstress values compared to constitutive model. Relative errors of (c) the magnetization\n(d) the magnetostriction and (e) the combination of the two over a parametric\nsweep ofKand\f. In parts (c)-(e) the red triangular markers are the locations of\nminimum combined error, while the yellow circular markers are locations of of minimum\nmagnetization or magnetostriction error. Data digitized from [12].\nRevisiting the anisotropy assumptions utilized in this model can help explain\nwhen the model is expected to accurately simulate experimental data. A relatively\nsmall restriction was placed on the isotropic magnetostrictive and Zeeman energies by\nassuming the principle stresses \u001b16=\u001b2=\u001b3, with a magnetic \feld Hk^e1. We note\nthis mimics the \feld / stress combinations applied in the experimental studies compared\nto above, and therefore should not decrease the accuracy of the model. Even for a non-\nisotropic magnetostrictive material, as long as the stress and \feld are parallel to the\nh100idirection the model accurately describes the magnetostrictive energy. Conversely,\na signi\fcant restriction was placed on the MCA by requiring it to be transversely\nisotropic. Due to this assumption the model is only capable of modeling Type I MH\ncurves. The crystalline structure of Terfenol-D and Galfenol are typically cubic with\nMCA of the form,\nfcubic=K1(m2\n1m2\n2+m2\n2m2\n3+m2\n3m2\n1) +K2(m2\n1m2\n2m2\n3): (25)\nWhile the use of cubic MCA is expected to produce a more accurate model, there is no\nknown closed form solution to the integral equations above once an MCA of this form\nis utilized. However, we can analyze the e\u000bect K1andK2have on magnetization curvesNonlinear One-Dimensional Constitutive Model for Magnetostrictive Materials 16\nFigure 6. MH curves under 50 MPa of compression with (a) Transversely Isotropic\nMCA,K=\u00004500J=m3, (b) Cubic MCA for Fe 79:1Ga20:9,K1=\u00001e3J=m3and\nK2= 1e4J=m3, (c) and Cubic MCA for Fe 81:6Ga18:4,K1= 3:5e4 J=m3andK2=\u00008e4\nJ=m3[59]. Inset graphs contain the probability density of the systems at Points 1 and\n2\nby relying on numerical integration (e.g., Riemann sums), to compare cubic MCA to\ntransversely isotropic.\nFigure 6 (a) shows a Type I MH curve produced using the proposed model under 50\nMPa of compression, and a Kvalue of -4500 (J/m3), simulating the Terfenol-D modeled\nabove. To understand how the MH curve saturates we examine the probability landscape\nof two points along the curve. For each point of interest, the probability density of the\nsystem in spherical coordinates is calculated using P(\u0012;H;\u001b;\f ) = exp (\u0000\fgL)=z. The\ninset of Figure 6 (a) plots the probability density against the possible magnetization\ndirections\u0012. Starting at Point 1, \u00160H= 15 mT in the negative zdirection, the applied\n\feld has moved the peak probability from \u0012=\u0019=2 a small amount towards \u0012=\u0019. As\n\u00160Hincreases to 20 mT and we transition along the MH curve to Point 2, the peak\nagain gradually shifts to the right. As the \feld is continually increased past Point 2, the\nlocation of peak probability continues to gradually shift towards \u0012=\u0019. It is this gradual\nchange in the probable direction of magnetization that produces Type I behavior. The\nonly noticeable change in the MH curve is the \fnal approach to saturation. This occurs\nwhen the location of peak probability is exactly at \u0012=\u0019, and instead of moving, it\nsimply becomes more and more probable (i.e., corresponding to an increasingly large\nenergy minima at that point).\nFigure 6 (b) shows an MH curve for Fe 79:1Ga20:9produced using K1=\u00001e3 J=m3\nandK2= 1e4 J=m3[59] and placing the material under 50 MPa of compression. For\nthis MCA energy minima occur in the h100ifamily of directions when h= 0 and\u001b= 0.\nApplying stress causes the minima to appear exclusively in the h100i,h\u0016100i,h010i, and\nh0\u001610idirections (i.e.,\u0006xand\u0006y). To plot the 2D probability landscape in a single\nline the inset of Figure 6 (b) shows P(\u0012) =P(\u0012;\u001e=\u0019=4). In the inset of Figure 6Nonlinear One-Dimensional Constitutive Model for Magnetostrictive Materials 17\n(b) the exact same trend as the inset of Figure 6 (b) is observed. Starting at Point 1,\n\u00160H= 15 mT, and moving to point 2, \u00160H= 20 mT, we see the same gradual shift in\nthe probable direction of magnetization which produces Type I behavior.\nFinally, 6 (c) shows an MH curve for Fe 81:6Ga18:4produced using K1= 3:5e4 J=m3\nandK2=\u00008e4 J=m3[59] and placing the material under 50 MPa of compression. Of\nnote the points of interest were speci\fcally chosen to highlight the Type II transition\nfrom concave up to down and again occur at \u00160H= 15 mT and 20 mT. Also as this\nis another cubic material, the single line plotted in the inset graph is the maximum\nprobability, P(\u0012) =P(\u0012;\u001e= 0). At Point 1 the direction with the highest probability\nis slightly to the right of \u0012=\u0019=2. However, as we move to Point 2 there is a signi\fcant\nchange in the probability landscape. The global minima abruptly shifts to \u0012=\u0019and\nan additional probability peak emerges. This abrupt change in the probable direction\nof magnetization is what causes the magnetization curve to change from concave up\nto down, and the presence of this second peak is what has kept the material from\nsaturating. Due to the assumption of transversely isotropic MCA the model in this\npaper is incapable of simulating Fe 81:6Ga18:4, or other materials with similar anisotropies\nthat produce abrupt changes in the probability landscape.\n4. Conclusion\nThis paper provides an analytical one-dimensional constitutive model for magnetostric-\ntion. Closed form analytical solutions were provided to calculate the average magneti-\nzation, magnetostriction, susceptibility, compliance, and piezomagnetic coupling coe\u000e-\ncient. Additionally, it was demonstrated that the analytical model maintains numerical\naccuracy over a large range of applied magnetic \felds and stress / anisotropy conditions.\nFinally, the model was used to simulate experimental data for three di\u000berent materials.\nThis comparison only required \ftting two model parameters to the data. Comparisons\nbetween the experimental and modeled results indicate that the model is capable of sim-\nulating Terfenol-D and is expected to also accurately describe certain cubic materials\nas long as they have probability landscapes that are well approximated as transversely\nisotropic.Nonlinear One-Dimensional Constitutive Model for Magnetostrictive Materials 18\n5. Appendix\n5.1. Low Anisotropy Series Expansion\nAs previously stated, the solutions become indeterminate when A= 0. By performing\na series expansion of exp ( Am2\n1) aboutA= 0 we not only provide a solution for when\nA= 0, but also obtain an accurate solution when h>>jAj. Expanding the exponential\naboutA= 0 and substituting it into the partition function we \fnd that,\nz=2\u0019=NX\nn=0zn=NX\nn=01Z\n\u00001(\u0000Am2\n1)n\nn!exp (hm1)dm1 (26)\nwhich has the general solution\nzn=(2n)!(\u0000A)n(\u0000h)\u00002n\nhn! \n(\u00001)4n+1\u00001 +2nX\nk=0(\u0000h)keh+ (\u00001)4n+1hke\u0000h\nk!!\n:(27)\nThis expression has been provided in terms of polynomial expansions, however slightly\nmore compact expressions can be obtained in terms of gamma functions. As the\nexpressions can become very lengthy, the authors utilized a computer algebra system\n(Mathematica) to simplify the expressions and export them for use in Matlab.\nUsing the thermodynamic relationships described above both the average material\nresponse and material properties can be found by taking partial derivatives of znwith\nrespect tohandAfor a desired level of accuracy, controlled by N. In the error analysis\nbelow, a value of N= 2 was utilized.\n5.2. Material Properties\nMagnetization and magnetostriction (Equations (6) and (7)) are proportional to hmi\nandhm\nmi. When restricted to one dimensional behavior, this can be simpli\fed to\nhMi=Mshmi (28)\nh\"mi=3\n2\u0015s\nm2\u000b\n(29)\nUsing this new notation the relationships in equations (8) - (10) can be rewritten as\nh\u001fi=\f\u00160M2\ns\u0002\nm2\u000b\n\u0000hmi2\u0003\n(30)\nhSmi=\f\u00123\u0015s\n2\u00132h\nm4\u000b\n\u0000\nm2\u000b2i\n(31)\nhqi=\fMs3\u0015s\n2\u0002\nm3\u000b\n\u0000hmi\nm2\u000b\u0003\n(32)\nThe de\fnitions for hmiandhm2iwere previously presented in equations (21)\nand (22). The only terms yet to be de\fned are the tensor productsNonlinear One-Dimensional Constitutive Model for Magnetostrictive Materials 19\nfhm\nm\nmi;hm\nm\nm\nmigsimpli\fed in one dimension as fhm3i;hm4igre-\nspectively. These two terms and are shown here.\n\nm3\u000b\n=8\n>>>>>>>>>>>>>>>><\n>>>>>>>>>>>>>>>>:1\n~z+8A7=2\u0010p\nA\u0000\n4A2\u0000\n1\u0000e\u00002h\u0001\n\u00002A\u0000\ne\u00002h(h\u00002) +h+ 2\u0001\n+\u0000\n1\u0000e\u00002h\u0001\nh2\u0001\u0001\n\u00006Ah+h3\n8A3 A> 0\n1\n~z\u00008A3\u0000\n4A2\u0000\n1\u0000e\u00002h\u0001\n\u00002A\u0000\ne\u00002h(h\u00002) +h+ 2\u0001\n+\u0000\n1\u0000e\u00002h\u0001\nh2\u0001\n+6Ah\u0000h3\n8A3 A< 0; \r+>0\n1\n~z\u00008A3\u0010\ne\u0000\r12\u0000\r22\u0000\u00001\n2\u0000\neh\u0000e\u0000h\u0001\n(6Ah+ 8(A\u00001)A+ 5h2)\n+1\n2\u0000\ne\u0000h+eh\u0001\nh(2A+ 3h)\u0001\ne\r12+h\u00003e\r22h(2A+h)\u0011\u0011\n+6Ah\u0000h3\n8A3A< 0; \r+<0\n(33)\n\nm4\u000b\n=8\n>>>>>>>>>>>>>>>>>>><\n>>>>>>>>>>>>>>>>>>>:1\n~z+8A4\u0000\nA\u0000\ne\u00002h+ 1\u0001\n(4A2\u00006A+h2)\n\u00001\n2\u0000\n1\u0000e\u00002h\u0001\nh(2A(2A\u00005) +h2)\u0001\n+12A2\u000012Ah2+h4\n16A4 A> 0\n1\n~z\u000016A4\u0000\ne\u00002h\u0000\n8A3\u0000\ne\u00002h+ 1\u0001\n\u00004A2\u0000\ne\u00002h(3\u0000h) +h+ 3\u0001\n+2Ah\u0000\ne\u00002h(h\u00005) +h+ 5\u0001\n\u0000\u0000\n1\u0000e\u00002h\u0001\nh3\u0001\u0001\n+12A2\u000012Ah2+h4\n16A4 A< 0; \r+>0\n1\n~z\u000016A4\u0010\ne\u0000\r22\u0000\n8A3\u0000\ne2h+ 1\u0001\n+ 4A2\u0000\nh\u0000e2h(h+ 3)\u00003\u0001\n\u00002Ah\u0010\n\u00006he\r22\u0000\r12+ 5e2h(h\u00001)\u0000h+ 5\u0011\n\u0000h3\u0010\n\u00006e\r22\u0000\r12+ 7e2h\u00001\u0011\u0011\u0011\n+12A2\u000012Ah2+h4\n16A4 A< 0; \r+<0\n(34)\n5.3. Error Surfaces\nThe following \fgures are the error surfaces generated by comparing the presented\nmodel to numerical integration. A grid of N= 100 logarithmically spaced \feld points\n10\u00002\u0014h\u0014106andN= 200 logarithmically spaced anisotropies 10\u00002\u0014j\u0006Aj\u0014106\nwere generated for Ntotal= 20;000 points. At each grid point numerical integration\nwas performed using Matlab's integral() function with relative and absolute errors\nof 10\u000012. The relative errors for each equation were calculated as jfnum\u0000feqnj=fnum,\nwherefis the parameter of interest.\nFigure 7 examines equation (19) for ~ zwhich notably includes the energy o\u000bsets\nshown in (20). Including these o\u000bsets changes the numerical value of the partition\nfunction, however only the slope of ~ zproduces observable quantities, so the shift has\nnegligible physical impact. The yellow triangular region, which has relative errors\nranging from 10% to 100%, is where the low anisotropy series expansion was employedNonlinear One-Dimensional Constitutive Model for Magnetostrictive Materials 20\nFigure 7. Logarithmically scaled relative error of the equation (19) for ~ zcompared\nto standard numerical integration with absolute errors 10\u000012\n(jAj=h < 10\u00007). While this error is quite high Figures 8 and 9 below show that\nthe low anisotropy expansion maintained low relative errors for the magnetization\nand magnetostriction, respectively. Outside the expansion region the equations for ~ z\nmaintains an average relative error near 10\u000014.\nFigure 8 compares equation (21) for hMito numerical integration. Over all\ntested \feld values the relative error ranged from only 10\u000016to 10\u00008showing that the\nmagnetization solutions maintain signi\fcant numerical accuracy for all applied \felds and\nstresses. The error starts increasing as h=jAjgrows, however once the low anisotropy\nexpansion is utilized the error returns to 10\u000013.\nFinally, Figure 9 compares equation (22) for h\"mito numerical integration. The\nmaximum observed error in this graph remains \u001410\u00003. While the error once again\nclimbs as both h=jAjgrows, the low anisotropy expansion prevents the error from\nclimbing any larger.Nonlinear One-Dimensional Constitutive Model for Magnetostrictive Materials 21\nFigure 8. Logarithmically scaled relative error of equation (21) for hMicompared to\nstandard numerical integration with absolute errors 10\u000012\nFigure 9. Logarithmically scaled relative error of equation (22) for h\"micompared to\nstandard numerical integration with absolute errors 10\u000012\nReferences\n[1] Claeyssen F, Lhermet N, Le Letty R and Bouchilloux P 1997 Journal of Alloys and Compounds\n25861{73 ISSN 0925-8388\n[2] Davino D, Giustiniani A, Visone C and Adly A 2011 Journal of Applied Physics 10907E509 ISSN\n0021-8979\n[3] Deng Z and Dapino M J 2018 Smart Materials and Structures 27113001 ISSN 0964-1726, 1361-\n665X\n[4] Lafont T, Gimeno L, Delamare J, Lebedev G A, Zakharov D I, Viala B, Cugat O, Galopin N,\nGarbuio L and Geo\u000broy O 2012 Journal of Micromechanics and Microengineering 22094009Nonlinear One-Dimensional Constitutive Model for Magnetostrictive Materials 22\nISSN 0960-1317\n[5] Li P, Liu Q, Zhou X, Xu G, Li W, Wang Q and Yang M 2021 Journal of Vibration and Control\n27573{581 ISSN 1077-5463\n[6] Narita F and Fox M 2018 Advanced Engineering Materials 201700743 ISSN 1438-1656\n[7] Wang L and Yuan F G 2008 Smart Materials and Structures 17045009 ISSN 0964-1726\n[8] Wang N j, Liu Y, Zhang H w, Chen X and Li Y x 2016 China Foundry 1375{84 ISSN 2365-9459\n[9] Zenkour A M and El-Shahrany H D 2020 Applied Mathematics and Mechanics 411269{1286 ISSN\n1573-2754\n[10] Domann J P and Carman G P 2017 Journal of Applied Physics 121044905 ISSN 0021-8979\n[11] Zhao X G and Lord D G 1998 Journal of Applied Physics 837276{7278 ISSN 0021-8979\n[12] Mahadevan A, Evans P G and Dapino M J 2010 Applied Physics Letters 96012502 ISSN 0003-6951\n[13] Carman G P and Mitrovic M 1995 Journal of Intelligent Material Systems and Structures 6673{\n683 ISSN 1045-389X\n[14] Wan Y, Fang D and Hwang K C 2003 International Journal of Non-Linear Mechanics 381053{\n1065 ISSN 0020-7462\n[15] Shi P, Jin K and Zheng X 2016 Journal of Applied Physics 119145103 ISSN 0021-8979\n[16] Zhang D G, Li M H and Zhou H M 2015 AIP Advances 5107201\n[17] Zhou H M, Li M H, Li X H and Zhang D G 2016 Smart Materials and Structures 25085036 ISSN\n0964-1726\n[18] Kim S, Kim K, Choe K, JuHyok U and Rim H 2020 AIP Advances 10085304\n[19] Atulasimha J 2006 Characterization and Modeling of the Magnetomechanical Behavior of Iron-\nGallium Alloys Ph.D. thesis University of Maryland URL https://drum.lib.umd.edu/handle/\n1903/3951\n[20] Evans P G and Dapino M J 2008 IEEE Transactions on Magnetics 441711{1720 ISSN 1941-0069\n[21] Armstrong W D 1997 Journal of Applied Physics 812321{2326 ISSN 0021-8979, 1089-7550\n[22] Dzialoshinskii I E 1957 Journal of Experimental and Theoretical Physics 2\n[23] Newnham R E 2005 Properties of Materials: Anisotropy, Symmetry, Structure (OUP Oxford)\nISBN 978-0-19-852075-7\n[24] Ibrahim M and Salehian A 2015 Journal of Intelligent Material Systems and Structures 261259{\n1271 ISSN 1045-389X\n[25] Yan Z 2018 Smart Materials and Structures 27015016 ISSN 0964-1726, 1361-665X\n[26] Sablik M J, Kwun H, Burkhardt G L and Jiles D C 1987 Journal of Applied Physics 613799{3801\nISSN 0021-8979\n[27] Li J and Xu M 2011 Journal of Applied Physics 110063918 ISSN 0021-8979\n[28] Wang Z D, Deng B and Yao K 2011 Journal of Applied Physics 109083928 ISSN 0021-8979\n[29] Datta S, Atulasimha J, Mudivarthi C and Flatau A B 2010 Journal of Magnetism and Magnetic\nMaterials 3222135{2144 ISSN 0304-8853\n[30] Datta S 2009 Quasi-Static Characterization and Modeling of the Bending Behavior of Single Crystal\nGalfenol for Magnetostrictive Sensors and Actuators Ph.D. University of Maryland, College Park\nUnited States { Maryland URL https://www.proquest.com/docview/304924991/abstract/\n3D4C6C6294454CCFPQ/1\n[31] Hubert O and Daniel L 2010 IEEE Transactions on Magnetics 46401{404 ISSN 1941-0069\n[32] Nan C W, Bichurin M I, Dong S, Viehland D and Srinivasan G 2008 Journal of Applied Physics\n103031101 ISSN 0021-8979\n[33] Hatipoglu G and Tadigadapa S 2015 Applied Physics Letters 107192406 ISSN 0003-6951\n[34] Viehland D, Wuttig M, McCord J and Quandt E 2018 MRS Bulletin 43834{840 ISSN 0883-7694,\n1938-1425\n[35] Bertotti G 1998 Hysteresis in Magnetism: For Physicists, Materials Scientists, and Engineers\n(Gulf Professional Publishing) ISBN 978-0-12-093270-2\n[36] Chikazumi S 2009 Physics of Ferromagnetism second edition ed International Series of Monographs\non Physics (Oxford, New York: Oxford University Press) ISBN 978-0-19-956481-1Nonlinear One-Dimensional Constitutive Model for Magnetostrictive Materials 23\n[37] Cullity B D and Graham C D 2009 Introduction to Magnetic Materials 2nd ed (Hoboken, N.J:\nIEEE/Wiley) ISBN 978-0-471-47741-9\n[38] O'Handley R C 2000 Modern Magnetic Materials: Principles and Applications (New York: Wiley)\nISBN 978-0-471-15566-9\n[39] Kardar M 2007 Statistical Physics of Particles (Cambridge University Press) ISBN 978-1-139-\n46487-1\n[40] Landau L D and Lifshitz E M 1976 Mechanics: Volume 1 (Butterworth-Heinemann) ISBN 978-0-\n7506-2896-9\n[41] Atulasimha J, Flatau A B and Cullen J R 2008 Journal of Applied Physics 103 014901 ISSN\n0021-8979\n[42] Smith R C, Dapino M J and Seelecke S 2003 Journal of Applied Physics 93458{466 ISSN 0021-\n8979\n[43] Evans P G and Dapino M J 2010 Journal of Applied Physics 107063906 ISSN 0021-8979\n[44] Wahi S K, Kumar M, Santapuri S and Dapino M J 2019 Journal of Applied Physics 125215108\nISSN 0021-8979\n[45] Evans P G and Dapino M J 2009 Journal of Applied Physics 105113901 ISSN 0021-8979\n[46] Evans P G and Dapino M J 2013 Journal of Magnetism and Magnetic Materials 33037{48 ISSN\n03048853\n[47] Atulasimha J and Flatau A 2011 Smart Materials and Structures 20043001\n[48] Clark A, Restor\u000b J, Wun-Fogle M, Lograsso T and Schlagel D 2000 IEEE Transactions on\nMagnetics 363238{3240 ISSN 1941-0069\n[49] Elhajjar R, Law C T and Pegoretti A 2018 Progress in Materials Science 97204{229 ISSN 0079-\n6425\n[50] Wun-Fogle M, Restor\u000b J B and Clark A E 2006 Journal of Intelligent Material Systems and\nStructures 17117{122 ISSN 1045-389X\n[51] Mo\u000bett M B, Clark A E, Wun-Fogle M, Linberg J, Teter J P and McLaughlin E A 1991 The\nJournal of the Acoustical Society of America 891448{1455 ISSN 0001-4966\n[52] Nayfeh A H, Younis M I and Abdel-Rahman E M 2005 Nonlinear Dynamics 41211{236 ISSN\n0924-090X, 1573-269X\n[53] Wang Q, Li X, Liang C Y, Barra A, Domann J, Lynch C, Sepulveda A and Carman G 2017 Applied\nPhysics Letters 110102903 ISSN 0003-6951, 1077-3118\n[54] Younis M, Abdel-Rahman E and Nayfeh A 2003 Journal of Microelectromechanical Systems 12\n672{680 ISSN 1941-0158\n[55] Cody W J 1993 ACM Transactions on Mathematical Software 1922{30 ISSN 0098-3500, 1557-7295\n[56] Raghunathan A, Melikhov Y, Snyder J E and Jiles D C 2009 Applied Physics Letters 95172510\nISSN 0003-6951\n[57] Talleb H, Do T A, Gensbittel A and Ren Z 2020 IEEE Transactions on Magnetics 561{4 ISSN\n1941-0069\n[58] McCabe J H 1974 Mathematics of Computation 28811{816 ISSN 0025-5718, 1088-6842\n[59] Ra\fque S, Cullen J R, Wuttig M and Cui J 2004 Journal of Applied Physics 956939{6941 ISSN\n0021-8979, 1089-7550"    },    {        "title": "2109.11122v1.The_free_energy_of_twisting_spins_in_Mn__3_Sn.pdf",        "content": "The free energy of twisting spins in Mn 3Sn\nXiaokang Li1;\u0003, Shan Jiang3;1, Qingkai Meng1, Huakun Zuo1, Zengwei Zhu1;\u0003, Leon Balents2;4and Kamran Behnia3\n(1) Wuhan National High Magnetic Field Center and School of Physics,\nHuazhong University of Science and Technology, Wuhan 430074, China\n(2)Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106-4030, USA\n(3) Laboratoire de Physique et d' \u0013Etude des Mat\u0013 eriaux\n(ESPCI - CNRS - Sorbonne Universit\u0013 e), PSL Research University, 75005 Paris, France\n(4) Canadian Institute for Advanced Research, Toronto, Ontario, Canada\n(Dated: September 24, 2021)\nThe magnetic free energy is usually quadratic in magnetic \feld and depends on the mutual ori-\nentation of the magnetic \feld and the crystalline axes. Tiny in magnitude, this magnetocrystalline\nanisotropy energy (MAE) is nevertheless indispensable for the existence of permanent magnets.\nHere, we show that in Mn 3Sn, a non-collinear antiferromagnet attracting much attention follow-\ning the discovery of its large anomalous Hall e\u000bect, the free energy of spins has superquadratic\ncomponents, which drive the MAE. We experimentally demonstrate that the thermodynamic free\nenergy includes terms odd in magnetic \feld ( O(H3) +O(H5)) and generating sixfold and twelve-\nfold angular oscillations in the torque response. We show that they are quantitatively explained by\ntheory, which can be used to quantify relevant energy scales (Heisenberg, Dzyaloshinskii-Moriya,\nZeeman and single-ion anisotropy) of the system. Based on the theory, we conclude that, in contrast\nwith common magnets, what drives the MAE in Mn 3Sn is the \feld-induced deformation of the spin\ntexture.\nAligned spins located on two adjacent atoms are af-\nfected by the anisotropic electrostatic forces connecting\ntheir orbital angular momenta [1]. This magnetocrys-\ntalline anisotropy energy (MAE), a consequence of the\nspin-orbit coupling, is remarkably small ( \u001860\u0016eV/atom\nin Co and\u00181\u0016eV/atom in Fe and Ni). Since it is the\noutcome of the competition between energies many or-\nders of magnitude larger, it is hard to calculate from \frst\nprinciples [2, 3].\nMn 3Sn, a noncollinear antiferromagnet with an inverse\ntriangle spin structure located on a breathing kagome\nlattice [4] has attracted much attention following the\nobservation of a large anomalous Hall e\u000bect(AHE) [5]\nwith a sizeable net Berry curvature near the Fermi\nlevel [6]. The discovery was followed by the observation\nof various counterparts of AHE, including the anoma-\nlous Nernst [7, 8] and the anomalous thermal Hall ef-\nfects [8{10], as well as the anomalous magneto-optical\nKerr e\u000bect [11, 12]. These are room-temperature e\u000bects\nrequiring a small magnetic \feld. Therefore, Mn 3Sn is\npotentially attractive in the \feld of antiferromagnetic\nspintronics [13{16] or as a Nernst thermopile [7, 17].\nThe peculiar spin texture of Mn 3Sn has been subject\nof several studies [18{23]. The magnetic Hamiltonian\nincludes Heisenberg and Dzyloshinskii-Moriya spin-spin\ninteraction terms dominating by far the small single-ion\nanisotropy term [19]. A study of torque magnetome-\ntry [24] quanti\fed the latter. Previous experiments have\ndocumented that magnetic domain walls are chiral [25]\nand host a topological Hall e\u000bect associated with a \fnite\nskyrmionic number [8].\nHere, we present a study of magnetization and angle-\ndependent magnetic torque in Mn 3Sn single crystals and\n\fnd an additional term in the magnetic free energy with\nan intriguingly delicate anisotropy. Magnetization, on\ntop of its zero-\feld (spontaneous) and \feld-linear com-ponents, has a previously undetected term, which is\nquadratic in magnetic \feld. This implies that the free\nenergy has a component proportional to the cube of the\nmagnetic \feld [26]. We show that, in contrast to or-\ndinary ferromagnets, this second-order magnetization is\nthe driver of the magneto-crystalline anisotropy. This\nconclusion is based on our torque magnetometry data,\nwhich uncovers a \feld-dependent MAE. The angular os-\ncillations of the torque signal in addition to its two-\nfold,K2, and sixfold, K6, components detected previ-\nously [24], have an additional twelve-fold, K12, compo-\nnent. The \feld dependence of the two intrinsic shape\nindependent terms ( K6/H3andK12/H5) establish\nthat they are caused by the second-order magnetization.\nBoth the six-fold and twelve-fold anisotropies are quan-\ntitatively reproduced by a theoretical calculation based\non the Hamiltonian of ref. [19]. Fits to the experimental\nquanti\fes the Zeeman energy, the Heisenberg (J) and the\nDzyaloshinskii-Moriya (DM) interaction term, and \fnally\nthe single-ion-anisotropy energy, the cost of misalign-\nment between spin lattice and crystal axes. The analy-\nsis demonstrates that the unusual features are driven by\n\feld-induced twisting of the spins, which result from the\nfrustrated competition of the above four energy scales,\nparticular to the inverse triangular spin structure.\nCentimeter-size Mn 3Sn single crystals were grown by\nthe vertical Bridgman technique [25]. As shown in\nFig. 1a, Mn atoms form a breathing kagome lattice. In\nthis inverse triangle spin lattice [4, 5], large magnetic\nmoments of the Mn atoms cancel each other. As seen\nin Fig. 1b-c, a weak, and mainly in-plane, ferromag-\nnetism persists, which may be caused by a slight devi-\nation of spin orientations from a perfect triangular align-\nment [4, 5]. A collection of six Mn atoms located in two\nadjacent layers constitutes the simplest unit of the spin\ntexture and has been treated as a magnetic octupole spinarXiv:2109.11122v1  [cond-mat.str-el]  23 Sep 20212\n-12-8-404812-300-200-1000100200300-\n1.2-0.8-0.40.00.40.81.2-30-20-100102030-12-8-404812-12-8-404812-\n12-8-404812-4-20240\n5 01 001 501 96-12-8-4048120\n5 01 001 501 96-4-2024\na \nM-χ/s615490H (mµB per f.u.)McMab M (mµB per f.u.) \n \n  \nMc (H // z) \nMab (H // xy-plane)T = 300 K \n/s6154902H2 (T2) /s615490H (T)  \n/s615490H (T) \n  M0+M2( ∝ O(H2))g\nfe\ndc\n \n \n \n  \nM0+M2( ∝ O(H2))b \n \n \n   \n \n  \n \n \n \nFIG. 1. A quadratic term in the magnetization of Mn 3Sn:(a) The spin texture of Mn 3Sn. Mn magnetic moments are\nlocating on an AB-stacked breathing kagome lattice. (b-c) In-plane, Mab, and the out-of-plane, Mc, magnetization in Mn 3Sn\nat 300 K, as a function of magnetic \feld swept from 14 T to -14 T. (d-e) The residual component of Mabafter subtracting\nits dominant linear term M1(/O(H)). It is plotted with the linear (d) and quadratic (e) x-coordinate respectively. There is\nan additional component, M2(/O(H2)). (f-g) The residual component of Mc. The sign of its second-order magnetization is\nopposite to Mab.\n02 4 6 8 1 012140501001502002500\n2 4 6 8 1 012144681012140\n3 06 09 06.06.57.07.516016517017518023456\ndcb\n \n/s615490H (T) \n/s615490H (T)T = 300 KM\nab(H // xy-plane) M (mµB per f.u.) \n  \n0° \n30° \n60° \n90°aM\nagnetic field HVSM quartz s\nample rodQ uartz wedge/s61553\nMn3Sn sample \n \n M-χ/s615490H (mµB per f.u.) \n0° \n30° \n60° \n90° \n M\n0+M2( ∝ O(H2))xy-plane \nM2 (mµB per f.u.) M1 (mµB per f.u.) M0 (mµB per f.u.)/s61553\n (°) 7\n%2 % /s615490H = 13 T40%\nFIG. 2. Anisotropy of the quadratic magnetization :\n(a) Experimental con\fguration for the angular magnetization.\n(b-c) Comparison of Maband it's residual after subtracting\nthe linear background, with four di\u000berent angles, 0\u000e, 30\u000e, 60\u000e\nand 90\u000erespectively. (d) The angle dependence of the M0,\nM1, andM2. Only the latter show angular oscillations with\na periodicity of 60 degrees.\ncluster [20].\nOur \frst new \fnding is shown in Fig. 1d-g. After sub-\ntracting a linear background ( M1(/O(B)), we \fnd that\nin addition to the spontaneous magnetization, M 0, thereis an additional \feld-dependent term (Fig. 1d) in mag-\nnetization. As seen in Fig. 1e, this term is quadratic in\nmagnetic \feld. Such a term is also present in the out-of-\nplane magnetization too, albeit with a smaller amplitude\nand an opposite sign (See Fig. 1f-g). We conclude that\nthe magnetization consists of three terms:\nMtotal=M0+M1(/O(H)) +M2(/O(H2)):(1)\nThe \frst term is the zero-\feld spontaneous magnetiza-\ntion associated with the weak residual ferromagnetism.\nThe second term ( M1) is the dominant linear magneti-\nzation resolved in previous studies of magnetization [5].\nThe third term, M2, not detected in previous studies is\nquadratic in \feld. It represents a second-order correction\nto the magnetization response [26{29]. Since the magne-\ntization is the partial derivative of the magnetic free en-\nergy with respect to the magnetic \feld ( M=@FM=@H),\na \fniteM2implies an additional term for the magnetic\nfree energy, which can be written as:\nFM=X\niM0;iHi+1\n2X\ni;j\u001fi;jHiHj+1\n3X\ni;j;kCi;j;kHiHjHk:\n(2)\nHere,Ci;j;kis a 3\u00023\u00023 tensor, which represents\nthe second odd term in the \feld dependence of the free\nenergy.\nThe angular variation of these three terms was inves-\ntigated by performing the angular dependent magneti-\nzation measurements, using the set-up shown in Fig. 2.\nRotation was achieved by changing the sharp angle of a3\n-90-60-300 3 06 09 0-1.0-0.50.00.51.0-\n90-60-300 3 06 09 0-6-3036  /s61556 (kJ/m3) \n  \n1 T \n2 T \n3 T \n4 T \n5 T \n6 TInsulating substrateI nsulating substratec\nb\nT  = 300 K, H  // xy-plane/s61553\n ( °) \n  \n7 T \n8 T \n9 T \n10 T \n11 T \n12 T \n13 TaH\nomemade capacitive torque m\nagnetometer:I\nnsulating substrateMetal cantileverM\netal plateSamplePhotograph of the experimental setup:x\ny-plane\nFIG. 3. Magnetic torque measurements: (a) The home-\nmade experimental setup and its photograph (top) with a\ncapacitive torque magnetometer, rotating the \feld in the xy-\nplane of the Mn 3Sn sample. (b) The angle dependent torque\nresponses for magnetic \felds up to 6 T. (c) The angle de-\npendent torque responses for magnetic \felds larger than 6 T.\nAs the \feld increases, the shape of oscillations evolve, indi-\ncating the emergence of an additional component. Hkx\ncorresponds to 0\u000e.\nquartz wedge held between the sample and the sample-\nholder. Fig. 2b and c show magnetization and its residual\nafter subtracting the linear background as a function of\n\feld for four di\u000berent angles in xy-plane.\nThe angle dependence of in-plane M0,M1, andM2are\nplotted in Fig. 2d. Angular variation is undetectable for\nM0andM1, but as large as 37 percent for M2and dis-\nplays a clear six-fold symmetry, with minima and max-\nima separated by 30 degrees. We conclude that the third\nterm of free energy F3/O(H3) has a sixfold anisotropic\ncomponent. In other words, M2=M2;0+M2;6cos(6\u001e).\nAt 13 T,M2;0= 4:4m\u0016B/f.u., andM2;6= 0:8m\u0016B/f.u.\nAs we will see below, the magnitude of the latter term\nis con\frmed with a higher precision by measurements\nof the magnetic torque, \u001c, which quanti\fes the angular\nderivative of the magnetic free energy( \u001c=@FM=@\u0012).\nFig. 3a shows a sketch and a photograph of the torqueset-up. The applied magnetic \feld rotates in the ab-\nplane of the Mn 3Sn sample, and the magnetic torque\nalong the c-axis is detected by the variation in the ca-\npacitance between two metal plates. Fig. 3b shows the\nangular dependence of the torque signal below 6 T resolv-\ning a clear six-fold symmetry. However, as the magnetic\n\feld increases further, a deviation becomes visible. As\nseen in Fig. 3c, when the magnetic \feld increases from 6\nT to 13 T, the shape of each oscillation evolves. Instead\nof being sine-like, it becomes sawtooth-like. When the\n\feld is along a high symmetry axis ( x;\u0000x;y;\u0000y), torque\nvanishes and the free energy is at its extremum, as one\nexpects. Remarkably, however, there is a widening dif-\nference between the slopes along the two high-symmetry\norientations. This is caused by the emergence of an ad-\nditional angle-dependent term.\nIn addition to the two-fold and six-fold terms detected\nby Duan et al. [24], our high-\feld data has an additional\nterm with twelve-fold symmetry. This can be seen in\nFig. 4a, which shows that the di\u000berence between the\nexperimental data and the sum of the two \frst terms\n(K2\u0001sin(2\u0001(\u0012+\u001e2))+K6\u0001sin(6\u0001(\u0012+\u001e6))). The residual\nhas a clear twelve-fold anisotropy ( K12\u0001sin(12\u0001(\u0012+\u001e12))).\nThe evolution of the \ftting parameters, K2,K6, andK12\nwith magnetic \feld is shown in Fig. 4d and e. K2(H) fol-\nlowsH3=2andK6(H) followsH3and remains the largest\nin the whole \feld range. Finally, K12(H) shows a steep\nincrease, which is close to H5at low \felds.\nThe strong \feld dependence of the torque compo-\nnents of Mn 3Sn is in sharp contrast with what is seen\nin ferromagnets[31], as con\frmed by our own data on\nbcc Fe (See the supplement[30]). Additionally, it implies\nthat a single magnetic \feld measurement (5 T in the\ncase of ref. [24], see Fig.4e) is not su\u000ecient to extract the\nmagnotcrystalline anisotropy. Let us now consider the\ninformation brought by these results on the components\nof magnetic free energy.\nThe two-fold term, K2, evolves much slower than H3\nand strongly depends on the sample aspect ratio (See\nFig. 4e). The \frst feature indicates that it is not caused\nby the second-order magnetization and the second fea-\nture implies a role played by sample geometry. K2can\nbe caused by a \fnite angle between the applied \feld and\nmagnetization, which can, in principle, occur in any sam-\nple with \fnite size.\nThe six-fold term, K6, shows a \feld dependence clearly\nlinking it to the \feld-cubic free energy ( F3/O(H3)),\ncon\frming what was deduced from measurements of\nangle-dependent magnetization. Both sets of data lead\nto the conclusion that there is a component of free en-\nergy with sixfold angular symmetry and cubic \feld de-\npendence (F(3;6)/H3cos6\u0012). A consistency check can\nbe done by comparing the magnitude of K6, the angle\nderivative and M2;6, the \feld derivative. At 13 T, K6is\n4700- 5600 Jm\u00003, implying F(3;6)\u00190:34\u00064\u0016eV=f:u:\nandM2;6is 0:8m\u0016Bper f.u. corresponding to F(3;6)\u0019\n0:20\u0016eV=f:u: . The list of all components of the free en-\nergy identi\fed by our experiments are given in Table I.4\n-90- 60- 300 3 06 09 0-8-4048-\n90- 60- 300 3 06 09 0-4-2024-\n90- 60- 300 3 06 09 0-8-40482\n1 02 00.0050.010.11100\n.010.11100\n.010.112\n1 02 00.0050.010.11T = 300 K, H // xy-plane  \n /s61556 (T = 300 K, H (= 13 T) // xy-plane) \n  \nFitting curve (K2⋅sin(2⋅(θ+φ2))+K6⋅sin(6⋅(θ+φ6)))  \n /s61556 - (K2⋅sin(2⋅(θ+φ2))+K6⋅sin(6⋅(θ+φ6))) \n  \nFitting curve (K12⋅sin(12⋅(θ+φ12)))eτ (kJ/m3)c\nb/s61553\n ( °)/s61556 (kJ/m3) \n \n τ   \n  \nFitting curve (K2⋅sin(2⋅(θ+φ2))+\nK6⋅sin(6⋅(θ+φ6))+K12⋅sin(12⋅(θ+φ12)))aH\n5H\n5H3H3/2d/s61549\n0H (T)H3/2H3 \n  \nK2 \nK6 \nK12 \n K\n2 \n# Rectangle \n# Square \n# Triangle \nDuan et al (2015)K6 \n  \nK12 \n \nFIG. 4. Components of the torque response: (a) Fit to the 13 T data with an expression, which has only a two-fold and\na six-fold term ( \u001c=K2\u0001sin(2\u0001(\u001e+\u001e2)) +K6\u0001sin(6\u0001(\u001e+\u001e6))). The mismatch is obvious. (b) The residual torque component\nafter subtracting the data and the previous \ft. It shows a clear twelve-fold symmetry. (c) A \ft which includes an additional\ntwelve-fold symmetry component ( K12\u0001sin(12\u0001(\u001e+\u001e12))). (d) Fitting parameters K2(H),K6(H) andK12(H) as a function of\nmagnetic \feld. K2(H) followsH3=2and is much smaller than K6(H).K6(H) followsH3.K12(H) followsH5at the beginning.\n(e) A comparison of K2(H),K6(H) andK12(H) in three di\u000berent samples with di\u000berent cross sections. Here, the sample with\nthe rectangular cross section has a larger aspect ratio ( lx=ly= 5.6), compared to the samples with square and triangle cross\nsections (lx=ly'1). The data reported by Duan et al. are also shown.\nComponent Gives rise to Amplitude at 13 T ( \u0016eV/Mn) Accounted by theory? Comments\nF(1;ab)/H M0 1.7 Yes residual FM\nF(1;c)/H M0 0.11 Canting? residual FM\nF(2;ab)/H2M1 21.4 Yes paramagnetism\nF(2;c)/H2M1 27.9 Yes paramagnetism\nF(3;0)/H3M2;0 0.37 No Field-induced correction to FM?\nF(3;6)/H3cos6\u0012 M2;6=K6 0.067/0.115 Yes Field-induced tilt of spins\nF(5;12)/H5cos12\u0012K12 0.018 Yes secondary correction to the tilt\nF(1:5;2)/H3=2cos2\u0012K2 0.028 No Boundary-related\nTABLE I. Components of the magnetic free energy in Mn 3Sn identi\fed by measurements of magnetization ( M0,M1andM2)\nand torque ( K2,K6andK12).\nLet us now show that theory provides a satisfactory\naccount of the existence and the amplitude of K6term as\nwell as the emergence and rapid growth of the secondary\nK12term with increasing magnetic \feld.\nFollowing Liu and Balents [19], the energy per mag-\nnetic unit cell (six spins) consists of the sum of four\nterms[30]: These are Heisenberg: 4 JP\niSi\u0001Si+1;\nDzyaloshinskii-Moriya (DM): 4 DP\ni^z\u0001Si\u0002Si+1; Single-\nion-anisotropy (SIA) : \u00002KP\ni(Si\u0001^ei)2; and Zeeman:\n\u00002\u0016P\niH\u0001Si. ForD> 0 and in absence of SIA and Zee-\nman terms, the ground state is an anti-chiral state with\nin-plane spins. A \fnite magnetic \feld will distort the spin\ntriangles. The distortion angles, \u0011i(See Fig.5 ) are small,because in our window of investigation ( H < 14T), one\nhasK\u001cJand\u0016H\u001cJ. Introducing a small parameter\nr\u001c1, withK=J; \u0016H=J ofO(r), one can expand \u00111;2in\na formal series in r:\n\u0011i=1X\nn=1\u0011i;nrn; (3)\nBy minimizing the energy terms over \u0011i;n, the small dis-\ntortions of the triangle at each order can be quanti\fed.\nThis leads to an expression for the free energy per unit\ncell. The \frst term is linear in magnetic \feld (See the5\n21 02 02x10-410-310-210-1100-\n2-1012-\n2-10120\n306090120150180-15-10-5051015-2-1012η3M\nHed S\nIAbZ\neemanD Mη2H ≠ 0S2S 3S1S\n3S 2/s61556\n (µeV/Mn)  \n/s615490H (T)  \nK6 (# Triangle) \nK6 (# Square) \n∂F(3,6)/∂/s61553 \nK12 (# Triangle) \nK12 (# Square) \n∂F(5,12)/∂/s61553S1H\n = 0η\n1H\neisenberga≠\nµ = 3 µBJ\n = 20.1 meVD\n/J = 0.18K\n/J = 0.0065H\nψ\nS\nample 3 T \n8 T \n13 T  Ψ(°) η3(°)c\nL\nag angle \nη2(°) η1 (°) \n \n   \n \n /s61553\n ( °)  \n \n   \n \n \nFIG. 5. Magneto-crystalline anisotropy driven by \feld-induced twist of non-aligned spins: (a) In absence of\nmagnetic \feld, the spin lattice is triangular. The magnetic \feld distorts the spin triangle (in white), which is no more\nisomorphic to the lattice triangle (in gray). The deformation angles \u0011iquantify the distortion. (b) Interaction between one\nspin and its immediate neighbors favor clockwise and anticlockwise twists. The Zeeman e\u000bect favors alignment of all spins\nwith magnetic \feld. Single-ion anisotropy causes in-equivalency between the two perpendicular orientations of the spin triangle\nwith respect to the lattice triangle. (c) When the magnetic \feld rotates, there is a lag angle  between the magnetic \feld and\nthe total magnetization . (d) The experimental K6andK12(symbols) compared to theoretical expectation (solid line) using\n\u0016= 3\u0016B,J= 20:1meV ,D=J = 0:18,K=J = 0:0065. (e) The deformation angles \u0011iand lag angle  at 3 T, 8 T and 13 T\npredicted by theory [30]. The three tilt angles follow the same pattern with an angular delay of 60\u000e. With increasing magnetic\n\feld, a secondary oscillation with a periodicity of 30\u000eemerges on top of a 60\u000eperiodicity.\nsupplement[30]):\nF(1;ab)=K\u0016H\nJ+p\n3D(4)\nThe quadratic term [30] has slightly di\u000berent expressions\nfor in-plane and out-of-plane orientations of magnetic\n\feld is:\nF(2;ab)=(\u0016H)2\n2J(1\u0000p\n3D\nJ)\nF(2;c)=(\u0016H)2\n2J(1\u0000Dp\n3J)(5)\nTherefore, one expects the quadratic free energy to be\nlarger for the out-of-plane orientation of the magnetic\n\feld, in agreement with what is seen experimentally (See\nTable I). For in-plane con\fguration, the \frst correction\nto the quadratic term has a cos6\u0012angle dependence. Its\namplitude is equal to:\nF(3;6)=(K+\u0016H)2((3J+ 7p\n3D)K+ 4p\n3D\u0016H )\n36(J+p\n3D)3(6)\nThe \feld dependence of this term is close to H3. As\none can see in Fig. 5d, this expression provides an ex-\ncellent account of the \feld and angular dependence ofthe experimentally observed K6. The next term has a\nsin2(6\u0012) angle dependence whose amplitude is equal to :\nF(5;12)=(K+\u0016H)2\n72(J+p\n3D)5\u0016HK((3J+ 7p\n3D)K2+\n2(J+ 4p\n3D)\u0016HK + 2p\n3D(\u0016H)2)2(7)\nThe \feld dependence of this term is close to H5and it\naccounts for the emergence of K12in the torque data and\nits \feld dependence. (See Fig. 5d).\nThe agreement between theory and experiment allows\nus to extract the energy scales of the system. Taking the\nmagnetic moment of each Mn atom to be \u0016= 3\u0016B, as\nreported by neutron di\u000braction studies [32, 33], we ex-\ntractedJ,D, andK. The results are summarized in\ntable II. It shows what is yielded by \ftting the torque\ndata with equations 6 and 7. Alternatively, one can use\nthe magnetization data and equations 4 and 5, the results\nare given in the second row of table II. As seen in the ta-\nble II,J=J1+J2= 20:1 meV, which is a third larger\nthan what is yielded by magnetization. The most plau-\nsible source of this di\u000berence is the existence of a \fnite\norbital contribution to magnetization [34] and assuming\na 30 percent orbital contribution to M 1would lead to\nconsistency. We note that our result is fairly close to the\nwhat has been reported by a study of magnon dispersion6\nby inelastic neutron scattering (18 meV)[21]. Finally,\nour study pins down the values for KandDand demon-\nstrates the capacity of torque magnetometry to quantify\nthe magnitude of Dzyaloshinskii-Moriya interaction in a\nfrustrated magnet.\nParameter J (meV) D/J K/J\nTorque 20.1(8) 0.18(2) 0.0065(5)\nMagnetization 13.8 0.18 0.0058\nTABLE II. The energy scales of Mn 3Sn extracted from the\ntorque and magnetization data using the theoretical expres-\nsions of the free energy.\nIn summary, we resolved di\u000berent components of the\nmagnetic free energy in Mn 3Sn. In addition to the dom-\ninant term, which is even in magnetic \feld, it includes\nodd terms with superquadratic \feld dependence and pre-\nsenting sixfold and twelvefold angular oscillations. We\nshowed that theory invoking a \feld-induced twist in the\norientation of in-plane spins can successfully explain the\npresence of these terms and their amplitude can be used\nto extract all energy scales of the system. The model alsomakes experimental predictions of speci\fc twists which\nmay be tested in the future in neutron scattering and\nanomalous Hall e\u000bect measurements. It is remarkable\nthat the simple localized spin description used here works\nin such quantitative detail for this highly itinerant mag-\nnet.\nThis work was supported by the National Sci-\nence Foundation of China (Grant No.51861135104 and\nNo.11574097) and The National Key Research and Devel-\nopment Program of China (Grant No.2016YFA0401704).\nK. B was supported by the Agence Nationale de la\nRecherche (ANR-18-CE92-0020-01; ANR-19-CE30-0014-\n04). S. J. acknowledges a PhD scholarship by the\nChina Scholarship Council (CSC). X. L. acknowledges\nthe China National Postdoctoral Program for Innovative\nTalents (Grant No.BX20200143) and the China Postdoc-\ntoral Science Foundation (Grant No.2020M682386). L.B.\nwas supported by the NSF CMMT program under Grant\nNo. DMR-2116515.\n*lixiaokang@hust.edu.cn\n*zengwei.zhu@hust.edu.cn\n[1] J. H. van Vleck, Phys. Rev. 52, 1178 (1937).\n[2] G. Daalderop, P. Kelly, and M. Schuurmans, Phys. Rev.\nB41, 11919 (1990).\n[3] S. V. Halilov, A. Y. Perlov, P. M. Oppeneer, A. N.\nYaresko, and V. N. Antonov, Phys. Rev. B 57, 9557\n(1998).\n[4] S. Tomiyoshi and Y. Yamaguchi, J. Phys. Soc. Jpn. 51,\n2478 (1982).\n[5] S. Nakatsuji, N. Kiyohara, and T. Higo, Nature 527,\n212 (2015).\n[6] H. Yang, Y. Sun, Y. Zhang, W.-J. Shi, S. S. Parkin, and\nB. Yan, New J. Phys. 19, 015008 (2017).\n[7] M. Ikhlas, T. Tomita, T. Koretsune, M.-T. Suzuki,\nD. Nishio-Hamane, R. Arita, etal., Nat. Phys. 13, 1085\n(2017).\n[8] X. Li, L. Xu, L. Ding, J. Wang, M. Shen, X. Lu, etal.,\nPhys. Rev. Lett. 119, 056601 (2017).\n[9] K. Sugii, Y. Imai, M. Shimozawa, M. Ikhlas, N. Kiy-\nohara, T. Tomita, etal., arXiv preprint arXiv:1902.06601\n(2019).\n[10] L. Xu, X. Li, X. Lu, C. Collignon, H. Fu, J. Koo, etal.,\nSci. Adv. 6, eaaz3522 (2020).\n[11] T. Higo, H. Man, D. B. Gopman, L. Wu, T. Koretsune,\nO. M. van't Erve, etal., Nature photonics 12, 73 (2018).\n[12] A. L. Balk, N. Sung, S. M. Thomas, P. F. S. Rosa, R. D.\nMcDonald, J. D. Thompson, etal., Appl. Phys. Lett.\n114, 032401 (2019).\n[13] L. \u0014Smejkal, Y. Mokrousov, B. Yan, and A. H. MacDon-\nald, Nat. Phys. 14, 242 (2018).\n[14] V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono,\nand Y. Tserkovnyak, Rev. Mod. Phys. 90, 015005 (2018).\n[15] M. Kimata, H. Chen, K. Kondou, S. Sugimoto, P. K.\nMuduli, M. Ikhlas, etal., Nature 565, 627 (2019).\n[16] H. Tsai, T. Higo, K. Kondou, T. Nomoto, A. Sakai,A. Kobayashi, etal., Nature 580, 608 (2020).\n[17] X. Li, Z. Zhu, and K. Behnia, Adv. Mater. 33, 2100751\n(2021).\n[18] J. Cable, N. Wakabayashi, and P. Radhakrishna, Phys.\nRev. B 48, 6159 (1993).\n[19] J. Liu and L. Balents, Phys. Rev. Lett. 119, 087202\n(2017).\n[20] M.-T. Suzuki, T. Koretsune, M. Ochi, and R. Arita,\nPhys. Rev. B 95, 094406 (2017).\n[21] P. Park, J. Oh, K. Uhl\u0013 \u0010\u0014 rov\u0013 a, J. Jackson, A. De\u0013 ak,\nL. Szunyogh, etal., npj Quantum Mater. 3, 1 (2018).\n[22] A. Zelenskiy, T. Monchesky, M. Plumer, and B. South-\nern, Phys. Rev. B 103, 144401 (2021).\n[23] S. Miwa, S. Iihama, T. Nomoto, T. Tomita, T. Higo,\nM. Ikhlas, etal., Small Science 1, 2000062 (2021).\n[24] T. Duan, W. Ren, W. Liu, S. Li, W. Liu, and Z. Zhang,\nAppl. Phys. Lett. 107, 082403 (2015).\n[25] X. Li, L. Xu, H. Zuo, A. Subedi, Z. Zhu, and K. Behnia,\nSciPost Phys. 5, 063 (2018).\n[26] D. Treves, Phys. Rev. 125, 1843 (1962).\n[27] G. Gorodetsky and D. Treves, Phys. Rev. 135, A97\n(1964).\n[28] N. Kharchenko, R. Szymczak, and M. Baran, J. Magn.\nMagn. Mater. 140, 161 (1995).\n[29] T. Liang, T. H. Hsieh, J. J. Ishikawa, S. Nakatsuji, L. Fu,\nand N. Ong, Nat. Phys. 13, 599 (2017).\n[30] See Supplemental Material for more details (2021).\n[31] C. D. Graham, Phys. Rev. 112, 1117 (1958).\n[32] P. Brown, V. Nunez, F. Tasset, J. Forsyth, and P. Rad-\nhakrishna, J. Phys. Condens. Matter 2, 9409 (1990).\n[33] S. Tomiyoshi, J. Phys. Soc. Jpn. 51, 803 (1982).\n[34] L. Sandratskii and J. K ubler, Phys. Rev. Lett. 76, 4963\n(1996).\n[35] R. M. Bozorth, Phys. Rev. 50, 1076 (1936).7\nSupplemental Material for \\The free energy of twisting spins in Mn 3Sn\"\nS1. SAMPLES AND METHODS\nMn 3Sn single crystals used in this work were cut to desired dimensions by a wire saw from a centimeter-size\nmother sample grown by the vertical Bridgman method [25]. A large-size sample (#1) was used for the magnetization\nmeasurements and three samples (#2, #3, #4) with di\u000berent cross-sections were used for the magnetic torque mea-\nsurements (See the table S1 for details). We also studied two bcc Fe samples, cut from the same commercial single\ncrystal, orientated along (110). One sample was cut to a thin-disk shape, with dimensions of 1 :3mm (diameter)\u0002\n0:12mm(thickness). Another sample was cut to a cuboid shape, with dimensions of 0 :39mm\u00020:36mm\u00020:6mm.\nAll magnetization measurements were performed in a physical property measurement system (Quantum design\nPPMS-16) with a vibrating sample magnetometer (VSM) option. All magnetic torque measurements were performed\nin a TeslatronPT (Oxford Instruments) with a homemade rotated probe equipped with the capacitive torque magne-\ntometer. A capacitance bridge (AH-2550A) was used to measure the variation of the capacitance between two metal\nplates proportional to change in the magnetic torque of the sample. Measurement of the rotation angle was calibrated\nby a Hall chip.\n#1 #2 #3 #4\nCross-section Square Triangle Square Rectangular\nlx=ly\u00181 1.15 1.2 5.6\nMass (mg) 30.0 3.12 4.4 1.15\nTechnique Magnetization Torque Torque Torque\nK2at 5T (kJ=m3)n 0.10 0.11 0.34\nTABLE S1. Details of Mn 3Sn samples used in this work.\nS2. REPRODUCIBILITY OF DATA\nAs seen in Fig. S1a, we repeated the torque measurements in three Mn 3Sn samples with di\u000berent cross-section.\nAll the samples show the same six-fold oscillation accompanied by a deviation at high \feld, suggesting the K6and\nK12are the intrinsic responses in Mn 3Sn. While in the sample #4 (with a rectangular cross-section) the two-fold\noscillation was more prominent. This indicates that the origin of K2is linked to the boundary geometry. As argued\npreviously [17, 25], there is a lag angle between the applied magnetic \feld and magnetization in Mn 3Sn and this lag\nangle increases with increasing aspect ratio of the sample [17, 25]. This provides an explanation for the larger K2in\nthe sample with the largest aspect ratio (Fig. 4e and see the table S1 for details).\nIn the case of the sample with a triangular cross section, we repeated the torque measurements at di\u000berent temper-\natures. The data is presented in Fig. S1b-c. When we compare the high \feld torque data at di\u000berent temperature,\nwe \fnd the K6andK12are robust from 300 K down to 100 K, as seen in Fig. S1b. But the signal collapsed suddenly\nat 50 K, nearing the transition temperature of the spin glass state [5], as seen in Fig. S1c. Our data suggests that\nthe angle-dependent magnetic free energy has little temperature dependence and implies that this is a property of the\ntriangular spin texture of the AF state which is destroyed at low temperature.\nS3. MAGNETIZATION AND MAGNETIC TORQUE IN BCC IRON\nIn order to compare with Mn 3Sn, we measured the magnetization of a disk-shape bcc Fe single crystal at 250 K,\nwith the \feld along (110) plane. As seen in Fig. S2a-b. Unlike Mn 3Sn, the dominant component in bcc Fe is the\nspontaneous magnetization, which saturates at 0.2 T, as seen in Fig. S2a. Focusing on the signal at high \feld region\nand having subtracted a linear background no quadratic magnetization is detectable (Fig. S2b).\nWe also measured the magnetic torque in the same crystal at 250 K, with the \feld rotating in (110) plane. Fig. S2c\nshows the photograph of the measurements. Fig. S2d-e show the angular torque responses in presence of di\u000berent\nmagnetic \felds. A clear four-fold oscillations combined with the two-fold component can be observed. Fig. S2f shows\nthe \ft of the data at 1 T. From which we extracted the amplitude of two-fold ( K2) and four-fold ( K4) components,8\n-505101520253035-\n90- 60- 300 3 06 09 0-4004080-\n90- 60- 300 3 06 09 0-10-50510152025/s61556\n (102 J/m3)/s61556 (102 J/m3)/s61553\n ( °) \n  \n300 K  250 K  200 K  150 K  100 Kbc\n 50 K \n Mn3Sn, H (= 11 T) // xy-planea /s61553\n ( °)/s61556 (kJ/m3) \n  \n# Rectangle (14 T) \n# Square (12 T) \n# Triangle (13 T)Mn3Sn, T = 300 K, H // xy-plane\nFIG. S1. Reproducibility of data: The magnetic torque responses in di\u000berent Mn 3Sn samples, with triangle, square, rectangular\ncross-section respectively (a). Temperature dependent magnetic torque responses in Mn 3Sn samples (# triangle), varying from\n300 K to 100 K (b), and 50K (c).\nand plotted them as a function of magnetic \feld in Fig. S2g. It can be seen that the dominant K4(H) begin to\nsaturate above 0.2 T, same to the literature [31], but the K2(H) still maintained a slow growth.\nComparison of the torque angular oscillations in bcc Fe and in Mn 3Sn reveals a fundamental di\u000berence. In Fe,\nthe toque (that is the angular dependence of the magnetic free energy) does not change with magnetic \feld (at least\nabove the coercive \feld). In Mn 3Sn, torque increases strongly with magnetic \feld ( /H3). In the \frst case, the\nangular dependence of magnetic free energy is caused by single-ion-anisotropy and is \feld independent. In the second\ncase, the larger the magnetic \feld, the larger the \feld-induced twist of the spins and the larger the energy cost of the\nrotation.\nWe also repeated the magnetization and magnetic torque of bcc Fe single crystal in another cuboid sample. As seen\nin Fig. S3, the magnetization responses have the higher saturation \feld ( Hs), about 0.5 T, and the in-plane curves are\nobviously separated, suggesting a larger anisotropy originated from the shape anisotropy. In contrast to the case in\nFig. S2, the torque responses in cuboid sample show a obvious two-fold oscillation, and the four-fold component only\nemerges above 1 T, as seen in Fig. S3. Moreover, the K2(H) is about ten times larger, imply the shape anisotropy\ncan have a remarkable contribution to the magnetic anisotropy in bcc Fe. However, it doesn't appear in Mn 3Sn case,\nas seen in Fig. S1.\nS4. COMPARISON WITH PREVIOUS STUDIES ON BCC IRON\nOur data is compatible with what was previously reported in the case of bcc Fe. However, there is a di\u000berence\nof notations. The parameters K iused in this work are distinct from those traditionally used in the case of cubic\nferromagnets. In the latter case, the free energy is written as\nE=K0+K\u0003\n1(\u000b2\n1\u000b2\n2+\u000b2\n2\u000b2\n3+\u000b2\n3\u000b2\n1) +K\u0003\n2(\u000b2\n1\u000b2\n2\u000b2\n3) +::: (S1)9\n06 01 201 802 40-20-10010200\n6 01 201 802 40-20-1001020300\n.11 5 1103006 01 201 802 40-20-1001020-\n2- 10 1 2 -3-2-101230\n.81 .21 .62 .02.062.082.102.122.142.162.18/s61556\n (kJ/m3)/s61556 (kJ/m3)/s61556 (kJ/m3)f/s61549\n0H (T)/s61556 (kJ/m3)  \n \n  \n0.13 T \n0.15 T \n0.2 T \n0.35 T  \n \n  \n0.5 T \n0.75 T \n1 T \n2 T \n3 Tb  /s61553 (°)g\ne\nd\nbcc Fe (thin disk sample), T = 250 K, H // (110) \n /s61553 (°) \n  \nK2 \nK4H1c \n \n Fitting curve (K2⋅sin(2⋅(θ+φ2))+K4⋅sin(4⋅(θ+φ6))) \n  \n1 TM\n - M1(∝H1)a \n  \n0° \n45° \n90°(110)M (µB per f.u.) \n/s615490H (T) \n \nFIG. S2. Magnetization and the magnetic torque in bcc Fe single crystal (thin disk sample): (a-b) Magnetization. Saturating\nat 0.2 T (a). No observable quadratic term (b). The demagnetizing factor is approach to 0, in the plane direction of the thin\ndisk. (c-g) The magnetic torque. The photograph of the measurements, the thin disk sample is \fxed with a quartz holder(c).\nThe angular torque responses at di\u000berent \feld, varying from 0.13 T to 0.35 T (d), 0.5 T to 3 T (e). The \ft of the data at 1 T,\nwith the formula ( \u001c=K2\u0001sin(2\u0001(\u0012+\u001e2)) +K4\u0001sin(4\u0001(\u0012+\u001e4))) (f). Field dependence of K2andK4(g).\nHere\u000biare direction cosines of the spontaneous magnetization with respect to the cubic axes [35]. How, when ( hkl)\n= (110), the torque can be written as \u001c=K\u0003\n1(2sin2\u0012+ 3sin4\u0012)=8 +K\u0003\n2(sin2\u0012+ 4sin4\u0012\u00003sin6\u0012)=64. Therefore,\nthere is a simple relationship between K\u0003\niand K i:K\u0003\n1=8\n5(4K2\u0000K4) andK\u0003\n2=64\n5(2K4\u00003K2).\nAs seen in Fig. S2f, K 4saturates to about 17 kJ/m3(1 kJ/m3= 10\u0001103ergs/cm3). On the other hand the saturated\nvalue for K 2can be estimates to be 11 kJ/m3. This would yield a K\u0003\n1and K\u0003\n2of 43.2 and 12.8 kJ/m3. As an see in\nthe table S2, these values are close to what was reported previously [31, 35].\nComponent Expression This work value (kJ/m3)Previously reported (kJ/m3)\n\u001c(This work) K2\u0001sin2\u0012+K4\u0001sin4\u0012[35] n n\n\u001c(Literature) K\u0003\n1\u0001(2sin2\u0012+ 3sin4\u0012)=8 +K\u0003\n2\u0001(sin2\u0012+ 4sin4\u0012\u00003sin6\u0012)=64n n\nK2 K\u0003\n1=4 +K\u0003\n2=64 11 (estimated) \u001812 at 2 T[31]\nK4 3K\u0003\n1=8 +K\u0003\n2=16 17 \u001816 at 2 T [31]\nK\u0003\n18\n5(4K2\u0000K4) 43.2 40-52.5 [35] or 48 \u00061 [31]\nK\u0003\n264\n5(2K4\u00003K2) 12.8 -17-29 [35] or 0\u00065 [31]\nTABLE S2. Torque's expressions and values for bcc Fe in (110) plane.10\n0.11 1 0110100-\n600 6 01 20-40-2002040-\n600 6 01 20-80-4004080-2- 10 1 2 -3-2-101231\n.01 .21 .41 .61 .82 .02.062.082.102.122.142.162.18/s61549\n0H (T)/s61556\n (kJ/m3)/s61556  (kJ/m3)ed\n/s61556\n (kJ/m3) \n /s61549\n0H (T) K2 \nK4H2bcc Fe (cuboid sample), T = 250 K, H // (110)c\n \n  \n0.2 T \n0.3 T \n0.5 T \n /s61553 (°) \n  \n1 T \n1.5 T \n2 T \n3 T \n4 Ta \n  \nLong side \nDiagonal \nShort side(110)M (µB per f.u.) \n M\n - M1(∝H1)\nFIG. S3. Magnetization and the magnetic torque in bcc Fe single crystal (cuboid sample). (a-b) Magnetization. Saturating\nat 0.5 T (a). No observable quadratic term (b). (c-e) The magnetic torque. The angular torque responses at di\u000berent \feld,\nvarying from 0.2 T to 0.5 T (c), 1T to 4 T (d). Field dependence of K2andK4(e).\nS5. THEORY OF ENERGY FROM MICROSCOPIC SPIN MODEL\nHere we extend the small anisotropy expansion of the earlier paper of Liu and Balents to higher order in the presence\nof an applied \feld. We consider the bulk energy of a uniform state. The basic unit of the structure is a set of 6 spins,\nforming a triangle in one layer and its counterpart in the neighboring layer. The spins in the second layer are assumed\nto be equal to their counterparts in the \frst layer. Then the energy depends upon three so far unspeci\fed spins Si\nwithi= 1;2;3. Then we write the energy perunit cell{i.e.per6spins as\nEu:c:=X\ni\u0002\n4JSi\u0001Si+1\u00002K(Si\u0001^ei)2+ 4D^z\u0001Si\u0002Si+1\u00002\u0016H\u0001Si\u0003\n; (S2)\nwhere we use the convention Si+3=Si. The factors account for the two copies of the triangles etc. We take the spins\nto be unit vectors so factors of spin length and magnetic moments should be absorbed into the above parameters.\nThe parameter J=J1+J2.\nWe assume K\u001cD\u001cJand\u0016H\u001cJ, andD> 0, and obtain an expansion of the ground state energy. For D> 0,\nthe ground state for K=\u0016H= 0 is an anti-chiral state with spins in the plane.\nA. Expansion to fourth order for in-plane \felds\nThe most important case is for an in-plane \feld. So take\nH=H(cos\u0012;sin\u0012;0): (S3)11\nWe can further assume in this case that the spins remain in the plane even for non-zero KandH, hence\nSi= (cos\u001ei;sin\u001ei;0): (S4)\nWe then write the angles as\n\u001e1=\u001e+\u00111; \u001e 2=\u001e\u00002\u0019\n3+\u00112; \u001e 3=\u001e\u00004\u0019\n3\u0000\u00111\u0000\u00112: (S5)\nHere\u001egives the order parameter angle, and \u00111;\u00112are small distortions of the triangle.\nTo be systematic, we introduce a small parameter r\u001c1, and letK!KrandH!Hr, and then expand \u00111;2\nin a formal series in rand minimize the energy order by order in r. This is e\u000bectively an expansion in K=J and\n\u0016H=J , which may safely be considered small parameters. There is a priori no need to assume D\u001cJas a second\nsmall parameter but since in reality it is small, it is sometimes convenient to simplify very cumbersome algebraic\nexpressions, and we will occasionally use it.\nThis procedure can be carried out at \fxed \u001e.\nSo assuming this condition, we can systematically write\n\u0011i=1X\nn=1\u0011i;nrn; (S6)\nand expand the full energy order by order in r. We successively minimize terms beginning at O(r2) in the energy\nover\u0011i;nwhich appear in these expressions. This determines the small distortions of the triangle at each order and\nresults in a fully determined expansion of the energy:\nEu:c:=1X\nn=0E(n)\nu:c:; (S7)\nwhere\nE(0)\nu:c:=\u00006J\u00006p\n3D; (S8)\nE(1)\nu:c:=\u00003K; (S9)\nE(2)\nu:c:=\u0000(\u0016H)2+K2+ 2\u0016HK cos(\u0012+\u001e)\n2\u0000p\n3D+J\u0001; (S10)\nE(3)\nu:c:=\u00001\n36(J+p\n3D)3h\n(3J+ 7p\n3D)K3cos(6\u001e) + 6(J+ 3p\n3D)\u0016HK2cos(5\u001e\u0000\u0012)\n+ 3(J+ 5p\n3D)(\u0016H)2Kcos(4\u001e\u00002\u0012) + 4p\n3D(\u0016H)3cos(3\u001e\u00003\u0012)i\n: (S11)\nThese expressions are a bit complicated and rather ugly at higher orders. Higher order terms are negligible for the\ne\u000bects we discuss here.\nThe energy should now be minimized over the order parameter angle \u001e. One can see that the third and second order\nterms have di\u000berent angular dependence, hence there is a competition between the two in determining this angle.\nTo proceed, we will assume that the second order term is dominant, being lower order. This is true unlessHbecomes\nvery small, because the second order term's angular dependence vanishes for H= 0. Comparing the coe\u000ecient of\ncos(\u001e+\u0012) from the second order term and the coe\u000ecient of cos(6 \u001e) from the third order term, we see this implies\nthe condition\n\u0016H\u001dK2=J: (S12)\nWhen this is true, (and recall we assumed \u0016H;K\u001cJ) the second order term is parametrically larger, so \u001emust be\nclose (but not equal!) to the minimum of this term. Hence we can write \u001e=\u0000\u0012+ , and we expect  \u001c1. Therefore\nwe expandE(2)\nu:c:to second order in  (it is quadratic around its minimum) and E(3)\nu:c:to \frst order in  , and minimize\nover . This leads to the \fnal expression for the energy\nEu:c:=\u00006J\u00006p\n3D\u00003K\u0000(\u0016H+K)2\n2(J+p\n3D)h\n1 +(3J+ 7p\n3D)K+ 4p\n3D\u0016H\n18(J+p\n3D)2cos(6\u0012)\n+\u0000\n(3J+ 7p\n3D)K2+ 2(J+ 4p\n3D)\u0016HK + 2p\n3D(\u0016H)2\u00012\n36(J+p\n3D)4\u0016HKsin2(6\u0012)i\n: (S13)12\nWe see that by including the small shift in the angle (  ), we generated a 12-fold harmonic (contained in the sin2(6\u0012)\nfactor. If we take the high \feld limit, \u0016H\u001dK(this is a stronger condition than Eq. (S12)) the expression simpli\fes\nfurther to\nEu:c:=\u00006J\u00006p\n3D\u00003K\u0000(\u0016H)2\n2(J+p\n3D)h\n1 +2p\n3D\u0016H\n9(J+p\n3D)2cos(6\u0012) +D2(\u0016H)3\n3(J+p\n3D)4Ksin2(6\u0012)i\n: (S14)\nIn solving for the energy, we obtain the angles \u00111;\u00112and :\n\u00111=\u0016H+K\n3(J+p\n3D)h\nsin 2\u0012\u0000\u0016H+ 2K\n6\u0016HK (J+p\n3D)2\u0010\n2p\n3D(\u0016H)2+ 2(J+ 4p\n3D)\u0016HK + (3J+ 7p\n3D)K2\u0011\ncos 2\u0012sin 6\u0012i\n;\n\u00112=\u0016H+K\n3(J+p\n3D)h\ncos(2\u0012+\u0019\n6)\n+\u0016H+ 2K\n6BK(J+p\n3D)2\u0010\n2p\n3D(\u0016H)2+ 2(J+ 4p\n3D)\u0016HK + (3J+ 7p\n3D)K2\u0011\ncos(2\u0012\u0000\u0019\n3) sin 6\u0012i\n;\n =B+K\n6\u0016HK (J+p\n3D)2\u0010\n2p\n3D(\u0016H)2+ 2(J+ 4p\n3D)\u0016HK + (3J+ 7p\n3D)K2\u0011\nsin 6\u0012: (S15)\nDespite the complexity, here are a few features:\n•AH2term appears \frst at order n= 2, and has no angular dependence. Because terms successively decrease\nwith increasing n, this is the dominant contribution to the H2term. It represents term in the magnetization\nlinear in \feld, i.e. just a susceptibility. The Din the denominator is subdominant, and so the susceptibility is\njust order 1 =J.\n•TheH3term appears \frst at order n= 3. Examining this expression we see that this term however is linear in\nDand vanishes for D= 0. The third order contribution has a pure cos 6 \u0012dependence.\n•The cos 12\u0012harmonic is formally of fourth order (e.g. in Eq. (S14) its coe\u000ecient is order H5=K). One can check\nthat no additional 12-fold contribution arises if we include the E(4)\nu:c:correction.\nMany of these features may match experimental observations. A few notable di\u000berences from theory are:\n•There are no 2-fold harmonics. They can appear only from e\u000bects that break the six-fold structural symmetry\nof the solid, e.g. \fnite size shape e\u000bects, or strain due to magnetostriction in the ordered state.\n•The energy is analytic in H. This means that the fractional power appearing experimentally in the two-fold\nanisotropy must arise from additional physics not included here, which is hardly surprising since the two-fold\nanisotropy itself is not present in this model.\n•TheH3term has a pure cos(6 \u0012) dependence on the angle, i.e. there is no angle-independent part. A small\nangle-independent part does arise at higher orders in the expansion but it is negligible. This means that the\nangle-independent H3term seen in experiment must arise from e\u000bects beyond the simple model studied here.\nB. Arbitrary spin angle\nIt is too cumbersome to carry out the full expansion to 6th order for applied \felds in arbitrary directions, but we\ncan do so for the \frst few orders. In this case, we can repeat similar manipulations but not assuming in-plane spins.\nSuppressing the details, we obtain the following energy tosecond order inH;K :\nEu:c:=\u00006J\u00006p\n3D\u00003K\u00001\n2J \n1\u0000p\n3D\nJ!\n(K+\u0016Hxy)2\n\u00001\n2J\u0012\n1\u0000Dp\n3J\u0013\n(\u0016Hz)2+O(r3): (S16)\nFrom this we can see that the linear in \feld magnetization is almost isotropic, i.e. the fractional di\u000berence in di\u000berential\nsusceptibility in the two directions is order D=J."    },    {        "title": "2109.12005v2.Impact_of_local_arrangement_of_Fe_and_Ni_on_the_phase_stability_and_magnetocrystalline_anisotropy_in_Fe_Ni_Al_Heusler_alloys.pdf",        "content": "Impact of local arrangement of Fe and Ni on the phase stability and\nmagnetocrystalline anisotropy in Fe-Ni-Al Heusler alloys\nVladimir V. Sokolovskiy1, Olga N. Miroshkina2;1, Vasiliy D. Buchelnikov1, and Markus E. Gruner2\n1Faculty of Physics, Chelyabinsk State University, Chelyabinsk 454001, Russia and\n2Faculty of Physics and Center for Nanointegration, CENIDE,\nUniversity of Duisburg-Essen, 47048 Duisburg, Germany\n(Dated: January 3, 2022)\nOn the basis of density functional calculations, we report on a comprehensive study of the in\ru-\nences of atomic arrangement and Ni substitution for Al on the ground state structural and magnetic\nproperties for Fe 2Ni1+xAl1\u0000xHeusler alloys. We discuss systematically the competition between\n\fve Heusler-type structures formed by shu\u000fes of Fe and Ni atoms and their thermodynamic stabil-\nity. All Ni-rich Fe 2Ni1+xAl1\u0000xtend to decompose into a dual-phase mixture consisting of Fe 2NiAl\nand FeNi. The successive replacement of Ni by Al leads to a change of ground state structure and\neventually an increase in magnetocrystalline anisotropy energy (MAE). We predict for stoichiomet-\nric Fe 2NiAl a ground state structure with nearly cubic lattice parameters but alternating layers of\nFe and Ni possessing an uniaxial MAE which is even larger than tetragonal L1 0-FeNi. This opens an\nalternative route for improving the phase stability and magnetic properties in FeNi-based permanent\nmagnets.\nPACS numbers:\nI. INTRODUCTION\nThe demand for permanent magnets for modern ap-\nplication technology is very high. Permanent magnets\nare used in automating and robotics industry and consti-\ntute a parts of an electric vehicles, wind turbines, drives,\nand storage and magnetic cooling devices1{7. Application\ngrade materials should exhibit particular intrinsic prop-\nerties, in \frst line a high saturation magnetization ( Ms)\nin combination with a large uniaxial anisotropy and a suf-\n\fciently high Curie temperature ( TC). But also extrinsic\nproperties must match, such as a micro/nanostructure\nthat preserves coercivity, high remanent magnetization\nand a high magnetic \rux density in combination with\nmechanical stability and corrosion resistance. Thermo-\ndynamic stability and ductility which a\u000bect the process-\ning of the material are also important demands.\nNowadays, the most widespread high-performance per-\nmanent magnets are based on rare earth elements.\nHowever, such materials are extremely expensive and\n\fnding inexpensive analogs consisting of abundant el-\nements with similar performance is one of the most\nurgent and challenging tasks in modern materials sci-\nence1,4,8{12. One of the most intensely discussed can-\ndidate is equiatomic FeNi13{19in the tetragonally or-\ndered L1 0structure (tetrataenite). It is characterized by\nMAE of\u00190:32\u00001:3 MJ/m3(Refs. 20{22) and might be\nconsidered as a rare-earth free competitor of Nd 2Fe14B\n(3:7\u00005:15 MJ/m3, see Refs. 23{25) , but hampered by a\nmuch smaller reduction with increasing temperature26.\nHowever, up to date, the synthesis of L1 0ordered\nFeNi is very challenging, because it becomes thermody-\nnamically stable at temperatures lower than the order-\ndisorder transition temperature, which is in the range\nof 200\u0000300oC. This leads to an extremely slow atomic\ndi\u000busion. Indeed, natural tetrataenite is found in mete-orites with a slightly Fe-rich (Fe: 50 :47\u00061:98 at.% and\nNi: 49:6\u00061:49 at.%) composition and a body centered\ntetragonal ( bct) structure with alternating layers of Fe\nand Ni along the elongated c-axis15,26. It has been shown,\nthat it is possible to grow L1 0-FeNi in the laboratory\nas thin \flms irradiated with neutrons below 593 K27,\nin a low temperature cyclic oxi-reduction process28, an\nalternate monatomic layer deposition technique29, crys-\ntallizing an amorphous alloy with equiatomic composi-\ntion at a crystallization temperature close to the order-\ndisorder transition temperature30. The preparation of\nL10-FeNi \flms on Au-Cu-Ni bu\u000ber-layer was studied by\nGiannopoulos et al.31. With their high-throughput mag-\nnetic characterization methods the authors revealed the\npresence of a hard magnetic phase with an out-of-plane\nanisotropy, while the MAE density increases from 0.12\nto 0.35 MJ/m3. Goto et al.32suggested to synthesize\nsingle-phase L1 0-FeNi powder with a high degree of or-\nder through nitrogen insertion and topotactic extraction.\nDespite these e\u000borts, bulk tetrataenite has not been\nobtained on a large scale yet. Therefore, scientist keep on\ntrying to \fnd new ways to overcome the obstacles in pro-\nducing L1 0-FeNi. Tian et al. worked on the optimization\nof the tetrataenite phase in terms of the order-disorder\ntransformation16, the e\u000bect of pressure17and alloying18\nby non-magnetic (Al and Ti) or magnetic (Cr and Co)\natoms. The e\u000bect of alloying was also investigated by\nother authors: Tuvshin et al.19showed that the addition\nof B can improve both the phase stability and increase\nMAE up to 2.6 MJ/m3. Izardar et al.33studied the inter-\nplay between chemical order and the magnetic properties\nand found that the reduction of chemical long-range or-\nder by 25% does not reduce the MAE signi\fcantly. How-\never, the central issues of proper growth conditions that\nstabilize the L1 0phase are still not fully resolved.\nOver the past decades, also full Heusler alloys of X2YZarXiv:2109.12005v2  [cond-mat.mtrl-sci]  31 Dec 20212\nstoichiometry were considered for permanent magnet ap-\nplications. The properties of these alloys can be tuned\n\rexibly depending on composition and structural defects.\nRecent theoretical studies34{36suggested Heusler alloys\nbased on Ni35,36, Fe34,36, Co34,36, Rh, Au, and Mn36as\npromising classes of hard magnetic materials.\nRecently Herper35revealed the e\u000bect of Y,Z, and\nfourth element doping as well as lattice deformation on\nMAE for Ni 2YZ(Y= Mn, Fe, Co and Z= B, Al, Ga,\nIn, Si, Ge, Sn). It was found that the phase stability and\nMAE of Ni 2CoZincreases by adding a fourth element\nfrom the main groups III and IV to the Zsublattice,\nwhile the addition of Fe to the Ysublattice does not\nhave a similar e\u000bect. Besides, small deviations from cu-\nbic structure lead to a doubling of the MAE in Ni 2FeGe.\nFor most systems considered in Ref. 35 , the MAE ex-\nhibits a quasilinear dependence on the tetragonality of\nthe lattice and changes its sign around c=a= 1. How-\never, for Ni 2FeGe, the MAE remains uniaxial for the c=a\nfrom 0.85 to 1.45. In this case, a small deviation from\nc=a= 1 leads to an increase in MAE from 1 to 2 MJ/m3.\nGao et al.36reported on the in\ruence of interstitial\natoms (H, B, C, and N) on the MAE for Fe-, Ni-, Co-, Rh-\n, Au-, and Mn-based Heusler alloys. Among tetragonal\nstructures, authors found 32 compositions with high en-\nergy of uniaxial anisotropy ( >0:4 MJ/m3) and 10 com-\npositions with high energy of in-plane anisotropy. It can\nbe noted that the addition of H atoms in the interstices\nleads to large MAE values (1 :5 MJ/m3). For Ni 2FeGa,\nMAE also increases from 0.23 MJ/m3to 1.43, 0.94, and\n0.56 MJ/m3with the addition of interstitial N, C, and\nH atoms, respectively. In contrast, MAE decreases with\nthe addition of interstitial atoms in Fe 2CoGa.\nFe2YZ(Y= Ni, Co, Pt) and Co 2YZ(Y= Ni, Fe, Pt),\nwhereZ=Al, Ga, Ge, In, Sn for both series, were consid-\nered by Matsushita et al.34. Among the 30 investigated\ncompositions, the cubic phase is found to be favorable in\n15 only. In particular, for Co 2NiZ, Co 2PtZ, and Fe 2PtZ,\nas well as for Fe 2NiGe and Fe 2NiSn, tetragonal structures\nprevail. For 15 tetragonal structures, the MAE is in the\nrange from\u000012 MJ/m3for Co 2PtAl to 5:19 MJ/m3for\nFe2PtGe. In case of compounds without Pt, the values\nof the MAE are signi\fcantly lower: from \u00002:38 MJ/m3\nfor Co 2NiGa to 1.09 MJ/m3for Fe 2NiSn. Note, for mag-\nnetically hard materials, negative values correspond to\nmoments oriented perpendicular to the tetragonal axis,\nleading to an easy plane, whereas positive MAE values\ndenote an easy axis which is the case of interest for ap-\nplication. In this regard, the authors have shown that\namong the considered alloys, MAE is positive in the case\nZelement is Ge or Sn. It is important to note that the\nabove mentioned theoretical studies considered the reg-\nular L2 1and inverse Heusler structures.\nAs FeNi represents one corner of the ternary Fe-Ni-\nAl phase diagram, these systems may be of interest as\npotential permanent magnets, as well. However, so far\nmainly stoichiometric Fe 2NiAl is reported in the litera-\nture, which does not provide a uniform picture. The-oretical studies34,37{39suggested that Fe 2NiAl possesses\nthe inverse cubic structure (Hg 2CuTi prototype), which\ncon\frms part of the experimental results37,40, but stands\nin con\rict with others. According to the direct synthesis\ncalorimetry of standard formation enthalpies conducted\nby Yin et al.41, Fe2NiAl crystallizes in a B2 structure in\nagreement with the high temperature phase diagram42.\nOther reports suggest, that Fe 2NiAl alloys are character-\nized by a solid solution decomposition into two isomor-\nphous body centered cubic ( bcc) phases, consisting of Fe-\nrich particles ( \f-phase) and a weak-magnetic NiAl-based\nmatrix (\f2-phase)43,44and as a result of the decompo-\nsition, a miscibility gap appears on the phase diagram\nat lower temperatures42,45,46. However, subsequent heat\ntreatment can improve the hard magnetic properties of\nFe-Ni-Al systems47{49. In the view, that the question\nabout the type of ordering in Fe 2NiAl is still unsettled,\nit is of fundamental interest to consider further types of\nordering motifs and extend this investigation to the Ni-\nexcess compositions with FeNi as the end point. In this\npaper, we therefore present a systematic \frst-principle\nstudy of structural, magnetic, and electronic properties\nof Fe 2Ni1+xAl1\u0000x(x= 0, 0.25, 0.5, 0.75, and 1) in depen-\ndence on a competing crystal ordering in austenite and\nmartensite phases. The paper is organized as follows: Af-\nter this introduction, Section II is devoted to the techni-\ncal details of the ab initio calculations. In Section III, the\nresults of our calculations are presented and discussed,\nwith emphasis on the energetic order of along the tetrag-\nonal transformation path in Sec. IIIA, the stability of\nthe cubic structure of Fe 2NiAl against decomposition in\nSec. IIIB and a comparison of the magnetic properties in-\ncluding MAE of the most stable structures in Sec. IIIC.\nSection IIID reports exchange constants and Curie tem-\nperatures, while the stability of o\u000b-stoichiometic at \fnite\ntemperatures is \fnally discussed in Sec. IIIE. Concluding\nremarks are given in Sec. IV.\nII. COMPUTATIONAL DETAILS\nStructural, magnetic, and thermodynamic properties\nof Fe-Ni-Al were calculated from \frst-principles using\nthe plane-wave basis set and the projector augmented\nwave (PAW) method implemented in the Vienna ab ini-\ntiosimulation package50(VASP). For the exchange and\ncorrelation functionals, the generalized gradient approx-\nimation (GGA) in the scheme of Perdew, Burke, and\nErnzerhof (PBE)51was applied; we employed PAW po-\ntentials with the following valence states: 3 p63d74s1for\nFe, 3d84s1for Co, 3p63d84s2for Ni, and 3 s23p1for Al.\nThe kinetic energy cut-o\u000b for the augmentation charges\nwas set to 1000 eV while the energy cut-o\u000b for plane wave\nbasis set was taken of 750 eV. The Brillouin zone inte-\ngration was performed using the \frst order method of\nMethfessel-Paxton with a smearing parameter of 0.1 eV\non a uniform Monkhorst-Pack 15 \u000215\u000215k-point grid.\nElectronic selfconsistency was assumed when the total3\nFIG. 1: The \fve considered structures for the Fe 2NiAl\nHeusler alloy with cubic lattice parameters: (a) Cu 2MnAl-\ntype Regular Heusler structure, (b) Hg 2TiCu-type inverse\nHeusler structure, (c) T#, (d) Tc, and (e) Tpstructures.\nT#, Tp, and Tc-structures can be obtained from the inverse\nstructure by various atomic shu\u000fes. For the Tpand Tcstruc-\ntures, originally suggested by Neibecker et al.52, the Ni atoms\nand part of Fe atoms are ordered into alternating planes and\ncolumns, correspondingly. T#can be conceived as chains of\nFe and Ni atoms, which are arranged perpendicular to each\nother in alternating planes.\nenergy di\u000berence between two consecutive steps reached\n10\u00007eV.\nThe alloys were modeled in a 16-atom cubic su-\npercell, which represents o\u000b-stoichiometric composi-\ntions Fe 2Ni1+xAl1\u0000xwith concentration steps of x=\n0:25, while still allowing for extensive calculations of\nphonon dispersions and the accurate determination of the\nMAE (see the supplemental Material, SM, in Ref. 53).\nThe atomic positions for the distorted con\fgurations\nwere considered fully relaxed when interatomic forces\nwere smaller than 10\u00002eV/\u0017A.\nWe considered \fve types of ordered atomic arrange-\nment depicted in Fig. 1. In the regular L2 1structure(space group Fm3m, No. 225, prototype Cu 2MnAl), Fe\natoms locate at equivalent 8 c(1/4, 1/4, 1/4) and (3/4,\n3/4, 3/4) sites while Al and Ni atoms occupy 4 a(0, 0,\n0) and 4b(1/2, 1/2, 1/2) sites, see Fig. 1(a). For the in-\nverse (FeNi)FeAl Heusler structure (space group F43m,\nNo. 216, prototype Hg 2TiCu), Fe atoms are placed at\nnon-equivalent 4 a(0, 0, 0) and 4 c(1/4, 1/4, 1/4) sites\nwhile Ni and Al are placed at 4 b(1/2, 1/2, 1/2) and\n4d(3/4, 3/4, 3/4) positions, respectively, c. f. Fig. 1(b).\nIn addition, we considered three structures related to the\ninverse Heusler type, with layer-wise and columnar order-\ning of Fe and Ni atoms on the 4 aand 4bsites. The T#\nstructure is characterized by columns of Fe and Ni atoms\nlocated at 4 aand 4bsites, which change their orientation\nfrom layer to layer as shown in Fig. 1(c). Tcconsists\nof alternating Fe and Ni atomic columns along the [001]\ndirection (z-axis) and can also be conceived as Fe and\nNi layers alternating along [110], c. f. Fig. 1(d). Tpis\ncomposed of layers of Fe and Ni atoms alternating along\n[001], see Fig. 1(e). The latter two structures were sug-\ngested in an earlier work on quaternary stoichiometric\nNiCoMn(Ga,Al) alloys52, where they proved competitive\nto the conventional and the inverse Heusler structure in\nFig. 1(a,b), in agreement with the experimental \fndings.\nTo determine the MAE for the fully relaxed struc-\ntures, we made use of the magnetic force theorem54{56:\nStarting from the wave functions obtained from accu-\nrate self-consistent collinear ground-state calculations\nwith a scalar-relativistic Hamiltonian, we performed non-\ncollinear calculations including the spin-orbit term cou-\npling in a non-self-consistent scheme, where the quantiza-\ntion axis was rotated into two di\u000berent orientations, [100]\nand [001], but the wave functions were kept \fxed other-\nwise. The MAE is then given by the energy di\u000berence\nbetween two spin moment directions57{60, as\n\u0001EMAE =Etot\n100\u0000Etot\n001;\nwhereEtot\n001andEtot\n100are the total energies with the re-\nspective spin orientations. For a positive value of MAE,\nthe out-of-plane spin con\fguration (easy axis) is energet-\nically favorable and vice versa the in-plane spin direction\n(easy plane) is preferred. For a deeper analysis of the de-\npendence of the MAE on the composition in the range x\nfrom 0 to 0.25, we carried out calculations using a 2 \u00022\u00021\nsupercell containing 64 atoms. Here, a 6 \u00026\u00026k-point\nmesh was taken into account. By this means we consid-\nered Fe 2Ni1+xAl1\u0000xwithx= 0, 0.0625, 0.125, 0.1875,\nand 0.25 in austenitic state.\nFinite-temperature properties were obtained from the\ncalculations of the vibrational density of states (VDOS)\nin the harmonic approximation within the direct (force-\nconstant) approach using the Phonopy61code. Depend-\ning on the composition and the respective primitive cells,\nwe employed 3\u00023\u00022, 3\u00023\u00023, 4\u00024\u00022, 4\u00024\u00024, and\n6\u00026\u00024 supercells to calculate Hellmann{Feynman forces\nfrom a set of non-equivalent atomic displacements using\nthe VASP code, to construct the force constant matrix\nof the system. From the eigenvalues of the resulting dy-4\nnamical matrix we obtained the VDOS and, \fnally, the\nlattice free energies.\nCurie temperatures were determined from the mean-\n\feld approximation to Heisenberg model parameterized\nfrom ab initio data. The Heisenberg exchange cou-\npling constants were obtained with the SPR-KKR pack-\nage62employing the PBE exchange-correlation func-\ntional. The calculations were performed for the most\nfavorable structures in austenite and martensite based\non the 16-atom supercells used in the VASP calculations.\nThe angular momentum expansion for the major compo-\nnent of the wave function, lmax, was restricted to d-states;\nthe energy convergence was set to 0.01 mRy. For self-\nconsistent cycles, the scattering path operator was cal-\nculated by the Brilloiun-zone integration assuming the\n573k-mesh grid with 4495 k-points.\nIII. RESULTS AND DISCUSSION\nA. Cubic to tetragonal distortion\nTo determine the equilibrium ground-state properties\nof Fe-Ni-Al, we apply a volume conserving tetragonal dis-\ntortion to the fully optimized cubic 16 atom unit cell of\nthe Heusler compound. From the calculations of the total\nenergy as a function of the volume and the c=aratio we\ndetermined the optimized lattice constants for all com-\npositions with the investigated crystalline structures in\naustenite (c=a\u00191) and martensite ( c=a6= 1), the data\nof the most important ones are summarized in Table I.\nFigure 2 displays the corresponding di\u000berence in total\nenergyE(c=a) as a function of the tetragonal distortion\nc=awith respect to the cubic inverse Heusler structure\nfor Fe 2Ni1+xAl1\u0000x(x= 0, 0.25, 0.5, 0.75, and 1.0) with\nthe di\u000berent types of atomic order shown in Fig. 1. Note\nthat for all cases the ferromagnetic (FM) con\fguration\nis found more favorable than potential ferri- or antifer-\nromagnetic arrangements of the magnetic moments. As\nfor some of the structures the cubic symmetry is already\nbroken forc=a= 1 { in particular for Tpand Tc, which\nare characterized by a layered atomic arrangement { the\nenergy landscape varies if we align the crystallographic c\naxis along [001] (i. e., the Cartesian z-axes) or [100] ( x-\naxis), while the deformation along [010] ( y-axis) remains\nequivalent to [100]. The corresponding binding curves are\nlabeled ascjjxandcjjzin Fig. 2, which may indeed show\nsigni\fcant di\u000berences. For realistic samples, where we\nmay expect anti-phase boundaries between grains with\ndi\u000berent orientations of the layering (or a polycrystalline\nsample), energies at c=a6= 1 can be averaged over for the\nalignment of calong all three Cartesian axes.\nWe \frst focus on the stoichiometric case Fe 2NiAl.\nIt can be seen from Fig. 2(a), that the layered Tpstruc-\nture withc=a\u00191, i. e., without signi\fcant additional\ntetragonal distortion of the unit cells, turns out to be\nthe ground state. This means that the austenite phase,\ncharacterized by c=a= 1, remains stable down to low-temperatures and a transformation to a martensite struc-\ntures withc=a6= 1 is suppressed.\nTpis\u001925 meV/atom lower in energy than T#, the\ninverse Heusler, and the Tcstructures, which are very\nclose in the energy to the regular L2 1Heusler structure.\nThe close competition between of Tp, T#, Tc, and in-\nverse structures hampers the chemical ordering of the\nsamples under typical experimental conditions and makes\nthe B2-type structure a likely outcome, rather than the\ninverse Heusler structure, as reported previously37,40.\nHowever, under favorable experimental preparation con-\nditions (applied stress, the rate of the cooling process or\nannealing time of a sample, etc.) it might be possible to\nsynthesize the Tpphase experimentally, for instance in\nepitaxial thin \flms by a sputtering procedure.\nA small deviation from the stoichiometry up to\nx= 0:25 keeps the Tpstructure in austenite as the\nmost favorable one, while the energy di\u000berence between\nother structures is reduced, which increases the competi-\nFIG. 2: Variation of the total energy as a function of tetrag-\nonal ratioc=afor Fe 2Ni1+xAl1\u0000xwith various atomic order\nin a cubic structure: (a) x= 0:0, (b)x= 0:25, (c)x= 0:5,\n(d)x= 0:75, and (e) x= 1:0. The binding surface E(c=a)\nis plotted with respect to the energy E0of the inverse cubic\nstructure.5\nTABLE I: Favorable crystal structure, lattice parameter a(in\u0017A), tetragonal distortion ratio c=a, and formation energy Eform\n(in eV/f.u.) of Fe 2Ni1+xAl1\u0000xas well as the total energy di\u000berence \u0001 Ec=a(in meV/atom) between the austenitic ( c=a\u00191)\nand martensitic states ( c=a > 1) with the lowest energy. The data are collected from the full geometric crystalline structure\noptimization. For the martensitic phase of Fe 2Ni1:5Al0:5, the data represent the average over individual tetragonal distortions\nof a 16-atom supercell along the three Cartesian axes63.\nCompositionAustenite Martensite\u0001Ec=aStructure a0c=a E form Structure atc=a E form\nx= 0:0 Tp5.744 0.992 -1.191 |\nx= 0:25 Tp5.728 0.987 -0.799 |\nx= 0:50 Inverse 5.699 0.999 -0.404 L1 0 5.004 1.475 -0.519 26.170\nx= 0:75 Inverse 5.677 1.000 -0.109 L1 0 5.016 1.446 -0.328 55.702\nx= 1:0 | bct-L10 5.019 1.424 -0.151\n| fct-L10a3.549 1.007\naFor better comparison with the ternary compositions, we list\nthe lattice parameters of L1 0-FeNi in terms of the bct tetrag-\nonal Heusler cell, and additionally with respect to the conven-\ntional (4 atoms) fct L1 0cell. The lattice parameters transform\nas:afct=abctp\n2and cfct=cbct\n2.\ntion between them. Apart from the regular Heusler struc-\nture, all ordering motifs of Fe 2NiAl and Fe 2Ni1:25Al0:75\nremain stable against a tetragonal distortion of the unit\ncell, see Figs. 2(a,b).\nIn turn, for compositions with x\u00150:5, the global en-\nergy minimum is obtained around c=a= 1:45 for the reg-\nular L1 0Heusler phase, see Figs. 2(c-e), while the binding\ncurvesE(c=a) for the other structures become remark-\nably \rat. The global minimum of the L1 0phase becomes\ndeeper as compared to the other structures, when we\napproach the binary FeNi compound. Figs. 2(c-e) show,\nthat only the tetragonal Tcstructure is in close proximity\nto the regular L1 0under the assumption that the tetrag-\nonal distortion is performed along zaxis. In this case,\nthe energy di\u000berence between the two phases is about\n6, 13, and 15 meV/atom for compositions with x= 0:5,\n0.75, and 1.0, correspondingly. The di\u000berence between\nenergy solutions for the Tcstructure with the tetrago-\nnalcaxis oriented along xandzincreases signi\fcantly\ntowards binary FeNi with c=a\u0019p\n2.\nIn the binary limit, the Tcstructure for FeNi consists\nof a stacking of mixed Fe/Ni layer in a staggered arrange-\nment in contrast to the ordered L1 0phase, in which Fe\nand Ni alternate layer-wise, and these ordering motifs\ncannot be transformed into each other by a simple site\nswapping process. Therefore, the close competition be-\ntween Tc-type FeNi distorted along zand L1 0tetrataen-\nite might be a reason for the di\u000eculties to synthesize\nL10-FeNi with a high degree of order in experiment.\nFor compositions between x= 0:5 and 0:75, the energy\ndi\u000berence \u0001 Ec=abetween the tetragonal regular L1 0-type\nHeusler phase and the most favorable stable cubic struc-\nture lies in the range of thermal energies (cf. Table I). Fol-\nlowing the empirical relation Tm\u0019kB\u0001Ec=a, which has\nbeen proven quite successful in predicting the marten-\nsitic transformation temperatures in Heusler alloys64, one\nmay be tempted to expect a structural transition around\nT= 650 K for x= 0:75 and around room temperatureforx= 0:5. However, as the \fnite temperature analysis\nin terms of the free energies in Sec. III E will show, con-\ntributions from mixing entropy are signi\fcant which can\ne\u000bectively inhibit such a transition and renders the above\nrelation useless for the prediction of a di\u000busive transition.\nB. Formation energies\nWe obtain insight into the structural stability of the\ncompounds in the austenitic phase from the formation\nand decomposition energies of the fully optimized con-\n\fgurations with respect to atomic positions and cell pa-\nrameters for each composition around c=a= 1. In a\n\frst step, we evaluate the formation energy ( Eform) in\nterms of the di\u000berence of the total energy of the ternary\ncompound and the weighted sum of the energies of the\nstable phases of the corresponding pure elements. This\nquantity is relevant e. g. for the preparation of thin \flms\nfrom the sputtering of elemental targets. A negative sign\nofEformpoints out the a stability of a compound under\nstudy against decomposition into its pure constituents.\nThis analysis reveals that the inverse structure at\nc=a = 1 keeps the cubic symmetry for all composi-\ntions, while Tpand T#structure remain almost cubic\n(c=a\u00190:98 or 0.99) for compositions with x= 0 and\n0.25. The Tcstructure also turns to be slightly tetrago-\nnally distorted with c=aabout 1.036 and 1.064 for x= 0\nand 0.25. With further increase in Ni excess, Tp, T#, and\nTcstructures deviate from the nearly cubic lattice vec-\ntors, which become clearly tetragonal. According to the\nlarge energy di\u000berence with respect to other structures,\nwe skip here the discussion of the regular L2 1ordering.\nThe ground state structure, lattice constants, and forma-\ntion energy found for all compositions in austenite and\nmartensite are listed in Table I. It should be noted that in\nthe case ofx= 1, the cubic structure with inverse atomic\noccupation is unstable due to a positive sign of Eform.6\nTABLE II: Calculated total magnetic moment ( \u0016totin\u0016B/f.u.), saturation magnetization Ms(in Am2/kg), and MAE (in\nmeV/f.u. and MJ/m3) for the most stable structures (austenite and martensite) of Fe 2Ni1+xAl1\u0000x. For FeNi, the magnetic\nproperties are also presented for the Tcstructure. Literature MAE values for L1 0-FeNi are mainly collected from theoretical\nstudies using di\u000berent methods (FLAPW-GGA65, VASP-GGA66, WIEN2k-GGA13,66, ASA-SPR-KKR-GGA13and FP-SPR-\nKKR-GGA14); the experimental ones marked with an asterisk (Refs. 20{22,67).\nCompositionAustenite Martensite\nStructure\u0016totMs MAEStructure\u0016totMs MAE\n[\u0016B/f.u.] [Am2/kg] [meV/f.u.] [MJ/m3] [ \u0016B/f.u.] [Am2/kg] [meV/f.u.] [MJ/m3]\nx= 0:0 Tp4.706 133.18 0.3075 1.0467 |\nx= 0:25 Tp4.856 132.10 0.0425 0.1468 |\nx= 0:50 Inverse 5.617 147.13 0.0 0.0 L1 0 5.722 149.87 0.12 0.415\nx= 0:75 Inverse 6.044 152.65 0.0 0.0 L1 0 6.11 154.42 0.21 0.7366\nx= 1:0 | L1 0 3.248 158.39 0.0825 0.5867\nTc3.255 158.72 -0.0487 -0.3463\n0.032650.2265\n0.078660.5666\n0.06913,660.4813,66\n0.11130.7713\n0.032140.2214\n0.32 - 1.3\u0003\nRegarding the proper design of annealing procedures\nused to establish chemical order and long-term stabil-\nity of the compounds, one may also be interested in the\nenergy of decomposition of the compound into possible\ncombinations with stable binary phases, which we will\nrefer to as Edec. In the following, we discuss as a par-\nticularly relevant case the relative stability of Fe 2NiAl\nagainst segregation into a dual phase-composite accord-\ning to the equation:\nEdec=EFe2NiAl\u0000\u0002\n2EFe+ENiAl\u0003\n; (1)\nwhereEFe2NiAl,EFe, andENiAlare the total energies of\nFe2NiAl, bccFe and bccNiAl, respectively. According to\nEq. (1), a negative value of Edecindicates again the sta-\nbility of a compound against a spinodal decomposition.\nFigure 3 illustrates the calculated Edecfor all consid-\nered structures of Fe 2NiAl.Edecturns out to be posi-\ntive for all structures under investigation, which implies\nFIG. 3: Decomposition energy Edecfor the stoichiomet-\nric Fe 2NiAl compound with di\u000berent crystalline structures\nagainst the segregation into a dual phase mixture of Fe and\nNiAl.a long-term instability with respect to the above decom-\nposition reaction. Nevertheless, the minimum value of\nEdec= 12:14 meV/atom found for Fe 2NiAl with the Tp\nstructure is rather small, corresponding to a temperature\nof around 100 K. As one can see from Fig. 3, this energy\ndi\u000berence is about ten times smaller than for the regular\nand two times smaller than for the inverse Heusler struc-\nture, respectively, meaning that the layered Tpstructure\nis signi\fcantly less prone to a decomposition into pure Fe\nand binary NiAl systems as compared to the other struc-\ntures. In fact, we veri\fed that Tp-Fe2NiAl is stable in\nterms of its total energy against decomposition into the\nmost relevant of binary and ternary compounds { except\nfor the Eq. (1); an comprehensive evaluation of the rel-\native stability of Tp, inverse and conventional Fe 2NiAl\nis presented in the SM.53Keeping in mind that decom-\nposition processes are signi\fcantly delayed at low tem-\nperatures and such small energies may be overcome by\nentropic contributions (which will be further discussed\nin Sec. III E), one may be optimistic that with an appro-\npriate scenario of preparation, Fe 2NiAl might be synthe-\nsized in a Tpordered structure and be su\u000eciently stable\nfor application purposes.\nC. Magnetic moments and magnetocrystalline\nanisotropy\nWe now turn to the magnetic properties of\nFe2Ni1+xAl1\u0000xin the lowest energy austenitic and\nmartensitic structures. In Table II, we present a sum-\nmary of the total magnetic moment, saturation magne-\ntization, and the MAE for the most preferred ordering\nmotifs. Of particular interest for a potential application\nas a permanent magnet is in \frst line { besides the total\nmagnetic moment, which is su\u000eciently large in all cases7\n{ the MAE as it determines the intrinsic coercivity of the\nmaterial. We can note that we obtain positive MAE val-\nues for all relevant structures (except for Tc-FeNi) which\ncorresponds to an uniaxial anisotropy as preferred in ap-\nplications. Our calculated MAE for L1 0-FeNi is compa-\nrable to previous theoretical studies and consistent with\nexperimental \fndings (see Table II). However, we predict\nthe largest MAE of \u00190.3 meV/f.u. (\u00191:05 MJ/m3) for\nTp-Fe2NiAl. Despite the (almost) cubic lattice vectors,\nthe alternating layers of Ni and Fe atoms result in a sig-\nni\fcant deviation from cubic symmetry (see Fig. 1(e)),\nwhich allows for a signi\fcant uniaxial anisotropy. For this\ncompound, the MAE is about 2 times larger than that\nfor the L1 0-FeNi. A hint for the origin of this increase\ncan be obtained from the element-resolved contributions\nspin-orbit coupling (SOC) term: For Tp-Fe2NiAl, the ab-\nsolute contributions from Fe and Ni turn out a factor of\n2\u00003 larger as compared to L1 0-FeNi, while the contri-\nbutions from Ni are negative in both cases, diminishing\nthe total MAE. This, however, becomes more relevant\nfor L1 0-FeNi, where all non-spinpolarized Al atoms are\nreplaced by Ni (for a complete discussion, see the SM53).\nAs a consequence, the replacement of Al by Ni excess\natoms successively reduces the MAE in the austenitic\nphase and it vanishes when the inverse Heusler struc-\nture becomes favored, consistent with its cubic symme-\ntry. In contrast, tetragonally distorted ternary L1 0-type\nFe2Ni1+xAl1\u0000xwithx= 0:5 and 0.75, which represents\nthe ground state in this composition range, demonstrates\na considerable MAE being comparable of L1 0-FeNi with\nonly little degradation. To obtain a better resolution\nin the range of low Ni excess, we performed additional\ncalculations for intermediate compositions with Ni ex-\ncess between x= 0 and 0.25 using a 64 atom supercell\nFe32Ni16+nAl16\u0000nwithn= 0, 1, 2, 3, and 4, which cor-\nresponds to Fe 2Ni1+xAl1\u0000xwithx= 0, 0.0625, 0.125,\n0.1875, and 0.25, respectively. Figure 4 shows the cor-\nresponding dependence of the MAE with respect to the\nNi excess concentration. Here we can con\frm a smooth\ndeviation of the MAE towards x= 0:25.\nCalculated MAE of L1 0-FeNi of about 0.59 MJ/m3is\nin a good agreement with the experimental data (Ta-\nble II). It is worthwhile to calculate MAE of Tc-FeNi\nsince the energy di\u000berence between the ground state L1 0\nand the tetragonally distorted Tcstructure is with about\n17 meV/atom comparatively small. MAE of Tc-FeNi is\nsmaller in its absolute value and has moreover a negative\nsign, which tells us that the preferred orientation of the\nmoments changes direction from the uniaxial easy axis\nto a cross-plane-direction. Thus, the competition of the\ndi\u000berent types of order might not only inhibit a perfect\nL10ordering { we must also expect a substantial impact\nregarding its hard-magnetic properties.\nIn addition to the MAE, we also calculated the or-\nbital moment anisotropy, given by the di\u000berence \u0001 \u0016l=\n\u0016001\nl\u0000\u0016100\nlbetween the orbital moments \u0016lin the re-\nspective crystallographic directions, see Table III. Ac-\ncording to Patrick Bruno's relation derived from a per-TABLE III: Total orbital magnetic moments \u0016lcalculated in\n[001] and [100] direction and their di\u000berence \u0001 \u0016l=\u0016[001]\nl\u0000\n\u0016[100]\nl (in\u0016B=f:u: ) for favorable structures of austenite and\nmartensite Fe 2Ni1+xAl1\u0000x.\nComposition Structure \u0016001\nl\u0016100\nl \u0001\u0016l\nAustenite\nx= 0:0 Tp0.113 0.101 0.012\nx= 0:25 Tp0.123 0.114 0.009\nx= 0:5 Inverse 0.151 0.151 0.0\nx= 0:75 Inverse 0.166 0.166 0.0\nMartensite\nx= 0:5 L1 0 0.155 0.157 -0.002\nx= 0:75 L1 0 0.175 0.164 0.011\nx= 1:0 L1 0 0.177 0.170 0.007\nturbative treatment of MAE in thin ferromagnetic \flms\nof cubic metals68, the MAE is proportional to the di\u000ber-\nence between orbital magnetic moments oriented along\neasy and hard axis. Hereby, the easy axis of the uniax-\nial anisotropy is parallel to the direction with the largest\norbital moment, which is accounted for by the reverse\norder of the crystallographic directions in our de\fnitions\nof the MAE and \u0001 \u0016l. Good agreement with this relation\nhas been found in particular for elemental systems, but\ncounter-examples have been reported for complex alloys\nand compounds (e. g., Refs. 69,70). We see that { except\nfor theL10martensite with x= 0:5 { for the relevant\nlowest energy structures indeed the easy axis coincides\nwith the direction of the largest orbital moment. Both,\nthe uniaxial MAE and \u0001 \u0016l, are maximal for stoichio-\nmetric composition and reduce with increasing Ni con-\ntent. However, as demonstrated in the inset of Fig. 4,\nthe functional dependence is di\u000berent indicating that the\nproportionality factor varies with composition. This indi-\ncates the presence of competing mechanisms connecting\norbital moments and MAE, which are for instance well\nknown from the related alloy FePt, where the proportion-\nality factor even changes sign71{73.\nIn a recent high precision \frst-principles investigation,\nthe decrease of the MAE with increasing disorder in FePt\ncould be related to the presence of a rather low concen-\ntration of anti-site defects74. One may expect that also in\nthe case of FeNi, the local environment contributes sub-\nstantially to the rather complex dependence of MAE on\ncomposition and type of chemical order, since as in the\ncase of FePt, the moment of the Ni-group element ex-\nhibits a predominantly itinerant (induced) character and\nis determined by the surrounding Fe moments, which are\nrather localized. Such itinerant character of Ni and a lo-\ncalized behavior of Mn is also a characteristic feature of\nNi in Ni-Mn-based Heusler alloys75.8\nFIG. 4: MAE of austenitic TpFe2Ni1+xAl1\u0000x(0\u0014x\u00140:25)\nas a function of Ni excess. The inset displays the orbital\nmoment anisotropy \u0001 \u0016lbehavior as a function of Ni content.\nD. Exchange coupling constants\nWe have already seen that the magnetic properties of\nthis compound are very sensitive to change in composi-\ntion and atomic arrangement. In this section, we investi-\ngate the e\u000bects of atomic arrangement in austenitic and\nmartensitic phases of Fe 2Ni1+xAl1\u0000xon the magnetic ex-\nchange interactions and Curie temperature in detail to\nunravel the impact of competing interatomic exchange\ninteractions of Fe and Ni atoms at di\u000berent sites, which\nare frequently observed in Heusler compounds. In Fig. 5\nwe compare the evolution of the magnetic exchange con-\nstantsJijof the ground state structures in the investi-\ngated compositions range. The exchange constants de-\n\fne the interactions between di\u000berent pairs of Fe and\nNi atoms as a function of the distance dbetween them\nin terms of a classical Heisenberg model Hamiltonian\nHmag=\u0000P\ni6=jJij~ ei\u0001~ ej, where~ eiand~ ejdescribe the\nunit vectors of the orientation of the magnetic spin mo-\nments at sites iandj. The calculations were performed\nfor the ground state structures obtained from the cor-\nresponding 16-atom supercell calculations. This implies\ntwo types of Fe and Ni atoms labeled as Fe 1, Fe 2and\nNi1, Ni2, which occupy structurally non-equivalent posi-\ntions: In case of the Tpstructure, four Fe 1atoms lie in\nthe same middle plane and four Fe 2atoms occupy the\nalternating tetrahedral sites, see Fig. 1(e). In the case of\ninverse cubic structure, four Fe 1and Fe 2are placed at\n4a(0, 0, 0) and 4 c(1/4, 1/4, 1/4) sites, respectively, see\nFig. 1(b). Finally, for the tetragonal regular structure,\nFIG. 5: Magnetic exchange parameters Jijfor the e\u000bective\nHeisenberg Hamiltonian between interacting transition metal\natoms for Fe 2Ni1+xAl1\u0000x(0\u0014x\u00141) with the most favor-\nable crystalline structures in austenite and martensite, i.e.,\n(a) Tpstructure of stoichiometric Fe 2NiAl withc=a= 0:992,\n(b) approximately cubic inverse Heusler structure for x= 0:5,\n(c) tetragonal regular L1 0Heusler structure for x= 0:75\nand (d) tetratenite L1 0-FeNi. The exchange constants are\ndepicted as a function of the interatomic distance dgiven\nin units of the respective lattice constant aof the supercell.\nPositive values refer to ferromagnetic (FM), negative values\nto antiferromagnetic (AFM) pair interactions. The images on\nthe right depict the corresponding supercell and inequivalent\ntypes of magnetic ions as used in the left column.9\nfour Fe 1and Fe 2atoms lie in the same bottom and top\nplane, correspondingly, see Fig. 1(a). As for Ni atoms,\nfour Ni 1are placed at regular Ni sites while Ni 2are the\nexcess Ni atoms, which substitute for Al.\nAs evident from the Figs. 5(a,b), the intra-sublattice\nmagnetic exchange constants reveal the largest FM cou-\npling between nearest Fe 1-Fe2pairs of atoms located\nat the distance d= (p\n3=4)a0in the Tpstructure,\nwhich quickly approaches to zero from the third coor-\ndination sphere outwards. The inter-sublattice exchange\ncouplings Fe 1(2)-Fe1(2)and Fe 1(2)-Ni1(2)are found to be\nsmaller su\u000eciently and demonstrate a damped oscillating\nbehavior with increasing distance. The larger \u0016totfor the\ncomposition with x= 0:25 can be related to the stronger\nFM coupling of Fe 1-Fe1and Fe 1-Ni2as compared to those\nforx= 0.\nIn the case of alloys with inverse cubic structure, which\nis shown in Fig. 5(b), the largest FM exchange interac-\ntion is observed between nearest Fe 1-Fe2pairs of atoms,\nsimilar to the Tpstructure. In Fig. 5(b) we observe a\nsplitting of Jijinto\u001922 meV and\u001914 meV between\nthe Fe 1and Fe 2atoms located at d= (p\n3=4)a0. The\nsmaller exchange coupling constant is found for two Fe 2\natoms, which have Al neighbors at d= 0:5a0. The in-\nteractions Fe 1(2)-Fe1(2)and Fe 1(2)-Ni1(2)show again sub-\nstantially smaller values except for Fe 1-Ni2and Fe 2-Ni1,\nwhich are two times higher than those of Tp. This \fnd-\ning relates to the larger values of \u0016totfor compositions\nwith inverse structure in comparison to the Tpstructure\n(see Table II).\nThe tetragonal distortion of the L1 0martensite with\na largec=aratio results in a clearly visible competition\nbetween FM and AFM exchange interactions of Fe pairs\nin the range of 0 :7< d=a < 1:2 as shown in Fig. 5(c).\nThis is most pronounced for Fe 1and Fe 2, which lie in\nthe parallel planes of atoms that are spaced by a distance\nc=2 apart. Please note that in this case the Fe atoms in\nboth layers (Fe 1and Fe 2) are equivalent from the chem-\nical point of view, but we keep the notation to visualize\nthe di\u000berence between in-plane and out-of-plane interac-\ntions. The nearest inter-sublattice Fe 1(2)-Fe1(2)pairs of\natoms reveal the strongest FM exchange coupling, which\nis comparable to the nearest intra-sublattice Fe 1-Fe2in-\nteractions in the cubic inverse and Tpstructures, while\ncross-plane interactions are signi\fcantly reduced. The\npresence of a number of negative Jijcould even lead to\nan e\u000bective decoupling of the layers, but the Fe-Ni in-\nteractions maintain the e\u000bective FM coupling between\nthe layers mediated by the Ni atoms. For the binary\nFeNi system shown in Fig. 5(d), we observe again very\nsimilar situation: A considerably weaker exchange cou-\npling between nearest Fe atoms in the plane, whereas\nthe cross-plane interaction between next-nearest Fe al-\nmost vanishes. The Fe-Ni interaction is slightly larger\ncompared to the case above and e\u000bectively strengthens\nthe FM ordering of the Fe layers. In both cases, we \fnd\nhints for a damped oscillatory evolution of the Fe-Fe in-\nteractions with increasing d.TABLE IV: Calculated Curie temperatures (in K) of austen-\nite and martensite phases of the Fe 2Ni1+xAl1\u0000xcompounds\nusing MFA.\nStructure x= 0:0x= 0:25x= 0:5x= 0:75x= 1:0\nTp1387 1266 | | |\nInverse | | 966 1024 |\nL10 | | 1140 1159 1191\nFrom the full set of Jijwe determined the Curie tem-\nperatures within the mean \feld approximation ( TMFA\nC),\nwhich are collected in Table IV for the most relevant\nstructures. For permanent magnet application of Heusler\nalloys, it is necessary to have both, a high Curie tem-\nperature and a large saturation magnetization. Indeed,\nwe \fnd that all considered compositions possess a rather\nhighTMFA\nC>1000 K. As can be also seen from Table IV,\nfor both cubic inverse and tetragonal L1 0structures,\nthere is only a slight increase of TMFA\nC with increasing\nNi content, since the di\u000berent trends in intra- and inter-\nsublattice Fe-Fe and Fe-Ni interactions are e\u000bectively bal-\nanced out. The largest value of TMFA\nC is obtained for the\nstoichiometric composition. The neglect of correlations\nclose to the phase transition in the MFA as compared\nto a full statistical treatment in the framework of Monte\nCarlo simulations and the missing treatment of itinerant\nmagnetism in localized spin models typically leads to an\nover-estimation of the Curie temperature, which easily\nreaches 20-30 % of the experimental value.76{81Therefore\nwe consider the calculated Curie temperature (and like-\nwise also the saturation magnetization) of the Fe 2NiAl\nHeusler compound in proper agreement with the experi-\nmental value of 1010 K (and Ms= 135 Am2/kg)82. If one\nreplaces Al by Ni in the stoichiometric Fe 2NiAl in the Tp\narrangement, one observes a slight decrease in TMFA\nC to-\nwardsx= 0:25, which results from the slight weakening\nof the nearest Fe 1-Fe2interaction and is much smaller\ncompared to the rather substantial drop in the MAE in\nthe same composition range.\nE. Finite temperature calculations\nAs discussed in Sec. III B, Fe-Ni-Al alloys near the sto-\nichiometric composition are prone to a phase separation\nby spinodal decomposition, see also Refs. 47{49. During\nthe slow cooling from the high-temperature ordered bcc\nphase, Fe 2NiAl becomes unstable against the segregation\nand decomposes into two bccphases consisting of FM Fe\nand a weak magnetic NiAl solid solution. To optimize the\nmagnetic properties, sophisticated heat treatment pro-\ncedures, including various temperatures, waiting times,\nand also magnetic \felds become important83. In this\nsection, we focus on the thermodynamic stability of the\nproposed structures in Fe 2Ni1+xAl1\u0000xby calculating the\n\fnite temperature free energy,\nF=Etot+Flat+Fel; (2)10\nwhich may help to develop further guidelines for the suc-\ncessful synthesis of the desired compound. Here, Etot\nrefers to the total energy for the respective structure\nfrom DFT calculations. Flatrepresents the lattice free\nenergy in the harmonic approximation derived from the\nVDOS of the respective structure calculated at the opti-\nmized lattice constant, which also includes the quantum\nmechanical zero point contribution to the lattice energy.\nFinally,Felis the electronic free energy, which contains\nthe change in internal energy Eel(T>0) at \fnite temper-\natures arising from the redistribution of electrons from\noccupied to unoccupied states according to the Fermi\ndistribution function and the corresponding entropy Sel.\nBoth are estimated with su\u000ecient accuracy within the\nSommerfeld approximation, which relates Felto the elec-\ntronic density of states at the Fermi level D(EF):\nFel(T) =Eel(T>0)\u0000TSel(T)\u0019\u0000\u00192\n6D(EF)k2\nBT2:(3)\nThe contribution Eel(T=0) is naturally included in Etot\nobtained from our DFT calculations, whereas we neglect\nin our work the \fnite temperature contributions to the\nfree energy from the magnetic degrees of freedom. This\ncan be considered an appropriate approximation at suf-\n\fciently low temperatures, where the magnetic moments\ncan be considered in an almost perfectly ordered arrange-\nment. In the vicinity of the Curie temperature, which is\nin our case in the order of 1000 K and thus rather high,\ncontributions from magnetic entropy may indeed become\nsigni\fcant. In turn, the accurate prediction of the mag-\nnetic entropy at low temperatures in terms of a classi-\ncal spin model is subject to di\u000eculties which may arise,\ne.g., from the violation of Nernst's theorem in classical\nsystems with continuous degrees of freedom. The solu-\ntion requires advanced modeling approaches (see, e. g.,\nRef. 84), which is beyond the scope of the present inves-\ntigation.\nReferring to Fig. 3 of Sec. III B, we will start our dis-\ncussion once again from the stoichiometric case. Fig-\nure 6 compares the stability of the four closely com-\npeting low energy structures of Fe 2NiAl, i. e., inverse\nHeusler, Tc, T#, and Tpwith respect to decomposi-\ntion on the pure Fe and binary NiAl at temperatures\nup toT= 1000 K, in terms of the free energy di\u000berence\nFdec=FFe2NiAl\u0000(2FFe+FNiAl). Including the con-\ntributions from zero point energy, all structures retain\ntheir energetic order derived from Etotwhich was shown\nin Fig. 3 and remain susceptible to the above mentioned\ndecomposition reaction. However, the \fnite temperature\ncontributions to the free energy inhibit this process: The\nstructure of Fe 2NiAl with the lowest energy, Tp, becomes\nstable against decomposition at T= 474 K, followed by\nT#atT= 643 K, whereas Tcand inverse Heusler are\npredicted to be stable above 739 and 750 K, respectively.\nThe relative stability of the ternary structures with re-\nspect to each other does not change with temperature,\nleaving Tpthe most probable candidate for a fully or-\ndered structure to be stabilized by an appropriate ther-\nFIG. 6: Decomposition of Fe 2NiAl into the pure Fe and\nbinary NiAl in terms of the respective free energy di\u000berence\nas a function of temperature.\nmodynamic procedure.\nSince Tpcorresponds to a free energy minimum in\nthe range of realistic annealing temperatures, we can\nnow discuss the stability of the o\u000b-stoichiometric Ni-\nexcess compounds Fe 2Ni1+xAl1\u0000xwith respect to a de-\ncomposition into the stoichiometric ternary compound\nTp-Fe2NiAl and binary L1 0-FeNi. This bears some anal-\nogy with experimental \fndings of a decomposition ten-\ndency in Mn-excess Ni 2Mn1+xZ1\u0000xHeusler alloys, for\nwhich a dual-phase system consisting of the L2 1-cubic\nNi2MnZphase and the L1 0-tetragonal NiMn phase has\nbeen reported85{89. In order to \fne-tune the temper-\nannealing conditions in the processing of the ternary sys-\ntem, it is important to have knowledge of the relative\nstability of particular ordering motifs at a given temper-\nature in terms of the mixing free energy Fmix:\nFmix=Faust=mart\nFe2Ni1+xAl1\u0000x\u0000TSmix\n\u0000\u0002\n(1\u0000x)FTp\nFe2NiAl+xFL10\nFeNi\u0003\n; (4)\nwhereFaust=mart\nFe2Ni1+xAl1\u0000xis the free energy of the ternary,\no\u000b-stoichiometric compositions under consideration in\naustenitic or martensitic phase; FTp\nFe2NiAl andFL10\nFeNi are\nthe corresponding free energies of Fe 2NiAl with the Tp\ncubic structure and FeNi with the L1 0tetragonal struc-\nture;Smixaccounts the mixing entropy arising from the\npartial disorder in the o\u000b-stoichiometric ternary system,\nwhere the excess Ni is distributed randomly over the four\nsublattices:\nSmix=\u00001\n4kB4X\ni=1\u0002\nxilnxi+ (1\u0000xi) ln(1\u0000xi)\u0003\n:(5)\nHerexiis the concentration of Ni on the ithsublattice\nandkBis the Boltzmann constant. Since these structures\nare fully ordered, Smix= 0 for stoichiometric Fe 2NiAl\nand FeNi and is thus left out in Eq. (4).11\nThe resulting behavior of the mixing free energy at dif-\nferent temperatures as a function of composition is de-\npicted in Fig. 7. For T= 0, it omits all \fnite temperature\ncontributions except for the zero point energy. For sto-\nichiometric Fe 2NiAl, where mixing entropy vanishes for\nall compounds under consideration, T#is in accordance\nwith Fig. 3 the closest competitor to the energetically\nmost favorable Tpstructure, which becomes closer in en-\nergy with increasing temperature. On the other side,\nregular L1 0-FeNi proves preferable to the Tcstructure\nwith all Al replaced by Ni in the range of realistic tem-\nperatures.\nFor the non-stoichiometric compositions, the impact of\ntemperature on Fmixincreases substantially due to the\npresence of the mixing entropy. This leads in particu-\nlar for the regular L1 0to a signi\fcant reduction of the\ndi\u000berence in free energy compared to a respective phase\nmixture of the stoichiometric compounds, which may sta-\nbilize L1 0-type solid solutions at the very Ni-rich content\nat high temperatures. Still, the o\u000b-stoichiometic com-\npounds remain unstable in general against this decompo-\nsition for all temperatures under consideration. A very\nsimilar trend was reported recently for o\u000b-stoichiometric\nNi-Mn-based Heusler compounds90,91. For the compo-\nsition with x= 0:25, we see a rather close competition\nbetween Tcand Tp(both with nearly cubic lattice pa-\nrameters) and L1 0martensite. The inverse Heusler struc-\nture was not considered here, since according to Fig. 2\nits energy minimum is clearly above the other candidates.\nIn turn, we observe that at x= 0:75 the Tcphase with\ntetragonal lattice parameters becomes with increasing\ntemperatures increasingly competitive with the marten-\nsitic L1 0phase.\nIn Fig. 8 we show the temperature dependence of the\nFIG. 7: The dependence of mixing free energy Fmixon\nthe Ni-excess concentration for selected FM austenite and\nmartensite structures of Fe 2Ni1+xAl1\u0000xat various temper-\natures in the range between 0 and 1000 K.\nFIG. 8: Temperature variation of the free energy of the\naustenitic Tpand Tcphase atx= 0:25 and martensitic Tcat\nx= 0:75. The free energy is plotted as the di\u000berence \u0001 Fmixto\nthe free energy of the regular L1 0structure at the respective\ncomposition.\nfree energy of Tp(atx= 0:25) and Tc(x= 0:25\nandx= 0:75) both at the respective optimum lattice\nparameters relative to the free energy of the tetrag-\nonal L1 0phase with the respective composition, i. e.,\n\u0001Fmix=FTp;Tc\nmix\u0000FL10\nmix. We see that for x= 0:25 the Tc\nstructure is more stable than L1 0for temperatures be-\nlowT= 800 K, while Tppreserves the lowest free energy\nin the entire temperature range. However, extrapolation\nof \u0001Fmiximplies that a crossover may occur somewhere\nbelowT= 1200 K and the temperature of the crossing\npoint may decrease further, if a larger amount of Al is\nsubstituted by Ni. On the Ni-rich side only Tcis compet-\nitive but its free energy remains considerably above the\nL10phase. Although here \u0001 Fmixdecreases with temper-\nature, we can not predict a stabilization at a reasonable\ntemperature. However, since the sign and the slope of\n\u0001Fmixchanges between x= 0:25 andx= 0:75 we might\nrather expect interesting cross-over e\u000bects between these\nstructures at intermediate compositions. To resolve the\ncorresponding phase transitions from \frst principles with\nsu\u000ecient accuracy, better structural models for the par-\ntially disordered phases are needed, which require signif-\nicantly larger supercells. As this severely increases the\ncomputational demands, the corresponding phonon cal-\nculations have been left for future investigations.\nIV. CONCLUSIONS\nThe substitutional series of the Heusler compounds\nFe2Ni1+xAl1\u0000x(0\u0014x\u00141) in austenitic and martensitic\nphases with di\u000berent kinds of atomic arrangements were\ninvestigated theoretically to address their phase stability\nas well as structural and magnetic properties including\nthe MAE. The present study was motivated by the po-\ntential importance of Fe 2NiAl and L1 0-FeNi for a perma-12\nnent magnet application. It predicts phase segregation\ntendencies in Fe 2NiAl and addresses the complexity of\nthe synthesis of FeNi with L1 0order which may arise in\npart from the presence of competing structures. We learn\nhow the phase stability and MAE can be controlled by\nexchanging the positions of Fe and Ni atoms, by consid-\nering three kinds of atomic arrangements which can be\nderived from the inverse Heusler structure of Fe 2NiAl by\nshu\u000fing Fe and Ni atoms on particular sites. To this end,\nthe formation and mixing energies of a series of compo-\nsitions with considered crystalline structures were calcu-\nlated at zero and \fnite-temperatures. Our calculations of\nthe saturation magnetization, exchange constants, and \f-\nnally Curie temperature con\frm that all considered com-\npounds exhibit su\u000eciently large values, as required for\npermanent magnet applications.\nFor the austenitic phase of Fe 2NiAl, we found that all\nrecently proposed structures denoted as Tc, T#and Tp\nare lower in energy compared to the inverse and regu-\nlar structures, which were mainly discussed so far in the\nliterature. The ground state corresponds to the ordered\nTpstructure, which was proposed recently as a stable\nlow temperature structure for some quarternary stoichio-\nmetric Heusler alloys52. It consists of alternating layers\nof pure Fe and pure Ni with mixed layers of Fe and Al\nin between and bcc-type coordination of the ions. This\nstructure was also found to possess the largest uniaxial\nMAE in the study, due to the intrinsic non-cubic arrange-\nment of chemical elements in the cell. The MAE reaches\na value of about 1.05 MJ/m3, exceeding the MAE of L1 0-\nFeNi. This is accompanied in the case of Tp-Fe2NiAl by\na larger anisotropy of the orbital magnetic moments. In\nthe Tpstructure, the partial substitution of Al by Ni\nyields a successive reduction of the MAE, which becomes\nnearly suppressed for Fe 2Ni1:25Al0:75. Further increase in\nthe o\u000b-stoichiometry leads to a crossover of the structural\nmotifs and at large Ni content \fnally to the appearance\nof tetragonal L1 0order. In the case of binary FeNi, the\nTcstructure with a staggered arrangement of Fe and Ni\nturns out to be rather close in energy to the L1 0phase.\nTc-FeNi possesses a planar MAE, with an absolute value\nonly about half as large as the one of L1 0-FeNi and the\nopposite sign.\nRegarding the decomposition of Fe 2NiAl into bcc-\nFe and NiAl, our total energy calculations reveal\nthat the Tpstructure has a positive formation energy\n(Eform\u001912:4 meV/atom) at T= 0 K being several\ntimes lower than other structures including the regular\nand inverse Heusler structure. Despite still lying above\nthe convex hull, it can be expected that Tp-Fe2NiAl can\nbe stabilized by a proper choice of annealing condition,since due to the contributions from vibrational entropy\ncalculated in the harmonic approximation, the free en-\nergy of the Tpphase becomes lower compared to a phase\nmixture of Fe and NiAl above T= 474 K, which is reason-\nable for typical temper-annealing procedures. The \fnite\ntemperature approach to study the segregation tendency\nin Ni-rich Fe 2Ni1+xAl1\u0000xhave shown that in particular\nthe contributions from mixing entropy render the free\nenergy quite sensitive to temperature. This reduces the\nfree energy o\u000bset to a phase mixture between stoichio-\nmetric Fe 2NiAl and FeNi substantially, which enhances\nthe probability to obtain o\u000b-stoichiometric phases. Nev-\nertheless, all o\u000b-stoichiometric compounds demonstrate\nthe tendency to decompose into a dual-phase mixture\nconsisting of Fe 2NiAl and FeNi at temperatures below\nT= 1000 K.\nIn general, we consider the fundamental understanding\nof the role of atomic arrangement in a crystalline struc-\nture in Heusler alloys of great interest for both the shape\nmemory and the permanent magnet community as this\nmight be employed to deliberately modify the phase sta-\nbility, structural and magnetic properties as well as MAE.\nWe believe that the Tpstructure in Fe 2NiAl and related\nmaterials could be interesting and promising candidate\nfor systems exhibiting a large intrinsic MAE, saturation\nmagnetization and high Curie temperature, suitable for\na potential application in low-cost hard magnets.\nV. ACKNOWLEDGMENTS\nThe authors dedicate this work to Prof. Dr. Peter En-\ntel, who passed away on Aug. 4, 2021. We are grate-\nful for the involved discussions, his brilliant ideas and\nsubstantial support which fostered many successful com-\nmon projects over the past decades, including this one.\nThis work was further supported by the RSF - Russian\nScience Foundation project No. 17-72-20022 (geometric\noptimization and mixing energy and Exchange param-\neters calculations). MAE calculations were performed\nwith the support of the Ministry of Science and Higher\nEducation of the Russian Federation within the frame-\nwork of the Russian State Assignment under contract\nNo. 075-00250-20-03. Phonon and mixing energy calcu-\nlations were performed on the MagnitUDE high perfor-\nmance computing system of the University of Duisburg-\nEssen (DFG INST 20876/209-1 and 20876/243-1 FUGG)\nwith \fnancial support from the Deutsche Forschungsge-\nmeinschaft (DFG, German Research Foundation) within\nTRR 270 (subproject B06), Project-ID 405553726.\n1R. Skomski and J. Coey, Scr. Mater. 112, 3 (2016).\n2R. Skomski, Permanent Magnets: History, Current Re-\nsearch, and Outlook (Springer International Publishing,\nCham, 2016), pp. 359{395.3K. Hono and H. Sepehri-Amin, Scr. Mater. 151, 6 (2018).\n4K. Skokov and O. Gut\reisch, Scr. Mater. 154, 289 (2018).\n5J. Mohapatra and J. P. Liu, Rare-Earth-free permanent\nmagnets: the past and future (Elsevier, 2018), vol. 27, pp.13\n1{57.\n6A. Kovacs, J. Fischbacher, M. Gusenbauer, H. Oezelt,\nH. C. Herper, O. Y. Vekilova, P. Nieves, S. Arapan, and\nT. Schre\r, Engineering 6, 148 (2020).\n7J. Coey, Engineering 6, 119 (2020).\n8M. Kuz'min, K. Skokov, H. Jian, I. Radulov, and O. Gut-\n\reisch, J. Phys. Condens. Matter. 26, 064205 (2014).\n9R. McCallum, L. H. Lewis, R. Skomski, M. Kramer, and\nI. Anderson, Annu. Rev. Mater. Res. 44, 451 (2014).\n10D. Niarchos, G. Giannopoulos, M. Gjoka, C. Sara\fdis,\nV. Psycharis, J. Rusz, A. Edstr om, O. Eriksson, P. To-\nson, J. Fidler, et al., JOM 67, 1318 (2015).\n11S. Hirosawa, J. Magn. Soc. Jpn. 39, 85 (2015).\n12D. Li, D. Pan, S. Li, and Z. Zhang, Sci. China Phys. Mech.\nAstron. 59, 617501 (2016).\n13A. Edstr om, J. Chico, A. Jakobsson, A. Bergman, and\nJ. Rusz, Phys. Rev. B 90, 014402 (2014).\n14M. Werwi\u0013 nski and W. Marciniak, J. Phys. D: Appl. Phys.\n50, 495008 (2017).\n15J. Cui, M. Kramer, L. Zhou, F. Liu, A. Gabay, G. Hadji-\npanayis, B. Balasubramanian, and D. Sellmyer, Acta Ma-\nterialia 158, 118 (2018).\n16L.-Y. Tian, H. Lev am aki, O. Eriksson, K. Kokko, \u0013A. Nagy,\nE. K. D\u0013 elczeg-Czirj\u0013 ak, and L. Vitos, Sci. Rep. 9, 1 (2019).\n17L.-Y. Tian, O. Eriksson, and L. Vitos, Sci. Rep. 10, 1\n(2020).\n18L.-Y. Tian, O. Gut\reisch, O. Eriksson, and L. Vitos, Sci.\nRep.11, 1 (2021).\n19D. Tuvshin, T. Ochirkhuyag, S. Hong, and D. Odkhuu,\nAIP Advances 11, 015138 (2021).\n20J. Paulev\u0013 e, D. Dautreppe, J. Laugier, and L. N\u0013 eel, Compt.\nrend. 254(1962).\n21J. Paulev\u0013 e, A. Chamberod, K. Krebs, and A. Bourret, J.\nAppl. Phys. 39, 989 (1968).\n22E. Poirier, F. E. Pinkerton, R. Kubic, R. K. Mishra,\nN. Bordeaux, A. Mubarok, L. H. Lewis, J. I. Goldstein,\nR. Skomski, and K. Barmak, J. Appl. Phys. 117, 17E318\n(2015).\n23R. Skomski and J. M. D. Coey, Permanent magnetism\n(Routledge, 2019).\n24Y. G. Pastushenkov, K. Skokov, N. Suponev, and\nD. Stakhovski, J. Magn. Magn. Mater. 290, 644 (2005).\n25Y. Otani, H. Miyajima, and S. Chikazumi, J. Appl. Phys.\n61, 3436 (1987).\n26L. H. Lewis, A. Mubarok, E. Poirier, N. Bordeaux, P. Man-\nchanda, A. Kashyap, R. Skomski, J. Goldstein, F. Pinker-\nton, R. Mishra, et al., J. Phys. Condens. Mat. 26, 064213\n(2014).\n27L. N\u0013 eel and J. Paulev\u0013 e, J. Appl. Phys. 35, 873 (1964).\n28E. Lima Jr and V. Drago, Phys. Stat. Sol. (A) 187, 119\n(2001).\n29T. Shima, M. Okamura, S. Mitani, and K. Takanashi, J.\nMagn. Magn. Mater. 310, 2213 (2007).\n30A. Makino, P. Sharma, K. Sato, A. Takeuchi, Y. Zhang,\nand K. Takenaka, Sci. Rep. 5, 1 (2015).\n31G. Giannopoulos, G. Barucca, A. Kaidatzis, V. Psycharis,\nR. Salikhov, M. Farle, E. Koutsou\rakis, D. Niarchos,\nA. Mehta, M. Scuderi, et al., Sci. Rep. 8, 1 (2018).\n32S. Goto, H. Kura, E. Watanabe, Y. Hayashi, H. Yanag-\nihara, Y. Shimada, M. Mizuguchi, K. Takanashi, and\nE. Kita, Sci. Rep. 7, 1 (2017).\n33A. Izardar and C. Ederer, Phys. Rev. Materials 4, 054418\n(2020).34Y. Matsushita, G. Madjarova, J. Dewhurst, S. Shallcross,\nC. Felser, S. Sharma, and E. Gross, J. Phys. D: Appl. Phys.\n50, 095002 (2017).\n35H. C. Herper, Phys. Rev. B 98, 014411 (2018).\n36Q. Gao, I. Opahle, O. Gut\reisch, and H. Zhang, Acta\nMater. 186, 355 (2020).\n37Y. Zhang, W. Wang, H. Zhang, E. Liu, R. Ma, and G. Wu,\nPhysica B: Condens. Matter 420, 86 (2013).\n38D. C. Gupta and I. H. Bhat, Mater. Chem. Phys. 146, 303\n(2014).\n39F. Dahmane, Y. Mogulkoc, B. Doumi, A. Tadjer, R. Khen-\nata, S. B. Omran, D. Rai, G. Murtaza, and D. Varshney,\nJ. Magn. Magn. Mater. 407, 167 (2016).\n40E. S. Popiel, W. Zarek, and M. Tuszy\u0013 nski, Nukleonika 49,\n49 (2004).\n41M. Yin, P. Nash, and S. Chen, Intermetallics 57, 34 (2015).\n42L. Zhang, J. Wang, Y. Du, R. Hu, P. Nash, X.-G. Lu, and\nC. Jiang, Acta Mater. 57, 5324 (2009).\n43A. Bradley and A. Taylor, Nature 140, 1012 (1937).\n44V. Menushenkov, M. Gorshenkov, E. Savchenko,\nI. Shchetinin, and A. Savchenko, Heat Treat. Met.\n59, 518 (2017).\n45S. M. Hao, T. Takayama, K. Ishida, and T. Nishizawa,\nMetall. Trans. A 15, 1819 (1984).\n46L. Zhang, Y. Du, H. Xu, C. Tang, H. Chen, and W. Zhang,\nJ. Alloys Compd. 454, 129 (2008).\n47K. Buschow and F. de Boer, Physics of Magnetism and\nMagnetic Materials , Focus on biotechnology (Springer US,\n2007).\n48V. Menushenkov, M. Gorshenkov, D. Zhukov,\nE. Savchenko, and M. Zheleznyi, Mater. Lett. 152,\n68 (2015).\n49V. Menushenkov, M. Gorshenkov, I. Shchetinin,\nA. Savchenko, E. Savchenko, and D. Zhukov, J. Magn.\nMagn. Mater. 390, 40 (2015).\n50G. Kresse and J. Furthm uller, Phys. Rev. B 54, 11169\n(1996).\n51J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.\nLett. 77, 3865 (1996).\n52P. Neibecker, M. E. Gruner, X. Xu, R. Kainuma, W. Petry,\nR. Pentcheva, and M. Leitner, Phys. Rev. B 96, 165131\n(2017).\n53See Supplemental Material at http://... for additional in-\nformation about MAE convergence test, crystal structures\nfor o\u000b-stoichiometric compositions, decomposition ener-\ngies, and SOC contribution to the MAE.\n54A. Liechtenstein, M. Katsnelson, and V. Gubanov, J. Phys.\nF: Met. Phys. 14, L125 (1984).\n55A. Liechtenstein, M. Katsnelson, and V. Gubanov, Solid\nState Commun. 54, 327 (1985).\n56A. I. Liechtenstein, M. Katsnelson, V. Antropov, and\nV. Gubanov, J. Magn. Magn. Mater. 67, 65 (1987).\n57J. Enkovaara, A. Ayuela, L. Nordstr om, and R. M. Niem-\ninen, Phys. Rev. B 65, 134422 (2002).\n58R. Umetsu, A. Sakuma, and K. Fukamichi, Appl. Phys.\nLett. 89, 052504 (2006).\n59M. E. Gruner, P. Entel, I. Opahle, and M. Richter, J.\nMater. Sci. 43, 3825 (2008).\n60A. Edstr om, M. Werwi\u0013 nski, D. Iu\u0018 san, J. Rusz, O. Eriksson,\nK. Skokov, I. Radulov, S. Ener, M. Kuz'Min, J. Hong,\net al., Phys. Rev. B 92, 174413 (2015).\n61A. Togo and I. Tanaka, Scr. Mater. 108, 1 (2015).\n62H. Ebert, D. K odderitzsch, and J. Min\u0013 ar, Rep. Prog. Phys.\n74, 096501 (2011).14\n63Due to the distribution of the atoms, a 16 atom cell of\nFe2Ni1:5Al0:5(Fe8Ni6Al2) possesses two inequivalent crys-\ntallographic directions along xoryandzaxes even with\ncubic lattice parameters a=b=c. Replacing two Al in\nthe inverse Heusler supercell, we necessarily obtain one\natomic layer which completely consists of Ni atoms. As a\nresult, a tetragonal distortion can be applied to the lattice\nparameters which are parallel or orthogonal to this layer.\nThe respective results are averaged.\n64M. Siewert, M. E. Gruner, A. Dannenberg, A. Chakrabarti,\nH. C. Herper, M. Wuttig, S. R. Barman, S. Singh, A. Al-\nZubi, T. Hickel, et al., Appl. Phys. Lett. 99, 191904 (2011).\n65R. Wu and A. Freeman, J. Magn. Magn. Mater. 200, 498\n(1999).\n66Y. Miura, S. Ozaki, Y. Kuwahara, M. Tsujikawa, K. Abe,\nand M. Shirai, J. Phys. Condens. Matter. 25, 106005\n(2013).\n67L. H. Lewis, F. E. Pinkerton, N. Bordeaux, A. Mubarok,\nE. Poirier, J. I. Goldstein, R. Skomski, and K. Barmak,\nIEEE Magn. Lett. 5, 1 (2014).\n68P. Bruno, Phys. Rev. B 39, 865 (1989).\n69F. Wilhelm, P. Poulopoulos, H. Wende, A. Scherz,\nK. Baberschke, M. Angelakeris, N. K. Flevaris, and A. Ro-\ngalev, Phys. Rev. Lett. 87, 207202 (2001).\n70C. Andersson, B. Sanyal, O. Eriksson, L. Nordstr om,\nO. Karis, D. Arvanitis, T. Konishi, E. Holub-Krappe, and\nJ. Dunn, Phys. Rev. Lett. 99, 177207 (2007).\n71I. V. Solovyev, P. H. Dederichs, and I. Mertig, Phys. Rev.\nB52, 13419 (1995).\n72P. Ravindran, A. Kjekshus, H. Fjellvaag, P. James,\nL. Nordstr om, B. Johansson, and O. Eriksson, Phys. Rev.\nB63, 144409 (2001).\n73M. E. Gruner, Phys. Stat. Sol. (A) 210, 1282 (2013).\n74M. Wolloch, D. Suess, and P. Mohn, Phys. Rev. B 96,\n104408 (2017).75J. Enkovaara, A. Ayuela, J. Jalkanen, L. Nordstr om, and\nR. M. Nieminen, Phys. Rev. B 67, 054417 (2003).\n76D. Garanin, Phys. Rev. B 53, 11593 (1996).\n77V. Sokolovskiy, V. Buchelnikov, M. Zagrebin, P. Entel,\nS. Sahoo, and M. Ogura, Phys. Rev. B 86, 134418 (2012).\n78M. Meinert, J. Phys. Condens. Matter 28, 056006 (2016).\n79B. Wasilewski, W. Marciniak, and M. Werwi\u0013 nski, J. Phys.\nD: Appl. Phys. 51, 175001 (2018).\n80X.-P. Wei and Y.-H. Zhou, Intermetallics 93, 283 (2018).\n81M. Zagrebin, M. Matyunina, O. Miroshkina,\nO. Pavlukhina, V. Sokolovskiy, and V. Buchelnikov,\nPh. Transit. 93, 43 (2020).\n82T. Saito and D. Nishio-Hamane, J. Appl. Phys. 124,\n075105 (2018).\n83M. Stanek, L. Wierzbicki, and M. Leonowicz, Arch. Metall.\nMater. 55, 571 (2010).\n84F. K ormann, A. Dick, T. Hickel, and J. Neugebauer, Phys.\nRev. B 81, 134425 (2010).\n85W. M. Yuhasz, D. L. Schlagel, Q. Xing, K. W. Dennis,\nR. W. McCallum, and T. A. Lograsso, J. Appl. Phys. 105,\n07A921 (2009).\n86W. Yuhasz, D. Schlagel, Q. Xing, R. McCallum, and T. Lo-\ngrasso, J. Alloys Compd. 492, 681 (2010).\n87T. Krenke, A. C \u0018 ak\u0010r, F. Scheibel, M. Acet, and M. Farle,\nJ. Appl. Phys. 120, 243904 (2016).\n88A. C \u0018 ak\u0010r, M. Acet, and M. Farle, Sci. Rep. 6, 1 (2016).\n89A. C \u0018ak\u0010r, M. Acet, U. Wiedwald, T. Krenke, and M. Farle,\nActa Mater. 127, 117 (2017).\n90P. Entel, M. E. Gruner, M. Acet, A. C \u0018 ak\u0010r, R. Arr\u0013 oyave,\nT. Duong, S. Sahoo, S. F ahler, and V. V. Sokolovskiy,\nEnergy Technol. 6, 1478 (2018).\n91V. V. Sokolovskiy, M. E. Gruner, P. Entel, M. Acet,\nA. C \u0018 ak\u0010r, D. R. Baigutlin, and V. D. Buchelnikov, Phys.\nRev. Materials 3, 084413 (2019)."    },    {        "title": "2110.15507v1.Two_types_of_magnetic_bubbles_in_MnNiGa_observed_via_Lorentz_microscopy.pdf",        "content": "1 \n Two types of magnetic bubbles in MnNiGa observed via Lorentz microscopy  \nHiroshi Nakajima1, Atsuhiro  Kotani1, Ken Harada1,2, and Shigeo  Mori1, * \n \n1Department of Materials Science, Osaka Prefecture University, Sakai, Osaka 599 -8531, Japan  \n2Center for Emergent Matter Science (CEMS), The Institute of Physical and Chemical Research \n(RIKEN), Hatoyama, Saitama 350 -0395, Japan  \n*E-mail:  mori@mtr.os akafu -u.ac.jp  \n \n \nABSTRACT  \nMagnetic bubbles are remarkable  spin structures that developed  in uniaxial magnets with strong  \nmagnetocrystalline anisotropy. Several contradictory reports have been published concerning  the \nmagnetic bubble structure in a metallic magnet MnNiGa: Biskyrmions  or type-II bubbles . Lorentz \nmicroscopy in polycrystalline MnNiGa  was used to explain the  magnetic bubble structure. \nDepending  on the connection  between the magnetic easy axis and the o bservation plane , two types \nof magnetic bubbles were formed . Magnetic bubbles with 180 ° domains were  formed if the easy \naxis was away from the direction perpendicular to the observation pla ne. The contrast of \nbiskyrmion  is reproduced by this form of a magnetic bubble. When the easy axis was approximately \nperpendicular to the observing plane , type -II bubbles were observed in the same specimen . The \nfindings  will fill a knowledge  gap between prior  reports on magnetic bubble s in MnNiGa.  \n \n \n  2 \n 1. Introduction  \nMagnetic bubbles are circularly rotating magnetic structures formed in  uniaxial magnets with  high \nmagnetocrystalline anisotropy. The structures have been known for more than 50 years  in \nferromagnets such as cobalt and hexagonal ferrites .1),2) Nevertheless , the recent finding  of magnetic \nskyrmions has rekindled interest in  magnetic bubbles , as magnetic bubbles have a topological \nnumber comparable with magnetic skyrmions.3),4) When  the spin structure spans  the entire  sphere, \nthe topological number  N is specified as N = 1. As a result of the interest, various unusual  magnetic \nbubbles have been discovered  in various  magnets , including  perovskite manganese oxides and \nlayered oxides .5)–7) Moreover , it is demonstrated that m agnetic bubbles  can be moved by a current .8) \nThe topological number is used to classify magnetic bubbles. A magnetic b ubble is classified as \ntype-I if it has a continuous rotating structure with N = 1. Type -II bubbles, conversely , have two \nparallel domain walls ( N = 0). Bloch lines are  formed as  two discontinuous points in type-II \nmagnetic bubble s.9) \nMnNi Ga is repo rted to form  magnetic bubbles.10) The material has  a layered Ni 2In-type \nstructure of hexagonal symmetry (space group P63/mmc ). According to the paper , the magnetic \nbubbles are biskyrmions, which are made up  of clockwise and counterclockwise bubbles with N = 2. \nRecent research, however, has called into question  the existence of biskyrmion. A Lorentz \nmicroscopy study reports that only type -I and type -II magnetic bubbles are found in MnNiGa and \nthat biskyrm ions can be explained by type -II magnetic bubbles.11) A further x -ray holography \nanalysis of MnNiGa  bubble formations determined that the magnetic bubbles are type -II.12) \nAdditio nally , Lorentz imaging research based on the transport of intensity equation (TIE) shows \nthat type -II bubbles resemble biskyrmions when TIE parameters  are incorrect.13) Although magnetic \ndomain st ructures observed by these authors are demonstrated to be type -II in MnNiGa , the contrast \nof biskyrmons10) is different from that predicted from the type -II bubble  structure because the \ncontrast has no Bloch lines. Moreover , the domain walls of a uniaxial magnet should contain  a pair \nof b right and dark Fresnel contrast lines in type-II magnetic bubbles. The domain walls of \nbiskyrmions , conversely , have just single bright or dark lines in the Fresnel image and no pair \nstructure is visible  in the contrast .10) \nWe observed magnetic domain s in MnNiGa  to understand the difference between the magnetic \nbubble structures reported in the previous studies10)–13). When  the magnetic easy axis of the c axis \nwas away from the direction perpendicular to the viewing  plane , magnetic bubbles with 180° \ndomains were created  (The plane is the surface formed during the thinning process). When  the TIE \nmethod was used on this structure, it created the biskyrmion contrast . Type -II magnetic bubbles \nwere also observed when the c axis was slightly inclined  from the perpendicular direction to the \nobserving  plane. These two structures were discovered  in various  grains of the same material . The \ndifferent crystallographic direction s concerning  the observation plane s should explain the \ndiscrepancy of the previou s studies.  3 \n 2. Experimental methods  \nArc melting was used to construct polycrystal  specimens of MnNiGa . The ingots of Ga and the \npowers of Mn and Ni were weighed on the basis of  the composition. They were melted in an argon \natmosphere  using an arc furnace.  They were sealed off in a vacuum with a quartz tube. Then , it was \nheated at 700 ℃ for 110  h. Finally, it was cooled to room temperature by being immersed in water. \nThe sp ecimens were confirmed by x -ray diffraction and energy -dispersive x -ray spectroscopy \n(EDS) of scanning electron microscopy.  The EDS results showed that the ratio of the specimen  was \nMn:Ni:Ga = 0.98 :1.10:1.00, which agrees with  the composition.  A vibrating sample magnetometer \nwas used to measure magnetization (Quantum Design, Inc.). Lorentz microscopy was conducted  \nusing a JEM -2100F transmission electron microscope (TEM ) (JEOL Co. Ltd., Japan). The \nmagnitude of the acceleration voltage was 200 kV. Magnetic domain walls were detected using \nFresnel imaging . A defocus value  f of Fresnel imaging  is defined as positive  in an overfocus \ncondition.  Small-angle electron diffraction  (SmAED)  pattern s were  observed to analyze  a magnetic \ndomain structure .14),15) The objective lens of the TEM was used to apply external magnetic fields.  \nThe image s based on TIE were calculated  using software (QPt).16) \n \n3. Results and discussion  \nFigure 1(a) sho ws the powder x -ray diffraction profile  of the  specimen  synthesized in this study . \nThe specimen can be indexed with the MnNiGa structure of the space group P63/mmc, whose \nstructure is illustrated in Figure 1(b). The temperature dependence of magnetization shows a \nferromagnetic transition at 350  K [Figure 1(c)] . These results are consistent with those of the \nprevious study.10) \nLorentz microscopy was used to examine  magnetic domains in the MnNiGa material . Figure \n2(a) illustrates  the Fresnel image at room temperature in the absence of a magnetic field. The inset \nelectron diffraction pattern reveals  that this location  has a zone axis of [ 22̅1]. The simulation of \nSupplementary Figure  1 indicates that the c axis almost points parallel to the specimen surface. \nHence , the c axis of the magnetic easy axis was tilted from the direction perpendicular to the \nobservation plane by more than 10°. Striped domains were formed in this grain although the \nremanent magnetic field of the objective lens  caused some magnetic bubbles.  A branching maze \npattern is typical of a uniaxial magnet17). Magnetization  points the in -plane directions in the \ndomains and the out -of-plane direc tions at the domain walls , as shown in the magnified image of \nFigure  2(b): The formation of 180° domains in this region is shown in  red arrows. The contrast of \nFresnel imaging  explains this: When the domain structure is 180° domains, single  bright or dark \nlines are observed  at domain walls , as shown in Figure  2(b). If the magnetization in  domains  points \nin out -of-plane directions , a pair of bright  and dark lines should form  at the walls. The SmAED \npattern  of Figure 2(c)  also supports the  180°  domain structure depicted by the arrows . The magnetic \ndeflection spots (  = 17.8 rad) correspond to a width of a pair of antiparallel domains  (d = 145 \nnm) because  these values sa tisfy Bragg ’s law  dsin = , where  = 2.51 pm is the wavelength of 4 \n electrons . When viewed from the direction parallel to the magnetic easy axis, a  uniaxial magnet \noften displays  striped domains with  out-of-plane magnetization . Nevertheless , as shown  in this area, \n180° domains arise  when the magnetic easy axis is  positioned  away from the direction \nperpendicular to the observation plane.  \n To confirm the formation process of magnetic bubbles, a magnetic field was applied in this \nregion. Figure 3 (a) shows the Fresnel image of magnetic b ubbles at 300 m T at the area where the \ntransition from the stripe to bubble structures occurred . When an external magnetic field was \ngenerated  parallel to the TEM ’s optical axis , domains parallel to the field increased while  domains \nantiparallel to the field decreased. The  antiparallel domains were then reduced  and shortened on the \nbasis of  the strength  of the external magnetic field. Consequently , the domains with a pair of bright \nand dark regions  were formed as magnetic bubbles  that the domains were pinched off . The \nmagnetization directions are represented  in Figure  3(b). Across the domain barriers, the \nmagnetization directions are antiparallel . The structure suggests  that the magnetic bubbles \noriginated  by shrinking  the 180° domains. A similar color map is generated  in the TIE mapping10), \nas shown in Figure  3(c). The magnetic bubb les formed by pinching off 180 ° domains depict the \ncontrast  of a pair of dark and bright half circles  in the Fresnel image as shown in Figure  3(d). On \nthe basis of  the above results, the contrast can be explained by the antiparallel magnetization of th e \nred arrows , as illustrated in Figure  3(e). \nAnother  grain with the c axis almost perpendicular to the viewing  plane was viewed to \ncompare the previous results. The electron diffraction pattern in Figure  4(a) showed the [001] zone \naxis by tilting a few degrees  in this location.  Figure 4 (b)–(d) illustrates  the formation of magnetic \nbubbles in magnetic fields. Note that  the condition of the external  magnetic field  direction remained \nunchanged: T he tilt of the specimen holder  was the same  between Figures  3 and 4 although  the \ndifferent grains had different directions of the c axis in terms of the observation plane.  Striped \ndomains were generated in the absence of a magnetic field . The magnetic field , conversely , \ngenerated magnetic bubbles with walls made up of  a pair of bright  and dark lines. Moreover , the \ncontrast is reversed at two points  known as  Bloch lines. Consequently , the images revealed  that the \nmagnetic bubbles are type -II. This type -II bubble structure is seen in the magnified image of Figure  \n4(e), which corresponds to the contrast schematic in Figure  4(f). Within the magnetic domains, the \nmagnetization points along the observation direction. The Bloch walls have in -plane ma gnetization \nas shown by the red arrows. This model of Figure  4(f) is different from that of Figure  3(e) in terms \nof the relationships between the magne tization direction and the observation plane . The type -II \nmagnetic bubbles are a typical characteristic of uniaxial magnets when the external magnetic fields \nare applied  from the direction  slight ly tilted from  the magnetic easy axis18). \nThe difference between these two structures can be better understood by looking at  \nthrough -focus images. Through -focus images of magnetic bubbles generated  by 180° domains, \nwhich reproduce the contrast of biskyrmions , are shown in Supplementary Figure  2. If the defocus \nvalue is reversed in  the images, the contrast of the magnetic bubbles is changed . The contrast \ndisappears in the in -focus f = 0 m. The contrast of the magnetic bubbles is higher if the defocus 5 \n value is larger. Noticeably, the magnetic domain contrast remained unchanged even if the defocus \nvalue is changed: Magnetic  bubbles are shown  by a pair of bright and dark patches  in the Fresnel \nimages of the large ( f = ±960 m) and small (f = ±190 m) defocus values . Through -focus \nimages of type-II magnetic bubbles (Supplementary Figure  3), on the other hand, demonstrate  a \ndefocus dependence of the Fresnel contrast. In the low defocus values  (f = ±190 m and ±380 \nm), the magnetic domain walls comprise a pair of bright and dark lines , as illustrated in Figure  4(f). \nWhen the defocus is large , the contrast of domain walls merges . For example, the blur contrast in \nthe Fresnel image of f = +1400  m is similar to  the magnetic domain contrast in Supplementary \nFigure  2. These findings show  that magnetic bubbles with 180° domains and type -II may be \ndifferentiated using  through -focus imaging.  \n In MnNiGa, there ha s been controversy in the existence of biskyrmions.11)–13) In the \nprevious studies , the authors claim that biskyrmions are the same as type -II magnetic bubbles.  They \ndemonstrated that the magnetic structure  observed in  a MnNiGa single crystal  can be \nexperimentally explained by type -II magnetic bubbles in Fresnel imaging and x-ray holograp hy. \nHowever, the contrast of biskyrmions in the prior study ’s Fresnel image10) appears  to be distinct  \nfrom that of type -II magnetic bubbles. We were able to see two types of magnetic bubbles in the \nspecimen  during our examination. The  angles formed by  the magnetic easy axis and the viewing  \nplane  determined them. Magnetic bubbles are 180° domains that are pinched off by the  external \nfield when  the easy axis is away from the direction perpendicular to the observing  plane, \nreproducing  the contrast of biskyrmion. The formation of 180 ° domains was reasonable because the \neasy axis was almost  parallel to  the observation plane. Conversely, type -II magneti c bubbles were \nalso observed when the c axis w as slightly tilted from the direction perpendicular to the observation \nplane . They were observed under  the same conditions , such as the direction of the external magnetic \nfields and the specimen tilt, although the directions of the c axis differed due to grain differences. In \na previous study10), the authors also demonstrated  that a s pecimen tilt of 25° caused biskyrmion  \ncontrast, most likely  because the tilt’s in-plane component of the magnetic field changed type -II \nmagnetic bubbles into 180° domains. However, the current  results show  that if a polycrystalline \nspecimen is observed, the specimen tilt is not required  to observe  the contrast of biskyrmion . The \nfindings of our observations  should help to explain the various assertions made in  previous studies.  \nNotably, the contrast of Figure  3 is derived from 180° domains that are pinched off by the magnetic \nfields. Hence , the contrast of Figure  3 should be f ormed in a ferroma gnetic phase under external \nmagnetic fields , which may explain the wide temperature range of the  contrast as reported .10) The \nreported 180° domains that are pinc hed off are frequent properties of uniaxial magnets, as observed \nin another uniaxial magnet ,19) when the magnetic easy axis is more than 10° away from the \nobserving  direction . \n4. Conclusions  \nLorentz microscopy was used in this research  to observe magnetic bubbles in MnNiGa. We were \nable to observe two sorts  of magnetic bubbles  on the basis  of the magnetic easy axis  directions . One 6 \n type is magnetic bubbles originating from  shortened  180°  domains, which reproduce the contrast of \nbiskyrmion. The other is type-II magnetic bubbles with two Bloch lines, which are common  in \nuniaxial magnets. The observations show  that the two types of magnetic bubbles can occur in \nMnNiGa, which explains  the contradiction in  previous research . \n \nReferences  \n1) P. J. Grundy, D. C. Hothersall, G. A. Jones, B. K. Middleton and R. S. Tebble, Phys. status \nsolidi 9, 79 (1972).  \n2) P. J. Grundy and S. R. Herd, Phys. status solidi 20, 295 (1973).  \n3) S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer,  R. Georgii and P. \nBöni, Science 323, 915 (2009).  \n4) N. Nagaosa and Y. Tokura, Nat. Nanotechnol. 8, 899 (2013).  \n5) A. Kotani, H. Nakajima, K. Harada, Y. Ishii and S. Mori, Phys. Rev. B 94, 024407 (2016).  \n6) D. Morikawa, X. Z. Yu, Y. Kaneko, Y. Tokunaga, T.  Nagai, K. Kimoto, T. Arima and Y. \nTokura, Appl. Phys. Lett. 107, 212401 (2015).  \n7) A. Kotani, H. Nakajima, K. Harada, Y. Ishii and S. Mori, Phys. Rev. B 95, 144403 (2017).  \n8) W. Jiang, P. Upadhyaya, W. Zhang, G. Yu, M. B. Jungfleisch, F. Y. Fradin, J. E. Pearson, Y. \nTserkovnyak, K. L. Wang, O. Heino nen and others, Science  349, 283 (2015).  \n9) K. Kurushima, K. Tanaka, H. Nakajima, M. Mochizuki and S. Mori, J. Appl. Phys. 125, \n053902 (2019).  \n10) W. Wang, Y. Zhang, G. Xu, L. Peng, B. Ding, Y. Wang, Z. Hou, X. Zhang, X. Li, E. Liu and \nothers, Adv. Mater. 28, 6887 (2016).  \n11) J. C. Loudon, A. C. Twitchett -Harrison, D. Cortés -Ortuño, M. T. Birch, L. A. Turnbull, A. \nŠtefančič, F. Y. Ogrin, E. O. Burgos -Parra, N. Bukin, A. Laurenson and others, Adv. Mater. \n31, 1806598 (2019).  \n12) L. A. Turnbull, M. T. Birch, A. Laurenson, N. Bukin, E. O. Burgos -Parra, H. Popescu, M. N. \nWilson, A. Stefan čič, G. Balakrishnan, F. Y. Ogrin and other s, ACS Nano 15, 387 (2020). \n13) Y. Yao, B. Ding, J. Cui, X. Shen, Y. Wang, W. Wang and R. Yu, Appl. Phys. Lett. 114, \n102404 (2019).  \n14) H. Nakajima, A. Kotani, K. Harada, Y. Ishii and S. Mori, Microscopy 65, 473 (2016).  \n15) H. Nakajima, A. Kotani, K. Harada and S. Mori, Microscop y 67, 207 (2018).  \n16) K. Ishizuka and B. Allman, J. Electron Microsc. (Tokyo). 54, 191 (2005).  \n17) H. Nakajima, H. Kawase, K. Kurushima, A. Kotani, T. Kimura and S. Mori, Phys. Rev. B 96, \n024431 (2017).  \n18) H. Nakajima, A. Kotani, K. Harada, Y. Ishii and S . Mori, Phys. Rev. B 94, 224427 (2016).  \n19) T. Nagai, M. Nagao, K. Kurashima, T. Asaka, W. Zhang and K. Kimoto, Appl. Phys. Lett. \n101, 162401 (2012).  \n  7 \n  \nFigure  1. (a) X-ray diffraction profile  in MnNiGa . The index is based on the P63/mmc structure. \nThe calculated lattice constants were listed. (b) Crystal structure of MnNiGa. (c) The temperature \ndependence of magnetization. The solid and hollow circles were measured in the field- and zero \nfield-cooling conditions, respectively.  \n \n8 \n  \nFigure 2. (a) Fresnel image  (overfocus f = +190 m). The inset is the electron diffraction pattern in \nthis area.  (b) Magnified image of the magnetic domains.  (c) Small-angle electron diffraction  pattern \nfrom  magnetic domains in MnNiGa . The camera length used  in the diffraction pattern was \napproximately 240 m.  \n \n \n \n9 \n  \nFigure  3. (a) Magnetic domains under the external magnetic field of 300 mT. The magnetic bubbles \nwere formed by pinching off the 180° domains.  The defocus value was f = −380 m (underfocus ). \n(b) Magnified  Fresnel  image of magnetic domains and (c) magnetization maps based on the \ntransport  of intensity equation. The arrows indicate the directions of magnetization.  In this area, the \nc axis was away from the observation direction : The angle  difference  was more than 10°. (d) \nMagnified image of magnetic bubbles.  The scale bar is 200 nm. (e) Schematic of the contrast of  a \nmagnetic bubble that is pinched off from 180° domains. The red arrows represent the magnetization \ndirection.  \n  \n10 \n  \nFigure  4. Formation of type -II magnetic bubbles.  (a) Electron diffraction pattern.  (b) Striped \nmagnetic domains with out a magnetic field. (c) The transition from the stri ped domains to magnetic \nbubbles at 1 80 mT. Further increase in the magnetic field at 240 mT increased magnetic bubb les in \n(d). In this area, the c axis was slightly tilted from the observation direction (approximately 3°). The \ndefocus value was f = −380 m (underfocus ). (e) Magnified image of magnetic bubbles.  The scale \nbar is 200 nm.  (f) Schematic of the contrast of  type-II magnetic bubb les. The red arrows represent \nthe magnetization  directions . The yellow arrowheads indicate Bloch lines.  \n \n \n \nSupplementary Figure  1. Comparison of the diffraction pattern and  simulation . (a) The \nelectron diffraction obtained from the area of Figure 2. (b) Simulation of an electron diffraction \npattern along  the [221] direction. The indexes are shown alongside the diffraction spots. The \nsimulation is based on the P63/mmc  structure. (c) The directions of the unit cell viewed along  the \n[221] direction. This simulation demonstrates that the c axis of the magnetic easy axis is parallel to \nthe specimen surface.  \n \n  \n \nSupplementary Figure  2. Through -focus images of magnetic bubbles comprising  180° domains \n(Biskyrmions) . The defocus value f is shown in each Fresnel image.  The area is the same as  in \nFigure 3 of the main text.  The applied magnetic field was 300 mT.  \n  \n \nSupplementary Figure  3. Through -focus images of type -II magnetic bubbles.  The defocus value \nf is shown in each Fresnel image.  The area is the same as in Figure 4 of the main text.  The applied \nmagnetic field was 180 mT. \n \n \n"    },    {        "title": "2111.02063v1.Anisotropic_magnetocaloric_effect_of_CrI___3____A_theoretical_study.pdf",        "content": "Anisotropic magnetocaloric e\u000bect of CrI 3: A theoretical study\nHung Ba Tran,1, 2, 3Hiroyoshi Momida,3Yu-ichiro Matsushita,1, 2Koun Shirai,3and Tamio Oguchi3, 4\n1Laboratory for Materials and Structures, Institute of Innovative Research,\nTokyo Institute of Technology, Midori-ku, Yokohama 226-8503, Japan\n2Quemix Inc., 2-11-2 Nihombashi, Chuo-ku, Tokyo 103-0027, Japan\n3Institute of Scienti\fc and Industrial Research, Osaka University, 8-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan\n4Center for Spintronics Research Network, Osaka University, Toyonaka, Osaka 560-8531, Japan\n(Dated: November 4, 2021)\nCrI3is considered to be a promising candidate for spintronic devices and data storage. We derived\nthe Heisenberg Hamiltonian for CrI 3from density functional calculations using the Liechtenstein\nformula. Moreover, the Monte{Carlo simulations with the Sucksmith{Thompson method were per-\nformed to analyze the e\u000bect of magnetic anisotropy energy on the thermodynamic properties. Our\nmethod successfully reproduced the negative sign of isothermal magnetic entropy changes when a\nmagnetic \feld was applied along the hard plane. We found that the temperature dependence of\nthe magnetocrystalline anisotropy energy is not negligible at temperatures slightly above the Curie\ntemperature. We clari\fed that the origin of this phenomenon is attributed to anisotropic magnetic\nsusceptibility and magnetization anisotropy. The di\u000berence between the entropy change of the easy\naxis and the hard plane is proportional to the temperature dependence of the magnetic anisotropy\nenergy, implying that the anisotropic entropy term is the main source of the temperature depen-\ndence of the free energy di\u000berence when magnetizing in a speci\fc direction other than the easy\naxis. We also investigated the magnetic susceptibility that can be used for the characterization of\nthe negative sign of the entropy change in the case of a hard plane. The competition of magne-\ntocrystalline anisotropy energy and external magnetic \feld at low temperature and low magnetic\n\feld region causes a high magnetic susceptibility as the \ructuation of magnetization. Meanwhile,\nthe anisotropy energy is suppressed at a su\u000ecient magnetic \feld applied along the hard axis, the\nmagnetization is fully rotated to the direction of the external magnetic \feld.\nI. INTRODUCTION\nThe development of Si-CMOS technology is a back-\nbone of electronic devices such as computers and\nsmartphones[1, 2]. The decrease in the transistor size has\nmany advantages, such as high e\u000eciency and low power\nconsumption. However, the size reduction is a di\u000ecult\ntask when it is close to the critical size[3]. Researchers are\nattempting to identify alternative methods to traditional\ncomputers, such as quantum and neuromorphic comput-\ners, which have many advantages over traditional com-\nputing hardware [4{7]. Spintronic devices are promising\nas an essential part of future computing devices because\nthey utilize the spin and charge of electrons[8]. Recent\nstudies on spin logic gates using magnetic materials with\nlayered structures have provided an opportunity for fu-\nture computing hardware with low power consumption\nand high speed[9].\nThe recent discovery of 2D materials also has potential\napplications in spintronic devices and data storage[9, 10].\nThey show a possible way to realize a device with\nlow power consumption, high e\u000eciency, compactness,\nand multi-functionality[9, 10]. Among several 2D ma-\nterials, van-der-Waals magnetic materials with layered\nstructures have attracted considerable attention because\nthey are easy to manufacture and have some unique\nproperties[9{11]. Layered magnetic materials have po-\ntential for application as data storage and memory\ndevices, where the layer's size can be reduced to a\nfew nanometers[9{11]. In this case, magnetocrystallineanisotropy energy (MAE) and interlayer exchange cou-\npling are vital for controlling the magnetic properties at\n\fnite temperatures. The MAE is the energy required to\nchange the direction of crystal magnetization from the\npreferred axis (easy axis) to another hard axis[12, 13].\nUnderstanding the behavior and origin of the tempera-\nture dependence of MAE provides many opportunities for\ncontrolling magnetic devices [12, 13]. Recent studies on\nperpendicular magnetic anisotropy in 2D materials have\nshown a clue to fabricate low-power consumption devices\nwith fast responses[9, 10]. Accurate prediction of MAE\nand its temperature dependence will enable the design of\ndata storage devices in spintronics[12, 13].\nCrI3is a compound of chromium and iodine, which is a\nmagnetic insulator of 2D materials [9{11]. Bulk CrI 3has\ntwo crystal structures: the low-temperature phase rhom-\nbohedral with space group R\u00163 and the high-temperature\nphase monoclinic with space group C2=m[14{17]. The\nmaterial is ferromagnetic (FM) below the Curie temper-\nature of 61 K and has uniaxial magnetic anisotropy with\nthe easy axis being the caxis and the hard plane being\ntheabplane[14{19]. CrI 3has been experimentally and\ntheoretically studied since it was applied to successfully\nmanufacture monolayer and multilayer structures from\nbulk[11, 14{19]. Although the MAE of bulk CrI 3was\ncalculated using density functional theory, the tempera-\nture dependence of MAE is not well understood yet[18{\n20]. The anisotropic magnetocaloric e\u000bect, which is the\nresponse of a magnetic anisotropy material with an exter-\nnal magnetic \feld, has been reported[19]. The peak-like\nmagnetization curves of CrI 3observed when applying aarXiv:2111.02063v1  [cond-mat.mtrl-sci]  3 Nov 20212\nmedium external magnetic \feld along the hard plane are\nquite unusual because the magnetization increases when\nthe temperature increases[18, 19]. In addition, alterna-\ntion of the sign from negative to positive for isothermal\nmagnetic entropy was observed when the magnetic \feld\nwas applied along the hard plane at low temperatures,\nwhich remains an open question[19].\nIn this study, the electronic structure and magnetic\nand magnetocaloric properties of bulk CrI 3with a rhom-\nbohedral structure were investigated using \frst-principles\ncalculations and Monte{Carlo simulations. The elec-\ntronic structure and magnetic exchange coupling con-\nstants of several magnetic con\fgurations were calculated.\nThe dependence of Curie temperature on the number of\nnearest neighbors was estimated using the mean-\feld ap-\nproximation (MFA) and Monte{Carlo method. In addi-\ntion, the magnetization versus magnetic \feld and tem-\nperature (M{HandM{T) curves, the temperature de-\npendence of MAE and the anisotropic isothermal mag-\nnetic entropy change were compared with the experimen-\ntal data from previous studies[18, 19]. To understand\nthe origin and characteristics of negative signs in the en-\ntropy change, we investigated the magnetic susceptibility\nand the relation between MAE and anisotropic magnetic\nentropy change. Our studies give a comprehensive un-\nderstanding of the anisotropic magnetocaloric e\u000bect with\nmagnetic anisotropy, where the temperature and mag-\nnetic \feld dependence of MAE is a key factor on the\nthermodynamic properties.\nII. METHODOLOGY\nThe electronic structure of bulk CrI 3was calculated\nusing DFT with the all-electron full-potential linearized\naugmented plane wave method as implemented in the Hi-\nLAPW package[21]. The experimental lattice structure\nwas assumed for the rhombohedral ( R\u00163) structure[18, 19].\nThe atomic positions were optimized by minimizing the\nresidual atomic forces to be less than 1 mRy/Bohr.\nThe generalized gradient approximation (GGA{PBE)\nwas used for exchange and correlation[22]. The cuto\u000b\nenergy was set as 20 Ry for the wavefunctions and 160\nRy for the potential. The magnetic anisotropy energy\nwas evaluated as the energy di\u000berence by rotating the\nquantization axis with the spin-orbit coupling (SOC) in\nthe second variation step. The convergence of MAE is\nachieved using a number of k-point meshes of 18 \u000218\u000218\nfor a primitive cell with rhombohedral axis.\nThe magnetic exchange coupling constants between\natoms in bulk CrI 3were calculated by DFT using\nthe Liechtenstein formula within the Korringa-Kohn-\nRostoker method as implemented in the Machikaneyama\npackage[23{25]:\nJij=1\n4\u0019ImZEF\ndETrn\n\u0001ti(E)T\"\nij\u0001tj(E)T#\nijo\n;(1)whereT\"\nijandT#\nijare the o\u000b-diagonal scattering path\noperators for spin-up and spin-down between atomic sites\niandj, respectively, and \u0001 ti(E) =t\"\ni\u0000t#\niwith the single\nsite t-matrixt\u001b\nifor spin\u001bat site i.\nThe classical Heisenberg model with uniaxial\nanisotropy is used as:\nHHeis=\u0000X\n<ij>Jm\nij\u0000 !Si\u0000 !Sj\u0000X\niku(\u0000 !eu\u0000 !Si)2\u0000g\u0016BX\ni\u0000\u0000!Hext\u0000 !Si;\n(2)\nwheregis the gyromagnetic ratio and \u0016Bis the Bohr\nmagneton. kudenotes the uniaxial anisotropy constant.\nThe \frst term expresses the exchange interactions be-\ntween spins at sites iand j. A spin tends to be paral-\nlel to the neighboring site when the magnetic exchange\ncoupling constant Jm\nijis positive. A spin prefers to be\nantiparallel with the neighboring spin for negative Jm\nij.\nThe second term is the interaction between the spin at\nsite iand the uniaxial anisotropy. euis the direction\nof the easy axis in the case of a positive ku. The last\nterm denotes the interaction of the spin at site iwith the\nexternal magnetic \feld. In an external magnetic \feld,\nspins tend to be parallel to the direction of the mag-\nnetic \feld to minimize energy. The size of the cell in\nthe Monte{Carlo simulation in CrI 3with hexagonal type\nis 20\u000220\u000220 unit cells containing 192000 atoms. The\nnumber of Monte{Carlo steps is 2000000, and the \frst\n1000000 steps are discarded. The Metropolis algorithm\nwas used to achieve thermal equilibrium in the Heisen-\nberg model.\nThe magnetic susceptibility was calculated as[26, 27]\n\u001f=@M\n@H=(\nM2\u000b\n\u0000hMi2)\nkBT(3)\nwhereMis magnetization, His the magnetic \feld, kBis\nthe Boltzamnn constant, and Tis temperature.\nThe MAE was estimated by integrating the magneti-\nzation versus the magnetic \feld ( M{H) curves as\nMAE =ZHext\n0fM(Hkeasy axis)\u0000M(Hkhard axis)gdH\n(4)\nwhere integration is considered until the saturation mag-\nnetization is achieved like the magnetic \feld equal to 7 T.\nThe MAE is estimated here as the area between the mag-\nnetization curves when applying magnetic \feld along the\neasy axis and hard plane in the integration. The temper-\nature dependence of the MAE can be estimated because\nthe strength of the magnetization and anisotropy \feld at\n\fnite temperatures is considered in the value and direc-\ntion of magnetization curves. The area between the two\nmagnetization curves|one is when the magnetic \feld\nalong the easy axis and the other is the magnetic \feld\nalong the hard plane|is considered as the energy cost\nfor magnetization in a certain direction other than the\npreferred axis.3\nFM𝒎𝐂𝐫=𝟑.𝟑𝟑𝝁𝐁𝒎𝐂𝐫=𝟑.𝟐𝟕𝝁𝐁Disordered Local MomentsEnergy (eV)Energy (eV)DOS (/eV per spin unit cell)𝒎𝐂𝐫=𝟑.𝟑𝟏𝝁𝐁AFM (intra-layer)AFM (inter-layer)𝒎𝐂𝐫=𝟑.𝟑𝟖𝝁𝐁 0 5 10 15 20−6−5−4−3−2−1 0 1 2 0 5 10 15 20−6−5−4−3−2−1 0 1 2 0 5 10 15 20−6−5−4−3−2−1 0 1 2 0 5 10 15 20−6−5−4−3−2−1 0 1 2\n 0 5 10 15 20−6−5−4−3−2−1 0 1 2 0 5 10 15 20−6−5−4−3−2−1 0 1 2 0 5 10 15 20−6−5−4−3−2−1 0 1 2 0 5 10 15 20−6−5−4−3−2−1 0 1 2(a)(b)(c)(d)\nEnergy (eV)Energy (eV)\nFIG. 1. The total density of states of rhombohedral CrI 3with various magnetic con\fgurations: (a) ferromagnetic, (b) disor-\ndered local moment, (c) antiferromagnetic with up and down magnetic moment in the same layer (AFM-intra-layer), and (d)\nantiferromagnetic with up and down magnetic moment in a di\u000berent layer (AFM-inter-layer).\nThe isothermal magnetic entropy change is estimated\nusing the Maxwell relation as[26, 27]\n\u0001SM(Hext;T) =ZHext\n0\u0012@M(H;T)\n@T\u0013\n;\n\u0018=NX\nj=0M(Hj;T+ \u0001T)\u0000M(Hj;T\u0000\u0001T)\n2\u0001T\u0001H;(5)\nThe isothermal magnetic entropy change is obtained by\nintegrating with the \fne mesh in the temperature and\nexternal magnetic \feld. \u0001 Tand \u0001Hare equal to 2 K\nand 0.2 T, respectively. Owing to the MAE, the isother-\nmal magnetic entropy changes strongly depend on the\ndirection of the applied magnetic \feld.\nIII. RESULTS AND DISCUSSION\nA. Electronic structure and magnetic exchange\ncoupling constants\nThe electronic structures of bulk CrI 3with ferromag-\nnetic (FM), disordered local moment (DLM) or param-\nagnetic (PM), and two antiferromagnetic (AFM) con\fg-\nurations are shown in FIG. 1. All electronic structuresare semiconducting with a similar local magnetic mo-\nment of the Cr-atom 3 :33 (\u0016B/atom) as the Cr3+state.\nThe magnetic moment of I is \u00000:13 (\u0016B/atom) in the\nFM state, which becomes zero in the other magnetic\ncon\fgurations by symmetry. Our results for the elec-\ntronic structure of DLM show a semiconducting struc-\nture, which is di\u000berent from previous studies[18, 19],\nwhich have showed a metallic structure. The discrep-\nancy may be caused by ignoring spin polarization in the\nprevious studies[18, 19]. The MAE of the FM state was\ncalculated to be 0.46 (meV/Cr-atom) between the easy\naxis (c-axis) and hard plane ( ab-plane) in the hexagonal\ncoordinates. This agrees with the previous theoretical\nwork in which the MAE was found to be 0.47 (meV/Cr-\natom) using the same exchange-correlation function[20].\nThe source of the strong MAE originates from the hy-\nbridization between the d-states of Cr and the pstates of\nI with a large SOC. The value of MAE at a low temper-\nature is 0.26 (meV/Cr-atom) in an experimental study\nby integrating the M{Hcurves, which is smaller than\nthe value of \frst-principles calculations[18]. The MAE\nof theoretical studies often tend to be higher than that\nof experimental studies and the value of the anisotropy\n\feld at low temperatures will be higher than that of the\nexperimental studies[18, 19].\nThe magnetic exchange coupling constants of several4\n(b)(c) 50 60 70 80 90 100 110 120 130\n 1 2 3 4 5 6 7 8TC (K)Nearest neighbourMonte−CarloExp(a)TC(Exp)                           61 KTC(MFA-FM)                207 KTC(MFA-DLM)             182 KTC(MFA-AFM-intra)    178 KTC(MFA-AFM-inter)    194 KTC(MC-FM)                   113 K𝑱𝐢𝐧𝐭𝐫𝐚𝟏𝐬𝐭𝟑.𝟗𝟔𝟓Å𝑱𝐢𝐧𝐭𝐞𝐫𝟏𝐬𝐭𝟔.𝟓𝟗𝟎Å𝑱𝐢𝐧𝐭𝐫𝐚𝟐𝐧𝐝𝟔.𝟖𝟔𝟕Å𝑱𝐢𝐧𝐭𝐞𝐫𝟐𝐧𝐝𝟕.𝟕𝟎𝟏Å𝑱𝐢𝐧𝐭𝐫𝐚𝟑𝐭𝐡𝟕.𝟗𝟐𝟗Åa= 7.701 Å\nCrI𝐚𝐛𝐜−3−2−1 0 1 2 3 4 5 6\n 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Jij (meV)distance d/aCr−Cr (FM)Cr−I (FM)I−I (FM)Cr−Cr (DLM)Cr−Cr (AFM−intra)Cr−Cr (AFM−inter)\nFIG. 2. (a) The crystal structure of CrI 3with several Cr{Cr pairs and the value of the Curie temperature of experimental and\nthe present study, which is obtained using MFA and Monte{Carlo simulations. (b) Magnetic exchange coupling constants as a\nfunction of distance over the lattice constant of the crystal structure in (a). (c) Curie temperatures of experimental work (red\nline) and Monte{Carlo simulations depend on the range of the nearest neighbor sites of Cr{Cr pairs (black line).\nmagnetic con\fgurations and their Curie temperatures in\nthe present calculations are shown in FIG. 2 (a) and (b).\nBecause only the FM phase has a \fnite local magnetic\nmoment at the I atom, the magnetic exchange coupling\nconstants of the Cr-I and I-I pairs are negligible in the\nDLM (PM) and AFM states. The large negative cou-\npling constants of the \frst nearest neighbor of the Cr-I\npair are consistent with the values of the magnetic mo-\nments of the Cr and I atoms because it minimizes the\ntotal energy of the FM state in the Heisenberg model.\nMoreover, the magnetic exchange coupling constants of\nthe Cr{Cr pairs are not sensitive to the choice of mag-\nnetic con\fgurations because the magnetic moment of the\nCr-atom is localized. The coupling constant of the \frst\nnearest neighbor between Cr atoms in the same layer,\nwhich has the largest positive value. This is the main\ncontribution to the intra-layer FM order. In addition,\nthe coupling constant of the second nearest neighbor in\nthe intra-layer is a small positive, while the third one is\nnegative (the distance is slightly larger than one lattice\nconstant). Meanwhile, the coupling constants of the \frst\nand second nearest neighbor Cr{Cr pairs in the di\u000berent\nlayers are small and positive, which is the origin of the\nstable ferromagnetic order observed in bulk CrI 3.\nCoupling constants up to 15 \u0017A are considered to pri-marily determine the Curie temperature within MFA.\nHowever, MFA usually overestimates the Curie tempera-\nture compared with experiment. The Curie temperature\nobtained by MFA was approximately three times larger\nthan the experimental value[18, 19]. The Monte{Carlo\nmethod was used to obtain a more accurate Curie tem-\nperature. In the Monte{Carlo method, we consider up\nto eight orders of the nearest neighbor of the Cr{Cr pair\nin the FM state. Although the value is much smaller\nthan the value of MFA, it is still higher than the value of\nthe experimental works. This contradicts previous the-\noretical studies, which show very good agreement with\nthe experimental value in the Curie temperature[28{30].\nIn the previous studies, the magnetic exchange coupling\nconstants were estimated by considering the total en-\nergy di\u000berence of several magnetic con\fgurations, which\nmakes it di\u000ecult to consider the long-range order of the\ncoupling constant[28{30]. The Curie temperature of the\nMonte{Carlo method as a function of the range of the\nmagnetic exchange coupling constants is shown in FIG.\n2 (c). The estimation of the Curie temperature strongly\ndepends on the range of magnetic interactions in the sim-\nulation. Less than the fourth nearest neighbor of Cr-Cr\npairs leads to an accidentally good agreement with the\nexperimental value. Although the Curie temperature was5\noverestimated, the long-range interaction of Cr-Cr pairs\nwas considered in the present study. Other e\u000bects, such\nas phonon need to be considered to achieve a more ac-\ncurate value for the Curie temperature of CrI 3in the\npresence of long-range magnetic interactions [31].\nB. Finite temperature MAE and anisotropic\nmagnetocaloric e\u000bect\nTheM{Hcurves at low temperatures, intermediate\ntemperatures, slightly above the Curie temperature, and\nhigh temperatures are shown in FIG. 3 (a). The area\nbetween the easy axis and hard plane is integrated to\nestimate MAE, where the e\u000bect of temperature on the\nmagnetization and magnetic anisotropy is considered. At\na low temperature (2 K) and zero external magnetic \feld,\nthe magnetization is along the easy axis and is perpen-\ndicular to the hard plane. However, the projection of\nmagnetization along the direction of the magnetic \feld\nincreases when a higher external magnetic \feld is ap-\nplied along the hard plane. The magnetization was fully\nrotated to the hard plane if the magnetic \feld was su\u000e-\ncient (higher than the anisotropy \feld). In contrast, the\nmagnetization anisotropy, de\fned as the di\u000berence in the\nsaturation magnetization of the easy axis and the hard\nplane at the zero external magnetic \felds, appears at an\nintermediate temperature. In addition, the slopes of the\ntwo magnetization curves (dashed and solid red lines in\nFIG. 3 (a)) which denote the di\u000berential magnetic sus-\nceptibility are similar when the external magnetic \feld\nis larger than the anisotropy \feld. As a result of mag-\nnetization anisotropy and di\u000berential magnetic suscep-\ntibility at intermediate temperatures, the estimation of\nMAE strongly depends on the strength of the external\nmagnetic \feld. If temperature is slightly higher than the\nCurie temperature, the magnetization at the zero exter-\nnal magnetic \feld equals zero. However, the application\nof an external magnetic \feld can lead to a \fnite magne-\ntization. The di\u000berential magnetic susceptibility of the\neasy axis is higher than that of the hard plane in the\ncase of a paramagnetic state. This is consistent with\nthe experimental work[18]. Note that we need to con-\nsider the relative temperature because the Curie tem-\nperature of our result is higher than the experimental\nvalue[18]. Moreover, the anisotropic magnetic suscepti-\nbility decreased rapidly with increasing temperature. At\nhigh temperatures, the disappearance of magnetization\nanisotropy and anisotropic magnetic susceptibility led to\na negligible MAE.\nThe temperature dependence of the magnetization\ncurves when a magnetic \feld is applied along the easy\naxis and hard plane is shown in FIG. 3 (b). At a\nweak external magnetic \feld (100 Oe), the magnetiza-\ntion is mainly along the easy axis when the temperature\nis lower than the Curie temperature. If the temperature\nincreases, there is a second-order magnetic phase tran-\nsition (FM-PM) characteristic of the Heisenberg model.\n100 Oe1 kOe10 kOe50 kOe(a)\n(c)(b)\n 0 0.1 0.2 0.3 0.4 0.5\n 50 100 150 200MAE (meV/f.u)T (K)Monte−Carlo methodf(T)=K0−(K0/TC).T 0 0.5 1 1.5 2 2.5 3 3.5\n 0 10 20 30 40 50 60 70m (µB/Cr)Hext (kΟe)\n 0 0.5 1 1.5 2 2.5 3 3.5\n 0 50 100 150 200m (µB/Cr)T (Κ)2 K75 K125 K175 Kdash: Hext|| absolid: Hext|| c\ndash: Hext|| absolid: Hext|| cFIG. 3. (a) M{Hcurves of several temperatures with the\nmagnetic \feld along the hard plane (the dashed line) and easy\naxis (the solid line). (b) Magnetization curves versus temper-\nature for various magnetic \feld strengths, where the magnetic\n\feld is along the hard plane (the dashed line) and easy axis\n(the solid line). (c) Temperature dependence of MAE by in-\ntegrating the M{Hcurves in Monte{Carlo simulations.\nAlthough the magnetization curve of the hard axis agrees\nwell with the experimental results, the magnetization\ncurve of the easy axis in our results is much higher than\nthat of the experimental one[18, 19]. When the mag-\nnetic \feld is equal to 10 kOe along the easy axis, the\nmagnetization at 0 K in the experimental work is 1.2\n(\u0016B/Cr-atom), while our result is 3.33 ( \u0016B/atom), which6\n(a)(b)(c)(d)Hext|| cHext|| ab\n−1 0 1 2 3 4 5\n 0 20 40 60 80 100 120 140 160 180 200−∆SM (J/kg.K)T (K)Hext=1 THext=2 THext=3 THext=4 THext=5 T−1 0 1 2 3 4 5\n 0 20 40 60 80 100 120 140 160 180 200−∆SM (J/kg.K)T (K)Hext=1 THext=2 THext=3 THext=4 THext=5 T\n 0 50 100 150 200 0 1 2 3 4 5 6 7 8−5 0 5 10 15 20 25 30 35 40 45\nT (K)Hext (T)−5 0 5 10 15 20 25 30 35 40 45 0 50 100 150 200 0 1 2 3 4 5 6 7 8−5 0 5 10 15 20 25 30 35 40 45\nT (K)Hext (T)−5 0 5 10 15 20 25 30 35 40 45𝑀(emu/g)𝑀(emu/g)\nFIG. 4. The isothermal magnetization depends on the temperature and strength of the magnetic \feld, with the direction of\n\feld along with the hard plane (a) and easy axis (c). The isothermal magnetic entropy changes as a function of the temperature\nand magnetic \feld strength for the hard plane (b) and easy axis (d).\nindicates saturation magnetization. There are multiple\nmagnetic domains in experimental studies if the mag-\nnetic \feld is insu\u000ecient to form a single magnetic do-\nmain. However, the size of the simulation box in our\nresults contains only a single magnetic domain. Increas-\ning the external magnetic strength to 1 kOe and 10 kOe\nleads to a large enhancement in the hard plane.\nPeak-like magnetization curves appear for an inter-\nmediate magnetic \feld (the blue dashed line in FIG. 3\n(b)). The temperature value of the peak is called T\u0003,\nwhich is slightly smaller than the Curie temperature\nand decreases with increasing strength of the magnetic\n\feld[18, 19]. It is only observed in the case of the hard\nplane and when an intermediate magnetic \feld is applied.\nIts origin is the decrease in the anisotropy \feld when the\ntemperature increases, so the magnetization projection\nin the direction of the magnetic \feld increases when the\ntemperature increases. This can be seen as the case of\nthe crossing of the dashed lines of 2 K and 75 K in FIG.\n3 (a). The magnetization at the intermediate tempera-\nture was larger than that at the low temperature when\nthe magnetic \feld was intermediate. This phenomenon\ndisappears when the magnetic \feld is higher than the\nanisotropy \feld at a low temperature. In the case of a\nmagnetic \feld equal to 50 kOe, a di\u000berence can be ob-\nserved in the slope of magnetization between the easyand hard planes. It only appears at a speci\fc tempera-\nture range from the intermediate temperature to slightly\nabove the Curie temperature. This is semi-quantitative\nin agreement with the experimental results[18, 19].\nFrom theM{Hcurves, the temperature dependence of\nthe MAE was estimated as FIG. 3 (c) by considering the\narea between the easy axis and the hard plane. The value\nof MAE at low temperatures is approximately equal to\nthe value of \frst-principles calculations (0.46 meV/Cr-\natom). The MAE decreased almost linearly when the\ntemperature increased in the low-temperature range.\nThe MAE curve becomes \ratter at higher temperatures,\nleading to a \fnite value slightly above the Curie tem-\nperature. This is due to magnetization anisotropy and\nanisotropic magnetic susceptibility, as discussed above.\nThis means that the magnetic anisotropy is still valid at\na temperature higher than the critical temperature, at\nwhich the anisotropic magnetic susceptibility and mag-\nnetization anisotropy are present. The e\u000bect of MAE was\nproposed as the origin of the anisotropic magnetocaloric\ne\u000bect in experimental studies[19]. This means that the\ne\u000bect of MAE is not negligible slightly above the Curie\ntemperature in the present study.\nThe isothermal magnetization curves and isothermal\nmagnetic entropy changes are shown in FIG. 4. The dif-\nference in magnetization and entropy change is due to7\ndash: Hext|| csolid: Hext|| ab𝜒(a.u)(a)(b)(c)−0.3−0.2−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6\n 0 1 2 3 4 5−∆SM (J/kg.K)Hext (T)−0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5\n 0 1 2 3 4 5 6 7 8−∆SM (J/kg.K)Hext (T)T= 20 KT= 40 KT= 60 KT= 80 KT= 100 KT= 120 K\n 0 1 2 3 4 5 6 7\n 0 50 100 150 200Hext (T)T (K)\n 0 2 4 6 8 10\nFIG. 5. The entropy change depends on the strength of the\nmagnetic \feld with the magnetic \feld along the hard plane\n(solid line) and the easy axis (the dashed line) at various\ntemperatures for large range (a) and small range (b). Note\nthat (b) is the inset region of (a) with a dashed purple line.\n(c) The magnetic susceptibility depends on the temperature\nand magnetic \feld.\nthe e\u000bect of the MAE. The magnetization at the zero ex-\nternal magnetic \feld is approximately zero for the hard\nplane case because the easy axis is perpendicular to the\nhard plane in uniaxial anisotropy. The magnetization in\nthe low magnetic-\feld and low temperature region is af-\nfected by the magnetic \feld and MAE in the case of hard\nplane. The higher applied magnetic \feld, the higher mag-\nnetization value. When the magnetic \feld is higher than\nthe anisotropy \feld, the magnetization is perpendicularto the easy axis. This leads to the disappearance of MAE\ne\u000bects for a su\u000ecient magnetic \feld applied. The mag-\nnetization curves become smooth after this region. If\nthe strength of the external magnetic \feld increases, the\nmagnetization can be rotated to the hard axis. The en-\ntropy change of the hard plane is quite di\u000berent from that\nin the easy axis case. Here, the entropy of the external\nmagnetic \feld along the easy axis is shown as a positive\npeak, where all values have a positive sign. Meanwhile,\nthe entropy change of the hard plane crosses the zero line.\nThe entropy change is negative at a magnetic \feld of 1\nT before the Curie temperature peak. When the mag-\nnetic \feld increases, the crossing temperature decreases.\nThis means that the material will be cooled by applying\na magnetic \feld along the hard plane in the adiabatic\nprocess for this temperature range. This behavior is un-\nusual because conventional magnetocaloric materials are\nheated and cooled by applying and removing the external\nmagnetic \feld in the adiabatic process[18, 19]. The origin\nof the negative sign in the entropy change curve of the\nhard plane comes from the peak, as in the magnetization\ncurve at intermediate temperatures and insu\u000ecient mag-\nnetic \felds. MAE tends to force the magnetization along\nthe easy axis, whereas the magnetic \feld is along the hard\nplane. Although the external magnetic \feld is constant,\nthe strength of MAE decreases when the temperature\nincreases, which leads to a reduction in the anisotropy\n\feld. If the MAE e\u000bect are weakened by temperature,\nthe magnetization can be rotated easily, leading to higher\nmagnetization along the direction of the magnetic \feld.\nOur results are in good agreement with the experimental\nworks[18, 19].\nThe entropy change as a function of the magnetic \feld\napplied to the easy axis, and the hard plane at some tem-\nperatures is shown in FIG. 5 (a) and (b). Note that the\nCurie temperature of our present work is higher than the\nexperimental value, so we use higher temperature to keep\nthe normalize temperature similar to that of the experi-\nmental works[19]. The entropy change curve of the easy\naxis and hard plane at the same temperature is distin-\nguished in the entire region of the applied magnetic \feld\n(up to 7.4 T). The entropy change of the easy axis at\nall temperatures is positive and increases almost linearly\nwith the magnetic \feld value. At the considered tem-\nperature, a higher temperature leads to a steeper slope\nof the entropy change. In contrast, when the tempera-\nture is smaller than the Curie temperature, the entropy\nchange of the hard plane (the solid line) exhibits similar\nbehavior. The entropy change decreases when the mag-\nnetic \feld increases and reaches a minimum and then\nincreases with a similar slope of the easy axis. The po-\nsition of the minimum entropy change decreases when\nthe temperature increase. The distance between the two\nentropy change curves (dashed and solid lines) increases\nwhen the magnetic \feld is smaller than the minimum and\nbecomes constant at a su\u000ecient applied magnetic \feld.\nHere, a temperature slightly higher than the Curie tem-\nperature is also considered with yellow line in FIG. 5 (a)8\nand (b). At this temperature, there is no minimum in\nthe entropy change curve of the hard plane. In addition,\nthe slopes of the two curves is di\u000berent over the entire\nmagnetic \feld range. This means that there is a di\u000ber-\nence in the entropy change between the hard plane and\neasy axis at slightly above Curie temperature.\nThe magnetic susceptibility of the hard plane case was\nconsidered in FIG. 5 (c) to understand the origin of the\nbehavior of the entropy change. The boundary of the two\nregions (the green and blue colors), which correspond to\ninsu\u000ecient and su\u000ecient magnetic \felds, can be clearly\nseen by considering the magnetic susceptibility. Mag-\nnetic susceptibility is the \ructuation of the magnetiza-\ntion value. It is large in the low temperature and low\nmagnetic \feld region, where there is competition between\nthe magnetic \feld and magnetic anisotropy. If the mag-\nnetic \feld is su\u000eciently higher than the anisotropy \feld,\nthe e\u000bect of MAE on the magnetization and magnetic\nsusceptibility becomes inferior because the direction of\nmagnetization is perpendicular to the easy axis. The\nboundary of the two regions in magnetic susceptibility\ncorresponds to the strength of the magnetic \feld of the\nminimum entropy change in the hard plane case in FIG.\n5 (a) and (b). It is high at low temperatures and de-\ncreases when the temperature increases and disappears\nabove the Curie temperature.\nC. Relations of MAE and anisotropic\nmagnetocaloric e\u000bect\nTo understand the relation between MAE and en-\ntropy change of the magnetocaloric e\u000bect, we de\fne the\nanisotropic magnetic entropy as the heat absorbed or re-\nleased by rotating the sample at a \fxed strength of the\nmagnetic \feld in the isothermal process as:\n\u0001SR\nM(Hext;T)\n=S(Hextkeasy;T)\u0000S(Hextkhard;T)\n= \u0001SM(Hextkeasy;T)\u0000\u0001SM(Hextkhard;T)\n=ZHext\n0\u001a@M(Hkeasy;T)\n@T\u0000@M(Hkhard;T)\n@T\u001b\n;\n=@EMAE\n@T\n(6)\nThe anisotropic magnetic entropy is related to the\nderivative of MAE, as shown in Eq. 6. This means\nthat the di\u000berence in entropy at a given temperature isslightly higher than that at the Curie temperature as yel-\nlow curves in FIG. 5 (a, b) is consistent with the \fnite\nMAE at these temperatures of FIG. 3 (c). In addition, at\ntemperatures below the Curie temperature, the estima-\ntion of MAE increases as the magnetic \feld increases and\nbecomes approximately constant if the magnetic \feld is\nlarger than the anisotropy \feld. This means that the\ndeepness of the minimum entropy change of the hard\nplane is directly related to the strength of the MAE,\nwhich decreases when the temperature increases. The\nMAE reduction at a \fnite temperature is related to the\nanisotropic entropy because the MAE is the di\u000berence\nin free energy, which includes the internal energy and\nentropy terms. Although our methods can successfully\nreproduce the experimental results and understand their\ncharacterization, a further study using these methods on\ncubic anisotropy and two ion anisotropy cases, is needed\nto overview MAE in the magnetocaloric e\u000bects.\nIV. SUMMARY\nThe electronic structure, magnetism, and magne-\ntocaloric properties of CrI 3were studied by combin-\ning \frst-principles calculations and Monte{Carlo simu-\nlations. The temperature dependence of the MAE, esti-\nmated by integrating the M{Hcurves as the Sucksmith-\nThompson method, is not negligible even above the Curie\ntemperature. Its origin is anisotropic magnetic suscep-\ntibility and magnetization anisotropy, which are con-\nsidered in our Monte{Carlo simulations. The relation-\nship between anisotropic magnetic entropy change and\n\fnite temperature magnetic anisotropy energy is clari-\n\fed in this study to understand the mechanism of the\nanisotropic magnetocaloric e\u000bect. Base on our results,\nthe anisotropic magnetic entropy is the main source of\ntemperature dependence of magnetic anisotropy energy\nwhere the entropy terms appears in the formulation of\nfree energy.\nACKNOWLEDGEMENTS\nThe authors would like to thank Tetsuya Fukushima\nfor his help with \frst-principles calculations. The au-\nthors would also like to thank Kunihiko Yamauchi for\nthe valuable discussions. This work was partly supported\nby the Di-CHiLD project (Grant No. J205101520). The\ncomputation in this work was partially performed using\nthe facilities of the Supercomputer Center, the Institute\nfor Solid State Physics, the University of Tokyo, and the\nCybermedia Center of Osaka University.\n[1] Frank, D. J.; Dennard, R. H.; Nowak, E.; Solomon, P. M;\nTaur, Y.; Wong H. P. Device scaling limits of Si MOS-FETs and their application dependencies Proc:IEEE\n2001 , 89, 259-288. DOI: 10.1109/5.9153749\n[2] Dennard, R. H.; Gaensslen, F. H.; Yu, H.; Rideout,\nV. L.; Bassous, E.; LeBlanc, A. R. Design of ion-\nimplanted MOSFETs with very small physical dimen-\nsions IEEE J :Solid\u0000State Circuits 1974 , 9, 256-268.\nDOI: 10.1109/N-SSC.2007.4785543\n[3] Thomas, S. Nanosheet FETs at 3 nm Nat:Electron :\n2018 , 1, 613. DOI: 10.1038/s41928-018-0179-9\n[4] Leuenberger, M. N.; Loss, D. Spintronics and quan-\ntum computing: switching mechanisms for qubits\nPhys:E:Low\u0000Dimen :Syst:Nanostruct :2001 , 10,\n452-457. DOI: 10.1016/S1386-9477(01)00136-9\n[5] Sengupta A.; Roy, K. Encoding neural and synaptic func-\ntionalities in electron spin: A pathway to e\u000ecient neu-\nromorphic computing Appl:Phys:Rev:2017 , 4, 041105-\n(23pp). DOI: 10.1063/1.5012763\n[6] Grollier, J.; Querlioz, D.; Stiles, M. D. Spintronic nan-\nodevices for bioinspired computing Proc:IEEE 2016 ,\n104, 2024-2039. DOI: 10.1109/JPROC.2016.2597152\n[7] Awschalom, D. D.; Bassett, L. C.; Dzurak, A. S.;Hu,\nE. L.; Petta, J. R. Quantum Spintronics: Engineer-\ning and Manipulating Atom-Like Spins in Semiconduc-\ntors Science 2013 , 339, 1174-1179. DOI: 10.1126/sci-\nence.1231364\n[8] Hirohata, A.; Yamada, K.; Nakatani, Y.; Prejbeanu,\nI.; Di\u0013 eny, B.; Pirro, P.; Hillebrands, B. Review\non spintronics: Principles and device applications\nJ:Magn :Magn :Mater :2020 , 509, 166711-(28pp). DOI:\n10.1016/j.jmmm.2020.166711\n[9] Ahn E. C. 2D materials for spintronic devices\nnpj 2D Mater :Appl:2020 , 4, 17-(14pp). DOI:\n10.1038/s41699-020-0152-0\n[10] Gibertini, M.; Koperski, M.; Morpurgo, A. F.;\nNovoselov, K. S. Magnetic 2D materials and het-\nerostructures Nat:Nanotechnol :2019 , 14, 408-419. DOI:\n10.1038/s41565-019-0438-6\n[11] Huang, B.; Clark, G.; Klein, D. R.; MacNeill, D.;\nNavarro-Moratalla, E.; Seyler, K. L.; Wilson, N.;\nMcGuire, M. A.; Cobden, D. H.; Xiao, D.; Yao, W.;\nP. Jarillo-Herrero, P.; Xu, X. Electrical control of 2D\nmagnetism in bilayer CrI3 Nat:Nanotechnol :2018 , 13,\n544-548. DOI: 10.1038/s41565-018-0121-3\n[12] Callen, H. B.; Callen, E. The present status of the tem-\nperature dependence of magnetocrystalline anisotropy,\nand the power law J:Phys:Chem :Solids 1966 , 27, 1271-\n1285. DOI: 10.1016/0022-3697(66)90012-6\n[13] Asselin, P.; Evans, F. L.; Barker, J.; Chantrell, R.\nW.; Yanes, R.; Chubykalo-Fesenko, O.; Hinzke, D.;\nNowak, U. Constrained Monte Carlo method and cal-\nculation of the temperature dependence of magnetic\nanisotropy Phys:Rev:B2010 , 82, 054415-(12pp). DOI:\n10.1103/PhysRevB.82.054415\n[14] Handy, L. L.; Gregory, N. W.; Structural Properties of\nChromium(III) Iodide and Some Chromium(III) Mixed\nHalides J:Am:Chem :Soc:1952 , 74, 891-893. DOI:\n10.1021/ja01124a009\n[15] Morosin, B.; Narath, A. X-Ray Di\u000braction and Nu-\nclear Quadrupole Resonance Studies of Chromium\nTrichloride J:Chem :Phys:1964 , 40, 1958-1967. DOI:\n10.1063/1.1725428\n[16] Pollini, I. Electron correlations and hybridization in\nchromium compounds Solid State Commun :1998 , 106,549-554. DOI: 10.1016/S0038-1098(98)00034-9\n[17] Wang, H.; Eyert, V.; Schwingenschl ogl, U. Elec-\ntronic structure and magnetic ordering of the semicon-\nducting chromium trihalides CrCl 3, CrBr 3, and CrI 3\nJ:Phys::Condens :Matter 2011 , 23, 116003-(8pp).\nDOI: 10.1088/0953-8984/23/11/116003\n[18] McGuire, M. A.; Dixit, H.; Cooper, V. R.; Sales, B. C.\nCoupling of Crystal Structure and Magnetism in the Lay-\nered, Ferromagnetic Insulator CrI 3Chem :Mater :2015 ,\n27, 612-620. DOI: 10.1021/cm504242t\n[19] Liu, Y.; C. Petrovic, C. Anisotropic magnetocaloric e\u000bect\nin single crystals of CrI 3Phys:Rev:B2018 , 97, 174418-\n(7pp). DOI: 10.1103/PhysRevB.97.174418\n[20] Gudelli, V. K.; Guo, G. Y. Magnetism and magneto-\noptical e\u000bects in bulk and few-layer CrI 3: a theoretical\nGGA + U study New J :Phys:2019 , 21, 053012-(16pp).\nDOI: 10.1088/1367-2630/ab1ae9\n[21] http://www.cmp.sanken.osaka-u.ac.jp/ ~oguchi/\nHiLAPW/index.html\n[22] Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized\nGradient Approximation Made Simple Phys:Rev:Lett:\n1996 , 77, 3865-3868. DOI: 10.1103/PhysRevLett.77.3865\n[23] Liechtenstein, A.; Katsnelson, M. I.; Antropov, V. P.;\nGubanov, V. A. Local spin density functional approach\nto the theory of exchange interactions in ferromagnetic\nmetals and alloys J:Magn :Magn :Mater :1987 , 67, 65-\n74. DOI: 10.1016/0304-8853(87)90721-9\n[24] http://kkr.issp.u-tokyo.ac.jp\n[25] Akai H.; Dederichs, P. H. Local moment disorder in ferro-\nmagnetic alloys Phys:Rev:B1993 , 47, 8739-8747. DOI:\n10.1103/PhysRevB.47.8739\n[26] Tran, H. B.; Fukushima, T.; Sato, K.; Makino, Y.;\nOguchi, T. Tuning structural-transformation tempera-\nture toward giant magnetocaloric e\u000bect in MnCoGe alloy:\nA theoretical study J:Alloys Compd :2021 , 854 157063-\n(9pp). DOI: 10.1016/j.jallcom.2020.157063\n[27] Tran, H. B.; Fukushima, T.; Makino, Y.; Oguchi,\nT. Magnetocaloric e\u000bect in MnCoGe alloys studied by\n\frst-principles calculations and Monte-Carlo simulation\nSolid State Commun :2021 , 323, 114077-(5pp). DOI:\n10.1016/j.ssc.2020.114077\n[28] Olsen, T. Theory and simulations of critical tem-\nperatures in CrI 3and other 2D materials: easy-\naxis magnetic order and easy-plane Kosterlitz{Thouless\ntransitions MRS Comm :2019 , 9, 1142-1150. DOI:\n10.1557/mrc.2019.117\n[29] Lu, X.; Fei, R.; Yang, L. Curie temperature of emerging\ntwo-dimensional magnetic structures Phys:Rev:B2019 ,\n100, 205409-(8pp). DOI: 10.1103/PhysRevB.100.205409\n[30] Liu, J.; Sun, Q.; Kawazoe, Y.; Jena, P. Exfoliat-\ning biocompatible ferromagnetic Cr-trihalide monolay-\nersPhys:Chem :Chem :Phys:2016 , 18, 8777-8784. DOI:\n10.1039/C5CP04835D\n[31] Tanaka, T.; Gohda, Y. Prediction of the Curie temper-\nature considering the dependence of the phonon free en-\nergy on magnetic states npj Comput :Mater :2020 , 6,\n184-(7pp). DOI: 10.1038/s41524-020-00458-5"    },    {        "title": "2111.05456v1.Design_of_soft_magnetic_materials.pdf",        "content": "Design of soft magnetic materials\nAnanya Renuka Balakrishna1,*and Richard D. James2\n1Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA 90089\n2Aerospace and Engineering Mechanics, University of Minnesota, Minneapolis, MN 55455\n*Corresponding author, Office: (213) 740-8762, Email: renukaba@usc.edu\nAbstract\nWe present a strategy for the design of ferromagnetic materials with exceptionally low\nmagnetic hysteresis, quantified by coercivity. In this strategy, we use a micromagnetic algo-\nrithm that we have developed in previous research and which has been validated by its success\nin solving the “Permalloy Problem”—the well-known difficulty of predicting the composition\n78.5% Ni of lowest coercivity in the Fe-Ni system— and by the insight it provides into the\n“Coercivity Paradox” of W. F. Brown. Unexpectedly, the design strategy predicts that cu-\nbic materials with large saturation magnetization msand large magnetocrystalline anisotropy\nconstantκ1will have low coercivity on the order of that of Permalloy, as long as the magne-\ntostriction constants λ100,λ111are tuned to special values. The explicit prediction for a cubic\nmaterial with low coercivity is the dimensionless number ( c11−c12)λ2\n100/κ1= 81 for/angbracketleft100/angbracketright\neasy axes. The results would seem to have broad potential application, especially to magnetic\nmaterials of interest in energy research.\nIntroduction\nA longstanding puzzle in materials science is understanding the origins of magnetic hysteresis in\nferromagnetic materials. Hysteresis in this domain refers to the differing behaviors obtained when\na demagnetized specimen is subject to an increasing magnetic field to saturation vs. that obtained\nwhen decreasing the field from saturation to zero. The effect is typically characterized by the\nfinal (absolute) value of the magnetic field after such a test, termed the coercivity. Informally,\nsoft magnets have low coercivity. This paper is concerned with the prediction of coercivity from\nmicromagnetic theory. An unexpected prediction of our study is that coercivity can be made\nvery small even in materials with large magnetocrystalline anisotropy constant, as long as the\nmagnetostrictive constants are tuned appropriately.\nAside from basic scientific interest on the origins of hysteresis and the traditional application to\ntransformers, a strategy for the discovery of new soft magnetic materials is desirable for rapid power\nconversion electronics, all-electric vehicles, and wind turbines, especially in cases where induction\nmotors are favored. Magnetic hysteresis has also become critical to the adoption of proposed\nspintronic and storage devices, as requirements for limiting energy consumption have moved to the\nforefront [1, 2]. These requirements impact a wide range of applications, from hand held electronic\ndevices to storage systems and servers at data centers. While our analysis is aimed primarily at\nbulk applications, the key idea that micromagnetic theory with magnetostriction, together with a\n1arXiv:2111.05456v1  [cond-mat.mtrl-sci]  9 Nov 2021well-chosen potent defect, can be used to predict hysteresis suggests a strategy for the lowering of\ncoercivity also in these small-scale applications. In fact, in certain film based devices, the likely\npotent defects, such as threading dislocations in epitaxial films, are often better characterized than\nin bulk material.\nCurrently, a widely accepted strategy to lower the hysteresis in cubic ferromagnetic alloys is\nbased on changing composition so as to reduce the magnitude of the anisotropy constant |κ1|. This\nhas the effect of flattening the graph of magnetocrystalline anisotropy energy vs. magnetization\nand reducing the penalty associated with magnetization rotation. Intuitively, this makes sense,\nas it apparently makes available additional low energy pathways of an alloy in a metastable state\non the shoulder of the hysteresis loop, as an applied field is being lowered. A related idea is the\nknown strategy of tuning the composition so as to be precisely at the point where two different\nsymmetries coincide, again leading to a flattening of the of the magnetocrystalline anisotropy energy\nvs. magnetization and the lowering of hysteresis. The latter is the strategy used by Clarke and\ncollaborators [3, 4] that led to the particular composition of Terfenol: Tb xDy1−xFe2,x= 0.3. We\nadd that modern research on these RFe 2cubic Laves phase materials has focused on the benefits\nof exploiting a nearby morphotropic phase boundary in these systems to enhance magnetostrictive\nresponse under small fields [5–7].\nHowever, these strategies cannot be the whole story behind coercivity. For example, in the\niron-nickel system a sharp drop in coercivity occurs at the Permalloy composition, at which κ1=\n−161Jm−3[8]. Tuning the magnetocrystalline anisotropy constant to zero in iron-nickel alloys\nin fact leads to an alloy composition with noticeably higher hysteresis. Similarly, in Sendust\n(Fe0.85Si0.096Al0.054alloy) the hysteresis is minimum when the magnetocrystalline anisotropy and\nmagnetostriction constants are close to zero, and not precisely at κ1= 0 [9]. Finally, there are quite\na few isolated examples of alloys which have very large magnetocrystalline anisotropy constants\nbut low coercivity: an example is the uniaxial phase of Ni 51.3Mn24.0Ga24.7having a uniaxial mag-\nnetocrystalline anisotropy constant of 2 .45×105Jm−3and coercivity less than 1 kAm−1in single\ncrystals [10]. Another example is the Galfenol alloys (Fe 1−xGaxfor 0.13≤x≤0.24) that have\nsmall magnetic hysteresis despite their very large magnetocrystalline anisotropy constants [11, 12].\nThese examples suggest that the magnetocrystalline anisotropy constant is the not the only factor\nthat governs hysteresis in magnetic alloys.\nThe precise role of magnetostriction constants on magnetic hysteresis is not well understood for\ntwo reasons: (a) Prior research has typically ignored the contribution of magnetoelastic interactions\non hysteresis, justified by the small values of the magnetostriction constants, and (b) mathematical\nmethods, such as the linear stability analysis, overestimate the coercivity of bulk alloys—by over\nthree orders of magnitude in some cases—as compared to measured experimental values [13, 14].\nThe latter is referred to as the “Coercivity Paradox” [13, 15]. These factors limit our understanding\nof how the balance between fundamental material constants—such as magnetocrystalline anisotropy\nand magnetostriction constants—affects magnetic coercivity.\nIn our recent work, we developed a computational tool based on micromagnetics, including\nmagnetoelastic terms, that is adapted to the prediction of coercivity in bulk magnetic alloys [18,\n19]. Micromagnetics, since its first postulation in 1963 by W.F.Brown Jr. [13], has been applied to\na wide range of problems in ferromagnets and forms the basis to several computational frameworks\n[20–22]. Our coercivity tool is based on the micromagnetics theory [18], however differs from earlier\nworks in the following ways: A key feature of this tool is that it uses a large but highly localized\ndisturbance—in the form of a N´ eel-type spike domain—to predict magnetic coercivity. N´ eel spike-\nlike domains are frequently observed to form around defects in various magnetic alloys (See [17, 23–\n26] for examples) and grow under an external field. In the absence of this spike-domain, the second\n2Figure 1: We model a 3D computational domain with a spike-domain microstructure. (a) The 3D\ncomputational domain is Ω embedded inside an oblate ellipsoid E, which is several times larger in size\nthan Ω. A spike-domain microstructure forms around a non-magnetic defect that grows under an\napplied field. (b) Spike-domain microstructures are commonly found in magnetic materials around\ndefects such as cavities and pores in Fe-film and Fe-Si crystal, respectively. [16, 17]. (Reprinted\nsubfigure (b) with permission from Ref. [16]. Copyright (2020) by the American Physical Society)\nvariation of the micromagnetic energy misses the energy barrier for magnetization reversal [13, 27].\nInstead, we model a large localized disturbance (i.e., compute a non-linear stability analysis) to\npredict coercivity on the shoulder of the hysteresis curve, see Fig. 1. Other features of our coercivity\ntool are that we account for magnetoelastic interactions, however small, in estimating coercivity of\nbulk magnetic alloys; and we use the ellipsoid and reciprocal theorems to accelerate our coercivity\ncalculations, see [18]. Our computations are fully three-dimensional with the magnetization evolving\nfrom (010) to (100) in all directions as one leaves the spike domain.\nTo arrive at the N´ eel spike as a reasonable description of a potent defect, we include in [18] a\nstudy of alternative nuclei, including rotated spikes and multiple spikes. These spike domains serve\nas nuclei that grow during magnetization reversal. In this previous study [18], we investigated the\nrole of defect geometry, defect orientation, and defect number on magnetic coercivity, and we found\nthat these features did not significantly affect coercivity when compared to fundamental material\nconstants. Mathematically, they can be interpreted as localized disturbances that, under the right\ncombination of material constants and suitable applied field, are able to surmount an energy barrier\n[28–31] Small, smooth disturbances of a homogeneous state are not able to surmount this barrier\nat realistic applied fields due to the dominance of domain wall energy at small scales, and, in our\nview, this is the essence of the Coercivity Paradox [18, 19]. Mathematically, the situation is that\nthe study of the second variation of the micromagnetic energy misses the important energy barrier,\nwhile it is captured as a large amplitude, but localized, disturbance.\nOur viewpoint is consistent with the thesis of Pilet [28] who examines the microscopic state on\nthe shoulder of the hysteresis loop (far away from where coercivity is measured) using magnetic\nforce microscopy, and finds good correlation with the presence of large localized disturbances. An-\nother feature of our tool is that we account for the magnetoelastic interactions—however small—in\nestimating coercivity. Our results are in a form that is amenable to alloy development, as in related\nsearches for low hysteresis phase transformations [32].\nAn alternative highly effective route to low hysteresis in magnetic materials is the synthesis of\n3nanocrystalline material [33]. For example, the powder synthesis and rapid solidification techniques\nhave resulted in nanocrystalline and amorphous magnetic alloys that offer low hysteresis and en-\nhanced permeability [2, 34, 35]. Here we do not specifically analyze this situation, but we think\nthis should be possible in the nanocrystalline case, though challenging due to the many grains that\nwould likely have to be considered.\nThe aim of the present work is to identify combinations of material constants, including satura-\ntion magnetization ( ms), magnetocrystalline anisotropy constant ( κ1), elastic moduli ( c11,c12,c44),\nand magnetostriction constants ( λ100,λ111) that give lowest coercivity in a cubic material. We do\nnot vary the exchange constant, which in practice does not vary much: we also give arguments\nwhy, when combined with the demagnetization energy, it should matter little. We find the striking\nresult that magnetostriction plays a critical role, and more importantly, coercivities on the order of\nthose found in very soft materials such as Permalloy are predicted to be possible in materials with\nlarge magnetocrystalline anisotropy constant, as long as the magnetostriction constants are tuned\nto special values. We find that magnetocrystalline anisotropy and magnetostriction constants play\na particularly important role, but neither has to be small.\nThe simulations are carried out using our newly developed coercivity tool. Details on our\nmicromagnetic algorithm are described in Methods. Specifically, we apply this tool in two studies:\nIn Study 1, we test the hypothesis that the tuning of magnetostriction constants, in addition to the\nmagnetocrystalline anisotropy constant, is necessary to reduce coercivity in magnetic alloys. Here we\nstudy how combinations of κ1andλ100affect coercivity, and ignore the contribution from λ111= 0.\nIn total, we compute coercivity values from N= 2,163 independent simulations. In Study 2, we\ntest our hypothesis that there exists a specific combination of material constants— κ1,λ100,λ111—at\nwhich magnetic coercivity is the lowest. Here, we compute coercivity by systematically varying\nthe magnetocrystalline anisotropy and the magnetostriction constants, in the ranges 0 ≤κ1≤\n2000Jm−3,−2000µ/epsilon1≤λ100≤2000µ/epsilon1, and 0≤λ111≤600µ/epsilon1respectively ( N= 605). Overall, our\ncomputations from over 2500 independent simulations show that the lowest coercivity is attained\nwhen the dimensionless number(c11−c12)λ2\n100\n2κ1≈81. Here,c11,c12are the elastic stiffness constants of\nthe soft magnet assuming a linear cubic relation. To our knowledge, this discovery of the balance\nbetween material constants at which magnetic hysteresis is small has not been proposed before.\nA theoretical analysis supporting the importance of this dimensionless number is given below.\nThis analysis further supports the idea that, in other situations, a second dimensionless number\nc12λ2\n100\n2κ1may become important, and tuning the stiffness constants c11andc12so that both of these\ndimensionless constants have particular values may be desirable [36].\nResults\nComputation of coercivity\nIn Study 1, we test our hypothesis that the magnetostriction constants, in addition to the mag-\nnetocrystalline anisotropy constant, is necessary to reduce coercivity in magnetic alloys. Fig. 2(a)\nshows a heat map of coercivity values as a function of the magnetocrystalline anisotropy constant κ1\nand the magnetostriction constant λ100. A key feature of this plot is that the coercivity is minimum\nalong a curve described by(c11−c12)λ2\n100\n2κ1≈81. As expected the coercivity is small when κ1→0 and\nthe magnetostriction constant is small λ100→0, see Inset A. However, surprisingly, we find that the\ncoercivity value is also small for non-zero magnetocrystalline anisotropy constants κ1>>0 with\nsuitable combinations of the magnetostriction constant, see Inset B. Note the rather large mag-\n4Figure 2: The coercivity map as a function of the magnetocrystalline anisotropy κ1and the mag-\nnetostriction constant λ100. (a) We carried out a total of N= 2163 computations with the\nmagnetostriction constant λ111set to zero. We find that minimum coercivity is achieved when\n(c11−c12)λ2\n100\n2κ1≈81. For comparison, the solid red-dot corresponds to the coercivity of the permalloy\ncomposition. Note, the parabolic relationship is a best fit polynomial to the spread of coercivity\ndata from our calculations. Alloys, such as the permalloy, lie close to the vertex of this parabola and\nwe interpret this as the shortest distance to the parabola with a dimensionless constant of 81. The\naxes (in red) indicates the crystallographic directions along which coercivity values were measured\nfor alloys with κ1>0 andκ1<0, respectively. The 3D surface contours of the coercivity values\nin (b) ‘inset A’ and (c) ‘inset B’ are shown. These plots show coercivity wells (regions of minimum\ncoercivity) as κ1→0 andκ1>>0, respectively. Note, in sub-figure (b), the 3D surface contour\nis discontinuous along κ1= 0—we attribute this discontinuity to the transformation of easy axes\nfrom [100] to [111] crystallographic direction.\n5netocrystalline anisotropy constants being considered, well outside the range associated to normal\nsoft magnetism. Furthermore, the minimum coercivity valleys are symmetric about the λ100= 0\naxis. This symmetric response arises from the even terms λ2\n100in the free energy expression. We\nobserve a similar lowering of coercivity in magnetic alloys with κ1<0 at suitable values of λ111\nmagnetostriction constant (see Supplementary Figure 2).\nFig. 2(b-c) are 3D surface plots of the inset regions ‘A’ and ‘B’ respectively. The “wells” in\nthese plots correspond to combinations of material constants at which coercivity is minimum. Fig.\n2(b) is a 3D plot of the inset region ‘A’—here, we note a discontinuity or a jump in coercivity values\natκ1= 0. This discontinuity is because of the change in easy axes for magnetic alloys with κ1>0\nandκ1<0. For example, we compute coercivities along [100] and [111] crystallographic directions\nfor magnetic alloys with κ1>0 andκ1<0, respectively. These alloys have different values of\nmagnetostriction constants along their easy axes. We believe that the computed rapid change of\ncoercivity at κ1= 0 is real, and arises from the anisotropy of these magnetostriction constants. Fig.\n2(c) shows the 3D surface plot of the inset region B. Here, coercivities comparable to that of the\npermalloy composition are achieved at large magnetocrystalline anisotropy values κ1≈1700Jm−3\nwith suitable combinations of the magnetostriction constant λ100≈1200µ/epsilon1.\nOverall, the findings from Study 1 contradict the general understanding of hysteresis in mag-\nnetism, i.e., the magnetocrystalline anisotropy constant needs to be near zero for small hysteresis.\nOur findings show that magnetic hysteresis (or coercivity value) is small not only when κ1→0 but\nalso whenκ1>>0, given suitable values of the magnetostriction constants. Fig. 2 shows that the\nmagnetostriction constant, λ100, in addition to the magnetocrystalline anisotropy constant κ1plays\nan important role in reducing coercivity in magnetic alloys.\nThus far, we investigated coercivity values by setting one of the magnetostriction constants\nλ111to be zero. In Study 2 we test the hypothesis that there exists a specific combination of\nmaterial constants, namely κ1,λ100,λ111, at which magnetic coercivity is the lowest. We compute\ncoercivities at every combination of κ1,λ100andλ111in the parameter space 0 ≤κ1≤2000Jm−3\nand−2000µ/epsilon1≤λ100≤2000µ/epsilon1and 0≤λ111≤600µ/epsilon1.\nFig. 3(a) shows the coercivity map as a function of κ1andλ100for increasing values of λ111. A\nkey feature here is that the minimum coercivity relationship κ1∝λ2\n100is unique at each value of the\nλ111magnetostriction constant. For example, the minimum coercivity valleys gradually widen and\nbecome asymmetric about λ100= 0, with increasing value of the λ111magnetostriction constant.\nWe identify combinations of material constants at which minimum coercivity is achieved, and then\nplot a 3D surface through these points in the κ1−λ100−λ111parameter space, see Fig. 3(b). This\n3D plot represents a surface through the material parameter space on which coercivity is small.\nTheoretical analysis\nThe results of our studies above can be understood in the following way. We begin from the free\nenergy that we have used in the simulations of micromagnetics in dimensional form [18],\n/integraldisplay\nΩ{∇m·A∇m+κ1(m2\n1m2\n2+ m2\n2m2\n3+ m2\n3m2\n1)\n+1\n2[E−E0(m)]·C[E−E0(m)]−µ0msHext·m}dx+1\n2/integraldisplay\nR3µ0|Hd|2dx. (1)\nwhere E=1\n2(∇u+ (∇u)T) and the energy is to be minimized, or minimized locally, over the\npair of functions u,m∈H1(Ω). Here, m(with components on the cubic axes m 1,m2,m3) has\nbeen previously non-dimensionalized so m·m= 1, and the demagnetization field satisfies the\n6Figure 3: The coercivity map as a function κ1,λ100andλ111. (a) The minimum coercivity is\nachieved along a parabolic relation,(c11−c12)λ2\n100\n2κ1≈p. Here, the value of coefficient pdepends on the\nvalue of the second magnetostriction constant λ111. For example, as λ111increases and the coercivity\nwell loses symmetry across the λ100= 0 axes. (b) The 3D surface plot identifies a combination of\nmaterial constants, namely κ1,λ100andλ111, at which the coercivity is small.\n7magnetostatic equations Hd=−∇ζm,∇·(−∇ζm+msm) = 0 on all of space, so Hdhas the\ndimensions of ms, as does Hext. Please note that mis extended to R3by making it vanish outside\nΩ. Typical accepted values from a large compositional space appropriate to Fig. 3(a) including\nmost of the Fe-Ni system are\nA∼10−11Jm−1, κ 1∼0−6×103Nm−2, λ 100∼0−2000×10−6, λ 111∼0−600×10−6,\nms∼106Am−1,µ0∼1.3×10−6NA−2, c 11,c12,c44∼10−24×1010Nm−2(2)\nA nondimensional form is obtained by dividing the micromagnetic energy by µ0m2\ns∼106Nm−2,\nchanging variables x/prime=1\n/lscriptx∈Ω/prime, where/lscriptis a typical length scale and x/primeis dimensionless. This\ngives a typical nondimensional values of the micromagnetic coefficients\nA\n/lscript2µ0m2s∼10−17,κ1\nµ0m2s∼10−3,(c11−c12orc44)(λ2\n100orλ2\n111)\nµ0m2s∼10−5−10−1,\n1\nmsHext∼10−3,1\nm2sH2\nd∼1 (3)\nThe range of magnetostrictive coefficients is consistent with Fig. 3(a), and for material constants\nwith a size range, we choose a typical intermediate value. A caveat with these numbers is the\nobservation made by Brown [27] (discussed also in [17]) that the formal linearization of geometrically\nnonlinear micromagnetics that gives Eq. 1 implies a possible mdependence of the elastic moduli\nc11,c12,c44. This dependence is usually neglected, as is done here. Note from Eq. 3 the wide ranging\nvalues of magnetostrictive energy and its relative importance generally.\nIt is seen from the nondimensionalized coefficients Eq. 3 that magnetostriction and demagne-\ntization energy are dominant, but it is important to observe that each can be made negligible by\nsuitable magnetization distributions. The magnetostrictive energy can be made to vanish by choos-\ning a magnetization that satisfies curl(curl E0(m))T= 0, that is, E0(m(x)) =1\n2(∇u+(∇u)T) is the\nsymmetric part of a gradient, while the demagnetization energy vanishes on divergence-free magne-\ntizations. Additionally, we note that the exchange energy is one of the smallest energy contributions\nand its contribution further decreases at increasing length scales. Although the exchange constant\ncould be affected by temperature, composition, and presence of defects, its order of magnitude\nA≈10−11Jm−1does not significantly affect large scale micromagnetic simulations. Consequently,\nwe do not vary the exchange constant in our calculations.\nWe first explain from a theoretical viewpoint why the N´ eel spike attached to a defect of the\ntype we have chosen is a particularly potent perturbation. A key observation comes from symmetry\nand holds also in the more general case of a geometrically nonlinear magnetoelastic free energy (see\nSection 6 of [37]). The observation is that domain walls involving a jump in the magnetization\nm+−m−/negationslash= 0, where m+andm−minimize the magnetocrystalline anisotropy energy density\nand satisfy the divergence free condition ( m+−m−)·n= 0 at an interface with normal n, have\nthe property that they give rise to strains that are perfectly mechanically compatible in the sense\nthatE0(m+)−E0(m−) =1\n2(a⊗n+n⊗a) for some vector a. The latter is the jump condition\nimplying the existence of a continuous displacement across the interface. These conditions hold\nalso for typical domain wall models with remote boundary conditions given by ( m+,E0(m+)) and\n(m−,E0(m−)). This argument applies not only to materials with cubic symmetry but also to many\nlower symmetry systems (see [37] for the precise statement of this result). .\nTo understand the relation between this symmetry argument and the N´ eel spike, we substitute\nthe form of CandE0(m) for cubic materials into Eq. 1. The latter is,\nE0(m) =3\n2\nλ100(m2\n1−1\n3)λ111m1m2λ111m1m3\nλ111m1m2λ100(m2\n2−1\n3)λ111m2m3\nλ111m1m3λ111m2m3λ100(m2\n3−1\n3)\n (4)\n8Figure 4: Growth of a spike-domain microstructure during a typical magnetization reversal. A\nspike-domain forms around a non-magnetic inclusion in the computation domain. This spike grows\ngradually on decreasing the external field (a-c). At the coercivity value, the spike-domain grows\nabruptly resulting in magnetization reversal (d-e). The hysteresis loop plots the average magneti-\nzation in the computation domain m1as a function of the applied field H ext.\nin the orthonormal cubic basis. We consider only the case κ1>0 corresponding to /angbracketleft100/angbracketrighteasy axes,\nfor which we have the most data above. (The case κ1<0 corresponding to /angbracketleft111/angbracketrighteasy axes is handled\nsimilarly.) Without loss of generality we also divide the whole energy by the dimensionless number\nκ1/µ0m2\ns. In addition, we define a new displacement ˆu(x) byu(x) =λ100ˆu(x) with corresponding\nstrain tensor E=λ100ˆE. Minimization of energy using uis equivalent to the same using ˆu, and\nthis equivalence applies also to local minimization or the relative height of an energy barrier. These\nchanges give the explicit nondimensional form of the free energy,\n/integraldisplay\nΩ/primeA\n/lscript2κ1mi,jmi,j+ (m2\n1m2\n2+ m2\n2m2\n3+ m2\n3m2\n1)\n+2c44λ2\n100\nκ1/bracketleftBigg/parenleftbigg\nˆ/epsilon112−3\n2λ111\nλ100m1m2/parenrightbigg2\n+/parenleftbigg\nˆ/epsilon113−3\n2λ111\nλ100m1m3/parenrightbigg2\n+/parenleftbigg\nˆ/epsilon123−3\n2λ111\nλ100m2m3/parenrightbigg2/bracketrightBigg\n+/parenleftbigg(c11−c12)λ2\n100\n2κ1/parenrightbigg/bracketleftBigg/parenleftbigg\nˆ/epsilon111−3\n2/parenleftbigg\nm2\n1−1\n3/parenrightbigg/parenrightbigg2\n+/parenleftbigg\nˆ/epsilon122−3\n2/parenleftbigg\nm2\n2−1\n3/parenrightbigg/parenrightbigg2\n+/parenleftbigg\nˆ/epsilon133−3\n2/parenleftbigg\nm2\n3−1\n3/parenrightbigg/parenrightbigg2/bracketrightBigg\n+c12λ2\n100\n2κ1/parenleftbigg\nˆ/epsilon111+ ˆ/epsilon122+ ˆ/epsilon133/parenrightbigg2\n−µ0ms\nκ1(He1m1+He2m2+He3m3)\n−µ0ms\n2κ1(Hd1m1+Hd2m2+Hd3m3) dx, (5)\nwhere we have used the magnetostatic equation to include the demagnetization term in the inte-\ngrand. Also, ˆ /epsilon1ij=1\n2(ˆui,j+ ˆuj,i) and for simplicity we have dropped the prime on x. However, Ω/prime\n9Figure 5: A 3D visualization of the needle domain. (a) This needle domain forms around an\nembedded defect in our micromagnetic computations. (b) The inset figure shows the needle domain\nwith distinct surfaces (marked by surface normal vectors n1,n2). The surface n1approximately\nsatisfies the strain compatibility condition, however, the surface n2is not compatible and contributes\nto non-zero elastic energy in the system. The magnetization is denoted by m.\nremains dimensionless.\nNow we make an observation about the structure of Eq. 5 relating to the symmetry argument\ngiven above. With κ1>0 as assumed, the two magnetizations m−= (1,0,0) and m+= (0,1,0)\nsatisfy ( m+−m−)·n= 0, where n=1√\n2(1,1,0). Therefore, by the symmetry argument described\nabove, E0(m+)−E0(m−) =1\n2(a⊗n+n⊗a) and it is indeed verified that that this holds with\na=3√\n2λ100\n2(−1,1,0). The values of the strains are given in Fig. 6(c). Importantly, these choices of\nm+,E0(m+) and m−,E0(m−) make the magnetocrystalline anisotropy energy and all three mag-\nnetostrictive terms in Eq. 5 vanish, and also give a locally divergence-free magnetization. Recalling\nthat the simulations are 3D (see Fig. 5), note that Fig. 6(c) can be confined to a slab between two\nsurfaces parallel to the plane of the page and, if m(x) =m−is assigned outside the slab, then these\nsurfaces are also pole-free. These facts support the potency (i.e., low energy barrier) provided by\nthe N´ eel spikes.\nAt the transient state, the magnetic poles on the tips of the spike domains are far apart and the\ngeometric features of the spike-domain have evolved to compatible interfaces. Similar symmetry\narguments can be made for transient state microstructures with multiple N´ eel spikes and/or other\ndefect geometries. In these cases the magnetic material constants dominate the energy barrier, and\nthe defect geometries appear to play a negligible role.\nThus we arrive at the following scenario typical of nucleation. When the applied field is large the\nmagnetocrystalline anisotropy energy of a large spike is disfavored, and spike is small and collapsed\nnear the defect, due to the dominating influence of (diffuse) domain wall energy at small scales. As\nthe applied field is decreased, the magnetocrystalline anisotropy and demagnetization energies favor\nthe growth of the spike. A local energy maximum is reached, beyond which an energy decreasing\npath is possible, leading to complete reversal of the magnetization.\nThe parabolic form of the locus of lowest coercivity points in Figs. 3(a) and 2(a) can now be\nunderstood heuristically. The terms involving the very small nondimensional exchange constant and\nvery large multiplier of the nondimensional demagnetization energy are the only terms that involve\n∇m. These are expected to compete within the diffuse domain walls present in these simulations,\n10Figure 6: Schematic of the metastability of the N´ eel spikes. (a) With large applied field the spikes\nare collapsed onto the nonmagnetic defect, stabilized by (diffuse) domain wall and demagnetization\nenergy. (b) As the applied field is decreased the spikes grow, so as to decrease magnetostrictive and\ndemagnetization energies. (c) The associated energy barrier is breached at small negative applied\nfields, and the spikes run to the boundary. The state (c) is transient, and precedes the full switch\nto the lower energy /angbracketleft¯100/angbracketrightmagnetization; the final state is like (a) except the spike domains are\npointing NE and SW (see Fig. 4e). (d) An example simulation of the spike domain at its transient\ndomain state during magnetization reversal. Labels ‘AA’, ‘BB’, ‘CC’ mark lengths across which we\nexamine the strain distribution in Fig. 7.\nbut otherwise lead only to a near divergence-free magnetization, as suggested by the inequality of\narithmetic-geometric means in the form\n/parenleftBigg/radicalbigg\nA\n/lscript2κ1mi,j−/radicalbiggµ0\n6κ1|∇ζm|δij/parenrightBigg/parenleftBigg/radicalbigg\nA\n/lscript2κ1mi,j−/radicalbiggµ0\n6κ1|∇ζm|δij/parenrightBigg\n≥0, (6)\nwhich implies, using the magnetostatic equations Hd=−∇ζmand∇·(−∇ζm+msm) = 0, that\nA\n/lscript2κ1mi,jmi,j−µ0ms\n2κ1Hd·m≥2/radicalBigg\nAµ0\n6/lscript2κ2\n1(divm)|∇ζm|. (7)\nIn the geometry being considered, the heuristic argument for the energy barrier suggests the\nshear strains in this geometry play a minor role, and this is supported by typical measurements of\nthe shear strains across the spike, Fig. 6(e), taken from the simulations. This observation, together\nwith the fact that the magnetization is near the easy axes [100] or [010] in region of the spikes\n(so thatmimj≈0, i/negationslash=j), indicates that the constant c44λ1002/κ1will play only a minor role in\ndetermining the energy barrier.\nGranted these approximations, the height of the energy barrier must then be affected mainly by\nthe remaining dimensionless constants\n(c11−c12)λ2\n100\n2κ1andc12λ2\n100\n2κ1. (8)\nAgain, from the simulations (Fig. 6(e)) tr εis quite small in comparison to the multiplier of ( c11−\nc12)λ2\n100, and so we expect the latter, ( c11−c12)λ2\n100/2κ1, to be the key dimensionless constant that\n11Figure 7: The strain and elastic energy distribution across the needle domain. (a-c) The strain\ndistribution ˆ /epsilon1ijacross lengths ‘AA’, ‘BB’, ‘CC’ (as marked in Fig. 6(d)) in the computational\ndomain. The strain distribution from the simulation, ˆ /epsilon111≈1, ˆ/epsilon122≈ˆ/epsilon133≈−0.5, is consistent\nwith the theoretical analysis in Fig. 6(a). The strains at the domain walls near the spike domain\nvary indicating the diffuseness of the walls. The shear strains are mostly zero throughout the\ncomputational domain. (d-f) A plot of the three polynomials f(ˆE,m) that accompany coefficients\n2c44λ2\n100\nκ1,(c11−c12)λ2\n100\n2κ1, andc12λ2\n100\n2κ1in Eq. 5. In line with our hypothesis the polynomials accompanying\n2c44λ2\n100\nκ1andc12λ2\n100\n2κ1make negligible energy contributions for magnetization reversal at the transient\nstate of the spike domain, and the polynomial accompanying(c11−c12)λ2\n100\n2κ1serves as an energy barrier\nfor magnetization reversal at the transient domain state.\n12governs the height of the barrier. In simulations, the term containing this constant accounted for\nup to 80% of the total magnetoelastic energy.\nThe value of this dimensionless constant giving the lowest energy barrier indicated by the sim-\nulations is\n(c11−c12)λ2\n100\n2κ1= 81.0056 (9)\nAs we have noticed previously [19], 78.5% Ni Permalloy does not exactly fall on the computed\nparabola given by Eq. 9. This may indicate that there are opportunities for lowering the coercivity\nof permalloy. It is possible that favorable heat treatments of permalloy do just that by modifying\nthe material constants present in Eq. 9. An alternative possibility is that, while Eq. 9 may be the\nprimary dimensionless constant affecting coercivity, both constants given in Eq. 8 may play a role,\nindicating that the fine tuning of elastic moduli according to their relative roles in the two constants\nEq. 8 may be a route to even lower coercivity.\nDiscussion\nAt present, the conventional approach to develop soft magnets is to reduce the magnetocrystalline\nanisotropy value to zero. Consequently the search for soft magnets is concentrated around the\nκ1→0 region. Our theory-guided prediction suggests that in addition to the coercivity well at\nκ1→0 region, there exists other regions along(c11−c12)λ2\n100\n2κ1≈81 at which coercivity is small. This\nanalytical relation between material constants gives greater freedom for alloy development and\nincreases the parameter space to discover novel soft magnets. In summary, our findings show that\nthe magnetostriction constants, in addition to the magnetocrystalline anisotropy constant, play an\nimportant role in governing magnetic coercivity. This was the case in study 1 when coercivity was\nminimum along a parabolic relation given by(c11−c12)λ2\n100\n2κ1≈81 withλ111= 0 in magnetic alloys. In\nstudy 2, we identified a 3D surface topology on which the coercivity is small. Below, we discuss some\nlimitations of our findings and then present its potential impact to the magnetic alloy development\nprogram.\nTwo features of this work limit the conclusions we can draw about the fundamental relationship\nbetween magnetic material constants. First, we compute coercivities assuming a simple defect\nstructure (i.e., two spike-domains formed around a non-magnetic inclusion) in an oblate ellipsoid.\nWhile defect geometry has been shown to have surprisingly small effect on the coercivity values [18],\nit is not known whether and how different defect geometries, and crystallographic texture of the\nmagnetic alloy would affect our predictions of the parabolic relation for minimum coercivity. Second,\nthe presence of mechanical loads, such as residual strains or boundary loads, could affect the delicate\nbalance between the magnetic material constants. The idealized stress-free conditions in our model\nis subject to the shortcomings associated with the presence of microstructural inhomogeneities and\nsurface conditions in bulk materials. Whether accounting for these inhomogeneities would yield\ncomparable results in experiments is an open question. With these reservations in mind we next\ndiscuss the impact of our findings to the alloy development program.\nA key feature of our work is that we demonstrate that the magnetostriction constants play\nan important role in governing magnetic coercivity. These findings contrast with prior research in\nwhich the magnetocrystalline anisotropy constant has been regarded as the only material parameter\nthat governs magnetic hysteresis, and the contribution from magnetostriction constants have been\nlargely neglected. Consequently, the commonly accepted norm in the literature is that magnetic\nalloys with small magnetocrystalline anisotropy constant have small coercivity, and magnetic alloys\n13with large magnetocrystalline anisotropy constant have large coercivity. However, by accounting for\nboth magnetocrystalline anisotropy and magnetostriction constants, we show that magnetic alloys\ndespite their large κ1values have small coercivities at specific combinations of the magnetostriction\nconstants.\nAnother significant feature of our work is that we identify a relationship(c11−c12)λ2\n100\n2κ1≈81\nbetween magnetic material constants at which the coercivity is small. This generic formula serves\nas a theoretical guide to the alloy development program, by suggesting alternative combinations\nof material constants—beyond κ1= 0—to develop soft magnets. While this relation is based on\n/angbracketleft100/angbracketrighteasy axes and might differ for alloys with different easy axes, our work is intended to guide\nthe search for soft magnetism in ordinary ferromagnets with large ms. In future work, we intend to\ninvestigate other easy axes, elastic stiffness constants, to further lower coercivity in magnetic alloys\n[36]. With the recent advances in atomic-scale engineering the compositions of magnetic alloys can\nbe tuned atom-by-atom [38, 39]. For these experimental approaches, our prediction of the material\nconstant formula could serve as a guiding principle, to engineer magnetic alloy compositions to small\nhysteresis. Overall, the fundamental relationship between material constants provides initial steps\nto experimentalists to discover soft magnets with high magnetocrystalline anisotropy constants.\nIn conclusion, the present findings contribute to a more nuanced understanding of how material\nconstants, such as magnetocrystalline anisotropy and magnetostriction constants, affect magnetic\nhysteresis. Specifically, magnetoelastic interactions have been regarded to play a negligible role\nin lowering magnetic coercivity. Given the current findings, we quantitatively demonstrate that\nthe delicate balance between magnetocrystalline anisotropy, magnetostriction constants and the\nspike domain microstructure (localized disturbance) is necessary to lower magnetic coercivity. We\npropose a mathematical relationship between material constants(c11−c12)λ2\n100\n2κ1≈81 at which minimum\ncoercivity can be achieved in a material with (100) easy axes. More generally, our findings serve\nas a theoretical guide to discover novel combinations of material constants that lower coercivity in\nmagnetic alloys.\nMethods\nMicromagnetics\nIn our coercivity tool, we use micromagnetics theory that describes the total free energy as a function\nof magnetization m, strain E, and magnetostatic field Hd:\nψ=/integraldisplay\nΩ{∇m·A∇m+κ1(m2\n1m2\n2+ m2\n2m2\n3+ m2\n3m2\n1) +1\n2[E−E0(m)]·C[E−E0(m)]−µ0msHext·m}dx\n+/integraldisplay\nR3µ0\n2|Hd|2dx. (10)\nThe form of Eq. 10 is identical to that used in our previous work, in which we detail the meaning\nof the specific terms, material constants, and normalizations [18]. For the present work, we note\nthat the exchange energy ∇m·A∇mpenalizes spatial gradients of magnetization. More generally,\nthe anisotropy energy for cubic alloys would have contributions from higher order energy terms\n(e.g.,κ2(m2\n1m2\n2m2\n3)), and these additional anisotropy coefficients are likely in general to contribute\nto our coercivity calculations. Although including these higher order anisotropy coefficients, e.g.\nκ2, could affect the easy axes of magnetization of the material. For example at κ1≈−κ2\n2other\n14crystallographic directions, such as [111] ,[110], and a family of irrational directions, would have\nsimilarly small magnetocrystalline anisotropy energy. This warrants a systematic investigation in a\nfuture study, especially with κ1,κ2,κ3near actual measured values. However, we do not think that\nthese higher order terms would significantly affect our coercivity calculations for the following rea-\nsons: First, in our computations, we model an oblate ellipsoid (pancake-shaped) that supports an\nin-plane magnetization. This ellipsoid geometry and the defect shape penalizes out-of-plane mag-\nnetization and thus m 3≈0 in our calculations of the magnetic hysteresis. Consequently, the energy\ncontribution from the higher order anisotropy term, κ2(m2\n1m2\n2m2\n3), is negligible in our computations.\nSecond, in the nondimensional form of the free energy (see Eq. (5)), the energy contribution from\nthe higher-order anisotropy term would scale asκ2\nκ1(m2\n1m2\n2m2\n3) with|m|= 1. This sixth-order energy\nterm is expected not to change the coercivity calculations significantly. We propose to investigate\nthe precise role of the higher-order anisotropy terms in a future study, however, as a first step\nwe investigate coercivity as a function of magnetocrystalline anisotropy κ1and magnetostriction\nconstantλ100. The elastic energy1\n2[E−E0(m)]·C[E−E0(m)] penalizes mechanical deformation\naway from the preferred strains, and the external energy, µ0Hext·maccounts for the mutual in-\nteraction between magnetization moment and the applied field. Finally, the magnetostatic energy\nµ0\n2|Hd|2computed in all of space R3penalizes the stray fields generated by the magnetic body in\nits surroundings.\nWe compute the evolution of the magnetization using an energy minimization technique, the\ngeneralized Landau-Lifshitz-Ginzburg equation, see Fig. 4:\n∂m\n∂τ=−m×H−αm×(m×H). (11)\nHere,H=−1\nµ0m2sδΨ\nδmis the effective field, τ=γmstis teh dimensionless time step, γis the\ngyromagnetic ratio, and αis the damping constant. We numerically solve Eq. 11 using the Gauss\nSiedel projection method [20], and identify equilibrium states when the magnetization evolution\nconverges,|mn+1−mn|<10−9. At each iteration we compute the magnetostatic field Hd=−∇ζm\nand the strain Eby solving their respective equilibrium equations:\n∇·(−∇ζm+msm) = 0 on R3(12)\n∇·C(E−E0) = 0 onE. (13)\nThe magnetostatic equilibrium condition arises from the Maxwell equations, namely ∇×Hd=\n0→Hd=−∇ζmand∇·B=∇·(Hd+msm) = 0.\nIn our calculations, we model a finite sized domain Ω centered around a non-magnetic defect Ω d,\nsee inset images in Fig. 4. This domain is several times smaller than the actual size of the ellipsoid\nE,see Fig. 1. We define the total demagnetization field as a sum of the local /tildewideH(x) and non-local\ncontributions H. The local contribution /tildewideH(x) varies spatially and accounts for the magnetostatic\nfields generated from defects and other imperfections inside the body. We calculate this local\ncontribution by solving ∇·(/tildewideH+ms/tildewidem) = 0 on Ω. The non-local contribution is computed as H=\n−Nmsm. Here, Nis the demagnetization factor matrix that is a tabulated geometric property of the\nellipsoid. We note, from the tabulated values in Ref. [40], the demagnetization factors for an oblate\nellipsoid (pancake-shaped) are N 11= N 22= 0,N33= 1. The mis the constant magnetization which\nis defined such that/integraltext\nΩm(x)dx= 0. This decomposition simplifies our computational complexity,\nbecause we now model a local domain Ω that is much smaller than modeling a domain in R3, and yet\naccount for the demagnetization contributions from both the body geometry and the local defects.\n15This decomposition is justified in the appendix of our previous paper [18]. Both the magnetostatic\nand mechanical equilibrium conditions in Eqs. 12–13 are solved in Fourier space, see Refs. [18, 21]\nfor further details.\nNumerical calculations\nIn the present work, we calibrate our micromagnetics model for the FeNi alloy with the following\nmaterial constants: A= 10−11Jm−1,ms= 106Am−2,µ0= 1.3×10−6NA−2,c11= 240.8GPa,c12=\n89.2GPa,c44= 75.8GPa. The values of κ1,λ100,λ111are systematically varied as detailed in the\nresults section. Here, note that magnetic alloys with positive and negative magnetocrystalline\nanisotropy constants, have their easy axes along {100}and{111}crystallographic directions, re-\nspectively. To accommodate this change of easy axes, we transform the energy potential from a\ncubic basis, and further details of this transformation are described in the appendix of [18].\nWe compute the coercivity of magnetic ellipsoids as a function of the magnetocrystalline anisotropy\nand magnetostriction material constants. In each computation, we model a domain of size 64 ×64×\n24 containing a defect with 16 ×16×6 grid points. We choose a grid size such that the domain walls\nspan across 3-4 unit cells. We initialize the computational domain with a uniform magnetization\nm= m 1, and force the magnetization inside the defect to be zero throughout the computation,\n|m|= 0, see Fig. 4. We apply a large external field along the easy axes, Hext>>0, and decrease\nit gradually in steps of δH= 0.25e1Oe. As we decrease the applied field a spike domain forms\naround the defect, see Fig. 1. This spike domain grows in size as the applied field is further lowered\nuntil a critical field value—known as the coercivity H c—at which the magnetization vector reverses.\nWe use this approach to predict the coercivity of the magnetic alloys at each combination of the\nmagnetocrystalline anisotropy and magnetostriction constants.\nSpecifically, in Study 1 and Study 2 we carry out a total of, n= 2,163 andn= 605, computations\nrespectively. In these computations, we systematically vary the material constants in the parameter\nspace range of 0≤κ1≤2000Jm−3,−2000µ/epsilon1≤λ100≤2000µ/epsilon1, and 0≤λ111≤600µ/epsilon1, respectively.\nOur investigation shows that the minimum coercivity is attained for a parabolic relation κ1∝λ2\n100,\nand a total of over 2,500 computations are necessary to confirm this relationship.\nData Availability . The authors declare that the data supporting the findings of this study are\navailable within the paper and its supplementary information files. Furthermore, additional data\nthat support the findings of this study are available from the corresponding author upon request.\nAcknowledgment . The authors acknowledge the Center for Advanced Research Computing at the\nUniversity of Southern California and the Minnesota Supercomputing Institute at the University\nof Minnesota for providing resources that contributed to the research results reported within this\npaper. The authors would like to thank Anjanroop Singh (University of Minnesota) for help in\nchecking some of the calculations. A.R.B acknowledges the support of a Provost Assistant Professor\nFellowship, Gabilan WiSE fellowship, and USC’s start-up funds. R.D.J acknowledges the support\nof a Vannevar Bush Faculty Fellowship. The authors thank NSF (DMREF-1629026) and ONR\n(N00014-18-1-2766) for partial support of this work.\nAuthor contribution . ARB and RDJ conceptualized the project, designed the methodology, and\nprocured funding. ARB worked on model development, theoretical analysis, and visualization of\ndata. RDJ worked on the theoretical calculations. Both authors were involved with the writing of\nthe paper.\nCompeting Interests . The authors declare that there are no competing interests.\n16References\n1. Fert, A. Nobel Lecture: Origin, development, and future of spintronics. Reviews of modern\nphysics 80,1517 (2008).\n2. Silveyra, J. M., Ferrara, E., Huber, D. L. & Monson, T. C. Soft magnetic materials for a\nsustainable and electrified world. Science 362(2018).\n3. Teter, J., Clark, A., Wun-Fogle, M. & McMasters, O. Magnetostriction and hysteresis for Mn\nsubstitutions in (Tb xDy1−x)(MnyFe1−y)1.95.IEEE transactions on magnetics 26,1748–1750\n(1990).\n4. Clark, A., Teter, J. & McMasters, O. Magnetostriction “jumps” in twinned Tb 0.3Dy0.7Fe1.9.\nJournal of applied physics 63,3910–3912 (1988).\n5. Yang, S. et al. Large magnetostriction from morphotropic phase boundary in ferromagnets.\nPhysical review letters 104, 197201 (2010).\n6. Bergstrom Jr, R. et al. Morphotropic Phase Boundaries in Ferromagnets: Tb 1−xDyxFe2Alloys.\nPhysical review letters 111, 017203 (2013).\n7. Hu, C.-C. et al. Room-temperature ultrasensitive magnetoelastic responses near the magnetic-\nordering tricritical region. Journal of Applied Physics 130, 063901 (2021).\n8. Bozorth, R. The permalloy problem. Reviews of Modern Physics 25,42 (1953).\n9. Takahashi, M., Nishimaki, S. & Wakiyama, T. Magnetocrystalline anisotropy and magne-\ntostriction of Fe-Si-Al (Sendust) single crystals. Journal of magnetism and magnetic materials\n66,55–62 (1987).\n10. Tickle, R. & James, R. D. Magnetic and magnetomechanical properties of Ni2MnGa. Journal\nof Magnetism and Magnetic Materials 195, 627–638 (1999).\n11. Atulasimha, J. & Flatau, A. B. A review of magnetostrictive iron–gallium alloys. Smart Ma-\nterials and Structures 20,043001 (2011).\n12. Clark, A. E., Wun-Fogle, M., Restorff, J. B. & Lograsso, T. A. Magnetostrictive properties of\nGalfenol alloys under compressive stress. Materials transactions 43,881–886 (2002).\n13. Brown, W. F. Micromagnetics 18(interscience publishers, 1963).\n14. Aharoni, A. & Shtrikman, S. Magnetization curve of the infinite cylinder. Physical Review\n109, 1522 (1958).\n15. Brown, W. F. Magnetostatic principles in ferromagnetism (North-Holland Publishing Com-\npany, 1962).\n16. Sch¨ afer, R. & Schinnerling, S. Tomography of basic magnetic domain patterns in ironlike bulk\nmaterial. Physical Review B 101, 214430 (2020).\n17. Hubert, A. & Sch¨ afer, R. Magnetic domains: the analysis of magnetic microstructures (Springer\nScience & Business Media, 2008).\n18. Balakrishna, A. R. & James, R. D. A tool to predict coercivity in magnetic materials. Acta\nMaterialia 208, 116697 (2021).\n19. Renuka Balakrishna, A. & James, R. D. A solution to the permalloy problem—A micromag-\nnetic analysis with magnetostriction. Applied Physics Letters 118, 212404 (2021).\n1720. Wang, X.-P., Garcıa-Cervera, C. J. & Weinan, E. A Gauss–Seidel projection method for mi-\ncromagnetics simulations. Journal of Computational Physics 171, 357–372 (2001).\n21. Zhang, J. & Chen, L. Phase-field microelasticity theory and micromagnetic simulations of\ndomain structures in giant magnetostrictive materials. Acta Materialia 53,2845–2855 (2005).\n22. Wang, J. & Zhang, J. A real-space phase field model for the domain evolution of ferromagnetic\nmaterials. International Journal of Solids and Structures 50,3597–3609 (2013).\n23. Otto, F. & Viehmann, T. Domain branching in uniaxial ferromagnets: asymptotic behavior of\nthe energy. Calculus of variations and partial differential equations 38,135–181 (2010).\n24. Otto, F. & Steiner, J. The concertina pattern. Calculus of variations and partial differential\nequations 39,139–181 (2010).\n25. D¨ oring, L., Ignat, R. & Otto, F. A reduced model for domain walls in soft ferromagnetic\nfilms at the cross-over from symmetric to asymmetric wall types. Journal of the European\nMathematical Society 16,1377–1422 (2014).\n26. Cinti, E. & Otto, F. Interpolation inequalities in pattern formation. Journal of Functional\nAnalysis 271, 3348–3392 (2016).\n27. Brown, W. F. Magnetoelastic interactions (Springer, 1966).\n28. Pilet, N. The relation between magnetic hysteresis and the micromagnetic state explored by\nquantitative magnetic force microscopy PhD thesis (University of Basel, 2006).\n29. Kn¨ upfer, H., Kohn, R. V. & Otto, F. Nucleation barriers for the cubic-to-tetragonal phase\ntransformation. Communications on pure and applied mathematics 66,867–904 (2013).\n30. Zhang, Z., James, R. D. & M¨ uller, S. Energy barriers and hysteresis in martensitic phase\ntransformations. Acta Materialia 57,4332–4352 (2009).\n31. Zwicknagl, B. Microstructures in low-hysteresis shape memory alloys: scaling regimes and\noptimal needle shapes. Archive for Rational Mechanics and Analysis 213, 355–421 (2014).\n32. Cui, J. et al. Combinatorial search of thermoelastic shape-memory alloys with extremely small\nhysteresis width. Nature materials 5,286–290 (2006).\n33. Thomas, S. V. et al. Nanocrystallites via Direct Melt Spinning of Fe 77Ni5.5Co5.5Zr7B4Cu for\nEnhanced Magnetic Softness. physica status solidi (a) 217, 1900680 (2020).\n34. McHenry, M. E., Willard, M. A. & Laughlin, D. E. Amorphous and nanocrystalline materials\nfor applications as soft magnets. Progress in materials Science 44,291–433 (1999).\n35. Soldatov, I., Andrei, P. & Schaefer, R. Inverted Hysteresis, Magnetic Domains, and Hysterons.\nIEEE Magnetics Letters 11,1–5 (2020).\n36. Ahani, N., Daware, A. & Balakrishna, A. R. Magnetoelastic interactions reduce hysteresis in\nsoft magnets. Manuscript in preparation (2021).\n37. James, R. D. & Hane, K. F. Martensitic transformations and shape-memory materials. Acta\nmaterialia 48,197–222 (2000).\n38. Ouazi, S. et al. Atomic-scale engineering of magnetic anisotropy of nanostructures through\ninterfaces and interlines. Nature communications 3,1–9 (2012).\n39. Kablov, E. et al. Bifurcation of magnetic anisotropy caused by small addition of Sm in\n(Nd 1−xSmxDy)(FeCo)B magnetic alloy. Journal of Applied Physics 117, 243903 (2015).\n40. Osborn, J. Demagnetizing factors of the general ellipsoid. Physical review 67,351 (1945).\n18Supplementary Information\n(Design of soft magnetic materials)\nAnanya Renuka Balakrishna1,*and Richard D. James2\n1Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA 90089\n2Aerospace and Engineering Mechanics, University of Minnesota, Minneapolis, MN 55455\n*Corresponding author, Email: renukaba@usc.edu\nSupplementary Methods\nIn our calculations, we compute coercivity for an oblate ellipsoid E(pancake shaped) under a\nsuitably oriented applied field. We apply the external magnetic field along the major axis of the\nellipsoid that is uniformly magnetized mexcept at the proximity of the defects. We account for the\neffect of the critically important, far-away poles on the boundary of the macroscopic body, while\ntreating their influence on the high-resolution computational domain near the defect, using the\nellipsoid and reciprocal theorems. In our numerical scheme, we use these theorems, to decompose\nthe magnetization into two: a constant magnetization m, and a spatially varying magnetization /tildewidem.\nThe latter magnetization is localized in the vicinity of the defect and decays away from it.\nSupplementary Figure 1: A representative micromagnetic simulation. (a) A schematic illustration\nof the 3D computational domain used in our numerical calculations. (b) The 3D microstructural\nevolution of the spike-domain under an applied magnetic field. (c) The spike-domain growth on a\ne1−e2plane through the 3D computational domain.\nThe ellipsoid geometry of the magnetic body is essential as a way of describing the poles on the\nboundary of the macroscale body without having to simulate the external fields arising due to these\npoles. As a consequence of this ellipsoid theorem, we model a 3D computational domain Ω >>E,\n1arXiv:2111.05456v1  [cond-mat.mtrl-sci]  9 Nov 2021Supplementary Figure 2: A heat map of the coercivity values as a function of the anisotropy\nconstantκ1and magnetostriction constant λ111. We set the magnetostriction constant along the\n<100>crystallographic directions λ100to be zero in these computations n= 903. We find that\nminimum coercivity is achieved for suitable combinations of magnetocrystalline anisotropy and\nmagnetostriction constants, κ1∝λ2\n111, for magnetic alloys with κ1<0.\nwith a nonmagnetic inclusion at its geometric centre. The size of this computational domain is\nchosen such that /tildewidem→mon the surface of the computational domain. Further details on how the\nmagnetization and the derivative fields are computed are described in Ref. [1], and for our purposes\nwe note that the equilibrium conditions and evolution equations Eqs. 11–13 are solved using the\nGauss-Siedel Projection method in Fourier space [1–3].\nSupplementary Figure 1 shows a representative calculation with the 3D evolution of the spike\ndomain microstructure during magnetization reversal. Please note that we model an oblate ellipsoid\nin our computations. This ellipsoid geometry of the magnetic body forces an in-plane magnetization,\nand consequently the domain evolution in our computations are primarily observed on the e1−e2\nplane with negligible changes in the third dimension.\nSupplementary Discussion\nIn our micromagnetic calculations, we primarily investigated coercivities of magnetic alloys with\neasy axes along <100>crystallographic directions (i.e., κ1>0). In this supplement we investigate\nwhether suitable magnetostrictions can lower coercivities of magnetic alloys with easy axes along\n<111>crystallographic directions (i.e., κ1<0). We model magnetic ellipsoids with with crystal-\nlographic directions [111] and [ 110] in-plane, and apply an external field along the [111] direction.\nWe transform the coordinate basis of the free energy function to suitable orientations (as described\n2in the Methods section) and further details of this transformation are given in Ref. [1]. We compute\ncoercivity by systematically varying the magnetocrystalline anisotropy and the magnetostriction\nconstants, in the ranges −2000≤κ1≤0Jm−3,= and−1050µ/epsilon1≤λ111≤1050µ/epsilon1, respectively.\nFurther details on the method, and specific values of the material constants used in our model\nare described in the methods section. Overall, our computations from over 903 independent cal-\nculations show that the lowest coercivity is attained when κ1∝λ2\n111, see Supplementary Figure 2.\nThis is consistent with our theoretical and analytical interpretation on the role of magnetostriction\nconstant in lowering coercivity for magnetic alloys with κ1>0.\nReferences\n1. Balakrishna, A. R. & James, R. D. A tool to predict coercivity in magnetic materials. Acta\nMaterialia 208, 116697 (2021).\n2. Wang, X.-P., Garcıa-Cervera, C. J. & Weinan, E. A Gauss–Seidel projection method for micro-\nmagnetics simulations. Journal of Computational Physics 171, 357–372 (2001).\n3. Zhang, J. & Chen, L. Phase-field microelasticity theory and micromagnetic simulations of do-\nmain structures in giant magnetostrictive materials. Acta Materialia 53,2845–2855 (2005).\n3"    },    {        "title": "2111.07110v1.Sensitive_electronic_correlation_effects_on_electronic_properties_in_ferrovalley_material_Janus_FeClF_monolayer.pdf",        "content": "arXiv:2111.07110v1  [cond-mat.mtrl-sci]  13 Nov 2021Sensitive electronic correlation effects on electronic pro perties in ferrovalley material\nJanus FeClF monolayer\nSan-Dong Guo1, Jing-Xin Zhu1, Meng-Yuan Yin1and Bang-Gui Liu2,3\n1School of Electronic Engineering, Xi’an University of Post s and Telecommunications, Xi’an 710121, China\n2Beijing National Laboratory for Condensed Matter Physics, Institute of Physics,\nChinese Academy of Sciences, Beijing 100190, People’s Repu blic of China and\n3School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, People’s Republic of China\nThe electronic correlation may have essential influence on e lectronic structures in some materi-\nals with special structure and localized orbital distribut ion. In this work, taking Janus monolayer\nFeClF as a concrete example, the correlation effects on its el ectronic structures are investigated\nby using generalized gradient approximation plus U(GGA+U) approach. For perpendicular mag-\nnetic anisotropy (PMA), the increasing electron correlati on effect can induce the ferrovalley (FV)\nto half-valley-metal (HVM) to quantum anomalous Hall (QAH) to HVM to FV transitions. For\nQAH state, there are a unit Chern number and a chiral edge stat e connecting the conduction and\nvalence bands. The HVM state is at the boundary of the QAH phas e, whose carriers are intrin-\nsically 100% valley polarized. With the in-plane magnetic a nisotropy, no special QAH states and\nprominent valley polarization are observed. However, for b oth out-of-plane and in-plane magnetic\nanisotropy, sign-reversible Berry curvature can be observ ed with increasing U. It is found that these\nphenomenons are related with the change of dxy/dx2−y2anddz2orbital distributions and different\nmagnetocrystalline directions. It is also found that the ma gnetic anisotropy energy (MAE) and\nCurie temperature strongly depend on the U. With PMA, taking typical U=2.5 eV, the electron\nvalley polarization can be observed with valley splitting o f 109 meV, which can be switched by\nreversing the magnetization direction. The analysis and re sults can be readily extended to other\nnine members of monolayer FeXY (X/Y=F, Cl, Br and I) due to sha ring the same Fe-dominated\nlow-energy states and electronic correlations with FeClF m onolayer. Our works emphasize the\nimportance of electronic correlation to determine the elec tronic state of some materials, and the\nelectronic correlation can induce exceptional phase trans ition.\nKeywords: Electroniccorrelation, Valleytronics, Magnet ic anisotropy Email:sandongyuwang@163.com\nI. INTRODUCTION\nIn recent years, the valley as a new degree of freedom\nof electrons for two-dimensional (2D) graphene-related\nmaterials has attracted intensive attention1–7. The local\nenergy extremes in the conduction band or valence band\narereferredtoasvalleys, andtheyprovideaneweffective\ndegreeoffreedom, in addition to conventionalchargeand\nspin. Withthebrokencentrosymmetry,thetwoinequiva-\nlent sublattices give rise to a degenerate but inequivalent\npair of valleys, which are well separated in the 2D hexag-\nonal Brillouin zone1. To achieve valley application, the\nexternal conditions, such as the optical pumping, mag-\nnetic field, magnetic substrates and magnetic doping8–11,\nhave been used to triggerpolarization. The FV materials\nmay be the best choice for valleytronics due to sponta-\nneous valley polarization12, and many FV materials have\nbeen predicted by the first-principle calculations13–20.\nThe FV materials possess magnetism, and generally\ncontain transition metal elements with localized delec-\ntrons, where the electronic correlations may have im-\nportant effects on the magnetic, topological, and val-\nley properties of these materials. Recently, the some of\nmonolayer FeXY (X/Y=F, Cl, Br and I) family, such as\nFeX2(X=Cl, Br and I) and FeClBr21–24, are predicted\nto be FV materials. Some differences can be found for\ntheir valley properties21,22,24, which is because different\nexchange-correlationfunctional is adopted. For example,for FeCl 2, the valley polarization appears at the valence\nbands by using GGA21,24, but that exists at the con-\nduction bands with HSE06 and GGA+ U(>1.9 eV)22,24.\nWithin GGA, it is found that the valley polarization can\nbe observedatthe valencebandsforFeX 2(X=Cl, Brand\nI)andFeClBr21,23,24. Thesemeanthattheelectroniccor-\nrelations have important effects on physical properties of\nmonolayer FeXY family. In fact, the different correla-\ntion strengths can make FeCl 2monolayer into different\nground states, like HVM and QAH states, with the as-\nsumption of PMA24. However, the correlation strength\nof a given material is fixed. So, it is very necessary for\ncalculating electronic properties of FeXY family to con-\nsider different strength of electronic correlation.\nIn light of these factors about monolayer FeXY family\nmentioned above, we take Janus monolayer FeClF as a\nconcrete example to investigate the correlation effects on\nits electronic structures by GGA+ Uapproach. Unlike\nprevious studies24(The lattice constants are optimized\nwith GGA, and the electronic structures are studied by\nGGA+U.), the lattice constants are optimized with var-\niedU, and the corresponding electronic structures and\nmagnetic properties are investigated. It is found that dif-\nferent correlationstrengths( U) alongwith different mag-\nnetic anisotropy (out-of-plane and in-plane) can drive\nthe system into different electronic states. For PMA,\nthe increasing Ucan induce the FV to HVM to QAH\nto HVM to FV transitions. For in-plane situation, there2\nFIG. 1. (Color online)The (a) top view and (b) side view of cry stal structure of Janus monolayer FeClF, and the rhombus\nprimitive cell is marked by the black frame. (c) The Brilloui n zone with high-symmetry points labeled.\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48/s51/s46/s49/s54/s51/s46/s49/s56/s51/s46/s50/s48/s51/s46/s50/s50/s51/s46/s50/s52/s51/s46/s50/s54/s51/s46/s50/s56/s97/s32/s40/s49/s48/s45/s49/s48\n/s32/s109/s41\n/s85 /s32/s40/s101/s86/s41\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48/s45/s49/s48/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48/s53/s48/s48/s54/s48/s48/s32\n/s32/s69/s40/s109/s101/s86/s41\n/s85 /s32/s40/s101/s86/s41\nFIG. 2. (Color online)For Janus monolayer FeClF, the lattic e\nconstants andenergydifferences(rectangle supercell) bet ween\nFM and AFM ordering as a function of U.\nare no special QAH states and prominent valley polariza-\ntion. These can be explained by considering the different\nFe-dorbital contributions, when including spin-orbital\ncoupling (SOC). However, calculated results show sign-\nreversibleBerrycurvaturewithincreasing Uforbothout-\nof-plane and in-plane magnetic anisotropy. Finally, it is\nprovedthat correlationstrengths have very important ef-\nfects on Curie temperature TCof FeClF. The TC(311 K)\nwithU=1.5 eV is five time that (63 K) with U=2.5 eV.\nOur works highlight the role of correlation effects in the\n2D FeXY family materials.\nThe rest of the paper is organized as follows. In the\nnext section, we shall give our computational details andmethods. In the next few sections, weshall presentstruc-\nture and stability, electronic structure and valley proper-\nties of Janus monolayer FeClF. Finally, we shall give our\ndiscussion and conclusion.\nII. COMPUTATIONAL DETAIL\nThe spin-polarized first-principles calculations are per-\nformed employing the projected augmented wave (PAW)\nmethod within density functional theory (DFT)25, as im-\nplemented in VASP code26–28. The exchange-correlation\neffect is treated by the GGA of Perdew-Burke-Ernzerhof\n(PBE-GGA)29. The energy cut-off of 500 eV and total\nenergy convergence criterion of 10−8eV are used in the\nstatic calculations. The force convergence criteria is set\nto be less than 0.0001 eV .˚A−1on each atom. The on-site\nCoulomb correlation of Fe atoms is considered within the\nGGA+Uscheme by the rotationally invariant approach\nproposed by Dudarev et al, in which only the effective\nU(Ueff) based on the difference between the on-site\nCoulombinteractionparameterandexchangeparameters\nismeaningful. TheSOCeffectisexplicitlyincludedinthe\ncalculationstoinvestigateMAEandelectronicstructures\nof FeClF monolayer. A vacuum space of more than 18 ˚A\nis used to avoid the interactions between the neighboring\nslabs. Thek-meshof24 ×24×1is usedtosamplethe Bril-\nlouinzoneforcalculatingelectronicstructuresandelastic\nproperties, and 12 ×24×1 Monkhorst-Pack k-point mesh\nfor the energies of ferromagnetic (FM) and antiferromag-\nnetic (AFM) states with rectangle supercell, as shown in\nFIG.1 of electronic supplementary information (ESI).\nThe elastic stiffness tensor Cijare calculated by using\nstrain-stressrelationship(SSR) method, where2Delastic\ncoefficients C2D\nijhavebeen renormalizedby C2D\nij=LzC3D\nij\nwith the Lzbeing the lengthofunit cell alongz direction.\nThe phonon dispersion spectrum is calculated with the\n5×5×1 supercell by using finite displacement method, as\nimplemented in the Phonopy code31. The 40 ×40 super-\ncell and 107loops are used to perform the Monte Carlo\n(MC) simulations, as implemented in Mcsolver code32.\nThe mostly localized Wannier functions including the d-3\n0 1 2 3\na/a00.00.51.01.52.0Gap /uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000c\n1.2 1.3 1.4 1.5 1.6 1.7\na/a00.000.050.100.150.200.250.30Gap /uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000cFV HVM HVM FV\nQAH\nFIG. 3. (Color online)For out-of-plane magnetic anisotrop y,\nthe energy band gap of Janus monolayer FeClF as a function\nofU(0-3 eV). The bottom plane is the enlarged view of the\ntop plane near the U=1.45 eV, and the phase diagram with\ndifferent Uis shown.\norbitals of Fe atom and the p-orbitals of Cl and F atoms\nare constructed on a k-mesh of 24 ×24×1 by the Wan-\nnier90 package33. The edge states are calculated with\nthe software package Wanniertools by the renormalized\neffective tight binding Hamiltonian34. The Berry curva-\ntures ofFeClF monolayerarecalculated directlyfrom the\ncalculated wave functions based on Fukui’s method35, as\nimplemented in the VASPBERRY code.\nIII. STRUCTURE AND STABILITY\nThe crystal structures of the Janus FeClF monolayer\nare plotted in Figure 1, along with Brillouin zone with\nhigh-symmetry points. It is clearly seen that the Fe-\nClF monolayer consists of Cl-Fe-F sandwich layer, which\ncan be built by replacing one of two Cl layers with F\natoms in FeCl 2monolayer. In experiment, Janus mono-\nlayerMoSSe can be achieved from MoS 2by replacing one\nof two S layers with Se atoms36,37. Due to broken ver-\ntical mirror symmetry, the space group of FeClF mono-\nlayer isP3m1 (No.156), which is lower than P¯6m2 of\nFeCl2monolayer (No.187). The FeClF (FeCl 2) has the\nsame symmetry with MoSSe (MoS 2). The lattice con-\nstantsais optimized with different U(0-3 eV), which isplotted in Figure 2. It is found that the aincreases with\nincreasing U, and ranges from 3.165 ˚A to 3.276 ˚A. To\ndetermine magnetic ground state, the energy differences\nbetween AFM and FM ordering as a function of Uare\nalso plotted in Figure 2. In considered Urange, the FM\norder is the most stable magnetic state. It is found that\nthe FM interaction decreases with increasing U, which\ncan produce important effects on Curie temperature of\nFeClF monolayer.\nTo demonstrate the stability of Janus FeClF mono-\nlayer, the phonon spectra and elastic constants are cal-\nculated by using GGA method. The phonon band dis-\npersions of FeClF monolayer calculated along the high-\nsymmetry directions of the Brillouin zone without imag-\ninary frequency modes are plotted in FIG.1 of ESI, sug-\ngesting its dynamical stability. To check the mechanical\nstability of monolayer FeClF, the elastic properties are\ninvestigated. Due to P3m1 space group, using Voigt no-\ntation, the elastic tensor can be reduced into:\nC=\nC11C12 0\nC12C11 0\n0 0 ( C11−C12)/2\n (1)\nThe independent C11andC12of FeClF monolayer are\n68.88 Nm−1and 22.21 Nm−1. These elastic constants\nsatisfy the Born criteria of mechanical stability38:C11>0\nandC11−C12>0,confirmingitsmechanicalstability. The\nYoungs moduli C2D, shear modulus G2Dand Poisson’s\nratiosν2DofFeClF monolayerare mechanicallyisotropic\ndue to hexagonal symmetry, and the corresponding val-\nues are 61.72 Nm−1, 23.34 Nm−1and 0.32.\nIV. OUT-OF-PLANE MAGNETIC\nANISOTROPY\nFirstly, we consider that the magnetocrystalline direc-\ntion of FeClF monolayer is along the out-of-plane. The\nout-of-plane FM maintains the horizontal mirror symme-\ntry, and breaks all possible vertical mirrors, which allows\na nonvanishing Chern number of the 2D system39. The\nevolutions of electronic band structures with Uare in-\nvestigated by GGA+SOC. The energy band gaps as a\nfunction of Uare plotted in Figure 3. The representative\nenergy band structures at different U values are shown\ninFigure 4. When U <1.4 eV, the gap decreases with\nincreasing U. However, with increasing U, the gap in-\ncreases for U >1.5 eV. Between U=1.4 eV and 1.5 eV,\nvery little gap can be observed, which may show nontriv-\nial topological properties.\nForU <1.4 eV, a remarkable valley polarization can\nbe observed in the valence bands at the -K and K points,\nand the -K point is polarized. However, for U >1.5 eV,\nthe noteworthy valley polarization occurs in the conduc-\ntion bands, and the K point is polarized. In these two\nregions, the monolayer FeClF is a FV material. Around\ntheU=1.40 eV, the band gap of -K point gets closed,\nand a narrow band gap still holds at K point, signifying4\n/uni0000012b /uni00000010/uni0000002e/uni00000030/uni0000002e /uni0000012b /uni00000030/uni000000ed/uni00000014/uni00000011/uni00000013/uni000000ed/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013/uni00000014/uni00000011/uni00000018/uni00000015/uni00000011/uni00000013Energy /uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000c\n0.00\n/uni0000012b /uni00000010/uni0000002e/uni00000030/uni0000002e /uni0000012b /uni00000030/uni000000ed/uni00000014/uni00000011/uni00000013/uni000000ed/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013/uni00000014/uni00000011/uni00000018/uni00000015/uni00000011/uni00000013Energy /uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000c\n1.00\n/uni0000012b /uni00000010/uni0000002e/uni00000030/uni0000002e /uni0000012b /uni00000030/uni000000ed/uni00000014/uni00000011/uni00000013/uni000000ed/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013/uni00000014/uni00000011/uni00000018/uni00000015/uni00000011/uni00000013Energy /uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000c\n1.40\n/uni0000012b /uni00000010/uni0000002e/uni00000030/uni0000002e /uni0000012b /uni00000030/uni000000ed/uni00000014/uni00000011/uni00000013/uni000000ed/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013/uni00000014/uni00000011/uni00000018/uni00000015/uni00000011/uni00000013Energy /uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000c\n1.45\n/uni0000012b /uni00000010/uni0000002e/uni00000030/uni0000002e /uni0000012b /uni00000030/uni000000ed/uni00000014/uni00000011/uni00000013/uni000000ed/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013/uni00000014/uni00000011/uni00000018/uni00000015/uni00000011/uni00000013Energy /uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000c\n1.50\n/uni0000012b /uni00000010/uni0000002e/uni00000030/uni0000002e /uni0000012b /uni00000030/uni000000ed/uni00000014/uni00000011/uni00000013/uni000000ed/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013/uni00000014/uni00000011/uni00000018/uni00000015/uni00000011/uni00000013Energy /uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000c\n2.50\nFIG. 4. (Color online)For out-of-plane magnetic anisotrop y, the energy band structures of Janus monolayer FeClF with Ubeing\n0.00 eV, 1.00 eV, 1.40 eV, 1.45 eV, 1.50 eV and 2.50 eV.\nFIG. 5. (Color online)For out-of-plane magnetic anisotrop y, topological edge states of Janus monolayer FeClF calcula ted along\nthe (100) direction with Ubeing 1.45 eV, including left (a) and right (b) edges.\nthe HVM, whose conduction electrons are intrinsically\n100% valley polarized24. The HVM can also be observed\naround the U=1.5 eV, but the band gap at K point dis-\nappears, and a narrow band gap exists at -K point. The\ngap closes, reopens, and then closes, which suggests a\npossible topological phase transition. Between the two\nHVMstates, aQAHinsulatorphasemayexist, whichcan\nbe characterized by chiral edge states. The edge states\nare calculated along (100) direction with U=1.45 eV,\nwhich is plotted in Figure 5. It is clearly seen that there\ndoes exist a chiral edge state connecting the conduction\nbands (-K/K valley) and valence bands (K/-K valley)for left/right edge. A single gapless chiral edge band\nmeans that the Chern number is equal to one ( C=1),\nwhich is consistent with integration over the Berry cur-\nvatures (See Figure 6), giving a nonzero Chern number\n(C=1). It is found that the QAH phase coexists with a\nvalley structure for both conduction and valence bands.\nWhen shifting the Fermi level into conduction/valence\nbands, the K/-K valley is polarized. In a word, the Fe-\nClF monolayer undergoes the FV, HVM, QAH, HVM\nand FV states with increasing U(seeFigure 3).\nBerry curvature is a powerful tool to investigate the\nvalley physics. The Berry curvature is calculated with5\n/s45/s75\n/s45/s50/s50/s45/s49/s55/s45/s49/s49/s45/s53/s46/s56/s45/s48/s46/s52/s54/s52/s46/s57/s49/s48/s49/s52\n/s75/s40/s49/s46/s48/s48/s41\n/s45/s75\n/s45/s48/s46/s56/s48/s57/s46/s50/s49/s57/s50/s57/s51/s57/s52/s57/s53/s57/s54/s54\n/s75/s40/s49/s46/s52/s53/s41\n/s45/s75\n/s45/s51/s46/s49/s45/s50/s46/s51/s45/s49/s46/s52/s45/s48/s46/s53/s49/s48/s46/s51/s54/s49/s46/s50/s50/s46/s49/s50/s46/s55\n/s75/s40/s50/s46/s53/s48/s41\nFIG. 6. (Color online)For out-of-plane magnetic anisotrop y, the calculated Berry curvature distribution of Janus mon olayer\nFeClF in the 2D Brillouin zone with Ubeing 1.00 eV, 1.45 eV and 2.50 eV.\n0.00 0.50 1.00 1.45 2.00 2.50 3.00\na/a00.00.51.01.5Gap /uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000c\nFIG. 7. (Color online)For in-plane magnetic anisotropy, th e\nenergy band gap of Janus monolayer FeClF as a function of\nU(0-3 eV).\nUfrom 0 to 3 eV, and the representative ones at dif-\nferentUvalues are plotted in Figure 6 . ForU=1.00\neV, the hot spots in the Berry curvature are around two\nvalleys, which have opposite signs and different magni-\ntudes. With increasing U, the FV state changes into the\nQAH state by a topological phase transition, connected\nby the HVM state. Within the range of U=1.4-1.5 eV,\nthe sign of Berry curvature at -K valley flips (For ex-\nampleU=1.45 eV in Figure 6). When Uspans the 1.5\neV, the QAH state changes into FV state with middle\nHVM state, resulting in the sign change of Berry curva-\nture at K valley (For example U=2.50 eV in Figure 6).\nThese mean that sign-reversible Berry curvature can be\nobserved with increasing U.\nForU <1.4 eV, the valley polarization is in the va-\nlence bands, but the valley polarization is in the conduc-\ntion bands for U >1.5 eV. For Ubetween 1.4 eV and 1.5\neV, the valley polarization can exit in both valence and\nconduction bands. To explain the peculiar effect of elec-\ntronic correlation and SOC on the band structure, the\nFe-d-orbital characters of energy bands for U=1.00 eV,\n1.45 eV and 2.50 eV are plotted in FIG.2 of ESI. For allconsidered U, the -K and K valleys in both valence and\nconduction bands are dominated by dz2ordx2−y2/dxy\norbitals. For U <1.4 eV, the -K and K valleys in va-\nlence bands are dominated by dx2−y2anddxyorbitals,\nand those in the conduction bands are mainly from the\ndz2orbitals. For U >1.5 eV, the opposite situation can\nbe observed. When Uis between 1.4 eV and 1.5 eV, the\n-K valley in the conduction bands and K valley in the\nvalence band are dominated by dx2−y2anddxyorbitals,\nand the -K valley in the valence bands and K valley in\nthe conduction band are dominated by dz2orbitals.\nThe valley polarization induced by SOC is due to the\nintra-atomic interaction:\nˆHSOC=λˆL·ˆS=ˆH0\nSOC+ˆH1\nSOC (2)\nin which λis the coupling strength, and ˆLandˆSare\nthe orbital angular moment and spin angular moment,\nrespectively. The magnetic exchange interaction results\nin that ˆH1\nSOC(the interaction of opposite spin states)\ncan be ignored. So, the ˆH0\nSOC(the interaction between\nthe same spin states) dominates the λˆL·ˆS. TheˆH0\nSOC\ncan be written as12,22,23:\nˆH0\nSOC=λˆSz‘(ˆLzcosθ+1\n2ˆL+e−iφsinθ+1\n2ˆL−e+iφsinθ)\n(3)\nwhereθandφare the polar angles of spin orientation.\nFor out-of-plane magnetization ( θ= 0◦),ˆH0\nSOCcan be\nreduced to:\nˆH0\nSOC=αˆLz (4)\nAt -K and K valleys, the group symmetry is C3h. Hence,\nthe orbital basis for -K and K valleys can be expressed\nas12,22,23:\n|φτ>=/radicalBig\n1\n2(|dx2−y2>+iτ|dxy>)\nor\n|φτ>=|dz2>(5)\nwhere the subscript τrepresent valley index ( τ=±1).\nTheresultingenergyatKand-Kvalleyscanbeexpressed\nas:\nEτ=< φτ|ˆH0\nSOC|φτ> (6)6\n/uni0000012b /uni00000010/uni0000002e/uni00000030/uni0000002e /uni0000012b /uni00000030/uni000000ed/uni00000014/uni00000011/uni00000013/uni000000ed/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013/uni00000014/uni00000011/uni00000018/uni00000015/uni00000011/uni00000013Energy /uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000c\n1.00\n/uni0000012b /uni00000010/uni0000002e/uni00000030/uni0000002e /uni0000012b /uni00000030/uni000000ed/uni00000014/uni00000011/uni00000013/uni000000ed/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013/uni00000014/uni00000011/uni00000018/uni00000015/uni00000011/uni00000013Energy /uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000c\n1.45\n/uni0000012b /uni00000010/uni0000002e/uni00000030/uni0000002e /uni0000012b /uni00000030/uni000000ed/uni00000014/uni00000011/uni00000013/uni000000ed/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013/uni00000014/uni00000011/uni00000018/uni00000015/uni00000011/uni00000013Energy /uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000c\n2.50\nFIG. 8. (Color online)For in-plane magnetic anisotropy, th e energy band structures of Janus monolayer FeClF with Ubeing\n1.00 eV, 1.45 eV and 2.50 eV.\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48/s45/s56/s48/s48/s45/s54/s48/s48/s45/s52/s48/s48/s45/s50/s48/s48/s48/s50/s48/s48/s32\n/s32/s77/s65/s69/s40 /s101/s86/s41\n/s85 /s32/s40/s101/s86/s41/s79/s117/s116/s32/s111/s102/s32/s112/s108/s97/s110/s101\n/s73/s110/s32/s112/s108/s97/s110/s101\nFIG. 9. (Color online) The MAE of Janus monolayer FeClF\nas a function of U(0-3 eV).\nIf the -K and K valleys are dominated by dx2−y2anddxy\norbitals, the valley energy difference |∆E|at -K and K\npoints (valley splitting) can be expressed as:\n|∆E|=EK−E−K= 4α (7)\nIf the -K and K valleys are mainly from the dz2orbitals,\nthe valley splitting |∆E|is given by:\n|∆E|=EK−E−K= 0 (8)\nSo, thesedifferentorbitalcomponentsof-KandKvalleys\nwith varied Udetermine the valley polarization distribu-\ntion.\nV. IN-PLANE MAGNETIC ANISOTROPY\nNext, we consider that the magnetocrystalline direc-\ntion of FeClF monolayer is along the in-plane. The elec-\ntronic band structures with different Uare calculated by\nGGA+SOC. The energy band gaps vs Uare shown in\nFigure 7, and the representative energy band structuresare plotted in Figure 8. It is found that the energy band\ngap firstly decreases, and then increases, when the Uin-\ncreases. The critical Uvalue is about 1.45 eV. Compared\nto out-of-plane magnetocrystalline direction, no special\nintermediate region exists. Figure 8 shows no observable\nvalley polarization in both the valence and conduction\nbands,whichcanbeexplainedby∆ E= 4αcosθ23. When\nthe magnetocrystalline direction of FeClF monolayer is\nalong in-plane direction ( θ=90◦), the valley splitting will\nvanish (∆ E=0). The Fe- d-orbital characters of energy\nbands for U=1.00 eV, 1.45 eV and 2.50 eV are plotted\nin FIG.3 of ESI. It is found that the distributions of Fe-\nd-orbital characters are akin to the cases of out-of-plane.\nAccording to FIG.4 of ESI, the sign-reversible Berry cur-\nvature can be observed, when the Uvalue strides over\nthe critical point (about U=1.45 eV). Calculated results\nshowthatthehotspotsintheBerrycurvaturearearound\ntwovalleyswith oppositesignsandalmostthe samemag-\nnitudes. To explore the topological properties of FeClF\nmonolayer at different U, we calculate the dispersion of\nthe edge state. The edge states with representative Uare\nplotted in FIG.5 of ESI (only left edge), which show that\nno chiral edge states traverse the bulk band gap. This\nmean that, for in-plane magnetocrystalline direction, no\nQAH states appear. So, the magnetocrystalline direction\nis very important to investigate electronic properties of\nFeClF monolayer.\nVI. MAGNETIC ANISOTROPY ENERGY AND\nANOMALOUS VALLEY HALL EFFECT\nThe magnetocrystalline direction of FeClF monolayer\ncan be regulated by external magnetic field. However,\nwe use MAE to determine intrinsic magnetic anisotropy\nof FeClF monolayer at different Uvalue. Within\nGGA+SOC+ U, the MAE can be calculated by EMAE=\nE(100)-E(001), which is plotted in Figure 9. The positive\nvalue means that the easy axis is perpendicular to the\nplane, while the negative value suggests the in-plane di-7\n/uni0000012b/uni00000010/uni0000002e/uni00000030/uni0000002e /uni0000012b/uni00000030/uni000000ed/uni00000014/uni00000011/uni00000013/uni000000ed/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013/uni00000014/uni00000011/uni00000018/uni00000015/uni00000011/uni00000013/uni00000015/uni00000011/uni00000018/uni00000016/uni00000011/uni00000013Energy /uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000c\n(a)\n/uni0000012b/uni00000010/uni0000002e/uni00000030/uni0000002e /uni0000012b/uni00000030/uni000000ed/uni00000014/uni00000011/uni00000013/uni000000ed/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013/uni00000014/uni00000011/uni00000018/uni00000015/uni00000011/uni00000013/uni00000015/uni00000011/uni00000018/uni00000016/uni00000011/uni00000013Energy /uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000c\n(b)\n/uni0000012b/uni00000010/uni0000002e/uni00000030/uni0000002e /uni0000012b/uni00000030/uni000000ed/uni00000014/uni00000011/uni00000013/uni000000ed/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013/uni00000014/uni00000011/uni00000018/uni00000015/uni00000011/uni00000013/uni00000015/uni00000011/uni00000018/uni00000016/uni00000011/uni00000013Energy /uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000c\n(c)\nFIG. 10. (Color online)For out-of-plane magnetic anisotro py with U=2.5 eV, the energy band structures of Janus monolayer\nFeClF (a) without SOC; (b) and (c) with SOC for magnetic momen t of Fe along the positive and negative z direction,\nrespectively. In (a), the blue (red) lines represent the ban d structure in the spin-up (spin-down) direction.\nFIG. 11. (Color online)(a) The schematics of anomalous val-\nley Hall effect under electron doping. (b) is the same as (a)\nbut with opposite magnetic moment. The green arrows rep-\nresent spin-down or spin-up states.\nrection. For U <1.15 eV, the FeClF monolayer possesses\nout-of-plane magnetic anisotropy. For U >1.15 eV, the\ndirection of the easy axis of FeClF monolayer is in plane.\nWe investigate the valley properties of FeClF monolayer\nwith representative U(2.5 eV)40,41, and its magnetocrys-\ntalline direction can be switched to out-of-plane by ex-\nternalmagneticfield. Thespin-polarizedbandstructures\nof monolayer FeClF without and with SOC are shown in\nFigure 10 . According to Figure 10 (a), a distinct spin\nsplitting can be observed in the band structures because\nof the exchange interaction, and the FeClF monolayer is\na direct band gap semiconductor with VBM and CBM\nprovided by the same spin-down. The valleys of -K and\nK are degenerate in energy for both conduction and va-\nlence bands. Figure 10 (b) shows that the SOC effect can\ninduce valley polarization in the conduction bands. Thevalleysplitting |∆E|is 109meV, and the energyofK val-\nley is lower than one of -K valley. Moreover, the valley\npolarization can be switched by reversing the magneti-\nzation direction with magnetic moment of Fe along the\nnegative z direction, which is confirmed by Figure 10 (c).\nBecause the low-energy bands at -K and K belong to the\nsame spin minority channel, the spin polarization of the\ncarriers is simultaneously switched.\nIt is also known that Berry curvature Ω( k) is associ-\nated with anomalous velocity υof Bloch electrons under\nan in-plane longitudinal electric field E:υ∼E×Ω(k)10.\nAccording to Figure 6, Berry curvature Ω( k) is charac-\nterized with unequal values and opposite signs for the -K\nand K valleys. With the reversed opposite magnetic mo-\nment, the valley polarization will also be reversed. The\ncorresponding values of the Berry curvatures at the K\nand -K valleys will be exchanged, but their sign remains\nunchanged. Under such condition, the anomalous valley\nHall effect can be observed in monolayer FeClF. When\nshifting the Fermi level between the K and -K valleys,\nthe spin-down electrons at K valley will accumulate on\nthe one side of the sample[ Figure 11 (a)]. By reversing\nthe magnetization direction, the spin-up electrons at -K\nvalley will gain opposite transverse velocities, moving to-\nwards another side [ Figure 11 (b)] of the sample due to\nits opposite Berry curvature.\nVII. CURIE TEMPERATURE\nWe also estimate the Curie temperature TCof mono-\nlayerFeClFatrepresentativeUvaluesbyMCsimulations\nwith the Wolf algorithm based on the Heisenberg model.\nThe effective classical spin model can be written as:\nH=−J/summationdisplay\ni,jSi·Sj−A/summationdisplay\ni(Sz\ni)2(9)\nwhereSi/Sj,Sz\ni,JandAare the spin vectors of each\nFe atom, the spin component parallel to the z direction,8\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48 /s53/s48/s48 /s54/s48/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s78/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32/s83\n/s84/s32/s40/s75/s41/s40/s97/s41\n/s48/s46/s48/s48/s46/s54/s49/s46/s50/s49/s46/s56/s50/s46/s52/s51/s46/s48/s51/s46/s54\n/s32/s51/s49/s49/s32/s75\n/s65/s117/s116/s111/s45/s67/s111/s114/s114/s101/s108/s97/s116/s105/s111/s110/s32/s40/s49/s48/s45/s50\n/s41\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s78/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32/s83\n/s84/s32/s40/s75/s41/s40/s98/s41\n/s48/s46/s48/s48/s46/s54/s49/s46/s50/s49/s46/s56/s50/s46/s52/s51/s46/s48/s51/s46/s54\n/s32/s54/s51/s32/s75\n/s65/s117/s116/s111/s45/s67/s111/s114/s114/s101/s108/s97/s116/s105/s111/s110/s32/s40/s49/s48/s45/s50\n/s41\nFIG. 12. (Color online)For Janus monolayer FeClF, the norma lized magnetic moment (S) and auto-correlation as a functio n\nof temperature with U=1.5 eV (Left) and 2.5 eV (Right).\nthe nearest neighbor exchange parameter and MAE, re-\nspectively. To extract J, we compare energies of the FM\n(EFM) and AFM ( EAFM) configurations of monolayer\nFeClF with rectangle supercell. The spin vector is nor-\nmalized ( |S|= 1), and the corresponding energies are\ngiven by:\nEFM=E0−6J−2A (10)\nEAFM=E0+2J−2A (11)\nwhereE0is the energy without magnetic coupling. Ac-\ncording to these equations, the Jcan be attained as:\nJ=EAFM−EFM\n8(12)\nThe calculated normalized JatU= 1.5 eV and 2.5 eV\nare 25.30 meV and 4.40 meV. The normalized magnetic\nmomentandauto-correlationofmonolayerFeClFvstem-\nperature are shown in Figure 12 , and the predicted TCis\nabout 311 K and 63 K. These mean that electron correla-\ntioncanproduceimportanteffects onCurietemperature,\nwhich is also can be understood by energy differences be-\ntween FM and AFM ordering vs U(seeFigure 2). The\nCurie temperature of FeClBr is predicted to be very high\n(651 K)23, which is due to adopt GGA method ( U=0\neV).\nVIII. DISCUSSION AND CONCLUSION\nThe importance of electron correlations on the elec-\ntronic properties of monolayer FeClF has been demon-\nstrated by the first-principles calculations. The different\ncorrelation strength can give rise to different electronic\nstates, like FV, HVM and QAH states. Although the\nmonolayer FeClF as a concrete example is investigated,the analysis and results in the work can be readily ex-\ntended to other members of monolayer FeXY (X/Y=F,\nCl, Br and I), including ten monolayers. This is because\nthe low-energy states and the electronic correlations are\ndominated by Fe atoms, and the rich correlation-driven\nphysics should be common amongthe FeXY family. Sim-\nilar phase diagram of monolayer FeCl 2with varied U\nhas been investigated24, where the out-of-plane magne-\ntocrystalline direction is always assumed. The sponta-\nneous valley polarization of monolayer FeClBr has also\nbeen studied by using GGA, and the MAE is 14 µeV/Fe\nwith out-of-plane direction23, which is lower than one\n(132µeV/Fe) ofFeClF within GGA. Thepredicted Curie\ntemperature of FeClBr monolayer is very high (651 K)23,\nwhich should be reduced with increasing Uaccording\nto our calculated results. The correlation-driven topo-\nlogical and valley states in septuple atomic monolayer\nVSi2P4have been revealed42, whose intermediate three\nlayers (VP 2) share the similar structure with FeXY fam-\nily. For a given material, the correlation strength should\nbe constant, which should be determined from experi-\nment. However, the rich electronic states can be real-\nized by tuning correlation effect, which can be achieved\nby applied strain. The electronic correlation depends on\nthe competition between kinetic and interactionenergies,\nand the strain can modify the bandwidth, adjust correla-\ntion effect42. The rich phase diagram in septuple atomic\nmonolayer VSi 2N4has been achieved by strain43, and\nsign-reversible valley-dependent Berry phase effects and\nQAH states have been investigated.\nIn summary, we have demonstrated the significance of\nelectron correlation in Janus monolayer FeClF, as a rep-\nresentative of the 2D FeXY material family. It is found\nthat different correlation strength (varied U) can result\nindifferentelectronicstateduetointerplaybetweenmag-\nnetic, correlation and SOC. The multiple transitions are\nobserved,includingthemagneticanisotropy,valleystruc-\nture, electroconductibility and electronic topology. For9\nPMA, there exists a QAH phase, whose boundary corre-\nsponds to the HVM state with fully valley polarized car-\nriers. For FV phases, the polarization can exist for both\nelectrons (conduction bands) and holes (valence bands),\nwhich depends the correlation strengths. The polariza-\ntion at -K and K valleys can be switched by reversing\nthe magnetization. The increasing Ucan induce sign-\nreversible Berry curvature for both out-of-plane and in-\nplane magnetic anisotropy. The magnetic interaction re-\nlatedwith Curietemperaturecanbe distinctly influenced\nby correlation strength. Our works provide a compre-\nhensive understanding of the correlation effects in the\n2D FeXY family, which can be spread to other 2D FVmaterials.\nACKNOWLEDGMENTS\nThis work is supported by Natural Science Basis Re-\nsearch Plan in Shaanxi Province of China (2021JM-\n456). We are grateful to the Advanced Analysis and\nComputation Center of China University of Mining and\nTechnology (CUMT) for the award of CPU hours and\nWIEN2k/VASP software to accomplish this work.\n1J. R. Schaibley, H. Yu, G. Clark, P. Rivera, J. S. Ross, K.\nL. Seyler, W. Yao, and X. Xu, Nat. Rev. Mater. 1, 16055\n(2016).\n2D. Xiao, G. B. Liu, W. Feng, X. Xu, and W. Yao, Phys.\nRev. Lett. 108, 196802 (2012).\n3D. Xiao, W. Yao, and Q. Niu, Phys. Rev. Lett. 99, 236809\n(2007).\n4G. Pacchioni, Nat. Rev. Mater. 5, 480 (2020).\n5W. Yao, D. Xiao, and Q. Niu, Phys. Rev. B 77, 235406\n(2008).\n6S. A. Vitale, D. Nezich, J. O. Varghese, P. Kim, N. Gedik,\nP. Jarillo-Herrero, D. Xiao, and M. Rothschild, Small 14,\n1801483 (2018).\n7C. Zhao, T. Norden, P. Zhang, P. Zhao, Y. Cheng, F. Sun,\nJ. P. Parry, P. Taheri, J. Wang, Y. Yang, T. Scrace, K.\nKang, S. Yang, G. Miao, R. Sabirianov, G. Kioseoglou, W.\nHuang, A. Petrou and H. Zeng, Nat. Nanotechnol. 12, 757\n(2017).\n88 H. Zeng, J. Dai, W. Yao, D. Xiao and X. Cui, Nat.\nNanotechnol. 7, 490 (2012).\n9D. MacNeill, C. Heikes, K. F. Mak, Z. Anderson, A.\nKorm´anyos, V. Z´ olyomi, J. Park and D. C. Ralph, Phys.\nRev. Lett. 114, 037401 (2015).\n10D. Xiao, M. C. Chang, and Q. Niu, Rev. Mod. Phys. 82,\n1959 (2010).\n11M. Zeng, Y. Xiao, J. Liu, K. Yang and L. Fu, Chem. Rev.\n118, 6236 (2018).\n12W. Y. Tong, S. J. Gong, X. Wan, and C. G. Duan, Nat.\nCommun. 7, 13612 (2016).\n13Y. Zang, Y. Ma, R. Peng, H. Wang, B. Huang, and Y. Dai,\nNano Res. 14, 834 (2021).\n14P. Zhao, Y. Ma, C. Lei, H. Wang, B. Huang, and Y. Dai,\nAppl. Phys. Lett. 115, 261605 (2019).\n15W. Du, Y. Ma, R. Peng, H. Wang, B. Huang, and Y. Dai,\nJ. Mater. Chem. C 8, 13220 (2020).\n16Z. Song, X. Sun, J. Zheng, F. Pan, Y. Hou, M. H. Yung,\nJ. Yang, and J. Lu, Nanoscale 10, 13986 (2018).\n17X. Y. Feng, X. L. Xu, Z. L. He, R. Peng, Y. Dai, B. B.\nHuang and Y. D. Ma, Phys. Rev. B 104, 075421 (2021).\n18Y. B. Liu, T. Zhang, K. Y. Dou, W. H. Du, R. Peng, Y.\nDai, B. B. Huang, and Y. D. Ma, J. Phys. Chem. Lett. 12,\n8341 (2021).\n19R. Peng, Y. Ma, X. Xu, Z. He, B. Huang, and Y. Dai,\nPhys. Rev. B 102, 035412 (2020).\n20J. Zhou, Y. P. Feng, and L. Shen, Phys. Rev. B 102,180407(R) (2020).\n21X. R. Kong, L. Y. Li, L. B. Liang, F. M. Peeters and X.\nJ. Liu, Appl. Phys. Lett. 116, 192404 (2020).\n22P. Zhao, Y. Dai, H. Wang, B. B. Huang and Y. D. Ma,\nChemPhysMater, In press (2021).\n23R. Li, J. W. Jiang, W. B. Mi and H. L. Bai, Nanoscale 13,\n14807 (2021).\n24H. Hu, W. Y. Tong, Y. H. Shen, X. Wan, and C. G. Duan,\nnpj Comput. Mater. 6, 1 (2020).\n25P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964);\nW. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).\n26G. Kresse, J. Non-Cryst. Solids 193, 222 (1995).\n27G. Kresse and J. Furthm¨ uller, Comput. Mater. Sci. 6, 15\n(1996).\n28G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).\n29J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett.\n77, 3865 (1996).\n30X. Wu, D. Vanderbilt and D. R. Hamann, Phys. Rev. B\n72, 035105 (2005).\n31A. Togo, F. Oba, and I. Tanaka, Phys. Rev. B 78, 134106\n(2008).\n32L. Liu, X. Ren, J. H. Xie, B. Cheng, W. K. Liu, T. Y. An,\nH. W. Qin and J. F. Hu, Appl. Surf. Sci. 480, 300 (2019).\n33A. A. Mostofia, J. R. Yatesb, G. Pizzif, Y.-S. Lee, I.\nSouzad, D. Vanderbilte and N. Marzarif, Comput. Phys.\nCommun. 185, 2309 (2014).\n34Q. Wu, S. Zhang, H. F. Song, M. Troyer and A. A.\nSoluyanov, Comput. Phys. Commun. 224, 405 (2018).\n35T. Fukui, Y. Hatsugai and H. Suzuki, J. Phys. Soc. Japan.\n74, 1674 (2005).\n36A. Y. Lu, H. Zhu, J. Xiao, C. P. Chuu, Y. Han, M. H.\nChiu, C. C. Cheng, C. W. Yang, K. H. Wei, Y. Yang, Y.\nWang, D. Sokaras, D. Nordlund, P. Yang, D. A. Muller,\nM. Y. Chou, X. Zhang and L. J. Li, Nat. Nanotechnol. 12,\n744 (2017).\n37J. Zhang, S. Jia, I. Kholmanov, L. Dong, D. Er, W. Chen,\nH. Guo, Z. Jin, V. B. Shenoy, L. Shi and J. Lou, ACS Nano\n11, 8192 (2017).\n38E. Cadelano and L. Colombo, Phys. Rev. B 85, 245434\n(2012).\n39X. Liu, H. C. Hsu and C. X. Liu, Phys. Rev. Lett. 111,\n086802 (2013).\n40Q. L. Sun, Y. D. Ma and N. Kioussis, Mater. Horiz. 7,\n2071 (2020).\n41J. Heyd, G. E. Scuseria and M. Ernzerhof, J. Chem. Phys.10\n118, 8207 (2003).\n42S. Li, Q. Q. Wang, C. M. Zhang, P. Guo and S. A. Yang,\nPhys. Rev. B 104, 085149 (2021).43X. Zhou, R. Zhang, Z. Zhang, W. Feng, Y. Mokrousov and\nY. Yao, npj Comput. Mater. 7, 160 (2021)."    },    {        "title": "2111.08556v1.Nature_of_Interfacial_Dzyaloshinskii_Moriya_Interactions_in_Graphene_Co_Pt_111__Multilayer_Heterostructures.pdf",        "content": "Nature of Interfacial Dzyaloshinskii-Moriya Interactions in Graphene/Co/Pt(111)\nMultilayer Heterostructures\nM. Blanco-Rey,1, 2G. Bihlmayer,3A. Arnau,4, 1, 2and J.I. Cerd´ a5\n1Departamento de Pol´ ımeros y Materiales Avanzados: F´ ısica, Qu´ ımica y Tecnolog´ ıa,\nFacultad de Qu´ ımica UPV/EHU, Apartado 1072, 20080 Donostia-San Sebasti´ an, Spain\n2Donostia International Physics Center, Paseo Manuel de Lardiz´ abal 4, 20018 Donostia-San Sebasti´ an, Spain\n3Peter Gr¨ unberg Institut and Institute for Advanced Simulation, Forschungszentrum J¨ ulich and JARA, 52425 J¨ ulich, Germany\n4Centro de F´ ısica de Materiales CFM/MPC (CSIC-UPV/EHU),\nPaseo Manuel de Lardiz´ abal 5, 20018 Donostia-San Sebasti´ an, Spain\n5Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, 28049 Madrid, Spain\n(Dated: November 17, 2021)\nDFT calculations within the generalized Bloch theorem approach show that interfacial\nDzyaloshinskii-Moriya interactions (DMI) at both interfaces of Graphene/Co n/Pt(111) multilayer\nheterostructures are decoupled for n≥3. Unlike the property of magnetocrystalline anisotropy\nfor this system, DMI is not affected by stacking defects in the Co layer. The effect of Graphene\n(Gr) is to invert the chirality of the vaccum/Co interfacial DMI, overall reducing the DMI of the\nheterostructure, which is nevertheless dominated by the strong spin-orbit coupling (SOC) of Pt. A\nspectral analysis in the reciprocal space shows that DMI at both the Gr/Co and Co/Pt interfaces\nhave the same nature, namely SOC-split hybrid bands of d-orbital character. This proves that a\nDMI model based on a single band, such the Rashba DMI model, is insuficient to describe the\nbehaviour of this family of Gr-capped 3 d/5dmetal heterostructures.\nI. INTRODUCTION\nWhen the magnetic exchange interactions are subject\nto sufficiently intense spin-orbit coupling (SOC) in an en-\nvironment with broken inversion symmetry, an antisym-\nmetric term appears that leads to canted and chiral orien-\ntation of spins, known as Dzyaloshinskii-Moriya interac-\ntion (DMI). Firstly observed for oxides [1] and modelled\nas orbital-magnetic-moment-dependent terms added to\nthe Anderson hamiltonian [2, 3], it was later generalized\nto metallic alloys with diluted magnetic impurities (Fert-\nLevy model) as a SOC correction of the RKKY exchange\n[4, 5].\nBecause of the broken inversion symmetry require-\nment, DMI is usually active at surfaces and interfaces,\nwhere it triggers complex chiral ordered spin structures\nat the nanoscale, such as cycloidal textures [6, 7] and\nskyrmions [8–10], and introduces asymmetry in the dis-\nplacement of domain walls [11]. The latter property has\nbeen exploited in synthetic magnets [12, 13].\nImproving the efficiency of the chirality, which is lost\nif symmetric exchange interactions dominate, has been\nidentified as a near-future challenge in the field of mag-\nnetic materials [14]. Asymmetric multilayering is used to\nenhance DMI, since interfaces contribute additively [15–\n17]. For example, in Ir/Co/Pt- [15] and Pt/Co/Ta-based\nheterostructures [18] skyrmions of diameter ∼100 nm\nhave been stabilized at room temperature. Contact be-\ntween a ferromagnet and a heavy non-magnetic phase\nis the obvious way to promote DMI, as hybridization\nwith the spin-orbit split 5 dbands facilitates the needed\nspin-flip electron excitations, hence Co/Pt has become\na paradigmatic system [19–21]. Alternatively, DMI\nstrength and handedness can also be manipulated by ad-\nsorption of light element atoms, , hydrogen [22, 23] andoxygen [24, 25], or capping with graphene (Gr) [23, 26]\nand hexagonal boron nitride [27]. This DMI variation\nupon adsorption stems from the charge density redistri-\nbution at the surface and it is correlated with the elec-\ntric dipole at the surfaces [24], where the correlation is\nendorsed by an analytical expression [28]. The manifes-\ntation of DMI through other more accessible properties\nof the system, such as electrostatic ones, has motivated\nthe search of DMI descriptors that allow its predictabil-\nity in systems of potential interest. For example, at a\n3d/5dinterface, it correlates with the spin moment mS\nof the 3datoms but not with the mSinduced at the in-\nterfacial 5datoms [19–21, 28, 29], neither with the spin\ndipole nor with the orbital magnetic moments [28]. Di-\nrect modification of DMI merely based in electric field\nmanipulation, by a STM tip, is questionable, as it may\nbe mistaken with a variation of the magnetic exchange\ncoupling strength [30].\nIn an attempt to classify DMI into types, it has been\nargued that Co/Pt and Gr/Co interfacial DMI are of\ndifferent nature, namely Fert-Levy-like and Rashba-like,\nrespectively [31]. In the former interface, the DMI en-\nergy contribution localizes at Pt, which has of strong\natomic SOC strength ξ. In the latter, Gr/Co, the afore-\nmentioned induced electrostatic potential change ∇Vis\nmodelled by a one-band Rashba hamiltonian in the pres-\nence of a Heisenberg exchange term [32–34]. These mod-\nels can be considered as limits of the strongly hybridized\nSOC-splitd-bands behavior.\nMotivated by recent experimental work on multilayer\nGr/Con/Pt(111) heterostructures [26, 35] and by the ex-\nisting density-functional theory (DFT) work on the indi-\nvidual interfaces of them, in this paper we show that that\nDMI does not adjust satisfactorily to either limit model.\nOur DFT calculations reveal that the Pt SOC dominatesarXiv:2111.08556v1  [cond-mat.mtrl-sci]  16 Nov 20212\noverall and that there are sizable contributions from Co\nSOC locally, although cancellations occur for Co at both\ninterfaces. The DMI chirality induced by the graphene\ncapping is opposite to and of the same order of magni-\ntude as that of the Co/Pt interface, in agreement with\nthe observation of Ref. [26]. We draw the important con-\nclusion that this result cannot be simply attributed to\nspecific states in the reciprocal space, i.e., a single-band\nRashba model cannot account for it.\nDuring pseudomorphic growth by intercalation in\nGr/Pt, Co attains a face tetragonal fccdistorted (fct)\nstacking. Therefore, the central regions of the slab are\nlocally centrosymmetric and DMI is solely an interfacial\neffect. In the present work we show that this regime is\nreached at Co 3thickness and that the DMI contributions\nof the stacking defects cancel out. Interestingly, this DMI\nbehaviour contrasts with that of the magnetocrystalline\nanisotropy of fctGr/Con/Pt(111), where a complex be-\nhaviour with Co nthickness is found that depends on the\ncompetition between the contributions from the fctbulk\n(in-plane) and twin boundary defects (out-of-plane) [35].\nThe paper is organized as follows: in Section II we\ndescribe the model structures of Gr/Co n/Pt used in\nthis work and the details of the DFT calculations based\non the generalized Bloch theorem with SOC. The re-\nsults and discussion section is split into a collection\nof thickness-dependent and layer-resolved DMI energies\n(section III A), as well as an analysis in the reciprocal\nspace (section III B), where the DMI contributions are\nenergy- and wavevector-resolved. Finally, conclusions are\ndrawn.\nII. THEORETICAL METHODS\nThe pseudovectors Dijcharacterize the DMI between\ntwo localized spin moments Si,Sj, which is expressed as\nthe hamiltonian term/summationtext\n/angbracketleftij/angbracketrightDij·Si×Sj. Fig. 1 shows\ntheD-vectors between the Co atom at the origin unit\ncell and its six closest Co atoms in the case of a mono-\nlayer. The 3-fold axes and mirror planes in the structure\nconstrain these D-vectors to be determined by two free\nparameters DyandDz[36], as indicated in the sketch.\nNote that the relative positions of the nearest Pt atom\nwith respect to each pair of Co neighbours alternates\nfrom right to left in the 3-fold symmetry and, thus, the\nD-vector out-of-plane component Dzalternates in sign.\nAs this work is restricted to relatively small angles be-\ntween spins, we will use an effective model of the energy\nthat maps all Co-Co interactions in the model slabs onto\na two-dimensional hamiltonian that depends on effective\nDyandDzparameters (see Fig. 1 and Supplemental Ma-\nterial Fig. ??).\nDFT calculations in the full-potential linearized aug-\nmented plane waves (FLAPW) formalism [37, 38] have\nbeen carried out with the code FLEUR [39]. PBE is\nthe chosen exchange and correlation functional [40] for\nthis work. We have used Co layers of one to five atom\nC\"Co\"\n-15-10-5 0 5 10 15\n-15-10-5 0 5 10 15y (Ang.)\nx (Ang.) N=12 \n-15-10-5 0 5 10 15\n-15-10-5 0 5 10 15y (Ang.)\nx (Ang.) N=32 \n-15-10-5 0 5 10 15\n-15-10-5 0 5 10 15y (Ang.)\nx (Ang.) N=8 N=32%N=12%N=8%(a)%(b)%\n(c)%(d)%(e)%M%K%q%ΓK’%z%x%y%FIG. 1. (a) Top view of a graphene-covered fctCo slab. For\nthe topmost Co plane, the central atom D-vectors of the DMI\nwith its in-plane nearest neighbours (red dashed segments) are\nindicated by thick arrows with 3-fold symmetry. The colour\ngradient from green to white depicts the Dzcomponent sign.\n(b) The spin spiral wavevectors used in this work belong to\nthe Γ Kdirection of the Brillouin zone, i.e. q=2π\na(1\nN,1\nN,0).\n(c-e) Examples of the spin orientations in a Co plane for three\nNvalues.\nthicknesses, pseudomorphic on a Pt substrate (lateral\nperiodicity 2.772 ˚A) of five atomic planes, with the re-\nlaxed interplanar spacings found in Ref. [35]. The ef-\nfect of the substrate thickness on the DMI is shown in\nthe Supplemental Material Fig. ??. For the basis set, a\n48×48×1 Monkhorst-Pack-point mesh [41] is used and\nplane wave expansion cut-offs of 4, 11 and 9.5 a.u. for the\nwavefunctions, density and potential, respectively. The\nlocal basis was constructed without local orbitals, with\nlmax= 6,8,10 for C, Co and Pt, respectively. The smear-\ning energy for the Fermi level determination was 0.03 eV.\nSuitable non-collinear spin structures are needed for\nthe model to show DMI. They are modelled as long-\nwavelength spin spirals in the generalized Bloch theorem\n(GBT) approach [42], which imposes a longer periodicity\nof the magnetization density described by a wavevector\nq. Although, in principle, the charge density can be self-\nconsistently converged for a given q, since the calcula-\ntions shown here involve long-wavelength spirals, we use\nthe GBT non-self-consistently to calculate new energies\nand electron wavefunctions as a perturbation of a q= 0\nground state (see Supplemental Material Fig. ??). Our\ncalculations on the minimal model system, a Gr/Co 1/Pt1\ntrilayer, show that this is a fair approximation in the low-\nqregime (see Supplemental Material Fig. ??). Finally,\nspin-orbit interactions are added also as a first-order per-\nturbation. The implementation of this procedure in the\nFLAPW code is described in Ref. [43].\nWe use flat spin spirals with a spin rotation axis ˆ sa,\nwhich yields a different energy in the presence of SOC.\nThe energy difference between axes pointing in opposite3\ndirections ˆs+\na(ˆs−\na) is the DMI energy for a given spiral\nwithqperiodicity. Our convention is that the spirals\nare anticlockwise (ACW) with respect to ˆ sa, so that\nˆs+\na(ˆs−\na) mean ACW (CW) or left-handed (right-handed)\nspirals. Therefore, if the energy difference ∆ Eˆsa\nDMI(q) =\nEDMI(q; ˆs+\na)−EDMI(q; ˆs−\na) is positive (negative), a CW\n(ACW) spiral is favoured. The FLAPW calculations of\nthis work are run for spirals with qvectors along the\nΓKdirection of the two-dimensional first Brillouin zone\nq=2π\na(1\nN,1\nN,0), direction in the Brillouin zone. As an\nexample, Fig. 1 shows the spins of a Co atomic plane in\nthe cases of N= 32,12,8 and ˆsa=Z. We usedNvalues\nbetween 8 and 48 (the latter is at the resolution limit\nmarked by our electron momentum kcalculation grid,\n48×48×1).\nIII. RESULTS AND DISCUSSION\nA. Additivity of interfacial DMI\nAt lowqvalues, the DMI energies for spins rotating in\nthe plane perpendicular to YandZaxes follow a nearly\nlinear inqand aq3dependence, respectively, as we de-\nscribe next. These trends are observed in Fig. 2 (red sym-\nbols and lines), which show ∆ Eˆsa\nDMI(q) for Gr/Co n/Pt5,\nn= 1−5, slabs in the low- qregime. Nevertheless, for\nthe in-plane spins case the energies are too low to ex-\ntract accurate quantitative results from a fit (note that\nthe energies are one order of magnitude smaller in this\nspin geometry). The evolution of ∆ EY\nDMI(q) with the Co\nlayer thickness (red line) shows a significant magnitude\nvariation, but no chirality change (i.e., no sign change).\nTheDyeffective parameter is extracted from the fit to a\n2D model with nearest neighbour interactions in a hexag-\nonal lattice with C3vsymmetry, given by the expression\n∆Eˆsa\nDMI(q) =\n4S2sinθˆsa·/bracketleftBig\n0,Dy(1 + 2 cosθ),2Dz(cosθ−1)/bracketrightBig\n(1)\nfor flat spin spirals q=2π\na(1\nN,1\nN,0), whereθ=2π\nN\nare the corresponding angles between spins S(assumed\nto have equal values). This expression has a linear be-\nhaviour in the low- qlimit for spins rotating in the XZ\nplane,\n∆EY\nDMI(q)≈12S2Dy2π\nN(2)\nWe use this equation to fit the DFT energies, so that\nthe resulting Dyvalues are to be interpreted as effective ,\nsince other contributions not considered in Eq. 1 also\nyield linear terms in qfor ˆsa=Y, such as inter-planar\nnearest neighbour bonds [36], second nearest and beyond\nneighbours.\nTo obtain Dy, we exclude the two highest qpoints\nof each Fig. 2 curve and use as spin moment value the\naverage over the Co atomic layers. In the central Co\n-5 0 5 10 15\n(a)∆ EDMI [meV]Y\n-5 0 5 10 15\n(b)∆ EDMI [meV]\n-5 0 5 10 15\n(c)∆ EDMI [meV]\n-5 0 5 10 15\n(d)∆ EDMI [meV]\n-5 0 5 10 15\n 0.0  0.1  0.2  0.3  0.4(e)∆ EDMI [meV]\nq [ΓK]-10123\nnCo = 1(f)Z\n-10123\nnCo = 2(g)\n-10123\nnCo = 3(h)\n-10123\nnCo = 3(i)\n-10123\n 0.0  0.1  0.2  0.3  0.4\nnCo = 5(j)\nq [ΓK]FIG. 2. DMI energy ∆ Eˆsa\nDMI(q) for spin spirals rotating in\nthe plane perpendicular to Y(panel column a-e) and Z(f-j)\naxes and Co layer thicknesses (panel rows) in three different\nheterostructures: Gr/Co n/Pt5(red), Co n/Pt5(green) and\nGr/Con(blue) with n= 1−5. In black, the contribution sum\nfrom Con/Pt5+Gr/Con. The spin spiral wavevector modulus\nqis given in fractional units of the distance Γ Kin the Brillouin\nzone (see Fig. 1b).\natomic planes we find mS(Co)/similarequal1.86µB. This value is\nslightly enhanced at the vacuum/Co and Co/Pt inter-\nfaces (1.94µBfornCo= 5), while graphene has a de-\nmagnetizing effect (1.57 µB) (see also the Supplemental\nMaterial Fig. ??bottom panels). Additionally, at the\ninterfacial Pt atoms a spin polarization of 0.27 µBis in-\nduced. This adds to ∆ EY\nDMI(q) a Co-Pt nearest neigh-\nbour contribution one order of magnitude smaller than\nthat of the Co-Co DMI interactions. The resulting Dy\nvalues, summarized in Fig. 3, have a very good agreement\nwith linear behaviour (the linear fit errors are ≤4%).\nTheDyvalues oscillate for nCo= 1−3 and converge\ntoDy/similarequal0.3 meVµ−2\nBafterwards. The fitted Dyvalue\nfor Co 1Pt5is in agreement with the literature calculated\nwith GBT and flat spin spirals [20, 44], too, which are\nclose to 0.5 meV µ−2\nB. This methodology tends to yield\nlarger energies than other electronic structure methods\n[44].\nFor spins rotating in the interface plane XY, the DMI\nenergy is a third order effect in q,\n∆EZ\nDMI(q)≈−4S2Dz/parenleftBig2π\nN/parenrightBig3\n(3)\nThis means that at this geometry the interfaces sustain4\n-0.2 0 0.2 0.4 0.6 0.8\n 0  1  2  3  4  5Dy [meV/ µB2]\nnGr/Con/Pt\nCon/Pt\nGr/Con\nCon/Pt + Gr/Con\nFIG. 3. In-plane components of the effective Dvector ob-\ntained from fits of the data in the left-hand column of Fig. 2 to\na linear law up to q= 0.25|ΓK|, where the averaged mS(Co)\nvalues over the Co layer have been used.\nan effective D-vector with an out-of-plane component Dz\n(note that a spin spiral with q-vector along Γ Mwould not\nallow to resolve Dz), although the magnitude of this ef-\nfect is small. The second column of Fig. 2 shows that,\nindeed, the energies are an order of magnitude smaller\nthan for spins rotating in the XZ plane. These data sets\ndo not allow for a good quality fit to a q3law, since the\nlargeξ(Pt) value magnify finite size effects. Slabs with a\nsingle Pt layer as substrate, which are shown in the Sup-\nplemental Material Fig. ??, have a smoother behaviour.\nIn them ∆EZ\nDMI(q) changes its sign, i.e., alternating chi-\nrality ofDz, as the Co layer grows beyond the monolayer\nthickness and tends toward small values when interfaces\nare decoupled at nCo= 3. A sizable non-zero Dzcom-\nponent can result in hybrid Bloch-N´ eel domain walls [45]\nand it has been postulated that it is responsible for an\nasymmetric skyrmion Hall effect [46]. In addition, the ef-\nfect of the nearest-neighbour interplanar interaction en-\nergy term for nCo= 2 is clearly distinguished in the\n∆EY\nDMI(q) curve.\nIn the remainder of the paper we focus the analysis\non theDycomponent only. To ascertain whether the\ncrossover at nCo= 3 is correlated with the additivity of\ninterfacial DMI, we have decomposed the heterostructure\nmodel slab into Co 1−5/Pt5and Gr/Co 1−5partial slabs\nwith the same atomic positions (green and blue lines and\nsymbols, respectively, in Fig. 2 and Fig. 3). In the follow-\ning, we call this form of analysis “partial slab decompo-\nsition” (PSD). The sum of the corresponding ∆ EY\nDMI(q)\nand effective Dy(black lines and open symbols) follows\nclosely the Gr/Co 2−5/Pt5values, while there is a large\ndifference for nCo= 1. Although this would suggest that\nthe Co/Pt and Gr/Co interfaces are already decoupled\natnCo= 2, their values still depend on the Co thickness.\nHence, effective decoupling does not occur until at least\nnCo= 3 In particular, for Gr/Co 2/Pt5the maximumvalueDy= 0.5 meVµ−2\nBis obtained.\nThe contribution of the Co/Pt interface accounts for\nmost of the ∆ EY\nDMI(q) energy in the heterostructure.\nImportantly, at the Gr/Co interface the chirality is op-\nposite (except in the GrCo bilayer) and of comparable\nmagnitude to that of Co/Pt. This type of analysis, how-\never, cannot determine if the effect of the Gr capping\nlayer is to invert the chirality of the vacuum/Co interface.\nFor this, we have calculated ∆ EY\nDMI(q) with atomic SOC\ncontributions selected ad hoc , a method we denote in the\nfollowing with the acronym ASOD (atomic spin-orbit de-\ncomposition). These results are shown in Fig. 4. This\nalternative method allows to assess the individual atom\ncontributions to interfacial DMI energy in a given system,\nsince it is additive in the atomic SOC strength ( ξ) by con-\nstruction. With ASOD we find that the contribution of\nthe interfacial Pt atomic plane dominates the whole DMI\neffect, showing similar energies in both Gr/Co 5/Pt5and\nCo5/Pt5slabs, whereas the Co plane in that same in-\nterface has a negligible contribution compared to Co Gr,\nas also reported in the literature for similar 3 d/5dinter-\nfaces [19, 20]. This is explained in part by the strength of\nSOC at Pt, which is one order of magnitude larger than\nat Co (ξCo= 74 meV and ξPt= 537 meV [47]). In the\ncase of a bare Co 5slab, the two outermost atomic planes\ncontribute with opposite chiralities and sizable values,\nnamely|Dy|= 0.175 meVµ−2\nB. Importantly, we find that\nthe Co contribution changes from positive to negative\nDyand it is nearly doubled in magnitude when it is in\ncontact with graphene in the Gr/Co 5slab (the contri-\nbution of graphene itself is negligible due to the small\nξCvalue). Therefore, graphene has the effect of reduc-\ning the net DMI of Gr/Co n(fct)/Pt heterostructures by\ninducing in the topmost Co plane an opposite chirality\nto that of the Co/Pt interface. This has been observed\nby MOKE microscopy in domain wall propagation ex-\nperiments, although the graphene effect on the DMI was\noverestimated there to be about one half of that of the\nCo/Pt interface (the reported spatial micromagnetic av-\nerages are 1.4 and -0.8 mJ/m2for the Co/Pt [12] and\nGr/Co [26] interfaces).\nWe have used two methods for the resolution of the\nDMI energy into its individual interfacial terms. The\nPSD method accounts for joint effect of (i) the SOC\nstrength and (ii) the electronic structure modification at\nthe interface when the constituents are brought together.\nIn the ASOD method only the relativistic effect is being\nprobed . The latter method is less realistic, but more in-\nformative. The importance of effect (ii) is manifested in\nthe DMI tuning by adsorption of light atoms. For ex-\nample, H adsorption on Ni/Co/Pd/W induces chirality\nchange [22], with the advantage that H uptake and des-\norption is a reversible process [22, 23]. DMI changes have\nalso been characterized during oxidation of 3 d/5dlayered\nsystems [24, 25]. This DMI behaviour is associated to a\ncharge density redistribution upon adsorption and, based\non this mechanism, electrostatic properties such as sur-\nface dipoles, work-functions and electronegativity have5\n-2-1 0 1 2\n     ∆EDMIY [meV](a) Co5\ntotal\nCov(top)\nCov(bottom) -3-2-1 0\n     (b) Gr/Co5\ntotal\nCoGrCov(bottom)\n 0 4 8 12\n0.0 0.1 0.2 0.3 0.4∆EDMIY [meV]\nq [ΓK](c) Co5/Pt5total\nPtCo CovCoPt\n 0 4 8 12\n0.0 0.1 0.2 0.3 0.4\nq [ΓK](d) Gr/Co5/Pt5total\nPtCoCoGrCoPt\nFIG. 4. Contributions to the DMI energy ∆ EY\nDMI(q) (spins\nrotating in the plane perpendicular to Yaxis and spiral\nwavevector along the Γ Kreciprocal direction) from the in-\ndividual interfacial atomic planes in Gr/Co 5/Pt5slab and its\nconstituents. The labels indicate the contributing atom and\nthe subindices indicate the interface the atom belongs to, Co v\nis the Co at the vaccum interface.\nbeen proposed as DMI descriptors [24, 28].\nTable I shows the perpendicular electric dipole pzof\nthe interfacial C, Co and Pt atoms of the three partial\nslabs in the limit cases nCo= 1 and 5. It is evaluated\naspz=−|e|/angbracketleftz/angbracketrightMT, where the average position is evalu-\nated as an integral over the charge density distribution\ninside the muffin-tin. For the interfaces at the top of\nthe slab,pz>0 (pz<0) means that the dipole points\noutward (inward), and viceversa for the interfaces at the\nslab bottom. For nCo= 1,pz(Co) depends strongly on\nthe interface type: the dipole points inward when coated\nwith graphene and outward when not. At the top of\nthenCo= 5 slab a sign inversion of pzdue to graphene\nis observed, too, alongside a reduction of the pzmag-\nnitude from 0.197 to 0.043 a.u. (a factor 4.6). At the\nburied Co/Pt interface, Pt has also the effect of reduc-\ning the dipole of the interfacial Co Ptatom by a factor\n3.25 with respect to Co v, but no sign inversion occurs.\nAs reported in Ref. [28], the pzare correlated with DMI\nenergies. In the present case of the slabs with nCo= 5,\nthere is agreement between the signs of the pzof interfa-\ncial Co atoms (Table I) and the signs of the contributions\nof these atoms to ∆ EY\nDMI (Fig. 4). However, there is no\nproportionality between the two magnitudes Note that\ndipoles and electronegativity are related to the D-vector\nby a non-linear analytical expression, which results in a\nlinear correlation between these properties for different\nadsorbed species [28].\nWe investigate next the influence on the DMI of the Co\nstacking, known to be a key factor to explain the mag-\nnetocrystalline anisotropy of Gr/Co/Pt heterostructures.\nWhen Co is pseudomorphically grown by intercalation\nin Gr/Pt(111), it results in a fctstacking rather than\nhcp[26, 48], with stacking defects scattered through-TABLE I. Electric dipole (atomic units) in the direction\nperpendicular to the interfaces inside the FLAPW muffin-\ntin spheres ( pz) for the interfacial atoms in the Gr/Co n/Pt5,\nCon/Pt5and Gr/Co nslabs with n= 1,5. The subindex in\nthe first column indicates the neighbouring atomic layer in\nthe interface. At graphene, the pzof the two sublattice C\natoms is averaged.\natom Gr/Co 1/Pt5 Co1/Pt5 Gr/Co 1\nC 0.014 0.014\nCo -0.111 0.153 -0.267\nPtCo -0.092 -0.088\natom Gr/Co 5/Pt5 Co5/Pt5 Gr/Co 5\nC 0.013 0.013\nCoGr(Cov) -0.043 (0.197) -0.043\nCoPt(Cov) -0.059 -0.060 (-0.195)\nPtCo -0.077 -0.077\nout, nucleating predominantly near the Pt(111) substrate\nsteps [35]. In a perfect fctheterostructure, DFT calcu-\nlations show that the magnetocrystalline anisotropy en-\nergy (MAE) follows a bulk-like behaviour with in-plane\nanisotropy starting at nCo= 8 (0.09 meV per Co atom,\nimposed by the lateral fctlattice strain) and the criti-\ncal thickness for perpendicular to in-plane switching at\nnc\nCo= 4. However, the experimental switch occurs later,\natnc\nCo/similarequal20 (about 4 nm), due to defects. The theory\nshows that each twin boundary contributes with nearly\n1 meV to out-of-plane anisotropy, as it introduces locally\nahcpstacked structure [35].\nFig. 5 shows atom-resolved ∆ EY\nDMI(q) for\nGr/Co 5/Pt5with Co in three different stackings\n(panels a-c): perfect hcp,fctstacking with a twin\nboundary defect at the middle Co plane, labelled tb3 in\nthe following, and perfect fct. As in the original fct\nheterostructure, with the hcpandtb3 stackings the large\nξPtdominates the net DMI, contributing similar energies\n(black symbols) In the case of a perfect fctgrowth,\nthe central region of the Co layer is centrosymmetric,\nand therefore will not contribute to DMI. Nevertheless,\nin this finite-thickness model, we observe that the\nCo atomic plane contributions oscillate around zero.\nOscillating values occur for the tb3 and thehcpslabs,\nwith larger energies in the latter. Note that for bare\nCo5slabs with the same geometries, i.e., in the absence\nof interfaces with Gr and Pt, cancellations at the Co\nplanes are almost total irrespective of the stacking\ntype (red symbols in Fig. 5(d)). There is an overall\nnegative contribution of the Co planes to the DMI\n(black symbols in Fig. 5(d)) that has its main origin\nat the Gr/Co interface. Sizable DMI occurs at the\nGr/Co and Co/Pt interfaces, while buried interfaces in\nCo that break inversion symmetry locally contribute\nalmost negligibly. This behaviour contrasts with that\nof the magnetic anisotropy, despite both properties\nsharing a common origin in the SOC, with MAE being6\nofξ2order at this symmetry and DMI being linear in\nξ. Therefore, in Gr/Co/Pt heterostructure DMI and\nmagnetocrystalline anisotropy will compete only close\nto the critical thicknesses, when the fctbulk limit and\nthe defect contributions compensate each other to give\na low MAE.\nB. Reciprocal space analysis\nSo far, in the literature, interfacial DMI has been dis-\ncussed in terms of two different mechanisms: a Rashba-\nlike behaviour in Gr/Co triggered by the surface potential\nchange∇Vinduced by graphene adsorption [31] or, al-\ntenartively, a Fert-Levy-like behaviour in Co/Pt, where\nPt SOC mediates the spin-flip of the Co itinerant elec-\ntrons. The aim of this section is to identify the nature\nof DMI at those interfaces based on information gath-\nered at the reciprocal space. The explicit dependence\nof DMI on each Bloch eigenstate /epsilon1nk(q) is too intricate\nto be analyzed by bare eye, due to the high density of\ndispersive bands in the Gr/Co n/Pt5models (see Supple-\nmental Material Fig. ??). Note that each band is subject\nto lateral shifts due to the spin spirals and to degeneracy\nliftings, mainly at crossings between bands, due to SOC\n(see Supplemental Material Fig. ??), as shown by San-\ndratskii for the CoPt bilayer [21]. Instead, in our analysis\nwe use quantities integrated in energies and in electron\nwavevectors k.\nTo analyze the spectral behaviour of the DMI chiral-\nity, we plot the corresponding energies integrated in k\nas a function of the number of electrons nefor each spin\nspiral vector q. This is similar to the MAEs in the force\ntheorem approach [47, 49]:\n∆EY\nDMI(ne;q) =/summationdisplay\nnk/epsilon1Y+\nnk(q)f/parenleftBig\n/epsilon1Y+\nnk(q)−/epsilon1Y+\nF(ne;q)/parenrightBig\n−/epsilon1Y−\nnk(q)f/parenleftBig\n/epsilon1Y−\nnk(q)−/epsilon1Y−\nF(ne;q)/parenrightBig\n(4)\nwhere qis a spin spiral wavevector along Γ K, the sum\nruns over the eigenvalues /epsilon1nkcalculated for opposite spin\nrotation axes ( Y+andY−) and the Fermi levels corre-\nspond to the filling up with neelectrons of the bands\nof each individual calculation with qandY+orY−.\nThe ∆EY\nDMI(ne;q) curves for Gr/Co 5/Pt5, Co 5/Pt5\nand Gr/Co 5slabs are qualitatively invariant with q(see\nFig. 6): the nodes in the curves are almost invariant and\npeaks only increase in amplitude with q, resulting in the\nlinear dependence observed in the previous section. This\nbehaviour occurs also in the less complex CoPt bilayer,\nwhere it has been explained [21] by the hybridization by\nthe spiral of electron states with wavevectors kandk±q,\nwhich have similar character regardless of q, giving rise\nto the dependence of Eq. 1 [50]. As observed in Fig. 6,\nthis scenario is not affected by the presence of the large\nnumber of additional bands of a thicker Pt substrate,\nwhich is the reciprocal space confirmation that the DMI\n(a)$(b)$(c)$\n(d)$C$Co$Pt$\n-2 0 2 4 6 8 10 12\n 0 0.1 0.2 0.3 0.46EDMIY [meV]\nq [KK]hcp\ntotalPtCo(1) (CoGr)Co(2)Co(3)Co(4)Co(5) (CoPt)-2 0 2 4 6 8 10 12\n 0 0.1 0.2 0.3 0.46EDMIY [meV]\nq [KK]tb3\n-2 0 2 4 6 8 10 12\n 0 0.1 0.2 0.3 0.46EDMIY [meV]\nq [KK]fct\n-3-2-1 0\n 0 0.1 0.2 0.3 0.46EDMIY [meV]\nq [KK]fct\nCo @ Gr/Co5/Pt5 - fctCo @ Gr/Co5/Pt5 - hcpCo @ Gr/Co5/Pt5 - tb3Co5 - fctCo5 - hcpCo5 - tb3-2 0 2 4 6 8 10 12\n 0 0.1 0.2 0.3 0.46EDMIY [meV]\nq [KK]hcp\ntotalPtCo(1) (CoGr)Co(2)Co(3)Co(4)Co(5) (CoPt)-2 0 2 4 6 8 10 12\n 0 0.1 0.2 0.3 0.46EDMIY [meV]\nq [KK]tb3\n-2 0 2 4 6 8 10 12\n 0 0.1 0.2 0.3 0.46EDMIY [meV]\nq [KK]fct\n-3-2-1 0\n 0 0.1 0.2 0.3 0.46EDMIY [meV]\nq [KK]fct\nCo @ Gr/Co5/Pt5 - fctCo @ Gr/Co5/Pt5 - hcpCo @ Gr/Co5/Pt5 - tb3Co5 - fctCo5 - hcpCo5 - tb3-2 0 2 4 6 8 10 12\n 0 0.1 0.2 0.3 0.46EDMIY [meV]\nq [KK]hcp\ntotalPtCo(1) (CoGr)Co(2)Co(3)Co(4)Co(5) (CoPt)-2 0 2 4 6 8 10 12\n 0 0.1 0.2 0.3 0.46EDMIY [meV]\nq [KK]tb3\n-2 0 2 4 6 8 10 12\n 0 0.1 0.2 0.3 0.46EDMIY [meV]\nq [KK]fct\n-3-2-1 0\n 0 0.1 0.2 0.3 0.46EDMIY [meV]\nq [KK]fct\nCo @ Gr/Co5/Pt5 - fctCo @ Gr/Co5/Pt5 - hcpCo @ Gr/Co5/Pt5 - tb3Co5 - fctCo5 - hcpCo5 - tb3\n-2 0 2 4 6 8 10 12\n 0 0.1 0.2 0.3 0.46EDMIY [meV]\nq [KK]hcp\ntotalPtCo(1) (CoGr)Co(2)Co(3)Co(4)Co(5) (CoPt)-2 0 2 4 6 8 10 12\n 0 0.1 0.2 0.3 0.46EDMIY [meV]\nq [KK]tb3\n-2 0 2 4 6 8 10 12\n 0 0.1 0.2 0.3 0.46EDMIY [meV]\nq [KK]fct\n-3-2-1 0\n 0 0.1 0.2 0.3 0.46EDMIY [meV]\nq [KK]fct\nCo @ Gr/Co5/Pt5 - fctCo @ Gr/Co5/Pt5 - hcpCo @ Gr/Co5/Pt5 - tb3Co5 - fctCo5 - hcpCo5 - tb3-2 0 2 4 6 8 10 12\n 0 0.1 0.2 0.3 0.46EDMIY [meV]\nq [KK]hcp\ntotalPtCo(1) (CoGr)Co(2)Co(3)Co(4)Co(5) (CoPt)-2 0 2 4 6 8 10 12\n 0 0.1 0.2 0.3 0.46EDMIY [meV]\nq [KK]tb3\n-2 0 2 4 6 8 10 12\n 0 0.1 0.2 0.3 0.46EDMIY [meV]\nq [KK]fct\n-3-2-1 0\n 0 0.1 0.2 0.3 0.46EDMIY [meV]\nq [KK]fct\nCo @ Gr/Co5/Pt5 - fctCo @ Gr/Co5/Pt5 - hcpCo @ Gr/Co5/Pt5 - tb3Co5 - fctCo5 - hcpCo5 - tb3 -2 0 2 4 6 8 10 12\n 0 0.1 0.2 0.3 0.46EDMIY [meV]\nq [KK]hcp\ntotalPtCo(1) (CoGr)Co(2)Co(3)Co(4)Co(5) (CoPt)-2 0 2 4 6 8 10 12\n 0 0.1 0.2 0.3 0.46EDMIY [meV]\nq [KK]tb3\n-2 0 2 4 6 8 10 12\n 0 0.1 0.2 0.3 0.46EDMIY [meV]\nq [KK]fct\n-3-2-1 0\n 0 0.1 0.2 0.3 0.46EDMIY [meV]\nq [KK]fct\nCo @ Gr/Co5/Pt5 - fctCo @ Gr/Co5/Pt5 - hcpCo @ Gr/Co5/Pt5 - tb3Co5 - fctCo5 - hcpCo5 - tb3FIG. 5. (a-c) Top panels: cross section of the Gr/Co 5/Pt5\nslabs with different Co stacking geometries: perfect hcp,fct\nwith a twin boundary introduced in the middle Co plane\n(tb3) and perfect fct. A red line indicates the stacking se-\nquences. Bottom panels: for each geometry, total DMI ener-\ngies (∆ EY\nDMI(q)) for spins rotating in the plane perpendicular\ntoY(black filled circles) and those obtained when SOC is ap-\nplied only to the indicated atomic planes (coloured filled cir-\ncles for Pt layer and empty for individual atomic planes). (d)\nCo layer contributions from Gr/Co 5/Pt5slabs (black sym-\nbols) compared to bare Co 5slabs with the three stacking\ntypes (red symbols).\nis localized within a very short spatial range (bondlength\ndistance) of the interface. At neutrality ( ne= 0) the\nknown positive ∆ EY\nDMI(0;q) values for Gr/Co 5/Pt5and7\n-20-10 0 10 20\n-100 -80 -60 -40 -20  0  20∆EDMIY [meV]\nne(a) Gr/Co5/Pt5\n-20-10 0 10 20\n-100 -80 -60 -40 -20  0  20∆EDMIY [meV]\nne(b) Co5/Pt5\n-4 0 4 8\n-40 -20  0  20q [ΓK]\n∆EDMIY [meV]\nne(c) Gr/Co5\n 0 0.1 0.2 0.3 0.4\n-20-10 0 10 20\n-4 -2  0  2∆EDMIY [meV]\nE-EF [eV] (no SOC)(d)Gr/Co5/Pt5Co5/Pt5Gr/Co5\nFIG. 6. (a-c) DMI energy (∆ EY\nDMI(q)) as a function of the\nband filling for the Gr/Co 5/Pt5, Co 5/Pt5and Gr/Co 5slabs.\nThe color scale bar indicates the qmagnitude. (d) The same\ncurves plotted as a function of energy (referred to the Fermi\nlevels of each slab calculated without SOC) for the particular\nq-point q= 0.25|ΓK|.\nCo5/Pt5, and negative for Gr/Co 5are retrieved. The\nsign of the DMI energy is kept under small variations of\nnearound the neutrality condition. When represented\nas a function of binding energies, ∆ EY\nDMI(ne;q) has the\nlast node before neutrality at the filling corresponding to\nEF−0.5 eV in the three slabs, as shown in the Fig. 6(d)\npanel for q=2π\na(1\n12,1\n12,0). This means that integration\nof eigenenergies in this energy window of 0.5 eV would\neffectively reproduce the net DMI, albeit non-zero DMI\ncontributions occur throughout the whole available en-\nergy spectrum. This is evident with a Pt substrate, since\nthe 5d−3dhybrid bandwidth spans several eV. How-\never, Fig. 6(c,d) shows the same qualitative behaviour\nfor Gr/Co 5, pointing out that there is a similar DMI\nmechanism here due only to the Co SOC contributions.\nNote that the 0.5 eV window coincidence for the three\nslabs is fortuitous, as it depends on the particular details\nof each band structure.\nWith focus on the DMI chirality inversion of the vac-\nuum/Co interface upon capping with graphene, we first\nverify that the ∆ EDMI(0;q) values reproduce the sign\nchange when evaluated with only the corresponding inter-\nfacial individual atomic SOC strength ξCo(see Fig. 7(a)).\nWe now turn to a k-resolved analysis of these quantities.\nFig. 7(b,c) show that DMI is not localized in the k-space,\nbut broad regions of the Brillouin zone (BZ) contribute\nwith opposite chiralities to the final net ∆ EDMI(0;q) in\nboth vacuum/Co and Gr/Co interfaces. Owing to the\nfound effective energy window, in Fig. 7(d,e) we restrict\nthek-resolved analysis to an integral over states within a\nwindow of 0.5 eV below the Fermi level. We observe that\nthe Co-C hybrid bands, with nearly conical dispersion at\ntheK,K/primespecial points yield a positive contribution to\n∆EDMI(0;q) (red spots at the BZ vertices), whereas the\nrest of the BZ contributes with negative values. In otherwords, the distinctive feature of the graphene adsorp-\ntion on the bandstructure, namely the conical band of\nmainly Co- dz2character and also small weight in dxz,yz\norbitals, actually contributes to a chirality opposite to\nthe observed one. The conclusion is that the DMI of\nthe Gr/Co interface cannot be attributed to individual\nCo-C hybrid bands near the Fermi level. For this rea-\nson, a model where the Dvector is estimated from a\nRashba hamiltonian αR(σ×k)zof a single band in a fer-\nromagnetically coupled environment [27, 31, 33, 34] is not\nsuitable for the Gr/Co interface. Instead, the contribut-\ning Co-C interactions extend to the whole spectrum. On\nthis basis we can state that the nature of interfacial DMI\nat Gr/Co and Co/Pt is the same, namely strongly hy-\nbridized SOC-split d-bands. We recall that Rashba band\nsplitting requires not only a ∇V, which can be indeed\nenhanced upon adsorption if this increases the asymme-\ntry of the charge distribution at the surface [51], but also\na substantial SOC strength ξ[52, 53].\nIV. CONCLUSIONS\nOur interface-resolved DFT study of DMI in\nGr/Con/Pt(111) heterostructures with varying Co layer\nthickness shows that the regime of additivity of interfa-\ncial DMI is reached already at n= 3 atomic planes and\nalso that the D-vectors have an almost negligible out-\nof-plane component. As the perpendicular magnetocrys-\ntalline anisotropy prevails at much larger nvalues [35],\na sizable DMI interaction to have spin canting and chi-\nral exchange effects are expected to be robust. However,\nunlike the magnetocrystalline anisotropy itself, interfa-\ncial DMI is insensitive to the internal structure of the\nCo layer. The observed DMI in domain wall propaga-\ntion suggests a comparable interface DMI strength but\nopposite sign at Gr/Co and Co/Pt interfaces [35]. This\nis confirmed by our calculations. Indeed, we find that the\ngraphene layer has the effect of inverting the chirality of\nthe vacuum/Co interface.\nThe Gr/Co and Co/Pt interfacial DMI has been clas-\nsified as being of different nature, namely, Rashba and\nFert-Levy mechanisms at Gr/Co and Co/Pt, respectively.\nOur study leads to the conclusion that this classifica-\ntion is subjective. Those models correspond to two lim-\niting cases of the same physics, as illustrated by the\nGr/Con/Pt system. The electrostatic dipole (a magni-\ntude identified as a DMI descriptor) at the vacuum/Co\nsurface is reversed and increases in magnitude upon cap-\nping with graphene. Nevertheless, this does not mean\nthat a Rashba SOC term is induced: a spectral resolu-\ntion of the DMI energy of Gr/Co reveals that it is not\nlinked to an individual band splitting. Instead, it results\nfrom the superposition of many hybridized bands with\nweight on the Co atoms, similarly to the other Co/Pt\ninterface.8\n-3-2-1 0 1 2 3 4 5\n-8 -4  0  4  8∆EDMIY [meV]\nne(a)Gr/Co5 (total)\nCoGrCov\n-0.8-0.4 0 0.4 0.8\n         ∆EDMIY \n [meV a.u.2]ky (a.u.-1)(b)-0.04-0.02 0 0.02 0.04\n     \n     (c)\n-0.8-0.40.00.40.8\n-0.8 -0.4 0.0 0.4 0.8ky (a.u.-1)\nkx (a.u.-1)(d)\n-0.8 -0.4 0.0 0.4 0.8\nkx (a.u.-1)(e)\nFIG. 7. (a) ∆ EDMI(ne) atq= 0.25|ΓK|for the Gr/Co 5slab\n(purple) compared with the curves calculated for SOC applied\nonly to the Co Grplane of that slab (green) and to Co vof the\nclean Co 5slab (blue) (note that the latter is centrosymmetric,\nthus the total ∆ EY\nDMI in the slab amounts to zero). (b-e) For\nthe same q-vector, ∆ EY\nDMI atne= 0 resolved in k-space.\nPanels (b,d) correspond to the Co Grinterfacial atomic plane\nand (c,e) to Co v(Co/vaccum interface). Data of panels (b,c)\nare calculated by integration over all the occupied bands. In\npanels (d,e), the integral energy range is restricted to a stripe\nof 0.5 eV below the Fermi level.ACKNOWLEDGMENTS\nPojects RTI2018-097895-B-C41 and\nPID2019-103910GB-I00, funded by\nMCIN/AEI/10.13039/501100011033/ and FEDER\nUna manera de hacer Europa ; GIU18/138 by Uni-\nversidad del Pa´ ıs Vasco UPV/EHU; IT-1246-19 and\nIT-1260-19 by Gobierno Vasco. Computational resources\nwere provided by the DIPC computing center.\n[1] I. E. Dzialoshinskii, Soviet Phys. JETP 5, 1259 (1957).\n[2] T. Moriya, Phys. Rev. Lett. 4, 228 (1960).\n[3] T. Moriya, Phys. Rev. 120, 91 (1960).\n[4] A. Fert and P. M. Levy, Phys. Rev. Lett. 44, 1538 (1980).\n[5] P. M. Levy and A. Fert, Phys. Rev. B 23, 4667 (1981).\n[6] M. Bode, M. Heide, K. von Bergmann, P. Ferriani,\nS. Heinze, G. Bihlmayer, A. Kubetzka, O. Pietzsch,\nS. Bl¨ ugel, and R. Wiesendanger, Nature 447, 190 (2007).\n[7] M. Schmitt, P. Moras, G. Bihlmayer, R. Cotsakis,\nM. Vogt, J. Kemmer, A. Belabbes, P. M. Sheverdyaeva,\nA. K. Kundu, C. Carbone, S. Bl¨ ugel, and M. Bode, Na-\nture Communications 10, 2610 (2019).\n[8] R. Wiesendanger, Nature Reviews Materials 1, 16044(2016).\n[9] A. Fert, N. Reyren, and V. Cros, Nature Reviews Mate-\nrials2, 17031 (2017).\n[10] A. N. Bogdanov and C. Panagopoulos, Nature Reviews\nPhysics 2, 492 (2020).\n[11] M. Heide, G. Bihlmayer, and S. Bl¨ ugel, Phys. Rev. B 78,\n140403 (2008).\n[12] Z. Luo, T. P. Dao, A. Hrabec, J. Vijayakumar,\nA. Kleibert, M. Baumgartner, E. Kirk, J. Cui,\nT. Savchenko, G. Krishnaswamy, L. J. Heyder-\nman, and P. Gambardella, Science 363, 1435 (2019),\nhttps://science.sciencemag.org/content/363/6434/1435.full.pdf.\n[13] A. Hrabec, Z. Luo, L. J. Heyderman, and P. Gam-9\nbardella, Applied Physics Letters 117, 130503 (2020).\n[14] E. Y. Vedmedenko, R. K. Kawakami, D. D. Sheka,\nP. Gambardella, A. Kirilyuk, A. Hirohata, C. Binek,\nO. Chubykalo-Fesenko, S. Sanvito, B. J. Kirby, J. Grol-\nlier, K. Everschor-Sitte, T. Kampfrath, C.-Y. You, and\nA. Berger, Journal of Physics D: Applied Physics 53,\n453001 (2020).\n[15] C. Moreau-Luchaire, C. Moutafis, N. Reyren, J. Sampaio,\nC. A. F. Vaz, N. Van Horne, K. Bouzehouane, K. Garcia,\nC. Deranlot, P. Warnicke, P. Wohlh¨ uter, J.-M. George,\nM. Weigand, J. Raabe, V. Cros, and A. Fert, Nature\nNanotechnology 11, 444 (2016).\n[16] H. Yang, O. Boulle, V. Cros, A. Fert, and M. Chshiev,\nScientific Reports 8, 12356 (2018).\n[17] M. Perini, S. Meyer, B. Dup´ e, S. von Malottki, A. Kubet-\nzka, K. von Bergmann, R. Wiesendanger, and S. Heinze,\nPhys. Rev. B 97, 184425 (2018).\n[18] S. Woo, K. Litzius, B. Kr¨ uger, M.-Y. Im, L. Caretta,\nK. Richter, M. Mann, A. Krone, R. M. Reeve,\nM. Weigand, P. Agrawal, I. Lemesh, M.-A. Mawass,\nP. Fischer, M. Kl¨ aui, and G. S. D. Beach, Nature Ma-\nterials 15, 501 (2016).\n[19] H. Yang, A. Thiaville, S. Rohart, A. Fert, and\nM. Chshiev, Phys. Rev. Lett. 115, 267210 (2015).\n[20] A. Belabbes, G. Bihlmayer, F. Bechstedt, S. Bl¨ ugel, and\nA. Manchon, Phys. Rev. Lett. 117, 247202 (2016).\n[21] L. M. Sandratskii, Phys. Rev. B 96, 024450 (2017).\n[22] G. Chen, M. Robertson, M. Hoffmann, C. Ophus, A. L.\nFernandes Cauduro, R. Lo Conte, H. Ding, R. Wiesen-\ndanger, S. Bl¨ ugel, A. K. Schmid, and K. Liu, Phys. Rev.\nX11, 021015 (2021).\n[23] B. Yang, Q. Cui, J. Liang, M. Chshiev, and H. Yang,\nPhys. Rev. B 101, 014406 (2020).\n[24] A. Belabbes, G. Bihlmayer, S. Bl¨ ugel, and A. Manchon,\nScientific Reports 6, 24634 (2016).\n[25] G. Chen, A. Mascaraque, H. Jia, B. Zimmermann,\nM. Robertson, R. L. Conte, M. Hoffmann, M. A.\nGonz´ alez Barrio, H. Ding, R. Wiesendanger, E. G.\nMichel, S. Bl¨ ugel, A. K. Schmid, and K. Liu, Sci-\nence Advances 6, 10.1126/sciadv.aba4924 (2020),\nhttps://advances.sciencemag.org/content/6/33/eaba4924.full.pdf.\n[26] F. Ajejas, A. Gud´ ın, R. Guerrero, A. Anad´ on Barcelona,\nJ. M. Diez, L. de Melo Costa, P. Olleros, M. A.\nNi˜ no, S. Pizzini, J. Vogel, M. Valvidares, P. Gargiani,\nM. Cabero, M. Varela, J. Camarero, R. Miranda, and\nP. Perna, Nano Letters , Nano Letters 18, 5364 (2018).\n[27] A. Hallal, J. Liang, F. Ibrahim, H. Yang, A. Fert, and\nM. Chshiev, Nano Letters 10.1021/acs.nanolett.1c01713\n(2021).\n[28] H. Jia, B. Zimmermann, G. Michalicek, G. Bihlmayer,\nand S. Bl¨ ugel, Phys. Rev. Materials 4, 024405 (2020).\n[29] S. Kim, K. Ueda, G. Go, P.-H. Jang, K.-J. Lee, A. Be-\nlabbes, A. Manchon, M. Suzuki, Y. Kotani, T. Naka-\nmura, K. Nakamura, T. Koyama, D. Chiba, K. T. Ya-\nmada, D.-H. Kim, T. Moriyama, K.-J. Kim, and T. Ono,\nNature Communications 9, 1648 (2018).\n[30] P.-J. Hsu, A. Kubetzka, A. Finco, N. Romming, K. von\nBergmann, and R. Wiesendanger, Nature Nanotechnol-\nogy12, 123 (2017).\n[31] H. Yang, G. Chen, A. A. C. Cotta, A. T. N’Diaye, S. A.\nNikolaev, E. A. Soares, W. A. A. Macedo, K. Liu, A. K.\nSchmid, A. Fert, and M. Chshiev, Nature Materials 17,605 (2018).\n[32] H. Imamura, P. Bruno, and Y. Utsumi, Phys. Rev. B 69,\n121303 (2004).\n[33] K.-W. Kim, H.-W. Lee, K.-J. Lee, and M. D. Stiles, Phys.\nRev. Lett. 111, 216601 (2013).\n[34] A. Kundu and S. Zhang, Phys. Rev. B 92, 094434 (2015).\n[35] M. Blanco-Rey, P. Perna, A. Gudin, J. M. Diez,\nA. Anadon, P. Olleros-Rodriguez, L. de Melo Costa,\nM. Valvidares, P. Gargiani, A. Guedeja-Marron,\nM. Cabero, M. Varela, C. Garcia-Fernandez, M. M.\nOtrokov, J. Camarero, R. Miranda, A. Arnau, and J. I.\nCerda, ACS Applied Nano Materials 4, 4398 (2021).\n[36] A. Cr´ epieux and C. Lacroix, Journal of Magnetism and\nMagnetic Materials 182, 341 (1998).\n[37] H. Krakauer, M. Posternak, and A. J. Freeman, Phys.\nRev. B 19, 1706 (1979).\n[38] E. Wimmer, H. Krakauer, M. Weinert, and A. J. Free-\nman, Phys. Rev. B 24, 864 (1981).\n[39]Fleur site: http://www.flapw.de.\n[40] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.\nLett. 77, 3865 (1996).\n[41] H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188\n(1976).\n[42] L. M. Sandratskii, Journal of Physics: Condensed Matter\n3, 8565 (1991).\n[43] M. Heide, G. Bihlmayer, and S. Bl¨ ugel, Physica B: Con-\ndensed Matter 404, 2678 (2009).\n[44] B. Zimmermann, G. Bihlmayer, M. B¨ ottcher, M. Bouhas-\nsoune, S. Lounis, J. Sinova, S. Heinze, S. Bl¨ ugel, and\nB. Dup´ e, Phys. Rev. B 99, 214426 (2019).\n[45] D. Lee, R. K. Behera, P. Wu, H. Xu, Y. L. Li, S. B.\nSinnott, S. R. Phillpot, L. Q. Chen, and V. Gopalan,\nPhys. Rev. B 80, 060102 (2009).\n[46] K.-W. Kim, K.-W. Moon, N. Kerber, J. Nothhelfer, and\nK. Everschor-Sitte, Phys. Rev. B 97, 224427 (2018).\n[47] M. Blanco-Rey, J. I. Cerd´ a, and A. Arnau, New Journal\nof Physics 21, 073054 (2019).\n[48] F. Ajejas, A. Anadon, A. Gudin, J. M. Diez, C. G.\nAyani, P. Olleros-Rodr´ ıguez, L. de Melo Costa, C. Nav´ ıo,\nA. Guti´ errez, F. Calleja, A. L. V´ azquez de Parga, R. Mi-\nranda, J. Camarero, and P. Perna, ACS Applied Materi-\nals & Interfaces 12, 4088 (2020).\n[49] G. H. O. Daalderop, P. J. Kelly, and M. F. H. Schuur-\nmans, Phys. Rev. B 50, 9989 (1994).\n[50] Some selection rules are found concerning the orbital\ncharacter|l, m/angbracketrightof the states with wavevectors kand\nk±q, When the q-dependence is perturbatively con-\nsidered in the analytical D-vector expression, a dipolar\nterm appears that couples different l-channels, as shown\nin Ref. [28]. If only the inversion symmetry breaking is\nconsidered, as in Ref. [23], only the couplings between\n|2, m/angbracketrightorbitals of suitable symmetry (determined by the\nSOC term in the hamiltonian [54]) survive.\n[51] O. Krupin, G. Bihlmayer, K. Starke, S. Gorovikov, J. E.\nPrieto, K. D¨ obrich, S. Bl¨ ugel, and G. Kaindl, Phys. Rev.\nB71, 201403 (2005).\n[52] S. LaShell, B. A. McDougall, and E. Jensen, Phys. Rev.\nLett. 77, 3419 (1996).\n[53] L. Petersen and P. Hedeg˚ ard, Surface Science 459, 49\n(2000).\n[54] E. Abate and M. Asdente, Phys. Rev. 140, A1303 (1965).Supplemental Material for: “Nature of Interfacial\nDzyaloshinskii-Moriya Interactions in Graphene/Co/Pt(111)\nMultilayer Heterostructures”\nM. Blanco-Rey,1, 2G. Bihlmayer,3A. Arnau,4, 1, 2and J.I. Cerd´ a5\n1Departamento de Pol´ ımeros y Materiales Avanzados: F´ ısica, Qu´ ımica y Tecnolog´ ıa,\nFacultad de Qu´ ımica UPV/EHU, Apartado 1072, 20080 Donostia-San Sebasti´ an, Spain\n2Donostia International Physics Center, Paseo Manuel\nde Lardiz´ abal 4, 20018 Donostia-San Sebasti´ an, Spain\n3Peter Gr¨ unberg Institut and Institute for Advanced Simulation,\nForschungszentrum J¨ ulich and JARA, 52425 J¨ ulich, Germany\n4Centro de F´ ısica de Materiales CFM/MPC (CSIC-UPV/EHU),\nPaseo Manuel de Lardiz´ abal 5, 20018 Donostia-San Sebasti´ an, Spain\n5Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, 28049 Madrid, Spain\n(Dated: November 17, 2021)\nS1arXiv:2111.08556v1  [cond-mat.mtrl-sci]  16 Nov 2021D1%D’1%D2%\n-2-1.5-1-0.5 0 0.5 1 1.5 2\n 0  0.2  0.4  0.6  0.8  1  1.2  1.4DMI geometric factor\nθ = 2π/N sinθ + sin2 θ \n sin2 θ - 2sin θ \n sinθ \n sin3 θ FIG. S1. Short-ranged DMI contributions on a 3-fold symmetric bilayer. Left:D-\nvectors at a fccbilayer for first ( D1) and second ( D2) in-plane nearest neighbours, and first\nnearest neighbours of consecutive atomic planes ( D/prime\n1). The colour gradient from green to white\ninD/prime\n1depicts the variation of the D1zcomponent sign. D2andD/prime\n1do not have a perpendicular\ncomponent. D2is 3-fold symmetric with no perpendicular component, i.e. there are two different\nvectors for the second nearest neighbour interaction. Right: for spin spirals defined by q-vectors\nq=2π\na(1\nN,1\nN,0), the in plane first nearest neighbours contribution is ∆ EY\nDMI, 1(θ) = 4S2D1y(sinθ+\nsin 2θ) and ∆EZ\nDMI, 1(θ) = 4S2D1z(sin 2θ−2 sinθ), whereSis the atomic spin, θ=2π\nN. Each\nsurface atom has three nearest neighbours at the plane below, which contribute as ∆ EY\nDMI, 1/primeθ) =\n2SS/primeD/prime\n1x√\n3 sinθ, whereS/primeis the atomic spin in the second plane. For the second in-plane nearest\nneighbours, the energy term is ∆ EY\nDMI, 2θ) = 2S1(D2x−˜D2x)√\n3 sin 3θ, noting that the interaction\nis three-fold symmetric. The figure shows the angle-dependence of the geometric factors in these\nenergy terms.\nS2-2 0 2 4 6 8 10 12\n 0  0.1  0.2  0.3  0.4∆ EDMIY [meV]\nq [ΓK](a) G/Co1/Ptn\nnPt=1\n       2\n       5\n-2 0 2 4 6 8 10 12\n 0  0.1  0.2  0.3  0.4∆ EDMIY [meV]\nq [ΓK](b) G/Co5/PtnFIG. S2. DMI energy dependence on the Pt substrate thickness. DMI energy ∆ EY\nDMI(q)\ndependence on the Pt substrate thickness for one (a) and five (b) Co layers. Here, too, we observe\nthe aforementioned substrate effect in the Co monolayer limit: there is a noticeable change in the\nGr/Co 1/Ptnbehaviour when a second Pt plane is added, that brings the energy close to the thick\nPt substrate limit (a). This crossover is not visible in the thick Co layer limit.\nS3 0 50 100 150 200 250 300 350 400 450\n 0.0  0.2  0.4E(q) [meV]\nq [ΓK](a) Gr/Con/Pt5\nnCo=1\n       2\n       3\n       4\n       5\n 0 50 100 150 200 250 300 350 400 450\n 0.0  0.2  0.4E(q) [meV]\nq [ΓK](b) Con/Pt5\n 0 50 100 150 200 250 300 350 400\n 0.0  0.2  0.4E(q) [meV]\nq [ΓK](c) Gr/ConFIG. S3. Energy dispersion of the spin spirals in Gr/Co n/Pt 5.Dependence on the Co\nlayer thickness of the spin spiral energy dispersion E(q) in Gr/Co nPt5slabs and their constituents\nCon/Pt5and Gr/Co n. These values are calculated for the q-vector along the Γ Kdirection without\nSOC and in a non-self-consistent scheme.\nS4-2 0 2 4 6 8 10 12\n 0.0  0.1  0.2  0.3  0.4∆EDMIY [meV]\nq [ΓK](b)\n NSCF\n  SCF\n-0.5 0 0.5 1 1.5 2 2.5\n 0.0  0.1  0.2  0.3  0.4∆EDMIZ [meV]\nq [ΓK](c) 0 10 20 30 40 50 60\n 0.0  0.1  0.2  0.3  0.4E(q) [meV]\nq [ΓK](a)\n NSCF\n  SCFFIG. S4. Effect of self-consistency in the spin spiral calculation of the Gr/Co 1/Pt 1\nslab. (a) Spin spiral energy dispersion in the absence of SOC for a Gr/Co 1/Pt1trilayer, with the\nq-vector along the Γ Kdirection. The red filled dots are calculated by imposing the non-collinear\nmagnetization density given by the Generalized Bloch Theorem (GBT) for each q-vector on the\ncollinear electron wavefunctions, i.e. the spin spiral energy is determined non-self-consistently\n(NSCF) or as a perturbation. The energies of optimized spirals after reaching self-consistency\n(SCF) in the Kohn-Sham equations are given by the black empty dots. At the low- qregime used\nin this work, agreement is good. (b,c) Comparison of the DMI energies ∆ Eˆsa\nDMI(q) for ˆsa=Y(b)\nandZ(c) calculated by adding SOC perturbatively to the NSCF and SCF spirals obtained above.\nWe observe that the agreement is quantitative, and thus the use of a perturbative approach to\nobtain the spinors from the collinear electron density of the system without SOC is justified in this\nqvalue range. Note that achieving self-consistency would be too computationally demanding for\nthe more realistic modes with five Pt planes as a substrate.\nS5-0.2-0.1 0 0.1 0.2(Gr)/Co5/Pt5\nC Pt Copz [meV]\n  Gr/Con/Pt5  Con/Pt5\n 0 0.5 1 1.5 2mS [µB]\n  Gr/Con/Pt5  Con/Pt5\n-4 0 4 8\n -30  -20  -10    0   10   20   30∆ρ (au-1)\nz (au)Gr/Con/Pt5   ∆ρ\n              ∆m-0.2-0.1 0 0.1 0.2(Gr)/Co1/Pt5\nC Pt Co \n  \n 0 0.5 1 1.5 2 \n  \n-4 0 4 8\n -30  -20  -10    0   10   20   30 \nz (au)FIG. S5. Dipoles, spin polarization and charge distribution at the interfaces. From top\nto bottom, perpendicular dipole pzin the muffin tin, spin moment in the muffin tin mSand charge\ndensity difference ∆ ρ(z) for the Gr/Co 5/Pt5(left column) and Gr/Co 5/Pt5(right) systems. Dots\nand squares show the data with and without graphene capping, respectively. In the bottom panels,\nthe ∆ρ(z) quantity (solid line) is calculated as the xy-integrated substraction of charge densities\nρ(Gr/Con/Pt5)−ρ(Gr)−ρ(Con)−ρ(Pt5), which accounts for charge redistribution at both Gr/Co\nand Co/Pt interfaces. The dashed line ∆ m(z) shows the magnetization density for ∆ ρ(z).\nS6-2 0 2 4 6 8 10 12\n 0  0.1  0.2  0.3  0.4∆EDMIY [meV]\nq [ΓK](a)\nnCo=1\n       2\n       3\n       4\n       5\n-2-1 0 1 2\n 0  0.1  0.2  0.3  0.4∆EDMIZ [meV]\nq [ΓK](b)FIG. S6. DMI energy for Gr/Co n/Pt 1slabs constituents. The filled dots show the de-\npendence of the DMI energy ∆ Eˆsa\nDMI(q) on the Co thickness for model slabs with a single Pt\nmonolayer. Each color corresponds to a thickness. The sum of the energies of the Gr/Co nand\nCon/Pt1slab constituents is indicated by empty squares to check interfacial DMI additivity at\neach thickness. On panel (a) for ˆ sa=Ythree observations are made: (i) additivity is not met\nforn= 1,2 yet; (ii) DMI energy values converge from n= 3 after oscillating; and (iii) the n= 2\ncurve has a maximum at q≈0.2|ΓK|due to the effect of the ∆ EY\nDMI, 1/primecontribution (see Fig. S1),\nwhich inverts the sign of the sin θleading term in the DMI energy. The same results are reported\nin the main paper with a Pt 5substrate. However, energies for n= 1,2 differ (they are about a\nfactor two larger in Gr/Co 1/Pt1than in Gr/Co 1/Pt5and a factor three smaller in Gr/Co 2/Pt1\nthan in Gr/Co 2/Pt5), proving that the ultrathin Co regime is strongly dependent on the electronic\nstructure of the Pt substrate and the DMI is not localized at the precise Co-Pt interface. Instead,\nit depends on the hybridization between Co and deeper Pt planes. In panel (b) for ˆ sa=Zwe\nobserve an energy change from n= 1 to thicker Co layers, i.e. the sign of Dzis inverted. Note\nthat this sign change is not obtained on the Pt 5substrate.\nS7(a) Gr/Co5/Pt1 \n-K Γq K-6-5-4-3-2-1 0 1 2E-EF (eV)\n01\nA(k,E;q) [arb.u.](b) ∆EDMIY(q) total\n-K Γq K-6-5-4-3-2-1 0 1 2E-EF (eV)\n-10-5 0 5 10\n∆EDMIY(k,E;q) [a.u.2]\n(c) Pt\n-K Γq K-6-5-4-3-2-1 0 1 2E-EF (eV)\n-10-5 0 5 10\n∆EDMIY(k,E;q) [a.u.2](d) CoPt\n-K Γq K-6-5-4-3-2-1 0 1 2E-EF (eV)\n-3-1.5 0 1.5 3\n∆EDMIY(k,E;q) [a.u.2](e) CoGr\n-K Γq K-6-5-4-3-2-1 0 1 2E-EF (eV)\n-3-1.5 0 1.5 3\n∆EDMIY(k,E;q) [a.u.2]FIG. S7.k-resolved spectral DMI energy for Gr/Co 5/Pt 1.(a) Spectral density of the\nGr/Co 5/Pt1system calculated without SOC for the q-vector q=2π\na(1\n24,1\n24,0) (q=0.125|ΓK|),\nindicated with a green arrow. The lateral splitting introduced by the spiral in the bands is observed.\n(b) Density of energy ∆ EY\nDMI(q). The actual DMI energy is the integral over occupied states, which\ninvolves many cancellations of positive and negative contributions. (c-e) The contributions of the\nindicated interfacial atoms to the density of panel (b). The more intense DMI contributions in\npanels (c,d) map the broad bands mainly originated from Pt( d) and Co(d) orbitals, centred at\nEF−3.5 andEF−1.5 eV, respectively. The Co Grinterfacial states show high intensity at the\nCo-C hybrid conical bands. Nevertheless, as in the other panels, cancellations of DMI energies of\nopposite sign attenuate the contribution of this band structure feature to the total DMI energy.\nS8"    },    {        "title": "2111.13004v1.Tunable_gigahertz_dynamics_of_low_temperature_skyrmion_lattice_in_a_chiral_magnet.pdf",        "content": "Tunable gigahertz dynamics of low-temperature skyrmion lattice in a chiral\nmagnet\nOscar Lee,1,\u0003Jan Sahliger,2Aisha Aqeel,2Safe Khan,1Shinichiro\nSeki,3Hidekazu Kurebayashi,1and Christian H. Back2, 4,y\n1London Centre for Nanotechnology, University College London, United Kingdom\n2Physik-Department, Technische Universit ¨at M ¨unchen, D-85748 Garching, Germany\n3Department of Applied Physics, The University of Tokyo, Bunkyo, Tokyo, 113-8656, Japan\n4Munich Center for Quantum Science and Technology (MCQST), D-80799 M ¨unchen, Germany\nAbstract\nRecently, it has been shown that the chiral magnetic insulator Cu 2OSeO 3hosts skyrmions in two sepa-\nrated pockets in temperature and magnetic field phase space. It has also been shown that the predominant\nstabilization mechanism for the low-temperature skyrmion (LTS) phase is via the crystalline anisotropy,\nopposed to temperature fluctuations that stabilize the well-established high-temperature skyrmion (HTS)\nphase. Here, we report on a detailed study of LTS generation by field cycling, probed by GHz spin dy-\nnamics in Cu 2OSeO 3. LTSs are populated via a field cycling protocol with the static magnetic field ap-\nplied parallel to the h100icrystalline direction of plate and cuboid-shaped bulk crystals. By analyzing\ntemperature-dependent broadband spectroscopy data, clear evidence of low-temperature skyrmion excita-\ntions with clockwise (CW), counterclockwise (CCW), and breathing mode (BR) character at temperatures\nbelowT= 40 K are shown. We find that the mode intensities can be tuned with the number of field-cycles\nbelow the saturation field. By tracking the resonance frequencies, we are able to map out the field-cycle-\ngenerated LTS phase diagram, from which we conclude that the LTS phase is distinctly separated from the\nhigh-temperature counterpart. We also study the mode hybridization between the dark CW and the BR\nmodes as a function of temperature. By using two Cu 2OSeO 3crystals with different shapes and therefore\ndifferent demagnetization factors, together with numerical calculations, we unambiguously show that the\nmagnetocrystalline anisotropy plays a central role for the mode hybridization.\n1arXiv:2111.13004v1  [cond-mat.mtrl-sci]  25 Nov 2021INTRODUCTION\nChiral magnets with large spin-orbit interaction and significant Dzyaloshinskii-Moriya inter-\naction (DMI) host different non-collinear spin configurations such as the helical, conical and\nskyrmion lattice configurations as well as the field polarized (ferrimagnetic) state in large applied\nmagnetic fields. Many chiral magnets share a common magnetic phase diagram with a skyrmion\nlattice phase stabilized by thermal fluctuation located in a small pocket in temperature and mag-\nnetic field space in the vicinity of the Curie temperature [1–7]. Furthermore, their excitation\ndynamics in the GHz regime has been theoretically predicted and experimentally shown in prior\nstudies [8–11].\nThe recent discovery of a novel low-temperature skyrmion (LTS) phase in various materials, in-\ncluding MnSi and Co 8ZnMn 4[12, 13], has consequently sparked renewed interest in the particular\nchiral magnetic insulator Cu 2OSeO 3[14–17]. It has been shown that in this material, by applying\nappropriate magnetic field and temperature protocols, a low-temperature skyrmion phase can be\nnucleated, which is distinct from the high-temperature skyrmion (HTS) phase and which seems\nto be stabilized by a combination of DMI and the magnetocrystalline anisotropy [14, 15, 18]. As\na consequence, it only appears if the externally applied magnetic field is oriented along a certain\ncrystallographic direction. Furthermore, in contrast to the HTS phase, the nucleation process for\nthe LTS is nontrivial: the metastable tilted conical phase seems to be a necessary prerequisite for\nthe formation of the LTS lattice [14–16]. A recent study by Halder et al. [15] has shown that,\nto a leading order, a cubic magnetic anisotropy term is responsible for stabilizing the LTS lattice\nand that the characteristics of the phases can be described rather well using a Ginzburg-Landau\nformalism adapted for chiral magnets with cubic anisotropy [19, 20]. The same formalism can\nbe extended to compute the spin excitations in both LTS and HTS lattice phases. The description\nabove suggests that one may influence the spatial configuration of the magnetization by control-\nling the strength of the magnetocrystalline anisotropy, e.g. by temperature.\nFrom the above studies, it is reasonable to conclude that the stabilization mechanisms responsi-\nble for the LTS phase originate from the magnetic anisotropies when the magnetic field is applied\nparallel toh100i. Moreover, recently, the dynamic modes of this new LTS phase have also been\nthoroughly investigated using low-temperature broadband ferromagnetic resonance experiments,\n2allowing in addition the identification of the low temperature elongated skyrmion phase [21].\nIn this paper, we address the dynamic modes in the LTS by applying broadband ferromagnetic\nresonance experiments as a function of temperature, T, and magnetic field, H. We report the\ninfluence of the number of field cycles on the intensity of the resonant spin excitations in the LTS\nphase and show that a large volume fraction of the investigated spin structure in bulk crystals\ncan indeed be converted into the LTS phase by appropriate field-cycling [14, 22]. Second, we\nobserve the low-lying breathing and counterclockwise modes and find evidence of the clockwise\ngyrational modes at higher frequencies [8, 9, 11]. Using the dynamic modes as experimental\nevidence for the LTS, we map out the phase diagram as a function of H and T. Finally, we\nsystematically trace the mode hybridization between a dark higher-order clockwise mode and\nthe breathing mode [23] as a function of temperature for two differently shaped crystals. We\nobserve that the demagnetization field contribution to the gap size is negligible, consequently\nrevealing the linear dependence on the strength of the magnetic anisotropies. Our experimental\nresults are accompanied by calculations based on a Landau-Ginzburg [14, 24] energy functional,\nwhich includes the cubic magnetic anisotropy term that effectively reproduces the experimentally\nobserved features. Discrepancies between the experimentally and theoretically observed size of the\nhybridization gap can be tentatively attributed to the influence of a gradient term of the exchange\ninteraction.\nEXPERIMENTAL PROCEDURES\nMaterial and phase diagram. Copper-oxoselenite, Cu 2OSeO 3is a chiral magnetic insulator with\nsmall Gilbert damping which crystallizes in a non-centrosymmetric cubic structure hosting 16\ncopper Cu2+ions per unit cell. The magnetization properties of the crystal are owed by the forma-\ntion of tetrahedral clusters by four Cu2+spins in a 3-up-1-down spin configuration, behaving as\na “spin-triplet” with the total spin S = 1. For the detailed visualization of the magnetic structure,\nsee Ref. [5]. The spin-spin interactions in the system are governed by the symmetric Heisenberg\ninteraction, which favours collinear spin structures, and an anti-symmetric Dzyaloshinskii-Moriya\ncomponent analogous to\u0000 !Si\u0002\u0000 !Sj, which make twist-like structures of neighbouring spins. Below\nTc\u001958 K, the system behaves like a typical chiral magnet, hosting various magnetic phases\nincluding helical, conical, ferromagnetic and skyrmions under the influence of applied magnetic\n3field due to a combined effect of Heisenberg and Dzyaloshinskii-Moriya (DM) interactions.\nTo put our experimental results into context, we first show the generic magnetic phase diagram\nof Cu 2OSeO 3for static magnetic fields applied parallel to the h100icrystallographic direction,\nas illustrated in Fig. 1(a). Above the transition temperature Tc\u001958 K, the system is param-\nagnetic. Below Tcand without applied field, the system is described by a multidomain helical\nphase. When the magnetic field exceeds a critical field value H c1, a phase transition into the\nconical phase occurs. By further increasing the magnetic field strength, the cone angle between\nthe spins and the field axis decreases, resulting in a fully polarized state at H c2. In addition to\nthese states, a topological nontrivial HTS phase exists in a small temperature and field pocket,\nclose toTc. This phase pocket can be extended considerably using various procedures, such as\napplying particular temperature-quenching techniques, applying pressure or using thin-film crys-\ntals where the thickness is of the order of several pitch lengths of the helical spiral ( \u001961 nm in\nCu2OSeO 3) [12, 13, 25–30]. Recently, it has been demonstrated that the second, low-temperature,\nskyrmion phase can be nucleated after traversing the so-called tilted conical phase and following a\nunique magnetic field-cycling protocol [21, 22]. To enter the tilted conical phase, a state where the\nhelix pitch tilts away from the field axis, the temperature needs to be decreased below T\u001930K.\nThen, as the applied field decreases further, the LTS phase emerges. Similarly, increasing the\nmagnetic field from zero magnetic field, the tilted conical phase may form upon increasing mag-\nnetic field, again leading to the LTS generation. In both cases, the volume fraction of the LTS can\nbe drastically increased by cycling the magnetic field within magnetic field boundaries where the\nLTS may be expected, as we will demonstrate below.\nExperimental setup. We used two differently shaped bulk Cu 2OSeO 3crystals for our experi-\nments to vary the demagnetization fields in the samples. One sample is a cuboid with dimensions\n1:6\u00021:6\u00021:0 mm3with all sides orientated along h100i. The corresponding demagnetizing\nfactors are 0.28, 0.28, and 0.44. The second sample is a 0.3 mm thick plate with approximate\ndimensions 1:9\u00021:4 mm2in the plane and a surface normal along the h100idirection. In this\ncase, the corresponding demagnetizing factors are \u00190.17, 0.12, and 0.71, with 0.71 being the\nvalue for the surface normal direction. The samples were carefully polished and placed on a\ncoplanar-waveguide (CPW) with a (001) surface facing down. The CPW with the sample on top\nis introduced into a closed-cycle cryostat, inserted into the gap of an electromagnet, as illustrated\n4in Fig. 1(b). The magnetic field is aligned normal to the plane of the CPW, i.e. along the h100i\ndirection of the Cu 2OSeO 3crystals. Since the samples are larger than the centre line (width\n1.25 mm) of the CPW, magnetization dynamics are excited with both in-plane and out-of-plane\noscillating magnetic fields, resulting in excitation of gyration and breathing modes respectively.\nWe have employed a microwave absorption spectroscopy to probe and identify these states, as\ndemonstrated by Y . Onose et al. [9] The microwave magnetic fields are generated in the range of\n1 - 6 GHz in steps of \u00193 MHz by connecting the output port of a vector network analyzer (VNA)\nto the CPW. We measure jS11jorjS21j, the reflection and the transmission signal, respectively, of\nthe microwave absorption spectra as a function of the applied field and subtract a background sig-\nnal by re-measuring the spectra in a high magnetic field well above Hc2, i.e. in the field polarized\nstate (see Supplementary Fig. S1 for details). The measurements were carried out in a temperature\nrange of T= 5\u000060K.\nField cycling protocol. After high-field cooling to the desired temperature from above the or-\ndering temperature, the applied field is first reduced from a large magnetic field H init, within the\nfield-polarized state to the upper boundary of the desired cycling window, H highwhere we subse-\nquently start the cycling procedure. In this process, the magnetic field is repeatedly decreased and\nthen increased between H highand H lowin steps of \u0001H. In this context, one complete cycle means\nthat the magnetic field is ramped down to H lowand subsequently increased again to its original\nvalue, i.e., H high!Hlow!Hhigh. Upon completing nnumber of cycles, the magnetic field is\neither decreased to zero-field, H decor increased to H init.\nFig. 1(c) illustrates a typical measurement profile of the field-cycling protocol for H init=\n200 mT, H high= 80 mT, H low= 40 mT and n= 100 cycles for the plate-shaped sample, recorded\nat 6 K, showing the normalized jS11j. Note that the colour depth reflects the intensity of the\ndetected dynamic modes. Each scan outside the cycling region refer to the measurements of jS11j\nat particular fields, separated by steps of 1 mT. For example, scans 0 and 1 correspond to single\nmeasurements at H initand H init+ 1 mT. In the cycling region, the data was recorded every 10 th\ncycle, thus separated by 10 mT. Therefore, each lower vertex of the triangular-shaped signal cor-\nresponds to a completion of 10 cycles.\n5As the field decreases from H initto H high, an almost linear slope assigned to the Kittel res-\nonance mode in the field-polarized regime is observed. Note that less prominent modes above\nand below the Kittel mode are standing spin waves across the whole Cu 2OSeO 3crystal, which\ncan be observed due to the low damping of the material at low temperatures [31]. The gradient\nchange around scan = 40 indicates the phase transition to the conical phase, exhibiting a frequency\nincrease with decreasing magnetic field. At around scan = 60, the spectrum reveals a significant\ndiscontinuity, which we attribute to the onset of the tilted conical phase [21]. In the cycling regime,\nsignatures of emergent resonant modes are observed at lower frequencies with their intensities in-\ncreasing with the number of cycles. Completing the cycling protocol, both gyration and breathing\nLTS modes are enhanced and clearly observed at low frequencies when further decreasing the field\ntowards zero magnetic field.\nRESULTS AND DISCUSSION\nLet us now discuss the magnetic excitations detected on the plate-shaped sample. Figs. 2(a - c)\nshow the combined excitation spectra of H incand H dec, measured for a various number of cycles\nnas a function of the magnetic field at T= 6 K. In the upper panel of Fig. 2, we show data\nwith n= 0, i.e. without field-cycling. In reminiscence of the conical state, two modes may be\ndistinguished, exhibiting an increase in absorption frequency for decreasing magnetic field. These\nresonance branches, which are expected to couple to the perpendicular excitation field with respect\nto the helix pitch vector, are therefore identified as \u0006Q modes [3, 21]. Their extracted resonance\npositions are marked by yellow dots. Next, we investigate the evolution of the excitation spectrum\nwith the number of field-cycles n= 100 (Fig. 2(b)) and n= 400 (Fig. 2(c)). We observe the\nemergence of additional low lying modes, which become more pronounced with the number of\ncycles and for, n= 400 unambiguously resemble the calculated resonances shown in Fig. 2(d).\nForn= 400 , a clear signature of a hybridization gap between the BR mode and a dark mode, not\nshowing up in the experiment, can be observed at approximately 0.5 Hc2, which corresponds to\n\u00160H\u001975mT.\nTo further highlight the evolution of the spectra under the influence of field cycling, we show\nin Fig. 2(e) the line-cuts at a fixed magnetic field at \u00160H= 90 mT, indicated by the dashed line in\nFig. 2(a - c) for a different number of field cycles. On the same note, we also show the evolution\n6of the mode amplitude and frequency frin Figs. 2 (f) - (g), respectively. The field is chosen to be\nhigh enough in order to minimize the effects of the mode hybridization occurring around 75 mT.\nWe notice that after their initial regime up to 500 cycles, both parameters steeply increase with\ncycle number up to around 4000 number of cycles and then start to saturate afterwards. Since\nthe remnants of the tilted conical phase strongly contribute to the absorption spectra for small\nnumbers of cycles ( <500), the fitted amplitudes and resonance positions might be inaccurate, as\nindicated by the shaded region.\nFrom similar measurements as a function of temperature shown in Figs. 3(a) and (b), we extract\nthe phase pockets of the LTS phase by monitoring the respective resonances for the plate-shaped\n(Fig. 3(a)) and the cuboid-shaped samples (Fig. 3(b)), and present the corresponding phase dia-\ngram in Fig. 3(c). In the plate-shaped sample, as depicted in Fig. 3(a), the LTS modes are observed\nbelow T= 24 K; however, the hybridization is no longer distinctive above T= 20 K. At 25K,\nthe LTS phase completely disappears while a subtle signal can still be seen at 24K. Therefore,\nwe extract a steep phase boundary. On the other hand, for the cuboid-shaped sample shown in\nFig. 3(b), we notice that the LTS phase can be observed up to T= 40 K, although the intensities of\nits modes are much lower than those of the LTS phase for the plate-shaped sample. In particular,\nthe hybridization for the cuboid sample occurs at a lower field, at around 50 mT.\nIn order to support these results theoretically, we employ the previously established phe-\nnomenological model for chiral magnets, which can be found in various publications [11, 14, 21].\nThe theoretical approach is based on the Ginzburg-Landau energy density, which consists of the\nfollowing contributions:\nF[M] =F0[M] +r0M2+UM4: (1)\nThe first energy term F0[M]represents exchange, Dzyaloshinsky-Moriya and dipolar interactions,\ncharacterized by their respective strengths, J,Dand\u001c. The shape of the samples under investiga-\ntion enters the latter by the corresponding demagnetization tensor Nwith tr(N) = 1. Additionally,\nthe Zeeman term, with magnetic field Band the cubic anisotropy, with anisotropy strength Kare\nincluded in F0[M]. The remaining terms are governed by the Ginzburg-Landau coefficients r0\nandU. WhileUis assumed to be constant and greater than zero, r0is set to be the measure of the\ndistance to the critical temperature Tcup to the first order and therefore r0\u0018T\u0000Tc. Rescaling\nthe theory reduces the number of parameters, leaving r0,N,KandBas variables. The dipolar\n7interaction strength was already determined in prior reports and is set to \u001c\u00190:88accordingly\n[11, 14].\nSince the focus of this work lies in the investigation of the LTS phase, we likewise limit our\nanalysis to the eigenmodes and eigenresonances of the topologically nontrivial skyrmion states. In\nthe following, the calculations are performed either for a spherical sample with demagnetization\nfactorsNi= 1=3representing the cuboid, or for a platelet sample with Nx= 0:17,Ny= 0:12\nandNz= 0:71. The temperature is set to r0= -1000, and the anisotropy constant to K = 0.0002,\ngiven in dimensionless units, if not stated otherwise. Our calculations revealed that the shape of\nthe sample under investigation does not affect the gap size significantly.\nIn Fig. 2(d), we show the calculated excitation spectrum normalized to Hc2. Calculations are\nperformed for the plate-shaped sample, with the magnetic field applied along the surface normal,\ni.e. parallel to theh100idirection. The field boundaries were chosen to cover the range in which\nthe skyrmion phase is energetically more favorable than the topological trivial states. Neverthe-\nless, it was already shown that the LTS phase may remain even in smaller fields, merging into\nthe elongated skyrmion phase, see Ref. [21]. The LTS phase is characterized by three distinct\nmodes of the skyrmion lattice, the counterclockwise (CCW) gyrational mode, which decreases in\nfrequency with decreasing magnetic field (violet), the breathing (BR) mode, which increases in\nfrequency with decreasing magnetic field (red) and the clockwise (CW) gyrational mode again\nwith decreasing frequency as a function of decreasing magnetic field (green). We find excellent\nagreement between the calculated excitation spectrum and the experimentally detected dynamic\nmodes after field cycling.\nA particular feature that becomes very clear from the theoretical analysis is the opening of\na gap for the BR mode (red) which hints to a hybridization and which is also observed in the\nexperiments. This hybridization feature is not observed in the HTS phase and will be discussed\nin the following. By introducing the cubic anisotropy term in our micromagnetic calculations, we\nobserve a hybridization of the breathing, clockwise and counterclockwise mode with additional\ndark skyrmionic modes, i.e. modes that cannot be observed in the experiment due to low spectral\nweight. These anti-crossings differ significantly in magnetic field position, spectral weight and\ngap size and the dominant hybridization observed also in our experiments is the one of the BR\n8mode. By means of real space images for different times t(see Supplementary Fig. S3 for details),\nthose dark modes are identified as higher-order clockwise modes. Based on the number of nodes,\nwe refer to these modes as Sextupole (6 nodes), Octupole (8 nodes) and Dectupole (10 nodes)\nCW mode. The following hybridizations are observed in our theoretical model: breathing mode -\nOctupole CW mode, CW mode - Dectupole CW mode, CCW mode - Sextupole CW mode. The\neffect on the mode characteristic is shown in the time evolution of the dynamic magnetization\ncomponent in Fig. 4. From this, it can clearly be seen that the individual modes additionally\nexhibit the excitation features of their hybridization counterpart.\nEven though more resonance branches are crossing each other, only the modes listed above\nlead to a repulsion due to the anisotropy. To verify these observations, we calculate the inner\nproduct, denoted as\nv\u000b\f\fv\f\u000b\n, of the normalized eigenstates v\u000bin momentum space at different\nmagnetic fields for K = 0 and K 6=0, respectively. Here, \u000band\findicate the eigenvalue indices.\nThe results are shown in Tab. I. The field values are chosen to be close to the centre of the observed\nhybridizations, in the order: CW – Dectupole CW, BR – Octupole CW, CCW - Sextupole CW.\nFrom the calculations, it is evident that the anisotropy does not affect the orthogonality between\nthe eigenstates CW – Octupole CW, BR – CCW, and BR – Sextupole CW. As linearly indepen-\ndent eigenstates, the crossing does not result in repulsion and hybridization, respectively. For the\nremaining inner products, the value changes significantly by introducing the additional energy\nterm. Approaching the respective intersection point increases the inner product further, indicating\na stronger interaction between the resonance modes.\nIn order to approach these findings from a phenomenological point of view, we further exam-\nine the real space images of the discussed resonance modes for K = 0, regarding their symmetry\ncharacteristics. From the visualization, we extract the number of symmetry axis mfor each mode,\nsummarized in Tab. II. While the CW and CCW modes only exhibit one symmetry axis, the num-\nbermincreases for the higher-order CW modes, in accordance with their node number. Due to\nthe axial symmetry of the breathing mode, we denote mto be infinity in this case. Considering\nthe hybridizing modes and their corresponding numbers of symmetry axis m, it follows that the\ncubic anisotropy imposes a symmetry selection rule, allowing hybridizations only between states\nwith axial symmetry and even m, or with even or odd number m, respectively.\n9TABLE I. Inner product calculated for various fields, with and without cubic anisotropy.\nH= 0.42Hc2,0H= 0.63Hc2,0H= 0.83Hc2,0\n\nv\u000b\f\fv\f\u000b\nK= 0K= 0.0002K= 0K= 0.0002K= 0K= 0.0002\nCW-Dectupole 0 0.015 0 0.010 0 0.002\nCW-Octupole 0 0 0 0 0 0\nBR-Octupole 0 0.040 0 0.117 0 0.047\nBR-CCW 0 0 0 0 0 0\nBR-Sextupole 0 0 0 0 0 0\nCCW-Sextupole 0 0.026 0 0.036 0 0.037\nTABLE II. Number of symmetry axis massigned to the respective modes.\nMode Number of symmetry axis m\nCCW 1\nCW 1\nSextupole CW 3\nOctupole CW 4\nDectupole CW 5\nBR 1\nNext, we are able to account for the size of the hybridization gap gquantitatively by extracting\nthe minimal frequency difference between the breathing and octupole mode branches from the\ntheoretical and experimental results. The calculated resonance frequencies, normalized by\n\u0017int\nc2,0=g\u00160\u0016B\n2\u0019~Hint\nc2,0 (2)\nare given in dimensionless units, with internal critical field Hint\nc2,0determined without cubic\nanisotropy. For small K, we assume Hint\nc2,0\u0019Hint\nc2, in order to obtain an estimate for the physical\nunits. The internal critical field is connected to the saturation magnetization Msby the internal\nsusceptibility \u001fint\nconin the conical phase [11, 32, 33],\nMs=Hint\nc2,0\u001fint\ncon: (3)\nIn Fig. 5, the calculated field dependence of the excitation frequencies (black dots) is shown for\nK = 0, K = 0.0002 and K = 0.0004. While the colors of the dots indicate the excitation geometry\n10with respect to the applied magnetic field direction (blue: in-plane, red: out-of-plane), the size\nof the dots reflects the spectral weight of the resonance modes. As already anticipated before, it\ncan be seen, that including the cubic anisotropy term into the model results in hybridizations of\ndifferent modes, which interaction strengths depend on the anisotropy value.\nIn order to extract the temperature dependence from our calculations quantitatively, it is suffi-\ncient to leave only the anisotropy value as a free parameter. The Ginzburg-Landau coefficient r0,\nthe measure for the temperature, enters our model as a scaling factor proportional to the saturation\nmagnetization Ms. Therefore, by extracting the temperature-dependent parameters Hint\nc2,0andMs,\nthe calculated spectra can be given in physical units at the corresponding temperature.\nThe anisotropy constant ~Kis determined with the help of the results obtained in [14, 15]. It\nwas found that, exceeding ~Kth\nc\u00190:0001 , the LTS phase would stabilize in the theoretical model.\nNote that in order to avoid confusion, the dimensionless anisotropy constant is marked by a tilde,\nhere. This value corresponds to the dimensionless value obtained from the experiments,\n~Kexp\nc=K\u001b;c\n\u00160Hint\nc2,0Ms\u00190:07 (4)\nwith the anisotropy constant K\u001b;cgiven in units of energy density [14]. Considering now the\nobserved anisotropy values in [15], the theoretical model can provide an estimate of the gap size,\nas a function of temperature.\nThe experimental data is evaluated the same way as described above. The resonance positions\nof the breathing and octupole mode in the vicinity of the hybridization were extracted from two\nLorentzian peak fits. Calculating the frequency difference \u0001fbetween both branches as a function\nof field, we define the minimum value as the gap size g. In Fig. 6(a) the fitting results are shown\nfor the measurements performed at 6 K. At this temperature, gobtained from \u0001f, given in the\ninset, reads 450 MHz.\nFor a conclusive temperature dependence of the gap size, we extend our analysis to the tem-\nperatures covered by our experiments. Our findings from both measurements and calculations are\ncollated in Fig. 6(b). In both crystals studied, we observe a similar trend qualitatively, namely a\ndecrease of gwith increasing temperature. In order to serve as a guide to the eye, a grey line is\n11added to the figure. Its intersection point is chosen to be consistent with the temperature-dependent\nanisotropy measurements from Halder et al. [15]. The theoretically estimated gap size, converted\nto physical units, is given by the blue line, reminiscent of the behaviour in the experimental data.\nIts slope, however, differs by approximately a factor of 2.3 with respect to the experiment. A small\ndeviation between experimental and theoretical data is induced by the conversion from dimension-\nless to physical units. However, this error is assumed to be small compared to the factor mentioned\nabove, leaving the origin of this discrepancy open. In order to further investigate these findings,\nwe included an additional exchange anisotropy term\nFe[M] =CX\nkX\n\u000bk2\n\u000bM\u000b\nkM\u000b\n\u0000k (5)\ninto the free energy functional. This anisotropy is allowed by the symmetry of the P213space\ngroup, and was already taken into consideration in [17, 34]. Depending on the sign of C, the helix\npitch aligns either along h111i(C < 0)orh100i(C > 0). Fixing all parameters apart from C, we\ncalculate the resonance spectra and therefore also the gap size, in the same manner as before, as a\nfunction of the anisotropy strength. The extracted values are summarized in (Fig. 7). The blue line\ncorresponds to the results already shown in Fig. 6(b), only including the cubic anisotropy. The red\ncurves (light red C > 0, dark redC < 0) are obtained for both cubic and exchange anisotropy,\nwhile the cubic anisotropy parameter was set to K= 0:0002 . For certain exchange anisotropy\nstrengths, independent of the sign of C, the gap reaches the experimentally observed value of\napproximately 0.6 GHz. In the case of positive C, the value increases with increasing anisotropy\nstrength. For negative C, the exchange anisotropy is counteracting the cubic anisotropy, leading\nfirst to a reduction of the gap size. Exceeding a certain value, the exchange anisotropy is prevail-\ning, causing the gap size to increase again. Due to energetically more favorable alignment either\nalong theh111iorh100idirections, the exchange anisotropy induces a distortion of the skyrmion\nlattice into an oblique and square lattice, depending on the external field. Similar observation,\ninduced by the cubic anisotropy in this case, were already reported in [14, 21]. Nevertheless,\ndespite this additional energy contribution, the resonance spectra are left essentially unchanged,\nas shown in Fig. 7(b) - (c). The calculations were performed for Fig. 7(b) K = 0.0002, C = 0 and\nFig. 7(c) K = 0.0002, C = 0.2, respectively, illustrating the significant impact on the hybridization\nprocess. These results suggest that further anisotropy terms [20, 21], like the one discussed above,\nmight play a non-negligible role and need to be considered for a comprehensive analysis of the\ntemperature dependence of the hybridization.\n12SUMMARY\nIn conclusion, we report a comprehensive study of the nucleation of the low-temperature\nskyrmion lattice that is stabilized by the magnetocrystalline anisotropy of the sample via a field-\ncycling protocol and systematically trace the characteristic hybridization in the breathing mode.\nBy employing two different sample geometries, we were able to remove contributions from the\ndemagnetizing field and investigate the dynamics of low-temperature skyrmion modes for various\ntemperatures. Our analysis of the hybridization gap highlights that the cubic magnetocrystalline\nanisotropy terms govern the origin of the hybridization following a particular symmetry selec-\ntion rule. In addition, we show that in the non-collinear skyrmion phase, the contribution of the\nexchange anisotropy to the excitation spectra can possibly be traced via the observation of the\nhybridization gap and detailed theoretical analysis. Note that this contribution can not be detected\nin collinear phases.\nACKNOWLEDGEMENT\nWe would like to acknowledge fruitful discussions with M. Garst. O. L. thanks the Lever-\nhulme Trust for financial support via RPG-2016-391. We further acknowledge JSPS Grants-In-\nAid for Scientific Research (Grant Nos. 18H03685, 20H00349, 21H04440), JST PRESTO (Grant\nNo. JPMJPR18L5) and Asahi Glass Foundation for funding. This work has also been funded\nby the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under SPP2137\nSkyrmionics, TRR80 (From Electronic Correlations to Functionality, Project No. 107745057,\nProject G9), and the excellence cluster MCQST under Germany’s Excellence Strategy EXC-2111\n(Project No. 390814868).\n13\u0003s.lee.14@ucl.ac.uk\nychristian.back@tum.de\n[1] S. M ¨uhlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, and P. B ¨oni,\nScience 323, 915 (2009).\n[2] X. Z. Yu, Y . Onose, N. Kanazawa, J. H. Park, J. H. Han, Y . Matsui, N. Nagaosa, and Y . Tokura, Nature\n465, 901 (2010).\n[3] S. Seki, J.-H. Kim, D. S. Inosov, R. Georgii, B. Keimer, S. Ishiwata, and Y . Tokura, Phys. Rev. B 85,\n220406 (2012).\n[4] T. Adams, A. Chacon, M. Wagner, A. Bauer, G. Brandl, B. Pedersen, H. Berger, P. Lemmens, and\nC. Pfleiderer, Phys. Rev. Lett. 108, 237204 (2012).\n[5] S. Seki, X. Z. Yu, S. Ishiwata, and Y . Tokura, Science 336, 198 (2012).\n[6] S. L. Zhang, A. Bauer, D. M. Burn, P. Milde, E. Neuber, L. M. Eng, H. Berger, C. Pfleiderer, G. van\nder Laan, and T. Hesjedal, Nano Lett. 16, 3285 (2016).\n[7] F. Qian, H. Wilhelm, A. Aqeel, T. T. M. Palstra, A. J. E. Lefering, E. H. Br ¨uck, and C. Pappas, Phys.\nRev. B 94, 064418 (2016).\n[8] M. Mochizuki, Phys. Rev. Lett. 108, 017601 (2012).\n[9] Y . Onose, Y . Okamura, S. Seki, S. Ishiwata, and Y . Tokura, Phys. Rev. Lett. 109, 037603 (2012).\n[10] Y . Okamura, F. Kagawa, M. Mochizuki, M. Kubota, S. Seki, S. Ishiwata, M. Kawasaki, Y . Onose, and\nY . Tokura, Nat. Commun. 4, 2391 (2013).\n[11] T. Schwarze, J. Waizner, M. Garst, A. Bauer, I. Stasinopoulos, H. Berger, C. Pfleiderer, and\nD. Grundler, Nat. Mater. 14, 478 (2015).\n[12] H. Oike, A. Kikkawa, N. Kanazawa, Y . Taguchi, M. Kawasaki, Y . Tokura, and F. Kagawa, Nat. Phys.\n12, 62 (2016).\n[13] K. Karube, J. S. White, N. Reynolds, J. L. Gavilano, H. Oike, A. Kikkawa, F. Kagawa, Y . Tokunaga,\nH. M. Rønnow, Y . Tokura, and Y . Taguchi, Nat. Mater. 15, 1237 (2016).\n[14] A. Chacon, L. Heinen, M. Halder, A. Bauer, W. Simeth, S. M ¨uhlbauer, H. Berger, M. Garst, A. Rosch,\nand C. Pfleiderer, Nat. Phys. 14, 936 (2018).\n[15] M. Halder, A. Chacon, A. Bauer, W. Simeth, S. M ¨uhlbauer, H. Berger, L. Heinen, M. Garst, A. Rosch,\nand C. Pfleiderer, Phys. Rev. B 98, 144429 (2018).\n14[16] F. Qian, L. J. Bannenberg, H. Wilhelm, G. Chaboussant, L. M. Debeer-Schmitt, M. P. Schmidt,\nA. Aqeel, T. T. M. Palstra, E. Br ¨uck, A. J. E. Lefering, C. Pappas, M. Mostovoy, and A. O. Leonov,\nSci. Adv. 4, eaat7323 (2018).\n[17] L. J. Bannenberg, H. Wilhelm, R. Cubitt, A. Labh, M. P. Schmidt, E. Leli `evre-Berna, C. Pappas,\nM. Mostovoy, and A. O. Leonov, npj Quantum Mater. 4, 11 (2019).\n[18] U. G ¨ung¨ord¨u, R. Nepal, O. A. Tretiakov, K. Belashchenko, and A. A. Kovalev, Phys. Rev. B 93,\n064428 (2016).\n[19] U. K. R ¨oßler, A. A. Leonov, and A. N. Bogdanov, J. Phys.: Conf. Ser. 303, 012105 (2011).\n[20] A. Bauer and C. Pfleiderer, in Topological Structures in Ferroic Materials: Domain Walls, V ortices and Skyrmions,\nedited by J. Seidel (Springer, 2016) p. 1.\n[21] A. Aqeel, J. Sahliger, T. Taniguchi, S. M ¨andl, D. Mettus, H. Berger, A. Bauer, M. Garst, C. Pfleiderer,\nand C. H. Back, Phys. Rev. Lett. 126, 017202 (2021).\n[22] D. Mettus, M. Halder, A. Chacon, A. Bauer, and C. Pfleiderer, unpublished (2021).\n[23] R. Takagi, M. Garst, J. Sahliger, C. H. Back, Y . Tokura, and S. Seki, Phys. Rev. B 104, 144410 (2021).\n[24] L. D. Landau and E. M. Lifshitz, Statistical Physics (Pergamon Press, 1980).\n[25] F. N. Rybakov, A. B. Borisov, S. Bl ¨ugel, and N. S. Kiselev, New J. Phys. 18, 045002 (2016).\n[26] Y . Okamura, F. Kagawa, S. Seki, and Y . Tokura, Nat. Commun. 7, 12669 (2016).\n[27] T. Nakajima, H. Oike, A. Kikkawa, E. P. Gilbert, N. Booth, K. Kakurai, Y . Taguchi, Y . Tokura, F. Ka-\ngawa, and T. Arima, Sci. Adv. 3, e1602562 (2017).\n[28] M. T. Birch, R. Takagi, S. Seki, M. N. Wilson, F. Kagawa, A. ˇStefan ˇciˇc, G. Balakrishnan, R. Fan,\nP. Steadman, C. J. Ottley, M. Crisanti, R. Cubitt, T. Lancaster, Y . Tokura, and P. D. Hatton, Phys. Rev.\nB100, 014425 (2019).\n[29] L. J. Bannenberg, R. Sadykov, R. M. Dalgliesh, C. Goodway, D. L. Schlagel, T. A. Lograsso, P. Falus,\nE. Leli `evre-Berna, A. O. Leonov, and C. Pappas, Phys. Rev. B 100, 054447 (2019).\n[30] H. C. Wu, K. D. Chandrasekhar, T. Y . Wei, K. J. Hsieh, T. Y . Chen, H. Berger, and H. D. Yang, Journal\nof Physics D: Applied Physics 48, 475001 (2015).\n[31] P. Che, I. Stasinopoulos, A. Mucchietto, J. Li, H. Berger, A. Bauer, C. Pfleiderer, and D. Grundler,\n“Confined dipole and exchange spin waves in a bulk chiral magnet with dzyaloshinskii-moriya inter-\naction,” (2021), arXiv:2104.06240 [cond-mat.mtrl-sci].\n[32] J. Waizner, Spin wave excitations in magnetic helices and skyrmion lattices, Ph.D. thesis, Universit ¨at\nzu K ¨oln (2016).\n15[33] M. Garst, J. Waizner, and D. Grundler, J. Phys. D 50, 293002 (2017).\n[34] P. Bak and M. H. Jensen, J. Phys. C 13, L881 (1980).\n[35] A. Savitzky and M. J. E. Golay, Anal. Chem. 36, 1627 (1964).\n[36] A. G. Gurevich and G. A. Melkov, Magnetization Oscillations and Waves (CRC Press, 1996).\n16H\nMAGNETMAGNETSample\nCPWhrf(b)\nT (K)\nμ0H (mT)Cu2OSeO3 H || <100>\n0 9003060\n15(a)\nHinit\nHn\nincHn\ndecHTS\nTilted\nHhighHlowConical\nLTS\nCycle\nElongated\nSkyrmionsconical\n1.06.0\n3.5-3.50|S11|\n-7\n0 115 230\nScanf (GHz)\nHc2T = 6 KField-Cycling(c)\nn = 0\nn = 100200\n0100μ0H (mT)HhighHlowHinitFIG. 1. Illustration of magnetization phases under field cycling and experiment schematic. (a) Mag-\nnetic phase diagram of Cu 2OSeO 3under field-cycling after high-field cooling. (b) Schematic representation\nof experimental setup (see the text for details). (c) (Top) Magnetic field as a function of scan number. (Bot-\ntom) Typical measurement profile of field-cycling protocol for n= 100 cycles at 6 K. Note the color depth\nencodes the intensity of the reflection signal, jS11j. The region indicated by the dashed red lines corresponds\nto the measurements during field-cycling, where the applied external magnetic field alternates linearly, as\nindicated in Fig. 1(a) and the top panel. ’Scan’ indicates a single frequency sweep at a fixed magnetic field\n\u00160H.\n17(f)\n(g)μ0H = 90mT T = 6K\n-2.50\nn = 0\nn = 400n = 100-Q+QKittel\nCCWf/fc2 |S11|\n-5T = 6K\nCW(a)\n(b)\n(c)\n(d)(e)\n0.050.100.15Amplitude (a.u)\n0 5 10 15 20\nNumber of cycles,n(×103)3.083.103.123.14fr(GHz)\nHc2\n3.02 3.12 3.220.910.920.930.94|S11|\nf (GHZ)n = 10000\n2.2 2.9 3.6\nf(GHz)0.25\n0.250.751.251.75|S11|\n0.10.20.30.40.51251020n x 103T = 6 K\nBRFIG. 2. Comparison of measured and calculated excitation spectra for the platelet sample and the\nevolution of field cycling. (a - c) Experimental data for microwave absorption spectra recorded at 6 K for\n0, 100, and 400 cycles, respectively. Fitted dots represent various magnetization phases. The dotted line is\ndrawn at\u00160H= 90 mT, i.e., the field value at which the plots shown in (e - g) are evaluated. (d) Calculated\nexcitation spectrum with anisotropy constant K= 0.0002. (e) Microwave reflection difference jS11jfor\nvarious number of cycles natT= 6 K. (f) The amplitude and the resonance frequency (g), as a function of\nfield cycling. The results are extrapolated from the fitting routine as shown by the inset ( n= 10000 cycles).\nNote that the shaded region indicates the background signal, where the LTS lattice is slowly nucleating. For\nfurther details, see S2.\n18T (K)�0H (mT)Cu2OSeO3 H || <100> high-field cooled150\n100\n50\n00 10 20 30 40CuboidPlatelet(c)(a)\n4 K 6 K 12 K\n16 K\n18K\n20 K\n22 K\n24 K\n25 K\n-7 -3.5 0\n|S11|\n \n200 0 10024\nμ0H (mT)f (GHz)\n100 0 100 024246\ng(b)\n6 K 12 K 18 K\n20K\n25K\n30 K\n35 K\n40 K\n45 K200 0 100 100 0 100 0\n-1 -0.5 0\n|S21|\nμ0H (mT)FIG. 3. Temperature evolution of microwave spectra for platelet and cuboid samples. Collective\nspin excitation spectra at various temperatures and corresponding magnetic phase diagram for the platelet\nand cuboid-shaped samples. (a) shows the reflection difference jS11jfor the platelet sample at various\ntemperatures for n= 800 . (b) Transmission difference jS21jfor the cuboid sample for n= 1000 cycles.\nNote that a clear hybridization gap, g, is observed for both samples, decreasing with increasing temperature.\n(c) Phase diagram elaborating the LTS phase boundaries for both platelet and cuboid samples.\n19FIG. 4. Calculated real-space images of skyrmions under resonant excitation of the six lowest-\nfrequency modes for K = 0.0002. The columns correspond to different characteristic moments during\nthe oscillation; notably, they are separated in time by one-fourth of the period \u001c. For clarity, only the\nnormalized dynamic magnetization component \u000eMz(t) is shown here. From top to bottom, the modes are\nidentified as the dectupole, clockwise gyration mode (CW), octupole, the breathing mode (BR), counter-\nclockwise (CCW) and the sextupole gyration.\n20FIG. 5. Field evolution of the resonance frequencies. Resonance frequencies as a function of the applied\nmagnetic field for increasing anisotropy strength. From left to right: K = 0, K = 0.0002, K = 0.0004.\nFrequencies are given in units of the internal critical field \u0017int\nc2,0=g\u00160\u0016B\n2\u0019~Hint\nc2,0(critical internal field Hint\nc2,0\nfor K = 0) and fields in units of transition field Hc2,0. The size of the dots represents the estimated spectral\nweight of the modes.\n20 40 60 80 100 120\n0H(mT)\n2.53.03.54.04.5f(GHz)\nΔfT = 6 K n = 20000 (a) (b)\ng\n50 60 70 80 90\nμ0H (mT)2007001200 Δf (MHz)\n0 10 20 30 40\nTemperature,T(K)0200400600g(MHz)\nTheory Platelet Cube\nFIG. 6. Field-cycling and temperature evolution of the hybridization gap. (a) Fitted frequency depen-\ndence of upper and lower Br modes as a function of field. The inset shows the frequency difference of the\ntwo modes, \u0001f, plotted as a function of field for the platelet sample at T= 6 K. (b) Temperature evolution\nof hybridization gap, g, for the platelet and cuboid samples. A smooth grey line is added as a guide to the\neye.\n21FIG. 7. Comparison of the effect of different anisotropy terms on the hybridization gap size. (a)\nTheoretically expected gap size as a function of anisotropy strength. The lines depict the results, calculated\nwith (light and dark red) and without (blue) exchange anisotropy. In the two cases (red lines), Kis set to\nbe a constant ( K= 0:0002 ). (b) - (c) Resonance frequencies spectra as a function of the external field, for\nK = 0.0002, C = 0 and K = 0.0002, C = 0.2.\n22SUPPLEMENTARY\nS.1 EXPERIMENTAL SETUP AND DATA PROCESSING\nThe magnetization dynamics of the low-temperature skyrmion phase can be determined by\nmeasuring the subtle microwave response of the magnetic system as a function of frequency\nand field. Thus, conventional magnetic spectroscopy techniques such as electron spin resonance\n(ESR) spectrometer cannot be employed. In ESR, the magnetic response to a system is probed by\nscanning the magnetic field at a fixed frequency. For this reason, we have performed our measure-\nments utilizing a broadband ferromagnetic resonance (BB-FMR) technique with a vector network\nanalyzer (VNA). The microwave line from the VNA is linked to the coplanar-waveguide (CPW)\nvia a semi-rigid cable. Then, the RF current, Icos(!t)is flown through the CPW which induces\na local magnetic field and directly couples with the magnetization, Min the sample. As the static\nmagnetic field His varied, frequencies are swept at a fixed microwave power (0 dBm). Therefore,\nthe microwave absorption in this reference is the change in reflection (transmission) power of the\nCPW that induces the driving field ( Hrf). The obtained spectra are denoted respectively as S11and\nS21for the response to reflection or transmission.\nFig. S1 (a) shows typical reflection spectra S11obtained directly from the VNA for the plate-\nshaped sample at T= 6K. For systematic measurement analysis and accurate determination of\nexcitation dynamics, the data was processed over three distinctive stages. Firstly, we carefully\nremove the excessive noise in the signal utilizing an algorithm involving a numerical filter. In order\nto preserve all necessary information from the data obtained, a lossless analytical Savitzky-Golay\nfilter [35] was employed, which the filter recalculates the signal from the convolution coefficients\nby fitting successive data points by a polynomial function using linear least squares. The filtered\nsignal,S0\n11is shown in Fig. S1(b). Secondly, the spectra in the region of interest were subtracted\nby a background signal ( \u00160H= 300 mT) obtained at a high magnetic field, as shown in Fig. S1(c).\nLastly, Fig. S1(d) shows the resulting signal after a full conversion into a linear scale, i.e., jS11j=\n10\u0001S11=10. The data were then fitted using multi-peak Lorentzian to extract the parameters of\ninterest. We have used the same systematic protocol for all data presented in this paper.\n23S.2 ADDITIONAL CYCLING DATA\nHere, we present additional spectra data obtained respectively, at 800 and 20000 field cycles as\nshown in Figs. S2(a - b) obtained at T= 6K. A vertical line is drawn to indicate the approximate\nfield (\u00160H\u001975 mT) where hybridization is clearly seen for both number of cycles. In order to\ninvestigate the evolution of the dynamic modes, we further plot the line-cut data at this field for\ndifferent numbers of field cycles in Fig. S2(c - g).\nOur results highlight that the intensity of the hybridization becomes more visible with the\nnumber of accumulated field cycles which must be connected to the volume fraction of the LTS\nphase. For example, a direct comparison between n= 100 andn= 400 illustrates clearer evidence\nof the hybridization gap in the LTS phase. Note that for n= 400 , the spectrum is dominated by\nthe excitations of the LTS phase in the field range of \u00160H\u001925\u0000125mT and only little weight\nis observed in the\u0006Q modes of the conical phase. Furthermore, the intensities of the LTS modes\nremain almost constant from n= 400 onwards even when the number of cycles is extended to the\nextreme case of n= 20000 .\nS.3 THEORETICAL MODEL\nThe theoretical analysis presented in this report draws upon the well established Ginzburg-\nLandau theory for chiral magnets [11, 14, 21]. The energy functional Fconsists of the following\nenergy terms,\nF[M] =X\nk \nr0Mk\u0001M\u0000k+J\n2(k\u0001k)(Mk\u0001M\u0000kg) +iDM\u0000k\u0001(k\u0002Mk)\nv+UX\nk2;k3;k4(Mk\u0001Mk2)(Mk3\u0001Mk4)\u000ek+k2+k3+k4;0\u0000B\u0001M0\n\u0000\u001cM0NM0+X\nk(k\u0001Mk)(k\u0001Mk)\nk\u0001k\n\u0000KX\nk;k2;k3;k4X\n\u000bM\u000b\nkM\u000b\nk2M\u000b\nk3M\u000b\nk4\u000ek+k2+k3+k4;0!\n:(6)\nThe parameters are the Ginzburg-Landau coefficients r0andU, the strength of the exchange\nJ, Dzyaloshinsky-Moriya D, dipolar\u001cinteractions, with demagnetization tensor N(tr(N) = 1),\nmagnetic field Band the cubic anisotropy constant K. It was already shown that the additional\n24anisotropy term is sufficient to stabilize the LTS phase [14] and to quantitatively reproduce the\nchanges in the resonance spectra, with respect to the high-temperature skyrmion phase [21].\nThe magnetization dynamics are described by the lossless Landau-Lifshitz equation of mo-\ntion [36]:\n@tM=\u0000\rM\u0002Beff (7)\nHere,\r=g\u0016B=~denotes the gyromagnetic ratio and Beff=\u0000\u000eF=\u000eMthe local effective mag-\nnetic field, given by the derivative of the total energy functional with respect to the magnetization.\nFor further analysis, the free energy Fand the magnetization Mare divided into a static and\ndynamic component,\nF=Fstat+F(t)\nM=Mmf+\u000eM(t):(8)\nInserting this ansatz into the equation of motion 7 and only keeping terms linear in F(t)and\n\u000eM(t), the resonance condition reads [32],\n!res=Im\u0002\nEigenvalues\u0000\nW\u0001\u0003\n(9)\nwithW=\rMmf\u0002\u001f\u00001\n0and\u001f\u00001\n0=\u000e2Fstat\n\u000eM2\f\f\f\nMmf.\nBy performing the calculation of Eq 9 in momentum space, we are able to determine the nor-\nmalized eigenvectors v\u000b\nj(k)and eigenfrequencies !\u000b(k)of the given matrix Wnumerically, with\ninternal indices j2[1;2;3]and momentum index k. A more detailed derivation can be found in\n[11, 14].\nBased on the rather large dimensions of the CPW, we assume the driving field to be homoge-\nneous in the performed calculations, therefore setting k= 0. The demagnetization factors of the\nsamples under investigation read approximately Nx= 0:17,Ny= 0:12andNz= 0:71for the\nplatelet and Ni= 1=3for a sphere, which we take as an approximation of the cuboid, where the\nexternal magnetic field is applied along the z-direction.\nNext to the breathing, clockwise and counterclockwise mode, we focus our study on three\nadditional modes, ranging in the same frequency regime. For the identification of the individual\n25modes, real space images of the normalized dynamics magnetization \u000eM zare calculated for dif-\nferent times t. The results are shown in Fig. S3. Based on the time evolution and the number\nof nodes, these modes are identified as higher-order clockwise excitations and are referred to as\nSextupole (6 nodes), Octupole (8 nodes) and Dectupole(10) mode.\nIn order to verify the repulsive character of the hybridizing modes, we calculate the inner\nproduct of the interacting resonance branches for K= 0andK6= 0. For a homogeneous driving\nfield, the internal product is given by [11],\n\nv\u000b\f\fv\f\u000b\n:=X\nj;K2LR\u0000\nv\u000b\nj(K)\u0001?v\f\nj(K) (10)\nwith reciprocal lattice LR. The results are presented in Table II of the main text, revealing the\neffect on the orthogonality of the eigenvectors of the hybridizing modes.\nFrom our calculations, we observe that neither the shape of the sample nor a finite kvector\nchanges the gap size, significantly (not shown here). In order to further investigate the discrepancy\nbetween the experimentally and theoretically obtained gap size we added an additional anisotropy\nterm\nFe[M] =CX\nkX\n\u000bk2\n\u000bM\u000b\nkM\u000b\n\u0000k (11)\ninto the free energy functional. Depending on the sign of the prefactor C, the helix pitch aligns\neither along theh111i(C < 0)or theh100i(C > 0)direction. In order to determine the effect\nof the exchange anisotropy on the hybridization gap size, we calculate the resonance spectra for\nvariousCvalues. For more details we refer the reader to the corresponding section in the main\ntext.\n267\n5\n3\n1\nS11 (dB)\n7\n5\n3\n1\nS11' (dB) |S11|T = 6K 300 mT 75 mT (a) (b)\n(c) (d)\nf (GHz) f (GHz)\nΔS11 (dB)\n1 2 3 4 5 63.5\n2.5\n1.5\n0.5\n0.5\n1 2 3 4 5 61.0\n0.8\n0.6\n0.4FIG. S1. Examples of microwave absorption spectra. (a) Transmission spectra, S11obtained from the\nVNA at T= 6K for the plate-shaped sample at \u00160H= 75 mT and \u00160H= 300 mT. (b) Numerically pro-\ncessed transmission spectra, S0\n11. (c) Microwave spectra difference, \u0001S11. (d) Microwave spectra difference\nafter linear conversion, jS11j.\n27(c)\n(d)\n(e)\n(f)\n(g)0.00.51.0\n0.00.5\n0.00.5\n0.00.5\n2 3 4 5\nf(GHz)0.00.5\n|S11|\n0 100 400 800 20000μ0H = 75mT T = 6K\nμ0H (mT)200 0 1001.03.56.01.03.56.0f (GHz)\nn = 20000n = 800(a)\n(b)\n-3.50|S11|\n-7-3.50|S11|\n-7f (GHz)FIG. S2. Evolution of measured excitation spectra for the platelet sample. (a - b) Experimental data\nfor normalized microwave absorption spectra, jS11j, recorded at 6 K for 800 and 20000 number of cycles,\nrespectively. (c - g) jS11jas a function of frequency measured at \u00160H= 75 mT (dotted vertical line in\n(a - b)) for different number of field cycles n, upon completing the full cycling protocol. Above 100 cycles,\nvarious dynamic modes, including CCW skyrmions (purple), Br skyrmions (red), and conical (orange), are\nclearly observed.\n28FIG. S3. Calculated real-space images of skyrmions under resonant excitation of the six lowest-\nfrequency modes for K = 0. The columns correspond to different characteristic moments during the oscil-\nlation; notably, they are separated in time by one-fourth of the period \u001c. For clarity, the normalized dynamic\nmagnetization component \u000eMz(t) is shown here. From top to bottom, the modes are identified as the dec-\ntupole, clockwise gyration mode (CW), octupole, the breathing mode (BR), counterclockwise (CCW) and\nthe sextupole gyration.\n29"    },    {        "title": "2112.04124v1.Lattice_dynamics_and_its_effects_on_magnetocrystalline_anisotropy_energy_of_pristine_and_hole_doped_YCo__5__from_first_principles.pdf",        "content": "Lattice dynamics and its effects on magnetocrystalline anisotropy energy of pristine and hole-doped\nYCo 5from first principles\nGuangzong Xing,1,2,\u0003Yoshio Miura,1and Terumasa Tadano1,2,y\n1Research center for Magnetic and Spintronic Materials,\nNational Institute for Materials Science, Tsukuba, Ibaraki, 305-0047, Japan\n2Elements Strategy Initiative Center for Magnetic Materials,\nNational Institute for Materials Science, Tsukuba, Ibaraki, 305-0047, Japan\n(Dated: December 9, 2021)\nWestudythelatticedynamicseffectsonthephasestabilityandmagnetocrystallineanisotropy(MCA)energy\nofCaCu 5-typeYCo 5atfinitetemperaturesusingfirst-principlescalculationsbasedondensityfunctionaltheory\n(DFT). Harmonic lattice dynamics (HLD) calculations indicate that YCo 5with 56 full valance electrons is\ndynamicallyunstableandthisinstabilitycanbecuredbyreducingthenumberofelectrons( Ne). Crystalorbital\nHamilton population analysis reveals that the observed phonon instability originates from the large population\nof antibonding states near the Fermi level, which is dominated by the Co atoms in the honeycomb layer. The\nantibonding state depopulates with decreasing Ne, resulting in stable phonons for hole-doped YCo 5withNe\n\u001455. We then evaluate the temperature-dependent MCA energy using both HLD and ab initio molecular\ndynamics (AIMD) methods. For the pristine YCo 5, we observe a very weak temperature decay of the MCA\nenergy, indicating little effect of lattice dynamics. Also, the MCA energies evaluated with AIMD at all target\ntemperatures are larger than that of the static hexagonal lattice at 0 K, which is mainly attributed to the\nstructural distortion driven by soft phonon modes. In the hole-doped YCo 5, where the distortion is suppressed,\na considerable temperature decay in MCA energy is obtained both in HLD and AIMD methods, showing that\nlattice dynamics effects on MCA energy are non-negligible.\nI. INTRODUCTION\nTheintermetalliccompoundCaCu 5-typeRECo 5(RE=rare\nearth)hasbeenextensivelystudiedinthepast,bothexperimen-\ntally and theoretically, due to its superior magnetic properties\nsuchashighCurietemperature,saturationmagnetization,and\nstrong coercivity [1–5]. One of the intrinsic magnetic prop-\nerties associated with high coercivity is magnetocrystalline\nanisotropy (MCA) energy, which depends only on the crystal\nstructuresandchemicalcompositions. MCAenergyisdefined\nas the ground state energy difference between different direc-\ntions of the magnetic field with respect to the crystal axes. In\nthepreviousstudies,itwasfoundbothexperimentallyandthe-\noreticallythatRECo 5(RE=Y,La,Ce,Sm)exhibitsauniaxial\nMCAenergythatresultsinanenergeticallyfavoredalignment\nofthemagneticmomentsalongthecrystallographic c-axis[6–\n12]. InRECo 5,thelargeMCAmainlyarisesfromtwoaspects:\nthe spin-orbit coupling of the itinerant 3delectrons at the Co\nsites, and the spin-orbit interaction of the localized 4felec-\ntrons at the RE site. Compared with 3delectrons, the MCA\noriginating from 4felectrons shows a strong temperature de-\npendence. The two contributions become comparable at fi-\nnite temperatures. For example, Alameda et al.reported the\nMCA constant of YCo 5was 7.4 MJ/m3at 4.2 K and slightly\ndecreased to 5.8 MJ/m3at room temperature [6]. The corre-\nspondingvaluesforthewell-knownSmCo 5are30MJ/m3and\n17 MJ/m3[12], respectively.\nAtheoreticalstudyofthetemperatureeffectonMCAenergy\niscrucialsincepermanentmagnetsareusuallyusedinahigh-\n\u0003XING.Guangzong@nims.go.jp\nyTADANO.Terumasa@nims.go.jptemperatureenvironment. Animportantphysicalfactoraffect-\ningMACatfinitetemperatureisthethermalfluctuationofthe\nspin moment. This effect has approximately been included in\ndensity functional theory (DFT) calculations with disordered\nlocalmoment(DLM)approach[13]. Forexample,Matsumoto\net al.[14] investigated the temperature dependence of MCA\nenergyandmagnetizationinhole-dopedYCo 5usingtheDLM\napproach. A DLM-based first-principles magnetization ver-\nsusfield(FPMVB)approachisintroducedbyPatrick etal.to\nstudy the temperature-dependent MCA energy of YCo 5and\nGdCo 5[15]. They found excellent agreement of temperature-\ndependentMCAenergycurvesinGdCo 5betweentheFPMVB\napproach and experiments.\nLattice dynamics is another important factor that can affect\nthe MCA energy at finite temperature [16, 17]. At elevated\ntemperatures, thermal excitation of phonons is expected to\nchange the electronic structures, which then influences the\nMCA energy. For example, Urru et al.[17] has recently re-\nported that the spin-reorientation transition of MnBi could be\nexplained by considering the vibrational free energy contri-\nbution calculated within the harmonic approximation (HA).\nThus, it is intriguing to study lattice dynamics effects on the\ntemperature-dependent MCA energy of permanent magnetic\nmaterials, particularly the CaCu 5-type RECo 5which displays\nthe same hexagonal lattice as MnBi.\nIn this work, we theoretically investigate the lattice dy-\nnamics and its effect on the MCA energy of YCo 5\u0000xat fi-\nnite temperature to study the MCA energy originating from\nthe itinerant states. By performing phonon calculations of\nYCo 5within the harmonic approximation, we show that the\nexperimentally-reported CaCu 5structure is dynamically un-\nstable. This phonon instability originates from the presence\nof large antibonding states near the Fermi level, which can be\nremoved by reducing the number of electrons ( Ne). We findarXiv:2112.04124v1  [cond-mat.mtrl-sci]  8 Dec 20212\nthatatleastoneelectronneedstoberemovedfromthesystem\ntostabilizephononsofYCo 5\u0000x. Forthehole-dopedYCo 5\u0000x,\nwe then evaluate the lattice dynamics effect on the MCA en-\nergy from the difference of the vibrational free energies com-\nputedwithdifferentspinorientations. Inthehigh-temperature\nregion, the phonon contribution to the MCA energy becomes\nsignificantandcomparabletotheMCAenergyat0Kobtained\nfrom DFT calculation. To evaluate the lattice dynamics effect\nin the undoped YCo 5, we also perform ab initio molecular\ndynamics (AIMD) simulations and find that the MCA energy\nhardly changeswith increasing temperature. Weattribute this\nweak temperature dependence mainly to structural distortion.\nThe structure of this paper is organized as follows. In the\nnext section, we describe our theoretical methods in detail.\nThe MCA energy both at 0 K and finite temperatures, and\ncomputational details are described in Secs. IIA, IIB, and\nIIC, respectively. In Sec. III, we show our main results. The\ndynamical instability of the CaCu 5-type YCo 5with 56 full\nvalance electrons and the microscopic origin of stabilization\nby reducing Ne, or hole-doping, are discussed in Sec. IIIA.\nIn Sec. IIIB, we evaluate the MCA energy both at 0 K and\nfinite temperatures using different approaches and discuss the\nrole of lattice dynamics in the temperature dependence of the\nMCA energy. Finally, we summarize this study in Sec. IV.\nII. METHODS\nA. MCA energy at 0 K from DFT calculation\nWeperformanoncollinearspin-orbitinteractioncalculation\nusing the well-known force theorem [18] to obtain the MCA\nenergy,KDFT\nuat 0 K, which is defined as\nKDFT\nu =EDFT\n?\u0000EDFT\nk; (1)\nwhereEDFT\n?andEDFT\nkare the sum of the occupied Kohn–\nSham eigenenergies with the magnetic moment ( m) being\naligned along the hard ([100] direction, m?c) and easy\n([001] direction, mkc) axes, respectively. The positive\nKDFT\nuindicates that the energetically favorable mis parallel\nto the crystallographic c-axis.\nIn order to understand the atomic site-dependent MCA en-\nergy, we carry out the second-order perturbation calculation.\nInthetight-bindingregime,theHamiltonianforspin-obitcou-\nplingisgivenbythesumofthecontributionsfromeachatomic\nsite,HSO=P\ni\u0018iLi\u0001Si,where Li(Si)isthesingle-electron\nangular (spin) momentum operator, and \u0018iis the spin-orbit\ncouplingconstantofatom i. Comparedwith3 dbandwidth,\u0018i\n(\u0018Co= 69.4 meV) is relatively small, which can be treated as\na perturbation term. The second-order perturbation energy is\nexpressed as\nE(2)=\u0000X\nkunoccX\nn0\u001b0occX\nn\u001bjhkn0\u001b0jHSOjkn\u001bij2\n\u000f(0)\nkn0\u001b0\u0000\u000f(0)\nkn\u001b;(2)\nwherejkn\u001biis the unperturbed state with energy \u000f(0)\nkn\u001b.k,\nn, and\u001brepresent the wave vector, band index, and spin,respectively. Theindex\"occ\"and\"unocc\"meansthesumover\nthe occupied and unoccupied states. jkn\u001bican be expanded\nas a sum of atomic orbitals, jkn\u001bi=P\ni\u0016ckn\ni\u0016\u001bji\u0016\u001bi, where\ntheatomicorbitalslabeledas \u0016andthecoefficient ckn\ni\u0016\u001bcanbe\nobtained by DFT calculations. Therefore, the site-dependent\nsecond-order contribution to the total energy with different\nspin processes and atomic orbitals can be calculated using\nfirst-principles calculations.\nTheMCAenergyat0K, KPT\nu,withinthesecond-orderper-\nturbationisdefinedas KPT\nu=E(2)\n?\u0000E(2)\nk,whereE(2)\n?(E(2)\nk)\naretotalenergiescalculatedbyEq.(2)withthemagnetization\nalonghard(easy)axisofCuCa 5-typeYCo 5. Thenthedecom-\nposedpartof KPT\nuatdifferentatomicsitewithspin-transition\nprocess is written as\nKPT\nu=X\niKi\n\u001b)\u001b0\n=X\niKi\n\")\"+Ki\n#)#+Ki\n\")#+Ki\n#)\":(3)\nThefirsttwotermsinEq.(3)arecalledspin-conservingterms\nand the rest are spin-flip terms, which originate from the spin\nscatteringprocessbetweentheoccupiedandunoccupiedstate\nnear the Fermi level. The detailed formulation of Ki\n\u001b)\u001b0is\ngiven in Ref. [19].\nB. Lattice dynamics contribution to MCA energy at finite\ntemperatures\nThe MCA energy at finite temperatures, Ku, can be eval-\nuated by the difference of Helmholtz free energies computed\nwith two different spin orientations, which is defined as\nKu(V;T) =F?(V;T)\u0000Fk(V;T); (4)\nwhereVis the volume of YCo 5. According to adiabatic\napproximation, Kuis considered as the sum of the following\nterms\nKu\u0019KDFT\nu(V0) +Kel\nu(V0;T) +Kphon\nu(V0;T)\n+Kmag\nu(V0;T) +KTE\nu(T);(5)\nwhereKDFT\nu(V0) is the MCA energy obtained from the DFT\ncalculation with the optimized volume V0,Kel\nu(V0;T) and\nKphon\nu(V0;T) represent the electronic and vibrational con-\ntribution to Ku, respectively. Kmag\nu(V0;T)is the magnon\ncontribution, which is not considered in the current study.\nKel\nufor the present system is very small and thus can be ig-\nnored. We roughly evaluated the thermal expansion term,\nKTE\nu(T)\u0019KDFT\nu(V(T))\u0000KDFT\nu(V0), using the mea-\nsured temperature-dependent lattice constants of YCo 5from\nRef. [20]. Since the change in the lattice constant cis neg-\nligible, we only change the in-plane lattice constant aas\na(T) =kT+a0, wherea0is the lattice constant optimized\nby DFT atT= 0K, andk\u00187\u000210\u00005Å K\u00001is the propor-\ntionalityconstantestimatedfromtheexperimentaltemperature\ndependence of the avalue [20].3\nThe vibrational MCA energy Kphon\nuis computed from the\ndifference between the vibrational free energies at different\nmagnetic orientations. In the HA, it becomes [17]\nKphon\nu =\u00001\n\flnZ0;?\nZ0;k(6)\nwhere\f= (kBT)\u00001istheinversetemperaturewith kBbeing\nthe Boltzmann constant, and Z0;mis the partition function of\nthe harmonic oscillator at the magnetic state mdefined as\nZ0;m=Y\nq\u00171\n2 sinh\u0010\n\f~!q\u0017;m\n2\u0011; (7)\nwith ~being the reduced Planck constant. !q\u0017;mis the har-\nmonic phonon frequency of the \u0017th branch at crystal momen-\ntumqinthemagneticstate m,whichcanbeobtainedbydiag-\nonalizing the dynamical matrix constructed from the second-\norderinteratomicforceconstants(IFCs)calculatedinthesame\nmagneticstate m. Ifthedifferenceof \u0001!q\u0017=!q\u0017;?\u0000!q\u0017;k\nis small, Eq. (6) can be approximated by a linear function of\n\u0001!q\u0017as\nKphon\nu\u0019~\n2X\nq\u0017[1 + 2n(!q\u0017;k)]\u0001!q\u0017;(8)\nwheren(!) = (e\f~!\u00001)\u00001istheBose–Einsteindistribution\nfunction. Since \u0001!q\u0017is very small in many cases, Eq. (8)\nis a reasonable approximation to Eq. (6) and will be used for\ncomputing the modal contribution to Kphon\nuin Sec. IIIB2.\nIn the classical limit ( ~!0), Eq. (8) reduces to Kphon\nu\u0019\n\f\u00001P\nq\u0017(\u0001!q\u0017=!q\u0017;k).\nIt is important to recall that the above harmonic lattice dy-\nnamics (HLD) method can be employed only when the struc-\nture is dynamically stable, namely, !q\u0017\u00150is satisfied for\nall phonon modes in the Brillouin zone. If the structure is\ndynamically unstable, which is the case for pristine YCo 5as\nwill be shown later, Eq. (7) becomes ill-defined and the HLD\nmethod breaks down. Hence, a beyond (quasi-)harmonic ap-\nproach is required for evaluating the vibrational contribution\nto the MCA energy of YCo 5. The limitation of the HLD\nmethod can be overcome by using anharmonic lattice dynam-\nicsmethods,suchastheself-consistentphonontheory[21–23]\nor temperature-dependent effective potential method [24], or\nab initiomolecular dynamics (AIMD) method. In this study,\nwe employ AIMD and evaluate the finite-temperature MCA\nenergy as\nhKuiMD(T) =1\nNNX\ns=1KDFT\nu(fRigs);(9)\nwhereKDFT\nu(fRigs)istheMCAenergycomputedatthe sth\nstructure snapshot fRigssampled from the AIMD trajectory\natthetargettemperature T,andNisthenumberofstructures\nsampled at each T. This approach can be used even for the\nsystems where an unstable phonon mode ( !2\nq\u0017<0)exits.\nBesides, contributions from the anharmonic terms of the po-\ntentialenergysurfaceandtheeffectofstructuraldistortionare\nincluded automatically.C. Computational details\nFirst-principle calculations in this study were performed\nby using the projector augmented wave (PAW) method [25],\nwithinthePerdew–Burke–Ernzerhof(PBE)generalizedgradi-\nentapproximation(GGA)[26],asimplementedintheVienna\nab initio simulation package (VASP) [27]. Lattice constants\nandatomicpositionsofCaCu 5-typeYCo 5werecarefullyopti-\nmizedwithakinetic-energycutoffof400eVfortheplane-wave\nexpansion, and the k-point mesh was generated automatically\ninsuchawaythatthemeshdensityinthereciprocalspacebe-\ncamelargerthan450Å\u00003. Forthestructuraloptimizationand\nphonon calculations, we used the Methfessel–Paxton smear-\ning method [28] with the width of 0.2 eV. On the other hand,\ntheMethfessel–Paxtonsmearingmethodwithasmallerwidth\nof 0.05 eV and tetrahedron method with the Blöchl correc-\ntion [29] was used for calculating the MCA energy. Note that\nkmesh density of\u00186000 Å\u00003was used to calculate KDFT\nu\n(KPT\nu) because the anisotropy energy is small, necessitating\na denser mesh. The initial local moments of 3 \u0016Band\u00000:3\n\u0016BweresetforCoandYatoms,respectively,forthecollinear\nspin-polarized calculations, including phonon calculations.\nThe harmonic phonon calculations were conducted by us-\ningthefinite-displacementmethod,asimplementedin /a.pc/l.pc/a.pc/m.pc-\n/o.pc/d.pc/e.pc[22, 30]. A 2\u00022\u00022supercell containing 48 atoms\nwas adopted for the phonon calculation. To obtain Kphon\nu\nusingEq.(6),weincorporatednoncollinearspin-orbitinterac-\ntion into the DFT calculation and estimated the second-order\nIFCs for two different spin orientations: m?candmkc.\nWe have confirmed that our implementation yields consistent\nresults with the density-functional perturbation theory imple-\nmentation [17] for MnBi. In addition, AIMD simulations\nwere carried out to evaluate hKuiMD(T)using Eq. (9). At\neach temperature, we performed a collinear AIMD run for\n5000stepswithatime-stepof2fsandextracted100structural\nsnapshots uniformly from the last 2500 steps. For each sam-\npled snapshot, we then calculated the difference of the total\nenergies between the two spin orientations. Finally, we esti-\nmated the average over the 100 snapshots using Eq. (9). We\nhave confirmed that 100 structures were sufficient to obtain\nconverged values of hKuiMD.\nIII. RESULTS AND DISCUSSION\nCaCu 5-type YCo 5displays a layered hexagonal structure\n(space group: P6=mmm) shown in Fig. 1(a). The cobalt\natoms are located either at the 2cor3gWyckoff sites, which\nrespectivelybelongtothehoneycombandkagomelayers. We\nlabel these inequivalent cobalt atoms as Co 2cor Co 3g. The\nsymmetryofYCo 5willbeloweredwithspin-orbitinteraction\nwhenthemagnetizationisalongthehardaxis([100]direction).\nThe 3-fold Co 3gwill be separated into two inequivalent cites,\nwhich have the corresponding multiplicities of 1 (denoted as\n3g1hereafter) and 2 (denoted as 3g2hereafter), respectively.\nTheoptimizedlatticeconstantsare a= 4:907Åandc= 3:942\nÅ, which agree reasonably well with the experimental values\nat 5 K (a= 4:936Å,c= 3:984Å, Ref. [20]) and at room4\nCo3g1Co3g2Co2cxyzY1a(a)\nxyz(b)\nFigure 1. (a) Crystal structure of CaCu 5-type YCo 5. The large\nspheresrepresentYatoms,andthesmalleronesareCoatoms. Wyck-\noff positions for different atoms are marked with the corresponding\natom colors. (b) Crystal structure of YCo 5with a 2\u00022\u00022 supercell.\nThearrowsindicatethedisplacementpatternofthethelowest-energy\nsoftphononmodeatLpoint[ q=(1\n2,0,1\n2)]. Thelengthofthearrows\nrepresents the relative magnitude of displacements. Cartesian axes\nare labeled as x,y, andz, respectively.\ntemperature( a= 4:950Å,c= 3:986Å,Ref.[20]; a= 4:921\nÅ,c= 3:994Å, Ref. [32]).\nA. Dynamical stability\nTheharmonicphonondispersioncurvesoftheCaCu 5-type\nYCo 5are shown in Fig. 2(a) by solid lines. While the previ-\nous experimental studies showed that the crystal structure of\nYCo 5is the CaCu 5type at and above room temperature, the\nharmonicphononofthepristineYCo 5turnsouttobeunstable.\nThelargestphononinstabilityoccursattheLpointsof q=(1\n2,\n0,1\n2),(0,1\n2,1\n2),and(1\n2,1\n2,1\n2). Thesecond-largestinstabilityis\nobserved at the M points of q=(1\n2, 0, 0), (0,1\n2, 0), and (1\n2,1\n2,\n0). Since the calculated phonon frequency is rather sensitive\ntothelatticeparameters,wealsocomputedthephonondisper-\nsioncurveswithvarious aandcvalues. Theresultsareshown\ninFig.S1ofthesupplementarymaterial(SM)[36]. Asshown\ninFig.2(b),thesmallest !2\nqvalue,whichappearseitheratthe\nL point or M point, is negative in the wide region of the ac\nplane,includingtheareaaroundtheexperimental (a;c)values\nreported in Refs. [20, 31–35]. While the phonons can be dy-\nnamicallystableintheupperleftregion( a.4:7Å,c&3:9Å)of the figure, the experimental and theoretical lattice param-\neters are located away from that region. Thus, the observed\ndynamicalinstabilityofYCo 5cannotexclusivelybeattributed\nto the slight errors in the GGA-PBE lattice parameters.\nThe soft modes at the L point and M point involve the in-\nplanedisplacementsofCo 2catomsaccompaniedbyrelatively\nsmall displacements of other atoms, as shown in Fig. 1(b).\nOnce the atoms are slightly displaced along the polarization\nvector of the soft mode, the six-fold rotational symmetry in\nthe honeycomb layer breaks, which decreases the total en-\nergy of the system. To understand the origin of the phonon\ninstability, we investigated the bonding nature of the nearest-\nneighboratomicpairsbycomputingtheCrystalorbitalHamil-\nton Populations (COHPs) [37] using the /l.pc/o.pc/b.pc/s.pc/t.pc/e.pc/r.pccode [38].\nThe\u0000COHP values computed with collinear magnetism are\nshown in Fig. 3(a), where the dash and solid lines represent\nthe results for the majority and minority spins, respectively.\nThemostnotablefeatureinFig.3(a)isthelargenegativepeak\nof\u0000COHPforthenearestCo 2c-Co2cbondattheFermilevel,\nrepresentingitsstrongantibondingnature. TheprojectedDOS\nin Fig. 3(b) shows that the antibonding state is formed by the\ndxyanddx2\u0000y2orbitals,whilethecontributionsfromtheother\norbitalsbothatCo 2candCo 3g(seeFig.S2intheSM[36])sites\nare negligible around the Fermi level. The large antibonding\ncontribution is energetically unfavorable. Hence, the system\nreacts in such a way that the population of the antibonding\nstate decreases at the Fermi level, which can be achieved by\ndistortingthestructureandtherebyliftingtheorbitaldegener-\nacy.\nThe large antibonding population at the Fermi level may\nbe reduced by hole doping, which is expected to improve the\nstability of the CaCu 5structure. To see the effect of hole\ndoping,wegraduallyreducedthenumberofvalenceelectrons,\nNe,fromtheoriginalvalueofthepristineYCo 5(Ne= 56)and\ncomputedthe\u0000COHPvaluesfortherelevantCo 2c-Co2cbond.\nAs shown in Fig. 3(c), the Fermi level lowers with reducing\nNe, and the peak position of the DOS in the minority spin\nstateshiftstothehigherenergy. Inaddition,thepopulationof\ntheCo 2c-Co2cantibondingstateattheFermileveldecreasesas\ndecreasingNe. Consequently,weobservedthatthephononsof\nthe CaCu 5structure became dynamically stable when Ne.\n55(see Fig. 2(a) for the Ne= 55case) and the soft-mode\nfrequency at the L point increased gradually with decreasing\nNe, as tabulated in Table I.\nOn the basis of the above analyses, we now discuss possi-\nble scenarios explaining the inconsistency between our com-\nputational result and experiments about the stability of the\nCaCu 5-type YCo 5. First, since the detailed structure analy-\nsesusingXRDmeasurementhavebeenreportedonlyatroom\ntemperature so far, there could be a lower-symmetry phase of\npristine YCo 5in the low-temperature region which is left to\nbe discovered. Should there be a structural phase transition,\nit would be second-order because no anomaly has been ob-\nservedinthelatticeparametersdownto5K[20]. Thesecond\nscenario is the possible off-stoichiometry in the experimental\nsamples. YCo 5isthemothercompoundfromwhichotherCo-\npoor and Co-rich phases, including YCo 3, Y2Co7, Y5Co19,\nand Y 2Co17, are derived by partial substitution [40]. Since5\n4.64.85.05.2a(˚A)3.84.04.2c(˚A)(b)Expt. (temperature)Expt. (pressure)Expt. (bulk)Expt. (thin film)°160°120°80040100!2q(meV2)°MK°ALHA02040Frequency (meV)Ne= 56Ne= 55(a)\nFigure 2. Phonon frequencies of CaCu 5-type YCo 5computed within harmonic approximation. (a) Phonon dispersion curves for the pristine\nYCo 5(Ne= 56, solid line) and the hole-doped system ( Ne= 55, dashed line). (b) Contour plot of the smallest !2\nqvalue for YCo 5(Ne\n= 56) computed with various lattice parameters. The experimental lattice parameters are shown with open symbols. Circles and crosses are\ntemperature- and hydrostatic pressure-dependence of lattice parameters from Ref. [20] and Ref. [31], respectively. Squares and diamonds are\nlattice parameters of bulk [32–34] and thin-film phases [35], respectively.\n−1 0 1−0.4−0.20.00.20.40.6−COHP(a)\nCo2c-Co 2c\nCo2c-Co 3g\nCo3g-Co 3g\nCo2c-Y\n−1 0 1\n/epsilon1−/epsilon1F(eV)−101PDOS (states/eV)Co2c(b)\ndxy,dx2−y2\ndyz,dxz\ndz2−1 0 1−10010DOS (states/eV)(c)\nNe= 56\nNe= 55\nNe= 54\n−1 0 1\n/epsilon1−/epsilon1F(eV)−0.50.00.5−COHP(d)\nCo2c-Co 2cNe= 56\nNe= 55\nNe= 54\nFigure3. (a)COHPcalculationofdifferentatomicbondsforYCo 5withNe= 56. (b)Projecteddensityofstates(PDOS)for3 dstatesatCo 2c\nsite. (c)Thetotaldensityofstates(DOS)and(d)COHPcalculationofCo 2c-Co2cbondsforYCo 5withNe=56,55,and54respectively. The\nsolid and dash lines in all panels represent minority- and majority-spin, respectively.\nthe formation energies of these derivatives are comparable to\nthatofYCo 5[41],theoff-stoichiometryofYCo 5+zcanoccur\nrather easily in the samples prepared by a high-temperature\ntreatment. Indeed, Pareti et al.[42] reported that YCo 5+z\ncould be synthesized for the range of \u00000:7< z < 2:3by\nperforming a quenching from high temperature. The Co-poor\noff-stoichiometryintroducesexcessholestothesystem,which\nare expected to decrease the population of the antibonding\nstateattheFermilevelandtherebyimprovethestabilityofthestructure. Third,thephononinstabilityofYCo 5mightappear\ndue to the limited accuracy of the present HLD calculation\nbased on GGA-PBE. While we have also observed the same\nphonon instability with other semilocal functionals such as\nPBEsol,moreaccuratetreatmentoftheelectroniccorrelation,\nfor example, by hybrid functionals or by the combination of\nthe dynamical mean-field theory and the DFT (DFT+DMFT)\nmay yield stable phonons. Moreover, the quantum fluctuation\nof atomic nuclei may help avoid the structural distortion to6\nTable I. Calculated Ne-dependence of the MCA energies ( KDFT\nu,\nKPT\nu,KVCA\nu) using different approaches (see the main text),\nanisotropy in the orbital moment ( \u0001mo), and the squared frequency\nof the soft mode at the L point ( !2\nL).\nNeMCA constant (meV/f.u.)\n\u0001mo(\u0016B/f.u.)!2\nL(meV2)KDFT\nuKPT\nuKVCA\nu\n56 0:383 0:177 0:383 0 :074\u000061:1\n55 2:947 2:451 2:757 0 :206 4 :7\n54 1:797 1:442\u00000:611 0 :153 79 :3\n53 0:356 0:426\u00001:407 0 :074 130 :1\n52\u00000:643\u00000:384\u00001:553 0 :028 156 :3\n51\u00001:069\u00000:833\u00001:331 0 :001 156 :1\n51 52 53 54 55 56\nNe−2024KDFT\nu(meV/f.u.)\nReducingNeVCA\nExpt. (Rothwarf et al.)\nExpt. (Alameda et al.)\nFigure4. CalculatedMCAenergy, KDFT\nu,asthefunctionofvalance\nelectron number obtained by manually reducing Neand VCA ap-\nproaches,respectively. Thecomputationalresultsarecomparedwith\ntheexperimentalvaluesreportedbyRothwarf etal.[39]andAlameda\net al.[6]. The dash lines are shown to guide the eye.\noccurandstabilizetheCaCu 5-typeYCo 5,akintothecaseofa\nsuperconductinghydride[43]. Answeringwhichofthesesce-\nnariosisthemostplausiblewouldbechallengingasitrequires\nadditional experimental and theoretical investigations, which\nare left for a future study.\nB. MCA energy of YCo 5\u0000x\nIn the following, we will focus on the MCA energy of pris-\ntine YCo 5(Ne= 56) as well as the hole-doped systems\n(Ne<56) at 0 K and finite temperatures. In the case of\nresults at 0 K, we compare the KDFT\nuwithKPT\nufor vari-\nousNevalues. In addition, we also calculated MCA energy,\nKVCA\nu, of the Y(Fe,Co) 5alloy using virtual crystal approx-\nimation (VCA) [44, 45] to compare with the experimental\nresults. Note the volume of Y(Fe,Co) 5alloy is fixed for the\nVCA calculation, similar to the case of manually reducing\nNe. The corresponding total number of valance electrons for\nthe VCA calculations, shown in the first column of Table I,\nis also labeled as Ne. In addition, the spin moment, namelym\u001b;Co,atdifferentCositesandanisotropyinorbitalmoment,\ndefined as \u0001mo=mo\nk\u0000mo\n?, are also studied. As for the\nfinite-temperatureeffects,wecalculatedtheHLDcontribution\nKphon\nu[Eq. (6)] for the dynamically stable cases and the en-\nsemble averagehKuiMD[Eq. (9)] for Ne= 54, 55, and 56\n(pristine YCo 5).\n1. MCA energy at zero kelvin\nFirst, we discuss the Nedependency of the MCA energy at\n0 K obtained from the different approaches mentioned above.\nThepresentcomputationalresultsareshowninFig.4andTa-\nble I. We also show the Nedependence of the spin magnetic\nmoments in Appendix A. As shown in Fig. 4 and Table I,\nall approaches gave similar trends in that positive MCA en-\nergy first increases, then decreases, and finally changes the\nsign to negative as Nedecreases. This trend is qualitatively\nconsistent with the MCA constants of Y(Fe xCo1\u0000x)5(x=\n0.00, 0.10, 0.16, and 0.26) alloys measured by Rothwarf et\nal.[39]. BothKDFT\nuandKPT\nuvalues of the pristine YCo 5\n(Ne= 56) are smaller than the experimental values of \u00183.6\nmeV/f.u. [6] and\u00183.20 meV/f.u. [46]. Similar underestima-\ntion problems have also been observed in the previous LDA-\norGGA-levelcalculations[7,47,48],whichhavesuccessfully\nbeen resolved either by considering the orbital polarization\ncontribution[47,49,50],byusingDFT+Uapproach[7],orby\nDFT+DMFT[48]. Nonetheless,sincethemainsubjectofthis\nwork is to study the lattice dynamics effects on the MCA en-\nergy,andtheGGA-PBEfunctionalgivesthereasonableresults\noftheNedependency(Fig.4),westillemploytheGGA-PBE\nfunctional in the study.\nFrom the results of the second-order perturbation calcula-\ntion, we found that the significant enhancement of the MCA\nenergy atNe= 55 originates from the spin conservation term\nKi\n#)#at the Co sites. Moreover, we have calculated the\nanisotropy of orbital moments \u0001mo(see Table I) and con-\nfirmedthattheBrunorelation( KDFT\nu/\u0001mo)[51,52]nicely\nholds forNe\u001554. More detailed discussion can be found in\nAppendix B.\n2. Lattice dynamics effects on the MCA energy at finite\ntemperature\nNext, we discuss the lattice dynamics effects on the MCA\nenergyofthepristineandhole-dopedYCo 5. Figure5(a)shows\nthetemperaturedependenceof Kphon\nu[Eq.(6)]calculatedfor\nthehole-dopedYCo 5(Ne\u001455),forwhichthephononsaredy-\nnamicallystableat0K.Inthe Ne\u001554region,theKphon\nuvalue\ngraduallydecreasesasthetemperatureraises. As Nedecreases\nfurther,thetemperaturedependencebecomesweaker,andthe\nsignoftheslopeeventuallychangesat Ne'53. Tounderstand\ntheoriginofthis Nedependence,wecomputedthemode-and\nenergy-decomposed Kphon\nuforNe= 55and 54, as shown in\nFig. 5(b), and (c), respectively. Here, the mode-decomposed\nvalue was defined as Kphon\nu;q\u0017=~\n2[1 + 2n(!q\u0017;k)]\u0001!q\u0017, and7\n−0.050.00 0.05\n∆F±(ω)\n−1.0−0.5 0.0 0.5 1.0Kphon\nu,qν(meV)\n−0.050.00 0.05\n∆F±(ω)\n−1.0−0.5 0.0 0.5 1.0Kphon\nu,qν(meV)\n0200 400 600 800 1000\nTemperature (K)−2.5−2.0−1.5−1.0−0.50.00.5Kphon\nu(meV/f.u.) Ne=55\nNe=54\nNe=53\nNe=52\nNe=51\nVCA(Ne=55.5)\nYFe5(Ne=51)\nFrequency (meV)\nFrequency (meV)Ne=55\nT=300KNe=54\nT=300K(a) (b) (c)\npositive negative positive negative\nFigure 5. (a) Calculated temperature dependence of Kphon\nu. Solid lines represent Kphon\nuof hole-doped YCo 5obtained by manually reducing\nNe, dashed line represents Kphon\nVCAof Y(Co,Fe) 5obtained by VCA approach with Ne= 55.5, and dotted line is the Kphon\nuvalue of YFe 5. (b)\nand (c) show the room-temperature ( 300K) mode- and energy-decomposed Kphon\nuof hole-doped YCo 5withNe=55, and 54, respectively.\nIn the mode-decomposed figures, the line color represents the sign of Kphon\nu;q\u0017, whose absolute value is represented by the linewidth.\nthe energy decomposition was performed by computing\n\u0001F\u0006(!) =1\nNqX\nq\u0017Kphon\nu;q\u0017\u000e(!\u0000!q\u0017;k)\u0012(\u0006\u0001!q\u0017);(10)\nwhere\u0012(x)isthestepfunctionwhichbecomes \u0012(x) = 1when\nx>0and0otherwise. \u0001F+(!)(\u0001F\u0000(!))includesthecon-\ntributionsfromphononmodesthatincrease(decrease) Kphon\nu.\nItisstraightforwardtoshowR1\n0d![\u0001F+(!) + \u0001F\u0000(!)]\u0019\nKphon\nu. It is remarkable in Fig. 5(b) that the soft phonon at\ntheLpointcontributesnegativelytotheMCAenergyatfinite\ntemperature. On the other hand, the lower transverse acous-\ntic mode along the \u0000-M line and the second-lowest optical\nmodearoundthe \u0000pointhaverelativelylargepositivecontribu-\ntions. Moreover, \u0001F\u0000(!)showsalargenegativepeakaround\n!\u001818meV, which can be attributed to the weakly disper-\nsivebranchesinthisenergyregion. Whenonemoreelectronis\nremovedfromthesystem,theantibondingnaturefurtherweak-\nens,andconsequently,theoverallphononfrequencyincreases,\nas shown in Fig. 5(c). This hardening decreases n(!q\u0017;k)in\nEq.(8),whichpartiallyexplainsthedisappearanceoftheclear\nstructures in the mode- and energy-resolved Kphon\nu. We also\nobserved the same qualitative change in the VCA calculation,\nwhich is shown in Fig. S3 of the SM [36].\nThe zero-temperature MCA energy, shown in Fig. 4, takes\nthe maximum value at around Ne= 55. By contrast, the lat-\ntice dynamics contribution tends to become more significant\nasincreasing Nefrom54to56. Hence,theeffectofthelattice\ndynamics on the MCA energy is expected to be larger in the\nlargeNeregion close to Ne= 56. Indeed, in the VCA cal-\nculation with Ne= 55:5, theKphon\nuvalue at 800 K is around\n\u00001:5meV/f.u. (dashedlineinFig.5(a)),whoseabsolutevalue\nis comparable to that of KDFT\nu = 2:4meV/f.u. Accordingly,\nwemayexpectthattheeffectofthelatticedynamicswouldbe-\ncome even more significant for the pristine YCo 5(Ne= 56).\nHowever, as Neapproaches 56, the phonon frequencies tendtosoften,especiallynoticeableattheLpoint,andtheeffectof\nphonon anharmonicity would be greater. The phonon anhar-\nmonicity is expected to influence the phonon frequencies at\nfinite temperatures and thereby the Bose–Einstein occupation\nfunction, which would affect the temperature-dependence of\nKphon\nu. Moreover, the (incipient) distortion of the hexagonal\nstructure, which is not considered in the HLD approach, may\nalso influence temperature dependence.\nTo understand the effect of the phonon anharmonicity and\nthe possible structural distortion on the MCA energy at fi-\nnite temperature, we compare the calculated hKuiMDvalues\nwithKtot\nu=KDFT\nu +Kphon\nuin Fig. 6(a). These two val-\nues should be similar in the high-temperature (classical) limit\nwhen phonon anharmonicity is weak. Besides, when there\nare no structural distortion, hKuiMD\u0019KDFT\nushould hold\nin theT!0limit because the present AIMD neglects the\nnuclear quantum effect, i.e., zero-point motion. Indeed, for\nNe= 54where the phonon frequencies are relatively higher\nthan theNe= 55case,hKuiMDandKtot\nuagree reasonably\nwellwitheachother. Also,the hKuiMDvalueat10Kis1.815\nmeV/f.u., which is in close agreement with KDFT\nu = 1:797\nmeV/f.u. (see Table I). By contrast, in the case of Ne= 55,\nthehKuiMDvalueat10Kis\u00182.24meV/f.u.,whichis \u001824%\nsmaller than KDFT\nu = 2:947meV/f.u. We also observed sim-\nilar discrepancy between the canonical average of the orbital\nmoment,h\u0001moiMD, at 10 K and the \u0001movalue obtained for\nthe perfectP6=mmmstructure shown in Table I (see Fig. S4\nintheSM[36]). Thesediscrepanciesindicatethatthehexago-\nnalP6=mmmstructure transformed to another lower-energy\nstructure in the present AIMD simulations at low tempera-\ntureseventhoughtheharmonicphononwasdynamicallystable\nwhenNe= 55(Fig. 2(a)). We have found that another struc-\nture possessing the P63=mcmsymmetry was slightly more\nstable than the P6=mmmphase by\u00180.25 meV/f.u., which\nexplains the observed discrepancy. Nevertheless, except for\nthe offset at T= 0K, the temperature decay of hKuiMDis8\n0 200 400 600 800\nTemperature (K)−0.50.00.51.01.52.02.53.0Ku(meV/f.u.)\n(a)\nKtot\nu-55\nKtot\nu-54\n/angbracketleftKu/angbracketrightMD-56/angbracketleftKu/angbracketrightMD-55\n/angbracketleftKu/angbracketrightMD-54\n0 200 400 600 800\nTemperature (K)−3.0−2.5−2.0−1.5−1.0−0.50.0Decay ofKu(meV/f.u.)(b)\nExpt.\nDLM\nKphon\nu-55\nKphon\nu-54Kphon\nVCA-55.5\n/angbracketleftKu/angbracketrightMD-56\n/angbracketleftKu/angbracketrightMD-55\n/angbracketleftKu/angbracketrightMD-54\nFigure 6. Temperature dependence of MCA energy evaluated with\ndifferent methods. (a) Filled symbols are hKuiMDvalues of YCo 5\nevaluated from AIMD with Ne= 56, 55, and 54, respectively. Solid\nlinesareKtot\nu(seetext)ofYCo 5withNe=55,and54,respectively.\n(b) Temperature decay of Kufor YCo 5withNe= 56, 55, 55.5, and\n54, respectively. Dash line is the Kphon\nVCAof Y(Co,Fe) 5(Ne= 55:5)\ndetermined using VCA method. The experimental data (black open\ncircles)andtheoreticalresults(redsolidcircles)basedondisordered\nlocal moment (DLM) method shown were taken from Ref. [6] and\nRef. [15], respectively. The error bars are the calculated standard\ndeviation obtained with the sampling structures.\n−4−2 0 2 40.20.40.60.81.01.2KDFT\nu(meV/f.u.)\nNe= 56\n−4−2 0 2 4\nQL(u1\n2˚A)−4004080Energy (meV/f.u.)\nFigure 7. Calculated MCA energy, KDFT\nu, (upper panel) and total\nenergy (lower panel) as the function of normal coordinate amplitude\nQLof the L-point soft mode for the pristine YCo 5. Note the total\nenergy has been offset by the corresponding value at QL= 0. The\ndash lines are shown to guide the eye.\nquite similar to that of Ktot\nuin the wide temperature range.\nHence, the effect of the anharmonicity on the MCA energy is\ninsignificant for Ne= 55.\nIt is interesting to see that, in the case of Ne= 56, the\nhKuiMDvalues were larger than KDFT\nu = 0:383meV/f.u.\nboth in low- and high-temperature ranges. Also, the hKuiMD\nvalue is almost temperature independent. To obtain deeper\ninsights into the mechanism behind them, we calculated theKDFT\nuvalueasafunctionofthenormalcoordinateamplitude\nQLof the L-point soft mode. As shown in Fig. 7, the sign\nof@2KDFT\nu=@Q2\nLatQL= 0is positive when Ne= 56.\nSince the L-point soft mode induces the structural distortion,\nit should affect the temperature dependence of hKuiMDmost\nsignificantly. Hence, we can approximately write\nhKuiMD\u0019Z1\n\u00001KDFT\nu(QL)PT(QL)dQL;(11)\nwherePT(QL)is the probability distribution of QLat tem-\nperatureT. In the low-temperature region, PT(QL)has a\npeak around either side of the double-well minima, where\nKDFT\nu(QL)becomes\u00181.0 meV/f.u. This value is in good\nagreement withhKuiMDat 10 K. In the high-temperature\nlimit,PT(QL)has a peak around QL= 0, i.e.,hQLi\u00190, but\nthe variance of the distribution hQ2\nLibecomes large. Thus,\nthehKuiMDvalue became larger than KDFT\nu(0)because of\n@2KDFT\nu=@Q2\nL>0. We note that, for Ne= 55and 54, the\nsignof@2KDFT\nu=@Q2\nLisnegative(seeFig.S5intheSM[36]),\nwhich is consistent with the negative sign of Kphon\nu;q\u0017shown in\nFigs. 5(b), (c).\nWealsoshowhKuiMD+TE=hKuiMD+KTE\nu(seeFig.S6\nintheSM[36])whichincludestheeffectofthermalexpansion\nevaluated approximately as explained in Sec. IIB. It is clear\nthat the effect of thermal expansion is insignificant except for\ntheNe= 54case at high temperatures, where the KTE\nuvalue\namounts to\u00000:3mev/f.u.\nFigure 6(b) compares the predicted temperature decay of\nthe MCA energy obtained from various computational meth-\nods, including the DLM result of Ref. [15], and that of the\nexperimentalresult. NotethatthedatashowninFig.6(b)have\nbeen offset by the corresponding value at the lowest tempera-\nture: 10 K forhKuiMD, and 0 K for the others. The present\ncalculationbasedonGGA-PBEpredictsthatlatticedynamics\nhardlyinfluences KuforthepristineYCo 5(Ne= 56)atfinite\ntemperatures mainly due to the presence of the structural dis-\ntortion. Therefore, it is reasonable to conclude that the lattice\ndynamics effect on the MCA energy is negligible for the pris-\ntineYCo 5andthespinfluctuationremainsthedominanteffect\nforyieldingthetemperaturedecayof Ku. However,whenthe\nstructural distortion is suppressed by hole-doping, the lattice\ndynamicseffectbecomesnon-negligibleinthe Ne&54region\nand therefore should be considered as well when comparing\ntheoretical results with experimental ones.\nIV. SUMMARY\nTo summarize, we theoretically studied the structural sta-\nbility and lattice dynamics effects on the MCA energy for\npristine and hole-doped YCo 5using first-principles methods\nbasedonDFT.TheCaCu 5-type(P6=mmm)structureofpris-\ntine YCo 5is predicted to be dynamically unstable at 0 K,\nand the soft phonon modes appear at L and M points. This\nphonon instability originates from the large population of an-\ntibonding states near the Fermi level which is formed by the\nnearest Co atoms in the honeycomb layer. We demonstrated\nthat the phonon instability can be removed by hole doping,9\nTable II. Calculated spin magnetic moment for different Co sites\n(m\u001b;Co). The spin magnetic moments of previous works (experi-\nmental or theoretical) are obtained from Refs. [7, 32, 47, 54–56]\nNem\u001b;Co(\u0016B)\nThis study Previous work\nCo2c Co3g Co2c Co3g\n56 1.498 1.5251.44a1.62b1.31a1.64b\n1.61c1.46d1.68c1.51d\n55 1.685 1.668\n54 1.833 1.810\n53 1.950 1.953\n52 2.061 2.100\n51 2.170 2.231 2.05e2.00e\naExperimental results, Ref. [54, 55]bDLM, Ref. [32]\ncGGA+U,Ref. [7]dFLAPW, Ref. [47]eYFe 5,GGA, Ref. [56]\nwhich depopulates the antibonding states. We also evaluated\nthe effect of lattice dynamics on the MCA energy based on\nharmonic lattice dynamics and AIMD methods by computing\nKphon\nuandhKuiMD, respectively. For the pristine YCo 5, we\nfound that lattice dynamics hardly influence the MCA energy\nat finite temperature. Also, the ensemble average hKuiMDat\nfinitetemperatureturnedouttobelargerthanthe KDFT\nucom-\nputed with the static P6=mmmlattice. We attributed these\nunique features to the dynamical distortion of the structure,\nwhich is induced by the soft phonon modes. By contrast, in\nthe hole-doped YCo 5where the dynamical distortion is sup-\npressed, a much larger temperature decay was observed both\ninKphon\nuandhKuiMD. While the lattice dynamic effect on\nthefinite-temperatureMCAenergyisnotassignificantasthat\nofthespinfluctuation,itseffectshouldnotbeneglectedwhen\nmakingaquantitativecomparisonbetweentheoryandexperi-\nments, particularly for the hole-doped YCo 5.\nACKNOWLEDGMENTS\nWe thank K. Masuda for the fruitful discussion and X. He\nfor assistance with the COHP analysis. This work was par-\ntiallysupportedbytheMinistryofEducation,Culture,Sports,\nScience and Technology (MEXT) as “Elements Strategy Ini-\ntiative Center for Magnetic Materials” (ESICMM, Project\nID: JPMXP0112101004) and as “Program for Promoting Re-\nsearches on the Supercomputer Fugaku” (DPMSD, Project\nID:JPMXP1020200307). Thefiguresofcrystalstructuresare\ncreated by the /v.pc/e.pc/s.pc/t.pc/a.pcsoftware [53].\nAPPENDIX A: MAGNETIC MOMENT\nTheNe-dependent spin magnetic moments at different Co\nsites,m\u001b;Co,arelistedinTableII.For Ne=56,thecalculated\nspin moments of Co 2cand Co 3gare 1.498 and 1.525 \u0016B, re-\nspectively,whichisslightlylargerthantheexperimentalvalues\nof1.44and1.31 \u0016B[54,55]. Tocomparewithprevioustheo-\nreticalresults,aselectionofresultsbasedondifferentmethods\nCo2cCo3g1Co3g2 Y−0.20.00.20.40.60.81.0Ki\nσ⇒σ/prime(meV/atom)(a)\nNe= 56↑⇒↑\n↓⇒↓↑⇒↓\n↓⇒↑\nCo2cCo3g1Co3g2 Y−0.20.00.20.40.60.81.0Ki\nσ⇒σ/prime(meV/atom)(b)\nNe= 55Figure 8. Decomposed atomic sites dependence of MCA energy,\nKi\n\u001b)\u001b0, obtained from second-order perturbation analysis in YCo 5\nwithNe= 56 (a) and 55 (b), respectively.\nissummarizedinTableII.Thesevaluesarespinmomentsob-\ntained by DLM method [32], GGA+U method [7], full poten-\ntial linearized augmented plane wave (FLAPW) method [47].\nOur results match well with the FLAPW values but differ\nslightly from the GGA+U and DLM results. The spin mo-\nments of both Co sites increase with decreasing Ne, which\ncan be explained by the DOS shown in Fig. S7(a), and (b) of\nthe SM [36]. When Neis decreased, the minority spin states\nshift to the higher energy, increasing spin splitting. Since the\nmajorityspinstatesarefullyoccupied,thedecreaseofminor-\nity spin states will lead to an increase of the net single spin\nstates,thenalargerspinmagneticmoment. Notethatthespin\nmagnetic moments for Ne= 51 is slightly larger than that of\nYFe 5withthevaluesof2.082,and2.033 \u0016BforFe 2candFe 3g\naccording to our calculations, which may be attributed to the\nnot fully occupied 3dmajority spin states (see Fig. S7(d) in\nthe SM [36]) in YFe 5(Ne= 51). As shown in Fig. S7(c) and\n(d)oftheSM[36],themajorityspinstatesofY(Fe,Co) 5eval-\nuated by the VCA approach will become partially occupied\nwhenNe\u001452, which is different from the DOS determined\nby reducing Neapproach.\nAPPENDIX B: VALIDITY OF THE BRUNO RELATION\nAccording to the Bruno relation, the MCA energy is pro-\nportional to the anisotropy in the orbital moment ( KDFT\nu/\n\u0001mo). To understand the validity of the Bruno relation with10\nNe,theatomicsitedependenceof Ki\n\u001b)\u001b0withdifferentspin-\ntransition processes are shown in Fig. 8 and Fig. S8 of the\nSM[36],respectively. Themaximum KPT\nuforNe=55origi-\nnatesmainlyfromtheenhancementofpositivespinconserva-\ntion termKi\n#)#both at Co 2c, Co 3g1, and Co 3g2sites. How-\never, positive Ki\n#)#decrease dramatically as NE decreases\n(see Fig. S8 in the SM [36]). The negative spin-flip term\nKi\n\")#of Co 2cand Co 3g1mainly contributes to KPT\nuwhen\nNe\u001452, resulting in a negative KPT\nu. Since the spin-flip\nterm is absent in the orbital magnetic moment, the Bruno re-lation holds for Ne\u001553and not for Ne\u001452. Although\nwe find a qualitatively similar KPT\nufor YCo 5withNe= 51\nand YFe 5with the corresponding values of \u00000:833meV/f.u.,\nand\u00000:919meV/f.u.,respectively,themechanismisdifferent\nfor these two systems. The negative Ki\n\")#(see Fig. S8(d) in\nthe SM [36]) at Co 2cand Co 3g1makes a main contribution\ntoKPT\nufor YCo 5withNe= 51, while negative Ki\n#)#(see\nFig. S9 in the SM [36]) at Fe 2cand Fe 3g1sites also makes a\nconsiderable contribution besides Ki\n\")#in YFe 5.\n[1] P. Larson, I. Mazin, and D. Papaconstantopoulos, Effects of\ndopingonthemagneticanisotropyenergyinSmCo 5\u0000xFexand\nYCo 5\u0000xFex, Phys. Rev. B 69, 134408 (2004).\n[2] H.Ucar,R.Choudhary,andD.Paudyal,Anoverviewofthefirst\nprinciples studies of doped RE-TM 5systems for the develop-\nment of hard magnetic properties, J. Magn. Magn. Mater. 496,\n165902 (2020).\n[3] P. Söderlind, A. Landa, I. L. M. Locht, D. Åberg, Y. Kvash-\nnin, M. Pereiro, M. Däne, P. E. A. Turchi, V. P. Antropov, and\nO. Eriksson, Prediction of the new efficient permanent magnet\nSmCoNiFe 3, Phys. Rev. B 96, 100404(R) (2017).\n[4] C.E.PatrickandJ.B.Staunton,Temperature-dependentmagne-\ntocrystallineanisotropyofrareearth/transitionmetalpermanent\nmagnetsfromfirstprinciples: ThelightRCo 5(R=Y,La-Gd)in-\ntermetallics, Phys. Rev. Mater. 3, 101401(R) (2019).\n[5] S.Kumar,C.E.Patrick,R.S.Edwards,G.Balakrishnan,M.R.\nLees, and J. B. Staunton, Torque magnetometry study of the\nspin reorientation transition and temperature-dependent mag-\nnetocrystallineanisotropyinNdCo 5,J.Phys.: Condens.Matter\n32, 255802 (2020).\n[6] J. M. Alameda, D. Givord, R. Lemaire, and Q. Lu, Co energy\nandmagnetizationanisotropiesinRCo 5intermetallicsbetween\n4.2 k and 300 k, J. Appl. Phys. 52, 2079 (1981).\n[7] M. C. Nguyen, Y. Yao, C.-Z. Wang, K.-M. Ho, and V. P.\nAntropov, Magnetocrystalline anisotropy in cobalt based mag-\nnets: A choice of correlation parameters and the relativistic\neffects, J. Phys.: Condens. Matter 30, 195801 (2018).\n[8] M.Bartashevich,T.Goto,R.Radwanski,andA.Korolyov,Mag-\nneticanisotropyandhigh-fieldmagnetizationprocessofCeCo 5,\nJ. Magn. Magn. Mater. 131, 61 (1994).\n[9] R. K. Chouhan and D. Paudyal, Cu substituted CeCo 5: New\noptimal permanent magnetic material with reduced criticality,\nJ. Alloy. Compd. 723, 208 (2017).\n[10] M.Bartashevich,T.Goto,M.Yamaguchi,andI.Yamamoto,Ef-\nfectofhydrogenonthemagnetocrystallineanisotropyofRCo 5,\nJ. Magn. Magn. Mater. 140-144, 855 (1995).\n[11] O. Gutfleisch, M. A. Willard, E. Brück, C. H. Chen, S. G.\nSankar, and J. P. Liu, Magnetic materials and devices for the\n21st century: Stronger, lighter, and more energy efficient, Adv.\nMater.23, 821 (2010).\n[12] H. Kirchmayr and E. Burzo, Magnetic properties of metals:\nCompounds between rare earth elements and 3d, 4d or 5d ele-\nments,LandoltBörnsteinNumericalDateandFunctionalRela-\ntionshipinScienceandTechnology.NewSeries 3,106(1990).\n[13] J. B. Staunton, L. Szunyogh, A. Buruzs, B. L. Gyorffy, S. Os-\ntanin, and L. Udvardi, Temperature dependence of magnetic\nanisotropy: An ab initio approach, Phys. Rev. B 74, 144411\n(2006).[14] M.Matsumoto,R.Banerjee,andJ.B.Staunton,Improvementof\nmagnetic hardness at finite temperatures: Ab initio disordered\nlocal-moment approach for YCo 5, Phys. Rev. B 90, 054421\n(2014).\n[15] C.E.Patrick,S.Kumar,G.Balakrishnan,R.S.Edwards,M.R.\nLees, L. Petit, and J. B. Staunton, Calculating the magnetic\nanisotropy of rare-earth–transition-metal ferrimagnets, Phys.\nRev. Lett. 120, 097202 (2018).\n[16] K. V. Shanavas, D. Parker, and D. J. Singh, Theoretical study\non the role of dynamics on the unusual magnetic properties in\nMnBi, Sci. Rep. 4, 1 (2014).\n[17] A. Urru and A. Dal Corso, Lattice dynamics eddffects on the\nmagnetocrystalline anisotropy energy: Application to MnBi,\nPhys. Rev. B 102, 115126 (2020).\n[18] G. H. O. Daalderop, P. J. Kelly, and M. F. H. Schuurmans,\nFirst-principlescalculationofthemagnetocrystallineanisotropy\nenergyofiron,cobalt,andnickel,Phys.Rev.B 41,11919(1990).\n[19] Y. Miura, S. Ozaki, Y. Kuwahara, M. Tsujikawa, K. Abe,\nand M. Shirai, The origin of perpendicular magneto-crystalline\nanisotropy in L10–FeNi under tetragonal distortion, J. Phys.:\nCondens. Matter 25, 106005 (2013).\n[20] A. Andreev, Thermal expansion anomalies and spontaneous\nmagnetostriction in rare-earth intermetallics with cobalt and\niron, Handb. Magn. Mater. 8, 59 (1995).\n[21] I. Errea, M. Calandra, and F. Mauri, Anharmonic free energies\nand phonon dispersions from the stochastic self-consistent har-\nmonic approximation: Application to platinum and palladium\nhydrides, Phys. Rev. B 89, 064302 (2014).\n[22] T. Tadano and S. Tsuneyuki, Self-consistent phonon calcula-\ntions of lattice dynamical properties in cubic SrTiO 3with first-\nprinciplesanharmonicforceconstants,Phys.Rev.B 92,054301\n(2015).\n[23] Y.Oba,T.Tadano,R.Akashi,andS.Tsuneyuki,First-principles\nstudyofphononanharmonicityandnegativethermalexpansion\nin ScF 3, Phys. Rev. Mater. 3, 033601 (2019).\n[24] O.Hellman,P.Steneteg,I.A.Abrikosov,andS.I.Simak,Tem-\nperature dependent effective potential method for accurate free\nenergy calculations of solids, Phys. Rev. B 87, 104111 (2013).\n[25] G. Kresse and D. Joubert, From ultrasoft pseudopotentials to\nthe projector augmented-wave method, Phys. Rev. B 59, 1758\n(1999).\n[26] J.P.Perdew,K.Burke,andM.Ernzerhof,Generalizedgradient\napproximation made simple, Phys. Rev. Lett. 77, 3865 (1996).\n[27] G. Kresse and J. Furthmüller, Efficient iterative schemes for\nab initio total-energy calculations using a plane-wave basis set,\nPhys. Rev. B 54, 11169 (1996).\n[28] M. Methfessel and A. T. Paxton, High-precision sampling for\nBrillouin-zone integration in metals, Phys. Rev. B 40, 361611\n(1989).\n[29] P. E. Blöchl, O. Jepsen, and O. K. Andersen, Improved tetrahe-\ndron method for Brillouin-zone integrations, Phys. Rev. B 49,\n16223 (1994).\n[30] T.Tadano,Y.Gohda,andS.Tsuneyuki,Anharmonicforcecon-\nstants extracted from first-principles molecular dynamics: Ap-\nplicationstoheattransfersimulations,J.Phys.: Condens.Matter\n26, 225402 (2014).\n[31] H.Rosner,D.Koudela,U.Schwarz,A.Handstein,M.Hanfland,\nI. Opahle, K. Koepernik, M. D. Kuz’min, K.-H. Müller, J. A.\nMydosh, and M. Richter, Magneto-elastic lattice collapse in\nYCo 5, Nat. Phys. 2, 469 (2006).\n[32] C. E. Patrick, S. Kumar, G. Balakrishnan, R. S. Edwards,\nM. R. Lees, E. Mendive-Tapia, L. Petit, and J. B. Staunton,\nRare-earth/transition-metalmagneticinteractionsinpristineand\n(Ni,Fe)-doped YCo 5and GdCo 5, Phys. Rev. Mater. 1, 024411\n(2017).\n[33] K.Nassau,L.Cherry,andW.Wallace,Intermetalliccompounds\nbetweenlanthanonsandtransitionmetalsofthefirstlongperiod:\nI–Preparation, existence and structural studies, J. Phys. Chem.\nSolids16, 123 (1960).\n[34] O.Moze,L.Pareti,A.Paoluzi,andK.H.J.Buschow,Magnetic\nstructure and anisotropy of Ga- and Al-substituted LaCo 5and\nYCo 5intermetallics, Phys. Rev. B 53, 11550 (1996).\n[35] S. Sharma, E. Hildebrandt, S. Sharath, I. Radulov, and L. Alff,\nYCo 5\u0006xthin films with perpendicular anisotropy grown by\nmolecularbeamepitaxy,J.Magn.Magn.Mater. 432,382(2017).\n[36] See Supplemental Material at [URL] for phonon dispersion\ncurveswithvariouslatticeconstants,projecteddensity-of-states\nof pristine YCo 5at Co 3gsite, mode- and energy-decomposed\nKphon\nudetermined with VCA, normal coordinate dependent\nKDFT\nuoftheL-pointsoftmode,totaldensity-of-statesofYCo 5\n(Y(Co,Fe) 5) obtained both by reducing Ne(VCA), and KPT\nu\nobtained from second-order perturbation analysis, respectively.\n[37] R. Dronskowski and P. E. Bloechl, Crystal orbital Hamilton\npopulations(COHP):Energy-resolvedvisualizationofchemical\nbonding in solids based on density-functional calculations, J.\nPhys. Chem. 97, 8617 (1993).\n[38] V.L.Deringer,A.L.Tchougréeff,andR.Dronskowski,Crystal\norbitalHamiltonpopulation(COHP)analysisasprojectedfrom\nplane-wave basis sets, J. Phys. Chem. A 115, 5461 (2011).\n[39] F. Rothwarf, H. Leupold, J. Greedan, W. Wallace, and\nD. K. Das, Magnetic anisotropy in the Th(Co (1\u0000x)Fex)5and\nY(Co (1\u0000x)Fex)5systems, Int. J. Magn. 4, 267 (1973).\n[40] M.Forker,A.Julius,M.Schulte,andD.Best,Mössbauerstudy\nofthehyperfineinteractionof57FeinY 1\u0000sCo5+2sandrelated\ncompounds, Phys. Rev. B 57, 11565 (1998).\n[41] T. Ishikawa, T. Fukazawa, G. Xing, T. Tadano, and T. Miyake,\nEvolutionary search for cobalt-rich compounds in the yttrium-\ncobalt-boron system, Phys. Rev. Mater. 5, 054408 (2021).\n[42] L.Pareti,M.Solzi,andG.Marusi,Phenomenologicalanalysisof\nthe magnetocrystalline anisotropy of the co sublattice in somerhombohedral and hexagonal intermetallic structures derived\nfrom the CaCu 5unit cell, J. Appl. Phys. 72, 3009 (1992).\n[43] I. Errea, F. Belli, L. Monacelli, A. Sanna, T. Koretsune,\nT. Tadano, R. Bianco, M. Calandra, R. Arita, F. Mauri, and\nJ. A. Flores-Livas, Quantum crystal structure in the 250-kelvin\nsuperconducting lanthanum hydride, Nature 578, 66 (2020),\n1907.11916.\n[44] L. Bellaiche and D. Vanderbilt, Virtual crystal approximation\nrevisited: Application to dielectric and piezoelectric properties\nof perovskites, Phys. Rev. B 61, 7877 (2000).\n[45] S. Steiner, S. Khmelevskyi, M. Marsmann, and G. Kresse, Cal-\nculation of the magnetic anisotropy with projected-augmented-\nwave methodology and the case study of disordered Fe 1\u0000xCox\nalloys, Phys. Rev. B 93, 224425 (2016).\n[46] J. Franse, N. Thuy, and N. Hong, Individual site magnetic\nanisotropyoftheironandcobaltionsinrareearth-ironandrare\nearth-cobalt intermetallic compounds, J. Magn. Magn. Mater.\n72, 361 (1988).\n[47] G.H.O.Daalderop,P.J.Kelly,andM.F.H.Schuurmans,Mag-\nnetocrystalline anisotropy of YCo 5and related RECo 5com-\npounds, Phys. Rev. B 53, 14415 (1996).\n[48] J.-X. Zhu, M. Janoschek, R. Rosenberg, F. Ronning, J. D.\nThompson, M. A. Torrez, E. D. Bauer, and C. D. Batista,\nLDA+DMFT approach to magnetocrystalline anisotropy of\nstrong magnets, Phys. Rev. X 4, 021027 (2014).\n[49] L. Steinbeck, M. Richter, and H. Eschrig, Itinerant-electron\nmagnetocrystallineanisotropyenergyofYCo 5andrelatedcom-\npounds, Phys. Rev. B 63, 184431 (2001).\n[50] M. Brooks, Calculated ground state properties of light actinide\nmetals and their compounds, Physica B+C 130, 6 (1985).\n[51] P. Bruno, Tight-binding approach to the orbital magnetic mo-\nment and magnetocrystalline anisotropy of transition-metal\nmonolayers, Phys. Rev. B 39, 865 (1989).\n[52] G.Autès,C.Barreteau,D.Spanjaard,andM.-C.Desjonquères,\nMagnetism of iron: From the bulk to the monatomic wire, J.\nPhys.: Condens. Matter 18, 6785 (2006).\n[53] K. Momma and F. Izumi, VESTA 3 for three-dimensional vi-\nsualization of crystal, volumetric and morphology data, J Appl\nCryst44, 1272 (2011).\n[54] J.SchweizerandF.Tasset,PolarisedneutronstudyoftheRCo 5\nintermetallic compounds.I. Thecobalt magnetisationin YCo5,\nJ. Phys. F: Met. Phys. 10, 2799 (1980).\n[55] A. Heidemann, D. Richter, and K. H. J. Buschow, Investigation\nofthehyperfinefieldsinthecompoundsLaCo 13,LaCo 5,YCo 5\nandThCo 5bymeansofinelasticneutronscattering,ZPhysikB\n22, 367 (1975).\n[56] F. Alzahraa Mohammad, S. Yehia, and S. H. Aly, A first-\nprinciple study of the magnetic, electronic and elastic prop-\nertiesofthehypotheticalYFe 5compound,PhysicaB 407,2486\n(2012)."    },    {        "title": "2112.08154v1.Effect_of_magnetocrystalline_anisotropy_on_magnetocaloric_properties_of_AlFe___2__B___2___compound.pdf",        "content": "E\u000bect of magnetocrystalline anisotropy on\nmagnetocaloric properties of AlFe 2B2compound\nHung Ba Tran,\u0003,y,z,{Hiroyoshi Momida,{Yu-ichiro Matsushita,y,zKazunori\nSato,x,kYukihiro Makino,?Koun Shirai,{and Tamio Oguchi{,k\nyLaboratory for Materials and Structures, Institute of Innovative Research, Tokyo Institute\nof Technology, Midori-ku, Yokohama 226-8503, Japan\nzQuemix Inc., Taiyo Life Nihombashi Building, 2-11-2, Nihombashi Chuo-ku, Tokyo\n103-0027, Japan\n{Institute of Scienti\fc and Industrial Research, Osaka University, 8-1 Mihogaoka, Ibaraki,\nOsaka 567-0047, Japan\nxDivisions of Materials and Manufacturing Science, Graduate School of Engineering,\nOsaka University, 2-1 Yamada-oka, Suita, Osaka 565-0871, Japan\nkCenter for Spintronics Research Network, Osaka University, Toyonaka, Osaka 560-8531,\nJapan\n?Daikin Industries, LTD., 1-1 Nishi-Hitotsuya, Settsu, Osaka 566-8585, Japan\nE-mail: tran.h.ag@m.titech.ac.jp\nAbstract\nIt is well known that the temperature dependence of the e\u000bective magnetocrys-\ntalline anisotropy energy obeys the l(l+1)=2 power law of magnetization in the Callen-\nCallen theory. Therefore, according to the Callen-Callen theory, the magnetocrystalline\nanisotropy energy is assumed to be zero at the critical temperature where the magneti-\nzation is approximately zero. This study estimates the temperature dependence of the\n1arXiv:2112.08154v1  [cond-mat.mtrl-sci]  15 Dec 2021magnetocrystalline anisotropy energy by integrating the magnetization versus magnetic\n\feld ( M{H) curves, and found that the magnetocrystalline anisotropy is still \fnite even\nabove the Curie temperature in the uniaxial anisotropy, whereas this does not appear\nin the cubic anisotropy case. The origin is the fast reduction of the anisotropy \feld,\nwhich is the magnetic \feld required to saturate the magnetization along the hard axis,\nin the case of cubic anisotropy. Therefore, the magnetization anisotropy and anisotropic\nmagnetic susceptibility, those are the key factors of magnetic anisotropy, could not be\nestablished in the case of cubic anisotropy. In addition, the e\u000bect of magnetocrystalline\nanisotropy on magnetocaloric properties, as the di\u000berence between the entropy change\ncurves of AlFe 2B2appears above the Curie temperature, which is in good agreement\nwith a previous experimental study. This is proof of magnetic anisotropy at slightly\nabove Curie temperature.\nIntroduction\nHeat-assisted magnetic recording is an important method for increasing the capacity and en-\nergy e\u000eciency of data storage.1,2Magnetic materials with strong magnetocrystalline anisotropy\nenergy (MAE) are required to maintain the stability of small-sized magnetic particles for a\nlong time. The other side is the di\u000eculty in changing the magnetic state using the write\nhead during the writing process. As the magnetocrystalline anisotropy energy depends on\ntemperature, applying heat can decrease the coercivity (magnetic \feld required to change the\nmagnetic state) of magnetic material.1,2Therefore, the temperature dependence of MAE is a\ncrucial characteristic of magnetic materials to determine the magnetic \feld needed to change\nthe magnetization. In early theoretical studies, the e\u000bective magnetocrystalline anisotropy\nconstant was found to be proportional to a polynomial of magnetization M3andM10in\nuniaxial and cubic anisotropy, respectively,.3Subsequently, the l(l+ 1)=2 power law (where\nlis the order of anisotropy constants) of magnetocrystalline anisotropy energy, the so-called\nCallen-Callen theory, was reproduced in previous studies by using the constrained Monte-\n2Carlo method.4,5If the magnetization is equal to zero at the Curie temperature, MAE should\nbe equal to zero.3{5However, \fnite magnetocrystalline anisotropy at slightly above the Curie\ntemperature has been observed in recent experimental studies.6,7\nThe application of magnetocrystalline anisotropy is the rotating magnetocaloric e\u000bect.8{14\nThe temperature and heat exchange of the anisotropic magnetic material can be controlled\nby rotating the sample (or changing the relative direction of the external magnetic \feld) at\na \fxed strength of the applied magnetic \feld. The origin of this e\u000bect is the modi\fcation of\nthe magnetocrystalline anisotropy on the magnetization when changing the relative direction\nof the external magnetic \feld. The isothermal magnetic entropy change has a peak at the\ncritical temperature because the derivative of magnetization diverges at the ferromagnetic-\nparamagnetic (FM-PM) transition.15,16If the magnetocrystalline anisotropy disappears at\nthe Curie temperature, the modi\fcation of the magnetocrystalline anisotropy only appears\non the low-temperature side of the entropy change peak. However, several experimental\nstudies have found that the isothermal magnetic entropy change is modi\fed not only on the\nlow-temperature side but also on the high-temperature side of the main peak when a magnetic\n\feld is applied in di\u000berent directions.8{14The magnetocrystalline anisotropy energy a\u000bects\nthe isothermal magnetic entropy change even above the Curie temperature in the anisotropic\nmagnetocaloric e\u000bect.\nIn recent years, AlFe 2B2has attracted much attention for future refrigeration with earth-\nabundant elements, room-temperature magnetic transitions, and strong magnetocrystalline\nanisotropy energy.17{21AlFe 2B2has an orthorhombic layered crystal structure with a space\ngroupCmmm as depicted in Figure 1. The Fe and B layers form a zigzag chain in the\nacplane. Furthermore, the Fe and B layers are separated by the Al plane. The Curie\ntemperature of the material is varied in the range of 274{320 K, which depends on the syn-\nthesis method.18AlFe 2B2has an easy ([100]), intermediate ([010]), and hard ([001]) magnetic\naxes.17The magnetocrystalline anisotropy constants (at 2 K) were reported in a previous\nexperimental study with K001=1.8 (MJ/m3) andK010=0.23 (MJ/m3).17The dependence of\n3magnetocaloric properties on the direction of an applied magnetic \feld has been reported in\nprevious experimental study.18However, to date, there has been no comprehensive theoret-\nical study on the rotating magnetocaloric properties of the material.\nab\ncAlFeB\ncba\nFigure 1: The crystal structures of orthorhombic AlFe 2B2. The gray, orange, and blue\nspheres indicate the Al, Fe, and B atoms, respectively.\nIn the present work, we \frst estimate the temperature dependence of the MAE in the\nuniaxial and cubic anisotropy cases by considering the energy cost to magnetize in some\nspeci\fc directions in simple-cubic lattice model. Moreover, as MAE is estimated as the\narea between two magnetization curves when applying magnetic \felds along the hard and\neasy axes, the ordinary Monte-Carlo method is considered. The magnitude of the projected\nmagnetization can be saturated when a the magnetic \feld is applied along the hard axis by\nincreasing the strength of the external magnetic \feld. The results are compared with the\nwell-known Callen-Callen theory for the uniaxial and cubic anisotropy cases. In addition, the\ntemperature dependence of the magnetocrystalline anisotropy constant and magnetocaloric\ne\u000bect in orthorhombic AlFe 2B2is also considered. This material has easy, intermediate, and\nhard axes, which can exhibit di\u000berent behavior in the temperature dependence of magnetic\n4anisotropy constants. In addition, the entropy change at a magnetic \feld of 2 T is compared\nwith an experimental work.18\nMethodologies\nThe electronic structure calculations and structure optimizations of orthorhombic AlFe 2B2\nwere performed using DFT using the full-potential linearized augmented plane wave (FLAPW)\nmethod.22The lattice constants ( a= 2:9168\u0017A,b= 11:033\u0017A, andc= 2:8660\u0017A) were used for\nAlFe 2B2.17The atomic position relaxation was carried out by optimizing all atomic forces\nof less than 1 mRy/Bohr. The generalized gradient approximation (GGA) was used for\nexchange and correlation.23Magnetocrystalline anisotropy energy is evaluated as the energy\nnecessary to rotate the quantization axis with spin orbit coupling (SOC) in the second vari-\nation steps.24The convergence of magnetocrystalline anisotropy energy was achieved using\na large number of k-point meshes of 48 \u000216\u000248.\nThe magnetic exchange constants ( Jij) of stoichiometric AlFe 2B2were evaluated using\nthe Liechtenstein formula in the Korringa-Kohn-Rostoker (KKR) method with coherent\npotential approximation (CPA):25{27\nJij=1\n4\u0019ImZEF\ndETrn\n\u0001ti(E)T\"\nij\u0001tj(E)T#\nijo\n; (1)\nwhereT\"\nijandT#\nijare o\u000b-diagonal scattering path operators for spin-up and spin-down be-\ntween atomic sites iandj, respectively, and \u0001 ti(E) =t\"\ni\u0000t#\niwith the single site t-matrix\nt\u001b\nifor spin\u001bat site i.\nThe Heisenberg model with uniaxial and cubic anisotropy terms was used to obtain the\n5magnetism and magnetocaloric properties at a \fnite temperature:4\nHHeis=\u0000X\n<ij>Jm\nij\u0000 !Si\u0000 !Sj\u0000X\niku(\u0000 !eu\u0000 !Si)2\n\u0000X\nikc(S4\nix+S4\niy+S4\niz)\u0000g\u0016BX\ni\u0000\u0000!Hext\u0000 !Si;(2)\nwheregis theg-factor,\u0016Bis the Bohr magneton, kuis the uniaxial anisotropy constant,\nandkcis the cubic anisotropy constant. The \frst term expresses the exchange interactions\nbetween spins at sites iand j. The spin tends to be parallel to the neighboring site when\nthe magnetic exchange coupling constant Jm\nijis positive. The spin is antiparallel to the\nneighboring spin for negative Jm\nij. The second term in Eq. (2) is the interaction between\nthe spin at site iand the uniaxial anisotropy, euis the direction of the easy axis in the case\nof a positive ku. The third term in Eq. (2) is the cubic anisotropy term with Sixas the\nxcomponent of Si. The \fnal term in Eq. (2) is the interaction of the spin at site iwith\nthe external magnetic \feld. At a \fnite external magnetic \feld, the spin tends to be parallel\nto the direction of the magnetic \feld to minimize the energy. The size of the simple-cubic\nlattice is assumed to be 80 \u000280\u000280 unit cells, and the \frst nearest neighbor of the exchange\ninteraction is considered. However, the size of the Monte-Carlo simulation box in the AlFe 2B2\ncompound is 30\u000230\u000230 unit cells with 270000 atoms. The number of Monte-Carlo steps\nis 200000, and the \frst 100000 steps are discarded. The Metropolis algorithm was used to\nachieve thermal equilibrium in the Heisenberg model.\nThe magnetocrystalline anisotropy energy was estimated by integrating the magnetiza-\ntion versus the magnetic \feld ( M{H) curves:\nMAE =ZHext\n0fM(Hkeasy axis)\u0000M(Hkhard axis)gdH (3)\nwhere the magnetic \feld is used to obtain the saturation of magnetization. The tempera-\nture dependence of magnetocrystalline anisotropy energy can be estimated using integration\n6because the area between the two magnetization curves can be measured as the energy cost\nto magnetize in a certain direction.\nThe magnetic susceptibility is calculated as\n\u001f=@M\n@H=(hM2i\u0000hMi2)\nkBT; (4)\nwherekBis Boltzmann constant. The magnetic susceptibility is the \ructuation of magneti-\nzation in the Monte-Carlo simulation.\nThe isothermal magnetic entropy changes (\u0001 SM) are estimated by integrating the deriva-\ntive of the magnetization ( M) based on the Maxwell relation15,16\n\u0001SM(Hext;T) =ZHext\n0\u0012@M(H;T)\n@T\u0013\n;\n\u0018=NX\nj=0M(Hj;T+ \u0001T)\u0000M(Hj;T\u0000\u0001T)\n2\u0001T\u0001H(5)\nThe isothermal magnetic entropy change is obtained by integrating with \fne mesh in tem-\nperature and external magnetic \feld. \u0001 Tand \u0001Hare taken as 2 K and 0.2 T, respectively.\nOwing to the magnetocrystalline anisotropy energy, the isothermal magnetic entropy changes\nstrongly depend on the direction of the applied magnetic \feld.\nResults and discussion\nUniaxial and cubic anisotropies in simple-cubic lattice model\nAs magnetic materials usually exhibit uniaxial or cubic anisotropy, a simple-cubic lattice with\nuniaxial and cubic anisotropy interaction was considered in the \frst part to provide a general\ncharacterization of uniaxial and cubic anisotropy. The magnetic exchange coupling constant\nof the \frst nearest neighbor, equal to 10 meV, was used in this simulation. The kuandkcwere\n7(a)\n(b)0 TC0.59 TC1.04 TC1.19 TC\ndash: Hext|| hard axissolid: Hext|| easy axisdash: Hext|| hard axissolid: Hext|| easy axis 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20MHext (T)\n 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20MHext (T)0 TC0.59 TC1.04 TC1.19 TCFigure 2: Magnetization-magnetic \felds ( M{H) curve at several temperatures when applying\nmagnetic \feld along the hard and easy axis of uniaxial anisotropy (a) and cubic anisotropy\n(b) of simple cubic lattice model. The magnetization was normalized to 1.\n80.25 meV, and the size of the magnetic moment was 2.0 \u0016B. The magnetization-magnetic \feld\n(M{H) curves at several temperatures in the uniaxial anisotropy case are shown in Figure 2\n(a). At 0 K, increasing the magnetic \feld along the hard axis leads to a linear enhancement\nof magnetization before saturating at HA= 2Ku=MSas the Stoner-Wohlfarth model for\nmagnetic reversal. When the temperature is increased, the saturation magnetization at\nzero external magnetic \feld is decreased. Furthermore, the anisotropy \feld, which is the\nrequired magnetic \feld to saturate the magnetization along the hard axis, is also decreased.\nThis results to a high order of dependence of MAE on magnetization rather than linear\ndependence. At an intermediate temperature, the magnetization curve when the magnetic\n\feld along the hard axis has a wing when the magnetic \feld approaches the anisotropy \feld.\nThis wing is related to the \fnite temperature e\u000bect on the direction of magnetization, and\nthe direction of magnetization is di\u000ecult along the hard axis. Furthermore, a high external\nmagnetic \feld is required to approach the saturation magnetization when a magnetic \feld\nis applied along the hard axis at these temperatures. If the magnetization curve of the\nhard axis is extrapolated to the limit of zero, by applying external magnetic \feld as the\nsaturation magnetization of the hard axis, there is a \fnite value of the di\u000berence in saturation\nmagnetization between the easy and hard axes, the so-called magnetization anisotropy. At a\ntemperature slightly above the Curie temperature, the magnetization curve indicates when\nthe magnetic \feld along the hard axis is not saturated. This means that magnetizing along\nthe hard axis in the uniaxial case is still harder than the magnetization along the easy axis\neven above the Curie temperature, which was also observed in the experimental study.6\nThis is due to the di\u000berence in the slope of the M{Hcurves in the easy and hard axes at a\n\fnite magnetic \feld applied, such as the anisotropic magnetic susceptibility. The magnetic\nsusceptibility is the \ructuation of the magnetization curves at a \fxed strength of the external\nmagnetic \feld. Therefore, the dependence of magnetic susceptibility on the direction of the\nexternal magnetic \feld at these temperatures implies that the external magnetic \feld leads\nto the enhancement of the anisotropic magnetic susceptibility. When the magnetization is\n9enhanced by the applied \feld at a temperature slightly higher than the Curie temperature,\nthe e\u000bect of magnetic anisotropy appears and enhances the \frst derivative of magnetization\nin the \feld, as the magnetic susceptibility.\nTheM{Hcurves at several temperatures in the cubic anisotropy case are shown in Figure\n2 (b). At 0 K, the magnetization along the hard axis at zero external magnetic \felds is non-\nzero because the hard and easy axes are not perpendicular in the cubic case. Moreover,\nit is understood that the anisotropy \feld is higher in this case than that in the case of\nuniaxial anisotropy. However, the anisotropy \feld is the same as 0.25 meV in both the cases.\nIf the temperature increases, the magnetization and anisotropy \felds also decrease as the\ntemperature a\u000bects the order parameter and MAE. However, the anisotropy \feld is decreased\nrapidly compared with the uniaxial case. The cubic anisotropy is a higher order of anisotropy\ninteraction than uniaxial anisotropy, as shown in Eq. (2). At high temperatures, the area\nbetween the two magnetization curves is diminishes to almost zero, in contrast to the uniaxial\ncase. This means that the magnetocrystalline anisotropy energy above the Curie temperature\nis approximately equal to zero. Because the magnetocrystalline anisotropy energy is related\nto the energy barrier in free energy, which contains the energy and entropy terms of cubic\nand uniaxial anisotropy, di\u000berent at temperatures above the Curie temperature. The high\nexternal magnetic \feld cannot enhance the anisotropic magnetic susceptibility compared\nwith the uniaxial case, as discussed above. One possibility for this phenomenon is the shape\nof the magnetic anisotropy energy surface of cubic anisotropy, which depends on the direction\nof magnetization, is isotropic and does not depend on the strength of the external magnetic\n\feld.\nThe temperature dependence of magnetization and normalized anisotropy constant of\nuniaxial and cubic anisotropies in a simple-cubic lattice model are shown in Figure 3. The\nmagnetization curve at zero external magnetic \felds was used to determine the e\u000bective\nanisotropy constant of the Callen-Callen theory. The results of the integration M{Hcurve\nof cubic anisotropy are in good agreement with M10, whereas the critical temperature of the\n10 0 0.2 0.4 0.6 0.8 1\n 0 0.2 0.4 0.6 0.8 1 1.2 1.4M, (Ku(T)/Ku(0)), (Kc(T)/Kc(0))\nT/TCMuniaxial(CallenïCallen)cubic(CallenïCallen)uniaxial(MC)cubic(MC)Figure 3: Temperature dependence of magnetization (black line), anisotropy constant of\nuniaxial (red line) and cubic (blue line) anisotropies of Callen-Callen model as M3andM10,\nrespectively. The integration of M{Hcurves for uniaxial (crossed purple points) and cubic\n(square green points) anisotropies using the Monte-Carlo method.\nuniaxial anisotropy constant is higher than the Curie temperature de\fned by the Callen-\nCallen theory. The uniaxial and cubic anisotropy constants of the Callen-Callen theory are\nobtained from the power law of magnetization. This means that when the magnetization\nbecomes approximately equal to zero at the Curie temperature, the e\u000bective anisotropy\nconstants are also assumed to be zero. The di\u000berent behavior between uniaxial and cubic\nanisotropy may be due to the shape of the magnetic anisotropic energy surface, which is\nthe energy that depends on the direction of magnetization with a \fnite external magnetic\n\feld. In uniaxial anisotropy, the external magnetic \feld enhances the anisotropic magnetic\nsusceptibility near the Curie temperature. It is preferred to magnetize along the easy axis\ncompared to the other axes. Therefore, the magnetization along the hard axis, which is\nthe integration of magnetic susceptibility, is slightly smaller than along the easy axis above\nthe Curie temperature. On the other hand, the anisotropic magnetic susceptibility of cu-\nbic anisotropy above the Curie temperature becomes approximately zero. Moreover, the\ncombination of cubic anisotropy and the external magnetic \feld is almost unchanged when\nmagnetizing along the hard and easy axes. This phenomenon leads to di\u000berent behaviors\nin the magnetocaloric e\u000bect of uniaxial and cubic anisotropy because the cubic anisotropy\n11energy disappears above the Curie temperature, while the uniaxial anisotropy energy is not\nnegligible at these temperatures.\nWe derived the relation of magnetization anisotropy and anisotropic magnetic suscepti-\nbility with MAE from the de\fnition of MAE and the magnetic susceptibility of Eq. (3) and\nEq. (4) as\n@EMAE\n@H\n=@F(Hextkeasy;T)\n@H\u0000@F(Hextkhard;T)\n@H;\n=M(Hextkeasy;T)\u0000M(Hextkhard;T)(6)\n@2EMAE\n@H2\n=@2F(Hextkeasy;T)\n@H2\u0000@2F(Hextkhard;T)\n@H2;\n=\u001f(Hextkeasy;T)\u0000\u001f(Hextkhard;T)(7)\nFrom the Eqs. (6) and (7), magnetization anisotropy is the \frst derivative while the\nanisotropic magnetic susceptibility is the second derivative of MAE on the magnetic \feld.\nThis means that their e\u000bects on MAE can be considered in our Monte-Carlo simulation as\nthe integration of M{Hcurves. The magnetization anisotropy and anisotropic magnetic\nsusceptibility lead to \fnite MAE of the uniaxial case and negligible MAE of the cubic case\nat a temperature slightly higher than the Curie temperature. Because the magnetization\nanisotropy is the integration of anisotropic magnetic susceptibility until the anisotropy \feld\nis reached, it only appears below the Curie temperature where the anisotropy \feld is \fnite.\nMoreover, because the anisotropy \feld of the cubic anisotropy decreases much faster than\nthe uniaxial anisotropy, the magnetization anisotropy may not appear in the cubic case.\n12Pristine AlFe 2B2\n−10−5 0 5 10 15 20 25\n 0.5 1 1.5 2 2.5 3 3.5 4Jij (meV)Distance d/aFe−Fe pairs(a)\n(b)\nFigure 4: (a) Magnetic exchange coupling constants of the Fe-Fe pairs as a function of Fe-Fe\ndistance (d=a) normalized with the lattice constant a. (b) Magnetization (purple line) and\nmagnetic susceptibility (green line) curves of the Monte-Carlo simulation at zero external\nmagnetic \feld.\nThe magnetic exchange coupling constants between Fe atoms of orthorhombic AlFe 2B2\nare shown in Figure 4 (a). The coupling constants exhibit a long-range behavior with os-\ncillation when the distance increases. The magnetic exchange coupling constants from the\n\frst to fourth nearest neighbors are positive. Furthermore, the coupling constants are in-\ncreased when the distance is increased. This is not an ordinary direct-exchange type of\nmagnetic interaction. The \frst, second, and third nearest neighbors are surrounded by B\natoms inside the Fe-B layer. Therefore, the p-state of B a\u000bects the coupling constants via a\n13super-exchange interaction, and the coupling constants increase when the distance increases\nin this case. On the other hand, the fourth nearest neighbor is a pair of Fe atoms in di\u000berent\nlayers separated by the Al plane. It leads to the highest value of the coupling constant of the\nfourth nearest neighbor as direct exchange interaction. The Curie temperature can be esti-\nmated from the magnetic exchange coupling constants using the mean-\feld approximation\n(MFA) and Monte-Carlo method. The Curie temperature obtained by MFA is 365 K, while\nthe value of the Monte-Carlo method is 199 K. The Curie temperature of the MFA is usually\noverestimated, whereas the Monte-Carlo method should be more precise. The magnetization\nand magnetic susceptibility of AlFe 2B2are shown in Figure 4 (b). The Curie temperature\nof the Monte-Carlo method is obtained as the divergence of the magnetic susceptibility.\nThe magnetization-magnetic \feld M{Hcurves of AlFe 2B2at several temperatures are\nshown in Figure 5 (a), (b), and (c). The magnetization curves obtained when applying\na magnetic \feld along the hard, intermediate, and easy axes at 2 K in the Monte-Carlo\nsimulation (the solid line) are in good agreement with the experimental results (the dash\nline).17The magnetocrystalline anisotropy energy can be estimated as the area between\ntwo magnetization curves, one for the easy axis and the other for the hard or intermediate\naxis. Applying an external magnetic \feld along the hard axis requires a large magnetic\n\feld to saturate the magnetization. The anisotropy \feld in the case of the hard axis of the\nMonte-Carlo method is slightly larger than the experimental results, whereas the anisotropy\n\feld of the intermediate axis is slightly smaller than the experimental value. When the\ntemperature is increased, the magnetization and anisotropy \feld are decreased. However,\nbelow the Curie temperature, the anisotropy \feld of the hard axis is decreased much faster\nthan that in the intermediate case. At 200 K (slightly above the Curie temperature of\n199 K), the magnetization is approximately zero at zero external magnetic \felds. However,\napplying an external magnetic \feld enhances the area between the magnetization curves at\nthis temperature. The area between the magnetization curves of the easy and intermediate\naxes is approximately equal to zero, whereas it is not negligible between the easy and hard\n142 K150 K\n200 K(a)(b)\n(c)(d) 0 0.5 1 1.5 2 2.5 3\n 0 1 2 3 4 5 6 7M (µB/f.u)Hext (Τ)[001](MC)[010](MC)[100](MC)[001](EXP)[010](EXP)[100](EXP) 0 0.5 1 1.5 2 2.5 3\n 0 1 2 3 4 5 6 7M (µB/f.u)Hext (Τ)[001][010][100]\n 0 0.5 1 1.5 2 2.5 3\n 0 1 2 3 4 5 6 7M (µB/f.u)Hext (Τ)[001][010][100]TC= 199 K 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8\n 0 50 100 150 200 250MAE (MJ/m3)\nT (K)K010(DFT)K001(DFT)K010(EXP)K001(EXP)K010(MC)K001(MC)Figure 5: (a), (b), (c) Magnetization-magnetic \feld ( M{H) curves of several temperature\nof Monte-Carlo method when magnetic \feld along the easy ([100]), intermediate ([010]),\nand hard ([001]) axis. The M{Hcurves at 2 K in the experimental study17are shown as\ndashed lines. (d) Temperature dependence e\u000bective magnetocrystalline anisotropy energy\nof the Monte-Carlo method and available data from \frst-principles calculations (0 K) and\nexperimental studies (2 K).17\naxes. This means that the magnetocrystalline anisotropy constant of the intermediate axis\nis approximately equal to zero, whereas the hard axis is not negligible at these temperatures.\nThe magnetocrystalline anisotropy energy constants of AlFe 2B2are shown in Figure 5\n(d). The anisotropy energy obtained using the present \frst-principles DFT calculations of\nthe hard axis is 1.5 MJ/m3, which is close to the experimental result of 1.8 MJ/m3at 2 K.17\nBesides, the value of the intermediate case of DFT is 0.16 MJ/m3, which slightly smaller\nthan the value of experimental study 0.23 MJ/m3.17By integrating the M{Hcurves in\nthe Monte-Carlo method, we can consider the temperature dependence of the magnetocrys-\n15talline anisotropy energy between the hard and intermediate axes with the easy axis. The\nanisotropy constant of the hard axis is decreased almost linearly as the temperature is in-\ncreased. Moreover, the uniaxial anisotropy in the previous section is still \fnite above the\nCurie temperature. On the other hand, the magnetocrystalline anisotropy constant of the\nintermediate axis is decreased slowly below the Curie temperature. Near the Curie tem-\nperature, the anisotropy constant decreased rapidly. Furthermore, it became approximately\nequal to zero at the Curie temperature. This means that the energy barrier between the\neasy and intermediate axes disappears at the Curie temperature.\nThe isothermal magnetization and entropy change curves are shown in the Figure 6.\nWhen a magnetic \feld is applied along the easy axis ([100]), the magnetization decreases\nwhen the temperature increases and becomes zero at the Curie temperature as the limit\nof the magnetic \feld being zero. By increasing the magnetic \feld, the magnetization is\nenhanced, and the transition temperature is shifted. From the magnetization curves, the\nisothermal magnetic entropy change can be estimated using Maxwell's relation.15,16The\nentropy change curves have a peak at the Curie temperature, where the derivative of the\nmagnetization diverges. The value of the peak is increased with increasing strength of the\nexternal magnetic \feld. On the other hand, the magnetic anisotropy became dominant in\nthe low-magnetic-\feld and low-temperature region in the case of the intermediate ([010]) and\nhard ([001]) axes. However, the di\u000berence in magnetization with the easy axis is decreased\nby increasing the temperature or magnetic \feld . At a su\u000ecient magnetic \feld at \fnite\ntemperatures, the magnetization looks similar to that in the easy axis case. Therefore,\nthe magnetocrystalline anisotropy energy leads to an extensive modi\fcation of the entropy\nchange at a low external magnetic \feld. The di\u000berence in the entropy change curves remain\nconstant at a su\u000ecient magnetic \feld. There is noise in the magnetization in the temperature\nrange near the Curie temperature due to the competition between magnetic anisotropy and\nthe external magnetic \feld, which tends to force the magnetization along its own direction.\nThis noise disappears at a su\u000eciently large magnetic \feld and high temperature. As a result\n16ï1 0 1 2 3 4 5 6 7\n 150 160 170 180 190 200 210 220 230 240 250ï6SM (J/kg.K)T (K)Hext=1 THext=2 THext=3 THext=4 THext=5 Tï1 0 1 2 3 4 5 6 7\n 150 160 170 180 190 200 210 220 230 240 250ï6SM (J/kg.K)T (K)Hext=1 THext=2 THext=3 THext=4 THext=5 Tï1 0 1 2 3 4 5 6 7\n 150 160 170 180 190 200 210 220 230 240 250ï6SM (J/kg.K)T (K)Hext=1 THext=2 THext=3 THext=4 THext=5 T(a)\n(d)(b)\n(e)(c)\n(f) 140 160 180 200 220 240 260 0 2 4 6 8 10 12 0 0.4 0.8 1.2 1.6 2\nT (K)Hext (T) 0 0.4 0.8 1.2 1.6 2\n 140 160 180 200 220 240 260 0 2 4 6 8 10 12 0 0.4 0.8 1.2 1.6 2\nT (K)Hext (T) 0 0.4 0.8 1.2 1.6 2\n 140 160 180 200 220 240 260 0 2 4 6 8 10 12 0 0.4 0.8 1.2 1.6 2\nT (K)Hext (T) 0 0.4 0.8 1.2 1.6 2M (µ!/f.u)M (µ!/f.u)M (µ!/f.u)Figure 6: Isothermal magnetization curves as a function of temperature and external mag-\nnetic \felds ( Hext) along (a) easy ([100]), (b) intermediate ([010]), and (c) hard ([001]) axes.\nThe isothermal magnetic entropy change (\u0001 SM) curves as a function of temperature for Hext\nalong the (d) easy ([100]), (e) intermediate ([010]), and (f) hard ([001]) axes.\n17of this noise in magnetization, the \u0001 SMcurves of the intermediate and hard axes become\nnoisy in the temperature range. Moreover, owing to the e\u000bect of MAE, the entropy change\nof the hard axis was the lowest at the magnetic \feld of 1 T. The di\u000berence in entropy change\nremain at a higher external magnetic \feld since the entropy change is the integration of\nmagnetization curves.\n(a)\n(b) 0 1 2 3 4\n 200 240 280 320 360−∆SM (J/kg.K)T (K)[100](EXP)[001](EXP)\n 0 1 2 3 4\n 200 240 280 320 360−∆SM (J/kg.K)T+∆T (K)[100](MC)[010](MC)[001](MC)\nFigure 7: Isothermal magnetic entropy change when magnetic \feld equal 2 T and along\ndi\u000berent axis of (a) experimental study18and (b) Monte-Carlo simulations. \u0001 Tis used as\n80 K to shift the entropy change peak for a simple comparison.\nThe isothermal magnetic entropy change when the magnetic \feld is 2 T in an experimen-\ntal study18and Monte-Carlo simulation are shown as Figure 7. \u0001 SMcalculated with the\nMonte-Carlo method is in semi-quantitatively good agreement with the experimental results.\nThe entropy change when applying a magnetic \feld along the easy axis is the highest because\n18the magnetization along the easy axis is assisted by the magnetic anisotropy. The entropy\nchange peak of the experimental results is wider than our result because the Curie temper-\nature by the Monte-Carlo method is smaller than the experimental value. In addition, the\ndiscrepancy between the two entropy change curves is due to the modi\fcation of the mag-\nnetic anisotropy on the magnetization curve. The gap between the entropy change curves of\nthe hard and easy axes is decreased when the temperature increased and disappears slightly\nabove the Curie temperature in both experimental and theoretical studies. This means that\nthe MAE is decreased when the temperature increased and \fnally disappears slightly above\nthe Curie temperature. Because the anisotropy constant of the intermediate axis is negligi-\nble above the Curie temperature, the di\u000berence in entropy change of the intermediate and\nthe easy axis is minor. Therefore, there is a nearly easy plane and hard axis at the Curie\ntemperature owing to the temperature dependence of the magnetic anisotropy.\nConclusion\nThe temperature dependence of the uniaxial and cubic anisotropy constants was consid-\nered using Monte-Carlo simulations for the classical Heisenberg model. The temperature\ndependence of the cubic anisotropy constant is found to be in good agreement with the\nCallen-Callen theory, while the critical temperature of the uniaxial anisotropy constant is\nslightly higher than the Curie temperature. In addition, the magnetism and magnetocaloric\nproperties of orthorhombic AlFe 2B2were reasonably predicted. The temperature depen-\ndence of the magnetocrystalline anisotropy energy of the hard K001and intermediate K010\ncases demonstrates di\u000berent behaviors. The good agreement in isothermal magnetic entropy\nchange emphasizes that the magnetic anisotropy a\u000bects the magnetization curves even above\nthe Curie temperature.\n19Acknowledgements\nThe authors thank Tetsuya Fukushima for his help with \frst-principles calculations. This\nwork was partly supported by the Di-CHiLD project (Grant No. J205101520). The compu-\ntation in this work was performed using the facilities of the Supercomputer Center, Institute\nfor Solid State Physics, University of Tokyo and the Cybermedia Center of Osaka University.\nReferences\n(1) Pan, L.; Bogy, D. B. Heat-assisted magnetic recording. Nat:Photonics 2009 , 3, 189-\n190.\n(2) Rottmayer, R. E.; Batra, S.; Buechel, D.; Challener, W. A.; Hohlfeld, J.; Kubuta, Y.; Li,\nL.; Lu, B.; Mihalcea, C.; Mount\feld, K.; Pelhos, K.; Peng, C.; Rausch, T.; Seigler, M.\nA.; Weller, D.; Yang X. M. Heat-Assisted Magnetic Recording. IEEE Trans:Magn:\n2006 , 42, 2417-2421.\n(3) Callen, H. B.; Callen, E. The present status of the temperature dependence of magne-\ntocrystalline anisotropy, and the l(l+ 1)/2 power law. J: Phys: Chem: Solids 1966 ,\n27, 1271-1285.\n(4) Asselin, P.; Evans, R. F. L.; Barker, J.; Chantrell, R. W.; Yanes, R.; Chubykalo-\nFesenko, O.; Hinzke, D.; Nowak, U. Constrained Monte Carlo method and calculation\nof the temperature dependence of magnetic anisotropy. Phys:Rev:B 2010 , 82, 054415\n(12pp).\n(5) Toga, Y.; Matsumoto, M.; Miyashita, S.; Akai, H.; Doi, S.; Miyake, T.; Sakuma, A.\nMonte Carlo analysis for \fnite-temperature magnetism of Nd 2Fe14B permanent magnet.\nPhys:Rev:B 2016 , 94, 174433 (9pp).\n(6) Fries, M.; Skokov, K. P.; Karpenkov, D. Y.; Franco, V.; Ener, S.; Gut\reisch, O. The\n20in\ruence of magnetocrystalline anisotropy on the magnetocaloric e\u000bect: A case study\non Co 2B.Appl:Phys:Lett: 2016 , 109, 232406 (5pp).\n(7) Booth, R. A.; Majetich, S. A. Crystallographic orientation and the magnetocaloric\ne\u000bect in MnP J:Appl:Phys: 2009 , 105, 07A926 (3pp).\n(8) Balli, M.; Jandl, S.; Fournier, P.; Vermette, J.; Dimitrov, D. Z. Unusual rotating\nmagnetocaloric e\u000bect in the hexagonal ErMnO 3single crystal. Phys:Rev:B 2018 , 98,\n184414 (12pp).\n(9) Hu, Y.; Hu, Q. B.; Wang, C. C.; Cao, Q. Q.; Gao, W. L.; Wang, D. H.; Du, Y.\nW. Large room-temperature rotating magnetocaloric e\u000bect in NdCo 4Al polycrystalline\nalloy.SolidStateCommun: 2017 , 250, 45-48.\n(10) Zhang, H.; Li, Y.; Liu, E.; Ke, Y.; Jin, J.; Long, Y.; Shen, B. Giant rotating mag-\nnetocaloric e\u000bect induced by highly texturing in polycrystalline DyNiSi compound.\nSci:Rep: 2015 , 5, 11929 (10pp).\n(11) Balli, M.; Jandl, S.; Fournier, P.; Gospodinov, M. Anisotropy-enhanced giant reversible\nrotating magnetocaloric e\u000bect in HoMn 2O5single crystals. Appl:Phys:Lett: 2014 , 104,\n232402.\n(12) Zhao, X.; Zheng, X.; Luo, X.; Ma, S.; Zhang, Z.; Liu, K.; Qi, J.; Zeng, H.; Rehman,\nS. U.; Ren, W.; Chen, C.; Zhong, Z. Giant rotating magnetocaloric e\u000bect enhanced\nby crystal electric \feld in antiferromagnetic ErNi 3Al9single crystal. J:AlloysCompd:\n2020 , 847, 156478 (7pp).\n(13) Jia, Y.; Namiki, T.; Kasai, S.; Li, L.; Nishimura, K. Magnetic anisotropy and large low\n\feld rotating magnetocaloric e\u000bect in NdGa single crystal. J: Alloys Compd: 2018 ,\n754, 44-48.\n21(14) Wu, Y. D.; Qin, Y. L.; Ma, X. H.; Li, R. W.; Wei, Y. Y. Large rotating magnetocaloric\ne\u000bect at low magnetic \felds in the Ising-like antiferromagnet DyScO 3single crystal.\nJ:AlloysCompd: 2019 , 777, 673-678.\n(15) Tran, H. B.; Fukushima, T.; Sato, K.; Makino, Y.; Oguchi, T. Tuning structural-\ntransformation temperature toward giant magnetocaloric e\u000bect in MnCoGe alloy: A\ntheoretical study. J:Alloys:Compd: 2021 , 854, 157063 (9pp).\n(16) Tran, H. B.; Fukushima, T.; Makino, Y.; Oguchi, T. Magnetocaloric e\u000bect in\nMnCoGe alloys studied by \frst-principles calculations and Monte-Carlo simulation.\nSolidStateCommun: 2021 , 323, 114077 (5pp).\n(17) Lamichhane, T. N.; Xiang, L.; Lin, Q.; Pandey, T.; Parker, D. S.; Kim, T. H.; Zhou, L.;\nKramer, M. J.; Bud'ko, S. L.; Can\feld, P. C. Magnetic properties of single crystalline\nitinerant ferromagnet AlFe 2B2.Phys:Rev:Materials 2018 , 2, 084408 (10pp).\n(18) Barua, R.; Lejeune, B. T.; Ke, L.; Hadjipanayis, G.; Levin, E. M.; McCallum, R.\nW.; Kramer, M. J.; Lewis, L. H. Anisotropic magnetocaloric response in AlFe 2B2.\nJ:AlloysCompd: 2018 , 745, 505-512.\n(19) Barua, R.; Lejeune, B. T.; Jensen, B. A.; Ke, L.; McCallum, R. W.; Kramer, M. J.;\nLewis, L. H. Enhanced room-temperature magnetocaloric e\u000bect and tunable magnetic\nresponse in Ga-and Ge-substituted AlFe 2B2.J:Alloys:Compd: 2019 , 777, 1030-1038.\n(20) Tan, X.; Chai, P.; Thompson, C. M.; Shatruk, M. Magnetocaloric E\u000bect in AlFe2B2:\nToward Magnetic Refrigerants from Earth-Abundant Elements. J: Am: Chem: Soc:\n2013 , 135, 9553-9557.\n(21) Mann, D. K.; Wang, Y.; Marks, J. D.; Strouse, G. F.; Shatruk, M. Microwave Synthesis\nand Magnetocaloric E\u000bect in AlFe 2B2.Inorg:Chem: 2020 , 59, 12625-12631.\n(22) http://www.cmp.sanken.osaka-u.ac.jp/ ~oguchi/HiLAPW/index.html\n22(23) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made\nSimple.Phys:Rev:Lett: 1996 , 77, 3865-3868.\n(24) Daalderop, G. H. O.; Kelly, P. J.; Schuurmans, M. F. H. First-principles calculation\nof the magnetocrystalline anisotropy energy of iron, cobalt, and nickel. Phys: Rev: B\n1990 , 41, 11919-11937.\n(25) Liechtenstein, A.; Katsnelson, M. I.; Antropov, V. P.; Gubanov, V. A. Local spin\ndensity functional approach to the theory of exchange interactions in ferromagnetic\nmetals and alloys. J:Magn:Magn:Mater: 1987 , 67, 65-74.\n(26) http://kkr.issp.u-tokyo.ac.jp\n(27) Akai H.; Dederichs, P. H. Local moment disorder in ferromagnetic alloys. Phys:Rev:B\n1993 , 47, 8739-8747.\n23"    },    {        "title": "2201.11269v1.Temperature_Dependence_of_Magnetocrystalline_Anisotropy_in_Itinerant_Ferromagnets.pdf",        "content": "Temperature  Dependence of Magnetocrystalline Anisotropy  in \nItinerant  Ferromagnets  \n \n \nDaisuke Miura*  and Akimasa Sakuma \n \nDepartment of Applied Physics, Tohoku University , Sendai 980 -8579 , Japan  \n \nWe theoretically investigated  magnetocrystalline anisotropy (MA)  \nat a finite temperature  𝑇  in ferromagnetic metals . Assum ing a \nRashba -type ferromagnet  with uniaxial MA , we define d the MA \nconstant s 𝐾u(𝑇)  derived  from several  different concepts . Our \npurpose was to examine  the equality between them  and to confirm a \npower law between 𝐾u(𝑇)  and magnetization  𝑀(𝑇)  in the form \nof 𝐾u(𝑇)𝐾u(0) ⁄ =[𝑀(𝑇)𝑀(0) ⁄ ]𝛼. We de monstrate that 𝛼 equals  \n2 in the itinerant -electron  limit  and increases with the localized \nfeature of electrons passing through  𝛼=3, predicted for the single -\nion MA in spin models.  \n \nMagnetocrystalline anisotropy (MA) is an important factor  affecting the performance of \nmagnetic materials ; thus, understanding  and control ling MA  are critical requirements  in the \nfield of  magnetics  [1]. Recent ly, MA, especially its temperature dependence,  has been actively \ninvestigated for rare-earth  (RE)  permanent magnets  [2], and the most recent studies have also \nfocused on  RE-free permanent magnets  [3]. In general,  the temperature dependence of the MA  \nconstant s (MACs)  of RE magnets  is complex  [4]. However, the basic mechanism  of this \ndependence  has been clearly  understood on the basis of a local moment picture , where the \nMACs of a material are defined in terms of the magnetization -direction dependence of the \nHelmholtz energy density of the material , which describe s the degree of MA  [5–7]. In this \npicture , a semi phenomenological interpretation can be found by focus ing on the interaction \nbetween the crystalline -electric field and the non -spherical localized -electron cloud  at each site. \nFor example, we have demonstrated  that the temperature dependence of the MACs of Nd-Fe-\nB magnets  can be accurately reproduced  by a power law of the Nd-site moment  [8,9] . Here , a \npower law  [8] extended from  the conventional Akulov –Zener –Callen –Callen  (AZCC)  law [10–\n12] must be used . In addition to such semi -phenomeno logical  treatment , theoretical \ninvestigations  for MA in finite temperatures within  the local moment picture  have  been  \nperformed on the basis  of the ligand -field theory  [7,13 –17], the Landau –Lifshitz –Gilbert \nequation  [18–20], the Monte -Carlo method  [21–24], and other techniques  [7,25,26] . However, \ncompared with the large number of theoretical studies on localized  electron systems, there are  \nfew reports on itinerant  systems, such a s RE-free permanent magnets.  In fact,  a trial of the 2 \n theoretical reproduction of the Curie temperature  of iron is still ongoing  [27]. Moreover, for  the \nMA of  Nd-Fe-B magnets, the contribution  from the Fe sublattice  is not understood  \ncompletely  [8,28,29] . In this regard , Y-Fe-B magnet s are often considered because Y is non -\nmagnetic , and several theoretical examinations of  its MA  have been reported  [28,29] . \nFurthermore, for itinerant  electron system s such as ferromagnetic metals, researchers have  \nattempted to understand the temperature -dependent MA on the basis of the temperature \ndependence of the magnetization.  Thiele et al.  [30] and Okamoto et al.  [31] experimentally \nshowed that in FeNiPt alloys and FePt, respectively, the first -order MAC 𝐾u(𝑇) is roughly \nproportional to 𝑀(𝑇)2, where 𝑀(𝑇) is the saturation magnetization at a temperature 𝑇. This \nsecond power law does not obey  the prediction of 𝑀(𝑇)3  as per the AZCC theory, which \nsuggests that the local moment picture is not appropriate  [32]. In such a  situation , the coherent \npotential approximation (CPA)  theory  in the disordered local moment  (DLM) picture  based on \nthe spin density functional theory  has provi ded a quantitative description of the finite \ntemperature magnetis m in itinerant ferromagnets  [33–36]. The DLM -CPA theory has \nsucceed ed at reproduc ing the second power law  in FePt [37,38] , and  it has also shown that other \nferromagnet ic metals  obey the second power law  for FePd  [38,39] , CoPt, Mn Al, and FeCo  [38], \nfrom  the first principles . Recently , the DLM -CPA theory has been  also employed to evaluate \nthe temperature  dependen ce of important factors in spintronic  devices , such as  the half \nmetallicity  of Co2MnSi  [40] and the Gilbert  damping constant of Fe  and FePt  [41].  \nVery recently,  we have demonstrated , on the basis of the DLM -CPA theory, that the zero-\nfield transverse spin susceptibility 𝜒⊥ of Rashba -type ferromagnets is almost constant in the \nrange of 𝑇<𝑇C (the Curie temperature) , and that 𝜒⊥ obeys the Curie –Weiss law for 𝑇>\n𝑇C [42]. As suggested  there in, this result  indicates 𝐾u(𝑇)∝𝑀(𝑇)2 rather than 𝑀(𝑇)3. This \nand the aforementioned first-principles result s imply that the second power law is widely valid  \nin ferromagnetic metals  with a uniaxial MA . Although we have referred to  a relation between  \nmagnetic susceptibility and  an MAC, we have  not yet discuss ed this in detail.  In this letter,  \nusing the Rashba -type ferromagnetic model  [42–46], we exam ine this relation and  demonstrate  \nits validity within  the DLM -CPA theory . In addition, we consider  the power law in \nferromagnetic metals .  \nWe now consider only uniaxial -MA ferromagnetic systems. Let the z and x axes in the \nCartesian coordinates be the easy and hard axes, respectively. We introduce three targ ets \nexamined as a uniaxial MAC: (1) 𝐾uMP evaluated from the magnetization process, (2) 𝐾uEXF \ndefined by the angle dependence of the Helmholtz energy density on the rotation of exchange 3 \n fields (EXFs), and (3) 𝐾̃uEXF obtained by a perturbative expansion of 𝐾uEXF with respect to a \nspin-orbit interaction (SOI).  \nDefinition of 𝐾uMP. The relationship  between 𝜒⊥(𝑇) and an MAC can be determined by \napplying the definition of 𝜒⊥(𝑇) to the Sucksmith –Thompson equation as follows  [42,47,48] . \n 𝐾uMP(𝑇):=μ0𝑀(𝑇)2\n2𝜒⊥(𝑇), (1) \nwhere μ0  is the permeability in vacuum. We assume that the higher -order MACs are \nnegligible; thus, Eq. (1) cannot be used for complex -MA materials such as RE permanent \nmagnets  [4]. This expression clarifies the temperature dependen ces of 𝑀(𝑇) and the MAC, \nthat is, we can interpret that 𝜒⊥(𝑇) plays a role in the correction for the second power law. \nAlthough Eq. (1) is a phenomenological expression, for a given Hamiltonian, one can \nmicroscopically obtain 𝜒⊥(𝑇)  within the DLM -CPA theory by numerically solving the \nintegral equation described in our previo us work  [42]. In this study, we employ the same model. \nAccordingly, we consider the electron system described by the tight -binding Rashba \nHamilto nian [49] with the EXF on a two -dimensional square lattice having a lattice constant \n𝑎 [42,44,46] . This Hamiltonian is defined by  \n ℋ{𝒆𝑗}≔ℋ0{𝒆𝑗}+𝒱. (2) \nHere, local EXFs are defined at each site in a crystal. The directions of the EXFs are represented \nby a configuration {𝒆𝑗}≔(𝒆1,𝒆2,…,𝒆𝑁) , where 𝒆𝑗  is the unit vector representing the \ndirection of the EXF located at a la ttice vector 𝑹𝑗, and 𝑁 is the total number of the sites. The \nfirst and second terms in Eq. (2) are respectively defined by  \n ℋ0{𝒆𝑗}≔∑𝜖𝒌𝑐𝒌†𝑐𝒌\n𝒌−𝛥ex∑𝒆𝑗⋅𝑐𝑗†𝛔̂𝑐𝑗𝑁\n𝑗=1, (3) \n 𝒱≔∑𝑐𝒌†𝑣̂𝒌𝑐𝒌\n𝒌, (4) \nwhere 𝜖𝒌≔−2𝑡(cos𝑘x𝑎+cos𝑘y𝑎) ; 𝑣̂𝒌≔𝜆(σ̂xsin𝑘y𝑎−σ̂ysin𝑘x𝑎) ;𝑐𝒌  is the \nannihilation operator of the electron with a wave vector 𝒌=(𝑘x,𝑘y)  in the spinor \nrepresentation; 𝑐𝑗≔𝑁−1/2∑𝑐𝒌exp(i𝒌⋅𝑹𝑗) 𝒌  , 𝛔̂≔(σ̂x,σ̂y,σ̂z)  for the Pauli matrices σ̂𝛼 \n(a quantity with the hat symbol is a 2×2 matrix); and 𝑡, 𝜆, and 𝛥ex are the strengths of the \nhopping integral, the Rashba SOI, and the local EXF, respectively.  \nDefinition of 𝐾uEXF . In the DLM scheme, the grand potential density is microscopically \nrepresented by  [50] 4 \n  𝑗𝛽𝜇=−1\n𝜋𝑉lim\n𝜂↘0∫ d𝐸 ℑTr⟨𝒢(𝐸𝜂,{𝒆𝑗})⟩𝑓𝛽(𝐸−𝜇)(𝐸−𝜇)∞\n−∞\n−1\n𝛽𝜋𝑉lim\n𝜂↘0∫ d𝐸 ℑTr⟨𝒢(𝐸𝜂,{𝒆𝑗})⟩{𝑓𝛽(𝐸−𝜇)ln𝑓𝛽(𝐸−𝜇)∞\n−∞\n+[1−𝑓𝛽(𝐸−𝜇)]ln[1−𝑓𝛽(𝐸−𝜇)]}+1\n𝛽𝑉⟨ln𝑊{𝒆𝑗}⟩, (5) \nwhere 𝐸𝜂≔𝐸+i𝜂; 𝒢(𝑧,{𝒆𝑗})≔(𝑧−ℋ{𝒆𝑗})−1for a complex energy 𝑧; Tr is taken over \nall one -electron states; 𝑓𝛽(𝑥)≔(e𝛽𝑥+1)−1for 𝛽≔(kB𝑇)−1; 𝜇 is the chemical potential; \n𝑉≔𝑎3𝑁  is the volume of the crystal; and ⟨…⟩≔Tr{𝒆𝑗}𝑊{𝒆𝑗}…≔\n∫d𝒆1d𝒆2⋯d𝒆𝑁𝑊{𝒆𝑗}…  denotes the EXF -configuration average integr ated on the unit \nspheres with respect to a probability distribution 𝑊{𝒆𝑗}. Within the CPA, we can represent this \nprobability distribution as  [51] 𝑊{𝒆𝑗}=∏ 𝑤(𝒆𝑗)𝑁\n𝑗=1  , where 𝑤(𝒆)≔\nexp[−𝛽Ω(𝒆)]∫d𝒆′exp [−𝛽Ω(𝒆′)] ⁄   and Ω(𝒆)≔𝜋−1lim\n𝜂↘0 ℑ∫ d𝐸∞\n−∞𝑓𝛽(𝐸−\n𝜇)Trspin ln{1̂−[−𝛥ex𝒆⋅𝛔̂−Σ̂(𝐸𝜂)]𝐺̂(𝐸𝜂)}  (Trspin  is taken in a 2×2  spin space). The \nCPA-Green function is given by 𝐺̂(𝑧)=𝑁−1∑[(𝑧−𝜖𝒌)1̂−𝑣̂𝒌−Σ̂(𝑧)]−1\n𝒌  , and Σ̂(𝑧)  is \ndetermined to satisfy the CPA condition  [42,46] . The one -body part of the Green function in Eq. \n(5) is calculated using the expression Tr⟨𝒢(𝑧,{𝒆𝑗})⟩=𝑁Trspin𝐺̂(𝑧) . To define the EXF -\ndirection dependence of 𝑗𝛽𝜇 , we can consider a rotated EXF given by 𝑅(Θ,Φ)∑⟨𝒆𝑖⟩𝑖 =\n∑Tr{𝒆𝑗}𝑊{𝑅(Θ,Φ)−1𝒆𝑗}𝒆𝑖 𝑖  , where the EXF direction before the rotation is represented by \n∑⟨𝒆𝑖⟩ 𝑖 and 𝑅(Θ,Φ) is a rotation matrix with respect to the rotation a ngles (Θ,Φ). Using this \nconfiguration of rotated EXFs, we define a rotated grand potential density  𝐽𝛽𝜇(Θ,Φ), which is \nexpressed as  \n 𝐽𝛽𝜇(Θ,Φ)≔𝑗𝛽𝜇 in replacement  of 𝑊{𝒆𝑗}→𝑊{𝑅(Θ,Φ)−1𝒆𝑗}, (6) \nwhere the rotation matrix acting on a vector is expressed in the Cartesian coordinates as follows : \n 𝑅(Θ,Φ)=(cosΘcosΦ −sinΦ sinΘcosΦ\ncosΘsinΦ cosΦ sinΘsinΦ\n−sinΘ 0 cosΘ). (7) \nWe introduce the angle -dependent Helmholtz energy density to fix the electron concentration  \nto 𝑛 . This energy density is defined by the Legendre transformation as 𝐹𝛽𝑛(Θ,Φ)≔\nmax\n𝜇[𝐽𝛽𝜇(Θ,Φ)+𝜇𝑛] . Further, with 𝛥𝐹𝛽𝑛(Θ,Φ)≔𝐹𝛽𝑛(Θ,Φ)−𝐹𝛽𝑛(0,0) , we define an 5 \n MAC by  \n 𝐾uEXF(𝑇)≔𝛥𝐹𝛽𝑛(𝜋2⁄,0), (8) \nwhere 𝛥𝐹𝛽𝑛(𝜋2⁄,0) represents the energy density needed to rotate the EXF from the easy \naxis to the hard axis.  \n Definition of 𝐾̃uEXF. Perturbatively expanding 𝛥𝐹𝛽𝑛(Θ,Φ) with res pect to 𝜆, we obtain  \n 𝛥𝐹𝛽𝑛(Θ,Φ)=−lim\n𝜂↘01\n𝑉∫d𝐸\n2𝜋𝑓𝛽(𝐸−𝜇0)∞\n−∞\n×ℑTr⟨[𝒢0(𝐸𝜂,{𝒆𝑗})𝒰(Θ,Φ)†𝒱𝒰(Θ,Φ)]2−[𝒢0(𝐸𝜂,{𝒆𝑗})𝒱]2⟩\n0+𝒪(𝜆4), (9) \nwhere quantities with the index 0 have 𝜆=0 in each evaluation, and 𝒰(Θ,Φ) is a unitary \noperator satisfying 𝒰(Θ,Φ)ℋ0{𝒆𝑗}𝒰(Θ,Φ)†=ℋ0{𝑅(Θ,Φ)𝒆𝑗}. Here, we assume that ⟨…⟩≃\n⟨…⟩0, neglecting the effect of 𝜆 on the EXF structure. To calculating the two -body part of the \nGreen functions according to Butler  [52] within the CPA, we have \nTr⟨[𝒢0(𝑧,{𝒆𝑗})𝒰(Θ,Φ)†𝒱𝒰(Θ,Φ)]2⟩\n0=Tr[⟨𝒢0(𝑧,{𝒆𝑗})⟩0𝒰(Θ,Φ)†𝒱𝒰(Θ,Φ)]2\n . This \nindicates that the vertex correction vanishes because of the symmetry of the Rashba SOI, i.e., \n𝑣̂−𝒌=−𝑣̂𝒌. Subsequently, the explicit form of the angle dependence is ob tained as  \n Tr⟨[𝒢0(𝑧,{𝒆𝑗})𝒰(Θ,Φ)†𝒱𝒰(Θ,Φ)]2−[𝒢0(𝑧,{𝒆𝑗})𝒱]2⟩\n0\n=sin2Θ𝜆2[Σ0+(𝑧)−Σ0−(𝑧)]2∑[𝑔𝒌+(𝑧)𝑔𝒌−(𝑧)]2\n𝒌sin2𝑘x𝑎, (10) \nwhere Σ0𝜎(𝑧)  and 𝑔𝒌𝜎(𝑧)  are the 𝜎 -spin elements in Σ̂0(𝑧)  and 𝑔̂𝒌(𝑧)≔[(𝑧−𝜖𝒌)1̂−\nΣ̂0(𝑧)]−1, respectively (these are diagonal). Thus, from the usual definition of the first -order \nMAC of 𝛥𝐹𝛽𝑛(Θ,Φ)=𝐾̃uEXFsin2Θ+𝒪(𝜆4), we have  \n 𝐾̃uEXF(𝑇)=−lim\n𝜂↘0 ℑ∫d𝐸\n2𝜋𝑓𝛽(𝐸−𝜇0)∞\n−∞\n×[Σ0+(𝐸𝜂)−Σ0−(𝐸𝜂)]2𝜆2\n𝑉∑[𝑔𝒌+(𝐸𝜂)𝑔𝒌−(𝐸𝜂)]2\n𝒌sin2𝑘x𝑎. (11) \nWe now present  a power law for the first -order MAC within the perturbative treatment. We \nintroduce a (nonperturbative) magnetization as  6 \n  𝑀̃(𝑇):=−𝑔μB\n2lim\n𝜂↘0 ℑ∫d𝐸\n𝜋𝑓𝛽(𝐸−𝜇0)∞\n−∞\n×[Σ0+(𝐸𝜂)−Σ0−(𝐸𝜂)]1\n𝑉∑𝑔𝒌+(𝐸𝜂)𝑔𝒌−(𝐸𝜂)\n𝒌, (12) \nwhere 𝑔 and 𝜇B are the g factor of an electron and the Bohr magneton, respectively. When \ncomparing Eqs. (11) and (12), using the Born approximation Σ0𝜎(𝑧)≃−𝜎𝛥ex⟨𝑒z⟩+\n𝐺0𝜎(𝑧)𝛥ex2(1−⟨𝑒z⟩2) and then approximating as Σ0+(𝐸𝜂)−Σ0−(𝐸𝜂)≃−2𝛥ex⟨𝑒z⟩+𝒪(𝛥ex2), \nwe obtain 𝐾̃uEXF(𝑇)𝐾̃uEXF(0) ⁄ ≃⟨𝑒z⟩2 and 𝑀̃(𝑇)𝑀̃(0) ⁄ ≃⟨𝑒z⟩. Thus, a second  power law is \nobtained as  \n 𝐾̃uEXF(𝑇)\n𝐾̃uEXF(0)≃(𝑀̃(𝑇)\n𝑀̃(0))2\n, (13) \nwithin the perturbative treatment up to the order of 𝜆2𝛥ex. Outside the perturbative range, this \nsecond  power law is not necessarily valid, as explained below.  \nFirst, we numerically exam the power law between 𝐾̃uEXF(𝑇) and 𝑀̃(𝑇). In the perturbative \ntheory with respect to 𝜆 , the model parameters are 𝑛  and 𝛥ex/𝑡 . We set electron \nconcentration  to 𝑛=0.27, which satisfies the condition of ferromagnetism with the u niaxial \nMA. Figure 1 shows the temperature dependences of 𝑘̃uEXF(𝑇)≔𝐾̃uEXF(𝑇)𝐾̃uEXF(0) ⁄   and \n𝛼̃(𝑇)≔ln𝑘̃uEXF(𝑇)/ln𝑚̃(𝑇) , where 𝑚̃(𝑇)≔𝑀̃(𝑇)𝑀̃(0) ⁄  , and 𝑚̃(𝑇)2 , 𝑚̃(𝑇)3 , and \n𝑚̃(𝑇)4 are shown as references for observing the power law. For 𝛥ex𝑡⁄=0.2 [Fig. 1(a)], the \nsecond  power law [ 𝛼̃(𝑇)≃2 ] can be clearly observed, but 𝛼̃(𝑇)  and its temperature \ndependency are found to increase with increasing 𝛥ex𝑡⁄ [Fig. 1(b) -(d)]. From these result s, \nwe suggest that the ratio of the exchange -splitting energy divided by the band width is an \nimportant factor for establishing the second  power law in metallic ferromagnetism. In other \nwords, we can consider that this tendency is consistent with the physi cally observed fact that \nitinerant systems are likely to exhibit 𝛼̃=2 whereas localized ones exhibit 𝛼̃>2. As shown \nin Fig. 1(d), the exponent 𝛼̃ exceeds the value 3 in the localized electron limit. At this point, \none can compare the spin model and the pre sent model. When considering a spin Hamiltonian \ndensity including the MA term of −∑𝐾𝑖(𝑆𝑖𝑧)2−∑ 𝐾𝑖𝑗𝑆𝑖𝑧𝑆𝑗𝑧\n𝑖≠𝑗 𝑖 , the Callen –Callen theory  [12] \npredicts the exponent 𝛼CC=(2𝐾int+3𝐾on)(𝐾int+𝐾on) ⁄  in the low -temperature limit  [53]. \nHere, 𝐾𝑖 and 𝐾𝑖𝑗 respectively represent single -ion and two -ion anisotropies; 𝑺𝑖 is the spin \noperator with a spin 𝑆  at site 𝑖 ; 𝐾on≔𝑆(𝑆−12⁄)∑𝐾𝑖𝑖 ; and 𝐾int≔(𝑆22⁄)∑ 𝐾𝑖𝑗 𝑖≠𝑗 . \nFrom this expression for 𝛼CC , we obtain 2<𝛼CC≤3  for 𝐾int/𝐾on>0 , as expected. \nFurthermore,  3<𝛼CC for 𝐾int𝐾on⁄ <0 [54]. These findings for the spin system indicate that \nitinerant ferromagnets, which involve both spin and or bital degrees of freedom, have the \npotential to realize a wide range of 𝛼̃ other than 2≤𝛼̃≤3, even with changing 𝛥ex𝑡⁄. \nNext, we consider an equality of the formulations of 𝐾uMP(𝑇) and 𝐾uEXF(𝑇). Although both 7 \n of them estimate the MA energy density of a material, their theoretical concepts are different. \n𝐾uMP(𝑇) is defined as the energy density required to magnetize  by an external magnetic field \nalong the hard axis. Accordingly, it is expressed by the magnetic susceptibility  as shown in Eq. \n(1). In contrast, 𝐾uEXF(𝑇) is defined as the energy density required to virtually rotate the EXF \nfrom the easy  axis to the hard axis in a fixed spin s tructure. Figure 2 shows four cases: (a) weak \nEXF and weak SOI, (b) strong EXF and weak SOI, (c) weak EXF and strong SOI, and (d) \nstrong EXF and strong SOI. The values of EXF and SOI are provided in the caption of Fig.  2. \nIn addition to the MACs, we also p lot the exponent calculated from 𝐾uEXF(𝑇), which is defined \nby 𝛼(𝑇)≔ln[𝐾uEXF(𝑇)/𝐾uEXF(0)]/ln[𝑀(𝑇)𝑀(0) ⁄ ] , where 𝑀(𝑇):=\n−𝑔μB\n2𝑎3lim\n𝜂↘0 ℑ∫d𝐸\n𝜋𝑓𝛽(𝐸−𝜇)∞\n−∞Trspinσ̂z𝐺̂(𝐸𝜂). As expected, all MACs match well in the weak -\n𝜆  cases (a) and (b), and there is a large difference between 𝐾uEXF(𝑇)  and 𝐾̃uEXF(𝑇)  in the \nstrong -𝜆 cases (c) and (d). Although 𝐾uMP(𝑇) and 𝐾uEXF(𝑇) seem to yield almost the same \nresult  in the all cases, 𝐾uMP(𝑇) slightly deviates from 𝐾uEXF(𝑇) in the strong -𝜆 cases. This \ndeviation implies that 𝐾uMP(𝑇)  and 𝐾uEXF(𝑇)  are not always consistent. In particular, \nalthough not shown in this letter, we observe that the difference between them at the zero \ntemperature increases when the chemical potential exist s at a highly half -metallic region in the \ndensity of states in the electronic structure. These temperature dependences differ at higher \ntemperatures, indicating that the half metallicity is lost as the temperature increases. In addition, \na noncollinear eff ect (NCE) between the directions of the magnetization and the EXF yields a \nmismatch between 𝐾uMP(𝑇) and 𝐾uEXF(𝑇) in general, as shown in our previous work on  the \nMA of  Dy-Fe-B magnets  [7] (in the present conditions, we have confirmed that the NCE is \nnegligible). Because such special behaviors may strongly depend on theoretical models, these \ncases should be studied in future work.  \n  In summary, we theoretically investigated the temperature -dependent MA in ferromagnetic \nmetals with a uniaxial MA using the Rashba model. We examined the temperature dependences \nof 𝐾uMP , 𝐾uEXF , and 𝐾̃uEXF  based on the Sucksmith –Thompson theory, the EXF -rotation \nconcept, and the perturbative theory with respect to the Rashba SOI, respectively. In addition, \nwe demonstrated the validity of the power law. Consequently, we confirmed that 𝐾̃uEXF \nfunctions well for a weak SOI and helps determine the effect of the strength of the EXF on its \ntemperature dependence. This enhances the understanding of the second  power law observed \nin ferromagnetic metals with a uniaxial MA. 𝐾uMP(𝑇)  and 𝐾uEXF(𝑇)  can also be used in a \nstrong -SOI case to estimate the MA, and the value s match well in the present conditions. 8 \n However, they are not always consistent, and further investigations are required to study this \ndiscrepancy.\n \nAcknowledgment  \nThis work was supported by JSPS KAKENHI Grant  Num bers JP17K14800,  JP19H056 12, \nand JP21K04624  in Japan.  \n \n*E-mail: dmiura 1222 @gmail.com  \n \n[1] S. Hirosawa, IEEE Trans. Magn. 55, 2100506 (2019).  \n[2] C. E. Patrick and J. B. Staunton, Phys. Rev. Mater. 3, 101401(R) (2019).  \n[3] J. Mohapatra and J. P. Liu, in Handbook of Magnetic Materials , edited by E. Brück (Elsevier, \nNorth Holland, 2018), p. 1.  \n[4] R. Skomski, P. Kumar, G. C. Hadjipanayis, and D. J. Se llmyer, IEEE Trans. Magn. 49, 3229 \n(2013).  \n[5] J. F. Herbst, Rev. Mod. Phys. 63, 819 (1991).  \n[6] M. Yamada, H. Kato, H. Yamamoto, and Y. Nakagawa, Phys. Rev. B 38, 620 (1988).  \n[7] D. Miura and A. Sakuma, J. Phys. Soc. Jpn. 88, 044804 (2019).  \n[8] D. Miura and A. Sakuma, AIP Adv. 8, 075114 (2018).  \n[9] D. Miura and A. Sakuma, Jpn. J. Appl. Phys. 58, 058002 (2019).  \n[10] N. Akulov, Z. Phys. 100, 197 (1936).  \n[11] C. Zener, Phys. Rev. 96, 1335 (1954).  \n[12] H. B. Callen and E. Callen, J. Phys. Chem. Solids 27, 1271 (1966).  \n[13] M. D. Kuz’min and A. M. Tishin, in Handbook of Magnetic Materials , edited by K. H. J. \nBuschow (Elsevier, Amsterdam, 2007), p. 149.  \n[14] R. Sasaki, D. Miura, and A. S akuma, Appl. Phys. Express 8, 043004 (2015).  \n[15] D. Miura, R. Sasaki, and A. Sakuma, Appl. Phys. Express 8, 113003 (2015).  \n[16] T. Yoshioka, H. Tsuchiura, and P. Novák, Phys. Rev. B 102, 184410 (2020).  \n[17] S. Yamashita, D. Suzuki, T. Yoshioka, H. Tsuchiu ra, and P. Novák, Phys. Rev. B 102, 214439 \n(2020).  \n[18] M. Nishino, I. E. Uysal, T. Hinokihara, and S. Miyashita, Phys. Rev. B 102, 020413(R) (2020).  \n[19] M. Nishino, I. E. Uysal, and S. Miyashita, Phys. Rev. B 103, 014418 (2021).  \n[20] M. Nishino, I. E. Uy sal, T. Hinokihara, and S. Miyashita, AIP Adv. 11, 025102 (2021).  \n[21] Y. Toga, M. Matsumoto, S. Miyashita, H. Akai, S. Doi, T. Miyake, and A. Sakuma, Phys. Rev. \nB 94, 174433 (2016).  \n[22] T. Hinokihara, M. Nishino, Y. Toga, and S. Miyashita, Phys. Rev. B 97, 104427 (2018).  \n[23] Y. Toga, M. Nishino, S. Miyashita, T. Miyake, and A. Sakuma, Phys. Rev. B 98, 054418 \n(2018).  9 \n [24] Y. Toga, S. Miyashita, A. Sakuma, and T. Miyake, Npj Comput. Mater. 6, 67 (2020).  \n[25] T. Miyake and H. Akai, J. Phys. Soc. Jpn. 87, 041009 (2018).  \n[26] S. Miyashita, M. Nishino, Y. Toga, T. Hinokihara, I. E. Uysal, T. Miyake, H. Akai, S. \nHirosawa, and A. Sakuma, Sci. Technol. Adv. Mater. 22, 659 (2021).  \n[27] T. Tanaka and Y. Gohda, Npj Comput. Mater. 6, 184 (2020).  \n[28] R. F. L. Evans, L . Rózsa, S. Jenkins, and U. Atxitia, Phys. Rev. B 102, 020412(R) (2020).  \n[29] R. Cuadrado, R. F. L. Evans, T. Shoji, M. Yano, A. Kato, M. Ito, G. Hrkac, T. Schrefl, and R. \nW. Chantrell, J. Appl. Phys. 130, 023901 (2021).  \n[30] J. U. Thiele, K. R. Coffey, M. F. Toney, J. A. Hedstrom, and A. J. Kellock, J. Appl. Phys. 91, \n6595 (2002).  \n[31] S. Okamoto, N. Kikuchi, O. Kitakami, T. Miyazaki, Y. Shimada, and K. Fukamichi, Phys. Rev. \nB 66, 024413 (2002).  \n[32] O. N. Mryasov, U. Nowak, K . Y. Guslienko, and R. W. Chantrell, Europhys. Lett. 69, 805 \n(2005).  \n[33] T. Oguchi, K. Terakura, and N. Hamada, J. Phys. F Met. Phys. 13, 145 (1983).  \n[34] A. J. Pindor, J. Staunton, G. M. Stocks, and H. Winter, J. Phys. F Met. Phys. 13, 979 (1983).  \n[35] J. Staunton, B. L. Gyorffy, A. J. Pindor, G. M. Stocks, and H. Winter, J. Magn. Magn. Mater. \n45, 15 (1984).  \n[36] B. L. Gyorffy, A. J. Pindor, J. Staunton, G.  M. Stocks, and H. Winter, J. Phys. F Met. Phys. 15, \n1337 (1985).  \n[37] J. B. Staunton, S. Ostanin, S. S. A. Razee, B. L. Gyorffy, L. Szunyogh, B. Ginatempo, and E. \nBruno, Phys. Rev. Lett. 93, 257204 (2004).  \n[38] N. Kobayashi, K. Hyodo, and A. Sakuma, Jpn. J. Appl. Phys. 55, 100306 (2016).  \n[39] J. B. Staunton, L. Szunyogh, A. Buruzs, B. L. Gyorffy, S. Ostanin, and L. Udvardi, Phys. Rev. \nB 74, 144411 (2006).  \n[40] K. Nawa, I. Kurniawan, K. Masuda, Y. Miura, C. E. Patrick, and J. B. Staunton, Phys. Rev. B \n102, 054424 (2020).  \n[41] R. Hiramatsu, D. Miura, and A. Sakuma, Appl. Phys. Express 15, 013003 (2022).  \n[42] D. Miura and A. Sakuma, J. Phys. Soc. Jpn. 90, 113601 (2021).  \n[43] S. E. Barnes, J. Ieda, and S. Maekawa, Sci. Rep. 4, 4105 (2014).  \n[44] K. W. Kim, K. J.  Lee, H. W. Lee, and M. D. Stiles, Phys. Rev. B 94, 184402 (2016).  \n[45] J. Ieda, S. E. Barnes, and S. Maekawa, J. Phys. Soc. Jpn. 87, 053703 (2018).  \n[46] A. Sakuma, J. Phys. Soc. Jpn. 87, 034705 (2018).  \n[47] W. Sucksmith and J. E. Thompson, Proc. R. Soc. L ondon, Ser. A 225, 362 (1954).  \n[48] A. S. Bolyachkin, D. S. Neznakhin, and M. I. Bartashevich, J. Appl. Phys. 118, (2015).  \n[49] T. Ando and H. Tamura, Phys. Rev. B 46, 2332 (1992).  \n[50] A. Deák, E. Simon, L. Balogh, L. Szunyogh, M. Dos Santos Dias, and J. B. Staunton, Phys. \nRev. B 89, 224401 (2014).  10 \n [51] H. Hasegawa, J. Phys. Soc. Jpn. 963 (1980).  \n[52] W. H. Butler, Phys. Rev. B 31, 3260 (1985).  \n[53] We notice that αCC depends on T even i n the simplest spin model with uniaxial  the \nuniaxial MA when measured more precisely  [15]. \n[54] Evans et al.[28] have examined these features more precisely using  the Green function method \nand constrained Monte -Carlo simulation beyond the Callen –Callen theory.  \n 012345\n0.00.51.0\n0.0 0.5 1.0\n012345\n0.00.51.0\n0.0 0.5 1.0\n012345\n0.00.51.0\n0.0 0.5 1.0\n012345\n0.00.51.0\n0.0 0.5 1.0\nFig.1Calculated temperature dependences of෨𝑘uEXF𝑇and𝛼(𝑇),\ndenoted bythesolid and dotted lines, respectively .The upper,\nmiddle, and lower dashed lines ineach figure represent 𝑚𝑇2,\n𝑚𝑇3,and𝑚𝑇4,respectively .(a)Τ𝛥ex𝑡=0.2\n𝑇/𝑇C(b)Τ𝛥ex𝑡=0.7\n(c)Τ𝛥ex𝑡=1.0\n(d)Τ𝛥ex𝑡=1.2𝛼\n𝛼\n𝛼\n𝛼෨𝑘uEXF෨𝑘uEXF෨𝑘uEXF\n෨𝑘uEXF012345\n0.00.51.0\n0.0 0.5 1.0\n012345\n0.00.51.0\n0.0 0.5 1.0(b)\nFig.2Calculated temperature dependences of𝐾uMP𝑇,𝐾uEXF(𝑇),\n෩𝐾uEXF𝑇,and𝛼(𝑇).The solid and dotted lines represent the\nMACs and𝛼(𝑇),respectively .Ineach case, the MACs are\nnormalized by෩𝐾uEXF0.The model parameters, Τ𝛥ex,𝜆𝑡,are\nsetas(a)(0.2,0.1),(b)(1.2,0.1),(c)(0.2,0.8),and(d)(1.2,0.8).𝑇/𝑇C෩𝐾uEXF\n෩𝐾uEXF𝛼\n𝛼𝛼𝛼\n𝐾uMP𝐾uMP,𝐾uEXF,෩𝐾uEXF𝐾uMP,𝐾uEXF,෩𝐾uEXF (a)\n(d)𝐾uMP,𝐾uEXF\n𝐾uEXF012345\n0.00.51.0\n0.0 0.5 1.0\n012345\n0.00.51.0\n0.0 0.5 1.0(c)"    },    {        "title": "2203.00817v1.Polaronic_Conductivity_in_Cr__2_Ge__2_Te__6__Single_Crystals.pdf",        "content": "arXiv:2203.00817v1  [cond-mat.mtrl-sci]  2 Mar 2022Polaronic Conductivity in Cr 2Ge2Te6Single Crystals\nYu Liu,1Myung-Geun Han,1Yongbin Lee,2Michael O. Ogunbunmi,3Qianheng Du,1,4Christie Nelson,5Zhixiang\nHu,1,4Eli Stavitski,5David Graf,6Klaus Attenkofer,5Svilen Bobev,3Liqin Ke,2Yimei Zhu,1and C. Petrovic1,4\n1Condensed Matter Physics and Materials Science Department ,\nBrookhaven National Laboratory, Upton, New York 11973, USA\n2Ames Laboratory, U.S. Department of Energy, Ames, Iowa 5001 1, USA\n3Department of Chemistry and Biochemistry, University of De laware, Newark, Delaware 19716, USA\n4Materials Science and Chemical Engineering Department,\nStony Brook University, Stony Brook, New York 11790, USA\n5National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, New York 11973, USA\n6National High Magnetic Field Laboratory, Florida State Uni versity, Tallahassee, Florida 32306-4005, USA\n(Dated: March 3, 2022)\nIntrinsic, two-dimensional (2D) ferromagnetic semicondu ctors are an important class of materials\nfor spin-charge conversion applications. Cr 2Ge2Te6retains long-range magnetic order in bilayer\nat cryogenic temperatures and shows complex magnetic inter actions with considerable magnetic\nanisotropy. Here, we performed a series of structural, magn etic, X-ray scattering, electronic, thermal\ntransport andfirst-principles calculation studies which r eveal that localized electronic charge carriers\nin Cr2Ge2Te6are dressed by surrounding lattice and are involved in polar onic transport via hopping\nthat is sensitive on details of magnetocrystalline anisotr opy. This opens possibility for manipulation\nof charge transport in Cr 2Ge2Te6- based devices by electron-phonon- and spin-orbit couplin g-based\ntailoring of polaron properties.\nI. INTRODUCTION\nFerromagnetic semiconductors are of great importance\nfor the newly-developing spintronics technology that al-\nlows for control of both charge and spin degrees of free-\ndom ofchargecarriers.1The mechanism ofintrinsic long-\nrange magnetic order in monolayers of 2D materials such\nas CrI 3and bilayerCr 2Ge2Te6might be exploited for de-\nsign of novel spin-related devices.2,3They can be stacked\ninto heterostructures4–9and their physical properties\nare highly tunable by carrier doping,10–13strain,14–16\nvacancies,17–21pressure22–26and electric field.27–30\nCr2Ge2Te6exhibits a paramagnetic(PM) to ferromag-\nnetic (FM) transition with the Curie temperature ( Tc) of\n61∼68 K in bulk, an out-of-plane easy axis and neg-\nligible coercivity.3,31TheTcis absent in monolayer but\npersists in bilayer Cr 2Ge2Te6of a relatively high value of\n30K.3Few-layerCr 2Ge2Te6crystalsholdanintrinsicFM\norderwithhightunabilitybygating,asrequiredofagood\nfunctional quantum material in spintronics.29Cr2Ge2Te6\nis proposed to be a two-dimensional (2D) Heisenberg fer-\nromagnet based on the spin wave theory,3but was also\nfound to follow a 2D Ising-like or tricritical mean-field\nmodel,32,33indicating a complex magnetic mechanism.\nCr2Ge2Te6has a semiconducting band gap of 0.2 ∼\n0.74 eV, which can be tuned into metallic by applying\npressure, organic ion intercalation, and vacancies.13,19,24\nIn a double-layer Cr 2Ge2Te6device subjected to elec-\ntrostatic gating metallic resistivity has been observed.34\nStrong spin-phonon coupling has been verified in\nisostructural Cr 2Si2Te6.35Yet, the electronic transport\nmechanism is still unclear in conducting Cr 2Ge2Te6sin-\ngle crystals.\nIn this letter we show that local bond distances ob-\ntained from X-ray absorption spectroscopy exhibit lit-tle change when compared to published average values,\nimplying little difference in the basic structural units of\neach atom. On the other hand, stacking of such units at\nlonger atomic distances results in stacking faults. We ob-\nserve relatively low values of thermal conductivity κ(T)\nbelow 300 K which is enhanced in magnetic field point-\ning to strong spin-phonon coupling. Moreover, electrical\nand thermal transportconduction revealpolaronictrans-\nport, commonly observed in oxide materials.36Due to\nstrongspin-orbitcouplingwhichisthesourceofmagnetic\nanisotropy, polaronic transport mechanism could allow\nfor tunable spin current in future spintronic devices.37–39\nII. RESULTS AND DISCUSSION\nTheobtainedsinglecrystalsexhibitsa2Dlayeredchar-\nacteristic with the cleave surface in the ab-plane and the\nc-axisisperpendiculartothissurface( Figure 1 a).40The\nlattice constant of about c= 20.57(2)˚A can be extracted\nby the Bragg’s law. Cr 2Ge2Te6crystalizes in a layered\nstructure with the R¯3hspace group, in which the lay-\ners stack along the (00l) direction ( Figure 1 b,c). There\nare different types of chemical bonds in crystal structure,\nincluding strong intralayer Cr-Te ionic bonds, Ge-Te co-\nvalent bonds, Ge-Ge dimers, and weak interlayer van\nder Waals (vdW) forces.41Each Cr is octahedrally sur-\nroundedbysix Te, andthe edge-sharingCrTe 6octahedra\nform a honeycomb network. Dimers of Ge-Ge occupy the\nvoid of each honeycomb, forming Ge 2Te6groups.\nFigure 1 d shows the atomic resolution high-angle an-\nnular dark-field (HAADF) scanning transmission elec-\ntron microscopy (STEM) image along the (00l) direc-\ntion. The Te atoms are hexagonally packed from the top\nview, while the Cr ions and Ge-Ge dimers occupy Te-2\noctahedral sites. The corresponding electron diffraction\npattern (inset in Figure 1 d) confirms the space group.\nThe chemical composition gives elemental ratio Cr : Ge\n: Te = 20 .09 : 20.05 : 59.86 (Figure 1 e).40We found\nthe presence of stacking faults in cross-section samples,\nas shown in Figure 1 f-j. There are at least about 8\nstacking faults in 300 nm length scale along the c-axis\n(Figure 1 g-j), which gives 8/(300 nm) = 0.027 nm−1\nstacking fault density. This represents the upper limit\nsince most of stacking faults are found near the original\ncrystal surface.\nThe FM arises from the super-exchange between the\nnearest-neighbor (NN) Cr linked by Te ligands through\nnearly-90◦angle, however the exchange interactions be-\nyond the NN spins are also crucial in determining the\nmagneticgroundstate.14Pressurecancompressthebond\nlengths of Cr-Cr and Cr-Te and tilt the Cr-Te-Cr an-\ngle from 90◦, resulting in an enhancement of antiferro-\nmagnetic (AFM) interaction.41This can also induce spin\nreorientation transition from an uniaxial to easy-plane\nanisotropy,22indicating that the magnetism is strongly\ndependent on Cr local atomic environment. To probe\nthis local environment we performed X-ray absorption\nnear edge structure (XANES) and extended X-ray ab-\nsorption fine structure (EXAFS) experiments. Figure\n2shows the normalized Cr, Ge, and Te K-edge XANES\nspectra and the Fourier transformmagnitudes of EXAFS\nspectra of Cr 2Ge2Te6, respectively. The XANES spectra\n(Figure 2 a-c) are close to that of Cr 2Te3with Cr3+and\nTe2+states.42The local environment of indicated atoms\nis revealed in the EXAFS spectra of Cr 2Ge2Te6mea-\nsured at room temperature ( Figure 2 d-f). In a single-\nscattering approximation, the EXAFS can be described\nby:43\nχ(k) =/summationdisplay\niNiS2\n0\nkR2\nifi(k,Ri)e−2Ri\nλe−2k2σ2\nisin[2kRi+δi(k)],\nwhereNiis the numberofneighbouringatomsat adis-\ntanceRifrom the photoabsorbing atom. S2\n0is the pas-\nsive electrons reduction factor, fi(k,Ri) is the backscat-\ntering amplitude, λis the photoelectron mean free path,\nδiis the phase shift, and σ2\niis the correlated Debye-\nWaller factor measuring the mean square relative dis-\nplacement of the photoabsorber-backscatter pairs. The\nmain peaks have been corrected by standards of the NN\nCr-Te [2.76(7) ˚A] and Ge-Te [2.45(13) ˚A] in the Fourier\ntransform magnitudes of EXAFS. We summarized the\nlocal bond distances from different central atom ranging\nfrom 1.5 to 4.5 ˚A in40. There are six Te NN for the Cr\natomic site with two different distances [2.76(7) ˚A and\n2.87(7)˚A]; the next NN is one Cr [3.95(25) ˚A], and then\nare six Ge with two different distances [4.11(25) ˚A and\n4.17(25) ˚A]. Ge atomic site has three NN Te [2.45(13)\n˚A] and one Ge [2.52(18) ˚A], and also three Te [3.84(18)\n˚A].31Experimental bond distances agree very well with\neach other when using different absorbing atom EXAFS\nspectra. The NN Te-Te distances are 3.81(90) ˚A and3.86(90) ˚A, and then there are three Te-Te [4.03(6) ˚A].\nThe peaks above 4.2 ˚A are due to multiple scattering\ninvolving different near neighbours and longer bond dis-\ntances. Overall, local structure-derived bond distances\n(Table I in40) are in agreement with bond distances from\nneutron powder Rietveld refinement of the average crys-\ntal structure.31\nFigure 3 a exhibits the temperature-dependent zero-\nfieldspecificheat Cp(T)ofCr 2Ge2Te6. Aλ-shapepeakis\nobserved at Tc= 66 K, corresponding well to the second-\norderPM-FMtransition. Thelowtemperaturepartfrom\n2 to 10 K can be well fitted by Cp(T) =γT+βT3+δT3/2\n[inset in Fig. 3(a)], where the first term is the Sommer-\nfeld electronic part, the second term is low-temperature\nlimit of Debye phonon part, and the third term is low-\ntemperature spin wave contribution.44The derived γ,β,\nandδare 9(5) mJ mol−1K−2, 2.52(4) mJ mol−1K−4,\nand11(3)mJmol−1K−5/2, respectively. TheDebyetem-\nperature Θ D= 198(1) K can be calculated from βusing\nΘD= (12π4NR/5β)1/3, whereN= 10 is the number of\natoms per formula unit.\nFigure 3 b shows the temperature-dependent in-plane\nthermal conductivity κ(T) of Cr 2Ge2Te6. In general, the\nκ(T) consists of the electronic part κeand the phonon\ntermκph, i. e., κ=κe+κph. Theκepart can be\nestimated from the Wiedemann-Franz law κe=L0T/ρ\nwithL0= 2.45×10−8W Ω K−2andρis the mea-\nsured electrical resistivity. The estimated κeis less\nthan 0.03 % of κ(T) due to the large electrical resis-\ntivity of Cr 2Ge2Te6, indicating a predominantly phonon\ncontribution. At 300 K, the value of κ(T) is about\n5.34 W K−1m−1, larger than that of 3 W K−1m−1\nfor Cr 2Si2Te6and, as expected due to the absence of\ndense grain boundaries, also larger when compared to\nCr2Ge2Te6polycrystal.17,35In the absence of magnetic\nfield above 100 K, the κ(T) shows weak temperature de-\npendence, different from phonon-dominated κph(T) cal-\nculated within Debye model (solid line in Figure 3 b)\nthat takes ∼1/Tbehavior at high temperatures.40The\ndifference is likely due to phonon scattering by magnetic\nfluctuations and strong spin-lattice coupling producing\nphonon glass.35With decreasing temperature, a rapid\nincrease occurs at Tcand a typical phonon peak was ob-\nserved at 25 K, however stacking faults have only minor\ninfluence on κ(T).40Strong positive field-dependent be-\nhavior of κ(T) inFigure 3 b is similar to RuCl 3where\nit was described within the model of phonon scattering\noffKitaev-Heisenbergexcitationsthatareeitherfraction-\nalized or incoherent type originating from strong mag-\nnetic anharmonicity.45Since Kitaev-type magnetic inter-\nactions have been discussed in Cr 2Ge2Te6it is likely that\nκ(T) in 9 T is connected with the same mechanism.46,47\nFigure 3 c presents the temperature dependence of in-\nplane electrical resistivity ρ(T) for Cr 2Ge2Te6, showing\nanobvioussemiconducting behaviorwith ρ300K= 0.45Ω\ncm. For resistivity mechanism we consider the thermally\nactivated model ρ(T) =ρ0exp(Eρ/kBT), the adiabatic\nsmall polaron hopping model ρ(T) =ATexp(Eρ/kBT)3\nwherekB= 8.617 eV K−1is the Boltzmann constant\nandEρis activation energy, and the Mott variable-range\nhopping(VRH)model ρ(T) =ρ0exp(T0/T)1/4.48Figure\n3d shows the fitting result of the adiabatic small polaron\nhopping model in two temperature ranges 380-150K and\n100-60 K. The extracted activation energy Eρis about\n118.9(3) meV for 380-150 K and 26.2(7) meV for 100-60\nK, respectively. Whereas the ρ(T) curve can not be well\nfitted with the VRH model it can be explained by the\nthermally activated model, i.e., plot of lnρvs 1/kBTand\nln(ρ/T) vs 1/kBToverlap with each other ( Figure 3 d).\nTemperature-dependent thermopower S(T) gives fur-\nther insight into electronic transport. The S(T) exhibits\npositive values in the entire temperature range of our\nmeasurementwith S300K= 582µVK−1(Figure 3 e), in-\ndicating dominant hole-type carriers. The value of S(T)\nat 300 K is quite similar to polycrystal, which argues\nfor the absence of phonon drag effect and dominant elec-\ntronic diffusion mechanism of thermal conduction.17The\nS(T) from 300 to 150 K can be fitted with the equation\nS(T) = (kB/e)(α+ES/kBT) (Figure3 f),48whereESis\nactivation energy and αis a constant. The derived acti-\nvation energy for thermopower is ES= 20(1) meV. This\nismuchsmallerthanthat forhightemperatureresistivity\nEρ= 118.9(3) meV, and also smaller when compared to\nactivation energy associated with low temperature resis-\ntivityEρ= 26.2(7) meV. The large discrepancy between\nESandEρtypically reflects the polaron transport mech-\nanism of carriers.\nAccording to the polaron model, the ESis the en-\nergy required to activate the hopping of carriers, while\ntheEρis the sum of the energy needed for creation of\ncarriers and activating the hopping of carriers.48There-\nfore, within the polaronhopping model the activation en-\nergyESis smaller than Eρ. Whereas the temperature-\ndependent thermopower S(T) shows a change of slope\nanomaly at Tcat 70 K, an additional change of slope\nis observed at 150 K, coincident with the rapid rise of\nelectrical resistivity ρ(T) on cooling through this tem-\nperature range ( Figure 3 c,e). This points to connec-\ntion between electrical transport and anisotropy of 2D\nmagnetic correlationsabove Tc, consistent with magnetic\nresonance experiments in low magnetic fields.49Close in-\nspection of S(T), however, reveals an additional peak\nin (40 - 50) K range on cooling below Tc(Figure 3 e).\nTo investigate this further, we turn to low-field magnetic\nsusceptibility and X-ray scattering measurements.\nFigure 4 a,b shows the temperature dependence of dc\nmagnetization M(T) for Cr 2Ge2Te6single crystal mea-\nsured at a low field H= 10 Oe applied in the ab-plane\nand along the c-axis, respectively. A PM-FM transi-\ntion was observed at Tc= 65 K. The M(T) is nearly\nisotropic above, while magnetic anisotropy is observed\nbelowTc. The value of M(T) forH/bardblcis larger than\nthat for H/bardblabbelowTc, confirming the easy caxis.\nThe splitting of ZFC and FC curves at low temperature\nmostly originatesfrom the anisotropic FM domain effect,\nin line with previous reports.50–53Figure 4 c,d shows thetemperature dependence of ZFC ac susceptibility mea-\nsured with oscillated ac field of 3.8 Oe and frequency of\n499 Hz. A sharp peak is observed at Tcin the real part\nm′(T) for both directions. An additional anomaly occurs\naround 40 K just like in the S(T) (Figure 3 e), as seen a\nbroad hump in both real part m′(T) and imaginary part\nm′′(T) (Figure 4 d) due to the interplay of magnetocrys-\ntalline anisotropy and external field.51The existence of\nanomalybelow Tcis confirmedby the temperatureevolu-\ntion of magnetic order parameter ( Figure 4 e) measured\nusing X-ray scattering. Below Tc, enhanced X-ray scat-\ntering was observed at the (006) Bragg peak at a photon\nenergy of 6 keV in the rotated, σ-to-π, scattering chan-\nnel. The energy and polarization dependencies of this\nX-ray scattering indicate its origin as resonant magnetic\nscattering due to dipole transitions from the Cr 1 sto\nunoccupied 4 pstates, which are spin-polarized by hy-\nbridization with Cr 3 dstates. This confirms influence of\nmagnetocrystalline anisotropy on transport behavior not\nonly above but also below magnetic Tc.\nStacking faults are found in variety of 2D vdW mate-\nrials such as Bi 2Te3or RuCl 3.54,55They originate from\ndislocations, core regions with large atomic displacement\nfrom the ideal position, and propagate as strain field in\nthecrystallatticeascovalentlybondedcrystalunitsstack\nwith vdW forces along the c-axis. Magnetic anisotropy\nin ordered state and local electronic structure may be af-\nfected by stacking faults55,56but also polaronic defects\ncan be created by stacking faults as a result of sub-\ncoordinated metal atoms and electron transfer via struc-\ntural sub-units.57\nWhereas surface Ge vacancies have been predicted to\ncontributeto enhanced electricalconductivity,20stacking\nfaults observed in structurally related material In 2Si2Te3\nwere associated with Ge(Si) pairs atomic positions that\ncouldmodulatethebandsneartheFermilevelviaGe(Si)-\nTe electron transfer that indirectly alters p-dhybridiza-\ntion between Cr 3 dand Teporbitals.20,58It was pro-\nposed that stacking faults lower local structure symme-\ntry at short range distances.58Good agreement of bond\ndistances Figure 2 d-f40with the averagestructure inter-\natomicdistances32imply that the stackingfault influence\non the unit cell symmetry is likely at the intermediate\nrange distances, calling for detailed total scattering anal-\nysis. Interestingly, stacking faults do not influence sig-\nnificantly either κ(T) mechanism nor magnetic stripe do-\nmains that are the source of skyrmionic bubbles.40,59–61\nStackingfaultsarevisibleincross-sectionalLorentzTEM\nsample at 13 K ( Figure 4 f). Note that dark and white\nlines parallel to the c-axis arising due to 180◦stripe\ndomains do not show significant changes upon crossing\nstacking faults (dark lines running perpendicular to the\nc-axis). This indicates that the faults do not perturb\nmagnetic state or topological magnetic spin textures.61\nOn the other hand, variationof crystaldistortion induces\nchanges in electrical transport as confirmed in an inde-\npendently grown crystal.40\nTo qualitatively understand the potential effects of4\nstructural distortion, we calculate the band structures\nand conductivities in distorted ( Figure 4 h)and undis-\ntorted structures.31,40The overall band structure of the\ndistorted structure share a great similarity with that of\nundistorted structure reported previously.62,63.Figure\n5a,b show scalar-relativistic band structure and partial\ndensity of states (PDOS) of the distorted structure cal-\nculated with a first-principles method. An indirect gap\nof 0.38 eV is obtained and that is reduced by spin-orbit\ninteraction down to 0.15 eV. The valence band maximum\n(VBM) is located at Γ-point in minority spin state; it is\nslightly off from the Γ-point in the majority spin chan-\nnel. The conduction band minimums (CBM) are located\nbetween TandH0in the majority spin channel and be-\ntweenH0andLin the minorityspin channel. Cr- dstates\npeakat -1eVbelowFermi level( EF)in the majorityspin\nchannel and 2 eV above Efin the minority spin chan-\nnel, corresponding to a spin splitting of ∼3 eV. There\nare flat bands along the Γ– Tdirection right below EF,\nwhich may significantly contribute to transport proper-\nties. Interestingly, the positions of these flat bands are\nsensitive to structure distortion. Figure 5 c,d compare\nthe band structures along Γ– Tdirection calculated us-\ning the distorted (crystal 1) and undisorted31structures.\nThedistortionslightlyshiftsthe flatbandsBD1and BD2\nabovethe moredispersiveband BD3alongthe Γ– Tpath.\nBandsBD1andBD2mostlyconsistofin-planeTe- pxand\nTe-pystates,40resulting in small exchange splitting and\nnegligible dispersion along the out-of-plane direction (Γ-\nTpath). In contrast, band BD3 consists of mainly Te- pz\nand Cr-dz2states, showing a much stronger dispersion\nalongkzand larger spin splitting. Considering the prox-\nimity of the flat bands near EF, their changesinduced by\nthe distortion can affect the transport properties signif-\nicantly, especially at higher temperatures and with hole\ndoping.\nTo explore the effects of bandstructure change on\ntransport, we calculate the conductivities for dis-\ntorted (crystal 1) and undistorted31structure using\nBoltzTraP.64Figure 5 e shows the calculated σxx, the\nin-plane component of electrical conductivity tensor, as\na function of temperature. The conductivities in both\nstructures increase with temperature, as expected for in-\ntrinsic semiconductors. The distorted structure has a\nhigher in-plane conductivity than the undistorted struc-\nture. The important contribution of BD1 and BD2 on\nconductivity can be further illustrated by calculating the\npredicted conductivities in the hole-doped case, as shown\ninFigure 5 f. The chemical potential µ2is lowered to\n0.05 eV below Ef, as denoted in Figure 5 c by green\ndash-doted line, to mimic the hole-doping effects. As is\nexpected, the conductivities of all cases increase at lower\ntemperatures. The σxxofdistortedstructure, however,is\nincreased more than 100 times. Therefore, first-principle\ncalculationsindicate that the localdistortionwith appro-priate doping can profoundly affect electronic and mag-\nneticpropertiesofCr 2Ge2Te6, leadingtoeverhighercon-\nductivity.\nIII. CONCLUSIONS\nIn summary, our results indicate strong influence\nof crystal structure distortion and magnetocrystalline\nanisotropy on electrical and thermal transport both\nabove and below Tcin 2D vdW magnetic Cr 2Ge2Te6\ncrystals. Electronic transport is dominated by polaronic\neffects commonly observed in oxide materials, confirm-\ning strong electron-phonon coupling. Moreover, inter-\nplay between spin-orbit electron-phonon coupling may\ntailor the spin polarization since the polaron could re-\ntain only one of the spin-polarized bands in its coherent\nspectrum.37This could be used in spin-orbitronic and\nmagneto-thermal devices that exploit current propaga-\ntion via spin-orbit torque mechanism.38,39\nIV. METHODS\nCrystals were synthesized from the self-flux method40.\nCrystals were cut in suitable geometry for in-plane re-\nsistivity measurements. Pulverized crystals were used\nXANES and EXAFS measaurement. Reference40de-\nscribes experimental methods in more details.\nACKNOWLEDGEMENTS\nWork at Brookhaven National Laboratory (BNL) is\nsupportedbytheOfficeofBasicEnergySciences, Materi-\nals Sciences and Engineering Division, U.S. Department\nof Energy under Contract No. DE-SC0012704. TEM\nsample preparation was carried out at the Center for\nFunctional Nanomaterials (BNL), which is supported by\ntheUS Department ofEnergy,OfficeofBasicEnergySci-\nences under Contract No. DE-AC02-98CH10886. This\nresearch used beamlines 4-ID and 8-ID of the National\nSynchrotron Light Source II, a U.S. Department of En-\nergy (DOE) Office of Science User Facility operated for\nthe DOE Office of Science by Brookhaven National Lab-\noratory under Contract No. DE-SC0012704. The work\ncarried out at the University of Delaware was supported\nby the U.S. Department of Energy, Office of Science, Ba-\nsic Energy Sciences, under Award DE-SC0008885. First\nprinciplecalculationsweresupportedbytheU.S. Depart-\nment of Energy, Office of Science, Office of Basic Energy\nSciences, Materials Sciences and Engineering Division.\nAmes Laboratory is operated for the U.S. Department of\nEnergybyIowaState Universityunder ContractNo. DE-\nAC02-07CH11358. A portion of this work was performed\nat the National High Magnetic Field Laboratory, which\nis supported by the NSF Cooperative Agreement No.\nDMR-1644779 and the State of Florida.5\n1S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M.\nDaughton, S. von Moln´ ar, M. L. Roukes, A. Y. Chtchelka-\nnova, and D. M. Treger, Science2001,294, 1488.\n2B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein,\nR. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A.\nMcGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-\nHerrero, and X. D. Xu, Nature2017,546, 270.\n3C. Gong, L. Li, Z. L. Li, H. W. Ji, A. Stern, Y. Xia, T.\nCao, W. Bao, C. Z. Wang, Y. Wang, Z. Q. Qiu, R. J. Cava,\nS. G. Louie, J. Xia, and X. Zhang, Nature2017,546, 265.\n4L. D. Alegria, H. Ji, N. Yao, J. J. Clarke, R. J. Cava, and\nJ. R. Petta, Appl. Phys. Lett. 2014,105, 053512.\n5X. Yao, B. Gao, M.-G. Han, D. Jain, J. Moon, J. W. Kim,\nY. Zhu, S.-W. Cheong, and S. Oh, Nano. Lett. 2019,19,\n4567.\n6J. Escolar, N. Peimyoo, M. F. Cracium, H. A. Fernandez,\nS. Russo, M. D. Barnes, and F. Withers, Phys. Rev. B\n2019,100, 054420.\n7M. Mogi, T. Nakajima, V. Ukleev, A. Tsukazaki, R.\nYoshimi, M. Kawamura, K. S. Takahashi, T. Hanashima,\nK. Kakurai, T. Arima, M. Kawasaki, and Y. Tokura, Phys.\nRev. Lett. 2019,123, 016804.\n8X. Li, J. Lu, J. Zhang, L. You, Y. Su, and E. Y. Tsymbal,\nNano Lett. 201919, 5133.\n9B. G. Park, J. Wunderlich, X. Marti, V. Holy, Y. Kurosaki,\nM. Yamada, H. Yamamoto, A. Nishide, J. Hayakawa, H.\nTakahashi, A. B. Shick and T. Jungwirth, Nat. Mater.\n2011,10, 347.\n10X. Tang, D. Fan, K. Peng, D. Yang, L. Guo, X. Lu, J. Dai,\nG. Wang, H. Liu, and X. Zhou, Chem. Mater. 2017,29,\n7401.\n11X. Tang, D. Fan, L. Guo, H. Tan, S. Wang, X. Lu, X.\nCao, G. Wang, and X. Zhou, Appl. Phys. Lett. 2018,113,\n263902.\n12C. Peng, G. Zhang, C. Wang, Y. Yan, H. Zheng, Y. Wang,\nM. Hu,Phys. Status Solidi RRL 2018,12, 1800172.\n13N. Wang, H. Tang, M. Shi, H. Zhang, W. Zhuo, D. Liu,\nF. Meng, L. Ma, J. Ying, L. Zou, Z. Sun, and X. Chen, J.\nAm. Chem. Soc. 2019,141, 17166.\n14N. Sivadas, M. W. Daniels, R. H. Swendsen, S. Okamoto,\nand D. Xiao, Phys. Rev. B 2015,91, 235425.\n15B. L. Chittari, D. Lee, N. Banerjee, A. H. MacDonald, E.\nHwang, and J. Jung, Phys. Rev. B 2015,101, 085415.\n16K. Zollner, P. E. Faria Jonior, and J. Fabian, Phys. Rev.\nB2019, 100, 195126.\n17D. Yang, W. Yao, Q. Chen, K. Peng, P. Jiang, X. Lu,\nC. Uher, T. Yang, G. Wang, and X. Zhou, Chem. Mater.\n2016,28, 1611.\n18Z. Hao, H. Li, S. Zhang, X. Li, G. Lin, X. Luo, Y. Sun, Z.\nLiu, and Y. Wang, Science Bulletin 2018,63, 825.\n19C. Song, X. Liu, X. Wu, J. Wang, J. Pan, T. Zhao, C. Li,\nand J. Wang, J. Appl. Phys. 2019,126, 105111.\n20C. Song, W. Xiao, L. Li, Y. Lu, P. Jiang, C. Li, A. Chen,\nand Z. Zhong, Phys. Rev. B 2019,99, 214435.\n21A. Milosavljevic, A. Solajic, B. Visic, M. Opacic, J. Pesic,\nY. Liu, C. Petrovic, Z. V. Popovic, N. Lazarevic, J. Raman\nSpectrosc. 202051, 2153.\n22Z. Lin, M. Lohmann, Z. A. Ali, C. Tang, J. Li, W. Xing,\nJ. Zhong, S. Jia, W. Han, S. Coh, W. Beyermann, J. Shi,\nPhys. Rev. Materials 2018,2, 051004.23Y. Sun, R. C. Xiao, G. T. Lin, R. R. Zhang, L. S. Ling,\nZ. W. Ma, X. Luo, W. J. Lu, Y. P. Sun, and Z. G. Sheng,\nAppl. Phys. Lett. 2018,112, 072409.\n24Z. Yu, W. Xia, K. Xu, M. Xu, H. Wang, X. Wang, N. Yu,\nZ. Zou, J. Zhao, L. Wang, X. Miao, and Y. Guo, J. Phys.\nChem. C 2019,123, 13885.\n25W. Ge, K. Xu, W. Xia, Z. Yu, H. Wang, X. Liu, J. Zhao,\nX. Wang, N. Yu, Z. Zou, Z. Yan, L. Wang, M. Xu, and Y.\nGuo,J. Alloys Compound. 2020,819, 153368.\n26E. Dong, B. Liu, Q. Dong, X. Shi, X. Ma, R. Liu, X.\nZhu, X. Luo, X. Li, Y. Li, Q. Li, and B. Liu, Physica B:\nCondens. Matter 2020,595, 412344.\n27W. Xing, Y. Chen, P. M. Odenthal, X. Zhang, W. Yuan,\nT. Su, Q. Song, T. Wang, J. Zhong, and S. Jia, 2D Mater.\n2017,4, 024009.\n28Y. Chen, W. Xing, X. Wang, B. Shen, W. Yuan, T. Su, Y.\nMa, Y. Yao, J. Zhong, Y. Yun, X. C. Xie, and S. Jia, ACS\nAppl. Mater. Interfaces 2018,10, 1383.\n29Z. Wang, T. Zhang, M. Ding, B. Dong, Y. Li, M. Chen, X.\nLi, J. Huang, H. Wang, X. Zhao, Y. Li, D. Li, C. Jia, L.\nSun, H. Guo, Y. Ye, D. Sun, Y. Chen, T. Yang, J. Zhang,\nS. Ono, Z. Han, and Z. Zhang, Nat. Nanotech. 2018,13,\n554.\n30K. F. Mak, H. Shan, and D. C. Ralph, Nature Reviews\nPhysics2019,1, 646.\n31V. Carteaux, D. Brunet, G. Ouvrard, and G. Andr´ e, J.\nPhys.: Condens. Matter 1995,7, 69.\n32Y. Liu and C. Petrovic, Phys. Rev. B 2017,96, 054406.\n33G. T. Lin, H. L. Zhuang, X. Luo, B. J. Liu, F. C. Chen, J.\nYan, Y. Sun, J. Zhou, W. J. Lu, P. Tong, Z. G. Sheng, Z.\nQu, W. H. Song, X. B. Zhu, and Y. P. Sun, Phys. Rev. B\n2017,95, 245212.\n34Ivan A. Verzhbitskiy, Hidekazu Kurebayashi, Haixia\nCheng, Jun Zhou, Safe Khan, Yuan Ping Feng and Goki\nEdaNature Electronics 2020,3, 460.\n35L. D. Casto, A. J. Clune, M. O. Yokosuk, J. L. Musfeldt,\nT. J. Williams, H. L. Zhuang, M. W. Lin, K. Xiao, R. G.\nHennig, B. C. Sales, J. Q. Yan, and D. Mandrus, APL\nMater.2015,3, 041515\n36T. T. M. Palstra, A. P. Ramirez, S-W. Cheong, B. R. Ze-\ngarski, P. Schiffer and J. Zaanen, Phys. Rev. B 1997,56,\n5104.\n37L. Covaci and M. Berciu, Phys. Rev. Lett. 2009,102,\n186403.\n38S. Watanabe, K. Ando, K. Kang, S. Mooser, Y. Vaynzof,\nH. Kurebayashi, E. Saitoh and H. Sirringhaus, Nat. Phys.\n2014,10, 308.\n39Vaibhav Ostwal, Tingting Shen and Joerg Appenzelle,\nAdv. Mater. 2020,32, 1906021.\n40See Supplementary information.\n41H. Ji, R. A. Stokes, L. D. Alegria, E. C. Blomberg, M.\nA. Tanatar, A. Reijnders, L. M. Schoop, T. Liang, R. Pro-\nzorov, K. S. Burch, N. P. Ong, J. R. Petta, and R. J. Cava,\nJ. Appl. Phys. 2013,114, 114907.\n42H. Ofuchi, N. Ozaki, N. Nishizawa, H. Kinjyo, S. Kuroda,\nandK.Takita, AIPConference Proceeding 2017,882, 517.\n43R.Prins andD.C.Koningsberger (eds.), X-rayAbsorption:\nPrinciples, Applications, Techniques of EXAFS, SEXAFS,\nXANES (Wiley, New York, 1988).\n44E. S. R. Gopal, Specific Heats at Low Temperatures\n(Plenum Press, New York, 1966).6\n45Richard Hentrich, Anja U. B. Wolter, Xenophon Zotos,\nWolfram Brenig, Domenic Nowak, Anna Isaeva, Thomas\nDoert, Arnab Banerjee, Paula Lampen-Kelley, David G.\nMandrus, Stephen E. Nagler, Jennifer Sears, Young-June\nKim, Bernd B¨ uchner, and Christian Hess, Phys. Rev. Lett.\n2018,120, 117204.\n46Changsong Xu, Hongjun Xiang and Laurent Bellaiche, npj\nComputational Materials 2018, 4, 57.\n47Changsong Xu, Junsheng Feng, Mitsuaki Kawamura,\nYouhei Yamaji, Yousra Nahas, Sergei Prokhorenko, Yang\nQi, Hongjun Xiang and L. Bellaiche, Phys. Rev. Lett.\n2020,124, 087205.\n48I. G. Austin, and N. F. Mott, Adv. Phys. 2001,50, 757.\n49J. Zeisner, A. Alfonsov, S. Selter, S. Aswartham, M. P.\nGhimire, M. Richter, J. van den Brink, B. B¨ uchner and V.\nKataev, Phys. Rev. B 2019,99, 165109.\n50X. Zhang, Y. L. Zhao, Q. Song, S. Jia, J. Shi, and W. Han,\nJpn. J. Appl. Phys. 2016,55, 033001.\n51S. Selter, G. Bastien, A. U. B. Wolter, S. Aswartham, and\nB. B¨ uchner, Phys. Rev. B 2020,101, 014440.\n52Y.LiuandC. Petrovic, Phys. Rev. Mater. 2019,3, 014001.\n53W. Liu, Y. Wang, Y. Han, W. Tong, J. Fan, L. Pi, N. Hao,\nL. Zhang, and Y. Zhang, J. Phys. D: Appl. Phys. 2020,\n53, 025101.\n54D. L. Medlin, N. Yang, C. D. Spataru, L. M. Hale and Y.\nMishin,Nature Comms. 2019,10, 1820.\n55Ichiro Yamauchi, Masatoshi Hiraishi, Hirotaka Okabe,\nSoshi Takeshita, Akihiro Koda, Kenji M. Kojima, RyosukeKadono and Hidekazu Tanaka, Phys. Rev. B 2018,97,\n134410.\n56Y. Liu, Y. Y. Li, S. Rajput, D. Gilks, L. Lari, P. L.\nGalindo, M. Weinert, V. K. Lazarov and L. Li, Nature\nPhysics 2014,10, 294.\n57Paulo R. Bueno, Ronald Tararan, Rodrigo Parra, Ed-\nnan Joanni, Miguel A. Ramirez, Willian C. Ribeiro, Elson\nLongo and Jose A. Varela, J. Phys. D: Appl. Phys. 2009,\n42, 055404.\n58R. Lefevre, D. Berthebaud, O. Lebedev, O. Perez, C. Cas-\ntro, S. Gascoin, D. Chateigner and F. Gascoin, J. Mater.\nChem. A 2017,5, 19406.\n59Xingxu Yan, Chengyan Liu, Chaitanya A. Gadre, Lei Gu,\nToshihiro Aoki, Tracy C. Lovejoy, Niklas Dellby, Ondrej\nL. Krivanek, Darrell G. Schlom, Ruqian Wu and Xiaoqing\nPan,Nature2021,589, 65.\n60K. Gianno, V. Sologubenko, M. A. Chernikov, H. R. Ott,\nI. R. Fisher and P. C. Canfield, Phys. Rev. B 2000,62,\n292.\n61Myung-Geun Han, Joseph A. Garlow, Yu Liu, Huiqin\nZhang, Jun Li, Donald DiMarzio, Mark W. Knight, Ce-\ndomir Petrovic, Deep Jariwala and Yimei Zhu. Nano Lett.\n2019,19, 7859.\n62Y. Lee, T. Kotani, and L. Ke, Phys. Rev. B 2020,101,\n241409.\n63G. Menichetti, M. Calandra, and M. Polini, 2D Mater.\n2019,6, 045042.\n64G. K. H. Madsen and David J. Singh, Comp. Phys. Comm.\n2006,175, 67.7\n/s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48 /s57/s48\n/s40/s48/s48/s49/s56/s41/s40/s48/s48/s49/s50/s41/s40/s48/s48/s54/s41\n/s32/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s50 /s113/s32 /s40/s176/s41/s40/s48/s48/s51/s41/s97\n/s98\n/s99/s102\n/s100/s101\n/s106/s104\n/s105\n/s103\nFIG. 1. Crystal structure and nanostructure characterizat ion of Cr 2Ge2Te6. a) 2θX-ray diffraction scan along the (00l)\ndirection. Inset shows optical image of Cr 2Ge2Te6single crystals. R¯3hunit cell of Cr 2Ge2Te6shown from b) the side view\nand c) the top view. d) High-angle annular dark-field (HAADF) scanning transmission electron microscopy (STEM) image\nobtained along the c-axis. Inset shows the electron diffraction pattern. e) X-ra y energy-dispersive spectrum of the selected\narea. HAADF STEM image f) of cross-sectional TEM sample imag ed along the a-axis. Dark-field TEM image g) obtained with\nthe reflection of stacking A (orange arrowed peak in h). Selec ted area electron diffractions (h and i) from the green-boxed area\n(h) and blue-boxed area (i). In (i) the Stacking A only exists . Atomic resolution HAADF STEM image j) from the green-boxed\narea. Two insets show the enlarged area showing the two differ ent stackings separated by a stacking fault, indicated with a\ndotted line. Two inclined yellow and red lines show the stack ing sequence of the Te/Ge columns along the c-axis.8\nFIG. 2. Local atomic structure. Normalized a) Cr, b) Ge, and c ) Te K-edge XANES spectra with insets depicting local atomic\nstructure. Fourier transform magnitudes of the EXAFS oscil lations (symbols) for d) Cr, e) Ge, and f) Te K-edge with the\nphase shifts correction. The model fits are shown as solid lin es. Insets show the corresponding filtered EXAFS (symbols) w ith\nk-space model fits (solid lines).\nFIG. 3. Thermodynamic, electronic and thermal transport pr operties. Temperature dependence of a) specific heat Cp(T),\nb) thermal conductivity κ(T) and c) electrical resistivity ρ(T) of our Cr 2Ge2Te6single crystals. Inset in a) shows the fitted\n(red line) low-temperature Cp(T) from 2 to 10 K (see text). Green solid curve in b) shows the fitt ing of the lattice thermal\nconductivity using the Debye model. The ln( ρ) vs 1/kBT(left axis) and ln( ρ/T) vs 1/kBT(right axis) d) fitted by adiabatic\nsmall polaron hopping model (see text). e) Temperature depe ndence of S(T)thermopower. f) The S(T) vs, 1000 /Tcurve fitted\nby polaron transport model (see text).9\nFIG. 4. Magnetocrystalline anisotropy. Temperature depen dence of zero field cooling (ZFC) and field cooling (FC) dc magn e-\ntization M(T) measured at H = 10 Oe for Cr 2Ge2Te6single crystal with a) H/bardblaband b)H/bardblc, respectively. Temperature\ndependence of ac susceptibility real part m′(T) and imaginary part m′′(T) measured with oscillated ac field of 3.8 Oe and fre-\nquency of 499 Hz applied c) in the ab-plane and d) along the c-axis, respectively. e) The integrated intensities of long itudinal\nscans through the (006) magnetic Bragg peak of Cr 2Ge2Te6. f) Lorentz TEM image of Cr 2Ge2Te6obtained perpendicular to\nthec-axis. at 13 K, showing 180◦stripe domains separated by the Bloch walls due to uniaxial a nisotropy (Ref. 61). Two\nstacking faults are visible as dark horizontal lines, which do not affect the stripe domain configuration. g) Unit cell of d istorted\nCr2Ge2Te6showing thermal ellipsoids from single crystal refinement a nd Ge-Ge bonds (red areas).\n/s40/s97/s41\n/s50/s50\n/s45/s50/s48\n/s45/s49/s45/s51\n/s71/s49\n/s84/s40/s99/s41\n/s48/s69/s32/s45/s32/s69\n/s70/s32/s40/s101/s86/s41/s32/s84/s101/s32 /s112\n/s32/s71/s101/s32 /s112\n/s50/s51/s68/s79/s83/s32/s40/s115/s116/s97/s116/s101/s115/s47/s101/s86/s47/s99/s101/s108/s108/s41/s49\n/s45/s50\n/s48/s45/s49\n/s45/s50/s45/s50/s45/s48/s46/s50/s48\n/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s52/s45/s48/s46/s50\n/s45/s48/s46/s54/s69/s45/s32/s69\n/s70/s32/s40/s101/s86/s41\n/s40/s100/s41/s69/s45/s32/s69\n/s70/s32/s40/s101/s86/s41\n/s84/s71/s49/s48/s48\n/s115\n/s120/s120/s47/s116 /s32/s91/s49/s48/s49/s55\n/s47/s87/s215 /s109 /s215/s115/s93/s50/s48\n/s40/s98/s41\n/s40/s102/s41\n/s84 /s72\n/s50/s76 /s72\n/s48/s70/s71/s83\n/s48/s66/s68/s50\n/s66/s68/s51/s32/s67/s114/s32 /s100\n/s66/s68/s49\n/s49/s48/s48 /s50/s48/s48/s48\n/s52/s48/s49/s48/s48\n/s48/s49/s50/s48\n/s50/s48/s53/s49/s48/s49/s53/s115\n/s120/s120/s47/s116 /s32/s91/s49/s48/s49/s55\n/s47/s87/s215 /s109 /s215/s115/s93\n/s84/s32/s40/s75/s41/s100/s105/s115/s116/s111/s114/s116/s101/s100\n/s99/s114/s121/s115/s116/s97/s108/s32\n/s117/s110/s100/s105/s115/s116/s111/s114/s116/s101/s100/s40/s101/s41\n/s45/s50/s48\n/s49/s48\n/s45/s49\n/s71 /s71/s71\nFIG. 5. (a) Scalar-relativistic band structure of Cr 2Ge2Te6with lattice distortion. Blue (red) color indicates the maj ority\n(minority) spin band. (b) The corresponding partial densit y of states (PDOS) projected on Te- p, Ge-p, and Cr- dstates. Scalar-\nrelativistic band structures calculated in (c) distorted a nd (d) undistorted structures. BD1, BD2, and BD3 are selecte d bands\nto analysis band characters (See Ref. 40). Orange and green d ash-dotted lines denote two chemical potentials, µ1 andµ2,\nrespectively, that are employed for conductivity calculat ions. Electrical conductivity as a function of temperature calculated\nwithout (e) and with (f) hole-doping in Cr 2Ge2Te6distorted and undistorted structures. The chemical potent ial is set to\n0.05 eV below VBM to mimic the hole-doped case. The red and blu e colors denote the distorted and undistorted structures,\nrespectively."    },    {        "title": "2203.04157v1.Breathing_skyrmions_in_chiral_antiferromagnets.pdf",        "content": "Breathing skyrmions in chiral antiferromagnets\nS. Komineas1, 2and P. E. Roy3\n1Department of Mathematics and Applied Mathematics,\nUniversity of Crete, 70013 Heraklion, Crete, Greece\n2Institute of Applied and Computational Mathematics, FORTH, Heraklion, Crete, Greece\n3Hitachi Cambridge Laboratory, Hitachi Europe Limited, Cambridge CB3 0HE, United Kingdom\n(Dated: March 9, 2022)\nBreathing oscillations of skyrmions in chiral antiferromagnets can be excited by a temporal mod-\ni\fcation of the Dzyaloshinskii-Moriya interaction or magnetocrystalline anisotropy strength. We\nemploy an adiabatic approximation and derive a formula for the potential that directly implies\nbreathing oscillations. We study the nonlinear regime and the features of larger amplitude oscilla-\ntions. We show that there is a maximum amplitude supported by the potential. As a consequence,\nwe predict theoretically and observe numerically skyrmion annihilation events due to excitation of\nlarge amplitude breathing oscillations. The process is e\u000ecient when the skyrmion is mildly excited\nso that its radius initially grows, while the annihilation event is eventually induced by the internal\nbreathing dynamics. We reveal the counter-intuitive property that the skyrmion possesses a nonzero\nkinetic energy at the moment of its annihilation. Finally, the frequency of small amplitude breathing\noscillations is determined.\nI. INTRODUCTION\nMagnetic skyrmions are topological textures with a\nswirling con\fguration of the magnetization stabilized by\nthe Dzyaloshinskii-Moriya (DM) interaction [1, 2]. They\nare commonly observed in chiral ferromagnetic \flms, typ-\nically extending a few or tens of nanometers laterally, and\ntheir nontrivial topology makes them robust against per-\nturbations. Skyrmions exhibit particle-like dynamics [3]\nwhich, together with their small size, lead to properties\nsuch as a low driving threshold current [4]. These qual-\nities have given rise to a range of proposed applications\nsuch as their use as the constituent information carriers\nin racetrack memory/logic devices [5{9], memristor ele-\nments in arti\fcial synapses for neuromorphic computing\narchitectures [10], in spintronics-based transistor device\nconcepts [11], and as spin-wave scatterers in magnonic\ncomputing and logic devices [12].\nMost of the existing work is focused on ferromagnetic\nskyrmions. On the other hand, skyrmions are expected\nto exist also in antiferromagnets (AFM) for essentially\nthe same reasons as in ferromagnets. Antiferromagnetic\nmaterials have advantages such as high operational fre-\nquencies in the THz range and robustness against ex-\nternal magnetic \feld perturbations. Furthermore, un-\nlike in ferromagents and despite their topological nature,\nskyrmions in AFMs do not present a Hall angle in their\ndynamical behavior. These properties have prompted\nproposals to replace ferromagnetic skyrmions by their an-\ntiferromagnetic counterparts [11, 13{18]. Recent experi-\nmental observations of stable skyrmions in AFMs [19, 20]\ngive further motivation for the study of AFM skyrmions\nfor future device implementations.\nThe presence of the chiral DM interaction is crucial\nfor the dynamics of solitons, in addition to its role in\ntheir stabilization. A skyrmion breathing mode arises\nin chiral ferromagnets [21] due to the breaking of the\nconservation of the total magnetization perpendicular tothe \flm. A similar mode exists also for antiferromagnetic\nskyrmions. In the latter case, independent oscillations of\nthe skyrmion radius and of its chirality are possible [22].\nWe are motivated by the emerging kinetic energy in the\nAFM continuum [23, 24] and we propose an e\u000bective po-\ntential for the skyrmion oscillation dynamics. This leads\nto a systematic approach in order to understand details\nof these dynamical modes, such as the frequency of os-\ncillations, by a combination of analytical and numerical\nmethods. The form of the e\u000bective potential leads to\nthe observation that an expansion of the skyrmion can\neventually lead to its annihilation by a subsequent col-\nlapse due to internal breathing dynamics. This counter-\nintuitive method shows that a large external force is not\nnecessary in order to annihilate the topological texture;\ninstead skyrmion annihilation is obtained by small con-\ntrolled changes of the DM or anisotropy parameters that\ncan be induced, e.g., by a voltage pulse [25], in combina-\ntion with internal dynamics.\nIt would be an ideal situation to have methods for\nskyrmion creation and annihilation by controlled small\nperturbations while at the same time the skyrmion re-\nmains robust to all usual perturbations. If this is\nachieved the potential for skyrmions as functional ob-\njects would be signi\fcantly enhanced. In this context,\nour method shows the way for controlled skyrmion anni-\nhilation by small perturbations. In addition, the success-\nful analytical arguments presented may be useful in stud-\nies for a method towards controlled individual skyrmion\ngeneration using only mild forces.\nSec. II discusses the energy in an antiferromagnet in\nthe discrete and in the continuum model. Sec. III shows\nhow skyrmion annihilation via breathing dynamics can\nbe achieved. In Sec. IV, we derive the e\u000bective potential\nfor breathing oscillations and give a theoretical descrip-\ntion of the annihilation dynamics. In Sec. V, we derive\nthe frequency of small amplitude breathing oscillation in\nthe case of skyrmions with a large and a small radius.arXiv:2203.04157v1  [cond-mat.mes-hall]  8 Mar 20222\nSec. VI discusses helicity oscillations for the skyrmion.\nSec. VII contains our concluding remarks.\nII. SKYRMION ENERGY\nWe consider a square lattice of spins in a material\nwith the usual exchange, perpendicular anisotropy and\nDzyaloshinskii-Moriya interaction. The magnetic energy\nof the lattice is\nEd=X\ni;jJnSi;j\u0001(Si+1;j+Si;j+1) +k\n2[1\u0000(Si;j)2\n3]\n+D[^e2\u0001(Si;j\u0002Si+1;j)\u0000^e1\u0001(Si;j\u0002Si;j+1)]\n(1)\nwhere the spin variables are assumed normalized jSi;jj=\n1. We will typically use, in numerical simulations, the\nparameter values\nJn= 2\u000210-21Joule; D = 0:047Jn; k = 0:01Jn:(2)\nThe equation of motion for the spins is\n@Si;j\n@t=\u0000\rSi;j\u0002Fi;j+\u000bSi;j\u0002@Si;j\n@t; (3)\nFi;j=\u00001\n\u00160\u0016s@Ed\n@Si;j\nwhere Fis the e\u000bective \feld, \r=ge\u0016B\u00160=\u0016h= 2:211\u0002\n105m A\u00001s\u00001is the gyromagnetic ratio, and \u000bis the\ndamping parameter. We choose the saturation magne-\ntization\u0016s= 4\u0016B, where\u0016Bis the Bohr magneton. The\nmaterial parameters have been chosen to resemble what is\nexpected for a range of antiferromagnetic oxides [26, 27].\nA continuum model is obtained if we consider a lat-\ntice of spin tetramers where the normalized N\u0013 eel vector\nn\u000b;\fis de\fned at tetramer sites ( \u000b;\f) [24]. The dis-\ntance between tetramer sites is de\fned to be 2 \u000fwhere\n\u000f\u0011p\nk=J is a small parameter. In the limit \u000f!0, a\ncontinuous N\u0013 eel vector \feld n=n(x;y;\u001c ) is obtained,\nwithn2= 1, where x;yand\u001care scaled space and time\nvariables [23, 24, 28, 29]. The energy in the continuum\nis (see also Refs. [30, 31])\nE=T+V\u0015 (4)\nwhere the kinetic energy is\nT=1\n2Z\n_n2d2x (5)\nand the dot denotes di\u000berentiation with respect to the\nscaled time \u001c, and the potential energy\nV\u0015=Eex+EDM+Ean (6)includes exchange, DM, and anisotropy contributions,\nEex=1\n2Z\n(@\u0016n)\u0001(@\u0016n)d2x\nEDM=\u0015Z\n\u000f\u0016\u0017^e\u0016\u0001(@\u0017n\u0002n)d2x\nEan=1\n2Z\n(1\u0000n2\n3)d2x(7)\nwith\u0015=D=pkJnbeing a scaled DM parameter, and\nall energy components being in units of Jn. Symbols\n@\u0016;@\u0017, with\u0016;\u0017= 1;2, denote di\u000berentiation with re-\nspect to (x;y), respectively, \u000f\u0016\u0017is the antisymmetric ten-\nsor, and the summation convention for repeated indices\nis adopted. Note that xis a scaled coordinate and actual\ndistances (in physical units) are given by\nax=\u000f; ay=\u000f (8)\nwhereais the distance between neighboring spins. The\nunit of time is\n\u001c0=\u0016s\nge\u0016B\u0016h\n2p2kJn= 0:373 ps (9)\nwhere the numerical value corresponds to the parameter\nvalues in Eq. (2) and \u0016s= 4\u0016B.\nThe potential energy V\u0015is identical in form to the en-\nergy for a ferromagnet with corresponding interactions.\nIt is thus known that the ground state is uniform (N\u0013 eel)\nfor\u0015<2=\u0019while it is the spiral for \u0015>2=\u0019[2]. Isolated\nskyrmions are localised excitations on a uniform back-\nground. On the other hand, the presence of the kinetic\nterm in Eq. (4) opens possibilities that are not there in\nferromagnets. We will study oscillations of the skyrmion.\nWe consider an axially-symmetric skyrmion con\fgura-\ntion for the N\u0013 eel vector, written conveniently in terms of\nthe spherical variables\n\u0002 = \u0002(r;t);\b =\u001e+\u001f(t) (10)\nwhere (r;\u001e) are polar coordinates and \u001fis an angle that\nmay be called the helicity. We will assume that \u001fmay\ndepend on time only or it is a constant. The kinetic\nenergy of Eq. (5) takes the form\nT=1\n2Z\u0010\n_\u00022+ sin2\u0002 _\u001f2\u0011\n2\u0019rdr: (11)\nThe exchange and anisotropy energies depend on \u0002 only,\nEex=1\n2Z\u0014\n(\u00020)2+sin2\u0002\nr2\u0015\n2\u0019rdr;\nEan=1\n2Z\nsin2\u0002 2\u0019rdr(12)\nwhere the prime denotes di\u000berentiation with respect to\nthe space variable r. The DM term is written as\nEDM=\u0015cos\u001f~EDM;\n~EDM=Z\u0012\n\u00020+cos \u0002 sin \u0002\nr\u0013\n2\u0019rdr:(13)3\nFIG. 1. The blue dotted line shows the potential energy V\u0015\nfor the static skyrmion versus its radius R=R(\u0015). The red\nsolid lines show the potential energy V\u00150(\u0002\u0015) given in Eq. (23)\nwith\u00150= 0:40;0:47;0:57 and corresponding equilibrium radii\nR0= 0:71;1:14;2:39. The radius Ris in scaled units and the\nactual length is given, as in Eq. (8), by aR=\u000f (it is\u000f= 0:1 for\nparameter values in Eq. (2)).\nThe static skyrmion pro\fle that minimizes the poten-\ntial energyV\u0015will be denoted by \u0002 \u0015(r) and it is obtained\nfor\u001f= 0. The skyrmion radius depends on the param-\neter\u0015. For small \u0015the skyrmion radius Ris small and\nfor\u0015!0 the radius R!0 and the energy takes the\nvalueE= 4\u0019[32{34]. The radius Rincreases with \u0015\nand it diverges, R!1 , for\u0015!2=\u0019[35, 36]. Fig. 1\nshows the numerically calculated energy from Eq. (6) of\na static skyrmion as a function of its radius R, de\fned to\nbe at the point where the magnetization points in-plane,\ni.e., \u0002 =\u0019=2.\nWe conclude this section by introducing the local mag-\nnetization vector mde\fned as the normalized mean value\nof the spins on tetramers [24, 37]. It is an auxiliary \feld\nin the continuum theory, given by\nm=\u000f\n2p\n2n\u0002_n: (14)\nA nonzero magnetization is obviously connected with dy-\nnamics. The kinetic energy in Eq. (5) is given in terms\nof the magnetization of Eq. (14) as\nT=4\n\u000f2Z\nm2d2x: (15)\nIn the following, we will make use of its discretized form\nT= 16X\n\u000b;\fm2\n\u000b;\f (16)that gives the kinetic energy as a sum over the lattice of\ntetramers.\nIII. SKYRMION ANNIHILATION VIA\nBREATHING\nThe fact that the energy of an in\fnitesimally small\nskyrmion is \fnite, E= 4\u0019, as shown in Fig. 1, sug-\ngests the possibility to annihilate the skyrmion by a \fnite\nforce. This idea will be combined with the dynamics of\nthe breathing mode that is known to exist in chiral mag-\nnets [21], as will be explained in the following.\nWe assume a material with parameter value \u00150that\ngives a static skyrmion with radius R0and energy V0. We\nfurther assume a method to expand this skyrmion and\nproduce one with a larger radius R>R 0and, naturally,\na larger energy V > V 0. We expect that the skyrmion\nenergy, when this is out of equilibrium, can be chosen\nV > 4\u0019ifRis large enough. Using such a large ra-\ndius skyrmion as an initial state, we anticipate that the\nbreathing dynamics can reduce the skyrmion radius down\ntoR!0 since the energy of the oscillator is larger than\nthe potential energy of a R!0 skyrmion.\nWe implement the above ideas in a numerical simula-\ntion of a spin lattice with 500 \u0002500 sites. We consider a\nstatic skyrmion for the parameter values in Eq. (2). We\npropagate in time the dynamical equations (3) including\ndamping with \u000b= 0:0025. The system is subjected to\na voltage pulse that initially modi\fes the DM parameter\nto the value D= 0:057Jn(\u0015= 0:57) as shown in Fig. 2a.\nThe evolution of the skyrmion radius and discrete energy\n(1) are shown in Fig. 2b. Four snapshot of the skyrmion\npro\fle (n3component) during the simulation are shown\nin Fig. 2c. At time t0, the initial skyrmion is shown.\nThe skyrmion expands during the voltage pulse, and it is\nshown at the end of the pulse at time t1. Subsequently,\nbreathing dynamics induces shrinking of the skyrmion,\nas shown at time t2. The skyrmion is eventually anni-\nhilated at time approximately 12 ps. After annihilation,\nlow amplitude waves in the form of radiation are found to\nspread radially, away from the original skyrmion center,\nas shown at time t3. In Fig. 2b, we see that the skyrmion\nradius is increasing during the voltage pulse. The pulse is\nswitched o\u000b at the time that the skyrmion has maximum\nradius. Then, the radius is decreasing due to breath-\ning dynamics until the skyrmion shrinks to a point and\nis annihilated. The energy is decreasing due to damp-\ning. There is a step-like increase of the energy when the\nvoltage pulse is switched o\u000b, due to the drop of the DM\nparameter. This brings the energy to a value greater than\n4\u0019. When annihilation of the skyrmion happens, the dis-\ncrete energy is somewhat lower than 4 \u0019(which is the pre-\ndiction of the continuum theory for minimum skyrmion\nenergy). Certainly, the prediction based on the contin-\nuum model is not expected to be quantitatively correct\nwhen the skyrmion is concentrated in a few lattice sites,\njust before annihilation. The energy dissipation contin-4\nFIG. 2. Skyrmion annihilation via breathing. (a) The param-\neterDis shown vs time. A square pulse of 7 ps is modulating\nthe DM parameter to the value D= 0:057Jn(\u0015= 0:57). Af-\nter the pulse is switched o\u000b, it is D= 0:047Jn(\u0015= 0:47). (b)\nThe total energy from Eq. (1) vs time is shown by a blue solid\nline. The skyrmion radius is shown by a red dashed line. It is\ngiven in units of lattice spacing a. (c) Spatial distribution of\nn3at time points ti(indicated on the energy graph). Snap-\nshott0shows the initial skyrmion (static pro\fle for parameter\nvalues of Eq. (2)), snapshot t1shows the skyrmion when the\npulse is switched o\u000b, snapshot t2shows the skyrmion before\nannihilation, snapshot t3shows the generated waves after the\nskyrmion has been annihilated. Only part of the simulation\nspace is shown.\nues faster after the skyrmion annihilation. We stop the\nsimulation when radiation reaches the boundaries of the\nnumerical mesh.\nWe proceed to a detailed study of the skyrmion anni-\nhilation dynamics. At the time that the skyrmion is con-\ncentrated in a point, the kinetic energy should be equal\ntoT=V\u00004\u0019>0. This seems incompatible with the in-\ntuitive expectation that _R!0 atR= 0. The apparent\ncontradiction can be resolved if we estimate the kinetic\nenergy for a skyrmion of small radius where the pro\fle is\napproximated by a Belavin-Polyakov (BP) con\fguration\n[32, 33]. A BP skyrmion with a time dependent radius is\ngiven by\ntan\u0012\u0002\n2\u0013\n=R(t)\nr: (17)\nThis is valid from r= 0 up to a distance r\u0018O(1=lnR)\nwhenR\u001c1 (this is based on Eq. (26) of Ref. [36]). The\nFIG. 3. Kinetic energy ( T) shown by a solid line for the\nannihilation event of Fig. 2. The skyrmion radius ( R) is also\nshown by a grey dashed line for comparison. The results\nare plotted up to the time point where V\u00194\u0019, after which\nthe continuum theory ceases to be valid. At this point, the\nskyrmion collapses while it can be clearly seen that T >0, in\nagreement with the theoretical prediction.\nkinetic energy is\nT=1\n2Z\n_\u000222\u0019rdr\u0019\u00004\u0019lnR_R2(18)\nwhere we have taken the limits in the integral (18) from\nr= 0 tor\u00181=lnRand we have only kept the domi-\nnant term for R!0. The skyrmion pro\fle decays expo-\nnentially for larger distances rand we thus neglect this\ncontribution to T. The quantity m=\u00008\u0019lnRcan be\nconsidered as the mass of the breathing skyrmion and\nit is diverging for R!0. This behavior is connected\nwith the well-known divergence of the integrated magne-\ntization for the BP skyrmion. A nonzero kinetic energy\nimplies\n_R\u00181p\n\u0000lnR; R!0: (19)\nBased on Eq. (19), we anticipate that _R!0 asR!0\nwhile at the same time the kinetic energy remains strictly\npositive due to the mass term. We \fnally mention that\nthe time it takes to achieve the annihilation via breath-\ning is \fnite as can be found by integrating in time _Rin\nEq. (19).\nFig. 3 shows the kinetic energy (16) for the numeri-\ncal simulation in Fig. 2 until the skyrmion is annihilated.\nThe skyrmion radius is shown in the same \fgure for com-\nparison. The kinetic energy starts from zero, reaches\na maximum and returns to a very small value as the\nskyrmion radius approaches a maximum. At that time,\nthe pulse is switch o\u000b and the skyrmion radius starts\ndecreasing rapidly until the skyrmion annihilates. The\nkinetic energy has a nonzero value at the time of annihi-\nlation. The numerical results con\frm the predictions of5\nthe previous paragraphs that are based on the continuum\nmodel.\nMethods usually employed for the creation or anni-\nhilation of complex topological textures involve driving\nthem at material boundaries or forcing them to shrink\ndown to the atomic size due to large \felds. By contrast,\nthe annihilation of the skyrmion described in this section\nis obtained due to a mild perturbation. Shrinking follows\nby the natural breathing dynamics and a singularity for-\nmation happens in a \fnite time interval. It should also be\nnoted that the nontrivial topology of the skyrmion is not\nan obstacle in the singularity formation and eventually\nin the annihilation process.\nThe proposed dynamics can be induced in various\nways. (i) One could temporarily increase the scaled pa-\nrameter\u0015(by modifying the DM or anisotropy param-\neters), as presented in Fig. 2. (ii) One might also de-\ncrease the parameter, thus initiating breathing, but, in\nthis case, a su\u000eciently low value should be maintained\nuntil the skyrmion annihilates. (iii) Alternatively, an ex-\nternal magnetic \feld would act as easy-plane anisotropy\nin an antiferromagnet e\u000bectively reducing the easy-axis\nanisotropy parameter and thus increasing the dimension-\nless parameter \u0015. This case would require a separate\nstudy due to the more involved dynamics introduced by\nthe external \feld [23, 24]. (iv) Modi\fcation of the kinetic\nenergy of the skyrmion could also give further alterna-\ntives in order to initiate annihilation dynamics. This\ncould be achieved by inducing magnetization in the an-\ntiferromagnetic lattice.\nIV. NONLINEAR BREATHING MODE\nWe proceed to a systematic study of the breathing dy-\nnamics in the linear and the nonlinear regime that will\nlead to quantitative predictions for the breathing and for\nthe annihilation dynamics. We consider, in this section,\nmodes with helicity \u001f= 0 and a dynamic skyrmion pro-\n\fle\n\u0002 = \u0002(r;t);\b =\u001e: (20)\nThe kinetic energy in Eq. (11) reduces to\nT=1\n2Z\n_\u000222\u0019rdr (21)\nand the potential energy is\nV\u0015(\u0002) =Eex+\u0015~EDM+Ean (22)\nwithEex;~EDM;Eande\fned in Eqs. (12), (13).\nWe aim to study breathing oscillations, that is, a pe-\nriodic change of the skyrmion pro\fle \u0002( r;t). We assume\na material with parameter \u00150. In order to invoke breath-\ning dynamics, we change the parameter to a value \u00156=\u00150\n(for example, by applying a voltage) and assume that theskyrmion pro\fle eventually relaxes to \u0002 \u0015. When the pa-\nrameter is restored to the value \u00150, breathing dynamics\nis initiated.\nIn order to make progress analytically, we make a sim-\nplifying assumption that is supported by numerical simu-\nlations. As the skyrmion radius changes (oscillates) dur-\ning breathing, we assume that the pro\fle adiabatically\nadjusts to the static skyrmion pro\fle \u0002 \u0015for the corre-\nsponding radius R=R(\u0015). Under this assumption, the\npotential energy that the skyrmion experiences during\nthe breathing motion is given by (22) applied for \u0015=\u00150,\nV\u00150(\u0002\u0015) =Eex+\u00150~EDM+Ean\n=V\u0015+ (\u00150\u0000\u0015)~EDM(23)\nwhere all terms are evaluated for \u0002 = \u0002 \u0015. The energy\ncomponents for each value of the parameter \u0015and the\ncorresponding value of radius Rcan be calculated nu-\nmerically.\nIn Fig. 1, we plot (by solid lines) the potential en-\nergy (23) for the eqilibrium pro\fles \u0002 \u0015as a function\nof their radius R. Three lines are plotted for the cases\n\u00150= 0:4;0:47;0:57, which correspond to static skyrmion\nsolutions with radius R0= 0:71;1:14;2:39 respectively.\nAs expected, every solid line has a minimum at the cor-\nresponding value of the radius.\nThe main features of the potential wells in Fig. 1 can\nbe anticipated. We \frst consider the case of small radius,\nthat is,R!0, or, equivalently, \u0015!0 [32, 33]. We have\nEex!4\u0019; E DM;Ean!0 and this gives V\u0015!4\u0019. We\nalso have ~EDM\u0018\u00008\u0019R, and thus Eq. (23) gives\nV\u00150!4\u0019;asR!0: (24)\nIn the case of large radius, we have V\u0015\u0018O(R\u00001) and\nEDM=\u00004\u0019RasR!1 , or, equivalently, \u0015!2=\u0019[36].\nThus, Eq. (23) gives\nV\u00150\u00184\u0019\u0012\n1\u0000\u0019\u00150\n2\u0013\nR; asR!1 (25)\nwhere we have used \u0015= 2=\u0019+O(R\u00002). Eqs. (24), (25)\ncon\frm the features of the potentials shown in Fig. 1 that\nare crucial for describing breathing dynamics.\nThe potential wells shown in Fig. 1 give rise to oscil-\nlating motion around the minimum V0=V\u00150(\u0002\u00150). We\ndemonstrate this by performing a simulation similar to\nthat in Sec. III, but we now retain the voltage pulse for a\nlonger time, 13 ps. We use a smaller damping parameter\n\u000b= 0:001 in order to demonstrate clearer the oscillat-\ning motion. Fig. 4 shows the skyrmion radius and the\nkinetic energy during the motion. An oscillating motion\nstarts in the potential well V\u00150=0:57. The skyrmion radius\nhas passed the maximum when the pulse is switched o\u000b.\nThen, a new oscillating motion sets in in the potential\nwellV\u00150=0:47. The oscillations eventually die out due to\ndamping. The kinetic energy Tis close to zero at both\nturning points of the oscillation, i.e., at the minimum and\nmaximum radii.6\nFIG. 4. We apply a square pulse that modi\fes the DM\nparameter, as in Fig. 2, but now with a duration of 13 ps.\nWe start the simulation with D= 0:057 (\u0015= 0:57) and we\nrestore toD= 0:047 (\u0015= 0:47) at the time marked by the\ndotted red line. The damping parameter is \u000b= 0:001. (a)\nSkyrmion radius vs time shows damped breathing oscillations.\n(b) Kinetic energy (16). The kinetic energy is close to zero at\nboth turning points of the oscillation.\nFig. 5 shows snapshots of the N\u0013 eel vector and the mag-\nnetization for the skyrmion. The N\u0013 eel vector nof the ini-\ntial skyrmion is shown in entry (a). The magnetization\nmis shown during shrinking in entry (b) and expansion\nin entry (c). It points azimuthally in the plane and gives\nopposite vectors during shrinking and expansion. This\nis expected as the magnetization in Eq. (14) for a time\ndependent pro\fle as in Eq. (20) is\nm=\u000f\n2p\n2_\u0002^e\u001e: (26)\nWe proceed to systematic simulations of large ampli-\ntude breathing oscillations. We set D= 0:047 (\u00150=\n0:47) and we simulate the conservative equations, i.e.,\nset\u000b= 0 in Eq. (3), for a range of initial pro\fles \u0002 \u0015.\nIf the energy EisV0< E < 4\u0019, we expect oscillating\nmotion between the two values of the radius Rmin; Rmax\nfor which the potential takes the value V\u00150=E. It is\nRmin< R 0< R maxwhereR0is the skyrmion radius at\nthe minimum of the potential. We observe almost per-\nfectly periodic motion that was veri\fed for many periods.\nFig. 6 shows numerical results for the amplitude of os-\ncillations, at \u00150= 0:47, for simulations initiated with\npro\fles \u0002 \u0015for a range of values of the parameter \u0015. The\nradius of the initial pro\fles are plotted by blue circles\nand they represent the one extremum of the oscillating\nmotion. Red circles give the radius of the skyrmion at the\nother extremum of the oscillation. The blue line shows\nthe skyrmion radius corresponding to the static pro\fle\n\u0002\u0015and the red line the radius for the partner con\fgura-\ntion with the same potential energy V. If we choose an\ninitial pro\fle \u0002 \u0015with\u0015>0:6, we have V > 4\u0019and the\nskyrmion annihilates during the shrinking phase, as dis-cussed in Sec. III. Thus, the numerical results are in very\ngood agreement with the assumption that the oscillating\nmotion is described by the potentials in Fig. 1.\nThe period of oscillation for a small amplitude is found\nto be approximately T= 20 in dimensionless units (or\nT= 7:5 ps when we use the time unit in Eq. (9)) that\ncorresponds to an angular frequency !b= 0:3. This is\nin agreement with the analytical result (35) given in the\nnext section. The frequency is decreasing for large ampli-\ntude oscillations. This is anticipated since the potential\nin Fig. 1 is slower than parabolic.\nWe return to the issue of skyrmion annihilation and we\ncan now expand upon the results of Sec. III. Based on the\npotential wells shown in Fig. 1, one can start breathing\ndynamics either pushing towards a radius smaller than\nthat at the potential minimum or to a larger radius (as\nwe have shown in Fig. 2). Furthermore, one can imag-\nine a combination of the two possibilities. That is, one\nmay modulate \u0015periodically around the value \u00150, thus\npushing the skyrmion radius to values smaller and larger\nthanR0periodically. Using the appropriate frequency\nfor the\u0015modulation, this is expected to lead to reso-\nnance, i.e., large amplitude oscillations for the skyrmion\nradius, and to its eventual annihilation. A small ampli-\ntude modulation of \u0015will be su\u000ecient for the resonance\nphenomenon.\nV. SMALL BREATHING OSCILLATIONS\nThe breathing motion with a small amplitude will be\nstudied analytically, based on the potential (23) plotted\nin Fig. 1. We will study separately the case of a skyrmion\nof large radius and a skyrmion of small radius.\nA. Skyrmions of large radius\nLet us consider a large value of \u00150so that the cor-\nresponding static skyrmion has a large radius. The\nskyrmion radius R(t) will vary with time during the\nbreathing motion. The skyrmion pro\fle is approximated\nas a domain wall centered at the position of the radius,\n\u0002 = 2 arctan\u0000\ne\u0000z\u0001\n; z\u0011r\u0000R(t) (27)\nand we assume that \u001f= 0 during the motion. The kinetic\nenergy (11) gives\nT=_R2\n2Z1\n0sech2(r\u0000R) (2\u0019rdr ):\nForR\u001d1, we may extend the lower limit to \u00001 (with\nan exponentially small error) and obtain\nT\u0019_R2\n2Z1\n\u00001sech2z(2\u0019Rdz ) = 2\u0019R_R2: (28)7\nFIG. 5. The N\u0013 eel \feld nand the magnetization \feld mof the skyrmion during breathing. (a) The N\u0013 eel \feld for the initial\nskyrmion. (b) The magnetization \feld during skyrmion expansion, at time t1indicated in Fig. 4. (c) The magnetization \feld\nduring skyrmion shrinking, at time t2. The magnetization points azimuthally in both cases as anticipated by Eq. (26).\nFIG. 6. For \u00150= 0:47 (D= 0:047), we show the maxi-\nmum and minimum radii of skyrmion breathing oscillations,\nwhen the initial skyrmion pro\fle is \u0002 \u0015. Lines show theo-\nretical results based on Eq. (22) and Fig. 1. Circles show\nresults of numerical simulations. The simulations are initi-\nated with skyrmion pro\fles at the blue circles (they give the\none extremum of the oscillation). The other extremum of the\noscillation is shown by red circles. (The radius Ris in scaled\nunits and the actual length is given, as in Eq. (8), by aR=\u000f ,\nwith\u000f= 0:1 for parameter values (2)).\nFor calculating the potential energy (23) we will use\nthe results [36]\nV\u0015=4\u00192\u00152\nR+O\u0000\nR\u00003\u0001\n; \u0015 2= 0:3057\nEDM=\u00004\u0019R+O\u0000\nR\u00001\u0001(29)\nwhereRis the radius of the static skyrmion for DM pa-rameter\u0015, approximated by [36]\n\u0015=2\n\u0019\u0000\u00152\nR2+O\u0000\nR\u00004\u0001\n: (30)\nOur objective is to evaluate the potential (23) around the\nradiusR0that corresponds to the parameter \u00150. We set\nR=R0(1 +\u000e) (31)\nand we use Eq. (30) to obtain\n\u00150\n\u0015\u00001\u0019\u0000\u0019\u00152\nR2\n0\u0012\n1\u00003\n2\u000e\u0013\n\u000e; \u000e\u001c1: (32)\nInserting Eq. (32) and (31) in Eq. (29), we have\nV\u0015\u00194\u00192\u00152\nR0(1\u0000\u000e+\u000e2)\n\u00150\u0000\u0015\n\u0015EDM\u00194\u00192\u00152\nR0\u0012\n\u000e\u00001\n2\u000e2\u0013\n:\nSubstituting the two last equations in the potential en-\nergy (23), we obtain the parabolic form\nV\u00150(\u0002\u0015) =4\u00192\u00152\nR0\u0012\n1 +1\n2\u000e2\u0013\n: (33)\nEqs. (28), (33) give the Lagrangian\nL= 2\u0019R3\n0_\u000e2\u00002\u00192\u00152\nR0\u000e2(34)\nwhich implies harmonic oscillations with angular fre-\nquency\n!b=p\u0019\u00152\nR2\n0\u0019r\u0019\n\u00152\u00122\n\u0019\u0000\u00150\u0013\n: (35)\nFig. 7 shows the results of numerical simulations for\nbreathing oscillations of small amplitude. For \u00150close\nto the value 2 =\u0019, the numerical results con\frm Eq. (35).\nThe dependence of the breathing frequency on R0has\nbeen obtained in Ref. [22], but the numerical factor in\nEq. (35) introduces a correction to that result. This mod-\ni\fcation originates in the corrected dependence of the ra-\ndius on the DM parameter given in Eq. (30) compared\nto an earlier result obtained in Ref. [35].8\nFIG. 7. The angular frequency for small amplitude breathing\noscillations as a function of the parameter \u00150. The angular\nfrequency goes to zero as \u00150approaches 2 =\u0019. The rate of\nconvergence is consistent with Eq. (35) within numerical ac-\ncuracy. Physical units for frequency !b=(2\u0019) are restored by\nmultiplying by 2 :68 THz.\nB. Skyrmions of small radius\nThe pro\fle of a skyrmion of small radius is approx-\nimated by a Belavin-Polyakov (BP) solution [32, 33].\nWe thus consider a BP skyrmion with a time dependent\nradius, as in Eq. (17). The kinetic energy is given in\nEq. (18) or, using Eq. (31),\nT\u0019\u00004\u0019R2\n0lnR0_\u000e2: (36)\nRegarding the potential energy, we use the formula\n(A5) derived in Appendix A. That is an improvement\nof the asymptotic results of Ref. [32]. The parameter \u0015\nand the skyrmion radius Rfor small radius are related\nby [32, 33]\n\u0015=\u0000RlnR: (37)\nUsing the latter, the potential (A5) is written as\nV\u00150(R) = 4\u0019\u0014\n1\u0000R(RlnR\u00002R0lnR0) +1\n2R2\u0015\n(38)\nthat is valid for small RandR0. Eq. (38) has a min-\nimum atR=R0as desired (note that the minimum\nis obtained thanks to Eq. (A5) while it would not have\nbeen possible to obtained by using previously available\nformulas). Inserting (31) in (38), we \fnd the quadratic\napproximation\nV\u00150\u00194\u0019\u0000\n1 +R2\n0lnR0\u0000R2\n0lnR0\u000e2\u0001\n: (39)\nEqs. (36) and (39) give the Lagrangian\nL=\u00004\u0019R2\n0lnR0(_\u000e2\u0000\u000e2): (40)We thus \fnd that small breathing oscillations for\nskyrmions of small radius have an angular frequency\n!b= 1: (41)\nNote that this is equal to the frequency (48) for helicity\noscillations discussed in the next Section.\nThe numerical results shown in Fig. 7 indicate that !b\ntakes a value of O(1) for skyrmions of small radius and it\nis thus consistent with the result in Eq. (41). We cannot\nsimulate the dynamics of skyrmions for very small \u0015due\nto their very small size and we thus do not present a\nprecise numerical result for !bas\u00150!0.\nVI. HELICITY OSCILLATIONS\nFor completeness, we consider an oscillation mode\nwhere the radial skyrmion pro\fle remains unchanged\nwhile the helicity depends on time\n\u0002 = \u0002(r);\b =\u001e+\u001f(t): (42)\nHowever, this assumption is not consistent with the equa-\ntions of motion derived from the energy functional. In-\ndeed, simulations show that Eq. (42) does not give a good\napproximation for the dynamical pro\fle. When we start\nfrom a static skyrmion pro\fle \u0002 \u0015and choose uniform he-\nlicity\u001f6= 0, we obtain a complicated motion that seems\nto combine breathing and helicity oscillations. Neverthe-\nless, Eq. (42) will prove its usefulness as it will lead to an\napproximation for the equation of motion for \u001fand to\nthe frequency of the observed oscillations. We therefore\nargue that this assumption is useful and we proceed to\nuse it in the following calculations.\nThe kinetic energy (11) reduces to the expression\nT=1\n2_\u001f2Z\nsin2\u0002 (2\u0019rdr ) = _\u001f2Ean (43)\nand the potential energy is\nV\u0015=Eex+\u0015cos\u001f~EDM+Ean (44)\nwhereEex;~EDM;Eandepend on \u0002 but not on \u001f, as seen\nin Eqs. (12), (13). Omitting terms independent of \u001f, the\nLagrangian is\nL=T\u0000V= _\u001f2Ean\u0000\u0015cos\u001f~EDM: (45)\nBy a standard scaling argument for the minimizer \u0002 \u0015, a\nvirial relation is obtained [38],\n2Ean+EDM= 0)\u0015~EDM=\u00002Ean: (46)\nThe latter is used in Eq. (45) to give\nL= 2Ean\u00121\n2_\u001f2+ cos\u001f\u0013\n(47)9\nLagrangian (47) describes a pendulum. For small \u001f\u001c1,\nit gives harmonic oscillations with angular frequency\n!h= 1 (48)\nthat is a period T= 2\u0019(orT= 2:3 ps when we use the\ntime unit in Eq. (9)). Large amplitude oscillations will\nhave a smaller frequency as in the case of a pendulum.\nThe value in Eq. (48) agrees with the results of numerical\nsimulations.\nNote that oscillations of helicity, for a small ampli-\ntude, are found to have the same frequency (48) as small\nbreathing oscillations for skyrmions of small radius (41).\nVII. CONCLUDING REMARKS\nWe have studied breathing oscillations of skyrmions in\nchiral antiferromagnets using analytical arguments and\ncalculations, within a continuum model, that are valid in\nthe nonlinear regime. The predictions are con\frmed and\nthe results are extended by systematic numerical simu-\nlations within the original discrete spin model. A signif-\nicant result is the prediction, con\frmed by simulations,\nthat the forced expansion of the skyrmion radius invokes\nbreathing oscillations that can lead to the skyrmion an-\nnihilation. This counter-intuitive process o\u000bers an al-\nternative to the typically employed methods of forced\nskyrmion suppression. It can prove to be a more conve-\nnient method as it only requires mild forcing. This is suf-\n\fcient because the annihilation is actually brought about,\nnot by the forcing itself, but by the invoked internal dy-\nnamics. Furthermore, the phenomenon is interesting also\nfrom a theoretical perspective because it involves the cre-\nation of a singularity in \fnite time. Finally, we give an\nanalytical calculation based on an energetic method for\nthe frequency of small amplitude oscillations.\nIn the development of the theoretical arguments, we\nhave given details about the kinetic energy of the con-\ntinuum model for an antiferromagnet. This is actually\nan emergent kinetic energy that originates in the ex-\nchange energy of the discrete model. We derive the sur-\nprising result that the kinetic energy can be tuned to a\nnonzero value at the point of the singularity formation\nand skyrmion annihilation.\nGiven the counter-intuitive dynamical behaviour of the\nbreathing skyrmion, it will be interesting to consider fur-\nther dynamical phenomena of this system. For exam-\nple, it is interesting to investigate the skyrmion domain\nwall velocity during breathing. If this could reach the\nmaximum velocity allowed for a traveling wall, that is\nvmax=p\n1\u0000(\u0019\u0015=2)2[39], then an instability of the N\u0013 eelstate will occur that may lead to the spontaneous forma-\ntion of a spiral.\nACKNOWLEDGEMENTS\nThis work was supported by the project \\Thunder-\nSKY\" funded by the Hellenic Foundation for Research\nand Innovation and the General Secretariat for Research\nand Innovation, under Grant No. 871.\nAppendix A: Approximation of the potential for\nsmall radius\nThe available asymptotic formulae for the energy at\nsmall\u0015are [32]\nV\u0015(\u0002\u0015) = 4\u0019\u0012\n1 +\u00152\nln\u0015\u0013\n+o\u0012\u00152\nln\u0015\u0013\n;\nEDM= 8\u0019\u00152\nln\u0015+o\u0012\u00152\nln\u0015\u0013\n:(A1)\nSubstituting in the potential energy (23), we obtain\nV\u00150= 4\u0019\u0014\n1 + (2\u00150\u0000\u0015)\u0015\nln\u0015\u0015\n: (A2)\nThe derivative of this potential is\nV0\n\u00150= 4\u0019\u0014\n2\u00150\u0000\u0015\nln\u0015\u00002\u00150\u0000\u0015\n(ln\u0015)2\u0015\n: (A3)\nWe haveV0\n\u00150(\u00150)6= 0, due to the term 1 =(ln\u0015)2, and\nthus no minimum is implied at \u0015=\u00150. The only way\nto obtain a formula that gives V0\n\u00150(\u00150) = 0 up to terms\n1=(ln\u0015)2is to write\nEex= 4\u0019\u0014\n1 +1\n2\u00152\n(ln\u0015)2\u0015\n; E DM= 8\u0019\u00152\nln\u0015;(A4)\nthat gives\nV\u00150(\u0015) = 4\u0019\u0014\n1 + (2\u00150\u0000\u0015)\u0015\nln\u0015+1\n2\u00152\n(ln\u0015)2\u0015\n:(A5)\nWe have\nV0\n\u00150= 4\u0019\u0014\n2\u00150\u0000\u0015\nln\u0015+ 2\u00150\u0000\u0015\n(ln\u0015)2\u0015\n+O\u0012\u0015\n(ln\u0015)3\u0013\n(A6)\nand thusV\u00150has the desired behaviour at \u0015=\u00150.\nFormula (A4) for Eexis an improvement over previous\nresults [36] and it is necessary in order to \fnd the oscil-\nlation frequency for breathing oscillations in Sec. V B.\n[1] A. Bogdanov and D. Yablonskii. Thermodynamically sta-\nble \"vortices\" in magnetically ordered crystals. the mixedstate of magnets. Zh. Eksp. Teor. Fiz. , 95:178{182, 1989.10\n[2] A. N. Bogdanov and A. Hubert. Thermodynamically sta-\nble magnetic vortex states in magnetic crystals. JMMM ,\n138:255{269, 1994.\n[3] Stavros Komineas and Nikos Papanicolaou. Skyrmion dy-\nnamics in chiral ferromagnets. Phys. Rev. B , 92:064412,\nAug 2015.\n[4] Wanjun Jiang, Gong Chen, Kai Liu, Jiadong Zang,\nSuzanne G.E. te Velthuis, and Axel Ho\u000bmann.\nSkyrmions in magnetic multilayers. Physics Reports ,\n704:1{49, 2017. Skyrmions in Magnetic Multilayers.\n[5] A. Fert, V. Cros, and J. Sampaio. Skyrmions on the\ntrack. Nature Nanotech , 8:152{156, 2013.\n[6] R. Tomasello, E. Martinez, R. Zivieri, L. Torres, M. Car-\npentieri, and G. Finocchio. A strategy for the design of\nskyrmion racetrack memories. Sci. Rep , 4:6784, 2014.\n[7] Wataru Koshibae, Yoshio Kaneko, Junichi Iwasaki,\nMasashi Kawasaki, Yoshinori Tokura, and Naoto Na-\ngaosa. Memory functions of magnetic skyrmions.\nJapanese Journal of Applied Physics , 54(5):053001, apr\n2015.\n[8] X. Zhang, M. Ezawa, and Y. Zhou. Magnetic skyrmion\nlogic gates: conversion, duplication and merging of\nskyrmions. Sci. Rep , 5:9400, 2015.\n[9] Shijiang Luo, Min Song, Xin Li, Yue Zhang, Jeongmin\nHong, Xiaofei Yang, Xuecheng Zou, Nuo Xu, and Long\nYou. Recon\fgurable skyrmion logic gates. Nano Lett. ,\n18:1180|-1184, 2018.\n[10] Kyung Mee Song, Jae-Seung Jeong, Biao Pan, Xi-\nchao Zhang, Jing Xia, Sunkyung Cha, Tae-Eon Park,\nKwangsu Kim, Simone Finizio, J org Raabe, Joonyeon\nChang, Yan Zhou, Weisheng Zhao, Wang Kang, Hyunsu\nJu, and Seonghoon Woo. Skyrmion-based arti\fcial\nsynapses for neuromorphic computing. Nat. Electron. ,\n3:148{155, 2020.\n[11] X. Zhang, Y. Zhou, and M. Ezawa. Antiferromagnetic\nskyrmion: Stability, creation and manipulation. Sci. Rep ,\n6:24795, 2016.\n[12] Kyoung-Woong Moon, Byong Sun Chun, Wondong Kim,\nand Chanyong Hwang. Control of spin-wave refraction\nusing arrays of skyrmions. Phys. Rev. Applied , 6:064027,\nDec 2016.\n[13] Joseph Barker and Oleg A. Tretiakov. Static and dy-\nnamical properties of antiferromagnetic skyrmions in the\npresence of applied current and temperature. Phys. Rev.\nLett., 116:147203, Apr 2016.\n[14] Chendong Jin, Chengkun Song, Jianbo Wang, and Qing-\nfang Liu. Dynamics of antiferromagnetic skyrmion driven\nby the spin hall e\u000bect. Appl. Phys. Lett. , 109:182404,\n2016.\n[15] Haiyan Xia, Chendong Jin, Chengkun Song, Jinshuai\nWang, Jianbo Wang, and Qingfang Liu. Control and ma-\nnipulation of antiferromagnetic skyrmions in racetrack.\nJournal of Physics D: Applied Physics , 50(50):505005,\nnov 2017.\n[16] Xiaofeng Zhao, Ruizhi Ren, Gang Xie, and Yan Liua.\nSingle antiferromagnetic skyrmion transistor based on\nstrain manipulation. Appl. Phys. Lett. , 112:252402, 2018.\n[17] Xue Liang, Guoping Zhao, Laichuan Shen, Jing Xia,\nLi Zhao, Xichao Zhang, and Yan Zhou. Dynamics of an\nantiferromagnetic skyrmion in a racetrack with a defect.\nPhys. Rev. B , 100:144439, Oct 2019.\n[18] Z. Jin, T. T. Liu, W. H. Li, X. M. Zhang, Z. P. Hou, D. Y.\nChen, Z. Fan, M. Zeng, X. B. Lu, X. S. Gao, M. H. Qin,\nand J.-M. Liu. Dynamics of antiferromagnetic skyrmionsin the absence or presence of pinning defects. Phys. Rev.\nB, 102:054419, Aug 2020.\n[19] H. Diego Gao, Shangand Rosales, Flavia A. G\u0013 omez Al-\nbarrac\u0013 \u0010n, Vladimir Tsurkan, Guratinder Kaur, Tom Fen-\nnell, Paul Ste\u000bens, Martin Boehm, Petr \u0014Cerm\u0013 ak, Astrid\nSchneidewind, Eric Ressouche, Daniel C. Cabra, Chris-\ntian R uegg, and Oksana Zaharko. Fractional antiferro-\nmagnetic skyrmion lattice induced by anisotropic cou-\nplings. Nature , 586:37{41, 2020.\n[20] Hariom Jani, Jheng-Cyuan Lin, Jiahao Chen, Jack Har-\nrison, Francesco Maccherozzi, Jonathon Schad, Saurav\nPrakash, Chang-Beom Eom, A. Ariando, T. Venkatesan,\nand Paolo G. Radaelli. Antiferromagnetic half-skyrmions\nand bimerons at room temperature. Nature , 590:74{79,\n2021.\n[21] Christoph Sch utte and Markus Garst. Magnon-skyrmion\nscattering in chiral magnets. Phys. Rev. B , 90:094423,\nSep 2014.\n[22] Volodymyr P. Kravchuk, Olena Gomonay, Denis D.\nSheka, Davi R. Rodrigues, Karin Everschor-Sitte, Jairo\nSinova, Jeroen van den Brink, and Yuri Gaididei. Spin\neigenexcitations of an antiferromagnetic skyrmion. Phys.\nRev. B , 99:184429, May 2019.\n[23] V. G. Bar'yakhtar, M. V. Chetkin, B. A. Ivanov, and\nS. N. Gadetskii. Dynamics of Topological Magnetic Soli-\ntons { Experiment and Theory . Springer, Berlin, 1994.\n[24] S Komineas and N Papanicolaou. Vortex dynam-\nics in two-dimensional antiferromagnets. Nonlinearity ,\n11(2):265, 1998.\n[25] Marine Schott, Laurent Ranno, Helene Bea, CLaire Bara-\nduc, Stephane Au\u000bret, and Anne Bernand-Mantel. Elec-\ntric \feld control of interfacial dzyaloshinskii-moriya inter-\naction in pt/co/alo xthin \flms. J. Magn. Magn. Mater. ,\n520:167122, 2021.\n[26] T. Archer, C. D. Pemmaraju, S. Sanvito, C. Franchini,\nJ. He, A. Filippeti, P. Delugas, D. Puggioni, V. Fioren-\ntini, R. Tiwari, and P. Majumdar. Exchange interac-\ntions and magnetic phases of transition metal oxides:\nBenchmarking advanced abinitio methods. Phys. Rev.\nB, 84:115114, 2011.\n[27] Priya Gopal, Riccardo De Gennaro, Marta Silva dos San-\ntos Gusamo, Rabih Al Rahal Orabi, Haihang Wang, Ste-\nfano Curtarolo, Marco Fornari, and Marco Buongiorni\nNardelli. Improved electronics structure and magnetic\nexchange interactions in transition metal oxides. J. Phys:\nCondens. Matter , 29:444003, 2017.\n[28] I. V. Bar'yakhtar and B. A. Ivanov. Nonlinear magneti-\nzation waves of an antiferromagnet. Sov. J. of Low Temp.\nPhys. , 5:361, 1979.\n[29] Helen V. Gomonay and Vadim M. Loktev. Spin transfer\nand current-induced switching in antiferromagnets. Phys.\nRev. B , 81:144427, Apr 2010.\n[30] A. N. Bogdanov and D. A. Yablonskii. Contribution to\nthe theory of inhomogeneous states of magnets in the re-\ngion of magnetic-\feld-induced phase transitions. mixed\nstate of antiferromagnets. Sov. Phys. JETP , 69:142,\n1989.\n[31] A. Bogdanov and A. Shestakov. Vortex states in antifer-\nromagnetic crystals. Phys. Solid State , 40:1350, 1998.\n[32] Stavros Komineas, Christof Melcher, and Stephanos Ve-\nnakides. The pro\fle of chiral skyrmions of small radius.\nNonlinearity , 33:2295{3408, May 2020.\n[33] Anne Bernand-Mantel, Cyrill B. Muratov, and Thilo M.\nSimon. Unraveling the role of dipolar versus11\nDzyaloshinskii-Moriya interactions in stabilizing compact\nmagnetic skyrmions. Phys. Rev. B , 101:045416, Jan 2020.\n[34] Anne Bernand-Mantel, Cyrill B. Muratov, and\nTheresa M. Simon. A quantitative description of\nskyrmions in ultrathin ferromagnetic \flms and rigidity\nof degree\u00061 harmonic maps from R2!S2.Archive for\nRational Mechanics and Analysis , 239(1):219{299, 2021.\n[35] S. Rohart and A. Thiaville. Skyrmion con\fne-\nment in ultrathin \flm nanostructures in the presence\nof Dzyaloshinskii-Moriya interaction. Phys. Rev. B ,\n88:184422, Nov 2013.\n[36] Stavros Komineas, Christof Melcher, and Stephanos Ve-nakides. Chiral skyrmions of large radius. Physica D:\nNonlinear Phenomena , 418:132842, 2021.\n[37] Stavros Komineas and Nikos Papanicolaou. Traveling\nskyrmion in chiral antiferromagnets. SciPost Phys. ,\n8:086, 2020.\n[38] A. Bogdanov and A. Hubert. The properties of isolated\nmagnetic vortices. Phys. Stat. Sol. B , 186:527, 1994.\n[39] Riccardo Tomasello and Stavros Komineas. Vortex prop-\nagation and phase transitions in a chiral antiferromag-\nnetic nanostripe. Phys. Rev. B , 104:064438, Aug 2021."    },    {        "title": "2203.05747v2.Hard_ferromagnetism_down_to_the_thinnest_limit_of_iron_intercalated_tantalum_disulfide.pdf",        "content": " 1   Hard ferromagnetism down to the thinnest limit of iron-intercalated tantalum disulfide Samra Husremović,a Catherine K. Groschner,a Katherine Inzani,b,c,† Isaac M. Craig,a Karen C. Bustillo,d Peter Ercius,d Nathanael P. Kazmierczak,a,‡ Jacob Syndikus,a Madeline Van Winkle,a Shaul Aloni,c Takashi Taniguchi,e Kenji Watanabe,f Sinéad M. Griffin,b,c D. Kwabena Bediakoa,g*  a Department of Chemistry, University of California, Berkeley, California 94720, United States b Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, United States c The Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States d National Center for Electron Microscopy, Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States  e Research Center for Functional Materials, National Institute for Materials Science, Tsukuba 305-0044, Japan f International Center for Materials Nanoarchitectonics, National Institute for Materials Science, Tsukuba 305-0044, Japan g Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States † Current position: School of Chemistry, University of Nottingham, Nottingham, NG7 2RD, United Kingdom ‡ Current position: Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, CA 91125, United States    * E-mail: bediako@berkeley.edu     2 ABSTRACT Two-dimensional (2D) magnetic crystals hold promise for miniaturized and ultralow power electronic devices that exploit spin manipulation. In these materials, large, controllable magnetocrystalline anisotropy is a prerequisite for the stabilization and manipulation of long-range magnetic order. In known 2D magnetic crystals, relatively weak magnetocrystalline anisotropy results in typically soft ferromagnetism. Here, we demonstrate that ferromagnetic order persists down to the thinnest limit of FexTaS2 (Fe-intercalated bilayer 2H-TaS2) with giant coercivities up to 3 tesla. We prepare Fe-intercalated TaS2 by chemical intercalation of van der Waals layered 2H-TaS2 crystals and perform variable-temperature quantum transport, transmission electron microscopy, and confocal Raman spectroscopy measurements to shed new light on the coupled effects of dimensionality, degree of intercalation, and intercalant order/disorder on the hard ferromagnetic behavior of FexTaS2. More generally, we show that chemical intercalation gives access to a rich synthetic parameter space for low-dimensional magnets, in which magnetic properties can be tailored by the choice of the host material and intercalant identity/amount, in addition to the manifold distinctive degrees of freedom available in atomically thin, van der Waals crystals.   3 INTRODUCTION The recent discovery of long-range magnetic ordering in the atomically thin layers of Cr2Ge2Te6,1 CrI3,2 and Fe3GeTe23 has spurred intense interest into the development of “two-dimensional” (2D) magnetic crystals. The experimental paradigm that has generated the most consistent success in this area has generally been to exfoliate layered van der Waals (vdW) crystals that are known to exhibit magnetic ordering in the bulk. In addition to serving as promising candidates for a new generation of compact, energy-efficient spin-based electronics,4–7 these 2D magnets are exceptional platforms for exploring fundamental principles pertaining to the stabilization of long-range magnetic order at low dimensions. According to the Mermin–Wagner–Hohenberg theorem, long-range magnetic order in 2D systems should not be stable in an isotropic model.8,9 However, intrinsic magnetocrystalline anisotropy (MCA)—which imparts a preferential spin orientation—plays an integral role in stabilizing the long-range magnetic order in 2D vdW magnets.10,11 Still, the overwhelming majority of ferromagnetic 2D crystals are soft, possessing weak coercive fields, which arise from relatively weak MCA.12 The development of new atomically thin materials with strong and predictable control over MCA is imperative for tailoring magnetism in 2D materials that may serve as testbeds for fundamental theory and pave the way towards ultralow power spintronic technologies. Transition metal dichalcogenides (TMDs) intercalated with open-shell transition metals comprise a family of materials with inherently tunable MCA, which makes them appealing targets for realizing 2D magnets with deterministic properties. The strength of MCA in intercalated TMDs is determined by the respective magnitudes of spin-orbit coupling (SOC) and unquenched orbital angular momentum (OAM) of the spin-bearing ions.13–16 While the SOC is increased when the host lattice or the intercalants comprise heavy elements, the OAM of spin-bearing ions is shaped by their oxidation states and coordination environments imposed by the interlayer galleries.13–18 However, low-dimensional analogues of these systems have not been magnetically characterized; strong interlayer interactions in the effectively ionic, bulk crystals of these intercalation compounds have, so far, precluded exfoliation of pristine, large 2D flakes suited for magnetic studies.19–22 Consequently, the manifold possibilities for designing a new class of 2D magnets with controllable MCA by exploiting the tunable vdW interface, host lattice, and intercalant identity/stoichiometry remain untapped.   4 In this work, we discover the persistence of long-range ferromagnetism in a transition metal-intercalated TMD, FexTaS2, down to the 2D limit. We show that chemical intercalation of iron between layers of the exfoliated, pristine host lattice 2H-TaS2 gives access to this intercalation compound with controlled dimensionality from few layers down to the thinnest limit, Fe-intercalated bilayer 2H-TaS2. We find very strong MCA in these 2D materials, associated with the high unquenched OAM of the trigonal antiprismatic (i.e. trigonally elongated pseudo-octahedral) Fe centers and SOC from the Ta-based host. This large MCA is reflected in magnetotransport measurements, which reveal that anisotropy-stabilized hard ferromagnetic behavior is retained in the 2D limit, producing giant coercive fields up to 3 T. Furthermore, we find that the degree of intercalation and order/disorder in intercalated species heavily influences the magnetoelectronic behavior, establishing this family of intercalation compounds as versatile platforms for designing 2D magnets. RESULTS AND DISCUSSION Fabrication of vdW heterostructures, chemical intercalation, and structural characterization Few-layer FexTaS2 was synthesized by intercalating iron into mechanically exfoliated 2H-TaS2 flakes using a procedure adapted from established methods for soft chemical intercalation of metals using molecular precursors containing zerovalent metal centers.23–27 In a typical procedure, 2H-TaS2 flakes were mechanically exfoliated onto SiO2/Si substrates and capped with electrically insulating hexagonal boron nitride (hBN) to prevent degradation of the TaS2 basal plane,28 while exposing one or more edges of the TMD flake. Under inert atmosphere, the SiO2/Si substrates bearing these vdW heterostructures are immersed in a 10 mM solution of Fe(CO)5 in toluene, acetone, or acetonitrile at 50 – 55 °C for about 24 hours as depicted in Figure 1a (See SI for experimental details). After washing with fresh solvent, samples were annealed under high vacuum (10–7 torr) at 350 °C for 30 minutes to promote ordering of intercalants (SI Figure 5).24   5 To examine the composition and structure of our FexTaS2 thin crystals, we obtained and high-resolution TEM (SI Figure 6), scanning TEM (STEM) combined with energy-dispersive X-ray spectroscopy (EDS) and differential phase contrast (DPC)-STEM with atomic resolution (See SI for experimental details). The c-axis HRTEM (SI Figure 6) data verify that the ab atomic periodicities match the expected values for bulk FexTaS2, and EDS measurements of cross-sectioned thin crystals confirm the incorporation of Fe into TaS2 after chemical intercalation (Figure 1b). The Fe intercalants occupy 6-coordinate interstitial sites between the layers of 2H-TaS2, as revealed by DPC-STEM of cross-sectioned FexTaS2 samples (Figure 1c, SI Figure 8, 10) displaying a prominent √3a × √3a superstructure in their Raman spectra (SI Figure 9). Further analysis of our STEM-DPC micrographs (See SI for details) shows  Figure 1. (a) Schematic illustration of iron intercalation into the hBN/2H-TaS2 heterostructure and formation of FexTaS2. (b) STEM-EDS scans perpendicular to the c-axis of a cross-sectional FexTaS2 sample (white region in STEM image). The dashed red box in the STEM image marks the area where EDS data was collected. (c) DPC-STEM micrograph of a cross-sectioned FexTaS2 sample along the !1010$ zone axis of the 2H-TaS2 lattice. The structure of Fe1/3TaS2 (ref  30) is overlaid with the micrograph. (d) Published crystal structures for bulk Fe0.25TaS2 (ref 29) and Fe0.33TaS2 (ref  30). (e)–(g) Pairs of Raman spectra and NBED patterns of three chemically intercalated FexTaS2 flakes on electron-transparent Si3N4 grids. In each case, Raman spectra and NBED data were measured in identical regions of the flake. The first order Bragg spots of the 2H-TaS2 host lattice and the Fe superlattices are marked in the NBED patterns. Raman spectra are normalized to the A1/A1g peak, and the spectral regions where phonon modes for different superlattices are expected32 are highlighted. The raw data are represented in light gray, while dark gray lines represent denoised traces.  \n 6 that the pseudo-octahedral Fe coordination environments are trigonally elongated by 8.0(6) %. The magnitude of this experimentally determined distortion is congruous with the ~8 % trigonal distortion predicted by our DFT calculations. The trigonally elongated pseudo-octahedral Fe coordination has also been reported for bulk FexTaS2 prepared via high-temperature chemical vapor transport crystal growth (Figure 1d).29,30  In bulk FexTaS2 analogs, depending on the stoichiometry, superlattices with 2a × 2a or √3a × √3a periodicity are observed (where a = the lattice constant of 2H-TaS2), with perfect superlattices at x = 0.25 and 0.33, respectively (Figure 1d).29–31 To examine if analogous intercalant ordering is present in our chemically treated few-layer materials, we measured both nanobeam electron diffraction (NBED) patterns and Raman spectra within 500 nm of each other on samples mounted on silicon nitride transmission electron microscopy (TEM) grids (See SI for experimental details). As shown in Figures 1e–g, NBED patterns of FexTaS2 samples prepared by chemical intercalation and annealing exhibit superlattice spots that are not present in 2H-TaS2 (SI Figure 5), and the accompanying Raman spectra display a new set of peaks in the low-frequency region (100 – 150 cm–1). These fingerprints of Fe ordering are consistent with previous literature on bulk FexTaS2.32,33 For example, the sample that displays sharp √3a × √3a superlattice spots in NBED data also exhibits an intense Fe-in-plane (Fe*) mode at 140 cm–1, which is characteristic of the √3a × √3a superstructure (Figure 1e).32 In contrast, as shown in Figures 1f,g, samples that evince two clear superstructures (both √3a × √3a and 2a × 2a) in NBED data are characterized by generally broader and less intense Raman features that span the energy ranges expected for each type of Fe superlattice ordering. For these dual-superlattice samples, sharp Fe-related Raman features may be absent (Figure 1g) or show a red-shift (Figure 1f) compared to the crystals that host a single, dominant Fe superstructure (Figure 1e). Further, a comparison of these data reveals that the sharpness of Fe-related NBED spots and Raman-active Fe* phonon modes appear to be correlated; samples with diffuse superlattice spots in NBED show broad Fe* features in Raman spectra (Figure 1g), while samples with sharp superstructure spots in NBED present more well defined Fe* peaks in Raman spectra (Figure 1e,f). These data demonstrate that chemical intercalation is a viable method for accessing FexTaS2 in the few-layer limit with the potential to vary stoichiometry and homogeneity. We find that all Raman spectra in Figure 1e–g display a red shift of the  7 A1g mode by 8–10 cm–1 from the expected value for 2H-TaS2.34,35 This shift is consistent with electron transfer from Fe to the host lattice,36,37 in agreement with our DFT calculations. For all calculated structures and magnetic configurations, we found that Fe converges to a +2 oxidation state, matching our electron energy loss spectroscopy (EELS) results (SI Figure 12) and published experimental data on bulk FexTaS2.38–40 We note that without annealing these chemically intercalated materials, we neither observe superlattice spots in NBED data nor significant changes to the Raman spectra (SI Figure 5). These observations suggest that a modest thermal treatment is required both for ordering of Fe in the vdW interface between TaS2 layers24 as well as charge transfer from the intercalants to the host lattice (see also SI Section 2.2).  Variable temperature transport measurements We interrogated the magnetoelectronic behavior of chemically intercalated FexTaS2 flakes by fabricating mesoscopic Hall bar devices for variable temperature magnetotransport measurements (Figure 2a) (See SI for details on device fabrication). In these experiments, we simultaneously monitored the longitudinal (ρxx) and transverse Hall resistivity (ρxy) as a function of temperature (T) and external magnetic field (H) to evaluate magnetic order in our materials via its effect on electronic transport. Figure 2 presents a representative set of results for a hBN/11-layer FexTaS2 heterostructure on a supporting SiO2/Si substrate (See SI for details on determining the layer count). Confocal Raman spectroscopy data (Figure 2b) acquired along the measurement channel of the Hall bar are consistent with a dominant √3a × √3a superstructure that gives rise to a strong peak at 139 cm–1, albeit showing evidence for the co-existence of a much less dominant 2a × 2a superlattice in a weaker broad feature spanning 100–135 cm–1. The co-existence of two superlattices can be a result of either stoichiometric inhomogeneities within the spot size (ca. 500 nm) of the Raman laser and/or deviations of x from the commensurate amounts of 1/4 and 1/3.20,41 These possibilities were evaluated by cross-sectioning devices with focused ion beam and using STEM-EDS to map the Fe–Ta ratio (see SI for experimental details). These measurements reveal an effectively consistent Fe:Ta value of 0.26 ± 0.02 along the Hall bar channel (Figure 2c).   8 To compare the Fe:Ta stoichiometry measured in thin flakes with the values determined for bulk (3D) crystals, we must consider that terminating Ta ions from the top and bottom TaS2 layers contribute substantially to the measured elemental composition of thin crystals. Consequently, the ratio of Fe to Ta should be lower in thin samples compared to their structurally equivalent 3D analogues. We can convert between the Fe:Ta values measured in thin samples (xt) possessing L number of layers and the stoichiometry of their compositionally equivalent bulk analogues (xb) using the following expression (see SI for details on the simple derivation):  𝑥!=𝑥\"/$1−1𝐿( (1) With this expression, we find that the equivalent bulk stoichiometry, xb = 0.29 ± 0.02 for our sample shown in Figure 2, confirming that the degree of intercalation represents a slight deviation from the commensurate  Figure 2. (a) Schematic of the hBN/11-layer FexTaS2 device on SiO2/Si. (b) Cumulative low-frequency Raman spectrum across the Hall bar channel of the device. (c) Representative cross-sectional STEM-EDS of the device. (d) Temperature-dependent longitudinal resistivity and its first derivative with respect to temperature of the hBN/11-layer FexTaS2 heterostructure. (e) Hall resistivity at 1.8 K with the ordinary Hall effect (OHE) highlighted. (f) Temperature dependence of anomalous Hall resistivity ρAHE (left), and magnetoresistance, MR (right), as a function of H. \n 9 value of x = 1/3. To facilitate comparison with the magnetotransport behavior of bulk crystals, all stoichiometries given for thin samples in this paper are converted to xb.  Temperature-dependent longitudinal resistance measurements revealed a phase transition upon cooling. An inflection in ρxx and corresponding peak in dρxx/dT at 31 K indicate that the sample underwent a phase transition at this temperature (Figure 2d).42 To probe the nature of this low-temperature phase, we measured the field-dependent Hall resistivity, ρxy, at 1.8 K as shown in Figure 2e. This Hall resistivity signal can be related to the magnetization (M) of the sample through the following expression: 43–45 𝜌#$=1𝑛𝑒𝜇%𝐻+𝜇%𝑅&\t𝑀 (2) where n is the carrier density, e is the elementary charge, µ0 is the vacuum permeability, H is the applied field, and Rs is the anomalous Hall coefficient. The first term of Equation (2), 𝜇%𝐻/𝑛𝑒, is referred to as the ordinary Hall effect (OHE), while the second term, 𝜇%𝑅&\t𝑀, is the anomalous Hall resistivity, ρAHE. Analysis of the former (See SI for details) uncovers that the major charge carriers in this 2D crystal are holes, with a carrier density of 2.0 × 1021 cm–3 (2.0 × 1027 m–3) at 1.8 K. These results, which are in agreement with measurements of bulk crystals (SI Figure 16), reveal the presence of hole pockets at the Fermi level after intercalation.13,41,43 Although the OHE reveals valuable information about the basic electronic structure of our materials, it does not convey any information on magnetism. Insight into the long-range magnetic ordering of few-layer FexTaS2 was obtained by extracting the magnetization-dependent ρAHE as well as measuring the magnetoresistance (MR), which reflects the evolution of scattering interactions between charge carriers and magnetic moments. The hysteretic behavior of ρAHE and MR displayed in Figure 2f reveal that the material is a hard ferromagnet with a very large coercive field (Hc) that reaches 2.0 ± 0.1 T at 1.8 K. To the best of our knowledge, this is the largest Hc value recorded among 2D ferromagnetic crystals to-date.42,46–50 The broadness of ρAHE and MR switching profiles is consistent with the presence of inhomogeneous MEIs arising from some magneto-structural disorder. This behavior is reminiscent of off-stoichiometric bulk FexTaS2, in which crystallographic defects (Fe vacancies or interstitials) weaken the coupling between neighboring Fe centers.20,31 The resulting inhomogeneous coupling of magnetic moments across the crystal  10 broadens the hysteresis curves and may engender MR through spin-disorder scattering.20 Indeed, we find that the maximum value of MR decreases as the temperature approaches the Curie temperature (TC), which is estimated to be about 31 K from the inflection in ρxx(T) shown in Figure 2d. This observed change in MR with temperature is consistent with a spin-disorder scattering mechanism.20 Interestingly, while we might expect fully closed hysteresis loops above this ordering temperature, Figure 2f instead shows that both MR and ρAHE traces retain their hysteretic behavior up to 40 K, consistent with the persistence of some long-range magnetic order above 31 K. This discrepancy in the apparent TC prompted us to further examine the magnetic ordering temperature.  Magnetic phase transitions and magnetocrystalline anisotropy (MCA) in thin FexTaS2 By fitting the temperature dependence of the remanent ρAHE signal to the power-law form 𝛼(1–𝑇/𝑇')(, it is possible to extract both the TC and the critical exponent (β) for the magnetic material (see SI for details).42 β provides insight into the nature of the magnetic phase transition in the crystal and enables us to identify its universality class, which characterizes the dimensionality of our system, the symmetry of its magnetization, and the range of spin–spin interactions.51,52 This analysis and the discontinuity in ρAHE(T) around 31 K before full hysteresis closure (Figure 3a), suggest the presence of two magnetic phase transitions at about 31 K and 41 K, which we discuss in detail later. The critical exponent \n Figure 3. (a) Temperature dependence of the remanent 𝜌!\"# for the 11-layer device of FexTaS2. Ferromagnetic (FM) and paramagnetic (PM) phases are highlighted. Dashed lines represent the least squares fit to the formula 𝛼(1–𝑇/𝑇$)%. (b) MR of the device with the external field parallel and perpendicular to the c-axis.  \n 11 of the low-temperature magnetic phase, β = 0.31 ± 0.04, closely matches the theoretical value for the 3D Ising model β = 0.325,53 which describes systems with uniaxial MCA. To experimentally examine the MCA, we compare the MR curves with the external field applied parallel to the c-axis (H || c) with those in which the external field is applied along the ab-plane (H || ab) in Figure 3b. The MR profile is hysteretic only for H || c, consistent with an easy-axis along c, and a strong out-of-plane MCA in few-layer FexTaS2.  Magnetic behavior, transport, and Fe disorder in the ultrathin limit of FexTaS2. The large MCA in FexTaS2 prepared by chemical intercalation raised the possibility of anisotropy-stabilized long-range ferromagnetism in ultrathin FexTaS2 with even fewer layers. Hysteretic magnetotransport features of intercalated 2-, 3-, and 5-layer TaS2 flakes (see SI Figure 4) confirm that hard ferromagnetism is retained down to the thinnest limit of FexTaS2 (Figure 4a). All measured devices (Figure 4b) exhibit hysteretic MR profiles only for H || c, confirming their large out-of-plane MCA (SI Figure 21). This observation of ferromagnetism and large MCA in few-layer Fe intercalated TaS2 is supported by Density Functional Theory (DFT) calculations (see SI for details). In these computations, we considered four √3a × √3a superlattices of FexTaS2 with L = 2, 3, 4, 5 TaS2 layers (see SI Figure 23). For all L-layer systems we find a high-spin ferromagnetic order with calculated Fe magnetic moments of 3.3–3.5 µB per site. For all calculated heterostructures and magnetic configurations, we find that Fe converges to a d6 manifold (i.e. an Fe2+ oxidation state), which aligns with literature on bulk FexTaS2. Consistent with our experiments, we find the easy-axis to be out-of-plane with a calculated MCA in the 5-layer case of 6 meV per formula unit (1.5 meV/Fe). The origin of this anisotropy is the large out-of-plane orbital moment, which was previously calculated to be of 1.0 µB/Fe in the bulk case.13 Notably, the large MCA in our crystals is expected to dominate over shape anisotropy, which is an important factor for thin films with smaller MCA.54  Additional magnetotransport measurements also reveal that these thin crystals have high carrier densities and hole carriers (SI Figure 16). However, despite these similarities, their magnetic behaviors are substantially different. The 5-layer Fe0.38(2)TaS2 crystal displays only slightly broadened ρxy and MR hysteresis curves, suggesting a small degree of inhomogeneous MEIs in this material. In contrast, the 3- 12 layer sample, Fe0.37(2)TaS2, which possesses a comparable stoichiometry to the 5L crystal, exhibits  substantially broader magnetization switching. These data indicate that stoichiometry may not be the primary determinant of magnetic order in 2D FexTaS2. Instead, dimensionality may play the dominant role in shaping magnetic properties. Indeed, the ρxy and MR features found in measurements of the 2-layer crystal, Fe0.58(14)TaS2 are comparable to those of the 3-layer crystal, despite a significant difference in stoichiometry. This strong influence of dimensionality on magnetic order may stem from its impact on intercalant ordering via electronic and/or structural effects. For example, measurements of bulk crystals of \na related system, FexTiS2, reveal that the ordering of intercalants in the plane is sensitive to the structure of Fe intercalants along the c-axis.55–57 By extension, we suggest that decreasing numbers of layers could result in a reduced tendency for ordering Fe centers in-plane. Moreover, in the low-dimensional limit, intercalation may produce distinct Fe intercalant structural phases that may not be present in bulk crystals,  Figure 4. (a) Magnetotransport measurements, (b) optical micrographs with heterostructure schematics, (c) cumulative Raman spectra, (d) temperature-dependent longitudinal resistivities and their first derivatives with respect to temperature and (e) temperature-dependent coercivities for the 5-, 3- and 2-layer devices. In (c), the Raman data is normalized to the A1/A1g peak. In (d), (𝜌(𝑇)/𝜌(300))/𝑑𝑇 maxima are marked on the 𝜌(𝑇)/𝜌(300) curves for the 3-layer and 5-layer devices. \n 13 a behavior that can also be theoretically understood from the Mermin–Wagner framework as applied to 2D crystallization processes.58  To evaluate the extent of this intercalant ordering in these ultrathin intercalation compounds, we used confocal Raman spectroscopy to probe the Fe superlattice modes in the intercalated 5-, 3- and 2-layer crystals. Figure 4c shows that the superlattice structures and order in these few-layer devices indeed differ appreciably. We find that the 5-layer sample is mostly ordered in the √3a × √3a superstructure, with a hallmark sharp and intense Fe* feature close to 140 cm–1.32 In the 3-layer device, the characteristic √3a × √3a and 2a × 2a superstructure modes are broader and closer in intensity, consistent with increased disorder and the presence of both superlattices, notwithstanding an effectively identical composition to the 5-layer crystal. Finally, the bilayer sample exhibits a substantially broad feature in the low-frequency region, pointing to a high level of intercalant disorder. Although sharp Fe-related modes are absent in this sample, the red-shifted A1 mode reveals that the TaS2 lattice has been electron-doped36,37 and STEM-EDS confirms the substantial intercalation of Fe. Taken together, we suggest that structural disorder, which is encouraged by reduced dimensionality, likely promotes the observed broadened magnetic responses in ultrathin FexTaS2. Traditionally considered an undesirable property of crystals, intercalant disorder plays a key, useful role in defining and manipulating the magnetic properties of magnetic intercalation compounds. In FexNbS2 crystals with x near 1/3, disorder results in a spin-glass phase that coexists with the antiferromagnetic state, and this disorder-induced spin glass enables the switching of antiferromagnetic domain orientations and electronic conductivity with small electrical pulses.59–63 In bulk FexTaS2 crystals, disorder has been demonstrated to increase ferromagnetic hardness, as defects can pin magnetic domains and increase the coercivity.64 Congruently, apparent intercalant disorder in our bilayer and 3-layer FexTaS2 crystals may, together with intrinsically large MCA, be one origin for the observed giant coercivities reaching 2.9 ± 0.3 T and 2.1 ± 0.4 T, respectively (Figure 4a). In contrast, the coercivity of the more ordered 5-layer FexTaS2 is lower, amounting to 1.1 ± 0.3 T at 1.8 K.  Notwithstanding the very large coercivities measured for these few-layer magnetic crystals, it is noteworthy that these coercive fields are slightly lower than those measured for analogous bulk FexTaS2.31  14 These discrepancies in Hc could suggest that the overall strength of MEIs may decrease in the few-layer limit. However, our analysis of ordering temperatures in these few-layer crystals reveals that the overall MEIs, while perhaps slightly weaker, may be quite comparable to those found in bulk crystals of similar composition. This insight was gleaned from a consideration of their Curie temperatures, determined from the temperature dependence of their longitudinal resistivities (Figure 4d) and coercivities (Figure 4e). Based on the inflections in ρxx and dρxx/dT shown in Figure 4d, the 5- and 3- layer crystals magnetically order below 25 K and 32 K, respectively. However, as we discuss in more detail below, the closure of their hysteresis, i.e., onset of Hc = 0, does not coincide with the inflections in ρxx (Figure 4e, SI Figure 18,19). For the 5-layer crystal, the magnetic hysteresis instead closes between 40 K and 60 K, about 20 K higher than the temperature of inflection in ρxx. The hysteresis observed in the 3-layer crystal instead closes below the ρxx inflection temperature (between 10 and 20 K), as observed in bulk Fe1/3TaS2.41 For the bilayer crystal, Figure 4d also shows that decreasing temperatures do not produce the characteristic sharp drop in ρxx below a magnetic ordering temperature, as observed in crystals with L ≥ 3 layers. Instead, ρxx exhibits an upturn in resistivity for T < 100 K to reach a low-temperature resistivity greater than the ρxx value measured at room temperature. This upturn in ρxx has not been observed in bulk crystals of FexTaS2 and may be a result of Kondo-like scattering or weak localization52,65–68 (SI Figure 22, SI Section 4.6), both of which can result in an increase in resistivity with decreasing temperature in metals.52 Because ρxx(T) for the bilayer sample does not exhibit any features typically attributed to a magnetic phase transition, we estimate TC as between 20 and 25 K based on the disappearance of hysteresis in ρxx over this temperature range (Figure 4e, SI Figure 17). In summary, these measured ordering temperatures for few-layer FexTaS2 are comparable to the TC for x = 0.33 bulk crystals (around 38 K),31,41 signifying that it is likely the intraplanar MEIs, which are retained in the 2D limit, that are the dominant coupling interactions in FexTaS2 flakes. Co-existing magnetic phases in few-layer FexTaS2 As described above, in all measured few-layer samples, there exist discrepancies between ordering temperatures determined from inflections in ρxx(T) and those measured from the closure of magnetic  15 hysteresis. Moreover, the temperature dependence of remanent ρAHE for the 5-layer FexTaS2 (SI Figure 20) appears to display more than one magnetic transition. This behavior, which was also observed for the 11-layer crystal (Figure 3a), again points to the presence of multiple co-existing ferromagnetic phases. Although one possible origin is superlattice inhomogeneity/disorder, these crystals display Raman spectra (Figures 2b, 4c) that are consistent with an overwhelmingly dominant √3a × √3a Fe superlattice. We therefore interrogated other possible origins of these magnetic transitions using DFT computations. Interestingly, in addition to the high-spin ground state, we also find metastable zero-spin and intermediate-spin solutions in our DFT calculations. For the 2-layer case, we find the zero-spin state is 461 meV per formula higher in energy than the high-spin (3.5 µB/Fe) case. We also find an intermediate spin state with 2.1 µB/Fe that is 273 meV per formula unit higher in energy than the high-spin case. In both cases, the magnetic moment reduction occurs with a decrease in the van-der-Waals gap—the interplanar S–S distance changes from 2.89 Å (high spin) to 2.83 Å (intermediate spin) to 2.56 Å (zero spin), with a corresponding reduction in Fe–S bond lengths. Along with this reduction in Fe moment and interlayer separation in going from the high- to intermediate-spin cases, we find that the Ta projected moment significantly increases from 0.03 µB per Ta in the high-spin case to 0.15 µB per Ta in the intermediate-spin case for those two Ta closest to the Fe. These Ta moments are antiferromagnetically coupled to the Fe moments. Our calculations suggest that the Ta acquires this net moment because of charge redistribution to the neighboring Ta sites with the reduction of the interlayer distance. We observe similar trends for the full L-layer series studied with changes in Fe magnetic moments resulting in modification of the interlayer separation and charge transfer to the neighboring Ta sites. We note, however, that in all cases we find a high-spin ground state.  These theoretical results point to other possible origins of the multiple magnetic phases observed in experiments, independent of intercalant/superlattice disorder. Firstly, we identify metastable spin states in our DFT calculations that are strongly dependent on the interlayer separation and the resulting Fe local environment. This strong structural dependence on the total magnetization could give rise to spin state transitions upon thermal lattice contraction. Secondly, we find that the Ta atoms adjacent to our Fe intercalants acquire a non-negligible moment (0.15 µB/Ta in the 2-layer case) depending on the interlayer separation. This moment on Ta could also result in a separate magnetic ordering temperature. Future  16 experimental and theoretical studies on 2D FexTaS2 spanning a greater range of stoichiometries, and degrees of intercalant ordering may reveal additional details on how magneto-structural disorder and lowered dimensionality affect the value of TC and competing long-range magnetic orders.  CONCLUSIONS We have demonstrated that long-range ferromagnetic order persists down to the thinnest limit of the magnetic intercalation compound FexTaS2. Combined structural, spectroscopic, and magnetoelectronic characterization indicates that both intercalant order/disorder and stoichiometry influence the magnetic properties of these low-dimensional materials. Nevertheless, overall, the high MCA expected from the coordination environment imposed by the vdW interface between TaS2 sheets appears to stabilize long-range, hard ferromagnetic ordering in samples of all thicknesses. More broadly, our results demonstrate a versatile platform for developing 2D magnets with magnetic properties that can be manipulated by modulating their intercalation amount, dimensionality, and structural homogeneity. This work opens the door to modulating the MCA and MEIs of intercalated TMDs by marrying the synthetic versatility of soft chemical intercalation23–27 with distinctive techniques developed for fabricating vdW architectures and controlling the emergent physics in atomically layered materials, such as layer count,2,69 deterministic assembly of vdW heterostructures,70,71 electrostatic gating,4,5,72 strain73,74 and the introduction of an interlayer twist.75,76 Exploring this constellation of tuning knobs is poised to produce novel 2D intercalation compounds that enable the exploration of exotic emergent magnetic phases as well as materials for low energy switching of magnetism. ASSOCIATED CONTENT Supporting information The Supporting information is available free of charge on the ACS Publications Website at: https://pubs.acs.org/doi/suppl/10.1021/jacs.2c02885/suppl_file/ja2c02885_si_001.pdf    17 Notes The authors declare no competing financial interest. ACKNOWLEDGEMENTS  The authors thank Oscar Gonzalez, Zhizhi Kong, Lilia Xie, and Steven Zeltmann for helpful discussions. This material is based upon work supported by the Air Force Office of Scientific Research under AFOSR Award No. FA9550-20-1-0007. We would like to thank Gatan, Inc. as well as P Denes, A Minor, J Ciston, C Ophus, J Joseph, and Ian Johnson at LBNL who contributed to the development of the 4D Camera used for STEM-DPC measurements. This research used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility located at Lawrence Berkeley National Laboratory, operated under Contract No. DE-AC02-05CH11231 using NERSC award ERCAP0020898 and ERCAP0020897. M.V.W. acknowledges support from a UCB Chancellor’s Fellowship and NSF Graduate Research Fellowship. Instrumentation used in this work was supported by grants from the W.M. Keck Foundation (Award # 993922), the Canadian Institute for Advanced Research (CIFAR–Azrieli Global Scholar, Award # GS21-011), the Gordon and Betty Moore Foundation EPiQS Initiative (Award #10637), and the 3M Foundation through the 3M Non-Tenured Faculty Award (#67507585). N.P.K. acknowledges support from the Hertz Fellowship and from the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1745301. Confocal Raman spectroscopy was supported by a DURIP grant through the Office of Naval Research under Award No. N00014-20-1-2599 (D.K.B.). Experimental and theoretical work at the Molecular Foundry, LBNL was supported by the Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. Computational works were carried out using supercomputing resources of the National Energy Research Scientific Computing Center (NERSC) and the TMF clusters managed by the High Performance Computing Services Group, at LBNL. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan, Grant Number JPMXP0112101001, JSPS KAKENHI Grant Number JP20H00354 and the CREST (JPMJCR15F3), JST.       18 ABBREVIATIONS MCA, magnetocrystalline anisotropy; OAM, orbital angular momentum; MEI, magnetic exchange interaction  REFERENCE (1) Gong, C.; Li, L.; Li, Z.; Ji, H.; Stern, A.; Xia, Y.; Cao, T.; Bao, W.; Wang, C.; Wang, Y.; Qiu, Z. Q.; Cava, R. J.; Louie, S. G.; Xia, J.; Zhang, X. Discovery of intrinsic ferromagnetism in two-dimensional van der Waals crystals. Nature 2017, 546, 265–269. (2) Huang, B.; Clark, G.; Navarro-Moratalla, E.; Klein, D. R.; Cheng, R.; Seyler, K. L.; Zhong, D.; Schjhmidgall, E.; McGuire, M. A.; Cobden, D. H.; Yao, W.; Xiao, D.; J.                                                                                                                                                                                                                                                                       hjjarillo-Herrero, P.; Xu, X. Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit. Nature 2017, 270–273. (3)  Deng, Y.; Yu, Y.; Song, Y.; Zhang, J.; Wang, N. Z.; Sun, Z.; Yi, Y.; Wu, Y. Z.; Wu, S.; Zhu, J.; Wang, J.; Chen, X. H.; Zhang, Y. Gate-tunable room-temperature ferromagnetism in two-dimensional Fe3GeTe2. Nature 2018, 563, 94–99. (4) Jiang, S.; Shan, J.; Mak, K. F. Electric-Field Switching of Two-Dimensional van Der Waals Magnets. Nat. Mater. 2018, 17, 406–410. (5) Huang, B.; Clark, G.; Klein, D. R.; MacNeill, D.; Navarro-Moratalla, E.; Seyler, K. L.; Wilson, N.; McGuire, M. A.; Cobden, D. H.; Xiao, D.; et al. Electrical Control of 2D Magnetism in Bilayer CrI3. Nat. Nanotechnol. 2018, 13, 544–548. (6) Jiang, S.; Li, L.; Wang, Z.; Mak, K. F.; Shan, J. Controlling Magnetism in 2D CrI3 by Electrostatic Doping. Nat. Nanotechnol. 2018, 13, 549–553. (7) Burch, K. S. Electric Switching of Magnetism in 2D. Nat. Nanotechnol. 2018, 13, 532. (8) Mermin, N. D.; Wagner, H. Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models. Phys. Rev. Lett. 1966, 17, 1133–1136.  (9) Halperin, B. I. On the Hohenberg-Mermin-Wagner theorem and its limitations J. Stat. Phys. 2019, 175, 521–529. (10)   Gibertini, M.; Koperski, M.; Morpurgo, A. F.; Novoselov, K. S. Magnetic 2D Materials and Heterostructures. Nat. Nanotechnol. 2019, 14, 408–419.   19  (11)  Gong, C.; Zhang, X. Two-Dimensional Magnetic Crystals and Emergent Heterostructure Devices. Science (80-. ). 2019, 363, eaav4450. (12) Zhang, S.; Xu, R.; Luo, N.; Zou, X. Two-Dimensional Magnetic Materials: Structures, Properties and External Controls. Nanoscale 2021, 13, 1398–1424. (13) Ko, K. T.; Kim, K.; Kim, S. B.; Kim, H. D.; Kim, J. Y.; Min, B. I.; Park, J. H.; Chang, F. H.; Lin, H. J.; Tanaka, A.; Cheong, S. W. RKKY Frromagnetism with Ising-like Spin States in Intercalated Fe1/4TaS2. Phys. Rev. Lett. 2011, 107, 247201.  (14) Dijkstra, J.; Zijlema, P. J.; van Bruggen, C. F.; Haas, C.; de Groot, R. A. Band-Structure Calculations of Fe1/3TaS2and Mn1/3TaS2, and Transport and Magnetic Properties of Fe0.28TaS2. J. Phys. Condens. Matter 1989, 1, 6363–6379. (15) Choe, J.; Lee, K.; Huang, C.-L.; Trivedi, N.; Morosan, E. Magnetotransport in Fe-Intercalated TS2: Comparison between Ti and Ta. Phys. Rev. B 2019, 99, 64420.  (16) Xie, L.; Husremovic, S.; Gonzalez, O.; Craig, I. M.; Bediako, D. K. J. Am. Chem. Soc. 2022. https://pubs.acs.org/doi/full/10.1021/jacs.1c12975. (17) Sirica, N.; Vilmercati, P.; Bondino, F.; Pis, I.; Nappini, S.; Mo, S.-K.; Fedorov, A. V; Das, P. K.; Vobornik, I.; Fujii, J.; Li, L.; Sapkota, D.; Parker, D. S.; Mandrus, D. G.; Mannella, N. The Nature of Ferromagnetism in the Chiral Helimagnet Cr1/3NbS2. Commun. Phys. 2020, 3, 65. (18) Obeysekera, D.; Gamage, K.; Gao, Y.; Cheong, S. wook; Yang, J. The Magneto-Transport Properties of Cr1/3TaS2 with Chiral Magnetic Solitons. Adv. Electron. Mater. 2021, 7, 1–8.  (19)  Cai, R.; Xing, W.; Zhou, H.; Li, B.; Chen, Y.; Yao, Y.; Ma, Y.; Xie, X. C.; Jia, S.; Han, W. Anomalous Hall Effect Mechanisms in the Quasi-Two-Dimensional van Der Waals Ferromagnet Fe0.29TaS2. Phys. Rev. B 2019, 100, 54430. (20)  Hardy, W. J.; Chen, C.-W.; Marcinkova, A.; Ji, H.; Sinova, J.; Natelson, D.; Morosan, E. Very Large Magnetoresistance in Fe0.28TaS2 Single Crystals. Phys. Rev. B 2015, 91, 054426.  (21)  Zheng, G.; Wang, M.; Zhu, X.; Tan, C.; Wang, J.; Albarakati, S.; Aloufi, N.; Algarni, M.; Farrar, L.; Wu, M.; Yao, Y.; Tian, M.; Zhou, J.; Wang, L. Tailoring Dzyaloshinskii–Moriya Interaction in a Transition Metal Dichalcogenide by Dual-Intercalation. Nat. Commun. 2021, 12, 3639.  (22)  Su, J.; Wang, M.; Liu, G.; Li, H.; Han, J.; Zhai, T. Air-Stable 2D Intrinsic Ferromagnetic Ta3FeS6 with Four Months Durability. Adv. Sci. 2020, 7, 2001722. (23)  Chen, K. P.; Chung, F. R.; Wang, M.; Koski, K. J. Dual Element Intercalation into 2D Layered Bi2Se3 Nanoribbons. J. Am. Chem. Soc. 2015, 137, 5431–5437.   20  (24)  Koski, K. J.; Wessells, C. D.; Reed, B. W.; Cha, J. J.; Kong, D.; Cui, Y. Chemical Intercalation of Zerovalent Metals into 2D Layered Bi2Se3 Nanoribbons. J. Am. Chem. Soc. 2012, 134, 13773–13779. (25)  Wang, M.; Williams, D.; Lahti, G.; Teshima, S.; Dominguez Aguilar, D.; Perry, R.; Koski, K. J. Chemical Intercalation of Heavy Metal, Semimetal, and Semiconductor Atoms into 2D Layered Chalcogenides. 2D Mater. 2018, 5, 45005. (26)  Motter, J. P.; Koski, K. J.; Cui, Y. General Strategy for Zero-Valent Intercalation into Two-Dimensional Layered Nanomaterials. Chem. Mater. 2014, 26, 2313–2317. (27)  Koski, K. J.; Cha, J. J.; Reed, B. W.; Wessells, C. D.; Kong, D.; Cui, Y. High-Density Chemical Intercalation of Zero-Valent Copper into Bi2Se3 Nanoribbons. J. Am. Chem. Soc. 2012, 134, 7584–7587. (28)  Navarro-Moratalla, E.; Island, J. O.; Mañas-Valero, S.; Pinilla-Cienfuegos, E.; Castellanos-Gomez, A.; Quereda, J.; Rubio-Bollinger, G.; Chirolli, L.; Silva-Guillén, J. A.; Agraït, N.; et al. Enhanced Superconductivity in Atomically Thin TaS2. Nat. Commun. 2016, 7, 11043. (29)  Morosan, E.; Zandbergen, H. W.; Li, L.; Lee, M.; Checkelsky, J. G.; Heinrich, M.; Siegrist, T.; Ong, N. P.; Cava, R. J. Sharp Switching of the Magnetization in Fe1/4TaS2. Phys. Rev. B. 2007, 75, 104401. (30)  Wijngaard, J. H.; Hass, C.; Devillers, M. A. C. Optical and Magneto-Optical Properties of Fe0.28TaS2. J. Phys. Condens. Matter. 1991, 3, 6913–6923. (31)  Chen, C. W.; Chikara, S.; Zapf, V. S.; Morosan, E. Correlations of Crystallographic Defects and Anisotropy with Magnetotransport Properties in FexTaS2 Single Crystals (0.23 ≤ x ≤ 0.35). Phys. Rev. B 2016, 94, 54406. (32)  Fan, S.; Neal, S.; Won, C.; Kim, J.; Sapkota, D.; Huang, F.; Yang, J.; Mandrus, D. G.; Cheong, S.-W.; Haraldsen, J. T.; Musfeldt, J. L. Excitations of Intercalated Metal Monolayers in Transition Metal Dichalcogenides. Nano Lett. 2021, 21, 99-106. (33)  Horibe, Y.; Yang, J.; Cho, Y.-H.; Luo, X.; Kim, S. B.; Oh, Y. S.; Huang, F.-T.; Asada, T.; Tanimura, M.; Jeong, D.; Cheong, S.-W. Color Theorems, Chiral Domain Topology, and Magnetic Properties of FexTaS2. J. Am. Chem. Soc. 2014, 136, 8368–8373. (34) Navarro-Moratalla, E.; Island, J. O.; Mañas-Valero, S.; Pinilla-Cienfuegos, E.; Castellanos-Gomez, A.; Quereda, J.; Rubio-Bollinger, G.; Chirolli, L.; Silva-Guillén, J. A.; Agraït, N.; Steele, G. A.; Guinea, F.; van der Zant, H. S. J.; Coronado, E. Enhanced Superconductivity in Atomically Thin   21  TaS2. Nat. Commun. 2016, 7, 11043. (35) Lin, D.; Li, S.; Wen, J.; Berger, H.; Forró, L.; Zhou, H.; Jia, S.; Taniguchi, T.; Watanabe, K.; Xi, X.; Bahramy, M. S. Patterns and Driving Forces of Dimensionality-Dependent Charge Density Waves in 2H-Type Transition Metal Dichalcogenides. Nat. Commun. 2020, 11, 2406. (36) Valencia-Ibáñez, S.; Jimenez-Guerrero, D.; Salcedo-Pimienta, J. D.; Vega-Bustos, K.; Cárdenas-Chirivi, G.; López-Manrique, L.; Galvis, J. A.; Hernandez, Y.; Giraldo-Gallo, P. Raman Spectroscopy of Few-Layers TaS2 and Mo-Doped TaS2 with Enhanced Superconductivity. 2022-01-24. arXiv. https://doi.org/10.48550/ARXIV.2201.09840. (accessed 2022-02-01). (37) Su, T.; Huang, J.; Wang, Q.; Zhang, X.; Zhou, L.; Tang, M.; Zhang, C.; Yuan, H.; Zhao, W.; Wang, Z.; Yuan, H.; Wang, X. Investigation of Phonon Modes in 2H-TaX2 (X = S/Se) Flakes with Electrostatic Doping. J. Appl. Phys. 2021, 130, 105302.  (38) Narita, H.; Ikuta, H.; Hinode, H.; Uchida, T.; Ohtani, T.; Wakihara, M. Preparation and Physical Properties of FexTaS2 (0.15 ≤ x ≤ 0.50) Compounds. Journal of Solid State Chemistry. 1994, pp 148–151. (39) Waszczak, J. V.; DiSalvo, F. J.; Eibschütz, M.; Mahajan, S.; Hull, G. W. Ferromagnetism in Metallic Intercalated Compounds FexTaS2 (0.20 ⩽ x ⩽ 0.34). J. Appl. Phys. 2003, 52, 2098–2100. (40) Eibschütz, M.; Disalvo, F. J.; Hull, G. W.; Mahajan, S. Ferromagnetism in Metallic FexTaS2 (x≈0.28). Appl. Phys. Lett. 1975, 27, 464–466. (41) Mangelsen, S.; Hansen, J.; Adler, P.; Schnelle, W.; Bensch, W.; Mankovsky, S.; Polesya, S.; Ebert, H. Large Anomalous Hall Effect and Slow Relaxation of the Magnetization in Fe1/3TaS2. J. Phys. Chem. C 2020, 124, 24984–24994. (42)  Fei, Z.; Huang, B.; Malinowski, P.; Wang, W.; Song, T.; Sanchez, J.; Yao, W.; Xiao, D.; Zhu, X.; May, A. F.; Wu, W.; Cobden, D. H.; Chu, J.-H.; Xu, X. Two-Dimensional Itinerant Ferromagnetism in Atomically Thin Fe3GeTe2. Nat. Mater. 2018, 17, 778–782. (43)  Pugh, E. M.; Lippert, T. W. Hall e.m.f. and Intensity of Magnetization. Phys. Rev. 1932, 42, 709–713. (44)  Pugh, E. M. Hall Effect and the Magnetic Properties of Some Ferromagnetic Materials. Phys. Rev. 1930, 36, 1503–1511.  (45)  Checkelsky, J. G.; Lee, M.; Morosan, E.; Cava, R. J.; Ong, N. P. Anomalous Hall Effect and Magnetoresistance in the Layered Ferromagnet Fe1/4TaS2: The Inelastic Regime. Phys. Rev. B 2008, 77, 14433.    22  (46) May, A. F.; Ovchinnikov, D.; Zheng, Q.; Hermann, R.; Calder, S.; Huang, B.; Fei, Z.; Liu, Y.; Xu, X.; McGuire, M. A. Ferromagnetism Near Room Temperature in the Cleavable van Der Waals Crystal Fe5GeTe2. ACS Nano 2019, 13, 4436–4442. (47) Zhang, Y.; Chu, J.; Yin, L.; Shifa, T. A.; Cheng, Z.; Cheng, R.; Wang, F.; Wen, Y.; Zhan, X.; Wang, Z.; et al. Ultrathin Magnetic 2D Single-Crystal CrSe. Adv. Mater. 2019, 31, 1900056. (48) Bonilla, M.; Kolekar, S.; Ma, Y.; Diaz, H. C.; Kalappattil, V.; Das, R.; Eggers, T.; Gutierrez, H. R.; Phan, M.-H.; Batzill, M. Strong Room-Temperature Ferromagnetism in VSe2 Monolayers on van Der Waals Substrates. Nat. Nanotechnol. 2018, 13, 289–293. (49) O’Hara, D. J.; Zhu, T.; Trout, A. H.; Ahmed, A. S.; Luo, Y. K.; Lee, C. H.; Brenner, M. R.; Rajan, S.; Gupta, J. A.; McComb, D. W.; et al. Room Temperature Intrinsic Ferromagnetism in Epitaxial Manganese Selenide Films in the Monolayer Limit. Nano Lett. 2018, 18, 3125–3131. (50) Kim, M.; Kumaravadivel, P.; Birkbeck, J.; Kuang, W.; Xu, S. G.; Hopkinson, D. G.; Knolle, J.; McClarty, P. A.; Berdyugin, A. I.; Ben Shalom, M.; et al. Micromagnetometry of Two-Dimensional Ferromagnets. Nat. Electron. 2019, 2, 457–463. (51)  Pathria, R. K.; Beale, P. D. 12 - Phase Transitions: Criticality, Universality, and Scaling. In Statistical Mechanics (Third Edition); Pathria, R. K., Beale, P. D., Eds.; Academic Press: Boston, 2011; pp 401–469. (52)  Blundell, S. Magnetism in Condensed Matter. Oxford University Press: Oxford; New York 2001; pp 119-120.  (53)  Fisher, M. E.; Ma, S.; Nickel, B. G. Critical Exponents for Long-Range Interactions. Phys. Rev. Lett. 1972, 29, 917–920. (54)  Spaldin, N. A. Magnetic Materials: Fundamentals and Applications; Cambridge University Press, 2010; pp 153. (55)  Chiew, Y. L.; Miyata, M.; Koyano, M.; Oshima, Y. Clarification of the Ordering of Intercalated Fe Atoms in FexTiS2 and Its Effect on the Magnetic Properties. Acta Crystallogr. Sect. B 2021, 77, 441–448. (56)  Negishi, S.; Negishi, H.; Sasaki, M.; Inoue, M. Montecarlo Simulation of Intercalated Atomic Distributions in Layered Dichalcogenide. Mol. Cryst. Liq. Cryst. Sci. Technol. Sect. A. Mol. Cryst. Liq. Cryst. 2000, 341, 69–74.   23  (57)  Negishi, S.; Negishi, H.; Sasaki, M.; Inoue, M. Simulation of Atomic Distribution of Guest Atoms in Layered 1T-TiS 2 Crystal Using Monte Calro Method and Its Effect on the Magnetic Properties and Local Structures. J. Phys. Soc. Japan 2000, 69, 2514–2530. (58)  Illing, B.; Fritschi, S.; Kaiser, H.; Klix, C. L.; Maret, G.; Keim, P. Mermin-Wagner Fluctuations in 2D Amorphous Solids. Proc. Natl. Acad. Sci. U.S.A. 2017, 114, 1856–1861. (59)  Haley, S. C.; Maniv, E.; Cookmeyer, T.; Torres-Londono, S.; Aravinth, M.; Moore, J.; Analytis, J. G. Long-Range, Non-Local Switching of Spin Textures in a Frustrated Antiferromagnet. 2021-11-18. arXiv preprint, arXiv:2111.09882. (accessed 2021-12-03). (60)  Wu, S.; Xu, Z.; Haley, S. C.; Weber, S. F.; Acharya, A.; Maniv, E.; Qiu, Y.; Aczel, A. A.; Settineri, N. S.; Neaton, J. B.; Analytis, J. G.; Birgeneau, R. J. Highly Tunable Magnetic Phases in Transition-Metal Dichalcogenide Fe1/3+δNbS2. Phys. Rev. X 2022, 12, 21003. (61)  Nair, N. L.; Maniv, E.; John, C.; Doyle, S.; Orenstein, J.; Analytis, J. G. Electrical Switching in a Magnetically Intercalated Transition Metal Dichalcogenide. Nat. Mater. 2020, 19 , 153–157.  (62)  Maniv, E.; Nair, N. L.; Haley, S. C.; Doyle, S.; John, C.; Cabrini, S.; Maniv, A.; Ramakrishna, S. K.; Tang, Y.-L.; Ercius, P.; Ramesh, R.; Tserkovnyak, Y.; Reyes, A. P.; Analytis, J. G. Antiferromagnetic Switching Driven by the Collective Dynamics of a Coexisting Spin Glass. Sci. Adv. 2021, 7, eabd8452. (63)  Little, A.; Lee, C.; John, C.; Doyle, S.; Maniv, E.; Nair, N. L.; Chen, W.; Rees, D.; Venderbos, J. W. F.; Fernandes, R. M.; Analytis, J. G.; Orenstein, J. Three-State Nematicity in the Triangular Lattice Antiferromagnet Fe1/3NbS2. Nat. Mater. 2020, 19, 1062–1067.  (64)  Choi, Y. J.; Kim, S. B.; Asada, T.; Park, S.; Wu, W.; Horibe, Y.; Cheong, S. W. Giant Magnetic Coercivity and Ionic Superlattice Nano-Domains in Fe0.25TaS2. Epl 2009, 86, 0–5. (65)  Amaladass, E. P.; Satya, A. T.; Sharma, S.; Vinod, K.; Srinivas, V.; Sundar, C. S.; Bharathi, A. Magnetization and Magneto-Transport Studies on Fe2VAl1−xSix. J. Alloys Compd. 2015, 648, 34–38. (66)  Jobst, J.; Kisslinger, F.; Weber, H. B. Detection of the Kondo Effect in the Resistivity of Graphene: Artifacts and Strategies. Phys. Rev. B - Condens. Matter Mater. Phys. 2013, 88, 1–6.  (67)  Liu, H.; Xue, Y.; Shi, J. A.; Guzman, R. A.; Zhang, P.; Zhou, Z.; He, Y.; Bian, C.; Wu, L.; Ma, R.; Chen, J.; Yan, J.; Yang, H.; Shen, C. M.; Zhou, W.; Bao, L.; Gao, H. J. Observation of the Kondo Effect in Multilayer Single-Crystalline VTe2 Nanoplates. Nano Lett. 2019, 19, 8572–8580.   24  (68)  Zhao, M.; Chen, B.-B.; Xi, Y.; Zhao, Y.; Xu, H.; Zhang, H.; Cheng, N.; Feng, H.; Zhuang, J.; Pan, F.; Xu, X.; Hao, W.; Li, W.; Zhou, S.; Dou, S. X.; Du, Y. Kondo Holes in the Two-Dimensional Itinerant Ising Ferromagnet Fe3GeTe2. Nano Lett. 2021, 21, 6117–6123. (69)  Wen, Y.; Liu, Z.; Zhang, Y.; Xia, C.; Zhai, B.; Zhang, X.; Zhai, G.; Shen, C.; He, P.; Cheng, R.; Yin, L.; Yao, Y.; Getaye Sendeku, M.; Wang, Z.; Ye, X.; Liu, C.; Jiang, C.; Shan, C.; Long, Y.; He, J. Tunable Room-Temperature Ferromagnetism in Two-Dimensional Cr2Te3. Nano Lett. 2020, 20, 3130–3139. (70)  Zhu, R.; Zhang, W.; Shen, W.; Wong, P. K. J.; Wang, Q.; Liang, Q.; Tian, Z.; Zhai, Y.; Qiu, C.; Wee, A. T. S. Exchange Bias in van Der Waals CrCl3/Fe3GeTe2 Heterostructures. Nano Lett. 2020, 20, 5030–5035. (71)  Dolui, K.; Petrović, M. D.; Zollner, K.; Plecháč, P.; Fabian, J.; Nikolić, B. K. Proximity Spin–Orbit Torque on a Two-Dimensional Magnet within van Der Waals Heterostructure: Current-Driven Antiferromagnet-to-Ferromagnet Reversible Nonequilibrium Phase Transition in Bilayer CrI3. Nano Lett. 2020, 20, 2288–2295. (72)  Verzhbitskiy, I. A.; Kurebayashi, H.; Cheng, H.; Zhou, J.; Khan, S.; Feng, Y. P.; Eda, G. Controlling the Magnetic Anisotropy in Cr2Ge2Te6 by Electrostatic Gating. Nat. Electron. 2020, 3, 460–465. (73)  Wang, Y.; Wang, C.; Liang, S.-J.; Ma, Z.; Xu, K.; Liu, X.; Zhang, L.; Admasu, A. S.; Cheong, S.-W.; Wang, L.; Chen, M.; Liu, Z.; Cheng, B.; Ji, W.; Miao, F. Strain-Sensitive Magnetization Reversal of a van Der Waals Magnet. Adv. Mater. 2020, 32, e2004533.  (74)  Zhong, J.; Wang, M.; Liu, T.; Zhao, Y.; Xu, X.; Zhou, S.; Han, J.; Gan, L.; Zhai, T. Strain-Sensitive Ferromagnetic Two-Dimensional Cr2Te3. Nano Res. 2022, 15, 1254–1259. (75)  Xu, Y.; Ray, A.; Shao, Y.-T.; Jiang, S.; Lee, K.; Weber, D.; Goldberger, J. E.; Watanabe, K.; Taniguchi, T.; Muller, D. A.; Mak, K. F.; Shan, J. Coexisting Ferromagnetic–Antiferromagnetic State in Twisted Bilayer CrI3. Nat. Nanotechnol. 2022, 17, 143–147. (76)  Song, T.; Sun, Q.-C.; Anderson, E.; Wang, C.; Qian, J.; Taniguchi, T.; Watanabe, K.; McGuire, M. A.; Stöhr, R.; Xiao, D.; Cao, T.; Wrachtrup, J.; Xu, X. Direct Visualization of Magnetic Domains and Moiré Magnetism in Twisted 2D Magnets. Science (80-. ). 2021, 374, 1140–1144.  25  "    },    {        "title": "2203.11794v1.A_strategic_high_throughput_search_for_identifying_stable_Li_based_half_Heusler_alloys_for_spintronics_applications.pdf",        "content": "A strategic high throughput search for identifying stable\nLi based half Heusler alloys for spintronics applications\nRohit Pathak1, Parul R. Raghuvanshi, Amrita Bhattacharya\u0003\nAb initio Materials Simulation Laboratory, Department of metallurgical engineering and\nmaterials science, Indian Institute of Technology, Bombay, Powai-400076, Mumbai,\nMaharashtra, India.\nAbstract\nIn this work, high throughput DFT calculations are performed on the alkali\nmetal-based half Heusler alloys; LiY pY0\n1\u0000pS (Y, Y0= V, Cr, Mn, Fe, Co, Ni\nandp= 0, 0.25, 0.5, 0.75, 1). Starting with 243 structural replica, systematic\n\flters are designed to select the energetically and vibrationally favorable com-\npositions by considering the contributions stemming from the magnetic align-\nments of the ions. Thereby, 26 dynamically stable magnetic compositions are\nidenti\fed, of which 10 are found to be ferromagnetic (FM), 4 antiferromagnetic\n(AFM) and 12 ferrimagnetic (FiM). 4 FM and 8 FiM ones are found to show\n100 % spin polarization. Further, tetragonal distortion is found to be present\nin 4 FM, 3 FiM and 4 AFM compositions, which indicates the possibility of\neasy-axis magnetocrystalline anisotropy. The ferromagnetic LiFe 0:5Mn0:5S and\nantiferromagnetic LiFeS are found to have the most prominent easy-axis mag-\nnetocrystalline anisotropy.\nKeywords: Highthroughput DFT, Spintronics, Half Heusler alloys,\nHalfmetalicity, Perpendicular magnetic anisotropy.\n\u0003Corresponding author\nEmail address: b_amrita@iitb.ac.in (Amrita Bhattacharya)\n1rohitphysics137@gmail.com\nPreprint submitted to Journal of Magnetism and Magnetic Materials March 23, 2022arXiv:2203.11794v1  [cond-mat.mtrl-sci]  22 Mar 20221. Introduction\nHeusler alloys comprise an exciting class of materials exhibiting versatile\nproperties [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] ranging from half metallicity[15,\n12], superconductivity[10, 16], magnetic properties[17, 2] etc. Naturally, they\nhave been used extensively in tunnel junction devices[18, 19, 20], solar cells[5,\n10], thermoelectrics[3, 21], and spintronics based applications[22, 23]. Since its\ndiscovery in the year 1903, this alloy class has an ever-escalating demand in the\nmaterial science research community [16, 24, 9, 25, 26, 27, 28, 18]. Heusler al-\nloys can be further majorly categorized as full Heusler (X 2YZ) and half Heusler\n(XYZ) depending upon the full or half occupancy of X site elements in the\nlattice [16, 14, 1, 10]. For spintronics applications, Heusler alloys, with X and\nY elements from transition metal family and Z element from the main group,\nhave been reported to be particularly promising [22, 29, 30, 23, 31, 9, 32, 19, 33,\n34, 35]. These materials are found to exhibit the key properties necessary for\nspintronics applications viz. half metallicity, high magnetization, and uniaxial\nmagnetocrystalline anisotropy [30, 24, 23, 22]. Apart from theoretical predic-\ntions [29, 36, 9, 32], there are plenty of reports of their experimental synthesis\n[10, 28, 14, 17, 37, 38, 39, 40, 41]. However, it is impossible to experimentally\ntune the elemental constituents stoichiometrically to achieve the best one with\nimproved properties pertaining to the enormous compositional space. In this\nregards, density functional theory (DFT) has emerged as a powerful computa-\ntional tool for materials discovery, which, if implemented in high throughput\nloops, not only leads to the identi\fcation of the e\u000ecient composition but also\nsheds light on the underlying phenomena [28, 9, 27, 18, 36, 42, 29]. As for in-\nstance, Faleev et al. performed high throughput ab initio DFT based study on\n3dand 4 dtransition metal (TM) based full Heusler alloys (X 2YZ) with the Z\nelement from the main group (i.e. from group III, IV, and V)[18]. Their search\nlead to the prediction of \u001819 potential candidates with high spin polarization\nand magnetocrystalline anisotropy. Ma et al. performed a high throughput ab\ninitio DFT based study on transition metal (TM) based half Heusler alloys,\n2XYZ (with X = Cr, Mn, Fe, Co, Ni, Ru, Rh; Y = Ti, V, Cr, Mn, Fe, Ni; Z\n= Al, Ga, In, Si, Ge, Sn, P, As, Sb) [9]. In their high throughput search they\npredicted several semiconducting ( \u001826), half-metallic ( \u001845) and near half-\nmetallic (\u001834) half Heusler alloys with negative formation energy. Recently\nAmudhavalli et al. explored Mn-based half Heusler alloys via DFT based \frst\nprinciple simulations, XYZ (X = Ir, Pt, Au; Y = Mn; Z = Sn, Sb) and predicted\nthe half-metallicity of all alloys at high pressure[43].\nWhile the TM based Heusler alloys are typically the most explored ones\n[29, 36, 9, 32, 10, 28, 14, 17, 37, 38, 39, 40, 41], recent studies reveal spintronics\npotential of Alkali metal based Heusler alloys. Zhang et al. studied Li-based\nhalf Heusler alloy viz. LiCrS and predicted one of its metastable phase possess\na signi\fcant magnetic moment with the Cr atom showing atomic-like moment\nof 5\u0016B[44]. Shakil et al. studied the magnetic and elastic properties of LiCrZ\n(Z = P, As, Bi and Sb) half Heusler alloys [45] and found these alloys to be\n100% spin-polarized with very high magnetization. Recent study of Wang et\nal.on alkali metal-based XCrZ (with X = Li, K, Rb, and Z = S, Se, Te) half\nHeusler alloys [46] revealed robust half metallicity (HM) in almost all alloys in\ntheir stable phases except for LiCrZ, which shows HM nature in only metastable\nphase. Alkali metal-based Heusler alloys have also been explored in the context\nof thermoelectric materials; for instance, Yadav et al. investigated LiYZ (Y =\nBe, Mg, Zn, Cd and Z = N, P, As, Sb, Bi) series and found that LiZnSb has a\ntypically large n-type power factor [47]. Although these studies hint towards the\npromise of Alkali metal based Heusler alloys for spintronics applications, but a\ndetailed systematic analysis of magnetic properties of Alkali and transition metal\nbased half Heusler alloys are hitherto absent in the literature.\nIn this work, high throughput DFT calculations are performed on LiY pY0\n1\u0000pS\n(Y, Y0= V, Cr, Mn, Fe, Co, Ni and p= 0, 0.25, 0.5, 0.75, 1) compositions to\nidentify the electronically and vibrationally stable ones for their spintronics ap-\nplications.\n3LiMnSLiCr0.5Fe0.5SLiFe0.5Mn0.5S\nLiMn0.5Cr0.5SLiFeS\n-0.95-1.05-1.15-1.25-1.0-1.1-1.2-1.3LiCrS\n(b)(a)\nLiY/Y SFigure 1: (a) The conventional cell of the most stable ( \f) phase of half Heusler composition.\nThe Li and S atoms fully occupy the speci\fc (4b and 4c) Wycko\u000b sites, while the 4a sites are\npartially occupied by the transition metal atoms (Y/Y0). (b) The formation energy plot of\nfew selected compositions originating from the parents LiMnS, LiFeS and LiCrS. The color\nscale shows the variation of the formation energy.\n2. Methodology\nThe conventional cell of a full Heusler alloy (X 2YZ) has a cubic lattice with\n4a (0, 0, 0), 4b (0.5, 0.5, 0.5), 4c (0.25, 0.25, 0.25) and 4d (0.75, 0.75, 0.75)\nWycko\u000b sites. In half Heusler (HH) alloy (XYZ) 4d sites remain vacant, leading\nto 1:1:1 stoichiometry. Depending upon the site occupancy by elements, these\nHH alloys may exist in three di\u000berent structural con\fgurations (viz. the \u000b,\f\nand\rphases). The details of the structures of the three phases are provided in\nthe Fig.1 of section S1 of supplementary information. Thus, the identi\fcation\nof energetically the most favorable structural con\fguration comprises the \frst\nstep. This is also one of the most lengthy steps, since for each of the LiY pY0\n1\u0000pS\ncompositions (Y, Y0= V, Cr, Mn, Fe, Co, and Ni; p= 0, 0.25, 0.75, and 1),\nthree di\u000berent structural replicas (for the \u000b,\fand\rphases) are considered\nwhile taking into account of spin polarization 5.\nOnce the ground state geometry is identi\fed, we calculate their respective\nformation energy using;\n4Ef=E(LiY pY0\n1\u0000pS)\u0000\u0016Li\u0000p\u0001\u0016Y\u0000(1\u0000p)\u0001\u0016Y0\u0000\u0016S (1)\nwhereby, E(LiY pY0\n1\u0000pS) is the ground state total energy of the respective\ncomposition. \u0016Li,\u0016Y,\u0016Y0, and\u0016Sare the chemical potential of the Li, Y, Y0\nand S elements, which is taken from their stable earth abundant bulk phases\n(the details of the phases taken as reference is provided in the section S2 of the\nsupplementary information). Thus, the compositions with a negative formation\nenergy is retained for further analysis while the remaining ones are discarded.\nHowever, a negative formation energy does not always ensure the dynamical\nstability of the composition. On the otherhand, the dynamical stability of the\ncomposition is extremely important for ensuring the formation of the composi-\ntion in laboratory. Thus, at this stage phonon calculations are performed under\nthe framework of density functional perturbation theory for energetically the\nmost stable compositions taking into account of spin polarization. The calcula-\ntion settings are carefully chosen to ensure that the dynamical instabilities (if\nany) are not an artifact arising from the choice of cell (see section 5 for details).\nThus, beyond this step only the dynamically stable compositions are retained\nfor magnetic structure analysis.\nFurther, the magnetic analysis is again explicity performed for the dynam-\nically stable compounds by comparing their energies in ferromagnetic (FM),\nantiferromagentic (AFM) and ferrimagnetic (FiM) con\fgurations. Thereby, the\nlowest energy con\fguration is identi\fed as the magnetic ground state. Further-\nmore, the percentage of spin polarization (SP)[48] is calculated for the FM and\nFiM con\fgurations, using the following relation;\nSP = \nDOS\"\u0000DOS#\nDOS\"+ DOS#!\nEF\u0002100 % (2)\nwhere, SP is the percentage of the ratio of di\u000berence in the density of ma-\njority spin (\") and minority spin ( #) states and the total density of spin states\nat the Fermi level E F. The compositions with 100% spin polarization (i.e., the\nHalf metallic ones) are particularly interesting because of their applications in\n5LiYpY 1-pS with ,  and   (243 structures)/uni03B1/uni03B2/uni03B3\nLiYpY 1-pS with -phases  (51 structures)/uni03B2\nVibrationally stable (26 structures)FMMagnetic structure analysis  (51 structures) \n3 tetr.\n8 HM\nAFM\nFiM\nFM (10 structures)\nAFM (4 structures)\nFiM (12 structures)\n4 tetr.\n3 tetr.\n4 HM  (1 tetr.)Figure 2: Flowchart of high throughput calculations showing the di\u000berent steps of the screen-\ning criteria designed for selecting compositions based on the stability, vibrational, and mag-\nnetic analysis.\nspintronics devices as spin \flters, magnetic tunnel junction etc [20].\nFinally, the magnetocrystalline anisotropy energy (MAE) is analyzed, which\nmay predominantly arise from the structural anisotropy arising from the tetrag-\nonal distortion due to the elemental substitutions in the compositions[12, 25].\nThe anisotropy constant K gives the estimation of the MAE in the crystal, which\nis calculated using;\nK =Ek\u0000E?\nV(3)\nwhere, Ekand E?are the total energies when the crystal is magnetized along\nthe in-plane and out-of-plane directions respectively for a crystal of volume V.\nIn order to calculate the Ekand E?, the global spin quantization axis is varied\nalong the three crystallographic orientation i.e. along the (100), (010), and (001)\n6axis, while incorporating the e\u000bect of spin orbit coupling (SOC) and the total\nenergy is compared for each case. Accordingly, the orientation with the lowest\ntotal energy is referred to as the easy axis for magnetic polarization.\nThereby, a composition with high spin polarization as well as high MAE is\nconcluded to be ideal for the spintronics applications[25, 18]. All these steps\ninvolved in high throughput screening are schematically shown in Fig. 2.\n3. Results\n3.1. Structural and stability\nAs discussed brie\ry in the method section 2, a pristine half Heusler alloy\nmay exists in three di\u000berent con\fgurations, i.e. the \u000b,\fand\r, depending\nupon the site occupancy of the X, Y, and Z elements. The site preference may\nalso change depending upon the substitution in LiY pY0\n1\u0000pS (Y, Y0= V, Cr,\nMn, Fe, Co, Ni and p = 0, 0.25, 0.5, 0.75, 1). Thus, for each of the di\u000berent\ncompositions all the three phases (i.e. the \u000b,\fand\r) are considered to generate\nthe di\u000berent structural replicas for the high throughput calculation. As for\ninstance, for generating the LiY 0:25Y0\n0:75S (LiY 0:75Y0\n0:25S) structures in one of\nthe given phase (say \u000b), the conventional lattice of LiY0S (LiYS) is considered\nand one of the four Y0(Y) atom is substituted with one Y (Y0) atom. However,\nin order to generate the intermediate con\fguration, LiY 0:5Y0\n0:5S, starting with\nLiY0:25Y0\n0:75S, one of the three remaining Y0atoms is substituted with one Y\natom iteratively and the lowest energy structure is identi\fed. Same process is\nrepeated for the \fand\rcon\fgurations. Thus, for each pair of elements at the Y\nsite, it took 15 di\u000berent structural replicas to identify one stable con\fguration.\nGiven that 15 di\u000berent pairs of elements are considered for the Y site, 225\nstructural replicas were generated for LiY pY0\n1\u0000pS and similarly 18 more for the\nparents (LiY0S /LiYS).\nAt the \frst step, spin polarized total energy calculations are performed on\nthe structural replicas. The \fcon\fguration, with X atom in the 4b, Y/Y0atom\nin the 4a and Z atom in the 4c position, is found to be more stable than the\n7XMΓZ0100200300400Frequency (cm-1)\nMΓZΓPRΓRX RM(a) (b) (c)\n(a)(b)(c)\nLi\nMnFeCrS\nXMΓZ0100200300400Frequency (cm-1)\nMΓZΓPRΓRX RM(a) (b) (c)\nXMΓZ0100200300400Frequency (cm-1)\nMΓZΓPRΓRX RM(a) (b) (c)Figure 3: The ground state magnetic ordering (top panel) and the corresponding phonon\nband structure (bottom panel) of selective representative magnetic compositions viz. (a)\nLiFe 0:5Mn0:5S (FM) (b) LiCr 0:25Mn0:75S (FiM) and (c) LiFeS (AFM).\nother two phases by more than 0.5 eV per formula unit (f.u.). Thereby, using\nthe\fcon\fgurations for all the given compositions and the bulk energies of the\nconstituent elements as reference, the formation energy of the compositions is\ncalculated. The formation energy of few of the selected ones, generated from\nLiFeS, LiMnS and LiCrS as parents, is given in the three phase plot of Fig. 1(b).\nAll compositions are found to have negative formation energy, while most of the\ncompositions in the three phase diagram are found to have negative formation\nenergy as compared to their parents i.e. LiMnS and LiCrS (shown in green),\nwhich indicates towards their formation in the laboratory.\nIt should be noted here that a spin polarized calculation may not be su\u000e-\ncient to predict the exact magnetic ground state of the composition. Since, the\ntotal energy contribution may marginally di\u000ber when di\u000berent spin alignments\n(i.e. in the FM, FiM and AFM) are explicitly taken into consideration. Thus,\n8for all of the 51 \fcon\fgurations, the magnetic ground state is calculated by\nvarying the spin alignments in di\u000berent FM, FiM and AFM spin con\fguration\nand thereby, lowest energy con\fguration is identi\fed as true magnetic ground\nstate of the composition (see Table.1 of section S3 of supplementary information\nfor the formation energy and magnetic properties of all the stable compounds).\nFurthermore, we perform phonon calculations on these ground state magnetic\nstructure and check their dynamical stability. Only 26 compositions are found\nto be dynamical stable (with no or very negligible negative phonon modes)(see\nFig. 2 of section S4 of the supplementary information for the dynamical stabil-\nity of all the FM compositions). These compositions are retained for further\ncalculation of the % of SP, MAE etc.\n3.2. Magnetic properties\nThe spin con\fguration of the dynamically stable ones reveals 10 FM, 12 FiM\nand 4 AFM compositions. We calculated the % SP in FM and FiM ones at the\nFermi level, which is found to be 100% in 4 FM and 8 FiM ones. The magnetic\nproperties i.e. the type of magnetic con\fguration, % SP and net magnetization\nper f.u. for all these dynamically stable compositions are provided in Table.1.\nFinally, we also check for tetragonal distortion in the structure of the mag-\nnetic compositions. 10 of these structures are found to be tetragonal. These are\nfurther taken in consideration for explicit calculation of the magnetocrystalline\nanisotropy energy (MAE). These intermediate compositions (see Table 1) are\nfound to show a wide variety of magnetic interactions including several exotic\nproperties such as half metallicity and MAE. The net magnetization is found\nto be dominant in Mn-Fe based ones (with highest of 3.5 \u0016Bin LiFe 0:5Mn0:5S)\nfollowed by Mn-Co (with highest of 2.99 \u0016Bin LiMn 0:5Co0:5S) and Fe-Co ones\n(with highest of 2.71 \u0016Bin LiFe 0:75Co0:25S). Several compositions are also found\nto show half metallicity (i.e. almost 100% spin polarization) while possess-\ning high moments per f.u. of >2muBviz. LiFe 0:5Mn0:5S, LiCo 0:5Mn0:5S,\nLiFe 0:25Co0:75S, LiV 0:25Mn0:75S, LiFe 0:75Co0:25S and LiFe 0:75Mn0:25S. Finally,\nsome of these compositions are also found to be tetragonal viz. LiFe 0:5Mn0:5S\n9Table 1: Details of the structural and magnetic properties of the 26 dynamically stable com-\npositions as obtained from high throughput search. The magnetic con\fguration (M Conf:),\ntetragonality (c/a), percentage of spin polarization (% SP) and net magnetization (M) are\nenlisted for each of these compositions.\nCompositions M Conf:M SP c/a\nLiFeS AFM 0 0 0.994\nLiCoS FM 1.99 89 1\nLiNiS FM 0.39 59 1\nLiCo 0:5Mn0:5S FM 2.99 96 1.009\nLiFe 0:5Co0:5S FM 2.36 54 0.998\nLiFe 0:5Mn0:5S FM 3.50 100 1.011\nLiCr 0:5Fe0:5S FiM 0.99 84 1.014\nLiCr 0:5Mn0:5S FiM 0.49 72 1.004\nLiFe 0:5V0:5S FiM 0.38 60 1.01\nLiCo 0:5Ni0:5S AFM 0 0 1.002\nLiCo 0:25Ni0:75S FM 0.57 37 1\nLiFe 0:25Co0:75S FM 2.22 100 1\nLiFe 0:25Ni0:75S FM 0.90 43 1\nLiCr 0:25Fe0:75S FiM 1 100 1\nLiCr 0:25Mn0:75S FiM 1.75 100 1\nLiV0:25Co0:75S FiM 0.5 100 1\nLiV0:25Fe0:75S FiM 1.25 100 1\nLiV0:25Mn0:75S FiM 2 100 1\nLiV0:25Ni0:75S FiM 0.25 100 1\nLiCo 0:25Mn0:75S AFM 0 0 1.009\nLiFe 0:75Co0:25S FM 2.71 100 1\nLiFe 0:75Mn0:25S FM 3.25 100 1\nLiCo 0:75Mn0:25S FiM 0.08 100 1\nLiCo 0:75Ni0:25S FiM 1.65 47 1\nLiNi 0:75Mn0:25S FiM 0.75 100 1\nLiFe 0:75Ni0:25S AFM 0 0 1.001\n10(b)(a)\nLiMnSLiCoS\nLiCo0.5Fe0.5S4.03.02.01.03.5\n0.02.51.50.5LiMn0.5Fe0.5S\nLiMn0.5Co0.5SLiFeS\nLiMnSLiCoSLiMn0.5Co0.5S0.6\n-0.20.00.20.40.8LiCo0.5Fe0.5SLiMn0.5Fe0.5S\nLiFeS\nFigure 4: The total (a) magnetization ( \u0016B/f.u.) and (b) magnetocrystalline anisotropy con-\nstant K (MJ/m3) plot of few selected compositions originating from the parents LiMnS, LiFeS\nand LiCoS. The color scale shows the variation in respective magnitudes.\nand LiCo 0:5Mn0:5S. The net magnetization and the corresponding MAE is pro-\nvided in Fig. 4 for a few selected compositions with interesting magnetic prop-\nerties viz. the ones generated from LiMnS, LiFeS and LiCoS as parents. The\ncompositions having both Mn and Fe are found to be highly magnetic (as shown\nin Fig. 4 a), while the Fe and Co/ Mn are found to show high in-plane magne-\ntocrystalline anisotropy (as shown in the dark blue region in Fig. 4 b).\nThe electronic band structure and density of states plots for three represen-\ntative compositions i.e. one from the FM, FiM and AFM category respectively\nis shown in Fig. 5. LiFe 0:5Mn0:5S (Fig. 5 a) is found to be half metallic FM with\n100% spin polarization, since only the down spin channel is found to be present\nat the Fermi level. Here, the spin on both the Mn (with moment of 3.8 \u0016Bper\natom) as well as the Fe (with moment of 2.7 \u0016Bper atom) atoms are found\nto be alligned in the same direction, which gives rise to a net uncompensated\nmoment of 3.5 \u0016Bper f.u. Intrestingly, the magnetic moment on Fe atoms in\nLiFe 0:5Mn0:5S is found to be higher as compared to that in the pure bulk Fe (2.2\n\u0016B). The distance between the magnetic atoms in this composition is compared\nwith its parents. Fe-Fe neighbor distance (3.864 \u0017A) is found to be the lowest\nin the AFM coupled parent LiFeS alloy, whereas the Mn-Mn neighbor distance\n11X M ΓZ R A-2-1012Energy (eV)\n-10 0 10 X M ΓZ R ASLi\nFeMn(a)\nΓX M Γ R X-2-1012Energy (eV)\n-15 0 15 ΓX M Γ R XSLi\nCrMn(b)\nΓ L-2-1012Energy (eV)\n-10 0 10 Γ LSLi\nFe\nW K W U W K UW(c)Figure 5: Electronic band structure of up (left) and down (right) spin components and their\ncorresponding density of states (middle) of one of the representative composition with ferro,\nferri and anti ferro ordering viz. (a) LiFe 0:5Mn0:5S, (b) LiCr 0:25Mn0:75S, and (c) LiFeS.\n(4.007 \u0017A) is found to be maximum in the FM coupled parent alloy LiMnS. The\nFe-Mn (3.953 \u0017A) neighbor distance in LiFe 0:5Mn0:5S is found to be intermediate\nto those in the two parents. This is also in accordance with the Bethe Slater\ncurve for transition metals [49], which states that the magnetic interaction in a\ngiven alloy varies as a function of neighbor distance and d orbital contribution.\nA bandgap of\u00190.23 eV is observed in up spin channel of LiFe 0:5Mn0:5S. To un-\nderstand the origin of the half metallicity, the orbital projected band structure\nof LiFe 0:5Mn0:5S composition is analysed using PyProcar[50]. The projection\n12of the d orbitals (t 2gand e gindividually) on the electronic band structure is\nanalyzed for the spin up and spin down channels separately. For the spin up\nchannel, the valence band maximum is found to be formed from the hybridiza-\ntion of t 2gorbitals of Mn and Fe (three orbitals converging at the valence band\nmaximum highlighted by red color in the left column of Fig. 6). In the con-\nduction band minimum of the spin up channel, no signi\fcant contribution is\nobserved from the d orbitals. For the spin-down channel, the t 2gand e gorbitals\nof Mn and t 2gof Fe are not found to make any signi\fcant contribution close to\ntheFermi level, while the e gorbitals of Fe (shown in green just below the Fermi\nlevel in the top right column of Fig. 6) makes signi\fcant contribution near the\nFermi level. The down spin is found to be completely metallic giving rise to the\nhalf metallic ferromagnetic characteristic of the composition. This composition\nalsoshowed afairly high MAE of0.1meV performula unitwhich corrsponds to\ntheanisotropy constant of0.36MJ/m3(see Fig.4 b) and the (001) axis is found\nto be the easy axis of magnetic polarization for FM spin alignment. However,\nthe orbital contribution to magnetism is found to be signi\fcantly low in both\nMn (0.035 \u0016B) as well Fe (0.11 \u0016B) atom as compared to the dominant spin\npart, which comprised the majority of the moment on each of the atom.\nOur high throughput search led to the identi\fcation of several FiM composi-\ntions. Out of these, few were also found to be half metallic with high anisotropy.\nOne of such ferrimagnetic (FiM) compositions viz. LiCr 0:25Mn0:75S is analyzed\nand presented as a representative case. It is found to be a FiM half metal with\nsigni\fcant up spin contribution and almost negligible down spin contribution at\nthe Fermi level (see Fig. 5 (b)). The moment on Cr atom (3.71 \u0016Bper atom)\nis found to be higher, but opposite to the moment on Mn atom (3.5 \u0016Bper\natom), which led to a net moment of 1.75 \u0016Bper f.u (as shown in Fig. 3 b).\nFurther, it is also found to be anisotropic with (001) is an easy axis of FiM\npolarization with MAE of0.017 meV performula unit andthecorrespond ing\nanisotropy constant of0.134 MJ/m3(as shown in Fig. 4 b). As expected the\norbital contribution to the moment of the Cr and Mn atoms is found to be fairly\nlower (of 0.012 \u0016Band 0.055\u0016Brespectively) as compared to the spin moment\n13Figure 6: The t 2gand e gorbital projections of the magnetic elements (Mn and Fe) in the\nferromagnetic composition LiFe 0:5Mn0:5S for the up spin (left) and down spin (right). The\ncolor scale denotes the weighted contribution stemming from the particular orbital.\n(of 3.71\u0016Band 3.5\u0016Brespectively). Despite of its FiM magnetic con\fguration,\nit is found to show a good magnetization of 1.75 \u0016B, with an out of plane MAE,\nmaking it a potential candidate for the spintronics application.\nAFM alloys with high MAE also have important spintronics applications viz.\nin spin transfer torque (STT) based magnetic random access memory (MRAM)\n[51, 52] etc [53]. Thus, as a representative case, the LiFeS alloy is analyzed and\npresented. In its stable AFM con\fguration, Fe atoms in every plane is found\nto possess equal (of 2.4 \u0016B) and unidirectional moments, which were cancelled\nby the moments of Fe atoms in adjacent plane (as shown in Fig. 3 c). This is\n14con\frmed by the \fnite, but equal and opposite contribution of the spin up and\nspin down channels at the Fermi level (as shown in Fig. 5 c). The (001) axis\nis found to be the easy axis with MAE of0.231 meV performula unit andan\nanisotropy constant of0.902 MJ/m3. The spin and orbital moment on the Fe\natoms are found to be 2.40 \u0016Band 0.096\u0016Brespectively.\nThe Curie temperature (T c) is also another important property for magnetic\napplication of an alloy and hence, is discussed hereby qualitatively. T cof an\nalloy mainly depends upon the interaction strength between the magnetic ions.\nThereby, in another words, T ccan be qualitatively predicted from the elemental\nconstituents and their uncompensated atomic magnetic moment in the solid.\nAmong the elemental solids, Fe, Co, and Ni are already known to show high\nTc. Most of magnetic compositions, enlisted in Table 1, also possess one of\nthese elements as constituent. Also, it is generally seen that in half Heusler\nalloys, with high magnetic moment ( >3muB), Tctends to be above room\ntemperature [44, 15]. This is in agreement with the T ccalculated using Monte\nCarlo Simulation by Sattar et al. on alkali metal based half Heusler alloys of\nformula XYZ, where X is alkali metal, Y is a 3d TM element and Z is a chalcogen\nelement. They found that T cin these compositions is generally higher than 300\nK Thus, from all these literature a crude idea on T cof LiY pY0\n1\u0000pS (Y, Y0= V,\nCr, Mn, Fe, Co, Ni and p = 0, 0.25, 0.5, 0.75, 1) compositions can be drawn to\nbe above room temperature.\n4. Conclusions\nWe employ \frst principles DFT calculations to design high throughput loops\nto screen stable half Heusler (HH) compositions viz. LiY pY0\n1\u0000pS (Y, Y0= V,\nCr, Mn, Fe, Co, Ni and p= 0, 0.25, 0.5, 0.75, 1) for spintronics applications.\nTaking into account of all possible geometric con\fgurations arising from the\nsite preference of the elements, a pool of 243 di\u000berent structures is generated.\nBy comparing their total energy obtained after incorporating the contribution\nstemming from the spin polarization, 51 low energy structures (belonging to the\n15\fphase) are selected. As this stage, the ground state energy of each of these\ncompositions is compared for di\u000berent spin alignments explicitly (ie. for ferro,\nferri and anti ferro) and the lowest energy structure is referred to be the mag-\nnetic ground state. An accurate phonon calculation is performed and 26 of the\ndynamically stable ones are retained. 10 FM, 12 FiM and 4 AFM compositions\nare screened to be dynamically stable, out of which 4 half metallic FM and 8\nhalf metallic FiM are identi\fed. A tetragonal distortion is observed in 3 FM\n(1 half metal), 3 FiM and 4 AFM leading to a magneto crystalline anisotropy\n(MAE) in the system. The ferromagnetic LiFe 0:5Mn0:5S composition is found\nto have the highest easy-axis anisotropy constant of 0.36 MJ/ m3with 100%\nspin-polarization with net magnetization of 3.5 \u0016B. The rest of the magnetic\nalloys are found to possess an anisotropy constant in the range of 0.001 MJ/ m3\n<K<0.36 MJ/m3. The present work provides a blueprint for the range of\nmagnetization, magneto crystalline anisotropy and spin polarization that can\nbe achieved in LiY pY0\n1\u0000pS compositions.\n5. Computational details\nWe performed \frst-principles density functional theory calculation within\nprojector augmented wave (PAW) method based on the plane-wave basis set as\nemployed in Vienna ab initio simulation package (VASP)[54, 55, 56, 57, 58, 59].\nThe PBE (Perdew, Burke, and Ernzerhof) corrected GGA (generalized gradient\napproximation) energy functional was used addressing the exchange and corre-\nlation interactions[60]. The PAW potentials [57, 56]for Li, Y, Y0, and S have\nbeen used to manifest the ion core and valence electron interaction. The kinetic\nenergy cuto\u000b for plane waves was set to 500 eV for the calculations. The opti-\nmized lattice constants were obtained by performing polarized spin calculations\nand relaxing the conventional unit cell within the conjugate gradient method.\nThe structure was relaxed till forces on each ion were less than 0.001 eV/ \u0017A,\nand total energy is converged within the convergence limit of 10\u00005eV. For each\nof the di\u000berent structures, the k-mesh was employed by checking in the total\n16energy convergence. Forcalculatingthemagnetic anisotropy, thesame k-mesh\nsizewasemployed which wasused forachiev ingtheconvergence inthetotal\nenergy ofthestruc tures. Incontrast, forDOS calculations, adenser k-mesh of\n27\u000227\u000227isemployed. For non-cubic structures, the k-points were chosen\nas per the inverse proportional ratio of the lattice constants. The spin-orbit\ncoupling was included in the total Hamiltonian while doing magnetocrystalline\nanisotropy calculation as it is fundamental origin of the anisotropy. Phonon\nband structure was plotted using the phonopy code based on the density func-\ntional perturbation theory [61, 62, 63]. The thermodynamic properties were\nobtained by using the phonon frequencies from the harmonic band structure.\nAcknowledgements\nAB acknowledges the funding received as IIT B seed grant project (RD/0517-\nIRCCSH0-043), SERB ECRA project (ECR/2018/002356) and BRNS regular\ngrant (BRNS/37098). The high performance computational facilities viz. Aron\n(AbCMS lab, IITB), Dendrite (MEMS dept., IITB), and Spacetime, IITB and\nCDAC Pune (Param Yuva-II) are acknowledged for providing the computational\nhours.\nReferences\nReferences\n[1] C. Felser, A. Hirohata, Heusler alloys, Springer, 2015.\n[2] P. J. Webster, Heusler alloys, Contemporary Physics 10 (6) (1969) 559{577.\n[3] G. J. Poon, Electronic and thermoelectric properties of half-heusler alloys,\nin: Semiconductors and semimetals, Vol. 70, Elsevier, 2001, pp. 37{75.\n[4] H. Lin, L. A. Wray, Y. Xia, S. Xu, S. Jia, R. J. Cava, A. Bansil, M. Z.\nHasan, Half-heusler ternary compounds as new multifunctional experimen-\ntal platforms for topological quantum phenomena, Nature materials 9 (7)\n(2010) 546{549.\n17[5] T. Gruhn, Comparative ab initio study of half-heusler compounds for op-\ntoelectronic applications, Physical Review B 82 (12) (2010) 125210.\n[6] Z. Bai, L. Shen, G. Han, Y. P. Feng, Data storage: review of heusler\ncompounds, in: Spin, Vol. 2, World Scienti\fc, 2012, p. 1230006.\n[7] D. Xiao, Y. Yao, W. Feng, J. Wen, W. Zhu, X.-Q. Chen, G. M. Stocks,\nZ. Zhang, Half-heusler compounds as a new class of three-dimensional topo-\nlogical insulators, Physical review letters 105 (9) (2010) 096404.\n[8] S. Chadov, X. Qi, J. K ubler, G. H. Fecher, C. Felser, S. C. Zhang, Tun-\nable multifunctional topological insulators in ternary heusler compounds,\nNature materials 9 (7) (2010) 541{545.\n[9] J. Ma, V. I. Hegde, K. Munira, Y. Xie, S. Keshavarz, D. T. Mildebrath,\nC. Wolverton, A. W. Ghosh, W. Butler, Computational investigation of\nhalf-heusler compounds for spintronics applications, Physical Review B\n95 (2) (2017) 024411.\n[10] T. Graf, S. S. Parkin, C. Felser, Heusler compounds|a material class with\nexceptional properties, IEEE Transactions on Magnetics 47 (2) (2010) 367{\n373.\n[11] B. Nanda, I. Dasgupta, Electronic structure and magnetism in half-heusler\ncompounds, Journal of Physics: Condensed Matter 15 (43) (2003) 7307.\n[12] C. Felser, L. Wollmann, S. Chadov, G. H. Fecher, S. S. Parkin, Basics and\nprospective of magnetic heusler compounds, APL materials 3 (4) (2015)\n041518.\n[13] F. Casper, T. Graf, S. Chadov, B. Balke, C. Felser, Half-heusler compounds:\nnovel materials for energy and spintronic applications, Semiconductor Sci-\nence and Technology 27 (6) (2012) 063001.\n[14] R. Helmholdt, R. De Groot, F. Mueller, P. Van Engen, K. Buschow,\nMagnetic and crystallographic properties of several c1b type heusler com-\n18pounds, Journal of magnetism and magnetic materials 43 (3) (1984) 249{\n255.\n[15] S. Wurmehl, G. H. Fecher, V. Ksenofontov, F. Casper, U. Stumm, C. Felser,\nH.-J. Lin, Y. Hwu, Half-metallic ferromagnetism with high magnetic mo-\nment and high curie temperature in co 2 fe si, Journal of applied physics\n99 (8) (2006) 08J103.\n[16] T. Graf, C. Felser, S. S. Parkin, Simple rules for the understanding of\nheusler compounds, Progress in solid state chemistry 39 (1) (2011) 1{50.\n[17] C. Wang, J. Meyer, N. Teichert, A. Auge, E. Rausch, B. Balke, A. H utten,\nG. H. Fecher, C. Felser, Heusler nanoparticles for spintronics and ferro-\nmagnetic shape memory alloys, Journal of Vacuum Science & Technology\nB, Nanotechnology and Microelectronics: Materials, Processing, Measure-\nment, and Phenomena 32 (2) (2014) 020802.\n[18] S. V. Faleev, Y. Ferrante, J. Jeong, M. G. Samant, B. Jones, S. S. Parkin,\nHeusler compounds with perpendicular magnetic anisotropy and large tun-\nneling magnetoresistance, Physical Review Materials 1 (2) (2017) 024402.\n[19] C. Felser, V. Alijani, J. Winterlik, S. Chadov, A. K. Nayak, Tetragonal\nheusler compounds for spintronics, IEEE transactions on magnetics 49 (2)\n(2013) 682{685.\n[20] C. Felser, G. H. Fecher, Spintronics: from materials to devices, Springer\nScience & Business Media, 2013.\n[21] P. R. Raghuvanshi, S. Mondal, A. Bhattacharya, A high throughput search\nfor e\u000ecient thermoelectric half-heusler compounds, Journal of Materials\nChemistry A 8 (47) (2020) 25187{25197.\n[22] S. Bader, S. Parkin, Spintronics, Annu. Rev. Condens. Matter Phys. 1 (1)\n(2010) 71{88.\n19[23] I. \u0014Zuti\u0013 c, J. Fabian, S. D. Sarma, Spintronics: Fundamentals and applica-\ntions, Reviews of modern physics 76 (2) (2004) 323.\n[24] K. Elphick, W. Frost, M. Samiepour, T. Kubota, K. Takanashi,\nH. Sukegawa, S. Mitani, A. Hirohata, Heusler alloys for spintronic devices:\nreview on recent development and future perspectives, Science and Tech-\nnology of Advanced Materials 22 (1) (2021) 235{271.\n[25] S. V. Faleev, Y. Ferrante, J. Jeong, M. G. Samant, B. Jones, S. S. Parkin,\nOrigin of the tetragonal ground state of heusler compounds, Physical Re-\nview Applied 7 (3) (2017) 034022.\n[26] C. M. Fang, G. De Wijs, R. De Groot, Spin-polarization in half-metals,\nJournal of Applied Physics 91 (10) (2002) 8340{8344.\n[27] H. C. Kandpal, G. H. Fecher, C. Felser, Calculated electronic and magnetic\nproperties of the half-metallic, transition metal based heusler compounds,\nJournal of Physics D: Applied Physics 40 (6) (2007) 1507.\n[28] A. O. Oliynyk, E. Antono, T. D. Sparks, L. Ghadbeigi, M. W. Gaultois,\nB. Meredig, A. Mar, High-throughput machine-learning-driven synthesis of\nfull-heusler compounds, Chemistry of Materials 28 (20) (2016) 7324{7331.\n[29] X. Li, J. Yang, First-principles design of spintronics materials, National\nScience Review 3 (3) (2016) 365{381.\n[30] S. A. Chambers, Y. K. Yoo, New materials for spintronics, MRS bulletin\n28 (10) (2003) 706{710.\n[31] C. J. Palmstr\u001cm, Heusler compounds and spintronics, Progress in Crystal\nGrowth and Characterization of Materials 62 (2) (2016) 371{397.\n[32] J. Ma, J. He, D. Mazumdar, K. Munira, S. Keshavarz, T. Lovorn,\nC. Wolverton, A. W. Ghosh, W. H. Butler, Computational investigation of\ninverse heusler compounds for spintronics applications, Physical Review B\n98 (9) (2018) 094410.\n20[33] C. Guillemard, S. Petit-Watelot, L. Pasquier, D. Pierre, J. Ghanbaja,\nJ. Rojas-S\u0013 anchez, A. Bataille, J. Rault, P. Le F\u0012 evre, F. Bertran, et al., Ul-\ntralow magnetic damping in co 2 mn-based heusler compounds: Promising\nmaterials for spintronics, Physical Review Applied 11 (6) (2019) 064009.\n[34] M. Meinert, J.-M. Schmalhorst, G. Reiss, Ab initio prediction of ferrimag-\nnetism, exchange interactions and curie temperatures in mn2tiz heusler\ncompounds, Journal of Physics: Condensed Matter 23 (3) (2010) 036001.\n[35] B. Balke, S. Wurmehl, G. H. Fecher, C. Felser, J. K ubler, Rational design\nof new materials for spintronics: Co2fez (z= al, ga, si, ge), Science and\nTechnology of advanced Materials.\n[36] Q. Gao, I. Opahle, H. Zhang, High-throughput screening for spin-gapless\nsemiconductors in quaternary heusler compounds, Physical Review Mate-\nrials 3 (2) (2019) 024410.\n[37] A. Ahmad, S. Mitra, S. Srivastava, A. Das, Size-dependent structural and\nmagnetic properties of disordered co2feal heusler alloy nanoparticles, Jour-\nnal of Magnetism and Magnetic Materials 474 (2019) 599{604.\n[38] A. Ahmad, S. Srivastava, A. Das, First-principles calculations and experi-\nmental studies on co2fege heusler alloy nanoparticles for spintronics appli-\ncations, Journal of Alloys and Compounds 878 (2021) 160341.\n[39] T. Kojima, S. Kameoka, A.-P. Tsai, The emergence of heusler alloy cata-\nlysts, Science and Technology of Advanced Materials 20 (1) (2019) 445{455.\n[40] L. Galdun, V. Vega, Z. Vargov\u0013 a, E. D. Barriga-Castro, C. Luna, R. Varga,\nV. M. Prida, Intermetallic co2fein heusler alloy nanowires for spintronics\napplications, ACS Applied Nano Materials 1 (12) (2018) 7066{7074.\n[41] B. Balke, G. H. Fecher, J. Winterlik, C. Felser, Mn 3 ga, a compensated\nferrimagnet with high curie temperature and low magnetic moment for spin\ntorque transfer applications, Applied physics letters 90 (15) (2007) 152504.\n21[42] J. Ballu\u000b, K. Diekmann, G. Reiss, M. Meinert, High-throughput screening\nfor antiferromagnetic heusler compounds using density functional theory,\nPhysical Review Materials 1 (3) (2017) 034404.\n[43] A. Amudhavalli, R. Rajeswarapalanichamy, K. Iyakutti, Structural, elec-\ntronic, mechanical and magnetic properties of mn based ferromagnetic half\nheusler alloys: A \frst principles study, Journal of Alloys and Compounds\n708 (2017) 1216{1233.\n[44] R. Zhang, L. Damewood, C. Fong, L. Yang, R. Peng, C. Felser, A half-\nmetallic half-heusler alloy having the largest atomic-like magnetic moment\nat optimized lattice constant, AIP Advances 6 (11) (2016) 115209.\n[45] M. Shakil, S. Hassan, H. Arshad, M. Rizwan, S. Gillani, M. Ra\fque, M. Za-\nfar, S. Ahmed, Theoretical investigation of structural, magnetic and elastic\nproperties of half heusler licrz (z= p, as, bi, sb) alloys, Physica B: Con-\ndensed Matter 575 (2019) 411677.\n[46] X. Wang, Z. Cheng, G. Liu, Largest magnetic moments in the half-heusler\nalloys xcrz (x= li, k, rb, cs; z= s, se, te): A \frst-principles study, Materials\n10 (9) (2017) 1078.\n[47] M. K. Yadav, B. Sanyal, First principles study of thermoelectric properties\nof li-based half-heusler alloys, Journal of Alloys and Compounds 622 (2015)\n388{393.\n[48] R. Soulen, J. Byers, M. Osofsky, B. Nadgorny, T. Ambrose, S. Cheng,\nP. R. Broussard, C. Tanaka, J. Nowak, J. Moodera, et al., Measuring the\nspin polarization of a metal with a superconducting point contact, science\n282 (5386) (1998) 85{88.\n[49] R. Cardias, A. Szilva, A. Bergman, I. Di Marco, M. Katsnelson, A. Licht-\nenstein, L. Nordstr om, A. Klautau, O. Eriksson, Y. O. Kvashnin, The\nbethe-slater curve revisited; new insights from electronic structure theory,\nScienti\fc reports 7 (1) (2017) 1{11.\n22[50] U. Herath, P. Tavadze, X. He, E. Bousquet, S. Singh, F. Mu~ noz, A. H.\nRomero, Pyprocar: A python library for electronic structure pre/post-\nprocessing, Computer Physics Communications 251 (2020) 107080.\n[51] I. A. Zhuravlev, A. Adhikari, K. Belashchenko, Perpendicular magnetic\nanisotropy in bulk and thin-\flm cumnas for antiferromagnetic memory ap-\nplications, Applied Physics Letters 113 (16) (2018) 162404.\n[52] P. Wadley, B. Howells, J. \u0014Zelezn\u0012 y, C. Andrews, V. Hills, R. P. Campion,\nV. Nov\u0013 ak, K. Olejn\u0013 \u0010k, F. Maccherozzi, S. Dhesi, et al., Electrical switching\nof an antiferromagnet, Science 351 (6273) (2016) 587{590.\n[53] E. Welbourne, Antiferromagnetic nanodiscs with perpendicular magnetic\nanisotropy for biological applications, Ph.D. thesis, University of Cam-\nbridge (2021).\n[54] J. Hafner, G. Kresse, The vienna ab-initio simulation program vasp: An\ne\u000ecient and versatile tool for studying the structural, dynamic, and elec-\ntronic properties of materials, in: Properties of Complex Inorganic Solids,\nSpringer, 1997, pp. 69{82.\n[55] J. Hafner, Ab-initio simulations of materials using vasp: Density-functional\ntheory and beyond, Journal of computational chemistry 29 (13) (2008)\n2044{2078.\n[56] P. E. Bl ochl, Projector augmented-wave method, Physical review B 50 (24)\n(1994) 17953.\n[57] G. Kresse, D. Joubert, From ultrasoft pseudopotentials to the projector\naugmented-wave method, Physical review b 59 (3) (1999) 1758.\n[58] G. Kresse, J. Furthm uller, E\u000ecient iterative schemes for ab initio total-\nenergy calculations using a plane-wave basis set, Physical review B 54 (16)\n(1996) 11169.\n23[59] G. Kresse, J. Furthm uller, E\u000eciency of ab-initio total energy calculations\nfor metals and semiconductors using a plane-wave basis set, Computational\nmaterials science 6 (1) (1996) 15{50.\n[60] J. P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation\nmade simple, Physical review letters 77 (18) (1996) 3865.\n[61] A. Togo, L. Chaput, I. Tanaka, G. Hug, First-principles phonon calculations\nof thermal expansion in ti 3 sic 2, ti 3 alc 2, and ti 3 gec 2, Physical Review\nB 81 (17) (2010) 174301.\n[62] X. Gonze, C. Lee, Dynamical matrices, born e\u000bective charges, dielec-\ntric permittivity tensors, and interatomic force constants from density-\nfunctional perturbation theory, Physical Review B 55 (16) (1997) 10355.\n[63] L. Chaput, A. Togo, I. Tanaka, G. Hug, Phonon-phonon interactions in\ntransition metals, Physical Review B 84 (9) (2011) 094302.\n24"    },    {        "title": "2204.04206v3.Sensitivity_of_the_MnTe_valence_band_to_orientation_of_magnetic_moments.pdf",        "content": "Sensitivity of the MnTe valence band to orientation of magnetic moments\nPaulo E. Faria Junior,1Koen A. de Mare,2, 3Klaus Zollner,1Kyo-hoon Ahn,3\nSigurdur I. Erlingsson,4Mark van Schilfgaarde,5, 6and Karel V\u0013 yborn\u0013 y3\n1Institute for Theoretical Physics, University of Regensburg, D{93040 Regensburg, Germany\n2Eindhoven University of Technology, Eindhoven NL{5612 AZ, Netherlands\n3FZU { Institute of Physics of the Czech Academy of Sciences, Cukrovarnick\u0013 a 10, Praha 6, CZ{16253\n4Department of Engineering, School of Technology,\nReykjavik University, Menntavegi 1, IS-102 Reykjavik, Iceland\n5National Renewable Energy Laboratory, Golden, CO 80401, USA\n6King's College London, Physics Department, Strand, London WC2R 2LS, United Kingdom\n(Dated: Feb14, 2022)\nAn e\u000bective model of the hexagonal (NiAs-structure) manganese telluride valence band in the\nvicinity of the A-point of the Brillouin zone is derived. It is shown that while for the usual an-\ntiferromagnetic order (magnetic moments in the basal plane) band splitting at A is small, their\nout-of-plane rotation enhances the splitting dramatically (to about 0.5 eV). We propose extensions\nof recent experiments (Moseley et al., Phys. Rev. Materials 6, 014404) where such inversion of mag-\nnetocrystalline anisotropy has been observed in Li-doped MnTe, to con\frm this unusual sensitivity\nof a semiconductor band structure to magnetic order.\nI. INTRODUCTION\nThe electronic structure of crystalline semiconductors\ncan be treated by various methods which di\u000ber greatly\nin their computational cost.1Among ab initio methods,\nGW is one of the most advanced approaches yet a numer-\nically rather expensive one.2A widely-used alternative is\ndensity functional theory (DFT) where the speed comes\nat the cost of worse performance (even if there are vari-\nous approaches to mitigate de\fciencies such as too small\ngaps) and yet faster options are available, of which tight-\nbinding approaches3andk\u0001pmodels4will be of interest\nhere. Such e\u000bective models need material parameters\n(such as on-site energies or hopping amplitudes) as an\ninput which can sometimes be of advantage because they\ncan be adjusted to \ft experiments. Also, they may o\u000ber\ninsight into mechanisms governing the band structure.\nAn archetypal example of an e\u000bective model is the\nKohn-Luttinger Hamiltonian5which has a wide range of\napplications to non-magnetic materials, including silicon\nand III-V semiconductors with \u0000 8manifold at the top of\nthe valence band (VB). Magnetism adds a new twist: for\nMn-doped GaAs, the host is described by this Hamilto-\nnian and the e\u000bect of ferromagnetic ordering is captured\nby a kinetic pdexchange term/^~ s\u0001~Swhere ^~ sis the spin\noperator (of the VB holes) and ~Sis the classical spin rep-\nresenting the Mn magnetic moments (usually treated on\nthe mean-\feld level). Such description of ferromagnetic\nsemiconductors6,7has been employed extensively in the\ncontext of spintronics8and now that antiferromagnetic\nspintronics9has become an active \feld, we hereby wish\nto contribute to its progress by presenting an e\u000bective\nmodel of hexagonal (NiAs-structure) MnTe which is a\nwell-established antiferromagnetic semiconductor, as ex-\nempli\fed by its T= 0 band structure in Fig. 1, with\na relatively high ( \u0019310 K) N\u0013 eel temperature. Typical\nsamples, both bulk and layers exhibit p-type conductiv-\nD. KRIEGNERet al.PHYSICAL REVIEW B96,2 1 4 4 1 8( 2 0 1 7 )\nL1¯100EMAE1¯100<EMAE0001\nc–planeL0001c–planeMnTeabc(b)(a)\nFIG. 1. Sketch of the atomic and possible magnetic structuresof antiferromagnetic hexagonal MnTe. (a) In-plane/cplane (groundstate) and (b) out-of-plane/c-axis (hard axis) orientation of themagnetic moments of Mn with the Néel vector⃗Lalong⟨1¯100⟩and⟨0001⟩are shown. The hexagonal basal plane, i.e., thecplane isindicated by a gray plane, while red, green, and blue arrows show thedirections of the unit cell axes.InP(111) [7], and Al2O3(0001) [16,25,26]s u b s t r a t e sa sw e l la son amorphous Si(111)/SiO2[27]. Due to lattice and thermalexpansion coefficient mismatch betweenα-MnTe and thesubstrates, films will experience strain that may affect the mag-netic properties such as MAs. For example [28], the dilutemagnetic semiconductor (Ga,Mn)As is known to have anin-plane MA under compressive strain and an out-of-plane MAfor tensile strain under suitable conditions. Here, we study theMAs in MnTe on different substrates, which cause differentstrain states. The knowledge of the easy axis directionsis crucial for transport phenomena modeling, which hasso far relied only on assumptions [7]. As far as the easyaxis directions are concerned, we confirm these assumptionsusing DFT+Ucalculations combined with experiments. Usingmagnetotransport, magnetometry, and neutron diffraction, wedetermine the easy axes to be along⟨1¯100⟩and show in whatrespect MAs are sensitive to epitaxy-induced strain.The paper is organized as follows. After introduction ofthe results of DFT+Ucalculations in Sec.II,w ed e s c r i b eo u rsamples structure and basic magnetometry characterizationin Sec.III.S e c t i o nIVpresents our neutron diffractionexperiments and Sec.Vcomplementary magneto-transportstudies. Further magnetometry experiments determining thespin-flop field are presented in Sec.VI.F i n a l l y ,w ec o n c l u d ein Sec.VII.II. MAGNETIC ANISOTROPY CALCULATIONSThe magnetic anisotropy energy (MAE) in antiferromag-nets comprises two main contributions: the dipole term and themagnetocrystalline anisotropy (MCA). In order to calculatethe latter, we use the relativistic version of the rotationallyinvariant DFT+Umethod [29], which takes into accountspin-orbit coupling, and nondiagonal in spin contributionsinto the occupation matrix. The full-potential linearizedaugmented plane-wave (FLAPW) [30] basis is used in theself-consistent total energy calculations. We useU=4e VandJ=0.97 eV parameters taken from a similar compound ofmanganese [31].The dipole term is a classical contribution from dipole-dipole interaction of localized magnetic moments [32]. Forcoherent rotations of the two AFM sublattices which strictlymaintain their antiparallel alignment, e.g., one that interpolatesbetween the two magnetic configurations shown in Fig.1,t h edipole term depends in general on the rotation angle. Thisdependence is absent for cubic crystals but present in MnTesince the crystal symmetry of the NiAs structure is lower.This causes the energy of the dipole-dipole interaction of thestructure in Fig.1(b),w i t hm a g n e t i cm o m e n t sa l i g n e da l o n gthec-axis, to be higher than that of any structure with magneticmoments oriented in the hexagonal basal plane (cplane), e.g.,Fig.1(a).For lattice constantsa=0.4134 nm andc=0.6652 nm[experimentally determined at 5 K / see Sec.III,F i g s .3(a)and3(b)], we obtain that Mn atoms carry the magnetic momentsof 4.27µB(spinMS=4.25µBplus orbitalML=0.02µBmagnetic moments). The energy difference of the two differentconfigurations shown in Fig.1from the dipole termEdipoleis calculated to be 0.135 meV per unit cell, favoring thealignment in thecplane. This contribution to MAE is onlyweakly dependent on strain or relevant lattice distortions andgives no anisotropy within thecplane.The DFT+Ucalculations of the MCA are much moreinvolved but, rather generally, a clear picture emerges of mod-erately large out-of-plane anisotropy and small anisotropieswithin thecplane. For the lattice constants quoted above,an energy difference between configurations in Figs.1(a)and1(b)of 0.11 meV per unit cell is calculated again favoringthe alignment in thecplane. The anisotropy within thecplane, defined as the energy difference between the magneticstructure in Fig.1(a)and one with magnetic moments rotatedby 90◦in thecplane, is small and at the edge of the accuracy(10µeV) of the calculation in this particular case.To model actual conditions in our experiments, we performzero-temperature calculations ofEMCAfor various choicesof lattice constants (see TableI). Adding the MCA to thedipole term, we can conclude that (a) the out-of-plane MAE istypically between 0.2 and 0.3 meV per unit cell (two formulaunits), favoring the moments within thecplane, and (b) theanisotropy within thecplane is typically an order of magnitudesmaller. For calculations under changingc/aratio shown inTableI,t h eM A Ew i t h i nt h ecplane is always smaller thanthe out-of-plane MAE (even for the extreme choice of latticeconstants withc=0.689 nm, see TableI,t h el a t t e ri sg r e a t e rthan 0.1 meV per unit cell), the MAE within thecplane exhibitsno clear trend upon unit cell deformation and it even changessign. In order to unambiguously determine anisotropieswithin thecplanes, it is therefore advisable to resort toexperiments.TABLE I. The total MAE,Edipole+EMCAin meV per unit cell fordifferent lattice parameters. The Néel vector directions with respectto the crystal are given as subscript of the energies, showing thepreferential magnetic moment orientation in thecplane.a(nm) 0.408 0.411 0.414 0.417 0.408 0.408c(nm) 0.670 0.670 0.670 0.670 0.650 0.689E[0001]−E[11¯20]0.20 0.24 0.23 0.22 0.28 0.12E[1¯100]−E[11¯20]−0.01 0.03 0.01 0.04 0.05−0.01214418-2−12−8−404\nKΓAKHMLEnergy [eV]MnTe in hexagonal NiAs structure}}Te p-statesd-MnFIG. 1. Band structure of MnTe calculated by QSGW for\n~L?z(in-plane orientation). Note the competing maxima of\nthe valence band at \u0000 and Apoints.\nity and we will therefore focus on its VB.\nThe magnetic structure of MnTe was established11long\nago (see Fig. 3) with a strong anisotropy favouring in-\nplane orientation of the magnetic moments and a weak\nresidual anisotropy within the plane.12Recently, Mose-\nley et al.13have found by neutron di\u000braction that, upon\ndoping by lithium, the magnetic moments rotate out of\nplane. They also noticed that in the density of states\n(DOS), signi\fcant changes occur and we use the e\u000bective\nmodel to explain how the VB responds to this change of\nmagnetic order (once spin-orbit interaction is taken into\naccount). Even if the Mn d-states lie14deep below the\nFermi level EFand seem too remote from the VB top16\nwhich is built dominantly from p-Te orbitals, we demon-arXiv:2204.04206v3  [cond-mat.mtrl-sci]  16 Feb 20232\nstrate that the combination of MnTe layered structure\nand relativistic spin-orbit interaction (SOI) lead to an\nunusual sensitivity of the electronic structure to the ori-\nentation of magnetic moments. In the next Section we\ndiscuss the competing VB maxima and we focus on the\none near A-point of the Brillouin zone (BZ) in Sec. III.\nWe conclude in Sec. IV.\nII. COMPETING VB MAXIMA\nOnce the SOI is taken into account, there arises a tight\ncompetition between valence band maxima close to A\nand \u0000 points of the BZ, see Fig. 2b. A long-standing\nconsensus17,18that the former prevails has recently been\nchallenged by Yin et al. [21] who claim that the VB top\noccurs in the vicinity of \u0000 point. To improve on the\npotentially less accurate DFT approach,21we employ\nthe Quasiparticle Self-Consistent GW approximation19\n(QSGW). The GW approximation, which is an explicit\ntheory of excited states is widely used to predict quasi-\nparticle levels with better reliability than density func-\ntionals. QSGW is an optimized form of the GW ap-\nproximation, where the starting Hamiltonian is gener-\nated within the GW approximation itself, constructed\nso that it minimizes the di\u000berence between the one-body\nand many-body Hamiltonians. As a by-product the poles\nof the one-body Green's function coincide with the poles\nof the interacting one: thus energy band structures have\nphysical interpretation as quasiparticle levels, in marked\ncontrast to DFT approaches (some examples are given in\nSec. 5.1 of the Supplementary Information) where the\nauxillary Hamiltonian has no formal physical meaning\n(in practice Lagrange multipliers of this Hamiltonian are\ninterpreted as quasiparticle levels). In practice QSGW\nyields high \fdelity quasiparticle levels in most materials\nwhere dynamical spin \ructuations are not strong.20\nBulk lattice constants of MnTe at room temperature\narea= 0:414 nm and c= 0:671 nm;13we show in\nFig. 2d that for such c=a= 1:621, the VB maximum\nclose to the A point safely prevails (\u0001 Eis the di\u000berence\nbetween energy of local VB maxima close to \u0000 and that\nclose toA). Most experiments nowadays are performed\nwith thin \flms of MnTe, however, and then lattice con-\nstants depend on the choice of substrate. Temperature-\ndependent data in Fig. 3 of Ref. 12 suggest that while\nsamples grown on SrF 2surface still fall into the same\nclass, low temperatures may e\u000bectively push the VB max-\nima close to the \u0000 point up and in particular, samples\ngrown on the InP substrate may exhibit the inverted\nalignment of the VB maxima.\nComparing the present QSGW results to DFT calcu-\nlations of Ref. 21, several remarks are in order. Lat-\ntice constants used in that reference (which correspond\ntoc=a= 1:57) have been obtained by structure opti-\nmisation in DFT rather than from experimental data.\nNext, the hybrid functional HSE06 may avoid the known\nproblem of underestimated gaps in DFT but this in it-(a) (b)\n−3−2−1012\n\u0000A K H A\n−3−2−1012\nΓA KHA\n(c) (d)\n\u0000A\nAKH\n1.58 1.59 1.6 1.61 1.62 1.63 1.64−0.2−0.100.1 a=7.7\na=7.8\na=7.9\nFIG. 2. Detailed view of the VB maximum and (c) the\nBrillouin zone of \u000b-MnTe corresponding to lattice constants:\n0.414 nm and c=a= 1:6208. Panels (a,b) show the di\u000berence\nbetween SOI ignored and included and (d) gives for the latter\ncase the energy di\u000berence between VB maximum in \u0000 and A\ndepending on the c=aratio (for several values of agiven in\na.u.); origin of this dependence is discussed in Sec. 5.2 of the\nSupplementary Material. Negative \u0001 Emeans that the VB\nmaxima around A prevail and all energies are given in eV.\nself does not guarantee a reliable description of \fner de-\ntails of the band structure (such as VB maxima align-\nment). Predicted valence and conduction bands are\nmore uniformly reliable in GW than in density-functional\nmethods. Moreover QSGW surmounts the problematic\nstarting-point dependence that plagues the usual im-\nplementations of the GW approximation and therefore\nQSGW is a better choice for our study than DFT. Re-\ngarding the subsequent derivation of an e\u000bective model\nfor the VB around \u0000 point,21we note as follows. The\nkz= 0 approximation is used; while this would be ap-\npropriate for very thin layers (say 5 nm), present exper-\niments12are more likely behaving like 3D bulk. Also,\nthe e\u000bective model (1) in Ref. 21 assumes a \fxed di-\nrection of the magnetic moments; to plot the exper-\nimentally relevant 'angular sweeps', the current direc-\ntion rather than N\u0013 eel vector is rotated which is, how-\never, not the actual experimental protocol. For systems\nwhere only the non-crystalline anisotropic magnetoresis-\ntance (AMR) occurs,22the two protocols are equivalent\nbut measurements in the Corbino geometry12prove this\nassumption false. Being aware of these issues, we strive\nto derive an e\u000bective model in the following which is free\nof these shortcomings captures the dependence on mag-\nnetic moments direction.\nA proper symmetry analysis of the crystal structure3\nof MnTe provides the non-symmorphic space group D4\n6h.\nOnce AF ordering is included the Mn atoms must be\ntreated as inequivalent since each Mn layer would have\nspins pointing in the opposite direction as shown in Fig. 3\nfor in-plane spins. Hence, the symmetry group is re-\nduced from D 6hto D 3dwithout SOI (see for instance\nSandratskii et al.23). Furthermore, the symmetry group\nwould also depend on the interplay of SOI and choice of\nthe AF direction since spins pointing in di\u000berent direc-\ntions behave di\u000berently under symmetry operations. For\nexample, in the out-of-plane AF con\fguration, the sym-\nmetry remains D 3dwhile for in-plane AF, either along\n[10\u001610] or [11 \u001620] directions, the symmetry group is re-\nduced C 2h. Besides the conceptual analysis of the sym-\nmetries, independent calculations using the WIEN2k and\nQuantum Espresso ab initio packages also provide the\nsame symmetry groups discussed above. Thus, for the\nparticular choice of in-plane AF the D 2hpoint group dis-\ncussed by Yin et al. should be replaced by C 2h.\nIII. EFFECTIVE MODELS\nSeveral attempts to describe the electronic structure\nof\u000b-MnTe in a simpli\fed way have been made so far.\nHere, the k\u0001papproach4,24is a common choice for\nsemiconductors25especially if only high-symmetry points\nin the BZ are of interest. Such a model for the VB top in\nA point was derived more than 40 years ago23and later\nextended to a tight-binding scheme.26The latter allows\nfor the description of the energy bands over the whole\nBZ but neither of these models allows to analyse the de-\npendence of electronic structure on the directions on Mn\nmagnetic moments. In the perfectly ordered AFM phase\n(as in Fig. 2d) and without SOI, the Bloch functions at\nthe top of the valence band in the A point transform\nas the two-dimensional irreducible representation E gor\n(\u0000+\n3) of the the D 3d. Including corrections up k2and no\nSOI (essentially given by Eq. 2 in Sandratskii et al.23)\none would obtain the following Hamiltonian:\nHkp;2\u00022=\u0012ak2\nx+bk2\ny+ck2\nz(a\u0000b)kxky\n(a\u0000b)kxkybk2\nx+ak2\ny+ck2\nz\u0013\n:(1)\nThe inverse e\u000bective masses (proportional to a;b;c ) im-\nply that Fermi surfaces (FSs) are, at this level of approx-\nimation, two prolate ellipsoids (both doubly degenerate)\ntouching at the point where they are pierced by the A\u0000\nline; other properties of this model and its parameters,\ne\u000bective masses, extracted from \fts to QSGW are given\nin Sec. 2 of the Supplementary Material. From the point\nof view of magnetism, this is a consequence of neglect-\ning the spin-orbit interaction. Once SOI is included, the\nband dispersion will depend on the direction of magnetic\nmoments. On the other hand, if higher order terms in\n~kwere included, the symmetry would be lowered and\nFSs would become warped and spin-split.27Consequent\nspin order in reciprocal space, being a hallmark28of so\ncalled altermagnetism, can lead to phenomena normally\n-5-4-3-2-1 0 1 2\nka=-π 0 πFIG. 3. NiAs structure (left) reduced to a toy model com-\nprising a one-dimensional chain of magnetic (B) and non-\nmagnetic (A) atoms; there are two of the latter, A aand A b\nper unit cell. Energies are shown in the units of tand \u0001=t= 1,\n\u0000\u000fd=t= 3 was chosen.\nunexpected in collinear compensated magnets such as the\nanomalous Hall e\u000bect.\nThe derivation of Eq. (1) is based solely on symmetry\narguments and entails neither any explicit information\nabout orbital composition of the corresponding Bloch\nstates nor any parametric dependence on magnetic or-\nder. In the following, we therefore \frst describe a toy\nmodel capturing the essence of interplay between mag-\nnetism and orbital degrees of freedom and next, we make\nuse of these insights to derive a realistic model of MnTe.\nA. Toy model\nConsider a 1D chain of alternating nonmagnetic (A)\nand magnetic (B) atoms depicted in Fig. 3 where only\nthe nearest neighbours couple (the amplitude being t).\nThe single-orbital-per-site tight-binding Hamiltonian as-\nsuming that the B-atom orbitals have on-site energies\n\u000fd\u0006\u0001 (where 2\u0001 is the exchange splitting) reads\nH1(\u0001) =0\nBB@0t 0te\u0000ika\nt \u000f d+ \u0001t 0\n0t 0t\nteika0t \u000fd\u0000\u00011\nCCA(2)\nin the basis of Bloch states with momentum kso thatka\nranging from\u0000\u0019to\u0019parametrises the BZ.\nThe toy model described by H1can be treated analyt-\nically (see Ref. [29], Sec. 1) and focusing on the bands\nof dominantly A-atom orbital composition, we observe a\ndownfolded cosine band of width downscaled by factor\n\u0019t=\u000fdin the largej\u000fdjlimit, i.e. remote B-atom dom-\ninated bands, as the dashed black dispersion in Fig. 3\ncon\frms. Atoms A aand A binteract only through the\nintermediate (magnetic) atoms Bwhich suppresses their\ne\u000bective coupling. There are two main observations to\nmake at this point. First, even if \u0001 \u001c\u000fdthere opens\na gap in the 'VB states' at the BZ edge (to make the\ngap better visible, we chose a larger value of \u0001 =\u000fdfor\nthe blue and red bands in Fig. 3). This allows for the in-\nsight that, inasmuch the atom A bis sandwiched between4\nspin-up (left) and spin-down neighbours (right), where\nthe exchange coupling is \u0001 and \u0000\u0001, their e\u000bect on the\nA-band (blue in Fig. 3 at the bottom right panel) does\nnot average out to zero. Next, an even more important\ninsight concerns the eigenstates of H1atka=\u0006\u0019.\nAt this point, we should point out that H1of (2) in\nfact only describes one of the two spin species; let us\ndenote it as up-spin and correspondingly, H1;\"=H1(\u0001).\nThe two states at ka=\u0006\u0019split by nonzero \u0001 turn out\nto be (jai\u0006jbi)\nj\"i wherejaiandjbirefer to orbitals\nof A aand A batoms, respectively. For the spin-down\nsector,H1;#=H1(\u0000\u0001) which leads to identical band\nstructure as in Fig. 3 whose eigenstates are nevertheless\nnot the same as for H1;\". The state degenerate with\n(jai\u0006jbi)\nj\"i is (jai\u0007jbi)\nj#i and thus, we arrive\nat the conclusion that, at the BZ edge, the VB states in\nour toy model come in two pairs (split by the gap) and\nwithout loss of generality, we now focus on the subspace\nspanned by the pair\n(jai+jbi)\nj\"i;(jai\u0000jbi)\nj#i: (3)\nUnlike the pairj\"i;j#i(without any orbital part), any\nlinear combination of the two states in (3) has a zero\nexpectation value of transversal spin operators ^ \u001bx, ^\u001by.\nThis can also be restated as h^~ \u001bijjz, or, easily generalised\nto the statement that the states (3) have the (expectation\nvalue of) spin parallel to the magnetic moments of atomsB. In this way, the direction of magnetic moments of the\natoms remote in energy from the VB top in\ruences the\ncurrent-carrying states close to the Fermi energy. In the\nfollowing, we denote the direction of spin in the basis\nstate (jai+jbi)\nj\"i by~Land it can be understood as the\nN\u0013 eel vector. In the following, we explore this in\ruence\nin the context of spin-orbit interaction; an alternative\npathway relies on spin disorder15(as it occurs for example\nat \fnite temperatures) and we outline an approach to it\nbased on the coherent potential approximation (CPA) in\nthe Supplementary material. It provides an alternative\ninterpretation of 'magnetic blue shift'16of the gap which\ndoes not rely on many-body e\u000bects.\nB. Extension to MnTe crystal\nThe previous argument can be extended to Te px,py\nstates which form the VB top near A. To account for \fne\ndetails of the band structure (as explained in Ref. 29,\nSec. 2), we also include the remote pzlevels (in A, they\nare\u00193 eV below the VB top, see Fig. 2a) whose dis-\npersion is dropped at this level of approximation. Also\nnote that the group of VB maxima close to \u0000 relies on\nTepzorbitals as explained in the Supplementary ma-\nterial. We will measure energy from the VB top (as it\nappears in the case of absent SOI) with EFdenoting the\nFermi energy and use two copies of Eq. (1) to describe\nthekx;y-dependent mixing of px,pyorbitals.\nDenoting the position of pzorbitals of tellurium by ez(ez<0,jez=EFj\u001d1), the full description of the VB close\nto A is provided by a block-diagonal 6 \u00026 matrix\nHkp=0\nBBBBB@ak2\nx+bk2\ny+ck2\nz(a\u0000b)kxky 0\n(a\u0000b)kxkybk2\nx+ak2\ny+ck2\nz0\n0 0 ez\nak2\nx+bk2\ny+ck2\nz(a\u0000b)kxky 0\n(a\u0000b)kxkybk2\nx+ak2\ny+ck2\nz0\n0 0 ez1\nCCCCCA(4)\nand the \frst and second 3 \u00023 block is written in the basis (3) whereas inside the blocks, the basis vectors are simply\njpxi;jpyi;jpzi. Since the matrix (4) does not explicitly depend on ~L(only its basis vectors are), we arrive at the\nconclusion that (when SOI is ignored) the band structure does not depend on the direction of Mn magnetic moments.\nIn the limitjezj!1 , the full model (4) combined with SOI breaks down into two decoupled 2 \u00022 blocks and\nsince we now have a microscopic understanding of the basis, one which contains the information about direction of\nMn magnetic moments, the SOI can now be evaluated. With \fnite ez, the dependence of band splitting in A can be\nbetter described as we explain in the following.\nC. Spin-orbit interaction\nWe are now in a position to explain the following be-\nhaviour of band structure calculated by relativistic ab ini-\ntiomethods. In panel (b) of Fig. 2, we could have already\nobserved the bands split by SOI and, compared to band\nwidths, such splittings were small (note that these split-\ntings cause the shift of the VB top away from A, the pointof high symmetry). Those calculations were done assum-\ning~Lkxand, at this level of detail, depend only little\non the direction of ~Las long as~L?zwhich is compati-\nble with MnTe being an easy-plane material.12However,\nwhen~Lkzis assumed in calculations, see Fig. 4, band\nsplittings become sizable. Restricting our discussion to\nTepx,pyorbitals combined into the states (3), this be-\nhaviour is linked to the directionality of Hso=\u0015~l\u0001~ \u001b5\n−6−4−2024\nK ΓA KH M L\nFIG. 4. MnTe band structure for ~Ljjz(energies in eV). Colour\ncoding: red Te, blue/green Mn.\nevaluated in the corresponding basis:\nHso;2\u00022=\u0012\n0i\u0015cos\u0012\n\u0000i\u0015cos\u0012 0\u0013\n(5)\nwhere~L\u0001^z= cos\u0012and~lis the orbital angular momen-\ntum operator. Clearly, SOI projected to such a restricted\nspace becomes ultimately ine\u000bective for the in-plane ori-\nentation of magnetic moments where \u0012=\u0019=2 (taking\ninto account also the jpziorbital,29small splittings at\nAin the in-plane con\fguration can nevertheless be also\naccounted for) whereas for \fnite ez, the full 6\u00026 matrix\nofHsomust be considered instead of (5). On the other\nhand, for out-of-plane magnetic moments, the splitting at\nA seen in Fig. 4 can be directly compared to eigenvalues\n\u0006\u0015ofHso;2\u00022.\nIV. DISCUSSION AND CONCLUSIONS\nAn e\u000bective model of the MnTe valence band around\nA point of the BZ depends on the level of detail needed:\nEq. (1) is a meaningful approximation to begin with but\nit cannot describe the band-dispersion dependence on thedirection of Mn magnetic moments; the six-band (or four-\nband, corresponding to jezj!1 limit) description us-\ning Eq. (4) combined with Hsoevaluated with respect\nto basis (3) times jpxi;jpyi;jpziis the reasonable next\nstep. On this level, the large sensitivity of the valence\nband at A to magnetic moment orientation can be ex-\nplained in terms of zero matrix elements of Hsobetween\n(jai+jbi)\nj!i and (jai\u0000jbi)\nj i wherea;brefer\nto the two Te atoms within unit cell of MnTe. Zoom-\ning into the details of the valence band smaller than\n\u0018100 meV would require adding further terms such as\nthose discussed on p. 8 of the supplementary information\nto Ref. [27]; on this level of approximation, phenomena\nsuch as the anomalous Hall e\u000bect or AMR can then likely\nbe successfully modelled.\nCalculations in Fig. 4 show that the splitting at A is\nassociated with reduction of band gap in agreement with\nDFT calculations.13This implies that not only angular-\nresolved photoemission (ARPES) could be used to con-\n\frm the sensitivity of MnTe band structure to the orien-\ntation of Mn magnetic moments but also optical absorp-\ntion measurements should reveal signatures of this e\u000bect.\nSuch experiments could also con\frm our results concern-\ning the competition of valence band maxima close to the\nA and \u0000 points of the Brillouin zone.\nV. ACKNOWLEDGEMENTS\nWe acknowledge assistance of Swagata Acharya with\nQSGW calculations and a preliminary KKR survey by\nAlberto Marmodoro; funding was provided by grants 22-\n21974S and EU FET Open RIA No. 766566, M.v.S.\nwas supported by the DOE-BES Division of Chemical\nSciences under Contract No. DE- AC36-08GO28308\nand P.E.F.Jr. acknowledges \fnancial support from the\nAlexander von Humboldt Foundation, Capes (Grant No.\n99999.000420/2016-06) and Deutsche Forschungsgemein-\nschaft SFB 1277 (Project No. ID314695032, subprojects\nB05, B07 and B11).\n1Fabien Tran et al., J. Appl. Phys. 126, 110902 (2019)\n2M. Grumet et al., Phys. Rev. B 98, 155143 (2018).\n3O.K. Andersen and O. Jepsen, Phys. Rev. Lett. 53, 2571\n(1984).\n4R. Winkler, Spin-orbit Coupling E\u000bects in Two-\nDimensional Electron and Hole Systems , (Springer, New\nYork, 2003).\n5J. M. Luttinger, Phys. Rev. 102, 1030 (1956).\n6M. Tanaka, Jpn. J. Appl. Phys. 60, 010101 (2021).\n7T. Jungwirth et al., Rev. Mod. Phys. 86, 855 (2014).\n8C.H. Marrows and B.J. Hickey, Phil. Trans. R. Soc. A 369,\n3027 (2011).\n9V. Baltz et al., Rev. Mod. Phys. 90, 015005 (2018).\n10Apart from the NiAs-type structure of MnTe, zinc-blende\nphase also exists which is AFM at low temperatures.\n11T. Komatsubara et al. '63, doi: 10.1143/JPSJ.18.35612D. Kriegner et al., Phys. Rev. B 96, 214418 (2017).\n13D.H. Moseley et al., Phys. Rev. Mat. 6, 014404 (2022).\n14H. Sato et al., Sol. St. Comm. 92, 921 (1994).\n15R. Baral et al., Matter, 5, pp. 1853-1864 (2022).\n16D. Bossini et al., New J. Phys. 22, 083029 (2020).\n17M. Podgorny and J. Oleszkiewicz, J. Phys. C: Sol. St. Phys.\n16, 2547 (1983).\n18More recent studies range from S. J. Youn et al., phys.\nstat. sol. (b) 241, 1411 (2004) to Mu et al., Phys. Rev.\nMat. 3, 025403 (2019) and this VB maxima alignment is\nalso consistent with our previous DFT+U calculations.12\n19T. Kotani et al., Phys. Rev. B 76, 165106 (2007).\n20M. van Schilfgaarde and Takao Kotani and S. Faleev, Phys.\nRev. Lett. 96, 226402 (2006).\n21G. Yin et al., Phys. Rev. Lett. 122, 106602 (2019).\n22A.W. Rushforth et al., Phys. Rev. Lett. 99, 147207 (2007).6\n23L.M. Sandratskii et al., phys. stat. sol. (b) 104, 103 (1981).\n24L. C. Lew Yan Voon, M. Willatzen, Thek\u0001pmethod:\nelectronic properties of semiconductors , (Springer, Berlin,\n2009).\n25P. E. Faria Junior et al., Phys. Rev. B 93, 235204 (2016).26J. Ma\u0014 sek, B. Velick\u0013 y and V. Jani\u0014 s, J. Phys. C: Solid State\nPhys. 20, 59 (1987).\n27Libor \u0014Smejkal, Jairo Sinova, and T. Jungwirth, Phys. Rev.\nX 12, 031042 (2022).\n28I. Mazin et al., Proceedings of Nat. Acad. Sci. USA 118\n(42) e2108924118.\n29Supplementary information."    },    {        "title": "2204.05073v1.DFT_calculation_of_intrinsic_properties_of_magnetically_hard_phase_L1__mathrm__0___FePt.pdf",        "content": "DFT calculation of intrinsic properties of magnetically hard phase L1 0FePt\nJoanna Marciniaka,b,∗, Wojciech Marciniaka,c, Mirosław Werwińskia\naInstitute of Molecular Physics, Polish Academy of Sciences, M. Smoluchowskiego 17, 60-179 Poznań, Poland\nbInstitute of Materials Engineering, Poznan University of Technology, Piotrowo 3, 60-965 Poznań, Poland\ncInstitute of Physics, Poznan University of Technology, Piotrowo 3, 60-965 Poznań, Poland\nAbstract\nDue to its strong magnetocrystalline anisotropy, FePt L1 0phase is considered as a promising magnetic recording media\nmaterial. Althoughthe magnetic propertiesof thisphasehave already been analyzed manytimesusingdensity functional\ntheory (DFT), we decided to study it again, emphasizing on full potential methods, including spin-polarized relativistic\nKorringa-Kohn-Rostoker (SPR-KKR) and full-potential local-orbital (FPLO) scheme. In addition to the determination\nof exact values of the magnetocrystalline anisotropy constants K1andK2, the magnetic moments, the Curie temperature,\nand the magnetostriction coefficient, we focused on the investigation of the magnetocrystalline anisotropy energy (MAE)\ndependence on the magnetic moment values using the fully relativistic fixed spin moment (FSM) method with various\nexchange-correlation potentials. We present nearly identical MAE( m) curves near the equilibrium point, along with\ndifferent equilibrium values of MAE and magnetic moments. For a magnetic moment reduced by about 10%, we\ndetermined a theoretical MAE maximum in the ground state (0 K) equal to about 20.3 MJm−3and independent of the\nchoice of the exchange-correlation potential form. These calculations allow us to understand the discrepancies between\nthe previous MAE results for different exchange-correlation potentials.\n1. Introduction\nIn 2011, the interest in rare-earth-free hard magnetic\nmaterials increased significantly due to the dramatic rise\nin prices of rare earth elements known as the rare earth\ncrisis [1]. Several articles summarize results of research\non magnetic materials free of rare earths, among oth-\ners, see Refs. [2] and [3]. One of the promising mate-\nrials is the ordered L1 0FePt alloy, for which measure-\nments at 4.2 K revealed an unusually high value of mag-\nnetocrystalline anisotropy constant K1of 1.92 meVf.u.−1\n(11.0 MJm−3) [4]. Another interesting property of the\nFePt L1 0phase, concerning permanent-magnet applica-\ntions, is the relatively high Curie temperature ( TC) of\nabout 775 K estimated for the bulk material from experi-\nments on nanoparticles [5]. In the literature, the crystal-\nlographic parameters of the tetragonal FePt L1 0structure\nvary:a= 3.85–3.88 Å, c= 3.74–3.79 Å, and the ratio\nc/a=0.966–0.981[6–10]. Valuesofthetotalspinmagnetic\nmoments in the range 3.06–3.24 µBper formula unit were\nobtained to date [9, 11–13]. The magnetostriction coeffi-\ncientλ001variesbetweenapproximately3.4 ×10−5[14]and\n6.4×10−5[15] for disordered films of FePt alloys. For com-\nparison, the highest value of magnetostriction coefficient,\nequal to 1.1–1.4×10−3, was recorded for the Terfenol-D\nalloy [16].\n∗Corresponding author\nEmail address: joanna.marciniak@ifmpan.poznan.pl (Joanna\nMarciniak)\nFigure 1: Crystal structure of FePt L1 0phase, in face-centered\ntetragonal (gray lines) and body-centered tetragonal (black lines)\nrepresentations. The latter was used for calculations in this work.\nVestacode was used for visualization [17].\nOur aim in the following work is twofold. Firstly, we\nwanttopresentathoroughbenchmarkstudyregardingthe\nbasic magnetic properties – magnetocrystalline anisotropy\nenergy (MAE), magnetocrystalline anisotropy constants\nK1andK2, magnetic moments [11, 12] and Curie tem-\nperature [18] – of this interesting, already well studied\nphase. To obtain these parameters, we utilize two den-\nsity functional theory (DFT) codes and a possibly broad\nspectrum of exchange-correlation functionals (described in\nSec. 2). Secondly and more importantly, we try to broaden\nthe understanding of the magnetic properties of the pre-\nsented alloy by fixed spin moment analysis and magne-\ntostrictioncoefficientcalculations, thoughthecomparative\nPreprint submitted to Elsevier April 12, 2022arXiv:2204.05073v1  [cond-mat.mtrl-sci]  11 Apr 2022experimental results of the latter present in literature are\nsparse [14], especially for the bulk alloy [15].\n2. Calculations’ details\nIn this work, ab initio calculations were performed us-\ning FPLO18, FPLO5 (full-potential local-orbital) [19, 20],\nand SPR-KKR 7.7.1 (spin-polarized relativistic Korringa-\nKohn-Rostoker) codes [21, 22].\nThe FPLO18 code was used for calculations necessary\nto determine spin and orbital magnetic moments, MAE,\nmagnetocrystalline anisotropy constants K1andK2, and\nthe magnetostriction coefficient. Dependence of MAE on\nthe spin magnetic moment was performed using the fixed\nspin moment (FSM) method.\nThe FPLO5 code was used as a base for the mate-\nrial’s Curie temperature calculations, and the SPR-KKR\ncode was used to obtain TC, MAE and magnetocrystalline\nanisotropy constants for cross-reference. The main rea-\nson for the use of FPLO5 and SPR-KKR, compared to\nFPLO18, is that both codes have the chemical disorder\nimplemented by means of the coherent potential approx-\nimation (CPA) [23], which allows utilizing the approach\ndescribed further to obtain Curie temperature with rela-\ntively low computational effort.\nThe high precision of FPLO is due, among other\nfactors, to the implementation of a full-potential ap-\nproach that does not incorporate shape approximation\ninto the crystalline potential and to the expansion of ex-\ntended states in terms of localized atomic-like basis or-\nbitals [20, 24]. The use of full potential is particularly\nessential for the accurate determination of such a subtle\nquantity as MAE, and the MAE results obtained for FePt\nusing this method are considered among the most accu-\nrate [12]. SPR-KKR 7.7.1 also has the ability for full po-\ntential calculation, which, oppositely to FPLO, does not\nlie in the basic principles of the code.\nAll calculations were performed after a proper opti-\nmization of the system geometry utilizing Perdew-Burke-\nErnzerhof (PBE) exchange-correlation functional [25] in\ncorresponding codes. The calculations were made for the\nbody-centered representation of the FePt L1 0structure,\nsee Fig. 1, belonging to the P4/mmmspace group, with\nFe atoms occupying the (0, 0, 0) position and Pt atoms\noccupying the (0.5, 0.5, 0.5) site. Lattice parameters of\na= 3.88/√\n2Å andc= 3.73Å, derived from face-centered\ntetragonal (fct) structure of Alsaad et al.[9], were used\nas initial values for geometry optimization. Final lattice\nparameters used in further calculations are: a= 2.74≈\n3.87/√\n2Å,c= 3.76Å anda= 2.74≈3.87/√\n2Å,\nc= 3.74Å, optimized with FPLO and SPR-KKR codes,\nrespectively, see Fig. 2.\nMain calculations were performed using the Perdew-\nBurke-Ernzerhof (PBE) (FPLO18 and SPR-KKR), Vosko-\nWilk-Nusair (VWN) (SPR-KKR) [26] and Perdew-\nWang 92 (PW92) (FPLO18 and FPLO5) [27] exchange-correlation potentials. Additional results of Curie temper-\nature and MAE dependence on the magnetic moment were\nobtained in FPLO5 and FPLO18, respectively. In both\ncodes, we used additionally von Barth-Hedin (vBH) [28]\nand Perdew-Zunger (PZ) [29] potentials together with the\nexchange only approximation.\nAll calculations in FPLO18 were performed in the fully\nrelativistic approach for a 56×56×42k-point mesh with\nenergy convergence criterion at the level of 10−8Ha and\nelectrondensityconvergencecriterionof 10−6,whichyields\nanaccuratevalueofthesystem’sMAEwithinareasonable\ncomputation time. Curie temperature calculations were\nperformed in the scalar-relativistic approach in CPA for a\n12×12×12k-point mesh and the same energy and density\nconvergence criteria.\nThe convergence criterion at the level of 10−6and\n5 000 k-points per reduced Brillouin zone, around 75 000\nk-points in a full Brillouin zone, were used to perform\nfullyrelativisticcalculationsinSPR-KKR.Allcalculations\nwere carried out in the full-potential (FP) approach with\nthe angular momentum cut-off criterion set at the level\noflmax= 5 (parameter NL = 6 in the SPR-KKR config-\nuration file) unless otherwise stated. The atomic sphere\napproximation (ASA) calculations in SPR-KKR were per-\nformed for comparison, and all computational parameters\nwere consistent between both approaches.\nThe Curie temperature of the system was determined\nbased on the disordered local moment (DLM) theory [30]\nand assuming that the energy difference between the ferro-\nmagnetic state (ordered spin magnetic moments) and the\nparamagnetic state (disordered spin magnetic moments –\nmodelled based on two types of CPA atomic sites with an-\ntiparallel magnetic moments) is proportional to the ther-\nmal energy needed for the ferromagnetic–paramagnetic\ntransition. In order to determine the Curie temperature,\nwe use the formula [31, 32]:\nkBTC=2\n3EDLM−EFM\nn, (1)\nwhereEDLMis the total energy of the DLM configuration,\nEFMis the total energy of the ferromagnetic configuration,\nnis the total number of magnetic atoms, and kBis the\nBoltzmann constant.\nMAE was determined by the formula:\nMAE =E(θ= 90◦)−E(θ= 0◦), (2)\nwhereθis the angle between the magnetization direction\nand the c axis. We determined the hard magnetic axis\nin the computational cell to be [1 1 0] ([1 0 0] in the\nstandard face-centered representation). However, differ-\nence between energy values in the x-yplane is insignifi-\ncant in our case and omitted in further study. In order\nto determine the magnetocrystalline anisotropy constants\nK1andK2, the dependence MAE( θ) was used, which for\na tetragonal cell takes the approximated form:\nMAE =K1sin2θ+K2sin4θ. (3)\n2 0 2 4 6 8 1055.0 55.5 56.0 56.5 57.0 57.5V (Å3)\n0.00.51.01.52.02.5\n0.955 0.960 0.965 0.970 0.975 0.980 0.985Energy (meV f.u.-1)\nc/a ratioFigure 2: Optimization of the FePt L1 0phase geometry in FPLO18\ncodeusingthePBEpotential. Energyminimumshiftedtothelowest\nobtained energy. The top figure (red line) depicts the energy versus\nvolumedependenceforfixed c/aratioequal to0.972, andthe bottom\nfigure (blue line) presents the energy versusc/aratio for constant\nvolume equal to 56.42 Å3(for fct cell). Obtained lattice parameters\narea= 3.87 Å and c= 3.76 Å, see the fct structure presented in\nFig. 1.\nIn order to calculate the magnetostriction coefficient\nλ001, MAE and energy dependence on the lattice parame-\ntercwere fit with linear and quadratic functions, respec-\ntively [33, 34], as presented in Fig. 5. The relation between\nenergy and MAE versusmagnetization angle θcan be pre-\nsented as follows:\nE(θ= 0◦) =αc2+βc+γ;\nE(θ= 90◦) =αc2+βc+γ+ MAE(c).(4)\nConsidering the lattice parameter elongation deriva-\ntive of the MAE for the optimized lattice parameter ( c=\n3.76Å), the magnetostriction coefficient λ001takes the\nform:\nλ001=−2\n3d(MAE)\ndc\nβ. (5)\n3. Results and Discussion\nBefore the actual calculations, the geometry of the\ncrystal structure was optimized. The procedure consisted\nof a simple least-squares third-order function fit to the\nE(V)dependency under a constant c/aratio and then\nthe same kind of fit to the energy dependence E(c/a)at\na constant volume corresponding to the minimum energy\n 0 5 10 15 20\n0 30 60 90 120 150 180 0  0.5  1  1.5  2  2.5  3Energy (MJ m-3)\nAngle θ (°)Angle θ (rad)\n 0  0.5  1  1.5  2  2.5  3Angle θ (rad)\nFPLO PBE\nSPR-KKR PBE ASA\nSPR-KKR PBE FP\nSPR-KKR VWN FP\nSPR-KKR VWN ASA\nSPR-KKR VWN ASA l max=3Figure 3: Energy of the L1 0FePt phase, as a function of the an-\ngle between magnetization direction and the c axis, calculated with\nvarious exchange-correlation potentials in FPLO18 and SPR-KKR\n7.7.1. Exact values of magnetocrystalline anisotropy constants K1\nandK2are presented in Tablee 1. SPR-KKR lmaxcutoff was set to\n5, unless stated otherwise.\ndevised before. For the FPLO18 with PBE, the equilib-\nrium volume of the fct cell Veqis equal to 56.42Å3and the\nequilibrium c/aratio is equal to 0.972, see Fig. 2. For the\nSPR-KKR code and PBE exchange-correlation functional,\navalue is lower by 0.5 h, whereascis smaller by 7.2 h\nthan that resultant from FPLO calculations. All results\nprovide very good agreement with previous experimental\nand calculations results [6–10].\nFor FPLO18-PBE, the spin magnetic moment ms=\n2.95µBand the orbital magnetic moment ml= 0.065µB\nwere observed on Fe atoms. On Pt atoms, these moments\nwerems= 0.22µBandml= 0.057µB. For values obtained\nusing methods other than the FPLO18 code with PBE, see\nTable 1. The sum of magnetic moments in the cell varies\nbetween 3.17 and 3.512 µB, a discrepancy of about 10%\nrelative to either of the values. The calculated magnetic\nmoments, presentedalsointheTable1, areconsistentwith\nthe values published previously [9, 11–13].\nAsmentioned in Sec.2, Curietemperatureswerecalcu-\nlated with the DLM method using Eq. 1 for all exchange-\ncorrelation potentials implemented in FPLO5 and for\nVWN and PBE exchange-correlation potentials in SPR-\nKKR. Two approaches were used to simulate the param-\nagnetic phase. In the first one, we assume no Pt contri-\nbution to the total magnetic moment and introduce coex-\nisting antiparallel spin orientations on the Fe atom only,\ni.e. Fe 0.5↑Fe0.5↓Pt. In the second approach, we introduce\ncoexisting antiparallel spin orientations on both Fe and Pt\natoms, i.e. Fe 0.5↑Fe0.5↓Pt0.5↑Pt0.5↓.TCvalues obtained\nusing both approaches are mostly identical, as in both\ncases, the Pt atoms are going to be in a non-magnetic\nstate (magnetic moments equal to or near 0 µB), as ex-\npected. All Pt moments are presented in Table 2.\n3Table 1: Spin magnetic moments ms, orbital magnetic moments ml, and total magnetic moments m(inµB), as well as magnetocrystalline\nanisotropy energy MAE (in MJm−3, and meVf.u.−1) and magnetocrystalline anisotropy constants K1andK2(in MJm−3) of the L1 0FePt.\nThe values of MAE, K1andK2in MJm−3from [11] were recalculated by authors. SPR-KKR lmaxcutoff was set to 5, unless otherwise\nstated.\nms,Feml,Fems,Ptml,Ptm K 1K2MAE (MJm−3) MAE (meVf.u.−1)\nFPLO PBE 2.95 0.065 0.22 0.057 3.28 15.3 0.39 15.66 2.76\nFPLO PW 92 2.85 0.067 0.24 0.057 3.21 – – 17.01 2.99\nFPLO PZ 2.81 0.067 0.24 0.057 3.17 – – 17.71 3.12\nFPLO vBH 2.81 0.067 0.24 0.057 3.17 – – 17.69 3.12\nFPLO Exchange only 3.07 0.066 0.22 0.056 3.41 – – 13.03 2.29\nSPR-KKR PBE ASA 3.07 0.072 0.33 0.046 3.51 16.4 0.77 17.14 2.99\nSPR-KKR PBE FP 2.98 0.064 0.33 0.047 3.42 16.7 0.47 17.17 3.00\nSPR-KKR VWN ASA 2.95 0.071 0.33 0.041 3.39 21.4 -0.13 21.29 3.72\nSPR-KKR VWN FP 2.89 0.066 0.34 0.046 3.33 17.9 0.76 18.63 3.25\nSPR-KKR VWN2.91 0.073 0.32 0.043 3.35 17.6 0.29 17.98 3.14ASAlmax= 3\nSPR-KKR VWN2.83 0.065 0.34 0.044 3.28 17.5 0.54 17.96 3.01ASAlmax= 3[11]\nVASP PBE [12] 2.83 0.056 0.39 0.044 3.32 15.3 0.74 15.7 2.74\nResults of calculated Curie temperatures vary consid-\nerably for different exchange-correlation potentials, from\nabout 430 K to 900 K. All of them are also presented\nin Table 2. Overall, the magnetic moments on Pt in the\nDLM stable states are minor, if any, and has a negligible\nphysical impact on the TCvalue. As shown in Table 2, in\nall considered cases the magnetic moment on Pt atom is\nbelow numerical accuracy. TC’s obtained with the VWN\npotential are low, suggesting that this form of exchange-\ncorrelation potential in the local density approximation is\nnot adequate to properly capture interatomic exchange in-\nteractions and thus TCvalues. On the other hand, FPLO\nexchange only approach seems to be sufficient. We were\nunable to obtain a stable DLM state using SPR-KKR with\nPBE exchange-correlation potential, so TCfor this case\nwas not included in Table 2. The results, except for the\none obtained in SPR-KKR VWN, agree well with the esti-\nmation from experiment by Rong et al.(775 K) [18]. How-\never, the DLM approach is known to overestimate values\nofTCconsiderably [22, 31], so slightly higher values could\nbe expected.\nThe magnetocrystalline anisotropy constants K1and\nK2are summarized in Table 1, and the results obtained\nare also shown graphically in Fig. 3, along the fits of Eq. 3\nfor each data series. These data series were performed\nfor the angle between the easy magnetization axis and de-\nfined magnetization direction in the /angbracketleft0;π/angbracketrightrange. Depen-\ndencies are symmetrical in relation to θ=π\n2, as expected.\nEq. 3 was fitted to the whole angular range to improve the\naccuracy of the fit. Presented values do not differ signifi-\ncantly, exceptforthecalculationswiththeSPR-KKRcode\nusing the VWN potential in ASA. This difference means\nthat the calculations using this potential depend relativelyTable 2: Curie temperatures and total magnetic moments on Pt\natomsinL1 0Fe0.5↑Fe0.5↓Pt0.5↑Pt0.5↓obtainedusingSPR-KKRand\nFPLO5 codes.\nTC(K)mPt(µB)\nSPR-KKR VWN ASA 426 0\nSPR-KKR VWN 595 0\nFPLO5 Exchange only 878 <10−6\nFPLO5 PW92 796 <10−6\nFPLO5 PZ 754 <10−6\nFPLO5 vBH 756 0\nstrong on the applied approach – ASA or FP, as men-\ntioned earlier. Out of other results, these from FPLO18\nand FP PBE SPR-KKR are the lowest, indicating better\nagreement to experiment. The difference between FPLO\nand SPR-KKR can be ascribed to the implemented basis.\nMoreover, our convergence tests suggested that the mini-\nmal useable angular momentum expansion in the system\nislmax= 4 (parameter NL = 5 in the SPR-KKR config-\nuration file), and lmax = 5, used in our work, gave yet\nfurther noticeable improvement. Taking into account the\nabove, we are convinced that LDA-ASA approach may be\ninsufficienttocapturethemagnetismofL1 0FePtproperly,\nwhich is exemplified by our result for SPR-KKR VWN ap-\nproach in a higher basis of lmax= 5.\nAll obtained magnetocrystalline anisotropy constants\n4 5 10 15 20\n2.6 2.8 3.0 3.2 3.4MAE (MJ m-3)\nMagnetic moment m (µB)Exchange only\nvon Barth Hedin\nPerdew and Zunger\nPerdew Wang 92\nPerdew Burke Ernzerhof 96Exchange only\nvon Barth Hedin\nPerdew and Zunger\nPerdew Wang 92\nPerdew Burke Ernzerhof 96Figure 4: Fixed spin moment MAE dependencies (lines) resultant\nfrom the calculations with different exchange-correlation potentials\nperformed in FPLO18. Equilibrium magnetocrystalline anisotropy\nvalues versusequilibrium total magnetic moments are marked with\nempty symbols.\nagree with earlier results of Khan et al. [K1=\n17.2MJm−3(3.01 meVf.u.−1),K2= 0.53MJm−3(0.09\nmeVf.u.−1)] and Wolloch et al.[K1= 15.3MJm−3(2.67\nmeVf.u.−1),K2= 0.74MJm−3(0.13meVf.u.−1)][11,12].\nHowever, the obtained magnetocrystalline anisotropy con-\nstants with K1values in range 15.3–21.4 MJm−3dif-\nfer significantly from the values obtained experimentally\n(K1= 11MJm−3) [4]. Origin of the difference can be\nconnected with imperfections of both, calculations and ex-\nperiments. Weaknesses of calculations are the impossibil-\nityofaccuratelydeterminationoftheexchange-correlation\npotential and difficulty in the exact evaluation of spin-\norbit interactions [35]. On the other hand, experiments\nwere performed on nanoparticles or bulk samples with in-\nhomogeneous microstructures that impact obtained prop-\nerties [36]. Thin films are deposited on substrates, which\ncan induce strain in the examined structure and introduce\nsome differences in properties [35]. The most important\nfact is that the ground state is not measured in experi-\nments [37], so these values cannot be directly compared.\nIt was checked by Daalderop et al.[35] and Strange et\nal.[37] that all the approximations in the intrinsic prop-\nerties calculations can not account for the discrepancy.\nThe combination of the FSM method with fully rela-\ntivistic calculations further allows testing the dependence\nof the MAE on the total magnetic moment, which allows\nfor determining a hypothetical maximum MAE depend-\ning on the magnetic moment for a given material. The\nMAE values obtained with various exchange-correlation\npotentials in FPLO18 code are 13.03–17.71 MJm−3and\ncorrelate with the total magnetic moment in the range\n3.17–3.41µBf.u.−1.\nTreating L1 0FePt FePt as a weak ferromagnet, MAE\nversusFSMdependencycouldbemappedtotheMAE ver-\nsustemperature dependency [4, 38], employing the Callen\nand Callen model [39]. Our analysis of the total Fe 3 dandPt5ddensityofstatesatthemajorityspinchannelpointed\nus towards a conclusion that treating L1 0FePt as a weak\nferromagnet can be, at least, ambiguous – and with the ar-\ntificially decreased total moment situation gets even more\ncomplex. Mryasov et al.have earlier shown that a precise\nclassification of FePt could be not so simple [40]. More-\nover, standard DFT as a zero kelvin model, omits a range\nof other temperature-dependent phenomena. Concluding,\nthe mentioned procedure applied to other magnetic mate-\nrials [38] and experimental data regarding FePt [4] could\nnot be reliably reproduced by the DFT method.\nFig. 4 presents nearly identical MAE( m) dependencies\nobtainedneartheequilibriumvaluewithvariousexchange-\ncorrelation potentials, except for the exchange only ap-\nproach. However, exact values of the equilibrium total\nmagnetic moment differ between the employed potentials.\nThis fact leads us to the conclusion, that observed wide\ndistribution of the calculated MAE values for L1 0FePt\nin[12]canbeexplainedbythefactthatdifferentexchange-\ncorrelation potentials lead to different values of magnetic\nmoment. Ontheotherhand, thereisasystempropertyin-\ndependent of the choice of exchange-correlation potential,\nindicated by all considered exchange-correlation potential\n(except exchange only approach), which is an MAE max-\nimum of about 20.3 MJm−3observed for magnetic mo-\nment of about 2.9 µB. In all cases, the equilibrium MAE\nresults obtained for the optimized structure lie to the right\nof the MAE maximum. Hence, the modifications leading\nto the reduction of the L1 0FePt magnetization should in-\ncrease the MAE to the hypothetical maximum value of\n20.3 MJm−3at 0 K. A reduction in magnetization on the\norder of a few percent can be achieved, for example, by re-\nplacing a few percent of Fe with related elements exhibit-\ning lower magnetic moment or completely non-magnetic,\nsuch as, for example, the 3 delements Ni, Ti or V. Alter-\nnatively, one can try to slightly increase the Pt content at\nthe expense of Fe or doping FePt with elements such as C,\nB and N, which should locate in the interstitial gaps.\nIt is worth noting that one would expect a monotonic\nincrease in magnetic moments with decreasing tempera-\nture, leading to an MAE maximum at a few kelvins [4].\nTaking this fact into account, for room temperature appli-\ncations, the goal should be to increase the magnetic mo-\nment of L1 0FePt, which should correspond to an increase\nof MAE.\nThe magnetostriction coefficient λ001, determined from\nthe Eq. 5, is 4.3×10−4. Magnetostriction coefficient λS\nequals 3.4×10−5–6.4×10−5for disordered FePt thin\nfilms [14, 15]. Hence, the calculated value is an order of\nmagnitude greater than that determined from the experi-\nments, which is an expected behaviour, as the calculations\nconducted by us relate to a perfect infinite single crystal.\nCompared to Terfenol-D (1.1-1.4 ×10−3[16]), the calcu-\nlated magnetostriction coefficient of the FePt L1 0struc-\nture is an order of magnitude lower.\n50.00.51.01.52.02.5\n3.72 3.74 3.76 3.78 3.802.42.52.62.72.82.93.0Energy (meV f.u.-1)\nMAE (meV f.u.-1)\nLattice parameter c (Å)Figure 5: Dependence of the calculated total energy and magne-\ntocrystalline anisotropy energy on the lattice parameter of L1 0FePt\nccalculated with FPLO18 code.\n4. Summary and Conclusions\nThe magnetocrystalline anisotropy constants K1and\nK2, magnetic moments, and Curie temperatures were cal-\nculated for L1 0FePt using full-potential methods and,\nwhere possible, compared with equivalent ASA calcula-\ntions. The results are in good agreement with the litera-\nture values of other calculations, though reproduce well-\nknown discrepancy between DFT calculations and exper-\niments. We have shown that the MAE results calcu-\nlated with different exchange-correlation potentials corre-\nlate with the magnetic moment values, which explains the\nobserved dispersion of the values determined so far. The\ncorrelation of MAE( m) FSM results near the equilibrium\nstate is exceptional. For the magnetic moment reduced\nby about 10%, we determined a theoretical maximum in\nMAE of 20.3 MJm−3, which is about 30% higher than\nthe theoretical equilibrium ground state value. We also\nderived the Curie temperature of L1 0FePt using multi-\nple exchange-correlation functionals in the DLM approach\nand found it to be in decent agreement with the experi-\nmental value. The evaluated magnetostriction coefficient\nλ001is in reasonable agreement with the measured values\nof disordered thin films of FePt alloy.\nAcknowledgements\nWe acknowledge the financial support of the Na-\ntional Science Centre Poland under the decision DEC-\n2018/30/E/ST3/00267. The computations were in the\nmain part performed on resources provided by the Poz-\nnan Supercomputing and Networking Center (PSNC). We\nthank Paweł Leśniak and Daniel Depcik for compiling\nthe scientific software and administration of the comput-\ning cluster at the Institute of Molecular Physics, Polish\nAcademy of Sciences. We thank Ján Rusz for valuable\ndiscussion and suggestions.References\n[1] K. Bourzac, The rare-earth crisis, Technol. Rev. (114) (2011)\n58–63.\n[2] M. D. Kuz’min, K. P. Skokov, H. Jian, I. Radulov, O. Gut-\nfleisch, Towards high-performance permanent magnets without\nrare earths, J. Phys. Condens. Matter 26 (6) (2014) 064205.\ndoi:10.1088/0953-8984/26/6/064205 .\n[3] R. Skomski, J. M. D. Coey, Magnetic anisotropy — How much\nis enough for a permanent magnet?, Scr. Mater. 112 (2016) 3–8.\ndoi:10.1016/j.scriptamat.2015.09.021 .\n[4] N. Hai, N. Dempsey, D. Givord, Magnetic properties of hard\nmagnetic FePt prepared by cold deformation, IEEE Trans.\nMagn. 39 (5) (2003) 2914–2916. doi:10.1109/TMAG.2003.\n815762.\n[5] C.-b. Rong, D. Li, V. Nandwana, N. Poudyal, Y. Ding, Z. L.\nWang, H. Zeng, J. P. Liu, Size-Dependent Chemical and Mag-\nnetic Ordering in L1 0-FePt Nanoparticles, Adv. Mater. 18 (22)\n(2006) 2984–2988. doi:10.1002/adma.200601904 .\n[6] P.Ravindran,A.Kjekshus,H.Fjellvåg,P.James,L.Nordström,\nB.Johansson, O.Eriksson, Largemagnetocrystallineanisotropy\nin bilayer transition metal phases from first-principles full-\npotential calculations, Phys. Rev. B 63 (14) (2001) 144409.\ndoi:10.1103/PhysRevB.63.144409 .\n[7] Z. Lu, R. V. Chepulskii, W. H. Butler, First-principles study\nof magnetic properties of L1 0-ordered MnPt and FePt alloys,\nPhys. Rev. B 81 (9) (2010) 094437. doi:10.1103/PhysRevB.81.\n094437.\n[8] T. J. Klemmer, N. Shukla, C. Liu, X. W. Wu, E. B. Sved-\nberg, O. Mryasov, R. W. Chantrell, D. Weller, M. Tanase, D. E.\nLaughlin, Structural studies of L1 0FePt nanoparticles, Appl.\nPhys. Lett. 81 (12) (2002) 2220–2222. doi:10.1063/1.1507837 .\n[9] A. Alsaad, A. A. Ahmad, T. S. Obeidat, Structural, elec-\ntronic and magnetic properties of the ordered binary FePt,\nMnPt, and CrPt 3alloys, Heliyon 6 (3) (2020) e03545. doi:\n10.1016/j.heliyon.2020.e03545 .\n[10] H. B. Luo, W. X. Xia, A. V. Ruban, J. Du, J. Zhang, J. P.\nLiu, A. Yan, Effect of stoichiometry on the magnetocrystalline\nanisotropy of Fe–Pt and Co–Pt from first-principles calculation,\nJ. Phys. Condens. Matter 26 (38) (2014) 386002. doi:10.1088/\n0953-8984/26/38/386002 .\n[11] S. Ayaz Khan, P. Blaha, H. Ebert, J. Minár, O. Šipr, Magne-\ntocrystalline anisotropy of FePt: A detailed view, Phys. Rev. B\n94 (14) (2016) 144436. doi:10.1103/PhysRevB.94.144436 .\n[12] M. Wolloch, D. Suess, P. Mohn, Influence of antisite defects and\nstacking faults on the magnetocrystalline anisotropy of FePt,\nPhys. Rev. B 96 (10) (2017) 104408. doi:10.1103/PhysRevB.\n96.104408 .\n[13] T. Burkert, O. Eriksson, S. I. Simak, A. V. Ruban, B. Sanyal,\nL. Nordström, J. M. Wills, Magnetic anisotropy of L1 0FePt\nand Fe 1-xMnxPt, Phys. Rev. B 71 (13) (2005) 134411. doi:\n10.1103/PhysRevB.71.134411 .\n[14] F. E. Spada, F. T. Parker, C. L. Platt, J. K. Howard, X-ray\ndiffraction and Mössbauer studies of structural changes and L1 0\nordering kinetics during annealing of polycrystalline Fe 51Pt49\nthin films, J. Appl. Phys. 94 (8) (2003) 5123–5134. doi:10.\n1063/1.1606522 .\n[15] J. Aboaf, T. McGuire, S. Herd, E. Klokholm, Magnetic, trans-\nport, and structural properties of iron-platinum thin films,\nIEEE Trans. Magn. 20 (5) (1984) 1642–1644. doi:10.1109/\nTMAG.1984.1063260 .\n[16] L. Sandlund, M. Fahlander, T. Cedell, A. E. Clark, J. B.\nRestorff, M. Wun-Fogle, Magnetostriction, elastic moduli, and\ncoupling factors of composite Terfenol-D, J. Appl. Phys. 75 (10)\n(1994) 5656–5658. doi:10.1063/1.355627 .\n[17] K. Momma, F. Izumi, VESTA : A three-dimensional vi-\nsualization system for electronic and structural analysis, J.\nAppl. Crystallogr. 41 (3) (2008) 653–658. doi:10.1107/\nS0021889808012016 .\n[18] C.-B. Rong, Y. Li, J. P. Liu, Curie temperatures of annealed\nFePt nanoparticle systems, J. Appl. Phys. 101 (9) (2007)\n09K505. doi:10.1063/1.2709739 .\n6[19] H. Eschrig, M. Richter, I. Opahle, Chapter 12 - Relativistic\nSolidStateCalculations, in: P.Schwerdtfeger(Ed.), Theoretical\nandComputationalChemistry, Vol.14ofRelativisticElectronic\nStructure Theory, Elsevier, 2004, pp. 723–776. doi:10.1016/\nS1380-7323(04)80039-6 .\n[20] K. Koepernik, H. Eschrig, Full-potential nonorthogonal local-\norbital minimum-basis band-structure scheme, Phys. Rev. B\n59 (3) (1999) 1743–1757. doi:10.1103/PhysRevB.59.1743 .\n[21] H. Ebert, B. L. Gyorffy, M. Battocletti, D. Benea, M. Kosuth,\nJ. Minar, A. Perlov, V. Popescu, The Munich SPR-KKR Pack-\nage, Version 7.7 (2017).\n[22] H. Ebert, D. Ködderitzsch, J. Minár, Calculating condensed\nmatter properties using the KKR-Green singles function\nmethod—recent developments and applications, Rep. Prog.\nPhys. 74 (9) (2011) 096501. doi:10.1088/0034-4885/74/9/\n096501.\n[23] P. Soven, Coherent-Potential Model of Substitutional Disor-\ndered Alloys, Phys. Rev. 156 (3) (1967) 809–813. doi:10.1103/\nPhysRev.156.809 .\n[24] H. Eschrig, 2. The Essentials of Density Functional Theory\nand the Full-Potential Local-Orbital Approach, in: W. Herg-\nert, M. Däne, A. Ernst (Eds.), Computational Materials Sci-\nence: From Basic Principles to Material Properties, Lecture\nNotes in Physics, Springer, Berlin, Heidelberg, 2004, pp. 7–21.\ndoi:10.1007/978-3-540-39915-5_2 .\n[25] J. P. Perdew, K. Burke, M. Ernzerhof, Generalized Gradient\nApproximation Made Simple, Phys. Rev. Lett. 77 (18) (1996)\n3865–3868. doi:10.1103/PhysRevLett.77.3865 .\n[26] S. H. Vosko, L. Wilk, M. Nusair, Accurate spin-dependent elec-\ntron liquid correlation energies for local spin density calcula-\ntions: Acriticalanalysis, Can.J.Phys.58(8)(1980)1200–1211.\ndoi:10.1139/p80-159 .\n[27] J. P. Perdew, Y. Wang, Accurate and simple analytic repre-\nsentation of the electron-gas correlation energy, Phys. Rev. B\n45 (23) (1992) 13244–13249. doi:10.1103/PhysRevB.45.13244 .\n[28] U. von Barth, L. Hedin, A local exchange-correlation potential\nfor the spin polarized case. i, J. Phys. C Solid State Phys. 5 (13)\n(1972) 1629–1642. doi:10.1088/0022-3719/5/13/012 .\n[29] J. P. Perdew, A. Zunger, Self-interaction correction to density-\nfunctional approximations for many-electron systems, Phys.\nRev. B 23 (10) (1981) 5048–5079. doi:10.1103/PhysRevB.23.\n5048.\n[30] J. Staunton, B. L. Gyorffy, A. J. Pindor, G. M. Stocks, H. Win-\nter, The “disordered local moment” picture of itinerant mag-\nnetism at finite temperatures, J. Magn. Magn. Mater. 45 (1)\n(1984) 15–22. doi:10.1016/0304-8853(84)90367-6 .\n[31] B. L. Gyorffy, A. J. Pindor, J. Staunton, G. M. Stocks, H. Win-\nter, A first-principles theory of ferromagnetic phase transitions\nin metals, J. Phys. F Met. Phys. 15 (6) (1985) 1337–1386.\ndoi:10.1088/0305-4608/15/6/018 .\n[32] L. Bergqvist, P. H. Dederichs, A theoretical study of half-\nmetallic antiferromagnetic diluted magnetic semiconductors, J.\nPhys. Condens. Matter 19 (21) (2007) 216220. doi:10.1088/\n0953-8984/19/21/216220 .\n[33] R. Wu, A. J. Freeman, Spin–orbit induced magnetic phenomena\nin bulk metals and their surfaces and interfaces, J. Magn. Magn.\nMater. 200 (1) (1999) 498–514. doi:10.1016/S0304-8853(99)\n00351-0.\n[34] M. Werwiński, W. Marciniak, Ab Initio study of magnetocrys-\ntalline anisotropy, magnetostriction, and Fermi surface of L1 0\nFeNi (tetrataenite), J. Phys. Appl. Phys. 50 (49) (2017) 495008.\ndoi:10.1088/1361-6463/aa958a .\n[35] G. H. O. Daalderop, P. J. Kelly, M. F. H. Schuurmans, First-\nprinciples calculation of the magnetocrystalline anisotropy en-\nergy of iron, cobalt, and nickel, Phys. Rev. B 41 (17) (1990)\n11919–11937. doi:10.1103/PhysRevB.41.11919 .\n[36] L.-W. Liu, C.-C. Hu, Y.-C. Xu, H.-B. Huang, J.-W. Cao,\nL. Liang, W.-F. Rao, Magnetic properties of L10FePt thin\nfilm influenced by recoverable strains stemmed from the polar-\nization of Pb(Mg 1/3Nb2/3)O3–PbTiO 3substrate, Chin. Phys.\nB 27 (7) (2018) 077503. doi:10.1088/1674-1056/27/7/077503 .[37] P. Strange, J. B. Staunton, B. L. Györffy, H. Ebert, First princi-\nples theory of magnetocrystalline anisotropy, Phys. B Condens.\nMatter 172 (1) (1991) 51–59. doi:10.1016/0921-4526(91)\n90416-C.\n[38] A. Edström, M. Werwiński, D. Iuşan, J. Rusz, O. Eriksson,\nK. P. Skokov, I. A. Radulov, S. Ener, M. D. Kuz’min, J. Hong,\nM. Fries, D. Y. Karpenkov, O. Gutfleisch, P. Toson, J. Fidler,\nMagnetic properties of (Fe 1-xCox)2B alloys and the effect of\ndoping by 5d elements, Phys. Rev. B 92 (17) (2015) 174413.\ndoi:10.1103/PhysRevB.92.174413 .\n[39] H. B. Callen, E. Callen, The present status of the temperature\ndependence of magnetocrystalline anisotropy, and the l(l+1)2\npower law, J. Phys. Chem. Solids 27 (8) (1966) 1271–1285. doi:\n10.1016/0022-3697(66)90012-6 .\n[40] O. N. Mryasov, U. Nowak, K. Y. Guslienko, R. W. Chantrell,\nTemperature-dependent magnetic properties of FePt: Effective\nspin Hamiltonian model, EPL 69 (5) (2005) 805. doi:10.1209/\nepl/i2004-10404-2 .\n7"    },    {        "title": "2204.07384v1.First_principles_Calculation_of_Magnetocrystalline_Anisotropy_of_Y_Co_Fe_Ni_Cu___5__Based_on_Full_potential_KKR_Green_s_Function_Method.pdf",        "content": "First-principles Calculation of Magnetocrystalline Anisotropy of\nY(Co,Fe,Ni,Cu) 5Based on Full-potential KKR Green's Function\nMethod\nHaruki Okumura and Hisazumi Akai\nInstitute for Solid State Physics, The University of Tokyo, Kashiwa 277-8581, Japan\nTetsuya Fukushima\nInstitute for Solid State Physics, The University of Tokyo, Kashiwa 277-8581, Japan and\nInstitute for AI and Beyond, The University of Tokyo,\nBunkyo-ku, Tokyo 113-8656, Japan\nMasako Ogura\nDepartment of Chemistry and Physical Chemistry,\nLMU Munich, Butenandtstrasse 11, D-81377 Munich, Germany\n(Dated: April 18, 2022)\n1arXiv:2204.07384v1  [cond-mat.mtrl-sci]  15 Apr 2022Abstract\nThe performance of permanent magnets YCo 5can be improved by replacing cobalt with other\nelements, such as iron, copper, and nickel. In order to determine its optimum composition, it is\nnecessary to perform systematic theoretical calculations in a consistent framework. In this study,\nwe calculated the magnetocrystalline anisotropy constant Kuof Y(Co 1\u0000x\u0000yFexCuy)3(Co1\u0000zNiz)2\non the basis of the full-potential Korringa-Kohn-Rostoker Green's function method in conjunction\nwith the coherent potential approximation. The calculated Kuof YCo 5was smaller than the\nexperimental value because of a missing enhancement due to orbital polarization. Although the\nvalue of Kuof Y(Co 1\u0000x\u0000yFexCuy)3(Co1\u0000zNiz)2was systematically underestimated compared to\ntheir experimental counterparts, the doping e\u000bect can be analyzed within a consistent framework.\nThe results have shown that YFe 3Co2has much higher Ku= 5:00 MJ =m3than pristine YCo 5\n(Ku= 1:82 MJ =m3), and that nickel as a stabilization element decreases Kuand magnetization in\nYFe 3(Co1\u0000zNiz)2. However, the anisotropy \feld of z\u00180:5 can compete with the value of YCo 5.\nI. INTRODUCTION\nPermanent magnets have been widely used in many applications, including magnetic\nmemory devices, wind power generators, electric vehicles, etc. As permanent magnets are\nalmost always used above room temperature, their Curie temperature ( TC) should be as\nhigh as possible. In addition, high magnetization ( Js) and high uniaxial magnetocrystalline\nanisotropy constants ( Ku) are also essential for high-performance permanent magnets. Fer-\nromagnetic transition metal elements like iron and cobalt usually exhibit high TCandJs.\nLarge axial anisotropy originates from rare-earth elements with a strong spin-orbit coupling\nof 4f-electrons.\nOne of the most extensively used permanent magnets is Nd-Fe-B alloy, which has high\nuniaxial anisotropy and coercivity [1, 2]; however, its limitation is low TCwithout the ad-\ndition of dysprosium (an expensive element) (Fig.4 in Ref.[3]). Another permanent magnet\nused is SmCo 5, which has high TC\u00181000 K and Ku\u001817:2 MJ=m3[4, 5]. Although its\nmagnetic properties do not decay even at high temperatures, its low-temperature magnetiza-\ntion is smaller than that of Nd 2Fe14B and Sm 2Fe17N3, as can be seen by the stoichiometric\ncomposition ratios of the transition metals and rare-earths. Although cobalt is more ex-\npensive than iron, various studies investigate improving the performance of SmCo 5because\n2samarium is less costly than neodymium and has high-temperature tolerance.\nYCo 5, which has the same crystal structure as SmCo 5, has moderate performance as\na magnet with experimental values of Ku\u00186:5 MJ=m3,TC= 978 K and magnetic mo-\nment of 8:3\u0016B/f.u. [4, 6{8]. The ferromagnetism and large anisotropy are mainly due to\ncobalt rather than yttrium, which does not have 4 f-electrons. Since yttrium is less ex-\npensive than neodymium, YCo 5magnets are preferred for mass production. In general,\nwhen the anisotropy comes mainly from the 3 d-electron of the cobalt sublattice instead of\n4f-electrons, the in\ruence of temperature on the magnetic properties is reduced. Magnetic\ninteractions between cobalt atoms are more signi\fcant than 3 d-4finteractions. In alloys\nwith 4f-electrons, the anisotropy rapidly decreases as temperature increases because of their\nweak magnetic interactions [9, 10]. In YCo 5, however, the experiment indicated that the\nanisotropy hardly changes in the vicinity of the room temperature [4, 7].\nIn the past decades, the anisotropy and magnetic properties of pristine YCo 5have been\ntheoretically calculated [11{24]. Some researchers replaced a fraction of cobalt with iron\nand/or copper to enhance the axial anisotropy of YCo 5, [18{24], and others investigated\nthe magnetic properties of YCo 5\u0000xNix[22, 25{28]. Landa et al: presented that proposed\nYFe 3(Co 1\u0000zNiz)2has higher energy products than YCo 5, where iron increases the magneti-\nzation and nickel thermodynamically stabilizes the 1-5 phases [28]. Many theoretical calcu-\nlations for both pristine and doped YCo 5are available; however, there are few systematic\nstudies within a consistent framework.\nIn this study, we have systematically calculated KuandJsof Y(Co 1\u0000x\u0000yFexCuy)3(Co 1\u0000zNiz)2\non the basis of the full-potential Korringa-Kohn-Rostoker (FPKKR) Green's function\nmethod [29]. The anisotropic potential is more realistic than spherical potentials, such\nas mu\u000en-tin potential or the potential obtained by the atomic sphere approximation. The\npotential in disordered alloys is treated in the framework of the coherent potential approxi-\nmation (CPA) [30, 31], within which the atomic sites of impurities are thought to be ideally\nrandomized.\nII. METHOD\nThe FPKKR method is based on Green's function formulation, where the multiple-\nscattering problem is solved [29]. The potential includes a non-spherical part. The space is\n3divided into Voronoi cells, and the CPA [30, 31] is employed to consider the disorder in the\npotentials. In contrast to the supercell method, the CPA enables the addition of arbitrary\nconcentrations of impurities to the host material without enlarging the unit cell.\nThe exchange-correlation functionals are chosen as the local density approximation (LDA)\nparameterized by Moruzzi-Janak-Williams [32] and the generalized gradient approximation\n(GGA) parameterized by Perdew-Burke-Ernzerhof [33]. Spin-orbit interaction is included\naslzszand the orbital polarization enhancement is not considered. In this study, therefore,\nthe orbital moment of cobalt atoms could be smaller than experimental values [34, 35] or\nother calculations with orbital polarization correction [18]. It is known that calculated\nKu's of YCo 5-based alloy may be systematically more diminutive than their experimental\ncounterparts.\nThe number of k-points in the \frst Brillouin zone is 2197. The maximum orbital angular\nmomentum cuto\u000b lmaxof each element was set to 3. The width of the energy contour for\ncomplex integration was set to 1.0 Ry, where 4 porbitals of yttrium are treated as the core.\nThe space group of hexagonal CaCu 5-type YCo 5is No.191 (P6=mmm ), and cobalt atoms\noccupy two inequivalent Wycko\u000b positions 3 g- and 2c-sites. From a crystallographic point\nof view, the spatial volume occupied by the 2 c-site is smaller than that of the 3 g-site. The\natomic radius of nickel (0.124 \u0017A) is smaller than that of cobalt (0.125 \u0017A), and thus nickel\ncan easily enter the 2 c-site [26, 36, 37]. In contrast, the atomic radii of iron and copper are\n0.126 \u0017A and 0.128 \u0017A, respectively. Hence, iron and copper can occupy the 3 g-site [36].\nWe considered 726 di\u000berent disordered compounds; Y(Co 1\u0000x\u0000yFexCuy)3(Co 1\u0000zNiz)2,\nwhere the concentration x;y, andzare varied by 0.1 under 0 \u0014x+y\u00141 and 0 \u0014x;y;z \u00141.\nWe optimized the lattice parameters of YCo 5, YCo 3Ni2, YFe 3Co2, YFe 3Ni2, YCu 3Co2, and\nYCu 3Ni2within LDA using the Vienna Ab initio Simulation Package (VASP) [38], and\nsummarized the lattice parameters in Table I. The relative change of the lattice parameter\nwith the addition of the third element is consistent with the relationship between the large\nand small atomic radii mentioned above. Lattice parameters for the remaining compounds\nwere determined using Vegard's law with optimized values of the above six materials. The\ndetermined lattice parameters are reasonable except for Y(Co 1\u0000zNiz)5(0:4<z), where the\nmeasured lattice parameters are not monotonically changed with increasing nickel content\n[27, 36]. Vegard's law is also applicable to the lattice constant of iron doping cases accord-\ning to the experimental values of YCo 5\u0000xFex(x <0:5) [39]. It is revealed that the lattice\n4TABLE I. Lattice parameters optimized by VASP with the LDA functional. The uniaxial mag-\nnetocrystalline anisotropy energy ( Ku), saturation magnetization ( Js), and anisotropy \felds ( Ha)\nof YCo 5and pseudo-binary alloys calculated using the full-potential KKR method. Except for\nYCu 3Ni2, all listed compounds are ferromagnetic.\na[\u0017A]c[\u0017A]c=a Volume [ \u0017A3]Ku[meV/f.u.] Ku[MJ/m3]Js[T]Ha[T]\nYCo 5 4.749 3.787 0.798 73.98 0.84 1.82 1.03 4.49\nYFe 3Co24.756 3.872 0.814 75.82 2.37 5.00 1.12 11.42\nYCu 3Co24.785 3.980 0.832 78.91 1.72 3.48 0.27 33.34\nYCo 3Ni24.724 3.834 0.812 74.11 0.49 1.05 0.65 4.11\nYFe 3Ni24.730 3.919 0.828 75.93 0.29 0.62 0.78 2.01\nYCu 3Ni24.760 4.027 0.846 79.01 | | 0.00 |\nconstant determined using LDA is typically underestimated owing to the overestimation of\nthe binding energy of 3 dorbitals. In addition to LDA calculations, we also performed GGA\ncalculations to con\frm the framework's accuracy.\nThe magnetocrystalline anisotropy energy is de\fned as Ku=E100\u0000E001, whereE100\n(E001) is the total energy of alloys magnetized along the [100] ([001]) direction in a hexagonal\nunit cell. The positive Kucorresponds to the uniaxial anisotropy. The dependence of\nthe calculated Kuon the lattice parameters is discussed in Sec. III A. The 3 g-site in the\nP6=mmm is separated into three inequivalent sites when the quantization axis is rotated by\n90 degrees.\nIII. RESULTS AND DISCUSSIONS\nA. YCo 5\nIn Table I, the optimized lattice volume of YCo 5in LDA was 73.9 \u0017A, which is close to\nthe volume of the \frst-order Lifshitz transition [40], but the calculated electronic structure\nwas in a high-spin state with a total magnetic moment of 6.52 \u0016B/f.u. (Js= 1:03 T). The\nmoment was smaller than the experimental values, which were in the range of 7.9-8.3 \u0016B/f.u.\n[4, 7, 41]. This might be probably due to the less orbital polarization in the calculation.\n5As mentioned in Refs.[24, 42], anisotropy is strongly a\u000bected by both the crystal structures\nand choice of exchange-correlation functional. Fig. 1 shows the lattice parameter dependence\nofKuusing FPKKR with the LDA functional, where both parameters aandc=aare changed.\nKuincreases as aincreases. The Kucalculated with optimized lattice parameters ( a= 4:749\n\u0017A,c=a= 0:798) was smaller than the Kucalculated with the experimental parameters ( a=\n4:948\u0017A,c=a= 0:803 [35]). Even though we used the experimental parameters, the calculated\nKuwas underestimated compared to the experimental Ku. This underestimation is related\nto the underestimated orbital polarization of cobalt [34, 35]. In the subsequent calculations,\n 1.5 2 2.5 3 3.5 4 4.5\n 4.75  4.8  4.85  4.9  4.95  5  5.05  5.1  5.15  5.2         Ku [MJ/m3]\na [Å]c/a=0.811\nc/a=0.803\nc/a=0.795\nc/a=0.798YCo5 (LDA)\nFIG. 1. Calculated Kuof YCo 5by full-potential KKR with the LDA functional as the function of\nthe lattice parameter a. The \flled circle is the Kucalculated using experimental c=a= 0:8034 [35],\nand the triangles are the Kucalculated using changed c=aby\u00061% from the experimental value.\nThe vertical dotted line is the experimental lattice constant, which was found to be a= 4:948\u0017A\n[35]. The open circle is the Kucalculated using the optimized lattice parameters shown in Table I.\n6FIG. 2. Calculated total and projected density of states of the transition metal 3 dorbital for\n(a) YCo 5, (b) YFe 3Co2, (c) YCu 3Co2, (d) YCo 3Ni2, (e) YFe 3Ni2, and (f) YCu 3Ni2using the\nfull-potential KKR method with the LDA functional. Fermi level is set to zero. It is shown that\nYCu 3Ni2is in paramagnetic states with Js= 0 T.\nwe used the optimized parameters and their interpolated values by using Vegard's law to\nperform the calculations in a consistent way.\nFig. 2(a) shows the calculated total density of states of YCo 5, along with the projected\ndensity of states for 3 dcomponents of cobalt atoms. In YCo 5, strong ferromagnetism was\nrealized, as in hcp cobalt. The shape of the density of states was similar to those reported in\n7-2-1 0 1 2 3 4 5 6 7\n 0  0.2  0.4  0.6  0.8  1\nx for iron contentY(Co1-xFex)3Co2LDA\nGGAKu [MJ/m3](a) (b)\n 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4\nY(Co1-xFex)3Co2LDA\nGGA\n 0  0.2  0.4  0.6  0.8  1\nx for iron contentJs [T]FIG. 3. (a) Calculated uniaxial anistoropy constant Kuand (b) magnetization Jsof\nY(Co 1\u0000xFex)3Co2as a function of iron content using the full-potential KKR method with LDA\nand GGA. Calculated values of YCo 5areKu= 1:82 (3.68) MJ =m3andJs= 1:03 (1.06) T for LDA\n(GGA).\nprevious studies [40, 43, 44]. In the minority spin states, the peak at the Fermi level mainly\ncontributes to cobalt 3 dstates. In Fig. 2 (b)-(f), the calculated total density of states for\ndoped YCo 5are shown together with the projected density of states of the 3 dorbital of\ntransition metals at both 2 c- and 3g-sites. Apart from YCu 3Ni2, all the compounds are\nin the ferromagnetic con\fguration with the \fnite magnetization listed in Table I. Fig. 2\n(c) and (d) show strong ferromagnetic character in spite of the addition of third elements\nin contrast to the cases of Fig. 2 (e) and (f). In YFe 3Ni2of Fig. 2(e), all cobalt has been\nsubstituted by other elements, and no distinct large exchange splitting can be identi\fed. In\nFig. 2(f), the Fermi level rises and almost all 3 d-bands are occupied (3 d8:76and 3d9:48for\nnickel and copper), leading to a non-magnetic state in YCu 3Ni2.\nB. Y(Co 1\u0000xFex)3Co2\nFig. 3(a) shows the calculated Kuof Y(Co 1\u0000xFex)3Co2as a function of iron content\nusing FPKKR with LDA and GGA. The LDA result shows two peaks of Kuatx\u00180:5 and\n8x\u00181:0. The increase in the anisotropy in low concentrations of iron ( x<0:5) is plausibly\nillustrated in the crystal \feld theory [45]. According to this model, cobalt atoms at the 3 g-\nsite contribute to the planer anisotropy and replacing them with iron atoms causes a large\nuniaxial anisotropy. This has been con\frmed for iron at the 3g-site is selectively replaced\nwith cobalt, as well as with uniform replacement at both the 2 c- and 3g-sites [23]. The \frst\npeak is also found for rigid band calculations [18], and the calculations employing virtual\ncrystal approximation [19].\nHowever, the experimental peak is found at around x= 0:17 [46], which is smaller than\nthe present calculation. The possible reasons for this discrepancy are threefold: (1) the site\npreference of iron atoms, (2) the di\u000berent lattice constants used in the present calculation,\nand (3) the choice of exchange-correlation functional. Firstly, the iron-site dependence of Ku\nin YCo 5\u0000xFex(x<0:1) is theoretically investigated in Ref. [23], where the Kuwith iron at\nthe 3g-site is more signi\fcant than that at the 2 c-site. The second is that the dependency of\nKuon the iron concentration is di\u000berent from that with larger lattice constants. The LDA\ncalculation with GGA optimized structures has a peak at x= 0:2 with the FPKKR method\n(not shown). A peak at x\u00180:2 is reported in the LDA calculation by using the lattice\nconstant obtained from experiments [23]. The third reason is evident from Fig. 3(a). In the\nGGA calculation, the peak found at x= 0:2 is moved to a lower concentration region than\nthe peak position of LDA. Hence, the peak position is located on the high iron concentration\nside only when the LDA calculation is performed using the lattice constants optimized with\nLDA.\nIn the GGA case of YCo 5, the absolute value of Ku= 3:68 MJ=m3(a= 4:91\u0017A,c=a=\n0:80) is higher than that of the LDA even when the expanded lattice constants are used,\nas shown in Fig. 1. In Fig. 3(a), it is shown that there is only one peak in the GGA\ncalculation. In the region above x= 0:2, the anisotropy constant monotonically decreases\nwith increasing iron concentration. From the GGA calculations, the uniaxial anisotropy of\nYFe 3Co2is weaker than that of YCo 5. The lower value of anisotropy in YFe 3Co2is also\ncon\frmed by earlier theoretical calculations using GGA [28].\nFig. 3(b) shows the calculated Jsas a function of iron content. Unexpectedly, in the LDA,\nJsdecreases with increasing iron until x= 0:4; the reason for this is not identi\fed. However,\nthis trend is not found in the GGA calculation, where the Jsmonotonically increases and is\nmore signi\fcant than Jsin the LDA case. Therefore, the decrease in magnetization in the\n9-2-1 0 1 2 3 4 5 6 7\n 0  0.2  0.4  0.6  0.8  1\ny for copper contentKu [MJ/m3]LDA\nGGA\nY(Co1-yCuy)3Co2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4\n 0  0.2  0.4  0.6  0.8  1\ny for copper contentJs [T]LDA\nGGA\nY(Co1-yCuy)3Co2(a) (b)\n 3.6 3.8 4 4.2 4.4\n 0.8  0.82  0.84  0.86  0.88  0.9  0.92  0.94  0.96  0.98  1\ny for copper contentKu [MJ/m3]FIG. 4. (a) Calculated uniaxial anisotropy constant Kuand (b) magnetization Jsof\nY(Co 1\u0000yCuy)3Co2as a function of copper content using the full-potential KKR method with\nLDA and GGA. The inset in (a) shows the detailed copper dependence of Kuat high copper\nconcentration using GGA.\nlow iron concentration is only seen for the LDA calculations and small lattice constants. In\nthe case of LDA, higher iron concentrations ( x>0:4) increase the magnetization, which can\nbe attributed to iron's more considerable spin magnetic moments, like Fe-Co binary alloys.\nIn Y(Co 1\u0000xFex)5system, we con\frmed that the peak of the Slater-Pauling curve appears\nwhen the number of d-electrons larger than that of YFe 3Co2. In addition, the peak has been\ntheoretically predicted in YCo 4Fe [20].\nC. Y(Co 1\u0000yCuy)3Co2\nIn Table I, the uniaxial anisotropy remains in the YCu 3Co2(y= 1:0), where cobalt atoms\nat the 3g-site are fully substituted for copper. From a crystallographic point of view, there\nare two layers: one includes the 3 g-site with copper, and the other includes yttrium and\nthe 2c-site with cobalt. Since the local moment of copper is negligibly small (0.001 \u0016B),\nthe ferromagnetism in the system is carried by the cobalt with the moment of 0.94 \u0016B.\nTherefore, the anisotropy of the material is mainly due to the magnetic interaction within\n10the layer containing the 2 c-site, not by the magnetic interaction along the perpendicular\ndirection.\nFig. 4(a) shows the calculated Kuof Y(Co 1\u0000yCuy)3Co2as a function of copper by using\nLDA and GGA. In the LDA, Kuof Y(Co 1\u0000yCuy)3Co2increases as copper increases until\nit takes maximum at y\u00180:6. This contradicts the fact that the experimental Kuof\nYCo 3Cu2(y= 0:67) is smaller than the Kuof YCo 5[47]. This discrepancy may be due\nto the preferential site occupancy of copper. In the low copper concentration, however, the\ntheoretical calculations revealed that a larger Kuis caused by adding copper to the 2 c-site\nrather than the 3 g-site [23]. Hence, the small experimental value is probably not explained\nby the site preference of copper alone.\nCompared to the LDA calculation, the peak in the GGA calculation is shifted to the lower\ncopper concentration, like the iron-doped case in Fig. 3(a). At y\u00180:67, the calculated Ku\nis still larger than that of pristine YCo 5(y= 0). As already mentioned, this enhancement\nof anisotropy contradicts the experimental results in the LDA cases [47]. In contrast, we\ncan estimate Kuof YCo 4Cu from the anisotropy \feld ( Ha) at 0 K, which is extrapolated\nfrom theHafrom the \fnite temperature [23, 47]. The experiment indicates that the Kuis\nenhanced by copper addition, at least in the low copper concentration region ( y\u00140:33).\nEven though the peak locations are di\u000berent, the present LDA and GGA calculations also\ncon\frm the increase in the anisotropy. The inset \fgure in Fig. 4(a) shows the detailed Ku\nat high copper concentrations from GGA. A minimum is observed at y\u00180:93, and the\nfurther increase in copper enhances the Ku. The calculated Ku= 3:6 MJ=m3aty= 0:93 is\na minimum value but is still comparable to the Kuof YCo 5. However, such material with\nhigh copper concentration is unsuitable for magnets because the magnetization becomes\nextremely small. The magnetization calculated for LDA and GGA, shown in Fig. 4(b),\ndecreases drastically as the copper content increases.\nD. YCo 3(Co1\u0000zNiz)2\nFig. 5(a) shows the Kuof YCo 3(Co 1\u0000zNiz)2as a function of nickel calculated using LDA\nand GGA. The chemical trend in LDA is almost the same as that of GGA except for the\nanomalous value at z= 0:6. The calculations show that a small amount of nickel enhances\ntheKu. A similar enhancement in low nickel concentration can be theoretically obtained\n11 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4\n 0  0.2  0.4  0.6  0.8  1\nz for nickel contentJs [T]\nLDA\nGGAYCo3(Co1-zNiz)2\n-2-1 0 1 2 3 4 5 6 7\n 0  0.2  0.4  0.6  0.8  1\nz for nickel contentKu [MJ/m3]\nLDA\nGGA YCo3(Co1-zNiz)2(a) (b)FIG. 5. (a) Calculated uniaxial anistoropy constant Kuand (b) magnetization Jsof\nYCo 3(Co1\u0000zNiz)2as a function of nickel content obtained using the full-potential KKR method\nwith LDA and GGA.\nin Gd(Co 1\u0000zNiz)5[48]. The theoretical calculation indicates that the nickel at the 2 c-site\nenhances the Kuin the gadolinium system. For yttrium, the enhancement of anisotropy has\nbeen analyzed using the band \flling model in the rigid band picture [12]. An enhancement\nin anisotropy by the electron addition is similar to the case of copper shown in Fig. 4(a).\nHowever, the experiment [37, 49] demonstrated the monotonic decrease of Kuas nickel con-\ntent increases. This discrepancy is attributed to the preferential occupation of nickel. As the\nnickel concentration increases, it occupies the 3 g-site as well. In fact, the neutron experiment\nindicates that the composition of x= 0:2 for Y(Co 1\u0000xNix)5is Y(Co 0:86Ni0:14)3(Co 0:71Ni0:29)2\n[37]. Therefore, the present assumption that the nickel occupied only the 2 c-site does not\nhold for higher nickel concentration.\nFig. 5(b) shows Jsof YCo 3(Co 1\u0000zNiz)2calculated by using LDA and GGA. The values\nofJsmonotonically decrease as the nickel content increases. This dependency corresponds\nto the slope of the right half of the Slater-Pauling curve, as in the cases of copper addition.\nThe present calculation underestimates the magnetization as compared to the experiment\n[27] but qualitatively reproduces the nickel concentration dependence of the experimental\nmagnetization.\n12In contrast to iron addition in Fig. 3(b), the calculated magnetization shows less signi\f-\ncant di\u000berences between GGA and LDA. This might be due to the existence of the signi\fcant\nstates near the Fermi level in the density of states of YCo 5shown in Fig. 2(a). Iron substi-\ntution shifts the Fermi energy to lower energy, which signi\fcantly changes the contribution\nof the signi\fcant part and a\u000bects the anisotropy in the case of LDA (Fig. 2(b)). In contrast,\nnickel substitution can suppress this e\u000bect, which increases the number of d-electrons and\ncompensates for the decrease in the number of d-electrons induced by iron doping. Thus,\nif there is some nickel ( e:g: z = 0:2 for Y(Co 1\u0000xFex)3(Co 1\u0000zNiz)2), it is expected that the\ndependency of Kuon iron concentration would be moderate (shown in Fig. S1(a) in supple-\nmental materials [50]). In GGA, the dominant states are in a deeper energy region (density\nof states with GGA in Fig. S7 in supplemental materials [50]). In the case of iron, therefore,\nthe calculated Jsof GGA are signi\fcantly di\u000berent from LDA cases.\nIt is worth mentioning the anomalous value of the LDA calculation at z= 0:6. In contrast\nto the calculation with higher nickel content [26], the spin-state transition (the \frst-order\nLifshitz transition) does not occur at z= 0:6. In the present LDA calculation, Fig. 2(a) and\n(d) indicate that both YCo 5and YCo 3Ni2are in the high-spin states. In the intermediate\nconcentration range, we also con\frmed that YCo 3(Co 1\u0000zNiz)2are still in the high-spin states\n(local density of states of 3 dorbitals in Fig. S3 in supplemental materials [50]). At z= 0:6,\nthe volume dependence of the magnetic moment also indicates that the system is in the high-\nspin state at the volume determined by Vegard's law (Fig. S4). In spite of no transitions,\nFig. 5(b) shows that the di\u000berential coe\u000ecient of magnetization with nickel concentration\nvaries atz= 0:6. The changes in the di\u000berential coe\u000ecients might be related to anomalies\nin theKu. It is necessary to calculate the contribution to the magnetocrystalline anisotropy\nfrom the band structure to investigate this, but this is beyond the scope of this paper.\nE. Magnetization vs anisotropy constant\nFig. 6 shows the calculated JsandKuof Y(Co 1\u0000x\u0000yFexCuy)3(Co 1\u0000zNiz)2by LDA except\nfor paramagnetic materials, such as YCu 3Ni2(the contour plots can be available in supple-\nmental materials [50]). The intersection of the solid lines represents the values of YCo 5. The\nmagnetization decreases by adding a small amount of the third element, but the anisotropy\nincreases. Fig. 7(a) shows the calculated Ha= 2Ku=Jsof doped YCo 5without copper as a\n13  x=0.0\n  y=0.0\n  z=0.0\n(x, z)=(0.0, 0.0)\n(x, y)=(0.0, 0.0)\n(y, z)=(0.0, 0.0)  All\n-1 0 1 2 3 4 5 6 7\n 0.2  0.4  0.6  0.8  1  1.2Ku [MJ/m3]\nJs [T]Y(Co1-x-yFexCuy)3(Co1-zNiz)2\n-2\n 0YCo5YFe3Co2\nYCu3Co2\nYCo3Ni2  x=0.1  z=0.1  y=0.1  z=0.7FIG. 6. Calculated magnetization Jsand anisotropy Kuconstant of alloys excluding a few paramag-\nnetic alloys, such as YCu 3Ni2, by the full-potential KKR method with LDA functional. The trian-\ngles correspond to all materials, the open diamonds to x= 0:0, the open squares to y= 0:0, the open\ncircles to z= 0:0, the \flled diamonds to ( x; z) = (0 :0;0:0), the \flled squares to ( x; y) = (0 :0;0:0),\nand the \flled circles to ( y; z) = (0 :0;0:0). The vertical and horizontal lines represent the Ku= 1:82\nMJ=m3andJs= 1.03 T of YCo 5, respectively.\nfunction of iron and nickel content. The Haof the disordered alloy is more signi\fcant than\nHa= 4:49 T of YCo 5in a relatively wide range.\n14FIG. 7. Calculated Ha= 2 Ku=Jsas functions of impurity contents by the full-\npotential KKR method with the LDA functional for (a) Y(Co 1\u0000xFex)3(Co1\u0000zNiz)2and (b)\nY(Co 1\u0000yCuy)3(Co1\u0000zNiz)2. The LDA calculation shows the Ha= 4:49 T of YCo 5(bottom left).\nFig. 7(b) shows the calculated Haof doped YCo 5without iron as a function of copper\nand nickel content. The copper doping also enhances Hafor the low nickel concentration\nregion because the Jsdrastically decreases when added copper. This agrees with the previous\ncalculation assuming no site-preference of copper [23]. As mentioned in Sec. III C, Hain\nhigher copper concentration is di\u000berent from the experimental result [47]. In addition to the\nsite preference of copper, another origin of the discrepancy is a way to calculate Ha= 2Ku=Js\nin the calculation. In the region of large copper concentration, where the magnetic moment\nis considerably low, it is not appropriate to determine the anisotropic magnetic \feld using\nthe formula Ha= 2Ku=Js. Consequently, the large Ha= 33:34 T of YCu 3Co2should not be\nobserved.\nIn Fig. 6, when nickel or copper alone is added, JsandKusimultaneously increase. The\nmost promising candidate is YFe 3Co2, located at the upper right of YCo 5in the \fgure;\nherein, both KuandJsare more signi\fcant than those of YCo 5. The 1-5 phases of YFe 3Co2\nare unstable because iron is unlikely to be dissolved in the 1-5 phases as a solid solution.\nHowever, in the previous theoretical calculations for the samarium case, it was shown that\nnickel stabilized the 1-5 phase and iron-rich SmCoFeNi 3was proposed [51].\nAdditionally, Landa et al: concluded that nickel stabilized YFe 3(Co 1\u0000zNiz)2from the\ncalculation of phase diagrams (CALPHAD) [28]. According to this result, higher nickel\n15 0 1 2 3 4 5\n 0  0.2  0.4  0.6  0.8  1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2\nz for Ni contentYFe3(Co1-zNiz)2Ku [MJ/m3]\nJs [T] Js \nKuJs (YCo5)\nKu (YCo5)FIG. 8. Calculated KuandJsof YFe 3(Co1\u0000zNiz)2as a function of nickel content using the full-\npotential KKR method with the LDA functional. Filled circle is Kuand open circle is Js. Arrows\nare values of YCo 5as a reference.\ncontent enhances the stabilization of the 1-5 phases. We calculated KuandJsusing the\nFPKKR to investigate the magnetic properties of YFe 3(Co 1\u0000zNiz)2. The results are shown\nin Fig. 8. The calculated Kudecreases as nickel content increases, which is in line with the\nearlier theoretical calculations by Landa et al: .\nFig. 8 indicates that nickel, as the stabilization element, will deteriorate both the Ku\nandJs. Therefore, the magnetic performance and stabilization for the 1-5 phases are in a\ntrade-o\u000b relation, and optimal nickel content should be determined for practical use. The\narrows represent the reference values for YCo 5in the \fgure, and the magnetization is greater\n16than that of YCo 5even when nickel is added up to z= 0:2. The anisotropy is also suggested\nto have a larger Kuthan that of YCo 5up toz= 0:5. From Fig. 7(a), we can expect the Ha\nof YFe 3(Co 1\u0000xNix)2to be equal to or higher than that of YCo 5. As originally mentioned in\nRef. [28], if the 1-5 phase can exist stably, YFe 3(Co 1\u0000xNix)2can be a useful magnet with\nbetter performance than YCo 5.\nIV. SUMMARY\nWe calculated the magnetocrystalline anisotropy and magnetization of Y(Co,Fe,Cu,Ni) 5\nin order to determine its optimal composition as a permanent magnet. The systematic\ncalculations are based on the full-potential KKR Green's function method combined with\nthe coherent potential approximation. The calculated anisotropy of YCo 5strongly depended\non lattice parameters. Optimized lattice parameters were used for consistency.\nThe results obtained using LDA indicate that the anisotropy constant Kuand magnetiza-\ntion are higher than YCo 5when the iron is added, and they reach a maximum in YFe 3Co2.\nHowever, nickel, used to stabilize the 1-5 phases, decreases the magnetization and anisotropy\nconstant. The magnetic anisotropy and magnetization exhibit higher values than YCo 5up\ntoz= 0:4 andz= 0:2. The calculated anisotropy \feld is expected to be larger than the\noriginal YCo 5up toz= 0:5.\nThe present systematic calculations shed light on the magnetic properties and anisotropy\nof YCo 5systems. The underestimation of the magnetocrystalline anisotropy constant is due\nto the absence of orbital polarization enhancement. The anisotropy constant obtained by the\nsystematic calculations in the YCo 5system is also essential for discussing the contribution\nof the cobalt sublattice to the anisotropy of the SmCo 5system.\n[1] M. Sagawa, S. Fujimura, N. Togawa, H. Yamamoto, and Y. Matsuura, J. Appl. Phys. 55,\n2083 (1984).\n[2] J. J. Croat, J. F. Herbst, R. W. Lee, and F. E. Pinkerton, J. Appl. Phys. 55, 2078 (1984).\n[3] J. M. D. Coey, Eng. 6, 119 (2020).\n[4] E. Tatsumoto, T. Okamoto, H. Fujii, and C. Inoue, J. Phys. Colloq. 32, C1 (1971).\n[5] J. M. D. Coey, IEEE Trans. Magn. 47, 4671 (2011).\n17[6] J. M. Alameda, D. Givord, R. Lemaire, and Q. Lu, J. Appl. Phys. 52, 2079 (1981).\n[7] H. P. Klein and A. Menth, AIP Conf. Proc. 18, 1177 (1974).\n[8] H. P. Klein, A. Menth, and R. S. Perkins, Physica B+C 80, 153 (1975).\n[9] T. S. Zhao, H. M. Jin, R. Gr ossinger, X. C. Kou, and H. R. Kirchmayr, J. Appl. Phys. 70,\n6134 (1991).\n[10] R. Skomski, J. Appl. Phys. 83, 6724 (1998).\n[11] L. Nordstrom, M. S. S. Brooks, and B. Johansson, J. Phys. Condens. Matter. 4, 3261 (1992).\n[12] G. Daalderop, P. Kelly, and M. Schuurmans, Phys. Rev. B 53, 14415 (1996).\n[13] M. Yamaguchi and S. Asano, J. Appl. Phys. 79, 5952 (1996).\n[14] J. X. Zhu, M. Janoschek, R. Rosenberg, F. Ronning, J. D. Thompson, M. A. Torrez, E. D.\nBauer, and C. D. Batista, Phys. Rev. X 4, 1 (2014), 1402.5543.\n[15] M. Sakurai, S. Wu, X. Zhao, M. C. Nguyen, C.-Z. Wang, K.-M. Ho, and J. R. Chelikowsky,\nPhys. Rev. Mater. 2, 084410 (2018).\n[16] M. C. Nguyen, Y. Yao, C. Z. Wang, K. M. Ho, and V. P. Antropov, J. Phys. Condens. Matter.\n30(2018).\n[17] M. Matsumoto, R. Banerjee, and J. B. Staunton, Phys. Soc. Jpn. Conf. Proc. 5, 011004\n(2015).\n[18] L. Steinbeck, M. Richter, and H. Eschrig, Phys. Rev. B 63, 22 (2001).\n[19] P. Larson and I. I. Mazin, J. Appl. Phys. 93, 6888 (2003).\n[20] P. Larson, I. I. Mazin, and D. A. Papaconstantopoulos, Phys. Rev. B 69, 2 (2004).\n[21] X. B. Liu, Z. Altounian, and M. Yue, J. Appl. Phys. 107(2010).\n[22] C. E. Patrick, S. Kumar, G. Balakrishnan, R. S. Edwards, M. R. Lees, E. Mendive-Tapia,\nL. Petit, and J. B. Staunton, Phys. Rev. Mater. 1, 1 (2017).\n[23] C. E. Patrick, M. Matsumoto, and J. B. Staunton, J. Magn. Magn. Mater. 477, 147 (2019).\n[24] A. Asali, J. Fidler, and D. Suess, J. Magn. Magn. Mater. 485, 61 (2019).\n[25] V. Crisan, V. Popescu, A. Vernes, D. Andreica, I. Burda, and S. Cristea, J. Alloys Compd\n223, 147 (1995).\n[26] H. Yamada, K. Terao, H. Morozumi, K. Terao, and H. Yamada, J. Phys. Condens. Matter.\n11, 483 (1999).\n[27] F. Ishikawa, I. Yamamoto, I. Umehara, M. Yamaguchi, M. I. Bartashevich, H. Mitamura,\nT. Goto, and H. Yamada, Physica B: Condens. Matter. 328, 386 (2003).\n18[28] A. Landa, P. S oderlind, E. E. Moore, and A. Perron, Appl. Sci. (Switz) 10, 1 (2020).\n[29] M. Ogura and H. Akai, J. Phys. Condens. Matter. 17, 5741 (2005).\n[30] H. Shiba, Prog. Theor. Phys. 46, 77 (1971).\n[31] P. Soven, Phys. Rev. B 2, 4715 (1970).\n[32] V. L. Moruzzi, J. F. Janak, and A. R. Williams, in Calculated Electronic Properties of Metals ,\nedited by V. L. Moruzzi, J. F. Janak, and A. R. Williams (Pergamon, 1978).\n[33] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996), erratum: Phys.\nRev. Lett.. 1997 Feb;78:1396{1396.\n[34] A. Heidemann, D. Richter, and K. H. Buschow, Z. Phys. B 22, 367 (1975).\n[35] J. Schweizer and F. Tasset, J. Phys. F. Met. Phys. 10, 2799 (1980).\n[36] Y. C. Chuang, C. H. Wu, and Y. C. Chang, J. Less. Common. Met. 84, 201 (1982).\n[37] J. Deportes, D. Givord, J. Schweizer, and F. Tasset, IEEE Trans. Magn. 12, 1000 (1976).\n[38] G. Kresse and J. Furthm uller, Comput. Mater. Sci. 6, 15 (1996).\n[39] F. Maruyama, H. Nagai, Y. Amako, H. Yoshie, and K. Adachi, Physica B: Condens. Matter.\n266, 356 (1999).\n[40] H. Rosner, D. Koudela, U. Schwarz, A. Handstein, M. Han\rand, I. Opahle, K. Koepernik,\nM. D. Kuz'min, K. H. M uller, J. A. Mydosh, and M. Richter, Nat. Phys. 2, 469 (2006).\n[41] W. Frederick and M. Hoch, IEEE Trans. Magn. 10, 733 (1974).\n[42] H. Ucar, R. Choudhary, and D. Paudyal, J. Magn. Magn. Mater. 496, 165902 (2020).\n[43] N. Plugaru, M. Valeanu, R. Plugaru, and J. Campo, J. Appl. Phys. 115, 23907 (2014).\n[44] E. Burzo, P. Vlaic, D. P. Kozlenko, N. O. Golosova, S. E. Kichanov, B. N. Savenko, A. Ostlin,\nand L. Chioncel, J. Mater. Sci. Technol. 42, 106 (2020).\n[45] K. Inomata, Phys. Rev. B 23, 2076 (1981).\n[46] J. J. M. Franse, N. P. Thuy, and N. M. Hong, J. Magn. Magn. Mater. 72, 361 (1988).\n[47] J. C. Tellez-Blanco, R. Grossinger, R. S. Turtelli, and E. Estevez-Rams, IEEE Trans. Magn.\n36, 3333 (2000).\n[48] A. L. Tedstone, C. E. Patrick, S. Kumar, R. S. Edwards, M. R. Lees, G. Balakrishnan, and\nJ. B. Staunton, Phys. Rev. Mater. 3, 1 (2019).\n[49] K. H. J. Buschow and M. Brouha, AIP Conf. Proc. 29, 618 (1976).\n[50] Supplemental Material are not available now.\n[51] P. S oderlind, A. Landa, I. L. Locht, D. \u0017Aberg, Y. Kvashnin, M. Pereiro, M. D ane, P. E.\n19Turchi, V. P. Antropov, and O. Eriksson, Phys. Rev. B 96, 1 (2017).\n20"    },    {        "title": "2204.09223v2.The_first__and_second_order_magneto_optical_effects_and_intrinsically_anomalous_transport_in_2D_van_der_Waals_layered_magnets_CrXY__X___S__Se__Te__Y___Cl__Br__I_.pdf",        "content": "The \frst- and second-order magneto-optical e\u000bects and intrinsically anomalous\ntransport in 2D van der Waals layered magnets Cr XY(X= S, Se, Te; Y= Cl, Br, I)\nXiuxian Yang,1, 2, 3Ping Yang,1, 2Xiaodong Zhou,1, 2Wanxiang Feng,1, 2,\u0003and Yugui Yao1, 2\n1Centre for Quantum Physics, Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurement (MOE),\nSchool of Physics, Beijing Institute of Technology, Beijing, 100081, China\n2Beijing Key Lab of Nanophotonics &Ultra\fne Optoelectronic Systems,\nSchool of Physics, Beijing Institute of Technology, Beijing, 100081, China\n3Kunming Institute of Physics, Kunming 650223, China\n(Dated: August 10, 2022)\nRecently, the two-dimensional magnetic semiconductor CrSBr has attracted considerable attention\ndue to its excellent air-stable property and high magnetic critical temperature. Here, we systemati-\ncally investigate the electronic structure, magnetocrystalline anisotropy energy, \frst-order magneto-\noptical e\u000bects (Kerr and Faraday e\u000bects) and second-order magneto-optical e\u000bects (Sch afer-Hubert\nand Voigt e\u000bects) as well as intrinsically anomalous transport properties (anomalous Hall, anoma-\nlous Nernst, and anomalous thermal Hall e\u000bects) of two-dimensional van der Waals layered magnets\nCrXY(X= S, Se, Te; Y= Cl, Br, I) by using the \frst-principles calculations. Our results show that\nmonolayer and bilayer Cr XY(X= S, Se) are narrow band gap semiconductors, whereas monolayer\nand bilayer CrTe Yare multi-band metals. The magnetic ground states of bilayer Cr XYand the\neasy magnetization axis of monolayer and bilayer Cr XYare con\frmed by the magnetocrystalline\nanisotropy energy calculations. Utilizing magnetic group theory analysis, the \frst-order magneto-\noptical e\u000bects as well as anomalous Hall, anomalous Nernst, and anomalous thermal Hall e\u000bects\nare identi\fed to exist in ferromagnetic state with out-of-plane magnetization. The second-order\nmagneto-optical e\u000bects are not restricted by the above symmetry requirements, and therefore can\narise in ferromagnetic and antiferromagnetic states with in-plane magnetization. The calculated\nresults are compared with the available theoretical and experimental data of other two-dimensional\nmagnets and some conventional ferromagnets. The present work reveals that monolayer and bilayer\nCrXYwith superior magneto-optical responses and anomalous transport properties provide an ex-\ncellent material platform for the promising applications of magneto-optical devices, spintronics, and\nspin caloritronics.\nI. INTRODUCTION\nAccording to the Mermin-Wagner theorem [1], the\nlong-range magnetic order is strongly suppressed in low-\ndimensional systems mainly due to the enhanced thermal\n\ructuations, which makes the symmetry-breaking order\nunsustainable. Nevertheless, recent experimental fabri-\ncations of atomically-thin ferromagnetic (FM) CrI 3[2]\nand Cr 2Ge2Te6[3] \flms, which are mechanically exfo-\nliated from their bulk van der Waals (vdW) materials,\nindicated that the long-range magnetic order can be es-\ntablished in two-dimensional (2D) systems in which the\nmagnetic anisotropy plays a vital role. Since then, 2D\nmagnetism has attracted enormous interest and more real\nmaterials were subsequently discovered, such as mono-\nlayer (ML) VSe 2[4, 5], Fe 3GeTe 2[6, 7], and CrTe 2[8].\nThe ML VSe 2, Fe3GeTe 2, and CrTe 2are FM metals with\nrelatively high Curie temperatures ( TC). TheTCof ML\nFe3GeTe 2is around 130 K [6, 7], and the TCof ML VSe 2\nand CrTe 2are even close to room-temperature [4, 5, 8],\nall of which are much larger than that of FM semicon-\nductors CrI 3(TC\u001845 K) [2] and Cr 2Ge2Te6(TC\u001825\nK) [3]. From the aspect of practical applications for 2D\nspintronics, it is imperative to search for magnetic semi-\n\u0003wxfeng@bit.edu.cnconductors with higher TC.\nA vdW layered material, CrSBr, was synthesized \ffty-\nyears ago [9], and its bulk form has been identi\fed to\nbe an A-type antiferromagnetic (AFM) semiconductor\nwith the N\u0013 eel temperature ( TN) of 132 K and electronic\nband gap of\u00181.5 eV [10, 11]. Recently, several theo-\nretical works predicted that ML CrSBr is a FM semi-\nconductor with high TCranging from 120 K to 300 K\nby di\u000berent model calculations [12{18]. Bulk CrSBr\ncan be mechanically exfoliated to ML and bilayer (BL),\nwhich were con\frmed to be FM ( TC\u0018146 K) and\nAFM (TN\u0018140 K), respectively, by a recent measure-\nment of second harmonic generation [19]. The ML and\nBL CrSBr have been paid much attention because their\nmagnetic critical temperatures are obviously higher than\nthat of CrI 3and Cr 2Ge2Te6. So far, some experimen-\ntal works appear to focus on the fascinating properties\nof 2D CrSBr, including tunable magnetism [11, 19{23],\nmagnon-exciton coupling [24], interlayer electronic cou-\npling [20], tunable electronic transport [21], and etc. The\nAFM semiconductor CrSBr was also used to introduce\nmagnetism in graphene/CrSBr heterostructure by con-\nsidering the proximity e\u000bect [25]. However, the magne-\ntotransport properties of 2D CrSBr, such as magneto-\noptical Kerr [26], Faraday [27], Sch afer-Hubert [28], and\nVoigt [29] e\u000bects (MOKE, MOFE, MOSHE, and MOVE)\nas well as anomalous Hall e\u000bect (AHE) [30], anomalousarXiv:2204.09223v2  [cond-mat.mtrl-sci]  9 Aug 20222\nNernst e\u000bect (ANE) [31], and anomalous thermal Hall\ne\u000bect (ATHE) [32], remain largely unexplored. Since the\nfamily materials of Cr XY(X= S, Se, Te; Y= Cl, Br,\nI) share the same crystal structure [15], atomically-thin\n\flms of Cr XYshould be easily obtained by the means of\nmechanical exfoliation, similarly to ML and BL CrSBr. A\nsystematic study on the magneto-optical e\u000bects (includ-\ning MOKE, MOFE, MOSHE, and MOVE) and intrin-\nsically anomalous transport properties (including AHE,\nANE, and ATHE) of ML and BL Cr XYwill contribute to\nfurther understanding of this class of 2D vdW magnets.\nIn this work, using the \frst-principles calculations, we\nsystemically investigate the electronic, magnetic, optical,\nmagneto-optical, and anomalous transport properties of\nML and BL Cr XY. We \frst reveal that CrS Yand CrSe Y\nare narrow band gap semiconductors, whereas CrTe Y\npresent multi-band metallic behaviors. Then, the mag-\nnetic ground state and the easy axis of magnetization for\nCrXY are con\frmed by magnetocrystalline anisotropy\nenergy (MAE) calculations. The results of MAE indicate\nthat the easy axes of CrTeBr and CrTeI point along the\nout-of-plane direction (i.e., z-axis), while the other seven\nfamily members of Cr XYprefer the in-plane magnetiza-\ntion. The BL CrSI, CrSeCl, and CrTeCl show strong in-\nterlayer FM coupling, whereas other six family members\nexhibit interlayer AFM coupling. Before calculating the\nmagneto-optical and anomalous transport properties, we\nprimarily analyze the shape of optical conductivity ten-\nsor by utilizing the magnetic group theory. The uncov-\nered symmetry requirements indicate that the \frst-order\nmagneto-optical e\u000bects (MOKE and MOFE) as well as\nthe AHE, ANE, and ATHE can only arise in the FM\nstate with out-of-plane magnetization. In order to ex-\nplore the FM and AFM states with in-plane magnetiza-\ntion, we further study the second-order magneto-optical\ne\u000bects (MOSHE and MOVE). The calculated \frst- and\nsecond-order magneto-optical e\u000bects as well as intrinsi-\ncally anomalous transport properties of Cr XYare com-\npared with other 2D magnets and some conventional fer-\nromagnets. Our results show that 2D Cr XYwith supe-\nrior magneto-optical responses and anomalous transport\nproperties provide an excellent material platform for the\npromising applications of magneto-optical devices, spin-\ntronics, and spin caloritronics.\nII. THEORY AND COMPUTATIONAL DETAILS\nThe magneto-optical e\u000bects, as a kind of fundamen-\ntal magnetotransport phenomena in condensed matter\nphysics, are usually considered to be a powerful probe\nof magnetism in low-dimensional systems [2, 3]. The\nMOKE and MOFE can be described as the rotation\nof the polarization planes of re\rected and transmitted\nlights after a linearly polarized light hits on the surface\nof magnetic materials, respectively. Here, we only con-\nsider the MOKE and MOFE in the so-called polar geom-\netry (e.g., both the directions of incident light and mag-netization are along the z-axis), as shown in Fig. 1(a).\nThe MOKE and MOFE are commonly regarded as \frst-\norder magneto-optical e\u000bects as their magnitudes are\nlinearly proportional to the strength of magnetization\nand their signs are odd in the direction of magnetiza-\ntion. On the other hand, if the magnetization lies on the\n2D atomic plane (i.e., xy-plane) and the angle between\nthe electric \feld ( E) of incident light and the direction\nof magnetization ( M) is 45\u000e, the polarization planes of\nre\rected and transmitted lights still rotate, which are\ncalled MOSHE and MOVE [Fig. 1(b)], respectively. The\nMOSHE and MOVE are considered to be second-order\nmagneto-optical e\u000bects as their magnitudes are quadratic\nto the strength of magnetization and their signs are even\nin the direction of magnetization.\nIt should be noted that in some FM and AFM materi-\nals, the \frst-order magneto-optical e\u000bects are prohibited\ndue to the symmetry requirements, but the second-order\nmagneto-optical e\u000bects can exist. Here, we take ML FM\nFe3GeTe 2and BL AFM Fe 3GeTe 2with in-plane mag-\nnetization [33] as examples to discuss this point. The\nmagnetic point group of ML FM Fe 3GeTe 2with in-plane\nmagnetization (when the spin points to the x-axis) is\nm0m20, in which the mirror plane Mxis perpendicular\nto the spin and is parallel to the z-axis. This mirror\nsymmetry operation reverses the sign of the o\u000b-diagonal\nelement of optical conductivity \u001bxy(\u0011\u001bz), thus indicat-\ning\u001bxy= 0. If the spin points to the y-axis, the magnetic\npoint group of ML FM Fe 3GeTe 2ism0m02, which con-\ntains a combined symmetry TMz(Tis the time-reversal\nsymmetry, andMzis a mirror plane perpendicular to the\nz-axis and parallel to the spin). This combined symme-\ntry operation reverses the sign of \u001bxy, also suggesting\n\u001bxy= 0. For BL AFM Fe 3GeTe 2, the magnetic point\ngroups are 20=mand 2=m0when the spin points to the\nx- andy-axes, respectively. Both the two magnetic point\ngroups have a combined symmetry TP(TandPare the\ntime-reversal and spatial inversion symmetries, respec-\ntively), which forces \u001bxy= 0. Overall, in the cases of\nML FM Fe 3GeTe 2and BL AFM Fe 3GeTe 2with in-plane\nmagnetization, the vanishing o\u000b-diagonal element of op-\ntical conductivity ( \u001bxy= 0) can not give rise to the \frst-\norder magneto-optical e\u000bects (MOKE and MOFE), refer\nto Eqs. (1){(3). In contrast, the second-order magneto-\noptical e\u000bects (MOSHE and MOVE) depend on both the\no\u000b-diagonal and diagonal elements, see Eqs. (4){(5). For\nthe in-plane magnetization along either x- ory-axis, the\ntwo diagonal elements are not equivalent ( \u001bxx6=\u001byy) due\nto the anisotropy of the in-plane optical response, which\nde\fnitively induces the second-order magneto-optical ef-\nfects regardless of whether the o\u000b-diagonal element is\nzero or not.\nThe Kerr and Faraday rotation angles ( \u0012Kand\u0012F),\nwhich are the de\rection of the polarization plane with re-\nspect to the incident light, can be used to quantitatively\ncharacterize the magneto-optical performance. Addition-\nally, the rotation angle \u0012K(F) and ellipticity \"K(F) are\nusually combined into the complex Kerr (Faraday) an-3\n/s113Κ/s113\nF/s113\nV/s113SH4\n5°M\nM EE(a)( b)/s295\nω''/s295ω''/s295ω'/s295\nω''/s295ω'/s295 ωx\ny\nz\nz xy\nxy\nS\niO2/SiS iO2/Si/s295ω\nFIG. 1. (Color online) Schematic illustration of magneto-optical Kerr and Faraday e\u000bects (a) as well as magneto-optical\nSch afer-Hubert and Voigt e\u000bects (b). ~!,~!0, and ~!00indicate the incident, re\rected, and transmission lights, respectively.\n\u0012K,\u0012F,\u0012SH, and\u0012Vrepresent the Kerr, Faraday, Sch afer-Hubert, and Voigt rotation angles, respectively. MandElabel the\nmagnetization and electric \feld of incident light, respectively. In (b), EandMform an angle of 45\u000eon thexyplane.\ngle [34{36],\n\u001eK=\u0012K+i\"K=i2!d\nc\u001bxy\n\u001bsubxx; (1)\nand\n\u001eF=\u0012F+i\"F=!d\n2c(n+\u0000n\u0000); (2)\nn2\n\u0006= 1 +4\u0019i\n!(\u001b0\u0006i\u001bxy); (3)\nwherecis the speed of light in vacuum, !is frequency\nof incident light, dis the thickness of thin \flm, and n\u0006\nare the refractive indices for the right- and left-circularly\npolarized light, respectively. \u001bsub\nxxis the diagonal element\nof the optical conductivity tensor for a nonmagnetic sub-\nstrate (e.g., SiO 2/Si, as shown in Fig. 1). In our cal-\nculations, only the interface between Cr XY and SiO 2\nlayers is considered, and the SiO 2layer is assumed to be\nthick enough. In this case, the Si layer is unimportant\nas it does not in\ruence the magneto-optical properties\nof Cr XY. The\u001bsub\nxxof the SiO 2layer can be written as\ni(1\u0000n2\nsub)!\n4\u0019, here thensubis the energy-dependent re-\nfractive index of SiO 2crystal (quartz) [37, 38]. \u001bxyis\nthe o\u000b-diagonal element of the optical conductivity ten-\nsor for a magnetic thin \flm and \u001b0=1\n2(\u001bxx+\u001byy) due\nto in-plane anisotropy.\nIn the case of MOSHE and MOVE, the rotation angle\n\u0012SH(V) and ellipticity \"SH(V) can be also combined into\nthe complex Sch afer-Hubert and Voigt angles. Consider-\ning the in-plane magnetization along the y-axis (Fig. 1),the complex Sch afer-Hubert angle can be expressed as\n\u001eSH=\u0012SH+i\"SH\n=\u0000i!d\nc(1\u0000n2\nsub)(n2\nk\u0000n2\n?)\n=\u0000i!d\nc(1\u0000n2\nsub)[\u000fyy\u0000\u000fxx\u0000\u000f2\nzx\n\u000fzz]; (4)\nand the complex Voigt angle is give by [39, 40]\n\u001eV=\u0012V\u0000i\"V\n=!d\n2ic(nk\u0000n?)\n=!d\n2ic[\u000f1\n2yy\u0000(\u000fxx+\u000f2\nzx\n\u000fzz)1\n2]: (5)\nHere,\u000f\u000b\f=\u000e\u000b\f+4\u0019i\n!\u001b\u000b\fwith\u000b;\f2x;y;z is the per-\nmittivity tensor (in cgs units), nkandn?are the refrac-\ntive indices of a magnetic thin \flm that are parallel and\nperpendicular to the direction of magnetization, respec-\ntively.\nFrom Eqs.(1){(5), one would know that the key to cal-\nculate the \frst- and second-order magneto-optical e\u000bects\nis the optical conductivity, which can be obtained by the\nKubo-Greenwood formula [41, 42],\n\u001bxy=\u001b1\nxy(!) +i\u001b2\nxy(!)\n=ie2~\nNkVX\nkX\nn;mfmk\u0000fnk\n\"mk\u0000\"nk\n\u0002h nkj^\u001dxj mkih mkj^\u001dyj nki\n\"mk\u0000\"nk\u0000(~!+i\u0011); (6)4\nwhere the superscripts 1 (2) indicates the real (imagi-\nnary) part of optical conductivity.  nkand\"nkare the\nBloch function and interpolated energy at the band index\nnand momentum k, respectively. fnk,V,Nk, ^vx(y),~!,\nand\u0011are the Fermi-Dirac distribution function, volume\nof the unit cell, total number of k-points for sampling the\nBrillouin zone, the velocity operators, the photon energy,\nand energy smearing parameter, respectively. \u0011was cho-\nsen to be 0.1 eV, corresponding to the carrier relaxation\ntime of 6.5 fs, which is in the realistic range for multilayer\ntransition metal chalcogenides [43, 44]. In this work, a\nsu\u000eciently dense k-mesh of 450\u0002390\u00021 is used to cal-\nculate the optical conductivity. We also tested a denser\nk-mesh of 500\u0002430\u00021 for the convergence of optical con-\nductivity, see Figs. S1(a) and S1(b) in Supplemental Ma-\nterial [45].\nThe AHE, which is featured by the transverse volt-\nage drop induced by a longitudinal charge current in the\nabsence of external magnetic \feld, is another fundamen-\ntal magnetotransport phenomenon in condensed mater\nphysics. The contributions to AHE can be distinguished\nto extrinsic (skew-scattering and side-jump) and intrinsic\nmechanisms [30]. The extrinsic AHE is dependent on the\nscattering of electrons o\u000b impurities or due to disorder,\nwhile the intrinsic AHE originates from the Berry phase\nnature of the electrons in a perfect crystal. In this work,\nwe only focus on the intrinsically anomalous transport,\ni.e., intrinsic AHE. The \frst-order magneto-optical ef-\nfects and intrinsic AHE share the similar physical origin,\nand the dc limit of the real part of the o\u000b-diagonal ele-\nment of optical conductivity, \u001b1\nxy(!!0), is nothing but\nthe intrinsic anomalous Hall conductivity (AHC). Based\non the linear response theory, the intrinsic AHC can be\ncalculated [46],\n\u001bxy=\u0000e2~\nVX\nn;kfnk\nn\nxy(k); (7)\nin whichfnkis the Fermi-Dirac distribution function, V\nis volume of the unit cell, and \nn\nxy(k) is the band- and\nmomentum-resolved Berry curvatures, given by\n\nn\nxy(k) =\u0000X\nn06=n2Im[h nkj^vxj n0kih n0kj^vyj nki]\n(\"nk\u0000\"n0k)2:(8)\nTo calculate \u001bxy, ak-mesh of 600\u0002520\u00021 was used. We\nalso tested the convergence of \u001bxyon a denser k-mesh of\n750\u0002650\u00021, see supplemental Fig. S1(c) [45].\nIn addition to the AHE, the transverse charge current\ncan also be induced by a longitudinal temperature gra-\ndient \feld, named ANE [31]. Additionally, a transverse\nthermal current arise under the longitudinal temperature\ngradient \feld, which is called ATHE or anomalous Righi-\nLeduc e\u000bect [32]. The magnetic materials that show\nlarge anomalous Nernst conductivity (ANC) and anoma-\nlous thermal Hall conductivity (ATHC) have great po-\ntentials to apply for spin caloritronics or thermoelectric\ndevices. The AHE, ANE, and ATHE are closely relatedto each other, and can be expressed through the anoma-\nlous transport coe\u000ecients in the generalized Landauer-\nB uttiker formalism [47{49],\nR(n)\nxy=Z1\n\u00001dE(E\u0000\u0016)n(\u0000@f\n@E)\u001bxy(E); (9)\nwhereE,\u0016, andfare energy, chemical potential, and\nFermi-Dirac distribution function, respectively. \u001bxyis\nthe intrinsic AHC calculated by Eq. (7) at zero tempera-\nture. Then, the temperature-dependent ANC ( \u000bxy) and\nATHC (\u0014xy) are written as\n\u000bxy=\u0000R(1)\nxy=eT; (10)\n\u0014xy=R(2)\nxy=e2T: (11)\nTo calculate \u000bxyand\u0014xy, the integral in Eq. (9) is set in\nthe range of [-1.0, 2.5] eV for semiconductors and [-1.0,\n1.0] eV for metals, with the energy interval of 0.0001 eV.\nThe \frst-principles density functional theory calcula-\ntions were performed by utilizing the Vienna ab initio\nsimulation package ( vasp ) [50, 51]. The projector aug-\nmented wave (PAW) method [52] was used and the gener-\nalized gradient approximation (GGA) with the Perdew-\nBurke-Ernzerhof parameterization [53] was adopted to\ntreat the exchange-correlation functional. Dispersion\ncorrection was performed at the van der Waals density\nfunctional (vdW-DF) level [54{56], where the optB86b\nfunctional was used for the exchange potential. For the\nML and BL structures, the vacuum layer with a thick-\nness of at least 15 \u0017A was used to avoid the interactions\nbetween adjacent layers. All structures were fully relaxed\nuntil the force on each atom was less than 0.001 eV \u0001\u0017A\u00001.\nTo describe the strong correlation e\u000bect of the dorbitals\non Cr atom, the GGA+U method [57] was used by taking\nappropriate parameters of U and J in the range of 4.05 \u0018\n4.40 eV and 0.80\u00180.97 eV [15], respectively. The values\nof U and J used for each compound are listed in supple-\nmental Tab. S1 [45]. To obtain accurate MAE, a large\nplane-wave cuto\u000b energy of 600 eV and a Monkhorst-\nPackk-mesh of 15\u000213\u00021 were used. The spin-orbit cou-\npling (SOC) was included in the calculations of MAE,\nmagneto-optical e\u000bects, and anomalous transport prop-\nerties. The maximally localized Wannier functions were\nconstructed by wannier90 package [41], and the d,p,\nandporbitals of Cr, X(X= S, Se, Te), and Y(Y=\nCl, Br, I) atoms were projected onto the Wannier func-\ntions, respectively. Totally, there are 44 and 88 Wannier\nfunctions for ML and BL Cr XY, respectively, which con-\nverged adequately the results of magneto-optical e\u000bects\nand anomalous transport properties. Similarly, the com-\nputational technique employing the maximally localized\nWannier functions can be in principle used to calculate\nthe physical quantities that are closely related to orbital\nangular momenta [58{61]. The isotropy software [62]\nwas used to identify magnetic space and point groups.\nThe work\row of the various codes used in our work is\ndepicted in supplemental Fig. S2 [45].5\nIII. RESULTS AND DISCUSSION\nA. Crystal, magnetic, and electronic structures\nFigure 2 shows the crystal structure of vdW layered\nmaterials Cr XY (X= S, Se, Te; Y= Cl, Br, I). One\ncan see that Cr XYhas a rectangular 2D primitive cell,\nin which a ML is made of two buckled planes of Cr X\nsandwiched by two Ysheets, and a BL is formed by two\nMLs with the AA stacking order referring to experimen-\ntal works [11, 19, 20, 63]. The relaxed lattice constants\nof ML and BL structures are collected in supplemental\nTab. S1 [45]. The calculated lattice constants of ML\nCrSBr,a= 3.54 \u0017A,b= 4.79 \u0017A, and thickness d= 8.00\n\u0017A, are in good agreement with the experimental results,\na= 3.50 \u0017A,b= 4.76 \u0017A [11], and d= 7.8\u00060.3\u0017A [19].\nThe magnetic anisotropy plays a vital role in stabiliz-\ning the long-range magnetic order of 2D systems. The\nMAE is de\fned as the di\u000berence of total energies be-\ntween di\u000berent magnetization directions (such as, along\nthex-,y-, andz-axis), which can be directly obtained\nby the \frst-principles calculations. To determine the\nmagnetic ground states, we calculate the MAE of ML\nand BL Cr XY. Since the FM ground state of ML Cr XY\nwas predicted by a recent theoretical work [15] and has\nbeen con\frmed by two experimental works [19, 20], here\nwe directly adopt the FM con\fguration as the ground\nstate of ML Cr XY. Thus, the magnetic ground states for\nML Cr XYcan also be determined by the Goodenough-\nKanamori rules, i.e., superexchange mechanism [64{66].\nThe magnetic exchange interactions can be judged by\nthe angle of the chemical bonds connecting the ligand\nand two nearest-neighboring magnetic atoms. Particu-\nlarly, a system prefers to be ferromagnetic if the angle\nequals 90\u000e. Taking ML CrSBr as an example, the bond\nangle of Cr-Br-Cr is 88.96\u000e, and the bond angles of Cr-\nS-Cr are 94.24\u000eand 96.58\u000e, suggesting the ferromagnetic\nground state. Our calculated MAE indicate that ML\nCrSI, CrSe Yprefer to the in-plane magnetization along\nthex-axis, ML CrSCl, CrSBr, and CrTeCl are in favor\nof in-plane magnetization along the y-axis, while ML Cr-\nTeBr and CrTeI take for the out-of-plane z-axis magne-\ntization. The evolution of the easy axis from in-plane\nto out-of-plane direction can be explained by the SOC\nstrength of the systems. The atomic SOC strength in-\ncreases gradually in the consequences of S !Se!Te\nand Cl!Br!I, and CrTeBr and CrTeI have the al-\nmost largest SOC strength among the other family com-\npounds. It leads to the MAE results that CrTeBr and\nCrTeI have the out-of-plane z-axis magnetization, while\nthe other family compounds have the in-plane x- ory-axis\nmagnetization. This is similar to the case of 2D mag-\nnetic materials Cr X3(X= Cl, Br, I) [67], in which the\nmagnetization direction changes from in-plane (CrCl 3) to\nout-of-plane (CrI 3) with the increasing of SOC strength.\nFor the BL Cr XY, we compare the total energy of FM\nand AFM states (see supplemental Fig. S3 [45]), and \fnd\nyx\nzy\nYCrXFIG. 2. (Color online) Top and side views of layered Cr XY\n(X= S, Se, Te; Y= Cl, Br, I). The blue, yellow, and brown\nspheres represent Cr, X, and Yatoms, respectively. The\nblack solid lines draw out the primitive cell. The experimental\nstacking order of bilayer structure is adopted.\nthat CrSI, CrSeCl, and CrTeCl exhibit the FM ground\nstate, while the other six family members prefer to be\nAFM state. The easy magnetization axes of BL Cr XY\nare identical to that of ML Cr XY. The above results are\nsummarized in supplemental Tab. S1 [45]. The easy mag-\nnetization axis of CrSBr is along the y-axis, which is in\nan accordance with the experimental reports [11, 19, 20].\nAfter obtaining the magnetic ground states of Cr XY,\nwe shall further ascertain the corresponding magnetic\ngroups as the group theory is a powerful tool to \fgure\nout the symmetry requirements for the magneto-optical\ne\u000bects and anomalous transport properties. The o\u000b-\ndiagonal elements of optical conductivity tensor [ \u001b\u000b\f(!)]\nfully determine the symmetry requirements of \frst-order\nmagneto-optical e\u000bects [see Eqs. (1){(3)]. At zero fre-\nquency limit, \u001b\u000b\f(!!0) is nothing but the intrinsic\nAHC [i.e., Eq. (7)], and therefore the symmetry require-\nments of AHE, ANE, and ATHE [refer to Eqs. (9){(11)]\nshould be the same as that of \u001b\u000b\f(!). Here, we only need\nto analyze the symmetry results of \u001b\u000b\f(!) under relevant\nmagnetic groups.\nSince the o\u000b-diagonal elements of optical conductivity\ntensor can be regarded as a pseudovector (like spin), its\nvector-form notation, \u001b= [\u001bx;\u001by;\u001bz] = [\u001byz;\u001bzx;\u001bxy],\nis used here for convenience. In 2D systems, only \u001bz\n(\u0011\u001bxy) is potentially nonzero, while \u001bx(\u0011\u001byz) and\u001by\n(\u0011\u001bzx) are always zero. It can be easily understood\nfrom Eq. (6) as the zero velocity along the z-axis (i.e.,\n^vz= 0, meaning that the electrons can not move along\nthe out-of-plane direction in 2D systems) de\fnitely gives\nrise to vanishing \u001byzand\u001bzx. For ML and BL FM Cr XY\nwith any magnetization directions ( x-,y-, orz-axis), the\nmagnetic space and point groups of are always identi-\n\fed to bePm0m0nandm0m0m[D2h(C2h) in Sch on\ries\nnotation], respectively. It is su\u000ecient for us to analyze\nmagnetic point group due to the translationally invari-\nance of\u001bxy[68]. Hence, we only discuss the magnetic\npoint group of m0m0m, in which there are one mirror6\n-202-\n202-\n2020\n5 1 0(a)E  (eV)C\nrSBr(\nb)E  (eV)C\nrSeBr(\nc)E  (eV)C\nrTeBrS\nY ΓX S  Tot \nCr d \nX  p \nBr pD\nOS\nFIG. 3. (Color online) The relativistic band structures and\norbital-decomposed density of states (in unit of states/eV per\ncell) for monolayer ferromagnetic Cr XBr (X= S, Se, Te) with\nout-of-plane magnetization.\nplaneMand two combined symmetries TM (Tis the\ntime-reversal symmetry). For the in-plane magnetiza-\ntion along the x-axis (y-axis), the mirror plane is Mx\n(My), which is perpendicular to the spin magnetic mo-\nment, such a mirror operation reverses the sign of \u001bxy,\nindicating that \u001bxyshould be zero. On the other hand,\nfor out-of-plane magnetization, the mirror plane is Mz\nand two combined symmetries are TMxandTMy. The\nspin magnetic moment is perpendicular to the Mz, which\ndoes not change the sign of \u001bxy. Moreover, the mirror\nplanesMxandMyare parallel to the spin, and both\nTMxandTMypreserve the sign of \u001bxy. A conclu-\nsion here is that \u001bxyis nonzero (zero) for ML and BL\nFM Cr XY with out-of-plane (in-plane) magnetization.\nIn the case of the BL AFM Cr XY, the magnetic space\nand point groups are Pm0m0n0andm0m0m0[D2h(D2) in\nSch on\ries notation] for out-of-plane magnetization, and\narePm0mnandmmm0[D2h(C2v) in Sch on\ries notation]\nfor in-plane magnetization. There is a combined sym-\nmetryTP(Pis the spatial inversion) in the two point\ngroups ofm0m0m0andmmm0, which forces \u001bxyto be\nzero. Overall, a nonzero \u001bxycan only exist in ML and BLFM Cr XYwith out-of-plane magnetization, which turn\nout to be our target systems for studying the MOKE,\nMOFE as well as AHE, ANE, and ATHE. It should be\nnoted that the out-of-plane magnetization is not the mag-\nnetic ground states for some family members of Cr XY,\nhowever, an applied external magnetic \fled can easily\ntune the spin direction, as already realized in layered\nCrSBr [20].\nThe \frst-order magneto-optical e\u000bects (MOKE and\nMOFE) can not exist in FM and AFM Cr XYwith in-\nplane magnetization, but it is not the case for the second-\norder magneto-optical e\u000bects (MOSHE and MOVE). Ac-\ncording to the Onsager relations, the diagonal elements of\nthe permittivity tensor are even in magnetization ( M),\ne.g.,\u000fyy(\u0000M) =\u000fyy(M). Furthermore, equations (4)\nand (5) show that the complex Sch afer-Hubert and Voigt\nangles depend on square of \u000fzx. Therefore, the MOSHE\nand MOVE must be even in M. In the case of FM and\nAFM Cr XY with in-plane magnetization, although \u000fzx\nis always zero, the MOSHE and MOVE do exist because\nof the nonvanishing \u000fyyand\u000fxx(the relation of \u000fyy6=\u000fxx\nalso satis\fes). Owing to this, we use the MOSHE and\nMOVE to characterize the in-plane magnetized ML FM\nand BL FM/AFM Cr XY. The complex Sch afer-Hubert\nand Voigt angles are calculated by orienting the spins\nalong they-axis, and the results of x-axis in-plane mag-\nnetization can be simply obtained by putting a minus\nsign, refer to Eqs. (4) and (5).\nThe relativistic band structures of ML FM and BL\nFM/AFM Cr XY with out-of-plane and in-plane mag-\nnetization are plotted in supplemental Figs. S4{S9 [45].\nThe results show that CrS Yand CrSe Yare narrow band\ngap semiconductors, while CrTe Yare multi-band metals.\nAll the band gaps of CrS Yand CrSe Yare summarized\nin supplemental Tab. S1 [45]. Here, we choose ML FM\nCrXBr (X= S, Se, Te) as representative examples to\ndiscuss the electronic properties of Cr XYfamily materi-\nals, as shown in Fig. 3. A clear trend can be seen that\nthe band gaps of Cr XBr become smaller and eventually\ndisappear with the increasing of atomic number (e.g., S\n!Se!Te). The orbital-decomposed density of states of\nML FM Cr XBr show that the dorbitals of Cr atom and\nporbitals of Xand Br atoms are dominant components\nnear the Fermi level. For CrSBr, the calculated band\ngaps of ML FM, BL FM, and BL AFM states are 1.34\neV, 1.22 eV, and 1.30 eV, respectively, which are in good\nagreement with that of bulk material ( \u00181.25 eV) [11].\nSince CrS Yand CrSe Yare narrow band gap semicon-\nductors, the electrons and holes could bound in pairs to\nform quasiparticle excitons. Taking CrSBr as an exam-\nple [20], the exciton binding energies for ML FM, BL FM,\nand BL AFM states are 0.5 eV, 0.37 eV, and 0.46 eV, re-\nspectively, which are apparently smaller than the exciton\nbinding energies of ML FM CrI 3(1.7 eV [69], 1.06 eV [70],\nand 0.84 [70]). It indicates that the relatively weak exci-\ntonic e\u000bects in CrS Yand CrSe Ymay not signi\fcantly\nin\ruence the optical and magneto-optical properties, in\ncontrast to CrI 3[69, 70].7\n-2-101-\n2-1010\n2 4 6 -2-1010\n2 4 6 02 4 6 /s113K (deg) ML (FM z-axis) \nBL  (FM z-axis)(a)C\nrSClC rSBrC rSIC\nrSeCl(b)/s113 K (deg)C\nrSeBrC rSeIC\nrTeCl(c)/s113 K (deg)P\nhoton energy (eV)CrTeBrP\nhoton energy (eV)CrTeIP\nhoton energy (eV)\nFIG. 4. (Color online) The magneto-optical Kerr rotation angles ( \u0012K) of monolayer (blue solid lines) and bilayer (pink solid\nlines) ferromagnetic Cr XY(X= S, Se, Te; Y= Cl, Br, I) with out-of-plane magnetization. The vertical dashed lines in (a)\nand (b) indicate the band gaps.\n-0.20.00.2-\n0.20.00.20\n2 4 6 -0.20.00.20\n2 4 6 02 4 6 CrSCl/s113F (deg) ML (FM z-axis) \nBL  (FM z-axis)(a)C\nrSBrC rSI(\nb)C\nrSeCl/s113F (deg)C\nrSeBrC rSeI(\nc)C\nrTeCl/s113F (deg)P\nhoton energy (eV)CrTeBrP\nhoton energy (eV)CrTeIP\nhoton energy (eV)\nFIG. 5. (Color online) The magneto-optical Faraday rotation angles ( \u0012F) of monolayer (blue solid lines) and bilayer (pink solid\nlines) ferromagnetic Cr XY(X= S, Se, Te; Y= Cl, Br, I) with out-of-plane magnetization. The vertical dashed lines in (a)\nand (b) indicate the band gaps.8\nTABLE I. The maximums of Kerr, Faraday, Sch afer-Hubert, and Voigt rotation angles ( \u0012K,\u0012F,\u0012SH, and\u0012V) in the moderate\nenergy range (1 \u00185 eV) as well as the corresponding incident photon energy ( ~!) for monolayer and bilayer Cr XY. Here,\u0012K\nand\u0012Fare calculated under the ferromagnetic states with out-of-plane magnetization, while \u0012SHand\u0012Vare calculated under\nferromagnetic or antiferromagnetic states with in-plane y-axis magnetization. For a better comparison, \u0012Fand\u0012Vare given in\ntwo units, deg and deg/ \u0016m (in the parentheses), which can be converted to each other by considering the e\u000bective thickness of\ncrystal structures along the z-axis.\n\u0012K(deg) ~!(eV)\u0012F(deg) (deg/ \u0016m) ~!(eV)\u0012SH(deg) ~!(eV)\u0012V(deg) (deg/ \u0016m) ~!(eV)\nCrSCl ML (FM) 0.25 4.01 0.04 (54.29) 4.11 -4.65 3.65 0.41 (559.34) 3.69\nBL (FM) 0.48 3.95 0.06 (33.93) 4.07 -9.30 3.66 0.85 (579.39) 3.70\nBL (AFM) -8.64 3.66 0.76 (513.05) 3.70\nCrSBr ML (FM) 0.56 3.56 0.06 (75.02) 3.66 -3.92 3.10 0.40 (496.28) 3.13\nBL (FM) 1.04 3.56 0.11 (68.76) 3.65 6.01 3.69 -0.80 (-502.03) 2.66\nBL (AFM) 9.87 2.24 -0.77 (-478.74) 2.28\nCrSI ML (FM) 0.64 3.02 0.08 (90.60) 3.28 5.90 3.30 -0.55 (-624.74) 3.37\nBL (FM) 1.00 3.06 0.12 (67.95) 3.41 8.63 3.37 -0.90 (-507.99) 3.41\nBL (AFM) 6.60 3.35 -0.77 (-436.75) 2.30\nCrSeCl ML (FM) 0.27 3.78 0.04 (49.52) 4.38 -3.66 3.40 0.31 (421.07) 4.76\nBL (FM) 0.51 3.74 0.08 (54.81) 4.41 6.99 2.82 -0.84 (-577.27) 2.91\nBL (AFM) 7.00 2.74 -0.76 (-524.16) 2.89\nCrSeBr ML (FM) 0.54 3.58 0.08 (101.94) 3.74 3.93 2.30 0.43 (552.31) 1.65\nBL (FM) 1.05 3.55 0.15 (95.57) 3.70 7.48 2.29 0.76 (484.44) 1.63\nBL (AFM) 10.27 2.45 0.87 (551.30) 1.63\nCrSeI ML (FM) 0.82 3.12 0.10 (114.72) 3.25 3.62 1.90 -0.44 (-504.61) 1.93\nBL (FM) 1.49 3.19 0.20 (114.72) 3.35 6.27 2.00 0.63 (362.78) 1.27\nBL (AFM) 5.17 1.92 -0.53 (-304.59) 1.99\nCrTeCl ML (FM) 0.45 4.26 0.10 (132.58) 4.36 8.54 2.66 -1.01 (-1342.27) 2.72\nBL (FM) 0.89 3.94 0.17 (114.92) 4.37 13.39 2.76 -1.67 (-1107.05) 2.85\nBL (AFM) 11.31 2.77 -1.50 (-994.53) 2.91\nCrTeBr ML (FM) 0.48 4.07 0.09 (110.91) 4.19 4.68 2.45 -0.57 (-731.54) 2.50\nBL (FM) 0.93 4.05 0.18 (112.56) 4.17 8.08 2.54 -1.07 (-680.64) 2.61\nBL (AFM) 10.15 2.66 -1.27 (-811.67) 2.70\nCrTeI ML (FM) 0.64 3.56 0.08 (93.56) 3.64 -2.53 4.28 0.32 (377.20) 4.48\nBL (FM) 1.17 3.61 0.19 (112.48) 3.76 -4.94 3.40 -0.46 (-269.45) 2.48\nBL (AFM) -7.25 3.39 -0.66 (-388.03) 3.05\nB. First-order magnetic-optical e\u000bects\nThe calculated optical conductivities ( \u001bxx,\u001byy, and\n\u001bxy) for ML and BL FM Cr XYwith out-of-plane magne-\ntization are displayed in supplemental Figs. S10{S13 [45].\nThe spectra of optical conductivity of ML and BL struc-\ntures have similar shape due to the weak vdW interac-\ntion [73]. After obtaining the optical conductivity ten-\nsors, we can calculate the \frst-order magneto-optic ef-\nfects, i.e., MOKE and MOFE, according to Eqs. (1){(3).\nThe Kerr and Faraday rotation angles ( \u0012Kand\u0012F) and\nthe corresponding ellipticities ( \"Kand\"F) as a function\nof photon energy are illustrated in Figs. 4{5 and supple-\nmental Figs. S14{S15 [45], respectively. From Figs. 4(a){4(b), one can observe that the \u0012Kof ML and BL CrS Y\nand CrSe Yare vanishing within the band gap due to\nthe semiconducting nature, while increase largely in high\nenergy region. The ML and BL CrTe Yexhibit distinct\n\u0012Kfrom zero frequency, suggesting the metallic property,\nas displayed in Fig. 4(c). The same features can also be\nfound in the Faraday rotation angles \u0012F[see Fig. 5]. The\nresultant MOKE and MOFE are thus consistent with the\nabove analysis about the electronic structures of Cr XY.\nEspecially, the peaks of \u0012Kand\u0012Ffor ML and BL\nCrXY appear at almost the same incident photon en-\nergy, as shown in Figs. 4 and 5. In the moderate energy\nrange (1\u00185 eV), the \u0012Kand\u0012Fare remarkably large.\nTaking Cr XBr as examples, the maximal values of \u0012K9\nTABLE II. The magneto-optical Kerr, Faraday, Sch afer-Hubert, and Voigt rotation angles ( \u0012K,\u0012F,\u0012SH, and\u0012V) for Cr XY\nand other 2D magnetic materials. For a better comparison, the absolute values of rotation angles are used here as the sign\nonly indicates the rotation direction with respect to the principle axis of incident light that may have di\u000berent de\fnitions in\nliteratures. Moreover, the rotation angles are given in a range due to the dependence on the number of layers for some 2D\nmaterials. The data marked by \"Exp\" are experimental measurements, while the others are computational results.\n\u0012K(deg) \u0012F(deg/\u0016m) \u0012SH(deg) \u0012V(deg)\nCrXY 0.25\u00181.49 33.93 \u0018132.58 2.53 \u001813.39 0.31 \u00181.67\nCrI3 0.29\u00182.86 [2]Exp50\u0018108 [71]\n0.60\u00181.30 [71]\nCrTe 2 0.24\u00181.76 [72] 30 \u0018173 [72]\nCrGeTe 3 0.0007\u00180.002 [3]Exp100\u0018120 [73]\n0.9\u00182.2 [73]\nGA 0.13 \u00180.81 [74] 9.3 \u0018137.8 [74]\nBP 0.02 \u00180.12 [74] 1.4 \u001818.9 [74]\nInS 0.34 [75] 84.3 [75]\nFenGeTe 2(n= 3, 4, 5) 0.74 \u00182.07 [33] 50.28 \u0018222.66 [33]\nCoO 0.007 [76]Exp\nCuMnAs 0.023 [77]Exp\n02C\nrSICrSBrMOSK (deg⋅eV) ML (FM z-axis) \nBL  (FM z-axis)CrSClC\nrSeICrSeBrCrSeClC\nrTeICrTeBrCrTeCl(a)0\n.00.5(b)M\nOSF (deg⋅eV)\nFIG. 6. (Color online) The magneto-optical strength of Kerr\n(a) and Faraday (b) e\u000bects for monolayer and bilayer Cr XY\nwith out-of-plane magnetization.\n(\u0012F) for ML Cr XBr (X= S, Se, Te) are 0.56\u000e(0.06\u000e),\n0.54\u000e(0.08\u000e), and 0.48\u000e(0.09\u000e), respectively. And the\nmaximal values of \u0012K(\u0012F) for BL Cr XBr (X= S, Se,Te) are 1.04\u000e(0.11\u000e), 1.05\u000e(0.15\u000e), and 0.93\u000e(0.18\u000e),\nrespectively. The maximal values of \u0012Kand\u0012Fas well\nas the corresponding photon energies for ML and BL\nCrXYare summarized in Tab. I. The Kerr rotation an-\ngles\u0012Kof ML and BL Cr XY are comparable or even\nlarger than that of conventional ferromagnets and some\nfamous 2D ferromagnets, e.g., bcc Fe (-0.5\u000e[35]), hcp Co\n(-0.42\u000e) [78], fcc Ni (-0.25\u000e) [79], CrI 3(ML 0.29\u000e, tri-\nlayer 2.86\u000e) [2], CrTe 2(ML 0.24\u000e, trilayer -1.76\u000e) [72],\nCrGeTe 3(BL 0.0007\u000e, trilayer 0.002\u000e[3]), gray arsenene\n(ML 0.81\u000e, BL-AA 0.13\u000e, BL-AC 0.14\u000e, here AA and\nAC are stacking patterns.) [74], blue phosphorene (ML\n0.12\u000e, BL-AA 0.02\u000e, BL-AB 0.03\u000e, BL-AC 0.03\u000e) [74],\nInS (ML 0.34\u000e) [75], and Fe nGeTe 2(n= 3, 4, 5) (0.74\u000e\n\u00182.07\u000e) [33]. In addition, the Faraday rotation angles\n\u0012Fof ML and BL Cr XYare also comparable or larger\nthan that of some famous 2D ferromagnets, such as CrI 3\n(ML 50\u000e=\u0016m, BL 75\u000e=\u0016m, trilayer 108\u000e=\u0016m) [71], CrTe 2\n(ML 30\u0018-173\u000e=\u0016m) [72], CrGeTe 3(ML 120\u000e=\u0016m, tri-\nlayer 100\u000e=\u0016m) [73], gray arsenene (ML 137.8\u000e=\u0016m, BL-\nAA 9.3\u000e=\u0016m, BL-AC 10.0\u000e=\u0016m) [74], blue phosphorene\n(ML 18.9\u000e=\u0016m, BL-AA 1.4\u000e=\u0016m, BL-AB 2.3\u000e=\u0016m, BL-\nAC 2.2\u000e=\u0016m) [74], InS (84.3\u000e=\u0016m) [75], and Fe nGeTe 2\n(n= 3, 4, 5) (50.28 \u0018222.66\u000e=\u0016m) [33]. The above\nmagneto-optical data are summarized in Tab. II for a\nbetter comparison.\nThe Kerr and Faraday rotation angles ( \u0012Kand\u0012F)\nat a given photon energy are obviously increasing with\nthe thin-\flm thickness of Cr XY, which can be easily seen\nfrom Tab. I and Figs. 4 and 5. In order to quantitatively\ndescribe the overall trend of Kerr and Faraday spectra,\nwe here introduce a physical quantity called magneto-10\noptical strengths (MOS) [75, 80, 81]\nMOSK=Z1\n0+~j\u0012K(!)jd!;\nMOSF=Z1\n0+~j\u0012F(!)jd!:(12)\nMOSKand MOS Fare plotted in Figs. 6(a) and 6(b),\nrespectively, from which one can observe that the MOKE\nand MOFE of BL Cr XYare nearly double that of ML\nCrXY. It is simply because that the MOKE and MOFE\nas the \frst-order magneto-optical e\u000bects are proportional\nto the magnetization, which is linearly dependent on the\nnumber of layers. This trend has also been observed in\n2D van der Waals magnets CrI 3[2] and Cr 2Ge2Te6[73].\nMoreover, a clear step-up trend of \u0012Kand\u0012Fcan be seen\nin ML and BL Cr XYby changing X(S!Se!Te) or\nY(Cl!Br!I). It can be understood that the MOKE\nand MOFE are also proportional to the SOC strength,\nwhich increases with the atomic number. This trend can\nalso be found in our previous work about hole-doped MX\n(M= Ga, In; X= S, Se, Te) monolayer, in which the\nMOKE and MOFE have a size consequence of In X >\nGaXdue to the larger SOC strength of In atom.\nC. Second-order magneto-optical e\u000bects\nThe \frst-order magneto-optical e\u000bects (MOKE and\nMOFE) are forbidden in ML and BL Cr XYwith in-plane\nmagnetization, while the second-order magneto-optical\ne\u000bects (MOSHE and MOVE) can arise without the re-\nstriction by symmetry. Here, we illustrate the MOSHE\nand MOVE of ML FM and BL FM/AFM Cr XYby tak-\ning the in-plane y-axis magnetization as an example, and\nthe results of the x-axis magnetization can be obtained by\nputting a minus sign (refer to Eqs. (4) and (5) and note\nthat\u000fzxequals zero due to the symmetry restriction). It\nhas to be mentioned here that due to the in-plane crystal\nanisotropy of Cr XY(a 2D rectangular primitive cell), the\nnatural linear birefringence exists. This phenomenon is\nactually not related to magnetism, and can not be mixed\ninto the second-order magneto-optical e\u000bects [82]. The\ndiagonal elements of permittivity tensor can be expanded\ninto a Taylor series in powers of magnetization M(up to\nsecond-order),\n\u000f\u000b\u000b(M) =\u000f(0)\n\u000b\u000b+\u000f(1)\n\u000b\u000bM+\u000f(2)\n\u000b\u000bM2; (13)\nwhere\u000f(0)\n\u000b\u000bis the part independent on magnetization, \u000f(1)\n\u000b\u000b\nand\u000f(2)\n\u000b\u000bare linearly and quadratically dependent on mag-\nnetization, respectively. According to the Onsager rela-\ntion,\u000f\u000b\u000b(\u0000M) =\u000f\u000b\u000b(M), we know \u000f(1)\n\u000b\u000b= 0. It turns\nout to be clear that \u000f(0)\n\u000b\u000bis responsible for the natural lin-\near birefringence originating from crystal anisotropy, and\n\u000f(2)\n\u000b\u000bis responsible for the MOSHE and MOVE originating\nfrom magnetism. We \frst calculate \u000f(0)\n\u000b\u000bin 2D nonmag-\nnetic Cr XYby forcing the magnetic moments to be zeroduring self-consistent \feld calculation, and then calcu-\nlate\u000f\u000b\u000b(the sum of \u000f(0)\n\u000b\u000band\u000f(2)\n\u000b\u000b) in 2D magnetic Cr XY.\nFinally,\u000f(2)\n\u000b\u000bcan be simply obtained by subtracting \u000f(0)\n\u000b\u000b\nfrom\u000f\u000b\u000b. Thus, we take \u000f(2)\n\u000b\u000binstead of\u000f\u000b\u000binto Eqs. (4)\nand (5). The corresponding optical conductivities \u001b(2)\n\u000b\u000b\nare shown in supplemental Figs. S16 and S17 [45].\nThe calculated Sch afer-Hubert and Voigt rotation an-\ngles (\u0012SHand\u0012V) of Cr XYare plotted in Figs. 7 and 8,\nand the corresponding ellipticities ( \"SHand\"V) are plot-\nted in supplemental Figs. S18 and S19 [45]. The spectra\nof\u0012SHand\u0012Vfor semiconducting CrS Yand CrSe Yare\nnegligibly small in the low energy range far below the\nband gaps. However, in contrast to the Kerr and Faraday\ne\u000bects,\u0012SHand\u0012Vtake on \fnite values near the band\nedges [e.g., Figs. 7(a) and 8(a)]. The reason is that \u0012SH\nand\u0012Vdepend on the diagonal elements of optical con-\nductivity, which are nonvanishing within the band gap\n(see Figs. S16 and S17 [45]). In the energy range larger\nthan the band gaps, \u0012SHand\u0012Voscillate frequently, as\nshown in Figs. 7(a,b) and 8(a,b), respectively. For metal-\nlic CrTe Y, the relatively larger \u0012SHand\u0012Vcan be found\nin Figs. 7(c) and 8(c), respectively. Taking CrSBr as an\nexample, the maximal values of \u0012SHfor ML FM, BL FM,\nand BL AFM states are -3.92\u000e, 6.01\u000e, and 9.87\u000eat the\nphoton energies of 3.10 eV, 3.69 eV, and 2.24 eV, respec-\ntively; the maximal values of \u0012Vfor ML FM, BL FM,\nand BL AFM states are 0.40\u000e, -0.80\u000e, and -0.77\u000eat the\nphoton energies of 3.13 eV, 2.66 eV, and 2.28 eV, respec-\ntively. The data of the largest \u0012SHand\u0012Vas well as\nthe related photon energies for all Cr XY family mem-\nbers are summarized in Tab. I. The \u0012SHand\u0012Vof Cr XY\nare larger than that of some 2D AFMs, for example, 10\nnm thick CoO thin-\flm ( \u0012SH\u00180.007 deg) [76] and 10\nnm thick CuMnAs thin-\flm ( \u0012V\u00180.023 deg) [77].\nFrom Eqs. (4) and (5), one can know that \u0012SH,\"SH,\n\u0012V, and\"Vare dominated by the two diagonal ele-\nments of permittivity tensor ( \u000fyyand\u000fxx) as the o\u000b-\ndiagonal element ( \u000fzx) is restricted to be zero by sym-\nmetry. By plugging the optical conductivity ( \u000fyy(xx)=\n1+4\u0019i\n!\u001byy(xx)) into Eq. (4), one shall \fnd \u0012SH/\u0000\u001b1\ndi\u000b=\n\u0000Re(\u001byy\u0000\u001bxx) and\"SH/\u0000\u001b2\ndi\u000b=\u0000Im(\u001byy\u0000\u001bxx).\nFigure 9 plots the real ( \u001b1\ndi\u000b) and imaginary ( \u001b2\ndi\u000b) parts\nof the di\u000berence between the diagonal elements of optical\nconductivities ( \u001byyand\u001bxx) for ML FM, BL FM, and BL\nAFM CrSBr. One can clearly observe that the spectral\nstructure of \u0012SHis determined by \u001b1\ndi\u000bonly di\u000bering by\nan opposite sign, comparing the middle panel of Fig. 7(a)\nwith Fig. 9(a). The spectral structure of \"SHis thus\ndetermined by \u001b2\ndi\u000b, comparing the middle panel of sup-\nplemental Fig. S18(a) [45] with Fig. 9(b). On the other\nhand, there is not such a clear relation between \u0012V(\"V)\nand\u001b1\ndi\u000b(\u001b2\ndi\u000b) since equation (5) involves the square\nroots of\u000fyyand\u000fxx, though the spectral structures of \u0012V\n(\"V) and\u001b1\ndi\u000b(\u001b2\ndi\u000b) seem to be similar somewhere.11\n-1001020-\n10010200\n2 4 6 -10010200\n2 4 6 02 4 6 /s113SH (deg) ML (FM y-axis) \nBL  (FM y-axis) \nBL  (AFM y-axis)C\nrSCl(a)C\nrSBrC rSI(\nb)C\nrSeCl/s113SH (deg)C\nrSeBrC rSeI(\nc)C\nrTeCl/s113SH (deg)P\nhoton energy (eV)CrTeBrP\nhoton energy (eV)CrTeIP\nhoton energy (eV)\nFIG. 7. (Color online) The magneto-optical Sch afer-Hubert rotation angles ( \u0012SH) of monolayer ferromagnetic (blue solid lines),\nbilayer ferromagnetic (dark pink solid lines), and bilayer antiferromagnetic (light pink solid lines) Cr XY with the in-plane\nmagnetization along the y-axis. The vertical dashed lines in (a) and (b) indicate the band gaps.\n-2-101-\n2-1010\n2 4 6 -2-1010\n2 4 6 02 4 6 /s113V (deg) \nML (FM y-axis) \nBL (FM y-axis) \nBL (AFM y-axis)CrSCl(a)C\nrSBrC rSI(\nb)C\nrSeCl/s113V (deg)CrSeBrC rSeI(\nc)C\nrTeCl/s113V (deg)P\nhoton energy (eV)CrTeBrP\nhoton energy (eV)CrTeIP\nhoton energy (eV)\nFIG. 8. (Color online) The magneto-optical Voigt rotation angles ( \u0012V) of monolayer ferromagnetic (blue solid lines), bilayer\nferromagnetic (dark pink solid lines), and bilayer antiferromagnetic (light pink solid lines) Cr XYwith the in-plane magnetization\nalong they-axis. The vertical dashed lines in (a) and (b) indicate the band gaps.12\n-4-20240\n2 4 6 -4-2024/s1151d\niff (1015 s-1)(a)(\nb)/s115 2d\niff (1015 s-1)P\nhoton energy (eV) ML \nBL (FM) \nBL (AFM)\nFIG. 9. (Color online) The real (a) and imaginary (b) parts\nof the di\u000berence between the two diagonal elements of optical\nconductivities ( \u001bdi\u000b=\u001byy\u0000\u001bxx) for monolayer ferromagnetic\n(blue solid lines), bilayer ferromagnetic (dark pink solid lines),\nand bilayer antiferromagnetic (light pink solid lines) CrSBr.\nD. Anomalous Hall, anomalous Nernst, and\nanomalous thermal Hall e\u000bects\nIn this subsection, we shall discuss the AHE as well\nas its thermoelectric counterpart, ANE, and its thermal\nanalogue, ATHE, for ML and BL FM Cr XYwith out-\nof-plane magnetization. The ANE and ATHE are eval-\nuated at 100 K below the Curie temperatures by the\ntheoretical prediction ( >108 K) [15] and experimental\nmeasurements ( >140 K) [19, 20]. The magnetic transi-\ntion temperature can be in principle increased via dop-\ning e\u000bects. For example, magnetic atom-doped 2D tran-\nsition metal dichalcogenides have been predicted to ex-\nhibit room-temperature ferromagnetism [83, 84]. Addi-\ntionally, the Curie temperature of trilayer Fe 3GeTe 2can\nbe improved to room-temperature by ionic-liquid gating\ntechnique [7].\nThe calculated intrinsic AHC ( \u001bxy) as a function of en-\nergy are displayed in Fig. 10. For semiconducting CrS Y\nand CrSe Y, the AHC is zero within the band gaps, sim-\nilarly to the magneto-optical Kerr and Faraday e\u000bects.\nFor metallic CrTe Y(Y= Cl, Br, I), \u001bxytakes the val-\nues of 0.14 e2=h, 0.75e2=h, and 1.05 e2=hfor ML FM\nstates, and of 0.35 e2=h, 0.98e2=h, and 1.56 e2=hfor\nBL FM states at the Fermi energy ( EF), respectively.\nThe magnetotransport properties like the AHE can bemodulated by electron or hole doping, which shifts the\nEFupward or downward, respectively. According to the\nrecent experimental works [7, 85], the doping carrier con-\ncentration for 2D materials can reach up to 1015cm\u00002\nvia the gate voltage. Then, the variation range of the EF\ncan be estimated in a reasonable range, see supplemental\nFig. S20 [45], by taking ML Cr XBr as examples. The en-\nergy ranges for ML and BL CrS Y, CrSe Y, and CrTe Yare\nreasonably given to be [-0.5, 2.0] eV, [-0.5, 1.0], and [-0.5,\n0.5] eV, respectively. In the above energy regions, \u001bxyin-\ncreases up to 1.93 e2=h, 1.54e2=h, and 1.57e2=hfor ML\nCrSBr, CrTeCl, and CrTeBr at -0.40 eV, -0.39 eV, and\n-0.12 eV by hole doping, respectively. And \u001bxyincreases\nup to 2.12e2=h, 1.88e2=h, and 2.24e2=hfor BL CrTeCl,\nCrTeBr, and CrTeI at -0.32 eV, -0.04 eV, and -0.16 eV\nby hole doping, respectively. On the other hand for elec-\ntron doping, \u001bxycan be reached to 1.60 e2=h, 2.12e2=h,\n3.61e2=hfor BL CrTeCl, CrTeBr, and CrTeI at 0.23 eV,\n0.14 eV, and 0.11 eV, respectively. It is obvious that\nthe metallic CrTe Yexhibit more prominent AHE than\nthe semiconducting CrS Yand CrSe Y. The largest \u001bxy\nfor CrTe Yare in the range of 544.46{827.95 S/cm (con-\nverted from the unit of e2=hby considering the e\u000bective\nthickness), which is larger than bulk Fe 3GeTe 2(140{540\nS/cm) [86], thin \flm Fe 3GeTe 2(\u0018400 S/cm) [87], thin\n\flm Fe 4GeTe 2(\u0018180 S/cm) [88], and is even compared\nwith bulk bcc Fe (751 S/cm [46], 1032 S/cm [89]).\nStrikingly, a peak of \u001bxyappears in BL CrS Yand\nCrSe Ynear the bottom of conduction bands, and grows\nlarger with the increasing of atomic number (Cl !Br\n!I), as displayed in Figs. 10(a) and 10(b). We inter-\npret the underlying physical reasons by taking BL CrSe Y\nas examples. At the energy position of the peaked \u001bxy,\nthere exists a band crossing in the band structure that\nis dominated by Cr dx2\u0000y2orbitals [see the left inset in\nFig. 10(b)]. The band crossing is a Weyl-like point, which\nis protected by the glide symmetry Mz\u001c(1\n2;1\n2;0). It pre-\nserves even after including SOC because the glide sym-\nmetry still exists. Pinning the EFat the band crossing\npoint, the Berry curvature exhibits hot spots around the\n\u0000 point [see the right inset in Fig. 10(b)], which de\fnitely\ngives rise to a large \u001bxy. With the increasing of atomic\nnumberY, the region of the hot spots of Berry curvature\nenlarges gradually, which enhances the peak of \u001bxy.\nThe ANE, the thermoelectric counterpart of AHE, is\na celebrated e\u000bect from the realm of the spin caloritron-\nics [90, 91]. At the low temperature, the ANC ( \u000bxy) is\nconnected to the energy derivative of \u001bxyvia the Mott\nrelation [92],\n\u000bxy=\u0000\u00192k2\nBT\n3e\u001b0\nxy(E); (14)\nwhich can be derived from Eq. (10). From the above\nequation, one can \fnd that a steep slope of \u001bxyver-\nsus energy generally leads to a large \u000bxy. For example,\nthere appear two opposite peaks of \u000bxyclose to the bot-\ntom of conduction bands for BL CrS Yand CrSe Y[see\nFigs. 11(a) and 11(b)], which exactly originates from the13\n-0.50 .00 .51 .01 .52 .0-101234-\n0.50 .00 .51 .01 .52 .0-0.50 .00 .51 .01 .52 .0-\n0.50 .00 .51 .0-101234-\n0.50 .00 .51 .0-0.50 .00 .51 .0-\n0.50 .00 .5-101234-\n0.50 .00 .5-0.50 .00 .5/s115xy (e2/h) ML (FM z-axis) \nBL  (FM z-axis)C\nrSCl(a)C\nrSBrC rSI(\nb)C\nrSeCl/s115xy (e2/h)\nΓ\nX Y\nC\nrSeBr\nΓX Ymaxm\nin\nC\nrSeI\nΓX Y\n(\nc)C\nrTeCl/s115xy (e2/h)E\n (eV)CrTeBrE\n (eV)CrTeIE\n (eV)\nFIG. 10. (Color online) The intrinsic anomalous Hall conductivities ( \u001bxy) as a function of energy for monolayer (blue solid lines)\nand bilayer (pink solid lines) Cr XYwith out-of-plane magnetization. The black dashed lines indicate the true Fermi energy\nfor metals or the top of valence bands for semiconductors. The blue and pink dashed lines indicate the bottom of conduction\nbands for monolayer and bilayer semiconducting CrS Y/CrSe Y, respectively. The right inset in (b) are momentum-resolved\nBerry curvature (in arbitrary unit) in 2D Brillouin zone. The left inset in (b) are orbital-projected band structures, in which\nthe violet solid triangles present the components of Cr dx2\u0000y2orbitals and the orange dashed lines label the energy position\nwhere the Berry curvature is calculated.\npositive and negative slopes of the peak of \u001bxyat the\nsame energy [see Fig. 10(a) and(b)]. For semiconduct-\ning CrS Yand CrSe Y,\u000bxyis unsurprisingly zero within\nthe electronic band gap. Among CrS Yand CrSe Y, ML\nCrSBr exhibits the excellent ANE under appropriate hole\ndoping, such as 2.59 AK\u00001m\u00001and -3.71 AK\u00001m\u00001at -\n0.38 eV and -0.44 eV, respectively. For metallic ML (BL)\nCrTe YwithY= Cl, Br, and I, the calculated \u000bxyat the\ntrue Fermi energy are 0.09 (0.09) AK\u00001m\u00001, 0.32 (0.36)\nAK\u00001m\u00001, and -0.58 (-1.13) AK\u00001m\u00001, respectively. By\ndoping holes or electrons, \u000bxycan reach up to -2.13 (0.94)\nAK\u00001m\u00001at -0.42 (-0.20) eV for ML (BL) CrTeCl, to -\n2.29 (-1.55) AK\u00001m\u00001at -0.14 (-0.12) eV for ML (BL)\nCrTeBr, and to 0.90 (-1.29) AK\u00001m\u00001at 0.16 (0.09) eV\nfor ML (BL) CrTeI, respectively. The calculated j\u000bxyjof\nML CrSBr, CrTeCl, and CrTeBr are considerably large\n(>2.0 AK\u00001m\u00001), which exceeds to bulk ferromagnets\nCo3Sn2S2(\u00182.0 AK\u00001m\u00001) [93] and Ti 2MnAl (\u00181.31\nAK\u00001m\u00001) [94], and even comparable with multi-layer\nFenGeTe 2(n= 3, 4, 5) (0.3{3.3 AK\u00001m\u00001) [33], sug-gesting that 2D Cr XYis an excellent material platform\nfor thermoelectric applications.\nThe ATHE, the thermal analogue of AHE, has been\nwidely used to study the charge neutral quasiparticles\nin quantum materials [95{101]. The calculated ATHC\n(\u0014xy) of ML and BL Cr XYare plotted in Fig. 12. Sim-\nilarly to\u001bxyand\u000bxy,\u0014xyis vanishing within the band\ngaps of CrS Yand CrSe Y. The calculated \u0014xyat the true\nFermi energy for ML (BL) CrTe YwithY= Cl, Br, and\nI are 0.02 (0.02) WK\u00001m\u00001, 0.09 (0.07) WK\u00001m\u00001, and\n0.12 (0.09) WK\u00001m\u00001, respectively. The ATHE can be\nfurther enhanced by doping appropriate holes or elec-\ntrons. For instance, \u0014xyincreases to its largest value\nof 0.15 WK\u00001m\u00001at -0.39 eV for ML CrSBr. In the\ncase of ML (BL) CrTe Y, the largest \u0014xyreaches up to\n0.14 (0.11) WK\u00001m\u00001, 0.13 (0.10) WK\u00001m\u00001, and 0.14\n(0.17) WK\u00001m\u00001at -0.37 (-0.34) eV, -0.09 (0.11) eV,\nand 0.08 (0.13) eV, respectively. The \u0014xyof 2D Cr XY\nare much larger than that of other vdW ferromagnets,\ne.g., VI 3(0.01 WK\u00001m\u00001) [101].14\n-0.50 .00 .51 .01 .52 .0-4-202-\n0.50 .00 .51 .01 .52 .0-0.50 .00 .51 .01 .52 .0-\n0.50 .00 .51 .0-4-202-\n0.50 .00 .51 .0-0.50 .00 .51 .0-\n0.50 .00 .5-4-202-\n0.50 .00 .5-0.50 .00 .5/s97xy (AK-1m-1) ML (FM z-axis) \nBL  (FM z-axis)C\nrSCl(a)C\nrSBrC rSI(\nb)C\nrSeCl/s97xy (AK-1m-1)C\nrSeBrC rSeI(\nc)C\nrTeCl/s97xy (AK-1m-1)E\n (eV)CrTeBrE\n (eV)CrTeIE\n (eV)\nFIG. 11. (Color online) The anomalous Nernst conductivities ( \u000bxy) calculated at 100 K as a function of energy for monolayer\n(blue solid lines) and bilayer (pink solid lines) Cr XYwith out-of-plane magnetization. The black dashed lines indicate the true\nFermi energy for metals or the top of valence bands for semiconductors. The blue and pink dashed lines indicate the bottom\nof conduction bands for monolayer and bilayer semiconducting CrS Y/CrSe Y, respectively.\n-0.50 .00 .51 .01 .52 .00.00.10.2-\n0.50 .00 .51 .01 .52 .0- 0.50 .00 .51 .01 .52 .0-\n0.50 .00 .51 .00.00.10.2-\n0.50 .00 .51 .0-0.50 .00 .51 .0-\n0.50 .00 .50.00.10.2-\n0.50 .00 .5-0.50 .00 .5/s107xy (WK-1m-1) ML (FM z-axis) \nBL  (FM z-axis)CrSCl(a)C\nrSIC rSBr(\nb)C\nrSeCl/s107xy (WK-1m-1)CrSeBrC rSeI(\nc)C\nrTeCl/s107xy (WK-1m-1)E\n (eV)CrTeBrE\n (eV)CrTeIE\n (eV)\nFIG. 12. (Color online) The anomalous thermal Hall conductivities ( \u0014xy) calculated at 100 K as a function of energy for\nmonolayer (blue solid lines) and bilayer (pink solid lines) Cr XY with out-of-plane magnetization. The black dashed lines\nindicate the true Fermi energy for metals or the top of valence bands for semiconductors. The blue and pink dashed lines\nindicate the bottom of conduction bands for monolayer and bilayer semiconducting CrS Y/CrSe Y, respectively.15\nIV. SUMMARY\nIn summary, we systematically investigated the band\nstructures, magnetocrystalline anisotropy energy, \frst-\nand second-order magneto-optical e\u000bects, and intrinsi-\ncally anomalous transport properties in monolayer and\nbilayer Cr XY(X= S, Se, Te; Y= Cl, Br, I) by employ-\ning the \frst-principles calculations. From the band struc-\ntures, monolayer and bilayer CrS Yand CrSe Yare iden-\nti\fed to be narrow band-gap semiconductors, whereas\nmonolayer and bilayer CrTe Yare multi-band metals.\nThe results of magnetocrystalline anisotropy energy show\nthat CrTeBr and CrTeI prefer to out-of-plane magnetiza-\ntion, whereas the other seven family members of Cr XY\nare in favor of in-plane magnetization. The magnetic\nground states for bilayer Cr XYare con\frmed, that is, bi-\nlayer CrSI, CrSeCl, and CrTeCl present strong interlayer\nferromagnetic coupling, whereas the other ones show in-\nterlayer antiferromagnetic coupling. The magnetic group\ntheory identi\fes the nonzero elements of optical conduc-\ntivity tensor, which in turn determines the symmetry of\nmagneto-optical Kerr and Faraday e\u000bects as well as the\nanomalous Hall, anomalous Nernst, and anomalous ther-\nmal Hall e\u000bects. The above physical e\u000bects can only exist\nin monolayer and bilayer ferromagnetic Cr XYwith out-\nof-plane magnetization. The magneto-optical strength\nfor the Kerr and Faraday e\u000bects of bilayer Cr XY are\nnearly two times that of monolayer Cr XYdue to the dou-\nbled magnetization, and the magneto-optical strength of\nboth monolayer and bilayer Cr XY increase distinctly\nwith the increasing of atomic number in XorYbe-\ncause of the enhanced spin-orbital coupling. The largest\nKerr and Faraday rotation angles in the moderate energy\nrange (1\u00185 eV) are found in bilayer CrSeI, which turns\nout to be 1.49\u000eand 0.20\u000e, respectively. The anomalous\ntransport properties of Cr XYare considerably large un-\nder proper hole or electron doping. For instance, the\nanomalous Hall, anomalous Nernst, and anomalous ther-\nmal Hall conductivities reach up to their maximums of3.61e2=h, -3.71 AK\u00001m\u00001, and 0.17 WK\u00001m\u00001in bi-\nlayer CrTeI, monolayer CrSBr, and bilayer CrTeI, re-\nspectively. In the case of in-plane magnetization, the\nsecond-order magneto-optical e\u000bects do exist in both fer-\nromagnetic and antiferromagnetic Cr XY, in contrast to\nthe \frst-order ones. The calculated Sch afer-Hubert and\nVoigt e\u000bects are considerable large in Cr XY. For exam-\nple, the largest Sch afer-Hubert and Voigt rotation angles\nare found to be 13.39\u000eand -1.67\u000ein bilayer ferromagnetic\nCrTeCl, and to be 11.31\u000eand -1.50\u000ein bilayer antiferro-\nmagnetic CrTeCl, respectively.\nOur results show that 2D van der Waals layered mag-\nnets Cr XY may be excellent material candidates for\npromising applications to magneto-optical devices, spin-\ntronics, and spin caloritronics. For example, the semi-\nconducting Cr XY can be used to make van der Waals\nmagnetic heterostructures, similarly to semiconducting\nferromagnetic CrI 3, which realizes the quantum anoma-\nlous Hall e\u000bect [102] and valley manipulation [103, 104].\nThe metallic Cr XYcan also be used to make heterostruc-\ntures, similarly to metallic Fe 3GeTe 2, which can increase\nthe critical temperatures of 2D magnetic materials [105],\nfabricate giant magnetoresistance (GMR) devices [106],\ndesign metal electrodes [107], and tailor the anomalous\ntransport properties [108].\nACKNOWLEDGMENTS\nThe authors thank Gui-Bin Liu, Chaoxi Cui, and\nShifeng Qian for their helpful discussion. This work\nis supported by the National Natural Science Founda-\ntion of China (Grant Nos. 11874085, 11734003, and\n12061131002), the Sino-German Mobility Programme\n(Grant No. M-0142), the National Key R&D Program\nof China (Grant No. 2020YFA0308800), and the Science\n& Technology Innovation Program of Beijing Institute of\nTechnology (Grant No. 2021CX01020).\n[1] N. D. Mermin and H. Wagner, Absence of ferromag-\nnetism or antiferromagnetism in one-or two-dimensional\nisotropic Heisenberg models, Phys. Rev. Lett. 17, 1133\n(1966).\n[2] B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein,\nR. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A.\nMcGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-\nHerrero, and X. Xu, Layer-dependent ferromagnetism\nin a van der Waals crystal down to the monolayer limit,\nNature 546, 270 (2017).\n[3] C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao,\nW. Bao, C. Wang, Y. Wang, Z. Q. Qiu, R. J. Cava,\nS. G. Louie, J. Xia, and X. Zhang, Discovery of intrin-\nsic ferromagnetism in two-dimensional van der Waals\ncrystals, Nature 546, 265 (2017).\n[4] M. Bonilla, S. Kolekar, Y. Ma, H. C. Diaz, V. Kalappat-til, R. Das, T. Eggers, H. R. Gutierrez, M.-H. Phan, and\nM. Batzill, Strong room-temperature ferromagnetism\nin VSe 2monolayers on van der Waals substrates, Nat.\nNanotech. 13, 289 (2018).\n[5] Z.-L. Liu, X. Wu, Y. Shao, J. Qi, Y. Cao, L. Huang,\nC. Liu, J.-O. Wang, Q. Zheng, Z.-L. Zhu, K. Ibrahim,\nY.-L. Wang, and H.-J. Gao, Epitaxially grown mono-\nlayer VSe 2: an air-stable magnetic two-dimensional ma-\nterial with low work function at edges, Sci. Bull. 63, 419\n(2018).\n[6] Z. Fei, B. Huang, P. Malinowski, W. Wang, T. Song,\nJ. Sanchez, W. Yao, D. Xiao, X. Zhu, A. F. May, W. Wu,\nD. H. Cobden, J.-H. Chu, and X. Xu, Two-dimensional\nitinerant ferromagnetism in atomically thin Fe 3GeTe 2,\nNat. Mater. 17, 778 (2018).\n[7] Y. Deng, Y. Yu, Y. Song, J. Zhang, N. Z. Wang, Z. Sun,16\nY. Yi, Y. Z. Wu, S. Wu, J. Zhu, J. Wang, X. H. Chen,\nand Y. Zhang, Gate-tunable room-temperature ferro-\nmagnetism in two-dimensional Fe 3GeTe 2, Nature 563,\n94 (2018).\n[8] X. Zhang, Q. Lu, W. Liu, W. Niu, J. Sun, J. Cook,\nM. Vaninger, P. F. Miceli, D. J. Singh, S.-W. Lian, T.-\nR. Chang, X. He, J. Du, L. He, R. Zhang, G. Bian, and\nY. Xu, Room-temperature intrinsic ferromagnetism in\nepitaxial CrTe 2ultrathin \flms, Nat. Commun. 12, 2492\n(2021).\n[9] H. Katscher and H. Hahn, Uber Chalkogenidhalogenide\ndes dreiwertigen Chroms, Naturwissenschaften 53, 361\n(1966).\n[10] O. G oser, W. Paul, and H. Kahle, Magnetic properties\nof CrSBr, J. Magn. Magn. Mat. 92, 129 (1990).\n[11] E. J. Telford, A. H. Dismukes, K. Lee, M. Cheng,\nA. Wieteska, A. K. Bartholomew, Y.-S. Chen, X. Xu,\nA. N. Pasupathy, X. Zhu, C. R. Dean, and X. Roy,\nLayered Antiferromagnetism Induces Large Negative\nMagnetoresistance in the van der Waals Semiconductor\nCrSBr, Adv. Mater. 32, 2003240 (2020).\n[12] N. Mounet, M. Gibertini, P. Schwaller, D. Campi,\nA. Merkys, A. Marrazzo, T. Sohier, I. E. Castelli, A. Ce-\npellotti, G. Pizzi, and N. Marzari, Two-dimensional\nmaterials from high-throughput computational exfoli-\nation of experimentally known compounds, Nat. Nan-\notech. 13, 246 (2018).\n[13] Z. Jiang, P. Wang, J. Xing, X. Jiang, and J. Zhao,\nScreening and Design of Novel 2D Ferromagnetic Mate-\nrials with High Curie Temperature above Room Tem-\nperature, ACS Appl. Mater. Interfaces 10, 39032 (2018).\n[14] Y. Guo, Y. Zhang, S. Yuan, B. Wang, and J. Wang,\nChromium sul\fde halide monolayers: intrinsic ferro-\nmagnetic semiconductors with large spin polarization\nand high carrier mobility, Nanoscale 10, 18036 (2018).\n[15] C. Wang, X. Zhou, L. Zhou, N.-H. Tong, Z.-Y. Lu,\nand W. Ji, A family of high-temperature ferromag-\nnetic monolayers with locked spin-dichroism-mobility\nanisotropy: MnN Xand CrCX(X= Cl, Br, I; C=\nS, Se, Te), Sci. Bull. 64, 293 (2019).\n[16] R. Han, Z. Jiang, and Y. Yan, Prediction of Novel 2D\nIntrinsic Ferromagnetic Materials with High Curie Tem-\nperature and Large Perpendicular Magnetic Anisotropy,\nJ. Phys. Chem. C 124, 7956 (2020).\n[17] H. Wang, J. Qi, and X. Qian, Electrically tunable high\nCurie temperature two-dimensional ferromagnetism in\nvan der Waals layered crystals, Appl. Phys. Lett. 117,\n083102 (2020).\n[18] K. Yang, G. Wang, L. Liu, D. Lu, and H. Wu, Triaxial\nmagnetic anisotropy in the two-dimensional ferromag-\nnetic semiconductor CrSBr, Phys. Rev. B 104, 144416\n(2021).\n[19] K. Lee, A. H. Dismukes, E. J. Telford, R. A. Wiscons,\nJ. Wang, X. Xu, C. Nuckolls, C. R. Dean, X. Roy, and\nX. Zhu, Magnetic Order and Symmetry in the 2D Semi-\nconductor CrSBr, Nano Lett. 21, 3511 (2021).\n[20] N. P. Wilson, K. Lee, J. Cenker, K. Xie, A. H. Dis-\nmukes, E. J. Telford, J. Fonseca, S. Sivakumar, C. Dean,\nT. Cao, X. Roy, X. Xu, and X. Zhu, Interlayer electronic\ncoupling on demand in a 2D magnetic semiconductor,\nNat. Mater. 20, 1657 (2021).\n[21] E. J. Telford, A. H. Dismukes, R. L. Dudley, R. A. Wis-\ncons, K. Lee, J. Yu, S. Shabani, A. Scheie, K. Watan-abe, T. Taniguchi, D. Xiao, A. N. Pasupathy, C. Nuck-\nolls, X. Zhu, C. R. Dean, and X. Roy, Hidden low-\ntemperature magnetic order revealed through magneto-\ntransport in monolayer CrSBr, arXiv:2106.08471.\n[22] D. J. Rizzo, A. S. McLeod, C. Carnahan, E. J. Telford,\nA. H. Dismukes, R. A. Wiscons, Y. Dong, C. Nuckolls,\nC. R. Dean, A. N. Pasupathy, X. Roy, D. Xiao, and\nD. N. Basov, Visualizing Atomically Layered Magnetism\nin CrSBr, Adv. Mater. 34, 2201000 (2022).\n[23] J. Cenker, S. Sivakumar, K. Xie, A. Miller, P. Thijssen,\nZ. Liu, A. Dismukes, J. Fonseca, E. Anderson, X. Zhu,\nX. Roy, D. Xiao, J.-H. Chu, T. Cao, and X. Xu, Re-\nversible strain-induced magnetic phase transition in a\nvan der Waals magnet, Nat. Nanotech. 17, 256 (2022).\n[24] Y. J. Bae, J. Wang, J. Xu, D. G. Chica, G. M. Diederich,\nJ. Cenker, M. E. Ziebel, Y. Bai, H. Ren, C. R. Dean,\nM. Delor, X. Xu, X. Roy, A. D. Kent, and X. Zhu,\nExciton-Coupled Coherent Antiferromagnetic Magnons\nin a 2D Semiconductor, arXiv:2201.13197.\n[25] T. S. Ghiasi, A. A. Kaverzin, A. H. Dismukes, D. K.\nde Wal, X. Roy, and B. J. van Wees, Electrical and\nthermal generation of spin currents by magnetic bilayer\ngraphene, Nat. Nanotech. 16, 788 (2021).\n[26] J. K. LL.D., On rotation of the plane of polarization by\nre\rection from the pole of a magnet, Phil. Mag. 3, 321\n(1877).\n[27] M. Faraday, Experimental researches in electricity.-\nNineteenth series, Phil. Trans. R. Soc. 136, 1 (1846).\n[28] R. Sch afer and A. Hubert, A new magnetooptic e\u000bect\nrelated to non-uniform magnetization on the surface of\na ferromagnet, phys. stat. sol. (a) 118, 271 (1990).\n[29] W. Voigt, Magneto-und elektrooptik, 3 (BG Teubner,\n1908).\n[30] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald,\nand N. P. Ong, Anomalous hall e\u000bect, Rev. Mod. Phys.\n82, 1539 (2010).\n[31] W. Nernst, Ueber die electromotorischen Kr afte, welche\ndurch den Magnetismus in von einem W armestrome\ndurch\rossenen Metallplatten geweckt werden, Ann.\nPhys. 267, 760 (1887).\n[32] T. Qin, Q. Niu, and J. Shi, Energy Magnetization and\nthe Thermal Hall E\u000bect, Phys. Rev. Lett. 107, 236601\n(2011).\n[33] X. Yang, X. Zhou, W. Feng, and Y. Yao, Strong\nmagneto-optical e\u000bect and anomalous transport in the\ntwo-dimensional van der Waals magnets Fe nGeTe 2(n=\n3, 4, 5), Phys. Rev. B 104, 104427 (2021).\n[34] Y. Suzuki, T. Katayama, S. Yoshida, K. Tanaka, and\nK. Sato, New magneto-optical transition in ultrathin\nFe(100) \flms, Phys. Rev. Lett. 68, 3355 (1992).\n[35] G. Y. Guo and H. Ebert, Band-theoretical investigation\nof the magneto-optical Kerr e\u000bect in Fe and Co multi-\nlayers, Phys. Rev. B 51, 12633 (1995).\n[36] P. Ravindran, A. Delin, P. James, B. Johansson, J. M.\nWills, R. Ahuja, and O. Eriksson, Magnetic, optical,\nand magneto-optical properties of MnX (X=As, Sb, or\nBi) from full-potential calculations, Phys. Rev. B 59,\n15680 (1999).\n[37] G. Ghosh, Dispersion-equation coe\u000ecients for the re-\nfractive index and birefringence of calcite and quartz\ncrystals, Opt. Commun. 163, 95 (1999).\n[38] An online database of refractive index,\nhttps://refractiveindex.info/.\n[39] N. Tesa\u0014 rov\u0013 a, T. Ostatnick\u0013 y, V. Nov\u0013 ak, K. Olejn\u0013 \u0010k,17\nJ.\u0014Subrt, H. Reichlov\u0013 a, C. T. Ellis, A. Mukherjee,\nJ. Lee, G. M. Sipahi, J. Sinova, J. Hamrle, T. Jung-\nwirth, P. N\u0014 emec, J. \u0014Cerne, and K. V\u0013 yborn\u0013 y, Systematic\nstudy of magnetic linear dichroism and birefringence in\n(Ga,Mn)As, Phys. Rev. B 89, 085203 (2014).\n[40] H.-C. Mertins, P. M. Oppeneer, J. Kune\u0014 s, A. Gaupp,\nD. Abramsohn, and F. Sch afers, Observation of the X-\nRay Magneto-Optical Voigt E\u000bect, Phys. Rev. Lett. 87,\n047401 (2001).\n[41] A. A. Mosto\f, J. R. Yates, Y.-S. Lee, I. Souza, D. Van-\nderbilt, and N. Marzari, wannier90: A tool for ob-\ntaining maximally-localised Wannier functions, Com-\nput. Phys. Commun. 178, 685 (2008).\n[42] J. R. Yates, X. Wang, D. Vanderbilt, and I. Souza,\nSpectral and Fermi surface properties from Wannier in-\nterpolation, Phys. Rev. B 75, 195121 (2007).\n[43] J. H. Strait, P. Nene, and F. Rana, High intrinsic mobil-\nity and ultrafast carrier dynamics in multilayer metal-\ndichalcogenide MoS 2, Phys. Rev. B 90, 245402 (2014).\n[44] N. Sivadas, S. Okamoto, and D. Xiao, Gate-\nControllable Magneto-optic Kerr E\u000bect in Layered\nCollinear Antiferromagnets, Phys. Rev. Lett. 117,\n267203 (2016).\n[45] See Supplemental Material at http://link.aps.org/xxx,\nwhich includes convergence test, structural information,\nand supplementary data and \fgures.\n[46] Y. Yao, L. Kleinman, A. H. MacDonald, J. Sinova,\nT. Jungwirth, D.-s. Wang, E. Wang, and Q. Niu, First\nprinciples calculation of anomalous Hall conductivity\nin ferromagnetic bcc Fe, Phys. Rev. Lett. 92, 037204\n(2004).\n[47] N. W. Ashcroft and N. D. Mermin, Solid state physics\n(SaundersCollege Publishing, Philadelphia, 1976).\n[48] H. van Houten, L. W. Molenkamp, C. W. J. Beenakker,\nand C. T. Foxon, Thermo-electric properties of quantum\npoint contacts, Semicond. Sci. Technol. 7, B215 (1992).\n[49] K. Behnia, Fundamentals of thermoelectricity (Oxford\nUniversity Press, 2015).\n[50] G. Kresse and J. Hafner, Ab initio molecular dynamics\nfor liquid metals, Phys. Rev. B 47, 558 (1993).\n[51] G. Kresse and J. Furthm uller, E\u000ecient iterative schemes\nfor ab initio total-energy calculations using a plane-wave\nbasis set, Phys. Rev. B 54, 11169 (1996).\n[52] P. E. Bl ochl, Projector augmented-wave method, Phys.\nRev. B 50, 17953 (1994).\n[53] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized\ngradient approximation made simple, Phys. Rev. Lett.\n77, 3865 (1996).\n[54] J. Klime\u0014 s, D. R. Bowler, and A. Michaelides, Chemical\naccuracy for the van der Waals density functional, J.\nPhys.: Condens. Matter 22, 022201 (2009).\n[55] K. Lee, E. D. Murray, L. Kong, B. I. Lundqvist, and\nD. C. Langreth, Higher-accuracy van der Waals density\nfunctional, Phys. Rev. B 82, 081101 (2010).\n[56] J. Klime\u0014 s, D. R. Bowler, and A. Michaelides, Van der\nWaals density functionals applied to solids, Phys. Rev.\nB83, 195131 (2011).\n[57] A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen,\nDensity-functional theory and strong interactions: Or-\nbital ordering in Mott-Hubbard insulators, Phys. Rev.\nB52, R5467 (1995).\n[58] T. Wo\u0013 zniak, P. E. Faria Junior, G. Seifert, A. Chaves,\nand J. Kunstmann, Exciton gfactors of van der Waals\nheterostructures from \frst-principles calculations, Phys.Rev. B 101, 235408 (2020).\n[59] T. Deilmann, P. Kr uger, and M. Rohl\fng, Ab Ini-\ntio Studies of Exciton gFactors: Monolayer Transition\nMetal Dichalcogenides in Magnetic Fields, Phys. Rev.\nLett. 124, 226402 (2020).\n[60] J. F orste, N. V. Tepliakov, S. Y. Kruchinin, J. Lind-\nlau, M. F. Victor Funk, K. Watanabe, T. Taniguchi,\nA. S. Baimuratov, and A. H ogele, Exciton g-factors in\nmonolayer and bilayer WSe 2from experiment and the-\nory, Nat. Commun. 11, 4539 (2020).\n[61] F. Xuan and S. Y. Quek, Valley Zeeman e\u000bect and Lan-\ndau levels in two-dimensional transition metal dichalco-\ngenides, Phys. Rev. Research 2, 033256 (2020).\n[62] H. T. Stokes, D. M. Hatch, and B. J.\nCampbell, ISOTROPY Software Suite.,\nhttps://stokes.byu.edu/iso/isotropy.php .\n[63] J. Klein, T. Pham, J. D. Thomsen, J. B. Curtis,\nM. Lorke, M. Florian, A. Steinho\u000b, R. A. Wiscons,\nJ. Luxa, Z. Sofer, F. Jahnke, P. Narang, and F. M.\nRoss, Atomistic spin textures on-demand in the van der\nWaals layered magnet CrSBr, arXiv:2107.00037.\n[64] P. W. Anderson, Antiferromagnetism. Theory of Su-\nperexchange Interaction, Phys. Rev. 79, 350 (1950).\n[65] J. B. Goodenough, An interpretation of the mag-\nnetic properties of the perovskite-type mixed crystals\nLa1\u0000xSrxCoO 3\u0000\u0015, J. Phys. Chem. Solids 6, 287 (1958).\n[66] J. Kanamori, Superexchange interaction and symmetry\nproperties of electron orbitals, J. Phys. Chem. Solids\n10, 87 (1959).\n[67] H. H. Kim, B. Yang, S. Li, S. Jiang, C. Jin, Z. Tao,\nG. Nichols, F. S\fgakis, S. Zhong, C. Li, S. Tian,\nD. G. Cory, G.-X. Miao, J. Shan, K. F. Mak, H. Lei,\nK. Sun, L. Zhao, and A. W. Tsen, Evolution of inter-\nlayer and intralayer magnetism in three atomically thin\nchromium trihalides, Proc. Natl. Acad. Sci. USA 116,\n11131 (2019).\n[68] X. Zhou, J.-P. Hanke, W. Feng, F. Li, G.-Y. Guo,\nY. Yao, S. Bl ugel, and Y. Mokrousov, Spin-order de-\npendent anomalous Hall e\u000bect and magneto-optical ef-\nfect in the noncollinear antiferromagnets Mn 3XN with\nX= Ga, Zn, Ag, or Ni, Phys. Rev. B 99, 104428 (2019).\n[69] M. Wu, Z. Li, T. Cao, and S. G. Louie, Physical ori-\ngin of giant excitonic and magneto-optical responses\nin two-dimensional ferromagnetic insulators, Nat. Com-\nmun. 10, 2371 (2020).\n[70] A. Molina-S\u0013 anchez, G. Catarina, D. Sangalli, and\nJ. Fern\u0013 andez-Rossier, Magneto-optical response of\nchromium trihalide monolayers: chemical trends, J.\nMater. Chem. C 8, 8856 (2020).\n[71] V. K. Gudelli and G.-Y. Guo, Magnetism and magneto-\noptical e\u000bects in bulk and few-layer CrI 3: a theoretical\nGGA + U study, New J. Phys. 21, 053012 (2019).\n[72] X. Yang, X. Zhou, W. Feng, and Y. Yao, Tun-\nable magneto-optical e\u000bect, anomalous Hall e\u000bect, and\nanomalous Nernst e\u000bect in the two-dimensional room-\ntemperature ferromagnet 1 T\u0000CrTe 2, Phys. Rev. B 103,\n024436 (2021).\n[73] Y. Fang, S. Wu, Z.-Z. Zhu, and G.-Y. Guo, Large\nmagneto-optical e\u000bects and magnetic anisotropy energy\nin two-dimensional Cr 2Ge2Te6, Phys. Rev. B 98, 125416\n(2018).\n[74] X. Zhou, W. Feng, F. Li, and Y. Yao, Large magneto-\noptical e\u000bects in hole-doped blue phosphorene and gray\narsenene, Nanoscale 9, 17405 (2017).18\n[75] W. Feng, G.-Y. Guo, and Y. Yao, Tunable\nmagneto-optical e\u000bects in hole-doped group-IIIA metal-\nmonochalcogenide monolayers, 2D Mater. 4, 015017\n(2017).\n[76] Z. Zheng, J. Y. Shi, Q. Li, T. Gu, H. Xia, L. Q. Shen,\nF. Jin, H. C. Yuan, Y. Z. Wu, L. Y. Chen, and H. B.\nZhao, Magneto-optical probe of ultrafast spin dynamics\nin antiferromagnetic CoO thin \flms, Phys. Rev. B 98,\n134409 (2018).\n[77] V. Saidl, P. N\u0014 emec, P. Wadley, V. Hills, R. P. Campion,\nV. Nov\u0013 ak, K. W. Edmonds, F. Maccherozzi, S. S. Dhesi,\nB. L. Gallagher, F. Troj\u0013 anek, J. Kune\u0014 s, J. \u0014Zelezn\u0013 y,\nP. Mal\u0013 y, and T. Jungwirth, Optical determination of\nthe N\u0013eel vector in a CuMnAs thin-\flm antiferromagnet,\nNat. Photon. 11, 91 (2017).\n[78] V. Antonov, B. Harmon, and A. Yaresko, Elec-\ntronic structure and magneto-optical properties of solids\n(Springer Science & Business Media, 2004).\n[79] P. M. Oppeneer, T. Maurer, J. Sticht, and J. K ubler,\nAb initio calculated magneto-optical Kerr e\u000bect of fer-\nromagnetic metals: Fe and Ni, Phys. Rev. B 45, 10924\n(1992).\n[80] W. Feng, J.-P. Hanke, X. Zhou, G.-Y. Guo, S. Bl ugel,\nY. Mokrousov, and Y. Yao, Topological magneto-\noptical e\u000bects and their quantization in noncoplanar an-\ntiferromagnets, Nat. Commun. 11, 118 (2020).\n[81] X. Zhou, W. Feng, X. Yang, G.-Y. Guo, and Y. Yao,\nCrystal chirality magneto-optical e\u000bects in collinear an-\ntiferromagnets, Phys. Rev. B 104, 024401 (2021).\n[82] E. Oh, D. U. Bartholomew, A. K. Ramdas, J. K. Fur-\ndyna, and U. Debska, Voigt e\u000bect in diluted magnetic\nsemiconductors: Cd 1\u0000xMn xTe and Cd 1\u0000xMn xSe, Phys.\nRev. B 44, 10551 (1991).\n[83] A. Ramasubramaniam and D. Naveh, Mn-doped mono-\nlayer MoS 2: An atomically thin dilute magnetic semi-\nconductor, Phys. Rev. B 87, 195201 (2013).\n[84] L. Sun, W. Zhou, Y. Liang, L. Liu, and P. Wu, Mag-\nnetic properties in Fe-doped SnS2: Density functional\ncalculations, Comput. Mater. Sci. 117, 489 (2016).\n[85] L. J. Li, E. C. T. O'Farrell, K. P. Loh, G. Eda,\nB.Ozyilmaz, and A. H. C. Neto, Controlling many-\nbody states by the electric-\feld e\u000bect in a two-\ndimensional material, Nature 529, 185 (2016).\n[86] K. Kim, J. Seo, E. Lee, K.-T. Ko, B. S. Kim, B. G. Jang,\nJ. M. Ok, J. Lee, Y. J. Jo, W. Kang, J. H. Shim, C. Kim,\nH. W. Yeom, B. I. Min, B.-J. Yang, and J. S. Kim,\nLarge anomalous Hall current induced by topological\nnodal lines in a ferromagnetic van der Waals semimetal,\nNat. Mater. 17, 794 (2018).\n[87] J. Xu, W. A. Phelan, and C.-L. Chien, Large Anoma-\nlous Nernst E\u000bect in a van der Waals Ferromagnet\nFe3GeTe 2, Nano Lett. 19, 8250 (2019).\n[88] J. Seo, D. Y. Kim, E. S. An, K. Kim, G.-Y. Kim, S.-Y.\nHwang, D. W. Kim, B. G. Jang, H. Kim, G. Eom, S. Y.\nSeo, R. Stania, M. Muntwiler, J. Lee, K. Watanabe,\nT. Taniguchi, Y. J. Jo, J. Lee, B. I. Min, M. H. Jo,\nH. W. Yeom, S.-Y. Choi, J. H. Shim, and J. S. Kim,\nNearly room temperature ferromagnetism in a magnetic\nmetal-rich van der Waals metal, Sci. Adv 6, eaay8912\n(2020).\n[89] P. N. Dheer, Galvanomagnetic E\u000bects in Iron Whiskers,\nPhys. Rev. 156, 637 (1967).\n[90] G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Spincaloritronics, Nat. Mater. 11, 391 (2012).\n[91] S. R. Boona, R. C. Myers, and J. P. Heremans, Spin\ncaloritronics, Energy Environ. Sci. 7, 885 (2014).\n[92] D. Xiao, Y. Yao, Z. Fang, and Q. Niu, Berry-phase ef-\nfect in anomalous thermoelectric transport, Phys. Rev.\nLett. 97, 026603 (2006).\n[93] S. N. Guin, P. Vir, Y. Zhang, N. Kumar, S. J.\nWatzman, C. Fu, E. Liu, K. Manna, W. Schnelle,\nJ. Gooth, C. Shekhar, Y. Sun, and C. Felser, Zero-Field\nNernst E\u000bect in a Ferromagnetic Kagome-Lattice Weyl-\nSemimetal Co 3Sn2S2, Adv. Mater. 31, 1806622 (2019).\n[94] J. Noky, J. Gayles, C. Felser, and Y. Sun, Strong\nanomalous Nernst e\u000bect in collinear magnetic Weyl\nsemimetals without net magnetic moments, Phys. Rev.\nB97, 220405 (2018).\n[95] Y. Onose, T. Ideue, H. Katsura, Y. Shiomi, N. Nagaosa,\nand Y. Tokura, Observation of the Magnon Hall E\u000bect,\nScience 329, 297 (2010).\n[96] M. Hirschberger, J. W. Krizan, R. J. Cava, and N. P.\nOng, Large thermal Hall conductivity of neutral spin ex-\ncitations in a frustrated quantum magnet, Science 348,\n106 (2015).\n[97] M. Hirschberger, R. Chisnell, Y. S. Lee, and N. P. Ong,\nThermal Hall E\u000bect of Spin Excitations in a Kagome\nMagnet, Phys. Rev. Lett. 115, 106603 (2015).\n[98] H. Doki, M. Akazawa, H.-Y. Lee, J. H. Han, K. Sugii,\nM. Shimozawa, N. Kawashima, M. Oda, H. Yoshida,\nand M. Yamashita, Spin Thermal Hall Conductivity of a\nKagome Antiferromagnet, Phys. Rev. Lett. 121, 097203\n(2018).\n[99] K. Sugii, M. Shimozawa, D. Watanabe, Y. Suzuki,\nM. Halim, M. Kimata, Y. Matsumoto, S. Nakatsuji, and\nM. Yamashita, Thermal Hall E\u000bect in a Phonon-Glass\nBa3CuSb 2O9, Phys. Rev. Lett. 118, 145902 (2017).\n[100] M. Akazawa, M. Shimozawa, S. Kittaka, T. Sakak-\nibara, R. Okuma, Z. Hiroi, H.-Y. Lee, N. Kawashima,\nJ. H. Han, and M. Yamashita, Thermal Hall E\u000bects\nof Spins and Phonons in Kagome Antiferromagnet Cd-\nKapellasite, Phys. Rev. X 10, 041059 (2020).\n[101] H. Zhang, C. Xu, C. Carnahan, M. Sretenovic, N. Suri,\nD. Xiao, and X. Ke, Anomalous Thermal Hall E\u000bect in\nan Insulating van der Waals Magnet, Phys. Rev. Lett.\n127, 247202 (2021).\n[102] J. Zhang, B. Zhao, T. Zhou, Y. Xue, C. Ma, and\nZ. Yang, Strong magnetization and Chern insulators in\ncompressed graphene/CrI3van der Waals heterostruc-\ntures, Phys. Rev. B 97, 085401 (2018).\n[103] D. Zhong, K. L. Seyler, X. Linpeng, R. Cheng,\nN. Sivadas, B. Huang, E. Schmidgall, T. Taniguchi,\nK. Watanabe, M. A. McGuire, W. Yao, D. Xiao, K.-\nM. C. Fu, and X. Xu, Van der Waals engineering of\nferromagnetic semiconductor heterostructures for spin\nand valleytronics, Sci. Adv. 3, e1603113 (2017).\n[104] K. L. Seyler, D. Zhong, B. Huang, X. Linpeng, N. P.\nWilson, T. Taniguchi, K. Watanabe, W. Yao, D. Xiao,\nM. A. McGuire, K.-M. C. Fu, and X. Xu, Valley Ma-\nnipulation by Optically Tuning the Magnetic Proximity\nE\u000bect in WSe 2/CrI 3Heterostructures, Nano Lett. 18,\n3823 (2018).\n[105] H. Wang, Y. Liu, P. Wu, W. Hou, Y. Jiang, X. Li,\nC. Pandey, D. Chen, Q. Yang, H. Wang, D. Wei, N. Lei,\nW. Kang, L. Wen, T. Nie, W. Zhao, and K. L. Wang,\nAbove Room-Temperature Ferromagnetism in Wafer-\nScale Two-Dimensional van der Waals Fe 3GeTe 2Tai-19\nlored by a Topological Insulator, ACS Nano 14, 10045\n(2020).\n[106] S. Albarakati, C. Tan, Z.-J. Chen, J. G. Partridge,\nG. Zheng, L. Farrar, E. L. H. Mayes, M. R. Field,\nC. Lee, Y. Wang, Y. Xiong, M. Tian, F. Xiang, A. R.\nHamilton, O. A. Tretiakov, D. Culcer, Y.-J. Zhao,\nand L. Wang, Antisymmetric magnetoresistance in van\nder Waals Fe 3GeTe 2/graphite/Fe 3GeTe 2trilayer het-erostructures, Sci. Adv. 5, eaaw0409 (2019).\n[107] Q. Wu, Y. S. Ang, L. Cao, and L. K. Ang, Design of\nmetal contacts for monolayer Fe 3GeTe 2based devices,\nAppl. Phys. Lett. 115, 083105 (2019).\n[108] Y. Shao, W. Lv, J. Guo, B. Qi, W. Lv, S. Li, G. Guo,\nand Z. Zeng, The current modulation of anomalous Hall\ne\u000bect in van der Waals Fe 3GeTe 2/WTe 2heterostruc-\ntures, Appl. Phys. Lett. 116, 092401 (2020)."    },    {        "title": "2205.00106v2.Controlled_Curie_temperature__magnetocrystalline_anisotropy__and_valley_polarization_in_2D_ferromagnetic_Janus_2H_VSeS_monolayer.pdf",        "content": "Controlled Curie temperature, magneto-crystal anisotropy , and valley polarization in\n2D ferromagnetic Janus 2H-VSeS monolayer\nCunquan Li1and Y ukai An1 ,∗\n1Key Laboratory of Display Materials and Photoelectric Devices,\nMinistry of Education, Tianjin Key Laboratory for Photoelectric Materials and Devices,\nNational Demonstration Center for Experimental F unction Materials Education,\nSchool of Material Science and Engineering, Tianjin University of T echnology, Tianjin, 300384, China\n(Dated: July 7, 2022)\nInspired by the successful synthesis of two-dimensional (2D) V-based Janus dichloride monolayers\nwith intrinsic ferromagnetism and high Curie temperature (T c), the electronic structure, spin-valley\nsplitting and magnetic anisotropy of Janus 2H-VSeS monolayers are investigated in detailed using\nfirst-principles calculations. The results show that the Janus 2H-VSeS monolayer exhibits a large\nvalley splitting of 105 me V, high T cof 278 K and good magnetocrystalline anisotropy (0.31 me V)\ncontributed by the in-plane dx2−y2/dxy orbitals of V atoms. The biaxial strain ( −8%< ε<8%) can\neffectively tune the magnetic moments of V atom, valley splitting ∆E, T cand MAE of Janus 2H-\nVSeS monolayer. The corresponding ∆E and T care adjusted from 72 me V to 106.8 me V and from\n180 K to 340 K, respectively . The electronic phase transition from bipolar magnetic semiconductor\n(BMS) to half-semiconductor (HSC), spin gapless semiconductor (SGS), and half-metal (HM) is\nalso observed due to the change of V 3d-orbital occupation. Due to the broken space- and time-\nreversal symmetry , the opposite valley charge carriers carry opposite Berry curvature, which leads\nto prominent anomalous Hall conductivity at the K and K′valleys. The maximum modulation\nof Berry curvature can reach to 45% and 9.5% by applying the biaxial strain and charge carrier\ndoping, respectively . The stable in-plane magnetocrystalline anisotropy and large spontaneous valley\npolarization make the ferromagnetic Janus 2H-VSeS monolayer a promising material for achieving\nthe spintronics and valleytronics devices.\nI. INTRODUCTION\nSince the successful discovery and synthesis of\ngraphene, [ 1] two-dimensional (2D) materials with hexag-\nonal lattice structures have attracted a great deal of in-\nterest due to their excellent physical and chemical prop-\nerties compared with the bulk counterparts. [ 2] As the\nmembers of 2D family , transition metal dichalcogenides\n(TMDs) MX 2(M=Mo, W; X=S, Se) with a central M\nsublayer sandwiched by two mirror-symmetric X sublay-\ners possess a tunable band gap (1-2 e V) and become a\ngood candidate for optoelectronic, valley electronic and\nnanoelectronic devices. [ 3,4] Due to the lack of inver-\nsion symmetry , TMDs possess a pair of inequivalent de-\ngeneracy energy valleys at the K and K′points in mo-\nmentum space and exhibit extraordinary quantum ef-\nfects, such as valley-spin locking and valley-spin Hall ef-\nfects. [ 5–7] The energy valley can be utilized and ma-\nnipulated, making it an excellent information carrier for\ncharge and spin to store information and perform logic\noperations. The biggest challenge in the development of\nvalleytronics is to achieve the manipulation and genera-\ntion of valley splitting by lifting the degeneracy between\nthe K and K′valleys. Until now, multiple pathways to\nlift the valley degeneracy have been performed, including\nthe circularly polarized light pumping, [ 8] external mag-\nnetic fields, [ 9] transition-metal (TM) atoms doping or\n∗ykan@tjut.edu.cnadsorption [ 10] and magnetic proximity effect. [ 11] How-\never, the generated valley splitting will disappear and\nreturn to the intrinsic paravalley states once the external\nfield is removed, which is impracticable for the appli-\ncation in valleytronics devices. Therefore, searching for\n2D materials with intrinsic valley splitting is crucial. Re-\ncently , the 2D ferrovalley monolayers (VS 2, [12] VSe 2[13]\nand VT e 2[14]) with intrinsic ferromagnetic ordering and\nspontaneous valley splitting have been successfully syn-\nthesized and attracted considerable attention due to their\nstrong magneto-valley coupling effects. The spontaneous\nvalley splitting in the V-based TMDs monolayers orig-\ninates from the magnetic interaction between the V 3d\nelectrons, which does not depend on the external fields\nand can achieve simultaneous manipulation of spin and\nvalley degrees of freedom.\nStable Janus group VI chalcogenides MXY (M=Mo,\nW; X, Y=S, Se, T e; X ̸=Y) are firstly proposed by Cheng\net al., [ 15] where the Janus MoSSe have been successfully\nsynthesized via experiments using the modified chemical\nvapor deposition methods. [ 16] At the same time, some\nJanus systems, including Janus graphene, [ 17] Janus\ngraphene oxide, [ 18] and group-III chalcogenides NN′X2\n(N, N′=Ga, In; N ̸=N′; X=S, Se, T e) [ 19] monolayers are\nalso systematically investigated, which exhibit extraor-\ndinary physical properties with tunable electric dipole\nand high carrier mobility . Additionally , 2D Janus TMDs\nwith intrinsic ferromagnetism, such as Janus MnXY (X,\nY=S, Se, T e; X ̸=Y) [ 20,21] and Cr(I, X) 3(X=Cl, I and\nBr) [ 22] monolayers have been proved to exhibit a large\nDzyaloshinskii-Moriya interaction behavior. [ 23] The ex-2\nperimental realization of 2D ferrovalley materials, espe-\ncially for the 2H-VSe 2monolayer, may provide an excit-\ning new platform for the Janus structure with intrinsic\nferromagnetic semiconductors. It is expected that the\nordered Janus 2H-VSeS monolayer can be grown by a\nsimilar modified CVD method as described above and in-\ntroduce new physical phenomena. In addition, the Janus\n2H-VSeS monolayer are firstly predicted by Du et al [ 24]\nand they comprehensively investigated its piezoelectricity\nand ferroelasticity properties. F urther, it is significant to\ngain greater insight into the intrinsic physical properties\nof Janus 2H-VSeS monolayer.\nIn this work, the geometric structure, kinetic stability ,\nCurie temperature (T c), magnetocrystalline anisotropy\nand valley splitting of Janus 2H-VSeS monolayers under\nvarious biaxial strains and charge carrier doping are sys-\ntematically discussed by the first-principles calculations.\nThe results show that the electronic structures of Janus\n2H-VSeS monolayer can be effectively modulated from\nbipolar magnetic semiconductor (BMS) to semiconduc-\ntor (HSC), spin-gap-free semiconductor (SGS) and half-\nmetal (HM), which results an obvious change in the mag-\nnetic moments of V atoms, valley splitting, band gap and\nTc. F urthermore, the Berry curvatures at the K- and K′-\nvalleys show the opposite signs due to the break of mirror\nsymmetry along with the out-of-plane, which can also be\neffectively modulated by the biaxial strain and charge\ncarrier doping. These results provide new insights for\nadjusting the electronic structure and magnetic proper-\nties of Janus 2H-VSeS monolayer, which is very helpful\nfor the design of new nanoelectronic devices.\nII. COMPUT A TIONAL METHODS\nThe first-principles calculations are performed using\nthe plane-wave basis Vienna Ab initio Simulation Pack-\nage (V ASP), [ 25–28] based on the density functional\ntheory (DFT). The generalized gradient approximation\n(GGA) functional in Perdew-Burke-Ernzerhof (PBE) is\nadopted. [ 29,30] Considering the strong-correlated cor-\nrection of V 3d localized electrons, the LSDA+U method\nis adopted, [ 31] where the effective onsite Coulomb inter-\naction parameter (U=2.00 e V) and exchange interaction\nparameter (J=0.87 e V) have been added to it. [ 32,33] A\nvacuum space of 18 Å in the z-direction is set to avoid the\nadjacent interactions. The cut-off energy is set at 500 e V\nand the criteria of energy and atom force convergence are\n10−6Å and 10−3e V/Å−1, respectively . The Brillouin\nzone is sampled with 16 ×16×1Γ-centered Monkhorst-\nPack grids. [ 34] The effect of van der W aals (vdW) cor-\nrections (DFT-D3 method) [ 35] are considered during\nthe structure optimizations and the V ASPKIT [ 14] code\nis used to process the V ASP data. The phonon spec-\ntrum and phonon density of states are calculated by the\nPHONOPY [ 36] based on density-functional perturba-\ntion theory (DFPT) and a 4 ×4×1 supercell is used to\ncalculate the Hessian matrix. The spin-orbit coupling(SOC) and non-collinear magnetism are considered in\nthe calculations of electronic structure. F or the calcula-\ntion of Berry curvature and anomalous Hall conductivity ,\nthe maximally localized W annier functions (ML WF s) are\nconstructed using the W ANNIER90 package. [ 37,38] In\nthe case of Janus 2H-VSeS monolayer, there are 22 bands\nin the energy range from about −6 to 4 e V, mainly formed\nby V d orbitals and X (X=Se, S) p orbitals. T en d or-\nbitals on atom V and six p orbitals on each atom X are\nchosen as the initial guess of W annier functions. After\nless than 400 iterative steps, the total W annier spreads\nare well converged down to 10−6Å2. A fine k-mesh\nof 30 ×30×1 is used in W ANNIER interpolation. The\nMonte Carlo simulation procedure MCSOL VER [ 39,40]\nbased on the W olff algorithm of the classical Heisenberg\nmodel is used to estimate the T c. In the specified tem-\nperature interval, we completely thermalize the system\nto equilibrium with 120,000 scans starting from the fer-\nromagnetic order, and all statistics are obtained from the\nnext 720,000 scans.\nIII. RESUL TS AND DISCUSSION\nIt has been reported that the trigonal prismatic (2H)\nphase is more energetically stable than the octahedral\n(1T) phase in the Janus VSeS monolayer. [ 24] Thus, the\ngeometry of Janus 2H-VSeS monolayer with a S-V-Se\nsandwich structure is fabricated by MedeA-V ASP mate-\nrials design software, as shown in Fig. 1(a). After the\noptimization of geometry , the corresponding V-Se and\nV-S bond lengths are 2.51 and 2.36 Å, respectively . The\noptimized lattice constant is 3.259 Å, which is between\nthose of 2H-VS 2(3.18 Å) and 2H-VSe 2(3.34 Å) mono-\nlayers. [ 13,41] F or the optimized geometry , the 2H-VSe 2\nand 2H-VS 2monolayers belong to the D 3hpoint group,\nwhile the Janus 2H-VSeS monolayer breaks the off-plane\nstructure symmetry and presents the C 3vpoint group.\nThe calculated average electrostatic potential along z-\naxis is rather asymmetric, as shown in Fig. 1(b). Obvi-\nously , the electrostatic potential of the S side is smaller\nthan that of the Se side, namely V S<VSe, implying a\nlarger electronegativity of the S atom. Each V atom\ncontributes a magnetic moment of 0.978 µBin the Janus\n2H-VSeS monolayer. Meanwhile, the difference of elec-\ntronegativity between the S and Se atoms can alter the\nlocal magnetic interaction, which induces a larger mag-\nnetic moment for the V atom. T o confirm the magnetic\nground states of Janus 2H-VSeS monolayer, the ferro-\nmagnetic (FM) and antiferromagnetic (AFM) states in\na 2×2×1 supercell are calculated. The total magnetic\nmoment is 4.0 µB for the Janus 2H-VSeS monolayer\nwith FM state, while the total magnetic moment is 0\nµBwith AFM state. The ferromagnetic stability energy\n(ΔE=E FM−AFM ) of 2 ×2×1 supercell is −260 me V<0,\nstrongly suggesting that the Janus 2H-VSeS monolayer\nis an intrinsic ferromagnetic material. As is known, the\nJanus 2H-VSeS monolayer suffers from the spin-wave, so3\nFIG. 1. (a) T op and side views of Janus 2H-VSeS monolayer. The red, yellow, and green spheres represent V, S and Se atoms,\nrespectively . (b) The average electric potential along z direction, (c) phonon spectrum for the Janus 2H-VSeS monolayer.\n(d) The spin-polarized band structures of Janus 2H-VSeS monolayer. The red and blue lines represent the spin-up( α) and\nspin-down( β) band structures. The insert is the first Brillouin zone containing the K and K′points. (e) The band structures\nof Janus 2H-VSeS monolayer with considering SOC. (f) The V d orbital-resolved band structure of Janus 2H-VSeS monolayer\nwith considering SOC, the purple and orange triangles denote the contribution from the dz2and dx2−y2/dxy orbitals of V\natom, respectively . The arrows between the VBM and the CBM denote the valley-selective optical transitions induced by left\nand right circularly polarized light σ+andσ−.\nthe non-magnetic (NM) ground state is a supercell with\nPeierls distortion, and the phonon dispersion is also con-\nsidered to be unstable due to the presence of imaginary\nfrequencies in the AFM state. Therefore, to estimate the\nstability of Janus 2H-VSeS monolayer, the phonon spec-\ntrum and phonon density of states with FM state are\nfinally calculated in a 4 ×4×1 supercell. As shown in Fig.\n1(c), the absence of significant imaginary phonon modes\nover the whole Brillouin zone strongly suggests that the\nJanus 2H-VSeS monolayer is dynamically stable, which\nis consistent with the previous report. [ 24] The highest\nfrequency is calculated to be 12.35 Thz, which is between\nthe reported values for the 2H-VSe 2(10.02 Thz) and 2H-\nVS2(13.7 Thz) monolayers. [ 41–43] The contribution at\nlow frequencies (0< ε<5 THz) mainly comes from the Se\natoms, in contrast to the V and S atoms which make the\nmain contribution at high frequencies (6< ε<12 THz), as\nshown in Fig. S1. It is maybe due to that the V (atomic\nweight: 50.94) and S (32.066) atoms are lighter compared\nto the Se (78.96) atoms. This can also remarkably affect\nthe thermal properties of Janus 2H-VSeS monolayer. At\nlow temperature, the thermal conductivity is mainly con-\ntributed by the phonons of low frequency Se atom. How-\never, at high temperature, the thermal conductivity is\nmainly contributed by phonons of higher frequency V/S\natoms. The spin-polarized band structure of Janus 2H-\nVSeS monolayer is shown in Fig. 1(d). One can see thatthe valence band maximum (VBM) occupied by spin-\nup (α) electrons is located at the Γ point of Brillouin\nzone, while the conduction band maximum (CBM) con-\ntributed by spin-down ( β) electrons is located at the M\npoint. Thus, the Janus 2H-VSeS monolayer is a bipolar\nmagnetic semiconductor (BMS) with an indirect energy\ngap of 0.407 e V. Clearly , there exists a Dirac valley con-\ntributed by the spin- αstates at the K and K′points, as\nshown by the two black lines closest to the valley gap in\nFig. 1(d). The two K and K′valleys are degenerate in\nenergy with identical valley gaps of 0.714 e V. The band\nstructures of Janus 2H-VSeS monolayer with considering\nSOC is shown in Fig. 1(e). It is obvious that the valley\nenergies at the K and K′points in the conduction band\nremain close, while the energy at the K′point is higher\nthan that at the K point in the valence band. Hence,\nthe valley degeneracy is broken, inducing a large valley\nsplitting ( ∆E=∆K−∆K′) of 105 me V, which is usually\nmanifested as two split peaks in the photoluminescence\nspectrum. Fig. 1(f) shows the V d orbital-resolved band\nstructure of Janus 2H-VSeS monolayer with considering\nSOC. Both the VBM and CBM at the K and K′points\nare mainly contributed by V atoms. At the two unequal\nK and K′valleys, the CBM is mainly contributed by\nthe in-plane V d x2−y2/dxy orbitals, while the VBM at\nthe K and K′points is dominated by the V d z2orbital.\nFig. S2(a) shows the corresponding 3D band structure4\nFIG. 2. (a)-(c) The V-d and Se(S)-p orbital-resolved MAEs of Janus 2H-VSeS monolayer. Dependence of (d) magnetic moment,\n(e) magnetic susceptibility , and (f) specific heat capacity on temperature for the Janus 2H-VSeS monolayer.\nof Janus 2H-VSeS monolayer with considering SOC. It\nis obvious that the energy of VBM at the Γpoint is the\nglobal maximum, while that at the K and K′points is\nonly a local maximum. At the same time, the energy of\nCBM at the K and K′points is also not globally minimal,\nand the global maximum is at the M point. 2D projected\nband structure of valence band and conduction band at\nthe k xky-plane is shown in Fig. S2(b)-(c). Clearly , the\nsix valleys within the Brillouin zone degenerate in energy\nwith respect to each other. This reflects that the SOC\neffect achieves large valley polarization in the Janus 2H-\nVSeS monolayer.\nConsidering the magnetic moment switching from the\nin-plane [100] axis to the out-of-plane [001] axis, the en-\nergy it spends is called magneto-crystal anisotropy en-\nergy (MAE). MAE( E[001]−E[100] ) can be defined by the\nfollowing equation:\nMAE = ξ2∑\no.u(\n|⟨o|Lx|u⟩|2− |⟨o|Lz|u⟩|2)\n/(Eu−Eo)\n(1)\nwhere the ξ,oandµare the SOC constants, and the en-\nergies of the occupied (E o) and non-occupied (E u) states,\nrespectively . E [001] and E [100] are the energies of the mag-\nnetization in the [001] and [100] directions, respectively .\nSo, the positive (negative) MAE value implies the in-\nplane (out-of-plane) easy-axis. One can see from Eqn(1)\nthat the orbital matrix element differences and energy\ndifferences together affect the MAE behavior, which can\nbe used to analyze the contribution of different inter-\norbit hybridization to the MAE. The V-d and Se(S)-p\norbital resolved MAEs for the Janus 2H-VSeS monolayer\nare shown in Figs. 2(a)-(c). The calculated total MAE\nvalue is 0.31 me V and the corresponding MAE values\ncontributed by the V-d, Se-p and S-p orbitals are 0.14,0.15 and 0.01 me V, respectively . These strongly suggest\nthat the Janus 2H-VSeS monolayer processes the in-plane\nmagnetic anisotropy (IMA), which is mainly contributed\nby the hybridized V d x2−y2and d xy orbitals as well as\nthe Se(S) p xand p yorbitals.\nThe T cof ferromagnetic materials can be well calcu-\nlated using the classical Heisenberg Monte Carlo (MC)\nmodel. In the calculation, the spin Hamiltonian including\nthe single-ion anisotropy can be described as following:\nH=−1\n2∑\n(i,j)[\nJxSx\niSx\nj+JySy\niSy\nj+JzSz\niSz\nj]\n−∑\niD(Sz\ni)2\n(2)\nF or the entire V-atom lattice, ( i,j) denote the individ-\nual V-atoms in the immediate vicinity , and itraverses\nthe entire lattice. J andD are the spin-exchange pa-\nrameter and magnetic anisotropy energy , respectively .\nJ>0 favors the ferromagnetic interactions and D>0 fa-\nvors the out-of plane easy axis. In the 4 ×4×1 supercell,\nthe energies of FM and AFM states and the magnetic ion\nanisotropy parameter D with considering SOC effect can\nbe calculated as following:\nEFM= E0−(1\n2×6×4)\nJ|S|2(3)\nEAFM= E0+ 4×(\n−1\n2×2 +1\n2×4)\nJ|S|2(4)\nJ=EAFM−EFM\n16|S|2(5)\nD=Eα\nFM−Ez\nFM\n4S2(6)5\nFIG. 3. (a) The Band structure of Janus 2H-VSeS monolayer\ncalculated by ML WF s. Calculated Berry curvature of Janus\n2H-VSeS monolayer (b) over 2D Brillouin zone and (c) along\nhigh symmetry lines. (d) Calculated anomalous Hall conduc-\ntivity of Janus 2H-VSeS monolayer. The two parallel dashed\nlines indicate the two valley extremes.\nwhere S(S=1/2) is the spin operator, and the αandz\nare the directions along the in-plane [100] and the out-\nof-plane [001] axis, respectively . Usually , the sub-nearest\nneighbor V-V interactions are negligible in this system.\nThus, only the nearest V-V interactions are considered\nin the calculation. The calculated exchanged parame-\nterJand the magnetic ion anisotropy parameter D are\n64 me V and −1.22 me V, respectively . Figs. 2(d)-(f) il-\nlustrates the dependence of magnetic moment, magnetic\nsusceptibility , and specific heat capacity on temperature\nfor the Janus 2H-VSeS monolayer. The simulated T cis\nas high as 278 K, which is close to the room temperature\n(300 K).\nIn the case of inversion symmetry breaking in the Janus\n2H-VSeS monolayer, the charge carriers also acquire a\nvalley-contrast Berry curvature. [ 44] The z-component of\nBerry curvature Ωz\nn(k)of the occupied state band ac-\ncording to the Kubo formula is calculated and can be\nexpressed as: [ 45,46]\nΩz\nn(k) =−∑\nn̸=m2 Im⟨Ψnk|vx|Ψmk⟩ ⟨Ψmk|vy|Ψnk⟩\n(Em−En)2 (7)\nwhere |Ψnk⟩is a calculated wave function of the Bloch\nstates at k points with the energy eigenvalue En, and\nυx(y)are velocity operators along xandydirections, re-\nspectively . T o ensure the accuracy of W annier basis func-\ntions, the tightly bound band structure is calculated us-\ning ML WF s, as shown in Fig. 3(a). The band structure\nis in high agreement with the DFT results [Fig. 1(e)],\nindicating that the resulting W annier basis functions are\nsufficiently localized to ensure the precision and accuracy\nFIG. 4. The dependence of (a) ferromagnetic stability energy\n∆E and (b) the band gap (E G) on the strains ε. The red,\nblue, and green lines indicate E G-up(α), E G-down( β), and\nEG, respectively . (c) The total MAE for the Janus 2H-VSeS\nmonolayer under various strains. (d) The magnetic moment\nof V atom and T cfor the Janus 2H-VSeS monolayers under\nvarious strains.\nof the calculation. Fig. 3(b)-(c) shows the calculated re-\nsults of Berry curvature along the high symmetry line\nand Brillouin zone, respectively . In the intrinsic ferro-\nmagnetic field of Janus 2H-VSeS monolayer, the Berry\ncurvature has the opposite signs at the K and K′points\nwith different absolute magnitudes. Meanwhile, the inte-\ngration of Berry curvature in the Brillouin zone gives the\ncontribution to the anomalous Hall conductivity , which\ncan be expressed as:\nσxy=−e2\nℏ∫\nBZd2k\n(2π)2Ωz\nn(k) (8)\nDue to the breaking of the time-reversal symmetry ,\na non-zero anomalous Hall conductivity calculated by\nW annierT ools appears, [ 47] as shown in Fig. 3(d). Since\nthe Hall currents from the K and K′valleys do not disap-\npear completely , a net charge current will be generated.\nDue to the presence of electric field E, the Berry curva-\nture drives an anomalous transverse velocity [ 48]:\nv=−e\nℏE×Ωn(k) (9)\nThis is an intrinsic contribution to the anomalous Hall\neffect. In our system, the charge carriers in the K and\nK′valleys have opposite transverse velocities due to the\nopposite sign of Berry curvature. As a result, the valley\ncurrents are generated. On the other hand, since the\nvalley current and spin current remain constant under\ntime reversal, valley Hall and spin Hall effects can appear\nin systems with constant time-reversal due to the broken\nreversal symmetry .\nSince the CBM and VBM are contributed by in-plane\ndx2−y2/dxy orbitals for the Janus 2H-VSeS monolayer,6\nFIG. 5. (a) Band structures of Janus 2H-VSeS monolayers under various strains ( ±4%) with SOC. (b) Dependence of ∆K,\n∆K′and valley splitting ∆E on the strain. (c) The Berry curvature of Janus 2H-VSeS monolayers at the strains of −4%,−2%,\n0%, 2%, and 4%, respectively .\nit is predicted that the magnetic and electronic proper-\nties are strongly related to in-plane biaxial strain. T o\nestimate the strain dependence of electronic properties,\nthe in-plane biaxial strain is applied by the relation of\nε=[(a-a 0)/a0]×100%, where a and a 0present the lat-\ntice constants of strained and unstrained Janus 2H-VSeS\nmonolayers. As shown in Fig. 4(a), the total energy\nof FM and AFM states show monotonically asymptotic\nchange with increasing the tensile strains ( ε=2, 4, 6, 8%)\nand the compressive strains ( ε=−2,−4,−6,−8%). The\nferromagnetic stability energy ∆E increases monotoni-\ncally with increasing the strains from −8% (165 me V)\nto 8% (318 me V). The distance between V atoms be-\ncomes smaller with continuously increasing compressive\nstrain, which leads to an increase in super-exchange in-\nteractions. However, the Janus 2H-VSeS monolayer con-\nsistently maintains the FM ground state under the var-\nious strains from −8% to 8%. It is clear from Fig.\n4(b) that the in-plane biaxial strain can effectively ad-\njust the electronic structures of Janus 2H-VSeS mono-\nlayer from the BMS character to Half-semiconductors\n(HSC), Spin gapless semiconductors (SGS), Half metal-\nlic (HM) characters. The unstrained Janus 2H-VSeS\nmonolayer exhibits clear BMS character with an indi-\nrect band gap (E G) of 0.423 e V, while the corresponding\nEG-α and E G-β values are 0.545 e V and 0.783 e V, re-\nspectively , as shown in Fig. 4(b). At the small strain\nrange ( −4.7%< ε<1.0%), the Janus 2H-VSeS monolayer\nstill maintains the BMS character. It is noted that the\nmonotonicity of E G-α and E G-βis very obvious at the\nstrain range ( −6.6%< ε<5.0%). In order to effectively\ndiscuss the change of spin- α and spin- β energy bands,\nthe difference of band edge between the spin- αand spin-\nβ electrons at the M and K points can be defined as\n∆ECBM −M−K=ECBM −K−α−ECBM −M−β. As the tensile\nstrain increases, the E Gand E G-αvalues show a mono-\ntonic linear decrease, while E G-β values show a mono-\ntonic linear increase. F or ε=0%, the ∆ECBM −M−Kand\nEVBM −Γ−α values are 0.1244 and −0.2382 e V, respec-\ntively . As the strain increases to ε=0.8 and 1.0%, the cor-\nresponding ∆ECBM −M−Kvalues are 0.0204 and −0.0004e V, respectively and the corresponding E VBM −Γ−α in-\ncreases to −0.1816 and −0.1211 e V. Therefore, it can\nconsider that the Janus 2H-VSeS monolayer is converted\nto the HSC character at ε=1%. As the strain contin-\nues to increase from 1% to 5.0%, the HSC character is\nstill remained. The linear fit is used as a function to\ndescribe its relationship. The E G and εcan be well\nfitted using the relationship: E G=0.5−10.06 ε. At the\nsame time, the E G-α and E G-βcan be also fitted with\nEG-α=0.51 −10.06 εand E G-β=0.82+6.8 ε, respectively .\nFigs.S3(c)-(f) shows the spin-polarized band structures\nof Janus 2H-VSeS monolayer under various strains. One\ncan see that the positions of CBM and VBM relative to\nthe F ermi level change regularly . The E VBM −Γ−αval-\nues under various strains are −0.0508 e V (2%), −0.1446\ne V (3%), −0.0559 e V (4%), 0 e V (5%), 0.0057 e V (5.1%)\nand the E CBM −K−αvalues are 0.2528 e V (2%), 0.0515\ne V (3%), 0.0413 e V (4%), 0.0049 e V (5%), 0.0015 e V\n(5.1%). Therefore, with ε=5.0%, only the spin- αelectron\nchannel is conductive, while the spin- β electron chan-\nnel is semiconductive, presenting a half-metallicity char-\nacter. Meanwhile, when the E VBM value equals 0 e V,\nthe Janus 2H-VSeS monolayer exhibit the SGS charac-\nter. When the tensile strain continues to increase, the E G\nand E G-αvalues are 0 e V, resulting that the Janus 2H-\nVSeS monolayer still maintains HM character. Under\nthe compressive range, the difference of band edge be-\ntween the spin- αand spin- βelectrons at the Γpoint can\nbe defined as ∆EVBM −Γ=EVBM −Γ−α−EVBM −Γ−β. At\nthe compressive strains ( −4.6%< ε<0%), the ∆EVBM −Γ\nvalue decreases from 0.3621 e V ( ε=0%) to 0.2585 e V\n(−1%), 0.1492 e V ( −2%), 0.0297 e V ( −3%), −0.0995\ne V (−4%) and −0.1809 e V ( −4.6%) e V. At the same\ntime, the difference of band edge between the spin- αand\nspin- βelectrons at the Γand K points can be defined as\n∆EVBM −Γ−K= E VBM −Γ−β-EVBM −K−α. Obviously , the\n∆EVBM −Γ−K value increases from −0.1932 e V ( ε=0%)\nto−0.1643 e V ( −1%), −0.1330 e V ( −2%), −0.0928 e V\n(−3%), −0.0451 e V ( −4%) and −0.0118 e V ( −4.6%).\nWhen ε=−4.7%, the E VBM −Γ−βand E VBM −K−αvalues\nare extremely close to each other, which will serve as the7\ntransition point from the BMS to HSC characters. The\nrelationship between the E G-α(EG-β) and εcan be ex-\npressed as: E G-α=0.57 −10.86 εand E G-β=0.79+11.28 ε\n(−4.6%< ε<1%), respectively . When the compressive\nstrain εis larger than −4.7%, the E VBM −Γ−βvalue in-\ncreases from −0.2013 e V to −0.0947 e V ( −5%), −0.0354\ne V (−6%), −0.0025 e V ( −6.4%), 0 e V ( −6.42%). Thus,\nε=−6.42% is the transition point from the HSS to SGS\ncharacters. Similar to the case of tensile strain, with the\nincrease of compressive strain, the Janus 2H-VSeS mono-\nlayer continues to maintain the HM character. There-\nfore, it can conclude that the biaxial strain can effectively\nadjust the Janus 2H-VSeS monolayer from the BMS to\nHSC, SGS, and HM characters.\nThe total MAE for the Janus 2H-VSeS monolayer un-\nder various strains is shown in Fig. 4(c). It is clear\nthat the Janus 2H-VSeS monolayer still remains the IMA\ncharacter under various strains. The total MAE de-\ncreases linearly as the tensile strain increases from 0%\nto 8%. However, under the compressive strain range, the\ntotal MAE firstly increases and reaches a maximum value\nof 0.5 me V at the compressive strain of −6%, then gradu-\nally decreases from −6% to 0%. Fig. S4 shows the V, Se\nand S atomic-layer-resolved MAE under various strains.\nThe MAE from the S and V atoms decrease linearly as\nthe strain increases from −8% to 8%, while that from the\nSe atom firstly increases to a maximum value at the strain\nof−6% and then gradually decreases, which is similar to\nthe total MAE behavior. Interestingly , the V and Se/S\natoms always maintain the IMA character throughout\nthe whole strain range. The main contribution of MAE\nin the Janus 2H-VSeS monolayer comes from the V and\nSe atoms, which is consistent with the fact that MAE\ncan be contributed by non-metallic atoms with stronger\nSOC. [ 49] Fig. S5(a)-(d) shows the V-d and Se/S-p or-\nbital resolved MAE for the Janus 2H-VSeS monolayers\nunder various strains, respectively . As the strain in-\ncreases from −8% to 8%, the positive MAE of Janus\n2H-VSeS monolayer still mainly comes from the matrix\nelement differences (d x2−y2/dxy orbitals) of V atom and\n(px/pyorbitals) of Se atom. While the matrix element\ndifference of Se (p z/pyorbitals) atom shows a transition\nfrom IMA to (perpendicular magnetic anisotropy) PMA\ncharacter. The matrix element difference of S (p x/py\norbitals) atom still makes rather small contribution to\nMAE. Hence, the matrix element differences (p x/pyand\npz/py orbitals) of Se atom and (d x2−y2/dxy orbitals) of V\natom dominate the MAE behavior for the Janus 2H-VSeS\nmonolayer under various biaxial strains. Fig. 4(d) shows\nthe magnetic moment of V atom and T cfor the Janus\n2H-VSeS monolayer under various strains. At the ten-\nsile strain ranges ( ε= 2, 4, 6, and 8%), the corresponding\nmagnetic moments of V atom are 1.024, 1.052, 1.088, and\n1.133 µB, respectively . At the compressive strain ranges\n(ε=−2,−4,−6, and −8%), the corresponding magnetic\nmoments of V atom are 0.968, 0.936, 0.899, and 0.861 µB,\nrespectively . It is obvious that the magnetic moment of\nV atom shows a monotonical increase with increasing the\nFIG. 6. The (a) MAE and (b) exchange energy J for the\nJanus 2H-VSeS monolayer with various charge carrier doping.\nThe (c) band structures and (d) Berry curvature for the Janus\n2H-VSeS monolayer with 0.05 holes and 0.05 electrons doping,\nrespectively . The F ermi level is set to zero.\nstrains from −8% to 8%. In fact, as the strain increases,\nthe bond length between the V atoms increases and the\nionic-bond interaction overcomes the covalent-bond in-\nteraction. As a result, the number of unpaired electrons\nin the V atom increases, leading to an increase in the\nmagnetic moment. This is consistent with the reported\nresults in the 2H-VS 2and 2H-VSe 2monolayers. [ 50] One\ncan see from Fig. 4(d) that as the strain increases from\n−8% to 8%, the corresponding T care 180 ( −8%), 189\n(−6%), 215 ( −4%), 247 ( −2%), 278 (0%), 292 (2%), 304\n(4%), 311 (6%) and 340 K (8%), respectively , exhibiting\na monotonical increase behavior. Apparently , this is due\nto the further enhancement of super-exchange interac-\ntion with the increase of strain. The dependences of the\nmagnetic moment of V atom and specific heat capacity\non the temperature under various strains are shown in\nFig. S6(a)-(b), and the transition behavior is consistent\nwith the above analysis.\nThe band structures of Janus 2H-VSeS monolayers un-\nder various biaxial strains ( ±4%) with considering SOC\nare shown in Fig. 5(a). One can see that applying strain\ncan effectively adjust the band structure of Janus 2H-\nVSeS monolayer accompanied by the bands up at the\ncompressive strain ( −4%) and the bands down at the\ntensile one (4%) with respect to the vacuum level. [ 51]\nTherefore, it is convinced that the valley degrees of free-\ndom can be well tuned by applying the strains. Fig.\n5(b) shows the dependence of ∆K,∆K′and valley split-\nting∆E on the biaxial strain for the Janus 2H-VSeS\nmonolayers. When the tensile strains ( ε=1, 2, 3, 4\nand 4.9%) are applied, the corresponding ∆K values are\n709, 660, 618, 580 and 552 me V, while the correspond-8\ning∆K′values are 604, 556, 514, 478 and 451 me V.\nF or the compressive strains ( ε=−1,−2,−3,−4%,−5%\nand−6%), the corresponding ∆K values are 827, 897,\n975, 1060, 1153 and 1260 me V, while the corresponding\n∆K′values are 721, 790, 868, 953, 1048 and 1151 me V.\nClearly , the ∆K and ∆K′values monotonously increase\nwith increasing the strains from −6% to 4.9%. Surpris-\ningly , ∆K and ∆K′values show a good quadratic de-\npendence on the strain, namely ∆K=36 ε2−6.02ε+0.7636\nand∆K′=37ε2−5.96ε+0.658. The valley splitting ∆E\nfirstly increases and reached a maximum value of 106.8\nme V at the strain of −4%, and then gradually decreases\nwith further increasing the strain from −3% to 5%.\nTherefore, the valley degrees of freedom of Janus 2H-\nVSeS monolayer can be well modulated by the biaxial\nstrain. The Berry curvature of Janus 2H-VSeS mono-\nlayers at the strains of −4%,−2%, 0%, 2%, and 4% is\nshown in Fig. 5(c). One can see that when the strain\nof 4% is applied, the maximum modulation of Berry cur-\nvature can reach 45%. The tensile (compressive) strain\ntends to increase (decrease) the Berry curvature and their\nmagnitude difference |Ωz\nn(K) + Ωz\nn(K′)|. Since the AHC\nis defined as the integral sum of Berry curvature in the\nBZ, and the Berry curvature is non-zero only around the\nK and K′points. The AHC is expected to be greatly\nenhanced under the tensile strain. In this way , one can\nadjust the transverse Hall voltage by applying the biaxial\nstrains.\nAs is well known, charge carrier doping can be consid-\nered to an effective method to modulate the electronic\nstructure and F ermi level of semiconductor. It has been\nreported that the electrical, magnetic and valley prop-\nerties of 2H-VS 2[52], 2H-VSe 2[13], and 2H-WSe 2[53]\nmonolayers can be well tuned by charge carrier doping,\nwhich can be also expected for the Janus 2H-VSeS mono-\nlayer due to the presence of unpaired electrons in the V\n3d orbitals.\nThe MAE and exchange energy J for the Janus 2H-\nVSeS monolayer with various charge carrier doping are\nshown in Figs. 6(a)-(b). It is obvious that the MAE\nshows a gradual decrease trend with increasing the hole\nor electron doping concentration. Interestingly , the elec-\ntron doping does not alter the IMA character of Janus\n2H-VSeS monolayer. On the contrary , the IMA charac-\nter is transformed into the PMA character when the hole\ndoping concentration is above −0.3e. On the other hand,\nthe exchange energy J still remains the positive value\nunder various electrons (holes) doping concentrations (0-\n0.5e), strongly suggesting that the transition from the\nFM to AFM states does not occur. The J firstly in-\ncreases to a maximum value of 69.2 me V at the hole\ndoping concentration of −0.1e and then gradually de-\ncreases with increasing the carrier doping concentration\nfrom −0.1e to 0.5e. Compared to the undoped case, the\nmaximum modulation of MAE and Jvalues can reach to\n82% (218%) and 98% (65%) by the electrons (holes) dop-\ning, respectively . The band structures of Janus 2H-VSeS\nmonolayer with holes (electrons) doping is shown in Fig.6(c). It is clear that the F ermi level shifts to the VBM\n(CBM) with introducing the hole (electrons) concentra-\ntion of 0.05e, implying that the charge carrier doping\ncan effectively modulate the band structures and position\nof the F ermi level. The lack of time-reversal symmetry\nand mirror symmetry makes the electronic structure of\nJanus 2H-VSeS monolayer more tunable compared with\nthe nonmagnetic TMDs. The control of valley freedom\nby charge carrier doping is also investigated and shown\nin Fig. 6(d). Obviously , for the Janus 2H-VSeS mono-\nlayer doped with holes (electrons), the peak intensity of\nBerry curvature Ωz\nn(k)at the K and K′points become\nweaker (stronger). The modulation of Ωz\nn(k)can reach\n3.8% (9.5%) by doping with the electron (hole) concen-\ntration of 0.05e, respectively . Compared to the biaxial\nstrain modulation, it is obvious that the charge carrier\ndoping has a smaller effect on the Ωz\nn(k).\nFIG. 7. Schematic of Janus 2H-VSeS monolayer for the val-\nleytronic device. The holes are denoted by “+” symbol.\nAs shown in Fig. 7, a valleytronic device for the Janus\n2H-VSeS monolayer can be constructed based on the cal-\nculated results of the Berry curvature Ωz\nn(k)and the\nanomalous Hall conductivity . With charge carrier dop-\ning, the F ermi level of Janus 2H-VSeS monolayer can\nbe modulated between the K-valley ( ∆K) and K′-valley\n(∆K′). As an example of proper hole doping, when an\nin-plane electric field E is applied, Janus 2H-VSeS mono-\nlayer is magnetised upwards and spin holes from the K′-\nvalley flow upwards and the accumulated holes generate\na charge Hall current that can be measured by a positive\nvoltage. On the contrary , while reversing the Janus 2H-\nVSeS monolayer or modulating magnetized downward,\nthe spin holes accumulated by flowing to the lower K-\nvalley can be measured as a negative voltage. This de-\nvice can be used as a spin filter, filtering out all carriers\nwith spin-up and spin-down tp move laterally , which will\nproduce a net Hall current. In summary , the Janus 2H-\nVSeS monolayer can be applied to anomalous Hall effect\nvalley electronics devices and become the basis for valley\nelectronics applications.9\nIV. CONCLUSIONS\nIn summary , the geometry , magnetic, electronic, and\nvalley properties of the Janus 2H-VSeS monolayer are\ninvestigated in detail using first-principles calculations.\nThe Janus 2H-VSeS monolayer exhibits an indirect band-\ngap semiconductor character with ferromagnetic Curie\ntemperature T cof 278 K, in-plane MA, and large spon-\ntaneous valley splitting of 105 me V. Interestingly , as the\nstrain increases from −8% to 8%, the magnetic moment\nof V atom and T cshow a monotonous increase trend,\nwhile the valley splitting and MAE firstly increase and\ngradually decrease. By analyzing the 3d orbital-resolved\nMAE of V atoms based on second-order perturbation the-\nory , the contributions to MAE mainly originate from the\nmatrix element differences between the d xy and d x2−y2orbitals of V atoms. The electronic phase transition from\nBMS to HSC, SGS, and HM is also observed for the Janus\n2H-VSeS monolayer. The strong SOC effect, Berry cur-\nvature, and anomalous Hall conductivity under broken\ninversion symmetry can also be efficiently tuned by ap-\nplying strain (modulation range: 45%) and charge car-\nrier doping (modulation range: 9.5%). Finally , a valley\nand spin filter device is designed based on the calcula-\ntion results. The versatility of Janus 2H-VSeS monolayer\nwith considerable MAE, high T c, and large valley split-\nting strongly suggests its potential application in two-\ndimensional spintronic devices.\nACKNOWLEDGEMENTS\nThis work was supported by Natural Science F ounda-\ntion of Tianjin City (Grant No. 20JCYBJC16540)\n[1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang,\nM. I. Katsnelson, I. Grigorieva, S. Dubonos, and\na. Firsov, Nature 438 , 197 (2005).\n[2] S. Stankovich, D. A. Dikin, G. H. Dommett, K. M.\nKohlhaas, E. J. Zimney , E. A. Stach, R. D. Piner, S. T.\nNguyen, and R. S. Ruoff, Nature 442 , 282 (2006).\n[3] K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz,\nPhys. Rev. Lett. 105 , 136805 (2010).\n[4] A. Splendiani, L. Sun, Y. Zhang, T. Li, J. Kim, C.-\nY. Chim, G. Galli, and F. W ang, Nano Lett. 10 , 1271\n(2010).\n[5] T. Olsen and I. Souza, Phys. Rev. B 92 , 125146 (2015).\n[6] H. Zeng, J. Dai, W. Y ao, D. Xiao, and X. Cui, Nat.\nNanotechnol. 7, 490 (2012).\n[7] D. Xiao, G.-B. Liu, W. F eng, X. Xu, and W. Y ao, Phys.\nRev. Lett. 108 , 196802 (2012).\n[8] K. F. Mak, K. He, J. Shan, and T. F. Heinz, Nat. Nan-\notechnol. 7, 494 (2012).\n[9] Q. Zhang, S. A. Y ang, W. Mi, Y. Cheng, and U. Schwin-\ngenschlögl, Adv. Mater. 28 , 959 (2016).\n[10] G. Aivazian, Z. Gong, A. M. Jones, R.-L. Chu, J. Y an,\nD. G. Mandrus, C. Zhang, D. Cobden, W. Y ao, and\nX. Xu, Nat. Phys. 11 , 148 (2015).\n[11] X. Liang, L. Deng, F. Huang, T. T ang, C. W ang, Y. Zhu,\nJ. Qin, Y. Zhang, B. Peng, and L. Bi, Nanoscale 9, 9502\n(2017).\n[12] C. Shen, G. W ang, T. W ang, C. Xia, and J. Li, Appl.\nPhys. Lett. 117 , 042406 (2020).\n[13] J. Liu, W.-J. Hou, C. Cheng, H.-X. F u, J.-T. Sun, and\nS. Meng, J. Phys.: Condens. Matter 29 , 255501 (2017).\n[14] C. W ang and Y. An, Appl. Surf. Sci. 538 , 148098 (2021).\n[15] Y. Cheng, Z. Zhu, M. T ahir, and U. Schwingenschlögl,\nEurophys. Lett. 102 , 57001 (2013).\n[16] A.-Y. Lu, H. Zhu, J. Xiao, C.-P . Chuu, Y. Han, M.-H.\nChiu, C.-C. Cheng, C.-W. Y ang, K.-H. W ei, and Y. Y ang,\nNat. Nanotechnol. 12 , 744 (2017).\n[17] F. Li and Y. Li, J. Mater. Chem. C 3, 3416 (2015).\n[18] A. C. de Leon, B. J. Rodier, Q. Luo, C. M. Hemmingsen,\nP . W ei, K. Abbasi, R. Advincula, and E. B. Pentzer, ACS\nnano 11 , 7485 (2017).[19] Y. Guo, S. Zhou, Y. Bai, and J. Zhao, Appl. Phys. Lett.\n110 , 163102 (2017).\n[20] J. Liang, W. W ang, H. Du, A. Hallal, K. Garcia,\nM. Chshiev, A. F ert, and H. Y ang, Phys. Rev. B 101 ,\n184401 (2020).\n[21] J. Y uan, Y. Y ang, Y. Cai, Y. W u, Y. Chen, X. Y an, and\nL. Shen, Phys. Rev. B 101 , 094420 (2020).\n[22] C. Xu, J. F eng, S. Prokhorenko, Y. Nahas, H. Xiang, and\nL. Bellaiche, Phys. Rev. B 101 , 060404 (2020).\n[23] I. Dzyaloshinsky , J. Phys. Chem. Solids 4, 241 (1958).\n[24] C. Zhang, Y. Nie, S. Sanvito, and A. Du, Nano Lett. 19 ,\n1366 (2019).\n[25] G. Kresse and J. F urthmüller, Comput. Mater. Sci. 6, 15\n(1996).\n[26] G. Kresse, J. F urthmüller, and J. Hafner, Phys. Rev. B\n50 , 13181 (1994).\n[27] G. Kresse and J. F urthmüller, Phys. Rev. B 54 , 11169\n(1996).\n[28] G. Kresse and D. Joubert, Phys. Rev. B 59 , 1758 (1999).\n[29] J. P . Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.\nLett. 77 , 3865 (1996).\n[30] P . Söderlind, O. Eriksson, B. Johansson, and J. Wills,\nPhys. Rev. B 50 , 7291 (1994).\n[31] P . Hohenberg and W. Kohn, Phys. Rev. 136 , B864\n(1964).\n[32] S. Grimme, J. Comput. Chem. 25 , 1463 (2004).\n[33] A. Liechtenstein, V. I. Anisimov, and J. Zaanen, Phys.\nRev. B 52 , R5467 (1995).\n[34] H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13 , 5188\n(1976).\n[35] S. Grimme, J. Antony , S. Ehrlich, and H. Krieg, J. Chem.\nPhys. 132 , 154104 (2010).\n[36] A. T ogo and I. T anaka, Scripta Mater. 108 , 1 (2015).\n[37] N. Marzari, A. A. Mostofi, J. R. Y ates, I. Souza, and\nD. V anderbilt, Rev. Mod. Phys. 84 , 1419 (2012).\n[38] A. A. Mostofi, J. R. Y ates, Y.-S. Lee, I. Souza, D. V an-\nderbilt, and N. Marzari, Comput. Phys. Commun. 178 ,\n685 (2008).\n[39] L. Liu, X. Ren, J. Xie, B. Cheng, W. Liu, T. An, H. Qin,\nand J. Hu, Appl. Surf. Sci. 480 , 300 (2019).10\n[40] L. Liu, S. Chen, Z. Lin, and X. Zhang, J. Phys. Chem.\nLett. 11 , 7893 (2020).\n[41] H. L. Zhuang and R. G. Hennig, Phys. Rev. B 93 , 054429\n(2016).\n[42] M. Esters, R. G. Hennig, and D. C. Johnson, Phys. Rev.\nB 96 , 235147 (2017).\n[43] H. Zhang, L.-M. Liu, and W.-M. Lau, J. Mater. Chem.\nA 1, 10821 (2013).\n[44] D. Xiao, W. Y ao, and Q. Niu, Phys. Rev. Lett. 99 , 236809\n(2007).\n[45] D. J. Thouless, M. Kohmoto, M. P . Nightingale, and\nM. den Nijs, Phys. Rev. Lett. 49 , 405 (1982).\n[46] Y. Y ao, L. Kleinman, A. MacDonald, J. Sinova, T. Jung-\nwirth, D.-s. W ang, E. W ang, and Q. Niu, Phys. Rev. Lett.\n92 , 037204 (2004).[47] Q. W u, S. Zhang, H.-F. Song, M. T royer, and A. A.\nSoluyanov, Comput. Phys. Commun. 224 , 405 (2018).\n[48] D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82 ,\n1959 (2010).\n[49] B. Y ang, X. Zhang, H. Y ang, X. Han, and Y. Y an, J.\nPhys. Chem. C 123 , 691 (2018).\n[50] Y. Ma, Y. Dai, M. Guo, C. Niu, Y. Zhu, and B. Huang,\nACS nano 6, 1695 (2012).\n[51] D. Zhang and S. Dong, Prog. Nat. Sci. 29 , 277 (2019).\n[52] N. Luo, C. Si, and W. Duan, Phys. Rev. B 95 , 205432\n(2017).\n[53] R. Mukherjee, H. Chuang, M. Koehler, N. Combs,\nA. Patchen, Z. Zhou, and D. Mandrus, Phys. Rev. Appl.\n7, 034011 (2017).— Supplementary Materials —\nSupplementary Materials: Controlled Curie temperature, magneto-crystal anisotropy ,\nand valley polarization in 2D ferromagnetic Janus 2H-VSeS monolayer\nCunquan Li1, Y ukai An1,∗\n1Key Laboratory of Display Materials and Photoelectric Devices,\nMinistry of Education, Tianjin Key Laboratory for Photoelectric Materials and Devices,\nNational Demonstration Center for Experimental F unction Materials Education,\nSchool of Material Science and Engineering, Tianjin University of T echnology, Tianjin, 300384, China\nFIG. S1. Phonon density of states for the Janus 2H-VSeS monolayer. The green, yellow, and red lines present Se, S and V\natom-projected phonon DOSs, respectively .\nFIG. S2. (a) 3D band structure of Janus 2H-VSeS monolayer with considering SOC. Different colors indicate different equiva-\nlence surfaces. (b) 2D Projected band structure of (b) valence band and (c) conduction band at the k xky-plane.2\nFIG. S3. The spin-polarized band structures of Janus 2H-VSeS monolayer at the strains of (a) −8%, (b) −5%, (c) −1%, (d)\n1%, (e) 5% and (f) 8%, respectively .\nFIG. S4. The V, Se and S atomic-layer-resolved MAE for the Janus 2H-VSeS monolayer under various strains.3\nFIG. S5. The V-d and Se(S)-p orbital resolved MAE for the Janus 2H-VSeS monolayer at the strains of (a) −8%, (b) −4%,\n(c) 4% and (d) 8%.\nFIG. S6. Dependence of (a) magnetic moment and (b) specific heat capacity on the temperature for the Janus 2H-VSeS\nmonolayer at the strains of −8%,−4%, 4% and 8%."    },    {        "title": "2205.02983v2.Strain_induced_Shape_Anisotropy_in_Antiferromagnetic_Structures.pdf",        "content": "Strain-induced Shape Anisotropy in Antiferromagnetic Structures\nHendrik Meer,1,\u0003Olena Gomonay,1Christin Schmitt,1Rafael Ramos,2, 3Leo\nSchnitzspan,1Florian Kronast,4Mohamad-Assaad Mawass,4Sergio Valencia,4\nEiji Saitoh,2, 5, 6, 7Jairo Sinova,1, 8Lorenzo Baldrati,1and Mathias Kl aui1, 9,y\n1Institute of Physics, Johannes Gutenberg-University Mainz, 55099 Mainz, Germany\n2WPI-Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan\n3Centro de Investigaci\u0013 on en Qu\u0013 \u0010mica Biol\u0013 oxica e Materiais Moleculares (CIQUS),\nDepartamento de Qu\u0013 \u0010mica-F\u0013 \u0010sica, Universidade de Santiago de Compostela, Santiago de Compostela 15782, Spain\n4Helmholtz-Zentrum Berlin f ur Materialien und Energie,\nAlbert-Einstein-Strasse 15, 12489 Berlin, Germany\n5Department of Applied Physics, The University of Tokyo, Tokyo 113-8656, Japan\n6Center for Spintronics Research Network, Tohoku University, Sendai 980-8577, Japan\n7Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan\n8Institut of Physics, Academy of Sciences of the Czech Republic, Praha 11720, Czech Republic\n9Center for Quantum Spintronics, Department of Physics,\nNorwegian University of Science and Technology, 7034 Trondheim, Norway\n(Dated: September 2, 2022)\nWe demonstrate how shape-dependent strain can be used to control antiferromagnetic order in\nNiO/Pt thin \flms. For rectangular elements patterned along the easy and hard magnetocrystalline\nanisotropy axes of our \flm, we observe di\u000berent domain structures and we identify magnetoelastic\ninteractions that are distinct for di\u000berent domain con\fgurations. We reproduce the experimental\nobservations by modeling the magnetoelastic interactions, considering spontaneous strain induced by\nthe domain con\fguration, as well as elastic strain due to the substrate and the shape of the patterns.\nThis allows us to demonstrate and explain how the variation of the aspect ratio of rectangular\nelements can be used to control the antiferromagnetic ground state domain con\fguration. Shape-\ndependent strain does not only need to be considered in the design of antiferromagnetic devices,\nbut can potentially be used to tailor their properties, providing an additional handle to control\nantiferromagnets.\nAntiferromagnetic materials (AFM) are promising can-\ndidates for fast, robust and energy-e\u000ecient spintronic\ndevices [1]. AFMs possess two or more magnetic sub-\nlattices, with vanishing net magnetization. This absence\nof magnetic stray \felds enables higher bit packing den-\nsity of AFMs devices compared to ferro(i)magnetic ma-\nterials (FMs), enhanced robustness against interfering\nexternal magnetic \felds and potentially THz switching\nspeeds [2]. Especially insulating antiferromagnetic mate-\nrials (iAFM) have emerged as a promising material class\nfor the development of low power devices, because their\nlow damping allows for the transport of spin currents over\nlong distances [3].\nCrucial for the implementation of AFMs as active spin-\ntronic devices is the control of the antiferromagnetic or-\nder. In recent years it has been established that current\npulses through an adjacent heavy metal layer can induce\na reorientation of the antiferromagnetic ordering in insu-\nlating AFMs [4{6]. For iAFMs with strong magnetostric-\ntion, the reorientation of the N\u0013 eel vector is dominated by\na thermomagnetoelastic switching and strongly depends\non the device geometry [7{9].\nFor FMs the device geometry and shape-induced con-\ntrol of the domains is a key tool for tailoring func-\n\u0003meer@uni-mainz.de\nyKlaeui@Uni-Mainz.detional device properties. In AFMs, conventional shape\nanisotropy caused by the magnetic dipolar interactions\nis not present, due to the absence of a demagnetization\n\feld. However, theoretical work on shape-induced phe-\nnomena in \fnite size antiferromagnets predicts an order-\ning of the antiferromagnetic domains, with long-range\nstrain \felds leading to the formation of shape-dependent\ndomain structures [10, 11]. It has been shown that the\nshape-induced domains in the FM layer of a AFM/FM\nbilayer can for instance be used to imprint an AFM vor-\ntex state into the adjacent AFM layer [12, 13]. How-\never, initial studies of patterning-induced e\u000bects in an-\ntiferromagnetic LaFeO 3could not observe any changes\nin the domain structure, after patterning di\u000berent ele-\nments with etching [14]. Later studies patterned elements\nvia an Ar+ion implantation-based patterning technique,\nwhich resulted in antiferromagnetic structures embedded\nin a non-magnetic layer [15, 16]. This technique led to\nthe observation of changes in the antiferromagnetic or-\ndering near the patterning edge for LaFeO 3[15, 17, 18]\nand more recently La 0:7Sr0:3FeO 3[16], interpreted as an\nedge e\u000bect near the edge for elements patterned along\nthe easy axis. Further studies have suggested the ex-\nploitation of this e\u000bect in exchange bias applications of\nAFM/FM heterostructures [19]. However, these previous\ninvestigations of patterning-induced modulations of the\nAFM order have been focused on the passive application\nof AFMs in AFM/FM bilayers. Considering the poten-arXiv:2205.02983v2  [cond-mat.mtrl-sci]  1 Sep 20222\ntial of antiferromagnets as active elements in spintronic\ndevices, it is important to investigate patterning- and\nshape-induced e\u000bects and in particular the control of the\ndomain con\fguration in AFMs without an adjacent FM\nlayer. It is crucial to not only understand patterning-\ninduced e\u000bects near the edge, but also the in\ruence of\nshape-dependent strain on the domain structure inside\na structured antiferromagnetic device. This e\u000bect would\nbe most suitable to tailor domain con\fgurations by the\nshape.\nThe prototypical collinear antiferromagnet NiO has\nbeen considered to be a promising candidate for an active\nelement in spintronic applications, in contrast to LaFeO 3,\ndue to the possibility of electrically controlling and read-\ning the AFM order [4, 20{22] and recent observations of\nultrafast currents in the THz regime in NiO/Pt bilayers\n[2, 23]. In addition, NiO exhibits a high N\u0013 eel tempera-\nture of 523 K in the bulk [24] and strong magneto-elastic\ncoupling [25]. The latter has been used extensively to ma-\nnipulate the AFM order of NiO by growth-induced strain\n[26{28], piezoelectric substrates exerting strain [29] and\nindirectly via current-induced heating leading to strain\n[9]. However, the e\u000bect of shape-dependent strain on the\ndomain structure of NiO thin \flms has not been explored.\nConsidering the application of AFMs with strong magne-\ntostriction like NiO or CoO in active spintronic devices,\nit is important to investigate how the geometry in\ruences\nthe antiferromagnetic domain con\fguration and how one\ncould use di\u000berent geometries to control the antiferro-\nmagnetic order.\nIn this work, we demonstrate the tailoring of the AFM\nground state domain con\fguration of NiO by shape-\ndependent strain. We study the N\u0013 eel vector orienta-\ntion in patterned elements by photoemission electron mi-\ncroscopy (PEEM) exploiting the x-ray magnetic linear\ndichroism (XMLD) e\u000bect for magnetic contrast. We \frst\nidentify and compare the shape-induced domain struc-\nture of elements oriented along di\u000berent axes before we\ntheoretically explore how shape-induced e\u000bects can ma-\nnipulate the antiferromagnetic ordering in di\u000berent ele-\nment geometries. Finally, we demonstrate how the modi-\n\fcation of the shape-dependent strain by variation of the\naspect ratio of our elements can be used to control the\nantiferromagnetic domain con\fguration, demonstrating\nthus a tool for the shape-induced control of future AFM\ndevices.\nI. RESULTS\nTo investigate shape-induced e\u000bects on the antiferro-\nmagnetic domain structure, we have grown an epitaxial\nNiO(10nm)/Pt(2nm) bilayer on an MgO(001) substrate\nand used Ar ion beam etching to pattern various\nelements with di\u000berent orientations. Similarly prepared\nbilayers of NiO and Pt are currently extensively used\nfor current-induced switching [4{9] and THz radiation\nexperiments [30]. As depicted in Fig.1a, we have etchedtrenches with a width of around 1 µm and a depth of\nabout 20 nm around the desired elements. Additionally,\nwe deposited about 1 :4 nm of ruthenium inside the\ntrenches to reduce the possibility of discharges during\nPEEM imaging [31]. To allow for a recon\fguration\nof the AFM domains, the sample was annealed after\npattering above its N\u0013 eel temperature for 10 minutes at\n550 K under vacuum. Measurements were carried out\nat the UE49-PGM/SPEEM at the BESSY II electron\nstorage ring operated by the Helmholtz-Zentrum Berlin\nf ur Materialien und Energie [32].\nWe \frst measured polarization-dependent absorption\nspectra around the Ni L3andL2edge (details of the\nmeasurement technique, see Appendix B) to verify the\nantiferromagnetic order of our \flms at room tempera-\nture. The XMLD contrast is proportional to the orien-\ntation of the N\u0013 eel vector. By studying the XMLD con-\ntrast dependence on the azimuthal angle \rand angle of\nthe beam-polarization !(Appendix B) we can identify\nFIG. 1. (a) XMLD-PEEM image of shape-induced NiO do-\nmains inside a rectangular shaped element, with its edges\noriented along the in-plane projection of the easy axis. (b)\nSketch of the experimental setup and orientation of the ob-\nserved di\u000berent N\u0013 eel vector directions [ \u00065\u00065 19] with re-\nspect to the polarization vector. An orientation along [5 5\n19] (yellow) is equally possible, but was not observed in this\nparticular element.3\nin Fig. 1a four antiferromagnetic domains present in a\nrectangular element of 10 µm\u00025µm whose long axis is\noriented along the [110] direction. Three di\u000berent levels\nof XMLD contrast are observed inside our element, in-\ndicating three types of domains, a larger domain in the\ncenter (domain 1 - blue), two originating from the short\nedges (domain 2 - green) and one narrow at the long\nedge (domain 3 - red). The directions of the N\u0013 eel vec-\ntor in the di\u000berent domains are depicted in Fig. 1b. We\ncan observe that the in-plane projection of the N\u0013 eel vec-\ntor in all domains is oriented orthogonal to the edges of\nthe element along [110] and [1 \u001610]. The formation of do-\nmain 3 (red) along the edge can be attributed to localized\nchanges of the anisotropy near the edge of the element,\nrelated to patterning-induced material property changes.\nHowever, the shape of domains 2 (green) and 1 (blue) in\nthe center of the element can not be understood by local\nchanges of the anisotropy at the edge of the element and\nwe need to model long-range magnetoelastic interactions\nto understand the domain con\fguration.\nNiO is known for its strong magnetoelastic coupling\nwhich is responsible for the creation of internal (magne-\ntoelastic) stresses in a magnetically ordered state. In a\nfree-standing homogeneously magnetically ordered sam-\nple, these stresses induce a pronounced spontaneous\nstrain (u0/10\u00004\u00003\u000110\u00003[33]) characterized by the\nstrain tensor ^ uspon\n0 whose components are related with\nthe components of the N\u0013 eel vector n. In a multilayer\nsystem the internal magnetoelastic stresses are comple-\nmented by the external stresses due to the clamping of\nthe antiferromagnetic layer by a nonmagnetic substrate\n[34]. In this case the resulting strain \feld can be split into\ntwo parts, spontaneous (or plastic) strains ^ uspon\n0[n(r)] as-\nsociated with the distribution of the N\u0013 eel vector (as in\nthe absence of a substrate) and additional, elastic strains\n^uelast: ^utot= ^uspon\n0+ ^uelast.\nTo calculate the elastic strain, we use an approach\nof elasticity theory with continuously distributed de-\nfects [35, 36]. In particular, we assume that in mag-\nnetic multilayers the defects originate from the incom-\npatibility between the spontaneous strain ^ uspon\n0and the\nnon-deformed (reference) state of a non-magnetic sub-\nstrate at the NiO/substrate interface, or from incom-\npatibility between spontaneous strain in neighboring do-\nmains. From the compatibility condition for the total\nstrain\"ijk\"lmn@j@mutot\nkn= 0 (where \"ijkis an antisym-\nmetric Levi-Civita tensor) we obtain a set of equations\nfor the elastic strains\n\"ijk\"lmn@j@muelast\nkn=\u0011il; (1)\nin which the incompatibility tensor \u0011il\u0011\u0000\"ijk\"lmn@j@muspon\nkn[n(r)] is calculated for a given\ndistribution of the N\u0013 eel vector.\nEquation (1) is similar to equations of electrostatics, in\nwhich the incompatibility ^ \u0011plays the role of the elastic\nor magnetoelastic charges and the elastic strains ^ uelast\ncorrespond to the potentials [37]. Moreover, similar to\nthe electric and magnetostatic stray \felds, the \feld of\nthe corresponding elastic strains is long-range and there-\nfore can stabilise an inhomogeneous distribution of the\nmagnetic vectors. The similarity with the equations of\nelectrostatics and magnetostatics allows for a qualitative\ninterpretation of the magnetoelastic e\u000bects in terms of\nmagnetoelastic charge distributions.\nHere, we consider some of the e\u000bects that reinforce our\nintuitive reasoning through modelling. The theoretical\ndescription of magnetic textures is based on minimizing\nthe total energy of a sample with respect to magnetic\nand elastic variables. The bulk energy of the NiO \flm,\nWbulk=Z\ndr(wmag+wm\u0000e+welas) (2)\nincludes magnetic, wmag, magnetoelastic, wm\u0000e, and\nelastic,welascontributions. The magnetic structure of\nNiO is described by the N\u0013 eel vector n(r) (jnj= 1), which\nis generally parameterized with two angles, 'and#, as\nshown in Fig. 1b.\nWe distinguish in our thin NiO \flms four types of T-\ndomains with the N\u0013 eel vector oriented along [ \u00065\u00065 19]\n[38]. The pairs with opposite orientation of projection\non the \flm plane have the same in-plane components of\nspontaneous strain and will be treated in the further dis-\ncussion as the same domain. Since the out-of-plane com-\nponent of the N\u0013 eel vector is the same in all domains, we\nconsider the in-plane angle 'as the only magnetic vari-\nable. We also neglect the magnetic homogeneity through-\nout the thickness of the NiO layer and consider the dis-\ntribution of the N\u0013 eel vector to be within the \flm plane\n(xy).\nWe start with a discussion of the origin of the domain\nstructure in thin \flms and patterned elements. A single-\ndomain continuous \flm of NiO is charged due to in-\ncompatibility strain charges homogeneously distributed\nat the interface with the non-magnetic substrate. The\ncharge density depends on the elastic and magnetoelas-\ntic properties of the interface and is localised in a thin\nlayer of the order of the exchange length (characteristic\nlength scale at which the N\u0013 eel vector decays inside the\nnonmagnetic region), see Appendix C. These magnetoe-\nlastic charges create additional homogeneous strain ^ uelast\nand their non-negative contribution to the energy of the\nNiO layer is proportional to the volume of the NiO.\nIn a multidomain sample with equally distributed do-\nmains of all types, the average strain incompatibility and\naverage charge density vanish. The local charge densityis still non-zero and contributes to the energy of the sam-\nple. However, this contribution is proportional to the av-\nerage domain volume. Hence, a small-scale multidomain4\nFIG. 2. (a) XMLD images of di\u000berent NiO(10)/Pt(2) rectangular devices with varying aspect ratio. The edges of the devices are\noriented along the in-plane projection of the easy axes. The arrows show the in-plane projection of the N\u0013 eel vector determined\nfrom the greyscale contrast. (b) Final equilibrium state of the magnetic texture after considering magneto-elastic interactions\nto simulate the domain distribution in di\u000berent aspect ratios. The color code indicates the direction of the N\u0013 eel vector.\nstructure is energetically favourable, with the domain size\nbeing limited by the positive energy contribution of the\ndomain walls (similar to the Kittel model in ferromagnets\n[39]). However, the formation of a new domain inside a\nsingle-domain region is blocked by a high energy barrier\nassociated with the coherent rotation of a large number\nof magnetic moments in the two sublattices. The en-\nergy barrier can be much lower at the sample surfaces\nand edges due to the additional contributions from sur-\nface energy and incompatibility charges at the element\ncorners [34].\nFirst, we consider the role of the surface magnetic\nanisotropy present in thin \flm NiO continuous \flms and\npatterned elements, which in our case favours alignment\nof the N\u0013 eel vector perpendicular to the surface. For this\nwe studied the evolution of the magnetic structure in the\npatterned elements with di\u000berent aspect ratio cut paral-\nlel to the in-plane projection of the easy magnetic axis\n(see Fig. 2a).\nIn this geometry the surface anisotropy induces the for-\nmation of the dark domains along [ 110] edges and bright\ndomains along [110] edges. The \fnal texture includes two\nclosure domains localised at the short edges and a large\northogonal domain that spreads between the two long\nedges. The closure domains grow from the edges due to\nmagnetoelastic forces that act to diminish the average\nmagnetoelastic charge of the sample. This growth is lim-\nited by an increase of the energy of the domain walls.\nThe size of the closure domains is of the order of the size\nof the short edge and depends on the aspect ratio of the\nsample (see Fig. 2b). It should be noted that in the\nabsence of magnetoelastic coupling, the closure domains\nwould be localised in the vicinity of the short edge within\nFIG. 3. Comparison between equilibrium states with (a) and\nwithout (b) consideration of magnetoelastic coupling.\na distance of several magnetic domain wall widths ( Fig.\n3), independent on the aspect ratio of the device. Our\nsimulations also show that the closure domains can be\nlocalised along the longer edges of the samples as well,\nas could be experimentally observed for larger patterned\ndevices. However, both con\fgurations (the one with the\nclosure domains along the short and the other along the\nlong edges) are observed in a \fnite range of aspect ra-\ntios (between 1/3 and 3) for which their energies have\ncomparable values. In this case the structure of the \fnal\nstate depends on the initial con\fguration and kinetics of5\nthe domain growth.\nTo better illustrate the e\u000bect of strain, we next inves-\ntigate elements oriented along the [100] and [010] axes,\nwhere even more conspicuous e\u000bects are expected. For\nelements oriented along the projection of the hard axes\nthe domains do not align along the edges of the element,\nbut instead are centered around the corners of the ele-\nments, see Fig. 4a. Inside the element we can observe a\nlarge green domain and two blue domains, which are lo-\ncated near the top left and bottom right corner. Outside\nof the element we observe a domain (green arrows) and\ntwo additional domains (blue arrows) located at the top\nright and bottom left corner, opposite to the domains in\nthe inner corners. In the case that the domain forma-\ntion is dominated by an alignment along certain crystal-\nlographic axes one would expect the same domains to be\npresent at the inside and outside edge of the element.\nHowever, this is not the case and we therefore need to\nconsider shape-dependent strain, in particular the role of\nincompatibility charges in the corners of the elements, to\nunderstand the origin of the domain structure.\nFor elements cut along the in-plane projection of the\nhard magnetic axis, the surface anisotropy favours an\norientation of the N\u0013 eel vector along a hard magnetic axis\nand does not set a preferable domain type. However,\nthe surface anisotropy sets orthogonal easy directions at\nFIG. 4. (a) Antiferromagnetic domain structure of a rectan-\ngular shaped element oriented along the in-plane projections\nof the hard axes. (b) Simulated equilibrium state of the mag-\nnetic texture. The color code and the arrows indicate the\ndirection of the N\u0013 eel vector.\nFIG. 5. Sketch of the di\u000berent direction of the rotation for\nstrain at inner and outer corners.\nneighboring edges and favours the formation of vortex-\nlike textures of the N\u0013 eel vectors in the vicinity of the sam-\nple corners. Such a rotation of the N\u0013 eel vector through\n90\u000eis associated with an inhomogeneous rotation of the\nspontaneous strain ^ uspon\n0 and creates an elastic vortex\nstructure { so-called disclinations [36, 40] { localized in\nthe corners. Each disclination is characterised by incom-\npatibility charges which have opposite sign in neighbor-\ning corners (see Fig. 5). These charges create a radi-\nally distributed \feld of elastic strain ^ uelast[36] that via\nmagnetoelastic coupling sets preferable directions for the\nN\u0013 eel vector along the bisectrices of the element. For the\nelements cut along the easy magnetic axis, this strain\ncouples with the orientation of the N\u0013 eel vector in the\ncenter of the domain walls and splits the energy of the\ndomain walls pinned to the neighboring corners. For the\nelements cut along the hard magnetic axis, it removes the\ndegeneracy between the bright and dark domains and fa-\ncilitates a formation of the domain of a certain type. In\nother words, the elastic strains lower the energy barrier\nfor a closure domain. Here, the closure domain starts\nto grow from the opposite corners, which have the same\nsign of rotation, as shown in Fig. 4(b). Interestingly,\nsuch magnetoelastic disclinations appear not only at the\ninner corners of the rectangular elements, but also at the\ncorners of the outer part of the element, where the spon-\ntaneous strain rotates in the opposite direction [41].\nHence, internal and external strain charges of the same\ncorner have opposite signs. As a result, the closure do-\nmains of the same type start to grow along the di\u000ber-\nent diagonals in the internal and external regions (see\nFig. 4(a) and (b)). Magnetoelastic disclinations (corner\ncharges) are present also in the elements cut along the\neasy magnetic axes. In this case, corresponding elastic\nstrains set a preferable direction of the N\u0013 eel vector inside\nthe domain walls near the corners. Our calculations show\na di\u000berence of domain wall width and pinning energy in\nthe neighboring corner as a result of the strain-induced\nanisotropy.\nIn addition, di\u000berent antiferromagnetic domains are\naccompanied by the deformation of the crystallographic6\nFIG. 6. Simulated evolution of the domain structure from the\ninitial state to the \fnal equilibrium state.\nstructure. Due to the need for mechanical equilibrium,\nthe creation of antiferromagnetic domains is accompa-\nnied by destressing e\u000bects [10, 42]. The total energy of a\npatterned element is here complemented by the destress-\ning energyWdestr, which describes the coupling with the\nnonmagnetic substrate and edge e\u000bects, and the surface\nenergyWsurfof the patterned edges. The surface energy\nis modelled in a way to favor an orientation of the N\u0013 eel\nvector parallel to the normal Nwith respect to the pat-\nterned edge:\nWsurf=\u0000KsurfI\n(n\u0001N)2d`: (3)\nHere, the constant Ksurf>0 parameterizes the surface\nenergy and `is the coordinate along the sample edge.\nThe destressing energy, Wdestr, is treated as an additional\ncontribution of the elastic strains ^ uelast, which maintains\nthe strain compatibility of the sample at the interface\nwith nonmagnetic substrate.\nTo demonstrate the role of incompatibility and the\ndestressing e\u000bects in the formation and stabilization of\nthe domain structure, we have calculated the evolution\nof the domain structure for elements along the in-plane\nprojection of the easy axes, starting from the almost ho-\nmogeneous state (domain 2, green) with small domains\n(domain 1, blue) localised at the long edges of the sam-\nple using di\u000berent values of the damping parameter (dif-\nferent rates of the energy losses). At the initial stage,\nthe closure domains (at the long edge), being pinned in\nthe corners, grow in size trying to reduce the average\nincompatibility of the sample (see Fig. 6). In case of\nslow (quasistatic) relaxation (large damping) the system\nevolves into a state with the closure domains 1 along the\nlong edges separated by domain 2. In the opposite case\nof small damping the closure domains merge and the \f-\nnal state corresponds to the closure domains of the type\n2 localised at the short edges.II. DISCUSSION\nBy investigating elements with di\u000berent orientations\nand aspect ratios etched into NiO/Pt bilayers, we identify\nlong-range strain to govern the shape-dependent forma-\ntion of antiferromagnetic domains. We observe a prefer-\nential orientation of the N\u0013 eel vector perpendicular to the\nedge of our devices due to patterning e\u000bects. Consistent\nwith our previous observations the lattice mismatch be-\ntween the MgO substrate and the NiO layer leads to a\nstabilization of di\u000berent T-domains with the largest out\nof plane components, due the strain in the out of plane\ndirection [38]. We investigate the domain structures of\nelements oriented along the projection of the easy and\nhard axes and can identify shape-dependent strain to be\nresponsible for the observed domain structures. We can\nreproduce our experimental observations by magnetoe-\nlastic modelling that accounts for the spontaneous strain,\ndue to the distribution of the N\u0013 eel vector, and elastic\nstrain due to contributions from the substrate and the\npatterning. In previous studies, the substrate-induced\nstrain is the same for the di\u000berent T-domains [38]. Here,\nhowever, the shear strain due to the domain formation\nvaries for the di\u000berent domains across the di\u000berent ge-\nometries. The strain that dominates the formation of\nthe domain structure does not arise from patterning in-\nduced strain, but from the strain that is associated with\nthe formation of each T-domain. The interactions be-\ntween these strains are responsible for the formation and\nstabilization of the equilibrium domain structure. Anal-\nogous to shape-anisotropy in ferromagnets, magnetoelas-\ntic interactions in antiferromagnets are long-range and\ncan be used to tailor the antiferromagnetic ground state\nof antiferromagnetic devices. For example, by choos-\ning the right size, aspect ratio and orientation one could\nuse shape-induced strain to control the antiferromagnetic\nground state in antiferromagnetic THz emitters to tailor\nand optimize their response [30]. In addition, the strain\nfrom the patterned device itself could be used in elec-\ntrical switching of antiferromagnets to support or hinder\nthe reorientation of the N\u0013 eel vector independent of the\nunderlying switching mechanism.\nFinally we note that long range magnetoelastic inter-\nactions in AFMs could also be a challenge for the devel-\nopment of antiferromagnetic data storage as reorienting\nthe N\u0013 eel order to switch bits will by the change in strain\na\u000bect neighbouring bits. However, by considering physi-\ncal separation, as in established bit-patterned media [43],\nthe non interacting nature of AFMs can be fully taken\nadvantage of due to the absence of magnetic stray \felds.\nIn summary, we identify how shape-dependent strain\ncan be used to control the antiferromagnetic ground state\nin NiO over several microns. Since magnetoelastic cou-\npling is signi\fcant for several other antiferromagnets such\nas CoO and Hematite, shape-induced strain can be con-\nsidered to be the antiferromagnetic equivalent of conven-\ntional shape-induced anisotropy in ferromagnets and pro-\nvide a unique means to control antiferromagnets.7\nACKNOWLEDGMENTS\nThe authors thank T. Reimer for skillful technical\nassistance. We thank HZB for the allocation of syn-\nchrotron radiation beamtime, we thankfully acknowledge\nthe \fnancial support by HZB. The work has bene\fted\nfrom insights gained from experiments that were per-\nformed at the CIRCE beamline at ALBA Synchrotron\nwith the collaboration of ALBA sta\u000b. L.B. acknowl-\nedges the European Union's Horizon 2020 research and\ninnovation program under the Marie Sk lodowska-Curie\nGrant Agreement ARTES No. 793159. L.B. and\nM.K. acknowledge support from the Graduate School\nof Excellence Materials Science in Mainz (MAINZ)\nDFG 266, the DAAD (Spintronics network, Project No.\n57334897 and Insulator Spin-Orbitronics, Project No.\n57524834), and all groups from Mainz acknowledge that\nthis work was funded by the Deutsche Forschungsgemein-\nschaft (DFG, German Research Foundation), TRR 173-\n268565370 (Project Nos. A01, A03, A11, B02, and B12)\nand KAUST (OSR-2019-CRG8-4048). J.S. additionally\nacknowledges the Alexander von Humboldt Foundation\nand O.G and J.S. acknowledge the EU FET Open RIA\nGrant No. 766566 and the Deutsche Forschungsgemein-\nschaft (DFG, German Research Foundation), TRR 288-\n422213477 (project A09) and the Grant Agency of the\nCzech Republic grant no. 19-28375X. R.R. also acknowl-\nedges support from the European Commission through\nthe Project 734187-SPICOLOST (H2020-MSCA-RISE-\n2016), the European Union's Horizon 2020 research\nand innovation program through the Marie Sklodowska-\nCurie Actions Grant Agreement SPEC No. 894006, the\nMCIN/AEI (RYC 2019-026915-I), the Xunta de Gali-\ncia (ED431B 2021/013, Centro Singular de Investigaci\u0013 on\nde Galicia Accreditation 2019-2022, ED431G 2019/03)\nand the European Union (European Regional Devel-\nopment Fund - ERDF). M.K. acknowledges \fnancial\nsupport from the Horizon 2020 Framework Programme\nof the European Commission under FET-Open Grant\nAgreement No. 863155 (s-Nebula) and from the Re-\nsearch Council of Norway through its Centers of Excel-\nlence funding scheme, project number 262633 \\QuSpin\".\nThis work was also supported by ERATO \\Spin Quan-\ntum Recti\fcation Project\" (Grant No. JPMJER1402)\nand the Grant-in-Aid for Scienti\fc Research on Innova-\ntive Area, \\Nano Spin Conversion Science\" (Grant No.\nJP26103005), Grant-in-Aid for Scienti\fc Research (S)\n(Grant No. JP19H05600) from JSPS KAKENHI, Japan.\nAppendix A: Sample preparation\nSimilar to previous studies [6, 38] we used re-\nactive magnetron sputtering to grow epitaxial\nMgO(001)//NiO(10nm)/Pt(2nm) thin \flms. Before\ndeposition the MgO(001) substrates were pre-annealed\nat 770\u000efor 2 h in vacuum. We then deposited NiO from\na Ni target at 430\u000eC and 150 W in an atmosphere of O 2(\row 3 sccm) and Ar (\row 15 sccm), before depositing\nthe platinum layer in situ at room temperature.\nAppendix B: XMLD signal\nWe observed X-ray magnetic linear dichroism (XMLD)\nin Fig. 7(a), calculated as I LH-ILV, and an absence of cir-\ncular magnetic dichroism (XMCD) in Fig. 7 (b), indicat-\ning purely antiferromagnetic ordering of our \flms. For\nsubsequent XMLD imaging we utilized the inversion of\nthe XMLD contrast at the Ni L 2edge at the energies\nE1=868.3 eV, E2=870.1 eV and calculated the XMLD\nimage by [I(E1)-I(E2)]/[I(E1)+I(E2)]. By studying the\nXMLD contrast dependence on the azimuthal angle \r\nand angle of the beam-polarization !we can identify in\nFig. 1(a) four di\u000berent antiferromagnetic domains to be\npresent in a rectangular element of 10 \u00025µm whose long\naxis is oriented along the [110] direction.\nNiO is a compensated collinear antiferromagnet with\ntwo magnetic sublattices whose magnetic structure is\nuniquely described by the N\u0013 eel vector n. In bulk NiO\ncrystals the exchange-striction leads to a rhombohedral\ncontraction along the h111iaxes. Thus, four twin T-\ndomains can be formed with the spins being ferromag-\nnetically coupled inside the f111gplanes and antiferro-\nmagnetically between the f111gplanes, due to the su-\nperexchange interaction. Within such a T-domain the\nspins can be oriented along three possible easy axes h11\u00162i,\nleading to three di\u000berent spin domains S-domains per T-\ndomain. There are a total of 12 possible domains in bulk\nNiO crystals [24, 44, 45]. However, in NiO thin \flms the\ngrowth induced strain from the substrate mismatch can\nlead to a preferential stabilization of S-domains parallel\nor perpendicular to the sample plane [27, 28]. In a previ-\nous study on similarly grown MgO//NiO(10)/Pt(2) thin\n\flms we observed that the epitaxial growth of NiO on\nMgO substrates results in compressive strain, which lead\nto a preferential out of plane component of the N\u0013 eel vec-\ntor [27, 28, 38]. Here, we have observed analogous con-\ntrast changes among the domains with varying azimuthal\nangle\rand orientation of the linear beam polarization\n!in Fig. 8. Thus, we can identify the orientation of the\nN\u0013 eel vector of our domains to be along the [ \u00065\u00065 19]\ndirections, in line with our previous \fndings [38].\nAppendix C: Modelling of the magnetic textures\nWe model these contributions according to the tetrago-\nnal crystallographic symmetry of the NiO \flm (deformed\ncubic crystal due to the substrate), so that the density\nof the magnetic energy is given by4\nwmag=1\n2A(r')2\u00001\n4HanMscos 4'; (C1)\nwhereAis the exchange sti\u000bness of NiO, Hanis the\nanisotropy \feld, and Ms=2 is the saturation magnetiza-8\nFIG. 7. (a) X-ray absorption spectrum and XMLD for linear\npolarized X-rays in the vicinity of the L 2edge. (b) X-ray\nabsorption spectrum and XMCD for circular polarized X-rays.\nFIG. 8. Simulated and experimental contrast of the di\u000berent\ndomains dependent on the beampolarization for (a) \r=\u000030\u000e\nand (b) \r= 45\u000e.\ntion of the magnetic sublattice. The densities of magne-\ntoelastic and elastic energy are\nwm\u0000e=\u0015meM2\ns[cos 2'(uxx\u0000uyy) + 2 sin 2'uxy]; (C2)\nwelas=1\n2\u0016\u0002\n(uxx\u0000uyy)2+ 4u2\nxy\u0003\n+\u0016\n2(1\u00002\u0017)(uxx+uyy)2;\nwhere\u0015meis the magnetoleastic parameter, \u0016is the shear\nmodulus,\u0017is the Poisson ratio. In Eq. (C2), we omit the\nout-of plane components of strain ( uzz,uxz,uyz) and\ndisregard the coupling between the N\u0013 eel vector and the\nisotropic strain ( uxx+uyy), since these e\u000bects are irrel-\nevant for the formation of the domain structure. The\ncoordinate axes xandyare aligned along the easy mag-\nnetic axis [1 10] and [110]. General expressions for these\ncontributions can be found in [38, 46].\nThe strain in the NiO layer is represented as a sum of\nspontaneous (plastic) strains and elastic strains, as de-\nscribed in the main part: ^ utot= ^uspon\n0+ ^uelast. The spon-\ntaneous strains are calculated by minimizing the elastic\nand magnetoelastic energy contributions (C2) for a given\nvalue of'(x;y):\nuspon\nxx=\u0000uspon\nyy=\u0000\u0015me\n\u0016M2\nscos 2';\nuspon\nxy=\u0000\u0015me\n\u0016M2\nssin 2': (C3)\nThe elastic strains ^ uelastare then calculated from the\ncompatibility equations\n\"ijk\"lmn@j@muelast\nkn=\u0011il; (C4)\nThe incompatibility arising at the interface between themagnetic and the nonmagnetic layer is calculated as\n\u0011xx=\u0000\u0011yy=\u0015me\n\u0016(@zMs)2cos 2';\n\u0011xy=\u0015me\n\u0016(@zMs)2sin 2': (C5)\nThe magnetic order, parameterized as Ms(z), vanishes\nat the scale of the exchange length \u0018exch(/several inter\natomic distances) , which is much smaller than the other\nlength scales in the system (domain wall width, thickness\nof the NiO layer, etc). We can assume that the incompat-\nibility charges (C5) are localized at the interface z= 0\nand substitute ( @zMs)2)M2\ns\u000e(z)=\u0018exch. For the elastic\nstrains ^uelastthat are created by incompatibility charges\n(C5), we use the Eshelby solution [37], so that\nuelas\nxx(r) =\u0000uelas\nyy(r) =u0\n4\u0019Z\ndr0cos 2'(r0)\njr\u0000r0j;\nuelas\nxy(r) =u0\n4\u0019Z\ndr0sin 2'(r0)\njr\u0000r0j; (C6)\nwhereu0\u0011\u0015meM2\ns=\u0016is the value of the spontaneous\nstrain.\nBy substituting Eqs. (C6) into Eq. (2) we get the ex-\npression for the destressing energy as\nWdestr=\u0014Z\ndrZ\ndr0cos 2 ['(r0)\u0000'(r)]\njr\u0000r0j: (C7)9\nHere,\u0014is a phenomenological coe\u000ecient that generally\ndepends on the elastic properties of both the magnetic\nand nonmagnetic layers and the interface. In a sim-\npli\fed model with isotropic magnetoelastic and elasticinteractions \u0014/\u0016u2\n0=(4\u0019). We calculate the distribu-\ntion of the magnetic variable '(r) from the following set\nof equations, which minimizes the energy of the sample\nWbulk+Wdestr:\n\u0000x2\nDW\u0001'+ sin 4'=2\u0014\nHanMsZ\ndr0sin 2 ['(r0)\u0000'(r)]\njr\u0000r0j+2\u0015meMs\nHan[(uxx\u0000uyy) sin 2'\u00002uxycos 2'];\n\u0001u+1\n1\u00002\u0017r(r\u0001u) = 2u0\u0012\n@x(cos 2') +@y(sin 2')\n@x(sin 2')\u0000@y(cos 2')\u0013\n(C8)\nwherexDW\u0011p\nA=H anMsis the domain wall width, and\nuis the displacement vector. The second and third equa-\ntions correspond to conditions of mechanical equilibrium(zero stresses) inside the NiO layer.\nEquations (C8) are complemented by the boundary\nconditions at the edges:\n\u0000AN\u0001r'+Ksurf\u0002\n(N2\nx\u0000N2\ny) sin 2'+ 2NxNycos 2'\u0003\n= 0;\n\u0000N\u0001ru+ 2u0\u0012\nNxcos 2'+Nysin 2'\nNxsin 2'\u0000Nycos 2'\u0013\n= 0: (C9)\nWhile the strain distribution within the \flm plane can\nbe calculated directly from the elasticity equations (C8),\nwe also introduce an approach based on the associated\nincompatibility component\n\u0011zz\u0011\u0000u0\u0002\n(@2\nx\u0000@2\ny) cos 2'\u00002@x@ysin 2'\u0003\n:(C10)\nThis component is mainly concentrated in the sample\ncorners, where it creates partial disclinations and facil-\nitates the formation of domains. To understand this\ne\u000bect, we consider the immediate vicinity of a corner.\nStrong surface anisotropy aligns the N\u0013 eel vector at the\nedges along Ndirection. Inside the sample the N\u0013 eel vec-\ntor rotates through 90\u000eto \ft the boundary conditions.\nDue to the magnetoelastic mechanism, such a rotation\ncreates a nonzero lattice curvature, which is described\nby the components of the bend-twist tensor [36]\nKzx=\u0000u0Yp\nX2+Y2; K zy=u0Xp\nX2+Y2; (C11)\nwith theXY coordinate frame centered at the corner.\nAccording to Eq. (C10) the associated incompatibility\n\u0011zz\u0019\u00062u0=(X2+Y2). The sign of \u0011zzdepends on the\ndirection of the rotation and is opposite for the neighbor-\ning corners, see Fig.5.\nSuch a distribution of lattice curvature and incompat-\nibility charge can be treated as an elastic defect, knownas a straight disclination line [47], with the Frank vec-\ntor (vector of disclination) \n = 2 \u0019u0pointing along z\ndirection. Similar disclinations of magnetoelastic origin\nthat appear at the junctions of several domain walls in\nferromagnets were described by Kleman et al. [36]. The\nelastic strain created by the disclination according to [36]\nare\nuelas\nxx\u0000uelas\nyy=\u0000u0\n2(1\u0000\u0017)X2\u0000Y2\nX2+Y2;\nuelas\nxy=\u0000u0\n2(1\u0000\u0017)XY\nX2+Y2: (C12)\nThe strain component uelas\nxy, which is maximal along bi-\nsectrixX=Y, sets the preferable direction of the N\u0013 eel\nvector either along or perpendicular to the bisectrix di-\nrection, depending on the sign of the incompatibility\ncharge. It should be mentioned that incompatibility\ncharges in the inner and outer corners have opposite signs\n(due to opposite direction of the rotation), thus favoring\nthe formation of domains of di\u000berent types.\nNumerical simulations of the textures were imple-\nmented by solving Eqs. (C8), (C9) with PDE Tools\nof Matlab2021 with the following parameters: xDW=\n100 nm,u0= 3\u000110\u00005,Han= 0:03 T,Ms= 106A/m,\n\u0016= 35 GPa. The equilibrium structure was calculated by\nthe relaxation from di\u000berent initial states with the loops\nfor self-correction of the destressing energy, see Fig. 6.\nTo obtain a multidomain structure, we always introduced\na domain wall into the system (either 180\u000eor 90\u000e).\n[1] V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono,\nand Y. Tserkovnyak, Rev. Mod. Phys. 90, 015005 (2018).[2] T. Kampfrath, A. Sell, G. Klatt, A. Pashkin, S. M ahrlein,10\nT. Dekorsy, M. Wolf, M. Fiebig, A. Leitenstorfer, and\nR. Huber, Nat. Photonics 5, 31 (2011).\n[3] R. Lebrun, A. Ross, S. A. Bender, A. Qaiumzadeh,\nL. Baldrati, J. Cramer, A. Brataas, R. A. Duine, and\nM. Kl aui, Nature 561, 222 (2018).\n[4] T. Moriyama, K. Oda, T. Ohkochi, M. Kimata, and\nT. Ono, Sci. Rep. 8, 14167 (2018).\n[5] X. Z. Chen, R. Zarzuela, J. Zhang, C. Song, X. F. Zhou,\nG. Y. Shi, F. Li, H. A. Zhou, W. J. Jiang, F. Pan, and\nY. Tserkovnyak, Phys. Rev. Lett. 120, 207204 (2018).\n[6] L. Baldrati, O. Gomonay, A. Ross, M. Filianina, R. Le-\nbrun, R. Ramos, C. Leveille, F. Fuhrmann, T. R. For-\nrest, F. Maccherozzi, S. Valencia, F. Kronast, E. Saitoh,\nJ. Sinova, and M. Kl aui, Phys. Rev. Lett. 123, 177201\n(2019).\n[7] G. P. Zhang, Y. H. Bai, T. Jenkins, and T. F. George, J.\nPhys. Condens. Matter 30, 465801 (2018).\n[8] L. Baldrati, C. Schmitt, O. Gomonay, R. Lebrun,\nR. Ramos, E. Saitoh, J. Sinova, and M. Kl aui, Phys.\nRev. Lett. 125, 077201 (2020).\n[9] H. Meer, F. Schreiber, C. Schmitt, R. Ramos, E. Saitoh,\nO. Gomonay, J. Sinova, L. Baldrati, and M. Kl aui, Nano\nLett. 21, 114 (2021).\n[10] H. V. Gomonay and V. M. Loktev, Phys. Rev. B 75,\n174439 (2007).\n[11] O. Gomonay, S. Kondovych, and V. Loktev, J. Magn.\nMagn. Mater. 354, 125 (2014).\n[12] J. Wu, D. Carlton, J. S. Park, Y. Meng, E. Arenholz,\nA. Doran, A. T. Young, A. Scholl, C. Hwang, H. W.\nZhao, J. Bokor, and Z. Q. Qiu, Nat. Phys. 7, 303 (2011).\n[13] J. Sort, K. S. Buchanan, V. Novosad, A. Ho\u000bmann,\nG. Salazar-Alvarez, A. Bollero, M. D. Bar\u0013 o, B. Dieny,\nand J. Nogu\u0013 es, Phys. Rev. Lett. 97, 067201 (2006).\n[14] S. Czekaj, F. Nolting, L. J. Heyderman, K. Kunze, and\nM. Kr uger, J. Phys. Condens. Matter 19, 386214 (2007).\n[15] E. Folven, Y. Takamura, and J. K. Grepstad, J. Electron\nSpectros. Relat. Phenomena 185, 381 (2012).\n[16] M. S. Lee, P. Lyu, R. V. Chopdekar, A. Scholl, S. T.\nRetterer, and Y. Takamura, J. Appl. Phys. 127, 203901\n(2020).\n[17] E. Folven, A. Scholl, A. Young, S. T. Retterer, J. E.\nBoschker, T. Tybell, Y. Takamura, and J. K. Grepstad,\nPhys. Rev. B 84, 220410(R) (2011).\n[18] E. Folven, Y. Takamura, and J. K. Grepstad, J. Electron\nSpectros. Relat. Phenomena 185, 381 (2012).\n[19] E. Folven, J. Linder, O. V. Gomonay, A. Scholl, A. Do-\nran, A. T. Young, S. T. Retterer, V. K. Malik, T. Ty-\nbell, Y. Takamura, and J. K. Grepstad, Phys. Rev. B 92,\n094421 (2015).\n[20] G. R. Hoogeboom, A. Aqeel, T. Kuschel, T. T. M. Pal-\nstra, and B. J. van Wees, Appl. Phys. Lett. 111, 052409\n(2017).\n[21] L. Baldrati, A. Ross, T. Niizeki, C. Schneider, R. Ramos,\nJ. Cramer, O. Gomonay, M. Filianina, T. Savchenko,\nD. Heinze, A. Kleibert, E. Saitoh, J. Sinova, and\nM. Kl aui, Physical Review B 98, 024422 (2018).\n[22] J. Fischer, O. Gomonay, R. Schlitz, K. Ganzhorn, N. Vli-\netstra, M. Althammer, H. Huebl, M. Opel, R. Gross,\nS. T. B. Goennenwein, and S. Gepr ags, Physical Review\nB97, 014417 (2018).\n[23] T. Moriyama, K. Hayashi, K. Yamada, M. Shima,\nY. Ohya, and T. Ono, Phys. Rev. Mater. 4, 074402(2020).\n[24] W. L. Roth, J. Appl. Phys. 31, 2000 (1960).\n[25] E. Aytan, B. Debnath, F. Kargar, Y. Barlas, M. M. Lac-\nerda, J. X. Li, R. K. Lake, J. Shi, and A. A. Balandin,\nAppl. Phys. Lett. 111, 252402 (2017).\n[26] A. Kozio l-Rachwa l, M. \u0013Sl\u0018 ezak, M. Zaj\u0018 ac, P. Dr\u0013 o_ zd_ z,\nW. Janus, M. Szpytma, H. Nayyef, and T. \u0013Sl\u0018 ezak, APL\nMater. 8, 061107 (2020).\n[27] S. Altieri, M. Finazzi, H. H. Hsieh, H. J. Lin, C. T. Chen,\nT. Hibma, S. Valeri, and G. A. Sawatzky, Phys. Rev.\nLett. 91, 137201 (2003).\n[28] D. Alders, L. H. Tjeng, F. C. Voogt, T. Hibma, G. A.\nSawatzky, C. T. Chen, J. Vogel, M. Sacchi, and S. Ia-\ncobucci, Phys. Rev. B 57, 11623 (1998).\n[29] A. Barra, A. Ross, O. Gomonay, L. Baldrati, A. Chavez,\nR. Lebrun, J. D. Schneider, P. Shirazi, Q. Wang,\nJ. Sinova, G. P. Carman, and M. Kl aui, Appl. Phys. Lett.\n118, 172408 (2021).\n[30] H. Qiu, L. Zhou, C. Zhang, J. Wu, Y. Tian, S. Cheng,\nS. Mi, H. Zhao, Q. Zhang, D. Wu, B. Jin, J. Chen, and\nP. Wu, Nature Physics 17, 388 (2021).\n[31] J. St ohr, A. Scholl, T. J. Regan, S. Anders, J. L uning,\nM. R. Scheinfein, H. A. Padmore, and R. L. White, Phys.\nRev. Lett. 83, 1862 (1999).\n[32] F. Kronast and S. Valencia Molina, J. large-scale Res.\nFacil. JLSRF 2, A90 (2016).\n[33] Y. S. Kiyotaka Nakahigashi, Nobuo Fukuoka, J. Phys.\nSoc. Jpn. 38, 1634 (1975).\n[34] H. Gomonay and V. M. Loktev, J. Phys. Condens. Matter\n14, 3959 (2002).\n[35] C. Teodosiu, Elastic Models of Crystal Defects (Springer\nBerlin Heidelberg, Berlin, Heidelberg, 1982).\n[36] M. Kl\u0013 eman and M. Schlenker, Journal of Applied Physics\n43, 3184 (1972).\n[37] J. D. Eshelby, Solid State Physics - Advances in Research\nand Applications 3, 79 (1956).\n[38] C. Schmitt, L. Baldrati, L. Sanchez-Tejerina,\nF. Schreiber, A. Ross, M. Filianina, S. Ding,\nF. Fuhrmann, R. Ramos, F. Maccherozzi, D. Backes,\nM.-A. Mawass, F. Kronast, S. Valencia, E. Saitoh,\nG. Finocchio, and M. Kl aui, Phys. Rev. Appl. 15,\n034047 (2021).\n[39] C. Kittel, Rev. Mod. Phys. 21, 541 (1949).\n[40] R. B. John A. Simmons, R. deWit, National Bureau of\nStandards , Vol. 1 (1970) p. 715.\n[41] K.-E. Kim, S. Jeong, K. Chu, J. H. Lee, G.-Y. Kim,\nF. Xue, T. Y. Koo, L.-Q. Chen, S.-Y. Choi, R. Ramesh,\nand C.-H. Yang, Nature Communications 9, 403 (2018).\n[42] E. V. Gomonay and V. M. Lotkev, Phys. Solid State 47,\n1755 (2005).\n[43] R. White, R. Newt, and R. Pease, IEEE Trans. Magn.\n33, 990 (1997).\n[44] W. L. Roth and G. A. Slack, J. Appl. Phys. 31, S352\n(1960).\n[45] G. A. Slack, J. Appl. Phys. 31, 1571 (1960).\n[46] O. Gomonay and D. Bossini, Journal of Physics D: Ap-\nplied Physics 54, 374004 (2021).\n[47] R. DeWit, Journal of Research of the National Bureau of\nStandards Section A: Physics and Chemistry 77A, 607\n(1973)."    },    {        "title": "2205.07549v1.Magnetic_neutron_scattering_from_spherical_nanoparticles_with_Neel_surface_anisotropy__Analytical_treatment.pdf",        "content": "Magnetic neutron scattering from spherical nanoparticles with\nNéel surface anisotropy: Analytical treatment\nMichael P. Adams,1,\u0003Andreas Michels,1,yand Hamid Kachkachi2,z\n1Department of Physics and Materials Science,\nUniversity of Luxembourg, 162A avenue de la Faiencerie,\nL-1511 Luxembourg, Grand Duchy of Luxembourg\n2Université de Perpignan via Domitia,\nLaboratoire PROMES CNRS UPR8521,\nRambla de la Thermodynamique, Tecnosud, F-66100 Perpignan, France\n(Dated: May 17, 2022)\n1arXiv:2205.07549v1  [cond-mat.mes-hall]  16 May 2022Abstract\nThemagnetizationprofileandtherelatedmagneticsmall-angleneutronscatteringcrosssectionof\nasinglesphericalnanoparticlewithNéelsurfaceanisotropyisanalyticallyinvestigated. Weemploya\nHamiltonianthatcomprisestheisotropicexchangeinteraction, anexternalmagneticfield, auniaxial\nmagnetocrystalline anisotropy in the core of the particle, and the Néel anisotropy at the surface.\nUsing a perturbation approach, the determination of the magnetization profile can be reduced to a\nHelmholtz equation with Neumann boundary condition, whose solution is represented by an infinite\nseries in terms of spherical harmonics and spherical Bessel functions. From the resulting infinite\nseries expansion, we analytically calculate the Fourier transform, which is algebraically related to\nthe magnetic small-angle neutron scattering cross section. The approximate analytical solution is\ncompared to the numerical solution using the Landau-Lifshitz equation, which accounts for the full\nnonlinearity of the problem.\nI. INTRODUCTION\nMagnetic small-angle neutron scattering (SANS) is a powerful technique for investigating\nspin structures on the mesoscopic length scale ( \u00181\u0000100 nm) and inside the volume of\nmagnetic materials [1, 2]. Recent SANS studies on magnetic nanoparticles, in particular\nemploying spin-polarized neutrons, unanimously demonstrate that their spin textures are\nhighly complex and exhibit a variety of nonuniform, canted, or core-shell-type configurations\n(see, e.g.Refs.[3–15]andreferencestherein). ThemagneticSANSdataanalysislargelyrelies\non structural form-factor-models for the cross section, borrowed from nuclear SANS, which\ndo not properly account for the existing spin inhomogeneity inside magnetic nanoparticles\nor nanomagnets (NM).\nProgress in magnetic SANS theory [16–25] strongly suggests that for the analysis of ex-\nperimental magnetic SANS data the spatial nanometer scale variation of the orientation and\nmagnitudeofthemagnetizationvectorfieldmustbetakenintoaccount, andthatmacrospin-\nbased models—assuming a uniform magnetization—are not adequate. The starting point\nfor a proper analysis of the scattering problem is a micromagnetic continuum expression for\n\u0003Electronic address: michael.adams@uni.lu\nyElectronic address: andreas.michels@uni.lu\nzElectronic address: hamid.kachkachi@univ-perp.fr\n2themagneticenergyofthesystem. Inthestaticcase, thisthenleadstotheso-calledBrown’s\nequations, a set of nonlinear partial differential equations for the magnetization along with\ncomplexboundaryconditionsonthesurfaceofthemagnet. FromtheseequationstheFourier\nimage and the magnetic SANS cross section may be obtained.\nIn this paper, we present an analytical treatment of the magnetic SANS cross section of\na spherical NM with Néel’s surface anisotropy [26]. The manuscript is organized as follows:\nIn Section II, we calculate the real-space spin structure of the spherical NM using classical\nmicromagnetic theory within the second-order perturbation approach. In Section III, we\ncompute the three-dimensional Fourier transform of the real-space spin structure, which di-\nrectly yields the magnetic neutron scattering cross section and the pair-distance distribution\nfunction. The analytical results are benchmarked by comparing them to numerical finite-\ndifference simulations using the Landau-Lifshitz equation of motion. Finally, Section IV\nsummarizes the main findings of this study.\nWe also make reference to our numerical study [27], where in contrast to the present\nanalytical work the full nonlinearity of the problem is considered.\nII. MICROMAGNETIC THEORY\nIn the static micromagnetic approach [28], the magnetic configuration of a system is\ndescribed by the continuous magnetization vector field M(r), which is subject to a con-\nstant magnitude kM(r)k=M0. The saturation magnetization M0is only a function of\ntemperature. The normalized magnetization vector field is then defined as\nm(r) =M(r)=M0= [mx(r);my(r);mz(r)]: (1)\nOur Hamiltonian for the NM includes the isotropic exchange interaction, the Zeeman energy,\na uniaxial magnetic anisotropy for spins in the core and Néel’s surface anisotropy for those\n3on the surface. In the continuum approach, it reads:\nH=\u0000AX\n\u000b2fx;y;zgZ\nVm\u000b\u0001m\u000bd3r\n\u0000M0B0Z\nVmd3r\u0000KcZ\nV(m\u0001eA)2d3r\n+AX\n\u000b2fx;y;zgI\n@Vm\u000brm\u000b\u0001nd2r\n\u0000Ks\n2X\n\u000b2fx;y;zgI\n@Vjn\u000bjm2\n\u000bd2r (2)\nwhereAis the exchange-stiffness constant, ris the Del operator, \u0001is the Laplace operator,\nB0=\u00160H0is a constant applied magnetic field, Kc>0denotes the uniaxial core anisotropy\nconstant, eAis a unit vector specifying the arbitrary core anisotropy axis, Ks>0is the\nNéel surface anisotropy constant [26] with\nn= [sin\u0012cos\u001e;sin\u0012sin\u001e;cos\u0012] (3)\nbeing the surface normal tothe boundaryof the NM [29, 30]. In(2), thetwo surface integrals\ntake into account the boundary conditions for the magnetization on the surface ( @V) of the\nNM of volume V, which result from the exchange interaction and the Néel term.\nFor small deviations from the homogeneous magnetization state, a perturbation approach\nis applicable. Let m0be the principal unit vector (average direction) associated with m(r)\nand let the vector function  (r)?m0describe the spin misalignment. One can then write:\nm(r) =m0p\n1\u0000k (r)k2+ (r);km(r)k= 1: (4)\nAssuming that  x; y; z\u001c1, the following second-order Maclaurin expansion in  is used\nto find an approximate closed-form solution for m(r):\nm(r)\u0018=m0+ (r)\u00001\n2k (r)k2m0; (5)\nwhere m0is taken as a known constant vector in subsequent calculations. By choosing the\northonormal vector base [31]\n8\n>>>>><\n>>>>>:g0=m0;\ng1=m0\u0002eA\nkm0\u0002eAk;\ng2=(m0\u0001eA)\u0001m0\u0000eA\nk(m0\u0001eA)\u0001m0\u0000eAk;(6)\n4the parametrization\n (r) = 1(r)g1+ 2(r)g2; (7)\nand by introducing the dimensionless coordinates \u0018=r=R(with\u0018=k\u0018k=r=R), where r\nis the position vector,\nr= [rsin\u0012cos\u001e;rsin\u0012sin\u001e;rcos\u0012]; (8)\nandRdenotes the radius of the NM, the minimization of the Hamiltonian (2) leads to the\nwell-known Helmholtz equation with Neumann boundary conditions on the unit sphere [29,\n30]:\n[\u0001\u0018\u0000\u00142\n\f] \f= 0; \f2f1;2g (9)\nd \f\nd\u0018\f\f\f\f\n\u0018=1=X\n\u000b2fx;y;zg\u001f\f\n\u000bjn\u000bj; (10)\nwhere the constants are defined as:\n\u00142\n1=m0\u0001b0+ 2kc(m0\u0001eA)2; (11)\n\u00142\n2=m0\u0001b0+ 2kc\u0002\n2(m0\u0001eA)2\u00001\u0003\n; (12)\n\u001f\f\n\u000b=ks(m0\u0001e\u000b)(g\f\u0001e\u000b); (13)\nwith the dimensionless quantities\nkc=R2Kc\nA; k s=RKs\nA; b0=R2M0\nAB0: (14)\nThee\u000b(with\u000b=x;y;z) in (13) denote the unit vectors of the Cartesian laboratory coordi-\nnateframe(inwhich nandraredefined). Weemphasizethatthereareonlytwoindependent\ndifferential equations for  , which is a consequence of the constraint km(r)k= 1.\nIn our graphical representations, we will frequently use the following values: kc= 0:1and\nks= 3:0, which (using R= 5 nmandA= 10\u000011J=m) correspond to Kc= 40 kJ=m3and\nKs= 6 mJ=m2[32, 33]. For M0= 1:7\u0002106A=m, the relation between b0(dimensionless)\nand the external field is B0= 4=17b0\u00021T.\nThe fundamental solution of the homogeneous Helmholtz equation (9) is well known [34,\n35]. Its nonsingular part can be expressed in spherical coordinates as an infinite series\n5in terms of spherical harmonics Y`m(\u0012;\u001e)and spherical Bessel functions of the first kind\njn(i\u0014\f\u0018),\n \f=1X\n`=0`X\nm=\u0000`c\f\n`mj`(i\u0014\f\u0018)Y`m(\u0012;\u001e): (15)\nThe imaginary number ‘ i’ in the argument of the spherical Bessel function is due to the\nnegativesignintheHelmholtzequation(9). Theexpansioncoefficients c\f\n`mareobtainedfrom\nthe Neumann boundary condition (10) using the method of least squares (see Appendix A).\nFrom there it is seen that the zero-order term with `= 0vanishes, which physically makes\nsense, since the spin misalignment in our model is caused by the Néel surface anisotropy\nand, thus, due to symmetry reasons there is no misalignment at the center of the NM, i.e.\n \f(\u0018= 0;\u0012;\u001e)\u00110. By contrast, the largest spin misalignment is found at the boundary of\nthe NM, i.e.\u0018= 1. Further, we find that the coefficients c\f\n`mvanish in the case of odd `and\nm, they are real-valued, and even with respect to the index m,i.e.c\f\n`m=c\f\n`;\u0000m. Taking these\nproperties into account, one can conveniently express the solution in terms of the associated\nLegendre polynomials Pm\n`(cos\u0012)with`= 2\u0017andm= 2\u0016[note that we use the convention\nthatY`m(\u0012;\u001e) =N`mPm\n`(cos\u0012)eim\u001e(p. 378 (14.30.1) in [36])]:\n \f=1X\n\u0017=1\u0017X\n\u0016=0a\f\n\u0017\u0016\u0007\u0017(\u0014\f\u0018)P2\u0016\n2\u0017(cos\u0012) cos(2\u0016\u001e); (16)\nwhere we define (compare p. 624–626 in [34])\n\u0007\u0017(\u001c) =j2\u0017(i\u001c) =p\u0019\n21X\ns=0(\u00001)\u0017(\u001c=2)2(s+\u0017)\ns!\u0000(2\u0017+s+ 3=2); (17)\nand the expansion coefficients are given by\na\f\n\u0017\u0016=2ksN2\u0017;2\u0016\n1 +\u000e\u0016;0g\f\u0001diag\u0002\nIx\n2\u0017;2\u0016;Iy\n2\u0017;2\u0016;Iz\n2\u0017;2\u0016\u0003\n\u0001m0\n\u0014\f\u00070\n\u0017(\u0014\f)(18)\nwith\nI\u000b\n`m=Z2\u0019\n0Z\u0019\n0Y\u0003\n`;m(\u0012;\u001e)jn\u000bjsin\u0012d\u0012d\u001e (19)\nand\nN`m=s\n2`+ 1\n4\u0019(`\u0000m)!\n(`+m)!: (20)\n6In (18),\u000e\u0016;0is the Kronecker delta function, diag[:::]denotes a 3\u00023diagonal matrix, and\n\u00070\n\u0017(\u001c)is the first-order derivative of (17) with respect to \u001c. For some small values of `and\nm, the exact solutions of the integrals I\u000b\n`mare listed in Appendix B.\nFrom (18) it is seen that the functions  \fdepend linearly on ks, such that for ks= 0the\nmagnetization of the NM is homogeneous (as expected). Since we assume that  x; y; z\u001c\n1, it is clear that the validity of our solution is restricted to a finite range 0\u0014ks\u0014ks;max.\nTaking only the terms with \u0017= 1into account (corresponding to `= 2), the remaining\n(second-order) expression reads:\n \f(\u0018;\u0012;\u001e )\u0019\u000015ks\n32\u00071(\u0014\f\u0018)\n\u0014\f\u00070\n1(\u0014\f)g\f\u0001V(\u0012;\u001e)\u0001m0; (21)\nwhere\nV(\u0012;\u001e) = diag2\n6664cos2\u0012\u00001=3\u0000sin2\u0012cos(2\u001e)\ncos2\u0012\u00001=3 + sin2\u0012cos(2\u001e)\n\u00002(cos2\u0012\u00001=3)3\n7775: (22)\nA reasonable approximation for small \u0014\fin (21) is obtained by taking into account the\nfirst two terms in the infinite series (17) for \u0007\u0017(\u001c). This results in the following expression\n[compare (21)]:\n\u00071(\u0014\f\u0018)\n\u0014\f\u00070\n1(\u0014\f)\u00191\n4\u00142\n\f\u00184+ 14\u00182\n\u00142\n\f+ 7: (23)\nIn the limit \u0014\f!0, this expression reduces to a quadratic function in \u0018\nlim\n\u0014\f!0\u001a\u00071(\u0014\f\u0018)\n\u0014\f\u00070\n1(\u0014\f)\u001b\n=\u00182\n2: (24)\nThe case of an infinite applied magnetic field B0, or of a strong uniaxial core anisotropy\n[compare (11) and (12)], corresponds to the limit\nlim\n\u0014\f!1\u001a\u00071(\u0014\f\u0018)\n\u0014\f\u00070\n1(\u0014\f)\u001b\n= 0; (25)\nwhich recovers the expected result of zero spin misalignment. Note that the limit \u0014\f!1\nis only obtained using all terms of the infinite series (17).\nOf particular interest is the behavior of  \fas a function of the radius Rof the NM.\nInspecting the Hamiltonian (2), it becomes clear that the surface anisotropy energy scales\nasR2, while the uniaxial core anisotropy energy scales as R3. Since the core and the surface\n7FIG. 1. Normalized effective energy potential of the Néel surface anisotropy as a function of\nthe Cartesian components of the average magnetization vector m0= [sin\fcos\u000b;sin\fsin\u000b;cos\f],\ncomputed via numerical integration of the surface contribution in (2) and by using the second-order\napproximation (21). Parameters are: eA= [0;0;1],b0=0,kc= 0:1, andks= 3:0. The minima of\ntheNéelsurfacecontributionareinthiscasealongthecubicspacediagonals m0= [\u00061;\u00061;\u00061]=p\n3,\nwhile the maxima correspond to the Cartesian axes \u0006ex;\u0006ey;\u0006ez. The effective energy potential\nhascubicsymmetryandisapproximatelyproportionaltoafunctionofthetype 'm4\n0;x+m4\n0;y+m4\n0;z\n(see also [29]).\nanisotropy act in opposite ways (trying to make the spin structure more homogeneous,\nrespectively, more inhomogeneous), we see that an increasing radius Rcorresponds to a\ndecreasing  \f. This behavior reflects the NM’s surface-area-to-volume ratio. With (21) it\nis not possible to make any prediction in this regard, because until this point we did not\ninclude the principal unit vector m0into the minimization of the Hamiltonian. Generally,\nm0is a function of ks,kc,b0, and eA.\nIn the special case when the uniaxial anisotropy axis and the applied magnetic field are\nboth directed parallel to the zaxis ( eA= [0;0;1]andb0= [0;0;b0]), the principal unit\n8FIG. 2. Comparison between the numerical solution using the Landau-Lifshitz equation (upper\nrow) and the second-order analytical solution (21) for k (\u0018)k=p\n 2\n1(\u0018) + 2\n2(\u0018)(lower row). ( a)\nand ( e) showk kon the boundary surface ( \u0018= 1), while ( b)\u0000(d) and ( f)\u0000(h) display selected\nplanar cuts in (\u0018x;\u0018y;\u0018z)space. The following parameters are used: eA=ez,b0= [0:4;0;0:4]\n(B0\u0018=133 mT),kc= 0:1,ks= 3:0, and m0= [sin\fcos\u000b;sin\fsin\u000b;cos\f]where\u000b= 0\u000eand\n\f= 40\u000e.\nmagnetization vector may be written as:\nm0= [1=p\n2 sin\f;1=p\n2 sin\f;cos\f]; (26)\nwhere\f2[0;arccos(1=p\n3)]. This choice is justified by the effective cubic symmetry of\nthe Néel anisotropy as shown in Fig. 1. This result was already predicted by Garanin and\nKachkachi [29]. The solutions for  1;2(\u0018;\u0012;\u001e )[using the particular m0(26)] then read:\n 1\u001915ks\n32\u00071(\u00141\u0018)\n\u00141\u00070\n1(\u00141)sin2\u0012cos(2\u001e) sin\f; (27)\n 2\u001915ks\n32\u00071(\u00142\u0018)\n\u00142\u00070\n1(\u00142)(1\u00003 cos2\u0012) sin\fcos\f: (28)\nInFig.2, theanalyticalsolution(21)(lowerrow)iscomparedtothenumericalsolutionbased\non the Landau-Lifshitz equation _m=\u0000\rm\u0002Be\u000b\u0000\u000bm\u0002(m\u0002Be\u000b)(upper row) [37], where\n\ris the gyromagnetic ratio, \u000bis the damping constant, and the dot denotes the first-order\n9FIG. 3. Real-space spin structure in the \u0018x-\u0018zplane computed using (4) and (21). Parameters are\nthe same as in Fig. 2. The external field B0\u0018=133 mTis applied in the \u0018x-\u0018zplane and inclined by\nan angle of \f= 40\u000erelative to the \u0018zaxis (compare to [29]).\ntime derivative (see our numerical study [27] for further details). Shown is the vector norm\nof the (\u0018)function scaled to its maximum value. From Fig. 2 it is seen that our analytical\napproximation is in qualitative agreement with the results from the numerical simulation.\nThe corresponding real-space spin structure m(\u0018)is displayed in Fig. 3, where the surface\nspin disorder becomes clearly visible.\nIt is also instructive to compare our solution (21) with that obtained using the Green\nfunction approach [29, 30]. In particular, for \u0018located close to the surface, where the\nmaximum spin misalignment with respect to m0occurs, the Green function method yields\nthe following approximate expression:\n \f(\u0018)\u0019\u000015ks\n32\u0014\n1\u0000\u00142\n\f\n14\u0015\n\u00182g\f\u0001V(\u0018)\u0001m0; (29)\nV(\u0018) =\u0000diag2\n6664\u00182\nx=\u00182\u00001=3\n\u00182\ny=\u00182\u00001=3\n\u00182\nz=\u00182\u00001=33\n7775: (30)\nThis expression is also found when (21) is expanded in \u0014\fat the surface of the NM ( \u0018= 1).\nWhile the infinite series approach using spherical harmonics and spherical Bessel functions\n10yields an exact solution of the Helmholtz equation, the Green’s function approach provides\nan approximate explicit expression of  \fin terms of the coefficients \u0014\f. Indeed, as was\nshown in [30], in the presence of core anisotropy, the Green function as the kernel of the\nHelmholtz equation is only obtained as a perturbative series in \u0014\f. As such, (29) is restricted\nto small values of \u0014\f,i.e.assuming that the core anisotropy and applied magnetic field are\nmuch smaller than the exchange coupling. This is manifest in (29) by the presence of the\nfactor 1\u0000\u00142\n\f=14which implies that the contribution of spin misalignment may diverge for\ntoo large\u0014\f(i.e.for a strong field and/or large core anisotropy).\nIII. MAGNETIC SANS CROSS SECTION\nThe quantity of interest in experimental SANS studies is the elastic magnetic differential\nscattering cross section d\u0006M=d\n, which is usually recorded on a two-dimensional position-\nsensitive detector. For the most commonly used scattering geometry in magnetic SANS\nexperiments, where the applied magnetic field B0kezis perpendicular to the wave vector\nk0kexof the incident neutrons (see Fig. 4), d\u0006M=d\n(for unpolarized neutrons) can be\nwritten as [1]:\nd\u0006M\nd\n(q) =8\u00193\nVb2\nH\u0010\njfMxj2+jfMyj2cos2\u0012q\n+jfMzj2sin2\u0012q\u0000(fMyfM\u0003\nz+fM\u0003\nyfMz) sin\u0012qcos\u0012q\u0011\n; (31)\nwhereVis the scattering volume and bH= 2:91\u0002108A\u00001m\u00001the magnetic scattering\nlength in the small-angle regime (the atomic magnetic form factor is approximated by 1,\nsince we are dealing with forward scattering); fM(q) = [fMx(q);fMy(q);fMz(q)]represents\nthe magnetization vector field M(r)in Fourier space, \u0012qdenotes the angle between qand\nB0, and the asterisk ‘ \u0003’ stands for the complex conjugate. Note that in the perpendicular\nscattering geometry, the Fourier components are evaluated in the plane qx= 0.\nThe Fourier transform of the three-dimensional magnetization vector field (with a tilde\nabove the symbol) is defined as follows\nfM(q) =1\n(2\u0019)3=2Z\nVM(r) exp (\u0000iq\u0001r)d3r; (32)\nM(r) =1\n(2\u0019)3=2Z\nVfM(q) exp (i q\u0001r)d3q: (33)\n11FIG. 4. Sketch of the perpendicular scattering geometry ( B0?k0). The scattering vector qcorre-\nsponds to the difference between the wave vectors of the incident ( k0) and scattered ( k1) neutrons.\nThe angle\u0012qspecifies the orientation of qon the detector. In the small-angle approximation, the\ncomponent of qalong k0is neglected.\nFor subsequent calculations, we introduce the following dimensionless quantities\n\u001d=qR; fM=(2\u0019)3=2\n4\u0019R3M0fM; (34)\nand we express the dimensionless scattering vector in spherical coordinates as\n\u001d= [\u001dsin\u0012qcos\u001eq;\u001dsin\u0012qsin\u001eq;\u001dcos\u0012q]: (35)\nNext, in (32) we use the following first-order approximation for the real-space magnetization\nvector m(\u0018)[see (5) and (7)]\nm(\u0018) =m0+2X\n\f=1g\f \f(\u0018): (36)\nAs shown in Appendix C, the final expression for the Fourier transform of the magnetization\nis then given by:\nfM(\u001d) =j1(\u001d)\n\u001dm0\n+2X\n\f=1g\f1X\n\u0017=1\u0017X\n\u0016=0(\u00001)\u0017a\f\n\u0017\u0016\u001a\f\n\u0017(\u001d)P2\u0016\n2\u0017(cos\u0012q) cos(2\u0016\u001eq); (37)\n12where\n\u001a\f\n\u0017(\u001d) =\u0000\u001dj2\u0017\u00001(\u001d)\u0007\u0017(\u0014\f)\u0000\u0014\f@\u0017(\u0014\f)j2\u0017(\u001d)\n\u001d2+\u00142\n\f; (38)\n@\u0017(\u001c) =p\u0019\n21X\ns=0(\u00001)\u0017(\u001c=2)2(s+\u0017)\u00001\ns!\u0000(2\u0017+s+ 1=2); (39)\nand\u0007\u0017(\u0014\f)is given by (17). The zero-order term /j1(\u001d)=\u001din (37) represents the form\nfactor of a homogeneously magnetized sphere [2]. In the limiting case of an infinite applied\nmagnetic field, which is equivalent to the limit \u0014\f!1, the additional terms [second line\nin (37)] vanish [compare (25)] and the spherical form factor remains. On the other hand, if\nks= 0, the additional terms also vanish because from the physical point of view, the Néel\nsurface anisotropy cancels and from (18) we know that the coefficients a\f\n\u0017\u0016are linear in ks.\nTaking only the terms with \u0017= 1into account and setting \u001eq=\u0019=2(\u001dx= 0), corresponding\nto the scattering geometry where the applied magnetic field B0kezis perpendicular to the\nwave vector k0kexof the incident neutrons (Fig. 4), the expression for fM(\u001d)can be\nwritten as [compare with (21)]:\nfM(\u001d)\u0019j1(\u001d)\n\u001dm0\n\u000015ks\n322X\n\f=1R\f(\u001d) (g\f\u0001V(\u0012q;\u0019=2)\u0001m0)g\f; (40)\nwhere the radial function is\nR\f(\u001d) =\u001dj1(\u001d)\n\u001d2+\u00142\n\f\u00071(\u0014\f)\n\u0014\f\u00070\n1(\u0014\f)\u0000j2(\u001d)\n\u001d2+\u00142\n\f@1(\u0014\f)\n\u00070\n1(\u0014\f): (41)\nR\f(\u001d)can be approximated for small \u0014\fand when only terms up to s= 1in the infinite\nseries (17) and (39) are kept:\nR\f(\u001d) =1\n4\u001dj1(\u001d)\n\u001d2+\u00142\n\f\u00142\n\f+ 14\n\u00142\n\f+ 7\u00007\n4j2(\u001d)\n\u001d2+\u00142\n\f\u00142\n\f+ 10\n\u00142\n\f+ 7: (42)\nFor small\u001d-values, one finds the following limit:\nlim\n\u001d!0R\f(\u001d) = 0; (43)\nwhich is consistent with\nZ\nV (r)d3r=0: (44)\n13This can be seen by inspecting the definition of the Fourier transform in (32). Note that\nforq!0the Fourier transform is proportional to the average of the magnetization vector\nfieldMand the maximum of this average is given by the homogeneous magnetization state.\nUsing this result, the \u001d!0limit for the first-order approximation in  of the Fourier\ntransform of the magnetization yields:\nlim\n\u001d!0fM(\u001d;\u0012q;\u001eq) =1\n3m0: (45)\nBeyond the linear approximation in  , a nonvanishing term appears in fMin the limit\n\u001d!0, which reduces the Fourier components relative to the homogeneous magnetization\nstate. In the second order in  , the result is [compare (5)]:\nlim\n\u001d!0fM(\u001d;\u0012q;\u001eq) =\u00141\n3\u00001\n2Z\nVk (\u0018)k2d3\u0018\u0015\n\u0001m0 (46)\nUsing (34) and\nd\u0006M\nd\n=16\u00192R6M2\n0b2\nH\nVSM; (47)\nthe dimensionless two-dimensional magnetic SANS cross section SM(\u001d;\u0012q)can be straight-\nforwardly obtained as [compare (31)]\nSM(\u001d;\u0012q) =jfMxj2+jfMyj2cos2\u0012q+jfMzj2sin2\u0012q\n\u0000(fMyfM\u0003\nz+fM\u0003\nyfMz) sin\u0012qcos\u0012q: (48)\nIn limitks!0, the resulting cross section from (37) is\nlim\nks!0SM(\u001d;\u0012q) =\u0012j1(\u001d)\n\u001d\u00132\u0000\nm2\n0;x+m2\n0;ycos2\u0012q\n+m2\n0;zsin2\u0012q\u00002m0;ym0;zsin\u0012qcos\u0012q): (49)\nThe relation (49) nicely demonstrates that, depending on the orientation of the uniformly\nmagnetized particle, different angular anisotropies become visible on the detector: For m0k\nex(i.e.m0y=m0z= 0), the scattering pattern is isotropic, while it exhibits a cos2\u0012q\n(sin2\u0012q) type shape when m0key(m0kez).\nFig.5showsSM(\u001d;\u0012q)alongwiththecontributionoftheindividualFouriercomponentsto\n(48). TheupperrowinFig.5presentstheresulttakingintoaccountonlythezero-orderterm\n(j1(\u001d)=\u001dm0) from (40), while in the lower row the second-order term ( \u0017= 1) is additionally\n14FIG. 5. Results for the two-dimensional Fourier components jfMxj2,jfMyj2,jfMzj2,CT =\n\u0000(fMyfM\u0003\nz+fM\u0003\nyfMz), and for the total magnetic SANS cross section SM(\u001d;\u0012q)[(48)] using ex-\npression (40). The upper row shows the results taking into account only the zero-order term in\n(40), which corresponds to the case of a homogeneously magnetized particle. The lower row displays\nthe results when the second-order term ( \u0017= 1) in (40) is taken into account. The parameters are:\neA=ez,b0= 0:1ez[B0\u0018=24 mT],kc= 0:1,ks= 3, and m0= [sin\fcos\u000b;sin\fsin\u000b;cos\f]. Since\nthe Néel surface anisotropy has effectively a cubic symmetry (see Fig. 1), we average SMover the\nangles\u000b= (45\u000e;135\u000e;225\u000e;315\u000e)and\f= 20\u000e. Logarithmic color scale is used.\nincluded. Since the zero-order term represents the case of a homogeneously magnetized NM,\nthis comparison provides useful insights about the impact of the Néel surface anisotropy\non the magnetic SANS cross section. In the case of a uniformly magnetized NM (upper\nrow) the Fourier components jfMxj2,jfMyj2, andjfMzj2are isotropic (rotational symmetry),\nwhile including the second-order terms (lower row) leads to an anisotropic behavior of the\ntransverse components jfMxj2andjfMyj2. The cross term ( CT) averages to zero for both\nsituations, and the dominating contribution to the magnetic SANS cross section is (for the\nparameters chosen in Fig. 5) given by the jfMzj2component. Therefore, it may be concluded\nthat the impact of the Néel surface anisotropy on SM(\u001d;\u0012q)is relatively small. By comparing\ntheSM(\u001d;\u0012q)from the upper and the lower row, it is seen that by including the Néel surface\nanisotropy the circular symmetry of the zeros of SM(deep blue colors) is broken. This\nfeature becomes more clearly visible by analyzing the azimuthal average of SM(\u001d;\u0012q), which\n15FIG. 6. Results for ( a) the azimuthally-averaged SANS cross section I(\u001d), (b) the pair-distance\ndistribution function P(\u0018), (c) the correlation function C(\u0018), and ( d) the quantities \u000fI(ks)(56)\nand\u000fP(ks)(57) (B0\u0018=24 mT). For the homogeneous case (blue curves in ( a) - (c)), the surface\nanisotropy is set to ks= 0, and for the inhomogeneous case (red curves ( a) - (c)), we use the\nsame parameters as in Fig. 5. The functional dependence of I(\u001d),P(\u0018), andC(\u0018)for the uniformly\nmagnetized particle all correspond to the analytically well-known cases, i.e.(51), (54), and (55).\nis readily computed as:\nI(\u001d) =1\n2\u0019Z2\u0019\n0SM(\u001d;\u0012q)d\u0012q: (50)\nIn the limit ks!0, the azimuthal average corresponding to (49) is:\nlim\nks!0I(\u001d) =\u0012j1(\u001d)\n\u001d\u00132km0k2+ (ex\u0001m0)2\n2: (51)\n16Moreover, we have also calculated the pair-distance distribution function,\nP(\u0018) =\u00182Z1\n0I(\u001d)j0(\u001d\u0018)\u001d2d\u001d; (52)\nand the correlation function\nC(\u0018) =P(\u0018)=\u00182: (53)\nInthelimit ks!0, thepair-distancedistributionandthecorrelationfunctioncorresponding\nto (51) are:\nlim\nks!0P(\u0018) =\u0019\u00182\n6\u0012\n1\u00003\u0018\n4+\u00183\n16\u0013km0k2+ (ex\u0001m0)2\n2; (54)\nlim\nks!0C(\u0018) =\u0019\n6\u0012\n1\u00003\u0018\n4+\u00183\n16\u0013km0k2+ (ex\u0001m0)2\n2: (55)\nThese functions are graphically displayed in Fig. 6. Due to the surface-anisotropy induced\nspin disorder, the form-factor maxima of I(\u001d)[Fig. 6( a)] are shifted to larger qvalues ( i.e.\nsmaller structures). Moreover, as already observed in numerical micromagnetic continuum\nsimulations [38, 39], the oscillations are damped for the case of surface spin disorder, which\nmimics the effect of a particle-size distribution or of instrumental resolution. In agreement\nwith this observation is the finding that the maximum of the P(\u0018)function [Fig. 6( b)]\nappears at smaller distances \u0018as compared to the homogeneous case. Likewise, due to spin\ndisorder, theC(\u0018)function [Fig. 6( c)] exhibits a larger amplitude [18].\nTo analyze the role of the surface anisotropy more quantitatively, we have computed the\nfollowing quantities, which describe the deviation of the one-dimensional SANS cross section\nand of the pair-distance distribution function from the homogeneous particle case:\n\u000fI(ks) =Z1\n0jI(ks= 0;\u001d)\u0000I(ks;\u001d)jd\u001d\nZ1\n0jI(ks= 0;\u001d)jd\u001d; (56)\n\u000fP(ks) =Z2\n0jP(ks= 0;\u0018)\u0000P(ks;\u0018)jd\u0018\nZ2\n0jP(ks= 0;\u0018)jd\u0018: (57)\nFig. 6( d) depicts both \u000fI(ks)and\u000fP(ks)as a function of ks. The difference is only of the\norder of a few percent, which suggests that the effect of surface anisotropy on the SANS\nobservables is relatively weak within the present analytical approximation; see the numerical\n17work [27], which takes into account the full nonlinearity of the micromagnetic equations.\nHowever, this is only true for the magnetic interactions considered here. Taking into account\nthe anisotropic and long-range dipole-dipole interaction and the asymmetric Dzyaloshinskii-\nMoriya interaction will very likely result in more inhomogeneous spin structures and in\nlarger deviations from the macrospin model [38–40]. Likewise, for NM of elongated shapes,\nthe surface anisotropy renders an additional first-order contribution to the effective energy\n[29], in addition to the second-order cubic contribution discussed above. This new shape-\ninduced contribution could also lead to an enhancement of the spin misalignment. The\nanalytical calculations presented here provide a general framework for future studies of\nmore complicated (anisotropic) magnetic interactions.\nIV. CONCLUSION\nWe have analytically computed the magnetization distribution and the ensuing mag-\nnetic small-angle neutron scattering (SANS) cross section of a spherical nanoparticle. Our\nmicromagnetic Hamiltonian takes into account the isotropic exchange interaction, an exter-\nnal magnetic field, a uniaxial anisotropy for the particle’s core, and Néel anisotropy on its\nboundary. The resulting Helmholtz equation has been solved by expanding the real-space\nmagnetization in terms of spherical Bessel functions and spherical harmonics. The central\nresults are the infinite series (16) and its second-order expansion (21) for the real-space\nmagnetization, and the corresponding Fourier transforms (37) and (40). Using these expres-\nsions, the two-dimensional magnetic SANS cross section SM(\u001d;\u0012q), the azimuthally-averaged\nSANS signalI(\u001d), and the correlation functions P(\u0018)andC(\u0018)were obtained and compared\nto the case of a homogeneous spin configuration (uniform magnetization vector field). The\nsignature of Néel’s surface anisotropy (of constant ks) has been identified in all of these func-\ntions. However, its effect is relatively small, even for large values of ks. Taking into account\nthe magnetodipolar and/or the Dzyaloshinskii-Moriya interaction, or shape asymmetry, will\nlikely result in configurations with stronger spin misalignment ( e.g.in vortex-type textures\nor skyrmions) and thereby in more prominent signatures in the SANS cross section and cor-\nrelation function. These interactions are beyond the scope of the current analytical approach\nand will be considered in our future (numerical) works [27].\n18ACKNOWLEDGMENTS\nMichael Adams and Andreas Michels thank the National Research Fund of Luxembourg\nfor financial support (AFR Grant No. 15639149).\n[1] S. Mühlbauer, D. Honecker, E. A. Périgo, F. Bergner, S. Disch, A. Heinemann, S. Erokhin,\nD. Berkov, C. Leighton, M. R. Eskildsen, and A. Michels, Rev. Mod. Phys. 91, 015004 (2019).\n[2] A.Michels, Magnetic Small-Angle Neutron Scattering: A Probe for Mesoscale Magnetism Anal-\nysis(Oxford University Press, Oxford, 2021).\n[3] S. Disch, E. Wetterskog, R. P. Hermann, A. Wiedenmann, U. Vainio, G. Salazar-Alvarez,\nL. Bergström, and T. Brückel, New J. Phys. 14, 013025 (2012).\n[4] K. L. Krycka, J. A. Borchers, R. A. Booth, Y. Ijiri, K. Hasz, J. J. Rhyne, and S. A. Majetich,\nPhys. Rev. Lett. 113, 147203 (2014).\n[5] K. Hasz, Y. Ijiri, K. L. Krycka, J. A. Borchers, R. A. Booth, S. Oberdick, and S. A. Majetich,\nPhys. Rev. B 90, 180405(R) (2014).\n[6] A.Günther,D.Honecker,J.-P.Bick,P.Szary,C.D.Dewhurst,U.Keiderling,A.V.Feoktystov,\nA. Tschöpe, R. Birringer, and A. Michels, J. Appl. Cryst. 47, 992 (2014).\n[7] T. Maurer, S. Gautrot, F. Ott, G. Chaboussant, F. Zighem, L. Cagnon, and O. Fruchart, Phys.\nRev. B89, 184423 (2014).\n[8] C. L. Dennis, K. L. Krycka, J. A. Borchers, R. D. Desautels, J. van Lierop, N. F. Huls, A. J.\nJackson, C. Gruettner, and R. Ivkov, Adv. Funct. Mater. 25, 4300 (2015).\n[9] A. J. Grutter, K. L. Krycka, E. V. Tartakovskaya, J. A. Borchers, K. S. M. Reddy, E. Ortega,\nA. Ponce, and B. J. H. Stadler, ACS Nano 11, 8311 (2017).\n[10] S. D. Oberdick, A. Abdelgawad, C. Moya, S. Mesbahi-Vasey, D. Kepaptsoglou, V. K. Lazarov,\nR. F. L. Evans, D. Meilak, E. Skoropata, J. van Lierop, I. Hunt-Isaak, H. Pan, Y. Ijiri, K. L.\nKrycka, J. A. Borchers, and S. A. Majetich, Sci. Rep. 8, 3425 (2018).\n[11] Y. Ijiri, K. L. Krycka, I. Hunt-Isaak, H. Pan, J. Hsieh, J. A. Borchers, J. J. Rhyne, S. D.\nOberdick, A. Abdelgawad, and S. A. Majetich, Phys. Rev. B 99, 094421 (2019).\n[12] P. Bender, D. Honecker, and L. F. Barquín, Appl. Phys. Lett. 115, 132406 (2019).\n19[13] M. Bersweiler, P. Bender, L. G. Vivas, M. Albino, M. Petrecca, S. Mühlbauer, S. Erokhin,\nD. Berkov, C. Sangregorio, and A. Michels, Phys. Rev. B 100, 144434 (2019).\n[14] D. Zákutná, D. Ni \u0014 z\u0014 nanský, L. C. Barnsley, E. Babcock, Z. Salhi, A. Feoktystov, D. Honecker,\nand S. Disch, Phys. Rev. X 10, 031019 (2020).\n[15] D. Honecker, M. Bersweiler, S. Erokhin, D. Berkov, K. Chesnel, D. A. Venero, A. Qdemat,\nS. Disch, J. K. Jochum, A. Michels, and P. Bender, Nanoscale Adv. 4, 1026 (2022).\n[16] D. Honecker and A. Michels, Phys. Rev. B 87, 224426 (2013).\n[17] A. Michels, S. Erokhin, D. Berkov, and N. Gorn, J. Magn. Magn. Mater. 350, 55 (2014).\n[18] D. Mettus and A. Michels, J. Appl. Cryst. 48, 1437 (2015).\n[19] S. Erokhin, D. Berkov, and A. Michels, Phys. Rev. B 92, 014427 (2015).\n[20] K. L. Metlov and A. Michels, Phys. Rev. B 91, 054404 (2015).\n[21] K. L. Metlov and A. Michels, Sci. Rep. 6, 25055 (2016).\n[22] A. Michels, D. Mettus, D. Honecker, and K. L. Metlov, Phys. Rev. B 94, 054424 (2016).\n[23] A. Michels, D. Mettus, I. Titov, A. Malyeyev, M. Bersweiler, P. Bender, I. Peral, R. Birringer,\nY. Quan, P. Hautle, J. Kohlbrecher, D. Honecker, J. R. Fernández, L. F. Barquín, and K. L.\nMetlov, Phys. Rev. B 99, 014416 (2019).\n[24] A. A. Mistonov, I. S. Dubitskiy, I. S. Shishkin, N. A. Grigoryeva, A. Heinemann, N. A. Sapo-\nletova, G. A. Valkovskiy, and S. V. Grigoriev, J. Magn. Magn. Mater. 477, 99 (2019).\n[25] V. D. Zaporozhets, Y. Oba, A. Michels, and K. L. Metlov, J. Appl. Cryst. xy, in press (2022).\n[26] L. Néel, J. Phys. Radium 15, 225 (1954).\n[27] M. P. Adams, A. Michels, and H. Kachkachi, J. Appl. Cryst. xy, abc (2022).\n[28] W. F. Brown Jr., Micromagnetics (Interscience Publishers, New York, 1963).\n[29] D. A. Garanin and H. Kachkachi, Phys. Rev. Lett. 90, 065504 (2003).\n[30] H. Kachkachi, J. Magn. Magn. Mater. 316, 248 (2007).\n[31] D. A. Garanin and H. Kachkachi, Phys. Rev. B 80, 014420 (2009).\n[32] X. Batlle, C. Moya, M. Escoda-Torroella, \u0012O. Iglesias, A. Fraile Rodríguez, and A. Labarta, J.\nMagn. Magn. Mater. 543, 168594 (2022).\n[33] R. C. O’Handley, Modern Magnetic Materials: Principles and Applications (Wiley, New York,\n2000).\n[34] H.J.WeberandG.B.Arfken, Essential Mathematical Methods for Physicists (AcademicPress,\nAmsterdam, 2003).\n20[35] K. F. Riley, M. P. Hobson, and S. J. Bence, Mathematical Methods for Physics and Engineering\n(Cambridge University Press, Cambridge, 2006).\n[36] F.W.J.Olver, D.W.Lozier, R.F.Boisvert,andC.W.Clark, NIST Handbook of Mathematical\nFunctions (Cambridge University Press, Cambridge, 2010).\n[37] G. Bertotti, Hysteresis in Magnetism (Academic Press, San Diego, 1998).\n[38] L. G. Vivas, R. Yanes, and A. Michels, Sci. Rep. 7, 13060 (2017).\n[39] L. G. Vivas, R. Yanes, D. Berkov, S. Erokhin, M. Bersweiler, D. Honecker, P. Bender, and\nA. Michels, Phys. Rev. Lett. 125, 117201 (2020).\n[40] S. A. Pathak and R. Hertel, Phys. Rev. B 103, 104414 (2021).\n[41] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products , Vol. 48 (Else-\nvier/Academic Press, Amsterdam, 2007).\n[42] J. D. Jackson, Classical Electrodynamics , 3rd ed. (Wiley, Hoboken, 1999).\nAppendix A: Solution of the Boundary Value Problem of the Helmholtz Equation\nThe coefficients c\f\n`min the fundamental solution (15) of the Helmholtz equation (9) must\nbedeterminedsuchthattheNeumannboundarycondition(10)issatisfied. Forthispurpose,\nwe use the method of least squares, where we make use of the orthogonality properties of\nthe spherical harmonics Y`m(\u0012;\u001e). The normal derivative of (15) at the surface of the NM\n(\u0018= 1) is:\nd \f\nd\u0018\f\f\f\f\n\u0018=1=1X\n`=0`X\nm=\u0000`c\f\n`m[j`(i\u0014\f\u0018)]0\n\u0018=1Y`m(\u0012;\u001e); (A1)\nwhere\n[j`(i\u0014\f\u0018)]0\n\u0018=1=d\nd\u0018j`(i\u0014\f\u0018)\f\f\f\f\n\u0018=1: (A2)\nOur goal is now to minimize the following error functional with respect to the coefficients\nc\f\n`m:\n\"[c\f\n`m] =Z2\u0019\n0Z\u0019\n0\f\f\f\f\f\fX\n\u000b2fx;y;zg\u001f\f\n\u000bjn\u000bj\n\u00001X\n`=0`X\nm=\u0000`c\f\n`m[j`(i\u0014\f\u0018)]0\n\u0018=1Y`m(\u0012;\u001e)\f\f\f\f\f2\nsin\u0012d\u0012d\u001e: (A3)\n21The minimum of this error functional is found from the condition that the partial derivatives\nof\"[c\f\n`m]with respect to the (c\f\nij)\u0003vanish:\n@\"[c\f\n`m]\n@(c\f\nij)\u0003= 0; (A4)\nwhere ‘\u0003’ denotes the complex conjugate. Using the orthogonality relation (A5) (p. 378\n(14.30.8) in [36]) and by defining the integral I\u000b\nijby (A6),\nZ2\u0019\n0Z\u0019\n0Y`m(\u0012;\u001e)Y\u0003\nij(\u0012;\u001e) sin\u0012d\u0012d\u001e =\u000e`i\u000emj; (A5)\nZ2\u0019\n0Z\u0019\n0Y\u0003\nij(\u0012;\u001e)jn\u000bjsin\u0012d\u0012d\u001e =I\u000b\nij; (A6)\nthe solution of (A4) can be written as:\nc\f\nij=1\n[jj(i\u0014\f\u0018)]0\n\u0018=1X\n\u000b2fx;y;zg\u001f\f\n\u000bI\u000b\nij: (A7)\nAlternatively, using again the indices `andm, expressing the coefficient \u001f\f\n\u000bas in (13), and\nby using the matrix-vector product, the coefficients c\f\n`mcan be more explicitly written as:\nc\f\n`m=ksg\f\u0001diag [Ix\n`m;Iy\n`m;Iz\n`m]\u0001m0\n[j`(i\u0014\f\u0018)]0\n\u0018=1: (A8)\nFor some low-orders of `ansm, the exact solutions of the integrals I\u000b\n`mare presented in\nAppendix B.\nFrom (A8) several conclusions can be drawn. First, the zero-order term ( `=m= 0) in\n(15) vanishes, which can easily be shown by rewriting the coefficient c\f\n00as:\nc\f\n00=ksp\u0019(g\f\u0001m0)\n[j0(i\u0014\f\u0018)]0\n\u0018=1= 0; (A9)\nwhich is due to the orthogonality of m0andg\fwith\f2f1;2g[see (6)]. Moreover, we see\nthat\n \f(\u0018= 0;\u0012;\u001e) = 0; (A10)\nwhich is a consequence of the behavior of the spherical Bessel functions of the first kind at\nthe origin. Since the j`with`\u00151are all vanishing at the origin \u0018= 0, this implies that the\ntotal \falso vanishes at the origin. Note that j0(0) = 1, but does not contribute to  \fdue\ntoc\f\n00= 0. From the physical point of view this make sense, since the spin misalignment is\n22caused by the Néel surface anisotropy and thus, due to symmetry reasons, there is no spin\ndisorder at the center of the spherical NM; the highest misalignment is found at its surface.\nSecond, in the table of Appendix B it is seen that the coefficients I\u000b\n`;mvanish for odd `or\nm, and thatI\u000b\n`;m=I\u000b\n`;\u0000m, so that the expansion coefficients also exhibit this symmetry\nc\f\n`;m=c\f\n`;\u0000m: (A11)\nTaking these properties into account, one can express the solution (15) more conveniently\nin terms of the associated Legendre polynomials Pm\n`(cos\u0012)with`= 2\u0017andm= 2\u0016[note\nthat we use the convention that Y`m(\u0012;\u001e) =N`mPm\n`(cos\u0012)eim\u001e(p. 378 (14.30.1) in [36])]:\n \f=1X\n\u0017=1\u0017X\n\u0016=0a\f\n\u0017\u0016\u0007\u0017(\u0014\f\u0018)P2\u0016\n2\u0017(cos\u0012) cos(2\u0016\u001e); (A12)\nwhere \u0007\u0017(\u001c)was defined in (17) and a\f\n\u0017\u0016in (18).\nThe infinite series in (18) is a consequence of the relation between the spherical Bessel\nfunctions of the first kind and the ordinary Bessel functions of the first kind (p. 262 (10.47.3)\nin [36])\njl(\u001c) =r\u0019\n2\u001cJl+1\n2(\u001c); (A13)\nand the well known series representation (p. 262 (10.47.3) in [36])\nJl(\u001c) =1X\ns=0(\u00001)s(\u001c=2)2s+l\ns!\u0000(l+s+ 1): (A14)\nAppendix B: Integral Coefficients\nThe integrals I\u000b\n`mcan be simplified, since both n\u000bandY\u0003\n`m(\u0012;\u001e)are separable functions\nin\u0012and\u001e. By rewriting the spherical harmonics in terms of complex exponentials and\nassociated Legendre polynomials and by using the definition of the Cartesian components\nof the surface normal vector nfrom (3) [note that we use the convention that Y`m(\u0012;\u001e) =\nN`mPm\n`(cos\u0012)eim\u001e(p. 378 (14.30.1) in [36]), the integrals are expressed as follows:\nIx\n`m=N`mK(1)\n`mU(1)\nm; (B1)\nIy\n`m=N`mK(1)\n`mU(2)\nm; (B2)\nIz\n`m=N`mK(2)\n`mU(3)\nm; (B3)\n23where\nK(1)\n`m=Z\u0019\n0Pm\n`(cos\u0012)jsin\u0012jsin\u0012d\u0012; (B4)\nK(2)\n`m=Z\u0019\n0Pm\n`(cos\u0012)jcos\u0012jsin\u0012d\u0012; (B5)\nU(1)\nm=Z2\u0019\n0e\u0000im\u001ejcos\u001ejd\u001e= (\u00001)jmj=22(1 + (\u00001)jmj)\n1\u0000m2+\u000ejmj;1; (B6)\nU(2)\nm=Z2\u0019\n0e\u0000im\u001ejsin\u001ejd\u001e=2(1 + (\u00001)jmj)\n1\u0000m2+\u000ejmj;1; (B7)\nU(3)\nm=Z2\u0019\n0e\u0000im\u001ed\u001e= 2\u0019\u000e0;m; (B8)\nN`m=s\n2`+ 1\n4\u0019(`\u0000m)!\n(`+m)!: (B9)\nThe integrals U(1)\nm,U(2)\nm, andU(3)\nmare solvable straightforwardly by using Euler’s formula for\nthe complex exponential and by splitting the region of integration according to the absolute\nvalues of the trigonometric functions. In the denominator of U(1)\nmandU(2)\nmwe included the\nKronecker delta \u000ejmj;1to take account of the cases m=\u00061. It is common to express the\nintegralsK(1)\n`mandK(2)\n`mby the substitution x= cos\u0012(dx=\u0000sin\u0012d\u0012):\nK(1)\n`m=Z1\n\u00001Pm\n`(x)p\n1\u0000x2dx; (B10)\nK(2)\n`m=Z1\n\u00001Pm\n`(x)jxjdx: (B11)\nUsingU(3)\nm= 2\u0019\u000e0;m, we need only to compute K(2)\n`;0such that the associated Legendre\npolynomials in K(2)\n`;mare reduced to the Legendre polynomials [with one index only (p. 352\nin [36])]. By considering the symmetry properties of the Legendre polynomials it becomes\nclear that the integral must vanish for odd `and can be simplified for even `in the following\nway:\nK(2)\n`;0= (1 + (\u00001)`)Z1\n0P`(x)xdx: (B12)\nThe closed-form of K(2)\n`;0is then found in terms of the Gamma function (p. 771 (7.126.1) in\n[41]):\nK(2)\n`;0=p\u0019(1 + (\u00001)`)\n4\u0000(3=2\u0000`=2)\u0000(`=2 + 2): (B13)\n24The overall solution for Iz\n`mis then written as:\nIz\n`m=\u0019(1 + (\u00001)`)p\n2`+ 1\n4\u0000(3=2\u0000`=2)\u0000(`=2 + 2)\u000e0;m: (B14)\nSince we only have to calculate K(1)\n`mfor evenm(sinceU(1)\nmandU(2)\nminclude the term\n1 + (\u00001)jmj), we may use the index m= 2\u0016, and by exploiting the parity of the associated\nLegendre polynomials Pm\n`(\u0000x) = (\u00001)`+mPm\n`(x), we find:\nK(1)\n`;2\u0016= (1 + (\u00001)`)Z1\n0P2\u0016\n`(x)p\n1\u0000x2dx: (B15)\nFrom these results it is seen that the integrals I\u000b\n`mwith\u000b2fx;y;zgvanish for odd `and\nm(note that in (B15) this is only the case for m= 2\u0016). For the remaining integrals K(1)\n2\u0017;2\u0016\nin (B15), where `= 2\u0017, we do not give an expression in closed form, but there must exist\none in terms of the Gamma function or the Beta function, since (p. 324 (3.251.2) in [41])\nZ1\n0xsp\n1\u0000x2dx=1\n2B\u0012s+ 1\n2;3\n2\u0013\n; (B16)\nwhere B(\u0001;\u0001)is the Beta function (Euler integral), and the associated Legendre functions\nP2\u0016\n2\u0017of even order and degree are true polynomials, as seen for example from the related\nRodrigues formula (p. 360 (14.7.14) in [36]). Since the remaining integrals are numerically\neasy to compute, we used Mathematica for the reconstruction of the analytically exact\nresults, which we present in the Table below for some low orders of `andm.\n25Ix\n`m=Z2\u0019\n0Z\u0019\n0Y\u0003\n`m(\u0012;\u001e)jsin\u0012cos\u001ejsin\u0012d\u0012d\u001e = (\u00001)jmj=22(1 + (\u00001)jmj)(1 + (\u00001)`)\n1\u0000m2+\u000ejmj;1s\n2`+ 1\n4\u0019(`\u0000m)!\n(`+m)!Z1\n0Pm\n`(x)p\n1\u0000x2dx\n`m0 \u00061 \u00062 \u00063 \u00064 \u00065 \u00066\n0p\u0019\n1 0 0\n2\u0000p\n5\u0019\n801\n8r\n15\u0019\n2\n3 0 0 0 0\n4\u00003p\u0019\n6401\n32r\n5\u0019\n20\u00001\n64r\n35\u0019\n2\n5 0 0 0 0 0 0\n6\u00005p\n13\u0019\n10240p\n1365\u0019\n20480\u00003\n1024r\n91\u0019\n20p\n3003\u0019\n2048\nIy\n`m=Z2\u0019\n0Z\u0019\n0Y\u0003\n`m(\u0012;\u001e)jsin\u0012sin\u001ejsin\u0012d\u0012d\u001e =2(1 + (\u00001)jmj)(1 + (\u00001)`)\n1\u0000m2+\u000ejmj;1s\n2`+ 1\n4\u0019(`\u0000m)!\n(`+m)!Z1\n0Pm\n`(x)p\n1\u0000x2dx\n`m0 \u00061 \u00062 \u00063 \u00064 \u00065 \u00066\n0p\u0019\n1 0 0\n2\u0000p\n5\u0019\n80\u00001\n8r\n15\u0019\n2\n3 0 0 0 0\n4\u00003p\u0019\n640\u00001\n32r\n5\u0019\n20\u00001\n64r\n35\u0019\n2\n5 0 0 0 0 0 0\n6\u00005p\n13\u0019\n10240\u0000p\n1365\u0019\n20480\u00003\n1024r\n91\u0019\n20\u0000p\n3003\u0019\n2048\nIz\n`m=Z2\u0019\n0Z\u0019\n0Y\u0003\n`m(\u0012;\u001e)jcos\u0012jsin\u0012d\u0012d\u001e =\u0019(1 + (\u00001)`)p\n2`+ 1\n4 \u0000 (3=2\u0000`=2) \u0000 (`=2 + 2)\u000em;0\n`m0 \u00061 \u00062 \u00063 \u00064 \u00065 \u00066\n0p\u0019\n1 0 0\n2p\n5\u0019\n40 0\n3 0 0 0 0\n4\u0000p\u0019\n80 0 0 0\n5 0 0 0 0 0 0\n6p\n13\u0019\n640 0 0 0 0 0\nTABLE I. Values of the integrals (19) for some small values of `andm.\n26Appendix C: Derivation of the Fourier Transform of the Magnetization\nThe Fourier transform of the magnetization vector field M(r)is written as:\nfM(q) =1\n(2\u0019)3=2Z\nVM(r) exp (\u0000iq\u0001r)d3r: (C1)\nInthesequel, wewillusedimensionlessquantities. Forthispurpose, wedefinethedimension-\nless scattering vector \u001d=qR, whereRis the radius of the nanomagnet, the dimensionless\nposition vector \u0018=r=R, and the dimensionless magnetization vector m=M=M0, where\nM0is the saturation magnetization. Substituting in (C1) results in\nfM(\u001d) =R3M0\n(2\u0019)3=2Z\nVm(\u0018) exp (\u0000i\u001d\u0001\u0018)d3\u0018: (C2)\nThe dimensionless Fourier transform fMis then defined as\nfM(\u001d) =1\n4\u0019Z\nVm(\u0018) exp (\u0000i\u001d\u0001\u0018)d3\u0018; (C3)\nwith\nfM=4\u0019R3M0\n(2\u0019)3=2fM: (C4)\nThe next step consists in calculating the Fourier integral of the first-order approximation\n(36) of the magnetization vector m. Since (36) is formulated in dimensionless spherical\ncoordinates \u0018;\u0012;\u001e, it is convenient to express the scattering vector \u001din spherical coordinates\nas well:\n\u0018= [\u0018sin\u0012cos\u001e;\u0018sin\u0012sin\u001e;\u0018cos\u0012]; (C5)\n\u001d= [\u001dsin\u0012qcos\u001eq;\u001dsin\u0012qsin\u001eq;\u001dcos\u0012q]; (C6)\nso that the plane-wave expansion of the complex exponential can be used [42]\nexp (\u0000i\u001d\u0001\u0018) = 4\u00191X\nk=0kX\nn=\u0000k(\u0000i)kjk(\u001d\u0018)Y\u0003\nkn(\u0012;\u001e)Ykn(\u0012q;\u001eq): (C7)\nThe Fourier integral (C3) is then expanded into the following infinite series:\nfM(\u001d) =1X\nk=0kX\nn=\u0000k(\u0000i)k^Mkn(\u001d)Ykn(\u0012q;\u001eq); (C8)\n27where\n^Mkn(\u001d) =Z2\u0019\n0Z\u0019\n0Z1\n0m(\u0018)jk(\u001d\u0018)Y\u0003\nkn(\u0012;\u001e)\u00182sin\u0012d\u0018d\u0012d\u001e: (C9)\nWe now use the infinite series (15) for  \fto express the first-order approximation of the\nmagnetization, which leads to\nm(\u0018) =m0+2X\n\f=1g\f1X\n`=0`X\nm=\u0000`c\f\n`mj`(i\u0014\f\u0018)Y`m(\u0012;\u001e): (C10)\nSincetheintegraltransform(C9)islinear, eachtermin(C10)canbeseparatelytransformed.\nFor the zero-order term, we obtain\nZ2\u0019\n0Z\u0019\n0Z1\n0m0jk(\u001d\u0018)Y\u0003\nkn(\u0012;\u001e)\u00182sin\u0012d\u0018d\u0012d\u001e =p\n4\u0019j1(\u001d)\n\u001dm0\u000ek;0\u000en;0:(C11)\nThis result is well known to the neutron-scattering community as the spherical form factor,\ncorresponding to a uniformly-magnetized spherical particle [2]. In the second step, we carry\nout the integration of the higher-order terms from (C10). The radial and the angular parts\nin the higher-order terms of (C10) are multiplicative, such that the volume integral (C9) is\nseparable into:\nA\f\n`k(\u001d) =Z1\n0j`(i\u0014\f\u0018)jk(\u001d\u0018)\u00182d\u0018; (C12)\nBkn\n`m=Z2\u0019\n0Z\u0019\n0Y`m(\u0012;\u001e)Y\u0003\nkn(\u0012;\u001e) sin\u0012d\u0012d\u001e: (C13)\nAs an intermediate result, the integral (C9) is then rewritten as\n^Mkn(\u001d) =p\n4\u0019j1(\u001d)\n\u001dm0\u000ek;0\u000en;0+2X\n\f=1g\f1X\n`=0`X\nm=\u0000`c\f\n`mA\f\n`k(\u001d)Bkn\n`m: (C14)\nThe integral (C13) directly corresponds to the orthogonality relation of the spherical har-\nmonics, and thereby we have Bkn\n`m=\u000e`k\u000emn(p. 378 (14.30.8) in [36]). Due to the term \u000e`k\ninBkn\n`m, and since Bkn\n`mandA\f\n`kare multiplicative in (C14), the following spherical Hankel\ntransform results:\nA\f\nkk(\u001d) =Z1\n0jk(i\u0014\f\u0018)jk(\u001d\u0018)\u00182d\u0018: (C15)\nIn order to calculate these integrals, we replace [using (A13)] the spherical Bessel functions\nof the first kind jk(\u0001)with the ordinary Bessel functions of the first kind J\u0013(\u0001). This yields:\nA\f\nkk(\u001d) =r\u0019\n2\u001dr\u0019\n2i\u0014\fW\f\nk(\u001d); (C16)\n28where\nW\f\nk(\u001d) =Z1\n0Jk+1\n2(i\u0014\f\u0018)Jk+1\n2(\u001d\u0018)\u0018d\u0018: (C17)\nis the Hommel integral. The result for the integral W\f\nkcan be found in the textbook by\nGradshteyn and Ryzhik (p. 664 (6.521.1) in [41]) and is written as\nW\f\nk(\u001d) =\u0000\u001dJk\u00001\n2(\u001d)Jk+1\n2(i\u0014\f)\u0000i\u0014\fJk\u00001\n2(i\u0014\f)Jk+1\n2(\u001d)\n\u001d2+\u00142\n\f: (C18)\nAgain, upon applying the relation (A13), we write the solution for A\f\nkk(\u001d)more convenient\nin terms of spherical Bessel functions of the first kind as\nA\f\nkk(\u001d) =\u0000\u001djk\u00001(\u001d)jk(i\u0014\f)\u0000i\u0014\fjk\u00001(i\u0014\f)jk(\u001d)\n\u001d2+\u00142\n\f: (C19)\nNow that the integrals are obtained, we have to substitute (C14) in (C8). The term on the\nfirst line of (C14) only accounts for k= 0andn= 0in (C8). Therefore, the contribution of\nthis term tofMisj1(\u001d)=\u001dm0, sinceY00(\u0012;\u001e) = 1=p\n4\u0019. SinceBkn\n`m=\u000e`k\u000emn, the final result\nis\nfM(\u001d) =j1(\u001d)\n\u001dm0+2X\n\f=1g\f1X\nk=0kX\nn=\u0000k(\u0000i)kc\f\nknA\f\nkk(\u001d)Ykn(\u0012q;\u001eq): (C20)\nUsing the properties of the coefficients c\f\nknstudied in Appendix A, we reformulate (C20)\nin terms of the associated Legendre polynomials, the indices k= 2\u0017andn= 2\u0016, and the\ncoefficients a\f\n\u0017\u0016:\nfM(\u001d) =j1(\u001d)\n\u001dm0+2X\n\f=1g\f1X\n\u0017=1\u0017X\n\u0016=0(\u00001)\u0017a\f\n\u0017\u0016\u001a\f\n\u0017(\u001d)P2\u0016\n2\u0017(cos\u0012q) cos(2\u0016\u001eq);(C21)\nwhere the radial function is given by\n\u001a\f\n\u0017(\u001d) =\u0000\u001dj2\u0017\u00001(\u001d)j2\u0017(i\u0014\f)\u0000i\u0014\fj2\u0017\u00001(i\u0014\f)j2\u0017(\u001d)\n\u001d2+\u00142\n\f: (C22)\nUsing once again (A13) and the well-known series (A14) for the Bessel functions of the first\nkind, we redefine the spherical Bessel functions of imaginary arguments [as they appear in\n(C22)] as:\n\u0007\u0017(\u001c) =j2\u0017(i\u001c) =p\u0019\n21X\ns=0(\u00001)\u0017(\u001c=2)2(s+\u0017)\ns!\u0000(2\u0017+s+ 3=2); (C23)\n@\u0017(\u001c) = ij2\u0017\u00001(i\u001c) =p\u0019\n21X\ns=0(\u00001)\u0017(\u001c=2)2(s+\u0017)\u00001\ns!\u0000(2\u0017+s+ 1=2); (C24)\n29such that it becomes clear that fM(\u001d)is a purely real-valued function. The radial function\nis then rewritten as\n\u001a\f\n\u0017(\u001d) =\u0000\u001dj2\u0017\u00001(\u001d)\u0007\u0017(\u0014\f)\u0000\u0014\f@\u0017(\u0014\f)j2\u0017(\u001d)\n\u001d2+\u00142\n\f: (C25)\nBy comparing this result to (C15), we find the following pair of spherical Hankel transforms\n\u001a\f\n\u0017(\u001d) =Z1\n0\u0007\u0017(\u0014\f\u0018)j\u0017(\u001d\u0018)\u00182d\u0018: (C26)\nBy comparing the result (C21) to (16) it becomes clear that the angular part is (due to the\northogonality relation of the spherical harmonics) shape invariant under Fourier transfor-\nmation, while the spherical Hankel transform (C26) of the radial function \u0007\u0017(\u0014\f\u0018)remains.\n30"    },    {        "title": "2205.10418v3.A_feasibility_analysis_towards_the_simulation_of_hysteresis_with_spin_lattice_dynamics.pdf",        "content": "A feasibility analysis towards the simulation of hysteresis with spin-lattice dynamics\nG. dos Santos,1, 2F. Rom´ a,3J. Tranchida,4S. Castedo,5L. F. Cugliandolo,6, 7and E. M. Bringa1, 2, 8\n1CONICET, Mendoza, 5500, Argentina\n2Facultad de Ingenier´ ıa, Universidad de Mendoza, Mendoza, 5500 Argentina∗\n3Departamento de F´ ısica, Universidad Nacional de San Luis, Instituto de F´ ısica Aplicada (INFAP),\nConsejo Nacional de Investigaciones Cient´ ıficas y T´ ecnicas (CONICET),\nChacabuco 917, D5700BWS San Luis, Argentina\n4CEA, DES, IRESNE, DEC, SESC, LM2C, F-13108 Saint-Paul-Lez-Durance, France.\n5Department of Physics and Astronomy, School of Natural Sciences,\nThe University of Manchester, Manchester M13 9PL, United Kingdom\n6Sorbonne Universit´ e, CNRS UMR 7589, Laboratoire de Physique Th´ eorique\net Hautes Energies, 4 Place Jussieu, 75252 Paris Cedex 05, France\n7Institut Universitaire de France, 1 rue Descartes, 75231 Paris Cedex 05, France\n8Centro de Nanotecnolog´ ıa Aplicada, Facultad de Ciencias, Universidad Mayor, Santiago, Chile 8580745\n(Dated: August 9, 2023)\nWe use spin-lattice dynamics simulations to study the possibility of modeling the magnetic hys-\nteresis behavior of a ferromagnetic material. The temporal evolution of the magnetic and mechanical\ndegrees of freedom is obtained through a set of two coupled Langevin equations. Hysteresis loops\nare calculated for different angles between the external field and the magnetocrystalline anisotropy\naxes. The influence of several relevant parameters is studied, including the field frequency, magnetic\ndamping, magnetic anisotropy (magnitude and type), magnetic exchange, and system size. The\nrole played by a moving lattice is also discussed. For a perfect bulk ferromagnetic system we find\nthat, at low temperatures, the exchange and lattice dynamics barely affect the loops, while the\nfield frequency and magnetic damping have a large effect on it. The influence of the anisotropy\nmagnitude and symmetry are found to follow the expected behavior. We show that a careful choice\nof simulation parameters allows for an excellent agreement between the spin-lattice dynamics mea-\nsurements and the paradigmatic Stoner-Wohlfarth model. Furthermore, we extend this analysis to\nintermediate and high temperatures for the perfect bulk system and for spherical nanoparticles, with\nand without defects, reaching values close to the Curie temperature. In this temperature range, we\nfind that lattice dynamics has a greater role on the magnetic behavior, especially in the evolution of\nthe defective samples. The present study opens the possibility for more accurate inclusion of lattice\ndefects and thermal effects in hysteresis simulations.\nKeywords Hysteresis, spin-lattice dynamics, ferromag-\nnetic, Fe\nI. INTRODUCTION\nFor basic research and applications, magnetic studies\nare of fundamental relevance as they are an important\ntool for revealing otherwise hidden structural, thermody-\nnamic and physicochemical properties of a material. Hys-\nteresis loops are usually measured to characterize the dy-\nnamic behavior of bulk samples that exhibit microstruc-\ntures like grain boundaries, defects, etc, and multiple do-\nmains [1, 2]. Furthermore, such measurements are also\nperformed for low-dimensional magnets such as magnetic\nnanoparticles (NPs) or thin films [3].\nThe power generated by a sample subject to an alter-\nnating magnetic field can be directly determined from\nthe shape of a hysteresis loop. In fact, the Specific Ab-\nsorption Rate (SAR), defined as the absorbed energy per\nunit of mass, is proportional to the area of this curve\n∗gonzalodossantos@gmail.com[4]. In magnetic hyperthermia [5–7], for instance, SAR\nprovides the heating efficiency of a specific type of mag-\nnetic nanoparticle. Furthermore, the magnetization of\nnanoparticles is of great interest for many other techno-\nlogical applications [8, 9]. Magnetization in small NPs is\nexpected to flip due to thermal fluctuations during the\ntime scale of a few seconds in typical experiments, giving\nrise to superparamagnetic behavior below some critical\nsize, which for Fe is close to 20 nm. However, hysteresis\nloops with ferromagnetic blocked behavior have been ob-\nserved for relatively simple Fe NPs, indicating complex\nmagnetic behavior, which is not fully understood [10–12].\nSome of this behavior might be related to lattice defects\n[11, 12]. It has been emphasized that defect engineer-\ning will allow novel future technological applications of\nmagnetic nanoparticles [13, 14], pointing out the need to\nmodel defective magnetic nanosystems.\nDifferent classical models are used to describe mag-\nnetic hysteresis [1, 15]. Possibly, the most popular one\nis that of Stoner-Wohlfarth (SW) [16, 17]. This model is\nrelatively simple to evaluate numerically, allowing easy\ncomparison with experimental results. However, the\nSW model assumes an important number of approxima-\ntions. Among them, it is considered that the material\nbehaves as a single magnetic domain. This is knownarXiv:2205.10418v3  [cond-mat.mtrl-sci]  7 Aug 20232\nas the macrospin approximation, which implies that be-\nlow a certain length scale, typically on the order of a\nfew nanometers, the internal structure of the material\nis neglected. In the SW model the Hamiltonian is the\nsum of a Zeeman term and an uniaxial anisotropy term.\nHysteretic behavior arises as a consequence of a simple\nzero-temperature dynamic protocol: as the external field\nvaries in time, the magnetization can either change con-\ntinuously while the system remains at an energy mini-\nmum, or it can jump abruptly to another value when that\nminimum becomes unstable. Usually the SW model is\nemployed to compare with experiments for a large collec-\ntion of randomly oriented NP, and obtain relevant phys-\nical quantities [10]. In addition, although in the original\nSW model thermal fluctuations were not considered, fur-\nther approaches have attempted to incorporate temper-\nature effects [18].\nFortunately, micromagnetic simulations allow one to\novercome many of these limitations [19, 20]. Thus, it\nis possible to study a system made up of a large num-\nber of interacting macrospins with a Hamiltonian in-\ncluding any complex energetic term. In particular, the\nequilibrium or nonequilibrium magnetization dynamics\nof such a model can be calculated by numerically solving\neither the stochastic Landau-Lifshitz-Gilbert (sLLG) or\nthe stochastic Landau-Lifshitz (sLL) phenomenological\nequations [21, 22]. From first principles it can be proved\nthat this same theoretical framework is valid to carry out\natomistic spin dynamic (ASD) simulations [23, 24]. This\nsimulation method has been successfully applied to the\nmodeling of hysteresis loops [25–28]. Another alterna-\ntive to include temperature effects in hysteresis loops is\nto use Metropolis Monte Carlo simulations for a fixed\nlattice [29, 30].\nMore recently, there has been increased interest in\nstudying more realistic models that take into account\nboth spin and lattice degrees of freedom, and the cou-\npling between them. Most of these methods are based\non coupled Spin Dynamics (SD) and Molecular Dynam-\nics (MD) simulations [31–35]. This is commonly known\nas spin-lattice dynamics (SLD) simulations. Other first-\nprinciple based methods combine atomistic spin dynam-\nics with ab initio molecular dynamics (ASD-AIMD) [36],\nbut this is limited to systems with a few atoms due to the\nhigh computational cost of calculating interatomic forces\nvia ab initio methods.\nSLD simulations are typically based on a Langevin ap-\nproach that allows one to describe the evolution of both\nlattice and spin degrees of freedom [37]. Although this\nmethod is very powerful, its implementation can be com-\nputationally demanding and, as the complexity of a sys-\ntem increases, the number of parameters that charac-\nterize it also increases. Therefore, the practical use of\nthis numerical scheme for calculation of hysteresis loops\nrequires a deep exploration. In this work, we use SLD\ncalculations with sLL spin dynamics to simulate the hys-\nteretic behavior of a ferromagnetic material. We use\nphysical parameters that are typical for bulk bcc iron.We focus on determining optimal simulation parameters\nthat allow one to obtain reliable results at low temper-\natures. Also, we explore how the hysteresis loops de-\npend on the different physical properties that character-\nize these magnetic systems: anisotropy, exchange, damp-\ning, and lattice vibrations. The paradigmatic SW model\nis taken as a reference to achieve this goal. Additional\nsimulations are carried out to analyze the importance\nof SLD simulation at higher temperatures. Finally, we\npresent NP hysteresis loops simulations, including defects\nlike the ones found in experiments.\nWe explore the possibility of using a high damping pa-\nrameter in order to reduce the relaxation time and speed\nup the calculations. The use of strong damping led us to\nrevise the definition of the noise-noise correlations in the\nsLL equations and find the correct parameter dependence\nthat lets the systems thermalize with their equilibrium\nenvironments at long times. In this work, we perform\nSLD simulations using the SPIN package of the software\nLAMMPS [37–39].In order to enable its use in this study\nwith relatively large damping, the software was modified\naccordingly.\nThe paper is organized as follows. In Sec. II, the the-\noretical models used and the simulation details are pre-\nsented. In Sec. III, the main results of this study are\nshown. A summary and the conclusions of the work are\ndrawn in Sec. V. In order to simplify several parametric\nstudies, as a first approach we run spin dynamics simula-\ntions with the positions of the atoms fixed at their ideal\nequilibrium values. The influence of the spin-lattice cou-\npling is explored and discussed in the last part of Sec.\nIII. Finally, in an Appendix we derive the Fokker-Planck\nequation associated to the SLD dynamics and from it, we\nfix the noise-noise correlations that ensure the asymptotic\napproach to equilibrium. We confirmed, with simulations\nnot shown here, that these do indeed take the systems to\nequilibrium even at large values of the damping factor,\nvery convenient to reduce the simulation time.\nII. METHODS\nIn Sec. II A we describe the main characteristics of the\nSLD model from LAMMPS[37–39], and in Sec. II B we\ngive some simulation details.\nA. SLD Model\nLet us consider an ensemble of Natoms each endowed\nwith a classical magnetic moment. The Hamiltonian of\nthe model is\nH=NX\ni=1|pi|2\n2mi+NX\ni,j,i̸=jV(rij) +Hmag. (1)\nThe first term in Eq. (1) accounts for the kinetic en-\nergy of the Natoms where piandmirepresent, respec-3\ntively, the linear momentum and mass of the i-th atom\n(mi= 55 .845 u for iron). The second term is the in-\nteratomic potential describing the interactions between\npairs of atoms at positions riandrjseparated by a dis-\ntance rij≡ |ri−rj|. As usual in molecular dynamics\nsimulations of metals, we use a classical embedded atom\nmodel (EAM) potential which describes well a broad\nspectrum of iron properties [40, 41]. The interatomic\ncutoff distance for this potential was set to 0 .57 nm.\nThe coupling among the spin and lattice degrees of\nfreedom is provided through the last term in Eq. (1), a\nmagnetic Hamiltonian defined as\nHmag=−µ0µH·NX\ni=0si−NX\ni,j,i̸=jJ(rij)si·sj+Hani.(2)\nHere, siis a classical unitary vector representing the spin\nof the i-th atom. The first term in Eq. (2) is the Zeeman\nenergy, the interaction of each spin with an external uni-\nform magnetic field H, where µ= 2.2µBis the atomic\nmagnetic moment for iron ( µBis the Bohr magneton)\nandµ0is the vacuum permeability constant. Note that\nthe Zeeman energy within LAMMPS is defined slightly\ndifferently than in Eq. (2), and a re-scaling had to be\napplied to recover our formulation. The second term is\njust a Heisenberg Hamiltonian describing the interaction\nbetween spins, where J(rij) is an interatomic distance-\ndependent exchange coupling which is defined as the fol-\nlowing Bethe-Slater curve [42, 43],\nJ(rij) = 4 α\u0010rij\nδ\u00112\u0014\n1−γ\u0010rij\nδ\u00112\u0015\ne−(rij\nδ)2\n×Θ(Rc−rij), (3)\nbeing Θ( Rc−rij) the Heaviside step function and Rcthe\ncutoff distance. The coefficients in Eq. (3) can be fitted\nwith ab-initio or experimental data. In the present work\nwe fit Eq. (3) to data by Ma et al. [44], as it was already\ndone in previous works [34, 45]. Specifically, we use the\nfollowing values: α= 96.0 meV, γ= 0.20,δ= 0.154 nm.\nIn addition, we choose Rc= 0.35 nm.\nFinally, the last term in Eq. (2) is responsible for com-\nputing the magneto-crystalline anisotropy. In this work\nwe consider one of two, either uniaxial Hunior cubic Hcub\nanisotropy. The corresponding expressions are given by\nHuni=−K1NX\ni=1(si·n)2(4)\nand\nHcub=NX\ni=1n\nK1[(si·n1)2(si·n2)2\n+(si·n2)2(si·n3)2\n+(si·n1)2(si·n3)2]\n−K2(si·n1)2(si·n2)2(si·n3)2o\n.(5)Here, the unit vectors n1,n2, andn3lie along the three\ncrystallographic directions [100], [010], and [001], respec-\ntively. In Eq. (4), nis also a unit vector that in general\ncould point along any of these axes. The first term in\nthe cubic anisotropy energy above is defined with differ-\nent sign within LAMMPS. K1andK2are the magneto\ncrystalline anisotropy constants which we set to K1= 35\nµeV/atom and K2= 3.6µeV/atom (equivalents to vol-\numetric anisotropies K1V= 470 kJ/m3andK2V= 46\nkJ/m3). In order to make comparisons, we have used the\nsame value of K1in both anisotropy equations. Since\nthis constant is positive (and also K2>0), then the easy\naxes of magnetization in Eqs. (4) and (5) are given by the\nunit vector defined above. Note that the values of K1and\nK2are ten times larger than those usually used to model\nbulk bcc iron [3]. However, such larger anisotropy mag-\nnitude has been considered for Fe nanoparticles [10, 46],\nand we also consider bulk values for selected runs. As\nwe will discuss later, using a large anisotropy in the sim-\nulations helps to quickly stabilize the magnetization of\nthe system. Increasing anisotropy for numerical reasons\nhas been employed for other magnetic simulations, for in-\nstance, to obtain domain walls with smaller widths [47]\nor to confine the magnetization dynamics to a plane [48].\nWe note that an improved implementation of the\nanisotropy would include a better description of the spin-\norbit coupling [49–51] and also a local variation of mag-\nnetic moments and couplings, for instance, depending\non atomic volume [52]. However, uniaxial and cubic\nanisotropies are often used to interpret experimental re-\nsults, for a number of models and simulations [10, 53–56].\nWe include a simple description of anisotropy within this\nspirit and, in our simulations, lattice and spin dynam-\nics are coupled through the distance-dependent exchange\nfunction J(rij) and by the spins Langevin thermostat set\nat the same temperature as the lattice (see below). In\naddition, to speed up our calculations we have neglected\nlong-range dipolar interactions in the magnetic Hamilto-\nnian. Since we simulate small systems (see below), this\napproximation should not affect the validity of our re-\nsults.\nThe core of this simulation method, lattice and spin\ncoupling, is contained in the following coupled Langevin\nequations [37],\ndri\ndt=pi\nmi, (6)\ndpi\ndt=NX\ni,j,i̸=j\u0014\n−dV(rij)\ndrij+dJ(rij)\ndrijsi·sj\u0015\neij\n−γL\nmipi+ξi, (7)\ndsi\ndt=1\n1 +λ2s[(ωi+ζi)×si+λssi×(ωi×si)].(8)\nIn Eq. (7), eijrepresents a unit vector along the line\nconnecting atoms iandj,γLis the lattice damping coef-\nficient, and ξ(t) is a random fluctuating force drawn from4\na Gaussian distribution with\n⟨ξ(t)⟩= 0,\n⟨ξa(t)ξb(t′)⟩= 2DLδabδ(t−t′), (9)\nwhere the aandbsubscripts indicate Cartesian vector\ncomponents, and the amplitude of the noise is\nDL=γLkBT. (10)\nHere, kBis the Boltzmann constant and Tthe thermo-\nstat temperature. Equation (7) describes the atoms dy-\nnamic, which is affected by the spins motion and by the\nfunctionality of the exchange function J(rij).\nAs shown in Eq. (8), the spin dynamics is modeled\nthrough the sLL equation. Here, ωi=−1\nℏ∂Hmag\n∂siis the\neffective field acting on spin iandλsis the spins damping\nparameter. ζ(t) is the stochastic field which is also drawn\nfrom a Gaussian probability distribution with\n⟨ζ(t)⟩= 0,\n⟨ζa(t)ζb(t′)⟩= 2DSδabδ(t−t′). (11)\nIn this case, the fluctuation-dissipation relation for the\nmagnetic degrees of freedom is\nDS=λs(1 +λ2\ns)kBT\nℏ. (12)\nIn the Appendix, we formally derive Eqs. (10) and (12).\nThese are the parameter dependencies of DLandDSthat\nallow for equilibration of the full system to a Boltzmann\ndistribution ∝e−βHwithHin Eq. (1), and the lattice\nand magnetic contributions specified below this equation.\nWe note that the Langevin equation presented in\nref. [37], Eq. (8), does not have the stochastic field\nin the relaxation term (the last term) and, following\nref. [37], we refer to it as the sLL equation (stochastic\nLandau-Lifshitz). When the stochastic field is added to\nboth effective field terms the equation is usually called\nthe sLLG equation (stochastic Landau-Lifshitz-Gilbert\n(sLLG). However, there is a small difference between the\ntypical sLL equation and Eq. (8): the latter has the\nGilbert factor included. Therefore, it is neither a typical\nsLL nor a typical sLLG equation. Still, once the noise\nparameters are well fixed, Eq. (8) also takes the system\nto thermal equilibrium at long enough times.\nB. Simulation details\nAs it was described in the previous subsection, in the\nSLD implementation of the software LAMMPS [37–39],\nthe evolution of the system is described by two coupled\nLangevin equations, one for the spins and another one\nfor the lattice degrees of freedom.\nEach Langevin equation has a damping term and a\nrandom force (or field) which are connected through the\n“fluctuation-dissipation” theorem, see Eqs. (10) and (12).We have chosen the damping constants equal to λs= 0.5\n(for the spin degrees of freedom) and λL= 1.0 s−1(for\nthe lattice). We use separate Langevin thermostats for\nthe lattice and spin subsystems, but both are set to the\nsame temperature. Most of the simulations were carried\nout at T= 10 K but, as indicated in some cases, others\nwere performed at higher temperatures.\nThe hysteresis loops were calculated applying an alter-\nnating magnetic field H, and averaging the curves over\nup to ten different cycles. At low temperatures, the in-\ndividual loops exhibit a small dispersion of a few percent\nwith respect to the average curve. A progressively larger\ndispersion is observed at higher temperatures, resulting\nin a dispersion of about 20% for the magnetization in\nthese cases. It is important to note that this variability,\nbeing inherent to the stochastic nature of the simulation\nmethod, does not alter the reported results nor the con-\nclusions drawn from them. Instead, it reflects the com-\nplexity of the system and the method ´s capacity to effec-\ntively capture temperature fluctuations, which provides\nvaluable insights into the behavior of the magnetization\ndynamics across the temperature range studied.\nThe field amplitude is varied discretely: it remains con-\nstant during certain simulation time, and then jumps to\nreach the value given by a function H=Hmaxcos(2 πft).\nHmaxis set to be larger than the expected saturation\nfield, and jumps do not have the same magnitude along\nthe entire field range, given that the simulation time at\neach field value is kept constant.\nTypical MD simulations use a time step of 1 fs. How-\never, to capture spin dynamics, the time step has been\nset in a range of 0.1 fs [37] to 10 fs [27]. Here we use 0.1 fs\nbut we have verified with several examples that using 1 fs\ndoes not change our results within the statistical spread.\nEach hysteresis cycle took around 2 −8 ns, giving MHz\nfrequencies. In particular, we have simulated field fre-\nquencies of f=f0,f0/2,f0/4, and f0/8, with f0= 500\nMHz. These values are equivalent to sweeping rates of\napproximately 2 .2×109T/s, 1 .1×109T/s, 0 .55×109\nT/s, and 0 .275×109T/s, respectively, with T /s repre-\nsenting Tesla per second. We note that in experiments,\nhysteresis loops are obtained using fields that change al-\nmost continuously, and the measurement of the magnetic\nmoment of a sample can take up to several microseconds.\nThose time scales are well beyond the feasibility of SLD,\nor of other simulation methods like ASD, since typically\nsimulations only reach nanosecond scale with sweep rates\nsimilar to ours [25–27]. Nevertheless, as we show in the\nnext section, the hysteresis loops quickly converge to a\nlimiting curve as the frequency is decreased.\nEfficient minimization techniques, such as the one in\nRef. [57] would lead the system into efficiently finding\nand falling into the lowest energy minima, for every value\nof the external field, meaning that the spins would be\naligned with the field as soon as the field direction is re-\nverted and therefore would not produce a hysteresis loop.\nIn addition, the use of efficient minimization techniques\nimplies that there is no real time scale associated with the5\nhysteresis loop frequency, which is important, for exam-\nple, when calculating energy balance as in hyperthermia\ncalculations.\nSimulations were run for different angles, ϕ= 0◦, 45◦,\nand 90◦, between the external field Hand the easy\nanisotropy axes. In practice, this was done by chang-\ning the uniaxial anisotropy axis directions while keeping\nHaligned in the [001] direction ( zaxis). For uniaxial\nanisotropy, for instance, for ϕ= 90◦or 45◦the unit vec-\ntornwas oriented along the [100] or [101] direction. Most\nsimulations where run with magnetic uniaxial anisotropy\naccording to Eq. (4).\nFor most cases we use cubic samples of dimensions\n(10×10×10)a3\n0(a0= 0.286 nm is the lattice param-\neter of bcc Fe), with periodic boundary conditions in all\ndirections. The resulting system contains 2000 atoms. To\nshow that this size is large enough to capture the main\nhysteresis properties of the model at low temperatures,\nwe also run a few simulations using a larger system with\n(15×15×15)a3\n0cells (6750 atoms), as shown in the Sup-\nplemental Material (SM). We have also run a few simula-\ntions at higher temperatures, cases in which the magnetic\nfluctuations increase significantly. In these instances the\nvolume of the system was set to (32 ×32×32)a3\n0(around\n65000 atoms) in order to avoid undesired finite-size ef-\nfects.\nFinally, we have to mention that the total magneti-\nzation Mas well as the components Mx,MyandMz,\nwill be expressed normalized to the ideal Fe bulk satura-\ntion magnetization Ms. The saturation magnetization is\ngiven by the maximum magnetic moment per unit vol-\nume. For bulk iron, in the volume of a bcc unit cell ( a3\n0)\nthere are 2 spins with magnetic moment µ= 2.2µBand,\ntherefore, Ms= 2×2.2µB/a3\n0≈1730 kA/m. In this way,\nM= 1.0 means M= 1730 kA/m.\nC. Nanoparticles simulation details\nAs it was mentioned in the introduction, we also per-\nformed hysteresis simulations of Fe NPs including lattice\ndefects. In this subsection, we give additional details re-\ngarding these simulations.\nWe have run hysteresis simulations for two different\nnanoparticles, a pristine and a defective nanoparticle, at\na temperature of T= 500 K. The pristine NP was built\nsimply by cutting a sphere with diameter 8 nm from the\nperfect bcc lattice. No further relaxation was carried out,\nto mimic usual frozen lattice simulations in atomistic spin\ndynamics.\nThe defective NP was built generating vacancies to the\npristine NP above by removing 33 .3% of the atoms, ran-\ndomly. After this, an energy minimization was performed\nto relax the defective lattice. The NP temperature was\nthen raised to 900 K using a 20 ps linear ramp, held at\n900 K during 20 ps, and then cooled-down to 500 K also\nwith a linear ramp. After that, the temperature was held\nat 500 K during 20 ps, resulting in a NP with a few smallvacancy clusters, since most vacancies were absorbed at\nthe surface. In order to include more extended defects,\nthis NP was deformed mechanically using a flat rigid in-\ndenter [58], to mimic the strain that a NP could expe-\nrience due to synthesis conditions. We used an indenter\nwith a repulsive constant K= 20eV\n˚A3(∼3 TPa), typi-\ncal of indentation in metals [59]. The indenter moved at\n200 m/s, which is higher than typical velocities used in\nindentation simulations, but much lower than the sound\nvelocity in the material and results are expected to be\nqualitatively similar. The NP is compressed up to a uni-\naxial strain of ∼20%, but expands sideways in such a\nway that the compressive volumetric strain at the maxi-\nmum indentation depth is lower than 5%. The indenter\nis then displaced upwards for unloading. The resulting\nNP was relaxed within the microcanonical NVE ensemble\nduring 10 ps. The final configuration used for the calcu-\nlation of hysteresis loops included small vacancy clusters\nand also a twin boundary, something expected in bcc NP\n[60], as seen in Figure 13 (f)-(g). The software Ovito [61]\nwas employed to render snapshots and to analyze the NPs\nmicrostructure and defects. Polyhedral Template Match-\ning (PTM) [62] was used to obtain the crystal structures\nand surface mesh tool [63] was used to analyze the NPs\nsurface and vacancies.\nVacancies imply a lower number of atoms that will pro-\nduce a smaller saturation magnetization, 10% smaller ne-\nglecting surface effects. NP topology changed from the\nroughly spherical shape of the pristine NP, including a\nmore faceted surface, as shown in Figure 13 (h). There\nare ordered terraces and facets in the roughly spherical\npristine NP. The defective NP shows surface disorder in\nFigure 13 (e), and large deviations from spherical shape\nin Figure 13 (i). From separate tests, dislocations do\nnot appear as stable for the NP size considered here.\nThe magnetic moments are assumed equal to the bulk\nvalue for these NP simulations. We note that the mag-\nnetic moment is expected to vary near defects, and recent\nstudies have explored variations near surfaces and vacan-\ncies [35, 45]. However, these effects would be small and\nare not expected to change the overall behavior of the\nhysteresis loops.\nFor both NPs, the hysteresis simulations were car-\nried out at T= 500 K, considering uniaxial magnetic\nanisotropy (Eq. (4)) with an anisotropy constant K1= 35\nµeV/atom which has been considered previously for Fe\nNPs [10, 46]. The external magnetic field is applied at\nzero degrees with respect to the anisotropy axis and it is\nvaried at a sweep rate of SR= 0.227×108T/s.\nIn these NP simulations the exchange function J(rij)\n(Eq. (3)) is fitted to the ab-initio data by Pajda et al. [64]\nusing the following fitting parameters: α= 25.498 meV,\nγ= 0.281, δ= 0.1999 nm. Previous simulations of Fe\nNPs with these parameters have shown excellent agree-\nment with experimental observations [34] and we note\nthat for this parametrization of J(rij), the Curie temper-\nature for an 8 nm NP is close to 600 K. The remaining\nparameters are the same as those of the bulk simulations.6\nIII. RESULTS AND DISCUSSION\nIn this section, we first analyze how the computational\nparameters, such as the simulation run time and the field\nfrequency, affect the final outcome of the calculations.\nThe aim is to establish optimal parameters that mini-\nmize the computational cost without affecting the reli-\nability of the calculations. Next, we focus on studying\nhow the hysteresis loops can change when different phys-\nical properties like anisotropy, exchange interaction, and\ndamping parameter, are modified. These studies are car-\nried out on the bulk samples described in the methods\nsection. In a first stage, we analyze these effects with-\nout coupling the lattice vibrations to the spins dynamics,\ni.e. we consider for most cases spin dynamics simulations\nwith the atoms fixed at their ideal-lattice positions. The\ninfluence of coupling to the lattice vibrations is analyzed\nin the final part of this section, where we run full SLD\nsimulations for both bulk and nanoparticles samples.\nA. Simulation time\nWe start analyzing the effect of the simulation run time\ntsim. This quantity represents the interval during which\nthe external field His held constant before increasing\nor decreasing its absolute value by a given amount. It\nis basically the total time of simulation used to calculate\neach individual discrete point of a hysteresis loop. After\nchanging the external field, the magnetic moments must\nrelax to a new stable configuration and therefore tsim\nmust be large enough to allow this process to occur. It\nis important, then, to determine an optimal value of this\nquantity that allows to achieve this goal.\nIn principle, one can imagine an individual spin pre-\ncessing around the field direction, with a frequency which\nincreases with the field magnitude. For Fe, the resulting\nLarmor frequency gives a period of 36 ps for a 1.0 T field,\nand several precession periods are needed to describe the\nmagnetization evolution. However, for damped dynam-\nics, the frequency remains the same but the spin spirals\ndown towards the field direction, allowing shorter sim-\nulations [23]. Figure 1 shows the time evolution of the\ncomponents of the normalized total magnetization along\nthe three Cartesian axes, Mx,My,Mz, corresponding to\ndirections [100], [010], and [001], respectively, during the\nsimulation of an entire loop for the case ϕ= 90◦. Unless\notherwise stated, for this and subsequent calculations we\nuse uniaxial anisotropy. As it can be noticed by inspec-\ntion of this figure, the applied field (green dashed curve\nand right axis) is maintained at a constant value for 90\nps before increasing or decreasing it to the next value, so\ntsim= 90 ps. We are interested here in the stabilization of\nthe magnetization along the field direction, Mz. During\nthe first moments after the field value is varied a fluctu-\n/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56 /s50/s48 /s50/s50/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s53 /s54 /s55 /s56 /s57/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s77\n/s105\n/s45/s48/s46/s56/s45/s48/s46/s52/s48/s46/s48/s48/s46/s52/s48/s46/s56\n/s40/s97/s41\n/s109\n/s48/s72/s32/s40/s84/s41\n/s40/s98/s41\n/s84/s105/s109/s101/s32/s40/s120/s57/s48/s32/s112/s115/s41/s77\n/s105/s32/s77\n/s120/s32 /s32/s77\n/s121/s32 /s32/s77\n/s122/s32 /s32/s60/s32/s77\n/s122/s62/s32 /s32/s70/s105/s101/s108/s100\n/s102/s32/s61/s32/s57/s48/s176/s32/s59/s32/s32 /s102\n/s48\n/s45/s48/s46/s56/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50\n/s109\n/s48/s72/s32/s40/s84/s41Figure 1. (Color online.) Time evolution of the components\nof the normalized total magnetization throughout an entire\ncycle (a) and half a cycle (b). The field value at each time\nis also included (green dashed line and right axis). Note that\neach field value is kept constant for 90 ps (9 ×105steps) of\nsimulation time. In (b), the time average value of Mz,⟨Mz⟩\n(blue dotted line), obtained from the last 30 ps of simulation,\nis also added for comparison.\nation period is observed, specially in the region around\nthe switching field, before Mzreaches a stable value af-\nter approximately 45 ps. This is better appreciated in\nFigure 1(b), where a zoom in the region around the first\nmagnetization switching is displayed. In this figure, we\nhave also included the average value of Mz,⟨Mz⟩, which\nis obtained in our simulations from the last 30 ps of each\nstep, where a well stabilized magnetization is observed.\nWe note that tsim= 90 ps is similar to simulation times\nused in micromagnetic and ASD calculations of hystere-\nsis loops [20, 25–27, 65, 66]. We have checked that longer\nsimulation times do not significantly affect the final re-\nsults, as it is shown in Figure S3 in the SM. It is crucial\nthat our magnetization dynamics is well described near\nthe switching field values. For fields of 0.5 T, the preces-\nsion period is ∼72 ps, and one would need a few ns of\nsimulation time for low damping. However, thanks to the\nhigh damping values discussed below, shorter simulation\ntimes can be employed.\nUsing tsim= 90 ps and frequency f0/4, we calculate the\nhysteresis loops for the cases ϕ= 0◦andϕ= 90◦. These\ncurves, along with snapshots showing typical spin config-\nurations at different stages of the process, are shown in\nFigure 2. As we can see, the system behaves qualitatively\naccording to what the SW model predicts, i.e. all spins\nare roughly in sync like a single macrospin [16, 17]. This\nis expected for low-temperature simulations, but devia-\ntions would occur at higher temperatures. Still, Figure 3\nshows that there is a roughly Gaussian distribution of\nspin values, an appreciable deviation from the simpler\nmacrospin assumption. The spin orientation histograms\nin this figure correspond to the state of the system de-\nscribed by the snapshot (c) of the ϕ= 90◦case of Figure 2\n(bottom right).7\n-1.0- 0.50 .00 .51 .0-1.0-0.50.00.51.0-\n1.0- 0.50 .00 .51 .0-1.0-0.50.00.51.0φ\n = 90°Mzµ\n0H (T)(a)(\nc)(\nd)(b)φ = 0°M\nzµ\n0H (T)(a)(\nb)(\nc)(\nd)\nφ\n = 0°\nφ  = 90°\nFigure 2. (Color online.) Top images: resulting hysteresis loops obtained at the converged frequency for ϕ= 0◦(top left) and\nϕ= 90◦(top right). Bottom images: snapshots of a fraction of the system, showing typical spin configurations at different\nstages of the hysteresis loop for the case ϕ= 0◦(bottom left) and ϕ= 90◦(bottom right). Each snapshot, (a), (b), (c) and\n(d), corresponds to the points marked on the hysteresis curves (upper images). In the snapshots, the atoms are represented\nas red spheres and the spins are colored according to their orientation along the field direction z. Each cubic cell represents a\nfraction of the system with a volume of (8 ˚A)3, extracted from the center of the original system.\nAnother observation is the behavior of the magnetiza-\ntion components around H≈0 (being Hthe modulus\nofH) for the case ϕ= 90◦. In Figure 1 we can see\nthat when this field takes values close to zero, Mz≈0\n(and also My≈0) while Mxtakes values close to sat-\nuration. This means that, at this stage, all spins are\naligned approximately along the xdirection (the easy\naxis of magnetization for the uniaxial anisotropy), show-\ning that the simulation reproduces well the expected be-\nhavior for ϕ= 90◦. The snapshots in Figure 2 (right\npanels) confirm this interpretation.\nSince these tests confirm that our simulations are con-\nsistent with some general physical properties of the sys-\ntem, from now on we set tsim= 90 ps.B. Hysteresis loops and the effect of field frequency\nAs it was argued in Section II, it is not possible to cal-\nculate hysteresis loops at typical very low experimental\nfrequencies using SLD simulations, due to intrinsic time\nlimitations of atomistic simulation approaches. Instead,\nwe are limited to working with high frequencies in the\norder of MHz. However, high frequency hysteresis loops\nappear to converge to a limit curve as the frequency is\ndecreased and, as the following analysis shows, this limit\nloop should not be so different from those measured ex-\nperimentally or calculated theoretically. Following this\nline of argumentation, we calculate the hysteresis loops\nfor different frequencies. In Figure 4, we show how the\narea of the hysteresis loops depends on the field frequency\nfor the cases with ϕ= 0◦andϕ= 45◦. Note that we ex-\nclude the case ϕ= 90◦from this analysis, since for this\nangle the curves do not present an hysteretic behavior.\nThe curves in Figure 4 exhibit a typical crossover from8\n/s78\n/s115\n/s120/s78\n/s115\n/s121/s78\n/s115\n/s122\nFigure 3. (Color online.) Histograms of individual spins ori-\nentations sx,sy,szalong the three axes. The histograms\ncorrespond to the state of the system described in the snap-\nshot (c) of Figure 2 for the case ϕ= 90◦.\na high frequency dynamic regime (for which the area of\nthe loops is large) to a low frequency dynamic regime (a\nweak variation of the area with field frequency) [67].\nGiven that the simulation time for a given field value\nis always the same and, in principle, long enough to cover\nspin relaxation for high damping, higher frequency would\nonly mean larger changes in the discrete field values. The\nlarger the field jumps, the larger the change across the\nenergy landscape, and the system might not adapt fast\nenough within the 90 ps of simulated time, decreasing\nspin switching probability and generating wider loops.\nThe magnitude of the jumps around the region of the\nswitching field are ∼0.3 T for f0and∼0.06 T for f0/4.\nTherefore, the stabilization of the frequency for ϕ= 0◦\noccurs when the field jumps are about one order of mag-\nnitude lower than the value of the converged coercive\nfield.\nForϕ= 45◦, we found convergence at a frequency of\nf0/2 while for the ϕ= 0◦case it is not until a frequency of\nf0/4 that the loop area reaches a stable value. At these\nfrequencies the area of the loops coincides (within the\nstatistical errors) with the predictions of the SW model.\nFigure S2 in the SM shows the corresponding hystere-sis curves calculated at several different frequencies for\nthe case ϕ= 0◦, evidencing convergence as frequency de-\ncreases. This is similar to results showing convergence as\nloop time increases in ASD simulations [68] for simula-\ntion times similar to our SLD runs. This analysis shows\nthat, although the convergence frequencies found in this\nstudy are much higher than those typically used in ex-\nperiments, they are low enough to let the magnetization\nequilibrate with the field direction at each field step. This\nmeans that, at these rates, our simulated hysteresis loops\nshould not present large discrepancies with experiments,\nas discussed for example by Westmoreland et al. [27].\n/s48/s46/s48/s48 /s48/s46/s50/s53 /s48/s46/s53/s48 /s48/s46/s55/s53 /s49/s46/s48/s48/s48/s46/s56/s49/s46/s48/s49/s46/s50/s49/s46/s52\n/s32/s102/s32/s61/s32/s48/s176\n/s32/s102/s32/s61/s32/s52/s53/s176/s65/s114/s101/s97/s32/s47/s32/s65/s114/s101/s97/s32/s83/s87\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s32 /s102/s32/s47 /s102\n/s48/s41\nFigure 4. (Color online.) Loop area vs field frequency for the\ncases with ϕ= 0◦andϕ= 45◦. The values of the area are\nnormalized by the area of the SW loop.\nIn Figure 5 we compare the hysteresis loops calculated\nforϕ= 0◦,ϕ= 45◦, and ϕ= 90◦at a single frequency\nf0/4, with those of the SW model [3]. As we can see,\na good agreement is observed in all cases. Nevertheless,\nit is important to stress that we do not expect that the\nsimulation curves should fit exactly the theoretical ones.\nUnlike the SW model (for which the macrospin approxi-\nmation is considered at T= 0), the SLD simulations are\ncarried out at low but finite temperature for many mag-\nnetic atomic moments which, along a hysteresis cycle, do\nnot rotate coherently in a perfect way (see Fig. 3). For\nexample, for the ϕ= 0◦case, we do not obtain a rect-\nangular curve because thermal fluctuations and also the\nexistence of many degrees of freedom, make it easier for\nthe magnetization to change its orientation at an exter-\nnal field value slightly lower than predicted by the SW\nmodel.9\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s102 /s32/s61/s32/s57/s48/s176/s102 /s32/s61/s32/s52/s53/s176/s77\n/s122\n/s109\n/s48/s72/s32/s40/s84/s41/s102 /s32/s61/s32/s48/s176\nFigure 5. (Color online.) Hysteresis curves calculated with\nSLD simulations (full lines) at different angles ϕcompared to\nthe theoretical ones according to the SW model (dashed lines).\nThe SLD curves correspond to the average curve obtained\nfrom many different cycles. In all cases a field frequency f=\nf0/4 was used.\nC. Anisotropy\nAccording to the SW model, the coercive field and,\ntherefore, the area of the hysteresis loop when the exter-\nnal field is applied along the easy axis ( ϕ= 0◦), is pro-\nportional to the anisotropy constant K1in Eq. (4). We\nhave explored this dependency in our SLD simulations by\nreducing and increasing the original anisotropy constant\nvalue (throughout this test called K∗\n1= 35 µeV/atom)\nby a factor of 10. Figure 6 (a) shows that the resulting\nsimulated loops qualitatively exhibit the expected behav-\nior, that is, the loop broadens (narrows) according to the\nincrease (reduction) of the anisotropy constant. This is\nfurther confirmed in Figure 6 (b) where it can be seen\nthat the loop area follows an approximately linear be-\nhavior as a function of K1/K∗\n1, deviating slightly from\nthe SW model prediction only for small values of this\nconstant. These deviations arise from the combination of\ntwo effects. On the one hand, this is due to the fact that\nsimulated loops are “rounded” near the point of magneti-\nzation reversal, deviating from perfect rectangular loops\nas explained above. The discrepancy with the SW model\narea becomes increasingly important as the loop narrows.\nOn the other hand, as it was mentioned in the methods\nsection, large anisotropy helps stabilizing the magnetiza-\ntion of the system [26]. For the case of low anisotropy\n(K1= 0.1K∗\n1) there is poor stabilization of Mzduring the\n90 ps of simulation of each field step (specially near the\nregion of magnetization switching), as seen in Figure S6.\nMuch longer simulation times would be required for low\nanisotropy values. Given that our smallest anisotropy is\nthe Fe bulk anisotropy, we could compare with some ex-\nperimental results. Magneto-optic Kerr effect measure-ments of Fe(100) appear to give a slightly lower coercive\nfield than our simulation estimate [69], although those\nmeasurements were obtained at higher temperatures and\nfor multi-domain samples.\nWe have also analyzed the influence of anisotropy\nsymmetry. Loops calculated with SLD simulations for\nsystems with uniaxial and cubic anisotropies, Eqs. (4)\nand (5), are shown in Figure 7 for the case ϕ= 0◦. We\ncompare these curves to the corresponding one for the\nSW model with uniaxial anisotropy. The reason is that\natT= 0 K both simulation curves should coincide with\nthe latter. Our results show that for ϕ= 0◦the hysteresis\nloop does not depend on the type of anisotropy. There-\nfore, the small differences between the curves shown in\nFigure 7 can be attributed to the effect of tempera-\nture, which is more pronounced in the case of cubic\nanisotropy. The area of the loop is slightly smaller for cu-\nbic anisotropy, similar to the results reported in Ref. [65].\nUsov and Peschany [70] considered the SW model for\nuniaxial and cubic anisotropies, for randomly oriented\nnanoparticles. They report that the maximum normal-\nized coercive field in both cases is 1, in agreement with\nour result. For a random collection of anisotropy orien-\ntations, the resulting coercive field for cubic anisotropy\nwas∼0.68 of the one for uniaxial.\nIt is more interesting to analyze the difference between\nuniaxial and cubic anisotropies, when the field is ap-\nplied in the [111] direction. For this field orientation,\nwe consider a SW model, the Hamiltonian of which has\nan arbitrary anisotropy term. We start by applying a\nstrong enough external field such that the Hamiltonian\nhas a single minimum with the magnetization Mpoint-\ning in the direction of H. Then, we decrease the in-\ntensity of the field using small jumps. At each step we\nuse a steepest descent method to try to escape from the\nenergy minimum, so that if it becomes unstable, a new\nminimum can be reached. Using this simple algorithm,\nwe have simulated the zero temperature dynamics of the\nSW model with both uniaxial and cubic anisotropies. As\nshown in Figure 8, in this case the corresponding hystere-\nsis loops for the SW model are very different. However,\nas expected, the loops calculated with the SLD simula-\ntions agree very well with the curves for the SW model\nfor both cubic and uniaxial anisotropies. In the case of\ncubic anisotropy, while the remanent magnetization for\nboth SW and SLD curves is ±1/√\n3, there is a quali-\ntative difference between them for field values greater\nthan the coercive one: a small hysteresis behavior present\nin the loop of the SW model is not well reproduced in\nthe SLD simulation. The reason for this is that the en-\nergy barrier separating these states is of approximately\n0.3µeV/atom, several orders of magnitude smaller than\nkBT∼8.6×10−3eV. Only a simulation at much lower\ntemperature would be able to reproduce this behavior.\nThe cubic case has a lower coercive field, which could\nbe somewhat expected from results for a collection of\nnanoparticles with random orientation [70].\nTo conclude this section, we mention that for a cou-10\nFigure 6. (Color online.) (a) Loops for different values of uniaxial anisotropy K1. (b) Loop area normalized to the area of the\nloop with K∗\n1= 35 µeV/atom, against K1/K∗\n1. The dashed line is the prediction of the SW model.\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s77\n/s122\n/s109\n/s48/s72/s32/s40/s84/s41/s32/s85/s110/s105/s97/s120/s105/s97/s108/s32/s45/s32/s83/s76/s68\n/s32/s67/s117/s98/s105/s99/s32/s45/s32/s83/s76/s68\n/s32/s85/s110/s105/s97/s120/s105/s97/s108/s32/s45/s32/s83/s87/s102/s32/s61/s32/s48/s176\n/s32 /s102\n/s48/s32/s47/s32/s52\nFigure 7. (Color online.) Hysteresis loops for uniaxial (red\nfull line) and cubic (blue circles) anisotropies for the ϕ= 0◦\ncase and frequencyf0/4. The results are compared to the SW\nprediction for uniaxial anisotropy (red dashed line).\nple of cases, we have tested a lattice-dependent Neel ´s\nanisotropy within the framework of Nieves et al. [52] and\nthe results obtained are similar to the ones for uniaxial\nor cubic anisotropy presented above.\nD. Exchange interactions\nThe exchange energy does not play a role in the SW\nresults, and within that framework there should be no\neffect of the exchange interaction on the simulated hys-\nteresis loops. To test this prediction within the SLD\nframework, we have run simulations with different val-\n/s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53/s45/s49/s48/s49/s77/s32/s97/s108/s111/s110/s103/s32/s116/s104/s101/s32/s102/s105/s101/s108/s100/s32/s100/s105/s114/s101/s99/s116/s105/s111/s110\n/s109\n/s48/s72/s32/s40/s84/s41/s32/s85/s110/s105/s97/s120/s105/s97/s108/s32/s45/s32/s83/s87 \n/s32/s67/s117/s98/s105/s99/s32/s45/s32/s83/s87 \n/s32/s85/s110/s105/s97/s120/s105/s97/s108/s32/s45/s32/s83/s76/s68\n/s32/s67/s117/s98/s105/s99/s32/s45/s32/s83/s76/s68/s32/s102/s32/s187/s32/s53/s52/s46/s55/s176\n/s32/s102\n/s48 /s32/s47/s32/s52Figure 8. (Color online.) Hysteresis loops for an external field\napplied along the [111] direction ( ϕ≈54.7◦) and frequency\nf0/4, for cubic and uniaxial anisotropy. The loops obtained\nwith SLD are compared to the predictions of an extension of\nthe SW model that incorporates cubic anisotropy (red short-\ndotted line), and to the SW with uniaxial anisotropy (green\ndot-dashed line).\nues of the function J(rij). In Figure 9, we compare the\ncurves obtained using the original exchange J(rij)∗, with\nthe ones obtained by increasing or reducing this inter-\naction by a factor of 10. As it can be seen, loops are\nnearly unaffected by these changes. Increasing the mag-\nnitude of the exchange interaction is expected to give\nhigher magnetization values and broader loops [71], but\nat low temperatures (as in this case) this effect is ex-\npected to be very small. Larger differences should be\nnoticed at higher temperatures where thermal fluctua-11\ntions start to play a major role in the magnetization be-\nhavior. Nevertheless, even at these low temperatures,\na few subtle effects can be noticed: (i) for the simula-\ntion with J(rij) = 0 .1×J(rij)∗a small but consistent\nreduction of the saturation magnetization is observed,\nand (ii) for J(rij) = 10 ×J(rij)∗the loop becomes more\nrectangular. This last effect is also observed in other\ntheoretical approaches based on the SW model which\ninclude exchange interactions through a mean field ap-\nproach as in Refs. [72, 73]. Lower/higher exchange will\nlead to lower/higher Curie temperature ( TC) and spin\nordering will be modified. For the same value of the ex-\nternal field, the histogram of spin values is narrower and\nwith a higher mean value for the higher exchange, facil-\nitating the macrospin behavior, leading to a larger satu-\nration field and helping with the sudden spin flip which\ncauses a more square loop.\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s49/s48/s49/s77\n/s122\n/s109\n/s48/s72/s32/s40/s84/s41/s32/s74/s40/s114\n/s105/s106/s41/s42\n/s32/s48/s46/s49/s120/s74/s40/s114\n/s105/s106/s41/s42\n/s32/s49/s48/s120/s74/s40/s114\n/s105/s106/s41/s42/s32/s102/s32 /s61/s32/s48/s176\n/s32/s102\n/s48/s32/s47/s32/s52\nFigure 9. (Color online.) Simulated hysteresis curves for the\ncaseϕ= 0◦considering different magnitudes of the exchange\ncoupling.\nE. Damping parameter\nThe value of the Gilbert damping parameter in the sLL\nequation is usually chosen to be in the range 0 .01−1. Ex-\nperimental estimates for Fe are around 0.001. However,\nthe possibility of increasing the spin damping values for\nnumerical convenience is often explored. In SLD simula-\ntions a value of λs= 0.1 was used in Ref. [34]. ASD [74]\nand micromagnetic [20, 65] simulations use values in the\nrange 0 .1−1.\nThe dynamic of a single spin in a magnetic field is well\nestablished, and there are studies discussing the role of\ndamping and simulation times [23]. However, obtaining\nspin relaxation times at a given damping for correlated\nmulti-spin systems at finite temperature will typically re-\nquire simulations. In general, as the damping increases,there is a faster energy loss, and the relaxation time re-\nquired for the system to reach a steady state decreases,\nwith spins aligned with the preferred orientation deter-\nmined by the external field and the anisotropy. There-\nfore, it is possible to use higher frequencies to calculate\nthe hysteresis loops, which become narrower as damp-\ning grows. In the same spirit, in most of the simulations\npresented here we use a larger damping value, λs= 0.5.\nUsing lower damping is possible but would require lower\nfrequencies as discussed below. The parameter λsis re-\nlated to the optimal value of the simulation run time tsim.\nIn micromagnetic simulations, it has been proposed to\nuse both very high damping and high loop sweep rates\n(SR), in order to achieve computational efficiency [20].\nThe rationale behind this proposal was as follows. The\nmagnitude of the coercive field is a function of the mea-\nsurement time, and the attempt frequency for spin flips.\nThis measurement time can be related to the SR, and\nthe flip frequency can be assumed to be proportional\nto the damping. Therefore, loops calculated with the\nsame value of SR /λswould have the same coercive field\nwhich gives the loop width. Based on this, Behbahani et\nal.were able to use high SR in their simulations to ob-\ntain hysteresis loops that closely match those obtained\nat very low SR, by employing extremely high damping\nvalues, and large time steps. Recently, this methodol-\nogy was successfully applied to simulate hysteresis of iron\noxide magnetic nanoparticles with application to hyper-\nthermia [66].\nIn Figure 10 we use a field frequency f0for the case\nwith ϕ= 90◦. For λs= 0.5 the hysteresis loop in (a) is\nreasonably close to the SW prediction, and the magne-\ntization versus time in (b) indicates reasonable conver-\ngence. However, for λs= 0.1, the loop in (a) is poorly\ndefined, as expected from the lack of convergence towards\nstable magnetization values shown at the top of panel (b).\nThe possibility of reducing the simulation time in mi-\ncromagnetic simulations by increasing both the SR and\nthe damping, leaving constant the ratio SR /λs, would\nbe equivalent, in our case, to keeping constant the ratio\nf/λs. However, SLD simulations at high damping might\ndepart from the desired dynamics and should be treated\nwith care. In our simulations, by choosing λs= 0.5, we\nwere able to set tsim= 90 ps and use f0/4 as discussed at\nthe beginning of this section. For lower damping values,\nsmaller frequencies would be required, significantly in-\ncreasing the computational costs. Following the previous\nscaling, for instance, for λs= 0.1, around 400 ps (4 ×106\nsteps) are required to stabilize the magnetization at each\nfield value, in contrast to the 90 ps (9 ×105steps) that\nare required for λs= 0.5. We have run some simulations\nfor such a low damping and nanosecond steps, and ver-\nified that this is indeed the case, explaining the lack of\nconvergence in Figure 10(a).12\nFigure 10. (Color online.) (a) Hysteresis curves for the case ϕ= 90◦with a field frequency f0and damping parameters λs= 0.1\n(dashed line) and λs= 0.5 (full line). The plot shows several cycles for each case. (b) Time evolution of the components of\nthe magnetization throughout the simulation of half a cycle for the cases considered in (a). The field value at each time is also\nincluded (green dashed line and right axis). Note that each field value is kept constant for 90 ps (9 ×105steps) of simulation\ntime.\nF. Coupling the lattice dynamics\nAll the results presented so far correspond to frozen-\nlattice simulations, meaning that the lattice vibrations\nare not included in the dynamics. We now analyze the\neffect of full SLD calculations actually coupling the spin\nand lattice degrees of freedom, which we refer to as\nmoving-lattice simulations. We analyze this effect for two\ndifferent systems, perfect bcc bulk samples and nanopar-\nticle samples including defects.\n1. Low temperature bulk simulations\nIn Figure 11 we compare the simulated hysteresis loops\nat T=10 K for ϕ= 0◦obtained with (moving-lattice) and\nwithout (frozen-lattice) spin-lattice coupling at different\nfrequencies. The former gives lower coercivity and nar-\nrower loops in comparison with the frozen-lattice case.\nThis effect is more pronounced for higher frequencies.\nThis is consistent with the fact that in a system that in-\ncorporates new degrees of freedom (in this case those of\nthe lattice), the energy barrier that must be overcome\nto reverse the magnetization should decrease. In other\nwords, when a reversal field is applied, a moving lattice\nprovides additional paths for the magnetization relax-\nation process. However, this effect is small at a frequency\nf0/4 where a stable magnetization is easily achieved for\nthe chosen sweep rate, showing that the lattice dynamics\ncontributions are not very important at these low tem-\nperatures ( T= 10 K). As it was previously shown for\niron nanoparticles at equilibrium, the lattice fluctuations\nwould become significant at temperatures higher than300 K [34, 45]. The equilibrium magnetization for a mov-\ning lattice is always smaller than that of a frozen lattice\nfor all temperatures [45]. In addition, in the present ap-\nproach, the anisotropy contributions do not depend on\nthe atomic positions nor the lattice temperature. Larger\ndifferences between moving and frozen lattice simulations\nmight be observed for Hamiltonians that include addi-\ntional terms which are sensitive to factors like atomic\nvolume [52, 75], or “effective” local electronic density [50].\n2. Intermediate and high temperatures bulk simulations\nIn this subsection, we focus on intermediate tempera-\ntures, above ∼1/3 of the Curie temperature and also\nhigh temperatures, close to TC. Lattice-induced spin\nfluctuations enter in this model through the interatomic\ndistance dependence of the exchange function J(rij) in\nthe magnetic Hamiltonian Eq. (2). At low temperatures,\nchanges in J(rij) due to lattice vibrations are negligible.\nTherefore, the coupled spin-lattice dynamics show little\ninfluence in hysteresis loops in that regime, as it was\nshown in Figure 11 and discussed in the previous sub-\nsection. To address the effect of the coupled spin-lattice\ndynamics at higher temperatures, we have run simula-\ntions for the case of uniaxial anisotropy with the easy\naxis aligned with the external field, ϕ= 0◦, at several\ntemperatures, in the range of T= 300 −1000 K. For the\nparametrization of J(rij) used in this work, TCis around\n1050 K, as discussed in Section II A. Given the large fluc-\ntuations observed at these temperatures, we consider for\nthese simulations a larger bulk system, including around\n65000 atoms, as mentioned in the Methods section, in or-\nder to avoid finite-size effects, and use longer simulation13\n/s102\n/s48/s77\n/s122\n/s32/s76/s97/s116/s116/s105/s99/s101/s32/s109/s111/s118/s105/s110/s103\n/s32/s76/s97/s116/s116/s105/s99/s101/s32/s102/s114/s111/s122/s101/s110\n/s102\n/s48 /s32/s47/s32/s50/s77\n/s122\n/s102\n/s48 /s32/s47/s32/s52/s77\n/s122\n/s109\n/s48/s72/s32/s40/s84/s41\nFigure 11. (Color online.) Comparison of the hysteresis loops\nobtained with (lattice moving) and without (lattice frozen)\ncoupling to the lattice dynamics at T= 10K. All the figures\ncorrespond the case ϕ= 0◦at different field frequencies, f0\n(top), f0/2 (middle) and f0/4 (bottom). The inset in the top\npanel shows histograms of the spins orientation along the field\ndirection zand correspond to the state of the system just\nbefore the beginning of the magnetization reversal marked\nwith circles in the respective curves.\ntimes (180 ps per point) in order to avoid the undesirable\nregime in which the magnetization cannot equilibrate to\nthe field direction before the next field increment.\nIn Figure 12, we compare the hysteresis loops ob-\ntained at these conditions under the two different sim-\nulation approaches, lattice-frozen and lattice-moving, for\nT= 500 K and T= 1000 K contrasted to the previous re-\nsults at T= 10 K. A small difference, consistent with the\nresults at 10 K and with the arguments outlined in the\nprevious subsection, can be observed between the lattice\nfrozen and lattice moving loops at T= 500 K. However,\nthis difference is within the margin of error given by the\nstandard deviation of the average by which the curves\nare obtained. This is because, at these relatively high\ntemperatures, the effect of increased lattice vibrations\non exchange and spin fluctuations is small compared to\nthe large fluctuations caused by the thermal noise in the\nstochastic field ζ(t) in Eq.(8). This is confirmed by the\nspin histograms in the inset of Figure S7 in the supple-\nmentary material, where a very similar dispersion of spinorientations is evidenced for both frozen and moving lat-\ntice cases. A more pronounced difference between the\nfrozen and moving lattice protocols is observed for the\nloops calculated at T= 1000 K. For the moving lattice\nsimulations, the Curie temperature is close to 1050 K (as\nit was mentioned above), while for the frozen lattice ones\nTC∼1150 K [45]. Consequently, in the moving lattice\nprocedure, the system is closer to a state where hysteresis\nshould vanish and, therefore, present loops with smaller\narea than those of the lattice frozen simulations.\nDespite the small differences obtained between the lat-\ntice moving and lattice frozen approaches, the effect of\ntemperature is well captured by the present simulation\nmethod and protocol. At higher temperatures, smaller\nloops are obtained, i.e., smaller coercivity and saturation\nmagnetization as the temperature is increased, following\nthe expected behavior.\nWe note that the SW model is sometimes applied at fi-\nnite temperatures, despite its lack of validity under those\nconditions [53], by re-scaling the saturation magnetiza-\ntion to match values at the desired temperature. If the\nsame procedure was applied to our case, there would be\nstrong disagreement between the model and the simula-\ntions, which represents a warning for such future com-\nparison between the SW model and experiments at finite\ntemperature.\n/s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s49/s48/s48/s48/s32/s75/s49/s48/s32/s75/s77\n/s122\n/s109\n/s48/s72/s32 /s40/s84/s41/s53/s48/s48/s32/s75\nFigure 12. (Color online.) Comparison of the hysteresis loops\natT= 10 K, T= 500 K and T= 1000 K for lattice frozen\nand lattice moving procedures. These results correspond to\nthe case ϕ= 0◦. For the simulations at T= 500 ,1000 K a\nsmaller sweep rate, SR= 0.5×108T/s (which corresponds\nto 180 ps of simulation time per point) was used.\n3. Nanoparticles simulations\nMagnetic nanoparticles are of technological interest,\nand their nanostructure will affect magnetization. For14\nexample, the microstrain state inside nanoparticles can\ngreatly modify their magnetic properties [76], including\ncases where lattice deformation can be driven by va-\ncancies [77]. As another example of defects influencing\nmagnetization, experiments suggest that defects might\nplay a role in Fe NPs remaining blocked, as opposed to\ndefect-free NP which would behave superparamagneti-\ncally [11, 12].\nMost atomistic spin dynamics simulations consider a\nfrozen perfect lattice, even for cases with defects, like\nNP surfaces [23, 78–80], interfaces [68, 81], and point de-\nfects [77]. Such approach allows for the use of discrete ex-\nchange values for 1st, 2nd, etc. nearest neighbors, greatly\nsaving computational time. There are isolated efforts to\ninclude lattice distortions for a frozen lattice, which re-\nquire the capability of handling magnetic interactions at\narbitrary distances and not just regular lattice positions.\nSpatial configurations from molecular dynamics can be\nused as input [82], or set with other criteria, like in the\nrecent study of a single dislocation dipole in bulk Cr [83].\nSometimes, defects have been emulated by more symmet-\nric configurations, like the the simulation of vacancies by\nweakly coupled ferromagnetic spin pairs, without consid-\nering lattice strain [77]. In addition, these codes will not\nallow for dynamical lattice effects, including evolution of\ndefects with time due to temperature, stress, and cou-\npling to magnetic degrees of freedom. For a NP, surface\nroughness might vary considerably with temperature.\nIn order to explore the role of lattice dynamics, we sim-\nulate hysteresis loops for pristine and defective Fe NPs\nsamples. In principle, one could also compare the mag-\nnetization of the defective NP with a frozen lattice with\nexactly the same NP with a moving lattice. Here we\nfocus on the extreme comparison of a pristine NP with\nfrozen lattice and a defective NP with moving lattice,\ni.e coupling the spin and lattice dynamics, given that we\nwould like to emphasize the potential of the method com-\npared to the current standard simulations that generally\nlack the inclusion of MD simulations configurations and\nemploy a fixed, ideal lattice structure. Figure 13 shows\nthe “perfect” NP used for the frozen lattice simulations,\nalongside the Fe NP with defects. It can be seen that\nthe defective NP has a non-spherical topology, with large\nplanar facets, and also includes a twin boundary, and a\nfew vacancy clusters. Figure 14 shows the histograms of\nmagnetization values over the entire simulation, covering\nseveral hysteresis cycles for both NPs. The differences\nbetween the two NPs is large, with the defective NP expe-\nriencing significantly larger and more frequent magneti-\nzation fluctuations. This suggests that, for a NP, defects\nwhich can be included in spin-lattice dynamics display\ndifferent qualitative and quantitative behavior than the\nabsence of defects. The larger fluctuations lead to nar-\nrower hysteresis loops, as qualitatively expected. Figure\n15 shows hysteresis loops for both NPs, with the defec-\ntive NP displaying lower saturation magnetization and\nlower coercive field, resulting in a significantly lower loop\narea, approximately one-quarter that of the perfect NP.We obtain a coercive field around 50 mT for the pristine\nfrozen NP and around 18 mT for the defective NP at 500\nK. This is for a single, oriented NP, with a diameter of 8\nnm. Loops from other experiments [11] show smaller co-\nercive fields, around 3 mT. However, our value for the our\ndefective NP shows very good agreement with an experi-\nmental value of around 20 mT for a collection of Fe NP of\nthe same size as in MD, at 300 K [10]. Furthermore, the\nexperimental coercive field is approximately half of what\nwe have obtained for the pristine NP. Given that the co-\nercive field of a random collection of NP is expected to\nresult in a coercive field of about half that for oriented\nNP [3], the agreement is reasonable. In addition, the hys-\nteresis loop of a prismatic Fe NP (210 ×210×15nm3)\ngives Hc∼75 mT for the field at 45◦, according to recent\nmicromagnetic simulations [84], also indicating that our\nsimulations obtain reasonable coercive field values. For\nexperimental measurement times of seconds, NP below\nsome critical size are expected to behave superparam-\nagnetically at high temperatures [3, 7], but for our NP\nsize and simulation times of ns, the NPs behave ferro-\nmagnetically. Recent experiments observed that, within\na collection of Fe NP with 5-20 nm diameter, many be-\nhaved with the expected superparamagnetism, but some\nremained in a blocked ferromagnetic state [11, 12]. De-\nfects were assumed to be the cause for the ferromagnetic\nbehavior, opposite to what we find in our simulations,\nwhere the defects considered here facilitate magnetiza-\ntion flips and decrease the coercive field. These prelimi-\nnary simulations of pristine and defective NPs highlight\nthe importance of SLD calculations. The introduction\nof defects, and the possibility that they can evolve and\nmigrate during the simulation time, can change the mag-\nnetic behavior of such samples.\nIV. DISCUSSION\nGiven the short timestep required to integrate the\natomic degrees of freedom, usually ∼0.1 fs - 1 fs, one\nneeds relatively long simulations to achieve a stable mag-\nnetization at a given value of the applied magnetic field.\nTypically, this would involve several precession periods.\nUsing the Larmor frequency for Fe gives a period of\n∼36 ps for a field of 1 T. In order to speed-up the spin-\ndynamics convergence, a Gilbert damping λs= 1.0 is\nused in most ASD simulations [25–28] and values much\nlarger than 1 have been used in micromagnetic simula-\ntions of hysteresis loops [20]. In our simulations, we\nfound that for a given damping, as frequency decreases,\nthe loop area also decreases until reaching a constant\nvalue. For bulk iron, and a Gilbert damping of 0 .5, we\nfound that frequencies of 125 MHz or lower provide such\na constant loop area, for simulation times of 90 ps for\neach field. Values between 30 and 200 ps (depending on\nthe system under study and the Gilbert damping em-\nployed) are also used for ASD simulations [25, 26, 28].\nFor the case in which the anisotropy axis and the field15\nFigure 13. (Color online) Snapshots of Fe nanoparticles employed in the hysteresis loop simulations. The upper panel ((a) -\n(d)) shows snapshots of the pristine NP (8 nm diameter spherical NP) while the lower panel ((e) - (h)) shows the corresponding\nsnapshots for the defective NP. In (a) and (e) the entire NP is shown, with atoms colored by PTM structure type. Surface\natoms are classified as “other” (unknown structure). In (b) and (f) a slice near the center of the NP is shown, also colored by\nPTM structure type. In (f), atoms in a twin boundary crossing the NP, and around a vacancy are also classified as “other”.\nFigures (c) and (g) show the same slice as (b) and (f), but atoms are colored according to their PTM atomic orientation along\nthe x-axis, to better show the twin rotation in the defective NP. Finally, in (d) and (h) a surface mesh is shown for the NP\nsurface, a few single vacancies, and vacancy clusters inside the defective NP.\ndirection are perpendicular, ϕ= 90◦, a converged loop\ncan be found at even higher frequencies. Agreement be-\ntween SLD and SW suggests that loops using this limiting\nfrequency would not be much different from the exper-\nimental or theoretical estimates. This result, partially\nsolves the problem related to the computational costs in\natomistic simulations of hysteresis loops, allowing to re-\nproduce experimental hysteresis loops, running SLD sim-\nulations at much higher frequencies than those typically\nused in experiments. Large computer clusters allow for\nsimulations of increasingly larger systems by distribut-\ning spatial scales, but time scale cannot be split amongst\ndifferent parallel processes. Therefore, new approaches\nfor accelerated dynamics [85], bypassing this time scale\nproblem, will be required for more efficient simulations\nof hysteresis loops with SLD.\nWhen atomic motion is coupled to spin dynamics we\nfound that, for the bulk simulations, the lattice dynam-\nics contributions depend on the loop frequency, but they\nare typically small at low and intermediate temperatures,\nbut more notorious at higher temperatures close to TC.\nAt low temperatures (10 K) atoms vibrations are negli-\ngible. At intermediate temperatures (300 −500 K), fluc-tuations in spin-spin exchange J(rij) produced by the\natomic motion are outweighed by the thermal fluctua-\ntions induced by the stochastic term in the sLL equation\nof the spin dynamics. In the case of the NP simula-\ntions we find significant differences, both qualitative and\nquantitative, between the hysteresis loops of a pristine\nNP with a frozen lattice and the loops foor a defective\nNP with a moving lattice, both with a spin temperature\nofT= 500 K.\nAlthough the effects of the coupled spin-lattice dynam-\nics are small for bulk bcc iron, in more complex systems\nthat exceed the scope of this work, there might be dynam-\nical effects that would result in different hysteresis loops\nif the lattice and spin dynamics are coupled. Also, future\nstudies could include a quantum thermostat to improve\nthe description of thermodynamic properties at low tem-\nperatures [86]. Equations of motion which would provide\nangular momentum conservation might be explored too\n[50, 51].\nSLD simulations could be applied to hysteresis loops\nfor other systems of technological interest, such as more\ncomplex nanoparticles [14], including core-shell nanopar-\nticles with realistic interfaces [87], magnetostrictive ma-16\n/s45/s48/s46/s54 /s45/s48/s46/s52 /s45/s48/s46/s50 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54/s48/s49/s50/s51/s52/s53/s54/s55/s67/s111/s117/s110/s116\n/s77\n/s122/s32/s68/s101/s102/s101/s99/s116/s105/s118/s101\n/s32/s80/s114/s105/s115/s116/s105/s110/s101\nFigure 14. (Color online.) Histograms of the magnetization\nalong the field direction ( Mz), for the defective and the pris-\ntine NPs. The histograms were built with the data obtained\ndirectly from the entire simulation and are normalized to unit\narea.\n/s45/s48/s46/s50 /s45/s48/s46/s49 /s48/s46/s48 /s48/s46/s49 /s48/s46/s50/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s32/s68/s101/s102/s101/s99/s116/s105/s118/s101/s32/s78/s80\n/s32/s80/s114/s105/s115/s116/s105/s110/s101/s32/s78/s80/s77\n/s122\n/s109\n/s48/s72/s32/s40/s84/s41\nFigure 15. (Color online.) Hysteresis loops comparing de-\nfective and pristine nanoparticles (NPs). The hysteresis loop\narea of the defective NP is nearly 4 times smaller than that\nof the pristine NP.\nterials with magnetic microstructures [88], compounds\nlike CrN, where the spin-lattice coupling is responsible for\nthe unusual temperature dependence of the thermal con-\nductivity [36], and bulk systems with defects [28, 45, 89–\n91]. Further examples include strained antiferromag-\nnetic materials that exhibit different responses to exter-\nnal fields [92] and other materials where strain affects\nthe magnetic properties [93, 94]. Finally, recent theoreti-\ncal studies suggest topological magnon phase transitions\ntuned by time-dependent strains in 2D materials [95]. Inbinary alloys like Fe-Cr the formation of stable precip-\nitates is known to occur at certain concentrations, and\nthis would dynamically affect magnetic properties [96],\nas well as magnetically driven phase transformation [97].\nMagnetic alloys can display complex microstructure, in-\ncluding dislocations, stacking faults, twins, etc., which\ngreatly affects their magnetic properties [98]. In particu-\nlar, High Entropy Alloys (HEA) are materials of techno-\nlogical interest, with a complex microstructure, including\ndislocations, twins, precipitates, and different crystallo-\ngraphic phases, which leads to a complex magnetic be-\nhavior [99], pressure-induced phase stabilities, and mag-\nnetovolume effects [100]. For all of the above cases, SLD\ncould provide information which can inform micromag-\nnetic and atomistic spin dynamics simulations.\nV. SUMMARY AND CONCLUSIONS\nIn this study, we apply a systematic approach toward\nincorporating Spin-Lattice Dynamics in the simulation\nof hysteresis phenomena. Each step of the process was\nthoroughly evaluated and optimized to ensure the accu-\nracy and reliability of the simulation results. We used\nSLD simulations to calculate the hysteresis loops of a\nbulk ferromagnetic system at different temperatures and\nthe hysteresis loops of bcc Fe NPs with the inclusion of\nstructural defects. SLD can include temperature effects\n(thermal spin fluctuations as well as lattice vibrations)\nand lattice defects in a relatively simple way, comple-\nmenting spin-dynamic or micromagnetic simulations.\nWe have also performed formal calculations to derive\nthe Fokker-Planck equation associated to the set of cou-\npled Langevin equations Eqs.(6)-(8). From this, we de-\nduce the correct expressions for the amplitudes of the\nwhite noise correlations that allow the system to reach\nan asymptotic stationary Gibbs-Boltzmann equilibrium\nstate.\nFrom our simulations at a temperature of 10 K, spins\nbehave almost like a macrospin, but there is some spread\nin the magnetization values, associated with thermal ef-\nfects and exchange interactions. The exchange energy,\nwhich is much larger than the Zeeman or anisotropy en-\nergies, does not significantly affect the hysteresis loops at\nlow temperatures. The coercive field depends linearly on\nthe magnitude of the uniaxial anisotropy, as in the SW\nmodel. Cubic anisotropy and uniaxial anisotropy pro-\nduce nearly the same loops for the external field aligned\nwith an easy anisotropy axis. However, there are impor-\ntant differences for the case of ϕ≈54.7◦misalignment,\nfor which the applied field lies along the [111] direction.\nFor this case, cubic anisotropy leads to a lower coercivity\nvalue.\nIn order to validate our low-temperature SLD results,\nwe compared to the Stoner-Wohlfarth model, which is\nwidely used in experimental and simulation studies. In\nthe SW model, there is no lattice, no temperature, and\nno exchange interactions. The model assumes uniax-17\nial anisotropy and Zeeman energy contributions, and\nspins in the volume are assumed to behave like a sin-\ngle macrospin. We compared the outcome of the SLD\nsimulations at T= 10 K to the predictions of the SW\nmodel for different angles between the external field and\nthe anisotropy axis. We analyzed the effects of several pa-\nrameters like simulation run time, field frequency, damp-\ning, exchange, anisotropy and lattice-vibrations. We\nfound that the low-temperature loops obtained with SLD\nagree very well with those predicted by the SW model,\nprovided that several parameters are chosen with care.\nOnce this agreement is established, validating our model,\nwe study the effect of the coupled spin-lattice dynam-\nics on the intermediate and high temperature hysteresis\nloops where the SW model is no longer valid.\nOur study employs several approximations, including\na classical Heisenberg Hamiltonian, but it can capture\nreasonably well the behavior of lattice defects that mod-\nify magnetic properties. In particular, we are able to\nmodel the high-temperature evolution of a nanoparti-\ncle with both point and extended defects, where such\ndefects lower the overall magnetization and significantly\nreduce the area of the hysteresis loop, compared to a typ-\nical frozen lattice approximation. This was qualitatively\nexpected, given that disorder hinders global magnetiza-\ntion and, under varying external fields, helps fluctuations\nwhich would reduce loop area. However, quantitative\nsimulations can certainly contribute to the understand-\ning of experimental results, and with the future design of\ntailored magnetization in NP.\nIn order to model complex materials of interest, it is\ndesirable to have reliable simulation techniques and pro-\ntocols for hysteresis processes that incorporate the effects\nof coupled lattice and spin dynamics and this study would\nbe a contribution in that direction. While most of our re-\nsults correspond to the already well-studied case of pure\nbulk Fe, our approach provides a framework for future\nstudies to efficiently use SLD for the study of hysteresis\nbehavior in magnetic materials in general.\nACKNOWLEDGMENTS\nWe thank Romina Aparicio for her valuable contri-\nbution through support simulations and helpful discus-\nsions. GDS and EMB thank support from a SIIP-\nUNCUYO-2022-2023 grant, from PICTO-UUMM-2019-\n00048 and from PIP 2021-2023 11220200102578CO. We\nthank Sergei Dudarev for sharing his valuable notes on\nspin Langevin damping. We thank computer run time\nin the cluster TOKO (toko.uncu.edu.ar). The authors\nthank support from IPAC-2019 grant from Sistema Na-\ncional de Computaci´ on de Alto Desempe˜ no (SNCAD),\nfor run time in the cluster Dirac http://dirac.df.uba.ar/.\nFR acknowledges financial support from CONICET (Ar-\ngentina) under Project No. PIP 112-202001-01294-CO\nand Universidad Nacional de San Luis (Argentina) un-\nder Project No. PROICO 03-2220. SC benefited froman Erasmus+ Grant during his stay at LPTHE, Paris.\nAppendix\nThe aim of this Appendix is to derive the Fokker-\nPlanck equation associated to the set of coupled Langevin\nequations, Eqs.(6)-(8), which rule the translational and\nrotational dynamics of the magnetic moments. From the\nFokker-Planck equation, we then deduce the amplitudes\nof the white noise correlations, Eqs.(10) and (12), that\nallow the system to reach an asymptotic stationary state\nof equilibrium Gibbs-Boltzmann form.\nFor the sake of completeness, we work with a Langevin\nequation for spins in which we add the stochastic field to\nboth the gyromagnetic and the relaxation terms, with a\nparameter Acontrolling the presence or absence of noise\nin the latter dissipative mechanism. In this way, for A=\n1 we obtain the sLLG equation while for A= 0 we recover\nour sLL Eq.(8) [22].\n1. Langevin formulation\nWe take a set of i= 1, . . . , N particles at positions\nri= (ri\nx, ri\ny, ri\nz), with momenta pi= (pi\nx, pi\ny, pi\nz) and\nmagnetic moments si= (si\nx, si\ny, si\nz). As in the main text,\nwe label a,b, orcthe space coordinates x,y, and z.\nSince the number of components of the magnetic mo-\nments are also three we use the labels a, b, c for them as\nwell. The magnetic moments are normalised such that\n|si|2= (si\na)2= 1 for each particle i. Here and in the fol-\nlowing we use Einstein summation notation over repeated\na, b, c indices; not over i, jparticle indices, and we write\nthe sums over these indices explicitly when needed.\nThe stochastic Langevin equations of motion Eqs.(6)-\n(8) are recast in the generic form [101]\ndtri\na=pi\na\nmi, (A.1)\ndtpi\na=NX\nj(̸=i)\u0014\n−∂V(rij)\n∂ria+∂J(rij)\n∂riasi·sj\u0015\n−γL\nmipi\na+ξi\na, (A.2)\ndtsi\na=gi\nabωi\nb+gi\nabζi\nb, (A.3)\nwhere dtdenotes the time derivative d/dt andωi\nb=\n−1\nℏ∂Hmag\n∂si\nbis the component bof the effective field act-\ning on spin i. The noises affecting the translation and\nmagnetic degrees of freedom, ξi\naandζi\na, are independent\nGaussian white random variables with zero mean and\ncorrelations\n⟨ξi\na(t)ξj\nb(t′)⟩= 2DLδabδijδ(t−t′), (A.4)\nand\n⟨ζi\na(t)ζj\nb(t′)⟩= 2Dsδabδijδ(t−t′). (A.5)18\nWe will fix the coefficients DLandDsbelow. The ma-\ntrices gi\nabandgi\nabare defined as\ngi\nab=1\n1 +λ2s[ϵabcsi\nc+λs\nsi(s2\niδab−si\nasi\nb)],(A.6)\ngi\nab=fi\nab+Ahi\nab, (A.7)\nwith\nfi\nab=1\n1 +λ2sϵabcsi\nc=1\n2(gi\nab−gi\nba), (A.8)\nhi\nab=λs\nsi1\n1 +λ2s(s2\niδab−si\nasi\nb)\n=1\n2(gi\nab+gi\nba). (A.9)\nHere, fandhare the antisymmetric and symmetric\nparts of g, respectively, and ϵabcis the completely an-\ntisymmetric Levi–Civita tensor. Note that for A= 0\nEq. (A.3) corresponds to the sLL Eq. (8). Instead, since\nfi\nab+hi\nab=gi\nab, forA= 1 we have that gi\nab=gi\naband it\nis then easy to show that Eq. (A.3) reduces to the well-\nknown sLLG equation. Importantly enough, we have re\ninserted the modulus of the local spins, si, since although\nit is fixed to be one, its variations with respect to the vari-\nous spin components are non-trivial and have to be taken\ninto account in the calculations that follow. For instance,\n∂s2\ni\n∂sia= 2sa,∂s2\ni\n∂(sia)2= 6 (A.10)\nfor each spin i. In the second equation a sum over the\nthree acomponents was implicit and led to a factor 3.\n2. The Fokker-Planck Equation\nLet us join the momenta, position and magnetic vari-\nables of all particles, each with three components, in a\nsingle α= 1, . . . , 9Ncomponent vector\ny= (p1, . . . ,pN,r1, . . . ,rN,s1, . . . ,sN). (A.11)\nStarting from the Chapman-Kolmogorov equation\nP(y, t+∆t) =Z\ndy0P(y, t+∆t|y0, t)P(y0, t),(A.12)\nwe obtain the Fokker-Planck description from the\nLangevin one by using the definition of P(y, t) via the\nDirac delta function,\nP(y, t+ ∆t|y0, t) =⟨δ(y−yξ,ζ(t+ ∆t))⟩.(A.13)\nThis compact Dirac delta notation indicates a product\nover all the 9 Ncomponents of the vector ywhich is\nforced to take the value given by the solution to the\nLangevin equations. The mean value ⟨. . .⟩is taken over\nthe two noise sources and yξ,ζ(t+ ∆t) is the solution\nto the Langevin equations (A.1)-(A.3) evaluated at timet+∆t, with initial condition y(t) =y0, which clearly de-\npends on the noises. We now expand yξ,ζ(t+∆t) around\ny0since the time increment ∆ tis small and the yincre-\nment ∆ yas well. Using y=y0+ ∆y,\nP(y, t+ ∆t|y0, t) = δ(y−y0)−∂α(δ(y−y0)⟨∆yα⟩)\n+1\n2∂α∂β(δ(y−y0)⟨∆yα∆yβ⟩)\n+O(∆t2), (A.14)\nwith summation over repeated α, β= 1, . . . , 9Nindices.\nCombining Eqs. (A.12) and (A.14), and integrating over\ny0, one gets\nP(y, t+ ∆t) = P(y, t)−∂α(⟨∆yα⟩P(y, t))\n+1\n2∂α∂β(⟨∆yα∆yβ⟩P(y, t))\n+O(∆t2). (A.15)\nTaking the limit for the differential of P,\n∂tP(y, t) = lim\n∆t→0P(y, t+ ∆t)−P(y, t)\n∆t, (A.16)\nand eliminating any term of higher order than ∆ tin\nthe right-hand-side of Eq. (A.15), we obtain the Fokker-\nPlanck equation\n∂tP(y, t) =−∂α\u0014⟨∆yα⟩\n∆tP(y, t)\u0015\n+1\n2∂α∂β\u0014⟨∆yα∆yβ⟩\n∆tP(y, t)\u0015\n.(A.17)\nThe next step is to calculate the averages ⟨∆yα⟩and\n⟨∆yα∆yβ⟩to leading order in ∆ tusing the Langevin\nEqs. (A.1)-(A.3) which read, in discrete time,\n∆ri\na≡ri\na(t+ ∆t)−ri\na(t) =pi\na\nmi∆t , (A.18)\n∆pi\na≡pi\na(t+ ∆t)−pi\na(t)\n=NX\nj(̸=i)\u0014\n−∂V(rij)\n∂ria+∂J(rij)\n∂riasi·sj\u0015\n∆t\n−γL\nmipi\na∆t+ξi\na∆t , (A.19)\n∆si\na≡si\na(t+ ∆t)−si\na(t)\n=gi\nabωi\nb∆t+gi\nabζi\nb∆t . (A.20)\nAll variables in the right-hand-sides of these equations\nare evaluated at the mid-point ymp= [y(t)+y(t+∆t)]/2\nsince we chose to work with the Stratonovich prescrip-\ntion, the unique scheme consistent with the conservation\nof the modulus of the magnetization, in the way we wrote\nthe equations [22, 101]. When working at O(∆t) we will\nbe able to, in some cases, replace this mid-point by the\ninitial one y0in the interval [ t, t+ ∆t].19\nThe averages of the position increments or the product\nof two position increments are\n⟨∆ri\na⟩=⟨pi\na∆t\nmi⟩=pi\na\nmi∆t+O(∆t2),(A.21)\n⟨∆ri\na∆rj\nb⟩=O(∆t2). (A.22)\nBesides, for the momentum and magnetisation compo-nents we obtain, respectively,\n⟨∆pi\na⟩=NX\nj(̸=i)\u0014\n−∂V(rij)\n∂ria+∂J(rij)\n∂riasi·sj\u0015\n∆t\n−γL\nmipi\na∆t+O(∆t3/2), (A.23)\n⟨∆pi\na∆pj\nb⟩= 2DLδabδij∆t+O(∆t3/2),\n(A.24)\nand\n⟨∆si\na⟩=gi\nabωi\nb∆t−2Ds1 + (Aλs)2\n(1 +λ2s)2si\na∆t\n+O(∆t3/2), (A.25)\n⟨∆si\na∆sj\nb⟩= 2Ds1 + (Aλs)2\n(1 +λ2s)2(s2\niδab−si\nasi\nb)δij∆t\n+O(∆t3/2). (A.26)\nThe averages over the cross products are sub-leading\nsince the noises ξi\naandζi\naare not correlated. Replacing\nnow Eqs. (A.21)-(A.26) into Eq.(A.17), we finally find\nthe explicit Fokker-Planck equation\n∂tP=−∂\n∂ria\u0012pi\na\nmiP\u0013\n−∂\n∂pia\n\nNX\nj(̸=i)\u0014\n−∂V(rij)\n∂ria+∂J(rij)\n∂riasi·sj\u0015\nP−γL\nmipi\naP\n\n+1\n2∂\n∂pia∂\n∂pi\nb(2DLδabP)\n−∂\n∂sia\u0014\ngi\nabωi\nbP−2Ds1 + (Aλs)2\n(1 +λ2s)2si\naP\u0015\n+1\n2∂\n∂sia∂\n∂si\nb\u0014\n2Ds1 + (Aλs)2\n(1 +λ2s)2\u0000\ns2\niδab−si\nasi\nb\u0001\nP\u0015\n, (A.27)\nwhere we recall that there is a sum over repeated aandbindices, and here also the iindex.\n3. Stationary solution\nWe now check whether the Gibbs-Boltzmann distribution Peq=Z−1e−H/kBT, where Zis the partition function,\nis a stationary solution of the Fokker-Planck Eq. (A.27). Such a stationary Pdoes not depend on time and should\nsatisfy ∂tP= 0, i. e.\n0 =−NX\nipi\na\nmi∂P\n∂ria−NX\ni̸=j\u0014\n−∂V(rij)\n∂ria+∂J(rij)\n∂riasi·sj\u0015∂P\n∂pia+ 3NX\niγL\nmiP+NX\niγL\nmipi\na∂P\n∂pia+DLNX\ni∂2P\n∂(pia)2\n+2λs\nsi1\n(1 +λ2s)NX\nisi\nbωi\nbP−NX\nigi\nabωi\nb∂P\n∂sia+ 6DsN1 + (Aλs)2\n(1 +λ2s)2P+ 2Ds1 + (Aλs)2\n(1 +λ2s)2NX\nisi\na∂P\n∂sia\n+Ds1 + (Aλs)2\n(1 +λ2s)2\"\n−6NP−4NX\nisi\na∂P\n∂sia+NX\ni\u0000\ns2\niδab−si\nasi\nb\u0001∂2P\n∂sia∂si\nb#\n, (A.28)20\nwhere we wrote the sum over iexplicitly. Now, considering that ∂αPeq=−Peq\nkBT∂αHand assuming that P=Peq, it\nfollows that the translational terms in Eq. (A.28) are\n0 =−P\nkBTX\ni̸=jpi\na\nmi\u0014∂J(rij)\n∂riasi·sj−∂V(rij)\n∂ria\u0015\n+P\nkBTX\ni̸=j\u0014\n−∂V(rij)\n∂ria+∂J(rij)\n∂riasi·sj\u0015pi\na\nmi\n+3PNX\niγL\nmi−γLP\nkBTNX\ni\u0012pi\na\nmi\u00132\n+DLP\nkBT\"\n−NX\ni3\nmi+1\nkBTNX\ni\u0012pi\na\nmi\u00132#\n. (A.29)\nThe first line automatically cancels out while the second line leads to the well-known dynamical fluctuation-dissipation\nrelation\nDL=γLkBT . (A.30)\nThis is the exact relationship as derived by Einstein which perfectly matches Eq. (10).\nOn the other hand, for the magnetic degrees of freedom and after trivial cancellations of terms we obtain\n0 = 2λs\nsi1\n(1 +λ2s)NX\nisi\nbωi\nbP−NX\nigi\nabωi\nb∂P\n∂sia\n+Ds1 + (Aλs)2\n(1 +λ2s)2\"\n−2NX\nisi\na∂P\n∂sia+NX\ni\u0000\ns2\niδab−si\nasi\nb\u0001∂2P\n∂sia∂si\nb#\n. (A.31)\nReplacing gi\nabby its explicit expression and the derivatives of P,∂P/∂si\na=βℏωi\naPand∂2P/∂si\na∂sj\nb= (βℏ)2ωi\naωj\nbP,\nwe have that\n0 = 2λs\nsi1\n(1 +λ2s)NX\nisi\nbωi\nbP−1\n1 +λ2sNX\ni[ϵabcsi\nc+λs\nsi(s2\niδab−si\nasi\nb)]ωi\nbωi\naℏ\nkBTP\n+Ds1 + (Aλs)2\n(1 +λ2s)2\"\n−2NX\nisi\naωi\naℏ\nkBTP+NX\ni\u0000\ns2\niδab−si\nasi\nb\u0001\nωi\nbωi\na\u0012ℏ\nkBT\u00132\nP#\n. (A.32)\nThe term proportional to ϵabc vanishes because\nϵabcωi\naωi\nbsi\nc=ωi·(ωi×si) = 0. Instead, the remaining\nterms cancel if we choose\nDs=λs(1 +λ2\ns)\n[1 + ( Aλs)2]kBT\nℏ, (A.33)\nwe have already set si= 1. This is the fluctuation-\ndissipation relation for the magnetic degrees of freedom.Note that for A= 0 we obtain Eq. (12). However, it is im-\nportant to highlight that Eq. (A.33) is different from the\none reported in Ref. [37], which is valid only for damp-\ning values λs≪1. We have checked, with simulations\nnot shown here, that this is the correct expression that\nallows one to use large values of the damping constant\nλsand let the magnetic degrees freedom equilibrate with\nthe environment.\n[1] G. Bertotti, Hysteresis in Magnetism (Academic Press,\nElsevier, London, 1998).\n[2] I. Mayergoyz, Mathematical Models of Hysteresis and their Applications\n(Academic Press, Elsevier, London, 2003).\n[3] B. D. Cullity and C. D. Graham,\nIntroduction to magnetic materials (John Wiley &\nSons, New Jersey, 2011).\n[4] B. Mehdaoui, A. Meffre, J. Carrey, S. Lachaize, L.-M.\nLacroix, M. Gougeon, B. Chaudret, and M. Respaud,\nOptimal size of nanoparticles for magnetic hyperther-\nmia: a combined theoretical and experimental study,Advanced Functional Materials 21, 4573 (2011).\n[5] Z. Shaterabadi, G. Nabiyouni, and M. Soleymani,\nPhysics responsible for heating efficiency and self-\ncontrolled temperature rise of magnetic nanoparticles in\nmagnetic hyperthermia therapy, Progress in biophysics\nand molecular biology 133, 9 (2018).\n[6] Q. A. Pankhurst, J. Connolly, S. K. Jones, and\nJ. Dobson, Applications of magnetic nanoparticles in\nbiomedicine, Journal of physics D: Applied physics 36,\nR167 (2003).\n[7] Z. Hedayatnasab, F. Abnisa, and W. M. A. W.21\nDaud, Review on magnetic nanoparticles for magnetic\nnanofluid hyperthermia application, Materials & Design\n123, 174 (2017).\n[8] A. Barman, S. Mondal, S. Sahoo, and A. De, Magne-\ntization dynamics of nanoscale magnetic materials: A\nperspective, Journal of Applied Physics 128(2020).\n[9] Z. Ma, J. Mohapatra, K. Wei, J. P. Liu, and S. Sun,\nMagnetic nanoparticles: Synthesis, anisotropy, and ap-\nplications, Chemical Reviews 123, 3904 (2021).\n[10] T. Ibusuki, S. Kojima, O. Kitakami, and Y. Shimada,\nMagnetic anisotropy and behaviors of fe nanoparticles,\nIEEE transactions on magnetics 37, 2223 (2001).\n[11] A. Balan, P. M. Derlet, A. F. Rodr´ ıguez, J. Bansmann,\nR. Yanes, U. Nowak, A. Kleibert, and F. Nolting, Di-\nrect observation of magnetic metastability in individual\niron nanoparticles, Physical review letters 112, 107201\n(2014).\n[12] A. Kleibert, A. Balan, R. Yanes, P. M. Derlet, C. A.\nVaz, M. Timm, A. F. Rodr´ ıguez, A. B´ ech´ e, J. Verbeeck,\nR. S. Dhaka, etal., Direct observation of enhanced mag-\nnetism in individual size-and shape-selected 3 d transi-\ntion metal nanoparticles, Physical Review B 95, 195404\n(2017).\n[13] X. Batlle, C. Moya, M. Escoda-Torroella, `O. Iglesias,\nA. F. Rodr´ ıguez, and A. Labarta, Magnetic nanopar-\nticles: From the nanostructure to the physical proper-\nties, Journal of Magnetism and Magnetic Materials 543,\n168594 (2022).\n[14] A. Lak, S. Disch, and P. Bender, Embracing defects and\ndisorder in magnetic nanoparticles, Advanced Science 8,\n2002682 (2021).\n[15] F. Liorzou, B. Phelps, and D. Atherton, Macroscopic\nmodels of magnetization, IEEE Transactions on Mag-\nnetics 36, 418 (2000).\n[16] E. C. Stoner and E. Wohlfarth, A mechanism of mag-\nnetic hysteresis in heterogeneous alloys, Philosophical\nTransactions of the Royal Society of London. Series A,\nMathematical and Physical Sciences 240, 599 (1948).\n[17] C. Tannous and J. Gieraltowski, The stoner–wohlfarth\nmodel of ferromagnetism, European journal of physics\n29, 475 (2008).\n[18] L. Lanci and D. V. Kent, Introduction of thermal acti-\nvation in forward modeling of hysteresis loops for single-\ndomain magnetic particles and implications for the in-\nterpretation of the day diagram, Journal of Geophysical\nResearch: Solid Earth 108(2003).\n[19] G. Bertotti, I. Mayergoyz, and C. Serpico,\nNonlinear magnetization dynamics in nanosystems\n(Elsevier, Oxford, 2008).\n[20] R. Behbahani, M. L. Plumer, and I. Saika-Voivod,\nCoarse-graining in micromagnetic simulations of dy-\nnamic hysteresis loops, Journal of Physics: Condensed\nMatter 32, 35LT01 (2020).\n[21] W. F. Brown, Thermal fluctuations of a single-domain\nparticle, Physical Review 130, 1677 (1963).\n[22] J. L. Garc´ ıa-Palacios and F. J. L´ azaro, Langevin-\ndynamics study of the dynamical properties of small\nmagnetic particles, Physical Review B 58, 14937 (1998).\n[23] R. F. Evans, W. J. Fan, P. Chureemart, T. A. Ostler,\nM. O. Ellis, and R. W. Chantrell, Atomistic spin\nmodel simulations of magnetic nanomaterials, Journal\nof Physics: Condensed Matter 26, 103202 (2014).\n[24] O. Eriksson, A. Bergman, L. Bergqvist, and J. Hellsvik,Atomistic Spin Dynamics (Oxford University Press,\nOxford, 2017).\n[25] D. B¨ ottcher, A. Ernst, and J. Henk, Atomistic magne-\ntization dynamics in nanostructures based on first prin-\nciples calculations: application to co nanoislands on cu\n(111), Journal of Physics: Condensed Matter 23, 296003\n(2011).\n[26] J. Alzate-Cardona, D. Sabogal-Su´ arez, R. Evans, and\nE. Restrepo-Parra, Optimal phase space sampling for\nmonte carlo simulations of heisenberg spin systems,\nJournal of Physics: Condensed Matter 31, 095802\n(2019).\n[27] S. C. Westmoreland, C. Skelland, T. Shoji, M. Yano,\nA. Kato, M. Ito, G. Hrkac, T. Schrefl, R. F. Evans,\nand R. W. Chantrell, Atomistic simulations of α-\nfe/nd2fe14b magnetic core/shell nanocomposites with\nenhanced energy product for high temperature perma-\nnent magnet applications, Journal of Applied Physics\n127, 133901 (2020).\n[28] S. Jenkins, R. W. Chantrell, and R. F. Evans, Atom-\nistic origin of the athermal training effect in granu-\nlar irmn/cofe bilayers, Physical Review B 103, 104419\n(2021).\n[29] T. Tajiri, H. Deguchi, M. Mito, K. Konishi, S. Miya-\nhara, and A. Kohno, Effect of size on the magnetic\nproperties and crystal structure of magnetically frus-\ntrated dymn 2 o 5 nanoparticles, Physical Review B\n98, 064409 (2018).\n[30] R. Essajai, Y. Benhouria, A. Rachadi, M. Qjani, A. Mz-\nerd, and N. Hassanain, Shape-dependent structural and\nmagnetic properties of fe nanoparticles studied through\nsimulation methods, RSC advances 9, 22057 (2019).\n[31] T. Shimada, K. Ouchi, I. Ikeda, Y. Ishii, and T. Kita-\nmura, Magnetic instability criterion for spin–lattice sys-\ntems, Computational Materials Science 97, 216 (2015).\n[32] P.-W. Ma, S. Dudarev, and C. Woo, Spilady: A parallel\ncpu and gpu code for spin–lattice magnetic molecular\ndynamics simulations, Computer Physics Communica-\ntions 207, 350 (2016).\n[33] X. Wu, Z. Liu, and T. Luo, Magnon and phonon dis-\npersion, lifetime, and thermal conductivity of iron from\nspin-lattice dynamics simulations, Journal of Applied\nPhysics 123, 085109 (2018).\n[34] G. Dos Santos, R. Aparicio, D. Linares, E. Miranda,\nJ. Tranchida, G. Pastor, and E. Bringa, Size-and\ntemperature-dependent magnetization of iron nanoclus-\nters, Physical Review B 102, 184426 (2020).\n[35] G. dos Santos, R. Meyer, R. Aparicio, J. Tranchida,\nE. M. Bringa, and H. M. Urbassek, Spin-lattice dynam-\nics of surface vs core magnetization in fe nanoparticles,\nApplied Physics Letters 119, 012404 (2021).\n[36] I. Stockem, A. Bergman, A. Glensk, T. Hickel,\nF. K¨ ormann, B. Grabowski, J. Neugebauer, and\nB. Alling, Anomalous phonon lifetime shortening in\nparamagnetic crn caused by spin-lattice coupling: a\ncombined spin and ab initio molecular dynamics study,\nPhysical Review Letters 121, 125902 (2018).\n[37] J. Tranchida, S. Plimpton, P. Thibaudeau, and A. P.\nThompson, Massively parallel symplectic algorithm for\ncoupled magnetic spin dynamics and molecular dynam-\nics, Journal of Computational Physics 372, 406 (2018).\n[38] A. P. Thompson, H. M. Aktulga, R. Berger, D. S.\nBolintineanu, W. M. Brown, P. S. Crozier, P. J. in’t\nVeld, A. Kohlmeyer, S. G. Moore, T. D. Nguyen, etal.,22\nLammps-a flexible simulation tool for particle-based\nmaterials modeling at the atomic, meso, and continuum\nscales, Computer Physics Communications 271, 108171\n(2022).\n[39] http://lammps.sandia.gov/doc/fix_nve_spin.html .\n[40] H. Chamati, N. Papanicolaou, Y. Mishin, and D. Papa-\nconstantopoulos, Embedded-atom potential for fe and\nits application to self-diffusion on fe (1 0 0), Surface\nScience 600, 1793 (2006).\n[41] NIST interatomic potentials repository, https:\n//www.ctcms.nist.gov/potentials/entry/\n2006--Chamati-H-Papanicolaou-N-I-Mishin-Y-Papaconstantopoulos-D-A--Fe/ .\n[42] T. Kaneyoshi, Introduction to amorphous magnets\n(World Scientific Publishing Company, Singapore,\n1992).\n[43] K. Yosida, D. C. Mattis, and K. Yosida,\nTHEORY OF MAGNETISM.: Edition en anglais,\nVol. 122 (Springer Science & Business Media, Berlin,\n1996).\n[44] P.-W. Ma, C. Woo, and S. Dudarev, Large-scale simula-\ntion of the spin-lattice dynamics in ferromagnetic iron,\nPhysical review B 78, 024434 (2008).\n[45] R. Meyer, G. dos Santos, R. Aparicio, E. M. Bringa,\nand H. M. Urbassek, Influence of vacancies on the\ntemperature-dependent magnetism of bulk fe: A spin-\nlattice dynamics approach, Computational Condensed\nMatter , e00662 (2022).\n[46] G. Pastor, J. Dorantes-D´ avila, S. Pick, and H. Dreyss´ e,\nMagnetic anisotropy of 3 d transition-metal clusters,\nPhysical review letters 75, 326 (1995).\n[47] J.-Y. Chauleau, T. Chirac, S. Fusil, V. Garcia,\nW. Akhtar, J. Tranchida, P. Thibaudeau, I. Gross,\nC. Blouzon, A. e. Finco, etal., Electric and antifer-\nromagnetic chiral textures at multiferroic domain walls,\nNature materials 19, 386 (2020).\n[48] A. Frej, C. S. Davies, A. Kirilyuk, and A. Stupakiewicz,\nPhonon-induced magnetization dynamics in Co-doped\niron garnets, Applied Physics Letters 123, 042401\n(2023).\n[49] M. Strungaru, M. O. Ellis, S. Ruta, O. Chubykalo-\nFesenko, R. F. Evans, and R. W. Chantrell, Spin-lattice\ndynamics model with angular momentum transfer for\ncanonical and microcanonical ensembles, Physical Re-\nview B 103, 024429 (2021).\n[50] W. Dednam, C. Sabater, A. E. Botha, E. B. Lom-\nbardi, J. Fern´ andez-Rossier, and M. J. Caturla, Spin-\nlattice dynamics simulation of the einstein–de haas\neffect, Computational Materials Science 209, 111359\n(2022).\n[51] J. R. Cooke III and J. R. Lukes, Angular momentum\nconservation in spin-lattice dynamics simulations, Phys-\nical Review B 107, 024419 (2023).\n[52] P. Nieves, J. Tranchida, S. Arapan, and D. Legut, Spin-\nlattice model for cubic crystals, Physical Review B 103,\n094437 (2021).\n[53] J. Wang, H. Duan, X. Lin, V. Aguilar, A. Mosqueda,\nand G.-m. Zhao, Temperature dependence of magnetic\nanisotropy constant in iron chalcogenide fe3se4: Excel-\nlent agreement with theories, Journal of applied physics\n112, 103905 (2012).\n[54] M. Goiriena-Goikoetxea, D. Mu˜ noz, I. Orue,\nM. Fern´ andez-Gubieda, J. Bokor, A. Muela, and\nA. Garc´ ıa-Arribas, Disk-shaped magnetic particles for\ncancer therapy, Applied Physics Reviews 7, 011306(2020).\n[55] J. Carrey, B. Mehdaoui, and M. Respaud, Simple mod-\nels for dynamic hysteresis loop calculations of magnetic\nsingle-domain nanoparticles: Application to magnetic\nhyperthermia optimization, Journal of applied physics\n109, 083921 (2011).\n[56] R. Hergt, R. Hiergeist, I. Hilger, W. A. Kaiser, Y. Lap-\natnikov, S. Margel, and U. Richter, Maghemite nanopar-\nticles with very high ac-losses for application in rf-\nmagnetic hyperthermia, Journal of Magnetism and\nMagnetic Materials 270, 345 (2004).\n[57] A. V. Ivanov, V. M. Uzdin, and H. J´ onsson, Fast and\nrobust algorithm for energy minimization of spin sys-\ntems applied in an analysis of high temperature spin\nconfigurations in terms of skyrmion density, Computer\nPhysics Communications 260, 107749 (2021).\n[58] J. Amodeo and L. Pizzagalli, Modeling the mechanical\nproperties of nanoparticles: a review, Comptes Rendus.\nPhysique 22, 35 (2021).\n[59] C. J. Ruestes, A. Stukowski, Y. Tang, D. Tramontina,\nP. Erhart, B. Remington, H. Urbassek, M. A. Meyers,\nand E. M. Bringa, Atomistic simulation of tantalum\nnanoindentation: Effects of indenter diameter, pene-\ntration velocity, and interatomic potentials on defect\nmechanisms and evolution, Materials Science and Engi-\nneering: A 613, 390 (2014).\n[60] E. R. Hopper, C. Boukouvala, D. N. Johnstone, J. S.\nBiggins, and E. Ringe, On the identification of twin-\nning in body-centred cubic nanoparticles, Nanoscale 12,\n22009 (2020).\n[61] A. Stukowski, Visualization and analysis of atomistic\nsimulation data with ovito–the open visualization tool,\nModelling and Simulation in Materials Science and En-\ngineering 18, 015012 (2009).\n[62] P. M. Larsen, S. Schmidt, and J. Schiøtz, Robust struc-\ntural identification via polyhedral template matching,\nModelling and Simulation in Materials Science and En-\ngineering 24, 055007 (2016).\n[63] A. Stukowski, Computational analysis methods in atom-\nistic modeling of crystals, Jom 66, 399 (2014).\n[64] M. Pajda, J. Kudrnovsk` y, I. Turek, V. Drchal, and\nP. Bruno, Ab initio calculations of exchange interac-\ntions, spin-wave stiffness constants, and curie temper-\natures of fe, co, and ni, Physical Review B 64, 174402\n(2001).\n[65] D. Aur´ elio and J. Vejpravova, Understanding magneti-\nzation dynamics of a magnetic nanoparticle with a dis-\nordered shell using micromagnetic simulations, Nano-\nmaterials 10, 1149 (2020).\n[66] R. Behbahani, M. L. Plumer, and I. Saika-Voivod, Mul-\ntiscale modelling of magnetostatic effects on magnetic\nnanoparticles with application to hyperthermia, Jour-\nnal of Physics: Condensed Matter 33, 215801 (2021).\n[67] M. I. Dolz, S. D. Calder´ on Rivero, H. Pastoriza, and\nF. Rom´ a, Magnetic hysteresis behavior of granular man-\nganite la0 .67ca0 .33mno3 nanotubes, Physcial Review B\n101, 174425 (2020).\n[68] P. Chureemart, R. Evans, R. Chantrell, P.-W. Huang,\nK. Wang, G. Ju, and J. Chureemart, Hybrid design\nfor advanced magnetic recording media: Combining\nexchange-coupled composite media with coupled gran-\nular continuous media, Physical Review Applied 8,\n024016 (2017).\n[69] J. Bansmann, M. Getzlaff, C. Westphal, and23\nG. Sch¨ onhense, Surface hysteresis curves of fe (110) and\nfe (100) crystals in ultrahigh vacuum—evidence of ad-\nsorbate influences, Journal of magnetism and magnetic\nmaterials 117, 38 (1992).\n[70] N. Usov and S. Peschany, Theoretical hysteresis loops\nfor single-domain particles with cubic anisotropy, Jour-\nnal of magnetism and magnetic materials 174, 247\n(1997).\n[71] H. Kachkachi, A. Ezzir, M. Nogues, and E. Tronc, Sur-\nface effects in nanoparticles: application to maghemite-\nfe o, The European Physical Journal B 14, 681 (2000).\n[72] D. L. Atherton and J. Beattie, A mean field stoner-\nwohlfarth hysteresis model, IEEE transactions on mag-\nnetics 26, 3059 (1990).\n[73] H.-w. Zhang, S.-y. Zhang, B.-g. Shen, and\nH. Kronm¨ uller, The magnetization behavior of\nnanocrystalline permanent magnets based on the\nstoner–wohlfarth model, Journal of magnetism and\nmagnetic materials 260, 352 (2003).\n[74] R. F. Evans, Atomistic spin dynamics, in\nHandbook of Materials Modeling: Applications: Current and Emerging Materials,\nedited by W. Andreoni and S. Yip (Springer, Cham,\n2020) pp. 427–448.\n[75] D. Perera, D. P. Landau, D. M. Nicholson, G. M. Stocks,\nM. Eisenbach, J. Yin, and G. Brown, Combined molec-\nular dynamics-spin dynamics simulations of bcc iron,\ninJournal of Physics: Conference Series, Vol. 487 (IOP\nPublishing, 2014) p. 012007.\n[76] B. Muzzi, E. Lottini, N. Yaacoub, D. Peddis, G. Bertoni,\nC. de Juli´ an Fern´ andez, C. Sangregorio, and A. L´ opez-\nOrtega, Hardening of cobalt ferrite nanoparticles by lo-\ncal crystal strain release: Implications for rare earth free\nmagnets, ACS Applied Nano Materials 5, 14871 (2022).\n[77] A. Lappas, G. Antonaropoulos, K. Brintakis, M. Vasi-\nlakaki, K. N. Trohidou, V. Iannotti, G. Ausanio,\nA. Kostopoulou, M. Abeykoon, I. K. Robinson, etal.,\nVacancy-driven noncubic local structure and magnetic\nanisotropy tailoring in fe x o- fe 3- δo 4 nanocrystals,\nPhysical Review X 9, 041044 (2019).\n[78] R. F. Evans, Atomistic spin dynamics (Springer, 2020)\npp. 427–448.\n[79] O. Hovorka, S. Devos, Q. Coopman, W. Fan, C. Aas,\nR. Evans, X. Chen, G. Ju, and R. Chantrell, The\ncurie temperature distribution of fept granular magnetic\nrecording media, Applied Physics Letters 101, 052406\n(2012).\n[80] M. Ellis and R. Chantrell, Switching times of nanoscale\nfept: Finite size effects on the linear reversal mechanism,\nApplied Physics Letters 106, 162407 (2015).\n[81] Z. Nedelkoski, D. Kepaptsoglou, L. Lari, T. Wen, R. A.\nBooth, S. D. Oberdick, P. L. Galindo, Q. M. Ramasse,\nR. F. Evans, S. Majetich, etal., Origin of reduced mag-\nnetization and domain formation in small magnetite\nnanoparticles, Scientific reports 7, 45997 (2017).\n[82] S. Westmoreland, R. Evans, G. Hrkac, T. Schrefl, G. Zi-\nmanyi, M. Winklhofer, N. Sakuma, M. Yano, A. Kato,\nT. Shoji, etal., Multiscale model approaches to the de-\nsign of advanced permanent magnets, Scripta Materialia\n148, 56 (2018).\n[83] B. Bienvenu, C. C. Fu, and E. Clouet, Interplay between\nmagnetic excitations and plasticity in body-centered cu-\nbic chromium, Physical Review B 107, 134105 (2023).\n[84] D. Sudsom, I. Juh´ asz Junger, C. D¨ opke, T. Blachowicz,\nL. Hahn, and A. Ehrmann, Micromagnetic simulationof vortex development in magnetic bi-material bow-tie\nstructures, Condensed Matter 5, 5 (2020).\n[85] S. J. Plimpton, D. Perez, and A. F. Voter, Parallel al-\ngorithms for hyperdynamics and local hyperdynamics,\nThe Journal of Chemical Physics 153, 054116 (2020).\n[86] C. Woo, H. Wen, A. Semenov, S. Dudarev, and P.-\nW. Ma, Quantum heat bath for spin-lattice dynamics,\nPhysical Review B 91, 104306 (2015).\n[87] S. D. Oberdick, A. Abdelgawad, C. Moya, S. Mesbahi-\nVasey, D. Kepaptsoglou, V. K. Lazarov, R. F. Evans,\nD. Meilak, E. Skoropata, J. van Lierop, etal., Spin cant-\ning across core/shell fe3o4/mnxfe3- xo4 nanoparticles,\nScientific Reports 8, 3425 (2018).\n[88] Y. He, J. Coey, R. Schaefer, and C. Jiang, Determina-\ntion of bulk domain structure and magnetization pro-\ncesses in bcc ferromagnetic alloys: Analysis of magne-\ntostriction in f e 83 g a 17, Physical Review Materials\n2, 014412 (2018).\n[89] D. Palanisamy, A. Kov´ acs, O. Hegde, R. E. Dunin-\nBorkowski, D. Raabe, T. Hickel, and B. Gault, Influ-\nence of crystalline defects on magnetic nanodomains in\na rare-earth-free magnetocrystalline anisotropic alloy,\nPhysical Review Materials 5, 064403 (2021).\n[90] L. Han, Z. Rao, I. R. Souza Filho, F. Maccari, Y. Wei,\nG. Wu, A. Ahmadian, X. Zhou, O. Gutfleisch, D. Ponge,\netal., Ultrastrong and ductile soft magnetic high-\nentropy alloys via coherent ordered nanoprecipitates,\nAdvanced Materials 33, 2102139 (2021).\n[91] M. Bersweiler, E. P. Sinaga, I. Peral, N. Adachi, P. Ben-\nder, N.-J. Steinke, E. P. Gilbert, Y. Todaka, A. Michels,\nand Y. Oba, Revealing defect-induced spin disorder in\nnanocrystalline ni, Physical Review Materials 5, 044409\n(2021).\n[92] J. van Rijn, D. Wang, B. Sanyal, and T. Banerjee, Strain\ndriven antiferromagnetic exchange interaction in srmno\n3 probed by phase shifted spin hall magnetoresistance,\narXiv preprint arXiv:2211.16888 (2022).\n[93] D. Li, S. Haldar, and S. Heinze, Strain-driven zero-field\nnear-10 nm skyrmions in two-dimensional van der waals\nheterostructures, Nano Letters 22, 7706 (2022).\n[94] Z. Shen, Y. Xue, Z. Wu, and C. Song, Enhanced\ncurie temperature and skyrmion stability in room tem-\nperature ferromagnetic semiconductor crise monolayer,\narXiv preprint arXiv:2207.11418 (2022).\n[95] N. Vidal-Silva and R. E. Troncoso, Time-dependent\nstrain-tuned topological magnon phase transition, Phys.\nRev. B 106, 224401 (2022).\n[96] P. Erhart, A. Caro, M. S. de Caro, and B. Sadigh, Short-\nrange order and precipitation in fe-rich fe-cr alloys:\nAtomistic off-lattice monte carlo simulations, Physical\nReview B 77, 134206 (2008).\n[97] C. Niu, C. R. LaRosa, J. Miao, M. J. Mills, and\nM. Ghazisaeidi, Magnetically-driven phase transforma-\ntion strengthening in high entropy alloys, Nature com-\nmunications 9, 1 (2018).\n[98] P. Zhao, L. Feng, K. Nielsch, and T. Woodcock,\nMicrostructural defects in hot deformed and as-\ntransformed τ-mnal-c, Journal of Alloys and Com-\npounds 852, 156998 (2021).\n[99] P. Kumari, A. K. Gupta, R. K. Mishra, M. Ahmad, and\nR. R. Shahi, A comprehensive review: Recent progress\non magnetic high entropy alloys and oxides, Journal of\nMagnetism and Magnetic Materials 554, 169142 (2022).\n[100] L. Liu, S. Huang, L. Vitos, M. Dong, E. Bykova,24\nD. Zhang, B. S. Almqvist, S. Ivanov, J.-E. Rubensson,\nB. Varga, etal., Pressure-induced magnetovolume effect\nin cocrfeal high-entropy alloy, Communications Physics\n2, 1 (2019).\n[101] C. Aron, D. G. Barci, L. F. Cugliandolo, Z. Gonza-\nlez Arenas, and G. S. Lozano, Magnetization dynam-\nics: path-integral formalism for the stochastic landau-\nlifshitz-gilbert equation, Journal of Statistical Mechan-\nics: Theory and Experiment , P09008 (2014).SUPPLEMENTARY MATERIAL\nThis supplementary material presents extra figures in-\ntended to expand the discussion and provide extra evi-\ndence and support the points discussed in the main text.\n1. Average curve\nFigure S1 presents an example of how the resulting\nhysteresis loops where obtained. The resulting curve was\nobtained by averaging several individual cycles.\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s77\n/s122\n/s109\n/s48/s72/s32/s40/s84/s41/s32/s32 /s102\n/s48/s32/s100/s105/s102/s102/s101/s114/s101/s110/s116/s32/s108/s111/s111/s112/s115\n/s32/s32 /s102\n/s48/s32/s32/s97/s118/s101/s114/s97/s103/s101/s32/s99/s117/s114/s118/s101\nFigure S1. (Color online.) Four different hysteresis loops for\ntheϕ= 0◦case at frequency f=f0plotted together with the\naverage curve (solid red line).\n2. Effect of frequency and loop convergence\nFigure S2 shows the hysteresis loops at different fre-\nquencies for the case in which the anisotropy axis and the\ndirection of the external field form an angle of ϕ= 0◦.\nThe plot evidences that the loops narrow when the fre-\nquency is decreased until the convergence frequency is\nfound and the resulting loop coincides with the Stoner-\nWohlfarth prediction (green curve).\n3. Increased simulation time\nAs it is argued in the main text, a simulation time of\ntsim= 90 ps for each field value is sufficient to produce\nreliable hysteresis loops. Figure S3 shows that increasing\nthis time by a factor of approximately 2 has no apprecia-\nble effect in the shape of the resulting curves.25\n/s45/s49/s46/s48/s48 /s45/s48/s46/s55/s53 /s45/s48/s46/s53/s48 /s45/s48/s46/s50/s53 /s48/s46/s48/s48 /s48/s46/s50/s53 /s48/s46/s53/s48 /s48/s46/s55/s53 /s49/s46/s48/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s77\n/s122\n/s109\n/s48/s72/s32/s40/s84/s41\nFigure S2. (Color online.) Hysteresis loops for the ϕ= 0◦\ncase at different field frequencies f0,f0/2,f0/4 and f0/8,\ncompared to the rectangular loop of the SW model (green\ncurve).\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s77\n/s122\n/s109\n/s48/s72/s32/s40/s84/s41/s32/s116\n/s115/s105/s109/s61/s32/s57/s48/s32/s112/s115\n/s32/s116\n/s115/s105/s109/s61/s32/s50/s48/s48/s32/s112/s115\nFigure S3. (Color online.) Hysteresis loops for the ϕ= 90◦\ncase and frequency f=f0/2 with different simulations times\n(tsim) for each point of the curve\n4. Size effects\nFigure S4 compares two individual hysteresis loops\nobtained for a bcc lattice in a cubic box of volume\nV= (10 a0)3(the system under study) and a larger sys-\ntem with V= (15 a0)3. The simulations correspond to\nthe case of ϕ= 0◦and frequency f0/4.\nFigure S5 compares the fluctuations of Mz, the compo-\nnent of the magnetization along the field direction, during\nseveral field steps for three different system volumes.\nBoth figures show that simulating a system with V=\n(10a0)3is sufficient to avoid large size-effects.\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s102/s32/s61/s32/s48/s176\n/s102\n/s48/s32/s47/s32/s52 /s32/s77\n/s122\n/s109\n/s48/s72/s32/s40/s84/s41/s32/s86/s32/s61/s32/s40/s49/s53/s97\n/s48/s41/s51\n/s32/s86/s32/s61/s32/s40/s49/s48/s97\n/s48/s41/s51Figure S4. (Color online.) Hysteresis cycles for the ϕ= 0◦\ncase and frequency f0/4 for different system sizes.\n/s50/s50 /s50/s52 /s50/s54 /s50/s56 /s51/s48 /s51/s50 /s51/s52/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s77\n/s122\n/s84/s105/s109/s101/s32/s40/s120/s57/s48/s32/s112/s115/s41/s32/s86/s32/s61/s32/s40/s53/s97\n/s48/s41/s51\n/s32/s86/s32/s61/s32/s40/s49/s48/s97\n/s48/s41/s51\n/s32/s86/s32/s61/s32/s40/s49/s53/s97\n/s48/s41/s51\n/s45/s48/s46/s57/s45/s48/s46/s56/s45/s48/s46/s55/s45/s48/s46/s54/s45/s48/s46/s53/s45/s48/s46/s52/s45/s48/s46/s51/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49\n/s32/s102/s32/s61/s32/s48/s176\n/s32 /s102\n/s48/s32/s47/s32/s52\n/s109\n/s48/s72/s32/s40/s84/s41\nFigure S5. (Color online.) Comparison of the fluctuations of\nthe component of the magnetization along the applied field di-\nrection Mzduring the time evolution of a simulation through-\nout half a cycle for different system sizes. The simulations\ncorrespond in all cases to the case of ϕ= 0◦and frequency\nf0/4. The field value at each time is also included (green\ndashed line and right axis).\n5. Anisotropy and magnetization fluctuations\nAs it was argued in the Methods section of the main\ntext, we have used mainly uniaxial anisotropy with an\nanisotropy constant equal to K∗\n1= 35 µeV/atom, ten\ntimes larger than the Fe bulk value. The main reason for\nthis is that, using a large anisotropy in the simulations\nhelps to quickly stabilize the magnetization of the system\nfor the ϕ= 0◦case, and therefore, less simulation time for\neach field value is required. Figure S6 shows this effect by\ncomparing the time-evolution of the components of the\nmagnetization for two different anisotropy intensities.26\n/s49 /s51 /s53 /s55 /s57/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s49 /s51 /s53 /s55 /s57/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s77\n/s105/s32/s77\n/s120/s32/s77\n/s121/s32/s77\n/s122\n/s75\n/s49/s61/s48/s46/s49/s75\n/s49/s42/s40/s97/s41\n/s40/s98/s41/s77\n/s105\n/s69/s120/s116/s101/s114/s110/s97/s108/s32/s102/s105/s101/s108/s100/s32/s115/s116/s101/s112/s115/s75\n/s49/s61/s49/s48/s75\n/s49/s42/s32/s102 /s32/s61/s32/s48/s176\n/s32/s102\n/s48/s32/s47/s32/s52\n/s32/s108\n/s115/s32/s61/s32/s48/s46/s53\nFigure S6. (Color online.) Comparison of the fluctuations of\nthe components of the magnetization for the simulations with\nϕ= 0◦, frequency f0/4 and different anisotropy constants,\nduring the same number of applied field steps. The Gilbert\ndamping in these simulations is λs= 0.5.6. Hysteresis loops and statistics at high\ntemperature\nFigure S7 shows the effect of spin-lattice coupling at\nT= 500 K. For this case, due to the increased spin fluctu-\nations, the simulations were run for a bcc Fe bulk system\nwith a volume of (32 ×32×32)a3\n0(65536 atoms), to avoid\nundesired finite-size effects. In addition, a smaller sweep\nrate of SR= 0.5×108T/s(which correspond to 180\nps of simulation time per point) was employed for this\nsimulations. The spin histograms shows that spin fluctu-\nations at this high temperature are comparable for both\napproaches.27\n/s45/s48/s46/s52 /s45/s48/s46/s51 /s45/s48/s46/s50 /s45/s48/s46/s49 /s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s78/s40/s83\n/s122/s41\n/s83\n/s122/s32/s77/s111/s118/s105/s110/s103\n/s32/s70/s114/s111/s122/s101/s110/s77\n/s122\n/s109\n/s48/s72/s32 /s40/s84/s41/s32/s76/s97/s116/s116/s105/s99/s101/s32/s109/s111/s118/s105/s110/s103/s32 /s32/s76/s97/s116/s116/s105/s99/s101/s32/s102/s114/s111/s122/s101/s110\nFigure S7. (Color online.) Hysteresis loops obtained at\nT= 500 Kunder the 2 approaches: lattice frozen and lat-\ntice moving. These results correspond to the case of ϕ= 0◦\nand a smaller sweep rate, SR= 0.5×108T/s(which cor-\nresponds to 180ps of simulation time per point). The curves\nwere obtained by averaging over 10 individual loops in each\ncase. The inset shows histograms of the spins orientation\nalong the field direction and corresponds to the state of the\nsystem just before the spin flip, indicated by the circle and\narrow. These histograms were obtained averaging over 5 dif-\nferent histograms."    },    {        "title": "2206.01286v2.Anisotropic_Gigahertz_Antiferromagnetic_Resonances_of_the_Easy_Axis_van_der_Waals_Antiferromagnet_CrSBr.pdf",        "content": "Anisotropic Gigahertz Antiferromagnetic\nResonances of the Easy-Axis van der Waals\nAntiferromagnet CrSBr\nThow Min Jerald Chamaz, Saba Karimeddinya,#, Avalon H. Dismukesb, Xavier\nRoyb, Daniel C. Ralphac*, Yunqiu Kelly Luoacdz*\naCornell University, Ithaca, NY 14850, USA\nbDepartment of Chemistry, Columbia University, New York, NY 10027, USA\ncKavli Institute at Cornell, Ithaca, NY 14853, USA\ndDepartment of Physics and Astronomy, University of Southern California, Los Angeles, CA\n90089, USA\n*Email: dcr14@cornell.edu\n*Email: yl664@cornell.edu, kelly.y.luo@usc.edu\nzT.M.J.C and Y.K.L contributed equally to this paper\nAbstract\nWe report measurements of antiferromagnetic resonances in the van der Waals\neasy-axis antiferromagnet CrSBr. The interlayer exchange \feld and magnetocrystalline\nanisotropy \felds are comparable to laboratory magnetic \felds, allowing a rich variety\nof gigahertz-frequency dynamical modes to be accessed. By mapping the resonance\nfrequencies as a function of the magnitude and angle of applied magnetic \feld we\nidentify the di\u000berent regimes of antiferromagnetic dynamics. The spectra show good\nagreement with a Landau-Lifshitz model for two antiferromagnetically-coupled sublat-\ntices, accounting for inter-layer exchange and triaxial magnetic anisotropy. Fits allow\n1arXiv:2206.01286v2  [cond-mat.mtrl-sci]  5 Aug 2022us to quantify the parameters governing the magnetic dynamics: at 5 K, the interlayer\nexchange \feld is \u00160HE= 0.395(2) T, and the hard and intermediate-axis anisotropy\nparameters are \u00160Hc= 1.30(2) T and \u00160Ha= 0.383(7) T. The existence of within-\nplane anisotropy makes it possible to control the degree of hybridization between the\nantiferromagnetic resonances using an in-plane magnetic \feld.\nKeywords: van der Waals magnet, antiferromagnetic resonance, triaxial magnetic\nanisotropy, interlayer exchange, microwave absorption spectroscopy\nvan der Waals (vdW) antiferromagnets (AFs), a new class of exfoliatable magnetic materials,\nprovide opportunities for future memory, logic, and communications devices because their magnetic\nproperties are highly tunable, for example by applied electric \felds1,2or strain3, they can possess\nlong-lived magnons4, and they can be straightforwardly integrated within complex heterostructures\nby mechanical assembly5,6. The strength of exchange coupling between vdW layers is typically much\nweaker than the direct exchange or superexchange coupling in ordinary 3D-crystal antiferromagnets.\nThis weak interlayer exchange yields antiferromagnetic resonances in the gigahertz range, rather\nthan the terahertz range that is more typical for antiferromagnets7{13. The resonance modes\nin vdW antiferromagnets can therefore be addressed and controlled with conventional microwave\nelectronics.\nHere, we use antiferromagnetic resonance measurements to map the gigahertz-frequency reso-\nnances in the vdW antiferromagnet CrSBr4,14{17. Unlike the only other two vdW antiferromagnets\nwhose modes have been characterized in detail previously (CrI 32,18and CrCl 319), CrSBr has a\nsigni\fcant triaxial magnetic anisotropy with an easy axis oriented within the vdW plane20. This\nanisotropy modi\fes the form of the antiferromagnetic resonances, and for a su\u000eciently strong mag-\nnetic \feld aligned with the anisotropy axis it induces a discontinuous transition that changes the\nmode spectrum abruptly. By measuring the resonances in CrSBr as a function of the angle and\nmagnitude of applied magnetic \feld and comparing to a Landau-Lifshitz (L-L) model, we iden-\ntify the resonance modes in the frequency range 1 - 40 GHz, make a quantitative determination\nof the interlayer exchange and anisotropy \felds, and show that the hybridization between modes\ncan be controlled with an in-plane magnetic \feld. These measurements provide the foundational\n2understanding needed for e\u000borts to utilize these modes within future spintronic devices.\nCrSBr is an A-type antiferromagnetic vdW semiconductor with intralayer ferromagnetic cou-\npling and interlayer antiferromagnetic coupling14,17,20. It has a bulk N\u0013 eel temperature ( TN) of 132\nK and an intermediate ferromagnetic phase with Curie temperature ( TC) in the range of 164 - 185\nK as measured using transport and optical methods14,15,17. Each vdW layer consists of two buckled\nrectangular planes of Cr and S atoms sandwiched between Br atoms. The layers are stacked along\nthec-axis through vdW interactions to form an orthorhombic structure (space group Pmmn ) (Fig.\n1a and Fig. 1b). In our measurements, we use millimeter-length, \rat, needle-like single crystals of\nCrSBr grown by a modi\fed chemical vapor transport approach (see growth details in the Methods\nsection). The long axis of the needle is oriented along the crystallographic a-axis and the \rat\nface corresponds to the vdW plane (supplementary information, Fig. S1). Previous magnetoresis-\ntance studies identi\fed a magnetic easy axis (N\u0013 eel axis) along the crystallographic b-axis (blue and\ngreen arrows in Fig. 1b and 1c), an intermediate axis along the a-axis, and a hard axis along the\nc-axis14,17.\nWe investigate the antiferromagnetic resonances of CrSBr by placing the crystal on a coplanar\nwaveguide and measuring the microwave absorption spectrum using a two-port vector network an-\nalyzer. For the measurements reported in the main text, we align the crystal with its long axis\n(i.e., the aaxis) perpendicular to the waveguide, such that the N\u0013 eel axis ^N(i.e., the baxis) is\nperpendicular to the in-plane RF \feld ( HRF). An external DC magnetic \feld is swept in-plane,\neither perpendicular ( H?), parallel ( Hk), or at intermediate angles relative to ^N(Fig. 1c). (See\nsupplementary information, Section I for spectrum acquisition details.) We use a low microwave\npower, -20 dBm, to avoid any signi\fcant sample heating and to measure the magnetic resonance\ninduced microwave absorption in the linear response regime. We verify that the spectra are insensi-\ntive to the incident power in this regime (supplementary information, section VIII). Representative\ntransmission spectra as a function of applied \feld at 5 K are shown in Fig. 2. Dark green features\nrepresent strong microwave absorption due to resonance modes. We observe two resonance modes\nin theH?con\fguration and show their dependence on magnetic \feld in Fig. 2a up to the maximum\napplied \feld of\u00060.57 T. Below, we will identify the two resonances in Fig. 2a as acoustic and op-\n3Figure 1: Crystal structure and experimental setup for the coplanar waveguide\nmeasurements. (a) Crystal structure of CrSBr as viewed along the out-of-plane c-axis,\nwith Cr, S and Br atoms represented by blue, yellow and red spheres respectively. (b)\nCrystal structure of two CrSBr vdW layers as viewed along the a-axis. Blue and green\narrows indicate the antiferromagnetic in-plane magnetic order with the N\u0013 eel vector ^N\naligned with the easy magnetic axis, which is the crystal b-axis. (c) Experimental\nschematic of a CrSBr crystal mounted on a coplanar waveguide for microwave absorption\nmeasurements (not to scale)21. Here, we show just 3 vdW layers of the bulk crystal for\nclarity. The long axis of the crystal ( a-axis) is aligned perpendicular to the signal line such\nthat the microwave magnetic \felds are perpendicular to the magnetic easy axis. The\npurple and green balls represent Cr atoms in adjacent layers, with magnetization polarized\nto the left (blue arrows) or right (green arrow) indicating the N\u0013 eel order. External \felds\ncan be applied either parallel ( Hk) or perpendicular ( H?) to ^N.\n4tical modes originating from an initial spin-\rop con\fguration in which the two spin sublattices are\ncanted away from the easy axis. In the Hkcase (Fig. 2b), two resonance features are also observed,\nbut with opposite signs of concavity ( d2f=dH2) compared to Fig. 2a. At approximately Hk=\u0006\n0.4 T in Fig. 2b there is a sudden transition beyond which only one mode is observed. A previous\nstudy associated a related transport signal as due to a spin-\rip transition, in which the external\n\feld overcomes the anisotropy \feld to abruptly align the two spin sublattices from an antiparallel\nto parallel orientation22. We will \fnd that the anisotropy parameter Hais slightly smaller than the\ninterlayer exchange \feld HE, so strictly speaking the discontinuity is likely a spin-\rop transition in\nwhich the N\u0013 eel vector initially reorients by 90\u000eand the spin-sublattices tilt to form a canted state\nwithin a narrow window of applied magnetic \feld, before eventually they saturate to a parallel\nalignment as a function of increasing \feld23.\nWe can identify the nature of the detected resonance modes by modeling them with two coupled\nL-L equations using a macrospin approximation for each spin sublattice. This is a reasonable\napproximation since the applied DC and AC magnetic \felds, as well as the interlayer exchange, are\nsmall perturbations relative to the intralayer ferromagnetic coupling20, so they should not a\u000bect\nthe net magnetization of individual atomic layers signi\fcantly. Denoting the magnetic-moment\ndirection of the two spin sublattices with the unit vectors ^m1and ^m2, we model the interlayer\nWeiss exchange \feld acting on ^m1(2)as\u0000HE^m2(1). The triaxial magnetic anisotropy can be\nmodelled by including the hard axis and intermediate axis anisotropy \felds \u0000Hc(^m1(2)\u0001^c)^cand\n\u0000Ha(^m1(2)\u0001^a)^a, whereHcandHaare constants and ^ cand ^aare unit vectors along the crystal's c\nandaaxes. The coupled two-lattice L-L equations can then be written as24\nd^m1(2)\ndt=\u0000\u00160\r^m1(2)\u0002(H\u0000HE^m2(1)\u0000Hc(^m1(2)\u0001^c)^c\u0000Ha(^m1(2)\u0001^a)^a): (1)\nHere\u00160is the permeability constant, \ris the gyromagnetic ratio, and His the externally applied\nmagnetic \feld. Since we will focus only on the relationship between the resonance frequency and\napplied magnetic \feld, but not the resonance linewidth, we omit a damping term.\nIn theH?con\fguration, since the external \feld is perpendicular to the N\u0013 eel axis the initial spin\n5𝜇!H∥(T)𝜇!H⊥(T)a)\nb)\nTransmission (dB)Transmission (dB)5 K\n5 Kf (GHz)f (GHz)RHLHOpticalAcoustic\nflopRHLHFM𝜒𝜒Figure 2: Microwave absorption spectra measured at 5 K as a function of\nmagnetic \feld applied perpendicular ( H?) or parallel ( Hk) to ^N.(a) Microwave\ntransmission (S 21) signal as a function of H?, magnetic \feld applied along the crystal a\naxis. (b) The corresponding spectra as a function of Hk, magnetic \feld applied along the\ncrystalbaxis. S 21values are shown relative to a \feld-independent subtracted background.\nDashed lines show a \ft to the results of the L-L model (Eqs. (2)-(8)). Diagrams on the\nright illustrate the form of some of the resonant modes.\n6con\fguration (before HRFis applied) is a spin-\rop state in which the two spin sublattices rotate\nsymmetrically towards H?as illustrated in Fig. 2a, with a tilt angle sin \u001f=H?=(2HE+Ha). Upon\nexcitation by HRF, assuming small damping, the magnetic sub-lattices will oscillate at frequencies\ncorresponding to the normal modes of the system. Taking terms to the \frst order in the excitation\namplitudes \u000e^m1(2), we calculate these frequencies by rewriting Eq. (1) as a 6 \u00026 matrix in the\n\u000em1(2)\nb;a;cbasis. Solving for the eigenvalues, and taking only the non-trivial positive solutions, we\nobtain the following resonance frequencies (details in supplementary information, section II).\n!1;?=\u00160\rs\n(H2\n?(2HE\u0000Ha) +Ha(Ha+ 2HE)2)(2HE+Hc)\n(Ha+ 2HE)2(2)\n!2;?=\u00160\rs\n((Ha+ 2HE)2\u0000H2\n?)Hc\nHa+ 2HE(3)\nThese equations should apply up to the \feld jH?j= 2HE+Haat which point the applied \feld drives\nthe spin sublattices fully parallel, which is a \feld beyond the range of our measurements. When\n\u00160Ha= 0 T, these modes simplify to the form of acoustic ( !1;?) and optical ( !2;?) modes previously\nreported for easy-plane antiferromagnets19,25. When\u00160Ha6= 0 T, the acoustic mode has a \fnite\nfrequency as H?goes to zero, instead of being proportional to H?. Even when \u00160Ha6= 0 T, the\nsample in this \feld geometry is symmetric under twofold rotation around the applied \feld direction\ncombined with sublattice exchange19. The acoustic and optical modes have opposite parity under\nthis two-fold rotation, so they remain unhybridized and they cross when they intersect (see the\n50 K and 100 K data in Fig. 3a,c), rather than undergoing an avoided crossing (supplementary\ninformation, section IV).\nIn theHkcon\fguration, the applied \feld is parallel to the N\u0013 eel axis and for small \felds\n(jHkj<p\n(2HE\u0000Ha)HaifHa<HE, orjHkj<HEifHa>HE) the spin sub-lattices initially stay\nantiparallel to each other and aligned along this preferred axis. By a matrix diagonalization proce-\ndure similar to the one discussed above, for this antiparallel con\fguration we obtain the resonance\n7frequencies\n!1;k=\u00160\rr\nH2\nk+Ha(HE+Hc) +HEHc\u0000q\nH2\nk(Ha+Hc)(Ha+ 4HE+Hc) +H2\nE(Ha\u0000Hc)2\n(4)\n!2;k=\u00160\rr\nH2\nk+Ha(HE+Hc) +HEHc+q\nH2\nk(Ha+Hc)(Ha+ 4HE+Hc) +H2\nE(Ha\u0000Hc)2:\n(5)\nThese modes correspond to linear superpositions of the left-handed (LH) and right-handed (RH)\nmodes previously reported for easy-axis antiferromagnets12,24. Here, the presence of triaxial anisotropy\n(Hc6=Ha) induces an avoided crossing at zero \feld (Fig. 2b) and causes the eigenmodes at zero\n\feld to be symmetric and antisymmetric combinations of the LH and RH modes26.\nIfHa< HE, in this model as a function of increasing Hkthe antiferromagnet undergoes a\nspin-\rop transition at Hk=p\n(2HE\u0000Ha)Hain which case the N\u0013 eel vector abruptly reorients to\nbe perpendicular to the applied \feld and each spin sublattice cants toward the \feld direction with\na tilt angle sin \u001f=Hk=(2HE\u0000Ha). For this con\fguration, we obtain the resonance frequencies\n!1;k;\rop=\u00160\rs\n(H2\nk(2HE+Ha)\u0000Ha(2HE\u0000Ha)2)(2HE+Hc\u0000Ha)\n(2HE\u0000Ha)2\n!2;k;\rop=\u00160\rs\n((2HE\u0000Ha)2\u0000H2\nk)(Hc\u0000Ha)\n2HE\u0000Ha:(6)\n(7)\nFor this case that Ha< HE, asHkis increased further the canting angle of the spin sublattices\nin the spin \rop state increases continuously, and eventually approaches \u001f=\u0019=2 reaching the\nfully-aligned spin state for Hk\u00152HE\u0000Ha. For the alternative case that Ha> HE, within this\nmodel the spin-\rop state is never stabilized, and the antiferromagnet makes a discontinuous spin-\n\rip transition directly from the anti-aligned state to the fully-parallel state at Hk=HE. Once the\nantiferromagnet is in the fully-parallel state for large Hk, in either case the calculated resonance\nfrequencies take the form\n!1;k;FM=\u00160\rq\n(Hk+Ha)(Hk+Hc) (8)\n8!2;k;FM=\u00160\rq\n(Hk+Ha\u00002HE)(Hk+Hc\u00002HE): (9)\nHowever, we do not expect either !2;k;\ropor!2;k;FMto be detectable in our measurements because\nthese modes are even as a function of 2-fold rotation about the Hkaxis, and the oscillating RF \feld\nhas only odd components for this con\fguration19.\nThe dashed lines in Fig. 2a represent \fts of the resonant modes for the H?con\fguration to Eqs.\n(2) and (3), and in Fig. 2b we present the analogous \fts of the modes for the Hkcon\fguration to Eqs.\n(4)-(6), (8) taking into account the predicted transition \felds. The \fts provide a good description\nof all the observed modes with a common set of \ft parameters. The values of the \ft parameters\nfor a simultaneous least-squares \ft to the data in Figs. 2a and 2b are an inter-layer exchange \feld\nstrength of \u00160HE= 0.395(2) T and anisotropy \feld strengths of \u00160Ha= 0.383(7) T along the\nin-plane intermediate axis and \u00160Hc= 1.30(2) T along the out-of-plane hard axis. As expected the\nexchange \feld is an order of magnitude smaller than the typical exchange in bulk antiferromagnets.\nIt is larger, however, than the value of 0.1 T measured in CrCl 3using similar techniques19. The\nmodes that are most prominent in the spectra correspond to the spin-\rop con\fguration for the\nH?geometry, the antiparallel spin con\fguration at low values of Hk, and the fully-parallel spin\nalignment at large values of Hk. However, because by our \fts Hais slightly smaller than HE, we do\nalso anticipate that there could in principle be a very small region corresponding to the spin-\rop\nstate in Fig. 2b.\nNext, we investigate the evolution of the resonant modes with temperature. As shown in Figs.\n2a,b and Figs. 3a-h, we observe qualitatively similar resonance features over the temperature range\nfrom 5 to 100 K. With increasing temperature, the modes shift to lower frequency and the magnetic\n\feld scales decrease for both the value of H?where the two modes become degenerate and for the\nvalue ofHkcorresponding to the discontinuous transition. These observations can be attributed to\ndecreasing values of all of the exchange and anisotropy parameters HE,HaandHcwith increasing\ntemperature. Figure 4 plots the values of HE,Ha, andHcextracted from simultaneous \fts of\nequations (2)\u0000(6) to theHkandH?transmission spectra for a series of temperatures from 5 to\n128 K. We observe a monotonic decrease in all three parameters.\n9𝜇!H∥(T)𝜇!H⊥(T)𝜇!H∥(T)𝜇!H⊥(T)𝜇!H∥(T)𝜇!H⊥(T)𝜇!H∥(T)𝜇!H⊥(T)\nc)d)50 Ka)b)\ne)f)g)h)100 K100 K\n140 K140 K\n180 K180 K\nTransmission (dB)f (GHz)\nf (GHz)f (GHz)\nf (GHz)f (GHz)\nf (GHz)f (GHz)\nf (GHz)acbacb\nH∥HRF\nHRF\nH⊥\nTransmission (dB)Transmission (dB)Transmission (dB)50 KFigure 3: Fitted absorption spectra at di\u000berent temperatures. (a, c, e, g)\nMicrowave transmission (S 21) signals as a function of H?, magnetic \feld applied along the\ncrystalaaxis, for the selected temperatures indicated. (b, d, f, h) S 21signals as a function\nofHkat the same selected temperatures. Dashed lines are \fts to Eqs. (2)-(8) with\nparameters HE,Ha, andHcthat are allowed to vary with temperature but are independent\nof the applied \feld magnitude and direction. Red and blue dotted lines represent the\npredicted spin-\rop and fully-parallel spin alignment transition \felds respectively.\n10Anisotropy and interlayer exchange parameters (T)T (K)TNTCFigure 4: Temperature dependence of the interlayer exchange and the anisotropy\n\feld strengths HE,Ha, andHc.a) Temperature dependence of the interlayer exchange\nHE(blue squares) and the in-plane easy-axis anisotropy parameter Ha(red circles). b)\nTemperature dependence of out-of-plane anisotropy parameter Hc. The black dashed lines\nindicate the estimated N\u0013 eel temperature TN\u0019132 K and the Curie temperature Tc\u0019160\nK, previously measured in magnetometry and magnetotransport measurements14,15,17.\n11The anisotropy parameter Hadecreases more quickly with increasing temperature than HE,\nwhich within our model should open up a larger window of \feld within which the spin-\rop con-\n\fguration is stabilized in the Hkgeometry. This regime corresponds to the one feature of our\nmeasurement which is not captured well by the \fts { see how the \feld dependence of the mea-\nsured frequency deviates from the prediction for the spin-\rop state for values of jHkjjust above\nthe discontinuous jumps for the 50 K and 100 K data in Figs. 3b and 3d. We take this as a hint\nthat our simple model may not fully capture the angular dependence of the exchange energy or the\nmagnetic anisotropy for large canting angles within the spin-\rop state.\nAt 140 K (Figs. 3e, f), the resonance spectra become quite di\u000berent compared to the lower-\ntemperature measurements, with only a single resonance mode observed, rather than two. The\nfrequency dependence can be \ft well to a ferromagnetic dependence (Eq. (8)) with the in-plane easy-\naxis anisotropy persisting to give slightly di\u000berent frequencies for H kvs. H?. This measurement\nis performed above the N\u0013 eel temperature of 132 K, so our interpretation is that this spectrum\ncorresponds to the intermediate ferromagnetic state observed previously by transport and optical\nsecond harmonic generation14,15,17. Finally, by 180 K (Figs. 3g, 3h) the anisotropy is no longer\nvisible and only a linear electron paramagnetic resonance signal remains, with a slope of 26.6(1)\nGHz/T and a very narrow linewidth27.\nWe have explored in more detail the relationship between the HkandH?modes by measuring\nthe evolution of the modes when the external \feld is applied at angles \u001ebetween the bcrystal axis\n(easy anisotropy axis) and the acrystal axis (intermediate anisotropy), where \u001e= 0\u000ecorresponds\nto theHkcon\fguration. As shown in Fig. 5a for data measured at 100 K, the convex-shaped\noptical and acoustic modes H?modes (\u001e= 90\u000e) \ratten out and evolve toward the concave-shaped\nHkmodes as\u001eis decreased. As this happens, the simple crossing between the modes present for\n\u001e= 90\u000eevolves into the avoided crossing visible for the 75\u000eand 60\u000edata. This hybridization in the\nmodes as\u001eis decreased from 90\u000eis analogous to the opening of an anti-crossing hybridization gap\nin CrCl 3due to breaking of 2-fold rotational symmetry by an out-of-plane applied magnetic \feld19,\nbut here the in-plane easy axis allows for two-fold rotational symmetry to be broken with an in-plane\n\feld oriented away from the H?direction (see further analysis in the supplementary information,\n12section IV and VII). As a function of decreasing \u001efrom 90\u000e, the discontinuity corresponding to\nthe spin-\rip transition \frst becomes visible at about \u001e= 60\u000eand is increasingly prominent at\nsmaller angles. The dashed lines in Fig. 5a represent a global numerical \ft to the spectra with the\nparameters (appropriate for 100 K) \u00160HE= 0.222(2) T, \u00160Ha= 0.147(2) T, and \u00160Hc= 0.625(7)\nT.\nLine scans showing the development of the anti-crossing gap in the spectra are shown in Fig.\n5b for small angles away from \u001e= 90\u000e. We quantify in Fig. 5c the strength of the mode coupling\nwith the parameter\u0001f\n2, half the frequency separation of the coupled resonance mode peaks. The\ncoupling parameter increases monotonically as the external \feld is rotated from 90\u000eto 55\u000e. For\n\u001e= 55\u000e, the coupling strength is\u0001f\n2= 4(1) GHz, while the full-width half maxima of the upper\nand lower modes areKu\n2\u0019\u00191:14 GHz andKl\n2\u0019\u00191:6 GHz. This satis\fes the condition\u0001f\n2>Kuand\n\u0001f\n2> Kl, indicating that the system is in the regime of strong magnon-magnon coupling, as has\npreviously been observed in other spin-wave systems19,26,28.\nIn summary, we report measurements of GHz-frequency antiferromagnetic resonance modes in\nthe van der Waals antiferromagnet CrSBr that are anisotropic with regard to the angle of applied\nmagnetic \feld relative to the crystal axes. The modes are well described by two coupled Landau-\nLifshitz equations for modeling the spin sublattices, when one accounts for interlayer exchange and\ntriaxial magnetic anisotropy present in CrSBr. The interlayer exchange and the anisotropy \feld\nstrengths were measured by \ftting the resonances for a series of temperatures from 5 to 128 K.\nAt 5 K we determine an interlayer exchange \feld HE= 0.395(2) T, and aniosotopy \felds Ha=\n0.383(7) T and Hc= 1.30(2) T. All three parameters weaken with increasing temperature. As the\nangle of an applied magnetic \feld is changed within the a-bcrystal plane, we observe a continuous\nevolution from unhybridized acoustic and optical modes with a simple mode crossing (for Hparallel\nto the crystal aaxis) to hybridized superpositions with an avoided crossing that is controllable by\nmagnetic \feld (as the angle of His tuned away from aaxis). This evolution of strong magnon-\nmagnon coupling can be understood as a consequence of breaking a two-fold rotational symmetry\nthat is present when the applied \feld is aligned along the crystal aaxis. Our characterization of\nantiferromagnetic resonances in an easily-accessible frequency range, and the understanding of how\n13ɸ= 90°ɸ= 75°ɸ= 60°\nɸ= 15°ɸ= 0°\n𝜇!H (T)f (GHz)\n𝜇!H (T)𝜇!H (T)f (GHz)\nTransmission (dB)ɸ= 30°ɸ= 45°\n𝜇!H (T)𝜇!H (T) 𝜇!H (T)𝜇!H (T)100 Ka)\nb)Transmission (dB)f (GHz)c)\"#$(GHz)\n𝜙(deg)Δf𝜇!H = 0.32 T\nFigure 5: Evolution of a mode crossing at 100 K as a function of orienting the\nin-plane applied magnetic \feld away from H?.a) Evolution of the absorption\nspectra at 100 K for intermediate angles of in-plane applied magnetic \feld between H?(\u001e\n= 90\u000e) andHk(\u001e= 0\u000e). The dashed lines are determined by \ftting the data for \u001e= 0\u000e\nand 90\u000eto eqs 2-6 to obtain the values of the \ft parameters HE,Ha, andHcdiscussed in\nthe text, and then calculating the resonance frequencies numerically for the intermediate\nangles. b) Line cuts of the absorption spectra near the avoided crossing for di\u000berent\napplied magnetic-\feld angles \u001e= 90\u000eto 55\u000e, for an applied \feld magnitude of \u00160H= 0.32\nT. Plots are o\u000bset vertically for clarity. c) Plot of coupling strength\u0001f\n2calculated as half\nthe separation between absorption resonances, as a function of \u001e.\n14the resonances can be tuned between uncoupled and strongly-coupled by adjusting magnetic \feld,\nsets the stage for future experiments regarding manipulation of the modes and the development of\ncapabilities like antiferromagnetic spin-torque nano-oscillators.\nCrystal growth and characterization\nCrSBr single crystals were synthesized using a modi\fed chemical vapor transport approach adapted\nfrom the original report by Beck29. Disulfur dibromide and chromium metal were added together\nin a 7:13 molar ratio to a fused silica tube approximately 35 cm in length, which was sealed under\nvacuum and placed in a three-zone tube furnace. The tube was heated in a temperature gradient\n(1223 to 1123 K) for 120 hours. CrSBr grows as black, shiny \rat needles. The long axis of bulk\nneedle crystals of CrSBr has been correlated to the a crystal axis by XRD experiments.\nSupporting Information\nExperimental methods, calculations of antiferromagnetic resonances from the Landau-Lifshitz equa-\ntion, full temperature series, symmetry analyses, data from a second crystal.\nAcknowledgements\nWe acknowledge inspiring discussions with Kin Fai Mak, Jie Shan, Joseph Mittelstaedt, Kihong\nLee, and Caitlin Carnahan. The research at Cornell was supported by the AFOSR/MURI project\n2DMagic (FA9550-19-1-0390) and the US National Science Foundation (DMR-2104268). T.M.J.C.\nwas supported by the Singapore Agency for Science, Technology, and Research, and Y.K.L. acknowl-\nedges the Cornell Presidential Postdoctoral Fellowship. The work utilized the shared facilities of\nthe Cornell Center for Materials Research (supported by the NSF via grant DMR-1719875) and\nthe Cornell NanoScale Facility, a member of the National Nanotechnology Coordinated Infras-\ntructure (supported by the NSF via grant NNCI-2025233), and it bene\fted from instrumentation\nsupport by the Kavli Institute at Cornell. Synthesis of the CrSBr crystals was supported as part\n15of Programmable Quantum Materials, an Energy Frontier Research Center funded by the U.S.\nDepartment of Energy (DOE), O\u000ece of Science, Basic Energy Sciences (BES), under award DE-\nSC0019443.\nAuthor Contributions\nT.M.J.C and Y.K.L devised the experiment and performed the measurements. T.M.J.C performed\nthe data analysis with assistance from Y.K.L.. S.K. assisted with experimental setup and L-L\nmodeling. A.H.D. synthesized the crystals under the supervision of X.R.. D.C.R. provided oversight\nand advice. T.M.J.C, Y.K.L, and D.C.R. wrote the manuscript. All authors discussed the results\nand the content of the manuscript.\nCompeting Interests\nThe authors declare no competing interests.\nPresent Addresses\n#S.K.: IBM T. J. Watson Research Center, Yorktown Heights, NY, 10598, USA\nReferences\n(1) Jiang, S.; Li, L.; Wang, Z.; Mak, K. F.; Shan, J. Controlling magnetism in 2D CrI 3by\nelectrostatic doping. Nature Nanotechnology 2018 ,13, 549{553.\n(2) Zhang, X. X.; Li, L.; Weber, D.; Goldberger, J.; Mak, K. F.; Shan, J. Gate-tunable spin waves\nin antiferromagnetic atomic bilayers. Nature Materials 2020 ,19, 838{842.\n(3) Cenker, J.; Sivakumar, S.; Xie, K.; Miller, A.; Thijssen, P.; Liu, Z.; Dismukes, A.; Fonseca, J.;\nAnderson, E.; Zhu, X.; Roy, X.; Xiao, D.; Chu, J.-H.; Cao, T.; Xu, X. Reversible strain-\n16induced magnetic phase transition in a van der Waals magnet. Nature Nanotechnology 2022 ,\n17, 256{261.\n(4) Bae, Y. J. et al. Exciton-Coupled Coherent Antiferromagnetic Magnons in a 2D Semiconduc-\ntor.2022 , 2201.13197. arXiv. https://arxiv.org/abs/2201.13197 (accessed July 28, 2022).\n(5) Zhong, D.; Seyler, K. L.; Linpeng, X.; Cheng, R.; Sivadas, N.; Huang, B.; Schmidgall, E.;\nTaniguchi, T.; Watanabe, K.; McGuire, M. A.; Yao, W.; Xiao, D.; Fu, K.-M. C.; Xu, X.\nVan der Waals engineering of ferromagnetic semiconductor heterostructures for spin and val-\nleytronics. Science Advances 2017 ,3, e1603113.\n(6) Geim, A. K.; Grigorieva, I. V. Van der Waals heterostructures. Nature 2013 ,499, 419{425.\n(7) Johnson, F. M.; Nethercot, A. H. Antiferromagnetic resonance in MnF 2.Physical Review\n1959 ,114, 705{716.\n(8) Sievers, A.; Tinkham, M. Far infrared antiferromagnetic resonance in MnO and NiO. Physical\nReview 1963 ,129, 1566{1571.\n(9) Kondoh, H. Antiferromagnetic Resonance in NiO in Far-infrared Region. Journal of the Phys-\nical Society of Japan 1960 ,15, 1970{1975.\n(10) Kampfrath, T.; Sell, A.; Klatt, G.; Pashkin, A.; M ahrlein, S.; Dekorsy, T.; Wolf, M.;\nFiebig, M.; Leitenstorfer, A.; Huber, R. Coherent terahertz control of antiferromagnetic spin\nwaves. Nature Photonics 2011 ,5, 31{34.\n(11) Baierl, S.; Mentink, J. H.; Hohenleutner, M.; Braun, L.; Do, T. M.; Lange, C.; Sell, A.;\nFiebig, M.; Woltersdorf, G.; Kampfrath, T.; Huber, R. Terahertz-Driven Nonlinear Spin Re-\nsponse of Antiferromagnetic Nickel Oxide. Physical Review Letters 2016 ,117, 197201.\n(12) Li, J.; Wilson, C. B.; Cheng, R.; Lohmann, M.; Kavand, M.; Yuan, W.; Aldosary, M.;\nAgladze, N.; Wei, P.; Sherwin, M. S.; Shi, J. Spin current from sub-terahertz-generated anti-\nferromagnetic magnons. Nature 2020 ,578, 70{74.\n17(13) Vaidya, P.; Morley, S. A.; Van Tol, J.; Liu, Y.; Cheng, R.; Brataas, A.; Lederman, D.; Del\nBarco, E. Subterahertz spin pumping from an insulating antiferromagnet. Science 2020 ,368,\n160{165.\n(14) Telford, E. J.; Dismukes, A. H.; Lee, K.; Cheng, M.; Wieteska, A.; Bartholomew, A. K.;\nChen, Y.-S.; Xu, X.; Pasupathy, A. N.; Zhu, X.; Dean, C. R.; Roy, X. Layered Antiferro-\nmagnetism Induces Large Negative Magnetoresistance in the van der Waals Semiconductor\nCrSBr. Advanced Materials 2020 ,32, 2003240.\n(15) Lee, K.; Dismukes, A. H.; Telford, E. J.; Wiscons, R. A.; Wang, J.; Xu, X.; Nuckolls, C.;\nDean, C. R.; Roy, X.; Zhu, X. Magnetic Order and Symmetry in the 2D Semiconductor\nCrSBr. Nano Letters 2021 ,21, 3511{3517.\n(16) Wilson, N. P.; Lee, K.; Cenker, J.; Xie, K.; Dismukes, A. H.; Telford, E. J.; Fonseca, J.;\nSivakumar, S.; Dean, C.; Cao, T.; Roy, X.; Xu, X.; Zhu, X. Interlayer electronic coupling on\ndemand in a 2D magnetic semiconductor. Nature Materials 2021 ,20, 1657{1662.\n(17) Telford, E. J. et al. Coupling between magnetic order and charge transport in a two-\ndimensional magnetic semiconductor. Nature Materials 2022 ,21, 754{760.\n(18) Shen, X.; Chen, H.; Li, Y.; Xia, H.; Zeng, F.; Xu, J.; Kwon, H. Y.; Ji, Y.; Won, C.; Zhang, W.,\net al. Multi-domain ferromagnetic resonance in magnetic van der Waals crystals CrI 3and\nCrBr 3.Journal of Magnetism and Magnetic Materials 2021 ,528, 167772.\n(19) MacNeill, D.; Hou, J. T.; Klein, D. R.; Zhang, P.; Jarillo-Herrero, P.; Liu, L. Gigahertz\nFrequency Antiferromagnetic Resonance and Strong Magnon-Magnon Coupling in the Layered\nCrystal CrCl 3.Physical Review Letters 2019 ,123, 047204.\n(20) Yang, K.; Wang, G.; Liu, L.; Lu, D.; Wu, H. Triaxial magnetic anisotropy in the two-\ndimensional ferromagnetic semiconductor CrSBr. Physical Review B 2021 ,104, 144416.\n(21) Momma, K.; Izumi, F. VESTA : a three-dimensional visualization system for electronic and\nstructural analysis. Journal of Applied Crystallography 2008 ,41, 653{658.\n18(22) G oser, O.; Paul, W.; Kahle, H. Magnetic properties of CrSBr. Journal of Magnetism and\nMagnetic Materials 1990 ,92, 129{136.\n(23) Baltz, V.; Manchon, A.; Tsoi, M.; Moriyama, T.; Ono, T.; Tserkovnyak, Y. Antiferromagnetic\nSpintronics. Reviews of Modern Physics 2018 ,90, 015005.\n(24) Ke\u000ber, F.; Kittel, C. Theory of Antiferromagnetic Resonance. Physical Review 1952 ,85,\n329{337.\n(25) Streit, P. K.; Everett, G. E. Antiferromagnetic resonance in EuTe. Physical Review B 1980 ,\n21, 169{182.\n(26) Liensberger, L.; Kamra, A.; Maier-Flaig, H.; Gepr ags, S.; Erb, A.; Goennenwein, S. T. B.;\nGross, R.; Belzig, W.; Huebl, H.; Weiler, M. Exchange-Enhanced Ultrastrong Magnon-Magnon\nCoupling in a Compensated Ferrimagnet. Physical Review Letters 2019 ,123, 117204.\n(27) Stanger, J.-L.; Andr\u0013 e, J.-J.; Turek, P.; Hosokoshi, Y.; Tamura, M.; Kinoshita, M.; Rey, P.;\nCirujeda, J.; Veciana, J. Role of the demagnetizing \feld on the EPR of organic radical magnets.\nPhysical Review B 1997 ,55, 8398{8405.\n(28) Chen, J.; Liu, C.; Liu, T.; Xiao, Y.; Xia, K.; Bauer, G. E.; Wu, M.; Yu, H. Strong Interlayer\nMagnon-Magnon Coupling in Magnetic Metal-Insulator Hybrid Nanostructures. Physical Re-\nview Letters 2018 ,120, 217202.\n(29) Beck, J. Uber Chalkogenidhalogenide des Chroms Synthese, Kristallstruktur und Magnetismus\nvon Chromsul\fdbromid, CrSBr. Zeitschrift f ur anorganische und allgemeine Chemie 1990 ,\n585, 157{167.\n19Supplement to “Anisotropic Gigahertz Antiferromagnetic\nResonances of the Easy-Axis van der Waals Antiferromagnet\nCrSBr”\nThow Min Jerald Cham,1,∗Saba Karimeddiny,1Avalon H. Dismukes,2\nXavier Roy,2Daniel C. Ralph,1, 3,†and Yunqiu Kelly Luo1, 3, 4, ∗\n1Cornell University, Ithaca, NY 14850, USA\n2Department of Chemistry, Columbia University, New York, NY 10027, USA\n3Kavli Institute at Cornell, Ithaca, NY 14853, USA\n4Department of Physics and Astronomy,\nUniversity of Southern California, Los Angeles, CA 90089, USA\n(Dated: August 5, 2022)\n∗These authors contributed equally\n†Correspondence email address: yl664@cornell.edu, kelly.y.luo@usc.edu, dcr14@cornell.edu\n1CONTENTS\nI. Acquisition of the microwave absorption spectra 3\nII. Calculation of antiferromagnetic resonances for CrSBr samples accounting for\ntriaxial anisotropy 4\nA. External field applied along the CrSBr intermediate axis ( H⊥) 5\nB. External field applied along the CrSBr easy axis ( H∥) 7\n1. Zero to small applied field, before spin-flop or spin-flip transition 9\n2. Spin-flop configuration 10\n3. Parallel orientation of the spin sub-lattices 11\nC. Spectra fitting 12\nIII. Effect of triaxial anisotropy on antiferromagnetic resonant modes 13\nIV. Resonance modes for weak magnetic fields at arbitrary in-plane angles relative to\nthe anisotropy axes 15\nV. Complete series of spectra at different temperatures 19\nVI. Spectra for a second crystal with its a-axis parallel to the coplanar waveguide\ncenter line 23\nVII. Selective excitation of modes from 2-fold rotation symmetry considerations 28\nVIII. Microwave power dependence 30\nReferences 31\n2I. ACQUISITION OF THE MICROWAVE ABSORPTION SPECTRA\nWe couple the CrSBr crystal to a ground-signal-ground coplanar waveguide (CPW) and\nperform RF (1 - 40 GHz) transmission measurements. For the data analyzed in the main\ntext, we aligned the long edge of the crystal (a-axis) perpendicular to the middle signal line\nand glued it in place using rubber cement, as shown in Fig. S1(a). We also took special care\nto align the vdW plane (CrSBr a-b crystal plane) parallel with the waveguide plane. We\nconnected the ends of the CPW to ports 1 and 2 of a Keysight 8722ES 40 GHz S-parameter\nvector network analyzer (VNA) using coaxial cables. The waveguide was then lowered into\na Janis He vapour flow cryostat and secured onto a brass sample stage containing a sample\nthermometer. The temperature was set with a Lakeshore 331 temperature PID controller.\nWe used a GMW model 5403 electromagnet, mounted on an stepper-motor-controlled in-\nplane rotation stage, to sweep the external field Hextparallel ( H∥) or perpendicular ( H⊥)\nwith respect to the N´ eel axis (crystal b-axis), or at angles in between.\nWe also performed transmission measurements on another crystal, for which we aligned\nthe long a-axis parallel to the middle signal line of the CPW, rather than perpendicular. We\nreport results from that sample in Section VI of this supplementary information.\nBefore acquiring spectra at each temperature, we wait for the temperature to stabilize and\nthen calibrate the VNA using open, short, and 50 Ω standards to account for temperature-\ndependent transmission factors. We measure the magnitude of the S 21transmission vector,\nsubtract a field-averaged background for each frequency, and perform Fourier Transform\nsmoothing and interpolation to remove any streak-like artifacts in the spectra. The trans-\nmission spectrum before and after the subtraction and smoothing process at 5 K are shown\nin Fig. S1(b) and (c).\n3f (GHz)𝜇!H∥(T)b)\nTransmission (dB)\n𝜇!H∥(T)f (GHz)a)\nd)c)\nTransmission (dB)Transmission (dB)f (GHz)𝜇!H∥\t= 0.1 TFIG. S1: Details of the microwave absorption measurements. a) CrSBr crystal with the a\naxis aligned perpendicular to the signal line of a ground-signal-ground coplanar waveguide.\nAbsorption spectra b) before and c) after the background subtraction and smoothing\nprocedure. d) Comparison of spectra linecuts at µ0H= 0.1 T with the field-averaged\nbackground. Plots are offset vertically for clarity.\nII. CALCULATION OF ANTIFERROMAGNETIC RESONANCES FOR CrSBr\nSAMPLES ACCOUNTING FOR TRIAXIAL ANISOTROPY\nTo model the antiferromagnetic resonance frequencies, we consider the coupled dynamics\nof the two spin sublattices, 1 and 2. In the approximation of small-angle precession, we\ncan write the magnetization orientation of each sublattice as a unit vector consisting of a\n4time-invariant equilibrium position plus a small time-varying part:\nˆm1(2)=ˆmeq\n1(2)+δˆm1(2)eiωt. (1)\nIn our analysis, we define the ˆ cdirection to be perpendicular to the van der Waal layers\nand along CrSBr hard axis ( c-axis); ˆ ato be along the CrSBr intermediate axis ( a-axis); and\nˆbto be along the CrSBr easy axis ( b-axis). We assume that the exchange interaction felt\nby one sublattice imposed by the other sublattice can be written as −HEˆm2(1). Therefore,\nthe general expression for the total effective magnetic field (including the external field,\nexchange, and first-order anisotropy terms) is\nH1(2)=Hext\nbˆb+ (Hext\na−Ha(meq\n1(2)a+δm1(2)a)eiωt)ˆa\n+ (Hext\nc−Hc(meq\n1(2)c+δm1(2)c)eiωt)ˆc−HEˆm2(1),(2)\nwhere HaandHcare anisotropy parameters along the CrSBr intermediate and hard axes\n[1, 2].\nThe general Landau-Lifshitz (L-L) equation without damping is written as:\ndˆm1(2)\ndt=−µ0γˆm1(2)×H1(2). (3)\nA. External field applied along the CrSBr intermediate axis ( H⊥)\nFor the case when the extermal field is applied along the aaxis, it induces a spin-flop\nconfiguration such that the equilibrium sublattice magnetizations ˆm1(2)stay in the b−a\nplane and tilt symmetrically toward ˆ a, as illustrated in Fig. S2 below. Therefore, ˆmeq\n1(2)can\nbe written as ( meq\n1(2)b,meq\n1(2)a,meq\n1(2)c) = (±cosχ,sinχ,0).\nSubstituting µ0Hext\nb=µ0Hext\nc= 0 T and µ0Hext\na=µ0H⊥into Eq. (2), the total effective\nmagnetic field can be written as\nH1(2)= (H⊥−Ha(meq\n1(2)a+δm1(2)a)eiωt)ˆa−Hc(meq\n1(2)c+δm1(2)ceiωt)ˆc−HEˆm2(1).(4)\nSubstituting in the equilibrium spin-flop configuration ˆmeq\n1(2)= (±cosχ,sinχ,0), Eq. (4)\n5𝒎\"H⊥bacintermediate axiseasy axishard axis𝜒𝜒1eq𝒎\"2eqFIG. S2: Configuration of antiferromagnetic sub-lattices with Hextalong the intermediate\naxis. This configuration is symmetric upon two-fold rotation about the ˆ a-axis and\ninterchange of the spin sub-lattices.\nbecomes\nH1(2)= (−HE(δm2(1)beiωt∓cosχ))ˆb+ (H⊥−Ha(δm1(2)aeiωt+ sin χ)−HE(δm2(1)aeiωt+ sin χ))ˆa\n+ (−Hcδm1(2)ceiωt−HEδm2(1)ceiωt)ˆc.\n(5)\nTherefore, writing the L-L equations (Eq. (3)) in matrix form and ignoring terms higher\nthan first-order in the precession amplitude:\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle−iw 0 γ[H⊥+ sin χ(Hc−Ha−HE)] 0 0 γsinχHE\n0 −iw −γcosχ(Hc+HE) 0 0 −γcosχHE\nγ[−H⊥+ sin χ(Ha+HE)]γcosχ(Ha+HE) −iw −γsinχHE γcosχHE 0\n0 0 γsinχHE −iw 0 γ[H⊥+ sin χ(−Ha+Hc−HE)]\n0 0 γcosχHE 0 −iw γ cosχ(HE+Hc)\n−γsinχHE −γcosχHE 0 γ[−H⊥+ sin χ(HE+Ha)]γcosχ(−Ha−HE) −iw/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0\n(6)\nTo obtain the equilibrium position, since the moments will remain in the b−aplane, we can\nignore the c-axis anisotropy and write the free energy (per unit sub-lattice magnetization)\nas\nF=−ˆm1·Hext−ˆm2·Hext+HEˆm1·ˆm2+1\n2Ha(ˆm1·ˆa)2+1\n2Hc(ˆm2·ˆc)2, (7)\nand then minimize with respect to χ. For an applied magnetic field perpendicular to the\neasy axis, this gives sin χ=H⊥/(2HE+Ha). We substitute this expression into the 6 ×6\n6eigenvalue equation (6) and solve it to get 6 eigenvalues. Only two of these are non-trivial\nand positive, corresponding to the resonance frequencies ω1(2),⊥:\nω1,⊥=µ0γ/radicaligg\n(H2\n⊥(2HE−Ha) +Ha(Ha+ 2HE)2)(2HE+Hc)\n(Ha+ 2HE)2(8)\nω2,⊥=µ0γ/radicaligg\n((Ha+ 2HE)2−H2\n⊥)Hc\nHa+ 2HE. (9)\nThese resonance modes are generalizations of the acoustic and optical modes previously\nreported for CrCl 3[3]. By considering the two-fold rotational symmetry of the system about\ntheaaxis, we find that the ω1,⊥mode is odd with respect to this symmetry and so it will\nbe excited by RF fields that are odd under a 2-fold rotation about ˆ a. The ω2,⊥mode is even\nand will be excited by RF fields that have an even component.\nAt large applied magnetic fields for which the two sublattices are driven completely par-\nallel (For H⊥≥2HE+Ha),ˆmeq\n1(2)= (0,1,0),χ→π/2, Eqs. (8) and (9) are no longer\napplicable, and we get the uniform ferromagnetic resonance mode\nω1,⊥,FM=µ0γ/radicalbig\n(H⊥−Ha)(H⊥−Ha+Hc) (10)\nand a lower-frequency mode\nω2,⊥,FM=µ0γ/radicalbig\n(H⊥−Ha−2HE)(H⊥−Ha+Hc−2HE) (11)\nIn the lower-frequency mode, δˆm1=−δˆm2so there is no net time-dependent magnetization,\nand for this reason we suggest that this mode will not couple to an external RF magnetic\nfield or produce any signal in a CPW measurement.\nB. External field applied along the CrSBr easy axis ( H∥)\nWhen an external magnetic field is applied along the CrSBr easy axis ( b-axis), there are\nthree possible equilibrium configurations, depending on the relative values of the H∥(the\napplied field), HE, and Ha. The two spin sublattices may remain antiparallel and aligned\nalong the easy axis, in which case according to Eq. (7) the energy (per unit sub-lattice\n7magnetization) of the state is FAP=−HE. The N´ eel vector may rotate perpendicular to\nthe applied field with the spin sublattices forming a spin-flop state with the canting angle\nsinχ=H∥/(2HE−Ha), resulting in the energy Fflop=−H2\n∥/(2HE−Ha)−HE+Ha.\nAlternatively, the two spin sublattices can be driven parallel to the applied magnetic field,\nwith the energy FP=HE−2H∥. The equilibrium configuration can be determined based on\nwhich state minimizes the energy. The field values corresponding to the various transitions\nare as follows: the spin-flop state becomes lower in energy than the antiparallel state when\nH∥>/radicalbig\n(2HE−Ha)Ha, the parallel state becomes lower in energy than the antiparallel state\nwhen H∥> H E, and the canting angle of the spin-flop state goes to π/2 to reach the parallel\nstate when H∥>2HE−Ha. This results in two possible scenarios for the evolution of the\nconfigurations of the antiferromagnet, as pictured in Fig.S3.\na)F\nH∥(T)mxH∥(T)𝜇!Ha= 0.05, 𝜇!HE= 0.1, 𝜇!Hc= 0.3𝜇!Ha= 0.2, 𝜇!HE= 0.15, 𝜇!Hc= 0.3b)FmxAPflopPAPPm1xm2xm1xm2x\nFIG. S3: Field-induced magnetic phase transitions for scenario (a) weak anisotropy:\nspin-flop transition at intermediate applied field H∥>/radicalbig\n(2HE−Ha)Ha(dashed red line)\nfollowed by a gradual transition into a parallel state when H∥>2HE−Ha(dashed blue\nline); and for scenario (b) strong anisotropy: spin-flip transition at H∥> H e(dashed green\nline).\n8I. If the intermediate-axis anisotropy is relatively weak ( Ha< H E), the antiferromag-\nnet is initially in the antiparallel state at low values of applied field, and as a function\nof increasing field it first makes a sudden, discontinuous transition to the spin-flop state at\nH∥=/radicalbig\n(2HE−Ha)Ha, and then the canting angle grows continuously until the fully-parallel\nstate is reached for fields H∥>2HE−Haor greater.\nII. If Ha> H E, as a function of increasing H∥there is no value of field for which the\nspin-flop state is stable. The antiferromagnet is initially in the antiparallel state for low\nvalues of applied field, but then makes a discontinuous transition to the fully-parallel state\nforH∥> H E.\nWe will calculate the resonance frequencies expected for all three states (antiparallel,\nspin-flop, and fully-parallel), and compare to our measurements. For CrSBr we will find\nthat Hais less than HE, but the values at most temperatures are rather close, so there\nis only a narrow range of field for which the spin-flop state is stabilized. Therefore the\nresonances that are most prominent in the measured spectra correspond to the antiparallel\nand fully-parallel states.\nThe same type of L-L analysis discussed above can be used to obtain analytic solutions for\nthe resonance modes corresponding to each of the possible equilibrium configurations. Since\nhere we are considering only applied magnetic fields parallel to ˆb, the easy-axis, µ0Hext\na=\nµ0Hext\nc= 0 T and µ0Hext\nb=µ0H∥. The total effective magnetic field given by Eq. (2)\nbecomes\nH1(2)=H∥ˆb−Ha(meq\n1(2)a+δm1(2)a)eiωt)ˆa−Hc(meq\n1(2)c+δm1(2)c)eiωtˆc−HEˆm2(1).(12)\n1. Zero to small applied field, before spin-flop or spin-flip transition\nHere the equilibrium orientations of both spin sublattices remain aligned with the ˆbeasy\naxis, so ˆmeq\n1(2)= (±1,0,0). With this, Eq. (12) reduces to\nH1(2)= (H∥−HE)ˆb−Haδm1(2)aeiωtˆa−Hcδm1(2)ceiωtˆc, (13)\n9Expanding the L-L equations (Eq. 3) and taking terms to the first order in the precession\namplitude gives the 6 ×6 eigenvalue equation\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle−iw 0 0 0 0 0\n0 −iw −γ(H∥+Hc+HE) 0 0 −HEγ\n0γ[H∥+Ha+HE)] −iw 0 HEγ 0\n0 0 0 −iw 0 0\n0 0 HEγ 0 −iw γ (−H∥+Hc+HE)\n0 −HEγ 0 0 γ(H∥−Ha−HE) −iw/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0.\n(14)\nSolving this, we again obtain two positive non-trivial resonance frequencies:\nω1,∥=µ0γ/radicalbigg\nH2\n∥+Ha(HE+Hc) +HEHc−/radicalig\nH2\n∥(Ha+Hc)(Ha+ 4HE+Hc) +H2\nE(Ha−Hc)2\n(15)\nω2,∥=µ0γ/radicalbigg\nH2\n∥+Ha(HE+Hc) +HEHc+/radicalig\nH2\n∥(Ha+Hc)(Ha+ 4HE+Hc) +H2\nE(Ha−Hc)2.\n(16)\nThese modes are linear superpositions of the left-handed (LH) and right-handed (RH) modes\ndepicted in Fig. 2 in the main text [2, 4] that would exist for an antiferromagnet with purely\nuniaxial anisotropy. The broken rotational symmetry about the field axis in a material with\ntriaxial anisotropy ( Hc̸=Ha) causes the LH and RH modes to be hybridized.\n2. Spin-flop configuration\nUsing the spin-flop configuration ˆmeq\n1(2)= (sin χ,±cosχ,0), Eq. (12) can be written as\nH1(2)= [H∥−HE(δm2(1)beiωt+ sin χ)]ˆb+ [−Ha(δm1(2)aeiωt±cosχ)−HE(δm2(1)aeiωt∓cosχ)]ˆa\n+ [−Hcδm1(2)ceiωt−HEδm2(1)ceiωt]ˆc\n(17)\n10Substituting these expressions into the L-L equations (Eq. (3)), removing the equilibrium\nterms, and ignoring the higher-order terms in the precession amplitudes gives:\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleiw 0 γcosχ(Hc−Ha+HE) 0 0 γcosχHE\n0 iw −γ[H∥+ sin χ(Hc−HE)] 0 0 −γsinχHE\nγ[cosχ(Ha−HE)]γ[H∥+ sin χ(Ha−HE)] iw −γcosχHE γsinχHE 0\n0 0 −γcosχHE iw 0 γ[cosχ(Ha−Hc−HE)]\n0 0 −γsinχHE 0 iw γ [−H∥+ sin χ(HE−Hc)]\nγcosχHE γsinχHE 0 γcosχ(HE−Ha)γ[H∥+ sin χ(Ha−HE)] iw/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0.\n(18)\nMinimizing free energy (Eq. (7)) with respect to χgives sin χ=H∥/(2HE−Ha), and solving\nfor the eigenvalues yields\nω1,∥,flop=µ0γ/radicaligg\n(H2\n∥(Ha+ 2HE)−Ha(Ha−2HE)2)(2HE+Hc−Ha)\n(Ha−2HE)2(19)\nω2,∥,flop=µ0γ/radicaligg\n((2HE−Ha)2−H2\n∥)(Hc−Ha)\n2HE−Ha. (20)\nwhere ω1,∥,flopcorresponds to an acoustic mode and ω2,∥,flopcorresponds to an optical\nmode.\n3. Parallel orientation of the spin sub-lattices\nFor applied fields H∥sufficiently strong that the two spin sublattices become parallel to\neach other, ˆmeq\n1(2)= (1,0,0),χ→π/2. Analogous to the results in Eqs. (10) and (11), we\nget the usual ferromagnetic mode\nω1,∥,FM=µ0γ/radicalig\n(H∥+Ha)(H∥+Hc) (21)\nand a lower frequency mode\nω2,∥,FM=µ0γ/radicalig\n(H∥+Ha−2HE)(H∥+Hc−2HE) (22)\n11C. Spectra fitting\nTo determine the exchange and anisotropy parameters, we fit the measured spectra to\nthe appropriate piece-wise continuous functions. For magnetic field ( H⊥) applied parallel to\ntheaaxis (the intermediate magnetic axis), the fitting function is\nf⊥(H⊥) =\n\n(8),(9) for H⊥<2HE+Ha\n(10) for H⊥>2HE+Ha(23)\nFor a magnetic field ( H∥) applied parallel to the baxis (the easy magnetic axis), if Ha< H E:\nf∥(H∥) =\n\n(15),(16) for H∥</radicalbig\n(2HE−Ha)Ha\n(19) for/radicalbig\n(2HE−Ha)Ha< H∥<2HE−Ha\n(21) for H∥>2HE−Ha(24)\nand for the alternative case if Ha> H E:\nf∥(H∥) =\n\n(15),(16) for H∥< H E\n(21) for H∥> H E.(25)\nAs shown in Fig. S4, the fit parameters obtained from the piecewise fitting procedure give\nHa< H Efor all temperatures, such that in our model the spin-flop configuration should be\nstabilized for the field range/radicalbig\nHa(2HE−Ha)< H < 2HE−Ha. At the base temperature of\n5 K, Hais only slightly less than HEso that there should be only a very small field window\nexhibiting the spin-flop configuration, and the spectra is almost equivalent to that of a spin-\nflip transition. With increasing temperature, the anisotropy strength decreases faster than\nthe exchange strength, resulting in a more prominent spin-flop regime.\n12𝜇!H∥(T)\n𝜇!H∥(T)\n𝜇!H∥(T)f (GHz)\n𝜇!H∥(T)f (GHz)\nTransmission (dB)Transmission (dB)Hflop=\t2𝐻!−𝐻\"𝐻! Hp= 2HE-Ha5 K50 K\n100 K128 K(2Ha-HE)HaFIG. S4: Spectra from Sample 1 of the main text, with dashed lines representing the fit\nfunctions discussed in the main text and supplementary information, Section II. The\nvertical red line represents the predicted spin-flop transition field, and the blue line\nrepresents the field where the canting angle in the spin-flop state should realize χ=π/2 to\nachieve fully-parallel spin alignment.\nIII. EFFECT OF TRIAXIAL ANISOTROPY ON ANTIFERROMAGNETIC RES-\nONANT MODES\nHere we analyze in more detail the effects of triaxial magnetic anisotropy on antiferromag-\nnetic resonance modes, comparing to both the easy-plane anisotropy case and the uniaxial\neasy-axis case.\n13For an easy-plane antiferromagnet, with external field applied within the easy plane, the\nnormal modes in the spin-flop configuration can be expressed as [3]\nω1,⊥=µ0γH⊥/radicalbigg\n1 +Hc\n2HE(26)\nω2,⊥=µ0γ/radicaligg\n(4H2\nE−H2\n⊥)Hc\n2HE, (27)\nwhere ω1,⊥corresponds to the acoustic mode and ω2,⊥the optical mode, e.g., as measured\npreviously in CrCl 3[3]. These expressions are equivalent to the H⊥modes given by Eqs.\n(8) and (9) when µ0Ha= 0 T. We plot these frequencies as solid lines in Fig. S5(a) using\nparameters appriate for CrSBr except that we set µ0Ha= 0 T. With the addition of an\nin-plane intermediate anisotropy energy along the a-axis ( µ0Ha̸= 0 T), both acoustic and\noptical modes increase in frequency (dashed lines in Fig. S5(a)). The intermediate anisotropy\naxis also causes the dispersion of the acoustic mode to deviate from a linear relation with\nfield and gaps this mode for low applied magnetic fields.\nFor an antiferromagnet with uniaxial anisotropy and external field along the preferred\naxis, the normal modes are [2]\nω1,∥=µ0γ/radicalig\nH2\n∥+ 2HcHE+H2\nc−2H∥/radicalbig\nHc(2HE+Hc) (28)\nω2,∥=µ0γ/radicalig\nH2\n∥+ 2HcHE+H2\nc+ 2H∥/radicalbig\nHc(2HE+Hc) (29)\nwhere ω1,∥andω2,∥correspond to the left-handed (LH) and right-handed (RH) modes, e.g.\nas measured previously in Cr 2O3[4]. These frequencies are equivalent to the H∥modes\ngiven by Eqs. (15) and (16), when Ha=Hc. The two modes described by Eqs. (28), (29) are\ndegenerate at µ0H∥= 0 T and disperse linearly at low field (solid lines in Fig. S5(b). Similar\nto gadolinium iron garnet [5], the presence of triaxial anisotropy for CrSBr ( Ha̸=Hc),\nbreaks rotational symmetry about the easy axis and couples the RH and LH modes to\nlift the degeneracy at zero applied field (dashed lines fitted to the measured spectrum in\nFig.S5(b)).\n14Transmission (dB)𝜇!H∥(T)𝜇!H⊥(T)f (GHz)a)b)f (GHz)Transmission (dB)5 K5 KFIG. S5: a) Comparison of resonance frequencies assuming easy-plane anistropy ( µ0Ha= 0\nT, solid lines) vs. the measured modes in CrSBr for magnetic field applied in-plane\nperpendicular to the easy axis at 5 K, along with fits (dashed lines) assuming triaxial\nanisotropy. b) Comparison of resonance frequencies assuming uniaxial easy-axis anisotropy\n(µ0Ha=µ0Hc= 0.7 T, solid lines) vs. the measured modes in CrSBr for applied magnetic\nfield parallel to the easy axis at 5 K, along with fits (dashed lines) assuming triaxial\nanisotropy. The fit parameters are µ0Ha= 0.383(7) T, µ0HE= 0.395(2) T, and µ0Hc=\n1.30(2) T.\nIV. RESONANCE MODES FOR WEAK MAGNETIC FIELDS AT ARBITRARY\nIN-PLANE ANGLES RELATIVE TO THE ANISOTROPY AXES\nHere we consider the evolution of the modes when the external field is applied at an\narbitrary angle ϕbetween the ˆband ˆaeasy and intermediate axes. In this case, Hext=\nH0(cosϕ, sinϕ, 0) and ˆ meq\n1(2)= (±cosχ1(2), sinχ1(2), 0).\nThe effective magnetic field acting on each sub-lattice is\nH1(2)= [H0cosϕ−HE(δm2(1)beiωt∓cosχ2(1))]ˆb\n+ [H0sinϕ−Ha(δm1(2)aeiωt+ sin χ1(2))−HE(δm2(1)aeiωt+ sin χ2(1))]ˆa\n+ [−Hcδm1(2)ceiωt−HEδm2(1)ceiωt]ˆc.(30)\n15bac𝜒2intermediate axiseasy axishard axisHext𝜙𝒎$2eq𝜒1𝒎$1eqFIG. S6: Equilibrium positions of sub-lattices with external field at an arbitrary angle ϕ\nbetween the easy and intermediate axes.\nAt equilibrium, the net force on each sub-lattice is zero [2]\nH0sin (ϕ−χ1)−HEsin (χ1+χ2)−Hasinχ1= 0\nH0sin (ϕ+χ2)−HEsin (χ1+χ2)−Hasinχ2= 0.(31)\nWe consider the small field regime where H0<√2HaHE, such that χ1,χ2are small. Using\nthe small angle approximation, the equilibrium positions χ1,2are given by the expression [2]\nχ1(2)=H0sinϕ(Ha∓H0cosϕ)\nH2\na+ 2HaHE−H2\n0cos2ϕ. (32)\nThe 6 ×6 matrix encapsulating the coupled L-L equations is\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle−iw 0M130 0 M16\n0−iw M 230 0 M26\nM31M32−iw M 34M350\n0 0 M43−iw 0M46\n0 0 M530−iw M 56\nM61M620M64M65−iw/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0. (33)\n16with the matrix elements\nM13=−γ((Ha−Hc) sinχ1+HEsinχ2−H0sinϕ)\nM16=γHEsinχ1\nM23=−γ(Hccosχ1+HEcosχ2+H0cosϕ)\nM26=−γHEcosχ1\nM31=−γ(−Hasinχ1−HEsinχ2+H0sinϕ)\nM32=−γ(−Hacosχ1−HEcosχ2−H0cosϕ)\nM34=−γHEsinχ1\nM35=γHEcosχ1\nM43=γHEsinχ2\nM46=−γ((Ha−Hc) sinχ2+HEsinχ1−H0sinϕ)\nM53=γHEcosχ2\nM56=γ(Hccosχ2+HEcosχ1−H0cosϕ)\nM61=−γHEsinχ2\nM62=−γHe cosχ2\nM64=γ(Hasinχ2+HEsinχ1−H0sinϕ)\nM65=−γ(Hacosχ2+HEcosχ1−H0cosϕ).(34)\nAt this point we use the small angle approximation for χ1,2in Eq. (32) and employ\nmatrix diagonalization to obtain an analytical solution for the resonance modes for the\nsmall-field regime. Using exchange and anisotropy field strengths obtained from the fits to\nthe experiment at ϕ= 0◦and 90◦at 100 K, we plot the evolution of the modes for a range\nof in-plane angles ϕ= 0◦to 90◦in Fig. S7a. We see that the upper and lower modes evolve\nsmoothly from the convex-paired optical and acoustic modes at ϕ= 90◦to the concave-\npaired RH and LH modes at ϕ= 0◦. In addition, fields applied at small angles away from\nϕ= 90◦cause the acoustic/optical mode crossing to become gapped. This is analogous to\nthe mode anti-crossing induced by a symmetry breaking out-of-plane field for the resonance\nmodes in easy-plane CrCl 3[3]. In our case, the two-fold rotational symmetry existing for\n17CrSBr with a field applied along the intermediate ˆ aaxis can be broken with an in-plane field\ncomponent along the easy ˆbaxis.\n𝜇!H (T)𝜇!H (T)\nf (GHz)\nFIG. S7: Resonance modes calculated for external field at an arbitrary angles ϕbetween\nthe easy and intermediate anisotropy axes, with the parameters (appropriate for 100 K)\nµ0HE= 0.222(5) T, µ0Ha= 0.147(4) T, and µ0Hc= 0.63(1) T. The acoustic and optical\nmodes evolve smoothly to the RH and LH modes as the external field is rotated from the\nintermediate axis ( ϕ= 90◦) to the easy axis ( ϕ= 0◦). For small angles away from the\nintermediate axis, we see a gap opening at the mode crossings, associated with the\nbreaking of the two-fold rotation symmetry.\nWhen the external field becomes comparable to the exchange and anisotropy, H0>\n√2HaHE, the small angle approximation no longer holds (main text, Fig. 5). Here, we\nnumerically calculate the equilibrium positions χ1,χ2by integrating the L-L equations for a\ngiven field direction and field strength. We then use these equilibrium values to evaluate the\nmatrix elements in Eq. (34) and diagonalize the matrix for the resonance mode frequencies.\n18V. COMPLETE SERIES OF SPECTRA AT DIFFERENT TEMPERATURES\nHere we present the complete series of H⊥andH∥spectra used to obtain the fit parameters\nfor the exchange and anisotropy fields from 5 - 160 K (Fig. 3 in the main text). Dashed\nlines represent fits, while the red and blue dotted lines represent the predicted spin-flop and\nfully-parallel spin alignment transition fields, respectively.\n𝜇!H∥(T)c)d)5 Ka)b)\ne)f)5 K\n25 K25 K\n50 K50 K\nTransmission (dB)Transmission (dB)Transmission (dB)𝜇!H∥(T)𝜇!H⊥(T)f (GHz)\nf (GHz)𝜇!H⊥(T)f (GHz)\nf (GHz)\n𝜇!H∥(T)𝜇!H⊥(T)f (GHz)\nf (GHz)acbacb\nH∥HRF\nHRF\nH⊥\n19k)l)100 Ki)j)\nm)n)100 K\n115 K115 K\n124 K124 K\nTransmission (dB)Transmission (dB)Transmission (dB)𝜇!H∥(T)𝜇!H⊥(T)f (GHz)\nf (GHz)𝜇!H∥(T)𝜇!H⊥(T)f (GHz)\nf (GHz)\n𝜇!H∥(T)𝜇!H⊥(T)f (GHz)\nf (GHz)g)h)75 K75 K\nTransmission (dB)𝜇!H∥(T)𝜇!H⊥(T)f (GHz)\nf (GHz)20s)t)133 Kq)r)\nu)v)133 K\n140 K140 K\n155 K155 K\nTransmission (dB)Transmission (dB)Transmission (dB)𝜇!H∥(T)𝜇!H⊥(T)f (GHz)\nf (GHz)𝜇!H∥(T)𝜇!H⊥(T)f (GHz)\nf (GHz)f (GHz)\nf (GHz)o)p)128 K128 K\nTransmission (dB)𝜇!H∥(T)𝜇!H⊥(T)f (GHz)\nf (GHz)\n𝜇!H∥(T)𝜇!H⊥(T)21y)z)160 Kw)x)\naa)ab)160 K\n180 K180 K\n220 K220 K\nTransmission (dB)Transmission (dB)Transmission (dB)𝜇!H∥(T)𝜇!H⊥(T)f (GHz)\nf (GHz)𝜇!H∥(T)𝜇!H⊥(T)f (GHz)\nf (GHz)f (GHz)\nf (GHz)𝜇!H∥(T)𝜇!H⊥(T)FIG. S8: Transmission spectra at all measured temperatures with fitted magnetic\nresonance modes. Red and blue dotted lines indicate the fitted spin-flop and ferromagnetic\nalignment fields respectively.\n22VI. SPECTRA FOR A SECOND CRYSTAL WITH ITS a-AXIS PARALLEL TO\nTHE COPLANAR WAVEGUIDE CENTER LINE\nWe have also measured the spectra of a second CrSBr crystal oriented with its long edge\n(a-axis) parallel to the center line of the coplanar waveguide (Fig. S9). In this configuration,\nfor magnetic fields applied parallel to the easy-magnetic-anisotropy baxis ( H∥) we measure\nsimilar resonances compared to those in the main text – two modes at low fields followed by\na discontinuous jump leading eventually to a ferromagnetic-like mode. For applied magnetic\nfields aligned along the intermediate-anisotropy aaxis ( H⊥), due to 2-fold rotational sym-\nmetry in this configuration (see supplementary information, Section VII), only the acoustic\nmode is excited (the one odd under the symmetry). Figure S10 shows the anisotropy and\nexchange field strengths as a function of temperature extracted from fits to these spectra.\nAll parameters are in reasonable agreement with those extracted for the sample analyzed in\nthe main text.\n23c)d)5 Ka)b)\ne)f)5 K\n25 K25 K\n45 K45 K\nTransmission (dB)\n𝜇!H∥(T)𝜇!H⊥(T)f (GHz)\nf (GHz)𝜇!H∥(T)𝜇!H⊥(T)f (GHz)\nf (GHz)\n𝜇!H∥(T)𝜇!H⊥(T)f (GHz)\nf (GHz)\nH⊥HRF\nH∥HRF\nbcabca\nTransmission (dB)Transmission (dB)24k)l)100 Ki)j)\nm)n)100 K\n115 K115 K\n125 K125 K𝜇!H∥(T)𝜇!H⊥(T)f (GHz)\nf (GHz)𝜇!H∥(T)𝜇!H⊥(T)f (GHz)\nf (GHz)\n𝜇!H∥(T)𝜇!H⊥(T)f (GHz)\nf (GHz)g)h)75 K75 K\nTransmission (dB)𝜇!H∥(T)𝜇!H⊥(T)f (GHz)\nf (GHz)\nTransmission (dB)Transmission (dB)Transmission (dB)25s)t)140 Kq)r)\nu)v)140 K\n160 K160 K\n180 K180 K𝜇!H∥(T)𝜇!H⊥(T)f (GHz)\nf (GHz)𝜇!H∥(T)𝜇!H⊥(T)f (GHz)\nf (GHz)f (GHz)\nf (GHz)o)p)132 K132 K\nTransmission (dB)𝜇!H∥(T)𝜇!H⊥(T)f (GHz)\nf (GHz)\nTransmission (dB)Transmission (dB)Transmission (dB)𝜇!H∥(T)𝜇!H⊥(T)FIG. S9: Transmission spectra at different temperatures for the second crystal, with a-axis\nparallel to the center line of the coplanar waveguide.\n26Anisotropy and interlayer exchange parameters (T)T (K)TNFIG. S10: Exchange and anisotropy fit parameters as a function of temperature for crystal\n2, with a-axis∥to the center line of the coplanar waveguide. The dashed line indicates the\nTNpreviously reported for bulk CrSBr [6, 7].\n27VII. SELECTIVE EXCITATION OF MODES FROM 2-FOLD ROTATION SYM-\nMETRY CONSIDERATIONS\nFor the H⊥configuration of crystal 2 with field applied along ˆ a(SI Section VI), we observe\nonly the acoustic modes, not the optical modes. This arises from symmetry considerations\npreviously discussed for CrCl 3[3], due to the system being symmetric under two-fold rotation\nabout the ˆ a-axis and exchange of the spin sublattices. Within same formalism used in ref.\n[3], the two-fold rotational symmetry still applies for a sample with triaxial anisotropy, with\nan intermediate anisotropy axis parallel to ˆ a. Including terms to first-order in precession\namplitudes, the applicable L-L equations are:\niωδm1=γmeq\n1×(Heqδm1+HEδm2+Hc(δm1·ˆc)ˆc+Ha(δm1·ˆa)ˆa) +τ1\niωδm2=γmeq\n2×(Heqδm2+HEδm1+Hc(δm2·ˆc)ˆc+Ha(δm2·ˆa)ˆa) +τ2.(35)\nHere,meq\n1(2),δmeq\n1(2)are the equilibrium and displacement vectors of each sublattice, respec-\ntively. τ1(2)are the torques due to the RF field HRFacting on each sublattice. Heqis\nthe magnitude of total effective magnetic field at the equilibrium condition: Heqmeq\n1(2)=\nH⊥ˆa−HEm2(1)−Ha(m1(2)·ˆa)ˆa. By balancing the torques at equilibrium, Heq=|Hˆa−\nHEm2−Ha(m1·ˆa)ˆa|=HE. Because the spin-flop configuration is 2-fold symmetric about\nˆa, C2aˆ meq\n1=ˆ meq\n2, where C 2ais an operator for a 2-fold rotation about the a axis, we can\ndecouple Eqs. (35) by defining two orthogonal modes that are linear combinations of the\nsublattice displacement vectors: δm±=δm1±C2aδm2, that obey the L-L equation\niωδm±=γmeq\n1×(Heδm±±HeC2yδm±+Hc(δm±·ˆc)ˆc+Ha(δm±·ˆa)ˆa) +τ± (36)\nUsing the equilibrium position condition previously obtained through minimizing the free\nenergy (Eq. (7)): sin χ=H⊥/(2HE+Ha), Eq. (36) can be expanded to give a 6 ×6 block\ndiagonal matrix, with upper left and lower right quadrants corresponding to the + and −\n28orthogonal modes respectively\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle−iw 0 −γHcsinχ 0 0 0\n0 −iw γH ccosχ 0 0 0\n0−γ(2HE+Ha) cosχ−iw 0 0 0\n0 0 0 −iw 0 −γ(2HE+Hc) sinχ\n0 0 0 0 −iw γ (2HE+Hc) cosχ\n0 0 0 2 γHEsinχ−γHacosχ −iw/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0.(37)\nSolving for the eigenvalues, we get the same solutions ω1,⊥andω2,⊥previously calculated\nfrom the coupled L-L equations (SI. IV, Eqs. (8,9)). We note that the acoustic mode is given\nby the −mode ( ω1,⊥=ω−) where δmeq\n1andδmeq\n2rotate in-phase, while the optical mode\nis given by the + mode ( ω2,⊥=ω+), where δmeq\n1andδmeq\n2rotate out-of-phase.\nThe absence of the ω2,⊥optical mode in our spectra can be understood as follows. In\ngeneral, we can split HRFinto components that are even or odd upon 2-fold rotation about\nˆa. For the even components, using ˆ meq\n1=C2aˆ meq\n2andHRF=C2aHRF, we can deduce that\nthe torques on sublattices 1, 2 also have even symmetry.\nτ1=ˆ meq\n1×HRF=C2aˆ meq\n2×C2aHRF\n=C2a(ˆ meq\n2×HRF) =C2aτ2(38)\nFrom the definitions of τ±, this gives τ−= 0 and τ+̸= 0. In other words, components of\nHRFthat are even with rotation about ˆ a, excite the optical mode but not the acoustic mode.\nA similar argument can be made for HRFcomponents that are odd with rotation about ˆ a,\nwhich gives τ1=−C2yτ2. This implies that τ+= 0 while τ−̸= 0 so only the acoustic mode\nis excited.\nFor the H⊥configuration of crystal 2, HRFonly has an in-plane component along the ˆb\naxis and an out-of-plane component along the ˆ caxis, both of which are odd with rotation\nabout ˆ a. Therefore, only the acoustic mode appears in the measured spectra. In the large\nfield regime for the H∥configuration, the system is 2-fold symmetric around the applied\nfield, this time along ˆb. Here, the in-plane component of HRFis even, while the out-of-plane\ncomponent is odd with rotation about ˆ a. Following the same arguments above, we deduce\nthat both the optical and the lower frequency ferromagnetic modes should be excited.\n29VIII. MICROWAVE POWER DEPENDENCE\nHere we present examples of spectra acquired using various applied microwave powers from\n-40 dB to -5 dB at 100 K. As there is negligible variation in the measured resonance modes\nwith power, we conclude that there is no significant sample heating from microwave excitation\nin our measurements. We use -20 dB for all measurements unless otherwise specified.\n𝜇!H⊥(T)f (GHz)f (GHz)𝜇!H⊥(T)𝜇!H⊥(T)𝜇!H⊥(T)f (GHz)\nf (GHz)𝜇!H⊥(T)a)b)c)\nd)e)-5 dB-10 dB-20 dB-30 dB-40 dB\nf (GHz)\nTransmission (dB)100 K\nTransmission (dB)\nFIG. S11: Absorption spectra at various applied microwave powers from -40 dB to -5 dB at\n100 K.\n30REFERENCES\n[1] C. Kittel, “Theory of antiferromagnetic resonance,” Physical Review 82, 565–565 (1951).\n[2] F. Keffer and C. Kittel, “Theory of antiferromagnetic resonance,” Physical Review 85, 329–337\n(1952).\n[3] David MacNeill, Justin T. Hou, Dahlia R. Klein, Pengxiang Zhang, Pablo Jarillo-Herrero,\nand Luqiao Liu, “Gigahertz frequency antiferromagnetic resonance and strong magnon-magnon\ncoupling in the layered crystal CrCl 3,” Physical Review Letters 123, 047204 (2019).\n[4] Junxue Li, C. Blake Wilson, Ran Cheng, Mark Lohmann, Marzieh Kavand, Wei Yuan, Mo-\nhammed Aldosary, Nikolay Agladze, Peng Wei, Mark S. Sherwin, and Jing Shi, “Spin current\nfrom sub-terahertz-generated antiferromagnetic magnons,” Nature 578, 70–74 (2020).\n[5] Lukas Liensberger, Akashdeep Kamra, Hannes Maier-Flaig, Stephan Gepr¨ ags, Andreas Erb,\nSebastian T. B. Goennenwein, Rudolf Gross, Wolfgang Belzig, Hans Huebl, and Mathias\nWeiler, “Exchange-enhanced ultrastrong magnon-magnon coupling in a compensated ferrimag-\nnet,” Physical Review Letters 123, 117204 (2019).\n[6] Evan J. Telford, Avalon H. Dismukes, Kihong Lee, Minghao Cheng, Andrew Wieteska, Amy-\nmarie K. Bartholomew, Yu-Sheng Chen, Xiaodong Xu, Abhay N. Pasupathy, Xiaoyang Zhu,\nCory R. Dean, and Xavier Roy, “Layered antiferromagnetism induces large negative mag-\nnetoresistance in the van der waals semiconductor CrSBr,” Advanced Materials 32, 2003240\n(2020).\n[7] Kihong Lee, Avalon H. Dismukes, Evan J. Telford, Ren A. Wiscons, Jue Wang, Xiaodong Xu,\nColin Nuckolls, Cory R. Dean, Xavier Roy, and Xiaoyang Zhu, “Magnetic Order and Symmetry\nin the 2D Semiconductor CrSBr,” Nano Letters 21, 3511–3517 (2021).\n31"    },    {        "title": "2206.05121v1.Topological_spiral_magnetism_in_the_Weyl_semimetal_SmAlSi.pdf",        "content": "Topological spiral magnetism in the Weyl semimetal SmAlSi\nXiaohan Yao,1Jonathan Gaudet,2, 3Rahul Verma,4David E. Graf,5Hung-Yu Yang,1Faranak Bahrami,1\nRuiqi Zhang,6Adam A. Aczel,7Sujan Subedi,8Darius H. Torchinsky,8Jianwei Sun,6Arun Bansil,9\nShin-Ming Huang,10Bahadur Singh,4Predrag Nikoli\u0013 c,11, 12Peter Blaha,13and Fazel Tafti1,\u0003\n1Department of Physics, Boston College, Chestnut Hill, MA 02467, USA\n2NIST Center for Neutron Research, Gaithersburg, Maryland 20899, USA\n3Department of Materials Science and Eng., University of Maryland, College Park, MD 20742-2115\n4Department of Condensed Matter Physics and Materials Science,\nTata Institute of Fundamental Research, Colaba, Mumbai 400005, India\n5National High Magnetic Field Laboratory, Tallahassee, FL 32310, USA\n6Department of Physics, Tulane University, New Orleans, LA 70118, USA\n7Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA\n8Department of Physics, Temple University, Philadelphia, PA 19122, USA\n9Department of Physics, Northeastern University, Boston, MA 02115, USA\n10Department of Physics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan\n11Department of Physics and Astronomy, George Mason University, Fairfax, Virginia 22030, USA\n12Institute for Quantum Matter at Johns Hopkins University, Baltimore, MD 21218, USA\n13Institute of Materials Chemistry, Vienna University of Technology, 1060 Vienna, Austria\nWeyl electrons are intensely studied due to novel charge transport phenomena such as chiral\nanomaly, Fermi arcs, and photogalvanic e\u000bect. Recent theoretical works suggest that Weyl elec-\ntrons can also participate in magnetic interactions, and the Weyl-mediated indirect exchange cou-\npling between local moments is proposed as a new mechanism of spiral magnetism that involves\nchiral electrons. Despite reports of incommensurate and non-collinear magnetic ordering in Weyl\nsemimetals, an actual spiral order has remained hitherto undetected. Here, we present evidence\nof Weyl-mediated spiral magnetism in SmAlSi from neutron di\u000braction, transport, and thermody-\nnamic data. We show that the spiral order in SmAlSi results from the nesting between topolog-\nically non-trivial Fermi pockets and weak magnetocrystalline anisotropy, unlike related materials\n(Ce,Pr,Nd)AlSi, where a strong anisotropy prevents the spins from freely rotating. We map the\nmagnetic phase diagram of SmAlSi and reveal an A-phase where topological magnetic excitations\nmay exist. This is corroborated by the observation of a topological Hall e\u000bect within the A-phase.\nI. INTRODUCTION\nIt is well known that the combination of magnetism\nand topology leads to novel electromagnetic phenomena\nsuch as extreme magnetoresistance, quantum anomalous\nHall e\u000bect, axion insulator, and chiral domain walls1{7.\nIt is less known, however, whether topological entities\nsuch as Dirac and Weyl fermions can directly partici-\npate in magnetic interactions and drive speci\fc types\nof magnetic ordering. Recent calculations have shown\nthat chiral Weyl electrons could couple to local magnetic\nmoments through the Ruderman-Kittel-Kasuya-Yosida\n(RKKY) mechanism, and mediate indirect Heisenberg\n(JijSi\u0001Sj), Kitaev (K\rS\r\niS\r\nj), and Dzyaloshinskii-Moriya\n(DM) interactions ( D\u0001Si\u0002Sj)8{11. The Weyl-mediated\nRKKY interactions provide a new mechanism for spi-\nral magnetic ordering and topological magnetic defects\n(hedghogs, skyrmions, and meron-antimeron pairs) that\nare in demand for high-density and high-speed memory\ndevices12{14.\nExperimental evidence of such a Weyl-mediated\nRKKY interaction has recently been found in NdAlSi15,\na Weyl semimetal with an incommensurate modulated\nspin density wave (SDW) whose wavelength is linked to\nthe nesting vector between two topologically non-trivial\nFermi pockets. A chiral transverse component is ob-served for this SDW, which is promoted by the Weyl-\nmediated DM interactions. However, a strong magne-\ntocrystalline anisotropy (MCA) in NdAlSi prevents the\nspins from developing a truly spiral order. Instead, a\ncommensurate ferrimagnetic order is ultimately stabi-\nlized at low temperatures15.\nIn this letter, we study SmAlSi, a true spiral mag-\nnetic counterpart of NdAlSi due to a much weaker MCA.\nWe use heat capacity, neutron di\u000braction, magnetization,\nHall e\u000bect, band structure calculations and Monte Carlo\nsimulations to reveal a spiral magnetic order in SmAlSi\nat low temperatures and attribute it to the Weyl-induced\nspin interactions in the absence of a signi\fcant MCA. We\nalso reveal a region in the phase diagram of SmAlSi that\nresembles the A-phase in helimagnets such as MnSi and\nGd3Ru4Al12, where skyrmions have been found16,17.\nII. METHODS\nSingle crystals of SmAlSi were grown using a self-\rux\nmethod18. A Bruker D8 ECO system was used for pow-\nder X-ray di\u000braction, and the FullProf suite was used for\nthe Rietveld re\fnement19. The electrical transport, heat\ncapacity, and magnetic susceptibility were measured us-\ning the Quantum Design PPMS Dynacool and MPMS-3\ninstruments. A piezoresistive technique was used for thearXiv:2206.05121v1  [cond-mat.mtrl-sci]  10 Jun 20222\n10110204812 Cm(J K1mol1)\n(a)\nCeAlSi\nT1\nTCEFSchottky\n10110204812 Cm(J K1mol1)\n(b)\nNdAlSi T1\nT2TCEF\n101102\nT(K)04812 Cm(J K1mol1)\n(c)\nSmAlSi\nT1\nT2TCEF\n0 100 200 300\nT(K)102\n101\n100101102\n(d)\nH= 100 OeNdAlSi\nSmAlSi\nCeAlSi\n0 4 812\nT(K)20253035 (cm3mol1)\nT2\nT1\nc/a\na\nc\nFIG. 1. Magnetic heat capacity is plotted as a function of\ntemperature in (a) CeAlSi, (b) NdAlSi, and (c) SmAlSi. The\nbroad Schottky anomaly is due to the crystal \feld levels and\nthe sharp peaks are magnetic transitions. The insets show\nan in-plane anisotropy in CeAlSi, out-of-plane anisotropy in\nNdAlSi, and spiral order in SmAlSi, consistent with the \u001fc=\u001fa\nratios plotted in (d). Whereas CeAlSi and NdAlSi show\nstrong in-plane and out-of-plane anisotropy, SmAlSi shows\n\u001fc=\u001faof order one.\nhigh-\feld magnetometry at the National High Magnetic\nField Laboratory. The neutron di\u000braction experiment\nwas performed with the \fxed-incident-energy (14.5 meV)\nneutron triple-axis spectrometer HB-1A in the High Flux\nIsotope Reactor (HFIR) at the Oak Ridge National\nLaboratory, and the SARAh software was used for the\nrepresentational analysis of the magnetic structure20.\nDensity functional theory (DFT) calculations with the\nfull-potential linearized augmented plane-wave (LAPW)\nmethod were implemented in the WIEN2k code21using\nthe Perdew-Burke-Ernzerhof (PBE) exchange-correlation\nfunctional22, spin-orbit coupling (SOC), and on-site\nCoulomb repulsion (Hubbard U)23. Monte Carlo sim-\nulations were performed using the Metropolis algorithm.\nIII. RESULTS AND DISCUSSION\nHeat Capacity. SmAlSi belongs to a family of RAlSi\nWeyl semimetals that have magnetic rare-earth (R3+)\nions in a non-centrosymmetric space group I41md(inset\nof Fig. 1 and Fig. S1)24{27. The key parameter that en-\nables spiral ordering in SmAlSi, unlike in other RAlSi\ncompounds, is a weak MCA that allows the spins to\nrotate nearly freely. The MCA can be quanti\fed us-\ning the heat capacity and magnetic susceptibility data.\nWe isolate the magnetic heat capacity ( Cm) of RAlSi inTABLE I. TCEFand \u0001 are extracted from the Schottky \fts\nin Fig. 1 (see Fig. S2). The softening of MCA is re\rected in\nthe systematic reduction of TCEF, \u0001, and the \u001fc=\u001faratio at\n2 K. The PrAlSi data are reproduced from Ref.24.\nMaterialTCEF(K) \u0001 (meV) T1(K)T2(K)\u001fc=\u001fa\nCeAlSi 91(7) 29(11) 8.2(2) N/A (188)\u00001\nPrAlSi 30(6) 18(9) 17.8(2) N/A 194\nNdAlSi 16(3) 6(2) 7.2(1) 3.3(1) 84\nSmAlSi 15(3) 5(2) 10.6(2) 4.6(2) 0.8\nFig. 1 by subtracting the phonon background using the\nnon-magnetic isostructural compound LaAlSi (Fig. S2).\nThe broad Schottky anomaly at TCEFin Figs. 1a,b,c is\ndue to crystal electric \feld (CEF) excitations, and the\nsharp peaks at T1andT2are due to magnetic order-\ning. We model the Schottky anomaly by assuming a dou-\nblet ground-state separated by excited multiplets in the\nKramers ions Ce3+, Nd3+, and Sm3+(Fig. S2)25. Our\nSchottky \fts (black dashed lines in Figs. 1a,b,c) yield\nCEF gaps \u0001 = 29(11), 6(2), and 5(2) meV in CeAlSi,\nNdAlSi, and SmAlSi, respectively. The systematic reduc-\ntion of \u0001 and TCEFin Table I shows a general softening\nof the MCA from R = Ce to Nd and Sm. This is also\nre\rected in the systematic change of the ratio between\nout-of-plane and in-plane magnetic susceptibility \u001fc=\u001fa\nat 2 K (Fig. 1d and Table I). Indeed, whereas CeAlSi has\nan easy plane with \u001fa=\u001fc\u0019200, and NdAlSi has an easy\naxis with\u001fc=\u001fa\u001980, SmAlSi has a negligible MCA with\n\u001fc=\u001fa\u00191 (inset of Fig. 1d), hence its propensity toward\nspiral ordering.\nNeutron Di\u000braction. It is challenging to probe the\nmagnetic ground-state of SmAlSi with neutrons because\nthe naturally occurring isotope149Sm has a large neutron\nabsorption cross-section ( \u001babs= 42;080 b) and Sm3+has\na small moment (0 :7\u0016B). Using154Sm (\u001babs= 8:4 b)\ncould help but this isotope is available only in oxide form,\nwhich is not suitable for growing intermetallic crystals.\nIn the absence of isotope-rich samples, we took advan-\ntage of the recent instrumentation upgrade of the HB-1A\nspectrometer where the beam \rux has been increased sig-\nni\fcantly, and aligned a single crystal of SmAlSi within\nthe (HHL) plane to probe the Bragg peaks along the\nhigh symmetry directions of a few Brillouin zone (BZ)\ncenters. The intensity of the nuclear Bragg peaks did\nnot increase below T1orT2ruling out a k= (0;0;0)\nferromagnetic (FM) order. Instead, we found antiferro-\nmagnetic (AFM) Bragg peaks at Q=\u00001\n3\u0000\u000e;1\n3\u0000\u000e;4\u0001\nand\u00001\n3\u0000\u000e;1\n3\u0000\u000e;8\u0001\n, which can be indexed by an incom-\nmensurate vector k=\u00001\n3\u0000\u000e;1\n3\u0000\u000e;0\u0001\nwith\u000e= 0:007(1)\n(Fig. 2a). The temperature dependence of an AFM peak\nintensity in the inset of Fig. 2a shows the onset of order\nparameter at T1= 10:6(2) K.\nThe magnetic ordering vector k=\u00001\n3\u0000\u000e;1\n3\u0000\u000e;0\u0001\nin\nSmAlSi is similar to the incommensurate ordering vector\nobserved in NdAlSi15, and indicates an SDW propagating\nalong the [1 ;1;0] direction. However, a large Ising MCA3\n[110](a)\na\nHelical 1\nc\n[1,-1,0]Helical 2\nc\n[1,-1,0]Cycloidal 1\nc\n[1,1,0](d)b\nCycloidal 2\nc\n[1,1,0]\n[110] [110] [110] [110] [110]Mixed 1 Mixed 2\nab\n(c)\n[110]ab[110]\nabcc\n1\n2\n3\n654\n1\n2\n31\n2\n31\n2\n31\n2\n31\n2\n31\n2\n3\n654\n654\n654\n654\n654\n654\nSmAlSi(b)\nFIG. 2. (a) Neutron di\u000braction scan along the ( HH8) direction in SmAlSi, which reveals a temperature dependent Bragg peak\natQ=\u00001\n3\u0000\u000e;1\n3\u0000\u000e;8\u0001\n. The top right inset is the temperature dependence of the Bragg peak intensity. Error bars correspond\nto one standard deviation. (b) View of the magnetic unit cell for a k=\u00001\n3;1\n3;0\u0001\nordering vector. Each color represents a\ndi\u000berent spin orientation on a particular Sm site. The black box represents the size of the structural unit cell. (c) 3D rendition\nof one possible spin structure. (d) Sketches of all six possible spiral orders in SmAlSi. The di\u000berent colors represent di\u000berent\nspin orientations as in (b).\nin NdAlSi prevents spiral ordering and drives a ferrimag-\nnetic up-down-down order with a \fnite FM component15,\nwhereas a weak MCA in SmAlSi (Fig. 1d) makes the spi-\nral ordering possible without an FM component.\nBy performing a magnetic representation analysis20,\nwe identi\fed six possible spin structures that yield a\nk=\u00001\n3\u0000\u000e;1\n3\u0000\u000e;0\u0001\nvector in SmAlSi and have constant\nmagnetization on each Sm site. We plot the six possi-\nble spin structures in Figs. 2b,c,d assuming \u000e= 0 (ideal\n120\u000eorder) for a simple visualization. Their magnetic\nunit cell is 3\u00023\u00021 times the structural unit cell and\ncontains up to 6 di\u000berent spins in an ABC-type structure\npropagating amongst both the (0 ;0;0) and\u0000\n0;1\n2;1\n4\u0001\nSm\nsites as shown in 2D and 3D, respectively, in Fig. 2b and\n2c. All six structures in Fig. 2d are spiral but with dif-\nferent chirality. First, there is a helical order where the\nspins are constrained in the plane perpendicular to the\n[1;1;0] propagation vector. Second, there is a cycloidal\norder where the spins are constrained in the plane paral-\nlel to the [1 ;1;0] propagation vector. Finally, there is a\nmixed spiral order where the spins are constrained within\nthe basal plane so they have a component that oscillates\nboth perpendicular and parallel to the propagation vec-\ntor. There are two copies of each type of spiral with in-\nphase and anti-phase ordering between the (0 ;0;0) and\u0000\n0;1\n2;1\n4\u0001\nprimitive Sm sites as shown in Fig. 2d. Of these\nsix possible con\fgurations, the \frst two (helical 1 and2) are consistent with the theoretical prediction that the\nDvector in the Weyl-induced DM interaction is (nearly)\nparallel to the bond direction when Weyl electrons have\n(nearly) spherically symmetric dispersion9.\nMagnetization. In all spiral con\fgurations of Fig. 2d,\nthe spins rotate 120\u000ewith respect to each other, such that\nno net magnetization is produced over an ABC cycle. For\nSmAlSi, however, the spins rotate with 120.24(4)\u000ewith\nrespect to each other due to the small incommensurabil-\nity (\u000e), so the magnetic unit cell is about 263 nm long\nwith no net magnetization. This is consistent with our\nM(H) curves in Figs. 3a,b that show the absence of hys-\nteresis near zero-\feld for both HkaandHkc. At \fnite\n\felds, however, we observe hysteretic metamagnetic tran-\nsitions when Hka(Fig. 3a) but not when Hkc(Fig. 3b).\nThe magnitude of M(H= 7 T) is only 0 :045\u0016Bwhich\nis an order of magnitude less than the expected value for\nSm3+(0.7\u0016B). The hysteretic metamagnetic transitions\nwith small amplitudes for Hkaand the strictly linear\nM(H) curves when Hkcare characteristics of materials\nwith topological skyrmion excitations, e.g. GaV 4(S,Se) 8\nand Gd 2PdSi 328{31.\nWe construct a magnetic phase diagram for SmAlSi\nbased on two sets of peaks in dM=dH as a function of \feld\n(black and red circles in Fig. 3c) and the \feld dependence\nofT1andT2peaks in the Cmdata (Fig. 3d). There are\n\fve phases in the phase diagram (Fig. 3e). Phase 1 is the4\n-6-4-20246\nH(T)-0.050.050.150.250.35M(arb. u.)\n(a)H|| a\nT= 11.0 K\n7.0\n6.0\n5.2\n5.0\n4.8\n4.0\n2.0\n-6-4-20246\nH(T)dM/dH(arb. u.)\n(c)H|| aT= 11.0 K\n7.0\n6.0\n5.2\n5.0\n4.8\n4.0\n2.0\n-6-4-20246\nH(T)-0.020.000.02M(arb. u.)\n(b)H|| c\nT= 2.0 K\n4.0\n4.8\n5.0\n5.2\n6.0\n7.0\n11.0\n0 5 10 15\nT(K)04812 Cm(J K1mol1)\n(d)\nH|| a\nT1\nT26\n7\n8\n9H= 1 T\n3\n4\n5\n0 4 8 12\nT(K)0481216H(T)\n(e)\nA-phase\nPhase 1 Phase 2Phase 3 Phase 4SmAlSi\n0 3 6 9\nH(T)-30-24-18-12-606xy(  cm)\n(f)\n= 0\n15\n30\n45\n60\n75\n90\nT= 5 K\nθ\nH|| a\nFIG. 3. (a) Magnetization curves with Hkaare plotted at several temperatures. The 2 K curve has correct values in units\nof\u0016Band the rest of the curves are shifted uniformly to improve visibility. (b) M(H) curves with Hkcare strictly linear.\n(c) Derivatives of the M(Hka) curves reveal two sets of peaks (black and red circles) that de\fne the A-phase boundaries. (d)\nMagnetic heat capacity as a function of temperature Cm(T) at several \felds with Hka. (e) The phase diagram of SmAlSi. The\n\flled magenta and blue squares correspond to T1andT2, respectively. The \flled black and red circles are from the panel (c).\nAll empty symbols are from the high-\feld data in Fig. S325. (f) The Hall data \u001axy(H) are shown at 5 K with the \feld pointing\nat several angles as indicated in the inset.\nclosest region to the zero-\feld ground-state investigated\nby neutron di\u000braction (Fig. 2). Its horizontal and vertical\nboundaries correspond, respectively, to the black peaks\nindM=dH andT2peaks in the Cmdata (Figs. 3c,d).\nPhase 2 is bound vertically by the T1andT2peaks in the\nCmdata (Fig. 3d). It is bound horizontally by the black\npeaks indM=dH (\flled black circles in Fig. 3c,e) and\nhysteresis loops in the high-\feld magnetic torque data\n(empty black circles in Fig. 3e). The high-\feld data are\npresented as Supplemental Material25(Fig. S3). Phase\n3 is horizontally bound by the black peaks in dM=dH\nand purple circles representing the saturation of the high-\n\feld torque signal (Fig. S3). It is vertically bound by the\nT2peaks in the Cmdata (blue squares in Fig. 3e) and\nsteps in the high-\feld torque data (open orange circles)25.\nThe two sets of dM=dH peaks (black and red symbols in\nFig. 3c,e) encompass an A-phase similar to the A-phase\nin materials such as GaV 4(S,Se) 8and Gd 2PdSi 3where\nskyrmions have been found28{31. Phase 4 is separated\nfrom the A-phase by the red dM=dH peaks in Fig. 3c\nand hysteresis loops in the high-\feld torque data (open\nred circles in Fig. 3e and Fig. S3).\nHall E\u000bect. Con\frming skyrmions in the A-phase of\nSmAlSi is highly challenging given that isotope-rich sam-\nples (with a small neutron absorption cross-section) are\ncurrently unavailable. However, we detect a topological\nHall signal \u001aT\nxywithin the temperature and \feld range of\nthe A-phase that indicates a multi- qnon-collinear spin\ntexture as observed in skyrmion lattices31,32. Figure 3fshows the total Hall signal \u001axy=\u001aO\nxy+\u001aT\nxymeasured\nwith the \feld directed at di\u000berent angles ( \u0012) with respect\nto the unit cell as indicated in the inset. The underly-\ning nonlinear \feld dependence of the ordinary Hall sig-\nnal\u001aO\nxy(H) results from multiband conduction33, which\nis \ftted to a three-band model (dashed lines in Fig. 3f).\nDetails of the three-band model are discussed in the Sup-\nplemental Material25(Fig. S4). We highlight the topo-\nlogical Hall signal \u001aT\nxy=\u001axy\u0000\u001aO\nxywith colors in Fig. 3f.\nNote that\u001aT\nxyis observed at T= 5 K between 4 and\n7 T, i.e. within the A-phase, and it is maximized when\nthe \feld is nearly aligned with the body diagonal of the\nmagnetic unit cell (45\u000e\u0014\u0012\u001475\u000e). It is lost at tempera-\ntures lower or higher than the A-phase boundaries. The\nemergence of an A-phase is facilitated by a Weyl-induced\nthree-spin \\chiral\" interaction S1\u0001(S2\u0002S3) whose cou-\npling strength is proportional to the applied magnetic\n\feld9. Since the chiral interaction acts as a chemical\npotential for skyrmions34, a \fnite \feld and/or a \fnite\ntemperature (which destabilizes conventional magnetic\norder) are needed for the A-phase.\nElectronic Structure. The observation of a spiral mag-\nnetic ground-state with a possible skyrmion phase in\nSmAlSi indicates sizable DM and chiral interactions. Re-\ncent calculations have shown that Weyl electrons could\nmediate such interactions by coupling to the local f-\nmoments through an RKKY mechanism8{11. The band\nstructure of SmAlSi in Fig. 4a has the necessary ingre-\ndients of the Weyl-mediated RKKY mechanism, namely5\n(a)Energy  (eV)2\n-8-6-4-20\nΓ ∑∑1 Γ X Z NSmAlSi\nTotal\nSmf\n4 8\nDOS Momentum\n(b)                                    (c)                                  (d) 𝒆−𝒉++𝑣𝑒−𝑣𝑒\n𝑞1\n𝑞2𝑞1\n𝑞2\nFIG. 4. (a) Band structure and density of states in SmAlSi\nwhereEFis marked by a dashed line. The f-bands are high-\nlighted in orange color. (b) Top and side views of the hole\npocket in the \frst BZ. (c) Same views of the electron pocket.\n(d) A map of the Weyl nodes where the circles, full triangles,\nand empty triangles represent Weyl nodes at kz= 0,kz>0,\nandkz<0, respectively.\n(i) localized f-bands well below the Fermi level ( EF),\n(ii) a small density of states (DOS) near EFfrom the\nlinearly dispersing bands with Weyl crossings, and (iii)\nintra-band k-space vectors (nesting vectors) that match\nthe magnetic ordering vectors and link topologically non-\ntrivial Fermi pockets. To verify the last point, we visual-\nize the Fermi surface of SmAlSi from DFT+SOC+ Ucal-\nculations in the \feld-driven FM state. It comprises one\nhole and one electron pocket (Figs. 4b,c) and 20 pairs of\nWeyl nodes (Fig. 4d). Details of the DFT calculations\nare presented in the Supplemental Material25, where the\nWeyl nodes are categorized into 5 groups ( W1toW5)\nbased on their symmetries and energies (Figs. S5 and\nS6). The closest and farthest Weyl nodes with respect\ntoEFareW1andW3nodes atE= 0 and 89 meV,\nrespectively (Fig. S6).\nAs shown in Fig. 4b,c, two momentum space vectors\nq1=\u00001\n3\u0000\u000e;1\n3\u0000\u000e;0\u0001\nandq2=\u00002\n3+\u000e;2\n3+\u000e;0\u0001\ncon-\nnect topologically non-trivial Fermi pockets. Both nest-\ning vectors are consistent with the observed spiral order-\ning vector k=\u00001\n3\u0000\u000e;1\n3\u0000\u000e;0\u0001\nin Fig. 2. Speci\fcally,\nq1andq2connect the W1andW5nodes atE= 0 and\n23 meV, respectively. The nesting e\u000bect in SmAlSi is en-hanced by the elongation of both the hole and electron\npockets along the L-direction (Figs. 4b,c). We con\frmed\nthe accuracy of our DFT calculations by comparing the\ntheoretical dimensions of the Fermi pockets to the fre-\nquency of the quantum oscillations in Fig. S725,35.\nAlthough similar nesting conditions can be found in\nthe band structures of CeAlSi, PrAlSi, and NdAlSi\n(Fig. S5)15,27,36, the strong MCA blocks spiral order-\ning in those systems. PrAlSi has an out-of-plane FM\norder while CeAlSi has an in-plane non-collinear FM or-\nder, and neither one shows evidence of DM interactions27.\nNdAlSi has an incommensurate AFM order at interme-\ndiate temperatures whose ordering vectors match the q1\nandq2nesting vectors15. The magnetic Bragg peaks as-\nsociated with the q1andq2vectors were respectively\nre\fned to a weak in-plane helical spin component and\na strong out-of-plane Ising component with a net FM\nmoment. The Ising character of the NdAlSi incommen-\nsurate order caused by a strong MCA dominates the\nWeyl-mediated RKKY interaction, and upon cooling, a\ncommensurate ferrimagnetic ground-state is ultimately\nreached. For SmAlSi, however, our observations of an in-\ncommensurate magnetism persisting down to the lowest\ntemperatures, a lack of ferromagnetism in zero \feld, and\na quasi-isotropic magnetocrystalline interaction suggest\nthat a perfect AFM spiral order could be stabilized by\nWeyl-mediated RKKY interactions.\nIV. CONCLUSION\nWe found a spiral magnetic order in the Weyl\nsemimetal SmAlSi whose periodicity is linked to the dis-\ntance in momentum space between Weyl Fermi pockets.\nReminiscent of the A-phase observed in skyrmion lat-\ntices, we observed a topological Hall e\u000bect in SmAlSi\nthat appears only at \fnite \felds and temperatures close\nto the onset of the magnetic order. Our observations are\nthus fully consistent with Weyl-mediated RKKY interac-\ntions and in-\feld chiral three-spin interactions that pro-\nmote Skyrmion lattices. For completeness, we present\na Monte Carlo (MC) simulation of the Weyl-mediated\nRKKY interactions as Supplemental Material25based on\nthe structural parameters of SmAlSi and reasonable es-\ntimates of the coupling constants. Our MC simulations\n(Figs. S8, S9, S10) provide a proof-of-principle demon-\nstration of how Weyl-mediate RKKY interactions lead\nto a spiral order in the ground-state (consistent with\nthe observed wave vector) and a multi- qmagnetic or-\nder at \fnite temperatures (consistent with the observed\nA-phase). In the absence of isotope-rich samples, our\nneutron di\u000braction data does not resolve all details of the\nmagnetic structure, especially at temperature between T1\nandT2.6\nACKNOWLEDGMENTS\nThis material is based upon work supported by the Air\nForce O\u000ece of Scienti\fc Research under award number\nFA2386-21-1-4059. S.M.H. is supported by the MOST-\nAFOSR Taiwan program on Topological and Nanostruc-\ntured Materials, Grant No. 110-2124-M-110-002-MY3 A\nportion of this research used resources at the High Flux\nIsotope Reactor, a DOE O\u000ece of Science User Facility\noperated by Oak Ridge National Laboratory. Any men-\ntion of commercial products is intended solely for fully\ndetailing experiments; it does not imply recommendation\nor endorsement by NIST. A portion of this work was per-\nformed at the National High Magnetic Field Laboratory,\nwhich is supported by the National Science FoundationCooperative Agreement no. DMR-1644779 and the state\nof Florida. P.N. acknowledges support from the Insti-\ntute for Quantum Matter, an Energy Frontier Research\nCenter funded by the U.S. Department of Energy, Of-\n\fce of Science, Basic Energy Sciences under Award No.\nDE-SC0019331. The work at Northeastern University\nwas supported by the US Department of Energy (DOE),\nO\u000ece of Science, Basic Energy Sciences Grant No. DE-\nSC0022216 and bene\fted from Northeastern University's\nAdvanced Scienti\fc Computation Center and the Discov-\nery Cluster and the National Energy Research Scienti\fc\nComputing Center through DOE Grant No. DE-AC02-\n05CH11231. The work at TIFR Mumbai was supported\nby the Department of Atomic Energy of the Government\nof India under Project No. 12-R&D-TFR-5.10-0100.\n\u0003fazel.tafti@bc.edu\n1Y. Tokura, K. Yasuda, and A. Tsukazaki, Nature Reviews\nPhysics 1, 126 (2019).\n2Z.-C. Wang, J. D. Rogers, X. Yao, R. Nichols, K. Atay,\nB. Xu, J. Franklin, I. Sochnikov, P. J. Ryan, D. Haskel,\nand F. Tafti, Advanced Materials 33, 2005755 (2021).\n3A. Bestwick, E. Fox, X. Kou, L. Pan, K. L. Wang,\nand D. Goldhaber-Gordon, Physical Review Letters 114,\n187201 (2015).\n4C. Hu, L. Ding, K. N. Gordon, B. Ghosh, H.-J. Tien, H. Li,\nA. G. Linn, S.-W. Lien, C.-Y. Huang, S. Mackey, J. Liu,\nP. V. S. Reddy, B. Singh, A. Agarwal, A. Bansil, M. Song,\nD. Li, S.-Y. Xu, H. Lin, H. Cao, T.-R. Chang, D. Dessau,\nand N. Ni, Science Advances 6, eaba4275.\n5C. Liu, Y. Wang, H. Li, Y. Wu, Y. Li, J. Li, K. He, Y. Xu,\nJ. Zhang, and Y. Wang, Nature Materials 19, 522 (2020).\n6Y. Sun, C. Lee, H.-Y. Yang, D. H. Torchinsky, F. Tafti,\nand J. Orenstein, Physical Review B 104, 235119 (2021).\n7B. Xu, J. Franklin, A. Jayacody, H.-Y. Yang, F. Tafti, and\nI. Sochnikov, Advanced Quantum Technologies 4, 2170033\n(2021).\n8P. Nikoli\u0013 c, Physical Review B 104, 024414 (2021).\n9P. Nikoli\u0013 c, Physical Review B 103, 155151 (2021).\n10S.-X. Wang, H.-R. Chang, and J. Zhou, Physical Review\nB96, 115204 (2017).\n11H.-R. Chang, J. Zhou, S.-X. Wang, W.-Y. Shan, and\nD. Xiao, Physical Review B 92, 241103 (2015).\n12Z. Wang, Y. Su, S.-Z. Lin, and C. D. Batista, Physical\nReview B 103, 104408 (2021).\n13J. Repicky, P.-K. Wu, T. Liu, J. P. Corbett, T. Zhu,\nS. Cheng, A. S. Ahmed, N. Takeuchi, J. Guerrero-Sanchez,\nM. Randeria, R. K. Kawakami, and J. A. Gupta, Science\n374, 1484 (2021).\n14S. Luo and L. You, APL Materials 9, 050901 (2021).\n15J. Gaudet, H.-Y. Yang, S. Baidya, B. Lu, G. Xu, Y. Zhao,\nJ. A. Rodriguez-Rivera, C. M. Ho\u000bmann, D. E. Graf, D. H.\nTorchinsky, P. Nikoli\u0013 c, D. Vanderbilt, F. Tafti, and C. L.\nBroholm, Nature Materials 20, 1650 (2021).\n16A. Neubauer, C. P\reiderer, B. Binz, A. Rosch, R. Ritz,\nP. G. Niklowitz, and P. B oni, Physical Review Letters\n102, 186602 (2009).\n17M. Hirschberger, T. Nakajima, S. Gao, L. Peng,\nA. Kikkawa, T. Kurumaji, M. Kriener, Y. Yamasaki,H. Sagayama, H. Nakao, K. Ohishi, K. Kakurai,\nY. Taguchi, X. Yu, T.-h. Arima, and Y. Tokura, Nature\nCommunications 10, 5831 (2019).\n18L. Xu, H. Niu, Y. Bai, H. Zhu, S. Yuan, X. He,\nY. Yang, Z. Xia, L. Zhao, and Z. Tian, (2021),\n10.48550/arXiv.2107.11957.\n19J. Rodr\u0013 \u0010guez-Carvajal, Physica B: Condensed Matter 192,\n55 (1993).\n20A. Wills, Physica B: Condensed Matter 276-278 , 680\n(2000).\n21P. Blaha, K. Schwarz, F. Tran, R. Laskowski, G. K. H.\nMadsen, and L. D. Marks, The Journal of Chemical\nPhysics 152, 074101 (2020).\n22J. P. Perdew, K. Burke, and M. Ernzerhof, Physical Re-\nview Letters 77, 3865 (1996).\n23V. I. Anisimov, J. Zaanen, and O. K. Andersen, Physical\nReview B 44, 943 (1991).\n24M. Lyu, J. Xiang, Z. Mi, H. Zhao, Z. Wang, E. Liu,\nG. Chen, Z. Ren, G. Li, and P. Sun, Physical Review\nB102, 085143 (2020).\n25See the Supplemental Material for details.\n26G. Chang, B. Singh, S.-Y. Xu, G. Bian, S.-M. Huang, C.-H.\nHsu, I. Belopolski, N. Alidoust, D. S. Sanchez, H. Zheng,\nH. Lu, X. Zhang, Y. Bian, T.-R. Chang, H.-T. Jeng,\nA. Bansil, H. Hsu, S. Jia, T. Neupert, H. Lin, and M. Z.\nHasan, Physical Review B 97, 041104 (2018).\n27H.-Y. Yang, B. Singh, J. Gaudet, B. Lu, C.-Y. Huang,\nW.-C. Chiu, S.-M. Huang, B. Wang, F. Bahrami, B. Xu,\nJ. Franklin, I. Sochnikov, D. E. Graf, G. Xu, Y. Zhao,\nC. M. Ho\u000bman, H. Lin, D. H. Torchinsky, C. L. Broholm,\nA. Bansil, and F. Tafti, Physical Review B 103, 115143\n(2021).\n28E. Ru\u000b, S. Widmann, P. Lunkenheimer, V. Tsurkan,\nS. Bord\u0013 acs, I. K\u0013 ezsm\u0013 arki, and A. Loidl, Science Advances\n1, e1500916.\n29Y. Fujima, N. Abe, Y. Tokunaga, and T. Arima, Physical\nReview B 95, 180410 (2017).\n30S. Bord\u0013 acs, A. Butykai, B. G. Szigeti, J. S. White, R. Cu-\nbitt, A. O. Leonov, S. Widmann, D. Ehlers, H.-A. K. von\nNidda, V. Tsurkan, A. Loidl, and I. K\u0013 ezsm\u0013 arki, Scienti\fc\nReports 7, 7584 (2017).\n31T. Kurumaji, T. Nakajima, M. Hirschberger, A. Kikkawa,\nY. Yamasaki, H. Sagayama, H. Nakao, Y. Taguchi, T.-h.7\nArima, and Y. Tokura, Science 365, 914 (2019).\n32M. Leroux, M. J. Stolt, S. Jin, D. V. Pete, C. Reichhardt,\nand B. Maiorov, Scienti\fc Reports 8, 15510 (2018).\n33H. Lin, Y. Li, Q. Deng, J. Xing, J. Liu, X. Zhu, H. Yang,\nand H.-H. Wen, Physical Review B 93, 144505 (2016).\n34P. Nikoli\u0013 c, Physical Review B 102, 075131 (2020).35P. M. C. Rourke and S. R. Julian, Computer Physics Com-\nmunications 183, 324 (2012).\n36H.-Y. Yang, B. Singh, B. Lu, C.-Y. Huang, F. Bahrami,\nW.-C. Chiu, D. Graf, S.-M. Huang, B. Wang, H. Lin,\nD. Torchinsky, A. Bansil, and F. Tafti, APL Materials\n8, 011111 (2020)."    },    {        "title": "2207.09164v1.Micromagnetic_simulation_of_neutron_scattering_from_spherical_nanoparticles__Effect_of_pore_type_defects.pdf",        "content": "Micromagnetic simulation of neutron scattering from spherical\nnanoparticles: E\u000bect of pore-type defects\nEvelyn Pratami Sinaga,1,\u0003Michael P. Adams,1Mathias\nBersweiler,1Laura G. Vivas,1Eddwi H. Hasdeo,1Jonathan\nLeliaert,2Philipp Bender,3Dirk Honecker,4and Andreas Michels1,y\n1Department of Physics and Materials Science,\nUniversity of Luxembourg, 162A Avenue de la Faiencerie,\nL-1511 Luxembourg, Grand Duchy of Luxembourg\n2Department of Solid State Sciences, Ghent University,\nKrijgslaan 281/S1, 9000 Ghent, Belgium\n3Heinz Maier-Leibnitz Zentrum, Technische\nUniversit at M unchen, D-85748 Garching, Germany\n4ISIS Neutron and Muon Source, Rutherford Appleton Laboratory, Didcot, United Kingdom\n(Dated: July 20, 2022)\n1arXiv:2207.09164v1  [cond-mat.mes-hall]  19 Jul 2022Abstract\nWe employ micromagnetic simulations to model the e\u000bect of pore-type microstructural defects\non the magnetic small-angle neutron scattering cross section and the related pair-distance distri-\nbution function of spherical magnetic nanoparticles. Our expression for the magnetic energy takes\ninto account the isotropic exchange interaction, the magnetocrystalline anisotropy, the dipolar in-\nteraction, and an externally applied magnetic \feld. The signatures of the defects and the role\nof the dipolar energy are highlighted and the e\u000bect of a particle-size distribution is studied. The\nresults serve as a guideline to the experimentalist.\nI. INTRODUCTION\nMagnetic small-angle neutron scattering (SANS) is the method of choice for studying\nspin structures on a mesoscopic length scale of typically 1 \u0000100 nm and inside the volume\nof magnetic materials [1, 2]. A growing number of experimental investigations on magnetic\nnanoparticles , in particular using polarized neutrons, unanimously suggest that the encoun-\ntered spin con\fgurations are highly complex and exhibit a variety of nonuniform, canted, or\ncore-shell-type textures (see, e.g., Refs. [3{16] and references therein). However, a problem\narises since the prototypical magnetic SANS data analysis is largely based on structural\nform-factor-type models for the cross section. These are borrowed from nuclear SANS and\ndo not properly account for the existing spin inhomogeneity inside magnetic nanoparticles.\nOn the other hand, analytical as well numerical computations of the magnetic SANS cross\nsection [17{27] strongly suggest that for the analysis of experimental magnetic SANS data\nthe spatial nanometer scale variation of the orientation and magnitude of the magnetiza-\ntion vector \feld must be taken into account; and that macrospin-based models|assuming\nauniform magnetization|are not adequate.\nTheoretical descriptions of magnetic SANS are based on Brown's static equations of\nmicromagnetics [28], which are a set of nonlinear partial di\u000berential equations for the mag-\nnetization along with complex boundary conditions on the sample's surface [29]. There-\nfore, closed-form analytical results for the SANS cross section are restricted to special\n\u0003Electronic address: evelyn.sinaga@uni.lu\nyElectronic address: andreas.michels@uni.lu\n2limiting cases such as the approach-to-saturation regime, where the governing equations\ncan be linearized [17, 21{23, 26, 27]. Recently, we have carried out numerical micromag-\nnetic computations to study the magnetic SANS cross section of microstructural-defect-free\nspherical nanoparticles during their transition from the single-domain to the multi-domain\nstate [30, 31]. The results for the magnetic SANS signal and correlation function have re-\nvealed pronounced di\u000berences as compared to the superspin model and provided guidance for\nthe experimentalist to identify nonuniform vortex-type spin structures inside nanoparticles.\nIn this work, we extend the numerical micromagnetic approach to include the e\u000bects of\nmicrostructural pore-type defects and of a particle-size distribution function. Defects in\nnanoparticles (e.g., surface anisotropy, vacancies, antiphase boundaries) are known for a\nlong time to give rise to spin disorder and in this way in\ruence the macroscopic magnetic\nproperties (see, e.g., Refs. [15, 32{35]). Therefore, \fnding their signature in the magnetic\nSANS cross section and correlation function is highly desirable.\nThe article is organized as follows: In Sec. II, we display the expressions for the magnetic\nSANS cross section and for the pair-distance distribution function. In Sec. III, we provide\ninformation on the micromagnetic simulations and on the implementation of the microstruc-\ntural defects. In Sec. IV, we present and discuss the results, with Sec. IV A focusing on the\nreal-space spin structures, the magnetization, and the SANS observables and Sec. IV B dis-\ncussing the e\u000bect of a particle-size distribution function. Finally, Sec. V summarizes the\nmain \fndings of this study and provides an outlook on future challenges.\nII. MAGNETIC SANS CROSS SECTION AND PAIR-DISTANCE DISTRIBU-\nTION FUNCTION\nThe quantity of interest is the elastic magnetic di\u000berential scattering cross section\nd\u0006M=d\n, which is usually recorded on a two-dimensional position-sensitive detector. For\nthe most commonly used scattering geometry in magnetic SANS experiments, where the\napplied magnetic \feld H0kezis perpendicular to the wave vector k0kexof the incident\nneutrons (see Fig. 1), d\u0006M=d\n (for unpolarized neutrons) can be written as [1]:\nd\u0006M\nd\n=8\u00193\nVb2\nH\u0010\njfMxj2+jfMyj2cos2\u0012\n+jfMzj2sin2\u0012\u0000(fMyfM\u0003\nz+fM\u0003\nyfMz) sin\u0012cos\u0012\u0011\n; (1)\n3whereVis the scattering volume, bH= 2:91\u0002108A\u00001m\u00001is the magnetic scattering length\nin the small-angle regime (the atomic magnetic form factor is approximated by 1, since we\nare dealing with forward scattering), fM(q) =ffMx(q);fMy(q);fMz(q)grepresents the Fourier\ntransform of the magnetization vector \feld M(r) =fMx(r);My(r);Mz(r)g,\u0012denotes the\nangle between qandH0, and the asterisk \\ \u0003\" marks the complex-conjugated quantity. Note\nthat in the perpendicular scattering geometry the Fourier components are evaluated in the\nplaneqx= 0 (compare Fig. 1). For a uniformly magnetized spherical particle with its\nsaturation direction parallel to ez, i.e.,Mx=My= 0, Eq. (1) reduces to:\nd\u0006M\nd\n=Vp(\u0001\u001a)2\nmag9\u0012j1(qR)\nqR\u00132\nsin2\u0012; (2)\nwhereVp=4\u0019\n3R3is the sphere volume, (\u0001 \u001a)2\nmag=b2\nH(\u0001M)2is the magnetic scattering-\nlength density contrast, and j1(qR) is the \frst-order spherical Bessel function. The well-\nknown analytical result for the homogeneous sphere case, Eq. (2), and its correlation function\n[see Eq. (4) below] serve as a reference for comparison to the nonuniform case.\nThe pair-distance distribution function p(r) can be computed from the azimuthally-\naveraged magnetic SANS cross section according to:\np(r) =r21Z\n0d\u0006M\nd\n(q)j0(qr)q2dq; (3)\nwherej0(qr) = sin(qr)=(qr) is the spherical Bessel function of zero order; p(r) corresponds to\nthe distribution of real-space distances between volume elements inside the particle weighted\nby the excess scattering-length density distribution; see the reviews by Glatter [36] and by\nSvergun and Koch [37] for detailed discussions of the properties of p(r). Apart from con-\nstant prefactors, the p(r) of the azimuthally-averaged single-particle cross section [Eq. (2)],\ncorresponding to a uniform sphere magnetization, equals (for r\u00142R):\np(r) =r2\u0012\n1\u00003r\n4R+r3\n16R3\u0013\n: (4)\nWe also display the correlation function c(r), which is related to p(r) via\nc(r) =p(r)=r2: (5)\nAs we will demonstrate in the following, when the particles' spin structure is inhomogeneous,\nthed\u0006M=d\n and the corresponding p(r) andc(r) di\u000ber signi\fcantly from the homogeneous\n4FIG. 1. Sketch of the scattering geometry assumed in the micromagnetic simulations. The applied\nmagnetic \feld H0kezis perpendicular to the wave vector k0kexof the incident neutron beam\n(k0?H0). The momentum-transfer or scattering vector qis de\fned as the di\u000berence between k0\nandk1, i.e., q=k0\u0000k1. SANS is usually implemented as elastic scattering ( k0=k1= 2\u0019=\u0015),\nand the component of qalong the incident neutron beam, here qx, is much smaller than the\nother two components, so that q\u0018=f0;qy;qzg=qf0;sin\u0012;cos\u0012g. This demonstrates that SANS\nprobes predominantly correlations in the plane perpendicular to the incident beam. The angle\n\u0012=\\(q;H0) is used to describe the angular anisotropy of the recorded scattering pattern on the\ntwo-dimensional position-sensitive detector. For elastic scattering, the magnitude of qis given by\nq= (4\u0019=\u0015) sin( =2), where\u0015denotes the mean wavelength of the neutrons and  is the scattering\nangle.\ncase [Eqs. (2) and (4)]. Due to the r2factor, features in p(r) at medium and large distances\nare more pronounced than in c(r).\nThe magnetic SANS cross section d\u0006M=d\n [Eq. (1)] depends on both the magnitude q\nand the orientation \u0012of the scattering vector qon the two-dimensional detector. The origin\nof its angular anisotropy ( \u0012dependence) is twofold [20]: (i) the trigonometric functions in\nEq. (1) are due to the dipolar interaction between the magnetic moment of the neutron and\nthe magnetization of the sample, while (ii) the Fourier components fMx;y;zmay additionally\ndepend on the angle \u0012via the intrinsic dipolar interaction between the magnetic moments\ncomprising the sample. Variation of the external magnetic \feld strength changes the relative\ncontributions of the fMx;y;ztod\u0006M=d\n and also their angular anisotropy [compare, e.g.,\nEq. (2)].\n5III. DETAILS ON THE MICROMAGNETIC SIMULATIONS\nThe micromagnetic computations of the spin structure of a single spherical nanopar-\nticle were performed using the GPU-based open-source software package MuMax3 (ver-\nsion 3.10) [38, 39]. This code allows the calculation of the space and time-dependent magne-\ntization of nano- and micron-sized ferromagnets. MuMax3 is based on a \fnite-di\u000berence\ndiscretization scheme of space using a two-dimensional or three-dimensional grid of or-\nthorhombic cells. In the micromagnetic simulations we have taken into account all four\nstandard contributions to the total magnetic Gibbs free energy, i.e., Zeeman energy EZin\nthe external magnetic \feld H0, dipolar (magnetostatic) interaction energy ED, energy of the\n(cubic) magnetocrystalline anisotropy Eani, and isotropic and symmetric exchange energy\nEex. The expressions for these energies are the following:\nEZ=\u0000\u00160Z\nVM\u0001H0dV; (6)\nED=\u00001\n2\u00160Z\nVM\u0001HDdV; (7)\nEani=Z\nVK1\u0000\nm2\nxm2\ny+m2\nym2\nz+m2\nxm2\nz\u0001\ndV; (8)\nEex=Z\nA\u0002\n(rmx)2+ (rmy)2+ (rmz)2\u0003\ndV; (9)\nwhere m(r) =M(r)=Msdenotes the unit magnetization vector with Msthe saturation mag-\nnetization, H0kezis the (constant) applied magnetic \feld, HD(r) is the magnetostatic\nself-interaction \feld, K1is the \frst-order cubic anisotropy constant, Ais the exchange-\nsti\u000bness constant, \u00160= 4\u0019\u000210\u00007Tm=A, and the integrals are taken over the volume of the\nsample. In the simulations, we used the following material parameters for iron (Fe): satura-\ntion magnetization Ms= 1700 kA=m, exchange-sti\u000bness constant A= 1:0\u000210\u000011J=m, and\na \frst-order cubic anisotropy constant of K1= 47 kJ=m3. These values result in a critical\nsingle-domain diameter of Dsd\ncr\u001872pAK 1=(\u00160M2\ns) = 13:6 nm [40].\nFigure 2 displays the model used in the micromagnetic SANS simulations. The sphere vol-\nume is \frst discretized into cubical cells with a size of 2 \u00022\u00022 nm3(\fnite-di\u000berence method).\nNext, we randomly assign defects into this structure, which are per de\fnition cells with a\nsaturation magnetization of Ms= 0. This might model e.g. nonmagnetic pore-type defects.\n6FIG. 2. Discretization of the spherical simulation volume Vinto cubical cells with a size of typically\n2\u00022\u00022 nm3(particle size: D= 40 nm). The black squares mark the randomly-chosen \\defect\"\ncells withMs= 0.\nAssuming the size of a single atom to be 0 :1 nm, such a hole comprises about 8000 atoms\nand represents, thus, a very strong perturbation in the magnetic nanoparticle structure.\nAll simulations were carried out by \frst saturating the nanoparticle by a strong external\n\feldH0, and then the \feld was decreased (in steps of typically 5 mT) following the major\nhysteresis loop. For a given volume concentration of defects ( xd\u00180\u000020 %), we perform,\nat each value of H0, micromagnetic simulations for typically N\u0018100 random orientations\nbetween the magnetic easy axis of the particle and H0; for each random particle orientation,\nthe defect distribution was randomly selected. All data shown in this paper, except the spin\nstructures in Fig. 3, correspond to an ensemble of randomly-oriented particles. Simulations\non 10-nm-sized single-domain particles (data not shown) yield the well-known values for the\nreduced remanence and coercivity predicted by the Stoner-Wohlfarth model [41].\nFor each step of H0and for each particular random easy-axis angle, we have obtained the\nequilibrium spin structure Mx;y;z(x;y;z ) by employing both the \\Relax\" and \\Minimize\"\nfunctions of MuMax3. The former solves the Landau-Lifshitz-Gilbert equation without\nthe precessional term and the latter uses the conjugate-gradient method to \fnd the con-\n\fguration of minimum energy. The translational invariance of the grid obtained with the\n\fnite-di\u000berence method enables the usage of the fast Fourier transformation technique for\nthe computation of the Cartesian Fourier components fMx;y;z(qx;qy;qz) ofMx;y;z(x;y;z ). We\nhave used the FFTW library [42] to compute and analyze the Fourier components of our\n7nanoscopic magnetic con\fgurations. These were then evaluated in the plane qx= 0 (corre-\nsponding to the scattering geometry shown in Fig. 1) and used in Eq. (1) to compute the\nmagnetic SANS cross section d\u0006M=d\n according to:\nd\u0006M\nd\n=NX\ni=1d\u0006M;i\nd\n; (10)\nwhered\u0006M;i=d\n represents (for \fxed xdandH0) the magnetic SANS cross section of a\nspherical particle with diameter Dand with a particular random easy-axis orientation \\ i\".\nEquation (10) implies that interparticle-interference e\u000bects are ignored in the simulations.\nLikewise, the e\u000bect of temperature has also not been taken into account.\nIV. RESULTS AND DISCUSSION\nA. Spin structure, magnetization, and magnetic SANS observables\nFigure 3 depicts the spin structures of 40-nm-sized Fe spheres for several defect con-\ncentrations xdand at an applied magnetic \feld of \u00160H0= 0:02 T ( H0kez). While the\ndefect-free case [Fig. 3(a)] exhibits a vortex-type spin con\fguration (reproducing the results\nfrom [31]), increasing xdresults in the progressive disordering (randomization) of the struc-\nture [Figs. 3(b) and (c)]. For the here-considered particle size of D= 40 nm, which is larger\nthan the single-domain size of Fe ( Dsd\ncr\u0018=13:6 nm), the vortex structure in Fig. 3(a) is clearly\na consequence of the dipolar interaction. Leaving out the dipolar energy in the simulations\n(i.e., setting ED= 0) results in a quasi single-domain state [Fig. 3(d)]. For xd= 0 % and\nsmall applied \felds, the vortex structure appears for particle sizes D&20 nm. Overall, we\nsee that (for D= 40 nm) adding defects changes the vortex-type spin structure signi\fcantly\ntowards more disordered spin con\fgurations.\nIn Fig. 4, we show the reduced hysteresis curves mz(H0) =Mz(H0)=M\u0003\nsfor various xd\u0018\n0\u000020 %. We note that the magnetization is here normalized by the defect concentration,\naccording to M\u0003\ns= (1\u0000xd)Ms. As expected, increasing xdresults in a reduction of the\nmagnetization, in particular around the remanent state and in the approach-to-saturation\nregime [compare Figs. 4(b) and (c)]. To be more quantitative, the reduced remanence\ndecreases from\u00180:11 atxd= 0 % to\u00180:04 atxd= 20 %, and the coercivity decreases from\n35 mT atxd= 0 % to 18 mT at xd= 20 %. The saturation \feld also increases from \u00180:48 T\n8(d)\n(a) (c) (b)\n1\n-10FIG. 3. Spin structures (snap shots) of 40-nm-sized Fe spheres with defect concentrations of\n(a)xd= 0 %, (b)xd= 5 %, and (c) xd= 15 %. (d) Spin structure of a defect-free Fe sphere ( xd=\n0 %) without taking into account the dipolar interaction in the energy minimization procedure\n(ED= 0). Applied magnetic \feld is \u00160H0= 0:02 T in (a)\u0000(d); initially all structures were\nsaturated. In the actual SANS simulations, the magnetic easy axis of the particle was randomly\nselected relative to H0. Here, in order to illustrate the e\u000bect of the defects, the easy axis was\nchosen the same in (a) \u0000(d).\nforxd= 0 % to\u00180:53 T forxd= 20 %. The inset in Fig. 4(a) depicts the e\u000bect of the dipolar\ninteraction (for xd= 0 %). For ED= 0, the shape of the mz(H0) loop is rectangular and\nagrees with the predictions of the Stoner-Wohlfarth coherent-rotation model, i.e., we \fnd a\nreduced remanence of \u00180:83 and a coercivity of 0 :33\u00022K1=Ms\u0018=18 mT [41]. With dipolar\ninteraction, two regions with a large hysteresis are seen, one at around 0 :5 T [Fig. 4(c)],\nwhich corresponds to the \frst deviation from the single-domain state, and one around the\nremanent state [Fig. 4(a)], which corresponds to the nucleation of the vortex-type structure.\nIn the following discussion of the SANS observables [ d\u0006M=d\n,p(r),c(r)], we concentrate on\nthe remanent state and on the high-\feld \\pocket\" at 0 :5 T.\nThe two-dimensional magnetic SANS cross sections along with the Fourier components\nforxd= 0 % and xd= 15 % are shown in Fig. 5(a) at the remanent state and in Fig. 5(b)\nfor\u00160H0= 0:5 T. Thed\u0006M=d\n in the remanent state are both horizontally elongated\n[Fig. 5(a)], and the Fourier components exhibit only little variation with the defect concen-\ntration, e.g., regarding their angular anisotropy. The functions jfMxj2,jfMyj2, andjfMzj2are\n(at 0 T) of comparable magnitude. Near saturation at 0 :5 T [Fig. 5(b)], d\u0006M=d\n is dom-\ninated by the isotropic jfMzj2Fourier component and exhibits the sin2\u0012anisotropy which\nis characteristic for an essentially saturated microstructure [compare Eq. (1)]. The contri-\nbution of the transversal Fourier components jfMxj2andjfMyj2tod\u0006M=d\n is much weaker\n9- 0.3 0 0.3- 0.500.501\n-1\n-1 0 1\n0.4 0 .6 0 .80 .80 .91 .0\nmz mz𝜇0𝐻0 (T) mz\n𝜇0𝐻0(T) 𝜇0𝐻0 (T) 0.5\n- 0.5\n- 0.5 0.51\n0 101\nED= 0ED≠ 0  \n1\n0%\n2%\n5%\n15%\n20%\n(c) (b)(a)FIG. 4. (a) Normalized magnetization curves mz(H0) of 40-nm-sized Fe spheres for di\u000berent defect\nconcentrations xd(see inset). (b) Zoom of mz(H0) around (a) H0= 0 and (c) around \u00160H0= 0:6 T.\nThe inset in (a) compares the magnetization curves of a defect-free 40-nm sphere with and without\nthe dipolar interaction energy ED.\nthan the longitudinal jfMzj2contribution. When increasing the defect concentration xdfrom\n0 % to 15 %,jfMxj2at 0:5 T changes its angular anisotropy, from isotropic to horizontally\nelongated, while the clover-leaf-type pattern of jfMyj2becomes more pronounced. The CTs\nchange their sign at the borders between the quadrants on the detector, e.g., in Fig. 5(a)\nwe see that CT < 0 for 0\u000e< \u0012 < 90\u000e,CT > 0 for 90\u000e< \u0012 < 180\u000e, and so on. We note\nthat theCTneeds to be multiplied with sin \u0012cos\u0012in order to obtain the corresponding\ncontribution to the magnetic SANS cross section [compare Eq. (1)]. We also emphasize that\ntheCTsin\u0012cos\u0012contribution to d\u0006M=d\n can be negative, in contrast to the other three\n10CT\ndΣM/dΩCT\n106\n103\n106\n103\n100100qy (nm-1) qy (nm-1)\nqz (nm-1) qz (nm-1) qz (nm-1) qz (nm-1) qz (nm-1) 0.0\n -0.60.6\n 0.0\n -0.60.6\n-0.6  0.0  0.6 -0.6  0.0  0.6 -0.6  0.0  0.6 -0.6  0.0  0.6 -0.6  0.0  0.6\n105\n0\n-105\n106\n103\n100106\n103\n100105\n0\n-105\nCTCT\n106\n103106\n103\n100100qy (nm-1) qy (nm-1)\nqz (nm-1) qz (nm-1) qz (nm-1) qz (nm-1) qz (nm-1)\n100103106 0.0\n -0.60.6\n 0.0\n -0.60.6\n-0.6  0.0  0.6 -0.6  0.0  0.6 -0.6  0.0  0.6 -0.6  0.0  0.6 -0.6  0.0  0.6\n105\n0105\n105\n0\n-105\n106\n103\n100106\n103\n100105\n0\n-105\ndΣM/dΩ(a)\n𝜇0H0(b)\n= 0.5T𝜇0H0 = 0T\nFIG. 5. (a) Two-dimensional Fourier components jfMxj2,jfMyj2,jfMzj2,CT=\u0000(fMyfM\u0003\nz+fM\u0003\nyfMz),\nand magnetic SANS cross section d\u0006M=d\n (in units of cm\u00001) of 40-nm-sized Fe spheres at \u00160H0=\n0 T. (upper row) Defect concentration xd= 0 %. (lower row) Defect concentration xd= 15 %.\nH0kezis horizontal in the plane. Logarithmic color scales are used except for the CTs, which are\ndisplayed using a linear color scale. Note that the respective Fourier components are multiplied\nwith the constant 8 \u00193V\u00001b2\nH(in order to have the same unit as d\u0006M=d\n), but not with the\ntrigonometric functions in the expression for d\u0006M=d\n [compare Eq. (1)]. (b) Same as (a), but for\n\u00160H0= 0:5 T.\ncontributions, which are strictly positive. Using the inequality jfMycos\u0012\u0000fMzsin\u0012j2\u00150, it\nis easily seen that the contribution CTsin\u0012cos\u0012is, however, always smaller than the sum of\nthe other terms (as it must be). The results in Fig. 5 underline that, generally, the Fourier\ncomponents in the magnetic SANS cross section [Eq. (1)] are anisotropic functions of the\nangle\u0012.\n11The angular anisotropy of the magnetization Fourier components is caused by the dipolar\ninteraction [43], which is a long-range, nonlocal, and anisotropic magnetic energy term.\nFigure 6 compares, at remanence, results for d\u0006M=d\n with and without the dipolar energy\nED. It is seen that d\u0006M=d\n is highly anisotropic (elongated along the horizontal direction)\nwhenEDis included in the computations [Fig. 6(a)], while it becomes weakly anisotropic\n(slightly elongated along the vertical direction) when ED= 0 [Fig. 6(b)]. For ED= 0, the\nspin structure remains essentially uniform throughout the magnetization process (compare\ninset in Fig. 4(a) and related text) and one observes the analytical sphere form factor results\n(thin black lines) for the azimuthally-averaged d\u0006M=d\n [Fig. 6(c)] and the pair-distance\ndistribution function p(r) [Fig. 6(d)]. The results in Fig. 6 emphasize the importance of\nconsidering complex dipolar-\feld-induced nonuniform spin textures for the understanding\nof magnetic SANS patterns. We emphasize, however, that the dipolar energy might be of\nminor relevance for smaller-sized (nearly uniformly magnetized) nanomagnets and in the\npresence of the Dzyaloshinskii-Moriya interaction, which may give rise to \rux-closure-type\nmagnetization patterns [44, 45].\nThe results for the (2 \u0019) azimuthally-averaged d\u0006M=d\n and for the correlation functions\np(r) andc(r) are displayed in Fig. 7 and corroborate the behavior found from the analysis of\nthe two-dimensional SANS cross sections, namely a weak dependence on xd. The signature\nof the vortex-type real-space spin structure in Fig. 3(a) is an oscillatory p(r) [see, e.g.,\nFigs. 6(d) and 7(b)], which appears to be rather stable against spin perturbations that\nare induced by hole-type defects. Related to that observation is the absence of a Guinier\nbehavior at low momentum transfers qand for small \felds [see, e.g., Fig. 7(a)]. Only with\nincreasing \feld strength (more uniform spin structure) is a Guinier-type behavior recovered.\nThis becomes also visible in Fig. 8(a), where the \feld dependence of d\u0006M=d\n is shown for\nxd= 15 %. The emergence of internal spin disorder leads to a shift of the characteristic form-\nfactor oscillations to larger q(smaller structures) and to the smearing of these features, in\nthis way mimicking the e\u000bect of a particle-size distribution and/or instrumental resolution\n[compare also to Figs. 6(c) and 7(a)]. With increasing \feld, the p(r) andc(r) in Fig. 8\napproach the analytical expressions for uniformly magnetized spheres [Eqs. (4) and (5)].\nHowever, regarding the last statement, one should be cautious and keep in mind that the\nmicrostructure of the defect-rich nanoparticles resembles a porous structure (defect cells\nhave a volume of 2 \u00022\u00022 nm3, see Fig. 2). Such magnetic holes represent a severe defect\n12 0.0\n -0.60.6\nqz (nm-1)qy (nm-1)\nqz (nm-1)\ndΣM/dΩ\ndΣM/dΩ\ndΣM/dΩ(× 10-4 nm-2)(c)\ndΣM/dΩ(cm-1)(d)p(r)(a)\nq (nm-1) r(nm)-0.6  0.0  0.6 -0.6  0.0  0.6106\n103\n100\n101\n100 102104106\n0 20 402\n02\nED= 0ED≠ 0  \ndΣM/dΩ dΣM/dΩ\n(b)FIG. 6. E\u000bect of the dipolar interaction on the SANS observables for D= 40 nm,xd= 2 %, and\n\u00160H0= 0 T. (a)d\u0006M=d\n withED(logarithmic color scale), (b) d\u0006M=d\n forED= 0 (logarithmic\ncolor scale), (c) azimuthally-averaged d\u0006M=d\n (log-log scale), and (d) pair-distance distribution\nfunctionp(r). The thin black lines in (c) and (d) are the analytical results for uniformly magnetized\nspheres [Eqs. (2) and (4)].\nin the microstructure with a large jump in the saturation magnetization and associated\nstray-\feld torques producing spin disorder. Therefore, at large \felds when the defect-rich\nnanoparticles are approaching a uniform magnetization state, it is not surprising that their\nc(r) at small and intermediate rdeviate slightly from the purely uniform case [compare\nFigs. 7(c) and 8(c)]. In particular the feature at small ris better resolved in c(r) than in\np(r); compare to Fig. 7(b), where at 0 :5 T the overall p(r) shape is preserved and only the\nmaximum is reduced at increased xd.\nThe behavior of the correlation function c(r) in the limit of r!0 is generally an in-\nteresting question, which provides information on the nature of the scattering contrast (see\n13c(r)dΣM/dΩ(cm-1)\nq(nm1)\nr(nm) r(nm)(× 10-6 nm-4) (× 10-4 nm-2)(a) (b) (c)p(r)\n0 20 402\n02\n0 20 40024\n= 0.5T= 0T 𝜇0𝐻0\n𝜇0𝐻0\n102\n101\n100101103105\nFIG. 7. (a) Azimuthally-averaged magnetic SANS cross section d\u0006M=d\n (log-log scale), (b) pair-\ndistance distribution function p(r), and (c) correlation function c(r) for di\u000berent applied magnetic\n\felds [0 T and 0 :5 T, see inset in (c)] and defect concentrations. Dashed lines are for xd= 0 %,\ndotted lines are for xd= 15 %, and the thin black lines correspond to the analytically-known\ndefect-free uniform case [azimuthally-averaged version of Eq. (2) for d\u0006M=d\n and Eqs. (4) and (5)\nfor, respectively, p(r) andc(r)].\n0 20 402\n02\n0 20 40024\n(× 10-4 nm-2) (× 10-6 nm-4)\n(b)p(r)(c)c(r)(a)\ndΣM/dΩ(cm-1)\nq(nm1) r(nm) r(nm)0T\n0.2T\n0.3T\n0.5T\n0.7T𝜇0𝐻0= \n𝜇0𝐻0= \n𝜇0𝐻0= \n𝜇0𝐻0= \n𝜇0𝐻0= \n102\n101\n100101103105\nFIG. 8. Field dependence [see inset in (c)] of (a) d\u0006M=d\n (log-log scale), (b) p(r), and (c)c(r) for\na defect concentration of xd= 15 %.\nthe discussion by Ciccariello, Goodisman, and Brumberger [46]). When sharp interfaces in\na sample separate homogeneous regions with a uniform (constant) scattering length density,\nthen the correlation function exhibits a \fnite slope at the origin (compare Eqs. (4) and (5)\nfor uniformly magnetized nanoparticles). This is the content of the well-known Porod law,\nwhich predicts a characteristic asymptotic q\u00004dependency of the SANS cross section [47].\nBy contrast, structures with a nonuniform scattering length density pro\fle, such as the\n14smoothly varying magnetization pro\fles M(r) of micromagnetics, are characterized by a\nc(r) that exhibits a zero slope at r= 0 and concomitant steeper power-law exponents of\nthe SANS cross section [2]. In the present case, we are dealing with a system with a poten-\ntially nonuniform magnetization on one side of the interface (inside the particle) and a zero\nmagnetization on the other side of the interface. The behavior of c(r) at small distances\nalso depends on the spin distribution in the vicinity of the surface and, therefore, on the\nsurface anisotropy and the related boundary conditions for the magnetization. This question\ndeserves a separate consideration and is beyond the scope of the present paper.\nB. E\u000bect of a particle-size distribution function\nIn SANS experiments on nanoparticles one always has to deal with a distribution of\nparticle sizes and shapes. The size of a particle has an important e\u000bect on its spin structure,\ne.g., smaller particles generally tend to be uniformly magnetized, whereas larger particles\nmay exhibit inhomogeneous spin structures [14]. It is therefore also of interest to study\nthe in\ruence of a distribution of particle sizes on the magnetic SANS observables [ d\u0006M=d\n,\np(r),c(r)]. This has been done using a lognormal distribution function, which is de\fned\nas [48]:\nf(D) =1p\n2\u0019Dlog\u001be\u00001\n2\u0012logD\u0000logD0\nlog\u001b\u00132\n; (11)\nwhereD0denotes the median and \u001bthe variance of the distribution withR1\n0f(D)dD= 1.\nFor the above function, the mean particle size D[\frst moment of f(D)] is related to the\nparameters of the distribution as D=D0e(log\u001b)2\n2. For given xdandB0, randomly-averaged\nmagnetic SANS cross sections were computed for particle diameters Dranging between\n10 nm\u0014D\u0014100 nm in binning intervals of \u0001 D= 2 nm. The magnetic SANS cross section\naveraged over the distribution, hd\u0006M=d\nif, is then computed as:\n\u001cd\u0006M\nd\n\u001d\nf=X\nkwkd\u0006M;k\nd\n; (12)\nwhere\nwk=Dk+\u0001D=2Z\nDk\u0000\u0001D=2f(D)dD (13)\n15denotes the weight of the size class Dk, which can be computed for given values of D0\nand\u001b;d\u0006M;k=d\n is the orientationally-averaged SANS cross section [compare Eq. (10)]\ncorresponding to Dk. Particle diameters outside of the above interval, i.e., smaller than\n10 nm and larger than 100 nm were not considered in our analysis ( wk= 0 forD < 10 nm\nandD> 100 nm).\nFigure 9 depicts the evolution of the azimuthally-averaged magnetic SANS cross section\nd\u0006M=d\n and of both correlation functions p(r) andc(r) with the width \u001bof the lognormal\ndistribution at zero \feld and at 1 :0 T (forD0= 40 nm and xd= 0 %). Note that the\np(r) andc(r) are (for each D0and\u001b) normalized to unity after the cross section has been\ncomputed according to Eq. (12). The d\u0006M=d\n [Figure 9(a)] exhibit the \\usual\" behavior\nknown e.g. from the study of instrumental broadening, namely a smearing of the form-factor\noscillations with increasing \u001b. The remaining oscillations of d\u0006M=d\n for small \u001bare more\npronounced in the high-\feld regime [Fig. 9(d)] than at remanence [Fig. 9(a)]. It is generally\nseen in Fig. 9(b) and (e) that the p(r) are more a\u000bected by the variation of \u001bthan thec(r)\n[Fig. 9(c) and (f)], which is related to the r2factor. For all values of \u001bdoes the oscillatory\np(r) behavior remain at zero \feld, and one observes a shift of the maximum of d\u0006M=d\n to\nlower momentum transfers with increasing \u001b[Fig. 9(a)]. The global minimum of p(r) at zero\n\feld shifts to larger distances with increasing \u001b, fromr\u0018=29 nm for\u001b= 1:1 tor\u0018=50 nm for\n\u001b= 1:6. Forxd6= 0, the behavior of the d\u0006M=d\n,p(r), andc(r) are qualitatively similar,\ndemonstrating the rather robust character of the oscillatory low-\feld feature in p(r).\nV. CONCLUSION AND OUTLOOK\nUsing micromagnetic computations we have investigated the e\u000bect of pore-type mi-\ncrostructural defects in spherical magnetic nanoparticles on their magnetic small-angle neu-\ntron scattering cross section d\u0006M=d\n and pair-distance distribution function p(r). The\nsimulations take into account the isotropic exchange interaction, the magnetocrystalline\nanisotropy, the dipolar interaction, and an externally applied magnetic \feld. Clearly, the\nd\u0006M=d\n andp(r) of nonuniformly magnetized nanoparticles cannot be described anymore\nwith the superspin model, which assumes a homogeneous spin microstructure. The dipolar\ninteraction is at the origin of many complex magnetization structures and related anisotropic\nscattering patterns. For small applied \felds and a not too small particle size (here, for\n16102\n101\n100101103105\n0 40 8001\n0 40 8001\np(r)\nc(r)102\n101\n100101103105\n0 40 8001\n0 40 8001\n(a)𝜇0𝐻0=0 T \np(r)\nc(r)1.1\n1.2\n1.3\n1.4\n1.5\n1.6=\n=\n=\n=\n=\n=\ndΣM/dΩ(cm-1)\n(d)\ndΣM/dΩ(cm-1)𝜇0𝐻0=1.0 T r(nm) r(nm)\nr(nm) r(nm)q(nm1)\nq(nm1)\n040 80\nD(nm)0.000.010.02f (D)\n(b) (c)\n(e) (f)FIG. 9. E\u000bect of a lognormal particle-size distribution on the azimuthally-averaged magnetic\nSANS cross section d\u0006M=d\n and on the correlation functions p(r) andc(r). Upper row [(a) \u0000(c)]\nis for\u00160H0= 0 T and the lower row [(d) \u0000(f)] is for\u00160H0= 1:0 T. Parameters are: D0= 40 nm,\nxd= 0 %, and \u001bvaries between 1 :1\u00001:6 [see inset in (c)]. For each \u001b, thep(r) andc(r) were\nnormalized to their respective maximum value at 0 T and 1 :0 T. The inset in (f) depicts the used\nsize-distribution function f(D) forD0= 40 nm and \u001b= 1:6. For each size class, 40 random particle\norientations were used to compute the averaged magnetic SANS cross section.\nD&20 nm), the dipolar energy results in a vortex-type spin structure and in a concomitant\noscillatory feature in the p(r) function. This characteristic signature appears to be rather\nstable against the here-used pore-type defects. The oscillatory p(r) shape also remains\nin the presence of a particle-size distribution function. At low \felds, deviations from the\nGuinier law and complicated real-space correlations are encountered. Within the present\nmodeling approach of defects|representing them by computational cells with zero satura-\ntion magnetization Ms|their e\u000bect on the SANS observables seems to be relatively small.\nThe dominating defect in spherical nanoparticles appears to be the outer surface of the\n17particles. A more realistic treatment of defects could be achieved by a more modest change\nin the material parameters, i.e., by reducing the saturation magnetization to a lower, but\nnonzero value and/or reducing the exchange interaction between the defect and the other\ncells [49]. Moreover, one could consider the inclusion of the magnetoelastic interaction into\nthe micromagnetic energy functional. Currently, this interaction is not implemented in most\nmicromagnetic codes, although recent research activities go into this direction [50]. Likewise,\nphenomenological models for surface anisotropy such as N\u0013 eel anisotropy, which give rise to\nadditional boundary conditions on the surface of the nanomagnet, are also not included in\nnumerical simulations of magnetic SANS. The micromagnetic approach to magnetic SANS\nconsist of \fnding, by means of magnetic-energy minimization, the three-dimensional vector\n\feld of the magnetization M(r). This represents a paradigm shift and is conceptually very\nmuch di\u000berent than the up-to-now used approach of \fnding a scalar function which describes\nthe structural saturation-magnetization pro\fle Ms(r) of the particle ensemble.\nACKNOWLEDGEMENTS\nEvelyn Pratami Sinaga, Michael P. Adams, and Andreas Michels acknowledge \fnancial\nsupport from the National Research Fund of Luxembourg (PRIDE MASSENA Grant and\nAFR Grant No. 15639149). Jonathan Leliaert is supported by the Fonds Wetenschappelijk\nOnderzoek (FWO-Vlaanderen) with senior postdoctoral research fellowship Bo. 12W7622N.\nThe simulations presented in this paper were carried out using the HPC facilities of the\nUniversity of Luxembourg ( https://hpc.uni.lu ).\n[1] S. M uhlbauer, D. Honecker, E. A. P\u0013 erigo, F. Bergner, S. Disch, A. Heinemann, S. Erokhin,\nD. Berkov, C. Leighton, M. R. Eskildsen, and A. Michels, Rev. Mod. Phys. 91, 015004 (2019).\n[2] A. Michels, Magnetic Small-Angle Neutron Scattering: A Probe for Mesoscale Magnetism\nAnalysis (Oxford University Press, Oxford, 2021).\n[3] A. Michels, F. D obrich, M. Elmas, A. Ferdinand, J. Markmann, M. Sharp, H. Eckerlebe,\nJ. Kohlbrecher, and R. Birringer, EPL (Europhysics Letters) 81, 66003 (2008).\n[4] S. Disch, E. Wetterskog, R. P. Hermann, A. Wiedenmann, U. Vainio, G. Salazar-Alvarez,\nL. Bergstr om, and T. Br uckel, New J. Phys. 14, 013025 (2012).\n18[5] K. L. Krycka, J. A. Borchers, R. A. Booth, Y. Ijiri, K. Hasz, J. J. Rhyne, and S. A. Majetich,\nPhys. Rev. Lett. 113, 147203 (2014).\n[6] K. Hasz, Y. Ijiri, K. L. Krycka, J. A. Borchers, R. A. Booth, S. Oberdick, and S. A. Majetich,\nPhys. Rev. B 90, 180405(R) (2014).\n[7] A. G unther, D. Honecker, J.-P. Bick, P. Szary, C. D. Dewhurst, U. Keiderling, A. V. Feoktys-\ntov, A. Tsch ope, R. Birringer, and A. Michels, J. Appl. Cryst. 47, 992 (2014).\n[8] T. Maurer, S. Gautrot, F. Ott, G. Chaboussant, F. Zighem, L. Cagnon, and O. Fruchart,\nPhys. Rev. B 89, 184423 (2014).\n[9] C. L. Dennis, K. L. Krycka, J. A. Borchers, R. D. Desautels, J. van Lierop, N. F. Huls, A. J.\nJackson, C. Gruettner, and R. Ivkov, Adv. Funct. Mater. 25, 4300 (2015).\n[10] A. J. Grutter, K. L. Krycka, E. V. Tartakovskaya, J. A. Borchers, K. S. M. Reddy, E. Ortega,\nA. Ponce, and B. J. H. Stadler, ACS Nano 11, 8311 (2017).\n[11] S. D. Oberdick, A. Abdelgawad, C. Moya, S. Mesbahi-Vasey, D. Kepaptsoglou, V. K. Lazarov,\nR. F. L. Evans, D. Meilak, E. Skoropata, J. van Lierop, I. Hunt-Isaak, H. Pan, Y. Ijiri, K. L.\nKrycka, J. A. Borchers, and S. A. Majetich, Sci. Rep. 8, 3425 (2018).\n[12] Y. Ijiri, K. L. Krycka, I. Hunt-Isaak, H. Pan, J. Hsieh, J. A. Borchers, J. J. Rhyne, S. D.\nOberdick, A. Abdelgawad, and S. A. Majetich, Phys. Rev. B 99, 094421 (2019).\n[13] P. Bender, D. Honecker, and L. F. Barqu\u0013 \u0010n, Appl. Phys. Lett. 115, 132406 (2019).\n[14] M. Bersweiler, P. Bender, L. G. Vivas, M. Albino, M. Petrecca, S. M uhlbauer, S. Erokhin,\nD. Berkov, C. Sangregorio, and A. Michels, Phys. Rev. B 100, 144434 (2019).\n[15] D. Z\u0013 akutn\u0013 a, D. Ni\u0014 z\u0014 nansk\u0013 y, L. C. Barnsley, E. Babcock, Z. Salhi, A. Feoktystov, D. Honecker,\nand S. Disch, Phys. Rev. X 10, 031019 (2020).\n[16] D. Honecker, M. Bersweiler, S. Erokhin, D. Berkov, K. Chesnel, D. A. Venero, A. Qdemat,\nS. Disch, J. K. Jochum, A. Michels, and P. Bender, Nanoscale Adv. 4, 1026 (2022).\n[17] D. Honecker and A. Michels, Phys. Rev. B 87, 224426 (2013).\n[18] A. Michels, S. Erokhin, D. Berkov, and N. Gorn, J. Magn. Magn. Mater. 350, 55 (2014).\n[19] D. Mettus and A. Michels, J. Appl. Cryst. 48, 1437 (2015).\n[20] S. Erokhin, D. Berkov, and A. Michels, Phys. Rev. B 92, 014427 (2015).\n[21] K. L. Metlov and A. Michels, Phys. Rev. B 91, 054404 (2015).\n[22] K. L. Metlov and A. Michels, Sci. Rep. 6, 25055 (2016).\n[23] A. Michels, D. Mettus, D. Honecker, and K. L. Metlov, Phys. Rev. B 94, 054424 (2016).\n19[24] A. Michels, D. Mettus, I. Titov, A. Malyeyev, M. Bersweiler, P. Bender, I. Peral, R. Birringer,\nY. Quan, P. Hautle, J. Kohlbrecher, D. Honecker, J. R. Fern\u0013 andez, L. F. Barqu\u0013 \u0010n, and K. L.\nMetlov, Phys. Rev. B 99, 014416 (2019).\n[25] A. A. Mistonov, I. S. Dubitskiy, I. S. Shishkin, N. A. Grigoryeva, A. Heinemann, N. A.\nSapoletova, G. A. Valkovskiy, and S. V. Grigoriev, J. Magn. Magn. Mater. 477, 99 (2019).\n[26] A. Malyeyev, I. Titov, C. Dewhurst, K. Suzuki, D. Honecker, and A. Michels, J. Appl. Cryst.\n55, 569 (2022).\n[27] V. D. Zaporozhets, Y. Oba, A. Michels, and K. L. Metlov, J. Appl. Cryst. 55, 592 (2022).\n[28] W. F. Brown Jr., Micromagnetics (Interscience Publishers, New York, 1963).\n[29] Note that the (linearized) analytical theories of magnetic SANS are based on Brown's static\nequations of micromagnetics, whereas numerical micromagnetic SANS simulations are ususally\nbased on the time-dependent Landau-Lifshitz-Gilbert equation.\n[30] L. G. Vivas, R. Yanes, and A. Michels, Sci. Rep. 7, 13060 (2017).\n[31] L. G. Vivas, R. Yanes, D. Berkov, S. Erokhin, M. Bersweiler, D. Honecker, P. Bender, and\nA. Michels, Phys. Rev. Lett. 125, 117201 (2020).\n[32] D. A. Garanin and H. Kachkachi, Phys. Rev. Lett. 90, 065504 (2003).\n[33] Z. Nedelkoski, D. Kepaptsoglou, L. Lari, T. Wen, R. A. Booth, S. D. Oberdick, P. L. Galindo,\nQ. M. Ramasse, R. F. L. Evans, S. Majetich, and V. K. Lazarov, Sci. Rep. 7, 45997 (2017).\n[34] A. Lappas, G. Antonaropoulos, K. Brintakis, M. Vasilakaki, K. N. Trohidou, V. Iannotti,\nG. Ausanio, A. Kostopoulou, M. Abeykoon, I. K. Robinson, and E. S. Bozin, Phys. Rev. X 9,\n041044 (2019).\n[35] A. Lak, S. Disch, and P. Bender, Adv. Sci. 8, 2002682 (2021).\n[36] O. Glatter, in Small Angle X-ray Scattering , edited by O. Glatter and O. Kratky (Academic\nPress, London, 1982) pp. 167{196.\n[37] D. I. Svergun and M. H. J. Koch, Rep. Prog. Phys. 66, 1735 (2003).\n[38] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, and B. Van Waeyen-\nberge, AIP Advances 4, 107133 (2014).\n[39] J. Leliaert and J. Mulkers, J. Appl. Phys. 125, 180901 (2019).\n[40] H. Kronm uller and M. F ahnle, Micromagnetism and the Microstructure of Ferromagnetic\nSolids (Cambridge University Press, Cambridge, 2003).\n[41] N. A. Usov and S. E. Peschany, J. Magn. Magn. Mater. 174, 247 (1997).\n20[42] M. Frigo and S. G. Johnson, Proceedings of the IEEE 93, 216 (2005), special issue on \\Program\nGeneration, Optimization, and Platform Adaptation\".\n[43] S. Erokhin, D. Berkov, N. Gorn, and A. Michels, Phys. Rev. B 85, 024410 (2012).\n[44] S. A. Pathak and R. Hertel, Phys. Rev. B 103, 104414 (2021).\n[45] T. K ohler, A. Feoktystov, O. Petracic, N. Nandakumaran, A. Cervellino, and T. Br uckel, J.\nAppl. Cryst. 54, 1719 (2021).\n[46] S. Ciccariello, J. Goodisman, and H. Brumberger, J. Appl. Cryst. 21, 117 (1988).\n[47] G. Porod, in Small Angle X-ray Scattering , edited by O. Glatter and O. Kratky (Academic\nPress, London, 1982) pp. 17{51.\n[48] C. E. Krill and R. Birringer, Philos. Mag. A 77, 621 (1998).\n[49] J. Leliaert, B. Van de Wiele, A. Vansteenkiste, L. Laurson, G. Durin, L. Dupr\u0013 e, and\nB. Van Waeyenberge, J. Appl. Phys. 115, 17D102 (2014).\n[50] F. Vanderveken, J. Mulkers, J. Leliaert, B. Van Waeyenberge, B. Sor\u0013 ee, O. Zografos, F. Ciub-\notaru, and C. Adelmann, Open Research Europe 1, 10.12688/openreseurope.13302.1 (2021).\n21"    },    {        "title": "2207.11472v1.Coexistence_of_Weyl_semimetal_and_Weyl_nodal_loop_semimetal_phases_in_a_collinear_antiferromagnet.pdf",        "content": "Coexistence of Weyl semimetal and Weyl nodal loop semimetal phases in a collinear\nantiferromagnet\nJie Zhan,1, 2Jiangxu Li,2Wujun Shi,3, 4Xing-Qiu Chen,1, 2,\u0003and Yan Sun1, 2,y\n1School of Materials Science and Engineering, University of Science and Technology of China, Shenyang, China.\n2Shenyang National Laboratory for Materials Science,\nInstitute of Metal Research,Chinese Academy of Sciences, Shenyang, China.\n3Center for Transformative Science, ShanghaiTech University, Shanghai 201210, China\n4Shanghai High Repetition Rate XFEL and Extreme Light Facility (SHINE),\nShanghaiTech University, Shanghai 201210, China\nAntiferromagnets (AFMs) with anomalous quantum responses have lead to new progress for the\nunderstanding of their magnetic and electronic structures from symmetry and topology points of\nview. Two typical topological states are the collinear antiferromagnetic Weyl semimetal (WSM) and\nWeyl nodal loop semimetal (WNLSM). In comparison with the counterparts in ferromagnets and\nnon-collinear AFMs, the WSMs and WNLSMs in collinear AFMs are still waiting for experimental\nveri\fcation. In this work, we theoretically predicted the coexistence of Weyl points (WPs) and\nWeyl nodal loops (WNLs) in transition metal oxide RuO 2. Owing to the small magnetocrystalline\nanisotropy energy, the WPs and WNLs can transform to each other via tuning the N\u0013 eel vector.\nMoreover, since the WPs are very close to Fermi level and the WNLs are even crossing Fermi level,\nthe topological states in RuO 2can be easily probed by photoemission and STM methods. Our\nresult provides a promising material platform for the study of WSM and WNLSM states in collinear\nAFMs.\nI. BACKGROUND AND INTRODUCTION\nRecently, the understanding of magnetic structures in\nantiferromagnets (AFMs) was refreshed according to the\nsymmetry constraint quantum transport properties and\ntopological band structures [1{18]. Generally, a mate-\nrial with magnetic order and zero net magnetic moment\nis considered as an AFM, in which the zero net moment\nmakes it not sensitive to external magnetic \feld. With\nthe progress of understanding in quantum responses\n(such as anomalous Hall e\u000bect (AHE) and spin-orbital\ntorque (SOT), et al.), some antiferromagnetic structures\nget intensive interest since they can host some quantum\nresponses that were believed to only exist in ferromag-\nnetic and ferrimagnetic counterparts with non-zero net\nmagnetizations [1{4].\nA typical example is the non-collinear AFM. Though\nthe theoretical prediction of AHE was as early as\n2001 [1], its experimental realization was only achieved\nafter around 15 years later [6]. After that, the non-\ncollinear AFM gets extensive attention due to their ex-\notic transport properties and topological band structures.\nThe observation of AHE [6, 7], anomalous Nernst e\u000bect\n(ANE) [19, 20] and magneto-optical responses [21, 22] in-\nspired the further understanding of collinear AFM. Ac-\ncordingly, the collinear AFM can be further classi\fed into\ntwo groups that with and without AHE [16].\nThough the existence of intrinsic AHE needs to break\ntime-reversal symmetry, most collinear AFMs can be\nviewed as two sublattices that are connected by the joint\n\u0003xingqiu.chen@imr.ac.cn\nysunyan@imr.ac.cn\nFIG. 1. Schematic of possible topological band structures\nin collinear antiferromagnets (AFMs). In the case of AFM1\nwith either PTorfTj\u001cgsymmetry, the linear touching points\nof electronic band structure are at least four-fold degenerate\nin the form of Dirac points. In the cases of AFM2 in the\nabsence of PTorfTj\u001cgsymmetries, both doubly degener-\nate Weyl points (WPs) and Weyl nodal loops (WNLs) are\nallowed. Their existence can be controlled via the magnetiza-\ntion orientation.\nspace group and time-reversal operations. Such symme-\ntries can lead to the cancellation of Berry curvature in\nthe whole Brillouin zone (BZ) and therefore the AHE is\nforbidden. Another consequence of the joint symmetry is\nthe spin degeneracy in band structures, with which the\nfour-fold Dirac points are allowed [23, 24] but that with\nlower-fold-degeneracy are forbidden by symmetry, see the\nup panel of Fig. 1arXiv:2207.11472v1  [cond-mat.mtrl-sci]  23 Jul 20222\nOn the other hand, a \fnite non-zero anomalous Hall\nconductivity (AHC) is symmetrically allowed if such\nkind of joint symmetry is absent and it is usually\nalong with band spin split. It allows topological band\nstructures with odd-number-fold degeneracy in collinear\nAFMs, where two typical doubly degenerate linear cross-\ning states are Weyl point (WP) and Weyl nodal loop\n(WNL) with non-zero Chern numbers [25{30]. Together\nwith spin-orbital coupling (SOC) and speci\fc magnetiza-\ntion orientation, the topological states can be transferred\nbetween WPs and WNLs, see the bottom panel in Fig. 1.\nWith the development of magnetic topological ma-\nterials, magnetic WSMs were experimentally veri\fed\nin ferromagnet Co 3Sn2S2[31{35], non-collinear AFM\nMn 3Sn/Ge [9, 36], as well as canted AFM YbMnBi 2[37].\nBesides, magnetic WNLSM was also directly observed in\nferromagnet Co 2MnGa [38] by ARPES measurements.\nIn comparison, despite both WPs and WNLs are al-\nlowed in collinear AFMs, none of them were experimen-\ntally observed, so far. The main reason maybe due to\nthe complicated metallic electronic structure with sev-\neral bands crossing Fermi level and WPs/WNLs far away\nfrom Fermi level, making them hard to detect in experi-\nmental measurements [12]. In this work, we \fnd the co-\nexistence of WPs and mirror symmetry protected WNLs\nin transition metal oxide RuO 2. With WNLs crossing\nFermi level and WPs only around 0.03 eV bellow Fermi\nlevel, it provides a model material platform for the direct\nphotoemission and STM measurements.\nII. METHOD\nTo analyze the topological properties, we calculated\nthe electronic band structure of RuO 2based on den-\nsity functional theory by full-potential local-orbital code\n(FPLO) with local basis [39]. Similar to previous calcu-\nlations, the correlation e\u000bect of Ru-4d orbitals are con-\nsidered by the e\u000bective on-site Hubbard U, with U=2\neV [16, 18, 40]. To calculate the Berry curvature related\ntopological invariants and surface states, we mapped\nthe Bloch wavefunctions into symmetry conserving max-\nimally projected Wannier functions [41] and constructed\ne\u000bective tight-binding model Hamiltonians by the Wan-\nnier functions overlap.\nIII. RESULTS AND DISCUSSION\nRuO 2is one of the most promising AFM AHE mate-\nrials [16, 42], in which the magneto-optical response can\nbe also manipulated by the orientation of magnetization\nand crystal chirality [18]. It is also the main motivation\nfor us to choose RuO 2as a typical example to study the\nWSM and WNL phases in collinear AFMs. RuO 2has\na rutile-type lattice structure belonging to space group\nP42=mnm (No.136), see Fig. 2(a). The symmetry is re-\nduced by the magnetic order, depending on the speci\fcspin orientation.\nThe magnetization is originated from the 4 dorbitals on\nRu sites, with easy axis along the [001] direction [43, 44].\nFig. 2(c-d) shows the energy dispersion of RuO 2with-\nout and with the consideration of SOC. In the absence of\njointPTsymmetry andfTj\u001cgsymmetry (with inversion\noperationP, time-reversal operation T, and fractional\ntranslation operation \u001c) that connect two magnetic sub-\nlattices, spin-up and spin-down channels are not degen-\nerate in the band structure, consistent with previously\nreports [16, 18]. Since the two Ru sites are connected by\na screw rotation operation S4z=fC4zj(1=2;1=2;1=2)g,\nthe bands from two spin channels are connected by a S4z\noperation.\nFIG. 2. Crystal and electronic structure of RuO 2. (a) Mag-\nnetic structure of RuO 2. (b) Three-dimensional Brillouin zone\n(BZ) for the tetragonal crystal lattice and its two-dimensional\nprojection in (001) surface. (c,d) Energy dispersion of RuO 2\nwithout and with the consideration of spin-orbital coupling\n(SOC).\nIn the case without considering SOC, the system fol-\nlows the spin rotation symmetry, and all the mirror sym-\nmetries and glide mirror symmetries are preserved. With\nthe protection of mirror plane m001, the band inversion\nin both spin-up and spin-down channels can form dou-\nbly degenerate nodal loops in the mirror invariant plane,\nkz= 0. As presented in Fig. 3(a), the nodal rings in\nkz= 0 plane can extend around half of the BZ in the\nkspace. Moreover, the nodal loops cut Fermi level with\nstrong dispersions, where the energy windows are in the\nrange of\u0018-0.3 eV to\u00180.2 eV, see the color bar labeled\nnodal loops in Fig. 3(a) and three dimensional energy\ndispersions in Fig. 3(b).\nIn addition, the other mirror symmetries of m110and\nm1\u000010also lead to nodal loops in kx\u0000ky=0 andkx+ky=0\nplanes, respectively, and they touch the nodal loops in\nkz=0 plane on the \u0000-M line. But the nodal loops in kx\u0006\nky=0 are around 0.15 eV above Fermi level and not easy\nto detect in experiments. Owing to the screw rotation\noperation that connects two Ru-sites, the nodal loops3\nFIG. 3. WPs and nodal loops in RuO 2. (a) Location of nodal loops from spin-up channel in the case without considering SOC.\nThe dashed circles represent the location of WPs after consideration of SOC. The red and blue colors for the dashed circles\nindicates the chirality of WPs. (b) Corresponding three dimensional energy dispersion of the nodal loop in kz=0 plane. (c)\nLocation of WPs and WNLs in the case of including SOC, with magnetization orientation along [110]. (d) Energy dispersion\nof the WNL (e) Fermi surface of RuO 2with energy lying at EW. (f) Energy dispersion around WP. The color bar is in the\nunit of eV, 0 eV is the Fermi level.\nfrom spin-up channel around the ( \u0000\u0019;\u0019; 0) corner and\nthat from spin-down channel around the ( \u0019;\u0019; 0) corner\nare linked by a C4zrotation operation.\nWith the inclusion of SOC, the existence and behavior\nof the original nodal rings depend on the speci\fc mag-\nnetization orientations, see the sketch in Fig. 1. When\nthe N\u0013 eel vector is perpendicular to the plane, the original\nnodal loops can be preserved if the wavefunctions of the\ntwo bands have opposite mirror eigenvalues. Otherwise,\nthe gapless linear crossing will be broken with opening\nband gaps, and WPs are allowed nearby. Meanwhile,\nwhen SOC is taken into consideration, the hybridization\nof two spin channels can generate additional WNLs and\nWPs.\nWe noted that previous ab\u0000initio calculations found\nthe magnetocrystalline anisotropy energy is only around\n2.76 meV/Ru [18]. Such small energy di\u000berence illus-\ntrates the easy tunability of the N\u0013 eel vector. Weak\ndoping in the form of Ru 1+xO2\u0000xand Ru 1\u0000xIrxO2can\nchange the N\u0013 eel vector to lying within (001) plane [16].\nRecently experimentally grown [001] oriented RuO 2thin\n\flms have the in-plane magnetization [42]. Therefore,\nmost studies about the electrical and optical responses\nfocused on the cases with in-plane N\u0013 eel vector. With tiny\nhole-doping, the AHC can reach up to \u0018300 S/cm [16],\na big value in all AMFs. In linear magneto-optical re-\nsponse, both Kerr rotation angle and Faraday rotationangle increase along with rotating N\u0013 eel vector away from\n[001] axis to (001) plane, and the peak value can reach\nup to\u00180.6 and\u00182.0 degree, respectively [18].\nConsidering the strong non-trivial quantum responses\nof RuO 2mainly happen in the cases with magnetization\nin (001) plane [16, 18], we will also focus on the same\nsituations. Taking magnetization along [110] as an ex-\nample, the magnetic structure breaks most of the mirror\nsymmetries, and only m110is left. Therefore the WNLs\nare only allowed in the (110) plane, namely, in the planes\nsatisfyingkx+ky= 0 orkx+ky=\u0019.\nAs presented in Fig. 3(c), one independent WNL exists\nin thekx+ky= 0 plane, which can extend around half\nof the BZ. With doubly degenerate linear crossing, the\nWNL hosts a \u0019Berry phase for the Wilson loop around\nit. Nearby the hinge of BZ with kx=ky=\u0019, the WNL is\nalmost \rat with very weak dispersion. A sharp dispersion\nappears away from the hinge, with energy window in the\nrange of -0.3 eV to 0.15 eV, crossing the Fermi level.\nThis WNL is constructed by the bands with opposite spin\norientations and does not exist in the situation without\nconsidering SOC.\nIn addition to the WNLs, we also \fnd one pair of\nindependent WPs with opposite chiralities in kz= 0\nplane. The WPs are originated from the nodal rings in\nFig. 3(a) in the case without including SOC, see the red\nand blue dashed circles. The [110] oriented magnetiza-4\ntion together with SOC break the original mirror symme-\ntrymz. Hence, the gapless nodal loops were broken with\nopening band gaps. Meanwhile, the WPs are formed on\nthe the original nodal loops, similar to the situation in\nCo3Sn2S2[31{33]. Considering the inversion symmetry,\nthere are four WPs near Fermi level in the 1st BZ in to-\ntal, see Fig. 3(a). Since the location of WP is close to\nthe Fermi cutting between the original nodal rings and\nFermi energy, this pair of WPs are very close to Fermi\nlevel, with an energy of E\u0018E0\u00000:03 eV, making them\nvery easy to detect by the surface measuring techniques,\nsuch as ARPES and STM. We further checked the Fermi\nsurface by \fxing the energy at WP and found that it\nalso presents as a linear touching of the Fermi surfaces.\nAs shown in Fig. 3(e), there are four Fermi surfaces at\nE=EW, consistent with the energy dispersion plotted\nin Fig. 2(d). Owing to the PTandfTj\u001cgsymmetry\nbreaking and strong local magnetization on Ru sites, all\nthe Fermi surfaces are with large spin split, except for\nsome high symmetry points and the special doubly de-\ngeneraced WPs. As presented in the upper-right corner\nof Fig. 3(e), the WP presents as a linear touching of one\nsmall red bubble and one large green bubble from original\nspin-up channel.\nCorrespondingly, the energy dispersion is strongly\ntilted near the WPs. The constant energy plane at EW\ncuts both electron and hole bands through the linear\ncrossing point, see Fig. 3(f). Therefore, it is a typical\ntype-II WP [45, 46]. Rotating the N\u0013 eel vector from [110]\nto [100] in (001) plane or from [110] to [001] in (1-10)\nplane, the positions of WPs are also shifting in both en-\nergy andkspaces. In some speci\fc angles, they are can-\nceled out in the situation of meeting the other WP with\nopposite chirality.\nFIG. 4. Surface states of RuO 2. (a) Fermi surface with\nenergy lying at WP. (b) Energy dispersion crossing one pair\nof WPs with opposite chirality. (c) Energy dispersion along\n\u0000\u0000M. Red and white color demonstrate the occupied and\nunoccupied states, respectively.\nA typical feature of the WSM is the existence of non-\nclosed topological surface Fermi arc [25]. To calculate thesurface state, we considered a half-in\fnite open bound-\nary condition along [001] direction by the Green's func-\ntion method [47, 48]. Putting the energy at the EW, one\ncan easily see a Fermi arc surface state connecting two\nWPs with opposite chiralities, see Fig. 4(a). Though the\nbulk magnetic structure of RuO 2has inversion symmetry,\nthe top or bottom surfaces does not follow it after cut-\nting bulk chemical bonding. Hence, these two Fermi arcs\nnear (\u0000\u0019=a;\u0019=a ) and (\u0019=a;\u0000\u0019=a) are not connected by\nany symmetries and their detailed shapes are di\u000berent.\nThe Fermi arc states are further con\frmed by the energy\ndispersion along the line crossing one pair of WPs. The\nsurface band starts from the bulk linear crossing point\n(W+) and ends at the other one (W-), see Fig. 4(b).\nIn addition to the above Fermi arc, one can see the\nother surface state almost parallel to the Fermi arc, which\nmerges into bulk states at the ends. At \frst glance, it also\nshows a Fermi arc feature. To see the physical origin of\nthis surface state, we plot the surface energy dispersion\nalong \u0000\u0000M, by crossing the connection of two WPs.\nAs presented in Fig. 4(c), both of them indeed belong\nto the same surface band and the constant energy EW\ncuts this band twice. Therefore, both of them are parts\nof the Fermi arcs. In principle, there should be an odd\nnumber of Fermi crossings with only one pair of WPs.\nBut because of the complicated bulk band overlap, there\nis not an indirect band gap between \u0000\u0000Mand some\nsurface states are merged into bulk, making the other\nFermi crossings not visible.\nIV. SUMMARY\nIn summary, we systematically understand the topo-\nlogical band structure in collinear AFM RuO 2from \frst-\nprinciple calculations and symmetry analysis. In absent\nof jointPTandfTj\u001cgsymmetries, WPs and WNLs can\ncoexist in RuO 2and they can transform to each other via\nmagnetization orientation. Furthermore, the WPs are\nonly around 30 meV below Fermi level and the WNLs cut\nFermi level. After the discovery of WSMs and WNLSMs\nin ferromagnets and non-collinear AFMs, RuO 2provides\na promising material platform for direct observation of\nWPs and WNLs in collinear AFMs.\nACKNOWLEDGMENTS\nThis work was supported by the National Science\nFund (Grant No. 51725103, 52188101) and Jiangxu\nLi acknowledges the support from the fellowship of\nChina Postdoctoral Science Foundation (Grant No.\n2021M700152).\n[1] R. Shindou and N. Nagaosa, Phys. Rev. Lett. 87, 116801\n(2001).[2] H. Chen, Q. Niu, and A. H. MacDonald, Phys. Rew.\nLett.112, 017205 (2014).5\n[3] J. K ubler and C. Felser, EPL (Europhysics Letters) 108,\n67001 (2014).\n[4] J. \u0014Zelezn\u0012 y, H. Gao, K. V\u0012 yborn\u0012 y, J. Zemen, J. Ma\u0014 sek,\nA. Manchon, J. Wunderlich, J. Sinova, and T. Jung-\nwirth, Physical review letters 113, 157201 (2014).\n[5] W. Feng, G.-Y. Guo, J. Zhou, Y. Yao, and Q. Niu,\nPhysical Review B 92, 144426 (2015).\n[6] S. Nakatsuji, N. Kiyohara, and T. Higo, Nature 527,\n212 (2015).\n[7] A. K. Nayak, J. E. Fischer, Y. Sun, B. Yan, J. Karel,\nA. C. Komarek, C. Shekhar, N. Kumar, W. Schnelle,\nJ. Ku bler, C. Felser, and S. S. P. Parkin, Sci. Adv. 2,\ne1501870 (2016).\n[8] P. Wadley, B. Howells, J. Zelezny, C. Andrews, V. Hills,\nR. P. Campion, V. Novak, K. Olejnik, F. Maccherozzi,\nS. S. Dhesi, S. Y. Martin, T. Wagner, J. Wunderlich,\nF. Freimuth, Y. Mokrousov, J. Kunes, J. S. Chauhan,\nM. J. Grzybowski, A. W. Rushforth, K. W. Edmonds,\nB. L. Gallagher, and T. Jungwirth, Science 351, 587\n(2016).\n[9] H. Yang, Y. Sun, Y. Zhang, W.-J. Shi, S. S. Parkin, and\nB. Yan, New Journal of Physics 19, 015008 (2017).\n[10] J. \u0014Zelezn\u0012 y, Y. Zhang, C. Felser, and B. Yan, Physical\nReview Letters 119, 187204 (2017).\n[11] W. Shi, L. Muechler, K. Manna, Y. Zhang, K. Koepernik,\nR. Car, J. Van Den Brink, C. Felser, and Y. Sun, Phys-\nical Review B 97, 060406 (2018).\n[12] N. J. Ghimire, A. Botana, J. Jiang, J. Zhang, Y.-S. Chen,\nand J. Mitchell, Nature Communications 9, 1 (2018).\n[13] J. Noky and Y. Sun, Applied Sciences 9, 4832 (2019).\n[14] Y. Zhang, T. Holder, H. Ishizuka, F. de Juan, N. Na-\ngaosa, C. Felser, and B. Yan, NATURE COMMUNI-\nCATIONS 10(2019).\n[15] H. Xu, J. Wei, H. Zhou, J. Feng, T. Xu, H. Du, C. He,\nY. Huang, J. Zhang, Y. Liu, et al. , Advanced Materials\n32, 2000513 (2020).\n[16] L. \u0014Smejkal, R. Gonz\u0013 alez-Hern\u0013 andez, T. Jungwirth, and\nJ. Sinova, Science advances 6, eaaz8809 (2020).\n[17] W. Feng, J.-P. Hanke, X. Zhou, G.-Y. Guo, S. Bl ugel,\nY. Mokrousov, and Y. Yao, Nature communications 11,\n1 (2020).\n[18] X. Zhou, W. Feng, X. Yang, G.-Y. Guo, and Y. Yao,\nPhysical Review B 104, 024401 (2021).\n[19] M. Ikhlas, T. Tomita, T. Koretsune, M.-T. Suzuki,\nD. Nishio-Hamane, R. Arita, Y. Otani, and S. Nakat-\nsuji, Nature Physics 13, 1085 (2017).\n[20] X. Li, L. Xu, L. Ding, J. Wang, M. Shen, X. Lu, Z. Zhu,\nand K. Behnia, Physical Review Letters 119, 056601\n(2017).\n[21] T. Higo, H. Man, D. B. Gopman, L. Wu, T. Koretsune,\nO. M. van't Erve, Y. P. Kabanov, D. Rees, Y. Li, M.-T.\nSuzuki, et al. , Nature Photonics 12, 73 (2018).\n[22] Z. Liu, H. Chen, J. Wang, J. Liu, K. Wang, Z. Feng,\nH. Yan, X. Wang, C. Jiang, J. Coey, et al. , Nature Elec-\ntronics 1, 172 (2018).\n[23] P. Tang, Q. Zhou, G. Xu, and S.-C. Zhang, Nature\nPhysics 12, 1100 (2016).\n[24] L. \u0014Smejkal, J. \u0014Zelezn\u0012 y, J. Sinova, and T. Jungwirth,\nPhysical review letters 118, 106402 (2017).\n[25] X. Wan, A. M. Turner, A. Vishwanath, and S. Y.\nSavrasov, Physical Review B 83, 205101 (2011).\n[26] H. Weng, C. Fang, Z. Fang, B. A. Bernevig, and X. Dai,\nPhys. Rev. X 5, 011029 (2015).[27] S.-M. Huang, S.-Y. Xu, I. Belopolski, C.-C. Lee,\nG. Chang, B. Wang, N. Alidoust, G. Bian, M. Neupane,\nC. Zhang, S. Jia, A. Bansil, H. Lin, and M. Z. Hasan,\nNature Communications 6, 1 (2015).\n[28] B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao,\nJ. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen,\nZ. Fang, X. Dai, T. Qian, and H. Ding, Phys. Rev. X 5,\n031013 (2015).\n[29] S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane,\nG. Bian, C. Zhang, R. Sankar, G. Chang, Y. Zhujun, C.-\nC. Lee, H. Shin-Ming, H. Zheng, J. Ma, D. S. Sanchez,\nB. Wang, A. Bansil, F. Chou, P. P. Shibayev, H. Lin,\nS. Jia, and M. Z. Hasan, Science 349, 613 (2015).\n[30] C. Fang, Y. Chen, H.-Y. Kee, and L. Fu, Physical Review\nB92, 081201 (2015).\n[31] E. Liu, Y. Sun, N. Kumar, L. Muechler, A. Sun, L. Jiao,\nS.-Y. Yang, D. Liu, A. Liang, Q. Xu, J. Kroder, V. Sus,\nH. Borrmann, C. Shekhar, Z. Wang, C. Xi, W. Wang,\nW. Schnelle, S. Wirth, Y. Chen, S. T. B. Goennenwein,\nand C. Felser, Nature Physics 14, 1125 (2018).\n[32] Q. Wang, Y. Xu, R. Lou, Z. Liu, M. Li, Y. Huang,\nD. Shen, H. Weng, S. Wang, and H. Lei, Nature com-\nmunications 9, 1 (2018).\n[33] Q. Xu, E. Liu, W. Shi, L. Muechler, J. Gayles, C. Felser,\nand Y. Sun, Physical Review B 97, 235416 (2018).\n[34] N. Morali, R. Batabyal, P. K. Nag, E. Liu, Q. Xu, Y. Sun,\nB. Yan, C. Felser, N. Avraham, and H. Beidenkopf, Sci-\nence365, 1286 (2019).\n[35] D. Liu, A. Liang, E. Liu, Q. Xu, Y. Li, C. Chen, D. Pei,\nW. Shi, S. Mo, P. Dudin, et al. , Science 365, 1282 (2019).\n[36] K. Kuroda, T. Tomita, M.-T. Suzuki, C. Bareille, A. Nu-\ngroho, P. Goswami, M. Ochi, M. Ikhlas, M. Nakayama,\nS. Akebi, et al. , Nature materials 16, 1090 (2017).\n[37] S. Borisenko, D. Evtushinsky, Q. Gibson, A. Yaresko,\nK. Koepernik, T. Kim, M. Ali, J. van den Brink,\nM. Hoesch, A. Fedorov, et al. , Nature communications\n10, 1 (2019).\n[38] I. Belopolski, K. Manna, D. S. Sanchez, G. Chang,\nB. Ernst, J. Yin, S. S. Zhang, T. Cochran, N. Shumiya,\nH. Zheng, et al. , Science 365, 1278 (2019).\n[39] K. Koepernik and H. Eschrig, Phys. Rev. B 59, 1743\n(1999).\n[40] S. L. Dudarev, G. A. Botton, S. Y. Savrasov,\nC. Humphreys, and A. P. Sutton, Physical Review B\n57, 1505 (1998).\n[41] K. Koepernik, O. Janson, Y. Sun, and J. Brink, arXiv\npreprint arXiv:2111.09652 (2021).\n[42] Z. Feng, X. Zhou, L. \u0014Smejkal, L. Wu, Z. Zhu, H. Guo,\nR. Gonz\u0013 alez-Hern\u0013 andez, X. Wang, H. Yan, P. Qin, et al. ,\narXiv preprint arXiv:2002.08712 (2020).\n[43] T. Berlijn, P. C. Snijders, O. Delaire, H.-D. Zhou, T. A.\nMaier, H.-B. Cao, S.-X. Chi, M. Matsuda, Y. Wang,\nM. R. Koehler, et al. , Physical review letters 118, 077201\n(2017).\n[44] Z. Zhu, J. Strempfer, R. Rao, C. Occhialini, J. Pelliciari,\nY. Choi, T. Kawaguchi, H. You, J. Mitchell, Y. Shao-\nHorn, et al. , Physical review letters 122, 017202 (2019).\n[45] A. A. Soluyanov, D. Gresch, Z. Wang, Q. Wu, M. Troyer,\nX. Dai, and B. A. Bernevig, Nature 527, 495 (2015).\n[46] Y. Sun, S. C. Wu, M. N. Ali, C. Felser, and B. Yan,\nPhys. Rev. B 92, 161107(R) (2015).\n[47] M. P. L. Sancho, J. M. L. Sancho, and J. Rubio, Phys.\nF: Met. Phys 14(1984).6\n[48] M. P. L. Sancho, J. M. L. Sancho, and J. Rubio, Phys.\nF: Met. Phys 15(1985)."    },    {        "title": "2207.13420v1.Importance_of_magnetic_shape_anisotropy_in_determining_magnetic_and_electronic_properties_of_monolayer___mathrm_VSi_2P_4__.pdf",        "content": "arXiv:2207.13420v1  [cond-mat.mtrl-sci]  27 Jul 2022Importance of magnetic shape anisotropy in determining mag netic and electronic\nproperties of monolayer VSi2P4\nSan-Dong Guo1, Yu-Ling Tao1, Kai Cheng1, Bing Wang2and Yee-Sin Ang3\n1School of Electronic Engineering, Xi’an University of Post s and Telecommunications, Xi’an 710121, China\n2Institute for Computational Materials Science, School of P hysics\nand Electronics,Henan University, 475004, Kaifeng, China and\n3Science, Mathematics and Technology (SMT), Singapore Univ ersity of\nTechnology and Design (SUTD), 8 Somapah Road, Singapore 487 372, Singapore\nTwo-dimensional (2D) ferromagnets have been a fascinating subject of research, and magnetic\nanisotropy (MA) is indispensable for stabilizing the 2D mag netic order. Here, we investigate mag-\nnetic anisotropy energy (MAE), magnetic and electronic pro perties of VSi 2P4by using the gen-\neralized gradient approximation plus U(GGA+U) approach. For large U, the magnetic shape\nanisotropy (MSA) energy has a more pronounced contribution to the MAE, which can overcome the\nmagnetocrystalline anisotropy (MCA) energy to evince an ea sy-plane. For fixed out-of-plane MA,\nmonolayer VSi 2P4undergoes ferrovalley (FV), half-valley-metal (HVM), val ley-polarized quantum\nanomalous Hall insulator (VQAHI), HVM and FV states with inc reasingU. However, for assump-\ntive in-plane MA, there is no special quantum anomalous Hall (QAH) state and spontaneous valley\npolarization within considered Urange. According to the MAE and electronic structure with fix ed\nout-of-plane or in-plane MA, the intrinsic phase diagram sh ows common magnetic semiconductor\n(CMS), FV and VQAHI in monolayer VSi 2P4. At representative U=3 eV widely used in references,\nVSi2P4can be regarded as a 2D- XYmagnet, not Ising-like 2D long-range order magnets predict ed\nin previous works with only considering MCA energy. Our findi ngs shed light on importance of\nMSA in determining magnetic and electronic properties of mo nolayer VSi 2P4.\nKeywords: Magnetic anisotropy, 2D ferromagnets, Phase tra nsition Email:sandongyuwang@163.com\nI. INTRODUCTION\nThere has been a tremendous interest in searching 2D\nmagnetic materials, which can interplay with other im-\nportant properties of materials such as FV, ferroelectric-\nity, piezoelectricity and QAH effects. According to the\nMermin-Wagner theorem, the long-range magnetic order\nat finite temperatureforthe 2DisotropicHeisenbergspin\nsystems is prohibited1. Experimentally, both the CrI 3\nmonolayer and Cr 2Ge2Te6bilayer show an easy out-of-\nplane magnetization2,3, while the 2D magnet CrCl 3has\nan easy in-plane magnetization4,5. The underlying mech-\nanism is that MA is responsible for the stable 2D mag-\nnetic order. With the presence of out-of-plane magne-\ntization, the long-range FM order can sustain at finite\ntemperature by opening a magnon gap to resist ther-\nmalagitations6. An easy-planeanisotropycan realizethe\nquasi-ordered topological vortex/antivortex pairs, which\ncan be described by the Berezinskii-Kosterlitz-Thouless\n(BKT) theory based on the 2D XYmodel7. Moreover,\nMAcanaffectthesymmetryof2Dsystems,andthenpro-\nduce important influence on their topological and valley\nproperties8–11, when the spin-orbital coupling (SOC) is\nincluded. For these 2D systems, the FV and QAH states\ncanexistwithfixedout-of-planemagneticanisotropy,but\nthese special states disappear for in-plane case9–11.\nThe magnetization direction can be determined by\nMAE, which mainly include MCA and MSA energies.\nCompared with SOC-induced MCA, the MSA due to\nthe magnetic dipole-dipole (D-D) interaction is often rel-\natively weak and is neglected. But for a 2D system\nwith weak MCA, the MSA may have an important con-tribution. The representative example is 2D magnet\nCrCl3, whichhasbeen confirmedtohaveaneasyin-plane\nmagnetization in experiment4,5. When only considering\nMCA energy, the CrCl 3shows an easy out-of-plane mag-\nnetization, which is contrary to experimental results12.\nWhenincludingMSAenergy,thetheoreticalresultsagree\nwell with experimental in-plane magnetization12. An-\nothercaseisCrSBrmonolayer13. TheMCAenergyshows\nthat the baxis is the easy one, and the aaxis would be\nhard. When including MSA energy, the MAE gives the\neasy magnetization baxis and hard caxis.\nRecently, the septuple-atomic-layer 2D MoSi 2N4and\nWSi2N4havebeen successfully synthesized by the chemi-\ncalvapordepositionmethod14. Subsequently, 2DMA 2Z4\nfamily with a septuple-atomic-layer structure has been\nestablished, which possess emerging topological, mag-\nnetic, valley and superconducting properties15. The\nVA2Z4of them possess magnetic properties, which have\nbeen widely investigated9,16–21due to their FV proper-\nties. The monolayer VSi 2P4is a typical example, which\nis predicted to haveout-of-planemagnetizationwith only\nconsidering MCA energy (6.38 µeV)17. In recent works,\nthe JanusmonolayerVSiGeN 4is predicted to be in-plane\nmagnetization with only considering MCA energy, and\nthe in-plane magnetization is enhanced, when including\nMSA energy (about -17 µeV)18,21. This implies that\nVSi2P4may be an in-plane 2D XYmagnet without the\nlong-range FM order. In this work, we investigate MCA\nenergy, MSA energy, MAE and electronic structures of\nVSi2P4as a function of U, and reveal the importance of\nMSA in determining its magnetic, topological and val-\nley properties. At representative U=3 eV9,15–21, VSi2P42\nFIG. 1. (Color online)For VSi 2P4monolayer, (a): top view and (b): side view of crystal struct ure. The primitive (rectangle\nsupercell) cell is shown by red (black) lines, and the AFM con figuration is marked with blue arrows in (a). (c): the first BZ\nwith high symmetry points.\ncan be regarded as a 2D- XYmagnet, which is differ-\nent from previous conclusion that the Ising-like 2D long-\nrange order magnet is predicted with only considering\nMCA energy9,17.\nThe rest of the paper is organized as follows. In the\nnext section, we shall give our computational details and\nmethods. In the next few sections, we shall present\nstructure and MAE, and electronic structures of VSi 2P4\nmonolayer. Finally, we shall give our conclusion.\nII. COMPUTATIONAL DETAIL\nBased on density-functional theory (DFT)22, the spin-\npolarized first-principles calculations are performed by\nemploying the projected augmented wave method, as im-\nplemented in VASP code23–25. We use popular general-\nized gradient approximation of Perdew-Burke-Ernzerhof\n(PBE-GGA)26as exchange-correlation functional. To\nconsider on-site Coulomb correlation of V atoms, the\nGGA+Umethod is used within the rotationally invari-\nant approach proposed by Dudarev et al27. To at-\ntain accurate results, we use the energy cut-off of 500\neV, total energy convergence criterion of 10−8eV and\nforce convergence criteria of less than 0.0001 eV .˚A−1\non each atom. A vacuum space of more than 32 ˚A\nis used to avoid the interactions between the neigh-\nboring slabs. We use Γ-centered 15 ×15×1 k-point\nmeshs in the Brillouin zone (BZ) for structure optimiza-\ntion and electronic structures calculations, and 9 ×16×1\nMonkhorst-Pack k-point meshs for calculating ferromag-\nnetic (FM)/antiferromagnetic (AFM) energy with rect-\nangle supercell. The SOC effect is explicitly included\nto investigate MCA, electronic and topological proper-\nties of VSi 2P4monolayer. The Berry curvatures are cal-\nculated directly from wave functions based on Fukui’s\nmethod28, as implemented in VASPBERRY code29,30.\nThe mostly localized Wannier functions are constructed\nfrom the d-orbitals of V atom, s-andp-orbitals of Si and\nGe,p-orbitals of N atoms by Wannier90 code31, and then\nedge states are calculated by employing the Wannier-\nTools package32.III. STRUCTURE AND MAGNETIC\nANISOTROPY ENERGY\nThecrystalstructuresofmonolayerVSi 2P4areplotted\ninFigure 1, along with the first BZ with high-symmetry\npoints. The primitive cell contains one V, two Si, and\nfour P atoms, which is stacked by seven atomic layers\nof P-Si-P-V-P-Si-P. Each V atom is coordinated with six\nP atoms, which forms a trigonal prismatic configuration.\nIn other words, this VP 2layer is sandwiched by two Si-P\nbilayers, and the crystal symmetry of VSi 2P4isP¯6m2\n(No.187) with broken inversion symmetry. The opti-\nmized lattice constants aof VSi 2P4monolayer is 3.486\n˚A, agreeing well with previous theoretical value9.\nBased on Goodenough-Kanamori-Anderson rules33,34,\nthe superexchange coupling between two V atoms is FM,\nbecausethe V-P-Vbondingangleis91 .68◦, which isclose\nto 90◦. To confirm this, the energy differences (per for-\nmula unit) between AFM and FM ordering as a function\nofUare also plotted in FIG.1 of electronic supplemen-\ntary information (ESI). Calculated results show that the\nFM state is always the magnetic ground state of VSi 2P4\nwithin considered Urange. It is found that energy dif-\nference between AFM and FM ordering around U=1.45\neV has a sudden jump, which is due to abrupt change\nof magnetic moment of V atom for AFM ordering (The\nmagnetic moment of V atom changes from 0.74 µBto 0\nµBto 0.079µBto 0.81µB, whenUvariesfrom 1.3eV to\n1.4 eV to 1.5 eV to 1.6 eV), and the small magnetic mo-\nment of V atom reduces the magnetic interaction energy.\nSimilar phenomenon can be observed, when energy dif-\nference between AFM and FM ordering is as a function\nof strain for VSiGeN 4and VSi 2P417,21.\nThe orientation of magnetization plays an impor-\ntant role on magnetic and electronic states of some 2D\nmaterials8–11. The MAE can be used to measure the\ndependence of the energy on the orientation of magne-\ntization. The MAE originates mainly from two parts:\n(1) MCA energy EMCA, which is an intrinsic property of\nthematerialcausedbytheSOC;(2)MSAenergy EMSA),3\n/s48 /s49 /s50 /s51 /s52/s45/s56/s48/s45/s54/s48/s45/s52/s48/s45/s50/s48/s48/s50/s48\n/s40/s97/s41/s111/s117/s116/s45/s111/s102/s45/s112/s108/s97/s110/s101\n/s73/s110/s45/s112/s108/s97/s110/s101/s32/s77/s67/s65/s32/s40 /s101/s86/s41\n/s85 /s32/s40/s101/s86/s41\n/s48 /s49 /s50 /s51 /s52/s45/s49/s52/s45/s49/s51/s45/s49/s50/s45/s49/s49/s45/s49/s48/s45/s57/s45/s56\n/s40/s98/s41\n/s85 /s32/s40/s101/s86/s41/s77/s83/s65/s32/s40 /s101/s86/s41\n/s48 /s49 /s50 /s51 /s52/s45/s56/s48/s45/s54/s48/s45/s52/s48/s45/s50/s48/s48/s50/s48\n/s73/s110/s45/s112/s108/s97/s110/s101/s111/s117/s116/s45/s111/s102/s45/s112/s108/s97/s110/s101\n/s85 /s32/s40/s101/s86/s41/s77/s65/s69/s32/s40 /s101/s86/s41\n/s40/s99/s41\nFIG. 2. (Color online) For VSi 2P4monolayer, MCA energy\n(a), MSA energy (b) and MAE (c) as a function of U.\nwhich is basically the anisotropic D-D interaction12,13:\nED−D=1\n2µ0\n4π/summationdisplay\ni/negationslash=j1\nr3\nij[/vectorMi·/vectorMj−3\nr2\nij(/vectorMi·/vector rij)(/vectorMj·/vector rij)] (1)\nwhere the /vectorMirepresentsthe local magnetic moments and\n/vector rijare vectors that connect the sites iandj. For the\nmonolayer with a collinear FM, when the spins lie in theplane, the Equation 1 can be written as:\nE||\nD−D=1\n2µ0M2\n4π/summationdisplay\ni/negationslash=j1\nr3\nij[1−3cos2θij] (2)\nwhereθijis the angle between the /vectorMand/vector rij. When\nthe spins are out of plane ( θij=90◦), theEquation 1 can\nfurther be simplified as:\nE⊥\nD−D=1\n2µ0M2\n4π/summationdisplay\ni/negationslash=j1\nr3\nij(3)\nTheEMSA(E||\nD−D−E⊥\nD−D)can be expressed as:\nEMSA=3\n2µ0M2\n4π/summationdisplay\ni/negationslash=j1\nr3\nijcos2θij (4)\nThe MSA tends to make magnetic moments of magnetic\nelements directed parallelto the surfaces, which can min-\nimize magnetostatic energy. Usually, the MCA energy\ndominates the MAE, and MSA energy can be ignored.\nHowever, MSA becomes significant to MAE for materi-\nals with weak SOC.\nFirstly, we calculate MCA energyby EMCA=E||\nSOC−\nE⊥\nSOC, which is obtained from GGA+ U+SOC calcula-\ntions. The EMCAas a function of Uis plotted in Fig-\nure 2(a). Calculated results show multiple transitions in\nthe MCA. With increasing U, theEMCAchanges from\nnegativevaluetopositivevalue, whichmeansatransition\nfrom in-plane to out-of-plane. At U=3.0 eV, the calcu-\nlatedEMCA(about 6 µeV) is consistent with available\nresult (6.38 µeV)17. Secondly, we calculate MCA energy\nbyEquation 4 , which depends on the crystal structure\nand local magnetic moment of V ( MV). TheMVas a\nfunction of Uis shown in FIG.2 of ESI. It is found that\nMVincreases (0.89 µB-1.15µB) with increasing U. The\nEMSAasafunctionof Uisshownin Figure2 (b), andthe\nEMSAchanges from -8 µeV to -14 µeV, when Uvaries\nfrom 0 eV to 4 eV. Calculated results show that the MSA\nis in favour of in-plane anisotropy. Finally, the MAE can\nbe obtained by EMAE=EMCA+EMSA, which is plotted\ninFigure 2 (c). When Uis between 2.03 eV and 2.31 eV,\nthe positive MAE means a preferred out-of-plane polar-\nization. When U>2.31 eV and U<2.03 eV, the in-plane\nanisotropy can be observed due to negative MAE.\nFor VA 2Z4, theU=3 eV has been adopted in previ-\nous calculations9,15–21. The VSi 2P4is predicted to be an\nout-of-plane ferromagnet (Ising-like 2D magnets) due to\nonlyconsideringMCAenergy17. However,ourcalculated\nresults show that VSi 2P4can be regarded as a 2D- XY\nmagnet like the CrCl 3monolayer4,5,12. The in-plane easy\nmagnetization means that there is no energetic barrier to\nthe rotationofmagnetizationinthe xyplane, andaBKT\nmagnetic transition to a quasi-long-range phase can be\nobserved at a critical temperature35,36. The critical tem-\nperature TBKT= 1.335J\nKBis obtained from the results\nof the Monte Carlo simulation for 2D-XY magnets on a4\ntriangularlattice37,38, whereJis the nearest-neighboring\nexchange parameter and KBis the Boltzmann constant.\nIn a FM configuration, each V has six neighbors with\nthe same spin, while, for AFM configuration, V has four\nneighbors with the opposite spin, and has two with the\nsame spin. Based on the Heisenberg spin model, the FM\n(EFM) and AFM ( EAFM) energy can be written as:\nEFM=E0−(6J+2A)S2(5)\nEAFM=E0+(2J−2A)S2(6)\nwhereE0is the total energy of systems without mag-\nnetic coupling, and Ameans the easy-axis single-ion\nanisotropy. Hence, Jcan be obtained directly from the\nenergy difference:\nJ=EAFM−EFM\n8S2(7)\nThe calculated Jis 16.07 meV ( S=1\n2) atU=3 eV, and\ntheTBKTis predicted to be 249 K.\nIV. ELECTRONIC STRUCTURES\nThemagnetizationisapseudovector,whichwillleadto\nthat the out-of-plane FM breaks all possible vertical mir-\nror symmetry of the system, while preserves the horizon-\ntal mirror symmetry. The spontaneous valley polariza-\ntion and a nonvanishing Chern number of 2D system can\nexist with the preserved horizontal mirror symmetry8.\nThus, the MA of 2D system is very important to decide\nits electronic state. Firstly, the out-of-plane MA is fixed\nwithin considered Urange. The energy band structures\nof VSi 2P4at some representative Uvalues are plotted in\nFIG.3 of ESI. The evolutions of total energy band gap\nalong with those at -K/K point and the valley splitting\nfor both valence and condition bands as a function of U\nare shown in Figure 3. Around about U=2.20 eV and\n2.40 eV, the total energy band gap of VSi 2P4is closed,\nand the HVM state can be achieved39, whose conduc-\ntion electrons are intrinsically 100% valley polarized. At\naboutU=2.20 eV (2.40 eV), the band gap of K (-K) val-\nley gets closed, while a band gap at -K (K) valley is kept.\nIn a trigonal prismatic crystal field environment, the\nV-dorbitals split into low-lying d2\nzorbital,dxy+dx2−y2\nanddxz+dyzorbitals. Calculated results show that\ndx2−y2+dxyordz2orbitals of V atoms dominate K and\n-K valleys of both valence and conduction bands, and the\nV-dorbital characters energy band structures at repre-\nsentative U=1.50 eV, 2.25 eV and 3.00 eV are plotted in\nFIG.4 of ESI. For U<2.20 eV, the dx2−y2+dxy/dz2or-\nbitals dominate K and -K valleys of valence/conduction\nbands (For example U=1.50 eV). With increasing Ube-\ntween 2.20 eV and 2.40 eV, the band inversion between\ndxy+dx2−y2anddz2orbitals at K valley can be observed\n(For example U=2.25 eV), which is accompanied by the\nfirst HVM state. When Uis larger than 2.40 eV, the1.0 1.5 2.0 2.5 3.0 3.5\nU/uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000c0.000.050.100.150.200.25Gap /uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000cA B C\n1.0 1.5 2.0 2.5 3.0 3.5\nU/uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000c0.00.20.40.60.8Gap /uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000cG−K\nGK\n1.0 1.5 2.0 2.5 3.0 3.5\nU/uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000c020406080Valley Splitting /uni00000003/uni0000000b/uni00000050/uni00000048/uni00000039/uni0000000c\nV\nC\nFIG. 3. (Color online)For VSi 2P4monolayer with fixed out-\nof-plane MA, Top panel: the global energy band gap; Mid-\ndle panel: the energy band gaps for -K and K valleys; Bot-\ntom panel: the valley splitting for both valence and conditi on\nbands as a function of U. The A and C regions mean in-plane,\nwhile the B region for out-of-plane.\nband inversion between dxy+dx2−y2anddz2orbitals oc-\ncurs at -K valley (For example U=3.00 eV), along with\nthe second HVM state. These lead to that the distribu-\ntions ofdx2−y2+dxyanddz2orbitals of 0 eV <U<2.20 eV\nare opposite to those of 2.40 eV <U<4 eV.\nThe distributions of Berry curvature are calculated,\nand they are plotted in FIG.5 at representative U=1.50\neV, 2.25 eV and 3.00 eV. For 0 eV <U<2.20 eV and 2.405\nFIG. 4. (Color online)For VSi 2P4monolayer with out-\nof-plane MA, the topological edge states at representative\nU=2.25 eV.\neV<U<4eV,the oppositesignsanddifferentmagnitudes\nfor Berry curvatures can be observed around -K and K\nvalleys. However,for2.20eV <U<2.40eV, theBerrycur-\nvatures around-Kand K valleys showthe samesigns and\ndifferent magnitudes. These indicate that the flipping of\nthe sign of Berry curvature will occur at K or -K valley.\nThe positive Berry curvature (For example U=1.50 eV)\nchanges into negative one (For example U=2.25eV) at K\nvalley, and then the negative Berry curvature (For exam-\npleU=2.25 eV) changes into positive one (For example\nU=3.00 eV) at -K valley. The twice flipping of the sign\nof Berry curvature is related with twice band inversion\nbetween dxy+dx2−y2anddz2orbitals.\nWith increasing U, twice gap close, band inversion and\nflipping ofthe signofBerrycurvaturesuggesttwice topo-\nlogical phase transition, and the QAH state may exist.\nTo confirm this, the edge states and Chern number are\ncalculated. Calculated results show that there are no\nnontrivial chiral edge states for 0 eV <U<2.20 eV and\n2.40 eV<U<4 eV. When Uis between 2.20 eV and 2.40\neV,theVSi 2P4isaQAHphase. Theedgestatesatrepre-\nsentative U=2.25eVareplotted in Figure4, which shows\na nontrivial chiral edge state connecting the conduction\nbands and valence bands, implying a QAH phase. The\npredicted Chern number C=-1 by integrating the Berry\ncurvature within the first BZ.\nFor 0 eV <U<2.20 eV, the valley splitting of valence\nband is observable, while the valley splitting of conduc-\ntion band can be ignored. However, for 2.40 eV <U<4\neV, the opposite situation can be observed with respect\nto the case of 0 eV <U<2.20 eV. For 2.20 eV <U<2.40\neV, the valley splitting of valence/condition bands de-\ncreases/increases from observable/ignored value to ig-\nnored/observable one with increasing U. In this re-\ngion, VSi 2P4is a VQAHI with spontaneous valley split-\nting and chiral edge states. These can be understood\nby the distributions of dx2−y2+dxyanddz2orbitals. If1.0 1.5 2.0 2.5 3.0 3.5\nU/uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000c0.000.050.100.150.200.250.30Gap /uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000cA B C\n1.0 1.5 2.0 2.5 3.0 3.5\nU/uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000c0.00.20.40.60.8Gap /uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000cG−K\nGK\n1.0 1.5 2.0 2.5 3.0 3.5\nU/uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000c−10−50510Valley Splitting /uni00000003/uni0000000b/uni00000050/uni00000048/uni00000039/uni0000000cV\nC\nFIG. 5. (Color online)For VSi 2P4monolayer with fixed in-\nplane MA, Top panel: the global energy band gap; Middle\npanel: the energy band gaps for -K and K valleys; Bottom\npanel: the valley splitting for both valence and condition\nbands as a function of U. The A and C regions mean in-\nplane, while the B region for out-of-plane.\ndx2−y2+dxy/dz2orbitals dominate -K and K valleys, the\nvalley splitting will be observable/ignored. The SOC\nHamiltonian only involving the interaction of the same\nspin states mainly contributes to valley polarization,\nwhich with out-of-plane magnetization can be simplified\nas40–42:\nˆH0\nSOC=αˆLz (8)6\nFIG. 6. (Color online) The phase diagram for monolayer\nVSi2P4with different Uvalues, including CMS, FV and\nVQAHI.\n/s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s69/s120/s116/s101/s114/s110/s97/s108/s32/s109/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41\n/s85 /s32/s40/s101/s86/s41/s32/s77/s105/s110\n/s32/s77/s97/s120\nFIG. 7. (Color online) The external magnetic field range to\novercome energy barrier from in-plane to out-of-plane as a\nfunction of U.\nwhereˆLz/αis to the zcomponent of orbital angular mo-\nmenta/coupling strength. For dx2−y2+dxyanddz2or-\nbitals and the wave vector symmetry at the K and -K\nvalleys, the basis functions are chosen as:\n|φτ>=/radicalBig\n1\n2(|dx2−y2>+iτ|dxy>)\nor\n|φτ>=|dz2>(9)\nTheresultingenergylevelatKor-Kvalleycanbedefined\nas:\nEτ=< φτ|ˆH0\nSOC|φτ> (10)\nwhere the subscript τ=±1 refers to valley index. If\ndx2−y2+dxyorbitals dominate -K and K valleys, the val-\nley splitting |∆E|is given by:\n|∆E|=|EK−E−K|= 4α (11)\nIf the -K and K valleys are mainly from dz2orbitals, the\nvalley splitting |∆E|is expressed as:\n|∆E|=|EK−E−K|= 0 (12)\nClearly, these results agree with the difference of the val-\nley polarizations in the conduction and valence bands\nwith different Ufrom DFT calculations.Subsequently, the in-plane MA is assumed to investi-\ngate electronic properties of VSi 2P4monolayer as a func-\ntion ofU. The total energy band gap along with those\nat -K/K point and the valley splitting for both valence\nand condition bands as a function of Uare plotted in\nFigure 5, and the representative energy band structures\nare shown in FIG.6 of ESI. With increasing U, the gap\ndecreases, and then increases. In considered Urange,\nthe gaps at -K and K points coincide completely, and\nthe valley splitting is always zero. These mean that no\nspontaneous valley polarization and QAH phase can ap-\npear in VSi 2P4. Fordx2−y2+dxy-dominated -K and K\nvalley, ∆ E= 4αcosθ42with general magnetization ori-\nentation, where θ=0/90◦means out-of-plane/in-planedi-\nrection. For in-plane one, the valley splitting of VSi 2P4\nbecome zero. Thus, VSiGeN 4monolayer is a common\nmagnetic semiconductor (CMS).\nAccording to the MAE and electronic structure dis-\ncussed above, the intrinsic phase diagram of VSi 2P4\nmonolayer is shown in Figure 6, and the electronic state\nincludes CMS, FV and VQAHI. For VA 2Z4, theU=3\neV has been widely used in previous calculations9,15–21.\nThe phase diagramshows that VSi 2P4intrinsically is not\na FV material. However, the FV states can be easily\nachieved by external magnetic field due to small MAE.\nThe external magnetic field to overcome energy barrier\nfrom in-plane to out-of-planeas afunction of Uis plotted\ninFigure 7 . AtU=3 eV, the energy barrier is equiva-\nlent to applying a external magnetic field of around 0.03-\n0.06 T. Although the Ucan not be directly tuned in\nexperiment, it can be regulated equivalently by strain,\nwhich has been confirmed in monolayer RuBr 211. For a\ngiven material, the correlation strength should be fixed.\nEven though VSi 2P4dose not belong to special QAH\nphase in the phase diagram, the QAH state can be re-\nalized by strain or external magnetic field and strain.\nStrain-induce QAH states have been predicted in many\n2D systems11,19,21,43.\nV. CONCLUSION\nIn summary, we have demonstrated that the inclusion\nofMSAcanresultinadifferentphasediagraminVSi 2P4.\nFor fixed out-of-plane situation, the VQAHI state can be\nobserved between two HVM states, which is related with\ntwice sign-reversible Berry curvature and twice band in-\nversions of dxy+dx2−y2anddz2orbitals at -K or K val-\nleys. For assumed in-plane situation, VSi 2P4is a CMS,\nand shows no spontaneous valley polarization. At repre-\nsentative U=3 eV9,15–21, VSi2P4is a CMS, which is dif-\nferent from previous FV material with only considering\nMCA energy9,17. However, spontaneous valley polariza-\ntion can be realized by small external magnetic field. It\nis possible to achieve QAH state in VSi 2P4by strain or\nexternal magnetic field and strain Our works can deepen\nour understanding of MSA in the V-based 2D MA 2Z4\nfamily materials.7\nACKNOWLEDGMENTS\nThis work is supported by the Natural Science Basis\nResearch Plan in Shaanxi Province of China (2021JM-456) and the National Natural Science Foundation of\nChina (No. 12104130). We are grateful to Shanxi Su-\npercomputing Center of China, and the calculations were\nperformed on TianHe-2.\n1N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133\n(1966).\n2B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein,\nR. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A.\nMcGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-\nHerrero and X. Xu, Nature (London) 546, 270 (2017).\n3C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao,\nW. Bao, C. Wang, Y. Wang, Z. Q. Qiu, R. J. Cava, S. G.\nLouie, J. Xia and X. Zhang, Nature (London) 546, 265\n(2017).\n4Z. Wang, M. Gibertini, D. Dumcenco, T. Taniguchi, K.\nWatanabe, E. Giannini, and A. F. Morpurgo, Nat. Nan-\notechnol. 14, 1116 (2019).\n5D. R. Klein, D. MacNeill, Q. Song, D. T. Larson, S. Fang,\nM. Xu, R. A. Ribeiro, P. C. Canfield, E. Kaxiras, R.\nComin, and J. H. Pablo, Nat. Phys. 15, 1255 (2019).\n6J. L. Lado and J. Fern´ andez-Rossier, 2D Mater. 4, 035002\n(2017).\n7J. M. Kosterlitz and D. J. Thouless, J. Phys. C. Solid State\nPhys.6, 1181 (1973).\n8X. Liu, H. C. Hsu, and C. X. Liu, Phys. Rev. Lett. 111,\n086802 (2013).\n9S. Li, Q. Q. Wang, C. M. Zhang, P. Guo and S. A. Yang,\nPhys. Rev. B 104, 085149 (2021).\n10S. D. Guo, J. X. Zhu, M. Y. Yin and B. G. Liu, Phys. Rev.\nB105, 104416 (2022).\n11S. D. Guo, W. Q. Mu and B. G. Liu, 2D Mater. 2D Mater.\n9, 035011 (2022).\n12X. B. Lu, R. X. Fei, L. H. Zhu and L. Yang, Nat. Commun.\n11, 4724 (2020).\n13K. Yang, G. Y. Wang, L. Liu , D. Lu and H. Wu, Phys.\nRev. B104, 144416 (2021).\n14Y. L. Hong, Z. B. Liu, L. Wang T. Y. Zhou, W. Ma, C.\nXu, S. Feng, L. Chen, M. L. Chen, D. M. Sun, X. Q. Chen,\nH. M. Cheng and W. C. Ren, Science 369, 670 (2020).\n15L. Wang, Y. Shi, M. Liu, A. Zhang, Y.-L. Hong, R. Li, Q.\nGao, M. Chen, W. Ren, H.-M. Cheng, Y. Li, and X.- Q.\nChen, Nature Communications 12, 2361 (2021).\n16Q. R. Cui, Y. M. Zhu, J. H. Liang, P. Cui and H. X. Yang,\nPhys. Rev. B 103, 085421 (2021).\n17X. Y. Feng, X. L. Xu, Z. L. He, R. Peng, Y. Dai, B. B.\nHuang and Y. D. Ma, Phys. Rev. B 104, 075421 (2021).\n18D. Dey, A. Ray and L. P. Yu, Phys. Rev. Materials 6,\nL061002 (2022).\n19Y. L. Wang and Y. Ding, Appl. Phys. Lett. 119, 193101\n(2021).\n20X. Zhou, R. Zhang, Z. Zhang, W. Feng, Y. Mokrousov and\nY. Yao, npj Comput. Mater. 7, 160 (2021).21S. D. Guo, W. Q. Mu, J. H. Wang, Y. X. Yang, B. Wang\nand Y. S. Ang, arXiv:2205.07012 (2022).\n22P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964);\nW. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).\n23G. Kresse, J. Non-Cryst. Solids 193, 222 (1995).\n24G. Kresse and J. Furthm¨ uller, Comput. Mater. Sci. 6, 15\n(1996).\n25G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).\n26J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett.\n77, 3865 (1996).\n27S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J.\nHumphreys and A. P. Sutton, Phys. Rev. B 57, 1505\n(1998).\n28T. Fukui, Y. Hatsugai and H. Suzuki, J. Phys. Soc. Japan.\n74, 1674 (2005).\n29H. J. Kim, https://github.com/Infant83/VASPBERRY,\n(2018).\n30H. J. Kim, C. Li, J. Feng, J.-H. Cho, and Z. Zhang, Phys.\nRev. B93, 041404(R) (2016).\n31A. A. Mostofia, J. R. Yatesb, G. Pizzif, Y.-S. Lee, I.\nSouzad, D. Vanderbilte and N. Marzarif, Comput. Phys.\nCommun. 185, 2309 (2014).\n32Q. Wu, S. Zhang, H. F. Song, M. Troyer and A. A.\nSoluyanov, Comput. Phys. Commun. 224, 405 (2018).\n33J. B. Goodenough, Phys. Rev. 100, 564 (1955).\n34J. Kanamori, J. Phys. Chem. Solid 10, 87 (1959).\n35K. Sheng, Q. Chen , H. K. Yuan and Z. Y. Wang, Phys.\nRev. B105, 075304 (2022).\n36J. L. Lado and J. Fern´ andez-Rossier, 2D Mater. 4, 035002\n(2017).\n37P. Jiang, L. Kang, Y.-L. Li, X. Zheng, Z. Zeng, and S.\nSanvito, Phys. Rev. B 104, 035430 (2021).\n38S. Zhang, R. Xu, W. Duan, and X. Zou, Adv. Funct.\nMater.29, 1808380 (2019).\n39H. Hu, W. Y. Tong, Y. H. Shen, X. Wan, and C. G. Duan,\nnpj Comput. Mater. 6, 129 (2020).\n40W. Y. Tong, S. J. Gong, X. Wan, and C. G. Duan, Nat.\nCommun. 7, 13612 (2016).\n41P. Zhao, Y. Dai, H. Wang, B. B. Huang and Y. D. Ma,\nChemPhysMater, 1, 56 (2022).\n42R. Li, J. W. Jiang, W. B. Mi and H. L. Bai, Nanoscale 13,\n14807 (2021).\n43H. Huan, Y. Xue, B. Zhao, G. Y. Gao, H. R. Bao and Z.\nQ. Yang, Phys. Rev. B 104, 165427 (2021)."    },    {        "title": "2208.02425v1.Electric_field_induced_magnetic_anisotropy_transformation_to_achieve_spontaneous_valley_polarization.pdf",        "content": "arXiv:2208.02425v1  [cond-mat.mtrl-sci]  4 Aug 2022Electric-field induced magnetic-anisotropy transformati on to achieve spontaneous\nvalley polarization\nSan-Dong Guo1, Xiao-Shu Guo1, Guang-Zhao Wang2, Kai Cheng1, and Yee-Sin Ang3\n1School of Electronic Engineering, Xi’an University of Post s and Telecommunications, Xi’an 710121, China\n2Key Laboratory of Extraordinary Bond Engineering and Advan ced Materials Technology of Chongqing,\nSchool of Electronic Information Engineering, Yangtze Nor mal University, Chongqing 408100, China and\n3Science, Mathematics and Technology (SMT), Singapore Univ ersity of\nTechnology and Design (SUTD), 8 Somapah Road, Singapore 487 372, Singapore\nValleytronics has been widely investigated for providing n ew degrees of freedom to future infor-\nmation coding and processing. Here, it is proposed that vall ey polarization can be achieved by\nelectric field induced magnetic anisotropy (MA) transforma tion. Through the first-principle calcu-\nlations, our idea is illustrated by a concrete example of VSi 2P4monolayer. The increasing electric\nfield can induce a transition of MA from in-plane to out-of-pl ane by changing magnetic anisotropy\nenergy (MAE) from negative to positive value, which is mainl y due to increasing magnetocrystalline\nanisotropy (MCA) energy. The out-of-plane magnetization i s in favour of spontaneous valley po-\nlarization in VSi 2P4. Within considered electric field range, VSi 2P4is always ferromagnetic (FM)\nground state. In a certain range of electric field, the coexis tence of semiconductor and out-of-plane\nmagnetization makes VSi 2P4become a true ferrovalley (FV) material. The anomalous vall ey Hall\neffect (AVHE) can be observed under in-plane and out-of-plan e electrical field in VSi 2P4. Our works\npave the way to design the ferrovalley material by electric fi eld.\nKeywords: Electric field, Magnetic anisotropy, Ferrovalle y Email:sandongyuwang@163.com\nI. INTRODUCTION\nThe manipulation of the carriers’ properties is key for\nsemiconductortechnology. Inadditiontochargeandspin\nof carriers, the valley, as a new degree of freedom, is\ncharacterized by a local energy extreme in the conduc-\ntion band or valence band. To encode, store and pro-\ncess information, two or more degenerate but inequiva-\nlent valley statesat the inequivalent kpoints should exist\nin a material1,2. Since the two-dimensional (2D) mate-\nrials, especially 2D 2H-phase transition metal dichalco-\ngenides (TMDCs), are booming, the valley-related filed\nis truly flourishing3–9. Nevertheless, realizing the valley\npolarization is indispensable to achieve valley applica-\ntion. To induce valley polarization, many methods have\nbeen proposed, such as optical pumping, magnetic field,\nmagnetic substrates and magnetic doping3–9. However,\nthese methods have some disadvantages that the intrin-\nsic energy band structures and crystal structures can\nbe destroyed. Recently, FV materials have been pro-\nposed, which possess intrinsic spontaneous valley polar-\nization introduced by their intrinsic ferromagnetism10.\nThe FV materials can overcome these shortcomings of\nthe extrinsic valley polarization materials. Many 2D ma-\nterials have been predicted to have spontaneous valley\npolarization10–24.\nHowever, many of predicted 2D FV materials in-\ntrinsically have no spontaneous valley polarization, be-\ncause their magnetization are preferentially in the layer\nplane17–19. To achieve valley polarization in these mate-\nrials, the externalmagnetic field is needed to adjust mag-\nnetization from in-plane to out-of-plane. Even though\npredicted 2D FV material possesses out-of-plane mag-\nnetization, the MAE is calculated with only considering\nFIG. 1. (Color online)(a): a 2D material with in-plane mag-\nnetization (blue arrows) shows no valley polarization. (b) :\nwhen an electric field (red arrow) is applied along out-of-pl ane\ndirection, the magnetization changes from in-plane to out- of-\nplane, and a valley polarization is induced.\nMCA, not including magnetic shape anisotropy (MSA).\nFor example, the easy magnetization axis of VSi 2P4is\nin-plane orientation with including MSA25, which is dif-\nferent from previous out-of-plane one with only consid-\nering MCA energy15. Thus, MSA is very important to\ndetermine MA of 2D materials with weak spin-orbital\ncoupling(SOC), like the representative2D magnetCrCl 3\nwith an easy in-plane magnetization in experiment26,27.\nOnlyconsideringMCAenergyshowsaneasyout-of-plane\nmagnetization for CrCl 3, but the theoretical results with\nincluding MSA energy agree well with experimental in-\nplane magnetization28.\nIn this work, we propose a way to achieve MA trans-\nformation from in-plane to out-of-plane by electric field,\nandthenrealizevalleypolarization(see Figure1). InFig-\nure 1(a), if a 2Dmaterial is with in-plane magnetization,\nno valley polarization can be observed. In Figure 1 (b),2\n/uni0000012b/uni00000010/uni0000002e/uni00000030/uni0000002e /uni0000012b/uni00000030/uni000000ed/uni00000014/uni00000011/uni00000013/uni000000ed/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013Energy /uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000c(a)\n/uni0000012b/uni00000010/uni0000002e/uni00000030/uni0000002e /uni0000012b/uni00000030/uni000000ed/uni00000014/uni00000011/uni00000013/uni000000ed/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013Energy /uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000c(b)\nFIG. 2. (Color online) For VSi 2P4monolayer, the energy\nband structures with in-plane (a) and out-of-plane (b) with -\nout applying electric field.\nwhen an electricfield isimposed alongout-of-planedirec-\ntion, the magnetization changes from in-plane to out-of-\nplane,andavalleypolarizationcanbeinduced. Recently,\n2D MA 2Z4family with a septuple-atomic-layer structure\nhas been established, and MoSi 2N4and WSi 2N4of them\nhave been synthesized by the chemical vapor deposition\nmethod29,30. Here, by the first-principle calculations, we\nuse a concrete example of VSi 2P4monolayer to illustrate\nour idea. Calculated results show that increasing electric\nfiledindeed canmakeMAchangefromin-planetoout-of-\nplane direction in VSi 2P4. The semiconductor properties\nand out-of-plane magnetization can coexist in a certain\nelectricfieldregion, implyingthat VSi 2P4becomesatrue\nFV material. Our findings can be extended to other 2D\nmaterials, and tune MA by electric field.\nII. COMPUTATIONAL DETAIL\nBased on density-functional theory (DFT)31, we per-\nform the spin-polarized first-principles calculations by\nemploying the projected augmented wave method, as\nimplemented in Vienna ab initio simulation package\n(VASP)32–34. The generalized gradient approximation\nof Perdew-Burke-Ernzerhof (PBE-GGA)35is adopted\nas exchange-correlation functional. To consider on-site\nCoulomb correlation of V atoms, the GGA+ Umethod\nin terms of the on-site Coulomb interaction of U= 3.0\neV15is used within the rotationally invariant approach\nproposed by Dudarev et al36, where only the effective\nU(Ueff) based on the difference between the on-site\nCoulomb interaction parameter and exchange parame-\nters is meaningful. The energy cut-off of 500 eV, total\nenergyconvergencecriterionof10−8eVandforceconver-\ngence criteria of less than 0.0001 eV .˚A−1on each atom\nare adopted to attain accurate results. To avoid the in-\nteractionsbetweentheneighboringslabs,avacuumspace\nof more than 32 ˚A is used. The Γ-centered 16 ×16×1 k-\npoint meshs in the Brillouin zone (BZ) areused for struc-\nture optimization and electronic structures calculations,\nand 9×16×1 Monkhorst-Pack k-point meshs for calcu-\nlating FM/antiferromagnetic (AFM) energy with rect-/s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53 /s48/s46/s50/s48 /s48/s46/s50/s53 /s48/s46/s51/s48/s45/s52/s48/s45/s50/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48/s49/s50/s48/s49/s52/s48\n/s69 /s32/s32/s40/s86/s47/s49/s48/s45/s49/s48\n/s109/s41/s40/s97/s41/s111/s117/s116/s45/s111/s102/s45/s112/s108/s97/s110/s101\n/s73/s110/s45/s112/s108/s97/s110/s101/s32/s77/s67/s65/s32/s40 /s101/s86/s41\n/s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53 /s48/s46/s50/s48 /s48/s46/s50/s53 /s48/s46/s51/s48/s45/s49/s52/s45/s49/s51/s45/s49/s50/s45/s49/s49\n/s40/s98/s41\n/s69 /s32/s32/s40/s86/s47/s49/s48/s45/s49/s48\n/s109/s41/s77/s83/s65/s32/s40 /s101/s86/s41\n/s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53 /s48/s46/s50/s48 /s48/s46/s50/s53 /s48/s46/s51/s48/s45/s52/s48/s45/s50/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48/s49/s50/s48/s49/s52/s48\n/s32/s69 /s32/s32/s40/s86/s47/s49/s48/s45/s49/s48\n/s109/s41/s73/s110/s45/s112/s108/s97/s110/s101/s111/s117/s116/s45/s111/s102/s45/s112/s108/s97/s110/s101/s77/s65/s69/s32/s40 /s101/s86/s41\n/s40/s99/s41\nFIG. 3. (Color online) For VSi 2P4monolayer, MCA energy\n(a), MSA energy (b) and MAE (c) as a function of E.\nangle supercell. The SOC effect is explicitly included\nto investigate MCA and electronic properties of VSi 2P4\nmonolayer. We calculate the Berry curvatures directly\nfrom wave functions based on Fukui’s method37by using\nVASPBERRY code38,39.3\n0.00 0.05 0.10 0.15 0.20 0.25 0.30\nE/uni00000003(V/10−10m)−0.3−0.2−0.10.00.10.2Gap /uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000cin-plane out-of-plane\n0.00 0.05 0.10 0.15 0.20 0.25 0.30\nE/uni00000003(V/10−10m)0255075100125Valley Splitting /uni00000003/uni0000000b/uni00000050/uni00000048/uni00000039/uni0000000c\nin-plane out-of-planeV\nC\nFIG. 4. (Color online)For VSi 2P4monolayer with fixed out-\nof-plane MA, the global energy band gap (Top panel) and the\nvalley splitting for both valence and condition bands (Bot-\ntom panel) as a function of E. The red vertical dashed line\ndistinguishes actual in-pane and out-of-plane.\nIII. STRUCTURE AND MAGNETIC\nANISOTROPY\nThe top and side views of crystal structure of VSi 2P4\nmonolayer are shown in FIG.1 of electronic supplemen-\ntaryinformation(ESI), alongwith thefirstBZwith high-\nsymmetry points. The V, Si and P atoms are packed in a\nhoneycomb lattice with a space group of P¯6m2 (No.187),\nwhich has broken inversion symmetry, allowing sponta-\nneous valley polarization. This crystal is composed of\ncovalently bonded atomic layers in the order of P-Si-P-\nV-P-Si-P along the zaxis. In other words, the mono-\nlayer VSi 2P4can be viewed as a MoS 2-like VP 2layer\nsandwiched in-between two slightly buckled honeycomb\nSiP layers. The optimized lattice constants aof VSi 2P4\n(3.486˚A) is good agreement with a previous report15.\nBecause the magnetization is a pseudovector, the out-\nof-plane FM breaks all possible vertical mirror symme-\ntry, but preserves the horizontal mirror symmetry, which\nallows the spontaneous valley polarization and a non-\nvanishing Chern number of 2D system. The VSi 2P4ispredicted to be an out-of-plane ferromagnet (a FV ma-\nterial) with only considering MCA energy15,25. However,\nthe VSi 2P4will become a 2D- XYmagnet(a non-FVma-\nterial), when including MSA energy25. The energy band\nstructures with both in-plane and out-of-plane magne-\ntization are plotted in Figure 2. It is clearly seen that\nVSi2P4is a FV material with out-of-plane magnetiza-\ntion, and it will become a common magnetic semicon-\nductor with in-plane case. Thus, just flipping the mag-\nnetic anisotropy through the external field, VSi 2P4will\nbecome a FV material. Next, we will flip the magnetic\nanisotropy from in-plane to out-of-plane through electric\nfield.\nFirstly, the magnetic ground state under the electric\nfield is confirmed, and the energy differences (per for-\nmula unit) between AFM and FM ordering as a function\nof electric field Eare plotted in FIG.2 of ESI. It is found\nthat the FM state is always ground state within con-\nsideredErange, and increasing Ecan reduce the FM\ninteraction.\nThe orientation of magnetization can affect mag-\nnetic, electronic and topological properties of 2D\nmaterials16,40–42. The orientation of magnetization can\nbedeterminedbyMAE,whichmainlyincludestwoparts:\n(1) MCA energy EMCAcaused by the SOC; (2) MSA en-\nergy (EMSA) due to the anisotropic dipole-dipole (D-D)\ninteraction28,43:\nED−D=1\n2µ0\n4π/summationdisplay\ni/negationslash=j1\nr3\nij[/vectorMi·/vectorMj−3\nr2\nij(/vectorMi·/vector rij)(/vectorMj·/vector rij)] (1)\nin which the /vectorMiand/vector rijrepresent the local magnetic\nmoments and vectors that connect the sites iandj.\nFor monolayer with a collinear FM ordering, the EMSA\n(E||\nD−D−E⊥\nD−D) can be expressed as:\nEMSA=3\n2µ0M2\n4π/summationdisplay\ni/negationslash=j1\nr3\nijcos2θij (2)\nwhere||and⊥mean that spins lie in the plane and out-\nof-plane, and θijis the angle between the /vectorMand/vector rij.\nTheEMSAdepends on the crystal structure and local\nmagnetic moment. To minimize magnetostatic energy,\nthe MSA tends to make spins directed parallel to the\nmonolayer. Usually, the MSA energy can be ignored,\nbut MSA becomes significant to MAE for materials with\nweak SOC.\nWithout inclusion of SOC, there is no link between\nthe crystalline structure and the direction of the mag-\nnetic moments. Thus, the MCA energy is calculated\nbyEMCA=E||\nSOC−E⊥\nSOCfrom GGA+ U+SOC cal-\nculations. The calculation includes two steps: (i) a\ncollinear self-consistent field calculation without SOC to\nobtain the convergent charge density; (ii) a noncollinear\nnon-self-consistent calculation of two different magneti-\nzationdirections(Themagnetizationdirectionsareinthe\nxyplane (E||\nSOC) and along the zaxis (E⊥\nSOC)) within4\n/uni0000012b /uni00000010/uni0000002e/uni00000030/uni0000002e /uni0000012b /uni00000030/uni000000ed/uni00000013/uni00000011/uni00000017/uni000000ed/uni00000013/uni00000011/uni00000015/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000015/uni00000013/uni00000011/uni00000017Energy /uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000c\n0.20\n/uni0000012b /uni00000010/uni0000002e/uni00000030/uni0000002e /uni0000012b /uni00000030/uni000000ed/uni00000013/uni00000011/uni00000017/uni000000ed/uni00000013/uni00000011/uni00000015/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000015/uni00000013/uni00000011/uni00000017Energy /uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000c\n0.21\n/uni0000012b /uni00000010/uni0000002e/uni00000030/uni0000002e /uni0000012b /uni00000030/uni000000ed/uni00000013/uni00000011/uni00000017/uni000000ed/uni00000013/uni00000011/uni00000015/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000015/uni00000013/uni00000011/uni00000017Energy /uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000c\n0.22\n/uni0000012b /uni00000010/uni0000002e/uni00000030/uni0000002e /uni0000012b /uni00000030/uni000000ed/uni00000013/uni00000011/uni00000017/uni000000ed/uni00000013/uni00000011/uni00000015/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000015/uni00000013/uni00000011/uni00000017Energy /uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000c\n0.24\nFIG. 5. (Color online)For VSi 2P4monolayer with fixed out-of-plane MA, the energy band struct ures at representative E=0.20,\n0.21, 0.22 and 0.24 V /10−10m.\nFIG. 6. (Color online)For VSi 2P4monolayer, schematic dia-\ngram of the AVHE along with the Hall voltage under an in-\nplane longitudinal electric field E. The out-of-plane electric\nfieldEis used to induce magnetization change from in-plane\nto out-of-plane. Upward small red arrows represent spin-up\ncarriers.\nSOC. Thenegative/positiveMAE correspondsto aneasy\naxis along the in-plane/out-of-plane direction. We show\nEMCAas a function of EinFigure 3 (a). Calculated\nresults show that the EMCAis positive within consid-\neredErange. When E<0.2 V/10−10m, theEMCAbasi-\ncally remains unchanged with about 6 µeV. For E>0.2\nV/10−10m, theEMCAhas a rapid increase, and reaches\nto 125µeV atE=0.3 V/10−10m. This implies that elec-\ntric filed can effectively tune EMCA. TheEMSAis cal-\nculated by Equation 2 . The local magnetic moment of\nV atom ( MV) as a function of Eis plotted in FIG.3 of\nESI. It is found that MVhas small change within con-\nsideredErange (1.08 µB-1.11µB). TheEMSAversusE\nis plotted in Figure 3 (b), and the EMSAchanges from\n-12.13µeV to -12.75 µeV. The MAE is calculated by\nEMAE=EMCA+EMSA, as shown in Figure 3 (c). When\nUis less than 0.2 V /10−10m, the in-plane anisotropy\ncan be observed due to negative MAE. When E>0.2\nV/10−10m, the positive MAE means a preferred out-of-\nplane magnetization. The MAE changes from negative\nvalue to positive value, implying a transition of MA from\nin-plane to out-of-plane induced by electric field.IV. ELECTRONIC STRUCTURES\nElectricfieldcanresultinsemiconductortometaltran-\nsition in bilayer MoSi 2N4and WSi 2N444. Thus, the elec-\ntronic structures of VSi 2P4under electric field are inves-\ntigated to confirm its semiconductor properties. With\nfixed out-of-plane magnetization, the evolutions of total\nenergy band gap and the valley splitting for both valence\nandcondition bandsasafunction of Eareplotted in Fig-\nure 4. At representative E, the energy band structures\nare plotted in Figure 5 . It is found that the gap de-\ncreases with increasing E, and a semiconductor to metal\ntransition is produced at E=0.22 V/10−10m. Combined\nwith the previous out-of-plane MA Eregion (E>0.20\nV/10−10m), the VSi 2P4is intrinsically a FV material for\nEbetween 0.20 V /10−10m and 0.22 V /10−10m.\nWithin considered Erange, the valley splitting of con-\nduction band is observable, while the valley splitting of\nvalence band can be ignored. This can be explained by\ndistribution of V- dorbitals. The V- dorbitals split into\nd2\nzorbital,dxy+dx2−y2anddxz+dyzorbitals in a trig-\nonal prismatic crystal field environment. According to\nFIG.4 of ESI, the dx2−y2+dxyorbitals dominate K and\n-K valleys of conduction bands, while the dz2orbitals\nmainly contribute to K and -K valleys of valence bands.\nSince the FM ordering breaks spin degeneracy between\nthe spin-up and spin-down bands, the SOC Hamiltonian\nonly involving the interaction of the same spin states\nmainly produces valley polarization. If dx2−y2+dxyor-\nbitals dominate the K and -K valleys, the valley splitting\n|∆E|can be expressed as45,46:\n|∆E|=|EK−E−K|= 4α (3)\nIf the -K and K valleys are mainly from dz2orbitals, the\nvalley splitting |∆E|is given:\n|∆E|=|EK−E−K|= 0 (4)\nwhereαandEK/E−Kare the SOC-related con-\nstant and the resulting energy level at K/-K valley.\nFordx2−y2+dxy-dominated -K and K valley with gen-\neral magnetization orientation, |∆E|=4αcosθ46, where\nθ=0/90◦for out-of-plane/in-plane magnetization. Thus,5\nthe valley splitting of VSi 2P4for both valence and con-\nduction bands will become zero, when VSi 2P4is with\nin-pane case.\nFIG.5 and FIG.6 of ESI present the calculated energy\nband structures and Berry curvatures of VSi 2P4under\nE=0.21 V/10−10m with SOC for magnetic moment of V\nalong the + zand−zdirections. FIG.5 (a) shows the\nvalley polarization, and the energy of K valley is higher\nthan one of -K valley. The valley polarization can be\nswitched by reversing the magnetization direction from\n+zto−zdirection (see FIG.5 (b) of ESI). The VSi 2P4\nunderE=0.21 V/10−10m is an indirect band gap semi-\nconductor with gap value of 26.4 meV. For the two situ-\nations, the opposite signs of Berry curvature around -K\nand K valleys can be observed. It is found that there are\nthe unequal magnitudes of Berry curvatures at -K and K\nvalleys. By reversing the magnetization, the magnitudes\nof Berry curvature at -K and K valleys exchange to each\nother, but their signs remain unchanged.\nUnder an in-plane longitudinal E, anomalous velocity\nυof Bloch electrons at K and -K valleys is related with\nBerry curvature Ω( k):υ∼E×Ω(k)47. An appropriate\nelectrondoping makesthe Fermi levelfall between the -K\nand K valleys of conduction band. When in-plane and\nout-of-plane electric fields are applied, the Berry curva-\nture forces the electron carriers to accumulate on one\nside of the sample, giving rise to an AVHE in monolayer\nVSi2P4(seeFigure 6). In fact, for example, when out-of-\nplane electric field E=0.24 V/10−10m, only -K valley of\nconduction band is conductive (see Figure 5), and AVHE\ncan be achieved with applied electric fields.V. CONCLUSION\nIn summary, we have demonstrated that the electric\nfield can result in a MA transformation in VSi 2P4. The\nelectric field can induce sudden change of positive MCA\nenergy, and has small effects on negative MSA energy.\nThis leads to sign change of MAE, giving rise to in-plane\nto out-of-plane transformation of MA. It is found that\nelectric field can induce semiconductor-metal phase\ntransition in VSi 2P4. However, for a certain electric\nfield region, semiconductor properties and out-of-plane\nmagnetization can coexist, confirming FV properties of\nVSi2P4. When in-plane and out-of-plane electric fields\nare applied, an AVHE in monolayer VSi 2P4can be\nachieved. Our findings can inspire more works of electric\nfiled-tuned MA.\nSUPPLEMENTARY MATERIAL\nSee the supplementary material for crystal structures;\nenergy difference between AFM and FM and local\nmagnetic moment of V atom as a function of E; the\nrelated energy band structures and Berry curvatures.\nConflicts of interest\nThere are no conflicts to declare.\nACKNOWLEDGMENTS\nThis work is supported by Natural Science Basis Re-\nsearch Plan in Shaanxi Province of China (2021JM-456).\nWe are grateful to Shanxi Supercomputing Center of\nChina, andthe calculationswereperformedonTianHe-2.\n1L. J. Sham, S. J. Allen, A. Kamgar, and D. C. Tsui, Phys.\nRev. Lett. 40, 472 (1978).\n2S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M.\nDaughton, S. von Molnr, M. L. Roukes, A. Y. Chtchelka-\nnova, and D. M. Treger, Science 294, 1488 (2001).\n3H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, Nat. Nan-\notechnol. 7, 490 (2012).\n4K. F. Mak, K. He, J. Shan, and T. F. Heinz, Nat. Nan-\notechnol. 7, 494 (2012).\n5A. Srivastava, M. Sidler, A. V. Allain, D. S. Lembke, A.\nKis, and A. Imamoglu, Nat. Phys. 11, 141 (2015).\n68 H. Zeng, J. Dai, W. Yao, D. Xiao and X. Cui, Nat.\nNanotechnol. 7, 490 (2012).\n7D. MacNeill, C. Heikes, K. F. Mak, Z. Anderson, A.\nKorm´anyos, V. Z´ olyomi, J. Park and D. C. Ralph, Phys.\nRev. Lett. 114, 037401 (2015).\n8C. Zhao, T. Norden, P. Zhang, P. Zhao, Y. Cheng, F. Sun,\nJ. P. Parry, P. Taheri, J. Wang, Y. Yang, T. Scrace, K.\nKang, S. Yang, G. Miao, R. Sabirianov, G. Kioseoglou, W.\nHuang, A. Petrou and H. Zeng, Nat. Nanotechnol. 12, 757\n(2017).\n9M. Zeng, Y. Xiao, J. Liu, K. Yang and L. Fu, Chem. Rev.118, 6236 (2018).\n10W. Y. Tong, S. J. Gong, X. Wan, and C. G. Duan, Nat.\nCommun. 7, 13612 (2016).\n11Y. B. Liu, T. Zhang, K. Y. Dou, W. H. Du, R. Peng, Y.\nDai, B. B. Huang, and Y. D. Ma, J. Phys. Chem. Lett. 12,\n8341 (2021).\n12Z. Song, X. Sun, J. Zheng, F. Pan, Y. Hou, M.-H. Yung,\nJ. Yang, and J. Lu, Nanoscale 10, 13986 (2018).\n13J. Zhou, Y. P. Feng, and L. Shen, Phys. Rev. B 102,\n180407(R) (2020).\n14P. Zhao, Y. Ma, C. Lei, H. Wang, B. Huang, and Y. Dai,\nAppl. Phys. Lett. 115, 261605 (2019).\n15X. Y. Feng, X. L. Xu, Z. L. He, R. Peng, Y. Dai, B. B.\nHuang and Y. D. Ma, Phys. Rev. B 104, 075421 (2021).\n16S. Li, Q. Q. Wang, C. M. Zhang, P. Guo and S. A. Yang,\nPhys. Rev. B 104, 085149 (2021).\n17H. X. Cheng, J. Zhou, W. Ji, Y. N. Zhang and Y. P. Feng,\nPhys. Rev. B 103, 125121 (2021).\n18K. Sheng, Q. Chen, H. K. Yuan and Z. Y. Wang, Phys.\nRev. B105, 075304 (2022)).\n19P. Jiang, L. L. Kang, Y. L. Li, X. H. Zheng, Z. Zeng and\nS. Sanvito, Phys. Rev. B 104, 035430 (2021).6\n20Y. Zang, Y. Ma, R. Peng, H. Wang, B. Huang, and Y. Dai,\nNano Res. 14, 834 (2021).\n21R. Peng, Y. Ma, X. Xu, Z. He, B. Huang, and Y. Dai,\nPhys. Rev. B 102, 035412 (2020).\n22W. Du, Y. Ma, R. Peng, H. Wang, B. Huang, and Y. Dai,\nJ. Mater. Chem. C 8, 13220 (2020).\n23R. Li, J. W. Jiang, W. B. Mi and H. L. Bai, Nanoscale 13,\n14807 (2021).\n24S. D. Guo, J. X. Zhu, W. Q. Mu and B. G. Liu, Phys. Rev.\nB104, 224428 (2021).\n25S. D. Guo, Y. L. Tao, K. Cheng, B. Wang and Y.-S. Ang,\narXiv:2207.13420 (2022).\n26Z. Wang, M. Gibertini, D. Dumcenco, T. Taniguchi, K.\nWatanabe, E. Giannini, and A. F. Morpurgo, Nat. Nan-\notechnol. 14, 1116 (2019).\n27D. R. Klein, D. MacNeill, Q. Song, D. T. Larson, S. Fang,\nM. Xu, R. A. Ribeiro, P. C. Canfield, E. Kaxiras, R.\nComin, and J. H. Pablo, Nat. Phys. 15, 1255 (2019).\n28X. B. Lu, R. X. Fei, L. H. Zhu and L. Yang, Nat. Commun.\n11, 4724 (2020).\n29Y. L. Hong, Z. B. Liu, L. Wang T. Y. Zhou, W. Ma, C.\nXu, S. Feng, L. Chen, M. L. Chen, D. M. Sun, X. Q. Chen,\nH. M. Cheng and W. C. Ren, Science 369, 670 (2020).\n30L. Wang, Y. Shi, M. Liu, A. Zhang, Y.-L. Hong, R. Li, Q.\nGao, M. Chen, W. Ren, H.-M. Cheng, Y. Li, and X.- Q.\nChen, Nature Communications 12, 2361 (2021).\n31P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964);\nW. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).\n32G. Kresse, J. Non-Cryst. Solids 193, 222 (1995).\n33G. Kresse and J. Furthm¨ uller, Comput. Mater. Sci. 6, 15(1996).\n34G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).\n35J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett.\n77, 3865 (1996).\n36S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J.\nHumphreys and A. P. Sutton, Phys. Rev. B 57, 1505\n(1998).\n37T. Fukui, Y. Hatsugai and H. Suzuki, J. Phys. Soc. Japan.\n74, 1674 (2005).\n38H. J. Kim, https://github.com/Infant83/VASPBERRY,\n(2018).\n39H. J. Kim, C. Li, J. Feng, J.-H. Cho, and Z. Zhang, Phys.\nRev. B93, 041404(R) (2016).\n40X. Liu, H. C. Hsu, and C. X. Liu, Phys. Rev. Lett. 111,\n086802 (2013).\n41S. D. Guo, J. X. Zhu, M. Y. Yin and B. G. Liu, Phys. Rev.\nB105, 104416 (2022).\n42S. D. Guo, W. Q. Mu and B. G. Liu, 2D Mater. 9, 035011\n(2022).\n43K. Yang, G. Y. Wang, L. Liu , D. Lu and H. Wu, Phys.\nRev. B104, 144416 (2021).\n44Q. Y. Wu, L. M. Cao, Y. S. Ang and L. K. Ang, Appl.\nPhys. Lett. 118, 113102 (2021).\n45P. Zhao, Y. Dai, H. Wang, B. B. Huang and Y. D. Ma,\nChemPhysMater, 1, 56 (2022).\n46R. Li, J. W. Jiang, W. B. Mi and H. L. Bai, Nanoscale 13,\n14807 (2021).\n47D. Xiao, M. C. Chang, and Q. Niu, Rev. Mod. Phys. 82,\n1959 (2010)."    },    {        "title": "2208.06745v2.2D_XY_ferromagnetism_with_high_transition_temperature_in_Janus_monolayer_V___2__XN__X___P__As_.pdf",        "content": "Two-dimensional XY ferromagnetism above room\ntemperature in Janus monolayer V 2XN (X = P, As)\nWenhui Wan,aBotao Fu,bChang Liu,cYanfeng Ge,aand Yong\nLiu\u0003a\naState Key Laboratory of Metastable Materials Science and Technology & Key\nLaboratory for Microstructural Material Physics of Hebei Province, School of\nScience, Yanshan University, Qinhuangdao, 066004, P. R. China\nbCollege of Physics and Electronic Engineering, Center for Computational Sciences,\nSichuan Normal University, Chengdu, China\ncInstitute for Computational Materials Science, Joint Center for Theoretical Physics\n(JCTP), School of Physics and Electronics, Henan University, Kaifeng, 475004,\nChina.\nE-mail: yongliu@ysu.edu.cn\nJuly 2022\nAbstract. Two-dimensional (2D) XY magnets with easy magnetization planes\nsupport the nontrivial topological spin textures whose dissipationless transport is\nhighly desirable for 2D spintronic devices. Here, we predicted that Janus monolayer\nV2XN (X = P, As) with a square lattice are 2D-XY ferromagnets by \frst-principles\ncalculations. Both the magnetocrystalline anisotropy and magnetic shape anisotropy\nfavor an in-plane magnetization, leading to an easy magnetization xy-plane in Janus\nmonolayer V 2XN. Resting on the Monte Carlo simulations, we observed the Berezinskii-\nKosterlitz-Thouless (BKT) phase transition in monolayer V 2XN with transition\ntemperature TBKT being above the room temperature. Especially, monolayer V 2AsN\nhas a magnetic anisotropy energy (MAE) of 292.0 \u0016eV per V atom and a TBKT of\n434 K, which is larger than that of monolayer V 2PN. Moreover, a tensile strain of 5%\ncan further improve the TBKT of monolayer V 2XN to be above 500 K. Our results\nindicated that Janus monolayer V 2XN (X = P, As) were candidate materials to realize\nhigh-temperature 2D-XY ferromagnetism for spintronics applications.\nKeywords : Magnetic semiconductor, Curie temperature, Correlation function, Strain\ne\u000bect.\n1. Introduction\nIt had been demonstrated that the long-range magnetic order can exist in two-\ndimensional (2D) ferromagnetic (FM) materials with an easy magnetization axis\nincluding 2D CrI 3, [1] Fe 3GeTe 2, [2] Cr 2Ge2Te6, [3] MnSe 2, [4] and VSe 2. [5] On thearXiv:2208.06745v2  [cond-mat.mtrl-sci]  23 Feb 20232\nother side, 2D FM materials with an easy magnetization plane, which are called 2D-\nXY FM materials, exhibit a quasi-long-range order that is described by Berezinskii-\nKosterlitz-Thouless (BKT) phase transition. [6] 2D-XY FM materials hold the formation\nof bound pairs of topological nontrivial meron and antimeron spin textures, which\nare promising for developing high-speed, low-power, and multi-functional spintronic\ndevices and energy-e\u000ecient nanoelectronic devices. [7] The material realization of 2D-\nXY ferromagnets with a high transition temperature ( TBKT) is a fasting-developing \feld\nand remains a challenge.[8]\n2D-XY ferromagnetism had been investigated in 2D hexagonal systems. In the\nexperiment, Tokmachev et al. observed that the sub-monolayer of Eu atoms self-\nassembled on the silicon surface, [9] graphene, [10] silicene, [11] and germanene [12]\nexhibited an easy magnetization plane. However, the corresponding transition\ntemperature TBKTof Eu-based systems are smaller than 20 K.[9, 10, 11, 12] Pinto et\nal. demonstrated a BKT phase transition in monolayer CrCl 3grown on graphene/6H-\nSiC(0001) substrate with a TBKTof 12.95 K. [13] On the theoretical side, Lu et al. [14]\nand Augustin et al. [15] investigated the spin textures and dynamics properties of\nmeron and antimeron in 2D-XY magnet monolayer CrCl 3. They found that the\neasy magnetization plane in CrCl 3are formed due to competition between the weak\nout-of-plane single-ion anisotropy and strong in-plane shape anisotropy, as well as\nthe exchange competition between di\u000berent neighbors, which leads to a low TBKTof\nmonolayer CrCl 3. [14, 15] Meanwhile, these topologically non-trivial spin textures are\nrobust to external static magnetic \feld. [14] Strungaru et al. reported the formation and\ncontrol of merons and antimerons in monolayer CrCl 3by the laser pulse via the two-\ntemperature model.[16] Besides CrCl 3, easy-plane magnetism has also been predicted\nin monolayer CrAs with a TBKTof 855 K, [17] monolayer VSi 2N4with aTBKTof 307\nK, [18] and monolayer FeX 2(X = Cl, Br, and I) with a maximum TBKTof 210 K.[19]\nHowever, the TBKT of these monolayers [17, 18, 19] has large uncertainty because\nthey were simply estimated by a classical XY model which only considers the nearest-\nneighboring magnetic coupling and constraints the spin vectors in the xy-plane. [20]\nThe 2D-XY ferromagnetism in 2D hexagonal systems motivates the search for easy\nmagnetisation plane in other 2D crystal systems. In 2022, Xuan et al. analyzed the\n166 stable tetragonal monolayer MX (M = transition metal, X = nonmetal) and found\nthat su\u000eciently strong M-d/X-p and M-d/M-d couplings can stabilize monolayer MX\nwith the square lattice without ferroelastic transition. [21] For example, monolayer VN\nadopted a square lattice, while monolayer VP and VAs exhibited a rectangular lattice\nwith ferroelasticity. [22, 23] However, it had not been reported that 2D square lattices\nhave 2D-XY ferromagnetism.\nSince the synthesis of Janus MoSSe [24] and WSSe, [25] the symmetry breaking\nin the vertical direction of 2D Janus structures can introduce extraordinary physics\nproperties. [26, 27, 28, 29, 30, 31] For example, Janus monolayer Cr(I, X) 3(X = Cl, Br)\nand Janus monolayer MnXTe (X = S, Se) exhibit strong spin-orbit coupling (SOC) and\nDzyaloshinskii-Moriya interactions (DMIs) due to the structural symmetry breaking3\nin the vertical direction, which leads to the intrinsic domain wall skyrmions or the\nmagnetic-\feld-induced bimerons. [32, 33] Compared to monolayer 2H-VSe 2which is not\nstable, Janus monolayer 2H-VSSe, 2H-VSeTe, and 2H-VSTe show a structural stability,\nan easy magnetization plane, and a TBKTof 106 K, 82 K, and 46 K, respectively. [34]\nThough monolayer CrI 3is an FM semiconductor with the perpendicular magnetic\nanisotropy, Janus monolayer Cr 2I3Cl3exhibits in-plane magnetic anisotropy. [29] Thus,\nconstructing Janus structures in 2D square systems may be a possible way to realize the\neasy-plane magnetism.\nIn this work, based on V-based tetragonal monolayers, we predicted that Janus\nmonolayer V 2XN (X = P, As) exhibited a 2D square lattice and 2D-XY ferromagnetism\nby \frst-principles calculations. We \frst identi\fed that the FM state was the magnetic\nground state for Janus monolayer V 2XN. Its structural stability was proved by\nthe formation energy, phonon dispersion, thermodynamic studies, and Born criteria.\nThen, the calculations of magnetic anisotropy energy (MAE) displayed that an easy\nmagnetization plane existed in monolayer V 2XN. TheTBKTwas estimated by critical\nbehavior of in-plane susceptibility by the Monte Carlo (MC) simulation based on the\nXXZ model. At last, we found that tensile strain can further enhance the TBKTbut\nhave a small in\ruence on the MAE.\n2. Computational method\nThe \frst-principles calculations were performed by the Vienna ab initio simulation\npackage (VASP) [35] with the projector augmented wave (PAW) [36] pseudopotentials\nand Perdew, Burke, and Ernzerhof (PBE) [37] exchange-correlation functionals. A\nvacuum layer of 20 \u0017A was adopted to prevent arti\fcial interaction between the adjacent\nperiodic images in the z-direction. The kinetic energy cuto\u000b was set to 550 eV. A k-point\nmesh of 12\u000212\u00021 with the Monkhorst-Pack scheme [38] was used for the Brillouin zone\n(BZ) of the primitive cell. The criteria of total energy and atomic force was 10\u00006eV and\n10\u00003eV/\u0017A, respectively. To appropriately describe the on-site Coulomb repulsion of the\nlocalizeddelectrons of V atoms, we applied a simpli\fed PBE+U approach proposed by\nDudarev et al. [39] An e\u000bective Hubbard term U f= U - J = 3 eV was adopted for V-d\norbitals. In previous works, U f= 2\u00184 eV [40, 41, 42, 22] for V-d orbitals had been\nadopted. We found that the di\u000berent values of U fwill not change our main conclusions.\nThe band structures were further checked by Heyd-Scuseria-Ernzerhof (HSE06) hybrid\nfunctional. [43] Through the convergence test, the MAE was calculated by a dense k\nmesh of 30\u000230\u00021 with the SOC e\u000bect being included. Phonon dispersion was calculated\nby a 5\u00025\u00021 supercell by the density functional perturbation theory implemented in\nphonopy codes. [44] A 4 \u00024\u00021 supercell was used to perform the molecular dynamics\n(MD) simulations in the canonical (NVT) ensemble, up to 5 ps with a time step of 1.0\nfs. The MC simulations were performed by a 180 \u0002180\u00021 supercell using MCsolver\ncode. [45] We run 2 \u0002105MC steps per site to reach the thermal equilibrium, followed\nby 8\u0002105MC steps per site for the averaging of magnetization and energies. We have4\n3.96 3.99 4.023.963.994.02\na (Å)b (Å)\n-27.257-27.255-27.253-27.251-27.249-27.247-27.245E(eV)\n(a)\n(b)(c)\nNV\nAs/P xy\nxzy\nFigure 1. (a, b) A top view and a side view of the crystal lattice of Janus monolayer\nV2XN (X = P, As). (c) The energy contour of Janus monolayer V 2PN as a function\nof lattice constants a;b. The lattice with a=b= 3:986\u0017A has the lowest energy.\nconsidered di\u000berent initial spin con\fgurations concluding spin along the one direction,\nin the one plane or random direction in 3D space. We got consistent \fnal results that\nthe spins in Janus monolayer V 2XN (X = P, As) tend to form merons and antimerons\nwhen the temperature is below the transition temperature T BKT.\n3. Results and discussion\nWe \frst calculated the intrinsic monolayer VN and VP. Monolayer VP adopted a\nrectangle lattice with the lattice constants of a= 4:219=4:453\u0017A andb= 3:420=4:443\n\u0017A by PBE/PBE+U calculations. Monolayer VN exhibited a square lattice with the\nlattice constant of a=b= 3:653\u0017A and 3:990\u0017A by PBE and PBE+U calculations,\nrespectively. These results agree well with previous works, [22, 23, 46] indicating the\nreliability of our calculations. Moreover, monolayer VN adopted a buckling and planar\nstructure in the PBE and PBE+U calculations, respectively [see Fig. S1]. Meanwhile,\nwe found that monolayer VN was a non-magnetic (NM) and FM metal in the PBE and\nPBE+U calculations, respectively. The large di\u000berence between the results in PBE and\nPBE+U calculations indicated the necessity to consider the on-site Coulomb repulsion\nof the localized delectrons in both the structural relaxation and electronic structures\nof V-based materials. In the following content, we only showed the results of PBE+U\ncalculations.\nNext, we calculated the energy of monolayer V 2XN (X = P and As) as a function\nof the lattice constant. Both monolayer V 2PN and V 2AsN adopted 2D square lattice as\nthe lowest-energy lattice structure [see Fig. 1(a-c) and Fig. S2(a)] The corresponding\nlattice constants are a= 3:986\u0017A and 3:989\u0017A, respectively. The point group and the\nspace group of monolayer V 2XN areC4vand P4mm (No. 99), respectively. There are\ntwo V atoms, an N atom, and an X (X = P and As) atom in the primitive cell. In\nPBE+U calculations, V atoms are located at the 2c (0.5, 0.0, 0.5) Wycko\u000b sites. The5\nJ1J2\n-6 -4 -2 0 2 4 6-109-108-107 FM         AFM1 \n AFM2    AFM3 \n AFM4E(ev)\n(%)AFM1                   AFM2                    AFM3                    AFM4             (a)\n(b) (c)\nJ3\n0 1000 2000 3000 4000 5000-522-520-518-516Energy (eV)\nTimes (fs)\n05101520\n M X (THz)\n\n(d) (e)\nFigure 2. (a) Possible AFM states of monolayer V 2PN with only V atoms being\nshown. The red and blue balls represent the V atoms with the up and down spins,\nrespectively. (b) FM state and exchange constants J 1, J2, and J 3. (c) The energy of\nof monolayer V 2PN in di\u000berent magnetic states as a function of biaxial strain. (d)\nThe phonon dispersion of monolayer V 2PN in the FM state. (e) Evolution of energy\nof monolayer V 2PN as a function of time at T = 300K.\nN and X atoms occupy the 1b (0.5, 0.5, 0.5179) and 1a (0.0, 0.0, 0.4313) Wycko\u000b sites\nfor monolayer V 2PN and the 1b (0.000, 0.500, 0.5176) and 1a (0.5, 0.0, 0.4216) Wycko\u000b\nsites for monolayer V 2AsN, respectively. Meanwhile, because both the P and As atoms\nmove away from the V-V plane, the V-V distance in monolayer V 2XN is smaller than\nthat of intrinsic monolayer VN. As a result, both the N side and X side of monolayer\nV2XN experience a compressive strain. We found that the crystal lattice and magnetic\nproperties of V 2PN and V 2AsN are similar except that the As atom has a larger buckling\nheight relative to the V-V plane and stronger SOC e\u000bect than that of the P atom. We\nwill mainly display the results of monolayer V 2PN and give the results of monolayer\nV2AsN in the end.\nTo determine the magnetic ground state of monolayer V 2PN, we considered it in\ndi\u000berent magnetic states including the NM state, FM state, and four anti-ferromagnetic\n(AFM) states by a 2 \u00022\u00021 supercell, as shown in Fig. 2(a, b). We did not display the NM\nstate as its much higher energy than other magnetic states. We de\fned the biaxial strain\nas\"= (a\na0\u00001), whereaanda0are unstrained and strained lattice constants, respectively.6\nAt\"\u0014\u00005%, an FM-to-AFM phase transition occurs [see Fig. 2(c)]. When \" >\u00005%,\nthe FM state is the magnetic ground state for monolayer V 2PN [see Fig. 2(c)]. The V-\nP-V and V-N-V bonding angles between the nearest V atoms of strain-free monolayer\nV2PN are shown in Table 1. The cation-anion-cation angle is close to 90.0\u000e, which favors\nan FM coupling between the V atoms according to Goodenough-Kanamori-Anderson\nsuperexchange theory. [47] The energy di\u000berence between the FM state and AFM states\nis 0.39 eV for monolayer V 2PN, indicating a high transition temperature. We also found\nthat the FM state was still the ground state of monolayer V 2PN for U f= 2 eV and 4 eV\n[see Fig. S3]. According to the Hund's rules, each V3+ion has two unpaired delectrons\nwith the same spin. The primitive cell with two V ions has a total magnetic moment\nof 4\u0016B. From the distribution of spin density, we found that the spin density locates\nmainly around the V atoms. Charge transfers from the V atoms to the N and P atoms\ndue to the electronegativity di\u000berence. Based on the Bader charge analysis, the P atom\nand N atom get 0.90 e and 1.47 e from V atoms, respectively. The magnetic moments\nof the V, P, and N ions in a primitive cell are 2.184, -0.224, and -0.255 \u0016B, respectively.\nTo check the stability of monolayer V 2PN, we calculated the formation energy by\nEf= (EV2PN\u00002EV\u0000EP\u0000EN)=4, whereEV2PN,EV,EP, andENare the energies of\nV2PN, V, P, and N atoms in their substance, respectively. 4 is the number of atoms in\nthe unit cell. The Efwas estimated as -0.29 eV/atom for monolayer V 2PN. Meanwhile,\nmonolayer V 2PN in the FM state was dynamically stable with the absence of imaginary\nfrequency in the phonon dispersion [see Fig. 2(d)]. Moreover, the MD simulation\nat 300 K showed that the \fnal crystal lattice of monolayer V 2PN was maintained\nwithout structure frustration, indicating its thermal stability [see Fig. 2(e)]. As last,\nthe calculated 2D e\u000bective elastic constants [see Table 1] satisfy the Born criteria [48]\nfor 2D square lattice: C 11>0, C 66>0, and C 11> |C 12|, indicating its mechanical\nstability. Thus, monolayer V 2PN has good structural stability.\nLiu et al. have recently reported the growth of monolayer VN on steel substrate\nvia physical vapor deposition technology. [49] Cheng et al. proved that monolayer VP\ncould be exfoliated from its bulk counterpart. [23] Similar to the experimental synthesis\nof Janus monolayer MoSSe [24] and WSSe, [25] one can replace one side of the P layer of\nmonolayer VP with the N atoms or replace one side of the N layer of monolayer VN with\nthe P atoms to prepare Janus monolayer V 2PN. V-N bonding is shorter and stronger\nthan V{P bonding because of the larger electronegativity between V and N elements.\nTherefore, it is more favorable to prepare the Janus V 2PN from monolayer VP because\nthe replacement of the P is an exothermic reaction.\nThe band structure of monolayer V 2PN is shown in Fig. 3(a). The spin-up bands\ncross the Fermi level while spin-down bands have a band gap. Thus, monolayer V 2PN\nis a half-metal. In the spin-down channel, the valence band maximum (VBM) and the\nconduction band minimum (CBM) are located at the M point and the \u0000 point of the\nBrillouin zone (BZ). The energy of the VBM is 1.02 eV lower than the Fermi level,\nwhich helps to prevent the thermally excited spin-\rip transition. The more accurate\nHSE06 hybrid functional gave similar band structures [see Fig. S4(a)], indicating the7\nTable 1. Lattice constants a(\u0017A), bonding angle (\u000e), and 2D e\u000bective elastic constants\nCij(N/m), the magnetocrystalline anisotropy (MCA), magnetic shape anisotropy\n(MSA) and MAE of monolayer V 2XN (X = P, As). The unit of anisotropy energy\nis\u0016eV per V atom.\na \\V-X-V \\V-N-VC11C12C66 MCA MSA MAE\nV2PN 3.986 71.2 88.2 75.11 22.19 48.36 58.8 57.8 116.6\nV2AsN 3.989 67.6 88.3 73.28 21.53 44.87 230.8 61.2 292.0\nrationality of U f= 3 eV in our work. The projected density of states (PDOS) near the\nFermi level was mainly contributed by the d-orbitals of the V atoms [see Fig. S5]. Due\nto theC4Vsymmetry, the pxandpxorbitals are degenerate. The dxzanddyzorbitals\nare degenerate. As shown in Fig. 3(b), the dx2\u0000y2,dxz+dyz, anddxyorbitals of the V\natoms dominated the PDOS at the Fermi level in the spin-up channel. The dyzanddxy\norbitals of the V atoms dominate the PDOS near the VBM and CBM in the spin-down\nchannel, respectively. There is an e\u000bective overlap between the PDOS contributed by\nthedxz+dyzorbitals of V atoms and px+pyorbitals of P and N atoms around the\nenergy of -3 eV in the spin-up channel and -2 eV in the spin-down channel. The overlap\nbetween the PDOS contributed by the V- dx2\u0000y2orbital and pzorbitals of P and N atoms\nare also large in the same energy zone, as shown in Fig. S5. That indicated a strong\np\u0000dcoupling between the V atom and the P or N atom, which stabilizes the square\nlattice of monolayer V 2PN and excludes the ferroelastic transition. [21]\nA sizable MAE is signi\fcant to maintain magnetic ordering against heat \ructuation.\nThere are two parts of contributions to the MAE. One is the magnetocrystalline\nanisotropy (MCA) energy ( EMCA) due to the SOC e\u000bect, the other one is the magnetic\nshape anisotropy (MSA) energy ( EMSA) due to the magnetic dipole-dipole interaction.\nFig. 3(c) displayed the MCA of monolayer V 2PN in the whole space. The energy of\nmonolayer V 2PN with magnetization along the in-plane direction is lower than that\nalong the out-of-plane direction. Moreover, the energy di\u000berence of monolayer V 2PN\nwith in-plane magnetization is less than 0.01 \u0016eV per V atom, thereby being regarded\nas isotropic in the xy-plane, as shown in Fig. 3(c). In the primitive cell, each V atom has\na directional distribution of spin density. However, the spin density of two neighboring\nV atoms is orthogonal to each other [see the inset of Fig. 3(d)], which may lead to the\nin-plane isotropic magnetic properties. We determined the EMCA of monolayer V 2PN\nby the energy di\u000berence between the system with magnetization along the z-direction\nandx-direction: EMCA =Esoc\nz\u0000Esoc\nx. As shown from Fig. 3(d), the EMCA is 58.8\u0016eV\nper V atom for monolayer V 2PN. The angular dependence of MCA can be expressed\nby [50]\nEMCA(\u0012) =E0+K1sin2\u0012+K2sin4\u0012: (1)\nHere\u0012is the azimuthal angle from the z-axis.E0is a constant. K1andK2are the\nanisotropy constants. By \ftting the EMCA ofxz-plane with Eq. 1, K1andK2were\nestimated as -58.805 and 0.003 \u0016eV for monolayer V 2PN. The quadratic term in Eq. 18\n030 60 90120 150 1800204060MCA (meV/V atom)\nq( ) xz\n xy\n-2-1012\nG M XE (eV)\nG spin-up\n spin-down(a) (b)\n(c) (d)\nx yz\n-1 0 1-2024PDOS (states/eV/cell)\nE (eV)dxy  dxz+dyz\n dx2-y2      dz2\nFigure 3. (a) band structure of monolayer V 2PN. Red and black lines are spin-\nup and spin-down bands, respectively. (b) Projected density of states (PDOS) of\nmonolayer V 2PN. The positive and negative represent the PDOS in the spin-up and\nspin-down channels, respectively. (c) Angular dependence of the magnetocrystalline\nanisotropy (MCA) of monolayer V 2PN with the direction of magnetization lying on\nthe whole space, (d) The MCA of monolayer V 2PN along di\u000berent the orientations of\nmagnetization in the xz- andxy-plane. The energy corresponding to the magnetization\nalong thex\u0000direction is set to be zero. The \u0012is the azimuthal angle from the z-axis.\nThe inset is the spin density with the isosurface as 0.03 eV/ \u0017A3.\ndominates the MAE of monolayer V 2PN.\nThe orbital-resolved MCA of monolayer V 2PN was displayed in Fig. 4. The V- d\norbitals make the main contribution to MCA. Based on the second-order perturbation\ntheory, [51] the EMCA =Esoc\nz\u0000Esoc\nxcan be expressed:\nEMCA =\u00182(1\u00002\u000e\u000b\f)X\no;\u000b;u;\fjho\u000bj^Lz\f\fu\f\u000b\nj2\u0000jho\u000bj^Lx\f\fu\f\u000b\nj2\n\"\u000b\nu\u0000\"\f\no: (2)\nwhere\u0018,^Lz(x),\"o, and\"uare the SOC strength, angular momentum operators, the\nenergy levels of occupied states and unoccupied states, respectively. \u000band\fare the spin\nindex \"+\" and \"-\". \u000e\u000b\fis 1 for\u000b=\fand 0 for\u000b6=\f. The expression of EMCA between\nour paper and original work [51] di\u000bers a minus due to the di\u000berent de\fnitions of EMCA.\nConsidering the denominator \"u\u0000\"oof Eq. 2, the electronic states near the Fermi level\ndominate the MCA. Therefore, we did not consider the MCA from the contribution of\nthe occupied state in the spin-down channel due to the large band gap. Meanwhile,\nthe summation in Eq. 2 is related to the PDOS of occupied and unoccupied states. We\nclassi\fed the dorbitals by magnetic quantum number m. We used the dm=0,djmj=1, and\ndjmj=2to represent dz2,fdxz,dyzg, andfdxy;dx2\u0000y2g, respectively. The matrix element9\np\nx        py            pzp\nx        py            pz0.01.02.03.04.05.0MAE (µev)2\n.42.1750 .025d\nxy      dyz      dz2      dxz      dx2-y2d\nxy      dyz      dz2      dxz      dx2-y2-0.207.816243240MAE (µev)3\n9.41\n.20\n.00\n.12514.75-0.17.0-\n0.20.0-0.05(a) V-p                                              (b)  V-d(\nc) P-p                                              (d) N-ppx        py            pzp\nx        py            pz-3.10.564.27.81115MAE (µev)8\n.6-\n3.05-0.05p\nx        py            pzp\nx        py            pz0.01.02.03.04.05.0MAE (µev)0\n.750\n.100.0\nFigure 4. (a) Orbital-resolved MCA of the monolayer V 2PN contributed by the\nhybridization between (a) the porbitals of the V atoms, (b) the dorbitals of the V\natoms, (c) the porbitals of the P atoms, and (d) the porbitals of the N atoms.\nin the numerator of Eq. 2 is nonzero for hmj^Lzjmiandhmj^Lxjm\u00061i. Combining the\nFig. 3(b) and 4(b), the EMCA is mainly and positively contributed byD\nd+\nx2\u0000y2\f\f\f^Lz\f\fd\u0000\nxy\u000b\ndue to that the dxyorbital in the CBM of spin-down bands is close to the Fermi level.\nThe hybridizationD\nd+\nx2\u0000y2\f\f\f^Lx\f\fd+\nyz\u000b\nand\nd+\nz2\f\f^Lx\f\fd+\nyz\u000b\nalso make positive contributions\nto theEMCA. All these parts lead to a positive EMCA for monolayer V 2PN, which favors\nthe in-plane magnetization of the V atom.\nCompared to MCA, the magnetic dipole-dipole interaction usually favors the\nmagnetization along the elongated direction of 1D or 2D materials.[52] We estimated\nthe MSA energy EMSA =Edipole\nz\u0000Edipole\nx of monolayer V 2PN with a classical magnetic\ndipole-dipole interaction model [see Fig. S6(a)]. The EMSA is also positive, indicating\nthat shape anisotropy also favors magnetization along the in-plane direction. Meanwhile,\ntheEdipoleis isotropic in the xy-plane [see Fig. S6(b)]. Thus, the MAE = EMCA+EMSA\nof monolayer V 2PN is isotropic in the xy-plane, indicating an easy magnetization xy-\nplane. Moreover, the mechanism of forming the easy-plane ferromagnetism in monolayer\nV2PN is di\u000berent from monolayer CrCl 3. Lu et al. have reported that the net e\u000bect of\ncompetition between the strong in-plane MSA and weak out-of-plane MCA leads to the\neasy magnetization plane of monolayer CrCl 3. [14]\nCompared to Janus monolayer V 2PN, theEMCA andEMSA of monolayer VN are\n-140.1\u0016eV per V atom and 36.9 \u0016eV per V atom, respectively [see Fig. S1(c)]. Thus,\nthe MAE of monolayer VN is negative, indicating that monolayer VN exhibit an easy\nmagnetization z-axis. Our result is contrasted with previous Kuklin's work, [46] due10\nto that we predicted a planar square lattice by PBE+U calculations while Kuklin\net al. predicted a bucking square lattice by PBE calculations [46] [see Fig. S1 (a,\nb)]. Thedjmj=1=fdxz;dyzgorbitals in the spin-up channel dominated the PDOS\naround the Fermi level [see Fig. S1(c)]. The orbital-resolved MAE shows that the\nnegative contribution ofD\nd+\njmj=1\f\f\f^Lz\f\f\fd+\njmj=1E\nis the main reason for the negative MCA\nof monolayer VN. As a result, the spins in monolayer VN prefers to be oriented along\nthez-direction through the normal FM phase transition.\nSince Janus monolayer V 2PN is a 2D-XY ferromagnet, it will occur a BKT phase\ntransition at low temperatures, [6] similar to monolayer CrCl 3. [14] Considering that\na true spin is a 3D vector, we adopted the XXZ model to estimate the transition\ntemperature TBKT:\nH=\u0000X\nhi;ji[Jx(Sx\niSx\nj+Sy\niSy\nj) +JzSz\niSz\nj]\u0000DX\ni(Sz\ni)2; (3)\nwhere Siis the spin vector on site iand is normalized to 1. Dis the single-ion\nanisotropy. We considered the exchange constants Jup to the third nearest neighbors\n[see Fig. 2(b)]. The energy of the FM and AFM states with magnetization along the\nz-axis were expressed as\nEz\nFM=E0\u00002J1;z\u00002J2;z\u00002J3;z\u0000D; (4)\nEz\nAFM1 =E0+ 2J2;z\u00002J3;z\u0000D; (5)\nEz\nAFM2 =E0+ 2J1;z\u00002J2;z\u00002J3;z\u0000D; (6)\nEz\nAFM3 =E0+ 2J3;z\u0000D: (7)\nThe energy corresponding to the magnetization along the x-axis can be obtained\nby replacing JzbyJxand dropping the single-ion anisotropy Din above equations.\nWe got eight equations to solve the exchange constants and single-ion anisotropy. The\nenergies of di\u000berent magnetic states can be obtained by \frst-principles calculations. By\nsubstituting the energies into Eq. 4 to 7, we obtained the JandD, as shown in Table 2.\nAll the exchange constants are positive indicating the FM coupling between the V ions\nat di\u000berent neighbors. The J3is much smaller than J1andJ2. TheDis negative, which\nindicates that spins of V ions prefer to be in-plane rather than out-of-plane in monolayer\nV2PN.\nTable 2. The exchange constants J(meV) and single-ion anisotropy D(meV) of\nmonolayer V 2PX (X = P, As).\nJ1;xJ1;zJ2;xJ2;zJ3;xJ3;z D\nV2PN 21.12 21.09 25.44 25.43 2.833 2.834 -0.007\nV2AsN 25.26 25.16 23.21 23.18 6.24 6.25 -0.008\nWe \frst made a qualitative estimation of TBKTof monolayer V 2PN. With 2D square\nlattice with the nearest-neighboring exchange constant J1being considered, the scaling11\nand renormalization-group calculations of XXZ model [53] gave the following form of\nTBKT:\nTBKT=4\u0019J1;x\nA+ In[(1\u0000\u0015)\u00001]: (8)\nwith\u0015=J1;z=J1;x. Cuccoli et al. predicted that the TBKT = 0:546J1;x=kBfor 2D\nsquare lattice with \u0015= 0:99 by MC calculations.[54] So we could estimate the constant\nAin Eq.8 to be about 18.410. On the other side, Gouva et al. proposed that the\nTBKT was scaled by (1 + 2 J2=J1) by including the second nearest-neighboring exchange\nconstantJ2. [55] By substituting the exchange constants from Table 2 into the Eq. 8,\nwe predicted the TBKTof monolayer V 2PN to be about 419.0 K which has exceeded the\nroom temperature.\nNext, we performed exact MC simulations. The out-of-plane averaged magnetic\nmomentshMziis much smaller than in-plane averaged magnetic moments hMxyiat any\ntemperature [see Fig. 5(a)]. The in-plane magnetization exhibit a large \ructuation at\nlow temperatures, which can be explained by the emergence of meron and antimeron\nspin textures. [14] The spin-spin correlation [14] C(r) =hS(0)\u0001S(r)iof V 2PN occurs\nan exponential decay to zero at a high temperature (500 K) at which V 2PN is in the\nparamagnetic phase [see Fig. 5(b)]. At a low temperature (100 K), C(r) exhibits an\nalgebraic decay continuously and indicates a quasi-long-range order through the BKT\nphase transition. [6]\nTheTBKT can not be determined by speci\fc heat CVdata as it displays not\na divergence but a maximum value at a temperature above TBKT for a 2D-XY\nferromagnet. [56, 54] Cuccoli. et al [54] de\fned the susceptibility as the average of\nthe squared magnetization,\n\u001f\u000b\u000b=1\nN<(XN\ni=1S\u000b\ni)2>; (9)\nand in-plane susceptibility as \u001fin= (\u001fxx+\u001fyy)=2. TheNis the number of magnetic\nmoments.h:::idenotes the time average of corresponding components. Such a de\fnition\nof\u001finretains its value in investigating the divergence of \u001finalso for \fnite lattice\nsimulations. [54] It has been proved that \u001findiverges exponentially as Tcis approached\nfrom above ( T!T+\nBKT):\n\u001fin\u0018eb(T\u0000TBKT)\u00001=2; (10)\nwherebis a non-universal constant. \u001finsatis\fes the form of\n\u001fin\u0018L2\u0000\u0011(11)\nforT\u0014TBKT, whereLis the size of the supercell. The exponent \u0011is a function of\ntemperature Tand satis\fes \u0011= 1=4 atTBKT. [54]\nWe \ftted the \u001finwith Eq. 10 and estimated the TBKTof monolayer V 2PN to be\nabout 395 K [see Fig. 5(c)], which is much higher than TBKT= 12:95 K of synthesized\nmonolayer CrCl 3. [13, 14] To justify this prediction of TBKT, we performed the MC\nsimulations by supercells with di\u000berent sizes L. We explored the \fnite-size scaling12\n30 60 90 120 150 1800.10.20.30.40.5\n 350 K    370 K   395 K\n 410 K    430KL7/4\nL\n400 500 600 700 8000.02.04.06.08.0  lnin~ (T-TBKT)-1/2\n       TBKT = 395 KInin\nT (K)\n0 100 200 300 4000.00.20.40.60.81.0Correlation C(r)\nd (Å) 100K\n 500 K\n0 200 400 600 8000.00.51.01.52.0<M> ()\nT (K) < Mz >\n < Mxy >(b) (a)\n(c) (d)\nFigure 5. (a) Temperature dependence of averaged magnetization hMxyiandhMzi\nin monolayer V 2PN. The error bars are the standard deviation. (b) The spin-spin\ncorrelation function C(r) of in monolayer V 2PN at T = 100 K and 500 K. (c) In-plane\nsusceptibility \u001finof monolayer V 2PN as a function of temperature. (e) \u001fin=L7=4vs\nlattice size Lof monolayer V 2PN at di\u000berent temperatures.\nbehavior of in-plane susceptibility \u001finby calculating the \u001fin=L7=4as a function of\nsupercell size L, as shown in Fig. 5(d). At TBKT= 395 K, the data \u001fin=L7=4is insensitive\nto sizeL, indicating the behavior of \u001fin\u0018L7=4. That is consistent with the exponent\n\u0011= 1=4 atTBKT[see Eq. 11]. [54] Meanwhile, the TBKTis also consistent with the\naforementioned rough estimation of TBKTby the renormalization-group calculations [see\nEq. 8]. [8] Thus, the estimation of TBKTis reliable.\nThe representative real-space arrangement of magnetic moments after MC\nsimulation is shown in Fig. 6 and Fig. S7 (a). The meron and antimeron spin\ntextures were observed for temperature T < T BKT [see Fig. 6(a)]. It was expected\nthat the transport of these topological nontrivial spin textures is dispersionless at\nlow temperatures, which can be applied in energy-e\u000ecient electronic devices. [14] The\nmagnetic moments prefer to lie in-plane around spin defects but along the out-of-plane\ndirection in the core of spin textures [see Fig. S7 (c ,d)], which is consistent with Wysin's\nprediction that perfect in-plane meron and antimeron is unstable and will develop out-\nof-plane spin component for 2D square lattices with \u0015=Jz=Jx>0:704. [57] Actually,\nthe\u0015has reached 0.99 for monolayer V 2PN [see Table 2]. As temperature increases,\nthe spin structures of spin textures were disturbed by thermal energy. The number of\nmerons or antimerons was decreased by thermal \ructuations [see Fig. 6(b) and Fig.\nS7(b)]. AtT\u0015TBKT, we only observed the paramagnetic states.13\n(a)   T = 32 K                                                     (b)  T =  191 K  \nAMM\nA\nFigure 6. The top view of the real-space arrangement of magnetic moments of V 2PN\nafter MC simulation at (a) T = 32 K and (b) T = 191 K. The length of the arrows\nrepresents the in-plane component of magnetic moments. The position of meron (M)\nand antimeron (A) are labeled.\nWe expanded the current calculations to monolayer V 2AsN. The FM state is the\nmagnetic ground state for monolayer V 2AsN with U fof 2\u00184 eV [see Fig. S2 (b)\nand S3 (b)]. The magnetic moments of the V, As, and N ions in a primitive cell are\n2.236, -0.267, and -0.259 \u0016B, respectively. Its formation energy Efis -1.20 eV/atom.\nIts structural stability can be seen from the Fig. S2 (c, d) and Table 1. The electronic\nstructures indicate that monolayer V 2AsN is also a half metal [see Fig. S2 (e, f) and S4\n(b)]. The CBM of monolayer V 2AsN in the spin-down channel is more far away from\nthe Fermi level than that of monolayer V 2PN. The estimation of MCA, MSA, MAE,\nexchange constants and TBKTare shown in Fig. S6(a), Fig. S8 (a, b), Table 1 and\nTable 2. Due to the larger SOC e\u000bect, monolayer V 2AsN possesses a larger MAE of 292\n\u0016eV per V atom, stronger exchange constants J, and a higher TBKTof 434 K than that\nof monolayer V 2PN.\nAt last, we investigated the strain e\u000bect on the magnetic properties of monolayer\nV2XN (X = P, As). As the FM state has higher energy than AFM states at \"=\u00005%\n[see Fig. 2(c)], we only varied the biaxial strain \"from -4% to 5%. We found that\ntensile strain increases the bond angles \\V-X-V and \\V-N-V in monolayer V 2XN (X\n= P, As), making them closer to 90\u000e. The FM coupling and the exchange constant J\nbetween V atoms was enhanced by tensile strain. As a result, the TBKTof monolayer\nV2PN and V 2AsN can be increased to 508.7 K and 531.3 K by a tensile strain of 5%,\nrespectively. That is also consistent with the fact that the energy di\u000berence between\nFM and AFM states increases with tensile strain increasing [see Fig. 2(c)]. On the\nother side, the compressive stain has a negative in\ruence on the exchange coupling\nbetween V atoms. The TBKTof monolayer V 2PN and V 2AsN decreased quickly with\nthe magnitude of compressive strain increasing. Compared to the TBKT, the MAEs did\nnot exhibit regular behavior with the temperatures changing. The EMSAdecreases as the\ndistance between atoms increases, while the EMCA has the opposite tendency [see Fig.\nS9]. Especially, the orbital resolved MAE shows that the contribution of hybridization14\n200300400500-\n4- 20 2 4 6 100200300(b)TBKT (K) \nV2AsN \nV2PN(a)MAE (µeV/V atom)ε\n (%) V2AsN \nV2PN\nFigure 7. The (a) transition temperature TBKT and (b) MAEs of monolayer V 2PN\nand V 2AsN as function of strain.\nD\nd+\nx2\u0000y2\f\f\f^Lz\f\fd\u0000\nxy\u000b\nis the main reason for such a irregular behavior [see Fig. S9]. The\nMAE of monolayer V 2PN and V 2AsN ranges from 271 :3\u0018321:9\u0016eV per V atom and\n88:5\u0018141:1\u0016eV per V atom under strain ranging from -4% to 5%, respectively. In\ngeneral, with a larger MAE and a higher T BKT, monolayer V 2AsN is better than V 2PN\nfor the material realization of the high-temperature 2D-XY ferromagnetism.\n3.1. Acknowledgments\nIn summary, based on \frst-principles calculations, we systemically investigated the\nstructural and magnetic properties of Janus monolayer V 2XN (X = P and As). The\nstrong electron correlation e\u000bect of dorbital of the V atoms was considered in both\nstructure relaxation and electron structures. Monolayer V 2XN has a square lattice with\ngood structural stability and FM half-metallic characters. The MAE is 116.6 \u0016eV/V\natom and 292.0 \u0016eV/V atom for monolayer V 2PN and V 2AsN, respectively. Both the\nmagnetocrystalline anisotropy and shape anisotropy favor an in-plane magnetization,\nleading to an easy magnetization xy-plane in monolayer V 2XN. MC simulations based\non the 2D XXZ model indicated that monolayer V 2XN occurs the BKT phase transition\nwith the emergence of topological non-trivial meron and antimeron spin textures.\nThrough the analysis of the critical behavior of in-plane susceptibility, the TBKT of\nmonolayer V 2PN and V 2AsN was estimated as 395 K and 434 K, respectively. A 5%\ntensile strain can increase the corresponding TBKTto 508.7 K and 531.3 K, respectively.\nOur work indicated that V-based Janus structures were candidate materials to realize\nthe high-temperature 2D-XY ferromagnetism for spintronic applications.15\n[1] Bevin Huang, Genevieve Clark, Efr\u0013 en Navarro-Moratalla, Dahlia R. Klein, Ran Cheng, Kyle L.\nSeyler, Ding Zhong, Emma Schmidgall, Michael A. McGuire, David H. Cobden, Wang Yao,\nDi Xiao, Pablo Jarillo-Herrero, and Xiaodong Xu. Layer-dependent ferromagnetism in a van\nder waals crystal down to the monolayer limit. Nature , 546(7657):270{273, 2017.\n[2] Zaiyao Fei, Bevin Huang, Paul Malinowski, Wenbo Wang, Tiancheng Song, Joshua Sanchez, Wang\nYao, Di Xiao, Xiaoyang Zhu, Andrew F. May, Weida Wu, David H. Cobden, Jiun-Haw Chu, and\nXiaodong Xu. Two-dimensional itinerant ferromagnetism in atomically thin Fe 3GeTe 2.Nat.\nMater. , 17(9):778{782, 2018.\n[3] Cheng Gong, Lin Li, Zhenglu Li, Huiwen Ji, Alex Stern, Yang Xia, Ting Cao, Wei Bao, Chenzhe\nWang, Yuan Wang, Z. Q. Qiu, R. J. Cava, Steven G. Louie, Jing Xia, and Xiang Zhang.\nDiscovery of intrinsic ferromagnetism in two-dimensional van der waals crystals. Nature ,\n546(7657):265{269, 2017.\n[4] Dante J. O'Hara, Tiancong Zhu, Amanda H. Trout, Adam S. Ahmed, Yunqiu Kelly Luo,\nChoong Hee Lee, Mark R. Brenner, Siddharth Rajan, Jay A. Gupta, David W. McComb,\nand Roland K. Kawakami. Room temperature intrinsic ferromagnetism in epitaxial manganese\nselenide \flms in the monolayer limit. Nano Lett. , 18(5):3125{3131, 2018.\n[5] Manuel Bonilla, Sadhu Kolekar, Yujing Ma, Horacio Coy Diaz, Vijaysankar Kalappattil, Raja\nDas, Tatiana Eggers, Humberto R. Gutierrez, Manh-Huong Phan, and Matthias Batzill. Strong\nroom-temperature ferromagnetism in VSe 2monolayers on van der waals substrates. Nat.\nNanotechnol. , 13(4):289{293, 2018.\n[6] J M Kosterlitz and D J Thouless. Ordering, metastability and phase transitions in two-dimensional\nsystems. J. Phys. C , 6(7):1181{1203, apr 1973.\n[7] J M Kosterlitz. The critical properties of the two-dimensional xy model. J. Phys. C: Solid State\nPhys. , 7(6):1046, mar 1974.\n[8] Jiesu Wang. Berezinskii-kosterlitz-thouless phase transition in a 2D-XY ferromagnetic monolayer.\nJ. Semicond. , 42(12):120401, dec 2021.\n[9] Andrey M. Tokmachev, Dmitry V. Averyanov, Alexander N. Taldenkov, Ivan S. Sokolov, Igor A.\nKarateev, Oleg E. Parfenov, and Vyacheslav G. Storchak. Two-dimensional magnets beyond\nthe monolayer limit. ACS Nano , 15(7):12034{12041, 2021.\n[10] Ivan S. Sokolov, Dmitry V. Averyanov, Oleg E. Parfenov, Igor A. Karateev, Alexander N.\nTaldenkov, Andrey M. Tokmachev, and Vyacheslav G. Storchak. 2D ferromagnetism in\neuropium/graphene bilayers. Mater. Horiz. , 7(5):1372{1378, 2020.\n[11] Andrey M. Tokmachev, Dmitry V. Averyanov, Oleg E. Parfenov, Alexander N. Taldenkov, Igor A.\nKarateev, Ivan S. Sokolov, Oleg A. Kondratev, and Vyacheslav G. Storchak. Emerging two-\ndimensional ferromagnetism in silicene materials. Nat. Commun. , 9(1):1672, 2018.\n[12] Andrey M. Tokmachev, Dmitry V. Averyanov, Alexander N. Taldenkov, Oleg E. Parfenov, Igor A.\nKarateev, Ivan S. Sokolov, and Vyacheslav G. Storchak. Lanthanide f7metalloxenes - a class\nof intrinsic 2D ferromagnets. Mater. Horiz. , 6(7):1488{1496, 2019.\n[13] Amilcar Bedoya-Pinto, Jing-Rong Ji, Avanindra K. Pandeya, Pierluigi Gargiani, Manuel\nValvidares, Paolo Sessi, James M. Taylor, Florin Radu, Kai Chang, and Stuart S. P. Parkin.\nIntrinsic 2D-XY ferromagnetism in a van der waals monolaye. Science , 374(6567):616{620, oct\n2021.\n[14] Xiaobo Lu, Ruixiang Fei, Linghan Zhu, and Li Yang. Meron-like topological spin defects in\nmonolayer CrCl 3.Nat. Commun. , 11(1):4724, 2020.\n[15] Mathias Augustin, Sarah Jenkins, Richard F. L. Evans, Kostya S. Novoselov, and Elton J. G.\nSantos. Properties and dynamics of meron topological spin textures in the two-dimensional\nmagnet crcl3. Nat. Commun. , 12(1):185, 2021.\n[16] Mara Strungaru, Mathias Augustin, and Elton J. G. Santos. Ultrafast laser-driven topological\nspin textures on a 2d magnet. npj Comput. Mater. , 8(1):169, 2022.\n[17] An-Ning Ma, Pei-Ji Wang, and Chang-Wen Zhang. Intrinsic ferromagnetism with high\ntemperature, strong anisotropy and controllable magnetization in the CrX (X = P, As)16\nmonolayer. Nanoscale , 12:5464{5470, 2020.\n[18] Qirui Cui, Yingmei Zhu, Jinghua Liang, Ping Cui, and Hongxin Yang. Spin-valley coupling in a\ntwo-dimensional VS 2iN4monolayer. Phys. Rev. B , 103(8):085421, feb 2021.\n[19] Michael Ashton, Dorde Gluhovic, Susan B. Sinnott, Jing Guo, Derek A. Stewart, and Richard G.\nHennig. Two-dimensional intrinsic half-metals with large spin gaps. Nano Lett. , 17(9):5251{\n5257, 2017.\n[20] Julio F. Fern\u0013 andez, Manuel F. Ferreira, and Jolanta Stankiewicz. Critical behavior of the two-\ndimensional XY model: A monte carlo simulation. Phys. Rev. B , 34:292{300, Jul 1986.\n[21] Xiaoyu Xuan, Wanlin Guo, and Zhuhua Zhang. Ferroelasticity in two-dimensional tetragonal\nmaterials. Phys. Rev. Lett. , 129:047602, Jul 2022.\n[22] Xiaoyu Xuan, Menghao Wu, Zhuhua Zhang, and Wanlin Guo. A multiferroic vanadium phosphide\nmonolayer with ferromagnetic half-metallicity and topological dirac states. Nanoscale Horiz. ,\n7:192{197, 2022.\n[23] Xuli Cheng, Shaowen Xu, Fanhao Jia, Guodong Zhao, Minglang Hu, Wei Wu, and Wei Ren.\nIntrinsic ferromagnetism with high curie temperature and strong anisotropy in a ferroelastic VX\nmonolayer (X=P, As). Phys. Rev. B , 104:104417, 2021.\n[24] Ang-Yu Lu, Hanyu Zhu, Jun Xiao, Chih-Piao Chuu, Yimo Han, Ming-Hui Chiu, Chia-Chin Cheng,\nChih-Wen Yang, Kung-Hwa Wei, Yiming Yang, Yuan Wang, Dimosthenis Sokaras, Dennis\nNordlund, Peidong Yang, David A. Muller, Mei-Yin Chou, Xiang Zhang, and Lain-Jong Li.\nJanus monolayers of transition metal dichalcogenides. Nat. Nanotechnol. , 12(8):744{749, 2017.\n[25] Yu-Chuan Lin, Chenze Liu, Yiling Yu, Eva Zarkadoula, Mina Yoon, Alexander A. Puretzky,\nLiangbo Liang, Xiangru Kong, Yiyi Gu, Alex Strasser, Harry M. Meyer, Matthias Lorenz,\nMatthew F. Chisholm, Ilia N. Ivanov, Christopher M. Rouleau, Gerd Duscher, Kai Xiao, and\nDavid B. Geohegan. Low energy implantation into transition-metal dichalcogenide monolayers\nto form Janus structures. ACS Nano , 14(4):3896{3906, 2020.\n[26] Dipesh B. Trivedi, Guven Turgut, Ying Qin, Mohammed Y. Sayyad, Debarati Hajra, Madeleine\nHowell, Lei Liu, Sijie Yang, Naim Hossain Patoary, Han Li, Marko M. Petri\u0013 c, Moritz Meyer,\nMalte Kremser, Matteo Barbone, Giancarlo Soavi, Andreas V. Stier, Kai M uller, Shize\nYang, Ivan Sanchez Esqueda, Houlong Zhuang, Jonathan J. Finley, and Sefaattin Tongay.\nRoom-temperature synthesis of 2D Janus crystals and their heterostructures. Adv. Mater. ,\n32(50):2006320, 2020.\n[27] Luojun Du, Taw\fque Hasan, Andres Castellanos-Gomez, Gui-Bin Liu, Yugui Yao, Chun Ning\nLau, and Zhipei Sun. Engineering symmetry breaking in 2D layered materials. Nat. Rev.\nPhys. , 3(3):193{206, 2021.\n[28] Fang Zhang, Wenbo Mi, and Xiaocha Wang. Spin-dependent electronic structure and magnetic\nanisotropy of 2D Ferromagnetic Janus Cr2I3X3 (X = Br, Cl) monolayers. Adv. Electron. Mater. ,\n6(1):1900778, 2020.\n[29] Rui Li, Jiawei Jiang, Xiaohui Shi, Wenbo Mi, and Haili Bai. Two-dimensional janus FeXY (X,\nY = Cl, Br, and I) monolayers: Half-metallic ferromagnets with tunable magnetic properties\nunder strain. ACS Appl. Mater. Interfaces , 13(32):38897{38905, 2021. PMID: 34370461.\n[30] Rui Li, Jiawei Jiang, Wenbo Mi, and Haili Bai. Ferroelectric polarization tailored interfacial\ncharge distribution to modify magnetic properties of two-dimensional Janus FeBrI/In 2S3\nheterostructures. Appl. Phys. Lett. , 120(16):162401, 2022.\n[31] Rui Li, Jiawei Jiang, Wenbo Mi, and Haili Bai. Room temperature spontaneous valley polarization\nin two-dimensional FeClBr monolayer. Nanoscale , 13:14807{14813, 2021.\n[32] Changsong Xu, Junsheng Feng, Sergei Prokhorenko, Yousra Nahas, Hongjun Xiang, and\nL. Bellaiche. Topological spin texture in Janus monolayers of the chromium trihalides Cr(I,\nX)3.Phys. Rev. B , 101(6):060404(R), 2020.\n[33] Jiaren Yuan, Yumeng Yang, Yongqing Cai, Yihong Wu, Yuanping Chen, Xiaohong Yan, and Lei\nShen. Intrinsic skyrmions in monolayer Janus magnets. Phys. Rev. B , 101(9):094420, 2020.\n[34] Dibyendu Dey and Antia S. Botana. Structural, electronic, and magnetic properties of vanadium-17\nbased Janus dichalcogenide monolayers: A \frst-principles study. Phys. Rev. Mater. , 4(7):074002,\n2020.\n[35] G. Kresse and J. Furthm uller. E\u000ecient iterative schemes for ab initio total-energy calculations\nusing a plane-wave basis set. Phys. Rev. B , 54:11169{11186, 1996.\n[36] G. Kresse and D. Joubert. From ultrasoft pseudopotentials to the projector augmented-wave\nmethod. Phys. Rev. B , 59:1758{1775, 1999.\n[37] John P. Perdew, Kieron Burke, and Matthias Ernzerhof. Generalized gradient approximation\nmade simple. Phys. Rev. Lett. , 77:3865{3868, 1996.\n[38] Hendrik J. Monkhorst and James D. Pack. Special points for brillouin-zone integrations. Phys.\nRev. B , 13:5188{5192, 1976.\n[39] S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton. Electron-\nenergy-loss spectra and the structural stability of nickel oxide: An LSDA+U study. Phys. Rev.\nB, 57:1505{1509, Jan 1998.\n[40] W. E. Pickett, S. C. Erwin, and E. C. Ethridge. Reformulation of the LDA+U method for a\nlocal-orbital basis. Phys. Rev. B , 58:1201{1209, Jul 1998.\n[41] Y. K. Wang, P. H. Lee, and G. Y. Guo. Half-metallic antiferromagnetic nature of La 2VTcO 6and\nLa2VCuO 6from ab initio calculations. Phys. Rev. B , 80:224418, Dec 2009.\n[42] F. Aryasetiawan, K. Karlsson, O. Jepsen, and U. Sch onberger. Calculations of hubbard U from\n\frst-principles. Phys. Rev. B , 74:125106, Sep 2006.\n[43] Jochen Heyd, Gustavo E. Scuseria, and Matthias Ernzerhof. Hybrid functionals based on a\nscreened coulomb potential. J. Chem. Phys. , 118(18):8207{8215, 2003.\n[44] Atsushi Togo and Isao Tanaka. First principles phonon calculations in materials science. Scr.\nMater. , 108:1{5, 2015.\n[45] Liang Liu, Xue Ren, Jihao Xie, Bin Cheng, Weikang Liu, Taiyu An, Hongwei Qin, and Jifan Hu.\nMagnetic switches via electric \feld in BN nanoribbons. Appl .Surf. Sci , 480:300{307, jun 2019.\n[46] Artem V. Kuklin, Svetlana A. Shostak, and Alexander A. Kuzubov. Two-dimensional lattices\nof VN: Emergence of ferromagnetism and half-metallicity on nanoscale. J. Phys. Chem. Lett. ,\n9(6):1422{1428, 2018. PMID: 29502418.\n[47] John B. Goodenough. Theory of the role of covalence in the perovskite-type manganites [La,\nM(II)]MnO 3.Phys. Rev. , 100(2):564{573, oct 1955.\n[48] Marcin Ma\u0013 zdziarz. Comment on `the computational 2D materials database: high-throughput\nmodeling and discovery of atomically thin crystals'. 2D Materials , 6(4):048001, jul 2019.\n[49] Z.X. Liu, Yong Li, X.H. Xie, Jun Qin, and Y. Wang. The tribo-corrosion behavior of monolayer\nVN and multilayer VN/C hard coatings under simulated seawater. Ceram. Int. , 47(18):25655{\n25663, 2021.\n[50] K. H. J. Buschow and F. R. de Boer. Physics of Magnetism and Magnetic Materials . Springer\nUS, 2003.\n[51] Ding sheng Wang, Ruqian Wu, and A. J. Freeman. First-principles theory of surface\nmagnetocrystalline anisotropy and the diatomic-pair model. Phys. Rev. B , 47(22):14932{14947,\njun 1993.\n[52] David C. Johnston. Magnetic dipole interactions in crystals. Phys. Rev. B , 93:014421, Jan 2016.\n[53] K.H. Fischer. Kosterlitz-thouless transition in layered high-tc superconductors. Physica C:\nSuperconductivity , 210(1):179{187, 1993.\n[54] Alessandro Cuccoli, Valerio Tognetti, and Ruggero Vaia. Two-dimensional XXZ model on a square\nlattice: A monte carlo simulation. Phys. Rev. B , 52:10221{10231, Oct 1995.\n[55] M. E. Gouv^ ea and A. S. T. Pires. The two dimensional classical anisotropic heisenberg\nferromagnetic model with nearest- and next-nearest neighbor interactions. Eur. Phys. J. B ,\n25(2):147{153, 2002.\n[56] Rajan Gupta and Clive F. Baillie. Critical behavior of the two-dimensional XY model. Phys.\nRev. B , 45:2883{2898, Feb 1992.\n[57] G. M. Wysin. Instability of in-plane vortices in two-dimensional easy-plane ferromagnets. Phys.18\nRev. B , 49:8780{8789, Apr 1994."    },    {        "title": "2208.08545v1.Magnetic__electronic__and_structural_investigation_of_the_strongly_correlated_Y___1_x__Sm__x_Co__5__system.pdf",        "content": "1 INTRODUCTION\nMagnetic, electronic, and structural investigation of the\nstrongly correlated Y1\u0000xSmxCo5system\nF . Passos1, D. Cornejo2, M. Fantini2, G. Nilsen3, V . Martelli1J. Larrea Jiménez1?\n1Laboratory for Quantum Matter under Extreme Conditions, Institute of Physics, University of\nSão Paulo, São Paulo, Brazil\n2Institute of Physics, University of São Paulo, São Paulo, Brazil\n3ISIS Pulsed Neutron and Muon Source, STFC Rutherford Appleton Laboratory, Harwell\nCampus, Didcot, UK\n* larrea@if.usp.br\nAugust 19, 2022\nAbstract\nSmCo5and YCo5are isostructural compounds both showing large magnetocrystalline anisotropy,\nwhere the former originates mainly from the crystal-electric field and magnetic interactions.\nWe investigate the contribution of both interactions by partially substituting Y by Sm in\nas-cast polycrystalline Y1\u0000xSmxCo5(with x=0, 0.1, 0.2, 0.3, and 0.4) and measuring their\nstructural, magnetic, and electrical properties through X-ray diffraction, magnetization, and\nelectrical transport measurements. Our results suggest an interplay between microstructure\nstrain in as-cast samples and the electronic and magnetic interactions.\n1 Introduction\nSmCo5alloys are well-known high-performance rare-earth permanent magnets which have been\nintensively studied in the last five decades due to their high magnetization saturation, high Curie\ntemperature, and high coercivity [1]. SmCo5has a hexagonal CaCu5-type crystal structure (space\ngroup P6 /mmm) and its excellent magnetic properties result from the large magnetocrystalline\nanisotropy originated mainly from the Sm localized 4 f-electrons interaction with the hexagonal\ncrystal field (CF) [2,3]. On the other hand, the isostructural compound YCo5also has a consider-\nably high magnetocrystalline anisotropy but it originates from the spin-orbit coupling of the large\norbital angular momentum of the Co 3 d-electrons [3,4].\nY1\u0000xSmxCo5is a promising candidate system to understand the role of the rare-earth localized\nf-electrons and the contribution of the transition metal itinerant d-electrons to the magnetocrys-\ntalline anisotropy and magnetic properties. Moreover, the different atomic ratios of Sm and Y\nallow to investigate the effect of changing the atomic spacing in the packed hexagonal crystal struc-\nture on the physical properties [4]. We have successfully synthesized polycrystalline Y1\u0000xSmxCo5\n(x=0, 0.1, 0.2, 0.3, and 0.4) solid solutions and investigated, for the first time, the influence of\nthe partial substitution of Y by Sm in the structural, magnetic, and electrical properties.\n1arXiv:2208.08545v1  [cond-mat.str-el]  17 Aug 20223 RESULTS AND DISCUSSION\n2 Experimental details\nPolycrystalline Y1\u0000xSmxCo5samples with x=0, 0.1, 0.2, 0.3, and 0.4 were synthesized by arc-\nmelting the stoichiometric amounts of high-purity parent materials under an Argon atmosphere.\nThe X-ray diffraction (XRD) of the samples were performed with the Cu-K\u000bradiation using a\nRigaku Ultima III diffractometer with a monochromator.\nThe magnetization vs magnetic field (M-H) hysteresis of each sample was measured using a\nvibrating-sample magnetometer (VSM) at room temperature with a maximum magnetic field of\n15 kOe. For the electrical transport measurements, the samples were polished to acquire a regular\nshape and their resistances were measured using the standard four-probe method at temperatures\nfrom 10 K to 290 K. The electrical contacts on the samples were made using silver conducting paint\nand 50\u0016m gold wires.\n3 Results and discussion\n3.1 Structural properties\nFigure 1 shows the XRD pattern of the as-cast Y1\u0000xSmxCo5(x=0, 0.1, 0.2, 0.3, and 0.4) samples\nand their respective Rietveld refinements. We can successfully recognized the diffraction peaks as-\nsociated with the CaCu5-type hexagonal crystal structure, besides a small Y2O3phase (see Fig. 1).\nFig. 2 shows the lattice parameters and the relative volume change as function of Sm substitution\nx, obtained by the Rietveld refinement of our XRD patterns.\nFigure 1: XRD patterns (circles) and Rietveld refinements (solid lines) of the polycrys-\ntalline Y1\u0000xSmxCo5samples (with x =0, 0.1, 0.2, 0.3 and 0.4). The Miller indexs corre-\nspond to the main peaks of the YCo5phase and the black dots indicate the peaks from\nthe Y2O3phase.\nWe observe that the lattice parameter c(Fig. 2b) monotonically increases with Sm substitution\nx, except for x<0.1. On the other hand, the lattice parameter a(Fig. 2a) does not show the\n23.2 Magnetic properties 3 RESULTS AND DISCUSSION\nFigure 2: Lattice parameters a and c, and unit cell volume of the Y1\u0000xSmxCo5samples\nfor different Sm doping xobtained from the Rietveld refinement method. Since this\nphase has the hexagonal P6 /mmm space group symmetry, the lattice parameter a and b\nare the same. Dashed lines are guide to eyes.\nlinear variation expected from the Vergard’s law, remaining almost constant up to x\u00190.2, with a\nsubsequent sizable rapid increase around x=0.3. This change in the lattice between x=0.2 and\nx=0.3 becomes noticeable also in the variation of volume as a function of doping (see Fig. 2c),\nindicating a structural deformation.\n3.2 Magnetic properties\nFigure 3 shows the M-H hysteresis loops of Y1\u0000xSmxCo5for each Sm substitution. The hysteresis\nloops suggest a ferromagnetic order for all compositions.\nFigure 3: Bulk magnetization as a function of the magnetic field of the Y1\u0000xSmxCo5\nsamples with different dopings of Sm measured at room temperature. The inset shows\na zoom to better visualize the coercivity of each sample.\nFrom the hysteresis loops we obtain the coercivity and the saturation magnetization for each\n33.3 Electrical transport measurements 3 RESULTS AND DISCUSSION\nsample. Fig. 4 shows the evolution of these properties with Sm substitution.\nFigure 4: (a) Saturation magnetization and (b) coercivity as a function of Sm doping\nin the Y1\u0000xSmxCo5samples. In (a) the dashed line is a linear fit to the data and in (b)\nthe dashed line is a guide to eyes.\nWe note that the saturation magnetization shows a linear decrease with the Sm substitution.\nOn the other hand, we observe that the coercivity first slightly decreases from x=0 to x=0.1, and\nthen starts to increase monotonically with the increasing of Sm substitution, in a very similar way\nto what we observed in the clattice parameter. This similar dependence of the coercivity and the\nclattice parameter suggests magneto-elastic coupling effects [5, 6]and can also be related to an\ninterplay between the change on the hexagonal crystal structure when substituting Y by Sm and\nthe (Y,Sm)Co5large uniaxial magnetocrystalline anisotropy along the c-axis [4], revealing as the\neasy-axis magnetization.\n3.3 Electrical transport measurements\nWe compare the temperature dependence of the electrical resistance normalized to its room tem-\nperature value for all the five samples, as shown in Fig. 5. We observe that for all samples the\nresistance shows a decrease with temperature very similar to the observed in other metals [7].\nWe can separate the contributions to the total resistivity as follows:\n\u001a(T) =\u001a0+\u001aee(T) +\u001amag(T) +\u001aph(T) (1)\nwhere\u001a0is the residual resistivity, \u001aeeis the contribution from the electron-electron interac-\ntion,\u001amagis the magnetic contribution from the electron-spin wave scattering, and \u001aphis the\ncontribution coming from the electron-phonon scattering processes. The dominant scattering\nmechanism varies significantly among materials and with the temperature range. In the case\nof metals at low temperatures, the strongly interacting electron-electron scattering is usually well\ndescribed by the Fermi liquid model which results in a T2dependence of the resistivity. On the\nother hand, for ferromagnets, the magnon modes quadratic dispersion relation !(k) =\u0001+Dk2\ngives rise to a magnetic contribution to the resistivity with the following temperature dependence\n43.3 Electrical transport measurements 3 RESULTS AND DISCUSSION\nFigure 5: Electrical resistances normalized at 290 K for polycrystalline Y1\u0000xSmxCo5.\nforkBT\u001c\u0001[8]:\n\u001amag(T) =b\u0001Te\u0000\u0001=T(1+2T\n\u0001) (2)\nwhere\u0001is the spin-wave gap, Dis the spin-wave stiffness, and b is a constant such that\nb/1=D2. Finally, the electron-phonon interactions usually has a \u001a/T5contribution at lower\ntemperatures, and a \u001a/Tcontribution at higher temperatures [9,10].\nThe attempt to include the T5contribution from the phonons in the fittings at lower tem-\nperatures (below 60 K) did not pay off, which indicates that phonon scattering is not significant.\nIndeed, it has already been suggested that the phonon contribution for RCo5systems in this tem-\nperature range is much smaller compared to the other contributions [9]and, therefore, can be\nneglected. Furthermore, for the YCo5sample, a quadratic dependence of the resistivity up to 60 K\nwas already observed previously [10]. Therefore, we perform a fitting of the electrical resistivity\nbelow 50 K for each sample including only the Fermi liquid and magnetic contributions using the\nfollowing equation:\n\u001a(T) =\u001a0+AT2+b\u0001T(1+2T\n\u0001)e\u0000\u0001=T(3)\nwhere Ais the quadratic parameter from the electron-electron scattering. Fig. 6 shows that\nthe fittings of eq. 3 have a good agreement with the experimental data up to 50 K and Fig. 7\nshows the values of all parameters obtained from the fittings of the resistivity data as a function\nof the Sm substitution.\nWe can note in Fig. 7a that the residual-resistance ratio (RRR) initially decreases almost lin-\nearly until x=0.2 followed by an increase at x=0.3. Besides that, the Aand bparameters (Figs.\n7b and c) show an overall tendency of decrease except at x=0.3 where all of them present an\nabrupt increase. This abrupt change agrees with the in-plane lattice parameters (see Fig. 2a) and\nunit cell volume (see Fig. 2c) changes at x=0.3, which suggests strain effects on the electronic\n54 CONCLUSION\nFigure 6: Temperature dependence of the difference between the electrical resistivity\nand the residual electrical resistivity ( \u001a0) for our polycrystalline samples Y1\u0000xSmxCo5.\nSolid lines are fittings according to eq. 3.\nand magnetic scattering mechanisms [6]. The decrease of the bparameter (Fig. 7c) means an in-\ncrease of the spin-wave stiffness D(b/1=D2) and, therefore, a hardening of the magnon modes\nwith increasing Sm substitution. Finally, in Fig. 7d we observe that the magnon gap has a maxi-\nmum around x=0.1 and then starts to decrease monotonically, which might indicate a magnetic\nanisotropy loss.\n4 Conclusion\nPolycrystalline Y1\u0000xSmxCo5as-cast samples were successfully synthesized, where the physical\nproperties were not dramatically affected by the presence of a small spurious yttrium /samarium\noxide phase. Our results indicate an abrupt change in the lattice parameter aand in the unit cell\nvolume (Fig. 2) at the same Sm substitution of x=0.3, which influences the electric and magnetic\nscattering process. In addition, similar dependence of lattice parameter cand the coercivity with\nthe Sm substitution suggests the important role that the f-electron orbitals plays on the magne-\ntocrystalline anisotropy and crystal electrical field, both underlying mechanisms to understand the\nmagnetic ground state in RCo5magnets. Finally, the electrical transport measurements indicated\nthe presence of both electron-spin and electron-electron interactions and an apparent hardening\nof the magnon modes with Sm substitution.\nOur results trigger further investigation at higher substitutions of Sm in order to understand\nthe role that strong correlation between itinerant and localized electrons plays on setting the\nmagnetic ground state in hard magnets.\n6REFERENCES REFERENCES\nFigure 7: Parameters obtained fitting the experimental with equation 3 for each\nY1\u0000xSmxCo5sample. (a) Residual-resistance ratio (RRR) obtained from \u001a0using\nRRR=\u001a(290 K) /\u001a0. (b) Quadratic parameter from the Fermi liquid contribution. (c)\nMagnetic contribution parameter where b/1=D2. (d) Spin-wave gap from the mag-\nnetic contribution.\nAcknowledgements\nFunding information FP acknowledges FAPESP grant number 2019 /24711-0. VM acknowl-\nedges the support of JP-FAPESP (2018 /19420-3). JLJ acknowledges support of JP-FAPESP (2018 /08845-\n3) and CNPq-PQ2 fellowship (310065 /2021-6).\nReferences\n[1]K. Strnat, G. Hoffer, J. Olson and W . Ostertag, A family of new cobalt-base permanent magnets\nmaterials , Journal of Applied Physics 38, 1001 (1967), doi:10.1063 /1.1709459.\n[2]P . Larson, I. I. Mazin and D. A. Papaconstantopoulos, Calculation of magnetic anisotropy\nenergy in SmCo5 , Phys. Rev. B 67, 214405 (2003), doi:10.1103 /PhysRevB.67.214405.\n[3]P . Larson and I. I. Mazin, Magnetic properties of SmCo5 and YCo5 , Journal of Applied Physics\n93, 6888 (2003), doi:10.1063 /1.1556154.\n7REFERENCES REFERENCES\n[4]L. Steinbeck, M. Richter and H. Eschrig, Itinerant-electron magnetocrystalline\nanisotropy energy of YCo5 and related compounds , Phys. Rev. B 63, 184431 (2001),\ndoi:10.1103 /PhysRevB.63.184431.\n[5]H. Rosner, D. Koudela, U. Schwarz, A. Handstein, M. Hanfland, I. Opahle, K. Koepernik,\nM. Kuz’min, K.-H. Müller, J. Mydosh et al. ,Magneto-elastic lattice collapse in YCo5 , Nature\nPhysics 2(7), 469 (2006), doi:10.1038 /nphys341.\n[6]R. L. Stillwell, J. R. Jeffries, S. K. McCall, J. R. I. Lee, S. T . Weir and Y. K. Vohra, Strongly\ncoupled electronic, magnetic, and lattice degrees of freedom in LaCo5 under pressure , Phys. Rev.\nB92, 174421 (2015), doi:10.1103 /PhysRevB.92.174421.\n[7]R. E. Peierls, Quantum Theory of Solids , Oxford: Oxford University Press,\ndoi:10.1093 /acprof:oso /9780198507819.001.0001 (2001).\n[8]J. Larrea J., M. B. Fontes, A. D. Alvarenga, E. M. Baggio-Saitovitch, T . Burghardt, A. Eichler\nand M. A. Continentino, Quantum critical behavior in a CePt ferromagnetic kondo lattice ,\nPhys. Rev. B 72, 035129 (2005), doi:10.1103 /PhysRevB.72.035129.\n[9]I. Mannari, Electrical resistance of ferromagnetic metals , Progress of Theoretical Physics 22,\n335 (1959), doi:10.1143 /PTP .22.335.\n[10]A. Kowalczyk, Transport properties of hexagonal YCo5 and YCo3B2 compounds , phys.\nstat. sol. (b) 218, 495 (2000), doi:10.1002 /1521-3951(200004)218:2 <495::AID-\nPSSB495 >3.0.CO;2-2.\n8"    },    {        "title": "2208.09210v1.Strain_Driven_Zero_Field_Near_10_nm_Skyrmions_in_Two_Dimensional_van_der_Waals_Heterostructures.pdf",        "content": "Strain-Driven Zero-Field Near-10 nm Skyrmions in Two-Dimensional van der Waals\nHeterostructures\nDongzhe Li,1,\u0003Soumyajyoti Haldar,2and Stefan Heinze2\n1CEMES, Université de Toulouse, CNRS, 29 rue Jeanne Marvig, F-31055 Toulouse, France\n2Institute of Theoretical Physics and Astrophysics, University of Kiel, Leibnizstrasse 15, 24098 Kiel, Germany\nMagnetic skyrmions – localized chiral spin structures – show great promise for spintronic applications. The\nrecent discovery of two-dimensional (2D) magnetic materials opened new opportunities for exploring such topo-\nlogical spin structures in atomically thin van der Waals (vdW) materials. Despite recent progress in stabilizing\nmetastable skyrmions in 2D magnets, their diameters are still beyond 100 nm and their lifetime, which is es-\nsential for applications, has not been explored yet. Here, using first-principles calculations and atomistic spin\nsimulations, we predict that compressive mechanical strain leads to stabilizing zero-field skyrmions with di-\nameters close to 10 nm in a Fe 3GeTe 2/germanene vdW heterostructure. The origin of these unique skyrmions\nis attributed to the high tunability of Dzyaloshinskii-Moriya interaction and magnetocrystalline anisotropy en-\nergy by strain, an effect which is shown to be general for Fe 3GeTe 2heterostructures with buckled substrates.\nBased on our first-principles parameters for the magnetic interactions, we calculate the energy barriers protect-\ning skyrmions against annihilation and their lifetimes using transition-state theory. We show that nanoscale\nskyrmions in strained Fe 3GeTe 2/germanene can be stable for hours at temperatures up to 20 K.\nKeywords: Spintronics, Magnetic Skyrmions, Strain, Dzyaloshinskii-Moriya interaction, van der Waals Heterostructures\nMagnetic skyrmions [1, 2] have attracted tremendous atten-\ntion due to their intriguing topological properties and promis-\ning applications for next-generation spintronics devices [3–8].\nThe Dzyaloshinskii-Moriya interaction (DMI), which prefers\na canting of the spins of adjacent magnetic atoms, is often\nrecognized as the key ingredient in forming such localized\nnoncollinear magnetic structures. The DMI originates from\nspin-orbit coupling (SOC) and relies on broken inversion sym-\nmetry. During the last decade, in order to obtain large inter-\nfacial DMI, first observed in ultrathin transition-metal films\n[9–11], a comprehensive effort has been denoted to develop\nferromagnetic/heavy metal (FM/HM) multilayers with strong\nSOC [12–15].\nIn 2017, the discovery of truly two-dimensional (2D) mag-\nnetic materials [16–18] opened up new opportunities for ex-\nploring novel magnetic phenomena in reduced dimensions.\nIn particular, two independent experimental groups reported\nthe first observation of skyrmions in the van der Waals (vdW)\nmagnets Cr 2Ge2Te6[19] and Fe 3GeTe 2[20]. After that, sev-\neral experimental groups observed stable skyrmion lattices in\n2D magnets vdW heterostructures such as FGT/WTe 2[21],\nFGT/Co/Pd multilayer [22], and FGT/Cr 2Ge2Te6[23]. These\nexciting measurements have broadened magnetic skyrmion\ninterface platforms from conventional transition-metal films\nand multilayers to vdW materials consisting of weakly bonded\n2D layers. Stabilizing skyrmions in 2D magnets has several\npotential advantages. These include avoiding pinning by de-\nfects due to high-quality vdW interfaces, the possibility of\nforming skyrmions with a minimum thickness (i.e., a single\nlayer limit), and easy control of magnetism via external stim-\nuli such as strain, electric field, light, or magnetic field. Up to\nnow, the reported skyrmion size in 2D vdW heterostructures is\nabove 100 nm [19–21] at least one order of magnitude larger\nthan the desired diameter ( <10 nm) required for memory ap-\nplications [3].\nFrom the theoretical point of view, during the lastthree years, atomistic spin models parametrized from first-\nprinciples calculations based on density functional theory\n(DFT) have been extensively used to explore the DMI in 2D\nmagnets. Several strategies have been proposed to achieve\nDMI in these materials by breaking inversion symmetry. Par-\nticularly, the family of Janus vdW magnets, which lacks in-\nversion asymmetry with different atoms occupying top and\nbottom layers, has been predicted to possess large enough\nDMI to generate skyrmions [24–31]. Additionally, it has been\nshown that when an electric field is applied perpendicular to\nthe CrI 3monolayer, the DMI emerges due to the breaking\nof inversion symmetry [32]. Another focus was designing\nfull vdW heterostructures consisting of magnetic and non-\nmagnetic vdW layers, tuning the DMI through the proximity\neffect [33, 34]. Moreover, metal FM/2D materials interfaces\nare predicted to possess significant DMI with strong magne-\ntocrystalline anisotropy energy (MAE) [35, 36]. The Néel-\ntype magnetic skyrmions observed in Fe 3GeTe 2crystals were\nattributed to the DMI due to the oxidized interfaces [37]. Fer-\nroelectrically controllable skyrmions [38, 39] have also been\nrecently proposed. However, previous studies were only fo-\ncused on the formation of isolated skyrmions and a quantifica-\ntion of individual skyrmion stability and lifetime in 2D mag-\nnets, crucial for all device applications, is so far missing in the\nliterature.\nIn this Letter, using first-principles calculations and atom-\nistic spin simulations, we demonstrate that a mechanical strain\ncan significantly enhance skyrmion stability and lifetime in\n2D vdW heterostructures, leading to zero-field near-10 nm\nskyrmions . We focus on Fe 3GeTe 2(FGT) vdW heterostruc-\ntures of current experimental interest with a high Curie tem-\nperature of about 230 K, which can be increased up to room\ntemperature by patterning [40] and electron doping [18]. The\norigin of these nanoscale skyrmions at zero magnetic fields\nis attributed to the highly tunable DMI and MAE by strain.\nWe show that the DMI is significantly enhanced by more thanarXiv:2208.09210v1  [cond-mat.mtrl-sci]  19 Aug 20222\n400% when applying a small compressive strain while the\nMAE is dramatically reduced to 25% of its original value.\nThe strained FGT heterostructure also displays a consider-\nable exchange frustration which can increase skyrmion sta-\nbility [41]. We show that the efficient strain-driven DMI and\nMAE control is general for FGT heterostructures with buck-\nled substrates. Finally, we calculate energy barriers between\nthe metastable skyrmion and the ferromagnetic ground state\nand quantify the stability of skyrmions by calculating their\nlifetimes as a function of magnetic field and temperature. Our\nwork demonstrates the possibility to stabilize skyrmions with\ndiameters on the order of 10 nm down to zero magnetic field\nin 2D vdW heterostructures.\nWe performed density functional theory (DFT) calculations\nusing two DFT codes which differ in their choice of basis\nset: The F LEUR code [42] is based on the full-potential lin-\nearized augmented plane wave (FLAPW) formalism while the\nQUANTUM ATK package [43] uses an expansion of electronic\nstates in a linear combination of atomic orbitals (LCAO). The\nformer ranks among the most accurate implementations of\nDFT, while the latter is computationally much less demand-\ning, therefore, useful for large-scale systematic or explorative\nstudies. For the DMI calculation, two different approaches are\nused: The generalized Bloch theorem (gBT) [44–46] and the\nsupercell approach [47], which are denoted in the following\nas FLAPW-gBT and LCAO-supercell. Computational details\nare given in Supporting Information (SI) Section I.\nThe total energies for different collinear and non-collinear\nspin structures obtained via DFT are used to parameterize the\nextended Heisenberg model which is given by:\nH=\u0000X\nijJij(mi\u0001mj)\u0000X\nijDij\u0001(mi\u0002mj)\n\u0000X\niKi(mz\ni)2\u0000X\niM(mi\u0001Bext)(1)\nwhere miandmjare normalized magnetic moments at po-\nsition RiandRirespectively. Mdenotes the size of the\nmagnetic moment at every site. The four magnetic interaction\nterms correspond to the Heisenberg isotropic exchange, the\nDMI, the magnetic anisotropy energy (MAE), and the exter-\nnal magnetic field, and they are characterized by the interac-\ntion constants Jij,Dij, andKi, andBext, respectively. Note\nthat our spin model is adapted to a collective 2D model by\ntreating three Fe layers of the FGT as a whole system, similar\nto a monolayer system. All magnetic interaction parameters\nare measured in meV/unit cell (uc).\nWe consider a vdW heterostructure where an FGT mono-\nlayer is deposited on germanene (denotes as FGT/Ge in the\nfollowing). As shown in Fig. 1(a), the FGT adopts the space\ngroups (194) P6 3/mmc and can be seen as a stack of three\nFe hexagonal lattices with an hcp stacking. In the following,\nthe top, center, and bottom (interface) atoms are denoted as\nFe3, Fe2, and Fe1, respectively. For the atomic relaxation, we\nemployed the generalized gradient approximation (GGA), ob-\ntaining a relaxed lattice constant of 4.00 Å for the FGT mono-\nFIG. 1. (a) Top view of the atomic lattice of the Fe 3GeTe 2mono-\nlayer. The dashed lines denote the 2D unit cell. (b) Side view of the\nFe3GeTe 2monolayer on germanene. (x;y;z )and (b1, b2) are the\nCartesian coordinates and crystallographic directions, respectively.\nDependence of (c) structural deformation \u0001z1,\u0001z2and (d) vdW\ngap,\u0001zgap, on biaxial strain for the FGT/Ge heterostructure.\nlayer, which is in good agreement with experimental data of\nabout 3.991\u00184.03 Å[48, 49]. Then, germanene is matched\nat the interface with the FGT with a lattice mismatch smaller\nthan 1%. In order to better describe the vdW gap ( \u0001zgap), we\ntook into account vdW interactions using semi-empirical dis-\npersion corrections as formulated by Grimme [50]. For mag-\nnetic exchange calculations, we used the local density approx-\nimation (LDA) without Hubbard Ucorrection since it yields\na magnetic moment of 1.76 \u0016B/Fe that compares well with\nexperiments, as also pointed out in Ref. [18, 51]. The lowest-\nenergy stacking configuration is the one where the Te atom\nis right above the center of the hexagonal ring of germanene\nwith an optimized vdW gap of about 2.86 Å (see Fig. 1b),\nwhich agree well with previous results [52].\nFor 2D materials, the lattice, electronic and magnetic prop-\nerties can be easily modified through interfacing with various\nsubstrates or fabricating stretchable heterostructures [53, 54].\nThis indicates the possibility of tuning magnetic interactions\nand even inducing distinct spin textures in FGT heterostruc-\ntures through strain engineering. We model the strain by ap-\nplying the in-plane biaxial strain along the lattice vectors, and\nwe fully relax structures for each strained lattice. The biaxial\nstrain is defined as \r= (a\u0000a0)=a0, whereaanda0are the\nstrained and unstrained lattice constants of FGT heterostruc-\ntures, respectively. Both compressive and tensile strains are\nconsidered ranging from \u00003% to 6.25%. It has been demon-\nstrated by ab initio phonon spectrum calculations that FGT is\nstable under such strains [55]. We emphasize that the value\nofa0= 4:0Å used in this work was evaluated in GGA,\nslightly higher than the lattice constant calculated in LDA,\na0= 3:91Å. If we use the latter as a reference, the largest\ncompressive strain used in this work becomes less than \u00001%.3\nTABLE I. Shell-resolved Heisenberg exchange constants ( Jn), Dzyaloshinskii-Moriya interaction constants ( Dn), and magnetocrystalline\nanisotropy energy ( K) obtained via DFT for a collective 2D spin model (i.e., three Fe atoms are treated as a whole) of FGT/Ge at \r= 0% and\n\r=\u00003%. A positive (negative) sign of Dndenotes a preference of CW (CCW) rotating cycloidal spin spirals. A positive (negative) sign of\nKindicates an easy in-plane (out-of-plane) magnetization direction.\nJ1J2J3J4J5J6J7J8D1D2D3D4D5D6D7K\n\r= 0% 22.87\u00000:21\u00001:780.43 1.25\u00000:310.00\u00000:150.23 0.09 0.00 0.00 0.00 0.00 0.00 \u00003:90\n\r=\u00003% 21.51\u00002:890.82 0.40\u00001:41\u00000:880.89\u00000:440.97 0.00\u00000:02\u00000:060.04 0.02 0.02\u00000:86\nFIG. 2. Strain-dependent energy dispersion of spin spirals for\nFGT/Ge obtained by FLAPW-gBT. (a) Energy dispersion ESS(q)\nwith respect to the FM state, EFM, of planar homogeneous spin\nspirals for FGT/Ge along the high symmetry path M-\u0000-K-M with-\nout SOC. The symbols (black circles for \r= 0% , red circles for\n\r=\u00003%) represent the DFT calculations in scalar-relativistic ap-\nproximation, while the solid lines are the fits to the Heisenberg ex-\nchange interaction. (b) The same as (a) but for the energy contri-\nbutionEDMI(q)of cycloidal spin spirals due to SOC. Note that posi-\ntive and negative energies represent a preference of counterclockwise\n(CCW) and clockwise (CW) spin configurations. (c) Zoom around\nthe FM state ( \u0000point), including the Heisenberg exchange, the DMI,\nand the MAE, i.e. E(q) =ESS(q) +EDMI(q) +K=2. The DMI\nleads to the local energy minima for CW rotating spin spirals, and\nthe MAE is responsible for the constant energy shift ( K=2) of the\nspin spirals with respect to the FM state.A remarkable structural modification is observed in Fig. 1c\nwhich shows the strain dependence of vertical, i.e., zcompo-\nnent, distances between Fe atoms of the different layers: Fe1-\nFe2,\u0001z1, and Fe2-Fe3, \u0001z2(see sketches in Figs. 1b,c). As\na negative strain is applied, \u0001z1increases rapidly, while \u0001z2,\nin contrast, decreases. Such a large geometrical change is due\nto the buckled structure of germanene, which introduces siz-\nable atomic forces under negative strains. We find the acting\nforce on the Ge atom (middle layer of FGT) is negative (i.e.,\ntowards to germanene layer) while the one on Fe2 is positive.\nThis leads to an increase in structural asymmetry, which might\nbe important for DMI modifications. In addition, the vdW gap\nin FGT/Ge decreases for both compressive and tensile strains\n(Fig. 1d), leading to an increase in hybridization strength at\nthe interface.\nNow we turn to the influence of strain on the ground state\nand magnetic interactions in FGT/Ge. Fig. 2 shows the calcu-\nlated energy dispersion E(q)of flat homogeneous spin spirals\nper unit cell using FLAPW-gBT. We first focus on the results\nin the scalar-relativistic approximation, i.e. without SOC (see\nFig. 2a). The energy dispersions are calculated along the high\nsymmetry directions \u0000M and \u0000KM of the 2D hexagonal Bril-\nlouin zone (BZ). The \u0000point corresponds to the FM state, the\nM point to the row-wise AFM state, and the K point to the\nNéel state with angles of 120\u000ebetween adjacent spins.\nBoth the FGT/Ge interfaces at zero strain ( \r= 0% ) and\nat a compressive strain ( \r=\u00003%) possess a minimum of\nE(q)at the \u0000point, i.e., the FM state, and they exhibit a large\nFM nearest-neighbor exchange constant, as clearly seen from\nthe large energy difference between the \u0000andM point (AFM\nstate). We note that the strained FGT/Ge heterostructure ( \r=\n\u00003%) displays a stronger exchange frustration which can be\nseen in Table I from the increased antiferromagnetic exchange\nconstants of beyond nearest neighbor (NN) shells. Therefore,\nthe NN approximation fails at large qfor strained FGT/Ge, as\ndemonstrated in Fig. S1 in SI.\nUpon including SOC, the DMI arises in systems with bro-\nken inversion symmetry. For free-standing FGT, the DMI in-\nvolving either Fe1 or Fe3 have opposite signs (i.e., chirality)\nbecause of the (001) mirror plane [56]. Upon incorporating\ngermanene in FGT, the inversion symmetry breaking at the\nFGT/Ge interface gives rise to an emergent DMI. The DMI\nfavors cycloidal spiraling magnetic structures with a unique\nrotational sense. For FGT/Ge a clockwise rotational sense is\nfavored as seen from the calculated negative energy contribu-\ntion due to SOC, EDMI(q), to the dispersion of cycloidal spin4\nFIG. 3. (a) Sketch of the structure of a monolayer FGT deposited on buckled 2D vdW substrates and calculated in-plane component of DMI\nas functions of biaxial strain \r. (b) The same as in (a) but for the free-standing FGT with structural deformations as in FGT/Ge. (c) The same\nas in (a) but for FGT/graphene heterostructure.\nspirals (Fig. 2b). In the vicinity of the \u0000point an isotropic\nDMI, expected for a system with C3vsymmetry, is apparent\nfrom the same slope in \u0000M and \u0000K directions.\nThe DMI constants obtained from a fit of the spin model,\nEq. (1), to the DFT energies (Table I) are dominated by the NN\nterm,D1, and the values rapidly decrease with distance. This\nindicates that the NN approximation fits very well in FGT het-\nerostructures. According to Moriya’s rule [57], the Dijcan\nbe expressed as Dij=d= =(^uij\u0002^z) +d?^z, where ^uijbeing\nthe unit vector between sites iandjand^zindicating normal\nto the plane. It has been shown that the out-of-plane com-\nponent of Dijplays a negligible role in forming skyrmions\nin 2D magnets [24, 31]. Therefore, we focus on its in-plane\ncomponent DMI, d= =. Thed= =(also known as microscopic DMI\ncoefficient) can be approximated by D1fitted to an effective\nNN DMI model presented in Fig. S1 in SI. The micromagnetic\nDMI coefficient Dcan be calculated using\nD=3p\n2d= =\nNFa2(2)\nwhereaandNFare the lattice constant and the number of\nferromagnetic layers, respectively.\nIn the equilibrium case ( \r= 0% ), we find a preference\nfor clockwise rotating spin spirals with d= == 0:25meV . The\ncorresponding micromagnetic DMI coefficient, is about jDj=\n0.36 mJ/m2. This value is comparable to the other two FGT\nheterostructures reported recently as promising interfaces to\nhost stable skyrmion states: FGT/In 2Se3[58] (\u00180.28 mJ/m2)\nand FGT/Cr 2Ge2Te6[23] (\u00180.31 mJ/m2). In the strained het-\nerostructure ( \r=\u00003%) the DMI increases by a factor of\nabout 4 to a value of d= == 1meV . Such a large DMI is compa-\nrable to state-of-the-art FM/HM interfaces such as Co/Pt (1.47\nmeV) [46], Pd/Fe/Ir(111) (1.0 meV) [59], and Fe/Ir(111) (1.7\nmeV) [11]. Its corresponding Dis 1.52 mJ/m2, which is morethan one order of magnitude higher than the critical DMI value\nof 0.1 mJ/m2[21] necessary for stabilizing skyrmions in FGT\nheterostructures. This indicates a possibility of strain control\nof magnetic skyrmions in these systems. In contrast, a tensile\nstrain (\r >0) has almost no effect on the DMI.\nThe inclusion of SOC also induces MAE. The easy magne-\ntization axis of FGT/Ge is out-of-plane and the values of the\nMAE areK=\u00003:90and\u00000:86meV/unit cell at \r= 0% and\n\r=\u00003%, respectively, i.e. the compressive strain decreases\nthe MAE to about 25% of its equilibrium value. The energy\ncontribution from MAE leads to an energy offset of K=2for\nspin spirals with respect to the FM state as can be seen in a\nzoom ofE(q)around \u0000for cycloidal spin spirals (Fig. 2c).\nThe FM state at zero energy is the magnetic ground state for\nunstrained as well as for strained FGT/Ge (Fig. 2c). However,\nthe effect of all interactions, i.e., the Heisenberg exchange,\nthe DMI, and MAE, upon applying a compressive strain is\nsignificant. The large reduction of the MAE shifts the energy\ndispersion close to the FM state. In the regime of small jqj,\nthe dispersion for \r= 0% shows an energy rise with q2while\nE(q)becomes extremely flat at \r=\u00003%. This indicates\nconsiderable exchange frustration in strained FGT/Ge which\nopens the possibility of metastable nanoscale skyrmions at\nzero magnetic field as reported in transition-metal ultra-thin\nfilms [60].\nTo demonstrate that the proposed strain tuning of DMI\nand MAE is quite general, we performed systematic studies\nfor two other FGT heterostructures with buckled substrate:\nFe3GeTe 2/silicene (FGT/Si) and Fe 3GeTe 2/antimonene\n(FGT/Sb). These calculations have been done using the\nLCAO-supercell approach since it is computationally less\ndemanding than FLAPW-gBT. The LCAO-supercell is in\nexcellent agreement with FLAPW-gBT for the DMI in\nFGT/Ge (see Fig. S2 in SI). Fig. 3a shows the variation of5\nFIG. 4. Skyrmion radius and energy barrier for skyrmion collapse\nin strained FGT/Ge ( \r=\u00003%) evaluated using magnetic inter-\naction parameters from DFT. (a) Skyrmion radius as a function of\nthe applied magnetic field. The skyrmion profile and spin texture at\nB= 0T andK=2are shown as insets. (b) Energy barriers of isolated\nskyrmions versus magnetic field strength. Calculated attempt fre-\nquenciesf0are shown on a logarithmic scale in the inset. Both radii\nand energy barriers are evaluated at different MAE values, namely K\n(green),K/2 (yellow),K/5 (purple),K/10 (red), and K/100 (blue),\nwhereK=\u00000:86meV is the MAE estimated by DFT.\nDMI under applied strain for three FGT interfaces. Clearly\nthe significant enhancement of DMI for a small compressive\nstrain is a robust effect which appears for all three interfaces\nwith buckled vdW substrates. Moreover, the amplitude of\nDMI for FGT/Sb is much stronger than for the other two\ninterfaces and exhibits a value of about 0.65 meV , even in the\nunstrained condition. We attribute this large increase to the\nmuch larger SOC constant of Sb which scales approximately\nasZ2, whereZdenotes the nuclear charge.\nWhen the FGT monolayer is deposited on a substrate, two\nsources modify its magnetic properties: (i) the structural de-formation of the FGT and (ii) the coupling of FGT states to\nthose of the substrate. To remove artificially the latter, we\nhave calculated the DMI for free-standing FGT but distorted\nas after deposition on germanene (Fig. 3b). We recover a sim-\nilar curve as in the case of FGT/Ge (Fig. 3a), confirming that\nthe enhancement of DMI is mainly attributed to the structural\ndeformation and not to interface hybridization. In contrast,\nwhen a flat vdW material such as graphene is used as the\nsubstrate, the DMI in FGT is almost constant with respect\nto strain (Fig. 3c). Note that due to the weak SOC effect in\ngraphene, we find a tiny DMI of about \u00000:04meV .\nSince the flat spin spiral energy dispersion with SOC in\nFGT/Ge under strain (Fig. 2c) indicates the possibility of sta-\nbilizing small magnetic skyrmions down to zero magnetic\nfield we performed atomistic spin simulations using the spin\nmodel described by Eq. 1 with the full set of DFT parame-\nters. Isolated skyrmions were obtained using spin dynamics.\nMinimum energy paths for the skyrmion collapse processes\ninto the ferromagnetic ground state were identified using the\ngeodesic nudged elastic band (GNEB) method [61] and the\ncollapse barriers for skyrmions were obtained from the sad-\ndle point along the path. Finally, we used harmonic transition\nstate theory (HTST) to quantify skyrmion stability by calcu-\nlating their lifetime [62–65] (for computational details see SI).\nIn our simulations, we create isolated skyrmions in the FM\nbackground based on the theoretical profile given in Ref. [2]\nand fully relax these spin structures by solving the damped\nLandau-Lifschitz equation self-consistently. We find no mag-\nnetic skyrmion emerging in FGT/Ge without applying strain\n(\r= 0% ). This is due to a very large out-of-plane MAE ( K)\nwith a moderate DMI amplitude. Interestingly, when a small\nstrain is applied ( \r=\u00003%), our spin dynamics simulations\npredict Néel-type skyrmions with a diameter (= 2 \u0002radius)\nof about 4 nm stabilized in the FM background at zero mag-\nnetic field (Fig. 4a). Since the MAE of FGT can be greatly\nsuppressed by doping or temperature [66–68], we present in\nFig. 4a the skyrmion radius with respect to the applied mag-\nnetic field under different values of the MAE. For skyrmion\nprofiles at different MAE values, please refer to Fig. S3.\nRemarkably, nanoscale skyrmions with radii ranging from\n2 nm to 11 nm are observed at small magnetic fields of less\nthan 0.5 T and the radii rapidly decrease with applied mag-\nnetic field as expected. In particular, at KandK=2, small\nskyrmions with radii of about 2 nm and 7 nm are obtained at\nzero magnetic field ( B= 0T, see inset of Fig. 4a for skyrmion\nprofile atK=2and Fig. S3 for profile at KandK=10). When\na small magnetic field is applied ( B < 0:5T), we obtain a\nsmall size skyrmion within sub-10 nm. Such small skyrmions\nare technologically desired for future skyrmionic devices. In\nstrained FGT/Ge, they originate from the extremely flat spin\nspiral curve close to the FM state (Fig. 2c). The interplay of\nexchange and DMI reduces the energy cost of a fast spin ro-\ntation while large MAE enforces the fast spin rotation. Note\nthat all previously reported magnetic skyrmions in 2D mag-\nnets were stabilized under finite magnetic fields.\nTo get insight into the stability of nanoscale skyrmions in6\nFIG. 5. The lifetime of skyrmions, \u001c, in strained FGT/Ge obtained in harmonic transition state theory based on the spin model with DFT\nparameters at a MAE of (a) K=2, (b)K=5, (c)K=10, and (d)K=100as a function of magnetic field and temperature.\nFGT/Ge, we have calculated the minimum energy path for\na transition between the metastable skyrmion and the FM\nground state using the GNEB method [61]. Surprisingly, for\na MAE ofK, although the 2 nm radius skyrmion can be sta-\nbilized at zero-field in spin dynamics simulations, its energy\nbarrier is almost zero (Fig. 4b), i.e. it is actually unstable. This\nhighlights the importance of going beyond conventional spin\ndynamics to establish stability of topological spin structures.\nFor a MAE of K=2, we already obtain a significant energy\nbarrier of around 40 meV protecting skyrmions from collapse\natB= 0 T (Fig. 4b). Upon further decreasing the MAE\nthe energy barrier is further enhanced. The skyrmion is an-\nnihilated via the radial symmetric mechanism in which the\nskyrmion shrinks symmetrically to the saddle point (SP) and\nthen collapses into the FM state [65]. As expected the DMI\nprotects the skyrmion state while the MAE prefers the FM\nstate and decreases the total barrier (Fig. S4 in SI). This im-\nmediately explains that the energy barrier increases upon de-\ncreasing the MAE. A second effect is an increase of the DMI\ncontribution due to the increased skyrmion radius at reduced\nMAE. The exchange frustration is not as large as in ultrathin\ntransition-metal film systems [41, 60], but it provides only a\nsmall negative contribution to the barrier (see Fig. S4 in SI).\nThe stability of skyrmions can be quantified by their\nlifetime,\u001c, which is given by the Arrhenius law \u001c=\nf\u00001\n0exp\u0010\n\u0001E\nkBT\u0011\n, where \u0001E,f0, andTare energy barrier, at-\ntempt frequency, and temperature, respectively. f0, obtained\nwithin harmonic transition state theory [62], depends strongly\non magnetic field and on the value of the MAE (inset of\nFig. 4b). This effect is similar to that observed in ultrathin\ntransition-metal films which can be traced back to a change of\nentropy with skyrmion radius and profile [64, 69].\nFrom the temperature and field dependence of the skyrmion\nlifetime (Fig. 5) we predict that isolated skyrmions in strained\nFGT/Ge are stable up to hours at a temperature ranging from\n12\u001828 K down to zero field depending on the MAE. There-\nfore, these skyrmions can be probed, e.g., by scanning tunnel-\ning microscopy or Lorentz transmission electron microscopy\nexperiments.\nTo summarize, we predict the formation of nanoscale –close to 10 nm diameter – skyrmions in 2D vdW heterostruc-\ntures down to zero magnetic field induced by mechanical\nstrain. In particular, our DFT calculations demonstrate that\na small compressive strain can enhance the DMI up to 400%\nwhile decreasing the MAE dramatically down to 25% of its\nvalue in Fe 3GeTe 2/Ge. Notably, such highly tunable mag-\nnetic interactions turn out to be general for FGT heterostruc-\ntures with buckled substrates (e.g., silicene, antimonene). Us-\ning atomistic spin simulations, we further explicitly calculate\nskyrmion energy barriers as well as lifetimes. We demonstrate\nthat near-10 nm are stable down to zero magnetic field with\nlifetimes of hours up to 20 K. Our findings thus show a route\nto realize sub-10 nm skyrmions in 2D vdW heterostructures at\nzero magnetic field.\nAcknowledgement\nWe thank Gustav Bihlmayer for valuable discussions.\nThis study has been (partially) supported through the grant\nNanoX n° ANR-17-EURE-0009 in the framework of the \"Pro-\ngramme des Investissements d’Avenir\". We gratefully ac-\nknowledge financial support from the Deutsche Forschungs-\ngemeinschaft (DFG, German Research Foundation) through\nSPP2137 \"Skyrmionics\" (project no. 462602351). Numeri-\ncal calculations were performed using HPC resources from\nCALMIP (Grant 2022-[P21023]).\nThe Supporting Information is available free of charge\nathttps://pubs.acs.org/ . Computational details\non DFT and atomistic spin simulations, effective nearest-\nneighbor model, comparison between the LCAO-supercell\nand FLAPW-gBT approaches, and minimum energy paths of\nskyrmion collapse with different energy decomposition.7\n\u0003dongzhe.li@cemes.fr\n[1] A. N. Bogdanov and D. Yablonskii, Zh. Eksp. Teor. Fiz 95, 178\n(1989).\n[2] A. N. Bogdanov and A. Hubert, Phys. Status Solidi 186(1994).\n[3] A. Fert, N. Reyren, and V . Cros, Nat. Rev. Mater. 2, 1 (2017).\n[4] K. Everschor-Sitte, J. Masell, R. M. Reeve, and M. Kläui, J.\nAppl. Phys. 124, 240901 (2018).\n[5] S. Luo, M. Song, X. Li, Y . Zhang, J. Hong, X. Yang, X. Zou,\nN. Xu, and L. You, Nano Lett. 18, 1180 (2018).\n[6] B. Göbel, I. Mertig, and O. A. Tretiakov, Phys. Rep. 895, 1\n(2021).\n[7] S. Li, W. Kang, X. Zhang, T. Nie, Y . Zhou, K. L. Wang, and\nW. Zhao, Mater. Horiz. 8, 854 (2021).\n[8] C. Psaroudaki and C. Panagopoulos, Phys. Rev. Lett. 127,\n067201 (2021).\n[9] M. Bode, M. Heide, K. V on Bergmann, P. Ferriani, S. Heinze,\nG. Bihlmayer, A. Kubetzka, O. Pietzsch, S. Blügel, and\nR. Wiesendanger, Nature 447, 190 (2007).\n[10] P. Ferriani, K. von Bergmann, E. Y . Vedmedenko, S. Heinze,\nM. Bode, M. Heide, G. Bihlmayer, S. Blügel, and R. Wiesen-\ndanger, Phys. Rev. Lett. 101, 027201 (2008).\n[11] S. Heinze, K. von Bergmann, M. Menzel, J. Brede, A. Kubet-\nzka, R. Wiesendanger, G. Bihlmayer, and S. Blügel, Nat. Phys.\n7, 713 (2011).\n[12] C. Moreau-Luchaire, C. Moutafis, N. Reyren, J. Sampaio,\nC. Vaz, N. Van Horne, K. Bouzehouane, K. Garcia, C. Deranlot,\nP. Warnicke, et al. , Nat. Nanotechnol. 11, 444 (2016).\n[13] A. Soumyanarayanan, M. Raju, A. Gonzalez Oyarce, A. K. Tan,\nM.-Y . Im, A. Petrovi ´c, P. Ho, K. Khoo, M. Tran, C. Gan, et al. ,\nNat. Mater. 16, 898 (2017).\n[14] O. Boulle, J. V ogel, H. Yang, S. Pizzini, D. de Souza Chaves,\nA. Locatelli, T. O. Mente¸ s, A. Sala, L. D. Buda-Prejbeanu,\nO. Klein, et al. , Nat. Nanotechnol. 11, 449 (2016).\n[15] W. Legrand, D. Maccariello, F. Ajejas, S. Collin, A. Vecchiola,\nK. Bouzehouane, N. Reyren, V . Cros, and A. Fert, Nat. Mater.\n19, 34 (2020).\n[16] C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y . Xia, T. Cao, W. Bao,\nC. Wang, Y . Wang, et al. , Nature 546, 265 (2017).\n[17] B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein,\nR. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A.\nMcGuire, D. H. Cobden, et al. , Nature 546, 270 (2017).\n[18] Y . Deng, Y . Yu, Y . Song, J. Zhang, N. Z. Wang, Z. Sun, Y . Yi,\nY . Z. Wu, S. Wu, J. Zhu, et al. , Nature 563, 94 (2018).\n[19] M.-G. Han, J. A. Garlow, Y . Liu, H. Zhang, J. Li, D. DiMarzio,\nM. W. Knight, C. Petrovic, D. Jariwala, and Y . Zhu, Nano Lett.\n19, 7859 (2019).\n[20] B. Ding, Z. Li, G. Xu, H. Li, Z. Hou, E. Liu, X. Xi, F. Xu,\nY . Yao, and W. Wang, Nano Lett. 20, 868 (2020).\n[21] Y . Wu, S. Zhang, J. Zhang, W. Wang, Y . L. Zhu, J. Hu, G. Yin,\nK. Wong, C. Fang, C. Wan, et al. , Nat. Commun. 11, 3860\n(2020).\n[22] M. Yang, Q. Li, R. Chopdekar, R. Dhall, J. Turner, J. Carlström,\nC. Ophus, C. Klewe, P. Shafer, A. N’Diaye, et al. , Sci. Adv. 6,\neabb5157 (2020).\n[23] Y . Wu, B. Francisco, Z. Chen, W. Wang, Y . Zhang, C. Wan,\nX. Han, H. Chi, Y . Hou, A. Lodesani, G. Yin, K. Liu, Y .-t.\nCui, K. L. Wang, and J. S. Moodera, Adv. Mater. 34, 2110583\n(2022).\n[24] J. Liang, W. Wang, H. Du, A. Hallal, K. Garcia, M. Chshiev,\nA. Fert, and H. Yang, Phys. Rev. B 101, 184401 (2020).[25] J. Yuan, Y . Yang, Y . Cai, Y . Wu, Y . Chen, X. Yan, and L. Shen,\nPhys. Rev. B 101, 094420 (2020).\n[26] Q. Cui, J. Liang, Z. Shao, P. Cui, and H. Yang, Phys. Rev. B\n102, 094425 (2020).\n[27] C. Xu, J. Feng, S. Prokhorenko, Y . Nahas, H. Xiang, and\nL. Bellaiche, Phys. Rev. B 101, 060404 (2020).\n[28] Y . Zhang, C. Xu, P. Chen, Y . Nahas, S. Prokhorenko, and\nL. Bellaiche, Phys. Rev. B 102, 241107 (2020).\n[29] Z. Shen, C. Song, Y . Xue, Z. Wu, J. Wang, and Z. Zhong, arXiv\npreprint arXiv:2109.00723 (2021).\n[30] Q. Cui, Y . Zhu, Y . Ga, J. Liang, P. Li, D. Yu, P. Cui, and\nH. Yang, Nano Lett. 22, 2334 (2022).\n[31] W. Du, K. Dou, Z. He, Y . Dai, B. Huang, and Y . Ma, Nano\nLett. 22, 3440 (2022).\n[32] J. Liu, M. Shi, J. Lu, and M. P. Anantram, Phys. Rev. B 97,\n054416 (2018).\n[33] W. Sun, W. Wang, H. Li, G. Zhang, D. Chen, J. Wang, and\nZ. Cheng, Nat. Commun. 11, 5930 (2020).\n[34] W. Sun, W. Wang, J. Zang, H. Li, G. Zhang, J. Wang, and\nZ. Cheng, Adv. Funct. Mater. 31, 2104452 (2021).\n[35] A. Hallal, J. Liang, F. Ibrahim, H. Yang, A. Fert, and\nM. Chshiev, Nano Lett. 21, 7138 (2021).\n[36] Z. Shao, J. Liang, Q. Cui, M. Chshiev, A. Fert, T. Zhou, and\nH. Yang, Phys. Rev. B 105, 174404 (2022).\n[37] T.-E. Park, L. Peng, J. Liang, A. Hallal, F. S. Yasin, X. Zhang,\nK. M. Song, S. J. Kim, K. Kim, M. Weigand, G. Schütz,\nS. Finizio, J. Raabe, K. Garcia, J. Xia, Y . Zhou, M. Ezawa,\nX. Liu, J. Chang, H. C. Koo, Y . D. Kim, M. Chshiev, A. Fert,\nH. Yang, X. Yu, and S. Woo, Phys. Rev. B 103, 104410 (2021).\n[38] Q. Cui, Y . Zhu, J. Jiang, J. Liang, D. Yu, P. Cui, and H. Yang,\nPhys. Rev. Research 3, 043011 (2021).\n[39] K. Dou, W. Du, Y . Dai, B. Huang, and Y . Ma, Phys. Rev. B\n105, 205427 (2022).\n[40] Q. Li, M. Yang, C. Gong, R. V . Chopdekar, A. T. N’Diaye,\nJ. Turner, G. Chen, A. Scholl, P. Shafer, E. Arenholz, et al. ,\nNano Lett. 18, 5974 (2018).\n[41] S. von Malottki, B. Dupé, P. Bessarab, A. Delin, and S. Heinze,\nScientific reports 7, 12299 (2017).\n[42] www.flapw.de.\n[43] S. Smidstrup, T. Markussen, P. Vancraeyveld, J. Wellen-\ndorff, J. Schneider, T. Gunst, B. Verstichel, D. Stradi, P. A.\nKhomyakov, U. G. Vej-Hansen, et al. , J. Phys. Condens. Matter\n32, 015901 (2019).\n[44] P. Kurz, F. Förster, L. Nordström, G. Bihlmayer, and S. Blügel,\nPhys. Rev. B 69, 024415 (2004).\n[45] M. Heide, G. Bihlmayer, and S. Blügel, Phys. B 404, 2678\n(2009).\n[46] B. Zimmermann, G. Bihlmayer, M. Böttcher, M. Bouhassoune,\nS. Lounis, J. Sinova, S. Heinze, S. Blügel, and B. Dupé, Phys.\nRev. B 99, 214426 (2019).\n[47] H. Yang, A. Thiaville, S. Rohart, A. Fert, and M. Chshiev,\nPhys. Rev. Lett. 115, 267210 (2015).\n[48] H.-J. Deiseroth, K. Aleksandrov, C. Reiner, L. Kienle, and\nR. K. Kremer, Eur. J. Inorg. Chem. 2006 , 1561 (2006).\n[49] B. Chen, J. Yang, H. Wang, M. Imai, H. Ohta, C. Michioka,\nK. Yoshimura, and M. Fang, J. Phys. Soc. Jpn. 82, 124711\n(2013).\n[50] S. Grimme, J. Antony, S. Ehrlich, and H. Krieg, J. Chem. Phys.\n132, 154104 (2010).\n[51] H. L. Zhuang, P. R. C. Kent, and R. G. Hennig, Phys. Rev. B\n93, 134407 (2016).\n[52] J. He, S. Li, A. Bandyopadhyay, and T. Frauenheim, Nano Lett.\n21, 3237 (2021).\n[53] C. Gong and X. Zhang, Science 363, eaav4450 (2019).8\n[54] X. Jiang, Q. Liu, J. Xing, N. Liu, Y . Guo, Z. Liu, and J. Zhao,\nAppl. Phys. Rev. 8, 031305 (2021).\n[55] X. Hu, Y . Zhao, X. Shen, A. V . Krasheninnikov, Z. Chen, and\nL. Sun, ACS Appl. Mater. Interfaces 12, 26367 (2020).\n[56] S. Laref, K.-W. Kim, and A. Manchon, Phys. Rev. B 102,\n060402 (2020).\n[57] T. Moriya, Phys. Rev. 120, 91 (1960).\n[58] K. Huang, D.-F. Shao, and E. Y . Tsymbal, Nano Lett. 22, 3349\n(2022).\n[59] B. Dupé, M. Hoffmann, C. Paillard, and S. Heinze, Nat. Com-\nmun. 5, 4030 (2014).\n[60] S. Meyer, M. Perini, S. von Malottki, A. Kubetzka, R. Wiesen-\ndanger, K. von Bergmann, and S. Heinze, Nat. Commun. 10,\n3823 (2019).\n[61] P. F. Bessarab, V . M. Uzdin, and H. Jónsson, Comput. Phys.\nCommun. 196, 335 (2015).\n[62] P. F. Bessarab, G. P. Müller, I. S. Lobanov, F. N. Rybakov, N. S.\nKiselev, H. Jónsson, V . M. Uzdin, S. Blügel, L. Bergqvist, and\nA. Delin, Sci. Rep. 8, 1 (2018).[63] S. Haldar, S. von Malottki, S. Meyer, P. F. Bessarab, and\nS. Heinze, Phys. Rev. B 98, 060413 (2018).\n[64] S. von Malottki, P. F. Bessarab, S. Haldar, A. Delin, and\nS. Heinze, Phys. Rev. B 99, 060409 (2019).\n[65] F. Muckel, S. von Malottki, C. Holl, B. Pestka, M. Pratzer, P. F.\nBessarab, S. Heinze, and M. Morgenstern, Nat. Phys. 17, 395\n(2021).\n[66] C. Tan, J. Lee, S.-G. Jung, T. Park, S. Albarakati, J. Partridge,\nM. R. Field, D. G. McCulloch, L. Wang, and C. Lee, Nat. Com-\nmun. 9, 1554 (2018).\n[67] Y .-P. Wang, X.-Y . Chen, and M.-Q. Long, Appl. Phys. Lett.\n116, 092404 (2020).\n[68] S. Y . Park, D. S. Kim, Y . Liu, J. Hwang, Y . Kim, W. Kim, J.-Y .\nKim, C. Petrovic, C. Hwang, S.-K. Mo, H.-j. Kim, B.-C. Min,\nH. C. Koo, J. Chang, C. Jang, J. W. Choi, and H. Ryu, Nano\nLett. 20, 95 (2020).\n[69] A. S. Varentcova, S. von Malottki, M. N. Potkina,\nG. Kwiatkowski, S. Heinze, and P. F. Bessarab, Npj Comput.\nMater. 6, 1 (2020)."    },    {        "title": "2209.00330v1.Two_dimensional_half_Chern_Weyl_semimetal_with_multiple_screw_axes.pdf",        "content": "arXiv:2209.00330v1  [cond-mat.mes-hall]  1 Sep 2022two-dimensional half Chern-Weyl semimetal with multiple s crew axes\nWei Xu,1,∗Jiawei Yi,1,∗Hao Huan,2Bao Zhao,2,3Yang Xue,1,†and Zhongqin Yang2\n1Department of Physics, East China University of Science and Technology, Shanghai 200237, China\n2State Key Laboratory of Surface Physics and Key Laboratory o f Computational Physical\nSciences (MOE) &Department of Physics, Fudan University, Shanghai 200433, China\n3Shandong Key Laboratory of Optical Communication Science a nd Technology,\nSchool of Physics Science and Information Technology, Liao cheng University, Liaocheng 252059, China\n(Dated: September 2, 2022)\nHalf topological states of matter and two-dimensional (2D) magnetism have gained much atten-\ntion recently. In this paper, we propose a special topologic al semimetal phase called a 2D half\nChern-Weyl semimetal (HCWS), which is a 2D magnetic Weyl sem imetal bound to the half Chern\ninsulator phase by symmetry, and the two phases can be conver ted to each other by manipulat-\ning the magnetization direction. We provide the symmetry co nditions to realize this state in 2D\nsystems with multiple screw axes. Tight-binding models wit h multiple basis and a predicted 2D\nmaterial, monolayer TiTe, are shown as the concrete example s for HCWSs. The TiTe monolayer was\nshown to have a high ferremagnetic Curie temperature (˜966 K ) as well as a Coulomb correlation-\nenhanced spin-orbit coupling (SOC), and further demonstra tes the effect of correlation-enhanced\nSOC on magnetocrystalline anisotropy energy and energy gap opening. Our work reveals a state\nwith switchable andspin-resolved half bodycharge current sas well as half boundarycharge currents,\nand will provide a platform for novel and high-performance t opological spintronics devices.\nIntroduction. —Two-dimensional (2D) half topological\nstates, possessing fully-spinpolarized topologically non-\ntrivial energy bands near Fermi level (E F), have gained\nmuch attention [1–9]. Two outstanding representatives,\n2D half Weyl semimetals [1–4] and half Chern insulators\n(HCIs) [5–9], with half-metallic linear bulk bands and\nhalf-metallic chiral edge states respectively, show poten-\ntialapplicationsinhigh-performanceandthinspintronics\ndevices. Unlikeinthe3Dcase,Weylpointsin2Dsystems\nmust be protected by additional crystal symmetries due\nto the loss of topological protection by dimension reduc-\ntion [2]. In addition, for ferromagnetic (FM) materials,\nthe crystal symmetry is related to the magnetization di-\nrection. Thus, the half Weyl points in 2D materials are\ngoverned by their magnetization direction, which gives\nsuch systems more diverse manipulation means based on\nregulating the magnetization direction.\nBesides the crystal symmetry, spin-orbit coupling\n(SOC) and electron correlation are also at the core of\n2D half topological states. Specifically, the long-range\nFM order in 2D half topological states is stabilized via\nSOC-induced magnetocrystalline anisotropy, lifting the\nMermin-Wagner restriction [10]. Electron correlation is\nalso a significant effect in 2D magnetic systems which\nmayenhancetheirSOC[11]. Thus,energygapopenedby\nSOC and magnetocrystalline anisotropy energy (MAE)\nare closely related to electron correlation. 2D half Weyl\nsemimetalsprovideanexcellentplatformforstudyingthe\nentangled physics of topological properties, crystal sym-\nmetry, SOC, and electron correlation.\nHere we draw attention to a peculiar class of\nWeyl semimetals called 2D half Chern-Weyl semimetals\n∗These two authors contributed equally to this work.\n†E-mail: xuey@ecust.edu.cn(HCWSs), whichare2DmagneticWeylsemimetalscapa-\nble of opening a nontrivial global energy gap character-\nized by Chern numbers by changing their magnetization\ndirection. As shown in Fig. 1, there are two different\ntransport channels in HCWSs that can both transport\nfully spin-polarized charge currents. The two channels\ncan be switched by manipulating the magnetization di-\nrection in HCWSs and the direction of spin in these two\nchannelsaredifferent, paralleltothe differentmagnetiza-\ntion directions. These novel properties show their great\napplication in spintronics.\nhalf Weyl points half Weyl points half Weyl points half Weyl points \nLead Lead \nFIG. 1. Schematic diagram of the two types of transport\nchannels of HCWSs. Both channels can transport fully spin-\npolarized charge currents, where the thick blue arrow indi-\ncates bulk transport channel and the thin blue arrows repre-\nsent edge transport channels. The spin orientations in them\nare different, denoted by red arrows. The green plane is the\nBZ, and the red and blue points on it are chiralities of half\nWeyl points, red for positive and blue for negative.\nBy generalizing nonmagnetic Dirac semimetals with\nmultiple screw axes [12–14] to magnetic systems, two\ntypes of hourglasslike, as well as nodal line 2D HCWSs\nare obtained. Futhermore, we predict a 2D magnets\nwith high ferromagnetic Curie temperature of 966 K,\nmonolayer TiTe, as a concrete candidates for realizingii\nthe 2D hourglasslike HCWSs. The two pairs of half\nWeyl points in TiTe are protected by a screw symmetry\nand a glide symmetry, respectively. The systems trans-\nform into HCIs after breaking the pristine nonsymmor-\nphic symmetries by rotating the magnetization direction.\nThe Coulomb correlation-enhanced and magnetization\ndirection-dependent SOC, MAE, and energy gap open-\ning are explicitly demonstrated in TiTe. Our work not\nonly reveals a state of matter and its interesting physical\nconnotationsbut alsoofferspromisingmaterialplatforms\nfor novel topological spintronics applications.\n2D HCWSs and HCIs from band inversion —We first\nanalyze the conditions needed for realizing 2D half Weyl\nsemimetals in the systems with multiple screw axes.\nThen give the requirements for non-trivial gap opening\nin these 2D half Weyl semimetals. For all 80 nonmag-\nnetic layer groups, there are total 11 layer groups with\ntwo screw axes along the axial direction [12]. Later, we\nwill focus on the layer groups with inversion symmetry\namong the above 11 layer groups and take the represen-\ntative case pmmn(No. 46) as our starting point. The\ngenerators of layer group pmmncontain two screw op-\nerations ( /tildewideC2x={C2x|a/2},/tildewideC2y={C2y|b/2}), and\nan inversion P. By including collinear FM and SOC, the\nspatialrotationandspin rotationarelockedglobally, and\nthe existence of the nonsymmorphic symmetries above\nwill depend on magnetization direction. In the following,\nwe discuss two typical FM cases with in-plane (x-axial)\nand out-of-plane (z-axial) magnetization vectors ( ˆm), re-\nspectively.\nCase 1—Ifˆmis along x direction, the time-reversal\nsymmetry Tand the glide mirror symmetry /tildewiderMz=\nP/tildewideC2x/tildewideC2yare broken, while the joint symmetry of these\ntwo symmetries is preserved, denoted as Θ = T/tildewiderMz.\nΘ leaves the X point ( kx=π,ky= 0) and Y point\n(kx= 0,ky=π) of Brillouin zone (BZ) invariant. On\nthe other hand, we have Θ2=e−i(kx+ky). Thus, at X\nand Y, Θ2=−1 will result in Krammers-like degener-\nacy.\nDue to the k-lines Γ −Xand Γ−Yare invariant under\n/tildewideC2xand/tildewiderMx=P/tildewideC2x, respectively. The eigenstates along\nΓ−Xand Γ−Ycan be labelled by the eigenvaluesof /tildewideC2x\n(±ie−ikx/2) and/tildewiderMx(±i), respectively. By considering\nthe following relations\nΘ/tildewideC2x=−eiky/tildewideC2xΘ\nΘ/tildewiderMx=−eikx/tildewiderMxΘ,(1)\nwe can draw a conclusion that the two bands evolved\nfrom a Krammers-like degenerate point at X (Y) will\ncarry the same eigenvalues of /tildewideC2x(/tildewiderMx) along Γ −X\n(Γ−Y) (details see Supplemental Material [15]). As\nshown in Figs. 2(c-e), if band inversion happens at the\nΓ point between two pairs of bands, two types of hour-\nglasslikeWeyl semimetals emerge, the hourglasslikeWeyl\nsemimetals with a pair of Weyl points (Figs. 2(d) and\n2(e)) and with two pairs of Weyl points (Fig. 2(c)).Noticing that the /tildewideC2xand/tildewiderMxdo not flip the x-direction\nspins, thus the above obtained Weyl fermions are all in\nsingle spin channel.\nX Γ YXY\nΓ\nX Γ YXY\nΓ\nX Γ YXY\nΓ(c) (d) (e) x-direction \nEnergy(eV) X Γ Y|Γ MXY\nΓ(b)\nXY\nΓ\nBrillouim ZoneM(a)\nz-direction \nEnergy(eV) \tC2x đMz\nj j\tMxđMz\nj j\nFIG. 2. (a) The first BZ with high-symmetry points and\nlines. The Γ-X (Y) lines are invariant under /tildewideC2x(/tildewiderMx) or\n/tildewiderMz(/tildewiderMz) when magnetization direction is parallel to x or z,\nrespectively. (b)-(e) The Schematic band structures for fo ur\ndistinctive half Weyl semimetal states of pmmnlayer group\nwith FM order along x and z directions, respectively. Red\nsolid lines and black dot lines denote the different eigenval ues\nof the symmetry operations on the high-symmetry lines given\nin (a). Each inset shows the band-crossing points inside the\nBZ.\nCase 2—Ifˆmis in the z direction, a similar discus-\nsion can be carried out using the preserved symmetries\n/tildewiderMz, Θx=T/tildewideC2x, and Θ y=T/tildewideC2y. Θxand Θ yleave\nthe boundaries of BZ X−MandY−Minvariant and\nresultinaKrammers-likedegeneracyalongthem, respec-\ntively. Meanwhile, eigenstates inside the whole BZ can\nbe denoted by eigenvalues of /tildewiderMz, which leaves the BZ\ninvariant. By considering the following relations\n/tildewiderMzΘx=−eikyΘx/tildewiderMz\n/tildewiderMzΘy=−eikxΘy/tildewiderMz,(2)\nwe make a conclusion that the two bands, evolved from a\nKrammers-likedegeneratepoint at boundariesof the BZ,\nwill carrythe different eigenvaluesof /tildewiderMzinside the whole\nBZ. Combining band inversion at the Γ point, this con-\nclusion will lead to a nodal line Weyl semimetal, shown\nin Fig. 2(b).\nObviously, the Weyl points come in pairs for all the\ncases above, satisfying the no-go theorem [16]. The\ngap opening at a 2D Weyl point would give a topologi-\ncal charge of ±1/2 [17, 18]. Moreover, the topological\ncharges of two Weyl points connected by inversion P\nmust be identical and will contribute to ±1 Chern num-\nbers for the case in Figs. 2(d) and 2(e) (Berry curvature\nΩisevenunder P). Toobtainednon-zeroChernnumbers\nfor the cases with multiple pairs of Weyl points (Figs.iii\n2(b) and 2(c)), additional symmetry need to be included,\nsuch as/tildewideC4z, to ensure that the topological charges of dif-\nferent Weyl points do not completely cancel out.\nTo show some examples, we built square lattice\nsymmetry-enforced tight-binding (TB) models [19] for\ndifferent symmetry-allowed d-orbital doublets with the\nsame spin. By considering in-plane and out-of-plane\nmagnetization, respectively, and by tuning the param-\neters of TB models, all possible half Weyl semimetals\nunder different basis are listed in Table I. The details of\nTB models are given in Supplemental Material [15].\nTABLE I. The possible half Weyl semimetal states obtained\nfrom symmetry-enforced TB models on a square lattice with\ndifferent FM orientations. The basis sets are listed in first\ncolumn.\nFMxaFMz\n(dxy,dz2) hourglasslike-Ibnodal line\n(dx2−y2,dz2) hourglasslike-I nodal line\n(dx2−y2,dxy) hourglasslike-I or II nodal line\n(dxz,dyz) hourglasslike-I or II\naThe subscripts x and z refer to the magnetization directions .\nbRoman numerals refer to the number of Weyl point pairs in an\nhourglasslike Weyl semimetal, I for one pair and II for two\npairs.\nmaterial examples —Next, we reveal a concrete ma-\nterial example of 2D HCWSs, monolayer TiTe, whose\ngeometric structure is displayed in Fig. 3(a). Its unit\ncell hasp4/nmmspace group (No. 129) symmetry and\ncomprises two Ti and two Te atoms. The crystal struc-\nture has two screw axes: /tildewideC2x={C2x|(a/2,0,0)}and\n/tildewideC2y={C2y|(0,a/2,0)}. The inversion center is lo-\ncated at the crossing point of the two screw axes. The\nfully optimized lattice parameters are a=b= 4.33\n˚Afor U=3 eV (Fig. 3(b)). The phonon spectra (Fig.\nS1(a)) without imaginary frequency mode manifests the\ndynamical stability of the monolayer TiTe. The ther-\nmal stability of monolayer TiTe is further confirmed\nby first-principles molecular-dynamics (MD) simulation\n(Fig. S1(b)), which shows that the structure remains\nintact at a temperature of 300 K after 10 ps.\nTABLE II. The relative total energy (meV per unit cell) of\nfour magnetic configurations shown in Fig. S2, magnetic cou-\npling parameters of J1andJ2(meV), anisotropy parameter D\n(meV per Ti atom), and critical temperature Tc(K) obtained\nfrom the Heisenberg model for TiTe monolayer with U=3 and\n5 (eV), respectively.\nFM AFM-N AFM-L AFM-Z J1J2DTc\nU=3 0 557.1 586.4 332.5 69.64 38.48 -0.37 966\nU=5 0 628.8 689.1 25.1 78.60 46.83 5.70 1148\nTo determine the magnetic ground state, the 2 ×2×1\nsupercell of the monolayer TiTe with four magnetic con-\nfigurations and different Hubbard U values (2-5 eV) are(a) (b)\n(c) (d)\n0 400 800 1200 1600 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Magnetization \nSpecific Heat \nTemperature (K) Normalized Magnetization (M/M s)\n01234Specific Heat (C v)U=2 \nU=3 \nU=4 \nLattice constant(Å) 4.1 4.2 4.3 4.4 4.5 4.6 050 100150200250E-E min (meV) \nC2y j\nC2x jy\nxz\nz\nyx\nTi Te \nα\nC2y j\nMAE(E-E[001] )(meV/Ti) \nFIG. 3. (a) Top and side view of 2D TiTe monolayer, the\nblue and pink balls represent Ti and Te atoms, respectively.\nDashedlines showunitcell. The redarrows indicate thescre w\naxes of /tildewideC2xand/tildewideC2y. The red dot represents the inversion\ncenter. (b) The total energies as a function of the lattice co n-\nstant for TiTe monolayer with U=2—3—4 eV, respectively,\nin which the minimum total energy at the equilibrium lattice\nconstant for each case is set as energy zero. (c) The MAE\nby rotating the spin within x-z and x-y planes, respectively .\n(d) The normalized average magnetic moment (blue curve)\nand specific heat (red curve) versus temperature for the TiTe\nmonolayer. The results in (c) and (d) are calculated with\nU=3 eV.\nconsidered (see Fig. S2). It is found that the monolayer\nTiTe maintains a FM ground state for all considered U\nvalues (Table II). The magnetic moment of TiTe is 2 µB\nper Ti atom from our DFT calculations. It can be well\nunderstood by analyzing the valence electron configura-\ntion of Ti atom (3 d24s2), which becomes Ti2+(3d2con-\nfiguration) after transfer 2 e−to Te atom, confirmed by\nthe Bader charge calculations [20]. Unpaired 2 e−elec-\ntrons prefer to have high spin alignment with 2 µBbased\nonHund’srules,whichcanbeseeninthepartialdensities\nof states (DOSs) of TiTe (Fig. 4(c)). The microscopic\nmechanism of FM ground state for TiTe is explained by\na Ti-Te-Ti FM super-exchange interaction according to\nthe Goodenough-Kanamori-Anderson rules [21–24], with\na bond angle of Ti-Te-Ti ( α= 98.63◦in Fig. 3(a)) close\nto the 90◦.\nThe MAE derived from the SOC plays a crucial role\nfor a 2D material keeping in a long-range FM ground\nstate [25]. The MAE of TiTe monolayer in the xz and\nxy planes as functions of the magnetization angle θis\ndepictedinFig. 3(c). Bycomparingtheenergydifference\nof the two planes, we find that the magnetic easy axes ofiv\n2\n1\n0\n-1 \n-2 \nM X Γ Y MLD3LD4 F3 \nF4 E-E /g3470(eV) dz2 \ndx2-y2 \ndxz/dyz \ndxy \nU=3 eV\nU=5 eVdxy \ndxz/dyz \ndx2-y2 \ndz2 \nE-E/g3470(eV)Density of states -2 024-2 024\n-4 43210 -3 -2 -1 (a) (b)\n(c) (d)0\n-1 1\nM X Γ Y Mz-direction\nx-directionE-E /g3470(eV) \n-π π -π/2 π/2 0-π \nkx-π/20π/2πkyM\nX ΓY800 \n0\nFIG. 4. (a) The orbital resolved band structures of TiTe\nmonolayer without SOC. The orbital projections are only ap-\nplied for the bands in spin-up channel and the spin-down\nbands are plotted with black dashed lines. (b) Band struc-\ntures with SOC of TiTe monolayer with x-magnetization and\nz-magnetization, respectively. Theirreduciblerepresen tations\nof the two bands crossing the E Ffor x-magnetization case are\ndisplayed. (c) The projected densities of states for the TiT e\nmonolayer with two U values, respectively, in which the dxz\nanddyzorbitals are degenerate. (d) Distribution of the Berry\ncurvatures for the TiTe monolayer with the magnetization\ndirection along the z axis.\nthe TiTe monolayerprefer the in-plane direction with the\nU = 3.0 eV. This trend of MAE remains consistent in the\nrange of U = 2 −3.4eV. With increasing U, the magnetic\nanisotropydirectionchangesfromthein-planetotheout-\nof-plane, and the critical U value is about 3.4 eV (Fig.\n5(c)). The mechanism for this interesting U-correlated\nMAE phenomenon will be explained latter. To evaluate\ntheTCof the TiTe monolayer, a Heisenberg model is\nbuilt as [26]\nH0=−J1/summationdisplay\n<i,j>Si·Sj−J2/summationdisplay\n<<i,j>>Si·Sj−D/summationdisplay\ni|Se\ni|2,\n(3)\nwhereSiis the spin vector, Se\niis the spin component\nalong the easy axis, Ji(i= 1,2) andDdenote the\nstrengths for exchange interaction and anisotropy, re-\nspectively. Both the first and second terms can be re-\ngarded as isotropic. Both JiandDcan be extracted\nfrom the first-principles calculations with the equations\nin Supplemental Material [15] and are listed in Table II.\nAs shown in Fig. 3(d), the Curie temperature for the FM\nstate can be estimated as TC≈966K, which is signifi-cantly higher than the room temperature and also than\nthat of the 2D star magnetic material of the CrI3 mono-\nlayer (45 K [27] or 95 K [28]).\nFor now, the magnetic ground state for TiTe mono-\nlayer is confirmed as FM, satisfying the prerequisites for\n2D HCWSs. By calculating the electronic band struc-\ntures of TiTe monolayer without SOC, one observes two\npairs of fully spin-polarized twofold band crossing points\nsymmetrically distributed on the x-axis and the y-axis\nin Fig. 4(a), which are related with each other by\n/tildewideC4z={C4z|(a/2,0,0)}and represent 2D Weyl points.\nThis half-semiconducting feature is robust to the change\nin Hubbard U (Fig. S3).\nConsidering the SOC effect, the stability of the 2D\nWeyl points depends on the magnetization directions.\nFor the x-axial magnetization, these half Weyl points can\nbe protected by preserved nonsymmorphic symmetries\n/tildewideC2xand/tildewiderMx, shown in Fig. 4(b), consistent with the\ndiscussion in Fig. 2(c). The marked different irreducible\nrepresentationsof LD3 and LD4 (F3 and F4) confirm the\nrobustness of the half Weyl points in TiTe with the in-\nplane-axial magnetization. As shown in Fig. 5(a), the\nedge state of TiTe starts from the Weyl point projection\non−X−Γ and ends at the projection of another Weyl\npoint on X−Γ, very similar to the Fermi arc in 3D Weyl\nsemimetals [18, 29–31].\n(b)\n0.6\n0\n-0.6Γ Γ X|-X\n90 \n0z\n30 60 120\n150\n180 \n210\n240\n2703003300 80 160x C=+2 \nC= -2(c)(a)\n0.6\n0\n-0.6E-E F(eV) \nΓ Γ X|-X\n(d)\n−2.502.55\n2.0 2.5 3.0 3.5 4.0 4.5 5.0\nU (eV) 00.1 0.2 0.3 0.4 0.7\n0.5\n0.3\n0.1MAE(meV/Ti) \nΔM orb(μ B)\nBand gap(eV) Band gap ΔM orb MAE z\nFIG. 5. (a) and (b) The edge states of a semi-infinite TiTe\nmonolayer cut along the [100] direction with x-magnetizati on\nand z-magnetization, respectively. (c) The MAE, orbital mo -\nment anisotropy (∆ Morb=|M[001]\norb| − |M[100]\norb|), and maxi-\nmal band gap as functions of U. (d) The band gap of the\nTiTe monolayer with the magnetization direction varying\nfromθ= 0◦toθ= 360◦in the xz plane. The green and\nblue areas indicate the monolayer with Chern numbers of -2\nand 2, respectively.\nTo characterize the low-energy band structures for thev\n2D HCWSs state, we construct a k·peffective model.\nWith x-axismagnetizationand SOC, the effective Hamil-\ntonian for half Weyl points in −X−Xis subjected to\nthe magnetic little co-group m′m2′of−X−X, with two\ngenerators, /tildewideC2xandT/tildewideC2y. The symmetry constraints\nare given by\n/tildewideC−1\n2xH(qx,qy)/tildewideC2x=H(qx,−qy)\n/parenleftBig\nT/tildewideC2y/parenrightBig−1\nH(qx,qy)/parenleftBig\nT/tildewideC2y/parenrightBig\n=H∗(qx,−qy),(4)\nwhereqis measured from the Weyl points in −X−X.\nIn the basis of the irreducible representations LD3 and\nLD4, we find that to linear order in q, the effective model\ntakes the form of the 2D Weyl model,\nH=A(τqx)σz+B(qy)σx+C(qx)σ0,(5)\nwhereA(τqx) =τ(c1qx−c2qx)+c3−c4,B(qy) =c5qy,\nC(qx) =c1qx+c2qx+c3+c4(ci=1−5are constant pa-\nrameters and τ=±for the Weyl points in X−Γ or\n−X−Γ respectively, σx−σzare the Pauli matrixes, and\nσ0is the unit matrix. Thus, the low-energy electrons in-\ndeed resemble a pair of 2D half Weyl fermions located in\n−X−X[32].\nConversely, by rotating x-axis magnetization to z-axis\nmagnetization, breaking the /tildewideC2xand/tildewiderMxwill remove the\nWeyl points and open an energy gap, as shown in Fig.\n4(b). The gap opening at a 2D Weyl point would induce\na finite Berry curvature Ω( q) =−2Im/an}bracketle{t∂qxuv|∂qyuv/an}bracketri}ht,\nwhere|uv/an}bracketri}htis the eigenstate of the valence band. The\nintegral of Berry curvature in a region around the Weyl\npoint gives a topological charge of ±1/2 [17]. Since the\nfour Weyl points are connected by preserved /tildewideC4z, their\ntopological charges have same sign, shown in Fig. 4(d),\nand contribute a non-zero Chern number ( C= 2), which\nsuggests that two chiral edge states appear in the bulk\ngap (Fig. 5(b)). The edge states are confirmed to be\nfully-spinpolarized by calculations, showing great poten-\ntial for dissipassionless spintronics.\nThe SOC effect of light 3 dtransition metal ions is usu-\nally small. While, a TiTe monolayer can open a band\ngap of up to 119 meV with U = 3 eV, and the band\ngap increases significantly as the U value increases (˜429\nmeV with U = 5 eV), as shown in Fig. 5(c). The same\nU-enhancement effect also appears in the MAE of TiTe\n(Fig. 5(c)). These phenomena suggested a significant\ncorrelation-enhancing SOC effect [11] in TiTe.\nCorrelation can enhance SOC by enlarging the ef-\nfective SOC parameters for both in-plane orbital\ngroup{dxy,dx2−y2}and out-of-plane orbital group\n{dxz/yz,dz2}in single spin channel [15]. For in-plane and\nout-of-plane groups, the enhanced effective SOC param-\netersλ/bardblandλ⊥are respectively given as\nλ/bardbl=λ+/vextendsingle/vextendsingle¯Lz/vextendsingle/vextendsingle\n4/planckover2pi13Ueff, λ⊥=λ+/vextendsingle/vextendsingle¯Lx/vextendsingle/vextendsingle\n3/planckover2pi13Ueff,(6)where the λis pristine SOC parameter, Ueff=U−\nJwithUandJare the on-site Coulomb repul-\nsion and exchange interaction parameters, and/vextendsingle/vextendsingle¯Li/vextendsingle/vextendsingle=\n/summationtext\nm∈occ./vextendsingle/vextendsingle/vextendsingle/an}bracketle{tm|ˆLi|m/an}bracketri}ht/vextendsingle/vextendsingle/vextendsingle,i=x or z is the expectation of ˆLi\nover the occupied states. The/vextendsingle/vextendsingle¯Lx/vextendsingle/vextendsingle(/vextendsingle/vextendsingle¯Lz/vextendsingle/vextendsingle) is also a func-\ntion ofλ⊥(λ/bardbl). Since the intrinsic SOC λis relatively\nsmall,/vextendsingle/vextendsingle¯Lx/vextendsingle/vextendsingle(/vextendsingle/vextendsingle¯Lz/vextendsingle/vextendsingle) can be expressed as a function of λ:/vextendsingle/vextendsingle¯Lx/vextendsingle/vextendsingle=κxλ(/vextendsingle/vextendsingle¯Lz/vextendsingle/vextendsingle=κzλ) and be solved self-consistently.\nAs shown in Fig. 4(a), the two crossing bands near\nEFmainly consist of set {dxy,dx2−y2}. The degeneracy\nof this set is lifted by SOC. According to the first-order\nperturbation, the splitting can be written as\n∆E= (2λ/planckover2pi12+Ueff\n2/planckover2pi1|¯Lz|)|cosθ|, (7)\nwhereθis the angle between the magnetic moment and\nz-axis. The equation above indicates that the band gap\nof TiTe is independent of in-plane rotation angle φ(the\nanglesθandφare illustrated in Fig. S5) and is zero with\nin-plane magnetization to the first order perturbation.\nThe U-enhancing band gap increases as θdecreases until\nit reaches the maximum at θ= 0, in good agreement\nwith the calculation results shown in Fig. 5(d).\nIt is interesting to note that as U increases, the easy\nmagnetizationaxisshifts fromin-planetoz-axis,followed\nby a rapid increase in MAE, as shown in Fig. 5(c). DOSs\nin Fig. 4(c) show that the occupied states are mostly dz2\nanddxyin spin-up channel and are separated from unoc-\ncupied states by large crystal field splitting, resulting in\nfirst-orderperturbation havingnegligible effect on single-\nion anisotropy, and therefore we considered second-order\nperturbation for correlation-enhanced MAE as\nMAE= (3\n4λ2\n⊥\n∆⊥−λ2\n/bardbl\n∆/bardbl)/planckover2pi14cos2θ, (8)\nwhere ∆ /bardbl=|ex2−y2−exy|and ∆ ⊥=|ez2−eyz|are en-\nergy splitting for the in-plane and out-of-plane orbitals,\nrespectively, whichcanbe estimated bythe weight-center\npositions of the DOSs [25, 33, 34]. One can see from\nthe equation above that for small U, the orbital moment\nquenched and the MAE are mainly controlled by the de-\nnominator. Because of the ∆ /bardbl>∆⊥for the small U\n(see Fig. 4(c)),/parenleftbigg\n3\n4λ2\n⊥\n∆⊥−λ2\n/bardbl\n∆/bardbl/parenrightbigg\n>0 leads to an in-plane\neasy magnetization. Increased correlation enhances or-\nbital polarization ∆ L=|Lz| − |Lx|(Fig. 5(c)), and\nalso improves ∆ ⊥to exceed ∆ /bardbl(Fig. 4(c)), resulting in/parenleftbigg\n3\n4λ2\n⊥\n∆⊥−λ2\n/bardbl\n∆/bardbl/parenrightbigg\n<0, hence, the easy magnetization shift-\ning to the z-direction.\nWe have some remarks before closing. First, the pro-\nposed 2D HCWSs here are protected by in-plane screw\nand glide symmetry. Hence, the Weyl points are no\nlonger protected by breaking these nonsymmorphic sym-\nmetries when the in-plane magnetization deviates fromvi\nthe in-plane axial. However, from equation (7), one can\nsee that the energy gap opening is independent of in-\nplane rotating angle φto the first-order perturbation.\nThe calculation results show a tiny gap (˜3.5 meV) when\na in-plane rotating applied, which means that the half\nWeyl semimetal properties are well maintained for any\nin-plane magnetization.\nSecond, the equation (8) indicates that the MAE of\nTiTe is also less relevant to in-plane rotating angle φ,\nwhich is consistent with the calculation results shown\nin Fig. 3(c). This property ensures that the in-plane\nmagnetization of TiTe, as well as the chirality of its Weyl\npoints, are easily modulated by the magnetic field.\nThird, the out-of-plane magnetization will open a fi-\nnite energy gap as shown in Fig. 5(d) and equation (7),\nwhose Chern number is related with the magnetization\ndirection. Thus, the chiralityofedgestates islockedwith\ntheir spin direction, enabling more diverse modulation of\nTiTe.\nIn summary, we have proposed a special topological\nstate: 2D HCWSs. It represents a half Weyl semimetal\nbound to a half Chern insulator by symmetry. We show\nthe symmetry conditions to realize this state in 2D sys-\ntems with multiple screw axes. Futhermore, we show\nexamples for 2D HCWSs based on TB models and a\nnew material TiTe, a 2D FM magnet with a high TC.\nCorrelation-enhanced SOC and its inherent relevance to\nthe MAE and energy gap opening in the TiTe are shown.\nThese results demonstrate a universal proposal for real-\nizing new low-power topological spintronics devices witheasy manipulation and diverse tunable tools.\nComputational methods —Our first-principles density\nfunctional theory (DFT) calculations are carried out\nwith the generalized gradient approximation proposed\nby Perdew, Burke, and Ernzerhof (PBE) [35], which is\nimplemented in the Vienna ab initio simulation package\n(VASP) [36]. The GGA+U method [37] is employed to\ndescribe the strongly correlated Ti-3 delectrons, unless\nexplicitly stated otherwise, all of the calculations were\ndone with U=3 eV [38, 39]. The plane-wave cutoff en-\nergywassetto500eVandthevacuumspaceismorethan\n15˚A to avoid the influence between two adjacent slabs.\nThe force convergecriterion wasless than 0.01 eV/ ˚A, the\nenergies were less than 10−6eV, and Monkhorst-Pack k-\npoint grids of 12 ×12×1 were adopted. The phonon\nspectra is calculated by using density functional pertur-\nbation theory (DFPT) implemented in PHONOPY code\n[40, 41] with 3 ×3×1 supercell. For the monolayer TiTe,\nthe hybrid functional HSE06 [42] is employed to verify\nband structure. The tight-binding model and the topol-\nogy characteristics of TiTe monolayer are calculated by\nwannier90[43] and wanniertools[44]. To estimate the TC\nof the FM states, the Monte Carlo simulations [45] are\nperformed on a 51 ×51 supercell with 106steps at each\ntemperature.\nACKNOWLEDGMENTS\nThis work was supported by National Natural Science\nFoundation of China under Nos. 11904101, 11604134,\n11874117, 12174059 and the Natural Science Foundation\nof Shanghai under Grant No. 21ZR1408200.\n[1]Xiaodong Zhou, Run-Wu Zhang, Zeying Zhang, Da-\nShuai Ma, Wanxiang Feng, Yuriy Mokrousov, and Yugui\nYao.J. Phys. Chem. Lett. , 10:3101–3108, 2019.\n[2]Jing-Yang You, Cong Chen, Zhen Zhang, Xian-Lei\nSheng, Shengyuan A. Yang, and Gang Su. Phys. Rev.\nB, 100:064408, 2019.\n[3]Run-Wu Zhang, Xiaodong Zhou, Zeying Zhang, Da-\nShuai Ma, Zhi-Ming Yu, Wanxiang Feng, and Yugui Yao.\nNano Lett. , 21:8749–8755, 2021.\n[4]N. L. P. Andrews, J. Z. Fan, R. L. Forward, M. C. Chen,\nand H.-P. Loock. Phys. Chem. Chem. Phys. , 19:73–81,\n2017.\n[5]Yang Xue, Bao Zhao, Yan Zhu, Tong Zhou, Jiayong\nZhang, Ningbo Li, Hua Jiang, and Zhongqin Yang. NPG\nAsia Mater , 10:e467–e467, 2018.\n[6]Hao Huan, Yang Xue, Bao Zhao, Guanyi Gao, Hairui\nBao, and Zhongqin Yang. Phys. Rev. B , 104:165427,\n2021.\n[7]Qilong Sun, Yandong Ma, and Nicholas Kioussis. Mater.\nHoriz., 7:2071–2077, 2020.\n[8]Yang Li, Jiaheng Li, Yang Li, Meng Ye, Fawei Zheng,\nZetao Zhang, Jingheng Fu, Wenhui Duan, and Yong Xu.\nPhys. Rev. Lett. , 125:086401, 2020.\n[9]Lei Liu, Hao Huan, Yang Xue, Hairui Bao, and Zhongqin\nYang.Nanoscale, to be published , 2022.[10]N. D. Mermin and H. Wagner. Phys. Rev. Lett. , 17:1133–\n1136, 1966.\n[11]Jiayu Li, Qiushi Yao, Lin Wu, Zongxiang Hu, Boya Gao,\nXiangang Wan, and Qihang Liu. Nat Commun , 13:919,\n2022.\n[12]Xiaotong Fan, Dashuai Ma, Botao Fu, Cheng-Cheng Liu,\nand Yugui Yao. Phys. Rev. B , 98:195437, 2018.\n[13]Moritz M. Hirschmann, Andreas Leonhardt, Berkay\nKilic, Douglas H. Fabini, andAndreas P. Schnyder. Phys.\nRev. Materials , 5:054202, 2021.\n[14]Akira Furusaki. Science Bulletin , 62:788–794, 2017.\n[15]Supplemental Materials , 2022.\n[16]H. B. Nielsen and Masao Ninomiya. Physics Letters B ,\n130:389–396, 1983.\n[17]Wang Yao, Shengyuan A. Yang, and Qian Niu. Phys.\nRev. Lett. , 102:096801, 2009.\n[18]Xiangang Wan, Ari M. Turner, Ashvin Vishwanath, and\nSergey Y. Savrasov. Phys. Rev. B , 83:205101, 2011.\n[19]Zeying Zhang, Zhi-Ming Yu, Gui-Bin Liu, and Yugui\nYao.Computer Physics Communications , 270:108153,\n2022.\n[20]Min Yu and Dallas R. Trinkle. The Journal of chemical\nphysics, 134:064111, 2011.\n[21]J.B. Goodenough. Interscience Monographs on Chem-\nistry, Inorganic Chemistry Section. Krieger, 1976.vii\nhttps://books.google.com/books?id=2sjdtAEACAAJ.\n[22]P. W. Anderson. Phys. Rev. , 115:2–13, 1959.\n[23]Junjiro Kanamori. Journal of Applied Physics , 31:S14–\nS23, 1960.\n[24]John B. Goodenough. Phys. Rev. , 100:564–573, 1955.\n[25]Joachim St¨ ohr and Hans Christoph Siegmann. Springer\nSeries in Solid-State Sciences. Springer, Berlin ; New\nYork, 2006. ISBN 978-3-540-30282-7.\n[26]Xuli Cheng, Shaowen Xu, Fanhao Jia, Guodong Zhao,\nMinglang Hu, Wei Wu, and Wei Ren. Phys. Rev. B , 104:\n104417, 2021.\n[27]Bevin Huang, Genevieve Clark, Efr´ en Navarro-\nMoratalla, Dahlia R. Klein, Ran Cheng, Kyle L. Seyler,\nDing Zhong, Emma Schmidgall, Michael A. McGuire,\nDavid H. Cobden, Wang Yao, Di Xiao, Pablo Jarillo-\nHerrero, and Xiaodong Xu. Nature, 546:270–273, 2017.\n[28]Wei-Bing Zhang, Qian Qu, Peng Zhu, and Chi-Hang\nLam.J. Mater. Chem. C , 3:12457–12468, 2015.\n[29]L. X. Yang, Z. K. Liu, Y. Sun, H. Peng, H. F. Yang,\nT. Zhang, B. Zhou, Y. Zhang, Y. F. Guo, M. Rahn,\nD. Prabhakaran, Z. Hussain, S.-K. Mo, C. Felser, B. Yan,\nand Y. L. Chen. Nature Phys , 11:728–732, 2015.\n[30]Hongming Weng, Chen Fang, Zhong Fang, B. Andrei\nBernevig, and Xi Dai. Phys. Rev. X , 5:011029, 2015.\n[31]Gang Xu, Hongming Weng, Zhijun Wang, Xi Dai, and\nZhong Fang. Phys. Rev. Lett. , 107:186806, 2011.\n[32]Gui-Bin Liu, Zeying Zhang, Zhi-Ming Yu, Shengyuan A.\nYang, and Yugui Yao. Phys. Rev. B , 105:085117, 2022.\n[33]Di Wang, Feng Tang, Yongping Du, and Xiangang Wan.Phys. Rev. B , 96:205159, 2017.\n[34]Myung-Hwan Whangbo, Elijah E. Gordon, Hongjun Xi-\nang, Hyun-Joo Koo, and Changhoon Lee. Acc. Chem.\nRes., 48:3080–3087, 2015.\n[35]John P. Perdew, Kieron Burke, and Matthias Ernzerhof.\nPhys. Rev. Lett. , 77:3865–3868, 1996.\n[36]G. Kresse and J. Furthm¨ uller. Phys. Rev. B , 54:11169–\n11186, 1996.\n[37]A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen. Phys.\nRev. B, 52:R5467–R5470, 1995.\n[38]Suzanne Lutfalla, Vladimir Shapovalov, and Alexis T.\nBell.J. Chem. Theory Comput. , 7:2218–2223, 2011.\n[39]Zhenpeng Hu and Horia Metiu. J. Phys. Chem. C , 115:\n5841–5845, 2011.\n[40]Atsushi Togo, Fumiyasu Oba, and Isao Tanaka. Phys.\nRev. B, 78:134106, 2008.\n[41]Atsushi Togo and Isao Tanaka. Scripta Materialia , 108:\n1–5, 2015.\n[42]Jochen Heyd, Gustavo E. Scuseria, and Matthias Ernz-\nerhof.J. Chem. Phys. , 118:8207–8215, 2003.\n[43]Arash A. Mostofi, Jonathan R. Yates, Young-Su Lee, Ivo\nSouza, David Vanderbilt, and Nicola Marzari. Computer\nPhysics Communications , 178:685–699, 2008.\n[44]QuanSheng Wu, ShengNan Zhang, Hai-Feng Song,\nMatthias Troyer, and Alexey A. Soluyanov. Computer\nPhysics Communications , 224:405–416, 2018.\n[45]R. F. L. Evans, W. J. Fan, P. Chureemart, T. A. Ostler,\nM. O. A. Ellis, and R. W. Chantrell. J. Phys.: Condens.\nMatter, 26:103202, 2014."    },    {        "title": "2209.04070v1.Understanding_magnetocrystalline_anisotropy_based_on_orbital_and_quadrupole_moments.pdf",        "content": "Understanding magnetocrystalline anisotropy based\non orbital and quadrupole moments\nYoshio Miura1;2;\u0003& Jun Okabayashi3\n1Research Center for Magnetic and Spintronic Materials, National Institute for\nMaterials Science (NIMS), Tsukuba 305-0047, Japan\n2Center for Spintronics Research Network (CSRN), Graduate School of Engineering\nScience, Osaka University, Machikaneyama 1-3, Toyonaka, Osaka 560-8531, Japan\n3Research Center for Spectrochemistry, The University of Tokyo, Bunkyo-ku, Tokyo\n113-0033, Japan\nE-mail:\u0003MIURA.Yoshio@nims.go.jp\nJune 2022\nAbstract.\nUnderstanding magnetocrystalline anisotropy (MCA) is fundamentally important\nfor developing novel magnetic materials. Therefore, clarifying the relationship between\nMCA and local physical quantities observed by spectroscopic measurements, such as\nthe orbital and quadrupole moments, is necessary. In this review, we discuss MCA\nand the distortion e\u000bects in magnetic materials with transition metals (TMs) based\non the orbital and quadrupole moments, which are related to the spin-conserving\nand spin-\rip terms in the second-order perturbation calculations, respectively. We\nrevealed that orbital moment stabilized the spin moment in the direction of the larger\norbital moment, while the quadrupole moment stabilized the spin moment along the\nlongitudinal direction of the spin-density distribution. The MCA of the magnetic\nmaterials with TMs and their interfaces can be determined from the competition\nbetween these two contributions. We showed that the perpendicular MCA of the face-\ncentered cubic (fcc) Ni with tensile tetragonal distortion arose from the orbital moment\nanisotropy, whereas that of Mn-Ga alloys originated from the quadrupole moment of\nspin density. In contrast, in the Co/Pd(111) multilayer and Fe/MgO(001), both the\norbital moment anisotropy and quadrupole moment of spin density at the interfaces\ncontributed to the perpendicular MCA. Understanding the MCA of magnetic materials\nand interfaces based on orbital and quadrupole moments is essential to design MCA\nof novel magnetic applications.\n1. Introduction\nMagnetocrystalline anisotropy (MCA), in which the internal energy varies on the\nmagnetization direction, is an important physical property of magnetic materials\n[1]. Perpendicular magnetization is required to enhance the thermal stability of\nmagnetization directions in permanent magnets and spintronic materials, used in\ninformation storage media and magnetoresistive random access memory (MRAM) [2, 3].arXiv:2209.04070v1  [cond-mat.mtrl-sci]  9 Sep 2022Understanding MCA based on orbital and quadrupole moments 2\nContrary to this, some studies suggest the importance of in-plane MCA of magnetic\nmaterials for applications such as a magnetic under-layer of a perpendicular recording\nmedium [4] and a spin torque oscillator for microwave-assisted magnetic recording\n(MAMR) [5, 6]. Furthermore, the development of soft magnets requires extremely small\nMCA under distortion to eliminate losses caused by magnetic hysteresis and circulating\nloops of current [7]. Hence, understanding MCA is necessary for the development\nof novel magnetic materials. Several experimental and theoretical studies have been\nconducted to determine the microscopic origin of MCA.\nTheoretically, MCA originates from spin-orbit interaction (SOI), which is a one-\nbody interaction between the electron's spin and its orbital motion and is caused by\nthe di\u000berence in the crystal symmetry because of the change in the magnetization\ndirection via SOI [8]. MCA of rare earth magnets arises from strong SOI of felectrons,\nwhich has been analyzed by the one-ion Hamiltonian, and understood as a competition\nbetween the crystalline electric \feld and SOI [9]. On the other hand, the MCA of\nmagnetic materials with TMs should be analyzed using electronic band structures\n[10, 11, 12, 13, 14, 15, 16, 17, 18, 19].\nFirst-principles density functional theory (DFT) calculations [20, 21, 22] are\nimportant for quantitatively obtaining MCA energies and electronic structures.\nHowever, owing to the small SOI of TMs, understanding the origin of MCA is di\u000ecult\nby simply analyzing the electronic structures by changing the magnetization directions.\nTherefore, a second-order perturbation analysis of SOI [23, 24, 25, 26, 27, 28, 29, 30, 31]\nwill be useful in deducing the physical origin of MCA.\nIn the second-order perturbation analysis of MCA, we can resolve MCA energies\nin terms of atoms, orbitals, and spins. By expanding the wavefunctions in DFT\ncalculations with local atomic orbitals and calculating the second-order perturbation\nterms of SOI using the localized basis sets, the contribution of speci\fc atoms, orbitals,\nand spins to MCA can be clari\fed quantitatively [32, 33]. In addition, the spin-\nconserving and spin-\rip terms in the second-order perturbation calculations of SOI are\nrelated to the orbital and quadrupole moments observed in spectroscopy, respectively\n[23, 29, 30]. Bruno theoretically showed that the spin-conserving term between\noccupied and unoccupied minority-spin states is related to the orbital moment and\nthat MCA energies are proportional to the di\u000berence in the orbital moments between\nthe perpendicular and in-plane magnetization directions [23]. This formula, known as\nthe Bruno relation, is widely used to estimate MCA energies from orbital moments\nobtained by spectroscopic measurements based on X-ray magnetic circular dichroism\n(XMCD) and X-ray magnetic linear dichroism (XMLD), using the magneto-optical sum\nrule [34, 35, 36, 37]. Furthermore, Wang, Wu, and Freeman showed that the spin-\rip\nterm between occupied majority-spin and unoccupied minority-spin states is related to\nthe quadrupole moment of spin density [24]. The quadrupole moment of spin density is\nthe same physical quantity with the magnetic dipole moment, which is extended to MCA\nanalysis with XMCD and XMLD measurements by van der Laan and St ohr [29, 30, 38].\nAccording to the Ref. [29], MCA energies ( EMCA) in the form of spin-conservingUnderstanding MCA based on orbital and quadrupole moments 3\nand spin-\rip terms are expressed as the orbital moment anisotropy and the magnetic\ndipole moment:\nEMCA\u0019\u0018\n4\u0001morb+21\n2\u00182\n\u0001excmT; (1)\nwhere \u0001morbandmTare the orbital moment anisotropy and magnetic dipole moment,\nrespectively, and \u0018and \u0001 excare the SOI constant and exchange splitting of the\ncorresponding materials, respectively. The orbital moment anisotropy is de\fned by\n\u0001morb=mperp:\norb\u0000minp:\norb, wheremperp:\norb andminp:\norbare the orbital moment with the\nperpendicular and in-plane magnetization, respectively.\nEq. (1) connects MCA energies to orbital and quadrupole moments measured by\nspectroscopy and is important for the microscopic understanding of MCA in realistic\nmaterials. According to Eq. (1), the positive and negative magnetic dipole moments\n(mT) contribute to a perpendicular MCA and an in-plane MCA, respectively. However,\nthere is a lack of intuitive understanding on the relationship between the magnetic\ndipole moments and MCA, i.e., why the positive (negative) mTcontributes to the\nperpendicular (in-plane) MCA. Because Eq. (1) is based on several approximations and\nassumptions, separately verifying this equation for various systems with XMCD and\nXMLD measurements is necessary. In addition, examining the conformity of theoretical\nresults by the second-order perturbation calculations based on DFT calculations with\nexperimental results is important.\nIn this review, in order to obtain intuitive understandings of MCA, we investigate\nthe dependence of MCA in magnetic materials with TMs on orbital and quadrupole\nmoments based on \frst-principle DFT and the second-order perturbation calculations,\ntogether with experimental veri\fcations. First, we show the detailed derivations\nof the relation between MCA energies and spectroscopic quantities through second-\norder perturbation calculations of SOI. The studies of Ref. [23] and [29] have been\nintended not to give the \fnal answers of MCA, but to give the important direction\nto understand MCA energy. To give a further important step in the right direction\nto understand MCA, intuitive understandings and classi\fcations of MCA for various\nsystem depending on the interface chemical bonding will be necessary. Thereafter, we\nreview collaborative work involving theoretical calculations and XMCD and XMLD\nmeasurements for understanding MCA of TM \flms.\nWe review also strain dependent MCA of bulk and interfaces, because they give\nrise to the precise determination of MCA characteristics.\nFirst, we discuss the MCA of fcc Ni with in-plane lattice distortion by the\npiezoelectricity of the underlayer BaTiO 3[39]. We \fnd that the anisotropy of the orbital\nmoments could describe the MCA of fcc Ni with tetragonal distortion. We then discuss\nthe MCA of Mn-Ga alloys, which shows a strong perpendicular MCA without a heavy\nelement. We clarify that the perpendicular MCA of Mn-Ga alloys is attributed to the\npositive magnetic dipole of Mn owing to the cigar-type distribution of the spin density\n[40]. Next, we discuss the MCA of the Co( t)Pd(111) multilayer showing perpendicular\nMCA when the thickness of the Co layer tis less than 0.6 nm [41]. We reveal that bothUnderstanding MCA based on orbital and quadrupole moments 4\nthe interfacial orbital moments anisotropy in Co and magnetic dipole moment in Pd are\ncrucial for the perpendicular MCA. Thereafter, we discuss the perpendicular MCA of\nFe/MgO(001) [42, 43], frequently investigated as an interface MCA of magnetic tunnel\njunctions (MTJs) used in MRAM applications. We \fnd that tetragonal distortions\nin MTJs, as well as in interface structures, signi\fcantly a\u000bect the contribution of the\norbital and quadrupole moments to MCA. This dependence of MCA on tetragonal\ndistortions may prove useful for the voltage control of MCA in future spintronic devices\n[44, 45, 46]. We will give an systematic discussion on MCA of bulk and interfaces and\nintuitive understandings of MCA based on the orbital and quadrupole moments.\n2. Second-order perturbation calculations of SOI\n2.1. MCA energies\nMCA energies are analyzed using second-order perturbation of SOI. The spin-orbit (SO)\nHamiltonian is given by\n^HSO=X\nI\u0018I~^L\u0001~^S;\nwhere~^L= (^Lx;^Ly;^Lz) and~^S= (^Sx;^Sy;^Sz) are the angular momentum and spin angular\nmomentum operators, measured in units of Dirac's constant ~, respectively; Iis the\natomic position index, and \u0018Iis the spin-orbit coupling strength at I.\u0018Ihas a localized\ncharacter and is given by\n\u0018I\u0011\u0018I(r) =~2\n2m2c21\nrdVI(r)\ndr; (2)\nwherecis the speed of light, mis the electron mass, ris the distance from the atomic\ncenter, and VI(r) is the potential between the electron and atomic nucleus [47]. We\nassume that H(0)is the one-electron Hamiltonian without SOI, satisfying the non-\nperturbative Schr odinger equation\nH(0)j~kn\u001bi=\u000f~kn\u001bj~kn\u001bi;\nwherej~kn\u001biis an unperturbed state of energy \u000f~kn\u001bwith indices ~k\u0000point~k, bandn, and\nspin\u001b. Using the eigenstate and eigenvalue, the variation in the total energy due to the\nsecond-order perturbation of SOI is given by\nE(2)=\u0000X\n~kunoccX\nn0\u001b0occX\nn\u001bjh~kn0\u001b0j^HSOj~kn\u001bij2\n\u000f~kn0\u001b0\u0000\u000f~kn\u001b: (3)\nj~kn\u001bican be expanded with an orthogonal basis of atomic orbitals labeled \u0016(or\u0015):\nj~kn\u001bi=X\nj\u0016c~kn\nj\u0016\u001bj\u0016\u001biei~k\u0001~Rj; (4)Understanding MCA based on orbital and quadrupole moments 5\nwhere~Rjis the atomic position at site jin the unit cell. We can obtain the second-\norder contribution of ^HSOto the total energy as a sum over terms depending on the\nspin transition processes, atomic orbitals, and atomic sites:\nE(2)=\u0000X\n\u001b\u001b0X\nII0\u0018I\u00180\nIX\n\u0015\u00150\u00160\u0016h\u0015\u001bj~^L\u0001~^Sj\u00150\u001b0ih\u00160\u001b0j~^L\u0001~^Sj\u0016\u001biG\u001b\u001b0\nII0(\u0015\u00150;\u00160\u0016);(5)\nwhereG\u001b\u001b0\nII0(\u0015\u00150;\u00160\u0016) is an integral of joint local density of states (LDOS) given by\nG\u001b\u001b0\nII0(\u0015\u00150;\u00160\u0016) =X\n~koccX\nnunoccX\nn0c~kn\u0003\nI0\u0015\u001bc~kn0\nI0\u00150\u001b0c~kn0\u0003\nI\u00160\u001b0c~kn\nI\u0016\u001b\n\u000f~kn0\u001b0\u0000\u000f~kn\u001b: (6)\nTo derive the Eq. (5), we assume that SOI acts only at the same atomic site and use\nthe relationP\njj0ei~k\u0001(~Rj\u0000~Rj0)\u0018(j~Rj\u0000~RIj) =\u0018I. Although SOI is e\u000bective within the same\natomic site, the joint LDOS G\u001b\u001b0\nII0(\u0015\u00150;\u00160\u0016) includes the electronic states for di\u000berent\natomic sites in the unit cell ( IandI0) because of the hybridization of atomic orbitals in\nthe crystal. The joint LDOS G\u001b\u001b0\nII0(\u0015\u00150;\u00160\u0016) can be calculated using \frst-principles DFT\ncalculations, and the coe\u000ecients c~kn\nj\u0016\u001bare obtained from the projections of unperturbed\neigenstate on each localized atomic orbital. We calculated G\u001b\u001b0\nII0(\u0015\u00150;\u00160\u0016) using the\nVienna ab initio simulation package (VASP) code [48, 49, 50]. Furthermore, we set the\nspin-orbit coupling constants de\fned by Eq. (2), which are used within VASP code,\nas follows: \u0018Mn(3d)= 41:5 meV ,\u0018Fe(3d)= 54:3 meV,\u0018Co(3d)= 69:4 meV,\u0018Ni(3d)= 87:2\nmeV,\u0018O(2d)= 24:3 meV,\u0018Mg(2p)= 47:5 meV,\u0018Ga(3p)= 35:4 meV,\u0018Pd(4d)= 187 meV.\nTo obtain MCA energies, its dependence on the magnetization direction must\nbe considered. Noncollinear magnetization can be introduced by expressing the spin-\nquantum axis of local basis sets j\u001bi(\u001b=\"or#) with a spin-1/2 rotation matrix according\nto K ubler's formulation [51, 52]\nj\"i=ei\u001e\n2cos\u0012\n2j~\"i+e\u0000i\u001e\n2sin\u0012\n2j~#i; (7)\nj#i=\u0000ei\u001e\n2sin\u0012\n2j~\"i+e\u0000i\u001e\n2cos\u0012\n2j~#i; (8)\nwhere\u0012and\u001eare the polar and azimuthal angles of magnetization with respect to the\nzandx-axes of the system, and zaxis is perpendicular to the plane of the tetragonal or\nhexagonal systems. j~\u001biindicates a spin state along the global spin-quantum axis, \fxed\nto thezaxis of the system.\nBy using Eqs. (7) and (8) in the matrix element h\u00160\u001b0j~^L\u0001~^Sj\u0016\u001bi, we obtain the\nfollowing expressions depending on the magnetization direction:\nh\u00160\"j~^L\u0001~^Sj\u0016\"i=1\n2[sin\u0012cos\u001eh\u00160j^Lxj\u0016i\n\u0000sin\u0012sin\u001eh\u00160j^Lyj\u0016i+ cos\u0012h\u00160j^Lzj\u0016i];\nh\u00160#j~^L\u0001~^Sj\u0016#i=\u00001\n2[sin\u0012cos\u001eh\u00160j^Lxj\u0016i\n\u0000sin\u0012sin\u001eh\u00160j^Lyj\u0016i+ cos\u0012h\u00160j^Lzj\u0016i];\nh\u00160\"j~^L\u0001~^Sj\u0016#i=1\n2[(cos\u0012cos\u001e\u0000isin\u001e)h\u00160j^Lxj\u0016iUnderstanding MCA based on orbital and quadrupole moments 6\n\u0000(cos\u0012sin\u001e+icos\u001e)h\u00160j^Lyj\u0016i\u0000sin\u0012h\u00160j^Lzj\u0016i];\nh\u00160#j~^L\u0001~^Sj\u0016\"i=1\n2[(cos\u0012cos\u001e+isin\u001e)h\u00160j^Lxj\u0016i\n\u0000(cos\u0012sin\u001e\u0000icos\u001e)h\u00160j^Lyj\u0016i\u0000sin\u0012h\u00160j^Lzj\u0016i];\nwhere we have used the eigenstate and eigenvalue relation between~^S= (^Sx;^Sy;^Sz)\nandj~\u001biand the orthonormal property of the eigenstate h~\u001bj~\u001b0i=\u000e~\u001b~\u001b0. Because we\nconsider the uniaxial MCA energies, the energy di\u000berence between the perpendicular\nmagnetization ( \u0012= 0 and\u001e= 0) and in-plane magnetization ( \u0012=\u0019\n2and\u001e= 0) should\nbe calculated. In this case, the matrix elements of SOI for each magnetization direction\nare given by\nh\u00160\"j~^L\u0001~^Sj\u0016\"i\u0012=0;\u001e=0=\u0000h\u00160#j~^L\u0001~^Sj\u0016#i\u0012=0;\u001e=0=1\n2h\u00160j^Lzj\u0016i; (9)\nh\u00160\"j~^L\u0001~^Sj\u0016\"i\u0012=\u0019\n2;\u001e=0=\u0000h\u00160#j~^L\u0001~^Sj\u0016#i\u0012=\u0019\n2;\u001e=0=1\n2h\u00160j^Lxj\u0016i; (10)\nh\u00160\"j~^L\u0001~^Sj\u0016#i\u0012=0;\u001e=0=h\u00160#j~^L\u0001~^Sj\u0016\"i\u0003\n\u0012=0;\u001e=0=1\n2h\u00160j^Lx\u0000i^Lyj\u0016i;(11)\nh\u00160\"j~^L\u0001~^Sj\u0016#i\u0012=\u0019\n2;\u001e=0=h\u00160#j~^L\u0001~^Sj\u0016\"i\u0003\n\u0012=\u0019\n2;\u001e=0=\u00001\n2h\u00160j^Lz+i^Lyj\u0016i:(12)\nSince we de\fne the MCA energy as positive for perpendicular magnetization ( \u0012= 0 and\n\u001e= 0), the MCA energy in the second-order perturbation calculations is given by\nE(2)\nMCA\u0011E(2)(\u0012=\u0019\n2;\u001e= 0)\u0000E(2)(\u0012= 0;\u001e= 0)\n=E\"\"\nMCA+E##\nMCA+E\"#\nMCA+E#\"\nMCA (13)\n=X\nII0\u0018I\u0018I0\n4X\n\u0015\u00150\u0016\u00160[h\u0015j^Lzj\u00150ih\u00160j^Lzj\u0016i\u0000h\u0015j^Lxj\u00150ih\u00160j^Lxj\u0016i]G\"\"\nII0(\u0015\u00150;\u00160\u0016)\n+X\n\u0015\u00150\u0016\u00160[h\u0015j^Lzj\u00150ih\u00160j^Lzj\u0016i\u0000h\u0015j^Lxj\u00150ih\u00160j^Lxj\u0016i]G##\nII0(\u0015\u00150;\u00160\u0016)\n\u0000X\n\u0015\u00150\u0016\u00160[h\u0015j^Lzj\u00150ih\u00160j^Lzj\u0016i\u0000h\u0015j^Lxj\u00150ih\u00160j^Lxj\u0016i]G\"#\nII0(\u0015\u00150;\u00160\u0016)\n\u0000X\n\u0015\u00150\u0016\u00160[h\u0015j^Lzj\u00150ih\u00160j^Lzj\u0016i\u0000h\u0015j^Lxj\u00150ih\u00160j^Lxj\u0016i]G#\"\nII0(\u0015\u00150;\u00160\u0016):(14)\nInE(2)\nMCA,h^Lyiterms in Eqs. (11) and (12) are automatically canceled out. E\u001b\u001b0\nMCA\nis the spin-resolved MCA energy, where E\"\"\nMCA andE##\nMCA are spin-conserving terms,\nandE\"#\nMCA andE#\"\nMCA are spin-\rip terms. It is important to note that the sign of the\nmatrix elements of ^Lzand^Lxare di\u000berent for the spin-conserving (positive) and spin-\rip\n(negative) terms. This means that the local electronic structures around the Fermi level\nhave an opposite contribution to MCA in the spin-conserving and spin-\rip processes.\nThe opposite sign between the spin-conserving and the spin-\rip terms arises from the\neigenvalue of ^Szfor the spin eigenstate j\u001bi, i.e., ^Szj\"i= +1\n2j\"iand ^Szj#i=\u00001\n2j\"i.\nThe spin-conserving term includes two eigenvalues with the same sign, wheres the spin-\n\rip term has two eigenvalues with the di\u000berent sign, leading the opposite contribution\nto MCA.Understanding MCA based on orbital and quadrupole moments 7\nTable 1. Nonzero matrix elements of the angular momentum operator\n~^L= (^Lx;^Ly;^Lz) for thedorbitals in Cartesian coordinates.\nhdx2\u0000y2j^Lxjdyzi= 1hdxyj^Lxjdzxi= 1hd3z2\u0000r2j^Lxjdyzi=p\n3\nhdx2\u0000y2j^Lyjdzxi= 1hdyzj^Lyjdxyi= 1hd3z2\u0000r2j^Lyjdzxi=p\n3\nhdx2\u0000y2j^Lzjdxyi= 2hdzxj^Lzjdyzi= 1\nIn Table 1, we show the nonzero matrix elements of the angular momentum operator\n~^L= (^Lx;^Ly;^Lz) for thedorbitals in Cartesian coordinates. The matrix elements of ^Lz\nare nonzero for dorbitals with the same magnetic quantum numbers for occupied and\nunoccupied states, such as dxy\u0000dx2\u0000y2anddyz\u0000dzx, contributing to the perpendicular\nMCA in the spin-conserving terms ( E\"\"\nMCA andE##\nMCA) and in-plane MCA in the spin-\n\rip terms ( E\"#\nMCA andE#\"\nMCA). On the other hand, the matrix elements of ^Lxare\nnonzero for dorbitals when the di\u000berence in magnetic quantum numbers between the\noccupied and unoccupied states is \u00061, contributing to the in-plane MCA in E\"\"\nMCA and\nE##\nMCA terms and the perpendicular MCA in E\"#\nMCA andE#\"\nMCA terms. These analyses\nenable us understand the relationship between the local electronic structures around\nthe Fermi level and their MCA contribution at each atomic site; This may be useful in\ndesigning new ferromagnetic materials with a strong perpendicular MCA for spintronics\napplications.\n2.2. Orbital moments\nThe \frst-order correction of wavefunctions for perturbation of SOI is given by\nj~kn\u001bi(1)=j~kn\u001bi+unoccX\nn0\u001b0occX\nn\u001bh~kn0\u001b0j^HSOj~kn\u001bi\n\u000f~kn0\u001b0\u0000\u000f~kn\u001bj~kn0\u001b0i: (15)\nThe orbital moment h^L\u0010i, which is the component of~^Lparallel to the direction\nof the spin quantum axis \u0010= (\u0012;\u001e), is given by as the expectation value of the\nangular momentum operator in Eq. (15). Because the expectation value of ^L\u0010for\nthe unperturbed states is zero, the orbital moment is given by\nh^L\u0010i'2X\n~kunoccX\nn0\u001b0occX\nn\u001bh~kn\u001bj^L\u0010j~kn0\u001b0ih~kn0\u001b0j^HSOj~kn\u001bi\n\u000f~kn0\u001b0\u0000\u000f~kn\u001b;\nwhere the factor of 2 comes from the Hermitian conjugate of the SOI Hamiltonian. We\nneglect the squared term of ^HSO.\nBy expandingj~kn\u001biwith the orthogonal basis given by Eq. (4), we obtained the\norbital moment at atomic site Iusing the following equation:\nh^L\u0010iI=\u00002\u0018IX\n\u001b\u001b0X\nI0X\n\u0015\u00150\u00160\u0016h\u0015\u001bj^L\u0010j\u00150\u001b0ih\u00160\u001b0j~^L\u0001~^Sj\u0016\u001biG\u001b\u001b0\nII0(\u0015\u00150;\u00160\u0016):Understanding MCA based on orbital and quadrupole moments 8\nWe note thath^L\u0010i=P\nIh^L\u0010iI. The sign of the orbital moment is determined by the\ntermh\u0015\u001bj^L\u0010j\u00150\u001b0i, which is equivalent to the~^Lcomponent parallel to the direction of\nthe spin quantum axis, and the following relation holds [23]:\nh\u0015\u001bj^L\u0010j\u00150\u001b0i= 2sgn(\u001b)\u000e\u001b\u001b0h\u0015\u001bj~^L\u0001~^Sj\u00150\u001b0i; (16)\nwhere sgn(\") = +1 and sgn(\") =\u00001. Due to\u000e\u001b\u001b0in Eq. (16), the spin-\rip terms do\nnot contribute to orbital moment. Thus, the orbital moment can be written only by the\nspin-conserving terms as follow:\nh^L\u0010iI=\u00004\u0018IX\n\u001bsgn(\u001b)X\nI0X\n\u0015\u00150\u00160\u0016h\u0015\u001bj~^L\u0001~^Sj\u00150\u001bih\u00160\u001bj~^L\u0001~^Sj\u0016\u001biG\u001b\u001b\nII0(\u0015\u00150;\u00160\u0016):\nFinally, the orbital moment with the spin moment along the zaxis (\u0012= 0 and\u001e= 0)\ncorresponding to mperp:\norb and the spin moment along the xaxis (\u0012=\u0019\n2and\u001e= 0)\ncorresponding to minp:\norbare given by\nh^LziI=\u0018IX\nI0X\n\u0015\u00150\u00160\u0016h\u0015j^Lzj\u00150ih\u00160j^Lzj\u0016i[G##\nII0(\u0015\u00150;\u00160\u0016)\u0000G\"\"\nII0(\u0015\u00150;\u00160\u0016)];\nh^LxiI=\u0018IX\nI0X\n\u0015\u00150\u00160\u0016h\u0015j^Lxj\u00150ih\u00160j^Lxj\u0016i[G##\nII0(\u0015\u00150;\u00160\u0016)\u0000G\"\"\nII0(\u0015\u00150;\u00160\u0016)]:\nTo obtain these equations, we used the relation in Eqs. (9){(10), and the eigenstates\nand eigenvalues of the spin angular momentum operator.\nIf we assume that the spin-\rip terms E\"#\nMCAandE#\"\nMCAand the spin-conserving term\nE\"\"\nMCAin Eq. (14) are negligible compared to the spin-conserving term E##\nMCA, the MCA\nenergy can be expressed as the sum of the orbital moment anisotropy at each atomic\nsite \u0001mI\norb:\nE(2)\nMCA\u00191\n4X\nI\u0018I[h^LziI\u0000h^LxiI] =1\n4X\nI\u0018I\u0001mI\norb; (17)\nThis is the Bruno relation, commonly used to connect MCA energies and the observed\norbital moments by XMCD.\nBecause the ferromagnetic materials with more than half elements as Fe, Co, and\nNi have fully occupied majority-spin states, the unoccupied majority-spin states around\nthe Fermi level are negligible. In this case, neglecting E\"\"\nMCA andE#\"\nMCA is reasonable\nfor MCA energies. However, there are cases in which the spin-\rip-term through the\nunoccupied minority-spin states E\"#\nMCA is not too small to be ignored compared to the\nspin-conserving term E##\nMCA, and the Bruno relation fails to describe MCA energies,\nboth quantitatively and qualitatively. If the energy depth of the occupied majority-spin\nstates is far from the Fermi level owing to strong exchange splitting, we can neglect\nthe spin-\rip term E\"#\nMCA because of the large denominator in Eq. (6), and the Bruno\nrelation well describes MCA energies.\n2.3. Quadrupole moments\nWang, Wu, and Freeman proposed that the spin-\rip term E\"#\nMCA can be related to\nquadrupole moments [24]. We start the formulation of MCA energies in Eq. (3) in theUnderstanding MCA based on orbital and quadrupole moments 9\nsecond-order perturbation. We then expand the unperturbed unoccupied eigenstates\nj~kn0#iusing the local atomic orbitals of Eq. (4) and retain the occupied eigenstates\nj~kn\"i.\nE\"#\nMCA=\u00001\n4X\n~kunoccX\nn0occX\nnX\nI\u00182\nI\u0014P\n\u00150\u00160c~kn0\u0003\nI\u00150#c~kn0\nI\u00160#h\u00150#j^Lzj~kn\"ih~kn\"j^Lzj\u00160#i\n\u000f~kn0#\u0000\u000f~kn\"\n\u0000P\n\u00150\u00160c~kn0\u0003\nI\u00150#c~kn0\nI\u00160#h\u00150#j^Lxj~kn\"ih~kn\"j^Lxj\u00160#i\n\u000f~kn0#\u0000\u000f~kn\"\u0015\n:(18)\nThe factor of1\n4comes from the eigenvalues of two spin angular momentum operators, and\nwe leaves brakets of spin states j\u001bito clearly specify the spin state in the second-order\nperturbation term. Here, we express a spin state by \u001bfor the global spin-quantum\naxis alongz-axis. Wang, Wu, and Freeman introduced two approximations to relate\nthe spin-\rip term and quadrupole moment. First, the replacement of the di\u000berence in\neigenvalues between the unoccupied and occupied states in the denominator with \u0001 exc\nis considered:\n\u000f~kn0#\u0000\u000f~kn\"\u0019\u0001exc; (19)\nwhere \u0001 excis the exchange splitting of ferromagnetic materials, corresponding to the\nrange of eigenvalues between the majority-spin occupied and minority-spin unoccupied\nstates. Second, the use of the completeness relation of brakets on occupied majority-spin\nstates:\noccX\nnjn\"ihn\"j\u0019 1: (20)\nThis relation holds if we consider ferromagnetic materials with more than half elements\nas Fe, Co, and Ni, where the majority-spin states are fully occupied. In this case, the\nsum of all occupied states is equivalent to the sum of all states.\nBy using Eqs. (19) and (20), the square of the expectation of the angular\nmomentum operator can be expressed as the expectation of the square of the angular\nmomentum operator:\nE\"#\nMCA\u0019\u00001\n4X\n~kunoccX\nn0X\nI\u00182\nIP\n\u00150\u00160c~kn0\u0003\nI\u00150#c~kn0\nI\u00160#[h\u00150#j^L2\nzj\u00160#i\u0000h\u00150#j^L2\nxj\u00160#i]\n\u0001exc:\nFurthermore, if we assume the tetragonal system which has the same lattice structure\nfor the in-plane xandyaxes, the following relation holds.\nh^L2\nxi=h^L2\nyi=1\n2h^L2\nx+^L2\nyi=1\n2h^L2\u0000^L2\nzi:\nThus, for tetragonal systems, we have\nE\"#\nMCA\u0019\u00001\n8X\nI\u00182\nIX\n~kunoccX\nn0P\n\u00160jc~kn0\nI\u00160#j2h\u00160#j3^L2\nz\u0000^L2j\u00160#i\n\u0001exc:Understanding MCA based on orbital and quadrupole moments 10\nHere, the matrix of h^L2iandh^L2\nziare diagonal for atomic-orbital basis set, and we use\nthe orthonormal property, h\u00150j\u00160i=\u000e\u00150\u00160. The operator 3 ^L2\nz\u0000^L2has the same form as\nthe z-component of intra-atomic quadrupole moment operator:\n^Qzz\u00112\n21(3^L2\nz\u0000^L2): (21)\nFinally, we can express the spin-\rip term of MCA energies related to the quadrupole\nmoments of the unoccupied minority-spin electron densities\nE\"#\nMCA\u0019\u000021\n16X\nI\u00182\nIX\n~kunoccX\nn0P\n\u00160jc~kn0\nI\u00160#j2h\u00160#j^Qzzj\u00160#i\n\u0001exc: (22)\nThis is an approximate expression of the spin-\rip term E\"#\nMCArelated to the quadrupole\nmoment. Based on the de\fnition of quadrupole moment in Eq. (21), oblate distributions\nof unoccupied minority-spin electrons along zaxis (cigar-like quadrupole), e.g., d3z2\u0000r2\norbital, gives a negative quadrupole moment, yielding perpendicular (positive) MCA\nthrough Eq. (22). In contrast, prolate distributions of unoccupied minority-spin\nelectrons along xyplane (pancake-like quadrupole), e.g., dx2\u0000y2anddxyorbitals, produce\na positive quadrupole moment, yielding an in-plane (negative) MCA.\nIn Eq. (22), the spin-\rip term of MCA energies E\"#\nMCAis described by the quadrupole\nmoments of the unoccupied minority-spin electrons. In addition, E\"#\nMCAcan be expressed\nby the quadrupole moments of occupied minority-spin electrons :\nE\"#\nMCA\u0019+21\n16X\nI\u00182\nIX\n~koccX\nnP\n\u0016jc~kn\nI\u0016#j2h\u0016#j^Qzzj\u0016#i\n\u0001exc; (23)\nwherePunocc\nn0=Pall\nn\u0000Pocc\nnand the quadrupole moment h^Qzzi#\nIincluding all states\n(occupied and unoccupied states) is zero, and h^Qzzi#\nI=P\n~kP\nnP\n\u0016jc~kn\nI\u0016#j2h\u0016#j^Qzzj\u0016#i.\nEq. (22) and Eq. (23) shows that the contribution of quadrupole moments to MCA\nenergies is opposite between the occupied and unoccupied states in the minority-spin\nelectrons. Then, we considered the quadrupole moments of spin density corresponding\nto the magnetic dipole moment mT. Again, we assumed that magnetic materials with\nmore than half elements as Fe, Co, and Ni have fully occupied majority-spin states.\nThis provides the following relationship:\nX\n~koccX\nnX\n\u0016jc~kn\nI\u0016\"j2h\u0016\"j^Qzzj\u0016\"i\u0019X\n~kallX\nnX\n\u0016jc~kn\nI\u0016\"j2h\u0016\"j^Qzzj\u0016\"i= 0:\nThus, we can write Eq. (23) using a intra-atomic magnetic dipole moment:\nE\"#\nMCA\u0019\u000021\n16X\nI\u00182\nI\n\u0001excX\n~koccX\nnX\n\u0016[jc~kn\nI\u0016\"j2h\u0016\"j^Qzzj\u0016\"i\u0000jc~kn\nI\u0016#j2h\u0016#j^Qzzj\u0016#i]\n=\u000021\n8X\nI\u00182\nI\n\u0001excX\n~koccX\nnX\n\u0016X\n\u001bjc~kn\nI\u0016\u001bj2h\u0016\u001bj^Qzz^Szj\u0016\u001bi\n= +21\n8X\nI\u00182\nI\n\u0001excmI\nT: (24)Understanding MCA based on orbital and quadrupole moments 11\nwhere the magnetic dipole moment and quadrupole moment of the spin density at each\natomic site is related by following equation:\nmI\nT\u0011\u0000h ^Qzz^SziI=\u00001\n2[h^Qzzi\"\nI\u0000h^Qzzi#\nI]: (25)\nFurthermore, we can rewrite the intra-atomic magnetic dipole moment using the spin\nmoment projected to each atomic orbitals m\u0016\nspinas follow:\nmI\nT=\u00001\n21(mpx\nspin+mpy\nspin\u00002mpz\nspin\n+ 6mdx2\u0000y2\nspin + 6mdxy\nspin\u00003mdyz\nspin\u00003mdzx\nspin\u00006md3z2\u0000r2\nspin ): (26)\nThe intra-atomic magnetic dipole moment mI\nTcan be observed by XMCD and XMLD\nmeasurements [38], and a positive (negative) mI\nTindicates the contribution of the spin-\n\rip term to the perpendicular (in-plane) MCA. Because the mI\nTterm is directly related\nto MCA energies, we do not need to consider the anisotropy of mI\nTto discuss its\ncontribution to MCA, unlike the orbital moments in Eq. (17). Furthermore, because\nthemI\nTterm in Eq. (24) is obtained using non-perturbed eigenstates, the mI\nTterm is\nirrelevant to SOI despite being derived from the second-order perturbation of SOI. In\nfact, we can obtain the mTusing Eqs. (25) and (26) without SOI. This is because the\napproximation in Eq. (20), which corresponds to neglecting the quantum uncertainty\nin the angular momentum operators,q\nhL\u0010i2\u0000hL2\n\u0010i\u00190.\nThe above treatment of angular momentum operators results in a classical\npicture of magnetic anisotropy, namely, the shape magnetic anisotropy (SMA) due to\nmagnetostatic dipole-dipole interactions. St ohr pointed out that there is a relationship\nbetween the magnetic dipole moment derived from the spin-\rip term of MCA energies\nand magnetostatic dipole-dipole interaction (see Appendix B in Ref. [30]). Furthermore,\nbecause Eq. (20) requires the occupied majority-spin and the unoccupied miniroty-spin\nstates inE\u001b\u001b0\nMCA, the magnetic dipole moment (quadrupole moment of spin density) in\nEq. (22)-(23) are related to the spin-\rip term E\"#\nMCA of MCA energies.\n2.4. Intuitive understanding of MCA based on orbital and quadrupole moments\nBased on discussions above, MCA energies can be presented by using the orbital moment\nanisotropy and the magnetic dipole moment (quadrupole moment of spin density) as\nfollows:\nEMCA\u0019X\nI1\n4\u0018I\u0001mI\norb+21\n8X\nI\u00182\nI\n\u0001excmI\nT: (27)\nA di\u000berence in the coe\u000ecient of the mI\nTterm by a factor of1\n4is noticed in this equation\ncompared to Eq. (1). The di\u000berence comes from the eigenvalues of the spin operator~^Sas\nmentioned in Eq. (18), because Wang, Wu and Freeman described the SOI Hamiltonian\nbyHSO=\u0018~^L\u0001~^\u001b[24], where ~^\u001bis the Pauli spin matrices (twice of~^S) and a factor1\n2\nis included in \u0018. Our formulations of the spin-\rip term of MCA energies related to the\nmagnetic dipole moments (quadrupole moments of spin-density) in Eq. (18)-(24) usingUnderstanding MCA based on orbital and quadrupole moments 12\nOrbital moment term\n<Lz> \n𝑑𝑑𝑥𝑥2−𝑦𝑦2,𝑑𝑑𝑥𝑥𝑦𝑦z\n<Lx> \n𝑑𝑑3𝑧𝑧2−𝑟𝑟2,𝑑𝑑𝑦𝑦𝑧𝑧,𝑑𝑑𝑧𝑧𝑥𝑥z<S>\nPerpendicular MCA\n<S>In-plane MCA\nz⇒Oblate distribution \nstableunstablePan-cake typeMagnetic dipole moment term (Quadrupole moment of spin density)\n𝑑𝑑𝑥𝑥2−𝑦𝑦2,𝑑𝑑𝑥𝑥𝑦𝑦⇒Prolate distribution\nzCigar type\nstable\nunstable𝑑𝑑3𝑧𝑧2−𝑟𝑟2,\n𝑑𝑑𝑦𝑦𝑧𝑧,𝑑𝑑𝑧𝑧𝑥𝑥𝑄𝑄zz<0(𝑚𝑚T>0) 𝑄𝑄zz<0(𝑚𝑚T>0) 𝑄𝑄zz=0(𝑚𝑚T=0)\n⇒Isotropic distribution\n(a)\n(b)\nNo difference\nFigure 1. (Color online)Schematic image on e\u000bects of the orbital moment anisotropy\nand quadrupole moment of spin density on MCA. (a)Spin moment can be stabilized\nalong the direction of the larger orbital moment due to SOI, described as the Bruno\nterm in Eq. (17). (b)Spin moment tend to be parallel to the longitudinal direction of\nthe spin-density distribution, characterized by the quadrupole moment ( intra-atomic\nmagnetic dipole moment) and described as van der Laan term in Eq. (24).\nHSO=\u0018~^L\u0001~^S(thus a factor of1\n2is not included in \u0018) are consistent with the formulation\nin St ohr's paper (see Eq. (27) in Ref. [30]).\nFurthermore, it is important to notice that Bruno term (orbital moment) and van\nder Laan term (quadrupole moment of spin density) [53] are not su\u000ecient to describe\nthe MCA, because they ignore perturbation terms involving unoccupied majority-spin\nstates[54]. In spite of the approximation, the Eq. (27) is still important to connect\nthe MCA energy with the local physical quantities observed by the spectroscopic\nexperiments. Following, we add an intuitive picture on MCA to the Bruno term and\nthe van der Laan term. Then, we would like to comment on the di\u000berence between Eq.\n(27) of the present paper and Eq. (28) of Ref. [29]. The Eq. (27) is the MCA energy\nformulated through the second order perturbation of SOI under the approximation of\nEq. (19) and (20). On the other hand, the latter is the energy due to the second-order\nperturbation derived from Eq. (9) of Ref. [29], and is not the MCA energy. Thus, to\nobtain the MCA energy, the energy di\u000berence between the perpendicular and in-plane\nmagnetization should be considered. In taking the energy di\u000berence, the ELSterm in\nEq. (28) of Ref. [29] will be canceled out between the perpendicular and in-plane\nmagnetization directions, because the ELSis independent of magnetization direction.\nThus, we ignore the ELSterm in Eq. (27) of the present paper.\nFigure 1 shows schematic images of the orbital and quadrupole moments'\ncontributions to MCA. The contribution of the orbital moment to MCA can be\nunderstood from the SOI Hamiltonian in second-order perturbation. Because of the\ninternal product term~^L\u0001~^Sin Eq. (3), an orbital moment parallel to the spinUnderstanding MCA based on orbital and quadrupole moments 13\nmoment is more favorable to stabilize the magnetization direction. This implies that\nthe spin moment aligns with a larger orbital moment. Thus, the orbital moment\nanisotropy \u0001 morbis proportional to the MCA energy. The oblate (pancake-like)\ndistributions of minority-spin electrons in xyplane cause the perpendicular orbital\nmoment (along zaxis), contributing to the perpendicular MCA, whereas the prolate\n(cigar-like) distributions along zaxis cause the in-plane orbital moment, contributing\nto the in-plane MCA (in xyplane). In contrast, the contribution of the quadrupole\nmoments of spin density (magnetic dipole moments) to MCA implies that the shape of\nthe spin-density distribution directly a\u000bects the MCA .\nIf the quadrupole moments ( Qzz) of the spin density de\fned by Eq. (21) is zero,\nthe spin-density distribution is spherical and does not contribute to MCA. However,\nifQzzis non-zero, the spin moment tends to orient in the longitudinal direction of\nthe spin-density distribution, and the prolate (cigar-type) distribution contributes to\nthe perpendicular MCA, whereas the oblate (pancake-type) distribution contributes to\nthe in-plane MCA. Therefore, the contributions to MCA from the two shapes of the\nspin-density distribution are opposite for the orbital moment and quadrupole moment\n(magnetic dipole moment). This is consistent with the opposite signs of the matrix\nelements of ^Lzand ^Lxfor the spin-conserving term (orbital moment anisotropy) and\nthe spin-\rip term (the quadrupole moment of spin density). Therefore, the MCA of\nmagnetic materials with TMs can be determined from the competition between the\norbital moment anisotropy and the anisotropy of the spin-density distribution (the\nquadrupole moment of spin-density).\nRecently, Suzuki et al. [55] formulated the expectation of the magnetic dipole\noperator by using wavefunctions with \frst-order perturbative corrections of SOI, and\ndirectly describe the spin-\rip term of perturbative MCA energies with the measurable\nphysical parameters related to the magnetic dipole moment. The formulation does not\nrequire the approximation and assumption such as Eqs. (19) and (20), and can apply\nmagnetic materials with small exchange splitting. However, an intuitive picture between\nthe MCA and SOI corrections of the magnetic dipole term is still unclear; thus, that is\na future work.\n3. Fcc Ni with in-plane distortions\nThe coupling between MCA and lattice distortion is a fundamental issue in magnetism.\nThe lattice distortion changes the symmetry of a crystal and produces di\u000berent electronic\nstructures a\u000becting magnetization directions. Therefore, magnetic properties and\nelectronic structures strongly couple with lattice structure and symmetry, known as\nthe inverse magnetostriction (magneto-elastic) e\u000bect. For example, fcc Ni does not have\nMCA in the cubic structure. However, the tensile and compressive tetragonal lattice\ndistortions with respect to the (001) plane cause the perpendicular and in-plane MCAs,\nrespectively.\nRecently, the magnetic properties of Ni/Cu multilayers on ferroelectric BaTiO 3Understanding MCA based on orbital and quadrupole moments 14\nTable 2. The orbital magnetic moment of Ni in Ni/Cu multilayer on BaTiO 3\nmeasured by XMCD for Ni Ledges in the normal incident (NI) and the grazing incident\n(GI) geometries, with and without electric \feld ( E) . The calculated orbital magnetic\nmoments of fcc Ni for the perpendicular ( \u0012= 0) and in-plane ( \u0012=\u0019=2) magnetization\ncon\fgurations with and without tensile distortion (2 %), corresponding to with and\nwithout electric \feld ( E) in the experiment.\nExperiment E= 0 kV/cm E= 8 kV/cm\nmorb(NI) 0.06\u0016B 0.04\u0016B\nmorb(GI) 0.04\u0016B 0.05\u0016B\nCalculation 2 % tensile strain without strain\nmorb(\u0012= 0) 0.0542\u0016B 0.0506\u0016B\nmorb(\u0012=\u0019=2) 0.0505\u0016B 0.0504\u0016B\n(b) \n-0.15 -0.10-0.05 0.00 0.05 0.100.15 0.20\n/g1831/g2897/g2887/g2885 ↓↓/g1831/g2897/g2887/g2885 ↑↓/g1831/g2897/g2887/g2885 ↓↑/g1831/g2897/g2887/g2885 ↑↑MCA energy [meV/atom] 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70\n-4 -3 -2 -1 0 1 2 3 4 5\nDistortion ratio ( a|| -a0)/ a0×100 [%] In-plane lattice constant a|| [angstrom] (a) \n∆/g1865/g2925/g2928/g2912 [μB/atom] /g1831/g2897/g2887/g2885 [×10 7erg/cm 3]\n-1.5 -1.0 -0.50.0 0.51.0 2.0 \n1.5 \n-0.006 -0.004 -0.002 0.000 0.002 0.004 0.006 0.008 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70\n-4 -3 -2 -1 0 1 2 3 4 5\nDistortion ratio ( a|| -a0)/ a0×100 [%] In-plane lattice constant a|| [angstrom] \n/g1831/g2897/g2887/g2885\n∆/g1865/g2925/g2928/g2912 \nFigure 2. (Color online)(a) DFT calculation results of MCA energies and the\norbital moment anisotropies of fcc Ni as a function of the in-plane lattice constant\nak(tetragonal distortion is de\fned as ( ak\u0000a0)/a0\u0002100), where a0= 3.524 \u0017A. The\ncrystal structure of fcc Ni for the conventional unit cell and the unit cell with 45\u000e\nin-plane rotation is also shown in the inset. (b) Spin-resolved MCA energies in the\nsecond-order perturbation of SOI from Eq. (13) for fcc Ni as a function of ak.\nsubstrate were controlled by the mechanical strain through the piezo electric e\u000bect . This\nstrain is induced by an applied electric \feld Eto BaTiO 3, switching the magnetization\nfrom the perpendicular to in-plane easy axis by tuning the lattice distortion with E[39].\nIn this experiment, without E, the Ni layer exhibited a tensile strain of 2 % through\nthe sandwiched Cu layers, and the Ni layer showed a perpendicular MCA. With E, the\nchange in the domain structures reduces the lattice constant of BaTiO 3and releases the\ntensile strain in the Ni layers, resulting in magnetization along the in-plane easy axis. In\naddition, a strain-induced change in the orbital magnetic moments of Ni was observedUnderstanding MCA based on orbital and quadrupole moments 15\nby XAS and XMCD measurements. The spectra for NiCu/BaTiO 3were measured for\nNiLedges in the normal incident (NI) and grazing incident (GI) geometries, with and\nwithoutE(8 kV/cm).\nTable 2 shows the observed orbital moments of Ni in the Ni/Cu multilayers on\nBaTiO 3by XMCD measurement. The orbital moments in the NI geometry correspond\nto those with the magnetization normal to the (001) plane, while the orbital moments\nin the GI geometry include half of the in-plane components owing to the grazing angle\n(60\u000e) from the sample surface normal (cos 60\u000e= 1=2 ). The values of the orbital moment\nanisotropy of Ni between the NI and GI geometries were estimated to be 0.02 \u0016B( for\nE= 0 and tensile strain = 2%) and 0.01 \u0016B( forE= 8 kV/cm and tensile strain =\n0). These results indicate that an applied Emodulates the orbital moments and their\nanisotropies, resulting in changes in MCA owing to lattice distortion.\nTo examine the relationship between MCA and lattice distortion in detail, we\nperformed second-order perturbation calculations for the MCA of fcc Ni with tetragonal\ndistortion in the (001) plane. Details of DFT calculations are presented in Ref. [39] .\nFigure 2(a) shows the MCA energy and orbital moment anisotropy of fcc Ni as a function\nof the distortion ratio ( ak\u0000a0)=a0\u0002100, where akis the in-plane lattice parameter, and\na0= 3:524\u0017A is the optimized lattice constant of the cubic fcc Ni in DFT calculations.\nThe inset of Fig. 2(a) shows the crystal structure and unit cell of distorted fcc Ni. The\nout-of-plane lattice parameter a?is fully relaxed for each ak.\nAs shown in Fig. 2(a), the MCA energy increases with an increasing distortion\nratio (i.e., tetragonal tensile strain), which is consistent with the experimental\nresults. Because the orbital moment anisotropy \u0001 morbmonotonically increases with\nan increasing distortion ratio, the change in the perpendicular MCA can be attributed\nto the change in the orbital moment anisotropy given by Eq. (17). Figure 2(b) shows\nthe dependence of the spin-resolved MCA energies in second-order perturbation from\nEq. (13) on the tetragonal strain. The spin-conserving term E##\nMCA has the largest\ndependence on the distortion ratio and is the main contributing factor to MCA. In\ncontrast, for the spin-\rip term E\"#\nMCA, the magnitude is smaller than the spin-conserving\nterm, andE\"#\nMCAhas the opposite dependence on the distortion ratio compared to MCA\nenergies.\nTo validate Eq. (27), we plotted the Bruno term from Eq. (17) in Fig. 3(a) and\nvan der Laan term from Eq. (24) as a function of the distortion ratio, where \u0001 Eexc\nwas set to 1 eV. By comparing Fig. 2(b) and Fig. 3(a), we found that the strain\ndependence of Bruno and van der Laan terms (on the distortion ratio) agrees with that\nof the spin-conserving term E##\nMCA and the spin-\rip term E\"#\nMCA, respectively, indicating\nthat the orbital moment anisotropy and the magnetic dipole (the quadrupole moment\nof spin density) successfully describe the MCA of distorted fcc Ni. In this case, the\norbital moment anisotropy is more responsible for the MCA of fcc Ni with tetragonal\ndistortion than the anisotropy of the quadrupole moment (the magnetic dipole term).\nTo further understand the MCA of fcc Ni, in Fig. 3(b), we show the square of the\nmatrix elements of the spin-conserving terms corresponding to LzandLxfor eachdUnderstanding MCA based on orbital and quadrupole moments 16\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 Contribution to MCA [meV/atom] 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70In-plane lattice constant a|| [angstrom] \nDistortion ratio ( a|| -a0)/ a0×100 [%] (b) \n/g1856/g3051/g3118/g2879/g3052/g3118/g1838/g3053/g1856/g3051/g3052 /g2870/g1833/g2898/g2919 ↓↓(a) \n-0.15 -0.10-0.05 0.00 0.05 0.100.15 0.20MCA energy [meV/atom] 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70\nDistortion ratio ( a|| -a0)/ a0×100 [%] In-plane lattice constant a|| [angstrom] \n-4 -3 -2 -1 0 1 2 3 4 5/g1831/g2897/g2887/g2885 ≈/g2022\n4∆/g1865/g2925/g2928/g2912 +21 \n8/g2022/g2870\n∆/g1831/g2915/g2934/g2913 /g1865/g2904\n-4 -3 -2 -1 0 1 2 3 4 5/g1856/g3051/g3118/g2879/g3052/g3118/g1838/g3051/g1856/g3052/g3053 /g2870/g1833/g2898/g2919 ↓↓\n/g1856/g3052/g3053 /g1838/g3053/g1856/g3053/g3051 /g2870/g1833/g3010/g3010 ↓↓/g1856/g3051/g3052 /g1838/g3051/g1856/g3053/g3051 /g2870/g1833/g2898/g2919 ↓↓\n/g1856/g2871/g3053/g3118/g2879/g3045/g3118/g1838/g3051/g1856/g3052/g3053 /g2870/g1833/g2898/g2919 ↓↓\nFigure 3. (Color online)(a) Bruno term (orbital moment anisotropy) expressed by\nEq. (17) and van der Laan term (magnetic dipole moment) expressed by Eq. (24)\nof fcc Ni as a function of ak. (b) Thedorbital contribution to the MCA energies of\ntetragonally distorted fcc-Ni as a function of the in-plane lattice constant ak, where\nG##\nNiis the joint local density of states expressed by Eq. (6).\norbital as a function of the distortion ratio. The matrix elements jhdxyjLzjdx2\u0000y2ij2G##\nNi\nin Eq. (14) show the main contribution to the perpendicular MCA, and their lattice\ndistortion dependence is similar to that of EMCA, i.e., it increases with increasing\ndistortion ratio, where the positive (negative) matrix elements indicate the contribution\nto the perpendicular (in-plane) MCA. We con\frm that the number of minority-spin\nelectrons in dx2\u0000y2increases owing to tensile distortion, whereas that in dxychanges\nnegligibly. This change in electron distribution in dx2\u0000y2under tensile distortion\nenhances the matrix element jhdxyjLzjdx2\u0000y2ij2G##\nNiand the orbital moment along the\nperpendicular direction, leading to the origin of the perpendicular MCA of fcc Ni under\ntensile distortion.\n4. Mn-Ga alloys\nIn Section 3, we introduced a bulk system with a perpendicular MCA originating from\nthe orbital moment anisotropy. We now discuss a perpendicular MCA arising from the\nquadrupole moment of the spin density. Recently, XMCD and XMLD measurements\nwere performed to detect the orbital and quadrupole moments in Mn-Ga binary alloys\nto clarify the origin of the perpendicular MCA of Mn 3\u0000\u000eGa alloys [40]. Mn 3\u0000\u000eGa is\none of the candidates that could overcome some of the problems in spintronics devices\nby reducing the energy consumption during magnetization reversal and enhancing\nthe thermal stability due to their strong perpendicular MCA with the ferrimagnetic\nproperty.\nSchematics of the crystal structures of L1 0-MnGa(\u000e= 2) and D0 22-Mn 3Ga(\u000e=0)\nare shown in Figs. 4(a) and (b), respectively. In Mn 3Ga, the MnIis located at the sameUnderstanding MCA based on orbital and quadrupole moments 17\nMn 3Ga(D0 22 )\nxyz(b) MnGa(L1 0)\nMn Mn \nMn Mn Mn Mn \nMn \nGa Ga Ga yz\nGa \nMn Mn Mn (a) \nx\n-0.05 0.00 0.05 0.10 0.15 0.20 \nL1 0-MnGa \nMn (c) MCA energy [meV/atom] /g1831/g2897/g2887/g2885 ↑↑\n/g1831/g2897/g2887/g2885 ↑↓/g1831/g2897/g2887/g2885 ↓↓\n/g1831/g2897/g2887/g2885 ↓↑D0 22 -Mn 3Ga \n-0.05 0.00 0.05 0.10 0.15 0.20 \nMn Ⅰ Mn Ⅱ(d) \nMCA energy [meV/atom] /g1831/g2897/g2887/g2885 ↑↑\n/g1831/g2897/g2887/g2885 ↑↓/g1831/g2897/g2887/g2885 ↓↓\n/g1831/g2897/g2887/g2885 ↓↑\n-2 -1 0 1 2LDOS [states/eV/atom] \nE-EF[eV] -2 -1 0 1 2\nE-EF[eV] -1.0 -0.5 0.0 0.5 1.0 LDOS [states/eV/atom] Mn II of D0 22 -Mn 3Ga Mn Iof D0 22 -Mn 3Ga d(3z2-r2)\nd(yz)\nd(x2-y2)\nd(xy)LDOS [states/eV/atom] \n-2 -1 0 1 2\nE-EF[eV] L1 0-MnGa d(3z2-r2)\nd(yz)\nd(x2-y2)\nd(xy)d(3z2-r2)\nd(yz)\nd(x2-y2)\nd(xy)\n-1.0 -0.5 0.0 0.5 1.0 \n-1.0 -0.5 0.0 0.5 1.0 (e) (f) (g) \nFigure 4. (Color online)Schematics of crystal structures and spin con\fgurations of\n(a)L1 0-MnGa and (b)D0 22-Mn 3Ga. Spin-resolved MCA energies in the second-order\nperturbation of the SOI for (c) Mn in L1 0-MnGa and (b) MnIand MnIIin D0 22-\nMn3Ga. Local density of states (LDOS) of each dorbital as a function of energy\nrelative to the Fermi energy EFfor (e) Mn in L1 0-MnGa, (f) MnIand (g) MnIIin\nD022-Mn 3Ga.\nTable 3. Calculated MCA energy EMCA, the spin moment mspin, the orbital moment\nanisotropy \u0001 morb, and the magnetic dipole moment mTof L1 0-MnGa and D0 22-\nMn3Ga with the experimental lattice constant [56, 57].\nCalculation L10-MnGa D022-Mn 3Ga\nEMCA 1.463 MJ/m31.856 MJ/m3\nmspin 2.667\u0016B -3.177\u0016B(MnI) 2.546\u0016B(MnII)\n\u0001morb 0.0014\u0016B -0.0030\u0016B(MnI) 0.0064\u0016B(MnII)\nmT 0.0493\u0016B 0.0180\u0016B(MnI) 0.0801\u0016B(MnII)Understanding MCA based on orbital and quadrupole moments 18\nD0 22 -Mn 3Ga L1 0-MnGa \n/g1856/g3051/g3118/g2879/g3052/g3118/g1838/g3053/g1856/g3051/g3052 /g2870/g1833/g2897/g2924 /g3097/g3097/g4594\n/g1856/g3051/g3052 /g1838/g3053/g1856/g3051/g3118/g2879/g3052/g3118/g2870/g1833/g2897/g2924 /g3097/g3097/g4594/g1856/g3053/g3051 /g1838/g3053/g1856/g3052/g3053 /g2870/g1833/g2897/g2924 /g3097/g3097/g4594\n/g1856/g3052/g3053/g1838/g3053/g1856/g3053/g3051 /g2870/g1833/g2897/g2924 /g3097/g3097/g4594\n/g1856/g3051/g3118/g2879/g3052/g3118/g1838/g3051/g1856/g3052/g3053 /g2870/g1833/g2897/g2924 /g3097/g3097/g4594\n/g1856/g3052/g3053/g1838/g3051/g1856/g3051/g3118/g2879/g3052/g3118/g2870/g1833/g2897/g2924 /g3097/g3097/g4594/g1856/g3053/g3051 /g1838/g3051/g1856/g3051/g3052 /g2870/g1833/g2897/g2924 /g3097/g3097/g4594\n/g1856/g3051/g3052 /g1838/g3051/g1856/g3053/g3051 /g2870/g1833/g2897/g2924 /g3097/g3097/g4594/g1856/g3052/g3053/g1838/g3051/g1856/g2871/g3053/g3118/g2879/g3045/g3118/g2870/g1833/g2897/g2924 /g3097/g3097/g4594/g1856/g2871/g3053/g3118/g2879/g3045/g3118/g1838/g3051/g1856/g3052/g3053 /g2870/g1833/g2897/g2924 /g3097/g3097/g4594\n/g2026/g2026/g4593=↓↓\n/g2026/g2026/g4593=↑↓\nContribution to MCA energy [meV/atom] -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 -0.5 0.0 0.5 /g2026/g2026/g4593=↓↓\n/g2026/g2026/g4593=↑↓\nContribution to MCA energy [meV/atom] Mn II Mn I\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 (a) (b) (c) \nFigure 5. (Color online)Matrix elements of the second order perturbation of SOI\nbetweend-orbitals of Mn atoms in L1 0MnGa and D0 22-Mn 3Ga for the spin-conserving\n(##) and the spin-\rip ( \"#) processes, where G##\nMnis the joint local density of states\nexpressed by Eq. (6). Positive and negative values indicate the contribution of the\nmatrix elements to the perpendicular and the in-plane MCA, respectively.\n(001) plane with Ga having negative spin moment, while MnIIis located between the\ntwo (001) MnI-Ga planes having positive spin moment. Detecting the element speci\fc\nquadrupole tensor Qzzis possible with the XLMD measurements and the sum rule in the\nMnLedges. According to the XMCD and XMLD measurements, MnGa and Mn 3Ga\nhad relatively large magnetic dipole moments mTof Mn (of the order of 0.01 \u0016B) while\nthe orbital moment anisotropies \u0001 morbwere negligibly small (less than 0.01) despite\nthe perpendicular MCA of MnGa and Mn 3Ga.\nTable 3 shows the calculated spin moments, orbital moment anisotropies \u0001 morb,\nand magnetic dipole moments mTof Mn in L1 0-MnGa and D0 22-Mn 3Ga with the\nexperimental lattice constant shown in Ref. [56] and Ref. [57], respectively. The\ndetails of DFT calculations are described in Ref. [40]. The calculated magnetic\ndipole moments are larger than the orbital moment anisotropies both in MnGa and\nMn3Ga, demonstrating consistency with the experimental results. This indicates that\nthe magnetic dipole moment related to the cigar-type quadrupole moment of the spin\ndensity plays an important role in the perpendicular MCA of these alloys. Figures\n4(c) and (d) show the spin-resolved MCA energies in the second-order perturbation\nof the SOI for MnGa and Mn 3Ga. We found that the spin-\rip term ( E\"#\nMCA) of Mn\nin MnGa and MnIIin Mn 3Ga are the main contributors to MCA energies. Since\nMn3Ga is a ferrimagnet, MnIand MnIIhave mutually opposite spin directions, where\nthe contribution of MnIto MCA is smaller than that of MnII. As discussed in Section 2,\nthe spin-\rip term E\"#\nMCAof the MCA energy is described by the magnetic dipole moment,\nindicating that the origin of the perpendicular MCA can be attributed to the cigar-type\nquadrupole moment of the spin density rather than the orbital moment anisotropies. InUnderstanding MCA based on orbital and quadrupole moments 19\n-0.05 0.00 0.05 0.100.15 0.200.25 MCA energy of  Mn        \nin MnGa [meV/atom] Bruno term van der Laan term \n-0.05 0.00 0.05 0.100.15 0.200.25 \n-6 -4 -2 0 2 4 6/g1831↓↑/g2897/g2887/g2885 /g1831↑↑/g2897/g2887/g2885 \n/g1831↓↓/g2897/g2887/g2885 /g1831↑↓/g2897/g2887/g2885 /g1831/g2897/g2887/g2885 \nDistortion ratio ( a|| -a0)/ a0×100 [%] -6 -4 -2 0 2 4 6\nDistortion ratio ( a|| -a0)/ a0×100 [%] (a) (b) MCA energy of  Mn        \nin MnGa [meV/atom] \nFigure 6. (Color online )(a) Spin-resolved MCA energies in the second-order\nperturbation of SOI for L1 0-MnGa shown in Eq. (13) as a function of ak. (b) Bruno\nterm shown in Eq. (17) and van der Laan term shown in Eq. (24) of L1 0-MnGa as a\nfunction of ak.\nFig. 4(e)-(g), we show the local density of states (LDOS) of L1 0-MnGa and D0 22-Mn 3Ga\nby the DFT calculation. In the LDOS of the MnIand MnIIsites, all orbital states were\nsplit through exchange interaction. However, exchange splitting was incomplete where\ncomplete spin splitting was required, in the Bruno formula, which enabled the transitions\nby spin mixing between occupied spin-up and unoccupied spin-down states.\nTo further understand how atomic orbitals contribute to the perpendicular MCA\nin L1 0MnGa and D0 22-Mn 3Ga, we shows in Fig. 5 the square of the matrix\nelements of LzandLxfor eachdorbital of Mn in the spin-conserving and spin-\rip\nterms. As shown in Fig. 5(a), the matrix elements that contribute positively to\nthe MCA energy of L1 0-MnGa (perpendicular MCA) are jhdyzjLxjd3z2\u0000r2ij2G\"#\nMnand\njhd3z2\u0000r2jLxjdyzij2G\"#\nMn, both of which are spin-\rip matrix elements. Although the\nmatrix elementjhdxyjLzjdx2\u0000y2ij2G##\nMnshows a large positive value, its spin-\rip term\njhdxyjLzjdx2\u0000y2ij2G\"#\nMnshows a large negative value, leading to a small contribution\nofdxyanddx2\u0000y2orbitals to the perpendicular MCA because each term is canceled\nout. For D0 22-Mn 3Ga, the matrix elements of MnIare much smaller than those of\nMnII. Furthermore, the spin-\rip matrix elements of Lxbetweend3z2\u0000r2anddyzof MnII\nhad large positive values, indicating the origin of the perpendicular MCA. Therefore,\nthe perpendicular MCA of L1 0MnGa and D0 22-Mn 3Ga originates from the cigar-type\ndistribution of spin density, especially because of d3z2\u0000r2anddyzorbitals, where the spin\nmoment of Mn tends to be oriented in the longitudinal (perpendicular) direction of the\nspin-density distributions.\nBecause the dependence of MCA on lattice distortion is an important physical\naspect, we investigated the change in MCA energies with tetragonal distortion. Figure\n6(a) shows MCA energies and MCA contributions of each spin transition process inUnderstanding MCA based on orbital and quadrupole moments 20\nTable 4. The spin moment mspin, the orbital moment anisotropy \u0001 morb, and the\nmagnetic dipole mTof interface Co and Pd in Co(4ML)/Pd(8ML)(111) multilayer\nobtained from the XMCD measurement and DFT calculations\nCo Pd\n\u0016B XMCD DFT XMCD DFT\nmspin 1.82 1.87 0.25 0.31\n\u0001morb 0.03 0.032 0.00 -0.001\nmT 0.0014 -0.0137 0.0014 0.0106\nthe second-order perturbation for Mn atom in L1 0-MnGa as a function of the in-plane\ndistortion ratio. We \fnd that the spin-\rip term E\"#\nMCA has the largest contribution\nto the perpendicular MCA. Whereas its dependence on the in-plane distortion ratio is\nweaker than that on the total MCA energy. In contrast, the spin-conserving term E##\nMCA\nexhibits a stronger dependence on in-plane distortion ratio and connects MCA energies\nto lattice distortion, which is consistent with the results for fcc Ni in Fig. 3.\nThe above behavior is also con\frmed in Fig. 6(b), where Bruno term in Eq. (17)\nand van der Laan term in Eq. (24) for Mn in L1 0-MnGa are plotted as a function\nof the distortion ratio. Here, we used \u0001 Eexc= 2 eV. These results indicate that the\norbital moment anisotropy is more sensitive to lattice distortion than the magnetic\ndipole moment (quadrupole moment of spin density), which can be understood in terms\nof an orbital-striction or an orbital-elastic e\u000bect as is discussed in Ref. [39].\n5. Co/Pd(111) multilayers\nSo far, we have discussed MCA in bulk materials. In this section, we discuss MCA for\ninterface systems. Among various interface combinations, the interaction between 3 d\nTMs and 4dor 5dTMs is considered for a study of the interface-driven MCA because\nof the large spin moment of 3 dTMs and the strong SOI in 4 dand 5dTMs.\nUltrathin Co/Pd(111) multilayers are typical arti\fcial nanosystems exhibiting\nan interface perpendicular MCA. The development of synthesized thin \flms with\nperpendicular-magnetization has led researchers to expect ultrahigh density recording\nmedia. Furthermore, using Co/Pd interfaces and multilayers, researchers have\ndemonstrated the photo-induced precession of magnetization [58, 59], creation\nof skyrmions using the interfacial Dzyaloshinskii-Moriya interaction [60], and\nmagnetization reversal using the spin-orbit torque [61]. Despite much interest in Co/Pd\ninterfaces, the mechanism of the perpendicular MCA and the role of Co and Pd sites were\nnot fully understood. To clarify the origin of the perpendicular MCA of Co/Pd(111),\nelement-speci\fc XMCD measurements for both Co and Pd were performed on ultrathin\n\flms of Co/Pd(111) multilayers [41].\nIn the experiments, ultra-thin multilayered samples of [Co (4 ML)/Pd (8 ML)(111)] 5Understanding MCA based on orbital and quadrupole moments 21\n-0.2 00.2 0.4 0.6 0.8 11.2 1.4 \n1 4 7 10 \n-1.0 -0.5 0.0 0.5 1.0 1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 \n/g1831↓↑/g2897/g2887/g2885 /g1831↑↑/g2897/g2887/g2885 /g1831↓↓/g2897/g2887/g2885 /g1831↑↓/g2897/g2887/g2885 Interface Co MCA energy [meV/atom] (b) \n/g1831↓↑/g2897/g2887/g2885 /g1831↑↑/g2897/g2887/g2885 /g1831↓↓/g2897/g2887/g2885 /g1831↑↓/g2897/g2887/g2885 Interface Pd MCA energy [meV/atom] (c) /g1831/g2897/g2887/g2885 \n/g1831/g2903/g2897/g2885 \nNumber of Co layer nCo (ML) /g1831/g2897/g2887/g2885 , /g1831/g2903/g2897/g2885 or /g1831/g2915/g2916/g2916 [MJ/m3](a) \nCo( nCo )/Pd(111) /g1831/g2915/g2916/g2916 =/g1831/g2897/g2887/g2885 −/g1831/g2903/g2897/g2885 nCo = 4 \nnCo = 4 \nFigure 7. (Color online)The dependence of the MCA energy ( EMCA), the SMA energy\n(ESMA), and the e\u000bective magnetic anisotropy energy ( Ee\u000b=EMCA\u0000ESMA) on the\nnumber of Co layer ( nCo) for Co(nCo)/Pd(8 ML)(111) multilayer. (b)Spin-resolved\nMCA energies in the second-order perturbation of SOI from Eq. (13) for interface Co\nand (c)Pd of Co(4 ML)/Pd(111).\nwith an out-of-plane easy axis were fabricated. A thicker Co monolayer (ML) case\nexhibit in-plane MCA because of large SMA. Then, the spectra of these samples were\nmeasured for the Pd M2;3and CoL2;3edges in the NI and GI geometries. Table 4\npresents the experimental values of mspin, \u0001morb, andmTfor the sample having out-of-\nplane magnetization together with those estimated from DFT calculations.\nFrom Table 4, a positive \u0001 morbfor Co sites and its negligible value for Pd sites were\nobserved in XMCD, indicating that the Bruno term (17) in the Co sites is the origin\nof the perpendicular MCA of Co/Pd(111). In contrast, a \fnite mTwas observed at\napproximately 0.01 \u0016Bfor both Co and Pd sites. Because the SOI constant of Pd atoms\n\u0018Pdis three times that of Co atoms \u0018Co, the contribution of van der Laan term (24) to\nthe perpendicular MCA for Pd atoms is three times that of Co atoms. Furthermore,\nthe observed \u0001 morbof Co was negligibly small for the sample with thicker Co layers\nhaving in-plane magnetization, indicating that \u0001 morbof the interface Co is responsible\nfor the perpendicular MCA.\nFig. 7(a) shows EMCA, the SMA energy ( ESMA =1\n2\u00160M2\ns), and the e\u000bective\nmagnetic anisotropy energy Ee\u000b=EMCA\u0000ESMAof Co(nCoML)/Pd(8 ML)(111), where\nMsis the total magnetization of Co/Pd(111). The in-plane lattice constants and out-of-\nplane atomic distances of the Co( nCo)/Pd(111) supercells were fully optimized for each\nthe number of Co layers ( nCo) by DFT calculations. Other details of DFT calculations\nare presented in Ref. [41]. EMCA (red line points) decreases with an increase in nCo,\nindicating that interface Co has the main contribution to MCA. Because the SMA\nenergy of Co/Pd(111) (blue line points) increases with increasing nCo, Co/Pd(111) has a\nnegativeEe\u000baroundnCo= 6\u00187 and prefers in-plane magnetization, which is consistent\nwith the experimental result.Understanding MCA based on orbital and quadrupole moments 22\n-0.200 -0.100 0.000 0.100 0.200 0.300 0.400 0.500 0.600 Bruno term van der Laan term MAE contribution [meV/atom] \nPd   Co   Co Co Co Pd  Pd Pd   Pd Pd Pd Pd Pd Co  Co Co Co  Pd \nzx\nFigure 8. (Color online)Bruno term shown in Eq. (17) and van der Laan term shown\nin Eq. (24) at each atomic site of Co and Pd in Co(4ML)/Pd(8ML)(111) multilayer.\nThe atomic structure of Co(4ML)/Pd(8ML)(111) is shown below the graph, where\neach atomic position along zaxis corresponds to the points in the graph.\nTo con\frm these results, we performed the second-order perturbation analyses of\nCo/Pd(111) multilayers. Figures 7(b) and (c) show the spin-resolved MCA energies\nin the second-order perturbation of the SOI for the interface Co and Pd atoms in\nCo(4ML)/Pd(111). We \fnd that for the interface Co, the spin-conserving term E##\nMCA\nis the main contributor to the perpendicular MCA. Whereas for the interface Pd, the\nspin-\rip term E\"#\nMCAshows the main contribution. Because E##\nMCAandE\"#\nMCAare related\nto \u0001morbandmT, these results are also consistent with the experimental results in\nTable 4.\nTo examine the layer-by-layer contributions to MCA energies of Co/Pd(111), we\nshow in Fig. 8 Bruno and van der Laan terms estimated from \u0001 morbandmTat\neach atomic site of Co(4ML)/Pd(8ML)(111). To obtain van der Laan term, we use\n\u0001exc= 4 eV. It is noted that the value of exchange splitting \u0001 exc= 4 eV is too large\nfor nonmagnetic element Pd. However, the \u0001 excis de\fned by Eq. (19) indicating the\nenergy range of eigenstate \u000f~k;n. Because the valence states of Pd in Co/Pd(111) shows\nbroad energy range corresponding to 4 \u00185 eV due to delocalized character of 4 delement\n(see Fig. 5(d) of Ref. [41]), we use 4 eV for \u0001 exc. As shown in Fig. 8, the Bruno term\nincreases only at the interface Co sites, whereas van der Laan term at the Pd sites. This\nindicates that an interface-driven perpendicular MCA originates from both the orbital\nmoment anisotropy and cigar-type quadrupole moment of spin density (corresponding to\npositive magnetic dipole moment). In Co/Pd(111), owing to the hybridization between\nthe Co 3dand Pd 4dorbitals, there is large proximity that induces SOI in the Co states\nand spin polarization in the Pd states, which leads to a large induced spin moment in\nPd of approximately 0.3 \u0016B.\nMoreover, the strong proximity e\u000bect in the Co/Pd(111) multilayer leads to the\norbital moment anisotropy in Co site \u0001 morband the magnetic dipole moment in PdUnderstanding MCA based on orbital and quadrupole moments 23\nsitemTcorresponding to the quadrupole moment of spin density. From the LDOS of\ninterfacial Co in the Co/Pd(111) multilayer, the Co 3 dxy, 3dx2\u0000y2, 3dyz, and 3dzxorbitals\ncontribute to the perpendicular MCA because of being dominant at the Fermi level in\nthe minority-spin states [41]. In addition, these states provide large spin-conserving\nmatrix elements of the interface Co, such as jhdx2\u0000y2jLzjdxyij2G##\nCoandjhdyzjLzjdzxij2G##\nCo,\nenhancing the perpendicular components through the Bruno term. Furthermore, we\nfound large spin-\rip matrix elements of the interface Pd, such as jhdyzjLxjdx2\u0000y2ij2G\"#\nPd,\ncontributing to the cigar-type distribution of spin density and the perpendicular MCA\nthrough van der Laan term.\n6. Fe/MgO(001) interface\nFinally, we discuss the MCA of Fe/MgO(001) interfaces in magnetic tunnel junctions\n(MTJs), which are important for spintronics applications. Fe/MgO-based MTJs exhibit\nlarge tunneling magnetoresistive (TMR) ratios of over 400 % at room temperature\nbecause of coherent tunneling of the highly spin-polarized \u0001 1evanescent states through\nthe MgO barrier [62, 63, 64]. To enhance the thermal stability of magnetization\ndirections at a \fnite temperature, MTJs with perpendicular magnetization are required.\nHowever, because body-centered cubic (bcc) Fe and FeCo alloys have cubic structures\nin bulk, we cannot expect a large perpendicular MCA in the bulk electrode regions\nof MTJs. Thus, the perpendicular MCA of the interface plays an important role in\nstabilizing the magnetization directions while preserving the large TMR ratios.\nMaruyama et al. reported perpendicular magnetization of a Fe/MgO(001) interface\nand a large voltage-controlled MCA (VCMA) change in a few atomic layers of Fe [44].\nIkeda et al. showed both a high TMR ratio of over 120 % at room temperature and\nthe perpendicular magnetization of CoFeB/MgO/CoFeB(001) MTJs when the CoFeB\nlayer was approximately 1.3 nm thick [65]. Furthremore, a large perpendicular MCA\nof 1.4 MJ/m3in Fe/MgO(001) was observed for 0.7 nm thick Fe layer with adsorbate-\ninduced surface reconstruction [66]. XMCD measurements were also performed for\nultrathin Fe/MgO(001), and the orbital moment anisotropy was dominant at the\nFe/MgO interface perpendicular to MCA; the contribution of quadrupole moments was\nsmall but \fnite at the lattice distorted interfaces [43].\nSeveral studies have discussed the origin of the perpendicular MCA of Fe/MgO(001)\nand the VCMA e\u000bect in terms of the orbital moment anisotropy and hybridization of\nthe Fe 3d3z2\u0000r2orbital with the O 2 pzorbital [67, 69, 68, 70, 71] from a theoretical\nperspective. However, the e\u000bects of the quadrupole moments (anisotropy of the spin-\ndensity distribution) on the perpendicular MCA and the correlation between the lattice\ndistortion and MCA have not been thoroughly discussed for Fe/MgO(001) [72, 42]. We\ncalculated the MCA energies of the Fe( nFe)/MgO(5 ML)(001) interface for various layer\nFe layer thickness nFeas a function of the in-plane lattice constant ak, whereakwas\nchanged from aFe= 2:8309 \u0017A toaMgO=p\n2 = 3:0043 \u0017A with an interval of approximately\n0.03\u0017A. The values of aFeandaMgOwere obtained using DFT calculations for bulk bcc FeUnderstanding MCA based on orbital and quadrupole moments 24\n1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 \n2.83 2.85 2.87 2.89 2.91 2.93 2.95 2.97 2.99 3.01 Interface dFe-MgO \nFe dFe MgO dMgO (001) Interlayer distance [Å] \nIn-plane lattice constant a|| [Å] Cubic Fe Cubic MgO \n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 \n2.83 2.85 2.87 2.89 2.91 2.93 2.95 2.97 2.99 3.01 \nIn-plane lattice constant a|| [Å] Fe(11ML) Fe(9ML) Fe(7ML) Fe(5ML) MCA energy [mJ/m 2](b) (a) \na|| \nFe Mg O\ndFe-MgO \ndFe dMgO Fe/MgO(001) \nz\nx\nFigure 9. (Color online)(a)The tetragonal distortion dependence of MCA energies\nof Fe(nFe)/MgO(5 ML)(001) for nFe=5, 7, 9 and 11 ML. (b)The optimized (001)\ninterlayer distance of Fe(11 ML)/MgO(5 ML)(001) interface, bulk MgO, and bulk Fe\nas a function of the in-plane lattice constant ak. The atomic structure of Fe/MgO(001)\ninterface is shown in the inset.\nand rock-salt MgO. In addition, we changed the number of Fe-layer nFefrom 5 ML to 11\nML in Fe/MgO(001) supercells. The out-of-plane atomic distances of the supercells with\neachakwere fully optimized by DFT calculations. Other details of DFT calculations\nwere the same as those in Ref. [42].\nFigure 9(a) shows the MCA energy of Fe/MgO(001) interface as a function of\nak. First, we found that the MCA energies of Fe/MgO(001) exhibited similar ak\ndependences, irrespective of nFe, although there were some di\u000berences. We con\frmed\nthat the Fe/MgO(001) supercells with nFe54 ML showed totally di\u000berent ak\ndependence compared to that with nFe=5 ML in Fig. 9(a). This indicates that nFe=5\nML is necessary to correctly describe the characteristics of the Fe/MgO(001) interface.\nThe MCA energy of Fe/MgO(001) in Fig. 9(a) exhibits a non-monotonic behavior with\nrespect to the change of the in-plane lattice constant; it gradually increases and reaches\na maximum around ak= 2:92\u0017A and then gradually decreases with increasing ak. This\nresult is consistent with the previous calculation results [72]. Fig. 9(b) shows the out-\nof-plane interlayer distance as a function of akfornFe= 11 ML. The result shows a\nmonotonic decrease with increasing ak, indicating that the non-monotonic behavior of\nMCA energies with tetragonal distortion do not originate from the structural properties\nof Fe/MgO(001).\nTo clarify the tetragonal distortion e\u000bect on MCA in Fe/MgO(001), we calculated\nthe spin-resolved MCA energies in second-order perturbation of the SOI EMCA\n\u001b\u001b0for\nthe interface Fe atom as a function of the in-plane lattice constant ak. Fig. 10(a)\nshows that the spin-conserving term EMCA\n## and spin-\rip term EMCA\n\"# mainly contribute\nto the distortion dependence of MCA. The spin-conserving term EMCA\n## exhibits non-\nmonotonic behavior for ak, whereas the spin-\rip term EMCA\n\"# monotonically increasesUnderstanding MCA based on orbital and quadrupole moments 25\n2.83 2.85 2.87 2.89 2.91 2.93 2.95 2.97 2.99 3.01 -0.05 0.00 0.05 0.100.15 0.200.25 0.300.35 0.40\n/g1831↓↓/g2897/g2887/g2885 \n/g1831↑↓/g2897/g2887/g2885 \n/g1831↓↑/g2897/g2887/g2885 /g1831↑↑/g2897/g2887/g2885 \nIn-plane lattice constant a|| [Å] MCA energy of interface Fe [meV/atom] Fe(11ML)/MgO(5ML)(001) \n-0.05 0.00 0.05 0.100.15 0.200.25 0.300.35 0.40\n2.83 2.85 2.87 2.89 2.91 2.93 2.95 2.97 2.99 3.01 van der Laan term Bruno term \nIn-plane lattice constant a|| [Å] MCA energy of interface Fe [meV/atom] Fe(11ML)/MgO(5ML)(001) (a) (b) \nFigure 10. (Color online)(a) Spin-resolved MCA energies in the second-order\nperturbation of SOI for Fe(11 ML)/MgO(5 ML)(001) interface shown in Eq. (13)\nas a function of ak. (b) Bruno term in Eq. (17) and van der Laan term in Eq. (24) of\nFe/MgO(001) interface as a function of ak.\nwith an increase in ak. The spin-conserving term EMCA\n## is slightly larger than the\nspin-\rip term EMCA\n\"# in Fe/MgO(001) interface.\nTo con\frm the relationship between the MCA energy and the orbital and\nquadrupole moments, we show in Fig. 10(b) Bruno and van der Laan terms of Fe\ninterface as a function of ak. To obtain van der Laan term, we use \u0001 exc= 4 eV.\nFig. 10(b) shows that the tetragonal distortion dependence of Bruno and van der Laan\nterms are consistent with those of the spin-conserving term EMCA\n## and the spin-\rip\ntermEMCA\n\"#, respectively. This means that the orbital moment anisotropy and magnetic\ndipole moment e\u000bectively describe the MCA of Fe/MgO(001) with tetragonal distortion.\nThe increase in van der Laan term (the spin-\rip term) with increasing akindicates\nthat Fe/MgO(001) interface has a cigar-type distribution of spin density with tensile\ndistortion, and the perpendicular MCA can be stabilized by orienting the spin moments\nalong the longitudinal direction of the spin-density distributions. Because the Bruno\nterm (the orbital moment anisotropy) of Fe/MgO(001) for ak=aFehas approximately\nthe same value as that for ak=aMgO, the increase in the perpendicular MCA owing\nto the tensile tetragonal distortion of Fe/MgO(001) is mainly caused by the additional\ncontribution of the quadrupole moment of spin density around the Fe interface. The\nXMCD and XMLD measurements for Fe/MgO(001) reported that the orbital moment\nanisotropy was dominant in Fe/MgO(001), with a \fnite contribution of the quadrupole\nmoment of spin density.\nOur calculation results suggest that the contribution of the anisotropy of quadrupole\nmoment is enhanced when the in-plane lattice constant is close to that of MgO. The\ncontribution of the spin-\rip term in EMCA is expected to be sensitive not only to the\nin-plane lattice constant, but also to the applied electric \feld. As a next step, we intend\nto clarify the relationship between the MCA and lattice distortion, together with theUnderstanding MCA based on orbital and quadrupole moments 26\nVCMA e\u000bects at various magnetic interfaces.\n7. Discussions\nConsidering above case studies, we discuss the modulation of orbital moments by local\nstrain. The orbital moments are strongly a\u000bected to the anisotropic local environments\nin the nearest sites through the hybridization. In the case of highly symmetric cubic\nstructure, orbital moments are completely quenched. When the stress can be applied\nalong some direction, the orbital hybridization along this strain direction is preferentially\nmodulated. Within the orbital sum rule in XMCD, the electron occupation in 3d\nstates which is modulated by SOI can be controlled by external strain, resulting in\nthe modulation of orbital moments [73]. Therefore, the relationship between strain and\norbital moments can be systematically formulated. Our studies in this review generalize\nthe orbital control by strain for some cases.\nUntil now, magneto-striction or magneto-elastic e\u000bect has been recognized as\nstrain e\u000bect in magnetism as a phenomenologically macroscopic understanding for\nsome materials which are quite essential for applications such as motor, actuator, and\nmobile devices [74]. However, the microscopic understanding considering the electronic\nstructures and orbital states has not been clari\fed yet. Recent studies using ultra-thin\n\flms strongly require more detailed analysis for strictive e\u000bect. Now, we develop novel\nconcept of orbital-strictive or orbital-elastic e\u000bect using the results of previous Sections.\nIn the case of Ni/Cu multilayers in Sec. 3, the orbital moments are formulated as\na linear relation with strain, which can be understood as orbital-elastic e\u000bect. A linear\nrelation originated from the enhancement of orbital moments in the spin-conserving\nelectron motion within the in-plane direction at the interfaces, which is categorized as\nthe case of Fig. 1(a). As shown in Fig. 2(a), orbital moments are related to in-plane\nstrain, resulting in the perpendicular MCA energy. In this case, the direction of strain\nand enhanced orbital moments are orthogonal in principle. Similar scenario can be\nadopted to the perpendicular MCA in Co-ferrite CoFe 2O4as orbital-elastic e\u000bect in\nCo2+site with large orbital moment [75]. Second type can be understood as quadrupole\ncases; the direction of strain becomes an easy axis. In this case, the contribution from\norbital moments is small and the charge distribution along elongated direction stabilizes\nthe MA. The relation between strain and MCA does not exhibit a linear and spin-\rip\ncontribution is dominant for the MCA, which is typical in the case of Mn 3\u0000\u000eGa because\nof the small contribution of orbital moment in Mn compounds. These two-types of\norbital-elastic e\u000bects can be categorized as a microscopic origin of strictive phenomena\nfrom the viewpoints of the electronic structures.\nOther cases for MCA can be also proposed using asymmetric Rashba-type SOI\n[76]. For example, Au(111) surface has a strong Rashba-type SOI, which induces the\nMCA on the deposited ultra-thin Fe layer. Since the potential pro\fle between Fe/Au is\ndi\u000berent, the interfacial electric gradient promotes the change of charge distributions,\nwhich is also detected by XMCD and other spectroscopic probes [77, 78]. Therefore, theUnderstanding MCA based on orbital and quadrupole moments 27\nMCA from asymmetric momentum space can be also developed as other orbital-strictive\nphenomena including a topological physics.\nAs discussed in Ref. [79] and [80], the of MCA energy obtained by the \frst-\nprinciples calculations often requires some scaling factors to compare the experimental\nresults. This can be attributed to many factors related to problems both theoretical\nand experimental points of view. In the calculations, the Bruno term requires to\nneglect the majority-spin spin-conserving term and the spin-\rip terms, which leads\nto the requirement of the scaling factors. In addition, the \frst-principles calculations\ndo not include the second Hund's law, which leads to the underestimation of the orbital\nmoment [32]. Furthermore, in the relationship between the spin-\rip MCA term and the\nmagnetic dipole moment, the exchange splitting \u0001 excacts as the scaling factor. On the\nother hand, in the experimental points of view, the MCA energy tends to be smaller\nthan the theoretical predictions due to problems of crystallinity, interface roughness and\ndegree of order of magnetic thin \flms, leading to the scaling factors in the Bruno and\nvan der Laan terms.\n8. Summary\nIn this review, we discussed the perpendicular MCA of magnetic materials and their\ninterfaces based on the orbital and quadrupole moments. First, we reviewed the\nrelationship among the orbital moments, quadrupole moments, and MCA based on the\ndetailed formulation of the second-order perturbation of the spin-orbit interaction. We\nargued that the orbital moment stabilizes the spin moment parallel to the direction with\na larger orbital moment, whereas the quadrupole moment stabilizes the spin moment\nalong the longitudinal direction of the spin-density distribution. These e\u000bects are\nexpressed as the spin-conserving Bruno term (related to the orbital moment anisotropy)\nand the spin-\rip van der Laan term (related to the anisotropy of the quadrupole moment\nand the magnetic dipole moment). We demonstrated that the contributions of dorbitals\nto these two e\u000bects are mutually opposite. In the Bruno term, in-plane dorbitals, such\nasdx2\u0000y2anddxy, provide a perpendicular MCA, whereas out-of-plane dorbitals, such as\nd3z2\u0000r2, prefer an in-plane MCA. In contrast, in van der Laan term, the out-of-plane and\nin-planedorbitals provide a perpendicular MCA and an in-plane MCA, respectively.\nThe MCA of magnetic materials and interfaces with TMs can be determined from the\ncompetition between these two contributions.\nWe then applied our formulations of MCA to various magnetic systems by\ncomparing the theoretical results with the XMCD and XMLD measurements. We\nshowed that fcc Ni with tensile tetragonal distortion shows perpendicular MCA arising\nfrom the Bruno term (the orbital moment anisotropy), while MnGa alloys such as\nL10-MnGa and D0 22-Mn 3Ga can be attributed to van der Laan term (the quadrupole\nmoment). Furthermore, the MCA of magnetic systems with interfaces was discussed.\nWe found that the perpendicular MCA of the Co/Pd(111) multilayer originates from\nthe orbital moment anisotropy of the interface Co and the anisotropy of the quadrupoleUnderstanding MCA based on orbital and quadrupole moments 28\nmoment at the interface Pd. Finally, we examined the tetragonal distortion dependence\nof MCA for Fe/MgO(001) interfaces, which are important systems as the perpendicularly\nmagnetized MTJs with high TMR ratios. The perpendicular MCA of Fe/MgO(001)\nexhibited a non-monotonic behavior with respect to the in-plane lattice constant, which\ncan be attributed to both Bruno term (the orbital moment anisotropy) and van der Laan\nterm (the quadrupole moment of spin density). These fundamental understandings\nof MCA will be essential in the theoretical design of novel magnetic materials and\ninterfaces, as well as for the control of MCA through lattice distortion and applied bias\nvoltage .\nAcknowledgements\nWe are grateful to S. Mitani, H. Sukegawa, and K. Masuda at NIMS, H. Yanagihara at\nthe University of Tsukuba, Y. Kota at Fukushima College, M. Shirai and A. Sakuma\nat Tohoku University for their valuable discussions on our work. For Sec. 3-5, we\nacknowledge several collaborators, H. Munekata at Tokyo Institute of Technology, T.\nTaniyama at Nagoya University, S. Mizukami and K. Z. Suzuki at Tohoku University\nwho prepared excellent samples and provided faithful comments. YM sincerely thanks Y.\nSuzuki of Osaka University for showing calculation notes regarding detailed derivations\nof the equations in Ref. [55]. This work was partly supported by the Grants-in-\nAid for Scienti\fc Research (Grant Nos. JP16H06332, JP20H00299, JP20H02190, and\nJP22H04966) from the Japan Society for the Promotion of Science(JSPS), Center for\nSpintronics Research Network (CSRN) of Osaka University, and Cooperative Research\nProject Program of the Research Institute of Electrical Communication (RIEC), Tohoku\nUniversity.\nReferences\n[1] D. Sander, J. Phys. Condens. Matter 16R603{R636 (2004).\n[2] B. Dieny and M. Chshiev, Rev. Mod. Phys. 89025008 (2017).\n[3] S. Bhatti, R. Sbiaa, A. Hirohata, H. Ohno, S. Fukami, and S. N. Piramanayagam, Mater. Today\n20, 530 (2017).\n[4] A. Hashimoto, S. Saito, and M. Takahashi, J. Appl. Phys. 99, 08Q907 (2006).\n[5] K. Yoshida, M. Yokoe, Y. Ishikawa, and Y. Kanai, IEEE Trans. Magn. 46, 2466 (2010).\n[6] M. Igarashi, Y. Suzuki, and Y. Sato, IEEE Trans. Magn. 46, 3738 (2010).\n[7] A. R. Balakrishna and R. D. James, npj Comp. Mat. 8, 4 (2022).\n[8] K. Yosida, Theory of Magnetism, Springer Series in Solid-State Sciences Vol. 122, ed. Fulde (1996)\n[9] J. Franse and R. Radwanski, Handbook of Magnetic Materials, Vol. 6, ed. K. H. J. Buschow (1991).\n[10] G. H. O. Daalderop, P. J. Kelly and M. F. H. Schuurmans, Phys. Rev. B 44, 12054 (1991).\n[11] A. Sakumra, J. Phys. Soc. Jpn, 633053 (1994).\n[12] P. Ravindran, A. Kjekshus, H. Fjellva, P. James, L. Nordstro, B. Johansson and O. Eriksson, Phys.\nRev. B 63, 144409 (2001).\n[13] J. B. Staunton, S. Ostanin, S. S. A. Razee, B. L. Gyor\u000by, L. Szunyogh, B. Ginatempo, and E.\nBruno, Phys. Rev. Lett. 93, 257204 (2004).\n[14] M. Sakamaki and K. Amemiya, Appl. Phys. Express 4, 073002 (2011).Understanding MCA based on orbital and quadrupole moments 29\n[15] T. Kojima, M. Mizuguchi, T. Koganezawa, K. Osaka, M. Kotsugi and K. Takanashi, J. J. Appl.\nPhys. 51, 010204 (2012).\n[16] Y. Kota and A. Sakuma, J. Phys. Soc. Jpn 81, 084705 (2012).\n[17] M. Kotsugi, J. Magn. Magn. Mater. 326, 235 (2013).\n[18] O. Sipr, S. Bornemann, H. Ebert, S. Mankovsky, J. Vackar, and J. Minar, Phys. Rev. B 88, 064411\n(2013).\n[19] S. Ueda, M. Mizuguchi, Y. Miura, J. G. Kang, M. Shirai, and K. Takanashi, Appl. Phys. Lett.\n109, 042404 (2016).\n[20] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).\n[21] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).\n[22] J. P. Perdew and A. Zunger, Phy. Rev. B 23, 5048 (1981).\n[23] P. Bruno, Phys. Rev. B 39, 865 (1989).\n[24] D. Wang, R. Wu, and A. J. Freeman, Phys. Rev. B 47, 14932 (1993).\n[25] M. Cinal, D. M. Edwards, and J. Mathon, Phys. Rev. B 50, 3754 (1994).\n[26] A. J. Freeman, O. N. Mryasov, D. S. Wang, and R. Wu, Mat. Sci. Eng. B31, 225 (1995).\n[27] D. S. Wang, R. Q. Wu, L. P. Zhong, and A. J. Freeman, J. Mag. Mag. Mat. 140-144 , 643 (1995).\n[28] H. A. D urr and G. van der Laan, Phys. Rev. B 54, R760 (1996).\n[29] G. van der Laan, J. Phys.: Condens. 103239 (1998) .\n[30] J. St ohr, J. Magn. Magn. Mater. 200, 470 (1999).\n[31] G. Autes, C. Barreteau, D. Spanjaard, and M.-C. Desjonqueres, J. Phys.: Condens. Matter 18,\n6785 (2006).\n[32] Y. Miura, S. Ozaki, Y. Kuwahara, M. Tsujikawa, K. Abe, and M. Shirai, J. Phys.: Condens.\nMatter 25, 106005 (2006).\n[33] Y. Miura, M. Tsujikawa, and M. Shirai, J. Appl. Phys. 113, 233908 (2013).\n[34] B. T. Thole, P. Carra, F. Sette, and G. van der Laan, Phys. Rev. Lett. 681943 (1992).\n[35] P. Carra, B. T. Thole, M. Altarelli, and X. Wang, Phys. Rev. Lett. 70, 694 (1993).\n[36] G. van der Laan, Phys. Rev. B 57, 5250 (1998).\n[37] G. van der Laan, Phys. Rev. Lett. 82, 640 (1999).\n[38] J. St ohr and H. K onig, Phys. Rev. Lett. 75, 3748 (1995).\n[39] J. Okabayashi, Y. Miura, and T. Taniyama, npj Quantum Materials 4, 21 (2019).\n[40] J. Okabayashi, Y. Miura, Y. Kota, K. Z. Suzuki, A. Sakumra, and S. Mizukami, Sci. Rep. 10,\n9744 (2020).\n[41] J. Okabayashi, Y. Miura and H. Munekata, Sci. Rep. 8, 8303 (2018).\n[42] K. Masuda and Y. Miura, Phys. Rev. B 98, 224421 (2018).\n[43] J. Okabayashi, Y. Iida, Q. Xiang, H. Sukegawa, and S. Mitani, Appl. Phys. Lett. 115, 252402\n(2019).\n[44] T. Maruyama, Y. Shiota, T. Nozaki, K. Ohta, N. Toda, M. Mizuguchi, A. A. Tulapurkar, T.\nShinjo, M. Shiraishi, S. Mizukami, Y. Ando, and Y. Suzuki, Nat. Nanotechnol. 4, 158 (2009).\n[45] S. Kanai, M. Yamanouchi, S. Ikeda, Y. Nakatani, F. Matsukura, and H. Ohno, Appl. Phys. Lett.\n101, 122403 (2012).\n[46] S. Miwa, M. Suzuki, M. Tsujikawa, K. Matsuda, T. Nozaki, K. Tanaka, T. Tsukahara, K. Nawaoka,\nM. Goto, Y. Kotani, T. Ohkubo, F. Bonell, E. Tamura, K. Hono, T. Nakamura, M. Shirai, S.\nYuasa and Y. Suzuki, Nat. Comm. 8, 15848 (2017).\n[47] M. E. Rose, relativistic electron theory, Chapter 7, Wiley (1961)\n[48] G. Kresse and J. Hafner, Phys. Rev. B 47, RC558 (1993).\n[49] G. Kresse and J. Furthmuller, Comput. Mater. Sci. 6, 15 (1996).\n[50] G. Kresse and J. Furthmuller, Phys. Rev. B 5411169 (1996).\n[51] J. K ubler, K-H. H ock, J. Sticht and A. R. Williams, J. Phys. F: Met. Phys. 18, 469 (1988).\n[52] K. Nakamura, T. Ito, A. J. Freeman, L. Zhong and J. Fernandez-de-Castro, Phys. Rev. B 67,\n014420 (2003).\n[53] The relationship between the spin-\rip term of the second order perturbation of SOI and theUnderstanding MCA based on orbital and quadrupole moments 30\nquadrupole moment of spin density is \frst proposed by Wang, Wu and Freeman [24]. However,\nsince van der Laan's works [28, 29], which later developed into the relationship with XMCD and\nXMLD measurements, have been more widely known, it would be easier for the reader to call it\nvan der Laan term. Therefore, we refer to the quadrupole terms as van der Laan terms in this\npaper as well.\n[54] In the derivation of Bruno term and van der Laan term, we assume that the magnetic materials\nwith more-than-half elements such as Fe, Co, and Ni have fully occupied majority-spin states.\nIn the calculation of the orbital and the magnetic dipole moment, however, we consider the\nunoccupied majority-spin states. Thus, the e\u000bects of partial occupation of the majority-spin\nstate of Fe, Co and Ni are included in the orbital and quadrupole moments of spin density.\n[55] Y. Suzuki and S. Miwa, Phys. Lett. A 383, 1203 (2019).\n[56] K. Z. Suzuki, R. Ranjbar, J. Okabayashi, Y. Miura, A. Sugihara, H. Tsuchiura and S. Mizukami,\nSci. Rep. 6, 30249 (2016).\n[57] B. Balke, G. H. Fecher, J. Winterlik and C. Felsera, Appl. Phys. Lett. 90, 152504 (2007).\n[58] C. Boeglin, E. Beaurepaire, V. Halt\u0013 e, V. L\u0013 opez-Flores, C. Stamm, N. Pontius, and H. A. D urr,\nJ.-Y. Bigot, Nature 465, 458 (2010).\n[59] K. Yamamoto, T. Matsuda, K. Nishibayashi, Y. Kitamoto, and H. Munekata, IEEE Trans. Mag.\n249, 3155 (2013).\n[60] S. D. Pollard, J. A. Garlow, J. Yu, Z. Wang, Y. Zhu and H. Yang, Nature Commun. 8, 14761\n(2017).\n[61] M. Jamali, Phys. Rev. Lett. 111, 246602 (2013).\n[62] W. H. Butler, X.-G. Zhang, T. C. Schulthess and J. M. MacLaren, Phys. Rev. B 63, 054416 (2001).\n[63] S. S. P. Parkin, C. Kaiser, A. Panchula, P. M. Rice, B. Hughes, M. Samant, and S.-H. Yang, Nat.\nMater. 3, 862 (2004).\n[64] S. Yuasa, T. Nagahama, A. Fukushima, Y. Suzuki, K. Ando, Nat. Mater. 3, 868 (2004).\n[65] S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. Hayakawa,\nF. Matsukura and H. Ohno, Nat. Mater. 9, 721 (2010).\n[66] J. W. Koo, S. Mitani, T. T. Sasaki, H. Sukegawa, Z. C. Wen, T. Ohkubo, T. Niizeki, K. Inomata,\nand K. Hono, Appl. Phys. Lett. 103, 192401 (2013).\n[67] M. Tsujikawa and T. Oda, Phys. Rev. Lett. 102, 247203 (2009).\n[68] K. Nakamura, T. Akiyama, T. Ito, M. Weinert, and A. J. Freeman, Phys. Rev. B 81220409R\n(2010).\n[69] M. K. Niranjan, C. G. Duan, S. S. Jaswal, and E. Y. Tsymbal, Appl. Phys. Lett. 96, 222504\n(2010).\n[70] H. X. Yang, M. Chshiev, B. Dieny, J. H. Lee, A. Manchon, and K. H. Shin, Phys. Rev. B 84,\n054401 (2011).\n[71] D. Yoshikawa, M. Obata, Y. Taguchi, S. Haraguchi, and T. Oda, Appl. Phys. Exp. 7, 113005\n(2014).\n[72] D. Odkhuu, Sci. Rep. 6, 32742 (2016).\n[73] J. Okabayashi, Progress in Photon Science II ed. K. Yamanouchi, S. Tunik, and V. A. Makarov p.\n471 Springer (2019).\n[74] S. Chikazumi, Physics of Ferromagnetism, Oxford University Press (2005).\n[75] J. Okabayashi, M. A. Tanaka, M. Morishita, H. Yanagihara, and K. Mibu, Phys. Rev. B 105,\n134416 (2022).\n[76] S. E. Barnes, J. Ieda and S. Maekawa, Scienti\fc Reports 4, 4105 (2014).\n[77] J. Okabayashi, S. Li, S. Sakai, Y. Kobayashi, T. Mitsui, K. Tanaka, Y. Miura, and S. Mitani, Phys.\nRev. B 103, 104435 (2021).\n[78] J. Okabayashi, S. Li, S. Sakai, Y. Kobayashi, K. Fujiwara, T. Mitsui, and S. Mitani, Hyper\fne\nInteractions 242, 59 (2021).\n[79] D. Weller, J. Stohr, R. Nakajima, A. Carl, M. G. Samant, C. Chappert, R. Megy, P. Beauvillain,\nP. Veillet, and G. A. Held, Phys. Rev. Lett. 75, 3752 (1995).Understanding MCA based on orbital and quadrupole moments 31\n[80] M. Cinal, Phys. Rev. B 105, 104403 (2022)."    },    {        "title": "2209.08477v1.Magnetocrystalline_Anisotropy_and_Magnetocaloric_Effect_Studies_on_the_Room_temperature_2D_Ferromagnetic_Cr__4_Te__5_.pdf",        "content": "Magnetocrystalline Anisotropy and Magnetocaloric Effect Studies on the Room-temperature 2D\nFerromagnetic Cr 4Te5\nShubham Purwar,1Susmita Changdar,1Susanta Ghosh,1and S. Thirupathaiah1,\u0003\n1Department of Condensed Matter and Materials Physics,\nS. N. Bose National Centre for Basic Sciences, Kolkata, West Bengal, India, 700106.\n(Dated: September 20, 2022)\nWe present a thorough study on the magnetoanisotropic properties and magnetocaloric effect in the layered\nferromagnetic Cr 4Te5single crystals by performing the critical behaviour analysis of magnetization isotherms.\nThe critical exponents b=0.485(3), g=1.202(5), and d=3.52(3) with a Curie temperature of TC\u0019340.73(4)\nK are determined by the modified Arrott plots. We observe a large magnetocrystalline anisotropy K u=330\nkJ/m3at 3 K which gradually decreases with increasing temperature. Maximum entropy change - DSmax\nMand\nthe relative cooling power (RCP) are found to be 2.77 J/kg { K and 88.29 J/kg , respectively near TCwhen\nthe magnetic field applied parallel to ab-plane. Rescaled - DSM(T,H) data measured at various temperatures\nand fields collapse into a single universal curve, confirming the second order magnetic transition in this system.\nFollowing the renormalization group theory analysis, we find that the spin-coupling is of 3D Heisenberg-type,\nfd : ng=f3 : 3g, with long-range exchange interactions decaying as J(r) = r{(d+ s)= r{4.71.\nI. INTRODUCTION\nInvestigation of the low dimensional magnetic materials\nwith ferromagnetic ordering at room temperature has gained\na lot of research interests in recent days due to their\npotential applications in the magnetic refrigeration and\nspintronic devices [1–4] for which the materials with strong\nmagentocrystalline anisotropy are crucial [5, 6]. In this\nregard, the van der Waals (vdW) magnets are of great interest\nfrom the fundamental science and advanced technological\npoint of view due to their peculiar 2-dimensional (2D)\nmagnetic properties [7–10] and strong magnetocrystalline\nanisotropy [11]. Usually, the Heisenberg-type ferromagnet\ndoes not exist with intrinsic long-range magnetic ordering at\nfinite temperature in the 2D limit due to dominant thermal\nfluctuations [12]. But the single domain anisotropy or the\nexchange anisotropy can remove these limitations and allow\nthe long-range magnetic ordering to obtain a 2D Ising-like\nferromagnet [13]. In this way, 2D ferromagnetism has\nbeen found experimentally in many vdW materials such\nas Cr 2Ge2Te6(TC\u001961 K) [14], Cr 2Si2Te6(TC\u001932\nK) [15], Fe 3GeTe 2(TC\u0019215 K) [9], and CrI 3(TC\u001945\nK) [13], but the long-range FM ordering far below the room\ntemperature limiting their usage in technology. However,\na very few systems such as MnP (T C\u0019303 K) show\nroom temperature 2D ferromagnetism in the bulk phase [16].\nThus, searching for new 2D room-temperature FM materials\nwith large magnetocrystalline anisotropy is crucial for the\nrealization of potential technological applications [1, 2].\nFurther, magnetocaloric effect (MCE) in room-temperature\nFM vdW materials with maximum entropy change - DSmax\nMis\nof recent research interests due to their potential applications\nin the environmental friendly magnetic refrigeration [17].\nThere exists quite a few 2D FM systems showing significant -\nDSmax\nMsuch as Fe 3{xGeTe 2(1.1J/kg{ K at 5 T) [18], Cr 5Te8\n(1.6J/kg { K at 5 T) [19], Cr 2Si2Te6(5.05 J/kg { K at 5 T),\n\u0003setti@bose.res.in\n(a)\nCr2Cr1Te\nbc\nCounts(a.u.)\n(002)\n(004)\n(008)\n(b)\n2θ(degree)\nCounts(a.u.)\nE (keV)\nAtomic %\nCr : 44.3\nTe : 55.7\nCrCr\nCr Te TeTeTeTe (c)FIG. 1. (a) X-ray diffraction pattern of Cr 4Te5single crystals. Inset\nin (a) shows photographic image of single crystals. (b). Schematic\ndiagram of Cr 4Te5crystal structure. (c) Energy dispersive X-ray\nspectroscopy and scanning electron microscopy image (inset) of\nCr4Te5single crystal showing the chemical composition and crystal\nmorphology, respectively.\nand Cr 2Ge2Te6(2.64 J/kg { K at 5 T) [20].\nA Recent theoretical study suggested that the layered\nCrxTeysystems are potential candidates to realize the much-\nanticipated room-temperature 2D ferromagnetism in the\nbulk [21]. Since then, variety of Cr xTeycompounds have\nbeen grown experimentally and studied for their peculiar\n2D ferromagnetism, including, CrTe [22], Cr 2Te3[23],arXiv:2209.08477v1  [cond-mat.mtrl-sci]  18 Sep 20222\nCr3Te4[24], Cr 4Te5[25], and Cr 5Te8[26]. Out of these,\nCr4Te5has been reported to show highest T C\u0019318 K among\nthis family of compounds, to date in the bulk form. Generally,\nCrxTeycompounds possess alternating stacks of CrTe 2layers\nintercalated by the Cr layers (excess) along the c-axis [27].\nImportantly, it is that the intercalated Cr concentration plays\na crucial role in the crystal structure formation, magnetic\nstructure, and transport properties of these compounds[28–\n31].\nIn this contribution, we present magnetic properties,\ncritical behaviour analysis, and anisotropic magnetocaloric\neffect associated with room temperature ferromagnetism in\nCr4Te5single crystals. Magnetization M(T) data suggest\nferromagnetic ordering at a T Cof\u0019338 K with strong\nmagnetic anisotropy below T Cbetween the field applied\nparallel to ab-plane and parallel to c-axis. As a result,\nwe find a large temperature dependent magnetocrystalline\nanisotropy (K u=330 kJ/m3) at 3 K with ab-plane as\nthe easy magnetization plane. Further, the universal\nscaling of magnetocaloric effect suggests a second order\nmagnetic phase transition in this system. The critical\nexponents derived from the magnetocaloric effect are in\ngood agreement with the values obtained from the critical\nanalysis. The renormalization group theory analysis suggests\n3D Heisenberg-type magnetic interactions in this system.\nII. EXPERIMENTAL DETAILS\nHigh quality single crystals of Cr 4Te5were grown by the\nchemical vapor transport (CVT) technique with iodine as a\ntransport agent. Stoichiometric mixtures of Cr (99.99%, Alfa\nAesar) and Te (99.99%, Alfa Aesar) powders were mixed\nin a molar ratio of 5:5, added with iodine (3 mg/cc), and\nsealed into an evacuated quartz tube under Ar atmosphere.\nThe tube was placed in a gradient-temperature horizontal\ntwo-zone tube furnace for about 15 days with one end,\ncontaining powder mixture (source), maintained at 1000o\nC and the other end kept at 820oC (sink), as per the\nprocedure described earlier [32]. The as-grown single crystals\nwere large in size ( 5\u00025mm2) and were looking shiny.\nPhotographic image of a typical single crystal is shown in the\ninset of Fig. 1(a). X-ray diffraction (XRD) was performed\non the single crystals using Rigaku X-ray diffractometer\n(SmartLab, 9kW) with Cu K aradiation of wavelength\n1.54059 ˚A. Surface morphology and elemental analysis of\nthe single crystals were done using the scanning electron\nmicroscopy (SEM) and energy dispersive X-ray spectroscopy\n(EDXS), respectively. The EDXS measurements suggest an\nactual chemical composition of Cr 3.98Te5. Hereafter, for\nconvenience, we use nominal composition formula of Cr 4Te5\nwherever required. Magnetic measurements [ M(T) and\nM(H) ] were carried out using physical property measurement\nsystem (9 Tesla-PPMS, DynaCool, Quantum Design).\n360270180900FC @ 4000 Oe\nZFC@ 4000 Oe\nFC @ 200 Oe\nZFC@ 200 Oe\n-6 -4 -2 0 2 4 6 3 K\n 50 K\n 150 K\n 300 K\n 370 K\n-80-4004080\n-2-1 0 123 K\n-80-4004080\n-6 -4 -2 0 2 4 63 K\n50 K\n150 K\n300 K\n370 K\n80\n60\n40\n20\n0\n360 270 180 90 0FC @ 4000 Oe\nZFC@ 4000 Oe\nFC @ 200 Oe\nZFC@ 200 OeH ǁ ab H ǁ c\nH ǁ ab H ǁ c(a) (b)\n(c) (d)M(emu/g) M(emu/g)T (K) T (K)\nH (Tesla) H (Tesla)H ǁ cH ǁ ab\nHǁc\nHǁabFIG. 2. Temperature dependent magnetization M(T) data measured\nin the zero-field-cooled and field-cooled modes with the magnetic\nfields (H = 200 and 4000 Oe) applied in the ab-plane (a) and along\nthec-axis (b). Field dependent magnetization M(H) data measured\nforHkab(c) and Hkc(d) at various sample temperatures.\nIII. RESULTS AND DISCUSSION\nFigure 1(a) shows X-ray diffraction (XRD) pattern of\nCr4Te5single crystal. In the XRD pattern, we find intensity\npeaks of (00l) Bragg plane, indicating the crystal growth\nplane along the c-axis. We further notice that the (00l) Bragg\npeak position is shifted to lower 2 qvalues compared to the\nother Cr xTeycompounds [33, 34] as excess Cr atoms are\nintercalated between two CrTe 2layers, leading to increase in\nthe lattice parameters [35]. Thus, the intercalated Cr atoms\n(Cr-vacancies) sandwiched between two CrTe 2layers create\nvan der Waals gap as schematically shown in Fig. 1(b).\nA. Magnetization Measurements\nTo explore the magnetic properties of Cr 4Te5,\nmagnetization as a function of temperature M(T) and\nfield M(H) were measured with the field applied parallel to\nab-plane ( Hkab) and c-axis ( Hkc). Figs. 2(a) and 2(b)\nshow M(T) of Cr 4Te5measured with magnetic fields of\n200 and 4000 Oe for both HkabandHkcorientations,\nrespectively. Paramagnetic (PM) to ferromagnetic (FM)\ntransition is clearly visible at a Curie temperature of TC\u0019\n338 K when measured with 200 Oe for both the orientations\nin zero-field-cooled (ZFC) and field-cooled (FC) modes.3\n50\n40\n30\n20\n10\n0\n6543210\nH (Tesla)H ll c310 K\n370 K100\n80\n60\n40\n20\n0Ku(kJ/m3\n)\n340 330 320 310\nT (K)1.2\n1.0\n0.8\n0.6\n0.4\n0.2\n0.0Ratio\n Ku (T) [ Ms/Ms(310 K)]\n [Ms/Ms(310 K)]3\n [Ms/Ms(310 K)]1050\n40\n30\n20\n10\n0M(emu/g)\n6543210\nH (Tesla)310 K\n370 KH ll ab(a)\n(b)80\n70\n60\n50\n40\n30\n20Ku(kJ/m3\n)\n340 330 320 310\nT (K)40\n36\n32\n28\n340330320310\nT (K)\nHs(T)\n(c)\n(d)\nFIG. 3. Magnetization isotherms measured around TCfor (a) Hk\naband (b) Hkc. (c) Temperature dependent magnetocrystalline\nanisotropy ( Ku). Bottom inset in (c) shows the saturation\nmagnetization M sand top inset in (c) shows saturation magnetic field\nHsestimated below TC. (d) Ratios of [M s/Ms(310 K)]n(n+1)/2\nwith n =1, 2, and 4 (right-axis) overlapped with magnetocrystalline\nanisotropy (left-axis).\nThe Curie temperature 338 K found in this study is higher\nthan any known composition in the Cr xTeyfamily in their\nbulk phase [25, 36]. Splitting in the M(T) curves between\nZFC and FC data at TCis attributed to the competition\nbetween FM and AFM phases at low temperatures [37, 38].\nInterestingly, as shown in the inset of the Fig. 2(b), below\nTC, the magnetization M(T) increases with decreasing\ntemperature for Hkab, while it decreases with decreasing\ntemperature for Hkc. This observation clearly hints at\na significant magnetic anisotropy in this system which is\ntemperature dependent [26].\nNext, Figs. 2(c) and 2(d) show the magnetization isotherms\nM(H) measured for HkabandHkcat various temperatures,\nrespectively. In contrast to the other Cr xTeysystems where\nthe easy-axis is along c-axis [37, 39, 40], Cr 4Te5has the\neasy-axis on the ab-plane as the magnetization saturation\noccurs at an applied field of 0.5 T for Hkab, while 1.9 T\nis required to attain the magnetization saturation for Hkc\nat 3 K. This observation is in agreement with a previous\nstudy on Cr 4Te5[25]. On the other hand, as can be seen\nfrom Figs. 2(c) and 2(d), the saturation magnetization ( Ms)\ngradually decreases with increasing temperature and at \u0019\n338 K the long-range magnetic order is suppressed in both\norientations. Inset in Fig. 2(d) demonstrates the magnetization\nisotherm comparison between HkabandHkcorientations\nat 3 K.B. Magnetocrystalline Anisotropy\nWe studied the magnetocrystalline anisotropy (K u) using\nthe Stoner-Wolfarth model [41] with the help of magnetization\nisotherms shown in the Figs. 3(a) and 3(b) for both\nHkaband Hkc, respectively, measured around T C.\nThe magnetocrystalline anisotropy is calculated by using\nthe formula 2Ku/Ms=m0Hs, where m0is the vacuum\npermeability, M sis the saturation magnetization, and H sis\nthe field at which saturation magnetization occur [42]. Both\nHsand M sare considered experimentally from the isotherms\nshown in Figs. 3(a) and 3(b) to obtain K u. Fig. 3(c) depicts the\ntemperature dependent K uestimated from the experimental\ndata. We find K u=78.11 kJ/m3at T = 310 K which gradually\ndecreases with increasing temperature and reaches to 23.73\nkJ/m3atTC(338 K). On the other hand, we estimate K u\n\u0019330 kJ/m3at 3 K that is much larger than the K uvalues\nof other 2D magnetic systems such as CrBr 3(\u001986 kJ/m3\nat 5 K) [43], Cr 2Ge2Te6(\u001920 kJ/m3at 2 K) [20], and\nCr2Si2Te6(\u001965 kJ/m3at 2 K) [20] and comparable to the\nKuvalues of CrI 3(300 kJ/m3at 5 K) [43] and Fe 4GeTe 2\n(250 kJ/m3at 80 K) [20]. Though Fe 3GeTe 2shows extremely\nlarge magnetocrystalline anisotropy of 1460 kJ/m3at 2 K [44],\nit’s Curie temperature is much below the room temperature\n(TC=220 K) [45]. Thus, our studied sample of Cr 4Te5having\nlarge K uvalue of\u0019330 kJ/m3above room temperature looks\npromising candidate from the technological point of view.\nSince the magnetic anisotropy expectation value <Kn>is\ndirectly proportional to Mn(n+1)/2\ns , as per the classical theory\nof magnetism [46, 47], we plotted [Ms(T)/M s(310)]n(n+1)/2\nfor n=1, 2, and 4. Here, n=1 represents intrinsic anisotropy,\nn=2 represents uniaxial anisotropy, and n=4 represents\ncubic anisotropy, giving rise the exponents 1, 3, and 10,\nrespectively. Thus, in Fig. 3(d), we plotted [Ms(T)/M s(310)] ,\n[Ms(T)/M s(310)]3, and [Ms(T)/M s(310)]10ratios as a\nfunction of temperature. Most importantly, in Fig. 3(d),\nthe overlapped K u(T) of Fig. 3(c) matches very well with\n[Ms(T)/M s(310)]3ratio, confirming the dominant uniaxial\nanisotropy in this system that is strongly temperature\ndependent. The temperature dependent K uis found to be\noriginated from the fluctuating local spin clusters those are\nactivated from the thermal energy [46, 47]. A recent study\non Cr 5Te8shows temperature dependent uniaxial anisotropy\nbut mixed with a small component of cubic anisotropy as\nwell [19]. Nevertheless, our observations suggest that Cr 4Te5\nis a stronger magnetocrystalline anisotropic system compared\nto it’s sister compound Cr 5Te8.\nC. Critical Behaviour Analysis\nScaling hypothesis suggests that the second order phase\ntransition in a magnetic system around T Ccan be described\nby a set of critical exponents and magnetic equations of the\nstate [48]. Thus, the spontaneous magnetization Mspbelow\nTC, inverse susceptibility c{1\n0above TC, and magnetization\nisotherms M(H) atTCare characterized by the critical\nexponents b,g, and d, respectively. Mathematical equations4\n2000\n1500\n1000\n500\n0\n2400 1600 800 0\nH/M (Oe g/emu)310 K\n 370 KArrott plot\nH ll ab\nM1/β\n(104\n(emu/g)1/β\nH ll ab\nM1/β\n(103\n(emu/g)1/β\nH ll ab\nM1/β\n(1012\n(emu/g)1/β\nH ll ab\n H ll ab\nM1/β(105(emu/g)1/β\nH ll ab(a) (b) (c)\n(e) (d) (f))\n)\n)))\nFIG. 4. (a) Arrott plot ( M2vsH/M ). Modified Arrott plot from (b) 3D XY model, (c) 3D Ising model, (d) 2D Ising model, (e) 3D Heisenberg\nmodel, and (f) Tricritical meanfield model.\ngoverning the critical exponents are given by,\nMsp(T) = M 0({e)bfore<0,T<Tc, (1)\nc{1\n0(T) = (h 0/m0)eg,fore>0,T>Tc (2)\nM = DH1/d,fore= 0,T = T c, (3)\nwhere e= (T– TC)/TCis the reduced temperature, M0,\nh0/m0, and Dare the critical amplitudes[49]. In order\nto obtain the critical exponents b,g, and das well as\nthe exact value of T C, we first did the Arrott-plot M2vs.\nH/M as shown in Fig. 4(a). However, this method does\nnot produce the required parallel linear curves at higher\nfields, particularly below TC[50]. Therefore, we used the\nmodified Arrott-plot (MAP) technique and plotted M1/bvs.\n(H/M)1/gfor various models around T C. Fig .4(b) shows\nMAP for 3D XY model ( b= 0.345, g= 1.316 ) [51], Fig. 4(c)\nshows MAP for 3D Ising model ( b= 0.325, g= 1.241 ) [52],\nFig. 4(d) shows MAP for 2D Ising model ( b= 0.125, g=\n1.75) [53], Fig. 4(e) shows MAP for 3D Heisenberg model\n(b= 0.365, g= 1.386 ) [52], and Fig. 4(f) shows MAP for\nTricritical meanfield model ( b= 0.25, g= 1) [54]. As can be\nseen from Figs. 4(b)-4(f), none of these models can actually\nproduce the required parallel linear curves at higher magnetic\nfields from their respective critical exponents, confirming\nthat a single theoretical model cannot properly describe themagnetic interactions present in Cr 4Te5. Thus, we used\nthe modified Arrott-plot technique in an iterative method\nin the vicinity of TCto produce parallel linear curves as\nshown in Fig. 5(a) [55, 56]. This procedure gives c{1\n0(T)\nandMsp(T) as the intercepts on the x-axis and y-axis,\nrespectively. Fig. 5(b) depicts Mspandc{1\n0plotted as a\nfunction of temperature derived from Fig. 5(a). From Eqs. 1\nand 2 we estimate the critical exponents b= 0.485(3) with\nTC= 340.73(4) K and g= 1.202(5) with TC= 340.85(4) K.\nThe estimated TCvalues are very close to the value of 338\nK obtained from the M(T) data [see Figs. 2(a-b)]. Further,\nthe critical exponents can be found more accurately using\nthe Kouvel-Fisher (KF) plots [57], governed by the below\nequations.\nMsp(T)[dM sp(T)/dT]{1= (T { T c)/b, (4)\nc{1\n0(T)[d c{1\n0(T)/dT]{1= (T { T c)/g, (5)\nFig. 5(c) depicts the Kouvel-Fisher plots fitted with the\nEqs. 4 and 5 below and above TC. From the KF method\nfitting we derive the critical exponents b= 0.479(6) with TC\n= 340.55(2) K and g= 1.196(6) with TC= 340.85(3) K. The\nvalues of bandgobtained from the KF method are in good\nagrement with the values obtained from the MAP method. In\naddition, the exponent dhas been calculated using the Widom\nscaling relation, d= 1 + (g/b)[58]. Thus, d= 3.478 and d5\nMsp(emu/g)\nχ0-1\n(Oeg/emu)\nH || ab(a)\nH || ab\n 40\n30\n20\n10\n0\n5 4 3 2 1 0Tc ≈340.5 K\nM(emu/g)\nH || ab\nH (Tesla)(d)H || ab(b)\n(c))\n M1/β\n(102\n(emu/g)1/β\nFIG. 5. (a) Modified Arrott plot of M1/bvs(H/M)1/gat high\nfield values. (b) Temperature dependent spontaneous magnetization\nMsp(T) (left-axis) and inverse initial susceptibility c{1\n0(T)\n(right-axis). (c) Kouvel-Fisher plot of temperature dependent\nMsp(T)[dM sp(T)/dT]{1(left-axis) and c{1\n0(T)[d c{1\n0(T)/dT]{1\n(right-axis). (d) Isothermal magnetization M(H) atTC\u0019340.5 K.\nInset in (d) shows M(H) plot in the log-log scale.\n= 3.497 are calculated from MAP and KF plots, respectively.\nImportantly, the dvalues obtained from both MAP and KF\nmethods are close to the value d= 3.52(3) obtained using\nthe critical isotherm (CI) following the Eq. 3 as shown in\nFig. 5(d). All the experimentally derived critical exponents\nare summarized in the Table II. Note here that, although the\ncritical exponents obtained from MAP and KF techniques\nare close, we used the critical exponents of MAP for further\ndiscussions. Therefore, the critical exponent b=0.485(3)\nobtained from this study is close to the meanfield-model value\nofb=0.5 and g=1.202(5) is close to the 3D Heisenberg-model\nvalue of g=1.386.\nFurther, to verify the accuracy of critical exponents and\ncritical temperature obtained from MAP and KF methods, we\nemployed scaling analysis [48]. The scaling equations of the\nstate for magnetic systems are given by,\nM(H, e) =ebf\u0006(H/eb+g) (6)\nAbove equation can be write as well,\nm = f\u0006(h) (7)\nwhere m = e{bM(H, e)is the normalized magnetization\nandh = H e{(b+g)is the normalized field. For the correct\nbandgvalues obtained from the MAP and KF methods,\nfollowing the scaling hypothesis, the scaled magnetization\nmand field hshould fall on two different branches of the\nuniversal curves when plotted using the Eq. 6 and Eq. 7.\nOne branch for T <TCand another branch for T >TC, as\nshown in Figs. 6(a) and 6(b). Since all the curves shown in\nFigs. 6(a) and 6(b) overlap with respective universal curves,the scaling hypothesis reaffirms the reliability of the derived\ncritical exponents using the MAP and KF methods.\nTABLE I. Obtained critical parameters following the renormalization\ngroup theory analysis\nfd : ng g b s d\nf2 : 1g 1.20 0.38 1.22 4.158\nf2 : 2g 1.20 0.41 1.19 3.927\nf2 : 3g 1.20 0.55 1.05 3.182\nf3 : 2g 1.20 0.405 1.79 3.963\nf3 : 3g 1.20 0.46 1.71 3.609\nNext, the renormalization group (RG) theory analysis\nsuggest that the long- and short- range exchange interactions\ndecay with the distance rasJ(r) = r{(d+ s)andJ(r) = e{r/b,\nrespectively [53]. Here, dis the spatial dimensionality, sis a\npositive constant, and bis spatial scaling factor. The relation\nbetween the critical exponent gandsis given by,\ng= 1 +4\nd(n + 2)\n(n + 8)Ds+8(n { 4)(n + 2)\n(n + 8)2d2\n\u0003[2(7n + 20)G(d/2)\n(n + 8)(n { 4)+ 1]Ds2(8)\nwhere Ds=s{ d/2 ,G(d/2) = 3 { 0.25 \u0003(d/2)2.nand\ndare the spin and spatial dimensionality of the system. By\nconsidering the experimental gvalue of 1.196(6), following\nthe RG theory, we have calculated the critical exponent b\nand local exponent sfor various sets of fd : ngvalues and\nare tabulated in Table I. From this, we find that the critical\nexponent banddobtained using the 3D Heisenberg-model of\nfd = 3 : n = 3gare close to the experimentally obtained values\nof the critical exponents. Thus, the long-range exchange\ninteractions in Cr 4Te5decays as J(r) = r{(d+ s)= r{4.71for\nd = 3 ands=1.71.\nD. Magnetocaloric Effect\nIn order to explore the magnetocaloric effect (MCE),\nwe analyzed the field dependent isotherms M(H) taken at\ndifferent temperatures [see Figs. 3(a) and 3(b)] in the vicinity\nofTCfor both Hkaband Hkcorientations. Magnetocaloric\neffect is an intrinsic property of a ferromagnetic system,\ncausing heating or cooling adiabatically under the applied\nmagnetic fields [59]. Thus, a magnetic entropy change\nDSm(T,H) is induced in the presence of magnetic fields\nwhich is represented by the formula,\nDSm(T,H) =ZH\no(¶S\n¶H)TdH =ZH\no(¶M\n¶T)HdH (9)\nwhere (¶S\n¶H)T= (¶M\n¶T)Hbased on Maxwell’s relation. In\ncase of magnetization measured at small discrete field and6\nm2\n(104\nemu/g)2\n250\n200\n150\n100\n50\n0M\nε\n-β\n(emu/g)\n20 15 10 5 0\nH\nε\n-(β+γ)\n (104\n kOe)T < T c\nT > T c(b) (a)\n))\nFIG. 6. (a) and (b) are the normalized magnetization M and m2\nplotted as a function of re-normalized field hbelow and above T C.\nSee the text for more details.\ntemperature intervals, DSm(T,H) could be written as\nDSm(T,H) =RH\n0M(T i+1,H)dH {RH\n0M(T i,H)dH\nTi+1{ Ti(10)\nFigs. 7(a) and 7(b) show - DSm(T,H) plotted as a function\ntemperature under various magnetic fields up to 5 T taken\nwith a step size of 1 T for both H kaband Hkcorientations,\nrespectively. All - DSm(T,H) curves show a maximum\nchange in entropy with a broad peak at around TCas seen\nfrom Figs. 7(a) and 7(b). Further, we observe that the value\nof -DSm(T,H) and RCP increase monotonically with field\nforHkabas shown in Fig. 7(c). Under an applied field of\n5 T, the maximum of - DSm(T,H) is about 2.78 J kg{1K{1\nfor Hkaband is about 2.58 J kg{1K{1for Hkc. These -\nDSm(T,H) values taken at 5 T are comparable to the other\n2D ferromagnetic systems such as Cr 2Ge2Te6(2.64 J kg{1\nK{1) [20] and Cr 5Te8(2.38 J kg{1K{1) [19], larger than the\nvalues of Fe 3{xGeTe 2(1.14 J kg{1K{1) [60] and CrI 3(1.56J\nkg{1K{1) [61], and smaller than the values of CrB 3(7.2 J\nkg{1K{1) [62] and Cr 2Si2Te6(5.05 J kg{1K{1) [20]. Further,\n-DSm(T,H) of the studied Cr 4Te5are in good agreement with\nthe earlier reported values of 2.58 J kg{1K{1and 2.42 J kg{1\nK{1for Hkaband Hkc, respectively [25].\nTo estimate the relative cooling power (RCP), we employed\nthe relation RCP = - DSmax\nM\u0002dTFWHM , where - DSmax\nM is\nthe maximum entropy change near TCanddTFWHM is the\nfull width at half maximum of the peak [63]. The calculated\nRCP in Cr 4Te5is of 88.29 J kg{1at around TCwith an\napplied field of 5 T parallel to ab-plane. The RCP value of\nCr4Te5obtained in this study is comparable to the RCP value\nobtained in the other 2D systems such as Cr 2Ge2Te6(87 J\nkg{1)[20], but smaller than the values obtained from Cr 5Te8\n(131.2 J kg{1) [19], CrI 3(122.6 J kg{1) [61], Cr 2Si2Te6(114\nJ kg{1) [20], Fe 3{xGeTe 2(113 J kg{1) [60], and CrBr 3(191.5\nJ kg{1) [62].\nIn addition, both - DSmax\nMand RCP are related by the power\nlaw of magnetic field as given below [63, 64],\n{DSmax\nM= aHp(11)\nRCP = bHq(12)where p and q are the exponents. At T=T C, they can be written\nas\np = 1 +b{ 1\nb+g(13)\nq = 1 +1\nd(14)\nTABLE II. Critical exponents of Cr 4Te5. The MAP, KF, and CI\nrepresent modified Arrott plot, the Kouvel-Fisher plot, and critical\nisotherm, respectively.\nTechnique b g d p q\n-DSmax\nM0.70\nRCP 1.27\nMAP 0.485 1.202 3.472 0.695 1.288\nKFP 0.479 1.196 3.515 0.683 1.285\nCI 3.52 1.284\nFig. 7(c) summarizes the field dependence of - DSmax\nMand\nRCP. The fit of - DSmax\nMby the Eq. 11 yields p=0.70. This is\nin very good agreement with the pvalue of 0.689(4) obtained\nfrom the Eq. 13. Similarly, the field dependence of RCP is\nfitted by the Eq. 12, yielding to q= 1.27 that is close to the\nvalue of 1.28(1) obtained from the Eq. 14 using the critical\nexponent of dderived from the magnetic isotherms. Fig. 7(d)\ndisplays the temperature dependence of pat 5 T. The p(T)\ncurve follows universal behaviour across TCas it reaches to\nthe value 1 for T <TC[65]. On other hand, well above TC,\npreaches to 2 as a consequence of the Curie-Weiss law [65].\nAt T = TC,p(T) has a minimum value of 0.7, consistent with\nthe minimum value of 0.75 as per the universal curve [64].\nOverall, temperature dependence of pperfectly follows the\nuniversal behaviour of a second order phase transition [64].\nNext, we performed the scaling analysis of MCE following\nthe procedure given by Franco et. al. [64, 66]. The scaling\nanalysis of - DSm(T,H) is constructed by normalizing the -\nDSm(T,H) curves with respect to the maximum of - DSmax\nM\n[DSm(T,H)\nDSmax\nM]. The reduced temperature ( q\u0007) is defined by\nchoosing two reference temperatures ( Tr1\u0014TCandTr2>\nTC), satisfying the condition,DSm(Tr1<Tc)\nDSmax\nM=DSm(Tr2>Tc)\nDSmax\nM=\nh. Here, h is a scaling constant having the values within the\nrange of 0<h<1. Then, the rescaled temperature q\u0007can be\nwritten as\nq{= (T C{ T)/(T r1{ TC),T\u0014TC (15)\nq+= (T { T C)/(T r2{ TC),T>TC (16)\nFollowing MCE scaling analysis, all the curves of\nDSm/DSmax\nMplotted as a function of reduced temperature qat\nvarious magnetic fields collapse into a single curve as shown\nin Fig.7(e). This, confirms the second order magnetic phase\ntransition in Cr 4Te5[67–69]. Moreover, it can be seen that7\n-ΔSMMax\n(J/kgK)\nRCP(J/kg)\n(c)\n(f)H ll ab(a)\n(b)1.2\n1.0\n0.8\n0.6\n0.4\n0.2\n4 2 0 -2 -4\nθ 1T  2T  3T\n 4T  5T360\n350\n340\n330\n320\n3 2 1\nH1/Δ\n (T1/Δ\n)Tr1\nTr2(e)H ll ab\n(d)H ll ab3.0\n2.5\n2.0\n1.5\n1.0\n0.5\n0.0\n360 345 330 315\nT (K) 1 T  2 T  3 T\n 4 T  5 TH ll ab\n0.3\n0.2\n0.1\n0.0-ΔSMR\n(J/kgK)\n360 345 330 315\nT (K)0.2 T\n0.4 T\n0.6 T\n0.8 T\n 2 T1.8\n1.6\n1.4\n1.2\n1.0\n0.8\n0.6p\n360 345 330 315\nT (K) 5 T\nFIG. 7. The calculated magnetic entropy - DSmas a function of temperature at different magnetic field. (a) H jjab, (b) Hjjc, (c) Field\ndependence of maximum magnetic entropy changes (- DSmax\nM) (left axis) and relative cooling power (RCP) (right axis), (d) The exponent p\nas function of temperature obtained from the fitting of field dependent isothermal magnetic entropy change at various temperature, (e) The\nnormalised magnetic entropy changes as a function of rescaled temperature qat different field, Inset shows T rvs H1/DwithD=b+g, (f)\nRotational magnetic entropy change (- DSR\nM) as a function of temperature.\nthe universal curve of MCE is independent of the applied\nmagnetic field and temperature as it generally determined\nby the intrinsic magnetization of the system [64]. Next, the\nreference temperatures Tr1andTr2linearly depends on H1/D\nwithD=b+gas shown in the inset of Fig. 7(e). The\nrotational magnetic entropy changes ( DSR\nM) can be calculated\nusing the formula DSR\nM(T,H) =DSm(T,H ab) {DSm(T,H c).\nFig. 7(f) depicts the temperature dependence of - DSR\nM(T,H) .\nFrom Fig. 7(f) we can find a maximum of - DSR\nM(T,H) at\naround T Cwhen derived at a field interval of 0.2 T. The\nmaxima of - DSR\nM(T,H) shifts to lower temperatures with\nincreasing field intervals and for 2 T of field interval no\nmaxima is found down to 315 K. This behaviour is generally\nfound in the systems with strong magnetic anisotropy [19].\nFinally, before concluding, we would like to compare our\nresults with the existing literature on the composition of\nCr4Te5. Till date there exits two studies on this system, (i) in\nthe bulk phase [25] and (ii) in thinfilm phase [70]. The Curie\ntemperature of 338 K observed from our M(T) study is higher\nthan the Curie temperature of \u0019320 K previously reported in\nboth the bulk and thinfilm phases. It is known that in Cr xTey\nsystems the Curie temperature depends on various parameters\nsuch as sample thickness, sample composition, and on the\nmethod of preparation [71, 72]. The critical exponents of\nb=0.485(3) and g=1.202(5) are close to the meanfield-likeand 3D Heisenberg-like interactions, respectively, while the\ncritical exponents of b=0.388 and g=1.29 as reported in\nRef. [25] in the bulk phase are found to be close to the 3D\nHeisenberg-like and meanfield-like interactions, respectively.\nOn the other hand, the critical exponent found in the thinfilm\nphase b=0.491 is close to the meanfield-like interactions,\nconsistent with our observations [70]. Such a mixed phase\ninteractions are reported in many other 2D Ferromagnets\nsuch as in Fe 0.26TaS2(b= 0.459, g= 1.205) [73] and\nCo0.22TaS2(b= 0.43, g= 1.15) [74]. Nevertheless,\nthe renormalization group theory analysis suggests the 3D\nHeisenberg-type magnetic interactions in the studied system\nwhen estimated using the experimental gvalue of 1.202.\nIV . SUMMARY\nTo summarize, we thoroughly studied the magnetic\nanisotropy and magnetocaloric effect associated with critical\nexponent analysis of Cr 4Te5. The critical exponents of\nb,g, and dobtained from the modified Arrot-plot and\nKouvel-Fisher techniques match very well. Scaling analysis\nof the magnetic entropy change confirms the second order\nmagnetic phase transition in this system. Strong uniaxial\nmagnetocrystalline anisotropy found in this system could8\nstabilize the long-range FM ordering within few layers.\nThe renormalization group theory analysis suggests 3D\nHeisenberg-type magnetic interactions in Cr 4Te5.V . ACKNOWLEDGEMENTS\nAuthors thank SERB (DST), India for financial support\n(Grant No. SRG/2020/000393). S.C. and S.G. acknowledge\nUniversity Grants Commission (UGC), India for the PhD\nfellowship.\n[1] Y . Deng, Y . Yu, Y . Song, J. Zhang, N. Z. Wang, Z. Sun, Y . Yi,\nY . Z. Wu, S. Wu, J. Zhu, et al. , Gate-tunable room-temperature\nferromagnetism in two-dimensional Fe3GeTe 2, Nature 563, 94\n(2018).\n[2] B. Wang, Y . Zhang, L. Ma, Q. Wu, Y . Guo, X. Zhang, and\nJ. Wang, MnX (X= P, As) monolayers: a new type of two-\ndimensional intrinsic room temperature ferromagnetic half-\nmetallic material with large magnetic anisotropy, Nanoscale 11,\n4204 (2019).\n[3] A. K. Geim and I. V . Grigorieva, Van der waals heterostructures,\nNature 499, 419 (2013).\n[4] J. T. Heron, M. Trassin, K. Ashraf, M. Gajek, Q. He, S. Y .\nYang, D. E. Nikonov, Y .-H. Chu, S. Salahuddin, and R. Ramesh,\nElectric-field-induced magnetization reversal in a ferromagnet-\nmultiferroic heterostructure, Phys. Rev. Lett. 107, 217202\n(2011).\n[5] H. Liu, Y . Zhang, O. E. Glukhova, G. Zhang, L. Wang, J. Zhao,\nand J. Gao, Interlayer hopping kinetics of vacancies in cri3\nlayers leading to monolayer/bilayer heterostructures, Advanced\nMaterials Interfaces 9, 2200626 (2022).\n[6] A. Soumyanarayanan, N. Reyren, A. Fert, and\nC. Panagopoulos, Emergent phenomena induced by spin–\norbit coupling at surfaces and interfaces, Nature 539, 509\n(2016).\n[7] C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y . Xia, T. Cao, W. Bao,\nC. Wang, Y . Wang, et al. , Discovery of intrinsic ferromagnetism\nin two-dimensional van der waals crystals, Nature 546, 265\n(2017).\n[8] K. L. Seyler, D. Zhong, D. R. Klein, S. Gao, X. Zhang,\nB. Huang, E. Navarro-Moratalla, L. Yang, D. H. Cobden, M. A.\nMcGuire, et al. , Ligand-field helical luminescence in a 2D\nferromagnetic insulator, Nature Physics 14, 277 (2018).\n[9] Z. Fei, B. Huang, P. Malinowski, W. Wang, T. Song, J. Sanchez,\nW. Yao, D. Xiao, X. Zhu, A. F. May, et al. , Two-dimensional\nitinerant ferromagnetism in atomically thin Fe3GeTe 2, Nature\nmaterials 17, 778 (2018).\n[10] M. Gibertini, M. Koperski, A. F. Morpurgo, and K. S.\nNovoselov, Magnetic 2d materials and heterostructures, Nature\nnanotechnology 14, 408 (2019).\n[11] H. L. Zhuang, P. R. C. Kent, and R. G. Hennig, Strong\nanisotropy and magnetostriction in the two-dimensional stoner\nferromagnet fe3gete 2, Phys. Rev. B 93, 134407 (2016).\n[12] N. D. Mermin and H. Wagner, Absence of ferromagnetism\nor antiferromagnetism in one- or two-dimensional isotropic\nheisenberg models, Phys. Rev. Lett. 17, 1133 (1966).\n[13] B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein,\nR. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A.\nMcGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-Herrero,\nand X. Xu, Layer-dependent ferromagnetism in a van der Waals\ncrystal down to the monolayer limit, Nature 546, 270 (2017).\n[14] Y . Liu and C. Petrovic, Critical behavior of quasi-two-\ndimensional semiconducting ferromagnet Cr2Ge2Te6, Phys.\nRev. B 96, 054406 (2017).[15] T. J. Williams, A. A. Aczel, M. D. Lumsden, S. E. Nagler, M. B.\nStone, J.-Q. Yan, and D. Mandrus, Magnetic correlations in the\nquasi-two-dimensional semiconducting ferromagnet CrSiTe 3,\nPhys. Rev. B 92, 144404 (2015).\n[16] X. Sun, S. Zhao, A. Bachmatiuk, M. H. R ¨ummeli, S. Gorantla,\nM. Zeng, and L. Fu, 2D Intrinsic Ferromagnetic MnP Single\nCrystals, Small 16, 2001484 (2020).\n[17] L. M. Moreno-Ram ´ırez and V . Franco, Reversibility of\nthe magnetocaloric effect in the bean-rodbell model,\nMagnetochemistry 7, 10.3390/magnetochemistry7050060\n(2021).\n[18] V . Y . Verchenko, A. A. Tsirlin, A. V . Sobolev, I. A.\nPresniakov, and A. V . Shevelkov, Ferromagnetic order, strong\nmagnetocrystalline anisotropy, and magnetocaloric effect in\nthe layered telluride Fe3GeTe 2, Inorganic Chemistry 54, 8598\n(2015), pMID: 26267350.\n[19] Y . Liu, M. Abeykoon, E. Stavitski, K. Attenkofer, and\nC. Petrovic, Magnetic anisotropy and entropy change in trigonal\nCr5Te8, Phys. Rev. B 100, 245114 (2019).\n[20] Y . Liu and C. Petrovic, Anisotropic magnetic entropy change in\nCr2X2Te6(X = Si and Ge ), Phys. Rev. Materials 3, 014001\n(2019).\n[21] Y . Zhu, X. Kong, T. D. Rhone, and H. Guo, Systematic\nsearch for two-dimensional ferromagnetic materials, Phys. Rev.\nMaterials 2, 081001 (2018).\n[22] T. Eto, M. Ishizuka, S. Endo, T. Kanomata, and T. Kikegawa,\nPressure-induced structural phase transition in a ferromagnet\nCrTe , Journal of Alloys and Compounds 315, 16 (2001).\n[23] F. Wang, J. Du, F. Sun, R. F. Sabirianov, N. Al-Aqtash,\nD. Sengupta, H. Zeng, and X. Xu, Ferromagnetic Cr2Te3\nnanorods with ultrahigh coercivity, Nanoscale 10, 11028\n(2018).\n[24] B. Hessen, T. Siegrist, T. Palstra, S. Tanzler,\nand M. Steigerwald, Hexakis (triethylphosphine)\noctatelluridohexachromium and a molecule-based synthesis\nof chromium telluride, Cr3Te4, Inorganic chemistry 32, 5165\n(1993).\n[25] L.-Z. Zhang, A.-L. Zhang, X.-D. He, X.-W. Ben, Q.-L. Xiao,\nW.-L. Lu, F. Chen, Z. Feng, S. Cao, J. Zhang, and J.-Y . Ge,\nCritical behavior and magnetocaloric effect of the quasi-two-\ndimensional room-temperature ferromagnet Cr4Te5, Phys.\nRev. B 101, 214413 (2020).\n[26] Y . Wang, J. Yan, J. Li, S. Wang, M. Song, J. Song, Z. Li,\nK. Chen, Y . Qin, L. Ling, H. Du, L. Cao, X. Luo, Y . Xiong,\nand Y . Sun, Magnetic anisotropy and topological Hall effect in\nthe trigonal chromium tellurides Cr5Te8, Phys. Rev. B 100,\n024434 (2019).\n[27] J. Dijkstra, H. H. Weitering, C. F. van Bruggen, C. Haas,\nand R. A. de Groot, Band-structure calculations, and magnetic\nand transport properties of ferromagnetic chromium tellurides\n(CrTe,Cr 3Te4,Cr2Te3), Journal of Physics: Condensed\nMatter 1, 9141 (1989).9\n[28] Z.-L. Huang, W. Bensch, D. Benea, and H. Ebert, Anion\nsubstitution effects on structure and magnetism in the\nchromium chalcogenide Cr5Te8—Part I: Cluster glass behavior\nin trigonal Cr(1+x)Q2 with basic cell (Q=Te, Se; Te:Se=7:1), J.\nSolid State Chem. 177, 3245 (2004).\n[29] Z.-L. Huang, W. Bensch, S. Mankovsky, S. Polesya, H. Ebert,\nand R. K. Kremer, Anion substitution effects on structure\nand magnetism of the chromium chalcogenide Cr 5Te8Part II:\nCluster-glass and spin-glass behavior in trigonal Cr(1+x) Q2\nwith basic cells and trigonal Cr(5+x) Q8 with superstructures\n(Q=Te, Se; Te:Se=6:2), J. Solid State Chem. 179, 2067 (2006).\n[30] J. Wontcheu, W. Bensch, S. Mankovsky, S. Polesya, H. Ebert,\nR. K. Kremer, and E. Br ¨ucher, Anion substitution effects on\nthe structure and magnetism of the chromium chalcogenide\nCr5Te8—Part III: Structures and magnetism of the high-\ntemperature modification Cr(1+x)Q2 and the low-temperature\nmodification Cr(5+x)Q8 (Q=Te, Se; Te:Se=5:3), J. Solid State\nChem. 181, 1492 (2008).\n[31] H. Ipser, K. L. Komarek, and K. O. Klepp, Transition metal-\nchalcogen systems viii: The CrTe phase diagram, Journal of\nthe Less Common Metals 92, 265 (1983).\n[32] T. Hashimoto, K. Hoya, M. Yamaguchi, and I. Ichitsubo,\nMagnetic properties of single crystals Cr2{dTe3, Journal of the\nPhysical Society of Japan 31, 679 (1971).\n[33] Y . Liu and C. Petrovic, Critical behavior of the quasi-two-\ndimensional weak itinerant ferromagnet trigonal chromium\ntelluride Cr0.62Te, Phys. Rev. B 96, 134410 (2017).\n[34] X. Zhang, T. Yu, Q. Xue, M. Lei, and R. Jiao, Critical behavior\nand magnetocaloric effect in monoclinic Cr5Te8, Journal of\nAlloys and Compounds 750, 798 (2018).\n[35] A. L. Coughlin, D. Xie, X. Zhan, Y . Yao, L. Deng, H. Hewa-\nWalpitage, T. Bontke, C.-W. Chu, Y . Li, J. Wang, et al. , Van der\nWaals Superstructure and Twisting in Self-Intercalated Magnet\nwith Near Room-Temperature Perpendicular Ferromagnetism,\nNano letters 21, 9517 (2021).\n[36] L.-Z. Zhang, Q.-L. Xiao, F. Chen, Z. Feng, S. Cao, J. Zhang,\nand J.-Y . Ge, Multiple magnetic phase transitions and critical\nbehavior in single crystal Cr5Te6, Journal of Magnetism and\nMagnetic Materials 546, 168770 (2022).\n[37] L.-Z. Zhang, X.-D. He, A.-L. Zhang, Q.-L. Xiao, W.-L. Lu,\nF. Chen, Z. Feng, S. Cao, J. Zhang, and J.-Y . Ge, Tunable\ncurie temperature in layered ferromagnetic Cr5+dTe8single\ncrystals, APL Materials 8, 031101 (2020).\n[38] Z.-L. Huang, W. Kockelmann, M. Telling, and W. Bensch, A\nneutron diffraction study of structural and magnetic properties\nof monoclinic Cr5Te8, Solid State Sciences 10, 1099 (2008).\n[39] J. Yan, X. Luo, G. Lin, F. Chen, J. Gao, Y . Sun, L. Hu, P. Tong,\nW. Song, Z. Sheng, W. Lu, X. Zhu, and Y . Sun, Anomalous hall\neffect of the quasi-two-dimensional weak itinerant ferromagnet\nCr4.14Te8, EPL (Europhysics Letters) 124, 67005 (2019).\n[40] R. Mondal, R. Kulkarni, and A. Thamizhavel, Anisotropic\nmagnetic properties and critical behaviour studies of trigonal\nCr5Te8single crystal, Journal of Magnetism and Magnetic\nMaterials 483, 27 (2019).\n[41] E. C. Stoner and E. Wohlfarth, A mechanism of magnetic\nhysteresis in heterogeneous alloys, Philosophical Transactions\nof the Royal Society of London. Series A, Mathematical and\nPhysical Sciences 240, 599 (1948).\n[42] B. D. Cullity and C. D. Graham, Introduction to magnetic\nmaterials (John Wiley & Sons, 2011).\n[43] N. Richter, D. Weber, F. Martin, N. Singh,\nU. Schwingenschl ¨ogl, B. V . Lotsch, and M. Kl ¨aui,\nTemperature-dependent magnetic anisotropy in the layered\nmagnetic semiconductors CrI3and CrBr 3, Phys. Rev.Materials 2, 024004 (2018).\n[44] N. Le ´on-Brito, E. D. Bauer, F. Ronning, J. D. Thompson,\nand R. Movshovich, Magnetic microstructure and magnetic\nproperties of uniaxial itinerant ferromagnet fe3gete2, Journal\nof Applied Physics 120, 083903 (2016).\n[45] N. Le ´on-Brito, E. D. Bauer, F. Ronning, J. D. Thompson,\nand R. Movshovich, Magnetic microstructure and magnetic\nproperties of uniaxial itinerant ferromagnet Fe 3GeTe 2, J. Appl.\nPhys. 120, 083903 (2016).\n[46] C. Zener, Classical theory of the temperature dependence of\nmagnetic anisotropy energy, Phys. Rev. 96, 1335 (1954).\n[47] W. Carr Jr, Temperature dependence of ferromagnetic\nanisotropy, Physical Review 109, 1971 (1958).\n[48] H. E. Stanley, Phase transitions and critical phenomena , V ol. 7\n(Clarendon Press, Oxford, 1971).\n[49] M. E. Fisher, The theory of equilibrium critical phenomena,\nReports on progress in physics 30, 615 (1967).\n[50] A. Arrott, Criterion for Ferromagnetism from Observations of\nMagnetic Isotherms, Phys. Rev. 108, 1394 (1957).\n[51] J. C. Le Guillou and J. Zinn-Justin, Critical exponents from field\ntheory, Phys. Rev. B 21, 3976 (1980).\n[52] S. Kaul, Static critical phenomena in ferromagnets with\nquenched disorder, Journal of magnetism and magnetic\nmaterials 53, 5 (1985).\n[53] M. E. Fisher, S.-k. Ma, and B. G. Nickel, Critical Exponents for\nLong-Range Interactions, Phys. Rev. Lett. 29, 917 (1972).\n[54] D. Kim, B. Revaz, B. L. Zink, F. Hellman, J. J. Rhyne, and\nJ. F. Mitchell, Tricritical Point and the Doping Dependence\nof the Order of the Ferromagnetic Phase Transition of\nLa1{xCaxMnO 3, Phys. Rev. Lett. 89, 227202 (2002).\n[55] A. Arrott and J. E. Noakes, Approximate Equation of State For\nNickel Near its Critical Temperature, Phys. Rev. Lett. 19, 786\n(1967).\n[56] A. K. Pramanik and A. Banerjee, Critical behavior\nat paramagnetic to ferromagnetic phase transition in\nPr0.5Sr0.5MnO 3: A bulk magnetization study, Phys. Rev. B\n79, 214426 (2009).\n[57] J. S. Kouvel and M. E. Fisher, Detailed magnetic behavior of\nnickel near its curie point, Phys. Rev. 136, A1626 (1964).\n[58] B. Widom, Degree of the critical isotherm, The Journal of\nChemical Physics 41, 1633 (1964).\n[59] V . K. Pecharsky and K. A. Gschneidner Jr, Magnetocaloric\neffect and magnetic refrigeration, Journal of Magnetism and\nMagnetic Materials 200, 44 (1999).\n[60] Y . Liu, J. Li, J. Tao, Y . Zhu, and C. Petrovic, Anisotropic\nmagnetocaloric effect in Fe 3{xGeTe 2, Sci. Rep. 9, 13233\n(2019).\n[61] Y . Liu and C. Petrovic, Anisotropic magnetocaloric effect in\nsingle crystals of CrI3, Phys. Rev. B 97, 174418 (2018).\n[62] X. Yu, X. Zhang, Q. Shi, S. Tian, H. Lei, K. Xu, and H. Hosono,\nLarge magnetocaloric effect in van der Waals crystal CrBr 3,\nFront Phys-beijing 14, 10.1007/s11467-019-0883-6 (2019).\n[63] K. Gschneidner Jr, V . Pecharsky, A. Pecharsky, and C. Zimm,\nRecent developments in magnetic refrigeration, in Materials\nscience forum , V ol. 315 (Trans Tech Publ, 1999) pp. 69–76.\n[64] V . Franco, J. Bl ´azquez, and A. Conde, Field dependence of the\nmagnetocaloric effect in materials with a second order phase\ntransition: A master curve for the magnetic entropy change,\nApplied physics letters 89, 222512 (2006).\n[65] V . Franco, R. Caballero-Flores, A. Conde, Q. Dong, and\nH. Zhang, The influence of a minority magnetic phase on\nthe field dependence of the magnetocaloric effect, Journal of\nMagnetism and Magnetic Materials 321, 1115 (2009).10\n[66] V . Franco, C. Conde, J. Bl ´azquez, A. Conde, P. ˇSvec,\nD. Jani ˇckovi ˇc, and L. Kiss, A constant magnetocaloric response\ninFeMoCuB amorphous alloys with different Fe/B ratios,\nJournal of applied physics 101, 093903 (2007).\n[67] H. Oesterreicher and F. Parker, Magnetic cooling near curie\ntemperatures above 300 k, Journal of applied physics 55, 4334\n(1984).\n[68] J. Y . Law, V . Franco, L. M. Moreno-Ram ´ırez, A. Conde, D. Y .\nKarpenkov, I. Radulov, K. P. Skokov, and O. Gutfleisch, A\nquantitative criterion for determining the order of magnetic\nphase transitions using the magnetocaloric effect, Nature\ncommunications 9, 1 (2018).\n[69] V . Franco, J. Law, A. Conde, V . Brabander, D. Karpenkov,\nI. Radulov, K. Skokov, and O. Gutfleisch, Predicting\nthe tricritical point composition of a series of LaFeSi\nmagnetocaloric alloys via universal scaling, Journal of Physics\nD: Applied Physics 50, 414004 (2017).\n[70] J. Wang, W. Wang, J. Fan, H. Zheng, H. Liu, C. Ma, L. Zhang,\nW. Tong, L. Ling, Y . Zhu, and H. Yang, Epitaxial growth and\nroom-temperature ferromagnetism of quasi-2D layered Cr 4Te5thin film, J. Phys. Appl. Phys. 55, 165001 (2022).\n[71] H. Li, L. Wang, J. Chen, T. Yu, L. Zhou, Y . Qiu, H. He,\nF. Ye, I. K. Sou, and G. Wang, Molecular Beam Epitaxy\nGrown Cr 2Te3Thin Films with Tunable Curie Temperatures\nfor Spintronic Devices, ACS Appl. Nano Mater. 2, 6809 (2019).\n[72] I. H. Lee, B. K. Choi, H. J. Kim, M. J. Kim, H. Y . Jeong,\nJ. H. Lee, S.-Y . Park, Y . Jo, C. Lee, J. W. Choi, S. W. Cho,\nS. Lee, Y . Kim, B. H. Kim, K. J. Lee, J. E. Heo, S. H. Chang,\nF. Li, B. L. Chittari, J. Jung, and Y . J. Chang, Modulating\nCurie Temperature and Magnetic Anisotropy in Nanoscale-\nLayered Cr 2Te3Films: Implications for Room-Temperature\nSpintronics, ACS Appl. Nano Mater. 4, 4810 (2021).\n[73] C. Zhang, Y . Yuan, M. Wang, P. Li, J. Zhang, Y . Wen, S. Zhou,\nand X.-X. Zhang, Critical behavior of intercalated quasi-van\nder Waals ferromagnet Fe0.26TaS2, Phys. Rev. Materials 3,\n114403 (2019).\n[74] Y . Liu, Z. Hu, E. Stavitski, K. Attenkofer, and C. Petrovic,\nMagnetic critical behavior and anomalous Hall effect in\n2H{Co 0.22TaS2single crystals, Phys. Rev. Research 3,\n023181 (2021)."    },    {        "title": "2210.11278v1.Large_orbital_magnetic_moment_in_VI3.pdf",        "content": "1 \n Large orbital magnetic moment in VI 3 \n \nDávid Hovančík1*, Cinthia Piamonteze2*, Jiří Pospíšil1, Karel Carva1, Vladimír Sechovsky1 \n1 Charles University, Faculty of Mathematics and Physics, Department of Condensed Matter \nPhysics, Ke Karlovu 5, 121 16 Prague 2, Czech Republic \n2Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland \nAbstract: The existence of the V3+- ion orbital moment is the open issue of the nature of \nmagnetism in the van der Waals ferromagnet VI 3. The huge magnetocrystalline anisotropy in \nconjunction with the significantly reduced ordered magnetic moment compared to the \nspin-only value provides strong but indirect evidence of a significant V orbital moment. We \nused the unique capability of X-ray magnetic circular dichroism to determine the orbital \ncomponent of the total magnetic moment and provide for the first time a direct proof of an \nexceptionally sizable orbital moment of the V3+ ion in VI 3. Our ligand field multiplet \nsimulations of the XMCD spectra in synergy with the results of DFT calculations agree with 2 \n the existence of two V sites with different orbital occupations and therefore different OM \nmagnitudes in the ground state. \n \nKeywords: 2D van der Waals magnet, x-ray magnetic circular dichroism, orbital moment, VI 3 \n \nIntroduction \nTwo-dimensional (2D) van der Waals (vdW) magnetic materials have recently attracted \never-increasing interest from materials researchers because their stable magnetic ordering, \neven in atomically thin monolayers, provides great opportunities for spintronic devices1-5. \nThe magnetic ordering in 2D relies on magnetic anisotropy since Mermin and Wagner6 have \nshown that long-range magnetic order cannot exist in the one- or two-dimensional isotropic \nHeisenberg model. As proposed by Bruno7 the magnetocrystalline anisotropy energy is \nproportional to the orbital moment (OM). In the extensively studied CrI 3, the electronic \nconfiguration of the Cr3+ ion which experiences an octahedral crystal field is 𝑡ଶ   ଷ, S = 3/2, L \n= 0. The small OM value below 0.1 B was confirmed by XMCD measurements8. As \nproposed by Kim et al.9, the magnetic anisotropy of CrI 3 comes from the iodine p orbital \nspin-orbit coupling (SOC) and the p-d covalence between Cr and I, hence, being relatively \nsmall. Here we focus on a Mott insulator VI 3, which began to be intensively investigated after 3 \n the papers of Son et al.10, Kong et al.11, and Tian et al.12 on ferromagnetism in bulk crystals \nwere published. The recent work of Lin et al.13 showing an anomalous increase of TC for one \nor few monolayers of VI 3 further boosted this interest. In VI 3 the electrons in a partially filled \n𝑡ଶ    level may possess an effective OM l = 114, which for the occupancy of two electrons \nleads to S=1, L=1 configuration. As S and L are antiparallel due to the less than half-filled V \nd-shell, one would expect a total magnetic moment of 1 B, if the picture is not affected by \nother effects quenching the orbital moment. Early publications on VI 310-12 showed huge \ndisagreement on the magnitude of the out-of-plane ( cR-axis; cR stands for c-axis in \nrhombohedral structure) magnetic moment (at 2 K and 5 T the values 2.5, 1.25, and 1.1 B/V, \nrespectively). Later, Liu et al. reported a moment of 1 B/V and proposed an unquenched \nOM15. This scenario was further supported by neutron diffraction experiment done by Hao et \nal.16, which found a magnetic moment of 1.2 B/V at 6 K. Electronic structure calculations \nhave found a ground state where the 𝑒ᇱ  orbital is half occupied leading to an unquenched \nOM17,18. Strong correlations represented by Hubbard U are sufficient to open a gap between \nSO split bands, so that this solution is semiconducting. On the other hand, another calculation \nsuggests different semiconducting ground state with fully occupied 𝑒ᇱ orbitals19, which leads \nto a negligible orbital moment. Direct evidence and quantification of an unquenched OM is \nstill lacking. 4 \n TX3 transition metal trihalides ( T =transition metal , X = Cl, Br, I) adopt two common \nlayered crystal structures with T3+ ions arranged in a honeycomb network at the edge-sharing \noctahedral coordination by six X ions (see Figure 2a). VI 3 undergoes a structural transition at \n79 K between the high-temperature rhombohedral 𝑅3ത and low-temperature monoclinic \n𝐶2/𝑚 variant11,20-23. Below 50 K it orders ferromagnetically with an easy-magnetization \ndirection tilted by ~ 40o from cR-axis16,24. A strong magnetic anisotropy is observed, which \nreinforces the proposition of an unquenched OM25. Around 36 K a second magnetic phase \ntransition was proposed separating the states with one (at temperatures between 36 and 50 K) \nand two inequivalent magnetically ordered V sites (at lower temperatures)26. In addition, a \nsymmetry lowering from monoclinic to a triclinic structure appears at 32 K upon cooling23,27. \nMagneto-transport measurements also show a qualitatively different ferromagnetic state \nbelow 40 K, compared to the one between Tc and 40 K28. \nMotivated by the multiple hints for non-zero OM in VI 3 we utilized the unique ability of \nX-ray magnetic circular dichroism (XMCD) to determine spin and orbital moments \nseparately 29,30 in an element-specific fashion. Our results confirm the existence of an \nexceptionally large OM. Ligand field multiplet simulations of the XAS and XMCD spectra \nare in qualitative agreement with the existence of two V sites with different orbital \noccupations and therefore different OM at 2 K. 5 \n  \n \nResults and discussion \nMagnetization measurements show a 1.23 B/V-ion along the easy magnetization direction, \nwell matching the neutron-experiment result16 (see SI, Figure S1). Angular dependence of 5-T \nmagnetization in the acR-plane at 2 K shown that the easy axis which is tilted ~ 40º from the \ncR-axis, also in agreement with previous publications16,24,31.   \nFigure 1a shows the V L 3,2 (2p1/2,3/2 → 3d, between 505 and 530 eV) helicity-averaged XAS \nof VI3 taken at 300K and 2K. The spectrum agrees well with the V3+ valence state published for \nV2O3 and the previous XAS published for VI 332,33. For the 2K spectrum besides the V L 3,2 \nabsorption edges, we detected another two resonances above 530eV which were identified as \nthe O K absorption edge (1 s → π*, 1s → σ) of the O 2 molecule34 see Figure 1b. The presence of \nO K edge XAS of different intensities was observed at each temperature below 90 K. We \nascribe this effect to the finite amount of frozen O 2 (Tboiling < 90K) coming from the chamber \nresidual pressure. The 90 K XAS measurement (see SI, Figure S2) shows an oxygen-free \nspectrum measured after the low-temperature data, clearly showing that the O 2 is not intrinsic \nto the sample surface.  6 \n  \nFigure 1. a) V L 2,3 XAS spectra of VI 3 measured at 300 K and 2 K after the removal of the step \nfunction (continuous lines). b) The XAS spectra in extended energy region and corresponding \nXAS integrals (dotted lines). c) XMCD spectrum at 2 K (continuous line) and corresponding \nintegral (dotted line). \nFigure 1c shows the V L 3,2 XMCD spectra derived as σ+(ω) - σ-(ω) measured at 2K and 5T \napplied parallel to the cR-axis. From the XMCD integral shown in figure 1c, one can notice \nthat the value of the integrated intensity is finite and positive. According to the \nmagneto-optical sum rules of the XMCD spectra29,30 (see SI, Eq (S1)) the total integral of the \nXMCD signal is proportional to the orbital magnetic moment. Therefore, the positive finite \nintegral undoubtedly confirms the nonzero out-of-plane ( cR-axis) OM component antiparallel \n7 \n to the spin moment, as expected. Applying the magneto-optical orbital sum rules to the \nspectra we obtain m orbital = 0.6 (1) B. The error bar takes into account the uncertainty in \ndetermining the XAS integral due to the oxygen contribution, as detailed in the SI. The spin \nsum rule cannot be applied to V spectra due to the large overlap of states from the L 3 and L 2 \nedge, which makes the calculation of a correction factor unfeasible35,36.  \n In VI 3 a trigonal distortion leads to a splitting of 𝑡ଶ    level into 𝑎ଵ    singlet and 𝑒ᇱ \ndoublet. Two possible ground states for 3 d2 configurations are: (i) lower energy of 𝑒ᇱ \ndoublet which would lead to the ground state with a fully occupied 𝑒ᇱ ଶ level and orbital \nsinglet with L= 0; (ii) electronic occupation of 𝑎ଵ   ଵ 𝑒ିᇱ ଵ levels where the 𝑒ᇱ doublet is \nfurther split by spin-orbit interaction (see Figure 2 b, c and d). In the second case, the electron \nin the 𝑒ିᇱ  level can be assigned an effective moment analogous to p electrons, l = 1, and the \nground state may be in the high OM state with L= 1.  8 \n  \nFigure 2. a) Crystal structure of VI 3 monolayer. The VI 6 cluster is emphasized. b) c) and d) \nCrystal field splitting for O h and D 3d (flattened/elongated octahedra) symmetry and \ncorresponding electron occupation.  \nIn the case of a trigonally distorted cubic lattice, one can decide which of the two above \ncases is energetically favorable knowing whether the cubic body diagonal (along C 3- axis) is \nelongated or shortened. In VI 3 the symmetry of more distant neighbors of V beyond the \noctahedral I cage differs significantly from the octahedral one (see Fig. 2), which may lead to \ncorrections (of trigonal symmetry) comparable to that of the small I cage distortion. \nTherefore, we have employed the electronic structure DFT (for details see SI, DFT) \ncalculations to get a more accurate picture. When both, the spin-orbit interaction and \ncorrelation effects in terms of Hubbard U are included, these calculations converge to two \n9 \n strikingly different solutions: either a state with quenched OM, typical for 3 d transition \nmetals, or a state with an exceptionally high OM17,18. The latter state is energetically \nfavorable, but the energy difference is small, approximately 5.6 meV according to our \ncalculation. Under some circumstances, the state with quenched OM may become preferred. \nFor the spin moment, all calculations predict values close to 2 B in agreement with Hund \nrules.  \n \nFigure 3. Calculated orbital-resolved DOS of VI 3 for majority spin V 3d electrons. a) solution \nwith high 3 d OM value b) solutions with quenched OM.   \nFor the solution with high OM, the orbital-resolved DOS for the majority spin (Figure 3 a) \nshows that the 𝑒ିᇱ  states are almost fully occupied while 𝑒ାᇱ  states are empty. Notably the \nrelevant bands are rather broad and 𝑎ଵ    character bands have a different evolution in \n10 \n k-space than 𝑒ᇱ bands. Nevertheless, one can say that 𝑎ଵ    states are generally energetically \nlower than 𝑒ᇱ as expected for this situation. On the other hand, the solution with the \nquenched OM shows a complete occupation of 𝑒ିᇱ  and 𝑒ାᇱ  states whereas, the 𝑎ଵ    orbital \nstates are above the Fermi level. This solution was found previously using the VASP code \nwith a smaller Hubbard U = 2 eV19. \nTo help understanding of the V ground state we have also performed ligand multiplet \nsimulation of the XAS and XMCD spectra incorporating the strong interaction between the \n2p core hole and 3 d electrons37. Given the broadness of some of the bands, the use of an ab \ninitio electronic structure to obtain parameters for the multiplet calculation has to be done \nwith caution. From the calculation performed without Hubbard U in order to look at single \nelectron energy levels we estimate 𝐸(𝑡ଶ    )−𝐸(𝑒) splitting 10𝐷 to be in the range of 1.3 \n- 1.8 eV. For the multiplet simulation, we have used 10𝐷 = 1.5 eV from this range. Other \nparameters and details of the simulation are described in the SI. 11 \n   \nFigure 4. X-ray absorption (left panel) and x-ray magnetic circular dichroism (right panel). The \ndata is plotted in black while the simulations are in blue, red, and green. a) and b) simulations \ncorresponding to 𝐷௧= −0.15 𝑒𝑉  and a quenched OM. c) and d) simulations corresponding \nto 𝐷௧= 0.3 𝑒𝑉  and an unquenched OM. e) and f) XAS and XMCD simulation is the \naverage of (a) and (b) simulations which corresponds to an OM of ~ 0.6 B. \n12 \n Figure 4 shows the measured spectra compared to simulations. Blue curves (Figure 4 a, b) \ncorrespond to a negative 𝐷௧ energy splitting which leads to 𝑒ᇱ orbital as the lowest in \nenergy. This simulation corresponds to 〈𝑆௭〉= −1, 〈𝐿௭〉= 0.05 for V in the ground state at 10 \nK. The red simulation (Figure 4 b, c), on the other hand, corresponds to a positive 𝐷௧ of the \n𝑡ଶ    levels with 𝑎ଵ    as the lowest energy orbital giving 〈𝑆௭〉= −1, 〈𝐿௭〉= 1.13. The largest \ndiscrepancy for the simulation with 𝑒ᇱ ଶ is an additional positive peak at the L 2 XMCD around \n520 eV. The 𝑎ଵ   ଵ 𝑒ିᇱ ଵ ground state simulation shows a different intensity ratio compared to the \ndata for the positive and negative peaks at the L 3-edge XMCD between 513 and 515 eV. The \n𝑎ଵ   ଵ 𝑒ିᇱ ଵ simulation gives an unquenched OM, in qualitative agreement with the experimental \nfinding, however, the size of m orbital is overestimated by a factor of 2, approximately. In the \ninelastic neutron scattering experiments, Lane et al.38 have found that their observations fit \nonly with a system composed of two simultaneously coexisting V sites with opposite trigonal \ndistortion, where one leads to the quenched OM while the other one leads to an unquenched \nOM. Various kinds of coexisting different stable states for V have already been suggested in \nother works17,26,39. The d SOC and crystal field parameters used for the simulations shown in \nFigure 4 are in close agreement with those used by Lane et al.. To simulate the co-existence of \ntwo V sites with opposite distortions we take the average (50% occupancy each) of the \nsimulations for 𝑒ᇱ ଶ and 𝑎ଵ   ଵ 𝑒ିᇱ ଵ ground state which is represented by the green curve in 13 \n Figure 4 e, f. The qualitative agreement between data and simulation is improved in \ncomparison to the single site simulations. Most importantly, the average OM from the two \nsites, which corresponds to 0.59 B is in very good agreement with the value obtained from the \nXMCD orbital sum rule. Therefore, our results support the picture of two coexisting \ninequivalent V magnetic sites.  \nRecently an ARPES investigation has revealed a significant occupation of 𝑎ଵ    orbitals, in \naddition to 𝑒ᇱ orbitals33. In their work, the authors attribute that to V2+ at the surface, although \ntheir XAS agrees with V3+. The argument for this proposition is based on a prediction that 𝑒ᇱ \nshould be fully occupied at the ground state and only the existence of V2+ could then explain \nthe observed 𝑎ଵ    occupation. The existence of a ground state with the 𝑎ଵ    orbital occupied \nand a high OM, as found here following previous works17,18,38 would be a reasonable \nexplanation of the ARPES results.  \nThe importance of both many-body effects as well as solid-state hybridization, and the \npossible presence of multiple V sites in this system represent a challenging task for the theory; \na more accurate description of the spectra may be achieved with the use of advanced \nconfiguration interaction techniques or the Bethe-Salpeter equation37,40. \n \n4. Conclusions 14 \n In summary, our XMCD results unequivocally demonstrate the existence of an unquenched \nOM of V in VI 3, thus resolving the long debate on this issue. Our findings are connected to \ntheoretical models of the V ground state in VI 3 , help us to reveal which orbitals are occupied, \nand explain the large magnetic anisotropy observed in this system. Using ligand field multiplet \nsimulations, we could show that the ground state of the V ion would agree with inelastic \nneutron scattering results where two V sites with opposite trigonal distortion were proposed. \n \nAcknowledgments \nThis work is a part of the research project GAČR 21-06083S which is financed by the Czech \nScience Foundation. The experiments were carried out in the Materials Growth and \nMeasurement Laboratory MGML (see: http://mgml.eu) which is supported by the program of \nCzech Research Infrastructures (project no. LM2018096). This project was also supported by \nOP VVV project MATFUN under Grant No. CZ.02.1.01/0.0/0.0/15_003/0000487. We thank \nA. Marmodoro for valuable discussions. This work was supported by the Ministry of \nEducation, Youth and Sports of the Czech Republic through the e-INFRA CZ (ID:90140). \n \n \nReferences: \n 15 \n  \n \n1 Duong, D. L., Yun, S. J. & Lee, Y. H. van der Waals Layered Materials: Opportunities \nand Challenges. ACS Nano  11, 11803-11830, doi:10.1021/acsnano.7b07436 (2017). \n2 Kumari, S., Pradhan, D. K., Pradhan, N. R. & Rack, P. D. Recent developments on 2D \nmagnetic materials: challenges and opportunities. Emergent Materials  4, 827-846, \ndoi:10.1007/s42247-021-00214-5 (2021). \n3 Gong, C. & Zhang, X. Two-dimensional magnetic crystals and emergent \nheterostructure devices. Science 363, 706 (2019). \n4 Ahn, E. C. 2D materials for spintronic devices. npj 2D Materials and Applications  4, \n17, doi:10.1038/s41699-020-0152-0 (2020). \n5 Miao, N. & Sun, Z. Computational design of two-dimensional magnetic materials. \nWIREs Computational Molecular Science  12, e1545, \ndoi:https://doi.org/10.1002/wcms.1545  (2022). \n6 Mermin, N. D. & Wagner, H. Absence of Ferromagnetism or Antiferromagnetism in \nOne- or Two-Dimensional Isotropic Heisenberg Models. Physical Review Letters  17, \n1133-1136, doi:10.1103/PhysRevLett.17.1133 (1966). \n7 Bruno, P. Tight-binding approach to the orbital magnetic moment and \nmagnetocrystalline anisotropy of transition-metal monolayers. Physical Review B  39, \n865-868, doi:10.1103/PhysRevB.39.865 (1989). \n8 Frisk, A., Duffy, L. B., Zhang, S., van der Laan, G. & Hesjedal, T. Magnetic X-ray \nspectroscopy of two-dimensional CrI3 layers. Materials Letters  232, 5-7, \ndoi:https://doi.org/10.1016/j.matlet.2018.08.005  (2018). \n9 Kim, D.-H.  et al. Giant Magnetic Anisotropy Induced by Ligand  LS Coupling in \nLayered Cr Compounds. Physical Review Letters  122, 207201, \ndoi:10.1103/PhysRevLett.122.207201 (2019). \n10 Son, S.  et al. Bulk properties of the van der Waals hard ferromagnet VI 3. Physical \nReview B  99, 041402, doi:10.1103/PhysRevB.99.041402 (2019). \n11 Kong, T.  et al. VI3-a New Layered Ferromagnetic Semiconductor. Advanced Materials  \n31, 1808074, doi:10.1002/adma.201808074 (2019). \n12 Tian, S.  et al. Ferromagnetic van der Waals Crystal VI3. Journal of the American \nChemical Society  141, 5326-5333, doi:10.1021/jacs.8b13584 (2019). \n13 Lin, Z.  et al. Magnetism and Its Structural Coupling Effects in 2D Ising Ferromagnetic \nInsulator VI3. Nano Letters  21, 9180-9186, doi:10.1021/acs.nanolett.1c03027 (2021). \n14 Khomskii, D. I. Transition Metal Compounds .  (Cambridge University Press, 2014). 16 \n 15 Liu, Y., Abeykoon, M. & Petrovic, C. Critical behavior and magnetocaloric effect in \nVI3. Physical Review Research  2, 013013, doi:10.1103/PhysRevResearch.2.013013 \n(2020). \n16 Hao, Y.  et al. Magnetic Order and Its Interplay with Structure Phase Transition in van \nder Waals Ferromagnet VI 3. Chinese Physics Letters  38, 096101, \ndoi:10.1088/0256-307x/38/9/096101 (2021). \n17 Yang, K., Fan, F., Wang, H., Khomskii, D. I. & Wu, H. VI3: A two-dimensional Ising \nferromagnet. Physical Review B  101, 100402, doi:10.1103/PhysRevB.101.100402 \n(2020). \n18 Sandratskii, L. M. & Carva, K. Interplay of spin magnetism, orbital magnetism, and \natomic structure in layered van der Waals ferromagnet VI3. Physical Review B  103, \n214451, doi:10.1103/PhysRevB.103.214451 (2021). \n19 Nguyen, T. P. T., Yamauchi, K., Oguchi, T., Amoroso, D. & Picozzi, S. Electric-field \ntuning of the magnetic properties of bilayer VI3: A first-principles study. Physical \nReview B  104, 014414, doi:10.1103/PhysRevB.104.014414 (2021). \n20 Son, S.  et al. Bulk properties of the van der Waals hard ferromagnet VI3. Physical \nReview B  99, 041402, doi:10.1103/PhysRevB.99.041402 (2019). \n21 Dolezal, P.  et al. Crystal structures and phase transitions of the van der Waals \nferromagnet VI3. Physical Review Materials  3, 121401, \ndoi:10.1103/PhysRevMaterials.3.121401 (2019). \n22 Kratochvílová, M.  et al. Crystal structure evolution in the van der Waals vanadium \ntrihalides. Journal of Physics: Condensed Matter  34, 294007, \ndoi:10.1088/1361-648x/ac6d38 (2022). \n23 Marchandier, T.  et al. Crystallographic and magnetic structures of the VI3 and LiVI3 \nvan der Waals compounds. Physical Review B  104, 014105, \ndoi:10.1103/PhysRevB.104.014105 (2021). \n24 Koriki, A.  et al. Magnetic anisotropy in the van der Waals ferromagnet VI 3. Physical \nReview B  103, 174401, doi:10.1103/PhysRevB.103.174401 (2021). \n25 Yan, J.  et al. Anisotropic magnetic entropy change in the hard ferromagnetic \nsemiconductor VI 3. Physical Review B  100, 094402, \ndoi:10.1103/PhysRevB.100.094402 (2019). \n26 Gati, E.  et al. Multiple ferromagnetic transitions and structural distortion in the van der \nWaals ferromagnet VI 3 at ambient and finite pressures. Physical Review B  100, 094408, \ndoi:10.1103/PhysRevB.100.094408 (2019). \n27 Doležal, P.  et al. Crystal structures and phase transitions of the van der Waals \nferromagnet VI 3. Physical Review Materials  3, 121401, \ndoi:10.1103/PhysRevMaterials.3.121401 (2019). 17 \n 28 Soler-Delgado, D.  et al. Probing Magnetism in Exfoliated VI3 Layers with \nMagnetotransport. Nano Letters  22, 6149-6155, doi:10.1021/acs.nanolett.2c01361 \n(2022). \n29 Thole, B. T., Carra, P., Sette, F. & van der Laan, G. X-ray circular dichroism as a probe \nof orbital magnetization. Physical Review Letters  68, 1943-1946, \ndoi:10.1103/PhysRevLett.68.1943 (1992). \n30 Carra, P., Thole, B. T., Altarelli, M. & Wang, X. X-ray circular dichroism and local \nmagnetic fields. Physical Review Letters  70, 694-697, \ndoi:10.1103/PhysRevLett.70.694 (1993). \n31 Valenta, J.  et al. Pressure-induced large increase of Curie temperature of the van der \nWaals ferromagnet VI 3. Physical Review B  103, 054424, \ndoi:10.1103/PhysRevB.103.054424 (2021). \n32 Caputo, M.  et al. Metal to insulator transition at the surface of V2O3 thin films: An \nin-situ view. Applied Surface Science  574, 151608, \ndoi:https://doi.org/10.1016/j.apsusc.2021.151608  (2022). \n33 De Vita, A.  et al. Influence of Orbital Character on the Ground State Electronic \nProperties in the van Der Waals Transition Metal Iodides VI3 and CrI3. Nano Letters  \n22, 7034-7041, doi:10.1021/acs.nanolett.2c01922 (2022). \n34 Yang, B. X., Kirz, J. & Sham, T. K. Oxygen K-edge absorption spectra of O2, CO and \nCO2. Physics Letters A  110, 301-304, \ndoi:https://doi.org/10.1016/0375-9601(85)90777-7  (1985). \n35 Teramura, Y., Tanaka, A. & Jo, T. Effect of Coulomb Interaction on the X-Ray \nMagnetic Circular Dichroism Spin Sum Rule in 3 d Transition Elements. Journal of the \nPhysical Society of Japan  65, 1053-1055, doi:10.1143/JPSJ.65.1053 (1996). \n36 Piamonteze, C., Miedema, P. & de Groot, F. M. F. Accuracy of the spin sum rule in \nXMCD for the transition-metal L edges from manganese to copper. Physical Review B  \n80, 184410, doi:10.1103/PhysRevB.80.184410 (2009). \n37 de Groot, F. M. F.  et al. 2p x-ray absorption spectroscopy of 3d transition metal \nsystems. Journal of Electron Spectroscopy and Related Phenomena  249, 147061, \ndoi:https://doi.org/10.1016/j.elspec.2021.147061  (2021). \n38 Lane, H.  et al. Two-dimensional ferromagnetic spin-orbital excitations in honeycomb \nVI3. Physical Review B  104, L020411, doi:10.1103/PhysRevB.104.L020411 (2021). \n39 Huang, C., Wu, F., Yu, S., Jena, P. & Kan, E. Discovery of twin orbital-order phases in \nferromagnetic semiconducting VI3 monolayer. Physical Chemistry Chemical Physics  \n22, 512-517, doi:10.1039/C9CP05643B (2020). 18 \n 40 Dvorak, M. & Rinke, P. Dynamical configuration interaction: Quantum embedding that \ncombines wave functions and Green's functions. Physical Review B  99, 115134, \ndoi:10.1103/PhysRevB.99.115134 (2019). \n \n  19 \n Supplementary Information - \nCRYSTAL GROWTH  \nThe single crystals of VI 3 were prepared by the CVT method directly from the \nstoichiometric ratio (1:3) of elements. All elements as well as the filing of the quartz ampoule \nwere performed in a glove box under Ar inert atmosphere. The degradation process of VI 3 \nwas described in detail by Kratochvilova et al.1. The chemical composition of single crystals \nwas verified by scanning electron microscopy (SEM) in combination with energy-dispersive \nX-ray detector (EDX). \n \nMAGNETIZATION MEASUREMENTS  \nWe performed detailed magnetization measurements of VI 3 bulk samples at 2 K to check the \ntotal value of V moment. The magnetization data was measured in magnetic field up to 7 T \nusing a SQUID magnetometer MPMS7T ( Quantum Design Inc. ). To probe the angular \ndependence of magnetization in acR-plane (at 2 K in 5 T) we used a homemade rotator with \nrotation axis orthogonal to the applied magnetic field. The angular-depended data was \nmeasured in relative units and then scaled to the data measured along the cR-axis and ab-plane. 20 \n The results are shown in Figure S1. The maximum magnetization value 1.23 B was found \nalong direction tilted by ~ 40º from the cR-axis.   \n \n \nFigure S1  - top: magnetization isotherms measured at 2 K bottom: angular dependent \nmagnetization. The angular dependence was scaled according to the absolute values of \nmagnetization measured along cR-axis and ab-plane (see blue and red dots).   \n \nXMCD sum rules \n21 \n     We used magneto-optical sum rules2,3 to analyze the direction and value of the projection \nof an orbital magnetic moment ( morbital). The spin sum rule requires a separation of the XMCD \ncontribution from the L 3 and L 2 edges. Correction factors of the spin sum rule have been \ncalculated for the late transition metals4,5. However, for Ti, V, and Cr the overlap is large and \nno correction factor can be properly calculated. Therefore, we focus only on the orbital sum \nrules given by the expression below: \n \n𝑷∙𝒎୭୰ୠ୧୲ୟ୪= −4\n3q\nr𝑛𝜇 (\n1) \n \nwhere 𝑷 is the x-ray propagation unit vector, q and r are defined in figure S2, 𝑛 stands for \nthe number of holes per atom in the 3 d orbital, 𝜇 is Bohr magneton. In expression (1) the \nquantity r is calculated from the XAS defined as the sum of individual spectra measured with \nleft and right circularly polarized X-rays. 22 \n  \nFigure S2 - top: XAS and bottom: XMCD measured at 90 K normal incidence, 5 T applied \nfield parallel to the X-ray direction. \nFigure S3 shows the XMCD, and corresponding integral measured at 2 K, 5 T, normal \nincidence. The integral value indicated is calculated as the average in the energy region \n540-580eV and the error bar the standard deviation. This data was used for the OM calculation \npresented in the manuscript. \n23 \n  \nFigure S3  - XMCD measured at 2 K, 5 T normal incidence and corresponding integral. The \nintegral value indicated is the average within the energy range 540-580 eV and the error bar is \nthe standard deviation. \n \nAs discussed in the manuscript, the XAS measured at low temperature presents an additional \ncontribution of the oxygen K-edge, which makes difficult a direct calculation of the XAS \nintegral. There are different ways estimating the XAS integral for the sum rule, which we \ndiscuss in sequence. Figure S4 shows the XAS spectra and corresponding integral for the 2 K \nand 300 K data. One way to calculate the XAS integral is simply to adjust the step function to \nthe 2K XAS spectrum around 527 eV subtract the step function from the XAS and take the \nintegral at this energy. The value of the XAS integral is 3.76 and the corresponding OM, if we \n24 \n use the XMCD integral from figure S3 and 𝑛= 8, is 0.67 B. The issue with this method is \nthat normalization of the step function cannot be done at a high-energy post-edge as it is \nusually done.  \n \nFigure S4  - XAS (continuous line), step function (dotted line) and corresponding integral after \nstep function removal (dashed line) for the spectrum measured at 2 K (green) and 300 K (dark \ngray) in normal incidence. The value of the respective integrals is given above the arrows at the \nenergy they were taken. \nA second method would be to take the integral of the data measured at 300 K, where the \noxygen spectrum does not interfere, and the step function can be normalized at high energy. In \nthis case the step function is normalized around 570eV and the corresponding integral after the \n25 \n step function removal is 4.81. The resulting OM is 0.53 B. This method is better than the \nprevious one because we can calculate the full XAS integral more precisely. The disadvantage \nis that the XAS at 300K is not identical to the one at 2K. As seen in figure S4, the relative ratio \nof L3 and L 2 peaks is different.  \nFor this reason, we suggest a third method, which is to estimate the difference in the XAS \nintegral, if the step function is normalized at 527 eV or at 570 eV and use that correction factor \nto estimate the correct XAS integral for the 2 K spectrum with the step normalized at 570 eV. \nThe correction factor is calculated using the 300 K data. As shown in figure S5 the XAS \nintegral of the 300 K data taken at 527 eV, with the step function normalized to the data at this \nenergy is 3.53. Therefore, the integral taken just above the L 2 edge corresponds to \n3.53/4.81*100=73% of the integral taken at 570 eV. Using this relative difference between the \n300 K integrals at two different energy points we can attempt to correct the 2 K XAS integral \nwhich can only be taken in a restricted energy range. We obtain: 3.76/0.73 = 5.12 as an \nestimate for the 2 K XAS integral value if there would be no Oxygen contribution to the \nspectrum. If we use this value for the XAS integral, we obtain an OM of 0.49 B. \nIn summary, we present three different methods to calculate the XAS integral and the OM. \nEach method has its advantages and disadvantages. The main message of these calculations is \nto show that the OM for vanadium in VI 3 is significantly large no matter which method we use. 26 \n In order to give a final number for the OM estimate which still describes the experimental \nproblems in its determination, we decided to take the average value of the three methods \nproposed and use the standard deviation among them as the error bar, which comes down to \nand OM of 0.6(1) B. \n \n \nFigure S5  - XAS measured at RT normal incidence (continuous line). Dotted line shows two \ndifferent options for the step function: aligning with the XAS at 527 eV or at 570 eV. Dashed \nlines show the corresponding integral, following the color of the step function. \n \nMULTIPLET SIMULATIONS \n27 \n X-ray absorption spectra and x-ray magnetic circular dichroism spectra were simulated using \nligand field multiplet simulations based on atomic Hartree-Fock calculations using Cowan's \ncode6 followed by chain symmetry proposed by Butler7. This approach has been used for the \nfirst time by Thole and van der Laan for the description of x-ray absorption spectra2. The local \nV symmetry used in the simulations was D 3d which is reduced to C 3 when the magnetic \nmoment is added. Slater Integrals were reduced by 50% from Hartree-Fock values. A crystal \nfield splitting (CFS) 10D q = 1.5 eV was used. The trigonal distortion D  was varied while D  \nwas kept to zero. In the text, we refer to the size of trigonal distortion using the energy splitting \nof the 𝑡ଶ     levels 𝐸൫𝑒ᇱ ൯−𝐸൫𝑎ଵ   ൯= 3𝐷ఙ+20 3⁄𝐷ఛ= 𝐷௧. The d-SOC used was 13 meV \nfollowing Lane et al.8, which corresponds to 50% of the atomic value. The p-SOC was \nincreased by 6% to fit with the experimental splitting of the L 3 and L 2 edges. \nThe energy levels in trigonal symmetry are related to the CFS parameters as shown in \nequations (2-4). \n \n𝐸(𝑒′)=−4×𝐷+𝐷ఙ+23ൗ×𝐷ఛ (2) \n𝐸(𝑎ଵ)=−4×𝐷−2×𝐷ఙ−6×𝐷ఛ (3) \n𝐸(𝑒)=6×𝐷+73ൗ×𝐷ఛ (4) \n 28 \n The choice of CFS parameters was based on DFT calculations and previous results, as from \nLane et al.8  \n \nXAS, XMCD MEASUREMENTS \nX-ray magnetic circular dichroism measurements were carried out using the EPFL-PSI \nend-station in the X-Treme beamline9, Swiss Light Source. The crystals were mounted inside a \nglove box and cleaved for a fresh surface. Then the crystals were transferred in an inert \natmosphere from the glove box to the load lock, where they were cleaved for a second time and \nthen pumped down. To improve conductivity for the total electron yield measurements, the \ncrystals were capped with a few nm carbon layer evaporated in situ in the ultra-high vacuum \nsample preparation chamber. The measurements shown were done with the x-rays \nperpendicular to the exfoliation plane (parallel to the c-axis in rhombohedral structure, cR).  \nFigure S6 shows two measurements done during the same beamtime with a time difference \nof 6.5 hours. These measurements show that no sign of change in the XAS nor XMCD is \nobserved due to long incidence time of the x-rays. 29 \n  \nFigure S6  - a) XAS and b) XMCD measurements done in the same crystal 6.5 hrs. apart in time \nshowing no sign of radiation damage. \n \nDFT CALCULATIONS \nDensity functional theory (DFT) calculations employed the full-potential linear augmented \nplane wave (FP-LAPW) method, as implemented in the band structure program ELK10. The \ngeneralized gradient approximation (GGA) parametrized by Perdew-Burke- Ernzerhof11 has \nbeen used to determine the exchange-correlation potential. SOC is known to play a key role in \nthe formation of OM and has been included in the calculation. DFT simulations utilizing \nGGA-PBE have already successfully described quasi-2D compound VI312. The full Brillouin \nzone has been sampled by 10 × 10 × 5 k-points and the convergence w.r.t. k-mesh density has \nbeen verified. An increased accuracy of expansion into spherical harmonics has been used with \nlmax=14. Since the material is known to be a Mott insulator, we have included the effect of \n30 \n electron-electron correlations in terms of the Hubbard correction term U = 4.3 eV 13,14 together \nwith Hund’s rule exchange parameter J = 0.8 eV, acting on 3 d electrons of V. Double counting \nwas treated in the fully localized limit. \n \n \n1 Kratochvílová, M.  et al. The surface degradation and its impact on the magnetic \nproperties of bulk VI3. Materials Chemistry and Physics  278, 125590, \ndoi:10.1016/j.matchemphys.2021.125590 (2022). \n2 Thole, B. T., Carra, P., Sette, F. & van der Laan, G. X-ray circular dichroism as a probe \nof orbital magnetization. Physical Review Letters  68, 1943-1946, \ndoi:10.1103/PhysRevLett.68.1943 (1992). \n3 Carra, P., Thole, B. T., Altarelli, M. & Wang, X. X-ray circular dichroism and local \nmagnetic fields. Physical Review Letters  70, 694-697, \ndoi:10.1103/PhysRevLett.70.694 (1993). \n4 Teramura, Y., Tanaka, A. & Jo, T. Effect of Coulomb Interaction on the X-Ray \nMagnetic Circular Dichroism Spin Sum Rule in 3 d Transition Elements. Journal of the \nPhysical Society of Japan  65, 1053-1055, doi:10.1143/JPSJ.65.1053 (1996). \n5 Piamonteze, C., Miedema, P. & de Groot, F. M. F. Accuracy of the spin sum rule in \nXMCD for the transition-metal L edges from manganese to copper. Physical Review B  \n80, 184410, doi:10.1103/PhysRevB.80.184410 (2009). \n6 Cowan, R. D. The Theory of Atomic Structure and Spectra .  (University of California \nPress, 1981). \n7 STEGER, E. Philip H. Butler. Point group symmetry applications. Methods and tables. \nplenum press New York and London, 1981 567 seiten, Preis: US $ 55.00. Crystal \nResearch and Technology  18, 404-404, doi: https://doi.org/10.1002/crat.2170180316  \n(1983). \n8 Lane, H.  et al. Two-dimensional ferromagnetic spin-orbital excitations in honeycomb \nVI3. Physical Review B  104, L020411, doi:10.1103/PhysRevB.104.L020411 (2021). \n9 Piamonteze, C.  et al. X-Treme beamline at SLS: X-ray magnetic circular and linear \ndichroism at high field and low temperature. Journal of Synchrotron Radiation  19, \n661-674,doi:10.1107/S0909049512027847 (2012). \n10 url: http://elk.sourceforge.net/ , <url: http://elk.sourceforge.net/ .> ( 31 \n 11 Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized Gradient Approximation Made \nSimple Physical Review Letters  78, 1396-1396, doi:10.1103/PhysRevLett.78.1396 \n(1997). \n12 Sandratskii, L. M. & Carva, K. Interplay of spin magnetism, orbital magnetism, and \natomic structure in layered van der Waals ferromagnet VI3. Physical Review B  103, \n214451, doi:10.1103/PhysRevB.103.214451 (2021). \n13 Anisimov, V. I., Zaanen, J. & Andersen, O. K. Band theory and Mott insulators: \nHubbard U instead of Stoner I. Physical Review B  44, 943-954, \ndoi:10.1103/PhysRevB.44.943 (1991). \n14 Anisimov, V. I., Aryasetiawan, F. & Lichtenstein, A. I. REVIEW ARTICLE: \nFirst-principles calculations of the electronic structure and spectra of strongly \ncorrelated systems: the LDA+ U method. Journal of Physics Condensed Matter  9, \n767-808, doi:10.1088/0953-8984/9/4/002 (1997). \n "    },    {        "title": "2210.11827v1.A_possible_electronic_state_quasi_half_valley_metal_in___mathrm_VGe_2P_4___monolayer.pdf",        "content": "arXiv:2210.11827v1  [cond-mat.mtrl-sci]  21 Oct 2022A possible electronic state quasi-half-valley-metal in VGe2P4monolayer\nSan-Dong Guo1, Yu-Ling Tao1, Zhuo-Yan Zhao1, Bing Wang2, Guangzhao Wang3and Xiaotian Wang4\n1School of Electronic Engineering, Xi’an University of Post s and Telecommunications, Xi’an 710121, China\n2Institute for Computational Materials Science, School of P hysics and Electronics,Henan University, 475004, Kaifeng , China\n3Key Laboratory of Extraordinary Bond Engineering and Advan ced Materials Technology of Chongqing,\nSchool of Electronic Information Engineering, Yangtze Nor mal University, Chongqing 408100, China and\n4Institute for Superconducting and Electronic Materials,\nUniversity of Wollongong, Wollongong 2500, Australia\nOne of the key problems in valleytronics is to realize valley polarization. Ferrovalley (FV) semi-\nconductor and half-valley-metal (HVM) have been proposed, which possess intrinsic spontaneous\nvalley polarization. Here, we propose the concept of quasi- half-valley-metal (QHVM), including\nelectron and hole carriers with only a type of carriers being valley polarized. The QHVM may re-\nalize separation function of electron and hole. A concrete e xample of VGe 2P4monolayer is used to\nillustrate our proposal through the first-principle calcul ations. To better realize QHVM, the electric\nfield is applied to tune related valley properties of VGe 2P4. Within considered electric field range,\nVGe2P4is always ferromagnetic (FM) ground state, which possesses out-of-plane magnetization by\ncalculating magnetic anisotropy energy (MAE) including ma gnetic shape anisotropy (MSA) and\nmagnetocrystalline anisotropy (MCA) energies. These out- of-plane FM properties guarantee intrin-\nsic spontaneous valley polarization in VGe 2P4. Within a certain range of electric field, the QHVM\ncan be maintained, and the related polarization properties can be effectively tuned. Our works pave\nthe way toward two-dimensional (2D) functional materials d esign of valleytronics.\nKeywords: Valley, Electric field, Magnetic anisotropy Emai l:sandongyuwang@163.com\nI. INTRODUCTION\nIn analogy to charge/spin of electronics/spintronics,\nthe valley has been recognized as an extra degree of free-\ndom for carriers, widely known as valleytronics1. The\nvalley is characterized by a local energy extreme in\nthe conduction band or valence band. In a material,\ntwo or more degenerate but inequivalent valley states\nshould exist to encode, store and process information2,3.\nFor nonmagnetic valleytronic materials, 2H-transition-\nmetal dichalcogenides (TMD) have attracted extensive\nattention4–10, because they have a pair of degenerate but\ninequivalent valleys and the large intervalley distance.\nHowever, these 2D materials lack spontaneous valley po-\nlarization. Although various strategies, such as optical\npumping, magnetic field, magnetic substrates and mag-\nneticdoping4–10, havebeenexecutedtoartificiallyinduce\nvalley polarization, these methods destroy the intrinsic\nenergy band structures and crystal structures.\nFortunately, FV semiconductor with spontaneous spin\nand valley polarizations has been proposed11, which\nbreaks time-reversal and space-inversion symmetry with\nperfect coupling of the valley with charge and spin. The\nFV semiconductors have been predicted in many 2D\nmaterials11–30, which can overcome these shortcomings\nof the extrinsic valley polarization materials. Recently,\nin analogy to half-metals in spintronics (see Figure 1 ),\nthe concept of HVM has been proposed31. The con-\nduction electrons of HVM are intrinsically 100% valley\npolarized and 100% spin polarized even when including\nspin-orbit coupling (SOC). The electron correlation ef-\nfect or strain can induce the FV semiconductor to HVM\ntransition in some special 2D materials19,31–34, which is\ngenerally related to topological phase transitions. How-\nFIG. 1. (Color online) Schematic diagram of the analogy be-\ntween ferromagnetic half-metal (a) and half-valley-metal (b).\nThe spin-up/spin-dn is equivalent to -K/K valley. In (a), on e\nspin channel is conducting, whereas the other is insulating .\nIn (b), one valley channel is metal, in which conduction elec -\ntrons are intrinsically 100% valley polarized, and the othe r is\ninsulator.\never, the HVM is just at one point, not a region of elec-\ntron correlationstrength orstrain, which may be difficult\nto achieve in experiment.\nIn this work, we propose a concept of QHVM, where\nelectron and hole carriers simultaneously exist with only\na type of carrier being valley polarized (see Figure 2 ).\nIn QHVM, the Fermi level slightly touches the conduc-\ntion band minima (CBM) and valence band maxima\n(VBM), and the Berry curvature in 2D Brillouin zone\n(BZ) only occurs around -K and K valleys with oppo-\nsite signs and unequal magnitudes. Under an in-plane\nlongitudinal electric field E, the nonzero Berry curvature\nΩ(k) makes the carriers of -K valley obtain the general\ngroup velocity v/bardbland the anomalous transverse velocity2\nFIG. 2. (Color online) Schematic diagram of quasi-half-val ley-metal: (a) the energy bandstructures, and the Fermi lev el slightly\ntouches the CBM and VBM. (b): the distribution of Berry curva ture in 2D BZ, and Berry curvature only occurs around -K\nand K valleys with opposite signs and unequal magnitudes. (c ): under an in-plane longitudinal electric field E, the electronic\ncarriers of -K valley turn towards one edge of the sample, and the hole carriers of Γ valley move in a straight line.\nv⊥35,36:\nv=v/bardbl+v⊥=1\n¯h∇kε(k)−e\n¯hE×Ω(k) (1)\nwherev/bardbl(v⊥) is along the electric field direction (per-\npendicular to the electric field and out-of-plane direc-\ntions). However, carriers of Γ valley only obtain the\ngeneral group velocity v/bardbl. The QHVM may be used to\nseparate electron and hole carriers.\nRecently, 2D MA 2Z4family with a septuple-atomic-\nlayer structure has been established37,38with diverse\nproperties from common semiconductor to FV semicon-\nductor to topological insulator to Ising superconductor,\nand MoSi 2N4and WSi 2N4of them have been achieved\nexperimentallybythechemicalvapordepositionmethod.\nTakingtheVGe 2P4monolayerasanexample,theQHVM\ncan be realized by electric filed. Within considered elec-\ntricfield range, the out-of-planeFM propertiesguarantee\nintrinsicspontaneousvalleypolarizationin VGe 2P4. The\nQHVM can be maintained in a certain range of electric\nfield, and the related polarization properties can be ef-\nfectively tuned. Our findings may be extended to other\nFV semiconductors, and the QHVM can be achieved by\nelectric field tuning.\nThe rest of the paper is organized as follows. In the\nnext section, we shall give our computational details and\nmethods. In the next few sections, we shall present\nstructure and stabilities, magnetic anisotropy (MA) and\nelectronic structures and electric field effects on physical\nproperties of VGe 2P4monolayer. Finally, we shall give\nour discussion and conclusion.\nII. COMPUTATIONAL DETAIL\nWithin density-functional theory (DFT)39, the spin-\npolarized first-principles calculations are performed by\nemploying the projected augmented wave method, as\nimplemented in Vienna ab initio simulation package\n(VASP)40–42. We use the generalized gradient approx-\nimation of Perdew-Burke-Ernzerhof (PBE-GGA)43as\nexchange-correlation functional. The on-site Coulomb\ncorrelation of V atoms is considered by the GGA+ Umethod in terms of the on-site Coulomb interaction\nofU= 3.0 eV, as widely used in V-based MA 2Z4\nfamily18,20–22. The GGA+ Uadopts the rotationally in-\nvariant approach proposed by Dudarev et al44, in which\nonly the effective U(Ueff) based on the difference be-\ntween the on-site Coulomb interaction parameterand ex-\nchange parameters is meaningful.\nTo attain accurate results, we use the energy cut-off\nof 500 eV, total energy convergence criterion of 10−8\neV and force convergence criteria of less than 0.0001\neV.˚A−1on each atom. A vacuum space of more than\n20˚A is used to avoid the interactions between the neigh-\nboring slabs. A Γ-centered 16 ×16×1 k-point meshs in\nthe BZ are used for these calculations of structure opti-\nmization, electronic structures and elastic constants, and\na 9×16×1 Monkhorst-Pack k-point meshs for calculat-\ning FM/antiferromagnetic (AFM) energy with rectangle\nsupercell, as shown in FIG.1 of electronic supplementary\ninformation(ESI)alongwiththefirstBZ.TheSOCeffect\nis explicitly included to investigate MCA and electronic\nproperties of VGe 2P4monolayer.\nThrough the direct supercell method, the interatomic\nforce constants (IFCs) using GGA+ Uare calculated\nwith the 5 ×5×1 supercell. Based on calculated har-\nmonic IFCs, the phonon dispersions are obtained by us-\ning Phonopy code45. The elastic stiffness tensor Cijare\ncarried out by using strain-stress relationship (SSR), and\nthe2Delasticcoefficients C2D\nijhavebeenrenormalizedby\nC2D\nij=LzC3D\nij, in which the Lzis the length of unit cell\nalong z direction. The Berry curvatures are directly cal-\nculated from wave functions based on Fukui’s method46\nby using VASPBERRY code47,48.\nIII. STRUCTURE AND STABILITY\nThe crystalstructuresofVGe 2P4monolayerareshown\ninFigure 3 . Different from experimentally synthesized\nMoSi2N4withα1phase37, theα2structure becomes fa-\nvorable for VGe 2P438. Despite this difference, both α2\nandα1structures lack inversion symmetry with a space\ngroup of P¯6m2 (No.187), which allows spontaneous val-3\nFIG. 3. (Color online) For VGe 2P4monolayer, (a): top view and (b): side view of crystal struct ure, and the primitive cell is\nshown by black lines in (a). (c): the phonon dispersions with FM order using GGA+ U(U= 3 eV).\n/s48 /s49 /s50 /s51 /s52 /s53 /s54 /s55 /s56/s45/s53/s57/s48/s45/s53/s57/s53/s45/s54/s48/s48/s45/s54/s48/s53/s45/s54/s49/s48\n/s84/s105/s109/s101/s32/s40/s112/s115/s41/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41\nFIG. 4. (Color online) For VGe 2P4monolayer, the varia-\ntion of free energy during the 8 ps AIMD simulation using\nGGA+U(U= 3 eV). Insets show the final structures (top\nview (a) and side view (b)) after 8 ps at 300 K.\nley polarization. The α2can be built by mirroringGe 2P2\ndouble layers of α1with respect to the vertical surface.\nThis crystalincludes sevencovalentlybonded atomic lay-\ners in the order of P-Ge-P-V-P-Ge-Palong the zaxis. In\nother words, a MoS 2-like VP 2layer is sandwiched be-\ntween two slightly buckled honeycomb GeP layers. The\noptimized lattice constants aof VGe 2P4is 3.56˚A, which\nagrees well with with previous theoretical value38. To\nobtain the magnetic ground state, the total energy dif-\nference between AFM and FM ordering by using rectan-\ngle supercell (FIG.1 of ESI) is calculated, and VGe 2P4\nprefers FM state.\nThe dynamical stability of the VGe 2P4monolayer is\nproved by analyzing the phonon spectra, as shown in\nFigure 3. There are eighteen optical and three acoustical\nphonon branches, corresponding to a total of twenty-one\nbranches due to seven atoms per cell. It is found that\nall phonon frequencies are positive in 2D BZ, confirm-\ning the dynamical stability of VGe 2P4at 0 K. Therefore,\nVGe2P4can exist as a free-standing monolayer. From\nthe application point of view, the mechanical stability\nof VGe 2P4is checked by elastic constants Cij. Due tohexagonal symmetry, the VGe 2P4has two independent\nelastic constants of C11andC12, and the calculated val-\nues areC11=182.68 Nm−1andC12=51.16 Nm−1. These\nCijof VGe 2P4satisfy the Born criteria of mechanical\nstability ( C11>0 andC11−C12>0)49, confirming its\nmechanical stability. The Ab initio molecular dynamics\n(AIMD) simulations are performed to assess the thermal\nstability of the VGe 2P4monolayer at room temperature.\nFigure 4 shows the free energy as a function of the sim-\nulation time, along with the snapshots of the geometric\nstructure at the end of AIMD simulation at 300 K. The\nfree energy is fluctuated around the equilibrium values\nwithout any sudden changes, and the small distortions\nin the final configuration are observed. This confirms\nthe thermodynamical stability of VGe 2P4at room tem-\nperature, suggesting the possible room temperature ap-\nplicability.\nIV. MAGNETIC ANISOTROPY AND\nELECTRONIC STRUCTURES\nThe magnetization of VGe 2P4can affect its symme-\ntry. For out-of-plane FM state, all possible vertical\nmirror symmetry is broken, but the horizontal mirror\nsymmetry is preserved, allowing the spontaneous valley\npolarization19,31–34. The MAE can be used to decide\nthe orientation of magnetization of VGe 2P4. The neg-\native/positive MAE means an easy axis along the in-\nplane/out-of-plane direction. The MAE mainly includes\nMCA energy EMCAand MSA energy ( EMSA). The\nSOC can produce a link between the crystalline struc-\nture and the direction of the magnetic moments, giving\nrise toEMCA. Based on GGA+ U+SOC calculations,\ntheEMCAcan be obtained by EMCA=E||\nSOC−E⊥\nSOC,\nwhere||and⊥mean that spins lie in the plane and\nout-of-plane. Firstly, a collinear self-consistent calcula-\ntion is performed to obtain the convergentchargedensity\nwithout SOC . Secondly, the convergentchargedensity is\nused to carry out noncollinear non-self-consistent calcu-\nlation of two different magnetization directions (in-plane\nand out-of-plane) within SOC. The EMSAis due to the4\nFIG. 5. (Color online) For VGe 2P4monolayer, the V- dx2−y2+dxy,dz2anddxz+dyz-orbital characters energy band structures\nwithout SOC (a, b) and with SOC (c) using GGA+ U(U= 3 eV). The (a) and (b) represent the band structure in the spi n-up\nand spin-dn directions.\n/s48/s46/s48/s48 /s48/s46/s48/s50 /s48/s46/s48/s52 /s48/s46/s48/s54 /s48/s46/s48/s56 /s48/s46/s49/s48/s45/s52/s48/s45/s50/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48/s49/s50/s48/s49/s52/s48\n/s73/s110/s45/s112/s108/s97/s110/s101/s111/s117/s116/s45/s111/s102/s45/s112/s108/s97/s110/s101\n/s32/s77/s67/s65\n/s32/s77/s83/s65\n/s32/s77/s65/s69/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41\nFIG. 6. (Color online) For VGe 2P4monolayer, the MCA en-\nergy, MSA energy and MAE as a function of EusingGGA+ U\n(U= 3 eV).\nanisotropic dipole-dipole (D-D) interaction50,51:\nED−D=1\n2µ0\n4π/summationdisplay\ni/negationslash=j1\nr3\nij[/vectorMi·/vectorMj−3\nr2\nij(/vectorMi·/vector rij)(/vectorMj·/vector rij)] (2)\nwhere the /vectorMirepresent the local magnetic moments of V\natoms, and vectors /vector rijconnects the sites iandj.\nFor a collinear FM monolayer, the Equation 2 for in-\nplane case can be expressed as:\nE||\nD−D=1\n2µ0M2\n4π/summationdisplay\ni/negationslash=j1\nr3\nij[1−3cos2θij] (3)\nwhereθijis the angle between the /vectorMand/vector rij. For out-of-\nplane situation, the Equation 2 can further be simplified\nas:\nE⊥\nD−D=1\n2µ0M2\n4π/summationdisplay\ni/negationslash=j1\nr3\nij(4)TheEMSA(E||\nD−D−E⊥\nD−D)can be written as :\nEMSA=−3\n2µ0M2\n4π/summationdisplay\ni/negationslash=j1\nr3\nijcos2θij (5)\nIt is clearly seen that the crystalstructure and localmag-\nnetic moment decide EMSA. Generally, the MSA tends\nto make spins directed parallel to the monolayer.\nFor V-based MA 2Z4,EMSAhas important effects on\ntheir magnetization direction17,52,53. For VSi 2P4mono-\nlayer, an out-of-plane ferromagnet is predicted, when\nonlyEMCAis considered18. However, the VSi 2P4will\nprefer in-plane case with including EMSA17. Calculated\nresults show that VSi 2P4is a FV semiconductor for out-\nof-plane case using GGA+ U(U= 3 eV), and it will be-\ncome a common magnetic semiconductor for in-plane\nsituation17,18. So, the EMSAisconsideredtodecidemag-\nnetization direction of VGe 2P4. Calculated results show\nthat the MAE is 19 µeV with EMCAandEMSAbeing\n31µeV and -12 µeV, and the positive MAE means that\nthe easy magnetization axis of VGe 2P4is out-of-plane.\nThe spin-polarized band structures of monolayer\nVGe2P4by using both GGA+ Uand GGA+ U+SOC are\ncalculated. In a trigonal prismatic crystal field environ-\nment, the V- dorbitals split into dz2orbital,dxy+dx2−y2\nanddxz+dyzorbitals, and these orbitalcharactersenergy\nband structures without SOC and with SOC are plotted\ninFigure 5. Based on Figure 5 (a) and (b), a distinct\nspin splitting can be observed due to the exchange in-\nteraction. It is found that VGe 2P4is an indirect band\ngap semiconductor of 0.091 eV, and its VBM and CBM\nare provided by the spin-dn and spin-up. The energies of\n-K and K valleys for both conduction and valence bands\nare degenerate, and no spontaneous valley polarization is\nobserved.\nThe spontaneous valley polarization is induced by\nSOC. Calculated results show observable valley splitting\nbetween -K and K valleys in the conduction bands, and\nneglectful valley splitting in the valence bands. This dif-5\n/uni0000012b /uni00000010/uni0000002e/uni00000030/uni0000002e /uni0000012b /uni00000030/uni000000ed/uni00000013/uni00000011/uni00000017/uni000000ed/uni00000013/uni00000011/uni00000015/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000015/uni00000013/uni00000011/uni00000017Energy /uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000c\n0.00\n/uni0000012b /uni00000010/uni0000002e/uni00000030/uni0000002e /uni0000012b /uni00000030/uni000000ed/uni00000013/uni00000011/uni00000017/uni000000ed/uni00000013/uni00000011/uni00000015/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000015/uni00000013/uni00000011/uni00000017Energy /uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000c\n0.05\n/uni0000012b /uni00000010/uni0000002e/uni00000030/uni0000002e /uni0000012b /uni00000030/uni000000ed/uni00000013/uni00000011/uni00000017/uni000000ed/uni00000013/uni00000011/uni00000015/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000015/uni00000013/uni00000011/uni00000017Energy /uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000c\n0.10\n/s45/s75\n/s45/s49/s50/s45/s57/s46/s52/s45/s54/s46/s55/s45/s51/s46/s57/s45/s49/s46/s50/s49/s46/s54/s52/s46/s52/s55/s46/s49/s57/s46/s57/s49/s51/s49/s53\n/s75/s40/s48/s46/s48/s48/s41\n/s45/s75\n/s45/s49/s50/s45/s57/s46/s52/s45/s54/s46/s55/s45/s51/s46/s57/s45/s49/s46/s50/s49/s46/s54/s52/s46/s52/s55/s46/s49/s57/s46/s57/s49/s51/s49/s53\n/s75/s40/s48/s46/s48/s53/s41\n/s45/s75\n/s45/s49/s50/s45/s57/s46/s52/s45/s54/s46/s54/s45/s51/s46/s57/s45/s49/s46/s49/s49/s46/s54/s52/s46/s51/s55/s46/s49/s57/s46/s56/s49/s51/s49/s53\n/s75/s40/s48/s46/s49/s48/s41\nFIG. 7. (Color online)For VGe 2P4monolayer, the energy band structures (top) and distributi on of Berry curvature in 2D BZ\n(bottom) at representative E=0.00, 0.05 and 0.10 V /˚A using GGA+ U(U= 3 eV).\nference can be explained by the distribution of V- dor-\nbitals. Accordingto Figure5,itisfoundthat dx2−y2+dxy\norbitals dominate -K and K valleys of conduction bands,\nwhile the ones of valence bands are mainly from dz2\norbitals. A simple perturbation theory can be used\nto further elucidate the essence of the valley polariza-\ntion with SOC Hamiltonian of out-of-plane magnetiza-\ntionˆH0\nSOC25,54:\nˆH0\nSOC=αˆLz (6)\nin which ˆLzandαare the orbital angular moment along\nzdirection andcoupling strength. In lightofthe orbitals’\ncontribution to -K and K valleys and their wave vector\nsymmetry, the basis functions are chosen as:\n|φτ\nc>=/radicalbigg\n1\n2(|dx2−y2>+iτ|dxy>) (7)\n|φτ\nν>=|dz2> (8)\nwhere the subscript c,νandτrepresent conduction\nbands, valence bands and valley index ( τ=±1). The\nenergy levels at -K and K valleys are then defined as:\nEτ=< φτ|ˆH0\nSOC|φτ> (9)\nAccording to the distribution of V- dorbitals, the valley\nsplitting in the bottom conduction and top valence bands\nare given by25,54:\n|∆Ec|=EK\nc−E−K\nc= 4α (10)|∆Eν|=EK\nν−E−K\nν= 0 (11)\nThe perturbation theory results are in excellent consis-\ntency with the first-principle calculations.\nAccording to Figure 5 (c), the VGe 2P4becomes a\nmetal with a negative gap ( Gcv) of -7 meV, when the\nSOC is included. The Fermi level slightly touches the\nCBM (-K valley) and VBM (Γ valley), but has no con-\ntact with K valley. Therefore, VGe 2P4monolayer is a\nQHVM. Todescribe propertiesofQHVM, twoothergaps\nare defined. One is the Fermi level minus the energy of\n-K valley in the conduction bands ( G−K), which can de-\nscribe concentration of electron carriers. Another is the\nenergy of K valley in the conduction bands minus the\nFermi level ( GK), which describes how easy the carriers\nof -K valley jump to K valley. The G−KplusGKequals\nto valley splitting in the conduction bands. For VGe 2P4,\ntheG−Kis only about 7 meV, so it needs to be regulated\nthrough the external field, for example electric field.\nV. ELECTRIC FIELD EFFECTS\nElectricfieldcaninduceasemiconductortometaltran-\nsition in bilayer MoSi 2N4and WSi 2N455. Moreover, the\nincreasing electric field can result in a transition of MA\nfrom in-plane to out-of-plane in VSi 2P456. In view of\nthese facts, electric field may tune physical properties\nof QHVM in VGe 2P4. Firstly, we confirm the magnetic\nground state under the electric field by the energy differ-\nences (per formulaunit) between AFM and FM ordering,6\n0.00 0.02 0.04 0.06 0.08 0.10\nE/uni00000003/uni0000000b/uni00000039/uni00000012Å/uni0000000c−0.3−0.2−0.10.00.10.2Gap /uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000c\nGCV\nGK\nG−K\n0.00 0.02 0.04 0.06 0.08 0.10\nE/uni00000003/uni0000000b/uni00000039/uni00000012Å/uni0000000c020406080100120Valley Splitting /uni00000003/uni0000000b/uni00000050/uni00000048/uni00000039/uni0000000cV\nC\nFIG. 8. (Color online)For VGe 2P4monolayer, the related\nband gap (top panel) and the valley splitting for both valenc e\nand condition bands (bottom panel) as a function of E.\nwhich are plotted in FIG.2 of ESI. Within considered E\nrange,theFMorderingisalwaysgroundstate. TheMAE\nalong with EMCAandEMSAas a function of Eare plot-\nted inFigure 6. It is found that the EMCAis positive\nwithin considered Erange, which firstly increases, then\ndecreases, and then increases with increasing E. The\nEMSAcan be obtained by Equation 5 . FIG.3 of ESI\nshows the local magnetic moment of V atom ( MV) as a\nfunction of E. It is found that the Ehasverylittle effects\nonMVwithin considered Erange, and the changeis only\n0.021µB, which leads to the small change of EMSAfrom\n-11.58µeV to -12.03 µeV. Finally, we calculate the MAE\nbyEMAE=EMCA+EMSA. It is clearly seen that EMAE\nandEMCAhave the same trend with respect to E. In\nconsidered Erange,EMAEis always positive, implying\nthat VGe 2P4possesses out-of-plane MA. This confirms\nspontaneous valley polarization in VGe 2P4within con-\nsideredErange.\nThe energy band structures and Berry curvature dis-\ntribution of VGe 2P4under electric field are investigated,\nand some representative ones are plotted in Figure 7 .\nThe evolutions of related gap ( Gcv,G−KandGK) and\nvalley splitting for both valence and condition bands as a/s48/s46/s48/s48 /s48/s46/s48/s50 /s48/s46/s48/s52 /s48/s46/s48/s54 /s48/s46/s48/s56 /s48/s46/s49/s48/s52/s55/s52/s56/s52/s57/s53/s48/s53/s49/s53/s50/s53/s51/s53/s52/s53/s53\n/s84\n/s67/s32/s40/s75/s41\nFIG. 9. (Color online)For VGe 2P4monolayer, the Curie tem-\nperature ( TC) as a function of E.\nfunction of Eare shown in Figure 8. With increasing E,\ntheG−Kincreases, which means that concentration of\nelectron carriers increases. However, the GKdecreases,\nwhich means that the carriers of -K valley more easily\njump to K valley. When E>0.09 V/˚A, the Fermi energy\nlevel simultaneously touches the -K and K valleys, and\nthe QHVM will disappear. It is found that the increasing\nEcan enhance valley splitting of conduction bands. So,\nthe electric field can effectively tune physical properties\nof QHVM in VGe 2P4. By applying an in-plane longitu-\ndinalEin VGe 2P4, anomalous velocity υof Bloch elec-\ntrons at -K valley is related with Berry curvature Ω( k)\n(seeFigure 7):υ∼E×Ω(k)35,36. The Berry curvature\nof -K valley forces the electron carriers to accumulate on\none side of the sample, and the hole carriers of Γ valley\nmove in a straight line. When an out-of-plane electric\nfields is applied, the carrier concentration can be effec-\ntively tuned.\nTo estimate Curie temperature TCof VGe 2P4, it can\nbe simply considered as a Ising spin system. The tran-\nsition temperature can be expressed as TC=2\n3J\nKBby\nusing the mean-field approximations (MFA)57, whereJ\nandKBare the nearest-neighboring exchange parameter\nand Boltzmann constant, respectively. The Jcan be ob-\ntained from the energy difference between AFM ( EAFM)\nand FM ( EFM) orderings. According to the FM and\nAFM configurations, the EFMandEAFMcan be writ-\nten as:\nEFM=E0−6JS2(12)\nEAFM=E0+2JS2(13)\nin which E0is the total energy of systems without mag-\nnetic coupling. The corresponding Jcan be attained:\nJ=EAFM−EFM\n8S2(14)\nTheTCvsEis plotted in Figure 9. Within Erange, the\npredicted TCis larger than 48 K.7\n/uni0000012b /uni00000010/uni0000002e/uni00000030/uni0000002e /uni0000012b /uni00000030/uni000000ed/uni00000013/uni00000011/uni00000017/uni000000ed/uni00000013/uni00000011/uni00000015/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000015/uni00000013/uni00000011/uni00000017Energy /uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000c\n0.00\n/uni0000012b /uni00000010/uni0000002e/uni00000030/uni0000002e /uni0000012b /uni00000030/uni000000ed/uni00000013/uni00000011/uni00000017/uni000000ed/uni00000013/uni00000011/uni00000015/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000015/uni00000013/uni00000011/uni00000017Energy /uni00000003/uni0000000b/uni00000048/uni00000039/uni0000000c\n0.10\nFIG. 10. (Color online)For VGe 2P4monolayer, the energy\nband structures at representative E=0.00 and 0.10 V /˚A using\nGGA+U(U= 4 eV).\nVI. DISCUSSION AND CONCLUSION\nFor a given material, the correlationstrength Ushould\nbe determined from future experiment. Here, the U=\n4 eV is also used to investigate related properties of\nVGe2P4to confirm that the QHVM can indeed be\nachieved by electric field tuning. FIG.4 of ESI shows\nthattheFMorderingisgroundstate, when E<0.19V/˚A.\nTheEMAE,EMCAandEMSAalong with the local mag-\nnetic moment of V atom ( MV) as a function of Eare\nplotted in FIG.5 and FIG.6 of ESI. When E<0.19 V/˚A,\nthe positive EMAEimplies that VGe 2P4has out-of-plane\nMA. The evolutions of related gap ( Gcv,G−KandGK)\nas a function of Eare shown in FIG.7 of ESI. The rep-\nresentative energy band structures at E=0.00 and 0.10\nV/˚A are plotted in Figure 10 . Without electric field,\nVGe2P4is a FV semiconductor with an indirect bandgap of 0.040 eV. The increasing Ecan induce FV semi-\nconductor to QHVM transition with critical Eof about\n0.05 V/˚A. However, when E>0.15 V/˚A, the QHVM will\ndisappear. For example E=0.10 V/˚A, the VGe 2P4is a\nQHVM with Gcv,G−KandGKbeing -0.133 eV, 0.056\neV and 0.060 eV. When correlation strength Uis in the\nreasonable range of 3 eV to 4 eV, the VGe 2P4can be\ntuned into QHVM by electric field.\nIn summary, we have proposed the concept of QHVM,\nwhich may be realized in monolayer VGe 2P4with dy-\nnamical, mechanical and thermal stabilities. The electric\nfield can be used tune the related physical properties of\nQHVM in VGe 2P4. For a certain electric field region,\nQHVM properties can be maintained, and the carrier\nconcentrationof-Kvalleycanbeeffectivelytuned. When\nin-plane and out-of-plane electric fields are applied, the\nelectron carriers of -K valley transversely drift and even-\ntually accumulate on one edge of the sample, while the\nhole carriers of Γ valley longitudinally move along the\nin-plane electric field direction. Our findings can inspire\nmore works to search for QHVM.\nConflicts of interest\nThere are no conflicts to declare.\nACKNOWLEDGMENTS\nThis work is supported by Natural Science Basis Re-\nsearch Plan in Shaanxi Province of China (2021JM-456).\nWe are grateful to Shanxi Supercomputing Center of\nChina, andthe calculationswereperformedonTianHe-2.\n1J. R. Schaibley, H. Yu, G. Clark, P. Rivera, J. S. Ross, K.\nL. Seyler, W. Yao and X. Xu, Nat. Rev. Mater. 1, 16055\n(2016).\n2L. J. Sham, S. J. Allen, A. Kamgar, and D. C. Tsui, Phys.\nRev. Lett. 40, 472 (1978).\n3S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M.\nDaughton, S. von Moln´ ar, M. L. Roukes, A. Y. Chtchelka-\nnova, and D. M. Treger, Science 294, 1488 (2001).\n4A. Srivastava, M. Sidler, A. V. Allain, D. S. Lembke, A.\nKis, and A. Imamoglu, Nat. Phys. 11, 141 (2015).\n5K. F. Mak, K. He, J. Shan, and T. F. Heinz, Nat. Nan-\notechnol. 7, 494 (2012).\n6H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, Nat. Nan-\notechnol. 7, 490 (2012).\n78 H. Zeng, J. Dai, W. Yao, D. Xiao and X. Cui, Nat.\nNanotechnol. 7, 490 (2012).\n8M. Zeng, Y. Xiao, J. Liu, K. Yang and L. Fu, Chem. Rev.\n118, 6236 (2018).\n9C. Zhao, T. Norden, P. Zhang, P. Zhao, Y. Cheng, F. Sun,\nJ. P. Parry, P. Taheri, J. Wang, Y. Yang, T. Scrace, K.\nKang, S. Yang, G. Miao, R. Sabirianov, G. Kioseoglou, W.\nHuang, A. Petrou and H. Zeng, Nat. Nanotechnol. 12, 757\n(2017).10D. MacNeill, C. Heikes, K. F. Mak, Z. Anderson, A.\nKorm´anyos, V. Z´ olyomi, J. Park and D. C. Ralph, Phys.\nRev. Lett. 114, 037401 (2015).\n11W. Y. Tong, S. J. Gong, X. Wan, and C. G. Duan, Nat.\nCommun. 7, 13612 (2016).\n12Y. B. Liu, T. Zhang, K. Y. Dou, W. H. Du, R. Peng, Y.\nDai, B. B. Huang, and Y. D. Ma, J. Phys. Chem. Lett. 12,\n8341 (2021).\n13Z. Song, X. Sun, J. Zheng, F. Pan, Y. Hou, M.-H. Yung,\nJ. Yang, and J. Lu, Nanoscale 10, 13986 (2018).\n14J. Zhou, Y. P. Feng, and L. Shen, Phys. Rev. B 102,\n180407(R) (2020).\n15P. Zhao, Y. Ma, C. Lei, H. Wang, B. Huang, and Y. Dai,\nAppl. Phys. Lett. 115, 261605 (2019).\n16S. D. Guo, J. X. Zhu, W. Q. Mu and B. G. Liu, Phys. Rev.\nB104, 224428 (2021).\n17S. D. Guo, Y. L. Tao, K. Cheng, B. Wang and\nY. S. Ang, J. Phys.: Condens. Matter (2022).\nhttps://doi.org/10.1088/1361-648X/ac9c3d\n18X. Y. Feng, X. L. Xu, Z. L. He, R. Peng, Y. Dai, B. B.\nHuang and Y. D. Ma, Phys. Rev. B 104, 075421 (2021).\n19S. Li, Q. Q. Wang, C. M. Zhang, P. Guo and S. A. Yang,\nPhys. Rev. B 104, 085149 (2021).8\n20Q. R. Cui, Y. M. Zhu, J. H. Liang, P. Cui and H. X. Yang,\nPhys. Rev. B 103, 085421 (2021).\n21Y. L. Wang and Y. Ding, Appl. Phys. Lett. 119, 193101\n(2021).\n22X. Zhou, R. Zhang, Z. Zhang, W. Feng, Y. Mokrousov and\nY. Yao, npj Comput. Mater. 7, 160 (2021).\n23H. X. Cheng, J. Zhou, W. Ji, Y. N. Zhang and Y. P. Feng,\nPhys. Rev. B 103, 125121 (2021).\n24W. Du, Y. Ma, R. Peng, H. Wang, B. Huang, and Y. Dai,\nJ. Mater. Chem. C 8, 13220 (2020).\n25R. Li, J. W. Jiang, W. B. Mi and H. L. Bai, Nanoscale 13,\n14807 (2021).\n26K. Sheng, Q. Chen, H. K. Yuan and Z. Y. Wang, Phys.\nRev. B105, 075304 (2022).\n27P. Jiang, L. L. Kang, Y. L. Li, X. H. Zheng, Z. Zeng and\nS. Sanvito, Phys. Rev. B 104, 035430 (2021).\n28K. Sheng, H. K. Yuan and B. K. Zhang, Nanoscale\n(2022).DOI: 10.1039/d2nr03860a\n29Y. Zang, Y. Ma, R. Peng, H. Wang, B. Huang, and Y. Dai,\nNano Res. 14, 834 (2021).\n30R. Peng, Y. Ma, X. Xu, Z. He, B. Huang, and Y. Dai,\nPhys. Rev. B 102, 035412 (2020).\n31H. Hu, W. Y. Tong, Y. H. Shen, X. Wan, and C. G. Duan,\nnpj Comput. Mater. 6, 129 (2020).\n32K. Sheng, B. K. Zhang, H. K. Yuan and Z. Y. Wang, Phys.\nRev. B105, 195312 (2022).\n33S. D. Guo, J. X. Zhu, M. Y. Yin and B. G. Liu, Phys. Rev.\nB105, 104416 (2022).\n34S. D. Guo, W. Q. Mu and B. G. Liu, 2D Mater. 9, 035011\n(2022).\n35X. Xu, W. Yao, D. Xiao and T. F. Heinz, Nat. Phys. 10,\n343 (2014).\n36D. Xiao, M. C. Chang, and Q. Niu, Rev. Mod. Phys. 82,\n1959 (2010).\n37Y. L. Hong, Z. B. Liu, L. Wang T. Y. Zhou, W. Ma, C.\nXu, S. Feng, L. Chen, M. L. Chen, D. M. Sun, X. Q. Chen,\nH. M. Cheng and W. C. Ren, Science 369, 670 (2020).\n38L. Wang, Y. Shi, M. Liu, A. Zhang, Y.-L. Hong, R. Li, Q.\nGao, M. Chen, W. Ren, H.-M. Cheng, Y. Li, and X.- Q.Chen, Nat. Commun. 12, 2361 (2021).\n39P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964);\nW. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).\n40G. Kresse, J. Non-Cryst. Solids 193, 222 (1995).\n41G. Kresse and J. Furthm¨ uller, Comput. Mater. Sci. 6, 15\n(1996).\n42G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).\n43J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett.\n77, 3865 (1996).\n44S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J.\nHumphreys and A. P. Sutton, Phys. Rev. B 57, 1505\n(1998).\n45A. Togo, F. Oba, and I. Tanaka, Phys. Rev. B 78, 134106\n(2008).\n46T. Fukui, Y. Hatsugai and H. Suzuki, J. Phys. Soc. Japan.\n74, 1674 (2005).\n47H. J. Kim, https://github.com/Infant83/VASPBERRY,\n(2018).\n48H. J. Kim, C. Li, J. Feng, J.-H. Cho, and Z. Zhang, Phys.\nRev. B93, 041404(R) (2016).\n49E. Cadelano and L. Colombo, Phys. Rev. B 85, 245434\n(2012).\n50X. B. Lu, R. X. Fei, L. H. Zhu and L. Yang, Nat. Commun.\n11, 4724 (2020).\n51K. Yang, G. Y. Wang, L. Liu , D. Lu and H. Wu, Phys.\nRev. B104, 144416 (2021).\n52S. D. Guo, W.-Q. Mu, J.-H. Wang, Y.-X. Yang, B. Wang\nand Y.-S. Ang, Phys. Rev. B 106, 064416 (2022).\n53D. Dey, A. Ray and L. P. Yu, Phys. Rev. Materials 6,\nL061002 (2022).\n54P. Zhao, Y. Dai, H. Wang, B. B. Huang and Y. D. Ma,\nChemPhysMater, 1, 56 (2022).\n55Q. Y. Wu, L. M. Cao, Y. S. Ang and L. K. Ang, Appl.\nPhys. Lett. 118, 113102 (2021).\n56S. D. Guo, X. S. Guo, G. Z. Wang, K. Cheng\nand Y. S. Ang, J. Mater. Chem. C (2022).\nhttps://doi.org/10.1039/D2TC03293G\n57L. Ke, B. N. Harmon, and M. J. Kramer, Physical Review\nB95, 104427 (2017)."    },    {        "title": "2210.14150v1.Tuning_of_topological_properties_in_the_strongly_correlated_antiferromagnet_Mn__3_Sn_via_Fe_doping.pdf",        "content": "Tuning of topological properties in the strongly correlated antiferromagnet Mn 3Sn via Fe doping\nAchintya Low, Susanta Ghosh, Susmita Changdar, Sayan Routh, Shubham Purwar, and S. Thirupathaiah\u0003\nDepartment of Condensed Matter and Materials Physics,\nS. N. Bose National Centre for Basic Sciences, Kolkata, West Bengal-700106, India.\nMagnetic topological materials, in which strong correlations between magnetic and electronic properties\nof matter, give rise to various exotic phenomena such as anomalous Hall effect (AHE), topological Hall\neffect (THE), and skyrmion lattice. Here, we report on the electronic, magnetic, and topological properties\nof Mn 3{xFexSn single crystals ( x=0, 0.25, and 0.35). Low temperature magnetic properties have been\nsignificantly changed with Fe doping. Most importantly, we observe that large uniaxial magnetocrystalline\nanisotropy that is induced by the Fe doping in combination with competing magnetic interactions at low\ntemperature produce nontrivial spin-texture, leading to large topological Hall effect in the doped systems at\nlow temperatures. Our studies further show that the topological properties of Mn 3{xFexSn are very sensitive to\nthe Fe doping.\nI. INTRODUCTION\nMagnetic topological materials are the illustrations of an\ninterplay between magnetic and electronic states of matter,\nproviding an important stage for illuminating several exotic\nphenomena such as the anomalous Hall effect (AHE) [1–\n3], the topological Hall effect (THE) [4–6], the skyrmionic\nlattice [7–9], and etc. On the other hand, the kagome\nlattice in which atoms are arranged in star-like formation\nanticipates a geometrical frustration, leading to noncollinear\nantiferromagnetic (AFM) ordering [10–15]. So far, several\nkagome intermetallics have been explored to a great extent\ndue to their potential magnetic topological properties. For\ninstance, Co 3Sn2S2is a magnetic Weyl semimetal showing\ngiant AHE in addition to chiral anomaly [16, 17], Mn 3Sn(Ge)\nare time-reversal symmetry broken Weyl semimetals, despite\nbeing antiferromagnets, show large AHE induced by the\nnonzero k-space Berry curvature [18–21], Fe 3Sn2which\nis a kagome ferromagnet generates skyrmionic bubbles in\naddition to the giant AHE [22, 23], YMn 6Sn6is a rare\nearth based kagome system showing several competing\nmagnetic orders and large THE [24], and Gd 3Ru4Al12posses\nlow temperature skyrmion lattice induced by the magnetic\nfrustration [25].\nSkyrmions, the vortex-like spin texture formation in the\nreal space, are topologically protected and characterized by\ntheir topological charge called the winding number [26].\nThe skymions pursuit futuristic technological applications\nin the high-density data storage devices [27, 28], fine\ncurrent controlled devices [29], and information processing\ndevices [30]. There exist several systems showing skyrmion\nlattice that is originated from different mechanisms. For\nexample, in noncentrosymmetric magnetic systems such\nas MnSi [8], FeGe [31], and FeCoSi [7] the skyrmion\nlattice formation was understood by the Dzyaloshinskii-\nMoriya interaction (DMI) under the strong spin-orbit\ncoupling [32, 33]. In the centrosymmetric magnetic\nsystems such as La 1{xSrxMnO 3[34] and Fe 3Sn2[22]\nthe competition between magnetic dipole interactions\n\u0003setti@bose.res.inand uniaxial magnetocrystalline anisotropy stabilizes the\nskyrmion lattice [35–37]. In addition, recent studies\nshow the existence of skyrmions in rare-earth based\nintermetallics Gd 2PdSi 3and Gd 3Ru4Al12due to the magnetic\nfrustration [25, 38].\nMn3Sn is a kagome itinerant antiferromagnet with a N ´eel\ntemperature of 420 K, has hexagonal crystal structure with\na space group of P6 3/mmc [18, 39]. Here, the Mn atoms\nform kagome network in the abplane of the crystal, showing\nchiral 120\u000einverse triangular spin structure stabilized by\nthe DM interaction [14, 40]. Due to a slight distortion in\nkagome lattice and as well off-stoichiometry of the system,\nusually, weak-ferromagnetism is present in this system [15,\n18]. Moreover, low-temperature magnetic structure depends\non the elemental ratio of Mn and Sn, annealing temperature,\nand crystal growth techniques. Thus, some studies report\nhelical spin structure in Mn 3Sn at low temperatures [41, 42]\nwhile the other studies realized spin-glass state below 50\nK [43, 44]. At room temperature, Mn 3Sn shows noncollinear\nantiferromagnetic ordering with 1200inverse triangular spin\nstructure [14, 15, 18, 40] leading to large anomalous Hall\neffect. Particularly, Mn 3Sn is of great research interest as\nthe triangular-coplanar magnetic order reshapes into a spiral-\nnoncoplanar magnetic ordering with a finite net magnetization\nalong the c-axis at a critical spin-reorientation transition\ntemperature (T SR\u0019260 K) [14, 18, 40] which is not found in\nMn3Ge [45]. As a result, the large AHE is suppressed below\nTSRin Mn 3Sn but not in Mn 3Ge [46].\nTheoretically, Mn 3Sn is not expected to show topological\nHall effect (THE) as the stoichiometric Mn 3Sn does not\npossess chiral-spin texture. However, there are few\nreports claiming the observation of a small topological Hall\nresistivity due to the field induced chiral-spin texture in\npolycrystalline Mn 3Sn [47] at low temperature and due to\ndomain wall formation in the single crystalline Mn 3Sn at\nroom temperature [48, 49]. On the other hand, Fe 3Sn sharing\nsimilar crystal structure of Mn 3Sn is a ferromagnetic metal\nwith an easy axis of magnetization in the abplane that can\nbe shifted to the c-axis with doping [50, 51]. Motivated\nfrom the polymorphic magnetic properties of Mn 3Sn and\nflexibility of manipulating the easy axis in Fe 3Sn, we have\nsubstituted Fe atoms at Mn sites of Mn 3Sn to enhance the\ntopological properties of Mn 3Sn as Fe substitution introducesarXiv:2210.14150v1  [cond-mat.mtrl-sci]  25 Oct 20222\nferromagnetism to the system.\nIn this work, single crystals of Mn 3{xFexSn ( x=0, 0.25,\nand 0.35) were systematically studied for their electrical\nresistivity, magnetic, and topological properties. While\nMn3Sn is found to be metallic in nature up to room\ntemperature with a spin-reorientation driven kink at 260 K,\nwith Fe doping the system shows magnetism induced metal-\ninsulator (MI) transition at 240 K for x=0.25 and 150 K\nforx=0.35. In addition to MI transition, x=0.35 system\nshows disorder induced resistivity upturn with a minima\nat T m=50 K. As for the magnetic properties, Mn 3Sn is\nfound to show a sudden drop in magnetization at a spin-\nreorientation transition temperature of 260 K and spin-glass-\nlike transition below 40 K. On the other hand, with Fe doping\nferromagnetic transition has been introduced alongside with\nenhanced magnetic anisotropy. Also, anisotropic anomalous\nHall resistivity has been induced at low temperatures with Fe\ndoping. Particularly, the out-of-plane Hall resistivity ( rzx)\nincreases with decreasing temperature for all the compositions\nfrom 300 K down to their respective magnetic transition\ntemperatures where a sudden change in Hall resistivity is\nnoticed. Though not much change in out-of-plane Hall\nresistivity is noticed with Fe doping at 2 K, the in-plane Hall\nresistivity ( rxy) is gigantically enhanced from -0.25 mW-cm\nto 48 mW-cm in going from x=0 to x=0.35. Along with the\nanomalous Hall resistivity, a large topological Hall resistivity\nalso is observed for both Fe doped systems at 2 K.\nII. EXPERIMENTAL DETAILS\nSingle crystals of Mn 3{xFexSn ( x=0, 0.25, and 0.35)\nwere prepared by self flux method [52–54]. First, Mn\n(Alfa Aesar 99.995% ), Fe (Alfa Aesar, 99.99% ), and Sn\n(Alfa Aesar 99.998% ) powders were taken with a ratio of\n(7-x) : x: 3, mixed thoroughly before inserting into\na preheated quartz tube, and sealed under partial Argon\npressure. The mixture was then heated up to 1000oC,\nslowly cooled down to 900oC, and then was air quenched to\nroom temperature by taking out the ampoule from furnace.\nIn this way, we obtained shiny hexagonal and rod shaped\nsingle crystals with a typical size of 1.5 mm \u00021 mm\n\u00021 mm. X-ray diffraction (XRD) measurements were\ndone on different surfaces of the single crystals using\nRigaku SmartLab equipped with 9 kW Cu K aX-ray source.\nElemental compositions of the crystals were calculated to\nbe Mn 2.97Sn1.03, Mn 2.74Fe0.26Sn, and Mn 2.64Fe0.36Sn using\nenergy dispersive X-ray spectroscopy (EDXS) technique.\nFor convenience we denote the compositions Mn 2.97Sn1.03,\nMn2.74Fe0.26Sn, and Mn 2.64Fe0.36Sn by x=0, x=0.25, and\nx=0.35, respectively, wherever applicable.\nElectrical resistivity measurements were carried out\nin the linear four-probe method and Hall measurements\nwere done in the measuring geometry shown in the\nschematic of Fig. 1(e). Magneto-transport and magnetization\nmeasurements were performed in Physical Properties\nMeasurement System (9T PPMS, DynaCool, Quantum\nDesign) using ETO and VSM options, respectively. Hall\n40 60 80\n20 40 60 80\n2θ(0)2θ(0)Intensity(a.u.) Intensity(a.u.)\n(0002)\n(0004)(011ത0)\n(022ത0)\n(033ത0)\n(044ത0)\nIxIzVy Vxab\nݖݕݔ[011ത0]\n[21ത1ത0][0001]\n࣋࢟࢞ ࣋࢞ࢠ\nHǁz HǁyMn\nSn(a) (c)\n(d)\n(b)\n(e)FIG. 1. (a) and (b) show typical XRD patterns of Mn 3Sn single\ncrystal taken from two different orientations as shown in the inset\nimages. (c) Schematic hexagonal lattice of Mn3Snin the abplane\nwhere Mn atoms form kagome geometry and Sn atoms sit in the\ncenter of hexagon. In (d), x,y, and z-axes correspond to [2¯110],\n[01¯10], and [0001] orientations, respectively. Magnetotransport\nmeasuring geometry is shown in (e), where rxyis measured with\ncurrent along the x-axis and external magnetic field applied along\nthez-axis to find Hall voltage along the y-axis and rzxis measured\nwith current along the z-axis and field applied along the y-axis to\nfind Hall voltage along the x-axis.\nresistivity was measured for two different orientations as\nshown in Fig. 1(e). rxystands for current along the x\ndirection and field along the zdirection where Hall voltage\nwas measured along the ydirection. Similarly rzxstands\nfor current along zdirection, field along the ydirection,\nand Hall voltage was measured along the xdirection. To\neliminate longitudinal voltage contribution due to any\npossible misalignment of the probes, the Hall resistivity\nwas calculated byr(H){ r({H)\n2. Temperature dependent Hall\nresistivity measurements were done using both positive (H)\nand negative (-H) applied fields and then calculated using the\nformular(T,H){ r(T,{H)\n2.\nIII. RESULTS AND DISCUSSIONS\nXRD patterns shown in Figs. 1(a) and 1(b) represent\nintensity reflections of (0002) and(01¯10)hexagonal planes,\nrespectively, taken from Mn 3Sn single crystal. Figs. 2(a),\n(c), and (e) depict zero-field out-of-plane resistivity ( rzz)\nmeasured between 2 and 300 K from Mn 3Sn, Mn 2.75Fe0.25Sn,\nand Mn 2.65Fe0.35Sn single crystals, respectively. Overall, a3\n0 100 200 3000.0000.8001.600\n0 100 200 3001.581.631.67\n  Fitted curve0.0000.0020.004\nH ^ z \nH // z\n0.400.801.20ρzz(mΩ-cm)\nT (K)0.981.041.11࢞=\n=࢞ .=࢞ .0.0000.0700.140\nT (K)M(μB/f.u.)࢞=\n=࢞ .\n=࢞ .\n(e)(d)\n(f)(c)(a) (b)\nߤܪ= 0.05ܶ\nTMI≈150KTMI≈245KTSR≈260K\nTC*=40KTSR≈260K\nTC≈150K\nTC≈245K\ndM/dT\nFC\nFIG. 2. Temperature dependent resistivity ( rzz) is measured\nwith current along the z-axis for Mn 3{xFexSn where (a) x=0, (c)\nx=0.25, and (e) x=0.35. (b), (d), (f) show temperature dependent\nmagnetization for x=0,x=0.25, and x=0.35, respectively, measured\nin the field-cooled (FC) mode. Zero-field-cooled (ZFC) data is\nshown in Fig.S6 of the Supplemental Material [61].\nmetallic nature of resistivity is observed from Mn 3Sn, except\nthat a kink has been noticed at TSR= 260K where a spin-\nreorientation (SR) transition from in-plane noncollinear AFM\norder to an out-of-plane spin-spiral structure occurs [42, 53].\nObservation of kink in the electrical resistivity of Mn 3Sn\nis consistent with earlier reports [39, 53, 54]. On the\nother hand, we observe a metal-insulator (MI) transition in\nMn2.75Fe0.25Sn at 240 K, which further reduced to 150 K\nin Mn 2.65Fe0.35Sn as shown in Figs. 2(c) and 2(e) with\nincreasing Fe concentration. Earlier, the authors found a MI\ntransition at 265 K in the case of Mn 2.8Fe0.2Sn [55]. In\naddition to the MI transition, in Mn 2.65Fe0.35Sn, we identify\nlow-temperature resistivity upturn with a minima at T m=50 K.\nThe resistivity upturn at low temperature could be due to (i)\nweak localization [56–58], (ii) Kondo effect [57, 59], or (iii)\nelastic electron-electron ( e { e) scattering [57, 60]. In order\nto elucidate the mechanism of resistivity upturn, we fit the\ndata with relevant formulae involved in the above mentioned\nthree mechanisms. We got the best fitting for the e { e\nscattering which takes the form r(T) = r0{aT1\n2+bT2, as\nFe doping induces disorder in addition to the electron carrier\npopulation. See the Supplemental Material [61] (see, also,\nreferences [15, 41, 42, 56, 57, 59, 60, 62, 63] therein) for more\ndetails.\nTo witness the effect of Fe doping on the magnetic\n-0.070.000.07\nH ^ z\nH // z\n-0.070.000.07\n  H^ z\n  H // z\n-0.1 0.0 0.1-0.0040.0000.004\n-0.100.000.10\n-0.1 0.0 0.1-0.0080.0000.008\n-6 -3 0 3 6-1.800.001.80\n-4 -2 0 2 4-0.150.000.15\n-0.1 0.0 0.1-0.010.000.01M(μB/f.u.)\n࢞=.࢞=.࢞=\n-0.850.000.85\n-2 0 2-0.20.00.2\nμ0H (Tesla) μ0H (Tesla)T=300 K T=5 K\n࢞=.࢞=.࢞=\n(e)(c)(a)\n(f)(d)(b)FIG. 3. (a), (c), and (e) show M(H) isotherms of Mn 3{xFexSn\nmeasured at 300 K for x=0,x=0.25, and x=0.35, respectively. (b),\n(d), and (f) show M(H) isotherms measured at 5 K for x=0,x=0.25,\nandx=0.35 respectively. Zoomed-in images of respective isotherms\nare show in the insets. Inset of (d) points to the asymmetric hysteresis\n(see the text for more details).\nproperties, we performed temperature dependent\nmagnetization M(T) with field applied parallel ( Hkz)\nand perpendicular ( H?z) to the z-axis. From the M(T)\ndata of Mn 3Sn as shown in Fig. 2 (b) we find a sudden\ndrop in the in-plane magnetic moment at T SR=260 K due\nto spin transformation from inverse triangular to helical or\nspiral configuration [53, 54]. Similarly, we find drop in the\nout-of-plane magnetic moment as well at T SR=260 K but not\nas large as the in-plane. Reducing the sample temperature to\nbelow 40 K, we observe increasing in-plane and out-of-plane\nmagnetic moments due to spin-glass-like transition [44, 53].\nWith Fe doping, in the case of x=0.25, isotropic increase in\nmagnetization is observed [see Fig. 2(d)] for both Hkzand\nH?zfield orientations with a ferromagnetic-like transition\nat a T C=240 K. With further lowering sample temperature\nthe in-plane magnetic moment gets saturated, while the out-\nof-plane magnetization show additional ferromagnetic-like\ntransition at T C\u0003=40 K, enhancing the anisotropy of the\nmagnetic structure at low temperatures. Further, with more\nFe doping, i.e., from x=0.35 we see a ferromagnetic-like\ntransition at TC=150 K for both orientations though the\nout-of-plane magnetization (1.6 mB/f.u. ) is much stronger4\n0.000.070.140.21\n0.000.080.160.24\n0 50 100 150 200 250 3000.001.302.603.90\nCoercivity(T)\nTemperature, T (K)H ǁ zH Ʇz\nH ǁ zH Ʇz\nH ǁ zH Ʇz࢞=\n=࢞ .\n=࢞ .\nFIG. 4. Coercivity plotted as a function of temperature for both H\nkzand H?z. Top panel shows the data from Mn 3Sn, middle panel\nshows the data from Mn 2.75Fe0.25Sn, and the bottom panel shows\nthe data from Mn 2.65Fe0.35Sn.\ncompared to in-plane magnetization (0.3 mB/f.u. ) at 2 K.\nDecreasing T Cwith increasing Fe concentration is consistent\nwith a previous report [64]. In conjunction, from the\nmagnetization data shown in Figs. 2(b), (d), and (f) and the\nelectrical resistivity shown in Figs. 2(a), (c), and (e) we find\nthat the metal-insulator transition occur nearly at the same\ntemperature of ferromagnetic transition in Mn 2.75Fe0.25Sn\nand Mn 2.65Fe0.35Sn, suggesting that ferromagnetism could\nbe a plausible origin of MI transition in these systems.\nFig. 3 depicts magnetization isotherms M(H) measured\nat various sample temperatures for both HkzandH?z\norientations. Figs. 3(a), 3(c), and 3(e) show isotherms of\nx=0, x=0.25, and x=0.35 samples, respectively, measured\nat 300 K and Figs. 3(b), 3(d), and 3(f) show isotherms\nofx=0, x=0.25, and x=0.35, respectively, measured at 5\nK (see Figs. S4 and S5 in the Supplemental Material [61]\nfor the isotherms at other temperatures). From the inset of\nFig. 3(a) we observe significant hysteresis for Mn 3Sn at 300\nK, similar to a previous report where a weak in-plane ( H?z)\nferromagnetism is observed with an effective magnetization\nof0.004 mB/f.u. [18]. Origin of this weak ferromagnetism\nis understood from the kagome lattice distortion or off-\nstoichiometry of the composition [18, 53]. On the other hand,\nthe out-of-plane magnetization ( Hkz) changes linearly with\napplied field like a typical AFM system. We further noticethat substituting Fe does not significantly affect the out-of-\nplane magnetic structure at room temperature, however the in-\nplane weak ferromagnetism increases with Fe concentration\nas demonstrated in the insets of Figs. 3 (a), (c) and (e), where\none can see increase in spontaneous magnetization from 4 \u0002\n10{3mB/f.u. forx=0, 8\u000210{3mB/f.u. forx=0.25 to 1\u0002\n10{2mB/f.u. forx=0.35. Here, doping with Fe mainly\nincreases the in-plane lattice distortion as Fe and Mn have\ndifferent atomic sizes, and there by enhancing the spontaneous\nmagnetization [65, 66].\nNext examining the isotherms measured at 5 K [see\nFigs. 3(b), (d), and (f))], from Mn 3Sn, the M(H) curves\nlook almost linear for both HkzandH?zexcept that\na small hysteresis due to spin-glass transition is observed.\nOn the other hand, from the x=0.25 and x=0.35 systems\nwe observe significant changes in the out-of-plane magnetic\nstructure with Fe doping as the coercivity and spontaneous\nmagnetization increase dramatically, while changes in the\nin-plane magnetic structure are minimal with Fe doping\nthough the M(H) loop modifies from linear to sigmoid-\nlike ingoing from x=0 to x=0.35. For x=0.35, the out-\nof-plane magnetization (1.8 mB/f.u. ) and coercivity (4 T)\nare extremely large compared to the in-plane magnetization\n(1mB/f.u. ) and coercivity (50 Oe), demonstrating an\nenhanced magnetic anisotropy with Fe doping. From a\ncloser observation on the M(H) curve of x=0.25, we find\nfield induced asymmetric M(H) loop [see inset of Fig. 3(d)],\nalso observed previously on the metamagnetic Heusler\nalloy Ni 50Mn35In15{xBx[67] and spin-glass SrRuO 3/SrIrO 3\nsuperlattice [68], possibly due to metastable magnetic phase\nchurned out by Fe doping.\nTo quantify the magnetic anisotropy induced by the Fe\ndoping, we plotted the coercivity measured at various sample\ntemperatures for both HkzandH?zorientations in Fig. 4.\nAs can be seen from the top panel of Fig. 4, for x=0,\nthere is not difference between the in-plane and out-of-plane\ncoercivities and almost constant from 300 K down to 50 K.\nHowever below 50 K, though isotropic, it increases drastically\nwith decreasing temperature. For x=0.25, again we find\nalmost constant and nearly zero in-plane and out-of-plane\ncoercivities from 300 K down to 50 K. Below 50 K, both in-\nplane and out-of-plane coercivities increase with decreasing\ntemperature. However, the out-of-plane coercivity ( Hkz) is\nalmost three times higher than the in-plane coercivity ( H?z)\nat 2 K. Similarly, in the case of x=0.35, the in-plane coercivity\nalways found to be zero for all the measured temperatures,\nwhile the out-of-plane coercivity starts increasing from\nzero at 100 K to 4 T at 2 K. Further, we estimate the\nmagnetocrystalline anisotropy energy density ( KU) using the\nStoner-Wohlfarth model [69, 70] by considering the z-axis as\nthe easy axis (uniaxial) of magnetization. Concentrating only\nat the low temperature (2 K) where the coercivity is maximum,\nuniaxial magnetocrystalline anisotropic energy density (K U)\nis calculated to be 1.2 \u0002104J/m3forx=0.25 and 5.3\n\u0002105J/m3forx=0.35. These values are significantly higher\ncompared to previously reported in-plane magnetocrystalline\nanisotropic energy density of \u0018103J/m3from Mn 3Sn [42].\nThe effect of Fe doping on the topological properties of5\n0 100 200 300-2.6-1.30.001530\n-3.6-2.4-1.2\n0 100 200 30002040-6.0-3.00.0\n-6.0-3.00.0\nT (K)ρzx(μΩ-cm)\nT (K)ρxy(μΩ-cm)\n(e)(b) (a)\n(c)\n(f)(d)࢞=\n=࢞ .=࢞ .࢞=\n=࢞ .=࢞ .ࣆ=ࡴ ࣆࢀ =ࡴ ࢀ\n150 K260 K\n115 KH ǁ y H ǁ zdρ/dTdρ/dT\n150 K\n40K50 100 150\n150 300\nFIG. 5. Out-of plane Hall resistivity ( rzx) plotted as a function of\ntemperature measured under the field of 1T from (a) x=0, (c) x=0.25,\nand (e) x=0.35. (b), (d), and (f) show in-plane Hall resistivity ( rxy)\nplotted as a function of temperature measured under the field of\n1T from x=0,x=0.25, and x=0.35, respectively. Insets of (d) and\n(f) show the first derivative of their Hall resistivity with respect to\ntemperature.\nMn3Sn is studied by performing Hall measurements at low\ntemperatures. As shown in Fig. 5, we measured out-of-\nplane [ rzx(T)] and in-plane [ rxy(T)] Hall resistivities as a\nfunction of temperature at an applied magnetic filed of 1 T.\nFrom Fig. 5(a) we can notice that rzxof Mn 3Sn increases\n(negatively) with decreasing temperature from 300 K down to\nTSR=260 K, and below rzxsuddenly becomes zero. On the\nother hand, from Fig. 5(b) we can see that rxyof Mn 3Sn is\nalmost constant from 300 K down to 2 K. Fig. 5(c) depicts rzx\nofx=0.25 in which we can notice an increase (negatively) in\nHall resistivity linearly with decreasing temperature down to\nTSR=115 K, and below this temperature rzxslightly increases\nto -3.6 mW-cm at 2 K. Fig. 5(d) depicts rxyofx=0.25 in which\nwe can notice slight increase (positively) in Hall resistivity\nwith decreasing temperature and reaches to 30 mW-cm at 2 K.\nInset in Fig. 5(d) represent drxy/dT having a minima at 40\nK, consistent with the ferromagnetic transition temperature of\nTC\u0003=40 K [see Fig. 2(d)]. Fig. 5(e) depicts rzxplotted for\nx=0.35 in which we can notice increase (negatively) in Hall\nresistivity linearly with decreasing temperature down to 2 K,\nexcept that a hump at T C=150 K is observed. Fig. 5(f) depicts\n-40-2002040\n-4-2024-4-2024\n-4.0-2.00.02.04.0\nμ0H (Tesla)ρzx(μΩ-cm)\n࢞=.࢞=.2K\n5K50K\n100K࢞=\n࢞=.࢞=.࢞=\n(e) (f)(c)(a)\n(d)(b)\nμ0H (Tesla)ρxy(μΩ-cm)H ǁ y H ǁ z\n150K2K\n5K50K\n100K\n150K\n-6 -3 0 3 6-4-2024\n-6 -3 0 3 6-54-2702754FIG. 6. Field dependent out-of-plane Hall resistivity rzxmeasured\nat different temperatures from x=0 (a), x=0.25 (c), and x=0.35\n(e). Curved blue arrows in (c) indicate field sweeping direction.\nSimilarly, field dependent in-plane Hall resistivity rxymeasured at\ndifferent temperatures from x=0 (b), x=0.25 (d), and x=0.35 (f).\nrxyofx=0.35 in which we notice increase (positively) in\nHall resistivity with decreasing temperature and gets saturated\nto 48 mW-cm at 2 K. Inset in Fig. 5(f) depicts drxy/dT\nshowing minima at the ferromagnetic transition temperature\nof T C=150 K [see Fig. 2(f)].\nFrom the Hall measurements shown in Fig. 5(a) it is clear\nthat the large Hall resistivity ( rzx) observed from Mn 3Sn at\nhigh temperatures is mainly driven by the nonzero k-space\nBerry phase [18, 21]. However, at low temperatures ( <\n260 K), the Berry phase disappears due to spin reorientation,\nand thus, rzxis negligible. Whereas the Berry phase is\nalways absent in the xyplane of Mn 3Sn,rxyis in general\nnegligible [18]. In this study, we notice T SR=115 K for\nx=0.25 while no spin-reorientation transition is observed\nforx=0.35. Worth to mention here that the authors have\nfound spin-reorientation transition at 125 K in the case of\nMn2.8Fe0.2Sn [55]. Thus, linear increase (negative) in rzx\nwith decreasing temperature down to 115 K for x=0.25 and\ndown to 2 K for x=0.35 is due to nonzero k-space Berry\nphase [see Figs. 5(c) and 5(e)]. Astonishingly, the in-plane\nHall resistivity ( rxy) has been gigantically enhanced with Fe\ndoping as high as 30 mW-cm for x=0.25 and 48 mW-cm for\nx=0.35 at 2 K. Note here that the in-plane Hall resistivity6\n(rxy) is due to the ferromagnetism induced by the Fe doping.\nField dependent Hall resistivity, rzx(H)andrxy(H), from\nall the samples measured at different temperatures are shown\nin Fig. 6. Here we mainly focus at low temperatures ( \u0014\n150 K) as the doping induced ferromagnetism significantly\nincreases the Hall resistivity at low temperatures. Since\nbelow 260 K, Mn 3Sn is already in the spin-spiral texture we\nfind only the ordinary Hall effect as the out-of-plane Hall\nresistivity ( rzx) linearly depends on the field [see Fig. 6(a)].\nSimilarly, we again find ordinary Hall effect from the in-\nplane Hall resistivity ( rxy) as well [see Fig. 6(b)]. Fig. 6(c)\ndepicts rzx(H) ofx=0.25 in which we observe anomalous\nHall resistivity (AHR) with a sudden jump near zero field.\nConsistent with asymmetric M(H) data of x=0.25 shown in\nFig. 3(d), asymmetric hysteresis in AHR is noticed at low\ntemperatures (2 and 5 K). As for rxy(H)ofx=0.25 shown in\nFig. 6(d), we observe sign reversed anomalous Hall resistivity\ncompared to rzx(H). Moreover, low temperature rxy(H)\ncurves show hysteresis in AHR resembling the M(H) data of\na typical ferromagnetic system [see Hkzdata in Fig. 3(d)].\nFig. 6(e) depicts rzx(H) ofx=0.35 in which we observe an\nanomalous Hall resistivity with a sudden jump near zero field,\nsimilar to x=0.25. However, unlike in x=0.25, we do not\nfind asymmetric hysteresis in rzx(H) at low temperatures.\nAlthough at zero field we observe a sudden jump in rzx, at\nhigher fields rzxis dominated by the ordinary Hall effect\nas it linearly depends on the field. Fig. 6(f) depicts rxy(H)\nofx=0.35 in which we observe sign reversed anomalous\nHall effect compared to rzx(H), again similar to the case\nofx=0.25. Moreover, the rxy(H) curves show hysteresis\nin AHR resembling the M(H)data of a typical ferromagnetic\nsystem [see Hkzdata in Fig. 3(f)]. From Figs. 6(d) and\n6(f) we observe increase in AHR hysteresis with decreasing\ntemperature in both x=0.25 and x=0.35 systems.\nThe Hall resistivity of a ferromagnetic metal can be\nexpressed as r(H) = rN(H) + rA(H) = m0R0H + R sM[1],\nwhere R0is normal Hall coefficient which is inversely\nproportional to carrier density ( R0= 1/nq ),Rsis anomalous\nHall coefficient, and Mis magnetization. However, we are\nunable to fit properly rxy(H)ofx=0.25 and x=0.35 using this\nformula at low temperatures as demonstrated in Figs. 7(a) and\n7(c). This is because rxy(H)has significant contribution from\nthe topological Hall resistivity rT\nxywhich can extracted using\nthe formula rT\nxy(H) = rxy(H) { rN(H) { rA(H)as plotted in\nFigs. 7(b) and 7(d) for x=0.25 and x=0.35, respectively. A\nmaximum rT\nxyof 5mW{cm has been derived at a critical field\nof 0.35 T from x=0.25 and 22 mW{ cm has been at a critical\nfield of 4 T from x=0.35 at 2 K. The maxima of rT\nxydecreases\nwith increasing temperature and disappears at around 150 K\ninx=0.25 [see Fig. 7(b)], while the maxima of rT\nxyinx=0.35\nbecomes negligible above 50 K [see Fig. 7(d)].\nThe topological Hall resistivity is generated by the itinerant\nelectrons by acquiring real space Berry curvature when\npassing through the nontrivial spin structure of a scalar spin\nchirality cijk= (dSi.[dSj\u0002dSk])[4, 71]. The nontrivial spin\nstructure is topologically protected and produces skyrmion\nlattice, characterised by the nonzero topological charge called\nthe winding number Q [72, 73]. By taking into account theskyrmionic picture, we can estimate the skyrmion density\n(nsk) using the relation Beff=f0nsk[5, 71, 74]. Here, f0is\nmagnetic flux quantum ( h/e) and Beffis the effective magnetic\nfield. Also, the topological Hall resistivity rTis directly\nrelated to BeffasrT\u0019PR0Beff[4, 5, 71, 75]. Here, Pis the\nlocal spin polarization of charge carrier ( P = mspon/msat) [4,\n71]. From the fittings shown in Fig. 7(a) we derived R0={1\u0002\n10{9m3/C and from the M(H) data shown in Fig. 3(d) we\nestimated P = 0.07 forx=0.25. Here, mspon=0.659 mB/f.u.\ntaken from Fig. 3(d) and msat=9mB/f.u. (3mB/Mn ) [18, 76].\nUsing these values, Beffis calculated to be 720 T. Also,\nthe skyrmion density is estimated to nsk\u00191.7\u00021017m{2\nwith a skyrmion size (helical period) of about lsk\u00192.4 nm .\nSimilarly, for Mn 2.65Fe0.35Sn, by taking the fitted value of\nR0= {1.5\u000210{9m3/C and calculated value of P = 0.19 ,\nthe estimated effective field Beff\u0019770 T which gives a\nskyrmion density nsk\u00191.9\u00021017m{2and the skyrmion\nsize lsk\u00192.2 nm . The skyrmion size is estimated using\nthe formula lsk= (h\nep\n3\n2Beff)1\n2[38]. For x=0.35, we used\nmspon=1.76 mB/f.u. taken from Fig. 3(f) and msat=9mB/f.u. (3\nmB/Mn ) [18, 76]. Note here that we presumably considered\nthe same atomic magnetic moment of 3 mB/atom for both\nMn and Fe [50]. Further, by assuming that \u001911.4% ( x=0.35)\nof Fe doing into Mn 3Sn does not significantly change the\nlattice parameters [77] we estimated the skyrmion density\nof9\u00021017m{2forx=0.25 and 12\u00021017m{2forx=0.35.\nskyrmion density values estimated from the crystal structure\ninformation are in good agreement with the values estimated\nfrom the topological Hall resistivity data.\nThe skyrmion sizes ( lsk), 2.4 nm from x=0.25 and 2.2\nnm from x=0.35, obtained in this study are comparable to\nthe skyrmion size of 2.49 nm observed from Gd 2PdSi 3[38].\nOn the other hand, the helical period of \u00191 nm is obtained\nfrom polycrystalline Mn 3Sn [47] which is a factor of 2.4\nsmaller compared to the size of Mn 2.75Fe0.25Sn. However,\nthe B20 systems in their bulk form show much longer-period\nhelical structures (skyrmions). For instance, MnSi shows a\nhelical period of\u001918nm[78], FeGe shows a helical period\nof\u001970nm[79], and Cu 2OSeO 3shows a helical period of\n\u001962nm[80]. Worth to mention here that all three above\nB20 systems show very large helical periods compared to our\nstudied samples of Mn 2.75Fe0.25Sn and Mn 2.65Fe0.35Sn.\nSeveral mechanisms are proposed to understand the\nstabilization of the skyrmion lattice in solids such as (i)\nDM interaction in noncentrosymmetric systems [14, 40,\n75], (ii) chiral domain wall induced skyrmion lattice [81–\n84], and (iii) uniaxial magnetocrystalline anisotropy in the\ncentrosymmetric systems [22, 34, 37, 85]. It is quite\npossible to tune the magnetic anisotropy by doping with alien\natoms [86]. For instance, doping Ni or Pd in Fe 3P at the Fe site\nconverts the in-plane anisotropy to the uniaxial out-of-plane\nanisotropy and thus the formation of antiskyrmions [87] and\nLi ion doped in Mott insulator La 2CuO 4acts as a vortex center\nfor the skyrmion [88]. Overall, our study demonstrates that Fe\ndoping into Mn 3Sn converts the in-plane magnetic structure\nof Mn 3Sn to a uniaxial anisotropic magnetic structure having\neasy magnetic axis in the z-axis. Hence, Fe doping generates\nsufficient uniaxial magnetocrystalline anisotropy in Mn 3Sn in7\n-8-4048\n-40-2002040-20-1001020-50-2502550\n-50-2502550-40-2002040\n-40-2002040\n-40-2002040-8-4048\n-8-4048\n-8-4048\nμ0H (Tesla)\nρxyT(μΩ-cm)\n-50-2502550\n-20-1001020\n-50-2502550\n-20-1001020ρxyT(μΩ-cm)ρxyρN+ρA\nρxyρN+ρA\nT=2K\nT=5K\nT=50K\nT=100K\nμ0H (Tesla) μ0H (Tesla) μ0H (Tesla)(a) (b) (c) (d)\nρN+ρA\nρxy\nρxyρN+ρA\nρxyρN+ρA-20-1001020\nT=2K\nT=5K\nT=50K\nT=100KρxyρN+ρA\nρxyρN+ρA\nρxyρN+ρAT=2K\nT=5K\nT=50K\nT=100KT=2K\nT=5K\nT=50K\nT=100KMn2.75Fe0.25Sn Mn2.65Fe0.35Sn\n-6 -3 0 3 6-40-2002040\n-6 -3 0 3 6-8-4048\n-6 -3 0 3 6-50-2502550\n-6 -3 0 3 6-20-1001020\nT=150K T=150KρxyρN+ρA\nρxyρN+ρA\nT=150K T=150Kρxy(μΩ-cm)\nρxy(μΩ-cm)\nFIG. 7. (a) and (c) are the field dependent in-plane Hall resistivity ( rxy) measured at various temperatures from x=0.25 and x=0.35,\nrespectively. The circles are experimental data and the solid black curves are fits to the equation rH=rN+rA=R0m0H+RSM (see the text\nfor more details). (b) and (d) show the derived in-plane topological Hall resistivity ( rTxy) from x=0.25 and x=0.35, respectively. See Fig.S7 in\nthe Supplemental Material [61] for the raw Hall and magnetoresistance (MR) data.\naddition to the geometrical frustration which help to stabilize\nthe low temperature skyrmion lattice.\nIV . SUMMARY\nIn summary, we have thoroughly examined the effect of Fe\ndoping on the electrical resistivity, magnetic, and topological\nproperties of Mn 3Sn. Low temperature magnetic structure has\nbeen significantly modified with Fe doping. Especially, we\nfind that Fe doping induces ferromagnetism below 240 K in\nx=0.25 and below 150 K in x=0.35. In addition, we observe\nferromagnetism driven metal-insulator transition in Fe doped\nsystems that is absent in Mn 3Sn. We further observe e { e\nscattering driven resistivity upturn at low temperature with\na resistivity minima at 50 K in x=0.35. Fe doping induces\nmagnetic anisotropy such way that the easy magnetisation\naxis shifts from in-plane in Mn 3Sn to the out-of-plane ( z-axis) in x=0.35. Thus, the large uniaxial magnetocrystalline\nanisotropy in addition to the competing magnetic interactions\nat low temperature produces nontrivial spin texture which is\nresponsible for the induced low temperature THE with Fe\ndoping. Further, the topological properties of Mn 3Sn are very\nsensitive to the Fe doping. That means, although Mn 3Sn does\nnot show THE at low temperatures, 8% ( x=0.25) of Fe doping\nshows topological Hall resistivity as high as 5 mW{ cm at a\ncritical field of 0.35T and 11.4% ( x=0.35) of Fe doping shows\ntopological Hall resistivity as high as 22 mW{ cm at a critical\nfield of 4T when measured at 2 K.\nACKNOWLEDGMENTS\nAuthors thank Science and Engineering Research Board\n(SERB), Department of Science and Technology (DST), India\nfor the financial support [Grant no. SRG/2020/000393].\n[1] R. Karplus and J. M. Luttinger, Hall Effect in Ferromagnetics,\nPhys. Rev. 95, 1154 (1954).[2] N. A. Sinitsyn, Semiclassical theories of the anomalous Hall\neffect, Journal of Physics: Condensed Matter 20, 0232018\n(2007).\n[3] F. W. Fabris, P. Pureur, J. Schaf, V . N. Vieira, and I. A.\nCampbell, Chiral anomalous Hall effect in reentrant AuFe\nalloys, Phys. Rev. B 74, 214201 (2006).\n[4] A. Neubauer, C. Pfleiderer, B. Binz, A. Rosch, R. Ritz, P. G.\nNiklowitz, and P. B ¨oni, Topological Hall Effect in the APhase\nof MnSi, Phys. Rev. Lett. 102, 186602 (2009).\n[5] N. Kanazawa, Y . Onose, T. Arima, D. Okuyama, K. Ohoyama,\nS. Wakimoto, K. Kakurai, S. Ishiwata, and Y . Tokura, Large\nTopological Hall Effect in a Short-Period Helimagnet MnGe,\nPhys. Rev. Lett. 106, 156603 (2011).\n[6] J. C. Gallagher, K. Y . Meng, J. T. Brangham, H. L. Wang,\nB. D. Esser, D. W. McComb, and F. Y . Yang, Robust Zero-Field\nSkyrmion Formation in FeGe Epitaxial Thin Films, Phys. Rev.\nLett. 118, 027201 (2017).\n[7] X. Yu, Y . Onose, N. Kanazawa, J. H. Park, J. Han, Y . Matsui,\nN. Nagaosa, and Y . Tokura, Real-space observation of a two-\ndimensional skyrmion crystal., Nature 465, 901 (2010).\n[8] T. Y . Nagaosa, N., Topological properties and dynamics of\nmagnetic skyrmions., Nature nanotechnology 8, 899 (2013).\n[9] S. M ¨uhlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch,\nA. Neubauer, R. Georgii, and P. B ¨oni, Skyrmion Lattice in a\nChiral Magnet, Science 323, 915 (2009).\n[10] R. Moessner and A. P. Ramirez, Geometrical frustration,\nPhysics Today 59, 24 (2006).\n[11] M. Nixon, E. Ronen, A. A. Friesem, and N. Davidson,\nObserving Geometric Frustration with Thousands of Coupled\nLasers, Phys. Rev. Lett. 110, 184102 (2013).\n[12] N. Qureshi, B. Ouladdiaf, A. Senyshyn, V . Caignaert,\nand M. Valldor, Non-collinear magnetic structures in the\nmagnetoelectric Swedenborgite CaBaFe 4O7derived by powder\nand single-crystal neutron diffraction, SciPost Phys. Core 5, 7\n(2022).\n[13] T. Asaba, Y . Su, M. Janoschek, J. D. Thompson, S. M.\nThomas, E. D. Bauer, S.-Z. Lin, and F. Ronning, Large\ntunable anomalous Hall effect in the kagome antiferromagnet\nU3Ru4Al12, Phys. Rev. B 102, 035127 (2020).\n[14] J. Liu and L. Balents, Anomalous Hall Effect and Topological\nDefects in Antiferromagnetic Weyl Semimetals: Mn3Sn/Ge ,\nPhys. Rev. Lett. 119, 087202 (2017).\n[15] T. Nagamiya, S. Tomiyoshi, and Y . Yamaguchi, Triangular spin\nconfiguration and weak ferromagnetism of Mn 3Sn and Mn 3Ge,\nSolid State Communications 42, 385 (1982).\n[16] E. Liu, Y . Sun, N. Kumar, L. Muechler, A. Sun, L. Jiao, S.-Y .\nYang, D. Liu, A. Liang, Q. Xu, et al. , Giant anomalous Hall\neffect in a ferromagnetic kagome-lattice semimetal., Nature\nphysics 14, 1125 (2018).\n[17] N. Morali, R. Batabyal, P. K. Nag, E. Liu, Q. Xu, Y . Sun,\nB. Yan, C. Felser, N. Avraham, and H. Beidenkopf, Fermi-\narc diversity on surface terminations of the magnetic Weyl\nsemimetal Co 3Sn2S2, Science 365, 1286 (2019).\n[18] S. Nakatsuji, N. Kiyohara, and T. Higo, Large anomalous Hall\neffect in a non-collinear antiferromagnet at room temperature.,\nNature 527, 212 (2015).\n[19] K. Kuroda, T. Tomita, M.-T. Suzuki, C. Bareille, A. Nugroho,\nP. Goswami, M. Ochi, M. Ikhlas, M. Nakayama, S. Akebi, and\nothers., Evidence for magnetic Weyl fermions in a correlated\nmetal., Nature materials 16, 1090 (2017).\n[20] A. K. Nayak, J. E. Fischer, Y . Sun, B. Yan, J. Karel,\nA. C. Komarek, C. Shekhar, N. Kumar, W. Schnelle,\nJ. K ¨ubler, C. Felser, and S. S. P. Parkin, Large anomalous\nHall effect driven by a nonvanishing Berry curvature in the\nnoncolinear antiferromagnet Mn 3Ge, Science Advances 2,\ne1501870 (2016).[21] H. Yang, Y . Sun, Y . Zhang, W.-J. Shi, S. S. P. Parkin,\nand B. Yan, Topological Weyl semimetals in the chiral\nantiferromagnetic materials Mn 3Ge and Mn 3Sn, New Journal\nof Physics 19, 015008 (2017).\n[22] Z. Hou, W. Ren, B. Ding, G. Xu, Y . Wang, B. Yang,\nQ. Zhang, Y . Zhang, E. Liu, F. Xu, W. Wang, G. Wu,\nX. Zhang, B. Shen, and Z. Zhang, Observation of Various\nand Spontaneous Magnetic Skyrmionic Bubbles at Room\nTemperature in a Frustrated Kagome Magnet with Uniaxial\nMagnetic Anisotropy, Advanced Materials 30, 1706306 (2018).\n[23] Q. Wang, S. Sun, X. Zhang, F. Pang, and H. Lei, Anomalous\nHall effect in a ferromagnetic Fe 3Sn2single crystal with a\ngeometrically frustrated Fe bilayer kagome lattice, Phys. Rev.\nB94, 075135 (2016).\n[24] N. J. Ghimire, R. L. Dally, L. Poudel, D. C. Jones, D. Michel,\nN. T. Magar, M. Bleuel, M. A. McGuire, J. S. Jiang,\nJ. F. Mitchell, J. W. Lynn, and I. I. Mazin, Competing\nmagnetic phases and fluctuation-driven scalar spin chirality in\nthe kagome metal YMn 6Sn6, Science Advances 6, eabe2680\n(2020).\n[25] M. Hirschberger, T. Nakajima, S. Gao, L. Peng, A. Kikkawa,\nT. Kurumaji, M. Kriener, Y . Yamasaki, H. Sagayama, H. Nakao,\net al. , Skyrmion phase and competing magnetic orders on a\nbreathing kagom ´e lattice., Nature communications 10, 1 (2019).\n[26] A. Fert, N. Reyren, and V . Cros, Magnetic skyrmions:\nadvances in physics and potential applications., Nature Reviews\nMaterials 2, 1 (2017).\n[27] S. Luo and L. You, Skyrmion devices for memory and logic\napplications, APL Mater. 9, 050901 (2021).\n[28] Y . Nii, T. Nakajima, A. Kikkawa, Y . Yamasaki, K. Ohishi,\nJ. Suzuki, Y . Taguchi, T. Arima, Y . Tokura, and Y . Iwasa,\nUniaxial stress control of skyrmion phase, Nature\ncommunications 6, 1 (2015).\n[29] G. Finocchio, F. B ¨uttner, R. Tomasello, M. Carpentieri,\nand M. Kl ¨aui, Magnetic skyrmions: from fundamental to\napplications, Journal of Physics D: Applied Physics 49, 423001\n(2016).\n[30] S. Luo, M. Song, X. Li, Y . Zhang, J. Hong, X. Yang, X. Zou,\nN. Xu, and L. You, Reconfigurable Skyrmion Logic Gates,\nNano Lett. 18, 1180 (2018).\n[31] X. Yu, N. Kanazawa, Y . Onose, K. Kimoto, W. Zhang,\nS. Ishiwata, Y . Matsui, and Y . Tokura, Near room-temperature\nformation of a skyrmion crystal in thin-films of the helimagnet\nFeGe, Nature materials 10, 106 (2011).\n[32] I. Dzyaloshinsky, A thermodynamic theory of “weak”\nferromagnetism of antiferromagnetics, Journal of Physics and\nChemistry of Solids 4, 241 (1958).\n[33] T. Moriya, Anisotropic Superexchange Interaction and Weak\nFerromagnetism, Phys. Rev. 120, 91 (1960).\n[34] X. Yu, Y . Tokunaga, Y . Kaneko, W. Zhang, K. Kimoto,\nY . Matsui, Y . Taguchi, and Y . Tokura, Biskyrmion states and\ntheir current-driven motion in a layered manganite., Nature\nCommunications 5, 1 (2014).\n[35] S. Hayami, S.-Z. Lin, and C. D. Batista, Bubble and skyrmion\ncrystals in frustrated magnets with easy-axis anisotropy, Phys.\nRev. B 93, 184413 (2016).\n[36] O. I. Utesov, Thermodynamically stable skyrmion lattice in a\ntetragonal frustrated antiferromagnet with dipolar interaction,\nPhys. Rev. B 103, 064414 (2021).\n[37] X. Yu, M. Mostovoy, Y . Tokunaga, W. Zhang, K. Kimoto,\nY . Matsui, Y . Kaneko, N. Nagaosa, and Y . Tokura, Magnetic\nstripes and skyrmions with helicity reversals, Proceedings of\nthe National Academy of Sciences 109, 8856 (2012).9\n[38] T. Kurumaji, T. Nakajima, M. Hirschberger, A. Kikkawa,\nY . Yamasaki, H. Sagayama, H. Nakao, Y . Taguchi, T. hisa\nArima, and Y . Tokura, Skyrmion lattice with a giant topological\nHall effect in a frustrated triangular-lattice magnet, Science\n365, 914 (2019).\n[39] S. Tomiyoshi, H. Yoshida, H. Ohmori, T. Kaneko, and\nH. Yamamoto, Electrical properties of the intermetallic\ncompound Mn 3Sn, Journal of Magnetism and Magnetic\nMaterials 70, 247 (1987).\n[40] P. Park, J. Oh, K. Uhl ´ıˇrov´a, J. Jackson, A. De ´ak, L. Szunyogh,\nK. H. Lee, H. Cho, H.-L. Kim, H. C. Walker, et al. , Magnetic\nexcitations in non-collinear antiferromagnetic Weyl semimetal\nMn3Sn, npj Quantum Materials 3, 1 (2018).\n[41] J. Cable, N. Wakabayashi, and P. Radhakrishna, A neutron\nstudy of the magnetic structure of Mn 3Sn, Solid State\nCommunications 88, 161 (1993).\n[42] T. F. Duan, W. J. Ren, W. L. Liu, S. J. Li, W. Liu, and Z. D.\nZhang, Magnetic anisotropy of single-crystalline Mn 3Sn in\ntriangular and helix-phase states, Applied Physics Letters 107,\n082403 (2015).\n[43] H. Ohmori, S. Tomiyoshi, H. Yamauchi, and H. Yamamoto,\nSpin structure and weak ferromagnetism of Mn 3Sn, Journal of\nMagnetism and Magnetic Materials 70, 249 (1987).\n[44] W. J. Feng, D. Li, W. J. Ren, Y . B. Li, W. F. Li, J. Li, Y . Q.\nZhang, and Z. D. Zhang, Glassy ferromagnetism in Ni3Sn-type\nMn3.1Sn0.9, Phys. Rev. B 73, 205105 (2006).\n[45] N. Kiyohara, T. Tomita, and S. Nakatsuji, Giant anomalous Hall\neffect in the chiral antiferromagnet Mn 3Ge, Phys. Rev. Applied\n5, 064009 (2016).\n[46] T. Chen, T. Tomita, S. Minami, M. Fu, T. Koretsune,\nM. Kitatani, I. Muhammad, D. Nishio-Hamane, R. Ishii,\nF. Ishii, R. Arita, and S. Nakatsuji, Anomalous transport due\nto Weyl fermions in the chiral antiferromagnets Mn 3X ( X=Sn,\nGe), Nat. Commun. 12, 572 (2021).\n[47] P. K. Rout, P. V . P. Madduri, S. K. Manna, and A. K.\nNayak, Field-induced topological Hall effect in the noncoplanar\ntriangular antiferromagnetic geometry of Mn3Sn, Phys. Rev. B\n99, 094430 (2019).\n[48] X. Li, L. Xu, H. Zuo, A. Subedi, Z. Zhu, and K. Behnia,\nMomentum-space and real-space Berry curvatures in Mn 3Sn,\nSciPost Physics 5, 063 (2018).\n[49] X. Li, C. Collignon, L. Xu, H. Zuo, A. Cavanna, U. Gennser,\nD. Mailly, B. Fauqu ´e, L. Balents, Z. Zhu, and K. Behnia, Chiral\ndomain walls of Mn 3Sn and their memory, Nat. Commun. 10,\n3021 (2019).\n[50] B. C. Sales, B. Saparov, M. A. McGuire, D. J. Singh, and\nD. S. Parker, Ferromagnetism of Fe 3Sn and Alloys, Sci. Rep.\n4, 10.1038/srep07024 (2014).\n[51] O. Y . Vekilova, B. Fayyazi, K. P. Skokov, O. Gutfleisch,\nC. Echevarria-Bonet, J. M. Barandiar ´an, A. Kovacs,\nJ. Fischbacher, T. Schrefl, O. Eriksson, and H. C. Herper,\nTuning the magnetocrystalline anisotropy of Fe3Sn by\nalloying, Phys. Rev. B 99, 024421 (2019).\n[52] P. C. Canfield and Z. Fisk, Growth of single crystals from\nmetallic fluxes, Philosophical Magazine B 65, 1117 (1992).\n[53] N. H. Sung, F. Ronning, J. D. Thompson, and E. D. Bauer,\nMagnetic phase dependence of the anomalous Hall effect in\nMn3Sn single crystals, Applied Physics Letters 112, 132406\n(2018).\n[54] J. Yan, X. Luo, H. Y . Lv, Y . Sun, P. Tong, W. J. Lu, X. B.\nZhu, W. H. Song, and Y . P. Sun, Room-temperature angular-\ndependent topological Hall effect in chiral antiferromagnetic\nWeyl semimetal Mn 3Sn, Applied Physics Letters 115, 102404\n(2019).[55] A. Low, S. Ghosh, and S. Thirupathaiah, Large Room-\nTemperature Pure Topological Hall Effect (THE) in Kagome\nAntiferromagnet Mn 3Sn, and Induced Giant Low-Temperature\nTHE with Fe Doping, arXiv:2112.15409 (2021).\n[56] B. L. Altshuler, D. Khmel’nitzkii, A. I. Larkin, and P. A.\nLee, Magnetoresistance and Hall effect in a disordered two-\ndimensional electron gas, Phys. Rev. B 22, 5142 (1980).\n[57] S. Dhara, R. R. Chowdhury, and B. Bandyopadhyay,\nObservation of resistivity minimum at low temperature in\nCoxCu1{x (x\u00180.17–0.76) nanostructured granular alloys,\nPhys. Rev. B 93, 214413 (2016).\n[58] A. S. Sukhanov, A. Heinemann, L. Kautzsch, J. D. Bocarsly,\nS. D. Wilson, C. Felser, and D. S. Inosov, Robust metastable\nskyrmions with tunable size in the chiral magnet FePtMo 3N,\nPhys. Rev. B 102, 140409 (2020).\n[59] J. Kondo, Resistance Minimum in Dilute Magnetic Alloys,\nProgress of Theoretical Physics 32, 37 (1964).\n[60] P. A. Lee and T. V . Ramakrishnan, Disordered electronic\nsystems, Rev. Mod. Phys. 57, 287 (1985).\n[61] See Supplemental Material at [URL will be inserted by\npublisher] for additional electrical transport, magnetic, and\nmagnetotransport data.\n[62] H.-Z. Lu and S.-Q. Shen, Weak localization and weak anti-\nlocalization in topological insulators, in SPIE Proceedings ,\nedited by H.-J. Drouhin, J.-E. Wegrowe, and M. Razeghi (SPIE,\n2014).\n[63] Y . Xu, J. Zhang, G. Cao, C. Jing, and S. Cao, Low-temperature\nresistivity minimum and weak spin disorder of polycrystalline\nLaCaMnO 3in a magnetic field, Phys. Rev. B 73, 224410\n(2006).\n[64] M. R. Felez, A. A. Coelho, and S. Gama, Magnetic properties of\nMn3{xFexSn compounds with tuneable Curie temperature by\nFe content for thermomagnetic motors, J. Magn. Magn. Mater.\n444, 280 (2017).\n[65] C. J. Fennie, Ferroelectrically Induced Weak Ferromagnetism\nby Design, Phys. Rev. Lett. 100, 167203 (2008).\n[66] A. Apostolov, I. Apostolova, S. Trimper, and J. Wesselinowa,\nAntiferroelectricity and weak ferromagnetism in rare earth\ndoped multiferroic BiFeO 3, Solid State Communications 300,\n113692 (2019).\n[67] T. Samanta, A. Us Saleheen, D. L. Lepkowski, A. Shankar,\nI. Dubenko, A. Quetz, M. Khan, N. Ali, and S. Stadler,\nAsymmetric switchinglike behavior in the magnetoresistance\nat low fields in bulk metamagnetic Heusler alloys, Phys. Rev.\nB90, 064412 (2014).\n[68] B. Pang, L. Zhang, Y . B. Chen, J. Zhou, S. Yao, S. Zhang,\nand Y . Chen, Spin-Glass-Like Behavior and Topological Hall\nEffect in SrRuO 3/SrIrO 3Superlattices for Oxide Spintronics\nApplications, ACS Applied Materials & Interfaces 9, 3201\n(2017), pMID: 28059493.\n[69] E. C. Stoner and E. P. Wohlfarth, A mechanism of magnetic\nhysteresis in heterogeneous alloys, Philosophical Transactions\nof the Royal Society of London. Series A, Mathematical and\nPhysical Sciences 240, 599 (1948).\n[70] J. M. D. Coey, Magnetism and Magnetic Materials (Cambridge\nUniversity Press, 2010).\n[71] K. S. Denisov, I. V . Rozhansky, N. S. Averkiev, and\nE. L ¨ahderanta, General theory of the topological Hall effect\nin systems with chiral spin textures, Phys. Rev. B 98, 195439\n(2018).\n[72] B. Dup ´e, M. Hoffmann, C. Paillard, and S. Heinze, Tailoring\nmagnetic skyrmions in ultra-thin transition metal films, Nature\ncommunications 5, 1 (2014).10\n[73] S. Heinze, K. V on Bergmann, M. Menzel, J. Brede,\nA. Kubetzka, R. Wiesendanger, G. Bihlmayer, and S. Bl ¨ugel,\nSpontaneous atomic-scale magnetic skyrmion lattice in two\ndimensions., Nature Physics 7, 713 (2011).\n[74] K. Zeissler, S. Finizio, K. Shahbazi, J. Massey, F. A.\nMa’Mari, D. M. Bracher, A. Kleibert, M. C. Rosamond,\nE. H. Linfield, T. A. Moore, et al. , Discrete Hall resistivity\ncontribution from N ´eel skyrmions in multilayer nanodiscs,\nNature Nanotechnology 13, 1161 (2018).\n[75] N. Kanazawa, M. Kubota, A. Tsukazaki, Y . Kozuka, K. S.\nTakahashi, M. Kawasaki, M. Ichikawa, F. Kagawa, and\nY . Tokura, Discretized topological Hall effect emerging from\nskyrmions in constricted geometry, Phys. Rev. B 91, 041122\n(2015).\n[76] P. J. Brown, V . Nunez, F. Tasset, J. B. Forsyth, and\nP. Radhakrishna, Determination of the magnetic structure of\nMn3Sn using generalized neutron polarization analysis, Journal\nof Physics: Condensed Matter 2, 9409 (1990).\n[77] Q. Recour, T. Mazet, and B. Malaman, Magnetic and\nmagnetocaloric properties of Mn 3{xFexSn2(0.1\u0014x\u00140.9),\nJ. Phys. Appl. Phys. 41, 185002 (2008).\n[78] Y . Ishikawa, K. Tajima, D. Bloch, and M. Roth, Helical spin\nstructure in manganese silicide MnSi, Solid State Commun. 19,\n525 (1976).\n[79] B. Lebech, J. Bernhard, and T. Freltoft, Magnetic structures of\ncubic FeGe studied by small-angle neutron scattering, J. Phys.\nCondens. Matter 1, 6105 (1989).\n[80] T. Adams, A. Chacon, M. Wagner, A. Bauer, G. Brandl,\nB. Pedersen, H. Berger, P. Lemmens, and C. Pfleiderer, Long-\nWavelength Helimagnetic Order and Skyrmion Lattice Phase in\nCu2OSeO 3, Phys. Rev. Lett. 108, 237204 (2012).[81] S. B. Gudnason and M. Nitta, Domain wall Skyrmions, Phys.\nRev. D 89, 10.1103/physrevd.89.085022 (2014).\n[82] R. Cheng, M. Li, A. Sapkota, A. Rai, A. Pokhrel, T. Mewes,\nC. Mewes, D. Xiao, M. De Graef, and V . Sokalski, Magnetic\ndomain wall skyrmions, Phys. Rev. B 99, 184412 (2019).\n[83] T. Nagase, Y .-G. So, H. Yasui, T. Ishida, H. K. Yoshida,\nY . Tanaka, K. Saitoh, N. Ikarashi, Y . Kawaguchi, M. Kuwahara,\nand M. Nagao, Observation of domain wall bimerons in\nchiral magnets, Nat. Commun. 12, 10.1038/s41467-021-23845-\ny (2021).\n[84] K. Yang, K. Nagase, Y . Hirayama, T. D. Mishima, M. B. Santos,\nand H. Liu, Wigner solids of domain wall skyrmions, Nat.\nCommun. 12, 10.1038/s41467-021-26306-8 (2021).\n[85] M. Preißinger, K. Karube, D. Ehlers, B. Szigeti, H.-A. Krug von\nNidda, J. S. White, V . Ukleev, H. M. Rønnow, Y . Tokunaga,\nA. Kikkawa, Y . Tokura, Y . Taguchi, and I. K ´ezsm ´arki, Vital\nrole of magnetocrystalline anisotropy in cubic chiral skyrmion\nhosts, npj Quantum Materials 6, 10.1038/s41535-021-00365-y\n(2021).\n[86] R. A. Beck, L. Lu, P. V . Sushko, X. Xu, and X. Li, Defect-\nInduced Magnetic Skyrmion in a Two-Dimensional Chromium\nTriiodide Monolayer, JACS Au 1, 1362 (2021).\n[87] K. Karube, L. Peng, J. Masell, M. Hemmida, H.-A. Krug von\nNidda, I. K ´ezsm ´arki, X. Yu, Y . Tokura, and Y . Taguchi, Doping\nControl of Magnetic Anisotropy for Stable Antiskyrmion\nFormation in Schreibersite (Fe,Ni) 3P with S 4symmetry,\nAdvanced Materials 34, 2108770 (2022).\n[88] I. Rai ˇcevi´c, D. Popovi ´c, C. Panagopoulos, L. Benfatto, M. B.\nSilva Neto, E. S. Choi, and T. Sasagawa, Skyrmions in a Doped\nAntiferromagnet, Phys. Rev. Lett. 106, 227206 (2011)."    },    {        "title": "2211.01732v1.Magnetism_in_Two_Dimensional_Ilmenenes__Intrinsic_Order_and_Strong_Anisotropy.pdf",        "content": "Magnetism in Two-Dimensional Ilmenenes:\nIntrinsic Order and Strong Anisotropy\nR.H Aguilera-del-Toro1\u00002\u0003, M. Arruabarrena2, A. Leonardo1\u00003,\nA. Ayuela1\u00002\n1Donostia International Physics Center (DIPC), 20018 Donostia, Spain\n2Centro de F\u0013 \u0010sica de Materiales - Materials Physics Center (CFM-MPC), 20018\nDonostia, Spain\n3EHU Quantum Center, Universidad del Pa\u0013 \u0010s Vasco/Euskal Herriko Unibertsitatea\nUPV/EHU, Leioa, Spain\nE-mail: rodrigo.aguilera@dipc.org\nOctober 2022\nAbstract. Iron ilmenene is a new two-dimensional material that has recently been\nexfoliated from the naturally-occurring iron titanate found in ilmenite ore, a material\nthat is abundant on earth surface. In this work, we theoretically investigate the\nstructural, electronic and magnetic properties of 2D transition-metal-based ilmenene-\nlike titanates. The study of magnetic order reveals that these ilmenenes usually present\nintrinsic antiferromagnetic coupling between the 3d magnetic metals decorating both\nsides of the Ti-O layer. Furthermore, the ilmenenes based on late 3d brass metals, such\nas CuTiO 3and ZnTiO 3, become ferromagnetic and spin compensated, respectively.\nOur calculations including spin-orbit coupling reveal that the magnetic ilmenenes have\nlarge magnetocrystalline anisotropy energies when the 3d shell departs from being\neither \flled or half-\flled, with their spin orientation being out-of-plane for elements\nbelow half-\flling of 3d states and in-plane above. These interesting magnetic properties\nof ilmenenes make them useful for future spintronic applications because they could\nbe synthesized as already realized in the iron case.\n1. Introduction\nTechnology for synthesizing two-dimensional materials has greatly improved in recent\nyears. Since the synthesis of graphene[1, 2], a large number of extensive systems\nonly a few atoms thick have been obtained. The study of these 2D materials has\nbrought new physical phenomena with countless applications into play, like their\nmagnetic properties[3, 4]. Obtaining magnetism in 2D isotropic crystals is forbidden\nin the Heisenberg model as explained by the Mermin-Wagner theorem[5]: the magnon\ndispersion is reduced with respect to their 3D counterparts, and it has an abrupt onset,\nwhich translates into low thermal agitation and a collapse of the spin order. However,\n2D systems with uniaxial magnetic anisotropy are able to withstand thermal agitation,\nallowing magnetic states in mono and multilayer materials.arXiv:2211.01732v1  [cond-mat.mtrl-sci]  3 Nov 2022Magnetism in Two-Dimensional Ilmenenes 2\nIn the past decades, the study of 2D magnetic properties has been performed\non epitaxially grown thin \flms, in which phenomena such as oscillating exchange\ncoupling[6, 7], giant magnetoresistance[8, 9] and Hall e\u000bect[10, 11] have been observed.\nNevertheless, the study of the intrinsic magnetic properties of these 2D systems is\nnovel, and most of the materials that have been synthesized are magnetic van der\nWaals crystals[3, 4]. In the last \fve years, non-van der Waals two-dimensional materials\nhave also been synthesized, mainly by exfoliating naturally occurring ores. By liquid\nexfoliation of natural iron ore hematite ( \u000b-Fe2O3), Balan et al.[12] synthesized a new\n2D material which called hematene. Contrary to its antiferromagnetic bulk, hematene\npresents ferromagnetic order. Similarly, other promising material for 2D magnets is\nilmenene [13], which has been synthesized using liquid phase exfoliation from titanate\nore ilmenite (FeTiO 3). Motivated by the synthesis of iron ilmenene, other similar 2D\nmaterials could be exfoliated from the bulk systems with their 3D counterpart ilmenite-\nlike structures.\nThe aim of the present work is then to characterize the ilmenene-like titanates\n(TMTiO 3, TM = V to Zn) to set a \frst theoretical basis for this exciting family of\nmaterials. Within the framework of the density functional theory, we systematically\nanalyze the crystalline structure of these compounds, \fnding that most of the ilmenenes\nexhibit triangular symmetry for TMs on both sides of a Ti-Ti hexagonal graphene-\nlike sublattice. In the chromium and copper ilmenenes, we \fnd structural distortions\nof Jahn-Teller origin. Our electronic structure calculations reveal that most of these\ncompounds are magnetic semiconductors which have TM layers antiferromagnetically\ncoupled. The calculations including the spin-orbit interaction show a strong magnetic\nanisotropy, with the magnetization being oriented out-of-plane (in-plane) below (above)\nhalf-\flling of the TM electronic 3d shell. The presence of out-of-plane anisotropy in\nsome of these compounds suggests potential applications for spintronics in thin layers\nand 2D materials.\n2. Model Details\nIn this work, we study the structural, electronic and magnetic properties of ilmenene-\ntype materials using the projector augmented wave method (PAW) implemented\nin the Vienna Ab-initio Software Package (VASP)[14, 15]. For the exchange and\ncorrelation potential we use the Perdew-Burke-Ernzerhof form of the generalized\ngradient approximation (GGA), with the formulation of Dudarev[16] for GGA+U. The\nHubbard U parameters for each element are chosen based on the available literature on\ntransition metal oxides[17], and are shown in Table S1 of the Supplementary Information\n(SI) with the considered valence states. A test calculation of the partial density of states\n(PDOS) of cobalt titanate using the HSE06 hybrid functional shows that including\na U parameter in the oxygen p-orbitals is needed to correctly describe the electronic\nstructure. This \fnding is in agreement with previous investigations concerning titanium\noxide and hematite[18, 19, 20, 21, 22, 23]. All calculations are performed with a well-Magnetism in Two-Dimensional Ilmenenes 3\nconverged plane-wave cuto\u000b energy of 800 eV, a gamma-centered 4x4x1 Monkhorst-Pack\nk-point mesh, and a Fermi smearing of 20 meV. Atomic coordinates are relaxed until\nforces in all directions were smaller than 0.5 meV/ \u0017A. An energy convergence criterion of\n10\u00007eV is used. Further tests using even a larger plane wave cuto\u000b up to 1000 eV and\n6x6x1 k-point mesh do not modify the presented results. Di\u000berences in local charges\nand local magnetic moments are univocally analyzed using the Bader method[24, 25].\nBy including the spin-orbit coupling, additional tests are performed to converge the\nmagnetocrystalline anisotropy energy with respect to the Brillouin Zone sampling.\nThe d-metal ilmenene sheets are obtained by cutting their respective bulk titanates\n(TMTiO 3, TM = V to Zn) in the hexagonal [001] direction. For the iron ilmenene\nFeTiO 3, transmission electron microscopy measurements con\frm the 2D structure in\nthis direction[13]. The layer structure is shown in Fig. 1. For all the compounds\nunder analysis, two di\u000berent layer-ilmenenes are tested: titanium and transition metal\nterminated ilmenene layers. This work focuses on the TM ended layers because they\nare found to be more stable for all the materials under analysis.\n3. Results and discussion\n3.1. Structural properties\nThe structure of the TM ilmenenes is graphically depicted from two viewpoints\nin Fig. 1. After structural relaxations, we \fnd that most of the compounds\nkeep the input symmetry, except for chromium and copper ilmenenes, which show\nstructural deformations of the perfect lattice due to the Jahn-Teller e\u000bect (see Fig.\n1(b)). The orange area in panel (a) denotes the chemical cell that reproduces the\ncrystalline structure of the ilmenenes when periodically repeated. We calculate magnetic\ncon\fgurations using a larger 2x2x1 magnetic cell. For chromium and copper titanates,\ndue to structural distortions, the unit cell and the magnetic cell coincide. In the distorted\ncompounds, the two in-plane lattice vectors become slightly di\u000berent: for the chromium\ncase a=10.57 \u0017A and b=10.00 \u0017A, and for the copper case a=10.44 \u0017A and b=10.21 \u0017A. Note\nthat these ilmenenes become anisotropic. Overimposed on the global lattice distortions,\nthe inner atomic distortions become more noticeable. For instance, the largest-smallest\ndistances are 6.0 \u0017A- 3.9 \u0017A and 5.4 \u0017A- 4.8 \u0017A for Cr and Cu ilmenenes, respectively. The\nlarge 1D anisotropy of these two Janh-Teller distorted ilmenenes is shown by the stripes\nof the green and violet areas in Fig. 1 (b). A summary of the interatomic distances of\nthe TM titanates is shown as a function of the elements across the period in the periodic\ntable in Fig. 1. We \fnd that the horizontal distance between transition metals in the\nsame layerlTM\u0000TMincreases from V to Mn, then decreases until Ni, and \fnally increases\nfor the brass metals Cu and Zn. The height distance between metals in di\u000berent layers\nhTM\u0000TMgenerally decreases. The distances involving Ti in the Ti-Ti and Ti-O bonds\nremain nearly constant, while the TM-O distances decrease. In essence, TM-ilmenenes\ncan be thought of as a solid network formed by Ti and O atoms on which the TM atomsMagnetism in Two-Dimensional Ilmenenes 4\nattach on both sides. For the Cr and Cu ilmenenes showing structural distortions, the\nTM-TM distances have two values because the triangular symmetry is broken, and the\nTi-O distances are split into several values as shown by the extra marks in the lower\npanel of Fig. 1.\nWe \fnd that in comparison with their bulk counterparts, TM ilmenenes are\ncompacted along the c direction, so that a two-dimensional hexagonal layer of titanium\nions is formed similar to graphene. The Ti-Ti sublattice in ilmenenes remains almost\nconstant with the distances varying within 3%. zThis compression for the iron\nilmenene has a minimum height about 2.91 \u0017A, in agreement with another work [13].\nThe theoretical value of 2.59 \u0017A for the interplanar space corresponding to the (11 \u001620)\nand ( \u00162110) lattice spacing compares well with the experimental one ( \u00182.53 \u0017A) [13],\nwhich is also supporting our calculations since the interatomic distances depend on the\nchosen U values. For cobalt titanates, a layer thickness of 4.03 \u0017A has been reported[26]\nin bulk, which decreases to 2.95 \u0017A in the ilmenene layer, while the horizontal distance\nbetween cobalt atoms in the same layer expands slightly from 5.06 \u0017A in bulk to 5.17 \u0017A\nin the layer. The signi\fcant changes in the height distances are due to the layer being\ndecorated with half of the TM atoms than in the corresponding bulk, in order to keep\nthe stoichiometry. It seems that the ilmenene layers depart largely from those found\nin ilmenites and not only their structural properties but their magnetic ones are thus\ndeserving a thoroughly study.\n3.2. Electronic properties: magnetic semiconductors\nWe now focus on the electronic structure of ilmenenes and \fnd that they show a gap\nopening that is linked to their stability. The calculated electronic band gaps range\nbetween 1.8 and 4 eV, as displayed in Fig. 2 (a), with values that are typical for\nsemiconductors. The gaps are increasing when going from V to Mn, and they are\noscillating for Fe, Co and Ni. In general, the values are smaller than the gap for TiO 2\nin the rutile bulk phase ( \u00183.2 eV). Note that the Mn and the Co ilmenenes remain\nwithin the same order of the TiO 2gap values. These trends follow the \flling of 3d TM\nelectronic levels as discussed below.\nThe TM atoms in the ilmenene compounds show local magnetic moments, which\nare calculated using the Bader method, and are shown in panel (b) of Fig. 2. The\ncalculated values are close to the ones expected by applying the Hund rules to isolated\nions. Furthermore, the local magnetic moment as a function of atomic number follows a\nSlater-Pauling-like rule, increasing from vanadium to manganese, then decreasing until\nit vanishes for zinc. Thus, the 3d ilmenenes can be taken as magnetic semiconductors.\nThe band structures and the PDOS projected onto the transition metal atoms\nare collected in the SI. From their analysis, an electronic \flling model for each TM\nilmenene is presented in Fig. 2(b). We \fnd that the orbitals can be classi\fed into\nzHowever, the chromium titanate layer does not show the fully compacting behavior to a \rat titanium\nsheet (see Fig. 1(b)).Magnetism in Two-Dimensional Ilmenenes 5\nthree groups: (i) out-of-plane dz2, (ii) in-plane dx2\u0000y2anddxy, and (iii) mixing in- and\nout- of plane contributions for dxzanddyz. Thedx2\u0000y2anddxyorbitals are highly\nhybridized, as the dxzanddyzorbitals are. Because the in-plane degenerated energy\nlevels of chromium and copper titanates are being partially occupied, they are held\nresponsible for the Jahn-Teller-like distortions in these structures, where a splitting of\nthese otherwise degenerated orbitals is observed under distortions.\nAll these electronic trends can be better understood following the orbital model\nin real space for each case, as shown in Fig. 2 (c). Below half-\flling, we show how\nthe magnetic moment increases in correlation with the gap values. For next elements,\nabove half-\flling, the Fe case becomes similar to the Ni one which has a weaker spin\ndistribution. In between, we \fnd the Co ilmenene with a large in-plane spin distribution\nhas also an out-of-plane component that cannot be neglected, an extra component that\nis related to the complexity added to this case when trying to write down a single spin\nhamiltonian. For brass metals, the local magnetization is nearly screened having a spin\ncompensated cloud around the brass metal centers. This model is going to be interesting\nin the discussion of magnetic anisotropies below. For instance, we note that the spin\ndistribution is not anisotropic for the case of Cr (Cu) having one-electron less below a\nhalf-\flled (\flled) 3d shell.\n3.3. Magnetic ordering\nWe next consider the magnetic behavior of the ilmenene-type materials, and calculate\nthe total energies of the magnetic con\fgurations shown schematically in Fig. 3.\nWe \fnd that, with the exception of ferromagnetic copper and spin-compensated\nzinc titanates, the ilmenenes show antiferromagnetism between the 2D TM layers in the\nso-called AFM-1 con\fguration.\nThe magnetic con\fgurations in Fig. 3 are used to \ft the Heisenberg Hamiltonian\nH=\u0000X\nijJij~Si\u0001~Sj (1)\nwhere the ~Siis the pseudospin of each isolated atomic species (e.g: ~S= 3=2 for Co),\nand theJiwithi= 1;2 are the inter-layer ( i= 1) and intra-layer ( i= 2) magnetic\ncouplings schematically depicted in 3. The magnetic couplings are computed from the\nenergy di\u000berences \u0001 E1=EFM\u0000EAFM 1, \u0001E2=EAFM 2\u0000EAFM 1.\nThe \ftted J ivalues are shown in Fig. 3 (b). The inter-layer coupling is the\nleading interaction because J 1is an order of magnitude larger than J 2. The inter-\nlayer coupling J 1is negative for most of the ilmenenes, that indicates a preference\nfor antiferromagnetism. The intra-layer coupling J 2is positive in all cases, and the\ncompounds favor intralayer ferromagnetism. Since the coordination of the TM atoms\non each layer side changes from being hexagonal in bulk to triangular in the layer,\nthe structural di\u000berences between bulk and two dimensional titanates mentioned above\nplay an important role in the magnetic ordering. It is noteworthy that the manganeseMagnetism in Two-Dimensional Ilmenenes 6\nilmenene even favors ferromagnetism within layers in contrast with its antiferromagnetic\nbulk counterpart.\n3.4. Magnetic anisotropy\nWe last study the magnetic anisotropy of d-metal ilmenenes. The spin-orbit interaction\ncouples the spin magnetic moment to the crystal lattice, which means that some spin\norientations are more stable than others. The magnetocrystalline anisotropy energy\n(MAE) is de\fned as the energetic di\u000berence between two magnetic con\fgurations with\ndi\u000berent spin orientations. We calculate the total energy of the ilmenenes for a number\nof spin orientations, and \fnd that, anisotropy-wise, ilmenenes can be classi\fed in two\ngroups: out-of-plane ilmenenes, in which the spin magnetic moment is aligned in the\nc-axis, and in-plane ilmenenes, with the spin aligned in any of the directions along TM-\nTi bonds projected in the plane (see Fig. 4). Due to symmetry around TM atoms,\nthere are six such equivalent directions in the abplane. The vanadium, chromium\nand manganese titanates in addition to copper ones have an out-of-plane anisotropy,\nand the magnetization in iron, cobalt and nickel titanates is oriented in-plane. These\nMAE trends agree with the electronic \flling model, in which below half-\flling TMs the\nelectrons near the Fermi level are those that include an out-of-plane component, and\nfor iron, cobalt and nickel titanates they are mainly in-plane.\nThe obtained MAE values range between several orders of magnitude from 10\u00002\nmeV up to tenths of meVs because the spin-orbit interaction varies greatly for the\ndi\u000berent ilmenenes. The chromium, iron and cobalt titanates have a strong MAE around\n5 meV, and in the case of the vanadium and nickel layers, this value is still large about\n0.6 and 0.13 meV, respectively. The manganese titanate has the smallest anisotropy ( \u0018\n0.04 meV) because the orbital magnetic moment is nearly fully quenched. This \fnding\nsummed to having the energetic di\u000berence between the magnetic AFM-1 and AFM-2\ncon\fgurations of the manganese titanate being very small (around 0.25 meV) points\nto non-collinear magnetic con\fgurations, a topic that might merit future experimental\ninvestigation on this speci\fc layer, but being beyond the actual scope of the paper.\nFurthermore, the iron ilmenene is also a particular 2D layer because it shows a strong\nin-plane anisotropy, contrary to the ilmenite bulk with a non-negligible out-of-plane\ncomponent.\nWe observe a noticeable trend for the magnetic anisotropy. Below half-\flling, the\nV and Cr ilmenenes exhibit out-of-plane anisotropy, while the half-\flled manganese\nilmenene has a very small MAE. Above half-\flling, most the compounds have an in-\nplane anisotropy. In fact, these trends in anisotropy follow the levels depicted in Fig.\n2 (c). For V and Cr ilmenenes, it shows that the spin density is perpendicular to the\nplane. The spin density in manganese ilmenene is isotropic, and this correlates with its\nlow MAE. Above half-\flling, the spin density of Fe, Co and Ni ilmenenes lies in-plane,\nin agreement with their in-plane MAE. The case of Cu ilmenene shows out-of-plane\nanisotropy even if the spin density lies in-plane, a \fnding that can be explained by theMagnetism in Two-Dimensional Ilmenenes 7\nout-of-plane changes in the orbitals when the spin-orbit coupling term is included with\nthe distortions.\nWe next bring the magnetic results on magnetic ilmenenes into contact with\nexperimental facts. Note that Fe ilmenenes can be obtained by exfoliation and deposited\non substrates for some experimental setups [13, 12]. Then, further measurements on\nthe magnetic properties of the Fe ilmenenes can be today performed in line with our\ntheoretical results. Furthermore, we have studied Co ilmenenes that could be interesting\nas they can behave in a similar way to study magnon physics in two dimensions as their\nbulk counterpart is already being used [27, 28, 26]. In fact, Cr ilmenenes seem to become\nkey as they are candidates to show out-of-plane anisotropy when in the form of ultrathin\nlayers. These \fndings are suggesting the future use of TM ilmenenes in spintronics\ndevices for injecting magnons and study magnetism in exfoliated 2D magnetic layers.\n4. Conclusion\nIn this work, we have analyzed the structural, electronic and magnetic properties of\nTM ilmenene-like systems. Our calculations reveal that most of the materials under\nanalysis present a triangular crystalline structure for TMs, with an ironed compression\nof the internal titanium ion layer with respect to bulk. The chromium and copper\nilmenenes exhibit notable structural distortions, which seem to have a Jahn-Teller\norigin. All the compounds studied have been found to be magnetic semiconductors\nwith band gaps in the range between 1.7 and 4 eV. The magnetic ground state is\nmainly antiferromagnetic between layers, and ferromagnetic and spin-compensated for\nCuTiO 3and ZnTiO 3, respectively. Furthermore, spin-orbit calculations revealed that\nTM ilmenenes can be divided into two groups: out-of-plane ilmenenes, for less than\nhalf-\flling of the 3d bands, and in-plane ilmenenes, above half-\flling of these levels,\nwith the spin aligned in the TM-titanium directions projected in the hexagonal plane.\nWe believe that this family of materials paves the way to other types of promising 2D\ncandidates with potential applications in the \feld of spintronics.\n5. Data availability statement\nAll data that support the \fndings of this study are included within the article (and any\nsupplementary \fles).\n6. Acknowledgement\nThis work has been supported by the Spanish Ministry of Science and Innovation with\nPID2019-105488GB-I00 and PCI2019-103657. We acknowledge \fnancial support by the\nEuropean Commission from the NRG-STORAGE project (GA 870114). The Basque\nGovernment supported this work through Project No. IT-1569-22. M.A was supported\nby the Spanish Ministry of Science and Innovation through the FPI PhD FellowshipMagnetism in Two-Dimensional Ilmenenes 8\nBES-2017-079677. R.H.A-T acknowledge the postdoctoral contract from the Donostia\nInternational Physics center.\n7. References\n[1] Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V and\nFirsov A A 2004 Science 306666{669\n[2] Geim A K and Novoselov K S 2007 Nature Materials 6183{191\n[3] Gong C, Li L, Li Z, Ji H, Stern A, Xia Y, Cao T, Bao W, Wang C, Wang Y et al. 2017 Nature\n546265{269\n[4] Huang B, Clark G, Navarro-Moratalla E, Klein D R, Cheng R, Seyler K L, Zhong D, Schmidgall\nE, McGuire M A, Cobden D H et al. 2017 Nature 546270{273\n[5] Mermin N D and Wagner H 1966 Phys. Rev. Lett. 17(22) 1133{1136\n[6] Nikolaev K R, Dobin A Y, Krivorotov I N, Cooley W K, Bhattacharya A, Kobrinskii A L, Glazman\nL I, Wentzovitch R M, Dahlberg E D and Goldman A M 2000 Phys. Rev. Lett. 85(17) 3728{3731\n[7] Gareev R R, B urgler D E, Buchmeier M, Olligs D, Schreiber R and Gr unberg P 2001 Phys. Rev.\nLett.87(15) 157202\n[8] Baibich M N, Broto J M, Fert A, Van Dau F N, Petro\u000b F, Etienne P, Creuzet G, Friederich A and\nChazelas J 1988 Phys. Rev. Lett. 61(21) 2472{2475\n[9] Binasch G, Gr unberg P, Saurenbach F and Zinn W 1989 Phys. Rev. B 39(7) 4828{4830\n[10] Li X W, Gupta A, McGuire T R, Duncombe P R and Xiao G 1999 Journal of Applied Physics 85\n5585{5587\n[11] Taylor J M, Markou A, Lesne E, Sivakumar P K, Luo C, Radu F, Werner P, Felser C and Parkin\nS S P 2020 Phys. Rev. B 101(9) 094404\n[12] Puthirath Balan A, Radhakrishnan S, Woellner C F and et al 2018 Nature Nanotech 13602{609\n[13] Puthirath Balan A, Radhakrishnan S, Kumar R, Neupane R, Sinha S K, Deng L, de los Reyes\nC A, Apte A, Rao B M, Paulose M, Vajtai R, Chu C W, Costin G, Mart\u0013 \u0010 A A, Varghese O K,\nSingh A K, Tiwary C S, Anantharaman M R and Ajayan P M 2018 Chemistry of Materials 30\n5923{5931\n[14] Kresse G and Furthm uller J 1996 Phys. Rev. B 54(16) 11169{11186\n[15] Kresse G and Joubert D 1999 Phys. Rev. B 59(3) 1758{1775\n[16] Dudarev S L, Botton G A, Savrasov S Y, Humphreys C J and Sutton A P 1998 Phys. Rev. B\n57(3) 1505{1509\n[17] Aguilera-Granja F and Ayuela A 2020 The Journal of Physical Chemistry C 1242634{2643\n[18] Lany S 2015 Journal of Physics: Condensed Matter 27283203 URL https://doi.org/10.1088/\n0953-8984/27/28/283203\n[19] Lany S, Raebiger H and Zunger A 2008 Phys. Rev. B 77(24) 241201 URL https://link.aps.\norg/doi/10.1103/PhysRevB.77.241201\n[20] Morgan B J and Watson G W 2009 Phys. Rev. B 80(23) 233102\n[21] Huang X, Ramadugu S K and Mason S E 2016 The Journal of Physical Chemistry C 1204919{\n4930\n[22] Jiang L, Levchenko S V and Rappe A M 2012 Phys. Rev. Lett. 108(16) 166403\n[23] Gou G Y, Bennett J W, Takenaka H and Rappe A M 2011 Phys. Rev. B 83(20) 205115 URL\nhttps://link.aps.org/doi/10.1103/PhysRevB.83.205115\n[24] Kutzelnigg W 1993 Angewandte Chemie International Edition in English 32128{129\n[25] Henkelman G, Arnaldsson A and J\u0013 onsson H 2006 Computational Materials Science 36354{360\nISSN 0927-0256\n[26] Arruabarrena M, Leonardo A, Rodriguez-Vega M, Fiete G A and Ayuela A 2022 Phys. Rev. B\n105(14) 144425 URL https://link.aps.org/doi/10.1103/PhysRevB.105.144425Magnetism in Two-Dimensional Ilmenenes 9\n[27] Elliot M, McClarty P, Prabhakaran D, Johnson R, Walker H, Manuel P and Coldea R 2021 Nature\nCommunications 123936\n[28] Yuan B, Khait I, Shu G J, Chou F C, Stone M B, Clancy J P, Paramekanti A and Kim Y J 2020\nPhys. Rev. X 10(1) 011062 URL https://link.aps.org/doi/10.1103/PhysRevX.10.011062\nFigure 1. Magnetic unit cell for transition metal ended ilmenene-like systems:\n(a) symmetric for most ilmenenes, and (b) distorted for chromium titanate CrTiO 3.\nThe color code of the atoms is as follows: TM (Cr) atoms in blue (purple), titanium\nin cyan, and oxygen in red. The orange area in panel (a) represents the smaller\nchemical cell. For chromium and copper titanates, the chemical and magnetic cells\ncoincide. Calculated interatomic distances: (c) horizontal TM-TM distance lTM\u0000TM,\n(d) layer height hTM\u0000TMand Ti-Ti distances, and (e) Ti-O and TM-O distances. For\nthe chromium and copper ilmenenes we present an average of the distances in their\ndistorted structures.Magnetism in Two-Dimensional Ilmenenes 10\nFigure 2. (a) Electronic band gaps for TM ilmenenes. Vertical lines separate\ndi\u000berent regions with TM below half-\flling, TM above half-\flling, and brass metals.\n(b) Calculated local magnetic moment around transition metal atoms. For each\ncompound, the electronic \flling model of the ground state is also shown. Red levels\nrepresent the in-plane dx2\u0000y2and dxyorbitals; green, the out-of-plane dz2orbital; and\nblue, the dxzand dyzorbitals with in- and out-plane components. (c) Orbital models\nof the local magnetization in 3d TMs and brass metals within ilmenenes. Blue and\ngreen regions denote spin polarized regions; grey, spin compensated ones.Magnetism in Two-Dimensional Ilmenenes 11\nFigure 3. (a) Magnetic ordering con\fgurations of TM ilmenenes: \\FM\"\nFerromagnetic, \\AFM-1\" antiferromagnetic by layers, \\AFM-2\" and \\AFM-3\"\nantiferromagnetic. (b) Couplings between both layer sides (J 1), and within the same\nside layer of TM ions (J 2).\nFigure 4. Magnetocrystalline anisotropy energy of the transition metal ilmenenes.\nIn blue, compounds with out-of-plane anisotropy; in red, with in-plane magnetic\nmoments. For the case of manganese titanate, the magnitude was enhanced by a\nfactor of 3 to make it visible in the shown range."    },    {        "title": "2211.11002v1.Spin_wave_spectra_in_antidot_lattice_with_inhomogeneous_perpendicular_magnetocrystalline_anisotropy.pdf",        "content": "Spin-wave spectra in antidot lattice with inhomogeneous\nperpendicular magnetocrystalline anisotropy\nM. Moalic, M.Krawczyk and M. Zelent\nInstitute of Spintronics and Quantum Information, Faculty of Physics, Adam\nMickiewicz University, Poznan, Poland\nAbstract\nMagnonic crystals are structures with periodically varied magnetic properties that are used to con-\ntrol collective spin-wave excitations. With micromagnetic simulations, we study spin-wave spectra in\na 2D antidot lattice based on a multilayered thin film with perpendicular magnetic anisotropy (PMA).\nWe show that the modification of the PMA near the antidot edges introduces interesting modifications\nto the spin-wave spectra, even in a fully saturated state. In particular, the spectra split in two types of\nexcitations, bulk modes with amplitude concentrated in a homogeneous part of antidot lattice, and edge\nmodes with an amplitude localized in the rims of reduced PMA at the antidot edges. Their dependence\non the geometrical or material parameters is distinct but at resonance conditions fulfilled, we found\nstrong hybridization between bulk and radial edge modes. Interestingly, the hybridization between the\nfundamental modes in bulk and rim is of magnetostatic origin but the exchange interactions determine\nthe coupling between higher-order radial rim modes and the fundamental bulk mode of the antidot\nlattice.\n1 Introduction\nMagnonic crystals (MCs), analogous to photonic crystals that have a periodic modulation of the refractive\nindex to control the propagation of electromagnetic waves, are magnetic structures with a periodicity of\nmaterial properties relevant to spin-wave (SW) propagation.[9, 25] Their main feature is the formation\nof bands of collective SW excitations and the separating them magnonic bandgaps, which is useful for\napplications to guide and control spin waves.[16, 12, 20, 26, 6] The basic types of MCs are artificial\ncrystals formed by periodic arrangement of two different materials.[12] Those can be two ferromagnetic\nmaterials, like in bicomponent MCs,[29] ferromagnetic and non-magnetic materials like in an array of\nferromagnetic dots in nonmagnetic matrix[13, 30, 27], or the inverse structures,[11, 17] i.e., an array of\nholes in a ferromagnetic matrix, antidot lattices (ADLs).1\nPeriodic modulation of SW-relevant parameters can be introduced not only in the patterning process[19,\n14] but also at a later stage. For instance, by ion irradiation of the ferromagnetic film or multilayers, to\nmodify the magnetic properties on the exposed areas.[3, 32, 8, 21, 34] Recently, another promising idea\nhas been explored, that is, the formation of periodicity by regular change of the magnetization orienta-\ntion in a homogeneous thin film.[35] The basic examples are the periodic stripe domains, which can be\nconsidered as 1D MCs[1, 28] or skyrmion lattices, which also form a kind of 2D MC.[7, 5, 31, 2] These\nstructures have an important property; their magnetization structure is reconfigurable and sensitive to the\n1Recently, the term of MC has been extended to the lattice of atomic spins, which form the spin-wave band structure at large,\nup to THz, frequency range.[36]\n1arXiv:2211.11002v1  [cond-mat.mes-hall]  20 Nov 2022external bias magnetic field; in this way, the magnonic band structure can be programmed after the man-\nufacturing of the device, depending on the actual needs.[35] Nevertheless, the structures combining both\npatterning and magnetization texture are not yet well explored, in particular in the context of MCs and\nmagnonics.\nIn recent studies, ADL based on multilayers with perpendicular magnetic anisotropy (PMA) has been\nindicated as a patterned system hosting periodic magnetization texture with interesting properties and\npotential applications.[22, 24, 18] The results of time-resolved magneto-optical Kerr effect (TR-MOKE)\nmicroscopy measurements of SW excitations in ADL based on [Co/Pd] 8multilayers were interpreted\nwith micromagnetic simulations.[22] To explain low frequency mode in the spectra, the authors assumed\nthat around the holes there is a rim with modified magnetic properties formed during the patterning\nprocess, i.e., Ga ions used in focused ion beam (FIB) in the patterning process penetrate larger area than\nthe antidots. In the rims around the antidots, the dose is sufficient to disturb the interfaces between Co\nand Pd, and so, the PMA, which originates from the interfaces in the multilayer. The absence or reduced\nPMA in the rims results in the in-plane alignment of the magnetization in the rims at remanence.[24] The\nMC composed of two areas of different magnetization orientations, out-of-plane in the bulk and in-plane\nin the rims, makes the SW spectra complex, with bulk modes localized in the ferromagnetic matrix, as\nin standard ADLs, and the additional modes localized in the rims, with possible hybrid excitations. Such\nstructures open the prospects for magnonic band structure with sub-bands sensitive to external stimuli in\ndifferent ways, e.g., different dependences on a magnetic field, the property, which can be exploited for\ndeveloping new phenomena and applications.\nIn this paper, we study the SW spectra in the ADL lattice based on multilayer thin film with PMA\nwith the rims around the antidots, where the PMA is reduced. In particular, we study the dependence of\nthe spectra on the rim anisotropy value, rim width and antidot diameter under a perpendicular magnetic\nfield, fully saturating the sample. We show that the variation of these properties affects mainly the edge-\nlocalized modes, but additionally may introduce or control a hybridization between the radial modes of\nthe rim and the bulk excitations of the ADL. This offers a possibility to control collective SWs by tuning\nmagnetic properties only in selected areas.\nThe paper is organized as follows. In the first section, we introduce the structure and the micro-\nmagnetic model used in the simulations. In Sec. 3 we present the simulation results of SW spectra in\ndependence on the anisotropy strength in the rim, antidot diameter and finally rim width, and interpret\nthe obtained results. In the last Sec. IV we summarise the results.\n2 Structure and methodology\nFor our numerical investigations, we selected a structuralized Co/Pd multilayer with high PMA (with the\nanisotropy constant Ku;bulk) and reduced anisotropy around the antidots, already investigated in Ref. [24].\nThe schematic of the unit cell of the studied structure is shown in Fig. 1. The structure is made up of 8\nrepetitions of Co (0.75 nm thickness) and Pd (0.9 nm) bilayers with a total thickness of 13.2 nm. Around\nthe antidot of diameter dis a rim of width wwith reduced PMA, Ku;rim. In our study, we will change these\nthree parameters. The antidots form a square lattice on the (x;y)plane with a lattice constant a=500 nm.\nTo maintain the same static magnetization configuration of the system, i.e. to maintain a full saturation of\nmagnetization in the out-of-plane direction, we perform numerical simulations using a sufficiently strong\nexternal magnetic field m0Hext=1 T directed along the zaxis.\nWe use micromagnetic simulations with Mumax3 software, [33, 15] which solves the Landau-Lifshitz-\nGilbert equation for magnetization vector M:\ndm\ndt=g\u001601\n1+a2(m\u0002Heff) +a\u00160(m\u0002(m\u0002Heff)); (1)\n2Figure 1: Schematic illustration of the investigated structure showing the Co/Pd supercell of ADL. Note\nthat the figure is not to scale. The white color around the hole represents the rim with the reduced PMA.\nwhere m=M=MSis the normalized magnetization, MSis the magnetization saturation, Heffis the ef-\nfective magnetic field acting on the magnetization, g=\u0000187 rad GHz/T is the gyromagnetic ratio, ais\nthe damping constant. The following components were considered in the effective magnetic field Heff:\ndemagnetizing field Hd, exchange field Hexch, uniaxial magnetocrystalline anisotropy field HKu, and ex-\nternal magnetic field, thermal effects were neglected. Thus, the effective field is expressed as:\nHeff=Hd+Hexch+Hext+HKu+hmf; (2)\nwhere the last term, hmfis a microwave magnetic field used for SW excitation. The exchange and\nanisotropy fields are defined as:\nHexch=2Aex\nm0MSDm;HKu=2Ku;bulk\nm0MS(u\u0001m)u; (3)\nwhere uis the unit vector that indicates the direction of anisotropy, Aexis the exchange constant.\nTo simplify the micromagnetic simulations of the multilayered ADL system, we used the effective\nthickness approach, which involves the effective, experimentally measured, values of the magnetic pa-\nrameters for a single discretization through the thickness of the magnetic multilayer. [23] In the simula-\ntions, we used a cell size of 500 =512\u00190:97 nm for the xandyaxis, and 13.2 nm for the zaxis. To mimic\na periodic structure, we used 32 repetitions of Mumax3 periodic boundary conditions, along the xand\nyaxes. The effective magnetic parameters for the Co/Pd multilayer are taken from Ref. [24], these are\nKu;bulk=4:5\u0002105J/m3,MS=0:81\u0002106A/m, Aex=1:3\u000210\u000011J/m. We used a low damping constant\na=1\u000210\u00007to simulate sharp SW spectra.\nTo obtain full SW spectra first, we relax the system, then we implement a special distribution of the\nexternal microwave magnetic field for SW excitation, hmf. Using the V oronoi tessellation algorithm, we\ndivided the ADL unitcell area into small random sub-areas, about 30 nm wide each, effectively acting\n3Figure 2: (a) SW spectrum dependent on the reduced anisotropy strength FKu. The intensity of the\nSW spectra has been presented in a logarithmic scale colormap. The black-dashed line represents the\nanalytically calculated ferromagnetic resonance frequency of the uniform thin film, Eq. (5). The blue-\ndashed-dotted line represents the resonance frequency of the fundamental mode in the isolated ring. The\nmarkers indicate the SW modes illustrated in the right panels (b-i). Here, the color saturation indicates the\nmode intensity and the hue indicates the phase of the mode, according to the diagram in the right bottom\ncorner. Inset (j) presents analogous dependence to panel (a), but with rim-matrix exchange interactions\ndisabled.\n4as SW point sources (nanoantennas). Each of these nanoantennas was assigned the same amplitude of a\nmicrowave magnetic field, but a random phase offset was used in the system’s excitation by an in-plane\nmagnetic field, along the diagonal of the ADL. To excite SWs in a wide frequency range we used a sinc\nfunction to apply a time modulation of the microwave field, which has a maximum amplitude of 0.5 mT,\na cutoff frequency of 40 GHz, and a sampling rate of 8.3 ps. The magnetic field pumps the system for 1\nns and stops before any data are recorded. The time-resolved, space-dependent magnetization was then\nsaved over 8.3 ns and processed by the Fast Fourier transform (FFT) to get 2D visualization of all SW\nmodes. To obtain the SW spectra, we applied a Hanning window to both the space and time dimensions of\nthe data before using the FFT. We then found for each frequency, the cell in the system that had the highest\namplitude and used it as the FFT amplitude in the spectra. This approach to calculate the SW spectra has\nthe benefit of not favoring the extended SW modes, as is in the case of homogeneous microwave field.\nTo refer to a particular SW mode, we use the Mmnbnotation, where we identify the radial n, azimuthal\nm, and bulk bnumbers, which represent the order of the modes in the rim along the radial and azimuthal\ndirections, and in the bulk, respectively. For bulk, by the mode order, we took the number of nodal points\nalong the line connecting neighboring antidots minus one. For instance, a homogeneous in space mode,\na fundamental mode, which has a pinned magnetization at the edges of the antidots, we will have the\norder number b=1. A bulk mode with one more nodal point between antidots, i.e., in total three nodes,\nis a second-order bulk mode with b=2. When a number is not applicable, we replace it with the ’-’\nsymbol. The radial mode order is defined as a number of amplitude quantizations in the rim along the\nradial direction, where n=1 means a fundamental mode with only partial pinning at the rim edges and no\nnodes between, then n=2 represents a mode with 1 node between the rim edges. For azimuthal modes,\nthe value of the number mdefines the number of full phase rotations along the circumference, where\nm=1 means a 2 pphase change along the circumference of the rim.\n3 Results\nInitially, we simulate the ADL with an antidot diameter of d=200 nm and a rim width of w=50\nnm depending on the anisotropy value in the rim. As the Haniis parallel to Hext, a local reduction of\nthe anisotropy field should result in a decrease of the resonant frequencies of SWs confined in these\nareas. Indeed, Fig. 2 (a) shows the dependence of the SWe spectrum on the reduced magnetocrystalline\nanisotropy factor FKu:\nFKu=Ku;rim\nKu;bulk; (4)\nwhich reveals two distinct groups of modes. One group with frequencies linearly decreasing as FKudoes,\nand the second one, only weakly dependent on the anisotropy in the rim, which corresponds to the ADL\nmodes in the rim and bulk, respectively. The representative eight modes are marked in Fig. 2 (a) and their\nin-plane spatial profiles are shown in Fig. 2 (b)-(i).\nFor an uniform distribution of anisotropy, FKu=1, the lowest frequency mode is a fundamental bulk\nmode M\u0000\u00001at the frequency 33.24 GHz, illustrated in Fig. 2 (h). The modes of higher frequencies\nare the higher-order bulk modes. The reduction of the anisotropy in the rims leads to the transition of\na homogeneous ADL to a structuralized bicomponent system with different magnetic parameters, and\nto isolate edge-localized radial and azimuthal modes. Therefore, with decreasing FKu, the frequencies\nof these two types of modes decrease, and crossover the bulk modes. At some FKuwe observe strong\nhybridization, in particular at crossings of the fundamental bulk mode M\u0000\u00001atFKu=0.86, 0.59, and\n0.21. The visualizations of modes presented in Fig. 2 (e) and (f) shows that these hybridizations are\nbetween the mode M\u0000\u00001and the radial modes of the rim: M1\u0000\u0000,M2\u0000\u0000, and M3\u0000\u0000(profile not shown).\nThe first two radial modes are shown in Fig. 2 (b) and (c), and their hybridizations with fundamental bulk\nmode in (e) and (f), respectively. Also, higher order bulk modes hybridize with the rim modes, although\n5Figure 3: (a) The spin-wave spectrum in dependence on the antidot diameter d. The green-dashed line\nrepresents the antidot diameter d=100 nm used in Fig. 2 and Fig. 4. (b-i) Visualization of spin-wave\nmodes for selected frequencies and antidot diameters marked in (a). The saturation indicates the intensity\nand the hue indicates the phase of the mode, according to the diagram in the right bottom corner.\nwith smaller strength. For example, the hybridization of the second-order radial mode M2\u0000\u0000with the\nsecond-order bulk mode M\u0000\u00002, is shown in Fig. 2 (i) as M2\u00002. The hybridization of the modes M\u0000\u00002and\nM14\u0000is shown in Fig. 2 (d), which clearly demonstrates the hybridization of the bulk modes also with the\nazimuthal modes of the rim.\nAt frequencies smaller than 33.24 GHz we see the parallel lines of different intensities. The intensive\nlines have already been identified as a radial modes of the rim; small intensity lines are azimuthal modes,\nfor example, at 16.56 GHz is the M12\u0000mode presented in Fig. 2 (g). The first-order radial mode in\nthe rim M1\u0000\u0000is analogous to the bulk fundamental mode but restricted in space to the rim. Thus, the\nM1\u0000\u0000andM\u0000\u00001are two fundamental excitations of our ADL. Their mode frequencies are influenced\nby the anisotropy, exchange, and magnetostatic fields. To analyze the role of individual interactions on\nthe coupling between the modes, we make a comparison of the full spectra Fig. 2 (a) with the spectra of\nsimplified structures.\nTo clarify the direct effect of the anisotropy field on the system, we calculate the ferromagnetic reso-\nnance frequency of a homogeneous film by varying the anisotropy field. We use Kittel’s formula:[10]\nf0=jgjm0(Hani+Hd+HKu)\n2p; (5)\nwith f0being the mode frequency, Hd=\u0000MS, and HKuas defined in Eq. (3). The function f0(FKu)\nis shown with black-dashed line in Fig. 2 (a). When FKu=1,f0and the resonance frequency for the\nsimulated ADL are only 400 MHz apart. This means that an ADL with an antidot diameter of a=200\nnm at full saturation has a minor effect on the resonant frequency of the fundamental bulk mode. However,\nasFKuis reduced, the fundamental mode smoothly transforms into a radial mode M1\u0000\u0000in the rim. At\nFKu=0:84 we observe a strong hybridization between both modes; see also the mode M1\u00001in Fig. 2\n(e). With further reduction of anisotropy in the rim, the frequency difference increases, reaching a stable\nfrequency separation of 6.20 GHz between f0and the M1\u0000\u0000mode. This difference between frequencies\n6Figure 4: (a) Spin-wave spectrum in dependence on rim width, w. The green-dashed line represents the\nrim width used in previous investigations. (b-i) Visualization of spin-wave modes at selected frequen-\ncies and antidot diameters marked in (a). The color saturation indicates the mode intensity and the hue\nindicates the phase of the mode, according to the diagram in the right bottom corner.\nis mainly due to shape anisotropy of the rim, which rises the rim mode frequencies.\nIf we compare the frequencies of the M1\u0000\u0000mode with those of the isolated ring (obtained from\nindependent micromagnetic simulations, keeping parameters of the rim), the green-dashed line in Fig. 2\n(a), we can see that the frequency of the fundamental mode of the rim is constantly lower by 3.59 GHz.\nThis means that the magnetostatic field from the ADL bulk and a side contact of the rim with the ADL,\neffectively lowers the internal demagnetizing field in the rim, lowering its mode frequencies.\nTo estimate the influence of the exchange interactions, in particular on the strength of the hybridiza-\ntions and a frequency of the excitations, we performed additional micromagnetic simulations. Namely,\nwe excluded the exchange interactions between the rim and bulk by introducing a 2 nm nonmagnetic\nspacer. The result is presented in Fig. 2 (j). We notice that the frequency of the fundamental bulk mode\nM\u0000\u00001atFKu=0:85 only slightly rises from 32.88 to 33.48 GHz. Thus, the hybridization strength has\nslightly decreased with the exchange decoupling of the rim from the bulk. For the hybridization of the\nM\u0000\u00001mode with higher order radial modes, at FKu=0:59, and 0.21, the frequency difference does not\nexceed 300 MHz in Fig. 2 (j), indicating very weakly coupled modes, when the exchenge between rim and\nbulk is excluded. This clearly indicates that these hybridizations are driven by the exchange interactions.\nThus, we can conclude that the hybridization of fundamental bulk mode with fundamental rim mode has a\nmagnetostatic character, while the hybridizations with higher order radial modes have a strong exchange\ncharacter.\nFor further analysis, we performed micromagnetic simulations of the ADL in dependence on the\nantidot diameter in the range from 200 to 0 nm (without antidot), with a 2 nm step; the other parameters\nare fixed, i.e., FKu=0 and w=50 nm. The results are shown in Fig. 3 (a). As expected, the diameter\nof the antidot only weakly affect the frequency of the fundamental bulk mode [see the mode M3\u00001and\nFig. 3 (f)] and the higher-order bulk modes [e.g., M2\u00002and Fig. 3 (i)]. Unexpectedly, we do not observe\navoid-crossings as dchanges, although many crossings between different types of modes are visible. On\nthe other hand, Fig. 3 (f) clearly indicates the hybrydisation occuring between the fundamental bulk mode\n7and the radial rim modes.\nThe frequency of the rim radial modes shows a slight decrease with decreasing d[see, the modes\nM1\u0000\u0000,M2\u0000\u0000, and M3\u0000\u0000and their profiles in Fig. 3 (b), (d), and (e), respectively]. But at an antidot\ndiameter of about 10 nm, the frequency of these modes starts to increase. It is caused by a change in the\ntopology of the ADL geometry. The structure changes from an array of antidots to a full ferromagnetic\nlayer. There are strong dipole interactions on opposite sides of the small antidot at this diameter range,\nwhich result in a change of the effective demagnetizing factors and an increase in mode frequency.[4] The\nfrequency of the second and higher-order azimuthal modes linearly increases at high rates, for example,\nthe frequency of the second-order azimuthal mode M12\u0000andM22\u0000[Fig. 3 (c) and (h)] increases from 6.5\nGHz at d=200 nm up to 10 GHz at 20 nm antidots. This is due to reduction in circumference of the\nantidot rim which contains the wavelength of the azimuthal mode to decrease, resulting in an increase in\nfrequency.\nThe other important parameter that affects the resonance spectrum of SWs is the width of the rim w.\nWe performed micromagnetic simulations with a varied width of the reduced anisotropy region from 0\nto 100 nm for a fixed antidot diameter ( d=200 nm) (see, Fig. 4). The results show that the width of\nthe rim significantly affects the frequency of radial and azimuthal modes [see, modes n=1\u0001\u0001\u00014 and their\nprofiles in Fig. 4 (b)-(e)], but having no significant effect on the frequency of the bulk modes (e.g., see the\nfrequency of the fundamental bulk mode Mn\u00001). As the rim area decreases, the frequency of azimuthal\nand radial modes in the rim increases, although at different rates. For example, the frequency of the\nmode M1\u0000\u0000increases from about 2.04 to 33.24 GHz, where the hybridization with the fundamental bulk\nmode M\u0000\u00001occurs; see the hybridized mode M1\u00001in Fig. 4 (f). Interestingly, there are hybridizations\nbetween the M\u0000\u00001mode and the radial modes Mn\u0000\u0000of the rim, but unexpectedly, the strength of these\nhybridizations increases with n.\n4 Conclusions\nWe have studied the SW spectrum in an ADL made of Pd/Co multilayers with PMA at full saturation\ninduced by a PMA and out-of-plane oriented external magnetic field. A characteristic feature of the\nstructure under investigation is a region around the antidots, a rim, in which the anisotropy was reduced.\nWe have shown that in such a system there are extended bulk modes, and radial and azimuthal excita-\ntions. The last two are confined to the rim area and they are very sensitive to variation of the local rim\nparameters. In particular, the frequency of the azimuthal modes increases with decreasing antidot size,\nwhile the dependence of the radial modes is not monotonous in that case. Both types of SWs decrease\nthe frequency with decreasing an anisotropy in the rim or with increasing a rim width.\nInterestingly, it is possible to achieve strong interaction between bulk modes of the ADL and the\nrim localized modes. Particularly strong hybridization has been found between the fundamental mode\nof ADL and radial excitation of the rim, reaching a value of 1.5 GHz. Importantly, such a coupling is\nrealized by magnetostatic interaction between fundamental modes of the bulk and the rim, while mainly\nby the local exchange coupling between the rim area and the bulk of ADL for hybridizations with higher\norder radial modes. This opens the way for designing magnonic crystals with hybridized localized and\nextended modes, scaled to the deep nanoscale and properties controlled locally at the rim areas. These\ninteresting properties can be especially useful when designing multifunctional SW devices.\n5 Acknowledgments\nThe research has received funding from the National Science Centre of Poland, Grant No. UMO–2020/37/B/ST3/03936.\nThe simulations were partially performed at the Poznan Supercomputing and Networking Center (Grant\nNo. 398).\n8References\n[1] C. Banerjee, P. Gruszecki, J. W. Klos, O. Hellwig, M. Krawczyk, and A. Barman. Magnonic band\nstructure in a Co/Pd stripe domain system investigated by Brillouin light scattering and micromag-\nnetic simulations. Physical Review B , 96(2):024421, 7 2017.\n[2] M. Bassotti, R. Silvani, and G. Carlotti. From the spin eigenmodes of isolated Néel skyrmions\nto the magnonic bands of skyrmionic crystals: a micromagnetic study as a function of the in-\nterfacial Dzyaloshinskii-Moriya interaction and the exchange constants. IEEE Magnetics Letters ,\n13:6101505, 2022.\n[3] R. L. Carter, J. M. Owens, C. V . Smith, and K. W. Reed. Ion-implanted magnetostatic wave reflective\narray filters. Journal of Applied Physics , 53(3):2655–2657, 3 1982.\n[4] G. Centała, M. L. Sokolovskyy, C. S. Davies, M. Mruczkiewicz, S. Mamica, J. Rychły, J. W. Kłos,\nV . V . Kruglyak, and M. Krawczyk. Influence of nonmagnetic dielectric spacers on the spin-wave\nresponse of one-dimensional planar magnonic crystals. Physical Review B , 100(22):224428, 12\n2019.\n[5] Z. Chen and F. Ma. Skyrmion based magnonic crystals. Journal of Applied Physics , 130:090901, 9\n2021.\n[6] A. V . Chumak, A. A. Serga, and B. Hillebrands. Magnonic crystals for data processing. Journal of\nPhysics D: Applied Physics , 50(24):244001, 5 2017.\n[7] S. A. Diáz, T. Hirosawa, J. Klinovaja, and D. Loss. Chiral magnonic edge states in ferromagnetic\nskyrmion crystals controlled by magnetic fields. Physical Review Research , 2(1):013231, 2 2020.\n[8] J. Fassbender, T. Strache, M. O. Liedke, D. Markó, S. Wintz, K. Lenz, A. Keller, S. Facsko,\nI. Mönch, and J. McCord. Introducing artificial length scales to tailor magnetic properties. New\nJournal of Physics , 11(12):125002, 12 2009.\n[9] Y . V . Gulyaev and S. A. Nikitov. Magnonic crystals and spin waves in periodic structures. Dokl.\nPhysics , 46:687, 2001.\n[10] C. Kittel. On the theory of ferromagnetic resonance absorption. Physical Review , 73(2):155–161,\n1948.\n[11] M. Kostylev, G. Gubbiotti, G. Carlotti, G. Socino, S. Tacchi, C. Wang, N. Singh, A. O. Adeyeye, and\nR. L. Stamps. Propagating volume and localized spin wave modes on a lattice of circular magnetic\nantidots. Journal of Applied Physics , 103(7):07C507, 2008.\n[12] M. Krawczyk and D. Grundler. Review and prospects of magnonic crystals and devices with repro-\ngrammable band structure. J. Phys.: Cond. Matter , 26(12):123202, 2014.\n[13] V . V . Kruglyak, P. S. Keatley, A. Neudert, R. J. Hicken, J. R. Childress, and J. A. Katine. Imag-\ning collective magnonic modes in 2D arrays of magnetic nanoelements. Physical Review Letters ,\n104(2):027201, 1 2010.\n[14] J. W. Lau and J. M. Shaw. Magnetic nanostructures for advanced technologies: fabrication, metrol-\nogy and challenges. Journal of Physics D: Applied Physics , 44(30):303001, 7 2011.\n[15] J. Leliaert, M. Dvornik, J. Mulkers, J. De Clercq, M. V . Miloševi ´c, and B. Van Waeyenberge. Fast\nmicromagnetic simulations on GPU - Recent advances made with mumax3. Journal of Physics D:\nApplied Physics , 51(12):123002, 3 2018.\n9[16] B. Lenk, H. Ulrichs, F. Garbs, and M. Münzenberg. The building blocks of magnonics. Physics\nReports , 507(4-5):107–136, 2011.\n[17] R. Mandal, P. Laha, K. Das, S. Saha, S. Barman, A. K. Raychaudhuri, and A. Barman. Effects\nof antidot shape on the spin wave spectra of two-dimensional Ni80Fe20 antidot lattices. Applied\nPhysics Letters , 103(26):262410, 12 2013.\n[18] S. Mantion and N. Biziere. Influence of Ga+ milling on the spin waves modes in a Co 2MnSi Heusler\nmagnonic crystal. Journal of Applied Physics , 131:113905, 3 2022.\n[19] J. Martin, J. Noes, K. Liu, J. Vicent, and I. K. Schuller. Ordered magnetic nanostructures: fabrication\nand properties. Journal of Magnetism and Magnetic Materials , 256(1-3):449–501, 1 2003.\n[20] S. A. Nikitov, D. V . Kalyabin, I. V . Lisenkov, A. Slavin, Y . N. Barabanenkov, S. A. Osokin, A. V .\nSadovnikov, E. N. Beginin, M. A. Morozova, Y . A. Filimonov, Y . V . Khivintsev, S. L. Vysotsky,\nV . K. Sakharov, and E. S. Pavlov. Magnonics: a new research area in spintronics and spin wave\nelectronics. Physics-Uspekhi , 58(10):1002, 2015.\n[21] B. Obry, P. Pirro, T. Brächer, A. V . Chumak, J. Osten, F. Ciubotaru, A. A. Serga, J. Fassbender,\nand B. Hillebrands. A micro-structured ion-implanted magnonic crystal. Applied Physics Letters ,\n102:202403, 5 2013.\n[22] S. Pal, J. W. Klos, K. Das, O. Hellwig, P. Gruszecki, M. Krawczyk, and A. Barman. Optically\ninduced spin wave dynamics in [Co/Pd] 8antidot lattices with perpendicular magnetic anisotropy.\nApplied Physics Letters , 105(16):162408, 10 2014.\n[23] S. Pal, B. Rana, O. Hellwig, T. Thomson, and A. Barman. Tunable magnonic frequency and damp-\ning in [Co/Pd] 8multilayers with variable Co layer thickness. Applied Physics Letters , 98(8):082501,\n2 2011.\n[24] S. Pan, S. Mondal, M. Zelent, R. Szwierz, S. Pal, O. Hellwig, M. Krawczyk, and A. Barman. Edge\nlocalization of spin waves in antidot multilayers with perpendicular magnetic anisotropy. Physical\nReview B , 101(1):014403, 1 2020.\n[25] H. Puszkarski and M. Krawczyk. Magnonic crystals – the magnetic counterpart of photonic crystals.\nSolid State Phenomena , 94:125, 2003.\n[26] J. Rychły, P. Gruszecki, M. Mruczkiewicz, J. W. Kłos, S. Mamica, and M. Krawczyk. Magnonic\ncrystals—Prospective structures for shaping spin waves in nanoscale. Low Temperature Physics ,\n41(10):745–759, 10 2015.\n[27] S. Saha, R. Mandal, S. Barman, D. Kumar, B. Rana, Y . Fukuma, S. Sugimoto, Y . Otani, and A. Bar-\nman. Tunable magnonic spectra in two-dimensional magnonic crystals with variable lattice symme-\ntry.Advanced Functional Materials , 23(19):2378–2386, 5 2013.\n[28] K. Szulc, S. Tacchi, A. Hierro-Rodríguez, J. Díaz, P. Gruszecki, P. Graczyk, C. Quirós, D. Markó,\nJ. I. Martín, M. Vélez, D. S. Schmool, G. Carlotti, M. Krawczyk, and L. M. Álvarez-Prado. Re-\nconfigurable magnonic crystals based on imprinted magnetization textures in hard and soft dipolar-\ncoupled bilayers. ACS Nano , 16(9):14168–14177, 9 2022.\n[29] S. Tacchi, G. Duerr, J. W. Klos, M. Madami, S. Neusser, G. Gubbiotti, G. Carlotti, M. Krawczyk,\nand D. Grundler. Forbidden band gaps in the spin-wave spectrum of a two-dimensional bicomponent\nmagnonic crystal. Phys. Rev. Lett. , 109(13):137202, 2012.\n10[30] S. Tacchi, F. Montoncello, M. Madami, G. Gubbiotti, G. Carlotti, L. Giovannini, R. Zivieri, F. Niz-\nzoli, S. Jain, a. O. Adeyeye, and N. Singh. Band diagram of spin waves in a two-dimensional\nmagnonic crystal. Physical Review Letters , 107(12):127204, 9 2011.\n[31] R. Takagi, M. Garst, J. Sahliger, C. H. Back, Y . Tokura, and S. Seki. Hybridized magnon modes in\nthe quenched skyrmion crystal. Physical Review B , 104(14):144410, 10 2021.\n[32] M. Urbánek, L. Flajšman, V . K ˇrižáková, J. Gloss, M. Horký, M. Schmid, and P. Varga. Research Up-\ndate: Focused ion beam direct writing of magnetic patterns with controlled structural and magnetic\nproperties. APL Materials , 6(6):060701, 6 2018.\n[33] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, and B. Van Waeyenberge.\nThe design and verification of MuMax3. AIP Advances , 4(10):107133, 10 2014.\n[34] A. Wawro, Z. Kurant, M. Jakubowski, M. Tekielak, A. Pietruczik, R. Böttger, and A. Maziewski.\nMagnetic properties of coupled Co/Mo/Co structures tailored by ion irradiation. Physical Review\nApplied , 9(1):014029, 1 2018.\n[35] H. Yu, J. Xiao, and H. Schultheiss. Magnetic texture based magnonics. Physics Reports , 905:1–59,\n4 2021.\n[36] K. Zakeri. Magnonic crystals: towards terahertz frequencies. Journal of Physics: Condensed\nMatter , 32(36):363001, 8 2020.\n11"    },    {        "title": "2211.11515v2.Inertial_dynamics_and_equilibrium_correlation_functions_of_magnetization_at_short_times.pdf",        "content": "1 \n INERTIAL DYNAMICS AND EQUILIBRIUM CORRELATION FUNCTIONS OF \nMAGNETIZATION AT SHORT TIMES  \nSergei V. Titov1, William J. Dowling2, Anton S. Titov3, Sergey A. Nikitov4, and Mikhail \nCherkasskii5,*, ‡  \n1Kotel’nikov Institute of Radioengineering and Electronics of the Russian Academy of \nSciences, Fryazino, Moscow Region 141190, Russia \n2Department of Electronic and Electrical Engineering, Trinity College Dublin, Dublin 2, \nIreland \n3The Moscow Institute of Physics and Technology (State University), Institutskiy per. 9, \nDolgoprudnyy, Moscow Region, 141701, Russia \n4Kotel’nikov Institute of Radioengineering and Electronics of the Russian Academy of \nSciences, Moscow 125009, Russia \n5Faculty of Physics, University of Duisburg-Essen, Duisburg 47057, Germany \nAbstract \nThe method of moments is developed and employed to analyze the equilibrium correlation \nfunctions of the magnetization of ferromagnetic nanoparticles in the case of inertial \nmagnetization dynamics. The method is based on the Taylor series expansion of the \ncorrelation functions and the estimation of the expansion coefficients. This method  \nsignificantly reduces the complexity of analysis of equilibrium correlation functions. \nAnalytical expressions are derived for the first three coefficients for the longitudinal and \ntransverse correlation functions for the uniaxial magnetocrystalline anisotropy of \nferromagnetic nanoparticles with a longitudinal magnetic field. The limiting cases of very \nstrong and negligibly weak external longitudinal fields are considered. The Gordon sum \nrule for inertial magnetization dynamics is discussed. In addition, we show that finite \nanalytic series can be used as a simple and satisfactory approximation for the numerical \ncalculation of correlation functions at short times. \n \n \n \n* Corresponding author: macherkasskii@hotmail.com  \n‡ Present address: Institute for Theoretical Solid State Physics, RWTH Aachen University, \n52074 Aachen, Germany \n  2 \n 1. Introduction \nSpin dynamics is of fundamental importance in spintronics and magnonics research. To \nincorporate ultrafast phenomena, more accurate mathematical models of the spin dynamics \nmust be developed. The prominent Landau-Lifshitz-Gilbert (LLG) equation, which adequately \naddresses picosecond spin precession, has been generalized to consider spin inertia manifesting \nas sub-picosecond nutation [1–23]. Although significant progress has been achieved in the \nstudy of ultrafast phenomena, the study of spin inertia has only just begun. \nThe inertial spin effects were introduced in the LLG equation by mesoscopic \nnonequilibrium thermodynamics theory [3], using the torque-torque correlation model [12], as \nmemory effects [4], with the classical Lagrangian approach [5,16,19] and by Dirac quantum \ntheory [14]. As a result of these investigations, the term with the second-order time derivative \nof magnetization was included in the LLG equation, which is then referred to as the inertial \nLandau-Lifshitz-Gilbert (ILLG) equation. This equation reads \n [ ][ ][ ] gat=´+´+´uHuuuuu&&&& , (1) \nwhere /SM =uM  is a unit vector along the magnetization M, SM=M is the saturation \nmagnetisation, H is the effective magnetic field, a is the Gilbert precession damping \ncoefficient, 5 112.210 raA dms g--× = × ´×  is the gyromagnetic ratio and τ is the inertial \nrelaxation time. The ILLG equation considers spin precession caused by Zeeman interaction, \nGilbert damping, which can be attributed to first order relativistic spin-orbit coupling, and \ninertia of spin originating due to second order relativistic spin-orbit effect [13,14]. The last \nterm in Eq. (1) is reminiscent of the dynamics of a massive particle, and causes nutation of \nmagnetization superimposing on the precession. Nutation resonance was recently detected at \nterahertz frequencies in NiFe, CoFeB and Co ferromagnetic films [1,2]  at a frequency, which \nis proportional to 1t-. In addition, it was found that inertia induces red-shift of the \nferromagnetic resonance [7,17,20]. The non-inertial LLG equation, used successfully at \ngigahertz frequencies, predicts the divergence of the integral spectral power, or in other words, \na plateau of the absorption coefficient at ultrahigh frequencies, which is equivalent to the \nnotorious plateau in the non-inertial Debye model of dielectric relaxation. Adding the inertial \nterm to the LLG equation solves this problem. \nTheoretical investigation of magnetization trajectories within the ILLG model \ndemonstrates the relatively complex behavior, which should be considered in the interpretation \nof the experimental results. For instance, the solutions for the non-damped limit of the ILLG 3 \n equation were obtained in terms of the Jacobi elliptic functions and elliptic integrals for a \nsingle-domain ferromagnetic nanoparticle [18]. Thus, the task of finding the analytical solution \nof the ILLG equation to calculate the correlation functions and the susceptibility is a formidable \none. For the damped case, numerical simulations were presented in Refs. [7,10,21]. However, \nsuch numerical studies provide an abundance of information, which is difficult to interpret \ndirectly. \nHere we present the method of moments employed for the analysis of the equilibrium \ncorrelation functions and the magnetic susceptibility of ferromagnetic nanoparticles \ndemonstrating inertial magnetization dynamics. The method is based on the Taylor series \nexpansion of the correlation functions of magnetization about the initial time, 0t=, and the \nanalytical estimation of the expansion coefficients (moments), namely [24–30] \n \n0()()n\nn\nn n\ntdSiCtdt==- . (2) \nAccording to the definition in Eq. (2) the moments are determined by the short time behavior \nof the correlation function ()Сt. Therefore, the moments imply information about the shape \nof the correlation function and particularly about its behavior on a short time interval. \nAccording to linear response theory, there is a relationship between the susceptibility and the \nequilibrium correlation function [26]. Consequently, the method of moments can be effectively \nused to analyze the complex shape of spectra. \nThe method of moments enables the analysis of spectra in terms of equilibrium \nmagnetization parameters (inertial relaxation time, temperature, direction cosines of the \nmagnetization vector) and the parameters of the free energy. In this method it is not required \nto solve the ILLG equation and to find time-dependent correlation functions. Thus, the \ndetermination of the spectral moments is a simpler problem than the calculation of spectral \nband contours. As a result, a natural interpretation of the magnetic susceptibility spectra can be \nobtained. \n2. Correlation functions for uniaxial magnetocrystalline anisotropy with a \nlongitudinal magnetic field \nMost ferromagnetic solids are magnetically anisotropic. Here we consider crystal \nsystems having a single axis of high symmetry (easy axis). The magnetocrystalline anisotropy \nof such crystals is called uniaxial anisotropy [31,32]. The total dimensionless free energy of a 4 \n uniaxial nanomagnet subject to a uniform d.c. magnetic field ZH=He  applied along the easy \naxis is [31] \n 2 (),coscosV\nkTJjsJxJ=-- , (3) \nwhere /()uKvkTs=  is the dimensionless anisotropy parameter , uK is the uniaxial anisotropy \nconstant, 0S /() vMHkTxm=  is the external d.c. magnetic field parameter, v denotes the \nvolume of the nanomagnet, kT is the thermal energy, and 721\n0410JAm m p---=×  is the \npermeability of free space in SI units. The uniaxial anisotropy is commonly used in treatments \nof the LLG equation [33–38] (Fig. 1). \n \nFIG. 1. Three-dimensional plot of the free energy (), VJj , Eq. (3), for 0s= and 0.7x= . \nThe magnetic susceptibility of anisotropic materials is a matrix known as the susceptibility \ntensor. Here we consider two different components of this complex tensor, namely, \nlongitudinal (along the easy axis) susceptibility cP and the transverse (perpendicular to easy \naxis) one c^ for uniaxial magnetocrystalline anisotropy with a longitudinal magnetic field \n \n0()()()()it\ngggg itedtwcwcwcwc¥\n-¢¢¢=-= ò,   ,g=^P. (4) \nAccording to linear response theory and the fluctuation-dissipation theorem [26], the \nsusceptibility gc and the equilibrium correlation function ()gCt  are connected via the \nfollowing expression \n ()(0)()ggg CiC cwww =- %, (5) \nwhere ()gCw% is the Fourier-Laplace transform of ()gCt  \n5 \n  \n0()()it\nggCCtedtww¥\n-=ò% . (6) \nThe temporal correlation functions are used for the description of the temporal evolution of \nmicroscopic variables, such as the components of magnetization of fine ferromagnetic \nparticles, and its influence on the value of the same microscopic variables later. These \ncorrelations are important in equilibrium systems, because a time-invariant macroscopic \nensemble can still have non-trivial temporal dynamics microscopically. \nNext, we analyze the longitudinal correlation function ()CtP for the longitudinal \ncomponent of magnetization ()()/S utMtM =PP  and the transverse correlation function \n()Ct^ for the transverse component of magnetization ()()/S utMtM^^= . These correlation \nfunctions are defined as \n 2()(0)()(0)Ctuutu =-PPPP , (7) \nand \n ()(0)()Ctuut^^^= , (8) \nwhere ()cos()utt J =P  and ()sin()cos()uttt Jj^=  are the longitudinal and transverse \ncomponents of a unit vector ( ) sincos,sinsin,cosJjJjJ =u  along the magnetization M. \nThe brackets  denote the equilibrium ensemble averages in the four-dimensional phase \nspace { }00\n00 0 0(0),(0),,sinxy t tJJjjwJwjJ= ===== & &  [39] \n () ()\n2\n0000\n00000\n00(,,,)sinstxyxyWddddpp\nJjwwJwwjJ¥¥\n-¥-¥×=×òòòò, (9) \nwhere  \n ()()\n2200\n00(,)/001\n00(,,,)xy VkT\nstxyWZehwhwJjJjww----=  (10) \nis the stationary distribution in the spin system [19], \n ()()\n2220000 00\n000\n00(,)expsinxyxyVZddddkTppJjhwhwJwwjJ¥¥\n-¥-¥éù=---êúëûòòòò (11) \nis the partition function, 00(,)VJj  is the free energy given by Eq. (3), and the inertial \nparameter is defined as [19] 6 \n  0\n2S vM\nkTmthg= . (12) \nThe stationary solution, Eq. (10), plays an important role in various applications such as escape \nrate theory [39]. The partition function Z, Eq. (11), can be calculated analytically and is given \nby \n 2\n22(,)eZBspxshs= , (13) \nwhere \n 22(,)\n22DD BeFeFxx xsxsxss\nss- éù +-æöæö=- êú ç÷ç÷èøèø ëû (14) \nand \n 22\n0()x\nxy\nDFxeedy-=ò (15) \nis the Dawson integral.  \nThe above correlation functions have been defined in the context of equilibrium statistical \nmechanics, and therefore the Boltzmann distribution is used in the calculations. The important \nfeature of the stationary Boltzmann distribution is that it contains the integral of motion for \nundamped motion under the exponent. Hence the correlation functions are calculated by the \naverage of trajectories of magnetization over the initial values of the phase space variables (the \nequilibrium distribution does not change in time). In the other words we average over all \npossible trajectories, which are determined by the initial conditions.  \nThe behaviour of the correlation functions ()gСt defined by Eqs. (7) and (8) is \ncompletely determined by the functions ()utP and ()ut^, which must be found from the \nsolution of the ILLG equation (1) with the effective field H expressed as [40] \n \n01\nSV\nvMm¶=-¶Hu. (16) \n3. Moments for longitudinal and transverse correlation functions \nThe correlation function ()gCt  can be expanded in the Taylor series with the coefficients, \nwhich are related to the time derivatives of the correlation function ()gCt  at 0t= [24,25,27] 7 \n  0\n1()()!n\ngg\ngn\nnitCtSSn¥\n==+å , (17) \nwhere \n0()()n\ngn\nng n\ntdSiCtdt==- . For realistic functions of free energy, the coefficients of the \npower series expansion are finite, so the series converges over some range of time. The \nevaluation of the moments g\nnS for magnetization correlation functions requires only the \ncalculation of the time derivatives at t = 0 of the functions ()utP and ()ut^, viz., \n 2\n0\n0()()(0)(0)n\ng gn\nnggn n\ntdutSiuudtd\n=æö\nç÷ =--ç÷èø. (18) \nTherefore the calculation of moments is reduced to the calculation of multiple integrals (see \nEq. (9)), while both the averaged functions and the exponential distribution function are \ndetermined by the coordinates and velocities taken at time t = 0 . Thus there is no need to solve \nthe ILLG equation (1) for finding the moments. \nAt the high-frequency limit ( w®¥ ), the spectrum of the correlation functions ()gCt , \nwhich are real functions of time, can be expanded as (see Eqs. (6) and (17)) \n ()1\n0g\nn\ng n\nniSCww¥\n+\n==-å%. (19) \nMoreover, using Eqs. (5) and (19), the spectrum ()gcw  can be expanded in powers of  1w- \n \n1()g\nn\ng n\nnScww¥\n==-å . (20) \nBoth the correlation functions and the susceptibility can be analyzed using the moments. \nHowever, in practical calculations it is difficult to determine all the moments, so the correlation \nfunction and susceptibility are approximated by a finite series (the summation index in Eqs. \n(17) and (20) max nn£ ). Thus, the series approximation with a finite number of terms \ncorresponds to the correlation function only at short times Eq. (17), and corresponds to the \nsusceptibility only at high frequencies Eq. (20).  8 \n 4. Analytical expressions for the moments in the case of uniaxial anisotropy with a \nlongitudinal magnetic field  \nHere we evaluate the moments g\nnS for the circular symmetric free energy Eq. (3). The \nsimilarity between the inertial magnetization dynamics and the rotation of a rigid body (like a \nsymmetric top) [19] imposes a restriction on the correlation functions. The time correlation \nfunctions of classical ensembles of rigid molecules are even functions of time [27]. This \nrestriction means that only the even moments 2g\nnS are non-zero. \nEquation (18) gives for the moments 2 2\n000coscos S JJ =-P and \n22\n000 sincos S Jj^=  after performing the ensemble average \n () ()\n2\n02\n2coshsinh 1sinh\n2,,SBBx\ns xx x\nsxsxs+-= +-P, (21) \n 2\n2\n0coshsin 111222(,)SBx\ns xx x\nssxs^éù - æö=-+-êúç÷èøêúëû. (22) \nThe higher the order of the moment, the more complicated the analytical expression for \nthe moment becomes. Next, for simplicity, we calculate the moments 2g\nnS for n = 1 and 2 only. \nNoting that  \n 0\n0 (0)sinx u wJ=-P& , (23) \n 00\n000 (0)coscossinxy u wJjwj^=-& , (24) \nand performing the ensemble average 2\n2 (0)(0)(0)g\nggg Suuu=-= &&& , we obtain the second-\norder moments  \n 2\n2\n2 2coshsinh 111222(,)SBx\ns xx x\nhssxséù - æö=-+- êúç÷èøêúëûP, (25) \n 2\n2\n2 2coshsin 111422(,)SBx\ns xx x\nhssxs^éù - æö=-++ êú ç÷èø êúëû. (26) \nThe weighted sum of the moments 2SP and 2S^ is given by \n 22 2121\n333SSh^+=P. (27) 9 \n The sum rule in Eq. (27) does not depend on the anisotropy s and field x parameters and \ncoincides with the sum rule for isotropic media [28]. \nTo evaluate the moments 4SP and 4S^ we need the time derivatives (0)uP&&  and (0)u^&&  \n(see Eq. (18)), which depend on the time derivative of the angular velocity components, xw \nand yw, viz., \n ()200\n00 (0)sincosxx u wJwJ=--P& && , (28) \n ()\n()\n200\n0000\n20000\n00000(0)coscossincos\ncotsinsincsccos.xx\nxyyyu wJjwJj\nwwJjwjwJj^ =-\n---& &&\n& (29) \nThe derivation of the expressions for xw& and yw& is given in Appendix A. By substituting Eqs. \n(A2) and (A3) for 0\nxw& and 0\nyw& into Eqs. (28) and (29), and performing the ensemble average \n2\n4 (0)g\ng Su=&& , we obtain the fourth moments, viz. \n 2222222 4\n4 4322\n22 22\n222(7/)2112/ 1121644\n1113cosh1sinh42242,(,)S\nBaaa\naaxhtxhth xshssst\nhh xxxsxxstssst\nxsé --+=--+++- êë\nù æöæö-+---+- ú ç÷ç÷èøèø ú +ú\nú\nûP\n (30) \n 2222222 4\n4 4322\n22 22\n222(7/)112/ 17\n416442\n11132coshsinh42242,(,)S\nBaaa\naaxhtxhth x\nhssst\nhh xxxsxxstssst\nxs^ é --+=--++ êë\nù æöæö+-+--+ ú ç÷ç÷èøèø ú +ú\nú\nû (31) \nwhere 222(1)ahah=+ . \nHigher-order moments can be obtained by analogy with the previous derivations. Due to \nthe length of the calculation, only 6gS and 8SP for the undamped ( 0a=) rotating \nmagnetization are given in Appendix B. Moreover, for 8SP we also put 0s=.  10 \n 5. Results and discussion  \nThe moments presented are not only a tool for estimating the correlation functions, but \nhave important physical meaning. In accordance with the Kramers-Kronig relation the moment \n0gS gives the static susceptibility 0 (0)g\ng S c¢= . For the physical interpretation of higher-order \nmoments 2g\nnS we consider the spectral  moments defined as [26,28] \n 21\n2\n02lim(/)gn\nngMid\ntwcwtwp¥\n-\n®¥¢¢ =+ ò, (32) \nwhere ()gcw¢¢-  is the imaginary part of the susceptibility given by Eq. (20). These spectral \nmoments 2g\nnM  coincide with the moments 2g\nnS under certain conditions. By substituting Eq. \n(20) into Eq. (32) we have 22ggMS= , while 2g\nnM  is not finite for 1n>. We note that all odd \nspectral moments 21g\nnM- are vanishing. However, if 20gS=, then 44ggMS=  and all 2g\nnM  are \nnot finite for 2n>, etc. Hence, it follows that the absorption coefficient ()()ggAwwcw ¢¢:  \ndecreases with increasing frequency as 2()n\ngAww-:  if 2g\nnM  is finite. Equation (32) for 1n= \nis the Gordon sum rule, which determines the finite value of the integral absorption (the area \nunder the absorption curve ()gAw). Gordon’s sum rule plays an important role in the theory \nof dielectrics. This rule as well as the spectral moments is a useful tool for extracting molecular \nparameters from the analysis of measured molecular spectra [41,42]. The same idea can be \nrealized in the case of magnetic media. For example, Eqs. (25), (26) and (32) for 1n= allow \none to estimate the free energy parameters, as well as the inertial parameter, by analyzing the \ntotal absorption in a magnetic medium under various conditions. \nIn the non-inertial case the static susceptibility is not changed. However, the odd \nmoments 21g\nnS+ are not vanishing now. For example, the first-order moments \n1 (0)(0)g\ngg Siuu=- &  are \n 2\n1 23coshsinh33 21242(,)NiSBxxxx s\ntssxsæö æö- ç÷ ç÷èøç÷ =---+\nç÷ç÷èøP, (33) 11 \n  2\n1 23coshsinh33 21442(,)NiSBxxxx s\ntssxs^æö æö- ç÷ ç÷èøç÷ =--+\nç÷ç÷èø, (34) \nwhere Nt is the free diffusion time \n 22\n0211\n2S\nNvM\nkTmtaha\nat g a=+=+. (35) \nWe have from Eq. (20) for the non-inertial case \n 2()(0)/()gg iCO cwww-=+&. (36) \nHence, the absorption coefficient ()()ggAwwcw ¢¢:  in the limit w®¥  tends to a constant \nand the area under the absorption curve is infinite in the non-inertial case. This is not the case \nfor the inertial magnetization dynamics (10gS=), where the absorption coefficient tends to \nzero at high frequencies leading to the correct physical result, namely, to a finite area under the \nabsorption curve (see Eq. (32) for 1n=). This phenomenon is well known in the theory of \ndielectrics, where considering the inertia of polar molecules ensures the transparency of the \nmedium at high frequencies. In the limit 0 a®  the first-order moments vanish 1 0gS® . \nThe non-inertial analogs of 2gS (compare with Eqs. (25) and (26)) are \n 422\n2 232\n22\n21915624822\n17152cosh1sinh242\n(,)NS\nBxxxstssss\nxxsxxxsss\nxsæ=--+-+- çè\nö æöæö-+--+ ÷ ç÷ç÷èøèø ÷+÷\n÷\nøP\n, (37) \n 4222\n2 22322\n22\n222139153288222\n617154coshsin\n(1\n1h242\n,)NS\nBxxxxstasssas\nxsxsxxxsaass\nxs^æ -=-+--+- ç\nè\nö æöæö-+++-+- ÷ ç÷ç÷èøèø ÷+÷\n÷æ\nè\nøö\nç÷ø\n. (38) \nIn the limit 0 a®  the moment 2SP vanishes, but the moment 2S^ does not.  \nAs we noted, the inertia does not affect the static susceptibility 0gS. On the contrary all \nmoments 2g\nnS with 1n³ depend on the inertial relaxation time t directly. Thus, the inertial 12 \n relaxation time t affects the area under the absorption curve 2gS via the scaling factor h. \nMoreover, all moments 2g\nnS with 2n³ depend on the damping parameter a. \nIn the limit 0 x®  and 0 s® , the moments tend to the isotropic case  \n 00 1/3 SS^==P, (39) \n 2\n22 1/(3) SS h^==P, (40) \nand \n ( )224\n44 2//(3) SSahth^==+P. (41) \nThe expressions for moments in the cases of a negligibly weak external field ( sx? ) (such \nthat the external field can be neglected) and in the opposite case of a very strong external field \n(xs? ) (such that the internal anisotropic potential can be neglected) are presented in \nAppendix C. In the latter case, the moments coincide with the moments derived in Ref. [43] \nfor a symmetric top molecule, which has an electric dipole moment m directed along the axis \nof symmetry of the top and rotates in an external uniform electric field E after the following \nsubstitution \n 2/(2)IkT h® ,   22//2zII ht ® ,   /()EkT xm® , (42) \nwhere I and zI are principal moments of inertia.  \nThe correlation functions can be calculated directly using Eqs. (7)-(12), but this requires \nnumerical solution of Eq. (1) and advanced numerical integration methods since the integrals \nin Eq. (9) are averages over all possible initial conditions. On the other hand, the correlation \nfunctions are well approximated by Eq. (17) at times shorter than /1th<. A comparison \nbetween numerical calculations and approximations of the correlation functions corresponding \nto inertial undamped ( 0 a® ) magnetization dynamics in a strong uniform external field \n(xs? ) is shown in Figs. 2 and 3. Thus, the moments can be used as a fairly simple and \nsatisfactory estimation of the numerical integration of the correlation functions at short times, \nthereby enabling the capture of fast oscillations. These fast oscillations are caused by the \ninertial term in the ILLG equation and are absent in the non-inertial case. A similar picture can \nbe seen in the magnetization trajectories, i.e. the ILLG equation predicts fast nutational \noscillations accompanying precessional motion of the magnetization, while the LLG equation \ndescribes precessional motion only [44]. \n 13 \n  \nFIG. 2. Longitudinal correlation function ()CtP for the field parameter 5x= (xs?  and \n0 a® ) and various values of the inertial relaxation time, namely: (blue) /10ht = , (orange) \n/15ht = , and (red) /20ht = .  Solid lines: numerical calculations Eqs. (1), (7), (9)-(12); \nsymbols: series approximation Eq. (17) with max 4 n=. \n \n \nFIG. 3. Transverse correlation function ()Ct^ for the inertial relaxation time /15ht = , \n0 a®  and various values of the field parameter xs? , namely: (blue) 5x=, (orange) \n15x= , and (red) 25x= . Solid lines: numerical calculations Eqs. (1), (8), (9)-(12); symbols: \nseries approximation Eq. (17) with max 3 n=. \n6. Conclusions  \nWe have presented the expressions for the moments up to the sixth order for the case of \na cyclic symmetry of the free energy of ferromagnetic nanoparticles. Magnetic particles with \nuniaxial anisotropy in an external field directed perpendicular to the easy axis [31], with \nbiaxial [31,45,46] and cubic anisotropies [47–50] can be investigated using the method \npresented. However, if circular symmetry of the magnetocrystalline anisotropy is broken, the \n14 \n moments can only be found in quadrature, and the expressions are quite complicated. In this \ncase, it is preferable to calculate the moments numerically.  \nBecause of the difficulties in extracting these moments from experimental spectral band \nshapes, it is unlikely that the complete expressions for moments higher than the fourth order \nwill be directly useful. However, knowledge of the second and fourth moments are useful for \ninvestigation of the static susceptibility, area under the absorption curve, and extraction of the \npotential parameters from the spectra. \nThe method presented can also be employed to describe the relaxation dynamics in the \nprocesses of magnetization or magnetization reversals. These dynamics correspond to a process \nof reaching thermodynamic equilibrium, and this process is described by correlation functions. \nFor example, the variation of the external field from +DHH  to H (DHH= ) results in a \ndisturbance of the equilibrium state of the spin system and the system relaxes to a new \nequilibrium state. Magnetic relaxation determines the linewidth of ferromagnetic, \nantiferromagnetic, and nutation resonances. Bloch [31] introduced the phenomenological \napproach that the evolution of the magnetization towards equilibrium was exponential, \nhowever embodying two distinct time constants: TP for the longitudinal component and T^ for \nthe transverse component. There is an intuitive connection between the time evolution of \ncorrelation functions and the time evolution of macroscopic systems: on average, the \ncorrelation function evolves in time in the same manner. The regression of microscopic thermal \nfluctuations at equilibrium follows the macroscopic law of relaxation of small nonequilibrium \ndisturbances.  As the values of microscopic variables separated by large timescales, should be \nuncorrelated, the evolution in time of a correlation function can be viewed from a physical \nstandpoint as the system gradually forgetting the initial conditions.  This concept is reflected in \nthe van Vleck–Weisskopf model, in which the longitudinal and transverse correlation functions \nare expressed as [25] \n ()()expun tCtCtTæö=- ç÷ç÷èøPP\nP,   ()()expun tCtCtT^^\n^æö=- ç÷\nèø, (43) \nwhere the correlation functions ()unCtP  and ()unCt^  correspond to the undamped dynamics of \nmagnetization. The model is commonly used to describe orientational relaxation in gases and \nliquids by considering inertial effects [25,51]. For the correlation functions given by Eq. (43) \nthe moment 1gS is not vanishing, viz., 15 \n  1 (0)/gun\ngg SiCT= . (44) \nHence the area under the absorption curve ()gwcw¢¢  is infinite for the van Vleck–Weisskopf \nmodel. The shortcomings of the van Vleck–Weisskopf model have been overcome in a more \ncomplex J-diffusion model [25,43]. For the J-diffusion model the first-order moment is \nvanishing.  \nNext application of the method of moments can be found in the calculation of the \ncumulants, which are used in statistical physics to describe complex distributions [52,53]. The \ncumulants g\nnK can be expressed via moments and vice versa. A very convenient form in terms \nof determinants reads [52] \n 1\n21\n321 1\n123100\n10\n21(1)det1\n11\n12g\ngg\nggg\ngn\nn\ngggg\nnnnnS\nSS\nSSSK\nnnSSSS-\n---æö\nç÷\nç÷\nç÷æöç÷ç÷=- ç÷ èø\nç÷\nç÷\nç÷ --æöæöç÷ ç÷ç÷ ç÷èøèø èøL\nL\nL\nMMMMO\nL,  \nwhere !\n()!!n n\nm nmmæö= ç÷- èø are the binominal coefficients. We anticipate that future work will \ndemonstrate the application of the moments for cumulant calculation in magnetic materials, \nwhere magnetization demonstrates inertial dynamics. Moreover, the presented method can also \nbe effectively used for non-Markovian systems, especially if stationary solutions of the master \nequation for the distribution function can be found in analytical form. \nIn summary, we note that it is difficult to evaluate all the moments, since the analytical \nexpression for the moments become more complicated as one goes to higher-order moments. \nThus, the correlation function and susceptibility are approximated by finite series, which are \nvalid only at short times. Even if the time series converged at larger times, one would seek an \nalternative method of evaluation of the correlation function for long times. However, the \nmoment method sharply reduces the complexity of correlation functions analysis. Indeed, the \nprecise expressions of these functions in Eqs. (7) and (8) necessitate the solution of the ILLG \nequation followed by averaging the dynamic functions over the initial conditions. However, \nthe moment method allows one to simplify this, i.e., one needs to find the average values of \nthe correlation functions over coordinates and velocities at the initial times only. Moreover, the 16 \n proposed method provides a basis to calculate the static susceptibility, integral reorientation \ntime, and other characteristics of ferromagnetic nanosystems. It is possible to investigate the \ndependence of these characteristics on the free energy parameters of ferromagnetic \nnanoparticles. \nAcknowledgments \nWe thank Y.P. Kalmykov for useful comments and suggestions. S.T. and A.T. express their \nacknowledgment to the Foundation for the Advancement of Theoretical Physics and \nMathematics “BASIS” (Grant ID 22-1-1-28-1). \nAppendix A: Time derivatives of the angular velocity components \nThe time derivatives of the angular velocity components, xw& and yw&, can be determined \nfrom the ILLG equation (1). This equation is used since it provides the relation between the \nhigh-order derivatives of coordinates and velocities. To find xw& and yw& we express the vector \nu and its time derivatives occurring in Eq. (1) in spherical polar coordinates {,,}reeeJj  as \n 1\n0\n0æö\nç÷=ç÷ç÷èøu ,   0\nsinJ\njJæö\nç÷=ç÷ç÷èøu & &\n&,   222\n2sin\ncossin\n2cossinJjJ\nJjJJ\nJjJjJæö--\nç÷=-ç÷\nç÷ +èøu&&\n&&& &&\n&&&&. (A1) \nBy substituting all these vectors into the vector Eq. (1), and cinsidering Eqs. (3) and (16), we \nderive two coupled equations expressing the initial values of the time derivatives of the angular \nvelocity components 0(0)xxww=&&  and 0(0)yyww=&&  via initial values of the angular velocity \ncomponents 00(0),(0)sin(0)xywJwjJ==& &  and angles 00,Jj  [19] \n () ( )\n20000\n000 211cot2cossin2xyyxawwwJwsJxJtth=+--+& , (A2) \n 00000\n01cotyxxyyawwwwJwtt=---& . (A3) \nAppendix B: High order moments  \nFor the calculation of the moments 6SP and 6S^, we need the time derivatives (0)uP&&&  and \n(0)u^&&& . By using Eqs. (A2) and (A3), we obtain for (0)uP&&&  and (0)u^&&& : 17 \n  () () ( )\n( )\n320000\n00 2\n002\n000 221(0)sinsin\nsin2sin5cos1,xxxy\nxxu wJwwwJt\nxswJwJJhh=++\n++-P&&&\n (B1) \n   ()()( )\n220000\n000 2\n020\n000000 2\n0202\n00000000 2221(0)sincoscos\n1(4sin1)coscossinsinsin2\n1cossinsin(5sin1)coscoscossin.xyyx\nxy\nxyu wwwjwJjt\nxwJjwJjJjht\nsssJJjwJJjwJjthhh^æö=++-ç÷èø\næö+-+- ç÷èø\n-+-+&&&\n (B2) \nBy using Eqs. (B1) and (B2) and performing the ensemble averaging, we obtain the moments \n,2\n6, (0) Su^\n^ =P\nP&&& , viz., \n ,,\n,6,\n6coshsinh\n(,)c sA ASABxxhxs^^\n^-^ éù +=+ êúëûPP\nPP, (B3) \nwhere \n  264242\n4322\n4424224222222\n424423222222\n24224222\n2426963453279692161625612864323232\n11711,416883288164\n2791511\n1616648244cA\nAxxxxxxsssssss\nhhxhhhxhxhhxhs\ntststtststststt\nxxxshhhxssstt^\n^-++-+--+-\n++-++-+--\n-+-+++-+=\n=2\n22\n53342232\n32422227,84\n53139313,8128323232488168sAhs\nstt\nxxxxxxshxhxhxhx\nsssssttstst^+\n--+-+= -++-- (B4) \n ( ) ( )\n( ) ( )4222\n2\n224\n42\n222\n2\n32\n223486132\n32\n3276\n2,8\n992,822\n332.21644cc\nssA\nA\nAA\nAAxxsssss\ns\nhxss h\ntst\nxshsst\nxxxshx\nst^\n^\n^+-+-+\n+-\n++-\n+++-\n--+--=\n=\n=P\nP\nP (B5) \nIn the isotropic case ( 0 s®  and 0 x® ) Eqs. (C8) and (C9) are simplified to yield  \n ( )44226\n66 4 )/ 6/(3 / SS hthth^++ ==P. (B6) 18 \n Here we also present the analytical expression for the moment ( )2(4)\n8 (0) Su=P\nP  for the \ncase 0s=, which is used in the calculations in Fig. 2. The moment 8SP can be obtained by \nanalogy with the derivation of the moment 6SP by noting that \n ( )\n( ) ( )\n(4)222\n2\n2\n21(0)sincos\n2sinsin22cos2.xyxx\nxxxyyxxu wwwJwJt\nxwwwwwJwJwJhæö=+++ç÷èø\n++++P&\n&&& (B7) \nThus, by using Eqs. (A2) and (A3), substituting Eq. (B7) into Eq. (18), and performing the \nensemble averaging, we obtain \n \n( )\n24\n2\n8 824\n2246\n211\n224618154246\n4384389012coth.Shhxhtt\nhhhhxxxxtttt--ì=+-+ í\nî\nü æö æö ï+--+-+-ç÷ ý ç÷èø ï èø þP\n (B8) \nAnalytical expressions for the moments of higher orders are quite complex, and their derivation \nis quite laborious. However, they can still be calculated numerically without much difficulty. \nAppendix C: Weak ( sx? ) and strong ( xs? ) external fields \nIf magnetic particles are placed in a very strong external field ( xs? ) (such that the \ninternal anisotropic potential can be neglected), the free energy in Eq. (3) becomes \n ()/()cosVkTJxJ =- . (C1) \nThe moments reduce to the following equations \n 22\n01cothS xx-=-+P, (C2) \n 12\n0 coth S xxx^--=- , (C3) \n 2 22coth1Sxx\nhx-=P, (C4) \n 2\n2 22coth1\n2Sxxx\nhx^ -+= , (C5) \n 2\n4 4211124coth Sahxhxtxéù æö æö=--- êú ç÷ ç÷èø èø ëûP, (C6) 19 \n  22\n4 4221114coth22Saahh xxhtxtx^éù æö æö æö=++-- êú ç÷ ç÷ ç÷èø êú èø èø ëû, (C7) \n ( )\n224\n211\n6 62241843048coth Shhhxxxxhttt--ìü æö ïï=-+++-+- íý ç÷ïï èø îþP, (C8) \n( )\n4242\n2211\n6 64242156412032416coth8Shhhhxxxxxhtttt^--ìü æö æö ïï=++--++-- ç÷ íý ç÷èø ïï èø îþ. (C9) \nIn the negligibly weak external field ( 0 x® ), the free energy in Eq. (3) becomes \n 2()/()cosVkTJsJ =- . (C10) \nThe equations for the moments simplify to yield  \n 0111\n2 ()DS\nF s ssæö=-ç÷\nèøP, (C11) \n 011124 ()DS\nF s ss^æö=+-ç÷\nèø, (C12) \n 2 211124 ()DS\nF hs ssæö=+- ç÷\nèøP, (C13) \n 2 211128 ()DS\nF hs ss^æö=-+ ç÷\nèø, (C14) \n 222\n4 421122/ 11124448 ()DS\nFaahshtshsts ssö +- ææö=---+++ ÷ çç÷èèø øP, (C15) \n 222\n4 424112/ 111214416 ()DS\nFaahsht\nhsts ss^ö -+ ææö=++-+ ÷ çç÷èèø ø, (C16) \n \n()422\n2\n26 4\n26 2\n41\n16\n,2798116111621281124\n279564(448)(/)4(/)\nDS\nFhhsssststs\nssshtht\nsshé æö æö+++++-++ ê ç÷ ç÷èø èø ë\n-+-++-+=\nù\nú\núûP\n (C17) 20 \n  \n( )\n()\n24\n24\n2246 6279411811\n.386411212 2\n27911\n168417(/)4(/)3\nDS\nFshhhsststs\nssshtht\nss^ é æöæö-----+-+- ç÷ç÷ êèøèø ë\n+++-++=\nù\nú\núû+ (C18) \n \nReferences \n[1] K. Neeraj et al., Inertial Spin Dynamics in Ferromagnets , Nat Phys 17, 245 (2021). \n[2] V. Unikandanunni, R. Medapalli, M. Asa, E. Albisetti, D. Petti, R. Bertacco, E. E. \nFullerton, and S. Bonetti, Inertial Spin Dynamics in Epitaxial Cobalt Films , Phys Rev \nLett 129, 237201 (2022). \n[3] M.-C. Ciornei, J. M. Rub í, and J.-E. Wegrowe, Magnetization Dynamics in the Inertial \nRegime: Nutation Predicted at Short Time Scales , Phys Rev B 83, 020410 (2011). \n[4] M. F ähnle, D. Steiauf, and C. Illg, Generalized Gilbert Equation Including Inertial \nDamping: Derivation from an Extended Breathing Fermi Surface Model , Phys Rev B \n84, 172403 (2011). \n[5] J.-E. Wegrowe and M.-C. Ciornei, Magnetization Dynamics, Gyromagnetic Relation, \nand Inertial Effects , Am J Phys 80, 607 (2012). \n[6] S. Bhattacharjee, L. Nordstr öm, and J. Fransson, Atomistic Spin Dynamic Method with \nBoth Damping and Moment of Inertia Effects Included from First Principles , Phys Rev \nLett 108, 057204 (2012). \n[7] E. Olive, Y. Lansac, and J. E. Wegrowe, Beyond Ferromagnetic Resonance: The \nInertial Regime of the Magnetization , Appl Phys Lett 100, 192407 (2012). \n[8] D. B öttcher and J. Henk, Significance of Nutation in Magnetization Dynamics of \nNanostructures , Phys Rev B 86, 020404 (2012). \n[9] T. Kikuchi and G. Tatara, Spin Dynamics with Inertia in Metallic Ferromagnets , Phys \nRev B 92, 184410 (2015). \n[10] E. Olive, Y. Lansac, M. Meyer, M. Hayoun, and J.-E. Wegrowe, Deviation from the \nLandau-Lifshitz-Gilbert Equation in the Inertial Regime of the Magnetization , J Appl \nPhys 117, 213904 (2015). \n[11] J.-E. Wegrowe and E. Olive, The Magnetic Monopole and the Separation between \nFast and Slow Magnetic Degrees of Freedom , Journal of Physics: Condensed Matter \n28, 106001 (2016). 21 \n [12] D. Thonig, O. Eriksson, and M. Pereiro, Magnetic Moment of Inertia within the \nTorque-Torque Correlation Model , Sci Rep 7, 931 (2017). \n[13] R. Mondal, M. Berritta, A. K. Nandy, and P. M. Oppeneer, Relativistic Theory of \nMagnetic Inertia in Ultrafast Spin Dynamics , Phys Rev B 96, 024425 (2017). \n[14] R. Mondal, M. Berritta, and P. M. Oppeneer, Generalisation of Gilbert Damping and \nMagnetic Inertia Parameter as a Series of Higher-Order Relativistic Terms , Journal of \nPhysics: Condensed Matter 30, 265801 (2018). \n[15] Y. Li, V. V. Naletov, O. Klein, J. L. Prieto, M. Mu ñoz, V. Cros, P. Bortolotti, A. \nAnane, C. Serpico, and G. de Loubens, Nutation Spectroscopy of a Nanomagnet \nDriven into Deeply Nonlinear Ferromagnetic Resonance , Phys Rev X 9, 041036 \n(2019). \n[16] S. Giordano and P.-M. D éjardin, Derivation of Magnetic Inertial Effects from the \nClassical Mechanics of a Circular Current Loop , Phys Rev B 102, 214406 (2020). \n[17] M. Cherkasskii, M. Farle, and A. Semisalova, Nutation Resonance in Ferromagnets , \nPhys Rev B 102, 184432 (2020). \n[18] S. V. Titov, W. T. Coffey, Y. P. Kalmykov, and M. Zarifakis, Deterministic Inertial \nDynamics of the Magnetization of Nanoscale Ferromagnets , Phys Rev B 103, 214444 \n(2021). \n[19] S. V. Titov, W. T. Coffey, Y. P. Kalmykov, M. Zarifakis, and A. S. Titov, Inertial \nMagnetization Dynamics of Ferromagnetic Nanoparticles Including Thermal \nAgitation , Phys Rev B 103, 144433 (2021). \n[20] M. Cherkasskii, I. Barsukov, R. Mondal, M. Farle, and A. Semisalova, Theory of \nInertial Spin Dynamics in Anisotropic Ferromagnets , Phys Rev B 106, 054428 (2022). \n[21] K. Neeraj, M. Pancaldi, V. Scalera, S. Perna, M. D’Aquino, C. Serpico, and S. Bonetti, \nMagnetization Switching in the Inertial Regime , Phys Rev B 105, 054415 (2022). \n[22] S. V. Titov, W. J. Dowling, and Y. P. Kalmykov, Ferromagnetic and Nutation \nResonance Frequencies of Nanomagnets with Various Magnetocrystalline \nAnisotropies , J Appl Phys 131, 193901 (2022). \n[23] J. Anders, C. R. J. Sait, and S. A. R. Horsley, Quantum Brownian Motion for Magnets , \nNew J Phys 24, 033020 (2022). \n[24] A. I. Burshtein and S. I. Temkin, Spectroscopy of Molecular Rotation in Gases and \nLiquids  (Cambridge University Press, Cambridge, 1994). \n[25] Y. P. Kalmykov and S. V. Titov, A Semiclassical Theory of Dielectric Relaxation and \nAbsorption: Memory Function Approach to Extended Rotational Diffusion Models of 22 \n Molecular Reorientations in Fluids , in Advances in Chemical Physics , Vol. 87 (2007), \npp. 31–123. \n[26] D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions  \n(CRC Press, 2018). \n[27] A. G. S. Pierre and W. A. Steele, Some Exact Results for Rotational Correlation \nFunctions at Short Times , Mol Phys 43, 97 (1981). \n[28] R. G. Gordon, Molecular Motion and the Moment Analysis of Molecular Spectra in \nCondensed Phases. I. Dipole ‐Allowed Spectra , J Chem Phys 39, 2788 (1963). \n[29] A. Abragam, The Principles of Nuclear Magnetism (1983, Oxford University Press, \nUSA) , Vol. 28 (1961). \n[30] Y. P. Kalmykov and S. V. Titov, Spectral Moments of the Rotational Correlation \nFunctions for the First- and Second-Rank Tensors of Asymmetric Top Molecules , Mol \nPhys 98, 1907 (2000). \n[31] W. T. Coffey, Y. P. Kalmykov, and S. V. Titov, Thermal Fluctuations and Relaxation \nProcesses in Nanomagnets  (World Scientific, Singapore, 2020). \n[32] A. Prabhakar and D. D. Stancil, Spin Waves: Theory and Applications  (Springer US, \nBoston, MA, 2009). \n[33] W. T. Coffey, D. S. F. Crothers, J. L. Dormann, L. J. Geoghegan, Yu. P. Kalmykov, J. \nT. Waldron, and A. W. Wickstead, Effect of an Oblique Magnetic Field on the \nSuperparamagnetic Relaxation Time , Phys Rev B 52, 15951 (1995). \n[34] W. T. Coffey, D. S. F. Crothers, Yu. P. Kalmykov, and S. V. Titov, Precessional \nEffects in the Linear Dynamic Susceptibility of Uniaxial Superparamagnets: \nDependence of the ac Response on the Dissipation Parameter , Phys Rev B 64, 012411 \n(2001). \n[35] Yu. P. Kalmykov and S. V. Titov, Complex Magnetic Susceptibility of Uniaxial \nSuperparamagnetic Particles in a Strong Static Magnetic Field , Physics of the Solid \nState 40, 1492 (1998). \n[36] E. E. C. Kennedy, Relaxation Times for Single-Domain Ferromagnetic Particles , in \nAdv. Chem. Phys. , Vol. 112 (2007), pp. 211–336. \n[37] H. Fukushima, Y. Uesaka, Y. Nakatani, and N. Hayashi, Switching Times of a Single-\nDomain Particle in a Field Inclined off the Easy Axis , J Appl Phys 101, 013901 \n(2007). 23 \n [38] H. Fukushima, Y. Uesaka, Y. Nakatani, and N. Hayashi, Dependence of Prefactor on \nthe Angle between an Applied Field and the Easy Axis for Single-Domain Particles , J \nMagn Magn Mater 323, 195 (2011). \n[39] W. T. Coffey and Yu. P. Kalmykov, The Langevin Equation, 4th Ed  (World Scientific, \nSingapore, 2017). \n[40] D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of \nAngular Momentum  (Singapore: World Scientific, 1998). \n[41] S. R. Jain and S. Walker, Far-Infrared Absorption of Some Organic Liquids , Journal \nof Physical Chemistry 75, 2942 (1971). \n[42] A. V. Naumov, Yu. G. Vainer, M. Bauer, S. Zilker, and L. Kador, Distributions of \nMoments of Single-Molecule Spectral Lines and the Dynamics of Amorphous Solids , \nPhys Rev B 63, 212302 (2001). \n[43] S. V. Titov, Y. P. Kalmykov, and W. T. Coffey, Extended Rotational Diffusion and \nDielectric Relaxation of Symmetrical Top Molecules in a dc Electric Field , J Chem \nPhys 118, 209 (2003).  \n[44] S. V. Titov, W. J. Dowling, Y. P. Kalmykov, and M. Cherkasskii, Nutation Spin \nWaves in Ferromagnets , Phys Rev B 105, 214414 (2022). \n[45] B. Ouari, S. V. Titov, H. el Mrabti, and Y. P. Kalmykov, Nonlinear Susceptibility and \nDynamic Hysteresis Loops of Magnetic Nanoparticles with Biaxial Anisotropy , J Appl \nPhys 113, 053903 (2013). \n[46] Y. P. Kalmykov, W. T. Coffey, S. V. Titov, J. E. Wegrowe, and D. Byrne, Spin-\nTorque Effects in Thermally Assisted Magnetization Reversal: Method of Statistical \nMoments , Phys Rev B 88, 144406 (2013). \n[47] A. J. Newell, Superparamagnetic Relaxation Times for Mixed Anisotropy and High \nEnergy Barriers with Intermediate to High Damping: 1. Uniaxial Axis in a 〈\n001〉\n \nDirection , Geochemistry, Geophysics, Geosystems 7, Q03015 (2006). \n[48] A. J. Newell, Superparamagnetic Relaxation Times for Mixed Anisotropy and High \nEnergy Barriers with Intermediate to High Damping: 2. Uniaxial Axis in a 〈\n111〉\n \nDirection , Geochemistry, Geophysics, Geosystems 7, Q03016 (2006). \n[49] Y. P. Kalmykov, S. V. Titov, and W. T. Coffey, Longitudinal Complex Magnetic \nSusceptibility and Relaxation Time of Superparamagnetic Particles with Cubic \nMagnetic Anisotropy , Phys Rev B 58, 3267 (1998). 24 \n [50] Y. P. Kalmykov, Y. L. Raikher, W. T. Coffey, and S. V. Titov, Precession-Aided \nMagnetic Stochastic Resonance in Ferromagnetic Nanoparticles with Cubic \nAnisotropy , Phys Rev B 71, 012415 (2005). \n[51] R. E. D. McClung, On the Extended Rotational Diffusion Models for Molecular \nReorientation in Fluids , Advances in Molecular Relaxation and Interaction Processes \n10, 83 (1977). \n[52] H. Risken, The Fokker-Planck Equation  (Springer, Berlin, 1989). \n[53] E. Barkai, R. Silbey, and G. Zumofen, Lévy Distribution of Single Molecule Line \nShape Cumulants in Glasses , Phys Rev Lett 84, 5339 (2000). \n "    },    {        "title": "2211.13528v1.Unravelling_the_role_of_Sm_4f_electrons_in_the_magnetism_of_SmFeO__3_.pdf",        "content": "Unravelling the role of Sm4felectrons in themagnetismof SmFeO 3\nDanila Amoroso,1Bertrand Dup´ e,2, 1and Matthieu J. Verstraete3,∗\n1Nanomat/Q-mat/CESAM, Universit´ e de Li` ege, B-4000 Sart Tilman, Belgium\n2Fonds de la Recherche Scientique (FNRS-FRS), B-1000 Brussels, Belgium\n3Nanomat/Q-mat/CESAM, Universit´ e de Li` ege, and European Theoretical Spectroscopy Facility, B-4000 Sart Tilman, Belgium\nMagnetic rare-earth orthoferritesRFeO 3host a variety of functional properties from multiferroic-\nity and strong magnetostriction, to spin-reorientation transitions and ultrafast light-driven manip-\nulation of magnetism, which can be exploited in spintronics and next-generation devices. Among\nthese systems, SmFeO 3is attracting a particular interest for its rich phase diagram and the high\ntemperature Fe-spin magnetic transitions, which combines with a very low temperature and as yet\nunclear Sm-spin ordering. Various experiments suggest that the interaction between the Sm and Fe\nmagnetic moments (further supported by the magnetic anisotropy), is at the origin of the complex\ncascade of transitions, but a conclusive and clear picture has not yet been reached. In this work, by\nmeans of comprehensiverst-principles calculations, we unravel the role of the magnetic Sm ions\nin the Fe-spin reorientation transition and in the detected anomalies in the lattice vibrational spec-\ntrum, which are a signature of a relevant spin-phonon coupling. By including both Sm-felectrons\nand non-collinear magnetism, wend frustrated and anisotropic Sm interactions, and a large mag-\nnetocrystalline anisotropy mediated by the SOC of the Sm-4felectrons, which drive the complex\nmagnetic properties and phase diagram of SmFeO 3.\nI. INTRODUCTION\nSamarium ferrite, SmFeO 3(SFO), belongs to the fam-\nily of magnetic perovskite oxides of the typeRMO3, with\nR= rare-earth andM= Cr, Mn or Fe (see Ref. [ 1] and\nreferences therein). These compounds host two dierent\nmagnetic sub-lattices at theAandBsites of theABO 3\nperovskite structure, giving rise to competing magnetic\ninteractions. This is of fundamental interest for the\nphysics of magnetism, and also has potential for appli-\ncations in functional materials and spintronics [ 2]. Two\ncentral phenomena are observed in these perovskite sys-\ntems: a temperature-dependent spin reorientation (SR)\nprocess and/or a magnetization compensation and rever-\nsal, which shows the importance of the entropy contribu-\ntion in the stabilization of magnetic state. In magnetic\nmaterials, heat is absorbed in magnonic excitations: the\npresence of rare-earth elements with high magnetic mo-\nments opens up the possibility to reach large magneto-\nelectric eect [ 3].\nAmong these compounds, SFO undergoes various\nphase transitions and shows fast magnetic switching [ 4–\n6]: at the N´ eel temperature (T N≃670 K), iron spins\norder antiferromagnetically along thea-axis of thePbnm\ncrystal structure, with an additional, weak ferromagnetic\n(wFM) component along thec-axis due to the spin cant-\ning; between 450 K and 480 K (T SR) a rotation of the\neasy axis associated to the wFM takes place, with a re-\norientation of the iron spins from thea-axis to thec-\naxis; below 100 K the net magnetization associated to the\ncanted spins monotonically decreases and reverses sign at\nvery low temperature, passing through a compensation\npoint (T comp≃4 K) with zero magnetization. Such a∗Matthieu.Verstraete@uliege.bebehavior at low temperature is interpreted as the appear-\nance of weak ferromagnetism associated to the Sm-spin\nsub-lattice, which would be opposite in direction with\nrespect to the Fe-wFM. Some experimental attempts to\ndetermine SFO magnetization at low temperature are\nreported, but a precise characterization of the Sm-spin\nproperties remains lacking. Diculties include the poor\nstability of the expected Sm-ordering and the strong Sm\nabsorption cross section for neutrons [ 7–9]. A so-called\ncluster-glass state (a spin-glass behaviour associated to\na magnetostatic excitation through a spin-phonon or a\nspin-lattice interaction) is also reported in the tempera-\nture region between 100 K and 200 K, where the net mag-\nnetization reaches its saturation [ 10]. Because of its high\nTSRand its rich phase diagram, SFO has attracted par-\nticular attention for exploitation in technological devices,\nand for property engineering ranging from magnetization\nand ferroelectricity enhancement by strain or oxygen va-\ncancies [ 11,12], to nanostructure fabrication [ 13,14] and\nSm-site doping [ 15,16]. Interestingly, SFO has been also\ninvestigated for gas sensing [ 17,18], with a recent pro-\nposal of an application for non-invasive colorectal cancer\nscreening [ 19].\nSimilarly to the case of otherRFeO 3systems, such\nas NdFeO 3[20,21] or TmFeO 3[22], the SR transition\nin SFO is considered to be strongly dependent on the\nmagnetic anisotropy related to theR-4felectrons. Nev-\nertheless, theoretical studies aimed at understanding the\nmicroscopic mechanisms driving the spin reorientation\ntransition in SFO remain limited due to the structural,\nelectronic, and magnetic complexity of the material [ 23–\n28]. In particular, the contribution of Sm 4felectrons\nand spin-orbit coupling (SOC) has often been neglected\ninrst-principles calculations, due to the complexity of\ntreatingfstates and non-collinear magnetism.\nIn this work, we report an investigation of the crys-\ntal and magnetic structures of SFO, exploiting density2\nfunctional theory (DFT) simulations and accounting ex-\nplicitly for the Sm-magnetism through the inclusion of\nthe Sm-4felectrons in the valence states. First, we show\nthat Sm-felectrons are essential for the good descrip-\ntion of the crystal structure (lattice parameters and inter-\natomic distances) by comparison to available experimen-\ntal data. Then, we provide estimates of the magnetic\ninteractions in terms of eective Heisenberg Fe-Fe, Sm-\nSm and Fe-Sm exchange couplings, and of the magnetic\nanisotropy by taking into account the SOC eect, and\nassociated non-collinear spin states for the Fe- and Sm-\nsubstructures. Finally, we report the results of phonon\ncalculations with a particular focus on the Raman active\nmodes in thePbnmcrystal structure. Our results sup-\nport the hypothesis of an active role for Sm magnetism\nin the experimentally observed anomalies in the Raman\nspectrum evolution as a function of temperature [ 29].\nTo the best of our knowledge this is therst time a\nmagnetic parametrization and magnetic anisotropy esti-\nmate have been carried out for SFO. Wend that it is\nthe strong magnetic anisotropy of the Sm 4felectrons\nwhich drives the high-temperature Fe spin reorientation,\nmediated by a small eective Sm-Fe magnetic exchange.\nII. COMPUTATIONAL DETAILS\nDFT calculations were performed using the projector\naugmented wave (PAW) method[ 30] as implemented in\nthe ABINIT simulation package [ 31–34]. The exchange-\ncorrelation potential was evaluated within the general-\nized gradient approximation (GGA) using the PBEsol\nfunctional [ 35]. The following orbitals were considered\nas valence states: O 2s2, 2p4; Fe 3s2, 3p6, 3d7, 4s1; Sm\n5s2, 5p6, 5d1, 6s2without Sm-felectrons (noted w/o\nSm-f) when treating the Sm-felectrons as core states,\nand with Sm-felectrons (noted w/ Sm-f) when taking\nSm-4f5electrons in the valence. The plane wave cuto\nenergy was set to 35 Ha and thek-mesh for the Brillouin\nzone (BZ) sampling of thePbnm20-atom cell to 6×6×4.\nThe self-consistent energy was converged below\n10−10eV for collinear calculations, without spin-orbit\ncoupling (SOC), and between 10−4and 10−6eV for cal-\nculations including SOC and non-collinear magnetism.\nNo symmetry constraints were taken into account\n(nsym=1). Wexed the Hubbard-U[ 36,37] correction\non the localized Fe-3dand Sm-4fstates to 4.5 eV and\n6 eV, respectively. DierentUvalues (5 or 7 eV on Sm-f\nor 4 eV on Fe-dstates) were explored and do not sub-\nstantially aect the estimate of the eective magnetic\ninteractions, as commented along the text.\nThe atomic structure and lattice parameters of the or-\nthorhombic 20-atom SFO-cell were fully optimized for\nboth the w/o Sm-fand the w/ Sm-fcases: in the w/o\nSm-fcase, we employedU(Fe-d) = 4.5 eV and collinear\nantiferromagnetic (AFM) G-type Fe-spin order; in the\nw/ Sm-fcase, we employedU(Sm-f)=6 andU(Fe-d)\n=4.5 eV and AFM G-type order for both the Sm andFe spin substructures. In the latter case, small Sm\nand Fe polar distortions along thec-crystallographic axis\nand additional antipolar Fe displacements along theb-\ncrystallographic axis appear, lowering the crystal symme-\ntry to theC 2v-Pbn2 1(n. 33) space group. Nevertheless,\ndistortions with respect to the associated centrosymmet-\nricD 2h-Pbnm(n. 62) structure were lower than 0.005 ˚A,\nand produce an energy gain lower than 0.1 meV/f.u.\nGiven typical DFT accuracies, we can not assert the low\nenergy phases are truly stable, and we decided to work\nin the SFO-Pbnmphase. Additionally, phonon frequen-\ncies calculated at theΓ-point of thePbnm-BZ through\nthe “nite dierence” approach using the Phonopy pack-\nage [ 38], do not reveal structural instabilities and are\nalmost the same (dierences below 1 cm−1) as those cal-\nculated in thePbn2 1structure. We also validated the\nvibrational energies and phonon eigenvectors of thenite\ndierence method by comparison with density functional\nperturbation theory (DFPT)[ 39,40].\nIn this work, all results concern calculations performed\nusing thePbnmstructures for both the w/o Sm-fand\nthe w/ Sm-fcases, unless stated otherwise.\nIII. CRYSTAL AND ELECTRONIC\nSTRUCTURES\nA. Structural properties\nSFO crystallizes in the commonD 2h−Pbnm(or\nPnma) orthorhombic perovskite structure [ 44], which\nhosts two interpenetrating pseudo-cubic Sm- and Fe-\nsubstructures. The crystal structure is depicted in Fig. 1.\nThe main structural distortions from the idealPm ¯3m-\ncubic cell are those related to the Sm-O bonds: the anti-\npolar o-centering of the samariumA-cations and the\noxygen atom rotations (a−a−c+in Glazer’s notation [ 42])\nproduce distorted SmO 12dodecahedra and tilted, corner-\nsharing FeO 6octahedra, respectively.\nThe calculated structural parameters, including unit\ncell lattice vectors and angles, interatomic distances,b/a\nandc/aratios were calculated and compared to exper-\nimental values, as shown in Table Iand Table SI. The\neect of the Sm-felectrons can already be seen in the\nstructural parameters: thealattice constant, in par-\nticular, increases leading to a bigger volume of the w/\nSm-funit cell compared to the w/o Sm-fone. Overall\nstructural parameters are very close and in good agree-\nment with available experimental data (T∼300 K in\nRefs. [ 41,45] and T∼100 K in Ref. [ 29]). Particularly,\nby analysing results from structural optimizations per-\nformed either in the w/o Sm-fcase (freezing Sm-felec-\ntrons in the core states) or in the w/ Sm-fcase (including\nthem in the valence), we found that in the latter case the\nagreement with X-ray diraction (XRD) measurements\nimproves signicantly. The relative error (RE) associ-\nated to our DFT results with respect to the experimen-\ntal data is overall reduced when taking into account the3\nParameters w/o Sm-f w/ Sm-f w/ Sm-f(1 GPa) Exp. 293 K[ 41]REw/o Sm-f REw/ Sm-f REw/ Sm-f(1 GPa)\n|a|(˚A) 5.36 5.41 5.40 5.40 -0.74% +0.17% -0.01%\n|b|(˚A) 5.62 5.60 5.59 5.60 +0.34% -0.06% -0.24 %\n|c|(˚A) 7.69 7.70 7.69 7.71 -0.21% -0.08% -0.27 %\n|b/a| 1.05 1.04 1.04 1.04 +1.09% -0.23% -0.23%\n|c/a| 1.44 1.42 1.42 1.43 +0.54% -0.25% -0.26%\nvolume( ˚A3)231.59 233.07 231.80 233.01 -0.61% +0.03% -0.52%\nTABLE I. Calculated lattice parameters for orthorhombicPbnmSFO either without (w/o Sm-f) or with (w/ Sm-f) the Sm-f\nelectrons as valence states. Room temperature experimental data from Ref. [ 41] and relative errors (RE) are also reported for\ndirect comparison.\nbac\nb\ncac(a)(b)\n(c)Sm 1'\nbacSm 2 Sm 1Sm 2'Sm 3'Sm 3Sm 4Sm 4'\nSm 2Fe1'Fe1Fe1''Fe2Fe4Fe3'Fe3Fe3''Fe1'Fe1Fe1''Fe2Sm\nFe\nO\nO1\nO3O2O4\nO5\nO6O7O8O9\nO9'O12O12'Fe3\nO2O4\nO9O3O1O10\nFIG. 1. Orthorhombic SFO structure.(a)Top view of the\nPbnmstructure, showing in-phase oxygen octahedra rota-\ntions around thec-axis (c+in Glazer’s notation [ 42]). Samar-\nium (Sm), Iron (Fe) and Oxygen (O) atoms are represented\nwith violet, green and red balls, respectively.(b)Sm and\nFe atomic arrangement, forming two distorted, pseudo-cubic,\nmagnetic substructures.(c)Sm-O (left) and Fe-O (right)\nbonds, forming distorted SmO 12polyhedra and FeO 6octa-\nhedra, respectively. Atomic labels guide identication of the\ndierent interatomic distances reported in Table SI. Atomic\nstructure were generated through the VESTA software [ 43].\nf-electrons (see RE reported in Tables I-SI).\nThe structural distortions of the ground state (shown\nin Fig. 1) produce inequivalent bonds and dierent dis-\ntances between neighboring magnetic cations, which cre-\nates inequivalent magnetic sites and could have a con-\nsequence in the extraction of shell-dependent magnetic\ninteractions. The six nearest neighbor Fe-Fe distances\nrange between 3.85 and 3.89 ˚A, with the shorter Fe 1-Fe3distance along theccrystallographic vector and the\nlonger distance for the in-plane Fe 1-Fe2pairs (a, bplane).\nAs highlighted in Fig. 1(b) and Fig. S5, the Sm pseudo-\ncubic cage is more distorted than the Fe one, and the\nsix nearest neighbor Sm-Sm distances range from 3.80\nto 3.99˚A. Both Sm-Sm and Fe-Fe pairs count twelve\nnext-nearest (NN) neighbors, with distances∈[5.19 :\n5.81]˚A and [5.41 : 5.60] ˚A, respectively. Values are re-\nported in Table SI.\nWe note that the length of theclattice vector isxed\nby the Fe 1-Fe3distance, whereas the length of the planar\naandbvectors is determined by the Sm 1-Sm 1′= Fe 1-\nFe1′and Sm 2-Sm 2′= Fe 1-Fe1′′distances, respectively\n(cfr.Fig. 1(b) and Table SI). Accordingly, one of the\neects of the Sm-felectrons on the structure is to intro-\nduce some in-plane anisotropic strain: the length of thea\nlattice vector is signicantly reduced w/o Sm-felectrons,\nwhereasbslightly increases;cis less aected (as shown\nin Table Iand Table SI). Similarly, Sm-f-related struc-\ntural anisotropy, mostly aecting thealattice strain, is\nalso reported for DyFeO 3[46], which suggests that this\ntrend is general. On the other hand, the overall volume\nw/ Sm-fagrees with that at room temperature although\nour DFT calculation is in principle done at 0 K. Due\nto thermal expansion, a reduction at low-T is expected\ninstead [ 29,45]. Therefore, to disentangle the eects of\nisotropic volume compression and anisotropic strain in\nthe dynamical properties reported in Sec. V, thePbnm\nw/ Sm-fstructure was also relaxed by applying a hy-\ndrostatic pressure of about 1 GPa, bringing the volume\ncloser to that w/o Sm-fand also closer to the expected\nlow temperature one.\nB. Electronic properties\nIn Fig. 2, we show the (collinear spin) electronic\npartial density of states (DOS) for antiferromagnetic\nSFO, in theG-type spin order, which is characterized\nby allrst-neighbor spins coupled antiferromagnetically\n(Fig.S4e). Spin-up and spin-down DOS channels as-\nsociated to one atom of each species are shown. The\nvalence bands (VB) can be divided into four main energy\nregions: (−7 :−6) eV with prominent Fe-3dstates and4\n1.000.500.000.501.00DOS (au)Fe (3d)\nFe (4s)\n1.000.500.000.501.00\n8 7 6 5 4 3 2 1 0 1 2 3\nDOS (au)\nEEfSm (4f)\n(eV)O (2p)Sm (5d)(a)\n(b)\nFIG. 2. Electronic density of state (DOS) for the antiferro-\nmagneticG-type Sm and Fe spin order:lleddandforbital\nstates of Fe (green color) and Sm (violet color) are mainly\nin the intervals of (-7:-6) eV and (-4.5:-3) eV, respectively;\np-states of O (red color) mainly occupy the top of the va-\nlence bands and hybridize with Fe-dstates. The conduction\nbands are characterized by empty Sm-fand Fe-dstates. Ver-\ntical dashed lines guide visualization of the dierent energy\nregions; shadow areas highlightp-d(O-Fe) and possibled-f\n(Fe-Sm) hybridization regions. The related electronic band\nstructure is shown in Fig. S1\na small O-2pstate contribution; (−6 :−4.5) eV with\nhybridizing Fe-4sand O-2pstates; (−4.5 :−3) eV with\nprominent localized Sm-4fstates, overlapping with\nsmall Fe-3dand O-2pstates contributions; (−3 : 0) eV\n(top of VB) characterized by hybridized Fe-4sand\nO-2pstates. The bottom of the conduction bands\n(CB) is characterized by the remaining empty Sm-4f\nspin-up states and the empty Fe-3dspin-down states.\nIn Figs. S1-S2, we show the total electronic DOS and\nband structure calculated in thePbnmstructure w and\nw/o Sm-f. The direct energy band gap at theX-point\n(0.5, 0.0, 0.0) of the Brillouin zone (BZ) is about 2.3 eV\nwithin DFT PBEsol+U. The Fe and O electronic states\nare not signicantly aected by the Sm states. In\nFig. S2, we compare the band structures obtained by\nincluding and excluding the Sm-4felectrons from the\nvalence states. The main eects are in the middle of the\nVB and bottom of the CB, related to the appearance of\nthe localizedf-states.\nIn the SI we show the (predictable) behavior of the 4f\nand 3dstates with the HubbardUparameters. No sub-\nstantial changes in the hybridization or spin-polarization\nare observed.\nIV.MAGNETIC PROPERTIES: COLLINEARMAGNETISMAND SOC EFFECT\nAs mentioned in the Sec. I, the Fe magnetic substruc-\nture of SFO experimentally undergoes a spin reorienta-\ntion (SR) transition,i.e.the rotation of the magnetiza-HSE Spin orderMAE\n(meV) Sm Fe∆E(meV/f.u.)\nJ1(Fe-Fe)≃5.9 FMz Gz ≃0\nJ1(Sm-Sm)≃0.0 FMz Gx ≃1\nJ2(Fe-Fe)≃0.2 FMx Gz ≃-44\nJ2(Sm-Sm)≃0.0 FMx Gx ≃-47\nJ1(Sm-Fe)≃0.0 Cz Gx ≃-1\nCx Gx ≃-43\nCx Gz ≃-46\nTABLE II. (Left side) Heisenberg magnetic exchange (HSE)\nforrst (J 1) and second (J 2) Fe-Fe and Sm-Sm neighbor in-\nteractions and Sm-FeJ 1interaction, given in meV per atom\n(NB: the values are also divided byS2, withS=5\n2). (Right\nside) Magnetocrystalline anisotropy (MAE, in meV/f.u.): en-\nergy dierences between various Sm and Fe magnetic orders\ntaking into account the spatial orientation of the spins. A\nschematic of some of the magnetic structures is shown in Fig. 3\n.\ntion vectors with respect to the crystallographic axis, be-\ntween 450 K and 480 K. Upon decreasing temperature,\nthe weak Fe-ferromagnetic (FM) moment rotates from\nthe longcaxis to the shortaaxis; the overall spin con-\nguration thus changes fromGxAyFM zto FM xCyGz,\nwith the AFM-G-type spin order remaining dominant.\nThis is related to the fact that, in SFO as in the other\nRFeO 3compounds, the strongest magnetic exchange in-\nteractions are related to the Fe-substructure, and they\ndetermine the main magnetic ordering andrelativeori-\nentation of the spins.\nThe importance of the Fe sublattice in the magnetic\nground state becomes clear in the DFT energies of mag-\nnetic congurations combining dierent spin orderings\nat the Sm- and the Fe- sites, as reported in Table SII:\nthe AFMG-type ordering, is the ground-state for the\nFe-spin substructure, independently of the magnetic or-\ndering considered for Sm-spins. On the other hand, the\nSm magnetic substructure shows a strong competition\nbetween ferromagnetic and antiferromagnetic congura-\ntions which are very close in energy.\nIn Table II, we report average Heisenberg spin ex-\nchange (HSE) estimated through an energy mapping\nmethod employing the DFT energy dierences from six\nmagnetic congurations among those listed in Table SII.\nThis Fe AFM nature is clear from the strongrst near-\nest neighbour exchange (J 1) of about 5.9 meV between\nFe sites. Beyond therst nearest neighbour interaction,\nthe exchange interactions fall rapidly to zero. In detail,\nwe consider the Heisenberg spin Hamiltonian of the type\nH=1\n2∑\ni=jJijSi·Sj, withJ ijthe isotropic Heisenberg\nexchange coupling between interacting spinsSat thei\nandjsites; here we useS=5\n2for both Sm and Fe ions.\nWe took into account sixrst- (J 1) and twelve second-\n(J2) neighbors for the Fe-Fe and Sm-Sm magnetic in-\nteractions, and eightrst-neighbor Sm-Fe interactions,\ntreating all of the magnetic pairs as structurally equiv-\nalent,i.e.neglecting variations in the inter-atomic dis-5\ntances related to the structural distortions described in\nthe previous Sec. III A . As anticipated, theJ 1exchange\ncoupling between Fe ions is strongly antiferromagnetic\nand is the leading term in the magnetic Hamiltonian of\nSFO;J 2is one order of magnitude smaller. On the other\nhand, Sm-related interactions are on average very small\n(close to zero and within the DFT accuracy), in line with\nthe quasi-degeneracy observed between the various FM\nand AFM Sm-spin orders.\nSuch very small Sm-Sm and Fe-Sm interactions reveals\na strong frustration in the Sm spin lattice. Furthermore,\nthe weak Sm-Fe exchange interaction shows more sensi-\ntivity than the Fe-Fe exchange to the spins order used\nfor the spin Hamiltonian parametrization (to the Fe-spin\norder in particular).J 1(Sm-Fe) can vary from negative\n(FM) to positive (AFM) values, but remains a small con-\ntribution to the total energy, on the order of 0.1 meV.\nThis leads to the approximately zero value reported in\nTable II. Similarly, the estimate of the weaker Sm-Sm\ninteractions also depends on the magnetic congurations\nused for the energy mapping. In particular, a weakly\nantiferromagneticJ(also of order 0.1 meV) is obtained\nwhen including only oneG-type Fe- spin conguration\namong the six used,i.e.when performing thetting\nwith higher energy spin congurations instead of those\nnearer the ground-state.\nSuchuctuations are not observed in the similarly\nsmallJ 2Fe-Fe interactions, which preserve magnitude\nand sign, remaining antiferromagnetic. These observa-\ntions suggest that the weak Sm-spin interactions depend\non the magnetic surroundings and that the interactions\ncould be dierent, either FM or AFM, for each magnetic\npair in the distorted SFO, giving rise to disorder and frus-\ntration in the Sm magnetic substructure [ 47,48]. Likely\nsupported by thermaluctuations and entropic contribu-\ntions, atnite temperature, such a behavior could justify\nthe reported cluster glass state in the low-T Sm-spin or-\ndering [ 10].\nNoteworthy, despite theU-induced shifts of the Sm-f\nand Fe-dstates in the VB and CB (Fig. S3), magnetic\nexchange energies (Fe exchange couplings, in particular)\nare not substantially aected by changing the Hubbard-U\ncorrection on the Sm-fand Fe-dstates, based on the tests\nwe performed and reported in Tables SII-SIII. The weak\ninuence of the Sm magnetic moment on the magnetic\nordering and the relative Fe-spin orientation is further\nsupported by calculations without the Sm-felectrons.\nFreezing the Sm-felectrons within the core states, and\ntherefore removing the magnetic contribution of Sm, does\nnot modify the magnetic interaction signicantly (last\ncolumn Table SII). This could be ascribed to the absence\nof strongf-dhybridization which could support super-\nexchange between the localized Sm-4fspins; in turn, this\nis also in line with the almost insensitivity of the Fe and\nO electrons to the presence of explicit Sm-felectrons\n(Sec. III B).\nSo far, we have discussed the isotropic magnetic\nexchange interactions, with results showing the muchz\nxSm\nSm(a)\n(b)\nc||za||xb||y\ncab\n0 meV\n-44 meV\nz Fe\nx,y,z Fex,y Sm -60 meV\nFIG. 3. Schematic view of(a) FMz-Gz,FMx-Gzand(b)\nFM xCy-FM xCyGznoncollinear Sm-Fe spin orders. Top view\nin(b)shows the little Sm-spin ferromagnetic canting. Energy\ngains (in meV/f.u.) from Table IIand Table IIIare also\nreported.\nweakerR-exchange couplings with respect to the stronger\nand dominant AFM Fe-Fe interaction in SFO. We now\nlook at the eect of the spin-orbit coupling in the\nmagnetic interactions, the magnetocrystalline anisotropy\n(MAE) in particular.\nIn Table II, we report the energy dierences between\nvarious spins states when including SOC in the DFT sim-\nulations. Particularly, werst looked at the energies as-\nsociated to theFM-GandC-Gcongurations when Sm\nand Fe spins are:i)collinear and aligned along either\nthexorzdirection, that provides an estimate of the to-\ntal MAE in the crystal;ii)noncollinear, with Sm-spins\naligned along thexdirection and the Fe-spins along the\nzdirection andvice versa, to estimate the MAE associ-\nated to the distinct magnetic substructures when com-\nparing energies with respect to the previous congura-\ntions. Schematic examples are shown in Fig. 3a. We per-\nformed these calculations by constraining the magnetic\nmoments along the wanted directions to avoid energetic\ncontributions from additional spontaneous spin compo-\nnents.\nSFO is characterized by a strong magnetic anisotropy\ndriven by the Sm ions. In fact, all magnetic orders with\nSm-spins aligned along thex-direction are lower in energy\nwith respect to congurations with Sm-spins along thez-\ndirection. At variance, Fe-spins experience a much lower\nenergy cost by having spins along thexorzdirection. We6\nalso checked the MAE,i.e.∆E(Gz-Gx), associated to\nthe Fe magnetic substructure when treatingf-electron\nas core states, still obtaining a very small anisotropy,\nsmaller than 1 meV/f.u.\nThen, we removed the DFT magnetization constraint\nand let the electronic system relax to its energy minimum\nconguration, starting fromGzorder for the Fe-spin sub-\nstructure and eitherFMxorCyorders for the Sm spins.\nIn both cases, the system spontaneously develops ad-\nditional spin components both in the Fe- and Sm-spin\nsubstructure: Fe-spins develop little antiferromagnetic\nC-type ordering along they-direction and weak ferromag-\nnetism along thex-direction, which gives rise to a little\nspins canting and to the observed net magnetization; Sm-\nspins develop additional C-type and ferromagnetic com-\nponents along theyandxdirections, respectively, ac-\ncording to the two initial congurations. Calculated total\nenergies and electrons moment components are reported\nin Table III. Thefstates of Sm display an important or-\nbital moment (OM) of opposite sign with respect to the\nmagnetic moment (MM) associated with theS=5\n2spin\nstate. Particularly, in the found lowest energy magnetic\nconguration, the FM xCy-FM xCyGz(Sm-Fe) congura-\ntion (Fig. 3b), the OM and MM components associated\nto the ferromagnetic order along thexdirection are of the\nsame order of magnitude, bringing to overall small mo-\nment and, in turn, to a Sm-FM xmoment competing with\nthe Fe-FM xmoment. Such delicate balance can be rather\nsensitive to any kind of small changes, which can be re-\nlated to the DFT approximations, such asU-correction\nor dierent exchange and correlation functionals, under-\nlying atomic positions or crystal structure. Moreover,\nat present, it is not that straightforward to identify a\nclear ground state from DFT simulations, because of the\npresence of various and competing metastable spin states\nassociated to dierent possible ways in occupying thef-\norbital states of Sm, hence in dening the spin-density\n(occupation) matrix between the correlated orbitals.\nNonetheless, our simulations put forward two main evi-\ndences, so far only envisioned upon experimental observa-\ntions:i)the major role played by the Sm-4felectrons in\nthe Fe-spin reorientation transition; we estimate a strong\nmagnetic anisotropy associated to the Sm ions, favoring\nin-plane orientation of spins.ii)spontaneous develop-\nment of canted-spin orders and non-collinearity between\nthe main magnetic ordering of the Sm- and Fe-spin sub-\nstructures, eventually favoring little and competing fer-\nromagnetic moments along thexdirection (ortheacrys-\ntallographic axis in thePbnmstructure).\nV. LATTICE DYNAMICS\nIn order to investigate possible spin-phonon coupling\neects, we calculated the phonon frequencies (ω) under\nvarious conditions and compered these results with low-\nT Raman active vibrations reported by M. C. Weber and\ncoauthors in Ref. [ 29]. Below the SR temperature, au-\na bc\nac\nb(a)\n(b)Ag(3) B1g(3)\nFIG. 4. Schematic view of the atomic pattern of distortion\nof the anomalous [ 29] Ag(3) and B 1g(3) phonon modes in the\nPbnmSFO structure, involving mainly:(a)antipolar motion\nof Sm-ions, FeO 6octahedra tilting and rotation around the\ncrystallographicc-axis;(b)octahedra rotation in theb-c(y-\nz) plane, respectively.\nthors report two kinds of deviations from typical thermal\nbehavior in the evolution of some phonon frequencies,\nω(T), associated with the rare-earth and oxygen atoms\ndisplacements (see Fig. 3 in Ref, [ 29]): an increase in the\nω(T) slope,i.e.phonons stiening, of the B 2g(1), A g(2)\nand B 3g(2) modes; decreasingω(T),i.e.phonons soften-\ning, for the A g(3) and B 1g(3) modes. Spectral changes\nare evident at T≃200 K, where a possible metastable\ncluster glass phase is observed [ 10]. Similar phonon\nanomalies have also been observed for other oxide per-\novskite systems, such as BiFeO 3[49] in the vicinity of\nTN, GdFeO 3at theR-spin ordering temperature [ 50] or\nTbMnO 3[51] and also in dierent compounds such as\nBiMn 3Cr4O12or CdCr 2S4associated with Cr-magnetism\nand the Cr-anion interactions [ 52,53]. Such features are\nconsidered a signature of strong spin-phonon coupling.\nIn Table IV, we report phonon frequencies for the\nRaman active modes calculated at the center (Γ) of\nthePbnm-Brillouin zone. The 60 phonon modes com-\nprises: B 1u+B 2u+B 3uacoustic modes, 8A usilent modes,\n7Ag+7B 1g+5B 2g+5B 3gRaman (RM) active modes,\n7B1u+9B 2u+9B 3uinfrared (IR) active modes [decompo-\nsition in thePnmasetting is : 7A g+5B 1g+7B 2g+5B 3g\nRM, 9B 1u+7B 2u+9B 3uIR [54]. Symmetry labels trans-\nformation from thePnmasetting to thePbnmis:\nB1g(u) →B 3g(u), B2g(u) →B 1g(u), B3g(u) →B 2g(u).].\nWe considered the relaxed atomic structure and lattice\nobtained either with or without the Sm-felectrons in the\nvalence states (cfr.Table I). For both structures we cal-\nculated vibrational energies w/ and w/o the Sm-fstates.\nWe also considered the third structure obtained by op-\ntimizing the crystal structure with thefelectrons but\napplying a hydrostatic pressure of about 1 GPa, in or-\nder to distinguish volume eects from anisotropic strain\neects (cfr.Table I). In this way, we can fully disentan-\ngle structural from Sm-f-induced eects on the observed\nspectra.\nInterestingly, theve reported anomalous modes7\nSm-order Fe-order∆E(meV/f.u.) Sm (Mx) Sm (My) Sm (Mz) Fe (Mx) Fe (My) Fe (Mz)\nFMxCyFM xCyGz ≃-57MM(µB) 4.85 -1.07 0.00 -0.01 -0.02 4.14\nOM(µB) -2.98 0.92 0.00 0.00 0.00 0.01\nFM xCyFM xCyGz ≃-60MM(µB) 0.77 -4.91 0.00 0.03 -0.02 4.14\nOM(µB) -0.67 3.03 0.00 0.00 0.00 0.01\nTABLE III. Noncollinear canted spin states with main ferromagnetic spin alignment alongx-direction (FMxCy) or main C-type\nantiferromagnetic order alongy-direction (FM xCy) for Sm. Bold characters highlight the main spin order for the two, Sm\nand Fe, magnetic substructures. Magnetic (MM) and orbital (OM) components of the spin moment for both Sm and Fe are\nreported. The reference for the energy dierence is theFMz-Gzspin order of Table II.\nw/o Sm-fstructure w/ Sm-fstructure 1 GPa structure\nSymmetry Sm-fcore Sm-fvalence Sm-fcore Sm-fvalence Sm-fvalence Exp.[ 29]\nAg(1) 110 112 104 107 107 110*\nB1g(1) 110 114 106 110 111 110\nB2g(1) 134 131 127 125 127 149\nAg(2) 139 145 127 134 136 146\nB3g(1) 150 150 152 151 152 161\nB1g(2) 160 162 149 152 154 158\nB3g(2) 236 244 210 218 220 241\nAg(3) 247 245 232 229 231 220\nB1g(3) 281 277 255 252 254 253\nB2g(2) 313 314 306 306 308 323\nAg(4) 319 322 306 310 312 319\nB1g(4) 345 345 344 342 345 –\nB3g(3) 354 353 356 353 357 354\nAg(5) 386 389 374 377 380 380\nAg(6) 417 415 420 419 424 421\nB3g(4) 426 424 427 426 432 426\nB2g(3) 425 421 433 429 435 433\nB2g(4) 449 445 446 443 446 456\nB1g(5) 460 460 446 446 449 463\nAg(7) 468 466 459 457 461 470\nB1g(6) 512 516 503 505 508 521\nB3g(5) 597 593 604 560 605 –\nB1g(7) 619 616 620 617 622 640\nB2g(5) 648 645 652 650 655 –\nTABLE IV. Phonon frequencies (in cm−1) of the Raman active modes in thePbnmcrystal structure, considering collinear\nG-type AFM magnetic ordering for the Fe atoms w/o Sm-f, and for both Sm and Fe spins sub-structures w/ Sm-f. In order,\nunderlying atomic structures arexed to: the one optimized w/o Sm-f; the one optimized w/ Sm-fand collinearG-type Sm\nand Fe spins order; to the one optimized taking into accountfstates and further hydrostatic pressure of about 1 GPa (cfr.\nTable I). Experimental data from Raman spectroscopy performed at 4 K have been provided by authors of Ref. [ 29]; * at 80 K\ntaken from Ref. [ 27]\n(violet-row in Table IV) exhibit strong coupling with the\nunderlying crystal structure: frequencies of the B 2g(1),\nAg(2) and B 3g(2) modes (which stien) are better re-\nproduced when considering structural distortions in the\nw/o Sm-fstructure. Frequencies of the A g(3) and B 1g(3)\nmodes (which soften) are well reproduced in the w/ Sm-\nfstructure. Moreover a direct spin-phonon coupling ef-\nfect is observed for theA g(2) andA g(3) modes, with\na further hardening and softening, respectively, induced\nby the magnetic Sm atoms. Both theA g(2) andA g(3)\nmodes involve in-plane Sm-antipolar motion, with addi-\ntional in-phase octahedra rotations and tiltings for theAg(3) mode (Fig. 4a). The involvement of Sm displace-\nments could justify a direct coupling between the vi-\nbration and the Sm spins. Moreover, we also observe\na possible interesting coupling with strain. In the w/o\nSm-fstructure, theastructural parameter is smaller\nthan in the w/ Sm-fstructure, producing a hardening of\nthese modes; at variance, the reduced volume alone does\nnot substantially aect the frequency. Similarly, also the\nB1g(3) mode displays a coupling to strain, but weaker\ndirect spin-phonon coupling than theA g(3) mode. The\nB1g(3) mode is in fact characterized by in-phase rota-\ntions but smaller Sm distortions than the twoA gmodes8\n(Fig. 4b).\nThe interpretation of the behavior of the B 2g(1) and\nB3g(2) modes is less direct: both modes display some\nstrain-phonon coupling, but weak spin-phonon coupling,\ndespite the fact that their distortion patterns involve mo-\ntion of the Sm ions (plus out-of-phase oxygen rotations\nfor B 3g(2)). The Sm motion in these two modes occurs\nalong theccrystallographic axis, which suggests a rela-\ntion to the anisotropic character of the Sm-related mag-\nnetic interactions.\nThe remaining Raman active modes are less sensitive\nto the structural distortions and coupling with Sm-f\nstates (within 10 cm−1), and they are in agreement with\nthe low-T data. Noteworthy, the measured vibration fre-\nquencies of the B 1g(5) and A g(7) (grey-row in Table IV)\nmodes are the only ones which match better with calcu-\nlations performed in the w/o Sm-fstructure.\nIn order to investigate coupling of phonons with SOC\nand Sm-magnetism, in Table SIV, we report frequencies\nof the Raman active modes for dierent Sm-magnetic or-\nders. Collinear antiferromagnetic or ferromagnetic Sm-\nspin orders do not aect phonons, whereas variations be-\ntween 3 and 5 cm−1are induced on the anomalous B 2g(1),\nB3g(2) and A g(3) modes by SOC eects, when consider-\ning the non-collinear FM xCy-FM xCyGz(Sm-Fe) cong-\nuration (Table III). Such eects are almost absent when\nexcluding Sm-felectrons.\nAdditionally, in Table SV, we report the silent and\ninfrared (IR) modes of thePbnmstructure. Unfortu-\nnately, no IR experimental spectra are available. Inter-\nestingly, the silent A u(1) mode and the B 3u(2), B 1u(2),\nB1u(4) and B 1u(5) IR modes show sensitivity to the Sm-\nfstates, whereas the B 3u(1), B 1u(1), A u(4), B 3u(4),\nB2u(4), B 3u(5), B 2u(6), B 1u(5) and B 3u(8) show coupling\nto strain.\nVI. DISCUSSION AND CONCLUSION\nBy means ofrst-principles calculations, we have in-\nvestigated structural, electronic, magnetic and dynami-\ncal properties of SmFeO 3unveiling spin-spin, spin-lattice\nand spin-phonon couplings eects driven by the magnetic\nSm ion. In particular, we performed comparative simu-\nlations with and without inclusion of thefelectrons in\nthe valence states, with the further inclusion of the spin-\norbit coupling. The energy mapping over various Sm and\nFe spins orders revealed a magnetic frustration of the Sm\nspin lattice due to very weak and eventually competing\nferromagnetic and antiferromagnetic Sm-Sm and Sm-Fe\nexchange interactions against robust antiferromagnetic\nFe-Fe exchange interactions. This supports the lack of\nSm magnetic ordering at high-temperature and possi-\nble disordered spin states at medium temperature. At\nvery low-temperature, the reduction of the energy cost\nrelated to thermaluctuations and entropy could allow\nfor the Sm spin moments ordering, stabilized by a large\nmagneto-crystalline anisotropy. In fact, we found outthat the Sm spins largely prefer to lay in the{a, b}plane\nof thePbnmcrystal structure, perpendicularly to the Fe\nspins, due to a strong spin-lattice coupling mediated by\nthe SOC of the Sm ions created by the 4felectrons. We\nthus identied the large anisotropy of the Sm magnetic\nions as a possible microscopic mechanism driving the ex-\nperimentally observed Fe spin reorientation transition in\nSmFeO 3. Furthermore, the small canting of both Sm\nand Fe spins associated with the non-collinear ground-\nstate, gives rise to a weak ferromagnetic moment along\nthea-axis. Interestingly, we found that Sm spins exhibit\nlarge orbital moment of opposite sign with respect to the\nmagnetic moment, which could be the origin of the mag-\nnetization reversal, hence anti-parallel Sm and Fe spins,\nobserved at low-T. Additional evidences of the magnetic\nactivity of the Sm-felectrons are also observed in the\nlattice vibrations. We pointed out in fact a direct cou-\npling between specic phonon modes and Sm spins, and\nalso further indirect evidences of the anisotropic charac-\nter of the Sm-felectronic interactions, which are in line\nwith experimentally reported anomalous vibrations. Fur-\nthermore, we also put forward interesting phonon-strain\ncouplings.\nThis work gives a deep insight into the underlying\nphysics of the rich SmFeO 3phase diagram and provide\nkey ingredients for future works, both from the theoret-\nical side, to investigate, for instance,nite-temperature\nand external perturbations eects through Monte Carlo\nand/or spin dynamics simulations, and from the exper-\nimental side to improve low-T characterization of the\nstructural and magnetic properties.\nACKNOWLEDGMENTS\nThis work was supported by the m-era.net project\nSWIPE funded by FNRS Belgium grant PINT-MULTI\nR.8013.20, and by computing time on the LUMI\nsupercomputer and on the Tier-1 supercomputer of\nthe F´ ed´ eration Wallonie-Bruxelles infrastructure (grant\nagreement n. 1117545), through the Consortium des\n´Equipements de Calcul Intensif. The authors acknowl-\nedge M.C. Weber and the SWIPE project partners, in\nparticular M. Guennou, G. Gordeev, M. Bibes, and\nL. Iglesias, for insightful discussions and scientic ex-\nchanges. D.A. acknowledges E. Bousquet and A. Sasani\nfor some technical support and implementation of the\nroutine for the orbital moment (OM) calculation in the\nABINIT package (release and publication in prepara-\ntion).9\n[1] E. Bousquet and A. Cano, Journal of Physics: Condensed\nMatter28, 123001 (2016).\n[2] D. Afanasiev, J. R. Hortensius, B. A. Ivanov, A. Sasani,\nE. Bousquet, Y. M. Blanter, R. V. Mikhaylovskiy, A. V.\nKimel, and A. D. Caviglia, Nature materials (2021).\n[3] R. Vilarinho, M. C.Weber, M. Guennou, A. C. Miranda,\nC. Dias, P. Tavares, J. Kreisel, A. Almeida, and J. A.\nMoreira, Scientic reports12, 9697 (2022).\n[4] Y. K. Jeong, J.-H. Lee, S.-J. Ahn, and H. M. Jang, Solid\nState Communications152, 1112 (2012).\n[5] L. G. Marshall, J.-G. Cheng, J.-S. Zhou, J. B. Goode-\nnough, J.-Q. Yan, and D. G. Mandrus, Phys. Rev. B86,\n064417 (2012) .\n[6] S. Cao, H. Zhao, B. Kang, J. Zhang, and w. Ren, Sci.\nRep.4, 5960 (2014) .\n[7] C.-Y. Kuo, Y. Drees, M. T. Fern´ andez-D´ıaz, L. Zhao,\nL. Vasylechko, D. Sheptyakov, A. M. T. Bell, T.W.\nPi, H.-J. Lin, M.-K. Wu, E. Pellegrin, S. M. Valvidares,\nZ.W. Li, P. Adler, A. Todorova, R. K¨ uchler, A. Steppke,\nL. H. Tjeng, Z. Hu, and A. C. Komarek, Phys. Rev.\nLett.113, 217203 (2014) .\n[8] X. Fu, X. Zeng, D.Wang, H. C. Zhang, J. Han, and\nT. J. Cui, Scientic Reports5, 14777 (2015).\n[9] X. Wang, J. Yu, X. Yan, Y. Long, K. Ruan, and X. Li,\nJournal of Magnetism and Magnetic Materials443, 104\n(2017).\n[10] M. K.Warshi, A. Kumar, A. Sati, S. Thota, K. Mukher-\njee, A. Sagdeo, and P. R. Sagdeo, Chemistry of Materials\n32, 1250 (2020).\n[11] M. Kuroda, N. Tanahashi, T. Hajiri, K. Ueda, and\nH. Asano, AIP Advances8, 055814 (2018).\n[12] H. Li, Y. Yang, S. Deng, L. Zhang, S. Cheng, E.-J. Guo,\nT. Zhu, H.Wang, J.Wang, M. Wu, P. Gao, H. Xiang,\nX. Xing, and J. Chen, Science Advances8, eabm8550\n(2022).\n[13] K. G. Abdulvakhidov, S. N. Kallaev, M. A. Kazaryan,\nP. S. Plyaka, S. A. Sadikov, M. A. Sirota, and S. V.\nZubkov, IOP Conf. Ser.: Mater. Sci. Eng.112, 012020\n(2016).\n[14] Q. Hu, B. Yue, D. Yang, Z. Zhang, Y.Wang, and\nJ. Liu, Journal of the American Ceramic Society105,\n1149 (2022).\n[15] J. Kang, Y. Yang, X. Qian, K. Xu, X. Cui, Y. Fang,\nV. Chandragiri, B. Kang, B. Chen, A. Stroppa, S. Cao,\nJ. Zhang, andW. Ren, IUCrJ4, 598 (2017).\n[16] E. Ateia, M. Arman, and E. Badawy, Appl. Phys. A\n125, 499 (2019).\n[17] M. Tomoda, S. Okano, Y. Itagaki, H. Aono, and\nY. Sadaoka, Sensors and Actuators B: Chemical97, 190\n(2004).\n[18] Z. Anaja, M. Naseri, and G. Neri, Sensors and Actua-\ntors B: Chemical304, 127252 (2020).\n[19] A. Gaiardo, G. Zonta, S. Gherardi, C. Malag` u, B. Fabbri,\nM. Valt, L. Vanzetti, N. Landini, D. Casotti, G. Cruciani,\nM. Della Ciana, and V. Guidi, Sensors20(2020).\n[20] L. Chen, T. Li, S. Cao, S. Yuan, F. Hong, and J. Zhang,\nJournal of Applied Physics111, 103905 (2012).\n[21] S. J. Yuan,W. Ren, F. Hong, Y. B.Wang, J. C. Zhang,\nL. Bellaiche, S. X. Cao, and G. Cao, Phys. Rev. B87,\n184405 (2013) .[22] U. Staub, L. Rettig, E. M. Bothschafter, Y.W.Windsor,\nM. Ramakrishnan, S. R. V. Avula, J. Dreiser, C. Pi-\namonteze, V. Scagnoli, S. Mukherjee, C. Niedermayer,\nM. Medarde, and E. Pomjakushina, Phys. Rev. B96,\n174408 (2017) .\n[23] M. Iglesias, A. Rodr´ıguez, P. Blaha, V. Pardo, D. Bal-\ndomir, M. Pereiro, J. Botana, J. Arias, and K. Schwarz,\nJournal of Magnetism and Magnetic Materials290-291,\n396 (2005), proceedings of the Joint European Magnetic\nSymposia (JEMS’ 04).\n[24] P. S¨ oderlind, P. E. A. Turchi, A. Landa, and V. Lordi,\nJournal of Physics: Condensed Matter26, 416001 (2014).\n[25] Y. Sun,W. Ren, S. Cao, H. Zhou, H. J. Zhao, H. Xu,\nand H. Zhao, Philosophical Magazine96, 1613 (2016).\n[26] V. Triguk, I. Makoed, and A. Ravinski, Phys. Solid State\n58, 2443–2448 (2016).\n[27] M. C.Weber, M. Guennou, H. J. Zhao, J. ´I˜ niguez, R. Vi-\nlarinho, A. Almeida, J. A. Moreira, and J. Kreisel, Phys.\nRev. B94, 214103 (2016) .\n[28] S. Ahmed, S. S. Nishat, A. Kabir, A. S. H. Faysal,\nT. Hasan, S. Chakraborty, and I. Ahmed, Physica B:\nCondensed Matter615, 413061 (2021).\n[29] M. C. Weber, M. Guennou, D. M. Evans, C. Toulouse,\nA. Simonov, Y. Kholina, X. Ma,W. Ren, S. Cao, M. A.\nCarpenter, B. Dkhil, M. Fiebig, and J. Kreisel, Nature\nCommunications13, 443 (2022).\n[30] P. E. Blochl, Phys. Rev. B50, 17953 (1994).\n[31] X. Gonze, J.-M. Beuken, R. Caracas, F. Detraux,\nM. Fuchs, G.-M. Rignanese, L. Sindic, M. Verstraete,\nG. Zerah, F. Jollet, M. Torrent, A. Roy, M. Mikami,\nP. Ghosez, J.-Y. Raty, and D. Allan, Computational\nMaterials Science25, 478 (2002).\n[32] X. Gonze, B. Amadon, P.-M. Anglade, J.-M. Beuken,\nF. Bottin, P. Boulanger, F. Bruneval, D. Caliste, R. Cara-\ncas, M. Cˆ ot´ e, T. Deutsch, L. Genovese, P. Ghosez, M. Gi-\nantomassi, S. Goedecker, D. Hamann, P. Hermet, F. Jol-\nlet, G. Jomard, S. Leroux, M. Mancini, S. Mazevet,\nM. Oliveira, G. Onida, Y. Pouillon, T. Rangel, G.-M.\nRignanese, D. Sangalli, R. Shaltaf, M. Torrent, M. Ver-\nstraete, G. Zerah, and J. Zwanziger, Computer Physics\nCommunications180, 2582 (2009).\n[33] X. Gonze, B. Amadon, G. Antonius, F. Arnardi,\nL. Baguet, J.-M. Beuken, J. Bieder, F. Bottin,\nJ. Bouchet, E. Bousquet, N. Brouwer, F. Bruneval,\nG. Brunin, T. Cavignac, J.-B. Charraud, W. Chen,\nM. Cˆ ot´ e, S. Cottenier, J. Denier, G. Geneste, P. Ghosez,\nM. Giantomassi, Y. Gillet, O. Gingras, D. R. Hamann,\nG. Hautier, X. He, N. Helbig, N. Holzwarth, Y. Jia,\nF. Jollet,W. Lafargue-Dit-Hauret, K. Lejaeghere, M. A.\nMarques, A. Martin, C. Martins, H. P. Miranda, F. Nac-\ncarato, K. Persson, G. Petretto, V. Planes, Y. Pouil-\nlon, S. Prokhorenko, F. Ricci, G.-M. Rignanese, A. H.\nRomero, M. M. Schmitt, M. Torrent, M. J. van Set-\nten, B. Van Troeye, M. J. Verstraete, G. Z´ erah, and\nJ.W. Zwanziger, Computer Physics Communications\n248, 107042 (2020).\n[34] M. Torrent, F. Jollet, F. Bottin, G. Z´ erah, and X. Gonze,\nComputational Materials Science42, 337 (2008).\n[35] J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov,\nG. E. Scuseria, L. A. Constantin, X. Zhou, and K. Burke,\nPhys. Rev. Lett.100, 136406 (2008) .10\n[36] V. I. Anisimov and O. Gunnarsson, Phys. Rev. B43,\n7570 (1991).\n[37] A. I. Lichtenstein, V. I. Anisimov, and J. Zaanen, Phys.\nRev. B52, R5467 (1995).\n[38] A. Togo and I. Tanaka, Scr. Mater.108, 1 (2015).\n[39] X. Gonze, J.-C. Charlier, D. Allan, and M. Teter, Phys.\nRev. B50, 13035 (1994) .\n[40] X. Gonze and C. Lee, Phys. Rev. B55, 10355 (1997).\n[41] E. N. Maslen, V. A. Streltsov, and N. Ishizawa, Acta\nCrystallographica Section B52, 406 (1996).\n[42] A. M. Glazer, Acta Crystallographica Section B28, 3384\n(1972).\n[43] K. Momma and F. Izumi, Journal of Applied Crystallog-\nraphy44, 1272 (2011).\n[44] R. L.White, Journal of Applied Physics40, 1061 (1969).\n[45] S. Chaturvedi, P. Shyam, A. Apte, J. Kumar, A. Bhat-\ntacharyya, A. M. Awasthi, and S. Kulkarni, Phys. Rev.\nB93, 174117 (2016) .\n[46] A. Sasani,First and second principles study of mag-\nnetic and multiferroic properties of rare-earth orthofer-\nrites, Ph.D. thesis, ULi` ege - Universit´ e de Li` ege, Li` ege,Belgium (09 December 2021).\n[47] P. Nordblad, inReferenceModule inMaterials Science\nand Materials Engineering(Elsevier, 2016).\n[48] E. Takeshi and S. J. Billinge, inUnderneath the Bragg\nPeaks, Pergamon Materials Series, Vol. 16, edited by\nT. Egami and S. J. Billinge (Pergamon, 2012) pp. 325–\n369.\n[49] R. Haumont, J. Kreisel, P. Bouvier, and F. Hippert,\nPhys. Rev. B73, 132101 (2006) .\n[50] A. Panchwanee, S. K. Upadhyay, N. P. Lalla, V. G. Sathe,\nA. Gupta, and V. R. Reddy, Phys. Rev. B99, 064433\n(2019) .\n[51] M. Schmidt, C. Kant, T. Rudolf, F. Mayr, A. A. Mukhin,\nA. M. Balbashov, J. Deisenhofer, and A. Loidl, The Eu-\nropean Physical Journal B71, 1434 (2009).\n[52] S. Kamba and al., submitted (2022).\n[53] K.Wakamura and T. Arai, Journal of Applied Physics\n63, 5824 (1988).\n[54] I. Smirnova, Physica B: Condensed Matter262, 247\n(1999)."    },    {        "title": "2212.07845v3.Exploration_of_all_3d_Heusler_alloys_for_permanent_magnets__an_ab_initio_based_high_throughput_study.pdf",        "content": "arXiv:2212.07845v3  [cond-mat.mtrl-sci]  21 Apr 2023Exploration of all-3d Heusler alloys for permanent magnets : anab initio based\nhigh-throughput study\nMadhura Marathe∗and Heike C. Herper\nDepartment of Physics and Astronomy, Uppsala university, 7 51 20 Uppsala, Sweden.\n(Dated: April 24, 2023)\nHeusler alloys have attracted interest in various fields of f unctional materials since their proper-\nties can quite easily be tuned by composition. Here, we have i nvestigated the relatively new class\nof all-3d Heusler alloys in view of its potential as permanen t magnets. To identify suitable candi-\ndates, we performed a high-throughput study using an electr onic structure database to search for\nX2YZ-typeHeusler systems with tetragonal symmetry and high m agnetization. For the alloys which\npassed our selection filters, we have used a combination of de nsity functional theory calculations\nand spin dynamics modelling to investigate their magnetic p roperties including the magnetocrys-\ntalline anisotropy energy and exchange interactions. The c andidates which fulfilled all the search\ncriteria served as input for the investigation of the temper ature dependence of the magnetization\nand determination of Curie temperature. Based on our result s, we suggest that Fe 2NiZn, Fe 2NiTi\nand Ni 2CoFe are potential candidates for permanent magnets with la rge out-of-plane magnetic\nanisotropy (1.23, 0.97 and 0.82MJ/m3respectively) and high Curie temperatures lying more than\n200K above the room temperature. We further show that the mag nitude and direction of anisotropy\nis very sensitive to the strain by calculating the values of a nisotropy energy for several tetragonal\nphases. Thus, application of strain can be used to tune the an isotropy in these compounds.\nI. INTRODUCTION\nDiscoveryofnewmagneticmaterialsisvitaltoimprove\nperformance of a range of applications from data storage\ntorenewableenergysources[1]. Permanentmagnetscon-\nstitute an essential component of electric generatorsused\nin wind turbines, and large amount of magnetic material\nisrequiredforeachgenerator[2, 3]. Mostcommonlyused\npermanent magnets in current devices typically contain\nrare-earth elements such as neodymium, samarium etc.\nwhich make these materials expensive and on top of that\ntheir mining is usually harmful to the environment [4].\nTherefore, alternative candidates for permanent magnets\narehighly soughtafter to improvethe overallsustainabil-\nity. We would like to note that there is an intrinsic limit\non the functional response of permanent magnets con-\nsisting of light elements determined by associated small\nspin-orbit coupling [5]. However, we expect such mag-\nnets could replace the rare-earth magnets in mid-range\napplications, and thus reduce the overall dependence on\nrare earth elements.\nWhat makes a material a good candidate for perma-\nnent magnets? – 1. its stability whether it can be syn-\nthesized in a given phase and structure: typically pre-\nferred symmetries are tetragonal or hexagonal crystal\nstructures so that there is a single well-defined easy axis,\n2. large magnetic moments, 3. large magnetocrystalline\nanisotropy with a preference for out-of-plane magnetiza-\ntion, and 4. Curie temperature above room temperature\nsothattheferromagneticstateisstableatworkingcondi-\ntions for devices. Currently the best strategy to acceler-\nate research into cost-effective and sustainable materials\n∗madhura.marathe@physics.uu.seis to use the high-throughput methods, that is, to comb\nthrough a large number of candidates availablein various\nstructural databases for alloys and compounds and then\nto perform electronic structure methods calculations to\ndetermine the required physical quantities.\nHeusler alloys are intermetallic compounds with L2 1\nstructure and typically with a chemical formula X 2YZ.\nConventionally X and Y elements are 3d transition met-\nals and Z belongs to either group III, IV or V (main\ngroup element). One advantage of these alloys is the\neasy tuning of its properties obtained by varying the\nconstituent elements, doping, site-disorder and/or strain\nmaking these useful for a multitude of functional proper-\nties from shape memory effects, half-metallicity to mag-\nnetocaloric response; for a general review on Heuslers see\nthe book edited by Felser and Hirohata [6]. Moreover, a\nnumberofstudieshavereportedlargemagnetocrystalline\nanisotropy in Heusler alloys and have also discussed pos-\nsibility of its tuning via interstitial doping, strain as well\nas local ordering of atoms [7–10], thus making Heusler\nalloys ideal potential candidates for permanent magnets.\nIn recent years, the conventional Heusler family has\nexpanded to include alloys in which all constituent ele-\nments are d-metals [11]. We mainly focus here on mag-\nnetic properties of this novel class of Heusler compounds\nin which Z belongs to 3d metals termed as all-3d Heusler\nalloys[12–16]. One of the first reports for this type of all-\n3d Heusler alloy is from Wei et al.[12] in which Ni-Mn\nalloys are doped with Ti to form the Heusler phase and\nadditionalCo-dopingmakesthematerialastronglyferro-\nmagnetic shape memory alloy. Further studies on their\nfunctional responses have reported giant exchange-bias\neffect [17], giant barocaloric effect [18] as well as large\nmagnetocaloric effect [19, 20]. First-principles calcula-\ntions have predicted occurrence of martensitic transfor-\nmation in all-3d Zn 2YMn (Y = Fe, Co and Ni) alloys [14]2\n!\"\n!!!\"# !$#\nFIG. 1. Unit cell for a typical Heusler alloys of the type X 2YZ\nshown for conventional 16-atom (a) and primitive 4-atom (b) .\nNote that the two unit cells are equivalent and blue dashed\nlines shown in (a) correspond to the primitive cell shown in\n(b). Here filling of atomic positions corresponds to the “sta n-\ndard” Heusler, whereas the “inverse” structure correspond s\nto exchange of atoms X and Y occupying positions (1\n2,1\n2,1\n2)\nand (1\n4,1\n4,1\n4).\nas well as high-spin polarization in Fe 2CrZ and Co 2CrZ\n(Z = Sc, Ti and V) alloys [16].\nIn this paper, we examine this new class of materi-\nals for potential permanent magnets because most of\nthe 3d metals are abundantly found and show strong\nmagnetic interactions, and there exist well-established\nroutes to make alloys in different phases from these\nmetals especially binary alloys of Ni-Fe and Fe-Co and\nternary alloys based on these alloys [21]. One of the well-\nstudied transition metal alloys is Fe-Co which has a high\nsaturation magnetization, but small magnetocrystalline\nanisotropy [21]. This alloy has the body-centered cubic\n(bcc) structure as the most stable phase, however first-\nprinciples calculations have predicted a large anisotropy\nfor Fe-Co in the body-centered tetragonal phase [22, 23].\nThis observation is confirmed by experiments in which\nstabilization of the tetragonal phase was achieved by ei-\nther epitaxial growth [24] or doping with another ele-\nment [25]. Typically bcc Fe-Co alloys are disordered,\nhowever ordered bulk or thin film conventional Heusler\nalloys of the form Fe 2CoZ [26, 27] and Co 2FeZ [28, 29] (Z\n= Si or Ga) have been synthesized and reported to have\nlarge Curie temperatures.\nThe main objective of our study is the calculation of\nthe relevant magnetic properties such as magnetocrys-\ntalline anisotropy and Curie temperatures for a selected\nset of all-3d Heuslers. It has been shown that strain\ncan be used to tune both the magnitude and the direc-\ntion of easy-plane axis in Heusler alloys [8, 30]. Thus,\nwe also explore possible tuning of the magnetocrystalline\nanisotropy for selected alloys via strain engineering. The\ndetails of our density functional theory calculations and\nMonte Carlo simulations are given in Sec. II. The selec-\ntion of suitable candidates is done using available repos-\nitory of materials and setting conditions on stability and\nmagnetic moments to filter out materials as described in\nSec.IIIA.Theresultsfromourcalculationsforstructural\nand magnetic properties are presented and discussed inSec. IIIB, then followed by analysis on how we can fur-\nther tune the properties by strain in Sec. IIIC. Finally\nwe conclude with a suggestion for a few potential per-\nmanent magnets and discuss general outlook for further\ntheoretical and experimental investigations in Sec. IV.\nII. COMPUTATIONAL DETAILS\nWe perform density functional theory (DFT) calcula-\ntions using the VASP code [31] to calculate the struc-\ntural and magnetic properties of Heusler alloys. We use\na 16-atom conventional unit cell for these calculations\nshown in Fig. 1(a). For these calculations, we use the\nenergy cutoff of 650eV and a k-mesh of 16 ×16×16.\nFurther we use the generalized-gradient approximation\nof the Perdew-Burke-Ernzerhof (PBE) type calculated\nwith the projected augmented-wave (PAW) method [32].\nTo calculate the magnetic anisotropy energy (MAE) and\nexchange interactions ( Jij), we make use of the full-\npotential linearized muffin-tin orbitals (FP-LMTO) code\nRSPt [33]. To reduce the computational effort, we use\na 4-atom primitive unit cell shown in Fig. 1(b) for these\ncalculations. Further a dense k-mesh of 36 ×36×36 is\nrequired to accurately capture smaller energy scale for\nmagnetic interactions. The MAE is calculated using the\nmagnetic force theorem [34] and is given by:\nEMAE=E[100]\nb−E[001]\nb, (1)\nwhereEα\nbis the fully-relativistic band energy for the\nmagnetization direction αcalculated from the self-\nconsistent scalar-relativistic potential. Within this def-\ninition, a positive EMAEcorresponds to the uniaxial\nanisotropy and therefore, indicates a material suitable\nas a permanent magnet. The exchange parameters are\ncalculated with the Liechtenstein-Katsnelson-Antropov-\nGubanov formula [35] as implemented in the RSPt code.\nUsing the calculated Jij’s, we map our system on a\nHeisenberg model given by spin Hamiltonian:\nH=−/summationdisplay\nijJijei.ej, (2)\nwhereeiis a unit vector representing local magnetic mo-\nmentmifor sitei. Note that the systems considered in\nthis study contain atoms with partially filled d-electrons\nand typically have long-range exchange interactions ex-\ntending over several lattice constants. It is essential that\nwe include these in our model to correctly estimate mag-\nnetic propertiesat finite temperatures. We haveincluded\npair-wise Jij’s for all neighbors jwithin 4.5 aradius from\nthe central site i[36]. We perform Monte Carlo (MC)\nsimulationsusingtheUppASDspindynamicscode[37]to\nobtain finite-temperature properties of the system. Our\nsimulation box has the dimensions 24 ×24×24and three\nensembles are used to reduce the statistical noise in our\ndata. At eachtemperature, 50,000steps areused to ther-\nmally equilibrate the system and statistical averages of3\n!\"#$#%&'(%)%*+$+),\n-&+*+\"$,.'/ 01'+&+*+\"$,'234 05\"6\n3(%4+'7)89(,.':/;'%\"1'::;\n3+%)4<'=>?@A'1%$%B%,+'C8)'48*(89\"1,'\n2D!E56'F<#4<'C#$'$<+'7#G+\"',$%)$#\"7'1%$%\n>8)'+%4<'\n48*(89\"1.'\n!\"#$\n%!\"!&HI!\"# $%\n!\"4&91+'%,'%''\n(8$+\"$#%&'4%\"1#1%$+D'J'>+K'\nL8K'M#'\n8)'N\"I\n!\"#$%O#,4%)1:\nP\n/\nQ\nR\n'#$\n('%#$\n#$$%O#,4%)1 S\n!\"#\nL%&49&%$+'N=-K'\n)&'(*+,L9)#+'$+*(+)%$9)+TO#,4%)1&'()*\n&+(,*\n&--*\n&)'*\nFIG. 2. Flowchart describing a detailed procedure to se-\nlect the systems for our study (steps 1–5) using the AFLOW\ndatabase followed by a quick summary of DFT + MC steps\nfor the selected systems (steps 6–7). The number of com-\npounds found after each step is given in brackets. For DFT\ncalculations in step 6, we ensured that there are no duplicat e\nentries and further restricted the pool to those alloys havi ng\nmagnetization larger than 1.0Tesla. Dashed arrows indicat e\nthat only key steps are included here.\nphysical quantities are taken over next 50,000 steps. A\nheating cycle is used to calculate the transition temper-\nature with a step size of 20K in the vicinity of magnetic\ntransition.\nIII. RESULTS AND DISCUSSION\nA. High-throughput systems selection\nWe used the electronic structure database\nAFLOW [38, 39] – containing more than 3.5 mil-\nlion material entries – to search for tetragonal Heusler\nalloys which are stable with negative formation of\nenthalpy ∆ Hand have magnetization larger than or\nequal to 1.0Tesla. We focused on X 2YZ type of alloys\nwith X = Mn, Fe, Co or Ni and both Y and Z belong\nto 3d group from Sc to Zn, that is, we included only\n3d transition metals in our search. Both standard and\ninverse Heusler structures (space group 139 and 119Compound a c/a MO MtotStatus\n˚A µB/f.u. Tesla\nMn2NiTi 5.97 1.48 AFM 0.0 0.0 ✗\nMn2TiZn 6.01 1.37 FM 4.73 1.01 ✳\nFe2CoNi 5.66 1.33 FM 7.34 1.88 /∈\nFe2CoTi 5.83 1.65 FM 5.18 1.22 ✔\nFe2CoV 5.70 1.51 FM 4.21 1.06 ✔\nFe2MnTi 5.81 1.01 FM 4.16 1.00 /∈\nFe2NiSc 6.03 1.63 FM 5.06 1.08 /∈\nFe2NiTi 5.85 1.57 FM 4.50 1.05 ✔\nFe2NiZn 5.76 1.47 FM 5.59 1.36 ✔\nCo2FeSc 5.97 1.01 FM 4.99 1.09 C\nCo2FeTi∗5.81 1.59 FM 4.33 1.03 C\nCo2FeZn 5.70 1.02 FM 4.91 1.24 C\nNi2CoFe 5.61 1.45 FM 5.62 1.49 ✔\nNi2FeCu 5.65 1.36 FM 3.92 1.01 /∈\nNi2MnCu 5.69 1.33 AFM 0.0 0.0 ✗\nNi2MnZn 5.77 1.22 AFM 0.0 0.0 ✗\nTABLE I. List of all “eligible” X 2YZ alloys extracted from\nAFLOW database including their structural parameters ob-\ntained from the database. All these compounds except\nCo2FeTi (marked with *) have the standard Heusler phase.\nThe fourth columns lists the magnetic ordering and the fifth\nand sixth columns list the total magnetic moment in differ-\nent units for each compound calculated using DFT. The last\ncolumn gives the status of the system with regards to further\nanalysis – ✗: AFM ordering; ✳: complex spin ordering; C:\ncubic structure energetically favored; /∈: small MAE values.\nAll these criteria exclude these alloys from further analys is,\nwhereas ✔indicates (meta-)stable tetragonal and FM phases\nwith large MAE values suitable for full analysis and are dis-\ncussed further in the next section.\nrespectively) are included.1A flowchart summarizing\nour selection procedure is depicted in Fig. 2. In Table I,\nwe have tabulated all the alloys which satisfied our\ninitial filtering criteria described in steps 1–4 and used\nfor DFT calculations in step 6 in Fig. 2. We also list\ncorresponding lattice constant for the cubic phase and\nc/avalues obtained from the database.\nWe further analyzed each of the shortlisted structures\nin the following way. First we performed DFT calcula-\ntions to check the stability of ferromagnetic (FM) order-\ning versus antiferromagnetic (AFM) ordering (step 6 in\nFig. 2). The corresponding low-energy spin ordering is\nlisted in column four of Table I. For the alloys found to\nbe stable in FM ordering, we havelisted the total magne-\ntizationMtotobtained from our calculations in columns\n1Note that here the exclusion of cubic Heusleralloys (space g roups\n225 and 216) from our search criteria can be a drawback becaus e\neven metastable tetragonal phases with negative ∆ Hwill be in-\ncluded in our list even when the cubic phase is the most stable\nphase. However, we examine for such a possibility.4\nFe2CoNi\nFe2CoTi\nFe2CoV\nFe2MnTiFe2NiSc\nFe2NiTi\nFe2NiZn\nCo2FeSc\nCo2FeTi\nCo2FeZn\nMn2TiZn\nNi2CoFe\nNi2FeCu-2-1012εMAE (MJ/m3)\n00.511.5κ(a)\n(b)\nFIG. 3. For all alloys which are found to be ferromagnetic\n(see Table I) MAE values (a) calculated using Eq. (1) and\nmagnetic hardness parameter κ(b) using Eq. (3) are given.\n5 and 6 of Table I. Our values agree well to those re-\nported in the AFLOW database, and all the systems in-\ndeed have large magnetic moments. We exclude those\nalloys for which AFM state is more stable (marked with\n✗) from further analysis.\nMost conventional Heusler alloys undergo a structural\nphase transition between cubic and tetragonalphases de-\npending on constituent elements and applied strain. We\nhave included those alloys which have a negative ∆ Hin\nthe tetragonal phase, but this does not ensure that the\nmoststablephaseisindeed tetragonal,and notcubic (see\nfootnote 1). Therefore, for the alloys with the FM or-\ndering, we calculated the structural stability against the\ntetragonal deformation. We keep the volume constant at\nthe reference structure obtained from the database as we\nvaryc/aratio from 0.8 to 1.7 for both standard and in-\nverse phases (discussed in more detail for a few selected\nalloys later). From this analysis, we confirm that for X =\nFe and Ni, tetragonalphasesconsidered areindeed stable\nin the T-phase or have a local minima and so could be\nstabilized by either strain or depositing on a substrate.\nFor X = Co, cubic structures are energetically far more\nfavorable which is similar to what is observed for tradi-\ntional Co-based Heusler alloys [40]. For Mn 2TiZn, the\nT-phase has a very shallow minima and also shows a\ntendency towards a more complex ferrimagnetic or AFM\nordering with varying c/aratio. Therefore, it is not suit-\nable material as a permanent magnet.\nIn the next step we calculated the magnetocrystalline\nanisotropy energy for these alloys using Eq. (1). The\ncorresponding results are given in Fig. 3(a). Further-\nmore using the calculated EMAEand the total magnetic\nmoments, we can estimate the magnetic hardness param-\neterκdefined as [41]:\nκ=/radicalBigg\n|EMAE|\nµ0M2s. (3)Here,µ0= 4π×10−7JA−2m−1is the vacuum magnetic\npermeability and Msis the saturation magnetization ex-\npressed in units of Am−1, such that κis a dimensionless\nquantity. Typically κ>∼1 is considered as a threshold\nfor hard magnets [41]. The calculated κvalues are given\nin Fig. 3(b). We find that Fe 2NiZn, Fe 2NiTi, Mn 2TiZn\nand Ni 2CoFe alloys have large and positive MAE val-\nues indicating a preference to have out-of-plane magne-\ntization. These alloys correspondingly show κclose to\n1 with the exceptional case of Ni 2CoFe with κ∼0.7\nwhich results from its much largermagnetization. There-\nfore, we would further investigate all these alloys except\nMn2TiZn because of its complex spin ordering. Two al-\nloys Fe 2CoTi and Fe 2CoV also have large but negative\nMAE which indicates in-plane magnetization is energet-\nically favored as well as κclose to 1. Therefore, these\ntwo alloys may also be interesting for applications other\nthan permanent magnets and are included in our subse-\nquent analysis. Ni 2FeCu alloy has a high κvalue of 0.8\nbut rather low EMAEof−0.55MJ/m3, and thus is not\nincluded as a potential candidate. All the remaining sys-\ntems have much smaller MAE values of ≤ |0.5|MJ/m3\nand lowκvalues therefore are not technologically viable.\nThe last column of Table I summarizes the final status of\nthe each system based on our initial analysis of suitable\ncandidates found in the AFLOW database.\nB. Functional properties of selected alloys\nWe first discuss in detail our structural analysis for\ntetragonal phases for the selected five alloys. We per-\nformed simulations for a number of tetragonal phases for\nboth standard and inverse structures to determine min-\nimum energy phases. These calculations are done such\nthat the total volume of the unit cell is kept constant at\nits referencetetragonalstructure. In Fig.4, wehaveplot-\nted the calculated total energy as a function of c/afor\nthe selected alloys. We shift the totalenergywith respect\nto that of the cubic standard phase of the corresponding\nalloy for easier comparison.\nFor Fe 2NiZn and Fe 2NiTi alloys, the standard tetrag-\nonal phase is energetically favorable, but this phase is\nvery close in energy to the inverse tetragonal phase as\nwellastothe cubicphase. ForFe 2NiZn, thestructureob-\ntainedfromAFLOW(redstarinFig.4)islowerinenergy\nby 1.87meV/atom compared to the local minima for the\ntetragonal inverse at c/a= 1.3 and higher in energy by\n3.99meV/atom as compared to the inverse cubic struc-\nture, whereas for Fe 2NiTi, the structure from AFLOW is\nthe most stable structure with the corresponding energy\ndifferences of 4.37meV/atom with the inverse tetragonal\nphase at c/a= 1.5 and 18.4meV/atom with the inverse\ncubic phase; these energy scales are clearly indicated in\nFig. 4f. Another interesting feature of the inverse phase\nenergy landscape for both the alloysis that from cubic to\ncompressive strain minimum it is almost flat with very\nsmall energy barrier to transform from one phase to an-5\n0.8 1 1.2 1.4 1.6 1.8\nc/a-0.8-0.400.4\nStandard\nAFLOW\nInverse0.8 1 1.2 1.4 1.6 1.8\nc/a-0.8-0.400.4\n0.8 1 1.2 1.4 1.6 1.8\nc/a-0.8-0.400.4∆E = E - Ecubstd (eV/f.u.)\n0.8 1 1.2 1.4 1.6 1.8\nc/a-0.8-0.400.4\n0.8 1 1.2 1.4 1.6 1.8\nc/a-0.8-0.400.4\n(a) (b) (c) (d) (e)-0.8-0.6-0.4-0.20.0+0.2\nTStandard\nCInverse\nTInverse(a) Fe2NiZn (b) Fe2NiTi\n(c) Fe2CoV (d) Fe2CoTi\n(e) Ni2CoFe\n(f) summary of \n  stable phases\nFIG. 4. Stability of tetragonal phases with respect to stand ard cubic phase: here the energy difference between a distort ed\nstructure and the cubic standard phase ∆ Eis plotted as a function of c/afor Fe 2NiZn (a), Fe 2NiTi (b), Fe 2CoV (c), Fe 2CoTi\n(d) and Ni 2CoFe (e). The results are shown for both standard and inverse phases along with the structure reported in the\nAFLOW database. (f) For easier comparison of energy scales, we summarize the ∆ Evalues for cubic and the tetragonal phases\nat the potential well. In all cases, tetragonal standard pha se (green bars) corresponds to the AFLOW structure.\nother. We have confirmed that the volume of inverse\nphase does not significantly vary from that of standard\nphase and small changes in volume do not affect the en-\nergy landscape. We note that these are zero tempera-\nture calculations, but at room temperature (equivalent\nto about 25meV) corresponding free energy landscape\nmay differ. Moreover alloy phases will depend and can\nbe controlled by synthesis conditions.\nIn contrast to these two alloys where standard and\ninverse phases show distinct behaviour, Fe 2CoV alloys\nshow almost no dependence on site occupancy by Fe\natoms either to form standard or inverse phase as shown\nin Fig. 4c. This implies a complete site-disorder if this\nalloy can be formed in the Heusler phase, but it is more\nlikely that it would form body-centered tetragonal phase\nas was observed in experiments [42]. Also overall the cu-\nbic phase is more stable, the tetragonal phase occurs as\na local minimum for this alloy. For Fe 2CoTi (Fig. 4d),\nthe standard T-phase is in metastable state compared\nto the cubic phase and the inverse cubic phase is overall\nenergetically favorable state. Despite their meta-stable\nstructures and in-plane magnetization, we consider both\nFe2CoZ alloys due to their high MAE values and also\nbecause it has been shown experimentally that dopingcan be used to control the sign of MAE for V-doped Fe-\nCo alloys [25]. For Ni 2CoFe (see Fig. 4e), the tetragonal\nphase from AFLOW is indeed the most stable phase ob-\nserved with the inverse tetragonal phase at c/a= 1.4\nlying 27.2meV/atom higher in energy.\nNext, we study finite temperature magnetic properties\nfor the selected five alloys in their tetragonal standard\nphase corresponding to the AFLOW structure. To map\neach system on the Heisenberg model, the pair-wise ex-\nchange interactions Jijwere computed for all the alloys\nunder study, see Fig. 5. Note that we have calculated\nthe pair-wise interactions between all four sites within\nthe unit cell up to a distance of 4.5 aeven though we\nhave only shown here only those interactions for which\nthe largest interaction was ≥1meV for brevity. As ex-\npected, we observethat the nearest-neighborinteractions\nare the largest in magnitude and positive, i.e. FM cou-\npling. Such strong nearest-neighbor interactions arise\nfrom the overlapping 3d orbitals. Often it yields that\nlarger values of Jijfor the nearest neighbor result in a\nhigh Curie temperature. However, note that the sign\nof the interactions changes from positive to negative de-\npending on the distance which implies a competition be-\ntween FM and AFM coupling within system which may6\n0 1 2 3 40102030\nFe-Fe\nFe-Ni\n3 3.5 4 4.5-0.0800.080.16\n0 1 2 30102030Jij (meV)Fe-Fe\nFe-Ni\nFe-Ti\n0 1 2 3-404812Fe-Fe\nFe-Co\nFe-V\nCo-V\n0 1 2 3\nDistance (a)0102030\nFe-Fe\nFe-Co\nCo-Ti\n0 1 2 301020Ni-NiNi-FeNi-CoCo-FeCo-CoFe-Fe(a) Fe2NiZn(b)\n(c) Fe2NiTi (d) Fe2CoV\n(e) Fe2CoTi (f) Ni2CoFe\nFIG. 5. Calculated pair-wise exchange interactions Jijfor Fe 2NiZn (a), Fe 2NiTi (c), Fe 2CoV (d), Fe 2CoTi (e) and Ni 2CoFe (f):\nit is plotted as a function of distance between the sites iandjgiven in units of lattice constant of that alloy. A positive Jijsign\ncorresponds to the ferromagnetic coupling between atoms at iandj, whereas a negative sign corresponds to the AFM coupling.\nNote that the scale is different for y-axis in each panel. In pa nel (b), we have zoomed in on slowly decaying interactions at\nlarger distances for Fe 2NiZn as an example; for other systems, we have truncated the p lots at 3afor clarity.\nimpact the overall temperature dependence.\nFor Z = Zn and Ti alloys, the interactions of Z with X\nand Y are typically negligible and would not contribute\nto the magnetic properties of the system. However, for\nZ = V (see Fig. 5d) we observe strong AFM coupling\nbetween nearest-neighbor Fe-V and Co-V of the same or-\nder of magnitude as leading FM coupling between Fe-Fe\nand Fe-Co. These trends can be understood by looking\nat the local atomic moments for each alloy tabulated in\nTable II. Therefore, for a uniform description for all the\nalloys we retain Jij’s for all four sites to describe each\nsystem. Also note that the Ni 2CoFe alloy differs from\nother alloy systems because this alloy contains only mag-\nnetic 3d metals and therefore results in all FM nearest\nneighbor interactions, whereas in the remaining alloys\nthere is an AFM coupling between Z and X similar to\nthat observed in conventional Heusler alloys.\nThe magnitude of Jijdecreases as the distance be-\ntween the sites increases for all iandjas is expected.\nHowever, the reduction in magnitude is rather slow and\neven at distances of around 3 a(corresponding to about\n15˚A), we observe significant non-zero Jijvalues. Use ofAlloy MXMYMZ\nFe2NiZn 2.59 0.56 −0.06\nFe2NiTi 2.32 0.36 −0.33\nFe2CoV 2.05 1.09 −0.79\nFe2CoTi 2.33 1.05 −0.35\nNi2CoFe 0.69 1.69 2.76\nTABLE II. Local moments on different atomic sites for the\nselected X 2YZ alloys are tabulated in µB. Note that the two\nsites occupied by atoms X for these alloys are equivalent and\nhave the same moment. Similar to the conventional Heusler\nalloys the first four alloys Z atom (Ti, V or Zn) has induced\nmoment with opposite sign which is also reflected in Jij’s\nshown in Fig. 5.\na dense k-mesh ensures that we do not see any noise. As\nan example, in panel (b) we have zoomed in on the tail\npart ofJij’s for Fe 2NiZn between 3-4.5 a. Note that the\nvalues are of the order of 0.1meV and vary between FM\nand AFM coupling. It has been reported that these oscil-\nlating long-range interactions originate from Ruderman-\nKittel-Kasuya-Yoshidainteractionsmediatedviaconduc-7\n0 200 400 600 800\nTemperature (K)0123456Mtot (µB)Fe2NiZn\nFe2NiTi\nFe2CoV\nFe2CoTi\nNi2CoFe\n0 400 8000246\n3.0a\n4.5a\nFe2NiZn\nFIG. 6. Curie temperature calculation: total magnetic mo-\nment as a function of temperature obtained from Monte Carlo\nsimulations is plotted for different alloys. The inset compa res\nthe corresponding data for Fe 2NiZn calculated by including\nonly short-range Jijinteractions within the model.\ntion electrons [43]. Similar behaviour is observed for all\nthe system considered here (not shown) and exclusion of\nthese large distance interactions can impact simulations\nof finite temperature properties as discussed later. Simi-\nlarlongtailsforexchangeinteractionshavebeenreported\nfor conventional Heusler alloys as well [36].\nTo obtain the finite temperature properties of the al-\nloys we fed the Jij’s to the Heisenberg model given in\nEq. (2). The resulting total magnetic moments of the\nsystemMtotas a function of temperature are plotted in\nFig. 6. As expected Mtotdecreases slowly as temper-\nature increases due to random orientations of magnetic\nmoment of each individual atom at different sites aris-\ning from thermal fluctuations. As a result, the material\nundergoes a second-order phase transition to a param-\nagnetic phase at high temperatures. Note that for the\nalloys under study magnetic transition occurs at much\nhigher temperature than the room temperature. The\ncalculated Curie temperature Tcwhich corresponds to\nthe peak in susceptibility from magnetization curves(not\nshown here) is highest for Ni 2CoFe at 740K followed by\nFe2NiTi (620K) and Fe 2CoTi (600K) and then Fe 2NiZn\n(520K) and Fe 2CoV (500K).\nBased solely on the largest observed FM interaction,\nwe wouldexpect that the largest TcforFe2NiTi, Fe 2NiZn\nor Fe2CoTi. However, this strong FM coupling is coun-\nterbalanced by AFM coupling among next nearest neigh-\nbors and the largest Tcis observed Ni 2CoFe for which all\nleading coupling terms are FM because of presence of\nall magnetic metals and AFM coupling is much smaller.\nThe strongest effect of such competition is observed for\nFe2CoV in which strong AFM coupling ofvanadium with\nFe and Co (Fig. 5d) reduces both Curie temperature and\ntotal magnetization at zero temperature.0.8 1 1.2 1.4 1.6\nc/a-3-2-1012εMAE (MJ/m3)\nStandard\nInverse\n0.8 1 1.2 1.4 1.6\nc/a-3-2-10125.76 5.42 5.15 5.85 5.51 5.23 4.92 5.00a (Å) a (Å)\n(a) Fe2NiZn (b) Fe2NiTi\nFIG. 7. Effect of tetragonality on magnetic anisotropy: we\nhave plotted here the calculated MAE for Fe 2NiZn (a) and\nFe2NiTi (b) as a function of c/aratio for both standard and\ninverse phases, and the corresponding in-plane lattice con -\nstantsaare indicated at the top of the plot. The AFLOW\nstructure is highlighted with a red star and the correspondi ng\nMAE values are those shown in Fig. 3.\nThese results correspond to the inclusion of Jij’s cor-\nresponding to all the neighboring sites within the dis-\ntance of 4.5 a. To establish the effect of long-range inter-\nactions on temperature-dependence, we have compared\nthe temperature-dependence of Mtotfor Fe 2NiZn when\nwe exclude Jij’s for sites at a distance larger than 3 aas\nshowninthe inset ofFig. 6. Note thatnearthe transition\ntemperature there is a lot of statistical noise in the data\nand the system does not undergo a smooth transition.\nSuch noise typically implies that the system has com-\npeting phases and is not able to reach equilibrium. For\nMC simulations, this may arise from either inadequate\nsampling or because “model material” does not describe\nthe “real” system completely. Additional tests showed\nthat the noise is not reduced by increasing either the\nsystem size, simulation time or the number of ensembles\nto improve the quality of the data. However, including\nlong-range interactions results in a smoother transition\nconfirming that the long-range nature of exchange inter-\nactions is essential to include in the Heisenberg model\nfor these systems. This noise can be explained by com-\npeting FM and AFM coupling present at larger distance\nshown in Fig 5(b). Therefore, we conclude that for ma-\nterials containing 3d-transition metals it is important to\ncheck convergenceof long-distanceinteractions for better\npredictive power of simulations.\nC. Tuning of MAE\nAs discussed at the beginning of Sec. IIIB, the alloys\nstudied here show two minima (either global or local)\nat the cubic phase and the tetragonal phase with cer-\ntain compressive strain. These two structural phases and\nthe standard and inverse phases arising from chemical8\nsite ordering are shown to have small energy differences,\ntherefore it should be possible to achieve these phases by\napplying strain or using different substrates for growing\nthin films. Typically for 3d elements, the d-orbitals lying\nnear the Fermi level contribute the most to MAE [44],\nand shape and position of d-orbitals is quite sensitive\nto structure, strain and chemical environment for the\nmetal, which in turn means that MAE too is sensitive\nto these changes. Therefore, next we examine the ef-\nfect of tetragonality and site-occupancy on the MAE val-\nues for the most promising alloys: Fe 2NiZn and Fe 2NiTi.\nNote that even though Ni 2CoFe has the highest Tcand\nis quite stable in its tetragonal phase compared to the\ncubic phase, its constituent elements tend to form binary\nalloys with several competing phases [21] making synthe-\nsis of Ni 2CoFe in Heusler structure very difficult. These\narguments based on existing binary phase diagrams are\nfurther confirmed by calculation of the phonon disper-\nsion for these three alloys to check for dynamic stability\nof each phase; see Supplemental Material appended at\nthe end for details. This alloy system therefore would re-\nquire more detailed study based on free energy analysis\nand is not considered for strain tuning.\nThe calculated MAE values as a function of c/aare\nplotted in Fig. 7 for both standardand inversephases. In\nagreement with earlier observations, we observe that the\nstrainleadstochangesinthe magnitudeofMAEandalso\nresultsinrotationoftheeasyplaneaxisfromout-of-plane\ntoin-planeaxisforbothstandardandinversephases. For\ninverse phases (blue diamonds), overall smaller |EMAE|is\nobserved for both strain regimes. Significantly for ap-\nplications, in the vicinity of stable AFLOW structures\n(shown with red star) values of MAE are comparable for\nstandard and inverse phases. This indicates that site dis-\norder should not affect the MAE for these alloys.\nFor both alloysin the standard phase, the MAE is neg-\native for tensile strains implying in-plane magnetization,\nits magnitude increases with increasing strain. However,\nwe observe highly non-monotonic trends in MAE as we\ngo from tensile ( c/a <1) to compressive ( c/a >1)\nstrain. For Fe 2NiZn standard phase, the tetragonal\nphases with c/abetween 1.1 to 1.5 have large positive\nMAE of >1MJ/m3which is very promising for appli-\ncations, whereas for Fe 2NiTi, positive MAE observed for\nthe ground state ( c/a= 1.57) reduces almost to zero\natc/a= 1.4 before going to a large positive value of\n1.2MJ/m3atc/a= 1.2. This trend implies that very ac-\ncurate control of growth or preparation condition would\nbe needed for potential applications for the latter alloy.\nFor conventional Ni-based Heusler alloys, the density of\nstates analysis based on occupation and position of d-\norbitals worked well [8], however for the alloys studied\nhere the correlation is more complex and not obvious.\nBased on the in-plane lattice constants for cubic and\ntetragonal phases (indicated as top x-axis in Fig. 7), we\nsuggest a few suitable substrates – GaAs having lattice\nconstant 4.00 ˚A for cubic phases or Cu with 3.61 ˚A for\ntetragonal phases at the local minima (see Table 1 inRef. [45]) – which have small lattice mismatch calculated\nwith respect to ap=a/√\n2 and therefore can be used to\ngrow the “correct” phase for tuning anisotropy.\nIV. SUMMARY AND CONCLUSIONS\nWe used a combination of high-throughput database\nsearch, density functional theory calculations and Monte\nCarlo simulations to study a new subclass of Heusler al-\nloys – all-3d Heuslers – as potential candidates for per-\nmanent magnets. After application of first filters with\na set of criteria we obtained about 20 systems with\ntetragonal symmetry and high magnetic moments in the\nAFLOW database. For these alloys, we performed a\nthorough examination of preferred easy-plane axis and\nCurie temperatures to find three potential candidates\n– Fe2NiZn, Fe 2NiTi and Ni 2CoFe – with magnetocrys-\ntalline anisotropy of the order of 1 MJ/m3with preferred\nout-of-plane magnetization and which remain ferromag-\nnetic for at least 200K above the room temperature. For\nFe2NiZn and Fe 2NiTi, the energy landscape is rather flat\nimplying strain can stabilize the tetragonal phase. We\nfurther showed for these alloys that strain engineering\nis a viable option to further tune their anisotropy and\nsite disorder between Fe and Ni would not reduce MAE\nsignificantly. A few conventional Heusler alloys show po-\ntential for spintronics applications, however, we obtain\nlow spin polarization values for these alloys (see Supple-\nmental Material appended at the end). Therefore, these\nthree alloys do not appear viable for spintronics applica-\ntions.\nWe also found two alloys – Fe 2CoV and Fe 2CoTi –\nwith high Curie temperature and large MAE, but with\nin-plane easy axis. These alloys are good candidates for\nfurther studies to examine whether alloying and/or dop-\ning can be used to rotate the easy axis. Moreover some\nof the alloys which did not pass our initial filters for per-\nmanent magnets showed some promising properties and\ncould be studied further for applications such as magne-\ntocalorics.\nOur study opens up a new class of rare-earth free per-\nmanent magnets with extensive potential possible via\nthin film growth for strain tuning. We would like to\nnote that due to absence of heavy metals, there is admit-\ntedly a limit to the largest anisotropy obtained within\nthis class of Heusler alloys. However, there exists a large\nmarket for intermediate applications of permanent mag-\nnets which do not require strong anisotropy. We believe\nthat usage of these magnets for such applications can re-\nsult in overall reduction in dependence on the rare-earth\nmetals.\nV. ACKNOWLEDGEMENTS\nWe acknowledge financial support from Olle Engkvist\nfoundation, StandUp, eSSENCE, the Swedish Strategic9\nResearch Foundation (SSF) (Grant EM16-0039),and the\nERC.Computationalresourcesandsupportwasprovided\nby the Swedish National Infrastructure for Computing\n(SNIC) atNSC(Link¨ oping)andPDC(KTHStockholm).MM also thanks Patrik Thunstr¨ om and Rafael Vieira\nfor fruitful discussions and technical support running the\nRSPt and UppASD codes.\n[1] O. Gutfleisch, M. A. Willard, E. Br¨ uck, C. H. Chen,\nS. G. Sankar, and J. P. Liu, Advanced Materials 23, 821\n(2011).\n[2] M. J. Kramer, R. W. McCallum, I. A. Anderson, and\nS. Constantinides, JOM 64, 752 (2012).\n[3] L. H. Lewis and F. Jim´ enez-Villacorta,\nMetallurgical and Materials Transactions A 44, 2 (2013).\n[4] G. Bailey, N. Mancheri, and K. Van Acker, Journal of\nSustainable Metallurgy 3, 611 (2017).\n[5] J. H. van Vleck, Phys. Rev. 52, 1178 (1937).\n[6] C. Felser and A. Hirohata, eds., Heusler alloys: Prop-\nerties, Growth, Applications , Springer series in Material\nScience No. 222 (Springer, 2016).\n[7] Y.-I. Matsushita, G. Madjarova, J. K. Dewhurst,\nS. Shallcross, C. Felser, S. Sharma, and E. K. U. Gross,\nJournal of Physics D: Applied Physics 50, 095002 (2017).\n[8] H. C. Herper, Phys. Rev. B 98, 014411 (2018).\n[9] Q. Gao, I. Opahle, O. Gutfleisch, and H. Zhang,\nActa Materialia 186, 355 (2020).\n[10] V. V. Sokolovskiy, O. N. Miroshkina, V. D. Buchelnikov,\nand M. E. Gruner, Phys. Rev. Mater. 6, 025402 (2022).\n[11] V. G. de Paula and M. S. Reis,\nChemistry of Materials 33, 5483 (2021).\n[12] Z. Y. Wei, E. K. Liu, J. H. Chen, Y. Li, G. D. Liu, H. Z.\nLuo, X. K. Xi, H. W. Zhang, W. H. Wang, and G. H.\nWu, Applied Physics Letters 107, 022406 (2015).\n[13] Z. Y. Wei, E. K. Liu, Y. Li, X. L. Han, Z. W. Du, H. Z.\nLuo, G. D. Liu, X. K. Xi, H. W. Zhang, W. H. Wang, and\nG. H. Wu, Applied Physics Letters 109, 071904 (2016),\nhttps://doi.org/10.1063/1.4961382.\n[14] Z. Ni, Y. Ma, X. Liu, H. Luo, H. Liu, and F. Meng,\nJournal of Magnetism and Magnetic Materials 451, 721 (2018).\n[15] Z. Ni, X. Guo, X. Liu, Y. Jiao, F. Meng, and H. Luo,\nJournal of Alloys and Compounds 775, 427 (2019).\n[16] K. ¨Ozdo˘ gan, I. V. Maznichenko, S. Ostanin,\nE. S ¸a¸ sıo˘ glu, A. Ernst, I. Mertig, and I. Galanakis,\nJournal of Physics D: Applied Physics 52, 205003 (2019).\n[17] S. Samanta, S. Ghosh, and K. Mandal,\nJournal of Physics: Condensed Matter 34, 105801 (2021).\n[18] A. Aznar, A. Gr` acia-Condal, A. Planes,\nP. Lloveras, M. Barrio, J.-L. Tamarit, W. Xiong,\nD. Cong, C. Popescu, and L. Ma˜ nosa,\nPhys. Rev. Materials 3, 044406 (2019).\n[19] S. Samanta, S. Ghosh, S. Chatterjee, and K. Mandal,\nJournal of Alloys and Compounds 910, 164929 (2022).\n[20] S. Samanta, S. Chatterjee, S. Ghosh, and K. Mandal,\nPhys. Rev. Materials 6, 094411 (2022).\n[21] E. P. Wohlfarth, ed., Ferromagnetic materials: A hand-\nbook on the properties of Magnetically ordered sub-\nstances, Vol. 2 (North-Holland Publishing company,\n1980) Chap. 2.\n[22] T. Burkert, L. Nordstr¨ om, O.Eriksson, and O.Heinonen ,\nPhys. Rev. Lett. 93, 027203 (2004).\n[23] K. Hyodo, Y. Kota, and A. Sakuma,\nJournal of the Magnetics Society of Japan 39, 37 (2015).[24] F. Yildiz, M. Przybylski, X.-D. Ma, and J. Kirschner,\nPhys. Rev. B 80, 064415 (2009).\n[25] T. Hasegawa, T. Niibori, Y. Takemasa, and M. Oikawa,\nScientific Reports 9, 5248 (2019).\n[26] Y. Du, G. Z. Xu, X. M. Zhang, Z. Y. Liu, S. Y.\nYu, E. K. Liu, W. H. Wang, and G. H. Wu,\nEPL (Europhysics Letters) 103, 37011 (2013).\n[27] A. K. Jana, M. M. Raja, J. A. Chel-\nvane, P. Ghosal, and S. N. Jammalamadaka,\nJournal of Magnetism and Magnetic Materials 521, 167528 (2021).\n[28] S. Wurmehl, G. H. Fecher, H. C. Kandpal,\nV. Ksenofontov, C. Felser, H.-J. Lin, and J. Morais,\nPhys. Rev. B 72, 184434 (2005).\n[29] P. J. Brown, K. U. Neumann, P. J.\nWebster, and K. R. A. Ziebeck,\nJournal of Physics: Condensed Matter 12, 1827 (2000).\n[30] H. C. Herper and A. Gr¨ unebohm,\nIEEE Transactions on Magnetics 55, 1 (2019).\n[31] G. Kresse and J. Furthm¨ uller, Computational Material s\nScience6, 15 (1996).\n[32] G.Kresse andD.Joubert,Phys. Rev. B 59, 1758 (1999).\n[33] J. M. Wills, M. Alouani, P. Andersson, A. Delin,\nO. Eriksson, and O. Grechnyev, Introduction to elec-\ntronic structure theory, in Full-Potential Electronic\nStructure Method: Energy and Force Calculations with\nDensity Functional and Dynamical Mean Field Theory\n(Springer Berlin Heidelberg, Berlin, Heidelberg, 2010)\npp. 25–34.\n[34] X. Wang, D. sheng Wang, R. Wu, and A. Freeman,\nJournal of Magnetism and Magnetic Materials 159, 337 (1996).\n[35] A. Liechtenstein, M. Katsnelson,\nV. Antropov, and V. Gubanov,\nJournal of Magnetism and Magnetic Materials 67, 65 (1987).\n[36] P. Entel, A.Dannenberg, M. Siewert, H.C. Herper, M. E.\nGruner, D. Comtesse, H.-J. Elmers, and M. Kallmayer,\nMetallurgical and Materials Transactions A 43, 2891 (2012).\n[37] O. Eriksson, A. Bergman, L. Bergqvist, and J. Hellsvik,\nAb-initio spin-dynamics: foundations and applications\n(Oxford University Press, 2017).\n[38] https://aflow.org/.\n[39] F. Rose, C. Toher, E. Gossett, C. Oses,\nM. B. Nardelli, M. Fornari, and S. Curtarolo,\nComputational Materials Science 137, 362 (2017).\n[40] S. Trudel, O. Gaier, J. Hamrle, and B. Hillebrands,\nJournal of Physics D: Applied Physics 43, 193001 (2010).\n[41] J.M.D.Coey,IEEE Transactions on Magnetics 47, 4671 (2011).\n[42] K. Takahashi, M. Sakamoto, K. Ku-\nmagai, T. Hasegawa, and S. Ishio,\nJournal of Physics D: Applied Physics 51, 065005 (2018).\n[43] E. S ¸a¸ sıo˘ glu, L. M. Sandratskii, and P. Bruno,\nPhys. Rev. B 77, 064417 (2008).\n[44] H. Takayama, K.-P. Bohnen, and P. Fulde,\nPhys. Rev. B 14, 2287 (1976).\n[45] L. Zhu and J. Zhao, Applied Physics A 111, 379 (2013).\n[46] A. Togo and I. Tanaka, Scripta Materialia 108, 1 (2015).10\nSUPPLEMENTARY DATA FOR “EXPLORATION\nOF ALL-3D HEUSLER ALLOYS FOR\nPERMANENT MAGNETS: AN AB INITIO\nBASED HIGH-THROUGHPUT STUDY”\nFor Fe 2NiZn, Fe 2NiTi and Ni 2CoFe – three alloys\nwhich show promising magnetic properties – we have\ndone additional analysis. The following results are ob-\ntained for the stable tetragonal phase found from den-\nsity functional theory calculations shown in Fig. 4 of the\nmanuscript.\nS1. PHONON BAND STRUCTURES\nWe used PHONOPY [46] code to obtain phonon dis-\npersion curves for the tetragonal phases of three Heusler\nalloys. The calculations were performed within the\nharmonic approximation using the finite displacement\nmethod. We used a 2 ×2×2 supercell constructed from\na conventional 16-atom unit cell and used displacement\nof 0.01˚A to calculate the forces with the VASP code [31].\nOther computational details are the same those used to\nperform structural analysis.\nThe calculated phonon frequencies along the high-\nsymmetry paths in the Brilloin zone are plotted in Fig. 8.\nWe indeed find that both Fe 2NiZn and Fe 2NiTi alloys\nare dynamically stable with no imaginaryfrequencies ob-\nserved in the phonon dispersion, whereas we get unsta-\nble phonon modes at the Γ point for Ni 2CoFe. Unstable\nΓ modes typically indicate that atoms prefer to displace\nwith respect to eachother. Thisis expected in the caseof\nNi2CoFe because its constituent elements have very sta-\nble binary phases and the Heusler tetragonal phase stud-\niedhereislikelytobeunstable. Theseresultsconfirmour\narguments based on binary phase diagrams of Fe-Co and\nFe-Ni, that this system would not be a suitable candidate\nas a permanent magnet. On the other hand, the dynamicstability of Fe 2NiZn and Fe 2NiTi corroboratesthat these\ntwo alloys are ideal permanent magnet candidates. Here,\nwe would like to note that we have not relaxed the struc-\nture of the systems to allow for shape changes, so there is\napossibilitythatthesystempreferssymmetryotherthan\ntetragonal as the ground state. However, a full analysis\nof all symmetries is beyond the scope of this paper.\nS2. DENSITY OF STATES\nFor Fe 2NiZn, Fe 2NiTi and Ni 2CoFe, we also exam-\nined the spin polarized density of states shown in Fig. 9.\nThis was calculated for the conventional 16-atom unit\ncell using the VASP package. For Fe 2NiZn and Ni 2CoFe,\nthe majority spin channel is almost completely occupied,\nwhereas for Fe 2NiTi there is a small pseudo-gap at the\nFermi level Ef. This typically can result in large spin\npolarization values defined by:\nσ=ρ↑−ρ↓\nρ↑+ρ↓, (4)\nwhereρ↑andρ↓correspond to the density of states\nat Fermi level for spin up and down electrons respec-\ntively. The calculated σvalues for Fe 2NiZn, Fe 2NiTi and\nNi2CoFe are about 69%, 56.6% and 40.7% respectively.\nThese low values result from the existence of sharp peaks\nof localized d-states in the minority channel. Because\nof the existence of multiple localized d-states near Ef,\nsmall changes in the structure, strain may result in shift-\ning of peaks such that the Fermi energy lies not at the\nmaximum, but at a minimum, and will result in spin po-\nlarization which is sensitive to external parameters. For\napplications, materials are required to have robust val-\nueswhichcanbesustainedoveralargeoperationalrange.\nBased on these results, we conclude that these Heusler\nalloys are not half-metallic in nature and therefore would\nnot be suitable for spintronics applications.11\nΓ X M0 Γ M\nWave vector012345678Frequency\n!\"#$%&!'()*\nΓ X M0 Γ M02468Frequency\n!+#$%&!'(,(\nΓ X M0 Γ M−6−4−202468Frequency\n!-#$'(!./%&\nFIG. 8. Calculated phonon band dispersion for the tetragona l Heusler alloys Fe 2NiZn (a), Fe 2NiTi (b) and Ni 2CoFe (c).\n-40040\n-40040total DOS (arb units)\n-10 -5 0 5 10 15\nE - Ef (eV)-40040(a) Fe2NiZn\n(b) Fe2NiTi\n(c) Ni2CoFe\nFIG. 9. Spin polarized density of states calculated for\nthe tetragonal Heusler alloys Fe 2NiZn (a), Fe 2NiTi (b) and\nNi2CoFe (c)."    },    {        "title": "2301.10148v2.Ru___2_x__Mn___1_x__Al_thin_films.pdf",        "content": "Ru2\u0000xMn 1+xAl thin \flms\nK.E. Siewierska,1,\u0003H. Kurt,2B. Shortall,1A. Jha,1N. Teichert,1\nG. Atcheson,1M. Venkatesan,1J.M.D. Coey,1Z. Gercsi,1and K. Rode1\n1School of Physics and CRANN, Trinity College Dublin, Ireland\n2Istanbul Medeniyet University, Department of Engineering Physics, Goztepe Istanbul 34700, Turkey\nThe cubic Heusler alloy Ru 2\u0000xMn1+xAl is grown in thin \flm form on MgO and MgAl 2O4sub-\nstrates. It is a highly spin-polarised ferrimagnetic metal, with weak magnetocrystalline anisotropy.\nAlthough structurally and chemically similar to Mn 2RuxGa, it does not exhibit ferrimagnetic com-\npensation, or large magneto galvanic e\u000bects. The di\u000berences are attributed to a combination of\natomic order and the hybridisation with the group 13 element Al or Ga. The spin polarisation is\naround 50 % to 60 %. There is a gap in the density of states just above the Fermi level in fully\nordered compounds.\nI. INTRODUCTION\nAntiferromagnetic spin electronics is attracting con-\nsiderable interest due to the relative abundance of an-\ntiferromagnetically coupled materials, the fast magneti-\nsation dynamics and the insensitivity to external mag-\nnetic \felds along with the absence of demagnetising\nforces. A major di\u000eculty is control and read-out of\nthe antiferromagnetic state. A compensated ferrimagnet\nwhere two antiferromagnetically coupled, but inequiva-\nlent, sublattices produce zero net moment at the com-\npensation temperature Tcomp potentially combines the\nadvantages of antiferromagnets,[1, 2] with those of a fer-\nromagnetic metal. The magnetisation can be controlled\nby an external \feld away from Tcomp, coupled with a\nhigh transport spin polarisation that facilitates reading\nthe magnetic state using magneto-optical Kerr e\u000bect[3{\n5], anomalous Hall e\u000bect (AHE),[6] and giant- or tunnel-\nmagnetoresistance[7] (GMR or TMR).\nIn 1995, van Leuken and de Groot [8] proposed that\nsome half-Heusler alloys (C1 bstructure) would exhibit\nmagnetic compensation, yet due to the inequivalent mag-\nnetic sublattices exhibit 100 % spin polarisation and thus\nbe half metallic. Subsequent work followed the em-\npirical Slater-Pauling rule,[9] M=Nv\u000018 for half-\nHeuslers and M=Nv\u000024 for full Heuslers. Nvis\nthe number of valence electrons, Mthe magnetisation\nin\u0016Bper formula unit ( \u0016Bf.u.\u00001). The D0 3phase of\nMn3Ga and Mn 3Al were suggested as possible compen-\nsated half-metallic ferrimagnets,[10] but neither crystal-\nlize in the D0 3structure in the bulk. There is a re-\nport on the growth of D0 3-structure Mn 3Al thin \flms\non GaAs substrates with a net moment of 0 :017\u0016Bf.u.\u00001\nand a Curie temperature of 605 K[11]. Disordered an-\ntiferromagnetic Mn 3Al \flms with a cubic structure and\na Ne\u0013 el temperature of 400 K were also grown on MgO\nsubstrates[12]. The equilibrium phase for Mn 2Ga and\nMn3Ga is hexagonal D0 19which can be changed to\ntetragonal D0 22by annealing at 400\u000eC.[13] On the other\nhand Mn 2Al and Mn 3Al crystallize in the cubic \f-Mn\n\u0003siewierk@tcd.iestructure and do not order magnetically but are spin\nglasses.[14] The materials can also be grown in the D0 19\nhexagonal structure when they are produced as epitaxial\nsputtered \flms on suitable seed layers[15].\nThe \frst experimental example of a compensated half\nmetallic ferrimagnet, was MRG, the near-cubic Mn-Ru-\nGa alloy with formula Mn 2RuxGa, discovered by Kurt\net al. [16] in 2014. MRG crystallises in space group F\u001643m\n(XA). The Ru concentration xallows tuning of Tcomp\nand also helps to stabilise the near-cubic structure. The\noriginal composition with x= 0:5 was initially thought\nto haveNv= 21 valence electrons and a half-\flled Ru\nsublattice, but a recent study[17] established that the\n\flms contained few vacant sites and Nv\u001924. Com-\npensated ferrimagnetism has also been demonstrated in\nbulk Mn 1:5FeV 0:5Al, a quaternary Heusler alloys with 24\nvalence electrons[18, 19]. Half-metallic compensated fer-\nrimagnetic materials, such as MRG, are of interest for\nspin orbit torque switching[20, 21] as well as magnetic\noscillations in the terahertz region for high-speed,[22, 23]\non-chip communications.[24, 25].\nThe magnetic and transport properties of MRG includ-\ning their dependence on the Ru content xare understood\nin a rigid band model. One sublattice (formed by states\noriginating from Mn in Wycko\u000b position 4 c) dominates\nthe band structure around the Fermi level and is cou-\npled antiferromagnetically to another (formed by Mn in\nposition 4a) whose states are su\u000eciently deep to not con-\ntribute signi\fcantly to the transport. In this model, the\nrole of Ru is to contribute extra electrons to the 4 csub-\nlattice and hence increase its moment and thus Tcomp.\nThe role of the group 13 element, Ga, is ignored. Al-\nbeit simplistic, this model explains most of the proper-\nties of MRG. Its experimental validation was provided\nby Siewierska et al. [17] who found an excellent linear\nrelation between the number of valence electrons and\nmagnetisation { one added valence electron increases the\nmagnetisation by one \u0016Bper formula unit.\nHere we change from Ga to Al to produce thin \flms of\nMRA, the Mn-Ru-Al Heusler alloy whose formula is best\nwritten Ru 2\u0000xMn1+xAl, and report optimised growth\nconditions, structural and magnetic properties along with\nmagnetotransport and spin polarisation measurementsarXiv:2301.10148v2  [cond-mat.mtrl-sci]  30 Jan 20232\nfor \flms crystallizing in space group Fm\u00163m(L21struc-\nture) with Al and Mn in the 4 a-4bplane and Ru oc-\ncupying the 8 ccentral cube. When x > 0, excess Mn\noccupies a fraction of the 8 cpositions, now split into\n4cand 4d. Note that 4 a;cand 4b;dare symmetrically\nequivalent. The comparison of the two compounds illus-\ntrates the importance of crystalline order in this group\nof Heusler alloys and highlights the importance of Ga.\nII. METHODOLOGY\nEpitaxial thin \flms of MRA were grown by DC mag-\nnetron sputtering, using our Shamrock sputtering sys-\ntem, on 10 mm\u000210 mm (100) SrTiO 3(STO), MgAl 2O4\n(MAO) and MgO substrates. Samples were co-sputtered\nin Argon from a Mn 2Al target (from Kojundo Chemi-\ncal Laboratory Co. Ltd, Japan) and a Ru target in a\nconfocal sputtering geometry onto the substrates main-\ntained at an optimized deposition temperature ( Tdep) we\nfound to be 425\u000eC. Prior to deposition, the back of the\nsubstrates was coated with Ti or Ta to ensure uniform\nabsorption of heat and keep Tdepconstant during de-\nposition. The \flms had an average thickness of 60 nm\nand were capped in-vacuo with a 2 nm layer of AlO x\ndeposited at room temperature in order to prevent oxi-\ndation. We note that this arrangement of targets does\nnot yield perfect Ru 2\u0000xMn1+xAl stoichiometry as a de-\ncrease in Mn also leads to decreased Al content. The sum\nof the amounts of Ru and Mn in the formula unit varies\nfrom 3:0 to 3:3 in our samples, and we therefore write\nthe formula as Ru 2\u0000xMn1+xAl where it is implicit that\nAl is sub-stoichiometric by up to 30 %.\nA Bruker D8 Discover X-ray di\u000bractometer with a cop-\nper K\u000bX-rays and a double-bounce Ge [220] monochro-\nmator (\u0015= 154:06 pm) was used to determine the di\u000brac-\ntion patterns and reciprocal space maps (RSM) of the\nthin \flms. The substrate (113) re\rection is in the same\nplane as the MRA (204) re\rection and it was used to cal-\nculate the lattice parameters. Low angle X-ray re\rectiv-\nity and symmetric X-ray di\u000braction patterns were mea-\nsured using a Panalytical X'Pert Pro ( \u0015= 154:19 pm)\ndi\u000bractometer. Film thickness and density were found\nby \ftting the interference pattern using X'Pert Re\rec-\ntivity software. The densities were used to deduce the\nstoichiometric ratios assuming full site occupation.[17]\nLow-\feld anomalous Hall e\u000bect (AHE) measurements\nwere performed in a 1 T GMW electromagnet system in\nambient conditions. Silver wires were cold-welded to the\nthin \flms with indium and the current used was 5 mA.\nHigher \feld measurements were performed in a supercon-\nducting magnet which had a maximum \feld of 4 T with\na cryostat for low-temperature magneto-transport. The\n\flms were contacted with silver paint. The 4-point Van\nder Pauw geometry was used to determine both the Hall\nand the longitudinal resistivity of the \flms.\nPoint contact Andreev re\rection (PCAR) measure-\nments were also performed in a Quantum Design PPMSusing a mechanically-sharpened Nb tip. The landing of\ntip onto the sample surface is carefully controlled by an\nautomated vertical attocubeTMpiezo stepper. Two hor-\nizontal steppers are used to move the sample laterally\nto probe a pristine area. The di\u000berential conductance\nspectra were \ftted using a modi\fed Blonder-Tinkham-\nKlapwijk model, as detailed elsewhere.[26, 27]\nMeasurements of the magnetisation with a magnetic\n\feld applied perpendicular or parallel to the surface of\nthe \flms were carried out using a 5 T Quantum Design\nSQUID magnetometer These data were corrected for the\ndiamagnetism of the substrate by linear subtraction.\nAb-initio calculations based on density functional the-\nory (DFT) were carried out using norm-conserving pseu-\ndopotentials and pseudo-atomic localized basis func-\ntions implemented in the OpenMX software package.[28]\nThe generalized gradient approximation (GGA-PBE) ap-\nproach was used for all calculations. The structure was\nfully relaxed to minimize interatomic forces. We used\na 16-atom supercell cell with 17 \u000217\u000217k-points to\nevaluate the total energies. Pre-generated pseudopoten-\ntials and pseudo-atomic orbitals with a cut-o\u000b radius of\n6, 7 and 7 atomic units (a.u.) were used for Mn, Ru and\nAl elements, respectively. An energy cut-o\u000b of 300 Ry\nwas used for numerical integrations. The convergence\ncriterion for the energy minimization procedure was set\nto 10\u00008Hartree. The spin orbit interaction (SOI) was\nturned o\u000b for the calculations.\nIII. RESULTS & DISCUSSION\nA. Structure\nWe \frst determine a suitable substrate for the growth\nof Ru 2\u0000xMn1+xAl thin \flms, and optimise the depo-\nsition temperature Tdep. MRG has a lattice parame-\nter ofa0\u0019600 pm and its epitaxial relation with the\nMgO substrate ensures that the in-plane [100] direction\nof MgO is parallel to the [110] direction of MRG, aMRG\u0019p\n2aMgO = 595:6 pm. We therefore explore the growth\nof MRA on MgO, SrTiO 3(STO,p\n2a= 552:3 pm) and\nMgAl 2O4(MAO,p\n2a= 571:8 pm).\nIn FIG. 1a (left column) we show X-ray di\u000braction pat-\nterns of Ru 1:9Mn1:1Al on MgO, MAO and STO. MRA\ndoes not crystallize on STO, while on both MAO and\nMgO the \flms are fully textured with the growth axis\nparallel to the [001] crystal direction. There are only\nminor di\u000berences between MgO and MAO, and we se-\nlect MgO as the preferred substrate. In FIG. 1a (right\ncolumn) the deposition temperature Tdepis varied from\n350\u000eC to 440\u000eC. Although a minor secondary phase\nis present in all samples, it is nearly suppressed at\nTdep= 425\u000eC. Finally, in FIG. 1b we show the patterns\nfor Ru 2\u0000xMn1+xAl \flms with x= 0:1;0:3;0:5 and 0:6.\nThe \flms are fully textured in the entire composition\nrange, with only a minor secondary phase whose asso-\nciated peaks are about 3 to 4 orders of magnitude less3\n102104106\n30 40 50 60 70S♣102104106\nS♣(002)(004)102104106\nS♣♢(002)(004)\n30 40 50 60 70S♣♢ ♢(002)\n(004)S♣♢ ♢(002)(004)S♣♢ ♢\n(a)(002)(004)\n2θ(◦)STOCountsMAOMgOTdep= 425◦C\n440◦C390◦C350◦CMgO [001]\nTdep= 425◦C\n102104106\nS♣\n♢ ♢(002)(004)\n102104106\n30 40 50 60 70S♣\n♢ ♢(002)(004)S♣\n♢ ♢(002)(004)\n30 40 50 60 70S♣\n♢ ♢\n(b)(002)(004)\nCountsx= 0.6\n2θ(◦)x= 0.5x= 0.3\nx= 0.1\nFIG. 1. X-ray di\u000braction patterns. (a) Ru 1:9Mn1:1Al de-\nposited on MgO, MAO and STO substrates at Tdep= 425\u000eC\n(left column) and on MgO with varying Tdep(right col-\numn). (b) Ru 2\u0000xMn1+xAl deposited at Tdep= 425\u000eC for\nx= 0:1;0:3;0:5 and 0:6. The substrate (002) re\rection\nis marked by `S', while the corresponding Cu K\fre\rection\n(\u0015= 139:23 pm) is marked by |. We also indicate a minor\nsecondary phase ( }).\nintense than the MRA peaks. A likely origin is a very\nsmall amount ( <1 %) of either a Ru-Mn[29] or Ru-Al\nbinary alloy in the \flm, or a Ru oxide at the interface\nbetween the \flm and the capping layer.\nThe RSM data was collected around the (113) re\rec-\ntion of MgO and shown in FIG. 2. The in-plane epitaxial\nrelationship of MgO and MRA is assumed to be MgO\n[100] parallel MRA [110], therefore we label the MRA\npeak as the (204) re\rection. We determined the in-plane\n(a) and out-of-plane lattice parameters ( c). The epitaxy\nimproves with increasing Ru content and the \flms are\nnear-cubic, with a maximal tetragonal distortion\u0000c\u0000a\na\u0001\n6.66.87.07.2\n123456log (Counts)6.66.87.07.2\n3.25 3.30 3.35 3.40 3.25 3.30 3.35 3.40qz/bardbl[001] (nm−1)x= 0.6 x= 0.6 x= 0.5 x= 0.5\nqx/bardbl[110] (nm−1)x= 0.3 x= 0.3 x= 0.1 x= 0.1FIG. 2. Reciprocal space maps around the MgO [113] (MRA\n[204]) re\rection from which we infer the in-plane lattice pa-\nrameter of MRA. The samples are ordered in the sample\nsurface plane.\nTABLE I. Crystal parameters of Ru 2\u0000xMn1+xAl. The lattice\nparameters in columns three and four are determined from\nreciprocal space mapping, see FIG. 2. The experimentally ob-\nserved ratio of S=F2\n002=F2\n004is compared to that calculated\nfor a fully ordered structure with the stoichiometry deduced\nfrom the \flm density. Mn fully occupies the 4 bposition, and\nRu most of the 8 c. The remaining 8 cis \flled by Mn. Al\noccupies only 4 a.\n#x a (pm)c(pm)S(exp)S(th)\nS08 0:10 596:80 600:53 0:25 0:26\nS11 0:10 596:95 601:54 0:23 0:26\nS13 0:10 597:17 600:79 0:23 0:26\nS16 0:23 595:87 601:92 0:23 0:25\nS12 0:25 596:05 602:31 0:20 0:25\nS03 0:36 596:82 602:19 0:19 0:24\nS09 0:49 600:81 602:12 0:24 0:23\nS20 0:55 598 600 0 :29 0:23\nS15 0:61 600:41 600:39 0:30 0:22\nS10 0:84 600:11 598:70 0:31 0:20\nof 1 % found for Ru 1:6Mn1:4Al. We summarize the struc-\ntural parameters of the samples in TABLE I. The degree\nof Ru order is estimated from the ratio S=F2\n002=F2\n004.\nL21-ordered (Fm\u00163m) Ru 2:0Mn1:0Al and Ru 1:0Mn2:0Al\nhaveS= 0:33 and 0:18, respectively. For the XA vari-\nants (F\u001643m), both Ru 2:0Mn1:0Al (Ru on 4 band 4d, Mn\nin 4c) and Ru 1:0Mn2:0Al (Mn on 4 band 4d, Ru occu-\npying 4c) haveS\u00190:02. Complete disorder among all\nelements in all positions suppresses the (002) re\rection\ncompletely and thus S= 0. We calculated the theoret-\nical ratioSusing the stoichiometries inferred from the\ndensities and an isotropic Debye-Waller factor of 0 :3\u0017A2\nfor all atoms. We \fnd that for x<0:5 the calculated and\nthe observed Sagree reasonably well, while for x>0:5,4\n-150-100-50050100150\n-15-10-5051015\nx= 0.10\n-150-100-50050100150\n-0.4 -0.2 0 0.2 0.4-40-2002040\nx= 0.49\nM(kA m−1)\nRxy(mΩ)M\nAHE\nµ0H(T)\nFIG. 3. Magnetometry and AHE recorded on two repre-\nsentative samples at 296 K. The applied magnetic \feld is\nperpendicular to the sample surface. In the upper panel, the\nAHE loop is plotted as a function of ( \u0000\u00160H) for easier com-\nparison with the magnetometry data. Both techniques indi-\ncate easy plane shape anisotropy with the saturation \feld (i.e.\nthe anisotropy \feld) near-equal to \u00160Ms. The magnetometry\ndata were corrected for the diamagnetism of the substrate by\nlinear subtraction of the high \feld slope, as was the normal\ncontribution to the transverse voltage.\nthe experimentally observed Sis higher than predicted\nby the fully-ordered model indicating a degree of disor-\nder. This disordered phase is likely to retain the overall\nstructure, but with increased Debye-Waller factors. A\nmuch-improved agreement with the observed ratios for\nx= 0:55;0:61 and 0:84 is obtained supposing the Debye-\nWaller factor is 0 :9\u0017A2.\nB. Magnetism and magnetotransport properties\nWe now turn to the magnetotransport properties of\nMRA. In FIG. 3 we show magnetometry and anomalous\nHall e\u000bect (AHE) loops of two representative samples,\nwhere the diamagnetic contribution from the substrate\nwas subtracted as a linear slope in the magnetometry\ndata, as was the high-\feld contribution to the transverse\nresistivity due to the normal Hall e\u000bect. The shape of\nthe hysteresis loops recorded using the two techniques\nare near-identical suggesting that the AHE re\rects the\nnet magnetisation of the sample. Both magnetometry\nand AHE indicate weak easy-plane anisotropy, which is\na result of the sample shape. A small contribution from\nperpendicular magnetocrystalline anisotropy due to the\n1 % tetragonal distortion of the unit cell is present and\ntherefore\u00160Han<\u0016 0Ms. We measured hysteresis loops\nof a Ru 1:9Mn1:1Al sample at di\u000berent temperatures in\norder to extract MsandHan. From these we infer K1\nas a function of temperature. Assuming that K1(T) =\nK1(0)\u0002(Ms(T)=M0)3we \fndK1(0) = 5:4 kJ m\u00003.TABLE II. Summary of Ru 2\u0000xMn1+xAl magnetic and trans-\nport properties. The magnetisation decreases monotonically\nwith increasing xforx >0:1. We note that the sign of \u001bxy\nchanges at x\u00190:25. In the approximation of a single species\nof carriers contributing to the transport, the carrier concen-\ntrationnof the low-xsamples (x/0:25) is approximately\ntwice that of the high- xsamples, while their mobility is half\nas great.\n#x Ms\u001bxx\u001bxyn \u0016\n(kA=m) (MS=m) (kS=m) (1028=m3) (mV m=s2)\nS08 0:10 137 0 :40\u00000:71 3:69 0:06\nS11 0:10 125 0 :41\u00000:81 2:18 0:10\nS13 0:10 150 0 :40\u00000:62 2:03 0:10\nS12 0:25 135 0 :42\u00001:38 2:76 0:08\nS16 0:23 135 0 :35 0:60 1:30 0:14\nS03 0:36 130 n/a n/a n/a n/a\nS09 0:49 107 0 :33 0:93 0:90 0:19\nS20 0:55 100 n/a n/a n/a n/a\nS15 0:61 76 0 :35 0:60 0:99 0:19\nS10 0:84 25 n/a n/a n/a n/a\nWeiss mean \feld theory was used to \ft the\ntemperature-dependent magnetisation data to obtain the\nexchange parameters and estimate the Curie temperature\nforx= 0:14. We take the Mn 4 bmoment to be 2 :66\u0016B\nfrom the DFT calculation (see FIG. 7) giving a sublat-\ntice magnetisation of 451 kA m\u00001. The Mn 8 c(now 4d)\nis then deduced from the di\u000berence between the net mag-\nnetisation at low temperature and that of the Mn 4 band\nRu 4csublattices to be \u00193:1\u0016Bper Mn or 92 kA m\u00001.\nThe choice of the quantum number Sin the Brillouin\nfunction of the localized mean \feld theory is always prob-\nlematic in a metal. It should somehow take account of the\nd-electron hybridisation with Al and Ru in our alloy. A\nchoice ofS=5=2for both Mn sites leads to an excellent\n\ft of the experimental data with three molecular \feld\ncoe\u000ecients nbb,nbdandnddand coordination numbers\nZbb,Zbd,Zdd= 12, 5:6, and 4:8 determined from the\ncomposition. Our \ft indicates that the material orders\njust above room temperature at TC= 353 K. Heisen-\nberg exchange energies were calculated to be Jbb,Jbd,\nJdd= 0:96,\u00002:40, and 1:33 meV per bond. The ratio of\nthe product of the coordination number and the Heisen-\nberg exchange energies is JbbZbb:JbdZbd:JddZddequal\nto 10 :\u000010 : 2:4. The exchange is a factor of two smaller\nthan for MRG, which explains the lower Curie temper-\nature. The net anisotropy constant is smaller than for\nMRG re\recting the shared crystal structure and the sim-\nilar degree of tetragonal distortion.\nFrom the observed transverse ( \u001axy) and longitudinal\n(\u001axx) resistivity we determine the carrier concentration\nand mobility assuming a single species of charge carriers.\nThe results are summarized in TABLE II. The magnetic\nordering temperature is above room temperature for x/\n0:6.\nThe sign of the transverse condictivity \u001bxy\u0019\u001axy=\u001a2\nxx\nchanges at x\u00190:26. Such a sign change is usually5\n-20020\n-20020\n-20020\n-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1\nRxy(mΩ)300 K\n250 K\nµ0H(T)200 K150 K\n100 K\n50 K\nFIG. 4. AHE recorded at 300, 250, 200, 150, 100 and 50 K\nfor sample S08. The sign of \u001bxychanges between 150 K and\n100 K, but the anisotropy \feld remains near constant, indi-\ncating that this change is due to a change in the nature (con-\ncentration and mobility) of the predominant charge carrier\nrather than crossing a temperature where magnetic compen-\nsation occurs.\nattributed to crossing the compensation point where\nMs\u00190. In FIG. 4 we plot the transverse resistance\nmeasured for Ru 1:9Mn1:1Al at 300, 250, 200, 150, 100\nand 50 K. As when changing x, its sign changes be-\ntween 150 K to 100 K. The anisotropy \feld, however,\nchanges monotonically from 0 :1 T to 0:2 T between room\ntemperature and 50 K without any divergence indicat-\ning anMsclose to zero. We therefore attribute the sign\nchange to a temperature-dependent \flling of two or more\npockets in the band structure near the Fermi energy,\nin agreement with the change of carrier concentration\nand their mobility shown in TABLE II. The change in\nthe anisotropy \feld due to shape anisotropy suggests the\nsaturation magnetisation increases by \u00190:9\u0016Bf.u.\u00001be-\ntween 300 K to 50 K, in agreement with low-temperature\nmagnetometry measured on other samples (not shown).\nThe results indicate that Ru 2\u0000xMn1+xAl is a ferrimag-\nnet where Ru in 8 cand Mn in 4 bform antiparallel fer-\nromagnetic sublattices. No compensation composition is\nfound in the range of xwe have investigated, nor is there a\ncompensation temperature due to the di\u000berent local envi-\nronments of Mn and Ru. Two (or more) species of charge\ncarriers with opposite spin dominate the Fermi level, and\ntheir concentrations and mobilities di\u000ber by around a\nfactor of two. Ru 2\u0000xMn1+xAl is not a half metal. In\nFIG. 5 we show the saturation magnetisation at 300 K as\na function of the number of valence electrons Nv. The\ndependence is non-linear, however for low Nv<26:3, the\nmagnetisation increases with 0 :6\u0016Bf.u.\u00001e\u00001, suggest-\ning a spin polarisation Paround 60 % in this region. The\ninsert in FIG. 5 shows point-contact Andreev re\rection\nrecorded on Ru 1:6Mn1:4Al. We \fnd the spin polarisation\n00.10.20.30.40.50.60.70.80.9\n25.5 26 26 .5 270.80.91.0\n-6 -4 -2 0 2 4 6\nµBf.u.−1\nNvGn\nBias ( mV)FIG. 5. Magnetisation of Ru 2\u0000xMn1+xAl as a function of\nnumber of valence electrons ( Nv). The solid line is a guide\nto the eye. For Nv/26:5 the magnetisation increases by\n\u00190:6\u0016Bf.u.\u00001e\u00001. The insert shows a point contact An-\ndreev re\rection (PCAR) spectrum for x= 0:25. The spin\npolarisation inferred from the \ft (solid line) is P= 52 %.\nP= 52 % in good agreement with that inferred from the\nplot ofMagainstNv.\nC. Comparison with density functional theory\nFinally we compare our results with ab initio calcu-\nlated magnetisation and density of states. We used a\n16-atom cell for all our calculations, and relaxed both\nlattice parameters ( a\u0019600 pm) and magnetic moments,\nbut kept the individual atoms at their assigned Wycho\u000b\npositions in agreement with the structural model inferred\nfrom the analysis of the x-ray data above. In FIG. 6 we\nshow the density of states (DoS) around the Fermi level\nfor a series of Ru 2\u0000xMn1+xAl samples with varying x.\nThe label on each panel corresponds to the content of the\n16-atom cell used for calculations, with Ru occupying the\n8cand Mn and Al the 4 band 4apositions, respectively.\nFor Ru 2:0Mn1:0Al, in particular, there is a gap in the spin\ndown DoS, but its onset is almost 1 eV above the Fermi\nlevel. With decreasing Ru (and increasing Mn) the gap\nmoves closer towards the Fermi level, but is simultane-\nously destroyed by the creation of supplementary states\nin the gap. None of the compositions are half metallic,\nalthough perfectly ordered Ru 1:0Mn2:0Al in the top left\npanel of FIG. 6 comes close. We have seen above that this\ncomposition disorders during growth. The bottom right\npanel of FIG. 6 illustrates the e\u000bect of Al substoichiome-\ntry. From these \fgures we estimate the spin polarisation\nof MRA to be around 50 %, in good agreement with the\nexperimental results.\nWe also show, in FIG. 7, the site- and ion-resolved\nmagnetic moments obtained by theory. The net moment6\n-40-20020\nRu8Mn 4Al4\n-40-20020\nRu7Mn 5Al4\n-40-20020\n-4 -3 -2 -1 0 1Ru6Mn 6Al4Ru5Mn 7Al4\nRu4Mn 8Al4\n-4 -3 -2 -1 0 1Ru7Mn 6Al3\nDoS ( eV−1)\nEnergy ( eV)\nFIG. 6. Density of states for Ru 2\u0000xMn1+xAl withxranging\nfrom 0 to 1. The composition Ru 7Mn6Al3was selected as it\ncorresponds closely to the experimental sample S16. It illus-\ntrates the change in the density of states due to the under sto-\nichiometry in Al, although neither the spin polarisation ( P\u0019\n50 %) nor the magnetic moment ( M\u00191:5\u0016Bf.u.\u00001) changes\nsigni\fcantly compred to the ordered version (Ru 7Mn5Al4).\n−3−2−101234\n0 0 .2 0 .4 0 .6 0 .8 1\nMoment ( µB)\nx(Ru2−xMn 1+xAl)DFT\nRu4c\nMn4bMn4d\nExp\nFIG. 7. Experimental and calculated net and sublattice\nmoments for xin the range 0 to 1. The net calculated and\nexperimental moments are given per formula unit. Note that\nthe experimental values have been shifted by 0 :9\u0016Bf.u.\u00001to\naccount for the higher magnetisation at T= 0 K.\ndecreases from\u00192\u0016Bf.u.\u00001to 1\u0016Bf.u.\u00001forxin the\nrange 0 to 1. Mn in the 4 bposition carries a strong\nmoment\u00193\u0016Bthat increases with increasing x, sug-\ngesting increased localisation of the 3 dbands. As xis\nraised above zero, some Mn \flls the 8 c(now 4d) position\nand it is coupled antiferromagnetically to Mn in 4 b, but\nferromagnetically to the remaining Ru (in 8 c/4c). Ru\ncarries a small moment in all compositions, but inter-\nestingly changes sign at x\u00190:75. In the \fgure we alsoplot the experimentally observed magnetisation, o\u000bset by\n0:9\u0016Bf.u.\u00001to account for the decrease of the magneti-\nsation between T= 0 K and 300 K (see above).\nIV. CONCLUSIONS\nWe \fnd that Ru 2\u0000xMn1+xAl crystallises on MgO and\nMgAl 2O4substrates in a tetragonally-distorted near-\ncubicFm\u00163mcrystal structure. For low values of x\n(<0:5), the structure remains essentially ordered with\nRu in the 8 cposition and Mn and Al in 4 band 4a, re-\nspectively, with the additional Mn \flling the sites vacated\nby Ru. As xis increased further, the crystal structure\nevolves towards F\u001643m, although order between the 4 c\nand 4dsites does not occur, and additionally at low x,\nthe 4a-4border is perturbed. All compositions are mag-\nnetically ordered and exhibit a high spin polarisation of\naround 50 %. Ab initio calculations agree well with the\nobserved crystal structure and magnetic mode. MRA is\nferrimagnetic, with Mn in the 4 bposition coupling anti-\nferromagnetically to Ru in 8 c. Two or more species of\ncharge carriers close to or at the Fermi level account for\nthe reduction of the spin polarisation and the change in\nthe sign of the Hall conductivity with xand T. In contrast\nto MRG, where the spin polarisation as inferred from the\nSlater-Pauling plot[17] is close to P= 100 %, the anoma-\nlous Hall angle ( \u001bxy=\u001bxx) of MRA is only \u00182‰, more\nthan an order of magnitude less than for MRG despite a\nhigher concentration of Ru.\nWhenxis increased above \u00190:6, MRA no longer ex-\nhibits AHE at room temperature although the crystal\nstructure is increasingly similar to that of MRG. We\ntherefore conclude that Ga, and not Ru, is promoting\nhigh conduction band spin orbit coupling in MRG and\nhelps to move the Fermi level into the spin gap by al-\nlowing Ga-Al antisites to form.[30] It can be argued that\nMRA retains inversion symmetry due to disorder, even\nwhenxis above\u00180:6. MRG has no centre of inversion\nbecause Ru and Mn order on the 4 dand 4cpositions, and\na topological contribution to the transverse conductivity\nis allowed by symmetry.\nAtx= 0, the Mn-Mn distance is 600 pm, and the\nMn orders ferromagnetically. The moment carried by\nRu is small, and it is unlikely that the magnetic order\nis due to the Ru-Mn AFM interaction. The question\nof the nature of this long-distance exchange interaction\narises. Several other Heusler alloys (Rh 2Mn(Pb;Sn;Ge)\nand Cu 2Mn(In;Sn;Al))[31, 32] where only one out of four\natoms per formula is manganese order ferromagnetically\nabove 300 K, and Au 4Mn[33] with a shorter \frst neigh-\nbour Mn-Mn distance of 404 pm has a Curie point at\n390 K. There, all the exchange interactions up to the\ntenth nearest-neighbours at 1000 pm are either ferromag-\nnetic or zero. Since di\u000berent elements are present in these\ndilute Mn-based ferromagnets, it is likely that long-range\nferromagnetic exchange is mediated by a p-band. The\nMn-Mn exchange (in Au 4Mn) only becomes oscillatory7\nand RKKY-like at Mn-Mn distances exceeding 1040 pm.\nIn summary, Ru 2\u0000xMn1+xAl is a ferrimagnetic Heusler\nalloy with high spin polarisation and magnetic order-\ning at room temperature. The absence of signi\fant\nspin-orbit coupling precludes perpendicular magnetic\nanisotropy as well as any notable magneto-resistive ef-\nfects. This contrast with MRG where Al is replaced by\nGa, is likely due to a combination of hybridisation of the\nMndandsstates with Ga pbands, and the presence, or\nabsence, of a centre of inversion.ACKNOWLEDGMENTS\nK.E.S. and J.M.D.C. acknowledge funding through\nIrish Research Council under Grant GOIPG /2016/308\nand in part by the Science Foundation Ireland under\nGrants 12/RC/2278 and 16/IA/4534. B.S. acknowl-\nedges funding from the Irish Research Council Grant\nEPSPG/2020/106G.A. A.J., G.A., Z.G., and K.R. ac-\nknowledge funding from `TRANSPIRE' FET Open Pro-\ngramme, H2020. N.T. was supported by the Euro-\npean Union's Horizon 2020 Research and Innovation Pro-\ngramme under Marie Sk lodowska-Curie EDGE Grant\n713567. H.K. and M.V. were funded through the ZEMS\ncontract, Science Foundation Ireland, grant 16/IA/4534.\n[1] L. Wollmann, A. K. Nayak, S. S. Parkin, and C. Felser,\nAnnual Review of Materials Research 47, 247 (2017).\n[2] T. Graf, C. Felser, and S. S. Parkin, Prog. Solid State\nCh.39, 1 (2011).\n[3] N. Teichert, G. Atcheson, K. Siewierska, M. N. Sanz-\nOrtiz, M. Venkatesan, K. Rode, S. Felton, P. Stamenov,\nand J. M. D. Coey, Phys. Rev. Materials 5, 034408\n(2021).\n[4] K. E. Siewierska, N. Teichert, R. Sch afer, and J. M. D.\nCoey, IEEE Trans. Mag. 55, 1 (2019).\n[5] C. Banerjee, N. Teichert, K. Siewierska, Z. Gercsi,\nG. Atcheson, P. Stamenov, K. Rode, J. M. D. Coey, and\nJ. Besbas, Nat. Commun. 11, 4444 (2020).\n[6] N. Thiyagarajah, Y.-C. Lau, D. Betto, K. Borisov,\nJ. M. D. Coey, P. Stamenov, and K. Rode, Appl. Phys.\nLett. 106, 122402 (2015).\n[7] K. Borisov, D. Betto, Y. C. Lau, C. Fowley, A. Titova,\nN. Thiyagarajah, G. Atcheson, J. Lindner, A. M. Deac,\nJ. M. D. Coey, P. Stamenov, and K. Rode, Appl. Phys.\nLett. 108, 192407 (2016).\n[8] H. van Leuken and R. A. de Groot, Phys. Rev. Lett. 74,\n1171 (1995).\n[9] I. Galanakis, P. Mavropoulos, and P. Dederichs, J. Phys.\nD Appl. Phys. 39, 765 (2006).\n[10] S. Wurmehl, H. C. Kandpal, G. H. Fecher, and C. Felser,\nJ. Phys. Cond. Mat. 18, 6171 (2006).\n[11] M. E. Jamer, Y. J. Wang, G. M. Stephen, I. J. McDon-\nald, A. J. Grutter, G. E. Sterbinsky, D. A. Arena, J. A.\nBorchers, B. J. Kirby, L. H. Lewis, B. Barbiellini, A. Ban-\nsil, and D. Heiman, Phys. Rev. Applied 7, 064036 (2017).\n[12] H.-W. Bang, W. Yoo, C. Kim, S. Lee, J. Gu, Y. Park,\nK. Lee, and M.-H. Jung, Applied Physics Letters 115,\n012402 (2019).\n[13] H. Kurt, K. Rode, M. Venkatesan, P. Stamenov, and\nJ. M. D. Coey, physica status solidi (b) 248, 2338 (2011),\n10.1002/pssb.201147122.\n[14] R. Zhang, Z. Gercsi, M. Venkatesan, A. Jha, P. Sta-\nmenov, and J. Coey, Journal of Magnetism and Magnetic\nMaterials 501, 166429 (2020).\n[15] X. Liu, I. Ohnuma, R. Kainuma, and K. Ishida, Journal\nof Phase Equilibria 20, 45 (1999).\n[16] H. Kurt, K. Rode, P. Stamenov, M. Venkatesan, Y.-C.\nLau, E. Fonda, and J. M. D. Coey, Phys. Rev. Lett. 112,\n027201 (2014).\n[17] K. E. Siewierska, G. Atcheson, A. Jha, K. Esien,R. Smith, S. Lenne, N. Teichert, J. O'Brien, J. M. D.\nCoey, P. Stamenov, and K. Rode, Phys. Rev. B 104,\n064414 (2021).\n[18] R. Stinsho\u000b, G. H. Fecher, S. Chadov, A. K. Nayak,\nB. Balke, S. Ouardi, T. Nakamura, and C. Felser, AIP\nAdv, 7, 105009 (2017).\n[19] R. Stinsho\u000b, A. K. Nayak, G. H. Fecher, B. Balke,\nS. Ouardi, Y. Skourski, T. Nakamura, and C. Felser,\nPhys. Rev. B 95, 060410 (2017).\n[20] J. Finley, C.-H. Lee, P. Y. Huang, and L. Liu, Advanced\nmaterials (Deer\feld Beach, Fla.) 31, e1805361 (2019).\n[21] J. Finley and L. Liu, Appl. Phys. Lett. 116, 110501\n(2020).\n[22] C. Banerjee, K. Rode, G. Atcheson, S. Lenne, P. Sta-\nmenov, J. M. D. Coey, and J. Besbas, Phys. Rev. Lett.\n126, 177202 (2021).\n[23] C. S. Davies, G. Bon\fglio, K. Rode, J. Besbas, C. Baner-\njee, P. Stamenov, J. M. D. Coey, A. V. Kimel, and A. Kir-\nilyuk, Phys. Rev. Research 2, 032044 (2020).\n[24] D. Betto, K. Rode, N. Thiyagarajah, Y.-C. Lau,\nK. Borisov, G. Atcheson, M. \u0014Zic, T. Archer, P. Stamenov,\nand J. M. D. Coey, AIP Adv. 6, 055601 (2016).\n[25] M. Chen, R. Mishra, Y. Wu, K. Lee, and H. Yang, Ad-\nvanced Optical Materials 6, 1800430 (2018).\n[26] P. Stamenov, J. Appl. Phys. 113, 17C718 (2013).\n[27] K. Borisov, C.-Z. Chang, J. S. Moodera, and P. Sta-\nmenov, Phys. Rev. B 94, 094415 (2016).\n[28] Openmx website.\n[29] M. Yin and P. Nash, Journal of Alloys and Compounds\n634, 70 (2015).\n[30] M. \u0014Zic, K. Rode, N. Thiyagarajah, Y.-C. Lau, D. Betto,\nJ. M. D. Coey, S. Sanvito, K. J. O'Shea, C. A. Fergu-\nson, D. A. MacLaren, and T. Archer, Phys. Rev. B 93,\n140202(R) (2016).\n[31] H. P. J. Wijn, Magnetic Properties of Metals: d-elements,\nAlloys and Compounds , edited by H. P. J. Wijn, Data\nin Science and Technology (Springer Berlin, Heidelberg,\n1991).\n[32] J. C. Suits, Phys. Rev. B 14, 4131 (1976).\n[33] Y. He, Z. Gercsi, R. Zhang, Y. Kang, Y. Skourski,\nL. Prendeville, O. Larmour, J. Besbas, C. Felser, P. Sta-\nmenov, and J. M. D. Coey, Phys. Rev. B 106, 214414\n(2022)."    },    {        "title": "2302.02145v2.Room_Temperature_Ferroelectricity__Ferromagnetism__and_Anomalous_Hall_Effect_in_Half_metallic_Monolayer_CrTe.pdf",        "content": "Room Temperature Ferroelectricity, Ferromagnetism, and Anomalous Hall Effect in\nHalf-metallic Monolayer CrTe\nImran Ahamed,1Atasi Chakraborty,2, 3Pushpendra Yadav,3Rik Dey,1\nYogesh Singh Chauhan,1Somnath Bhowmick,4and Amit Agarwal3,∗\n1Department of Electrical Engineering, Indian Institute of Technology, Kanpur 208016, India\n2Institut f¨ ur Physik, Johannes Gutenberg Universit¨ at Mainz, D-55099 Mainz, Germany\n3Department of Physics, Indian Institute of Technology, Kanpur 208016, India\n4Department of Materials Science and Engineering,\nIndian Institute of Technology, Kanpur 208016, India\n(Dated: October 25, 2023)\nTwo-dimensional materials hosting ferroelectricity and ferromagnetism are crucial for low-power\nand high-speed information processing technologies. However, intrinsic 2D multiferroics in the\nmonolayer limit are rare. Here, we demonstrate that monolayer CrTe, obtained by cleaving the\n[002] surface, is dynamically stable multiferroic at temperatures beyond room temperature. We\nshow that it orders ferromagnetically with significant in-plane magnetocrystalline anisotropy, and\nit is a half-metal featuring a large half-metal gap. Remarkably, the broken inversion symmetry and\nbuckled geometry of monolayer CrTe make it a ferroelectric with a large spontaneous out-of-plane\npolarization and significant magnetoelectric coupling. In addition, we demonstrate polarization or\nelectric field-induced tunability of the anomalous Hall effect, accompanied by substantial bandstruc-\nture modulation. Our findings establish monolayer CrTe as a room-temperature multiferroic with\ngreat potential for applications in spintronics and ferroelectric devices.\nI. INTRODUCTION\nAtomically thin 2D materials have attracted signifi-\ncant interest for diverse applications, including electron-\nics1–5, optoelectronics6–8, energy storage9, sensors10–13,\nand catalysis14–16. However, one challenging frontier\nhas been the quest for intrinsic 2D magnets with high\nCurie temperatures suitable for room-temperature mag-\nnetic applications. Several 2D materials have been pro-\nposed so far, with both the intrinsic and extrinsic origin\nof magnetism17–25. Among them, some of the 2D mate-\nrials like CrI 326, Cr 2Ge2Te627, VSe 228, and Fe 3GeTe 229\nhave shown promise. However, they often fall short in\nterms of their transition temperatures or the strength of\nmagnetic response.\nA more significant challenge is finding multiferroic 2D\nmaterials manifesting more than one ferroic property si-\nmultaneously. One of the most desired combinations is\ncoexisting ferroelectric (FE) and ferromagnetic (FM) or-\ndering, having a substantial electromagnetic coupling.\nAlthough such materials hold great potential for ad-\nvanced information storage and processing, offering com-\npact form factors and low dissipation30, they are scarce in\nthe monolayer limit30–32. 2D multiferroics provide excit-\ning possibilities for exploring novel fundamental physics\nvia their electrically tunable structural, electronic, mag-\nnetic, and topological properties32–34. Additionally, they\nserve as versatile building blocks for creating custom-\ndesigned stacked heterostructures with tailored function-\nalities32.\nHere, we predict monolayer CrTe as a remarkable in-\ntrinsic room-temperature multiferroic featuring in-plane\nferromagnetism and out-of-plane ferroelectricity. Our in-\nterest in CrTe stems from the successful growth of amulti-layer CrTe via chemical vapor deposition, revealing\nroom-temperature ferromagnetic behavior35. We show\nthat monolayer CrTe is a half-metal with a substantial\ngap for one spin channel while metallic for another. Our\ninvestigation into its magnetic and ferroelectric proper-\nties, within density functional theory (DFT) combined\nwith Monte Carlo simulations, demonstrates the persis-\ntence of its robust ferromagnetism and ferroelectricity be-\nyond room temperature35. Additionally, monolayer CrTe\nhosts an anomalous Hall effect, indicating its topological\nnature. The unique out-of-plane FE polarization, which\nan external electric field can reverse, originates from its\ninversion-symmetry-breaking buckled structure. Sponta-\nneous polarization facilitates electric field tunability of its\nstructural, electronic, and magnetic characteristics36–38.\nThese intrinsic multiferroic attributes in monolayer CrTe,\ncombined with a significant electromagnetic coupling and\nanomalous Hall effect, present exciting opportunities for\ndesigning novel functionalities30,33,39.\nII. EXFOLIATION ENERGY AND\nSTRUCTURAL STABILITY\nWe first demonstrate the feasibility of exfoliating\nmonolayer CrTe from its bulk form and assess its stabil-\nity through vibrational phonon mode calculations. Bulk\nCrTe has five different phases. The crystal structure\nof the five phases of bulk CrTe are MnP, NaCl, NiAs,\nWurtzite, and ZnS-type,35,40–42, as shown in Fig. S1 of\nthe Supplemental Material (SM)43. We find that the\nNiAs-type bulk CrTe is the most stable ground state [see\nTable S1 of SM43]. We construct a single-layer geometry\nby extracting it from the NiAs-type bulk CrTe along the\n[002] direction [see Fig. 1(a)]. For ease of notation, inarXiv:2302.02145v2  [cond-mat.mes-hall]  24 Oct 20232\n0.40.6\n0.20.81.01.21.4\nFIG. 1. Structure, exfoliation, and stability of monolayer CrTe. (a) NiAs-type CrTe supercell displaying the [002] monolayer\nwithin the box. (b) The exfoliation energy (1.37 J/m2) of the (002)-oriented monolayer CrTe is estimated by separating one\nlayer from the rest of the four layers. The inset shows the side and top views. (c) The phonon dispersion curve of (002)-oriented\nmonolayer CrTe without significant negative frequencies indicates its dynamic stability.\nthe rest of the manuscript, we refer to the (002)-oriented\nCrTe monolayer as monolayer CrTe. To establish the\nfeasibility of exfoliation of monolayer CrTe, we calcu-\nlate its exfoliation energy ( Eex). For this, we consider\na five-layered CrTe system in equilibrium and separate\none layer by increasing its distance from the rest of the\nlayers, as shown in the inset of Fig. 1(b). We estimate\nthe exfoliation energy using the expression44,\nEex=Ed−deqn−Edeqn\nA. (1)\nHere, A,deqandEdeqnrepresents the in-plane surface\narea, equilibrium layer separation in a bulk crystal, and\nthe total energy of the five-layer slab in equilibrium, re-\nspectively. In Eq. (1), Ed−deqn is the total energy of the\nsystem when one monolayer of the five-layer thick slab\nis at d−deqdistance from the rest. The details of the\nfirst principles-based density functional theory calcula-\ntions are presented in SM43. We estimate the exfoli-\nation energy of monolayer CrTe to be 1.37 J/m2[see\nFig. 1(b)]. This is comparable in magnitude to the ex-\nfoliation energy of GeP 3and InP 345,46. The non-vdW\nnature of bulk CrTe24leads to the larger exfoliation en-\nergy of CrTe monolayer compared to other known vdW\nmaterials such as graphene, MoS 2, phosphorene, SnP 3,\nGdTe 3, GeS, GeTe and Cu(Cl, Br)Se 244,47.\nTo ascertain the dynamical stability of the free-\nstanding monolayer, we check that its phonon disper-\nsion has no imaginary component. For this, we per-\nform the phonon dispersion calculation through the lin-\near response method within density functional perturba-\ntion theory (DFPT)48as implemented in the PHONOPY\ncode49. We show the calculated phonon dispersion of\nmonolayer CrTe in Fig. 1(c). The monolayer has the ex-\nact stoichiometry as bulk CrTe, containing one Cr and\none Te atom per unit cell. Thus, the phonon dispersion of\nmonolayer CrTe has six modes with three acoustic and\nthree optical branches, as highlighted in Fig. 1(c). Wefind that monolayer CrTe has positive phonon frequen-\ncies throughout the Brillouin zone, indicating that it is\ndynamically stable. The possibility of exfoliating mono-\nlayer CrTe in a stable form and its dynamical stability\nhas not been demonstrated earlier.\nIII. MAGNETISM AND GROUND STATE\nANISOTROPY IN MONOLAYER CrTe\nThe Cr atoms form a local trigonal network with the\nneighboring Te within the monolayer geometry, as de-\npicted schematically in Fig. 2(a). The crystal field split-\nting of dorbitals within an ideal trigonal network is\nshown in Fig. 2(b). However, Cr atoms are not in the\nplane of Te atoms, leading to a breaking of degeneracy\nin the Jahn-Teller active orbitals of the ideal trigonal\nnetwork. This can allow for both ferromagnetic and an-\ntiferromagnetic (AFM) ordering. Schematic representa-\ntions of the hopping mechanisms for both FM and AFM\nground states are depicted in Fig. 2(c) and (d), respec-\ntively.\nTo identify the magnetic ground state of the system, we\nperform self-consistent total energy calculations for dif-\nTABLE I. The relative energies of different spin configura-\ntions such as FM, AFM-1, and AFM-2 with spins aligned\nalong x,y, and zdirection, with respect to the ground\nstate spin configuration. In the last column, we report\nthe exchange interaction parameter Jalong different axes,\nconsidering a nearest neighbor Heisenberg model.\nSpin\ndirectionEFM\n(meV)EAFM −1\n(meV)EAFM −2\n(meV)J\n(meV)\nx\ny\nz0.00\n0.04\n13.29781.82\n779.33\n776.72777.67\n782.97\n776.658.14\n8.12\n7.953\ndxz dyzd3z2-1dxydx2-y2TrigonalFM AFM\nExchange pathways\n(a)\n(b)(c) (d)\n(e) (f) (g)\nsite-1 site-2\nsite-1 site-2\nFIG. 2. (a) The trigonal environment of Cr formed with nearest neighbor Te atoms. (b) The crystal field splitting of d\norbitals within an ideal trigonal environment. Schematic representation of (c) ferromagnetic and (d) antiferromagnetic exchange\nmechanisms between two neighboring Cr sites. The arrow indicates the hopping paths. (e-g) The energy dispersion indicates\nhalf-metallic nature and density of states (DOS). The states near the Fermi energy have large contributions from Cr- dstates.\nferent magnetic configurations. Our density functional\ntheory (DFT) based calculations reveal that the ferro-\nmagnetic (FM) state is lower in energy than the anti-\nferromagnetic (AFM) states with an energy difference of\n∼780 meV [see Table I]. We consider FM and two dif-\nferent AFM spin configurations in a 2 ×2 supercell to\ncalculate the exchange parameters [see Fig. S3 in SM43].\nThe calculated total moment per Cr atom is 4 µB, con-\nsistent with a high spin configuration as shown in Fig. 2\n(c) and (d).\nThe electronic band structure and density of states\n(DOS), calculated using the generalized gradient approx-\nimation (GGA) with the Hubbard Ucorrection, or the\nGGA+ Uscheme, with U= 2 eV, show that the FM state\nof monolayer CrTe is half metallic [see Fig. 2(e) and (g)].\nHalf metallicity is also found in NiAs-type bulk CrTe (see\nFig. S2 in SM). The up-spin channel has a finite contribu-\ntion at the Fermi level, as seen in Fig. 2(e). However, the\ndown-spin channel is insulating with a large direct band\ngap of 2.11 eV; see Fig. 2(g). We find the states near the\nFermi energy are dominated by contributions from the\nCr-dorbital along with negligible hybridization from the\nTe-pstates [see Fig. 2(f)]. We estimate the half-metal\nenergy gap for monolayer CrTe to be Ehmg∼0.79 eV,\nwhich is three times higher than that of the bulk ( Ehmg∼0.24 eV). The Ehmgof 0.79 eV for CrTe-monolayer is\ncomparable to that of 1T-TaN 2monolayer ( Ehmg∼0.72\neV)50. This 100% spin polarization at the Fermi energy\nwith a large half-metallic gap in monolayer CrTe makes\nit an exceptional candidate for potential applications in\nspin-filter and spin transport devices, spin-selective opti-\ncal excitations and opto-magnonic devices.\nSince long-range magnetic ordering in low dimen-\nsional systems is supported by the magnetocrys-\ntalline anisotropy, we have calculated magnetocrystalline\nanisotropy energy (MAE) for the ground state FM state.\nThe MAE ( θ, ϕ) is defined as the change in energy when\nspin orientation [ ˆS(θ, ϕ),θandϕbeing polar and az-\nimuthal angle] of the system is varied from its ground\nstate direction (easy axis). The MAE for monolayer\nCrTe is presented in Fig. 3(a) and (b). The energy is\nlowest when the out-of-plane z-component of the spin\nvanishes, and monolayer CrTe possesses an easy plane\nanisotropy with MAE(90o,0o) = -3.3 meV per unit cell\n[see Fig. 3(a)]. The anisotropy in the xyplane is negli-\ngible [see Fig. 3(b)]. To put this in perspective, we note\nthat the MAE of CrTe monolayer is comparable to or\nhigher than that of several other well studied 2D fer-\nromagnetic monolayers, such as Hf 2Te (2.2 meV/u.c)51,\nFe3GeTe 2(2.76 meV/u.c)52, CrI 3(1.6 meV/u.c)53, Fe3P4\nφxyS\n-5-10\n0°30°6090°120\n150\n180\n210\n240270300330° °\n°\n°\n°\n°°°°\nzx\nS\n-\nθ5-10\n0°30°6090°120\n150\n180\n210\n240270300330° °\n°\n°\n°\n°°°°(b) (a) (c) (d)\nFIG. 3. Magnetocrystalline anisotropy plots in (a) xzand (b) xyplane, calculated using DFT (red data points), and showing\nin-plane orientation to be energetically favorable. The data points enclosed in green circles are used to determine the anisotropic\nexchange parameters. Classical Monte Carlo simulation predicts a magnetic phase transition temperature Tcaround 400 K\nbased on (c) magnetic moment vs. temperature and (d) specific heat ( Cv) vs temperature plots. The solid black line is the\nbest-fit curve, serving as a visual aid.\n(2.45 meV/u.c)54, Fe 2Si (1.15 meV/u.c)55, MnAs (0.56\nmeV/u.c)56, CrPbTe 3(1.37 meV/u.c)57.\nIV. PARAMAGNETIC TO FERROMAGNETIC\nPHASE TRANSITION\nTo estimate the transition temperature of the FM\nphase in CrTe, we map the system to the nearest neigh-\nbor ( nn) Heisenberg model using the ground state en-\nergies obtained for different magnetic configurations [see\nSM43section-IV for details]. Since the second neighbor’s\ndistance (7.21 ˚A) is substantially larger than the nearest\nneighbor’s (4.16 ˚A), we restrict our spin model to the first\nneighbor. Furthermore, we find that the calculated single\nion anisotropy term [ ∝(Sα\ni)2, with α=x, y, z ] for the\nCrTe monolayer is minimal and can be safely neglected\n[see SM section-IV for details]. We obtain the effective\nspin model to be,\nH=−1\n2X\ni̸=j[Jx\nnnSx\niSx\nj+Jy\nnnSy\niSy\nj+Jz\nnnSz\niSz\nj].(2)\nHere, Jα\nnn(where α=x, y, z ) represent the nnexchange\ninteractions along the α- axis. The values of these ex-\nchange parameters along the three axes are tabulated\nin the last column of Table I. As a consistency check,\nwe compare the MAE( θ, ϕ) values obtained using the ex-\nchange interaction parameters from Eq. (2) with the DFT\ncalculated MAE. We find that MAE values obtained from\nthe model Hamiltonian [solid blue curve in Fig. 3(a) and\n(b)] agree well with the DFT calculated values [red data\npoints]. Motivated by this, we use the obtained magnetic\nmodel to determine the Curie temperature.\nTo estimate the magnetic transition temperature ( Tc),\nwe do classical Monte Carlo (MC) simulations with the\nnearest neighbor spin Hamiltonian specified by Eq. (2).\nWe start with randomly oriented Heisenberg spins on\na system of 100 ×100 spins, allowing all spins to rotatefreely in three-dimensional space. We choose 107relax-\nation steps to reach equilibrium at each temperature,\nthen calculate the average energy value and moment over\nthe next 105steps. We present the total spin per Cr site,\n⟨S⟩, as a function of temperature in Fig. 3(c). For calcu-\nlating the temperature dependence of the specific heat,\nwe used the following expression,\nCv≈⟨E2⟩ − ⟨E⟩2\nT2. (3)\nWe display the variation of the calculated Cvper site\nwith temperature in Fig. 3(d). Both spin and specific\nheat plots show a clear signature of paramagnetic to fer-\nromagnetic transition, with Tc∼400K[Fig. 3(c) and\n(d)]. Earlier experimental work reported the Tcof thin\nfilms of CrTe crystals of thickness 100 nm, 69 nm, and\n37 nm crystals to be ∼340,∼370, and ∼360 K, respec-\ntively24. Our calculations and these results strongly sug-\ngest that layered samples of CrTe are room-temperature\nferromagnets down to the monolayer limit.\nHaving shown the stability and the possibility of room\ntemperature ferromagnetism in monolayer CrTe, we now\nfocus on demonstrating the existence of ferroelectricity\nin monolayer CrTe in the next section.\nV. OUT-OF-PLANE FERROELECTRICITY IN\nMONOLAYER CrTe\n2D materials with no inversion symmetry can have\nan out-of-plane polar axis and show electric polarization\nleading to ferroelectricity36,38. The buckled nature of the\nmonolayer CrTe, highlighted in the inset of Fig. 1(b),\nmotivates the exploration of its FE properties. Using\nthe optimized buckled monolayer of CrTe, we study the\nFE properties utilizing the first-principle Berry phase\nmethod58–61. The change in buckling height between\nCr and Te planes (denoted by δ) changes the electric5\nFIG. 4. (a) The polarization dependence of the relative total energy showing the spontaneous polarization Psand polarization\nswitching barrier height EG. (b) The buckled CrTe honeycomb monolayer’s side view shows polarization’s dependence on\nbuckling height δ. (c) The nearest neighbor dipole-dipole interaction energy of monolayer CrTe is calculated by mean-filed\ntheory showing the harmonic fit. (d) The polarization-electric field curve (S-curve) of the FE monolayer is obtained from the\ndata of (a). (e) The variation of the magnetic moment of Cr-atom with the electric field.\ndipole, resulting in the variation of the FE polarization.\nFigure 4(a) illustrates the variation of the relative total\nenergy of the monolayer CrTe with the change in the\nout-of-plane FE polarization. The variation of the total\nenergy with polarization has the form of an anharmonic\ndouble well potential (w-curve) with two minima corre-\nsponding to the top and bottom panels of Fig. 4(b). The\nmaximum energy point in the w-curve corresponds to the\nplanar monolayer structure shown in the middle panel of\nFig. 4(b). The two minima points correspond to the neg-\native and positive out-of-plane FE polarization, achieved\nby the inversion of the buckling.\nWe quantify the FE behavior of monolayer CrTe using\nthe Landau-Ginzburg (LG) equation. The LG equation\nfor the free energy ( G) as a function of polarization ( Pi)\nper unit cell is62–64\nG=X\ni\"\nA\n2P2\ni+B\n4P4\ni+C\n6P6\ni#\n+D\n2X\n⟨i,j⟩(Pi−Pj)2.(4)\nHere, iindicates the unit cells, and jindicates the near-\nest neighbor unit cell of the ithunit cell. The free en-\nergy is in units of meV per unit cell, and Pis in units\nof 10−10C/m. We estimate the coefficients A,B, and\nCby fitting the DFT data with the Eq. (4). We ob-tain the values of the coefficients A,B, and Cto be\n−61.71 meV/(10−10C/m)2, 1.51 meV/(10−10C/m)4\nand−0.006 meV/(10−10C/m)6, respectively. The spon-\ntaneous polarization and potential barrier are Ps= 7×\n10−10C/m and EG= 729 meV, respectively. The value\nofPsfor monolayer CrTe is higher than that of some\nknown 2D FEs such as MX(M= Ge, Sn; X= S, Se)62,δ-\n(As,Sb,Bi)N65, group III–V monolayers (III = Ga, In; V\n= P, As, Sb)38, other 2D honeycomb binary compounds37\nand some elemental group V monolayers66. The energy\nbarrier for polarization switching is higher than most of\nthe known 2D FE materials and comparable to the value\nin two-dimensional In 2Se367.\nTheS-curve, illustrated in Fig. 4(d), captures the po-\nlarization dependence on the electric field, and it is de-\ntermined by E=dG/dP13,68. The blue shaded region\nin Fig. 4(a) and Fig. 4(d) is the region of negative ca-\npacitance. The capacitance Cis proportional to the\nslope dP/dE . This region of negative capacitance in the\nS-curve is difficult to observe experimentally in ferro-\nelectrics because of the unstable nature of the material\naround P= 0 (i.e., paraelectric region). The S-curve\ngives the coercive field Ecof±0.27 V/nm corresponding\nto the field required to switch the polarization direction.\nThe FE transition temperature is dictated by the aver-6\nStructure-1 Structure-2\nStructure-4 Structure-5(a)\n(d)\n(b)\n(e)\nE-E\nF (eV)\nM Γ Γ K M Γ Γ KM Γ Γ K M Γ Γ K\n<Sx>2\n1\n0\n-1\n-2\nStructure-3\n(c)\nM\nΓ Γ KE-E\nF (eV)2\n1\n0\n-1\n-2M Γ Γ K\nStructure-7\n(f)\n-200 0 200 400 -200 0 200 -100 0-200 0 200 -200 0 200 -250 0 250\nFIG. 5. Impact of applied electric field on the electronic band structure and the spin polarization (left) and anomalous\nHall conductivity σxy(right) for different atomic arrangements. The color scale in the bandstructure plots indicates the sx\npolarization of the itinerant spins. Panel (a) is the equilibrium structure with δ= 1.8˚A which gradually decreases to δ= 0\n(zero polarization state), as shown in panels (b) - (e), resulting in a significantly reduced AHC. Panel (f) is the same as (a)\nbut buckled in the opposite direction.\nage dipole-dipole interaction strength between the near-\nest neighboring unit cells or the parameter Din Eq. (4).\nWe calculate Dby taking a supercell of the dipole and\nfitting the relative total energy with a classical harmonic\noscillator-like parabolic function D62,69as indicated in\nEq. (4) and shown in Fig. 4(c). We obtain Dto be 7.13\nmeV/(10−10C/m)2. We can estimate the transition tem-\nperature of the FE from the expression62Tc∼DP2\ns/kB.\nWe estimate the Tcfor monolayer CrTe to be much higher\nthan room temperature (around 4000 K). This indicates\nthat monolayer CrTe is a room-temperature multiferroic,\nsupporting an out-of-plane FE order and an in-plane fer-\nromagnetic half-metallic state.\nThe simultaneous presence of ferroelectricity and fer-\nromagnetism in monolayer-CrTe motivates the study of\nthe magnetoelectric (ME) coupling and the anomalous\nHall effect. A finite ME coupling indicates the possibil-\nity of tuning i) the magnetic state by applying an electric\nfield and ii) the polarization state by applying a mag-\nnetic field. More interestingly, time reversal symmetry\nbroken magnetic systems can also host anomalous Hall\neffect (AHE). The AHE can be tuned in such a scenario\nby applying a vertical electric field. We explore both\nthese effects in the following two sections.VI. MAGNETOELECTRIC COUPLING IN\nMONOLAYER CrTe\nTo understand the ME coupling, we analyze the elec-\ntric field data, polarization value, and the magnetic mo-\nment of the Cr-atom corresponding to the respective po-\nlarization state. In the stable region of the FE, the vari-\nation of the magnetic moment of the Cr-atom in the pos-\nitive as well as the negative polarization state is shown\nin Fig. 4(e). We find that the ME coupling is positive\nfor the up-polarization state and negative for the down-\npolarization state based on the slope of the curve. The\nME coupling coefficient αis proportional to the varia-\ntion of the magnetic moment (∆ m) with the change in\nthe applied electric field (∆ E), or α=µ0∆m\n∆E. Here, µ0\nis the permeability of free space. The linear ME coupling\ncoefficient is calculated for the linear region of Fig. 4(e)\nfor an electric field ranging from −0.3 to +0 .3 V/nm. We\nobtain the value of αto be +9 .4 ps/m and −9.4 ps/m\nfor the positive and negative Pzstate, respectively. This\nsignificant value of the ME coupling αis of the same or-\nder of magnitude as that found in bulk CaMnO 3and in\nthe prototypical ME material - bulk Cr 2O370,71.\nVII. ELECTRICALLY TUNABLE ANOMALOUS\nHALL CONDUCTIVITY\nThe anomalous Hall conductivity in 2D time reversal\nsymmetry broken materials is the Hall response without7\na magnetic field72. It is given by the Brillouin zone (BZ)\nintegral of the out-of-plane Berry curvature of a given\nband (Ωz\nn) weighted by its occupation factor73,74,\nσAHC\nxy=−e2\nℏX\nnZ\nBZdk\n(2π)2fn(k)Ωn,z(k). (5)\nHere, fn(k) = 1 /[1 +e(En(k)−µ)/(kBT)] is the equilibrium\nFermi function of the system with chemical potential µ\nand temperature T. The Berry curvature of the nthband\nin a system can be calculated using59,\nΩz\nn=−2ImX\nm̸=n⟨ψmk|∂kxH|ψnk⟩⟨ψnk|∂kxH|ψmk⟩\n(En−Em)2.\n(6)\nHere, EnandEmare the energy eigenvalues of the nth\nandmthband, respectively.\nWe highlight the vertical electric field-induced modu-\nlation of the electronic band structure, spin polarization\nof the itinerant electrons, and the AHC in Fig. 5. The\nvertical electric field controls the polarization state or\nthe buckling of the monolayer. This is reflected in the\nmodulation of electronic properties, spin polarization,\nand the AHC. The electric field-induced modulation of\nthe magnetic moments of the ferromagnets is shown in\nFig. 4(e). This makes monolayer CrTe a versatile plat-\nform with electrically tunable physical properties, includ-\ning the AHC. This can be very useful for electronic, spin-\ntronic, and optoelectronic devices.\nVIII. CONCLUSION\nIn summary, we demonstrated monolayer CrTe to be\na dynamically stable room temperature multiferroic at\nthe atomic scale. It exhibits fascinating properties, fea-\nturing both half-metallicity with 100% spin polarization\ninduced by in-plane ferromagnetism and strong out-of-\nplane ferroelectricity. These properties make monolayer\nCrTe a promising candidate for various applications. Its\nhalf-metallic nature makes monolayer CrTe well-suited\nfor spintronic applications like spin injection and spin fil-tering. The presence of ferroelectricity makes it an excel-\nlent choice for FE memory devices and tunnel field-effect\ntransistors.\nMore interestingly, we establish the existence of a ro-\nbust ME coupling along with the presence of an anoma-\nlous Hall state in monolayer CrTe. The coexistence of\nanomalous Hall and half-metallicity indicates the pres-\nence of topologically protected edge states that are 100%\nspin polarized. These edge states can carry dissipation-\nless charge and spin currents in nano-ribbon and other\nconfined geometries of monolayer CrTe. The multiferroic\ncharacteristics and strong ME coupling of CrTe mono-\nlayers offer exciting prospects for magnetic sensors, in-\nformation devices, and high-density data storage appli-\ncations. Additionally, CrTe provides a unique atomic\nmonolayer platform with electric field tunable electronic,\nmagnetic, anomalous Hall, and ferromagnetic properties.\nThis opens up several exciting directions for further stud-\nies.\nACKNOWLEDGMENTS\nIA acknowledges IIT Kanpur for providing the Post-\ndoctoral Fellowship. AC acknowledges the Science and\nEngineering Research Board (SERB), India, for Na-\ntional Postdoctoral Fellowship (PDF/2021/000346) and\nthe Alexander von Humboldt Foundation, Germany, for\npartial financial support. Authors acknowledge finan-\ncial support from SERB (Grant No. CRG/2021/003687,\nMTR/2019/001520, CRG/2018/002400), Swarnajayanti\nFellowship (Grant No. DST/SJF/ETA-02/2017-\n18), Department of Science and Technology, India\n(DST/NM/TUE/QM-6/2019(G)-IIT Kanpur). Au-\nthors acknowledge the National Supercomputing Mission\n(NSM) for providing computing resources of “PARAM\nSanganak” at IIT Kanpur, which is implemented by C-\nDAC and supported by the Ministry of Electronics and\nInformation Technology (MeitY) and Department of Sci-\nence and Technology (DST), Government of India. The\nauthors also acknowledge the HPC facility provided by\nCC, IIT Kanpur.\n∗amitag@iitk.ac.in\n1L. Li, Y. Yu, G. J. Ye, Q. Ge, X. Ou, H. Wu, D. Feng,\nX. H. Chen, and Y. Zhang, Nature Nanotechnology 9,\n372 (2014).\n2B. Radisavljevic, A. Radenovic, J. Brivio, V. Giacometti,\nand A. Kis, Nature Nanotechnology 6, 147 (2011).\n3K. Nandan, B. Ghosh, A. Agarwal, S. Bhowmick, and\nY. S. Chauhan, IEEE Transactions on Electron Devices\n69, 406 (2022).\n4S. Sinha, P. C. Adak, A. Chakraborty, K. Das, K. Deb-\nnath, L. D. V. Sangani, K. Watanabe, T. Taniguchi, U. V.\nWaghmare, A. Agarwal, and M. M. Deshmukh, Nature\nPhysics 18, 765 (2022).5A. Chakraborty, K. Das, S. Sinha, P. C. Adak, M. M. Desh-\nmukh, and A. Agarwal, 2D Materials 9, 045020 (2022).\n6Q. H. Wang, K. Kalantar-Zadeh, A. Kis, J. N. Coleman,\nand M. S. Strano, Nature Nanotechnology 7, 699 (2012).\n7X. Ren, Z. Li, Z. Huang, D. Sang, H. Qiao, X. Qi, J. Li,\nJ. Zhong, and H. Zhang, Advanced Functional Materials\n27, 1606834 (2017).\n8P. Yadav, B. Khamari, B. Singh, K. V. Adarsh, and\nA. Agarwal, Journal of Physics: Condensed Matter 35,\n235701 (2023).\n9R. Wang, X. Li, Z. Wang, and H. Zhang, Nano Energy\n34, 131 (2017).\n10P. Wan, X. Wen, C. Sun, B. K. Chandran, H. Zhang,8\nX. Sun, and X. Chen, Small 11, 5409 (2015).\n11Z. Hu, L. Zhang, A. Chakraborty, G. D’Olimpio, J. Fujii,\nA. Ge, Y. Zhou, C. Liu, A. Agarwal, I. Vobornik, D. Farias,\nC.-N. Kuo, C. S. Lue, A. Politano, S.-W. Wang, W. Hu,\nX. Chen, W. Lu, and L. Wang, Advanced Materials ,\n2209557 (2023).\n12A. Chakraborty, J. Fujii, C.-N. Kuo, C. S. Lue, A. Politano,\nI. Vobornik, and A. Agarwal, Phys. Rev. B 107, 085406\n(2023).\n13A. Priydarshi, Y. S. Chauhan, S. Bhowmick, and A. Agar-\nwal, Journal of Applied Physics 131, 034101 (2022).\n14X. Ren, J. Zhou, X. Qi, Y. Liu, Z. Huang, Z. Li, Y. Ge,\nS. C. Dhanabalan, J. S. Ponraj, S. Wang, J. Zhong, and\nH. Zhang, Advanced Energy Materials 7, 1700396 (2017).\n15D. Deng, K. S. Novoselov, Q. Fu, N. Zheng, Z. Tian, and\nX. Bao, Nature Nanotechnology 11, 218 (2016).\n16C. Tan, X. Cao, X.-J. Wu, Q. He, J. Yang, X. Zhang,\nJ. Chen, W. Zhao, S. Han, G.-H. Nam, M. Sindoro, and\nH. Zhang, Chemical Reviews 117, 6225 (2017).\n17S. Bhowmick, A. Medhi, and V. B. Shenoy, Phys. Rev. B\n87, 085412 (2013).\n18P. Rastogi, S. Kumar, S. Bhowmick, A. Agarwal, and Y. S.\nChauhan, IETE Journal of Research 63, 205 (2017).\n19V. Kochat, A. Apte, J. A. Hachtel, H. Kumazoe, A. Kr-\nishnamoorthy, S. Susarla, J. C. Idrobo, F. Shimojo,\nP. Vashishta, R. Kalia, A. Nakano, C. S. Tiwary, and\nP. M. Ajayan, Advanced Materials 29, 1703754 (2017).\n20A. Kumar, A. Strachan, and N. Onofrio, Journal of Ap-\nplied Physics 125, 204303 (2019).\n21H. Ren and G. Xiang, Nanomaterials 12, 3451 (2022).\n22C. B. Crook, C. Constantin, T. Ahmed, J.-X. Zhu, A. V.\nBalatsky, and J. T. Haraldsen, Scientific Reports 5, 12322\n(2015).\n23A. Hallal, F. Ibrahim, H. Yang, S. Roche, and M. Chshiev,\n2D Materials 4, 025074 (2017).\n24H. Wu, W. Zhang, L. Yang, J. Wang, J. Li, L. Li, Y. Gao,\nL. Zhang, J. Du, H. Shu, and H. Chang, Nature Commu-\nnications 12, 5688 (2021).\n25Y. Guo, S. Zhou, and J. Zhao, J. Mater. Chem. C 9, 6103\n(2021).\n26B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein,\nR. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A.\nMcGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-\nHerrero, and X. Xu, Nature 546, 270 (2017).\n27Z. Wang, T. Zhang, M. Ding, B. Dong, Y. Li, M. Chen,\nX. Li, J. Huang, H. Wang, X. Zhao, Y. Li, D. Li, C. Jia,\nL. Sun, H. Guo, Y. Ye, D. Sun, Y. Chen, T. Yang, J. Zhang,\nS. Ono, Z. Han, and Z. Zhang, Nature Nanotechnology 13,\n554 (2018).\n28M. Bonilla, S. Kolekar, Y. Ma, H. C. Diaz, V. Kalappattil,\nR. Das, T. Eggers, H. R. Gutierrez, M.-H. Phan, and\nM. Batzill, Nature nanotechnology 13, 289 (2018).\n29Y. Deng, Y. Yu, Y. Song, J. Zhang, N. Z. Wang, Z. Sun,\nY. Yi, Y. Z. Wu, S. Wu, J. Zhu, J. Wang, X. H. Chen, and\nY. Zhang, Nature 563, 94 (2018).\n30N. A. Spaldin and R. Ramesh, Nature Materials 18, 203\n(2019).\n31F. Matsukura, Y. Tokura, and H. Ohno, Nature Nanotech-\nnology 10, 209 (2015).\n32Q. Song, C. A. Occhialini, E. Erge¸ cen, B. Ilyas,\nD. Amoroso, P. Barone, J. Kapeghian, K. Watanabe,\nT. Taniguchi, A. S. Botana, S. Picozzi, N. Gedik, and\nR. Comin, Nature 602, 601 (2022).\n33Y. Tokura, S. Seki, and N. Nagaosa, Reports on Progressin Physics 77, 076501 (2014).\n34A. Narayan, A. Cano, A. V. Balatsky, and N. A. Spaldin,\nNature Materials 18, 223 (2018).\n35Y. Liu, S. K. Bose, and J. Kudrnovsk´ y, Phys. Rev. B 82,\n094435 (2010).\n36H. S ¸ahin, S. Cahangirov, M. Topsakal, E. Bekaroglu,\nE. Akturk, R. T. Senger, and S. Ciraci, Phys. Rev. B\n80, 155453 (2009).\n37D. Di Sante, A. Stroppa, P. Barone, M.-H. Whangbo, and\nS. Picozzi, Phys. Rev. B 91, 161401 (2015).\n38R. Gao and Y. Gao, physica status solidi (RRL) – Rapid\nResearch Letters 11, 1600412 (2017).\n39A. Mahajan and S. Bhowmick, The Journal of Physical\nChemistry C 127, 11407 (2023).\n40Z. Charifi, D. Guendouz, H. Baaziz, F. Soyalp, and\nB. Hamad, Physica Scripta 94, 015701 (2018).\n41N. Kang, W. Wan, Y. Ge, and Y. Liu, Frontiers of Physics\n16, 63506 (2021).\n42H. Moulkhalwa, Y. Zaoui, K. O. Obodo, A. Belkadi,\nL. Beldi, and B. Bouhafs, Journal of Superconductivity\nand Novel Magnetism 32, 635 (2019).\n43Supplemental Material has details of the i) Computational\nmethods, ii) Bulk magnetic phases of CrTe, iii) Impact\nof Coulomb interactions on the ground state, and iv) the\nModel calculation of the effective spin Hamiltonian.\n44B. Ghosh, S. Puri, A. Agarwal, and S. Bhowmick, The\nJournal of Physical Chemistry C 122, 18185 (2018).\n45Y. Jing, Y. Ma, Y. Li, and T. Heine, Nano Letters 17,\n1833 (2017).\n46N. Miao, B. Xu, N. C. Bristowe, J. Zhou, and Z. Sun, Jour-\nnal of the American Chemical Society 139, 11125 (2017).\n47I. Ahamed, Y. S. Chauhan, S. Bhowmick, and A. Agarwal,\nComputational Materials Science 216, 111869 (2023).\n48S. Baroni, P. Giannozzi, and A. Testa, Phys. Rev. Lett.\n58, 1861 (1987).\n49A. Togo and I. Tanaka, Scripta Materialia 108, 1 (2015).\n50J. Liu, Z. Liu, T. Song, and X. Cui, J. Mater. Chem. C 5,\n727 (2017).\n51J.-J. Liao, Y.-Z. Nie, X. guang Wang, Q. lin Xia, R. Xiong,\nand G. hua Guo, Applied Surface Science 586, 152821\n(2022).\n52H. L. Zhuang, P. R. C. Kent, and R. G. Hennig, Phys.\nRev. B 93, 134407 (2016).\n53B. Yang, X. Zhang, H. Yang, X. Han, and Y. Yan, Applied\nPhysics Letters 114, 192405 (2019).\n54S. Zheng, C. Huang, T. Yu, M. Xu, S. Zhang, H. Xu,\nY. Liu, E. Kan, Y. Wang, and G. Yang, The journal of\nphysical chemistry letters 10, 2733 (2019).\n55Y. Sun, Z. Zhuo, X. Wu, and J. Yang, Nano letters 17,\n2771 (2017).\n56B. Wang, Y. Zhang, L. Ma, Q. Wu, Y. Guo, X. Zhang,\nand J. Wang, Nanoscale 11, 4204 (2019).\n57I. Khan and J. Hong, Nanotechnology 31, 195704 (2020).\n58R. D. King-Smith and D. Vanderbilt, Phys. Rev. B 47,\n1651 (1993).\n59M. V. Berry, Proceedings of the Royal Society of London.\nA. Mathematical and Physical Sciences 392, 45 (1984).\n60R. D. King-Smith and D. Vanderbilt, Phys. Rev. B 49,\n5828 (1994).\n61R. Resta, Rev. Mod. Phys. 66, 899 (1994).\n62R. Fei, W. Kang, and L. Yang, Phys. Rev. Lett. 117,\n097601 (2016).\n63A. D. Bruce, Advances in Physics 29, 111 (1980).9\n64J. C. Wojde l and J. ´I˜ niguez, Phys. Rev. B 90, 014105\n(2014).\n65Y. Zhang, T. Ouyang, C. He, J. Li, and C. Tang,\nNanoscale 15, 6363 (2023).\n66C. Xiao, F. Wang, S. A. Yang, Y. Lu, Y. Feng, and\nS. Zhang, Advanced Functional Materials 28, 1707383\n(2018).\n67W. Ding, J. Zhu, Z. Wang, Y. Gao, D. Xiao, Y. Gu,\nZ. Zhang, and W. Zhu, Nature Communications 8, 14956\n(2017).\n68M. Hoffmann, F. P. G. Fengler, M. Herzig, T. Mittmann,\nB. Max, U. Schroeder, R. Negrea, P. Lucian, S. Slesazeck,and T. Mikolajick, Nature 565, 464 (2019).\n69M. Soleimani and M. Pourfath, Nanoscale 12, 22688\n(2020).\n70E. Bousquet and N. Spaldin, Phys. Rev. Lett. 107, 197603\n(2011).\n71E. Bousquet, N. A. Spaldin, and K. T. Delaney, Phys.\nRev. Lett. 106, 107202 (2011).\n72R. Karplus and J. M. Luttinger, Phys. Rev. 95, 1154\n(1954).\n73M.-C. Chang and Q. Niu, Phys. Rev. Lett. 75, 1348 (1995).\n74D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82,\n1959 (2010)."    },    {        "title": "2302.05146v2.Correlation_driven_topological_transition_in_Janus_VSiGeP2As2.pdf",        "content": "   \n \n  \nCorrelation -driven topological transition in Janus VSiGeP 2As 2 \n \nGhulam Hussain1, Amar Fa khredine2, Rajibul Islam1, Raghottam M. Sattigeri1, Carmine Autieri1,* and Giuseppe \nCuono  1 \n1 International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnik ów \n32/46, PL -02668 Warsaw, Poland  \n2 Institute of Physics, Polish Academy of Sciences, Aleja Lotnik ów 32/46, PL -02668 Warsaw, Poland  \n* Correspondence: autieri@magtop.ifpan.edu.pl  \n \nAbstract: The appearance of intrinsic fer romagnetism in 2D materials opens the possibility of \ninvestigati ng the interplay between magnetism and topology . The magnetic anisotropy energy \n(MAE) describing the easy axis for magnetizatio n in a particular direction is an  important yardstick  \nfor nanoscale applications. Here,  the first-principles approach is used to investigate the electronic \nband structures,  the strain dependence of MAE in pristine VSi 2Z4 (Z=P, As) and its Janus phase \nVSiGeP 2As 2 and the evolution of the topology as a function of the Coulomb interaction . In the \nJanus phase the compound presents a breaking of the mirror symmetry, which  is equivalent to \nhaving an electric field, and the system can be  piezoelectric.  It is revealed that all three monolayers \nexhibit ferromagnetic ground state ordering, which is  robust even under  biaxial strains. A large \nvalue of coupling J is obtained, and this, together with the magnetocrystalline anisotropy , will \nproduce a large critical temperature. We found an  out-of-plane (in-plane) magnetization for VSi 2P4 \n(VSi 2As 4), wh ile in -plane magnetization for VSiGeP 2As 2. Furthermore, we observed a \ncorrelation -drive n topological transition  in the Janus  VSiGeP 2As 2. Our analysis of t hese emerging \npristine and Janus -phased magnetic semiconductors open s prospects for stu dying  the interplay \nbetween magnetism and topology in two -dimensional materials . \nKeywords: Correlation -driven topological transition; vanadates; density functional theory; 2D \nferromagnetism  \n \n \n1. Introduction  \nSince the observation of intrinsic  ferromagnetism in two -dimensional layered materials \n(2D) such as CrGeTe 3 [1] and CrI 3 [2], the fields of magnetism and spintronics have \nreceived tremendous research attention in the 2D limit [3-14]. The atomically thin 2D \nmagnetic materials are considered ideal syste ms, where the magnetic and spin -related \nfeatures  can effectively be controlled and m odulated via proximity effects, electric field, \nmagnetic field, strain, defects, and optical doping [15-22]. Unlike bulk  materials, where \nmagnetic ord ering is possible without magn etic anisotropy, long -range    \n \n magnetic  ordering in layered  2D materials  is no t conceivable in systems deprived of \nmagnetic anisotropy, which is necessary to balance out thermal fluctuations [23]. Due to \nthe fact that magnetic anisotropy is primarily caused by spin -orbit coupling (SOC) effects \n[24], it becomes a crucial characteristic. Furthermore, spintronic devices such as magnetic \ntunnel junctions and spin valves show enhanced performance based on 2D magnetic \nstructures with substantial magnet ic anisotropy [25-27]. It has been demonstrated that \nstrain engineering is an effective method of tuning the magnetic, electronic, and optical \ncharac teristics of materials [28-33]. \nThe recently discovered new family of 2D layered materials MA 2Z4, where M, A and Z \nrepresent  the transition metal atoms (Mo, W, Hf, Cr, V) , IV-elements  (Si, Ge) , and \nV-elements (N, As, P), respectively [34], have sparked intense interest in different studies \n[35-44]. These layered materials exhibit outstanding mechanical, electronic, magnetic, \nand optical properties [38, 44 -58]. It was shown that in the Janus phases of these \ncompounds , the breaking of th e mirror symmetry brings Rashba -type spin -splitting  \n[59-62] and that this, together with the large valley splitting , can give an important \ncontribution to semiconductor valleytronics and spintronics. In the present work,  the \nstructural, electronic  and magnetic  properties  of pris tine VSi 2Z4 (Z=P, As) and their Janus \nphase VSiGeP 2As 2 are explored. We found ferromagnetic o rdering in these systems, and \ntheir magnetic anisotropy energy (MAE) reveal s a strong dependency on the biaxial \nstrain. Besides, an out -of-plane direction is found as an easy axis for the magnetization of \nVSi 2P4, while an in-plane direction is favored in V Si2As 4 and V SiGeP 2As 2. In the Janus \nphase , the compound presents breaking of the mirror symmetry,  this can give \npiezoelectric properties , and it  is equivalent to hav ing an electric field , which can \nmanipulate magnetism and produce skyrmions  in 2D materials  [63, 64] . Intriguingly, \nthere occurs  a topological phase transition from  a trivial to topol ogically non -trivial state \nin V SiGeP 2As 2 monolayer , when the Hu bbard U parameter is increased.  Our \ninvestigation  of these compounds  open s prospects for studying their intrinsic magnetism , \nthe interplay between magnetism and topology in two -dimensional materials  and spin \ncontrol in spintronic s. \n \n2. Computational details  \nFirst -principles re lativistic approach based on density functional theory (DFT) using \nVienna Ab Initio  Simulation Package (VASP)  [65, 66]  is employed . The Perdew–Burke–\nErnzerhof  (PBE) formalism in the framework of generalized gradient approximation  \n(GGA)  was use d to include the electron exchange -correlation [67]. Als o, we implemented \nthe projector -augmented wave s cheme to resolve the Kohn -Sham eq uations through the \nplane -wave basis set. An energy cutoff of 500 eV is taken into account for the expansion \nof wave func tions. The Monkhorst –Pack scheme is applied for k-point sampling with \n15×15×1 k-point mesh. The lattice constants were optimized at the  PBE level. The \noptimized lattice constant for the Janus V SiGeP 2As 2 structure is 3.562  Å, which is \nbetween those of VSi 2P4 (3.448  Å) and VSi 2As 4 (3.592  Å) monolayers . In addition, the   \n \n convergence criterion for force is taken as 0.0001 eV/Å, while 10−7 eV of energy tolerance \nis considered for the lattice relaxation. Also, the number of electrons treated as valence is \n41. In examining the dynamical stability, a 4×4×1 supercell of VSiGeP 2As 2 monolayer is \ntaken for calculating the phonon dispersion  using t he PHONOPY code [68]. The GGA+U \nroutine , along with SOC , is executed , and the strong ly correlated correction intended for \nV-3d is considered throughout the calculations . The values of  the Hubbard parameter \nused for the d -orbitals of V are U = 4 eV for VSi 2P4, and 2 eV for VSi 2As 4 and VSiGeP 2As 2, \nand the Hund  coupling  JH is set at 0.87 eV. The main source of SOC in this compound is \nAs; the value  of SOC for As was e stimated to be  0.164 eV  [69, 70] . \n3. Results and discussion  \nThe monolayered VSi 2Z4 (Z=P, As) 2D materials cryst allize in a hexagonal geometry with \nP m2 (No. 187) as the space group. These structures are seven -atom thick mono layered \nsystems ; the atoms are strongly bonded together with the order as Z -Si-Z-V-Z-Si-Z for \npristine and P -Si-P-V-As-Ge-As in the case of  the Janus phase. Figure 1a shows the \npristine VSi 2P4, VSi 2As 4, and Janus VSiGeP 2As 2 structures. The VSi 2Z4 (Z=P, As)  \nmonolayers have broken inversion symmetry while protecting the mirror -plane \nsymmetry with respect to V plane. Also, the primitive cell with side and top views is \nshown for the Janus VSiGeP 2As 2 phase in Fig.  1a, which presents the breaking of mirror \nsymmet ry with regard to the V atom.  This  is equivalent to an electric field , and the \nsystem can show  piezoelectric ity. The optimized lattice constants for VSi 2P4 and VSi 2As 4 \nmonolayers are 3.448 Å and 3.592  Å, respectively , whereas , for the Janus V SiGeP 2As 2 \nstructure , it is 3.562  Å. Fig. 1b presents the  2D Brillouin zone with the high -symmetry \npoints indicated by red letters. Fig. 1c  shows  the schematic representation for the \ntopological transition as a function of onsite Coulomb intera ction in VSiGeP 2As 2 \nmonolayer .  \n \n \n \n \n \n \n \n \n \n \nFigure 1 (a) Side view  of VSi 2P4 monolayer, side and top view s for Janus phase  VSiGeP 2As 2 prim i-\ntive cell . (b) 2D Brillouin zone with the high -symmetry points indicated by red letters. (c) Schema t-\nic representation for the topological phase transition as a function of onsite Coulomb intera ction \nobserved in VSiGeP 2As 2 monolayer.  \n  \n \n  \n \n \n \nThe stabilities of pristine VSi 2Z4 (Z=P, As) monolayers and Janus VSiGeP 2As 2 structure \nwere studied through the cohesive energi es and the phonon dispersion . We computed \nthe cohesive energies per atom ( Ec); for VSi 2Z4, Ec = [EVSi2Z4  – (EV + 2ESi + 4EZ)]/7, where the \nenergy terms EVSi2Z4 , EV, ESi, EZ represent the total energies of VSi 2Z2 monolayer and that \nof V, Si  and Z atoms, respectively. Similarly, for the Janus VSiGeP 2As 2, it can be written \nas Ec = [EVSiGeP2As2  – (EV + ESi + EGe + 2EP + 2EAs)]/7. The values of E c were calcul ated as -3.25, \n-2.60 and -2.92 eV/atom for VSi 2P4, VSi 2As 4 and VSiGeP 2As 2. These are relatively higher as \ncompared to recently reported MoSiGeP 2As 2 (-2.77 eV/atom), WGeSiP 2As 2 (-2.84) [62], \nand other transition -metal based 2D Janus materials such as MoSSe, WSSe (−2.34 eV, \n-2.06 eV) [71]. Here, the phonon dispersion  for VSiGeP 2As 2 is calculated along the high \nsymmetry directions of the Brillouin zone (K -Г-M-K) with the method of finite diff erence \nimplemented in  the Phonopy code.  In Fig. 2a , we show  the phonon dispersion  of \nVSiGeP 2As 2 revealing no imaginary frequency modes, thus dynamically stable. The \npristine monolayers VSi 2Z4 (Z=P, As) are already reported to be dynamically stable [9, 31] . \nThe large values of cohesive energies E c, and the dyn amical stability established from \nphononic spectra can promise their experimental realization.  \nThe electron ic configuration for an unbonded V atom is 3d34s2. However, the V atom in \nVSi 2Z4 (Z=P, As) is trigonal -prismatically coordinated with six Z atoms. Such type of \ncrystal field divides the 3d orbitals into dz2, d yz/dxz, and dxy/dx2- y2 as reported in MoS 2 for \nMo atoms, which requires that dz2 orbital should be occupied first [72]. The V a tom d o-\nnates four electrons to neighboring Z atoms, with one electron remaining , giving rise to \nV4+ valence state. With this one unpaired electron in dz2 a magnetic moment of 1 μ B is ex-\npected according to Hund’s  rule and Pauli exclusion principle. Our DFT calculations i n-\ndeed revealed a magnetic moment of ~1 μ B per formula unit  for VSi 2Z4 (Z=P, As) and Ja-\nnus VSiGeP 2As 2 structures. In addition, the total energies of two distinct magnetic co n-\nfigurations were evaluat ed in order to determine  the magnetic ground state. For  the a n-\ntiferromagnetic (AFM) configuration, the magnetic moments were made antiparallel to \nnearest neighbors, instead , all magnetic moments were  initialized in the same direction \nin the ferromagnetic ( FM) configuration. In both instances, the spin ori entations were \noff-plane. Fig. 2b  depicts these two common magn etic orderings with a 2×2×1 supercell  \nfor which the total energies and magnetic moments of FM and AFM configurations were \ncalculated, respectively . For the 2×2×1 supercell,  a magnetic moment of ~4.0 μ B is re-\nvealed for both the pristine and Janus phases in the FM state, while 0 μ B is observed with \nAFM alignment. Moreover, the energy difference between the FM and AFM states (E FM −   \n \n EAFM) indicated negative energies, strongly suggesti ng intrinsic ferromagnetism in VSi 2Z4 \n(Z=P, As) monolayers and their Janus structure.  The optimized lattice constan ts a o, ene r-\ngy difference between FM and AFM alignments and the easy axis for the magnetization \nfor VSi 2Z4 (Z=P, As) and Janus phase are reported in Table 1 . We also computed the a v-\nerage electrostatic potential profiles along the z -axis for the pristine and the Janus phase. \nAs indicated in Fig. 2c,d the profiles are symmetric for VSi 2Z4 (Z=P, As), however in the \ncase of Janus VSiGeP 2As 2, the  calculated average electrostatic potential is rather a sym-\nmetric with a work function difference, ΔΦ of 0.35 eV (Fig. 2 e). \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nTable 1 Optimized lattice constants a o, energy difference s between FM and AFM alignments and \nthe easy axis for the magnetization.  \n \n \n \n Material  ao (Å) [EFM − E AFM] (eV)  Easy axis  \nVSi 2P4 3.448  -0.143  Out-of-plane  \nVSi 2As 4 3.592  -0.202  Inplane  \nVSiGeP 2As 2 3.562  -0.210  Inplane  \nFigure 2 (a) The phonon dispersion  for Janus VSiGeP 2As 2 monolayer indicating no imaginary fr e-\nquencies. (b) Two magnetic configurations FM and AFM, considered to evaluate the magnetic \nground state. The planar average electrostatic potential energy of  (c) VSi 2P4, (d) VSi 2As 4, and (e) \nJanus VSiGeP 2As 2 monolayers.  The work function difference ΔΦ is est imated to be 0.3 5 eV  for the \nJanus phase.   \n  \n \n  \nThe transition metal based 2D materials host degenerate energy valleys (at K, Kʹ points of \nBrillouin zone) owing to  a lack of inver sion symmetry. Such energy vall eys can be m a-\nnipulated and utilized in valley -spin Hall effects and valley -spin locking [73-75]. Gene r-\nating and controlling the valley polarization b y making the K, Kʹ valleys non -degenerate \nis a big challenge in valleytronics.  There are multiple means to lift this valley degeneracy \nbetween the K, Kʹ valleys and consequently generate the valley polarization. However, \nwhen an external magnetic field is removed , the polarization disappears.  In general, the \n2D monolayers preserve the long -range ferromagnetic ordering due to the intrinsic an i-\nsotropy . Specifically in V -based TMDs, the spontaneous valley polarization results from \nthe magnetic interaction amon g the V -3d electrons, which  is independent of external \nfields and enables the modulation  of spin and valley degrees of freedom. We therefore \ninvestigate the orbital -projected band structures of VSi 2Z4 (Z=P, As) and Janus VS i-\nGeP 2As 2 monolayers as shown  in Fig. 3 . As illustrated, all the three structures reveal \nnondegenerate energy values at the K and Kʹ valleys, and as a result  they  show different \nenergy band gaps at the two valleys . The valley polarization is defined as  [5], ΔEv/c = EKʹv/c \n− EKv/c, where EK,Kʹv/c represent  the energies of electronic  band edges at K /Kʹ valleys, cor-\nrespondingly. In the case of VSi 2P4 , using this definition, w e found  a valley polarization  \nof 76.6 meV  in the bottom  conduction  band,  while the top valence bands at K/K ʹ valleys \nremain almost degenerate with valley polarization of -3.9 meV . By contrast, for VSi 2As 4, \nthe valley polarization is -8.2 meV  in the bottom conduction band, whereas that of the top \nvalence band it is calculated to be ~88 meV. On the other hand,  in the Janus phase,  the \nbottom conduction bands at K/K’ remain almost degenerate in energy with valley p o-\nlarization of -5 meV and 73.3 m eV in the top valence bands. This reveals that intrinsic \nferromagnetism is much more efficient in creating valley polarization. In addition, the \nconduction band minimum (CBM) in VSi 2P4 is composed of V -dxy and V -dx2- y2 states at \nboth K and Kʹ points, whil e the valence band maximum (VBM) is majorly composed of  \nV-dz2 orbitals. On the other hand, this orbital composition becomes reverse for pristine \nVSi 2As 4 and Janus VSiGeP 2As 2 i.e. V -dz2 orbitals contribute  to the CBM, while V -dxy and \nV-dx2- y2 form the VBM.  \n \n \n \n \n \n \n \n \nFigure 3 Orbitally resolved electronic band structures of (a) VSi 2P4, (b) VSi 2As 4, and (c) VS i-\nGeP 2As 2 Janus structure. The V -3d orbitals are represented by different colors, where the size of \nthe colored dot describes the contribution from particular orbitals . The contribution decreases as \nthe size of the colored dot decreases.  The values of  the Hubbard parameter used for the d -orbitals \nof V are U = 4 eV for VSi 2P4, and 2 eV for VSi 2As 4 and VSiGeP 2As 2.  \n  \n \n  \n \n \n \nWe study the dependence of magnetic fe atures of the VSi 2Z4 and Janus VSiGeP 2As 2 on \nthe biaxial strain. The energy difference between the FM and AFM configuration (E FM − \nEAFM), which determin es the magnetic ground for the ma terial, is illustrated in Fig. 4  as a \nfunction of compressive and tensile strains. All systems retain the FM orderings under \ndifferent biaxial strains and do not show any phase transition from FM to AFM state with \nthe applied strain. The strain , in this instance , is defined as follows:  \n   (    \n  )      \nHere, ‘a o’ designates the lattice constant at  a strainless state, and ‘a’ represents  the \nstrained lattice constant. The exchange parameter ‘J’ , by taking into account the nearest \nneighbor exchange intera ctions , can be written as [28]: \n    (        \n | ⃗ | ) \nWhere |  | = ½, since the electronic configuration 3d34s2 becomes 3d1 after losing four \nelectrons. The energy differences  between the FM and AFM alignment s can be  easily  \ncalculated using DFT ground state formalism, which can be u sed to compute the \nHeisenberg exchange parameter ‘J’. The large value of ‘J’, togheter with the \nmagnetocrystalline anisotropy will produce a large critical temperature.  \n \n \n \n \n \n \nT \nh\n  \nThe magnetic anisotropy energy (MAE) is used to determine the easy axis for magnet i-\nzation direction . It is defined as  the energy difference  between the  out-of-plane and \nin-plane spin alignments i.e. MAE = E ⊥ - E||. Consequently, a negative MAE will indicate  \nan out-of-plane easy -axis (perpendicular direction for magnetization), while positive \nvalue s of MAE will indicate  an in-plane easy -axis ( magnetization parallel to the plane \ndirection). The MAE is originated because of the reliance of magnetic attributes on a \nspecific crystallographic direction.  Classically, dipole –dipole interactions are believed to \nFigure 4 Strain dependence of energy differences between two magnetic configurations (FM and \nAFM ) for  (a) VSi 2P4, (b) VSi 2As 4, and (c) VSiGeP 2As 2 Janus structures. The values of Hubbard p a-\nrameter used for the d -orbitals of V are U = 4 eV for VSi 2P4, and 2 eV for VSi2As 4 and VSiGeP 2As 2. \n  \n \n be the origin of MAE, nonetheless quantum mechanically, the main cause lies in SOC \n[29]. For that reason, SOC  effects should be considered  in the evaluation of MAE . Thus, \nnon-collinear calculations with SOC taken into account are c arried out to evaluate the \ntotal energies (E ⊥, E ||) for the corresponding magnetization directions. We found MAE \nvalue s of -4 μeV for VSi 2P4 and 53 μeV in VSi 2As 4, indicating out-of-plane and in -plane \nmagnetizations, respectively. Similarly, an  in-plane magnetization is confirmed in ViS i-\nGeP 2As 2 with an MAE value of 48 μeV. The direction of magnetization is essential to a t-\ntain spontaneous valley polarization [76].The effect of biaxial strai n on MAE for all the \nmonolayer  systems is p resented in Fig. 5 . One can see  how the MAE is influenced by the \ntensile and compressive strains. For VSi 2P4, the MAE decreases for either strain dire c-\ntions, with persistent out -of-plane easy axis for magnetization  as shown in Fig. 5a. On the \nother hand, the in -plane easy axis in VSi 2As 4 is found tunable , it can be transformed to \nout-of-plane direction by applying some critical tensile or compressive strains as ind i-\ncated in Fig. 5 b. Likewise, an out -of-plan e magnetization can be achieved in Janus ViS i-\nGeP 2As 2 monolayer at ℇ =1.5 %  as shown in Fig. 5 c. Shaded r egions show  the tuning of \neasy axis for the magnetization direction.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nNext, we show  the electronic band structures of Janus ViSiGeP 2As 2 monolayer by varying \nonsite Coulomb interaction  known as the Hubbard parameter ‘U’ and by taking the SOC \neffect in consideration. Clearly , the CBM at the K/Kʹ valleys is made up of V -dz2 orbitals \nwhen U=2 eV is in the strain -free state, whereas the VBM is composed of V-dxy and V -dx2- \nFigure 5 MAE  as a function of biaxial strain calculated for two magnetic configurations ([001], \n[100]) (a) VSi 2P4, (b) VSi 2As 4, and (c) VSiGeP 2As 2 Janus structures. Shaded regions indica te the \nmodulation of  the easy axis. The brown region is for an out-of-plane easy -axis, while  the cyan r e-\ngion indicates an in -plane easy-axis. The values of  the Hubbard parameter used for the d -orbitals \nof V are U = 4  eV for VSi 2P4, and 2 eV for VSi 2As 4 and VSiGeP 2As 2.  \n  \n \n y2 states.  Upon increasing the Hubbard parameter ‘U’, the V -dz2 orbitals come  down in \nenergy, while the dxy/dx2- y2 states  go up in energy. When U reaches 2.8 eV, the system \nbecomes gapless at the Kʹ point, although gapped at the K valley. The gapless natu re of \nthe band structure at Kʹ displays Weyl -like linear dispersion. Further raising the U, the \nelectronic band gap becomes smaller and smaller at K valley. Conversely, at the Kʹ valley \nthe band gap opens again with a band inversion exchanging the orbital contributions of \nthe valence and conduction b ands as compared to the band structure at U=2 eV. \nConsequently, a topological phase transition  occurs  between U=2.8 and U=3.1 eV , lead ing \nto the emergence of the quantum anomalous Hall phase [5]. At U=3.1 eV, the band gap \ncloses at the K point and starts to reopen at 3.2 eV , with another band inversion achieved \nat K valley. At U=3.2 eV, w e have a band inversion at both K and K’ ; as a result, the Janus \nstructure is rest ored to the trivial ferrovalley insulating phase.  The orbitally -projected \nband structure at U=3.6 eV compl ies with all these behaviors. The evolution of band gaps \nand topological phases as a function of the Coulomb repulsion  at both K/Kʹ valleys is \nsummarized in Fig. 6 h. As indicated, the trend of band gaps at the two valleys are quite \nsimilar ; they begin  to diminish, then they go  to zero, and finally they reopen  by \nincreasing U. Since the band gap is smaller at Kʹ than at K valley (when U=2 eV), the \ncritical Hubbard parameter U necessary for closing the band gap is not similar ; it is U=2.8 \neV and 3.1 eV, respectively.  While usually the Coulomb repulsion kills the  topological \nproperties, in this case the Coulomb repulsion is necessary to observe the topological \nphase.  Additionally, the range of U where  the topological phase appears is between 2.8 \nand 3.1 eV , that is a realistic physical range for the Coulomb repuls ion of 3d electrons . \nMoreover, the orbital characters at the K/K ʹ points of the Brillouin zone are investigated \nas shown in schematic Fig. 7, which reveals the splitting of the energy levels of d orbitals \nin a trigonal prismatic crystal field environment. Here, only the middle layer containing \nV ions is displayed since the nonmagnetic top and bottom layers of these monolayers do \nnot contribute to the spin density distribution.  \n   \n \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n4\n. \nConclusions  \nIn conclusion, based on first principles calculations, we present a detailed and \ncomprehensive study o f pristine VSi 2Z4 (Z=P, As) and Janus VSiGeP 2As 2 monolayers. In \nthe Janus phase , the compound shows  breaking of the mirror symmetry, that is \nFigure 7  A schematic for the evolution of d orbitals  of the spin up-subsector  as a function of Hu b-\nbard parameter U for Janus VSiGeP 2As 2 monolayer at K/K ʹ valleys. At U=2 eV and U=3.6 eV , the \nsystem is in the trivial ferrovalley insulating phase , while at U=3 eV , it is in the topological phase.  \nFigure 6 (a-g) Orbitally -resolved electronic band structures of Janus VSiGeP 2As 2 monolayer with \nPBE+U and SOC included  under different Hubbard parameter values U. The size of  the colored dot \nis pr oportional to the weight of the corresponding orbitals. (h) B and gaps for the two K/K ʹ valleys.  \n  \n \n equivalent to hav ing an electric field, and the system can be  piezoelectric.  After exploring \ntheir structural stability through ground state energies and phononic spec tra, the \nelectronic, magneti c and topological features are investigated. It is observed that these \nstructures exhibit ground -state ferromagnetic ordering that persists at any tensile and \ncompressive strains. In addition, VSi 2P4 shows  -4 μeV MAE with out -of-plane easy -axis, \nwhich incr eases with the atomic number of pnictogens ; for instance , in VSi 2As 4 the MAE \nincrease s dramatically to 53 μeV with in-plane magnetization direction. Likewise, an  \nin-plane ma gnetization is established in V SiGeP 2As 2 with an MAE value of 48 μeV . \nBesides, we analyzed the effect of strain on the magnetic properties such as MAE, which \nrevealed strong dependence on the biaxial strain.  \nWe investigated how the topology of VSiGeP 2As 2 evolves as a function of the Coulomb \ninteraction , and we observed  the topological phase in the physical range of Hubbard  U \nfor 3d electrons . Our analysis of these emerging pristine and Janus -phased magnetic \nsemiconductors open s prospects for studying the interplay between magnetism and \ntopology in two -dimensional materials.  \n \n \nAuthor Contributions: Conceptualization, C.A., G. C., G. H.; Data curation, G.H.; Investigation, \nG.H.; Methodology,C. A., G.C., G. H.; Writing —original draft, C. A., G. C., G. H.;Writing —review \n& editing, C. A., G. C., G. H., A. F., R. I., R. S., supervi sion C. A., G. C. All authors have read and \nagreed to the published version of the manuscript  \n \nFunding: The work is supported by the Foundation for Polish Science through the International  \nResearch Agendas program co -financed by the European Union within  the Smart Growth Oper a-\ntional Programme  (Grant No. MAB/2017/1) . \n \nInstitutional Review Board Statement: Not applicable.  \n \nInformed Consent Statement: Not applicable.  \n \nData Availability Statement: The data that support the findings of this study are available  on re-\nquest from the corresponding author  \n \nAcknowledgments:  This work is supported by the Foundation for Polish Science through the \ninternational research agendas program co -financed by the European Union within the smart \ngrowth operational program  (Grant No. MAB/2017/1).  A.F. was supported by the Polish National \nScience Centre under Project  No. 2020/37/B/ST5/02299.  We acknowledge the access to the \ncomputing facilities of the Interdisciplinary Center of Modeling at the University of Warsaw, \nGrants No. G75 -10, No. GB84 -0, No. GB84 -1 and No. GB84 -7. We acknowledge the access to the \ncomputing facilities of the Poznan Su percomputing and Networking Center Grant No. 609.  \n  \nConflicts of Interest:  The authors declare no conflict of interest.    \n \n  \n1. Gong, C., et al., Discovery of intrinsic ferromagnetism in two -dimensional van der Waals crystals.  Nature, 2017. 546(7657): p. 265 -269. \n2. Huang, B., et al., Layer -dependent ferromagnetism in a van der Waals crystal down to the monolayer limit.  Nature, 2017. 546(7657): p. \n270-273. \n3. Li, Y., et al., Modulated Ferromagnetism and Electric Polarization Induced by Surfac e Vacancy in MX2 Monolayers.  The Journal of \nPhysical Chemistry C, 2022.  \n4. Tiwari, S., et al., Computing Curie temperature of two -dimensional ferromagnets in the presence of exchange anisotropy.  Physical \nReview Research, 2021. 3(4): p. 043024.  \n5. Sheng, K. , et al., Strain -engineered topological phase transitions in ferrovalley 2 H− RuCl 2 monolayer.  Physical Review B, 2022. \n105(19): p. 195312.  \n6. Li, C. and Y. An, Tunable magnetocrystalline anisotropy and valley polarization in an intrinsic ferromagnetic Ja nus 2 H -VTeSe \nmonolayer.  Physical Review B, 2022. 106(11): p. 115417.  \n7. Yin, Y., et al., Emerging versatile two -dimensional MoSi $ _2 $ N $ _4 $ family.  arXiv preprint arXiv:2211.00827, 2022.  \n8. Islam, R., et al., Tunable spin polarization and electronic structure of bottom -up synthesized MoSi 2 N 4 materials.  Physical Review B, \n2021. 104(20): p. L201112.  \n9. Feng, X., et al., Valley -related multiple Hall effect in monolayer V Si 2 P 4.  Physical Review B, 2021. 104(7): p. 075421.  \n10. Autieri, C., et al., Limited Ferromagnetic Interactions in Monolayers of MPS3 (M= Mn and Ni).  The Journal of Physical Chemistry C, \n2022. 126(15): p. 6791 -6802.  \n11. Basnet, R., et al., Controlling magnetic exchange and anisotropy by non -magnetic ligand substitution in layered MPX 3 (M= Ni, Mn; X= \nS, Se).  Phys. Rev. Research, 2022.  4,( 023256).  \n12. Liu, W., et al., A three -stage magnetic phase transition revealed in ultrahigh -quality van der Waals bulk magnet CrSBr.  ACS nano, \n2022. 16(10): p. 15917 -15926.  \n13. Liu, X., et al., Canted  antiferromagnetism in the quasi -one-dimensional iron chalcogenide BaFe 2 Se 4.  Physical Review B, 2020. \n102(18): p. 180403.  \n14. Liu, P., et al., Strain -driven valley states and phase transitions in Janus VSiGeN4 monolayer.  Applied Physics Letters, 2022. 121(6): p. \n063103.  \n15. Deng, Y., et al., Gate -tunable room -temperature ferromagnetism in two -dimensional Fe3GeTe2.  Nature, 2018. 563(7729): p. 94 -99. \n16. Gibertini, M., et al., Magnetic 2D materials and heterostructures.  Nature nanotechnology, 2019. 14(5): p. 408 -419. \n17. Huang, B., et al., Electrical control of 2D magnetism in bilayer CrI3.  Nature nanotechnology, 2018. 13(7): p. 544 -548. \n18. Burch, K.S., Electric switching of magnetism in 2D.  Nature Nanotechnology, 2018. 13(7): p. 532 -532. \n19. Tian, Y., et al., Optically driven magnetic phase transition of monolayer RuCl3.  Nano Letters, 2019. 19(11): p. 7673 -7680.  \n20. Zhao, Y., et al., Surface vacancy -induced switchable electric polarization and enhanced ferromagnetism in monolayer metal trihalides.  \nNano Letters, 2018. 18(5): p. 2943 -2949.  \n21. Song, X., F. Yuan, and L.M. Schoop, The properties and prospects of chemically exfoliated nanosheets for quantum materials in two \ndimensions.  Applied Physics Reviews, 2021. 8(1): p. 011312.  \n22. Jiang, S., et al ., Exchange magnetostriction in two -dimensional antiferromagnets.  Nature Materials, 2020. 19(12): p. 1295 -1299.  \n23. Mermin, N.D. and H. Wagner, Absence of ferromagnetism or antiferromagnetism in one -or two -dimensional isotropic Heisenberg \nmodels.  Physical Review Letters, 1966. 17(22): p. 1133.  \n24. Lado, J.L. and J. Fernández -Rossier, On the origin of magnetic anisotropy in two dimensional CrI3.  2D Materials, 2017. 4(3): p. 035002.  \n25. Gurney, B., et al., Spin valve giant magnetoresistive sensor materials fo r hard disk drives , in Ultrathin Magnetic Structures IV . 2005, \nSpringer. p. 149 -175. \n26. Prinz, G.A., Magnetoelectronics.  science, 1998. 282(5394): p. 1660 -1663.    \n \n 27. Kryder, M.H., Magnetic thin films for data storage.  Thin solid films, 1992. 216(1): p. 174-180. \n28. Webster, L. and J. -A. Yan, Strain -tunable magnetic anisotropy in monolayer CrCl 3, CrBr 3, and CrI 3.  Physical Review B, 2018. \n98(14): p. 144411.  \n29. Singla, R., et al., Curie temperature engineering in a novel 2D analog of iron ore (hematene)  via strain.  Nanoscale Advances, 2020. \n2(12): p. 5890 -5896.  \n30. Hussain, G., et al., Strain modulated electronic and optical properties of laterally stitched MoSi2N4/XSi2N4 (X= W, Ti) 2D \nheterostructures.  Physica E: Low -dimensional Systems and Nanostructur es, 2022. 144: p. 115471.  \n31. Zhang, J., et al., Prediction of bipolar VSi 2 As 4 and VGe 2 As 4 monolayers with high Curie temperature and strong magnetocrystalline \nanisotropy.  Physical Review B, 2022. 106(23): p. 235401.  \n32. Hussain, G., et al., Electron ic and optical properties of InAs/InAs0. 625Sb0. 375 superlattices and their application for far -infrared \ndetectors.  Journal of Physics D: Applied Physics, 2022. 55(49): p. 495301.  \n33. Dey, D., A. Ray, and L. Yu, Intrinsic ferromagnetism and restrictive th ermodynamic stability in MA 2 N 4 and Janus VSiGeN 4 \nmonolayers.  Physical Review Materials, 2022. 6(6): p. L061002.  \n34. Hong, Y. -L., et al., Chemical vapor deposition of layered two -dimensional MoSi2N4 materials.  Science, 2020. 369(6504): p. 670 -674. \n35. Cao, L., et al., Two-dimensional van der Waals electrical contact to monolayer MoSi2N4.  Applied Physics Letters, 2021. 118(1): p. \n013106.  \n36. Bafekry, A., et al., MoSi2N4 single -layer: a novel two -dimensional material with outstanding mechanical, thermal, e lectronic and \noptical properties.  Journal of Physics D: Applied Physics, 2021. 54(15): p. 155303.  \n37. Mortazavi, B., et al., Exceptional piezoelectricity, high thermal conductivity and stiffness and promising photocatalysis in \ntwo-dimensional MoSi2N4 famil y confirmed by first -principles.  Nano Energy, 2021. 82: p. 105716.  \n38. Jian, C. -c., et al., Strained MoSi2N4 monolayers with excellent solar energy absorption and carrier transport properties.  The Journal of \nPhysical Chemistry C, 2021. 125(28): p. 15185 -15193. \n39. Wang, Q., et al., Efficient Ohmic contacts and built -in atomic sublayer protection in MoSi2N4 and WSi2N4 monolayers.  npj 2D \nMaterials and Applications, 2021. 5(1): p. 1 -9. \n40. Yu, J., et al., High intrinsic lattice thermal conductivity in monolaye r MoSi2N4.  New Journal of Physics, 2021. 23(3): p. 033005.  \n41. Cui, Z., et al., Tuning the electronic properties of MoSi2N4 by molecular doping: a first principles investigation.  Physica E: \nLow-dimensional Systems and Nanostructures, 2021. 134: p. 114873.  \n42. Bafekry, A., et al., Tunable electronic and magnetic properties of MoSi2N4 monolayer via vacancy defects, atomic adsorption and \natomic doping.  Applied Surface Science, 2021. 559: p. 149862.  \n43. Yao, H., et al., Novel Two -Dimensional Layered MoSi2Z4 (Z=  P, As): New Promising Optoelectronic Materials.  Nanomaterials, 2021. \n11(3): p. 559.  \n44. Lv, X., et al., Strain modulation of electronic and optical properties of monolayer MoSi2N4.  Physica E: Low -dimensional Systems and \nNanostructures, 2022. 135: p. 11496 4. \n45. Pham, D., Electronic properties of a two -dimensional van der Waals MoGe 2 N 4/MoSi 2 N 4 heterobilayer: Effect of the insertion of a \ngraphene layer and interlayer coupling.  RSC Advances, 2021. 11(46): p. 28659 -28666.  \n46. Yuan, G., et al., Highly Sen sitive Band Alignment of Graphene/Mosi2n4 Heterojunction Via External Electric Field.  Available at SSRN \n4052449.  \n47. Cao, L., et al., Two-dimensional van der Waals electrical contact to monolayer MoSi.  2021.  \n48. Chen, R., D. Chen, and W. Zhang, First -princ iples calculations to investigate stability, electronic and optical properties of fluorinated \nMoSi2N4 monolayer.  Results in Physics, 2021. 30: p. 104864.  \n49. Bafekry, A., et al., Band -gap engineering, magnetic behavior and Dirac -semimetal character in the MoSi2N4 nanoribbon with armchair \nand zigzag edges.  Journal of Physics D: Applied Physics, 2021. 55(3): p. 035301.    \n \n 50. Ray, A., et al., Inducing Half -Metallicity in Monolayer MoSi2N4.  ACS omega, 2021. 6(45): p. 30371 -30375.  \n51. Ng, J.Q., et al., Tunable ele ctronic properties and band alignments of MoSi2N4/GaN and MoSi2N4/ZnO van der Waals heterostructures.  \nApplied Physics Letters, 2022. 120(10): p. 103101.  \n52. Xu, J., et al., First -principles investigations of electronic, optical, and photocatalytic properti es of Au -adsorbed MoSi2N4 monolayer.  \nJournal of Physics and Chemistry of Solids, 2022. 162: p. 110494.  \n53. Bafekry, A., et al., Effect of electric field and vertical strain on the electro -optical properties of the MoSi2N4 bilayer: A first -principles \ncalcul ation.  Journal of Applied Physics, 2021. 129(15): p. 155103.  \n54. Li, Q., et al., Strain effects on monolayer MoSi2N4: Ideal strength and failure mechanism.  Physica E: Low -dimensional Systems and \nNanostructures, 2021. 131: p. 114753.  \n55. Hussain, G., et al., Exploring the structural stability, electronic and thermal attributes of synthetic 2D materials and their \nheterostructures.  Applied Surface Science, 2022. 590: p. 153131.  \n56. Islam, R., et al., Fast electrically switchable large gap qu antum spin Hall states in MGe $ _2 $ Z $ _4$.  arXiv preprint \narXiv:2211.06443, 2022.  \n57. Sheoran, S., et al., Coupled Spin -Valley, Rashba Effect, and Hidden Spin Polarization in WSi2N4 Family.  The Journal of Physical \nChemistry Letters, 2023: p. 1494 -1503.  \n58. Islam, R., et al., Switchable large -gap quantum spin Hall state in the two -dimensional M Si 2 Z 4 class of materials.  Physical Review B, \n2022. 106(24): p. 245149.  \n59. Van Thiel, T., et al., Coupling charge and topological reconstructions at polar oxide  interfaces.  Physical review letters, 2021. 127(12): \np. 127202.  \n60. Smaili, I., et al., Janus monolayers of magnetic transition metal dichalcogenides as an all -in-one platform for spin -orbit torque.  Physical \nReview B, 2021. 104(10): p. 104415.  \n61. Zhao, X. , et al., Topological properties of Xene turned by perpendicular electric field and exchange field in the presence of Rashba \nspin-orbit coupling.  Journal of Physics: Condensed Matter, 2022.  \n62. Hussain, G., et al., Emergence of Rashba splitting and spin -valley properties in Janus MoGeSiP2As2 and WGeSiP2As2 monolayers.  \nJournal of Magnetism and Magnetic Materials, 2022. 563: p. 169897.  \n63. Dou, K., et al., Theoretical Prediction of Antiferromagnetic Skyrmion Crystal in Janus Monolayer CrSi2N2As2.  ACS nano, 20 22. \n64. Laref, S., et al., Topologically stable bimerons and skyrmions in vanadium dichalcogenide Janus monolayers.  arXiv preprint \narXiv:2011.07813, 2020.  \n65. Kresse, G. and D. Joubert, From ultrasoft pseudopotentials to the projector augmented -wave method . Physical review b, 1999. 59(3): p. \n1758.  \n66. Kresse, G. and J. Furthmüller, Efficient iterative schemes for ab initio total -energy calculations using a plane -wave basis set.  Physical \nreview B, 1996. 54(16): p. 11169.  \n67. Perdew, J.P., K. Burke, and M. Er nzerhof, Generalized gradient approximation made simple.  Physical review letters, 1996. 77(18): p. \n3865.  \n68. Togo, A. and I. Tanaka, First principles phonon calculations in materials science.  Scripta Materialia, 2015. 108: p. 1 -5. \n69. Cuono, G., et al., Spin–orbit coupling effects on the electronic properties of the pressure -induced superconductor CrAs.  The European \nPhysical Journal Special Topics, 2019. 228(3): p. 631 -641. \n70. Wadge, A.S., et al., Electronic properties of TaAs2 topological semimetal inve stigated by transport and ARPES.  Journal of Physics: \nCondensed Matter, 2022. 34(12): p. 125601.  \n71. Li, F., et al., Electronic and optical properties of pristine and vertical and lateral heterostructures of Janus MoSSe and WSSe.  The \njournal of physical che mistry letters, 2017. 8(23): p. 5959 -5965.  \n72. Yan, S., et al., Enhancement of magnetism by structural phase transition in MoS2.  Applied Physics Letters, 2015. 106(1): p. 012408.    \n \n 73. Ominato, Y., J. Fujimoto, and M. Matsuo, Valley -dependent spin transport in monolayer transition -metal dichalcogenides.  Physical \nReview Letters, 2020. 124(16): p. 166803.  \n74. Ahammed, R. and A. De Sarkar, Valley spin polarization in two -dimensional h− M N (M= Nb, Ta) monolayers: Merger of valleytronics \nwith spintronics.  Physica l Review B, 2022. 105(4): p. 045426.  \n75. Cui, Q., et al., Spin-valley coupling in a two -dimensional V Si 2 N 4 monolayer.  Physical Review B, 2021. 103(8): p. 085421.  \n76. Liu, P., et al., Strain -driven valley states and phase transitions in Janus VSiGeN4 mo nolayer.  arXiv preprint arXiv:2206.00892, 2022.  \n "    },    {        "title": "2302.07330v1.A_theoretical_perspective_on_the_modification_of_the_magnetocrystalline_anisotropy_at_molecule_cobalt_interfaces.pdf",        "content": "A theoretical perspective on the modi\fcation of the magnetocrystalline anisotropy at\nmolecule-cobalt interfaces\nAnita Halder,1Sumanta Bhandary,1David D. O'Regan,1Stefano Sanvito,1and Andrea Droghetti1,∗\n1School of Physics and CRANN Institute, Trinity College Dublin, The University of Dublin, Dublin 2, Ireland\nWe study the modi\fcation of the magnetocrystalline anisotropy (MCA) of Co slabs induced\nby several di\u000berent conjugated molecular overlayers, i.e., benzene, cyclooctatetraene, naphthalene,\npyrene and coronene. We perform \frst-principles calculations based on Density Functional Theory\nand the magnetic force theorem. Our results indicate that molecular adsorption tends to favour\na perpendicular MCA at surfaces. A detailed analysis of various atom-resolved quantities, accom-\npanied by an elementary model, demonstrates that the underlying physical mechanism is related\nto the metal-molecule interfacial hybridization and, in particular, to the chemical bonding between\nthe molecular C pzand the out-of-plane Co dz2orbitals. This e\u000bect can be estimated from the\norbital magnetic moment of the surface Co atoms, a microscopic observable accessible to both\ntheory and experiments. As such, we suggest a way to directly assess the MCA modi\fcations at\nmolecule-decorated surfaces, overcoming the limitations of experimental studies that rely on \fts of\nmagnetization hysteresis loops. Finally, we also study the interface between Co and both C 60and\nAlq3, two molecules that \fnd widespread use in organic spintronics. We show that the modi\fcation\nof the surface Co MCA is similar upon adsorption of these two molecules, thereby con\frming the\nresults of recent experiments.\nI. INTRODUCTION\nSurface magnetic properties dominate in thin \flms,\nmultilayer structures and nanoparticles of 3 dtransition\nmetals, resulting in a rich physics1,2that is of critical\nimportance for many technological areas, such as high-\ndensity magnetic recording3and spintronics4. Most\nof these properties can be further modi\fed through\nthe adsorption of molecules5. The formation of strong\ncovalent bonds between a molecule and a transition\nmetal surface leads to the hybridization of the molecular\norbitals with the 3 delectronic bands6{8. This a\u000bects the\nsurface magnetic moments9{11, spin-polarization12{15,\nand electron correlation16. Furthermore, the hybridiza-\ntion can increase the strength of the magnetic exchange\ninteraction between the surface atoms17,18, thus con-\ntributing to the magnetic hardening observed in recent\nexperiments19{22. Selected molecules can even lead to\na ferromagnetic response in otherwise non-magnetic Cu\nnanostructures23. These electronic e\u000bects are some-\ntimes further modi\fed through surface reconstruction\ncontrolled by temperature annealing24,25, a feature that\no\u000bers additional possibilities for materials engineering.\nAmong the surface properties that can be drastically\nmodi\fed through molecular adsorption there is also the\nmagnetic anisotropy. This determines the \\easy\" and\n\\hard\" directions of magnetization of thin \flms26,27\nand, therefore, their magnetization switching dynamics\nand thermal stability. Bairagi et al.28reported that\nthe deposition of buckyball C 60molecules on epitaxial\nCo hcp(0001) thin-\flms causes an enhancement of the\nperpendicular magnetic anisotropy. The same group\nlater extended the study, considering another popular\nmolecule in spintronics, namely Alq 3, and comparing\nultrathin Co \flms supported on gold and platinum (Ref.29). They found a considerable enhancement of the\nperpendicular magnetic anisotropy in all samples and\nconcluded that this e\u000bect is rather general in the case\nof chemisorbed molecules on Co. Along the same lines,\nBenini et al. recently suggested that a similar magnetic\nanisotropy enhancement could be one of causes for the\nhuge magnetic hardening measured in polycrystalline\nCo \flms interfaced with C 60and Gaq 3(Ref. 22). It\nmust be noted, however, that in their experiment the\nmagnetic \feld was swept in the \flms plane rather than\nout-of-plane. The magnetic anisotropy is, in general,\ndetermined by a combination of several extrinsic and\nintrinsic factors26,27. Nonetheless, in all the mentioned\nexperiments, the magnetocrystalline anisotropy (MCA)\nis regarded as the main contribution that is modi\fed\nupon adsorption. The MCA stems from the spin-orbit\ncoupling (SOC) and is related to the atomic structure\nand bonding of a material. Several works18,30{33have\nalready shown that the SOC-induced spin-texture of\nmetallic or seminconducting surfaces can be dramatically\na\u000bected by the chemical bonding with adsorbates of\nvarious kinds. In 3 dferromagnetic \flms with a molecular\noverlayer, this e\u000bect is expected to give rise to MCA\nchanges.\nThe modi\fcation of surface magnetic properties\nin thin \flms are generally rationalized by means of\n\frst-principles calculations based on Density Functional\nTheory (DFT). The results for Co(0001) show that\nthe contribution of the out-of-plane Co dz2orbitals to\nthe surface MCA favours in-plane magnetization. This\nin-plane contribution is completely suppressed upon C 60\nadsorption, when the Co dz2orbitals hybridize with\nthe Cpzorbitals of C 60(Ref. 28 and 29). In contrast,\nthe contributions to the MCA of the other 3 dCo or-\nbitals, which favour out-of-plane magnetization, remain\nuna\u000bected by C 60. Hence, the perpendicular MCA isarXiv:2302.07330v1  [cond-mat.mtrl-sci]  14 Feb 20232\nenhanced at the C 60/Co interface. The DFT results\nqualitatively support the experimental observations.\nHowever, a quantitative correlation between interfacial\nMCA and molecule-metal hybridization remains to be\nestablished, and systematic investigations on model\nsystems are needed to address this issue. Notably, the\ncomparison between DFT and experimental results\nappears very problematic, when going beyond phe-\nnomenological considerations. On the one hand, the\norbital-resolved MCA, computed by DFT and used in\nprevious studies as a descriptor for the e\u000bect, is a micro-\nscopic quantity that is not directly measurable. On the\nother hand, in experiments, the MCA is estimated from\nmagnetization hysteresis loops as an average macro-\nscopic quantity. In principles, the experiments could\nbe modelled through multi-scale approaches combining\nDFT and atomistic spin dynamics simulations34,35, but,\nin practice, the level of complexity and the number\nof features (e.g., disorder, temperature, etc.) that\nsuch simulations should consider, are out-of-reach. To\novercome this gap between theory and experiments and\nto proceed towards a quantitative understanding of the\nMCA modi\fcation at molecule-Co interfaces, we then\nneed to identify a MCA-related microscopic observable,\nwhich can be directly calculated from \frst-principles\nas well as measured in experiments with comparable\naccuracy.\nIn this work, we explain how the chemical bonding\nbetween the Co dz2and the molecular C pzorbitals\nfavours perpendicular MCA at molecule-Co interfaces.\nThe physics is \frstly described through an elementary\nmodel and then analyzed at the quantitative level by\nmeans of \frst-principles DFT calculations for several\nprototypical systems, namely benzene, cyclooctate-\ntraene, naphthalene, pyrene, and coronene molecules on\nCo. The results con\frm overall the phenomenological\narguments used so far to interpret the experimental\nresults in Refs. 22, 28, and 29, and provide a strong\nindication that the modi\fcation of the MCA is general\nfor Co with chemisorbed conjugated molecules. Going\nbeyond the results of previous studies, we examine the\nkey electronic parameters determining the magnitude\nof the interfacial MCA. In particular, we highlight\nthe importance of the energy splitting between the\nin-plane Co dx2\u0000y2orbitals, which are not a\u000bected\nby the bonding with molecules, and the dz2orbitals,\nwhich are instead strongly hybridized with molecular C\natoms. When, upon molecular adsorption, the dz2to\ndx2\u0000y2energy splitting increases, we \fnd an enhanced\nperpendicular interface MCA. An additional, important\nobservation stemming from our DFT calculations is that\nthe e\u000bect of the pz-dz2hybridization on the MCA can be\nquantitatively estimated through the orbital magnetic\nmoment, which is a microscopic observable and can be\nmeasured in experiments. As such, we suggest a way\nto directly assess our \frst-principles predictions to \fll\nthe current gap between theoretical and experimentalstudies. Finally, we complete our work by comparing the\nadsorption of the two experimentally studied molecules,\nnamely C 60and Alq 3, on Co. We show that the\nMCA modi\fcation is quite similar in the two cases, in\nagreement with experimental observations22,29.\nThe paper is organized as follows. We begin in Sec.\nII by introducing an elementary model describing the\nMCA at conjugated molecule/Co interfaces. We then\ncontinue by discussing the results of our DFT calcu-\nlations. After providing the computational details in\nSec. III, we describe in great detail the interface be-\ntween Co and benzene, which is one the simplest aro-\nmatic molecules. Speci\fcally, in Sec. IV A, we analyse\nvarious atom-resolved quantities, which help us to under-\nstand the basic electronic properties and their quantita-\ntive correlation with the MCA. We then extend our study\nto consider the interface with other conjugated molecules\nand to examine the e\u000bect of the chemical reactivity (in\nSec. IV B) and of molecular coverage (in Sec. IV C).\nFinally, we discuss the results for C 60/Co and Alq 3/Co\ninterfaces in Sec. IV D, and we conclude in Sec. V.\nΔdxydxzdyzdz2dx2−y2\nΔ/2\nE\nΔdxydxzdyz σ∗\n(pz−dz2)dx2−y2\nΔ/2\nEσ(pz−dz2)\nΔ'a)\nb)\nFIG. 1. (a) Energy diagram for the crystal-\feld 3 dorbital\nsplitting at a Co(001) surface. We assume that the dz2and\ndx2\u0000y2orbitals remain degenerate and that the energy dif-\nference, \u0001, between dxyand doublet ( dxz,dyz) is twice the\nenergy di\u000berence between ( dxz,dyz) and (dz2,dx2\u0000y2). In\npanel (b) the same energy diagram is now modi\fed by the\nadsorption of a conjugated molecule. This displays the bond-\ning,\u001b, and anti-boding, \u001b\u0003, states due to the hybridization of\nthe Codz2orbital with the pzorbital, which originally form\nthe\u0019molecular system.\nII. A SIMPLE MODEL FOR SURFACE MCA\nThe origin of the MCA is the SOC, whose Hamiltonian\nis written as ^HSOC =\u0018^L\u0001^S=~2, where ^Land ^Sare\nrespectively the orbital and spin angular momentum\noperator, and \u0018is the SOC constant. For a ferromag-\nnetic thin \flm, the MCA energy EMCA is de\fned as the\nenergy di\u000berence calculated between the magnetization3\nparallel (k) and perpendicular ( ?) to the \flm surface,\ni.e.,EMCA =Ek\u0000E?. Positive (negative) EMCA implies\nout-of-plane (in-plane) MCA. Throughout this work, we\nassume a Cartesian frame of reference with the \flm's\nperpendicular direction being along the z-axis.\nIn 3dtransition-metals, the SOC constant \u0018is rela-\ntively small compared to other characteristic energies,\nsuch as the band width, the crystal \feld splitting, and\nthe exchange coupling. Thus, EMCA can be evaluated\nfrom the SOC-induced ground state energy corrections\nwithin second-order perturbation theory as26,36,37\nEMCA =X\no;ujh oj^HSOCj uij2\nk\u0000jh oj^HSOCj uij2\n?\n\u000fo\u0000\u000fu:(1)\nHerej oiandj uiare the unperturbed occupied and\nunoccupied states with energies \u000foand\u000fu, respectively.\nThese can have either the same or di\u000berent spin. We set\nthe spin quantization axis along the zdirection, which\nis normal to the \flm. The matrix elements in Eq. (1)\ndescribe virtual transitions, which can promote in-plane\nor out-of-plane anisotropy. They strongly contribute\nto the total EMCA when they couple occupied and\nunoccupied states close to the Fermi level so that the\ndenominator in Eq. (1) is small. In crystalline \flms,\nj oiandj uiare Bloch states. However, to capture\nsome qualitative features of the MCA in a simple\nway, we consider here a model proposed in Ref. [38]\nusing atomic 3 dorbitals and ignoring electronic band\nformation. The matrix elements in Eq. (1) can then be\ncalculated analytically39and are reported in Sec. S1 of\nthe supplementary material (SM). Based on these, one\ncan then derive some simple rules, which will be used to\ninfer the direction of the MCA39{41. Since each dorbital\nis characterized by its magnetic quantum number min\naddition to the spin quantum number, one has that:\n(i) the transitions between orbitals with m= 0 (dz2)\nandjmj= 1 (dyzanddxz) in the same spin channel or\nbetween orbitals with m= 0 (dz2) andjmj= 2 (dx2\u0000y2\nanddxy) in di\u000berent spin channels favour in-plane MCA;\n(ii) the transitions between orbitals with m= 0 andjmj\n= 1 in di\u000berent spin channels and between orbitals with\njmj= 2 in the same spin channel favour out-of-plane\nMCA.\nWe consider the simplest case of a Co fcc(001) surface\n(the treatment can be easily generalized to other sur-\nfaces). The crystal \feld splits the \fve dorbitals into a\nsinglet (dxy), a doublet ( dxz,dyz) and two further singlets\n(dz2anddx2\u0000y2). Since generally the separation between\ndz2anddx2\u0000y2is small, we assume that these orbitals\nremain degenerate. In addition, to simplify the calcula-\ntion, we set the energy di\u000berence, \u0001, between dxyand\nthe doublet ( dxz,dyz) to be twice the energy di\u000berence\nbetween (dxz,dyz) and (dz2,dx2\u0000y2). This is displayed\nin the energy level diagram of Fig. 1(a). Since (fcc) Co\nis a strong ferromagnet and the majority-spin band isnearly fully occupied42, majority-spin states do not sub-\nstantially contribute to the MCA energy. Therefore, we\nonly take into account transitions between spin-down oc-\ncupied states to the spin-down unoccupied ones, and we\napply the simple rules mentioned above to infer their con-\ntribution. Speci\fcally, since Co has two electrons in the\nspin-downdshell, thedxyorbital is fully \flled, the dxz\nanddyzare half-\flled and the dz2anddx2\u0000y2are empty.\nThus the dominant transitions in Eq. (1) are, on the one\nhand, from dxytodx2\u0000y2favouring out-of-plane MCA,\nand, on the other hand, from dxytodxzand fromdyzto\ndz2anddx2\u0000y2(weighted by a factor 1 =2 to account for\nthe half-\flling of dxzanddyz) favouring in-plane MCA.\nNeglecting degeneracies, the total EMCA turns out to be\nequal to\nEMCA =\u000011\u00182\n12\u0001; (2)\nand the MCA is therefore in-plane ( EMCA<0) for our\nchoice of parameters.\nNow, when a conjugated organic molecule is\nchemisorbed on the Co thin \flm, the pzatomic or-\nbitals, originally forming the \u0019molecular orbitals, hy-\nbridize with the minority Co dz2states, giving rise to\nmolecule-metal hybrid bonding and antibonding states6.\nThe bonding \u001bstate has a predominant pzcharacter and\nis situated at low energies, while the antibonding \u001b\u0003state\nhasdz2character and appears at much higher energies as\nshown in Fig. 1(b). The modi\fcation of the MCA energy\ninduced by molecular adsorption can then be computed\nvia Eq. (1) considering these additional hybrid states. In\nour simple model, we indicate as \u00010the energy separa-\ntion between the doublet ( dxz,dyz) and the antibonding\nstate, and we neglect the low-energy bonding state, which\nis mostly localized on the molecule rather than at the\nmetal surface. Furthermore, since the anti-bonding state\nhas mostly dz2character we approximate j\u001b\u0003i\u0019jdz2i\nin the evaluation of the transition matrix elements. In\npractice, we assume that the only e\u000bect of the absorption\nis to shift the dz2orbital up in the energy by an amount\n(\u00010\u0000\u0001=2). Hence, the resulting MCA energy is modi\fed\nand becomes\nEMCA =7\u00182\n12\u0001\u00003\u00182\n4\u00010: (3)\nNotably, by analyzing this equation we see that, if \u00010>\n9\u0001=7, the Co surface MCA will switch from in-plane\n(EMCA<0) to out-of-plane ( EMCA>0) in the hy-\nbrid Co-molecule system. The reason is that, for a large\nenough \u00010, the contribution of the transition from dyzto\ndz2favouring in-plane MCA in Eq. (1) is drastically sup-\npressed. This remarkably simple result indicates that, in\nprinciple, not just the magnitude, but also the direction\nof the MCA can be modi\fed through molecular adsorp-\ntion. In spite of the simplicity of our model and the\nheuristic choice of the crystal-\feld splitting parameters,\nwe will show in the following that the key qualitative4\nfeatures of the MCA at organic-ferromagnetic metal in-\nterfaces are largely captured by such atomic-like picture.\nIII. COMPUTATIONAL DETAILS\nDFT calculations are performed using the projector\naugmented wave (PAW)43method as implemented in\nthe Vienna Ab-initio Simulation Package (VASP)44.\nThe generalized gradient approximation (GGA) in\nthe formulation by Perdew, Burke, and Ernzerhof\n(PBE)45is chosen for the exchange-correlation func-\ntional. The tetrahedron integration method46with a\nkinetic-energy cuto\u000b of 600 eV is employed. An energy\nconvergence criterion equal to 10\u00007eV for total energy\ncalculations and 10\u00003eV for ionic relaxations is adopted.\nIn order to describe the thin-\flm geometry, we\nconsider \fnite slabs comprising a few Co fcc(001) or\nhcp(0001) layers as speci\fed in the following sections\nand in the SM. Each slab is separated by about 8 \u0017A of\nvacuum along the z-direction (orthogonal to the surface)\nfrom its periodic image. The lattice constant of fcc Co\nis \fxed to the experimental value of 3.54410 \u0017A as from\nRef. [47]. The molecule-Co systems are modeled with\nthe molecule adsorbed on one side of the slab. During\nthe interface geometry optimization the two bottom\nlayers of the slab are maintained \fxed, while all the\nother atoms are allowed to relax until the ionic forces\nare lower than 0.001 eV/ \u0017A.\nThe MCA energies are calculated by using the mag-\nnetic force theorem48{50followings two steps. Firstly,\nwe carry out a scalar relativistic collinear charge self-\nconsistent calculation to obtain the charge density.\nThen, we use that charge density as input in non-\ncollinear calculations performed non-self-consistently\nand including SOC, where the magnetization vector\nis oriented along di\u000berent directions. In particular,\nthe MCA energy is given by the di\u000berence of the\ntotal band energies calculated with the magnetiza-\ntion parallel and perpendicular to the slab surface,\nEMCA =Eband;k\u0000Eband;?. This method has already\nbeen applied in several previous works to estimate the\nMCA energies of di\u000berent ferromagnetic multilayer\nsystems (see, for example, Refs. 51{56), and the results\ngenerally agree well with the values obtained from fully\nself-consistent total-energy calculations56,57. For the\nsmallest considered systems, we also directly veri\fed\nthat force theorem and total energy calculations give\ncomparable results up to 0.1 meV per supercell.\nThek-point sampling is performed using a Monkhorst-\nPack (MP) grid. All calculations are converged with\nrespect to the number of k-points. We \fnd that a 12 \u000212\nk-point mesh in the two-dimensional Brillouin zone is\nenough to reach an accuracy below 1 meV on the MCA\nenergy of slabs with 2 \u00022 in-plane periodicity. Althoughthe MCA energy is a very small quantity (of the order\nof one meV or even less), and the DFT results are quite\nsensitive to the details of the calculations and to the\nexchange correlation functional58{60, we mostly focus on\ngeneral trends, which are expected to be well captured\nby DFT calculations.\nOrbital magnetic moments are calculated without the\norbital polarization contribution61or without any e\u000bec-\ntive orbital polarization corrections62. In case of Co this\ncontribution is negligible63. The SOC matrix elements\nare de\fned as64\nEI;mm0\nSOC =X\nnkX\n\u001b\u001b0Plm\u001b\nnkh\u001eI;lm\u001bj^HSOCj\u001eI;lm0\u001b0iPlm0\u001b0\u0003\nnk:\n(4)\nHere,\u001eI;lm\u001bis the PAW partial wave of an atom Iwith\norbital quantum number l, magnetic quantum number m\nand spin index \u001b. The index of the occupied Kohn-Sham\nelectronic bands is n, andPlm\u001b\nnkis the projection of the\nn-th Kohn-Sham pseudo-orbital onto a PAW projector\nsited on the atom I. The atomic SOC energy is then\nobtained as EI\nSOC=P\nm;m0EI;mm0\nSOC (see Ref. 64). The\ncontribution to the MCA energy of the I-th atom,EI\nMCA,\nis estimated from the di\u000berence of the atomic SOC ener-\ngies with the magnetization in-plane and out-of-plane,\nEI\nMCA/\u0001EI\nSOC=EI\nSOC;k\u0000EI\nSOC;?; (5)\nwhere the spin quantization axis is taken perpendicular\nto the surface (i.e. along the zdirection). For most of\nthe considered systems, the sum of \u0001 EI\nSOCover all atoms\nis found to be twice the MCA energy obtained from the\nmagnetic force theorem, i.e., EMCA\u00190:5P\nI\u0001EI\nSOC.\nBased on the discussion of Ref. [65], this indicates that\nthe SOC is e\u000bectively a second-order perturbative con-\ntribution to the system non-relativistic Hamiltonian and,\ntherefore, the interpretation of the results based on Eq.\n(1) is well justi\fed.\nIV. DFT RESULTS\nA. Interface between benzene and Co\nIn order to illustrate how the physics introduced\nthrough the elementary model emerges in real systems,\nwe start by performing DFT calculations for an interface\nbetween benzene (Bz) and Co. This can be considered\nas a prototypical model system, since Bz is the simplest\naromatic\u0019-conjugated molecule that chemisorbs on\nreactive 3dferromagnetic surfaces66,67. We consider\nfcc Co and select the same fcc(001) surface as in the\natomic model. Note that Co fcc(001) \flms have been\nextensively studied for spintronics68{73. They represent\na convenient system to study interface e\u000bects, as the\nbulk contribution to the total MCA vanishes for large\n\flms due to the cubic symmetry. Nevertheless, since5\nFIG. 2. DFT results for the Co and Bz/Co slabs. (a) Side view of the Bz/Co slab. Brown, white, and yellow spheres indicate\nC, H, and Co atoms, respectively. (b) MCA energy of the bare Co slab, of the Bz/Co slab, and of the distorted slab [bare\nCo (-Bz)] obtained by removing the molecule from the Bz/Co slab. (c) DOS projected over the 3 dorbitals of the Co atom\nthat is labeled 36 in (a). The Fermi energy is at 0 eV. The black and cyan lines are for the bare Co slab and the Bz/Co slab,\nrespectively. The red curve in the bottom panel is the DOS projected over the Bz C pzorbital, which hybridizes with the dz2\norbital. (d) \u0001 EI\nSOCevaluated for all atoms. The red, orange and black dots represent the values for the Co slab, the distorted\nCo slab, and the Bz/Co slab. Atoms 1-8 are at the slab bottom surface, atoms 33-40 are at the top surface, and atoms 41-52\nare the Bz atoms. The Co atoms in the \frst and second layer of the Bz/Co slab, for which \u0001 EI\nSOCis positive, are highlighted\nwith orange and light yellow colors in the slab side view.\nseveral recent experiments mentioned in the introduction\nhave focused on hcp(0001) surfaces28, in the SM (Sec.\nS2) we also report some additional calculations for\nCo hcp(0001) slabs. The physics driven by molecular\nadsorption is similar, at the atomic scale, for Co hcp\nand fcc surfaces and, as such, the discussion and the\nconclusions presented in the following sections will be\nqualitatively valid for both cases.\n1. Electronic structure\nWe study a 5-layer Co slab with a 2 \u00022 in-plane\nsupercell. Results for slabs of di\u000berent thickness are\npresented in the SM (Sec. S3). Bz is chemisorbed on the\ntop slab surface in the hollow position [see Fig. 2(a)],\nadopting a non-planar structure, where the H atoms aresituated slightly above the C ones. This is common for\nconjugated molecules on ferromagnetic surfaces6,12,71,74.\nThe length of the C-Co bonds, which are formed by the\ntwo C atoms directly on top of two Co ones, is 2.04 \u0017A,\nwhile the length of the other C-Co bonds is 2.17 \u0017A.\nThe electronic structure of the Co surface and the\nmodi\fcations induced by molecular adsorption are qual-\nitatively similar to those described by our elementary\nmodel. In Fig. 2(c) we plot the density of states (DOS)\nprojected over the 3 dorbitals of a Co atom forming a\nbond with the molecule across the interface [speci\fcally,\nthe atom considered is labeled as 36 in Fig. 2(a)]. The\nblack and the cyan lines are for the bare Co slab and\nfor the Bz/Co slab, respectively. For the bare surface,\nthe majority spin (up) states are almost fully occupied,\nwhile, the minority spin (down) DOS crosses the Fermi\nlevel. The crystal-\feld splitting is qualitatively compa-6\nrable to the level splitting assumed in the diagram of\nFig. 1(a), although the Co states are now broadened\ninto bands. Consistently, dxyis the orbital with the\nlowest energy, and most of the corresponding minority\nDOS is located below the Fermi energy, while dyzand\ndxzform a doublet. Their minority DOS is identical and\ncentered across the Fermi level. Thus, they are almost\nhalf-\flled. Finally, dz2anddx2\u0000y2have high energies,\nand the minority DOS is centered above the Fermi level.\nUpon Bz adsorption, there is a signi\fcant change in\nthe Co electronic structure. By analyzing the DOS pro-\njected over the Co dz2orbital and the C pzorbital of\nBz [red line in Fig. 2(c)], we can identify the formation\nof the Co-Bz pz-dz2bonding and antibonding states, in\nparticular in the minority spin channel, as assumed in\nour elementary model. The minority bonding state has\nmostlypzcharacter and is quite localized in energy at\nabout\u00004:5 eV below the Fermi level. In contrast, the\nminority anti-bonding state with dominating dz2char-\nacter, is very broad in energy, and the DOS extends\nover an energy range as large as 7 eV crossing the Fermi\nlevel. We note that the DFT calculations predict some\nmolecule-metal charge transfer and spin density redistri-\nbution leading to a reduction of the DOS spin splitting\ncompared to the bare surface case. As a result, the cen-\nter of thedz2minority state is shifted below rather than\nabove the Fermi energy. These material-speci\fc features\nwere not included in the elementary model. However,\nwe \fnd that the e\u000bect of the molecule-Co hybridization\non the MCA is similar to that described in Sec. II, as\nexplained in the following.\n2. MCA energy\nThe change of the slab MCA due to the adsorption of\nBz is summarized in Fig. 2(b). The MCA energy EMCA\nis equal to\u00005:6 meV for the bare Co slab, indicating\nan in-plane easy axis, whereas it is drastically reduced\nto\u00000:6 meV upon molecular adsorption. Such a drop\nin the MCA energy may result from the molecule-Co\nelectronic hybridization, as discussed above, but it\nmay also derive from the displacement of the surface\natoms following the Bz adsorption. This displacement\nwas indeed found to induce large changes in the bands\nspin-texture of some metallic surfaces32. In order to\nunderstand which of the two e\u000bects is dominant for\nour speci\fc system, we remove the molecule from the\noptimized Bz/Co slab and consider the remaining Co\nlayers without re-relaxing their atomic positions [bare\nCo(-Bz) in Fig. 2(b)]. In the case of this \\distorted\nslab\", we \fnd that EMCA retains a value of \u00004:4 meV,\nsimilar to that of the original bare Co slab. Hence, we\nconclude that the atomic displacement has only a minor\nin\ruence on the Co MCA energy, leaving the molecule-\nmetal hybridization as the main e\u000bect responsible for\nthe MCA modi\fcation.\nFIG. 3. In-plane versus out-of-plane di\u000berence \u0001 EI;mm0\nSOC =\nEI;mm0\nSOC;k\u0000EI;mm0\nSOC;?(in meV) for the 3 dorbitals of the Co atom\n36, whose DOS is shown in Fig. 2.\nThe drastic MCA reduction induced by Bz adsorption\nmight, at \frst thought, be attributed to a complete\nsuppression of the MCA of all interfacial Co layers. This\nis however not the case. The physics is more complex,\nas we shall understand through a careful microscopic\nanalysis. Speci\fcally, we compute \u0001 EI\nSOCde\fned in Eq.\n(5), which is proportional to the atom-resolved MCA\nenergy,EI\nMCA. The results are plotted in Fig. 2(d) as a\nfunction of the atom index. We label the Co atoms at\nthe bottom (top) slab surface as 1-8 (32-40). In the bare\nCo slab (red dots), \u0001 EI\nSOCis negative for all atoms and\nthe largest values ( \u00190:33 meV) are found at the surface,\nas we expect for a fcc slab. In the distorted slab (orange\ndots), these values get slightly enhanced (reduced) at the\ntop (bottom) surface but remain negative. In contrast,\nin the Bz/Co slab (black dots), the changes with respect\nto the bare Co slab become dramatic. Now, \u0001 EI\nSOCis\npositive and increases in magnitude ( \u00190:48 meV) for\nthose Co atoms, which bind to the molecule, and also for\nsome of the atoms in the second layer underneath the\ninterface [cf. Fig 2(b)]. Hence, we understand that the\nreduction of the slab MCA upon Bz adsorption is due to\na balance between positive and negative contributions\ntoEMCA, which respectively come from the Co atoms\nhybridized with the molecule and the rest of the Co\natoms. Notably, if we consider only the surface MCA\nenergy estimated by summing the \u0001 EI\nSOCcontributions\nfor the Co atoms at the slab top surface, we will \fnd\nthat this quantity switches from negative to positive\nafter molecular adsorption. The magnetization of the\ninterfacial Co layer would more favorably point along\nthe perpendicular direction, while the magnetization of\nrest of the slab prefers to lay in-plane, as in the bare\nCo slab. In practice, however, because of the exchange\ninteraction between the surface and the other layers,\nwhich is much stronger than the MCA, the slab will\nbehave as a system with a MCA resulting from the\naverage of the atomic contributions.\nThe e\u000bect of the molecule-Co hybridization and,\nin particular, the positive sign of the MCA energy of\nthose Co atoms that bind to the molecule, are the7\nsame, qualitatively, as predicted by the elementary\nmodel. This can be seen in detail, by analysing the\nSOC matrix elements EI;mm0\nSOC of Eq. (4), which describe\ntransition amplitudes between two Co 3 dorbitals\n(labelled through the quantum numbers mandm0as\nfor Ref. [75]). In particular, we here take the di\u000berence\n\u0001EI;mm0\nSOC =EI;mm0\nSOC;k\u0000EI;mm0\nSOC;?between the matrix\nelements calculated with in-plane and out-of-plane\nmagnetization. We then proceed along the lines of Sec.\nII investigating what transitions gets suppressed because\nof the hybridization. SinceP\nm;m0\u0001EI;mm0\nSOC = \u0001EI\nSOC\n(see Sec. III), positive (negative) values of \u0001 EI;mm0\nSOC\nfor a Co atom Icorrespond to out-of-plane (in-plane)\ncontributions to the MCA of that atom.\nFig. 3 displays the values of \u0001 EI;mm0\nSOC for the various\ncombinations of the 3 dorbitals of the atom 36, which\nis strongly hybridized with Bz and whose DOS was\ndiscussed above. Positive (negative) values are in red\n(blue). In the case of the bare Co surface, \u0001 EI;mm0\nSOC is\nnon-zero and positive for the transition between dxyand\ndx2\u0000y2, whereas it is negative for the transitions con-\nnectingdxytodxzanddyztodz2ordx2\u0000y2. These latter\ntransitions overall dominate, favouring in-plane MCA.\nIn contrast, in the case of the Bz-Co interface \u0001 EI;mm0\nSOC\ncompletely vanishes for the transition connecting dyzto\ndz2. The remaining dominant contribution to \u0001 EI\nSOC\nis positive, due to the dxytodx2\u0000y2transition, which\nremains uncompensated by any negative contributions.\nConsequently, the out-of-plane MCA is favoured over\nthe in-plane as predicted by the elementary model in the\nlimit of large \u00010.\n3. Orbital moments and Bruno's model\nThe prediction that the MCA of Co slabs changes upon\nBz adsorption has so far relied on an analysis of the SOC\nenergies and of the SOC matrix elements. These quanti-\nties can be readily computed in theory, but are not mea-\nsurable. Therefore, the mechanism for the MCA mod-\ni\fcation proposed here can not be veri\fed in the lab.\nIn order to overcome this limitation, here we show that\nthe MCA is also related to the orbital magnetic mo-\nment of the Co atoms, \u0016I\nl;\u000b=h^LI\n\u000bi\u0016B=~, where\u0016Bis\nthe Bohr magneton, Ilabels the atom, and \u000b=x;y;z .\nNotably,\u0016I\nl;alpha is an observable directly accessible by\nexperiments76, for example, through X-ray magnetic cir-\ncular dichroism (XMCD)77. Speci\fcally, under the as-\nsumption that the majority spin band is fully occupied\n(valid for Co), Bruno showed in Ref. [36] that the per-\nturbative expression of the MCA energy, Eq. (1), can be\nrewritten as\nEMCA =\u0000\u0018~\n4\u0016BX\nI(\u0016I\nl;k\u0000\u0016I\nl;?); (6)\n00.050.1\n00.050.1\n8 16 24 32 40 48 56\nAtom index-0.75-0.5-0.2500.250.50.75EMCA (meV)Bare Co (DFT)\nBz/Co (DFT)\nBare Co (Bruno model)\nBz/Co (Bruno model)µl (µB)Out-of-plane\nIn-plane(a)\n(c)(b)\nOut-of-plane\nIn-planeFIG. 4. Orbital magnetic moments and atomic MCA energy\nEI\nMCA. (a)\u0016I\nl;?and (b)\u0016I\nl;kfor all atoms. Atoms 1-8 are at\nthe slab bottom surface, atoms 33-40 are at the top surface,\nand atoms 41-52 belong to Bz. The red (black) dots represent\nthe values for the bare Co (Bz/Co) slab. (c) Atom resolved\nMCA energy calculated by DFT and compared to the values\nobtained from Bruno's model, i.e., EI\nMCA =\u0000\u0018~\n4\u0016B(\u0016I\nl;k\u0000\u0016I\nl;?).\n(Red: DFT for Co. Black: DFT for Bz/Co. Green: the\nBruno's model for Co. Blue: the Bruno's model for Bz/Co).\nwhere\u0016l;k(?)denotes the component of orbital mo-\nment of atom Ialong the in-plane (out-of-plane) spin\nmagnetization direction. E\u000bectively, Bruno's model\nindicates that the easy axis of magnetization coincides\nwith the direction of maximum orbital magnetic moment\ndetermined by the bonding and the crystal \feld76.\nThe calculated \u0016I\nl;kand\u0016I\nl;?are depicted in Fig. 4(a)\nand 4(b), respectively. In the middle of the slab (atoms\n8-32), we see that \u0016I\nl;k\u0019\u0016I\nl;?and the values are compa-\nrable to the atomic orbital magnetic moment of bulk Co,\n0:075\u0016B(Ref. [58]), excluding orbital polarization63.\nIn contrast, at the bare Co surface layers, the atomic\norbital moments get substantially enhanced as reported\nin previous studies78, and, moreover, \u0016I\nl;k(\u00190:125\u0016B)\nbecomes larger than \u0016I\nl;?(\u00190:1\u0016B) and approximately\ntwice the Co bulk value. These results are further\ndrastically modi\fed when we consider the interface with\nthe molecule. In this case, \u0016I\nl;?(\u00190:095\u0016B) remains\nsomewhat similar to the corresponding value for the\nbare Co surface, while \u0016I\nl;kis drastically reduced to\nabout 0:06\u0016Bfor the Co atoms bonded to the molecule.\nAs a consequence, by using Eq. (6), we expect that the\nsurface MCA energy will be negative, thus favouring\nin-plane MCA, for bare Co surfaces, whereas it will\nbe positive, thus favouring out-of-plane MCA, for the8\nBz-Co interface. This is indeed what we \fnd.\nFig. 4(c) compares the layer-resolved MCA energy\ndiscussed in the previous section with the results\nobtained by using Bruno's model with the Co atomic\nSOC value \u0018= 86 meV39,79. For the bare Co surface,\nthe result from the model (red curve) and the DFT\ncalculations (green curve) follow the same pattern,\ndespite some quantitative di\u000berences. We see that\nthe largest contribution to the MCA energy is due to\nsurface atoms with enhanced orbital moments. The\nquantitative discrepancies between the DFT results\nand Bruno's model are consistent with previous re-\nports for metallic slabs80. In contrast, in the case of\nthe Bz-Co interface, we \fnd a remarkable agreement\nbetween the model and the DFT atomic MCA energy\n(blue) for those atoms bonded to the molecule. This\ndemonstrates that the modulation of the surface layer\nMCA can indeed be explained in terms of variations of\nthe Co atoms' orbital moments induced by hybridization.\nThe reduction of the in-plane orbital magnetic mo-\nments is not a feature speci\fc of the Bz-Co interface\nbut is a rather general e\u000bect at interfaces between fer-\nromagnetic transition metals and conjugated molecules,\nas we will further discuss in the following sections (see\nSec. IV D). It can be understood in a rather simple way\nby extending a qualitative argument used in Ref. [76]\nto explain orbital moment quenching at surfaces. The\norbital moment of a free delectron is perpendicular to\nthe motion of the electron. For a thin-\flm geometry,\nthe in-plane electron motion will be disrupted due to\nthe Coulomb repulsion from the neighboring atoms. In\ncontrast, the electron motion perpendicular to the sur-\nface will be less perturbed due to the loss of neighbors.\nHence, the orbital moment normal to the plane gets\npartially quenched, while that along the plane does not.\nUpon the adsorption of the molecule on the top of the\nslab, the orbital motion perpendicular to the surface will\nbe disrupted due to the bonding of the surface atoms\nwith the molecule. As a consequence, the in-plane orbital\nmagnetic moment gets reduced. This is a remarkable\ne\u000bect, which, to our knowledge, has not been explicitly\ndiscussed in the literature about molecule-Co interfaces.\nIt can be measured via XMCD experiments79, which\nare common for the interface characterization81{86, and\nit does represent a \fngerprint of the pz-dz2hybridization.\nTo date, experiments dedicated to the magnetic\nanisotropy at hybrid molecule/Co interfaces have relied\non measurements of magnetization hysteresis loops of\nmolecule-covered thin \flms22,28,29. However, hystere-\nsis loops may be a\u000bected by many factors besides the\nMCA. For example, at the electronic level, an important\ncontribution may come from the modi\fcation of the Co\nintra- and inter-layer exchange interaction induced by the\nbonding with the molecules and by the resulting surface\nrelaxation, as suggested in Refs. 17, 18, and 87. Mea-surements of the orbital moments, as proposed here, will\nprovide a de\fnite way to isolate the MCA contribution\nfrom others that do not depend on the SOC. The results\nwill therefore drive a considerable progress towards un-\nderstanding the magnetic properties of hybrid molecule-\nmetal thin \flms.\nB. Interface between cyclooctatetraene and Co\nHaving understood the mechanism leading to the\nMCA modi\fcation at conjugated molecule-Co interfaces,\nwe now investigate how the e\u000bect can be further con-\ntrolled by selectively changing the molecular chemical\nproperties. Thus, we consider the same Co slab as in the\nprevious section, but we replace Bz with cyclooctate-\ntraene (Cot), C 8H8. This is a more reactive molecule,\nwhich binds strongly to 3 dtransition metal surfaces and\nthus results in a larger hybridization6. The adsorption\ngeometry is shown in the inset of Fig. 5(a), where the\nmolecule appears in a hollow position with four C-C bond\non top of Co atoms. Like Bz, Cot adopts a nonplanar\nstructure with the H atoms situated slightly above the\nC ones, and with an average C-Co bond length of 2.07 \u0017A.\nFig. 5(b) depicts the DOS projected over the 3 d\norbitals of a Co atom at the bare Co surface (black line)\nand of a Co atom underneath a Cot C-C bond (cyan\nline). In the bottom panel, which is dedicated to the\ndz2orbital, we also include the DOS projected over the\nCpzorbital of Cot for a more complete analysis of the\ninterface electronic structure. We see that the results\nfor the bare surface are the same as in Fig. 2(c), but\nthe DOS of the Co atoms bonded to the molecule is\nsigni\fcantly altered. We clearly recognize the formation\nof thepz-dz2bonding and anti-bonding states like in\nthe Bz/Co case. However, since Cot is more reactive\nthan Bz and, therefore, more hybridized with Co, the\npz-dz2antibonding state is greatly broadened and its\ncenter is moved towards higher energies compared to the\nBz/Co case. On the one hand, the majority-spin DOS\nextends slightly across the Fermi energy. On the other\nhand, the minority spin dz2DOS becomes negligible\nnear the Fermi level, and the most distinct peaks are in\nthe energy region from about 0.8 eV to 1.5 eV above the\nFermi level. Thus, the energy splitting between the dz2\norbital and the other Co 3 dorbitals increases after Cot\nadsorption. This is in good qualitative agreement with\nthe elementary model of Sec. II and, referring to Fig. 1,\nwe estimate that \u00010is as large as\u00182 eV at the Cot/Co\ninterface. As a consequence, the SOC-induced transition\nfromdyztodz2, which promotes in-plane MCA at the\nclean Co surface, is entirely suppressed because of the\nhybridization with the molecule.\nThe atom-resolved MCA estimated through \u0001 EI\nSOC\nis negative for all atoms in the bare Co slab, whereas it\nbecomes positive for the Co atoms directly bonded to9\nFIG. 5. MCA at the Cot/Co interface. (a) \u0001 EI\nSOCevaluated\nfor all atoms. Atoms 1-8 are at the slab bottom surface,\natoms 33-40 are at the top surface, and atoms 41-56 belong\nto Cot. The left-hand side inset shows the molecule and the\nCo atoms to which it binds to. The right-hand side one shows\nthe MCA energy of the bare Co slab (same as in Fig. 2), of\nthe Cot/Co slab, and of the \\distorted\" slab (calculated by\nremoving the Cot molecules from the optimized Cot/Co slab).\n(b) DOS projected over the 3 dorbitals of a surface Co atom\nbonded to Cot. The Fermi energy is at 0 eV. The black and\ncyan lines are for the bare Co slab and the bare Cot/Co slab,\nrespectively. The red curve in the bottom panel is the DOS\nprojected over the C pzorbital of Cot.the molecule. As for Bz/Co, these atoms would \fnd it\nenergetically favourable to align their magnetic moments\nalong the perpendicular direction. Yet, this e\u000bect is\nquantitatively larger here than that in Bz/Co. The\nvalue of \u0001EI\nSOCof the hybridized Co atoms is 0 :65 meV\nfor Cot/Co versus 0:48 meV for Bz/Co. Furthermore,\nalthough the number of these hybridized atoms at the\nCot/Co interface is the same as in Bz/Co, their overall\ncontribution is so important for Cot that the MCA of\nthe entire slab switches from in-plane to out-of-plane. In\nfact, the total slab MCA energy goes from \u00005:57 meV\nto 0:71 meV after molecular adsorption [see inset in Fig.\n5(a)].\nThe results for Cot/Co and their comparison with\nthose for Bz/Co demonstrate the dramatic impact that\nthe chemical properties of the adsorbate molecules can\npotentially have on the Co slab. Our predictions may\nbe readily veri\fed experimentally. More importantly,\nwe hope that they will stimulate additional experimental\nstudies, where the perpendicular MCA of molecular cov-\nered Co thin \flms is varied, and eventually maximized,\nby chemically tuning the reactivity of the molecules with\nthe \flm surfaces.\nFIG. 6. MCA modi\fcation for a slab covered with a series of\naromatic hydrocarbon molecules. The total (surface) MCA\nenergies, represented by the blue (magenta) bars, are esti-\nmated by the sum of \u0001 EI\nSOC for all the slab (top surface)\natoms.\nC. Dependence of the MCA on the molecular\ncoverage\nThe calculations presented in the previous sections\nhave demonstrated that the MCA modi\fcation at\nhybrid interfaces is mostly a \\local\" phenomenon at the10\nmetal atoms forming covalent bonds with the adsorbed\nconjugated molecules. Therefore, we expect that the\ne\u000bect will be further enhanced by increasing the surface\nmolecular converge, namely the number of C-metal\nbonds across a molecule/Co interface. This is indeed the\ncase as we now demonstrate.\nWe consider a Co slab, where the top surface is cov-\nered with various aromatic hydrocarbon molecules com-\nprising a di\u000berent number of fused Bz rings: naphtha-\nlene (2 Bz rings), pyrene (4 Bz rings) and coronene (6\nBz rings). The slab total MCA energy as well as the\ntop surface MCA energy estimated by the SOC energy\n\u0001ESOC, are compared in Fig. 6 (note that the slab in-\nplane unit cell is now larger than the one used in the\nprevious sections). The trend emerging from the \fgure\nis clear. The total MCA energy has the largest nega-\ntive value for the bare Co slab. The absolute magnitude\nof the MCA energy then decreases as a function of the\nnumber of Bz rings in the molecular adsorbate. This ef-\nfect appears even more dramatic when we inspect only\nthe relative variation of the top surface MCA energy, es-\ntimated by summing the \u0001 EI\nSOCcontributions over the\ntop surface layer atoms. The sign of such surface MCA\nswitches from negative to positive for pyrene, and the\nmagnitude becomes quite large in the case of the coro-\nnone/Co interface, where most of the surface Co atoms\nform a covalent bond with the molecule. This is a sharp\ntrend, which could be easily veri\fed experimentally, al-\nthough we are not aware of any study for the hydrocarbon\nseries. We note, however, that a perpendicular surface\nMCA has been reported for graphene-coated Co \flms88.\nThe physics for such system is evidently the same as for\ncoronene/Co, and our phenomenological arguments can\nbe used to interpret those experimental results as well.\nOverall, our \fndings demonstrate the possibility to engi-\nneer thin \flms with large out-of-plane MCAs via surface\nmolecular functionalization.\nFIG. 7. Optimized geometry of (a) Alq 3and (b) C 60de-\nposited on top of a 4-layer Co slab. Yellow, maroon, blue,\nred, and light pink spheres represent Co, C, Al, O and H\natoms, respectively.D. Interfaces with C 60and Alq 3\nThe results obtained so far provide a detailed un-\nderstanding of the mechanism leading to the MCA\nmodi\fcations of Co surfaces upon molecular adsorption.\nFurthermore, they suggest some potential routes to\nenhance the e\u000bect. Now, we consider two of the most\nexperimentally studied molecule-metal interfaces for\nspintronics, namely Co/C 60and Co/Alq 3, showing that\nthe same concepts learnt so far, apply also these rather\ncomplex systems.\nFIG. 8. MCA at the C 60/Co interface. (a) \u0001 EI\nSOCevaluated\nfor all atoms. Atoms 1-18 are at the slab bottom surface,\natoms 55-72 are at the top surface, and atoms 73-132 belong\nto C 60. (b) DOS projected over a surface Co atom in the bare\nCo slab (black line), and DOS projected over the 3 dorbitals\nof the Co atoms 68 (cyan line) and 67 (orange line) in the\nC60/Co slab. The Fermi energy is at 0 eV.11\n1. C 60/Co interface\nCo thin \flms with a C 60overlayer are the \frst systems\nfor which an enhanced anisotropy was experimentally\nreported28,29. We model the interface through a 3 \u00023\nin-plane slab with one molecule in contact with the top\nsurface. We consider only 4 Co layers to reduce the\ncomputational cost, since the system is considerably\nlarger than Bz/Co and Cot/Co. Similarly to Ref. [10],\nwe compare four di\u000berent adsorption geometries named\nhexagon, pentagon, hexagon-hexagon, and hexagon-\npentagon. In the \frst two cases, C 60has either a\nhexagonal or pentagonal face on top of the Co surface.\nIn the hexagon-hexagon case, the molecule bonds to Co\nthrough the edge between two hexagonal faces, whereas\nin the hexagon-pentagon geometry, it bonds through the\nedge between one hexagonal and one pentagonal face.\nThe lowest energy con\fguration is the hexagon-pentagon\none shown in Fig. 7(b). Its energy is 335 meV lower than\nthe energy of the least stable con\fguration, pentagon,\nand about 200 meV lower than the energy of the second\nmost stable con\fguration, hexagon. Similar results\nwere also obtained in Ref. [28] (for hcp Co slabs) and\nin Ref. [29] (for fcc Co slabs). In the following, we\nwill only discuss the results for the hexagon-pentagon\ncon\fguration. Although the details of the C 60/Co\nelectronic structure depend on the adsorption geome-\ntry, we \fnd that the overall physics is similar for all cases.\nThe Co surface MCA is strongly a\u000bected by C 60. The\n\u0001EI\nSOC is plotted in Fig. 8 for all the atoms in the\nbare Co and in the C 60/Co slabs (black and red points\nrespectively). The values for the Co atoms, which form\na strong covalent bond with the molecule, change sign,\nswitching from negative in the bare Co case to positive\nin the C 60/Co case. Speci\fcally, \u0001 EI\nSOChas the largest\npositive value (\u00190:4 meV) for the Co atom, labelled\n68, which is closer to, and therefore more strongly hy-\nbridized with, C 60. For the other Co atoms underneath\nthe molecule, \u0001 EI\nSOC has somewhat smaller values,\nvarying with the Co-C bond length. Finally, \u0001 EI\nSOC\nremains negative for the surface atoms that are not\nbonded at all with C 60. Overall, the order of magnitude\nof the atomic MCA energy EI\nMCA\u00190:5\u0001EI\nSOC of the\nCo atoms hybridized with the molecules is similar to\nthat obtained by Bairagi et al. in Fig. 3(c) of Ref.\n[28], although our results are for fcc Co(001), whereas\nRef. [28] considers hcp Co(0001). The MCA energy\naveraged over the interfacial layer is of the order of\n0.1 meV=atom, which was found consistent with the\nexperimental measurement of Ref. [28].\nThe relation between the Co-C hybridization and\nthe MCA energy can be understood by examining the\nC60/Co electronic structure and following the very same\nreasoning as in the previous sections. For instance, Fig.\n8(b) compares the dz2-projected DOS of two Co surface\natoms bonded to the molecule, namely atom 68 that,as noted above, has the largest positive atomic MCA\nenergy, and atom 67 that has instead a negative and\nconsiderably smaller (in absolute value) MCA energy.\nThe DOS of the Co atom 68 is much broader than that of\natom 67, indicating a much stronger chemisorption. For\natom 68 in the minority spin channel, the C-Co bonding\nand antibonding states are recognized respectively at\nabout -3 eV below the Fermi level and 1 eV above\nthe Fermi energy. The picture overall resembles the\ndz2-projected DOS for Cot in Sec. IV B. Accordingly, we\n\fnd an enhanced positive MCA energy. Conversely, for\natom 67, the formation of the bonding and antibonding\nstates is much less marked and, in the minority channel,\nthedz2-projected DOS is large near the Fermi level.\nHence, the MCA energy remains negative. Even for\nthis complex interface, the overall picture for the MCA\nmodi\fcation is fully consistent with that developed\nstarting from our elementary model.\n00.050.10.15\n18 36 54 72 90 108 126\nAtom index00.050.10.15µl (µB)Out-of-plane\nIn-plane(a)\n(b)\nFIG. 9. Orbital magnetic moments (a) \u0016I\nl;?and (b)\u0016I\nl;kfor\nall atoms in the Co/C 60slab. The red (black) dots represent\nthe values for the bare Co (C 60/Co) slab. Atoms 1-18 are at\nthe slab bottom surface, atoms 55-72 are at the top surface,\nand atoms 73-132 belong to C 60.\nThe impact of the pz-dz2hybridization can be further\nappreciated by noting that it induces a substantial\nspin redistribution in the DOS. In particular, the spin\nsplitting of the Co 3 dstates and, hence, also the Co\nspin magnetic moments \u0016I\ns, are reduced at the interface.\nThe calculated \u0016I\nsis equal to 1 :895\u0016Bfor the surface\natoms in the bare slab. In contrast, \u0016I\nsbecomes equal to\n1.59\u0016Band 1.25\u0016B, respectively, for atoms 67 and 68\nat the C 60/Co interface. A similar reduction has been\nobtained in other theoretical studies28and measured\nexperimentally10.\nThe atomic MCA or \u0001 EI\nSOCare not quantities accessi-\nble in experiments, as already observed in Sec. IV A. Yet,\nlike in the case of Bz, they are directly correlated with\nthe orbital magnetic moment, a measurable microscopic\nobservable. In Fig. 9 we compare the orbital magnetic12\nmoment of each atom for the bare Co and the C 60/Co\nslab with in-plane and out-of-plane spin magnetization,\n\u0016I\nl;kand\u0016I\nl;?. The \frst and most striking \fnding is that\n\u0016I\nl;kand\u0016I\nl;?are drastically reduced for those Co atoms\nthat are strongly bonded to C 60. For example, atom\n68 is found to have the lowest orbital magnetic moment\namong all Co atoms. The atomic orbital magnetic mo-\nment, therefore, shows a similar trend to the spin mag-\nnetic moment (although the former is one order of mag-\nnitude smaller than the latter). The hybridization tends\nto quench both of them. Despite that, as anticipated,\nthe di\u000berence \u0001 \u0016I\nl=\u0016I\nl;k\u0000\u0016I\nl;?is approximately pro-\nportional to \u0001 EI\nSOCat the surface. \u0001 \u0016lis negative and\nequal to approximately \u00000:02\u0016Bfor all Co atoms of the\nbare surface. In contrast, upon molecular absorption, it\nswitches to positive values ranging from about 0 :005\u0016Bto\n0:02\u0016Bfor those interface atoms that are more strongly\nhybridized with the molecule. The measurement of such\ne\u000bect by means, for instance, of XMCD will provide a\nstrong validation for the proposed mechanism for the\nMCA modi\fcation at C 60/Co interfaces and complement\nthe previous studied based on the \ft of the magnetization\nhysteresis loops.\n2. Alq 3/Co interface\nAlq3is the most popular molecular material inves-\ntigated for spin transport89{91. Understanding the\nproperties of its interface with Co \flms, used as elec-\ntrodes in spin-valve devices, has been an outstanding\nproblem for over a decade in organic spintronics13,92.\nThe molecule presents an Al atom octahedrally coordi-\nnated with the oxygen and nitrogen atoms of the three\n8-hydroxyquinoline ligands. The electronic structure\nof the molecule93{97and of its interface with Co71,72\nhas been already studied in detail by means of DFT\ncalculations in previous works, where results were also\ncompared with spectroscopic measurements. However,\nto date, there are no \frst-principles calculations for\nthe MCA of Alq 3/Co interfaces. Recent experiments29\nreported that Alq 3, similarly to C 60, on various Co thin\n\flms favours a perpendicular MCA. We con\frm these\nobservations here, showing that the underlying physics is\nindeed rather similar for Alq 3/Co and C 60/Co interfaces.\nIn order to model the Alq 3/Co interface, we adopt\nthe slab described in Ref. [72] and shown in Fig. 7(a)98.\nTwo of the molecule's quinoline ligands are chemisorbed\nalmost parallel to the surface, whereas the third one\nstands vertically. Similarly to all the systems studied\nso far, we \fnd that \u0001 EI\nSOC becomes positive and, on\naverage, equal to about 0.25 meV/atom for the surface\nCo atoms underneath the molecule, whereas \u0001 EI\nSOC\nremains negative for the surface Co atoms not bonded\nwith Alq 3. These results are displayed in Fig. 10(a),\nwhere the values for the clean Co slab and the Alq 3/Co\nFIG. 10. MCA at the Alq 3/Co interface. (a) \u0001 EI\nSOCevalu-\nated for all atoms. Three Co atoms are shown as represen-\ntative cases. Atoms 1-32 are at the bottom surface of the\nslab, atoms 97-128 are at the top surface in contact with the\nmolecule, and atoms 129-182 belong to the molecule. (b) DOS\nprojected over the out-of-plane 3 dorbitals for the three Co\natoms highlighted in panel (a).\none are respectively represented as black and red dots.\nThe di\u000berent values of \u0001 EI\nSOC for the various atoms\nat the interface re\rect the strength of their hybridization\nwith the molecule. Similarly to the other interfaces\nstudied, the hybridization leads to the formation of\nCo-molecule antibonding states with dz2character,\nwhich are sharply de\fned in particular in the minority\nspin channel. For instance, in Fig. 10(b), we plot the\norbital-projected DOS for three di\u000berent Co atoms (red,\norange, and cyan curve) and we compare it with the DOS\nof the clean Co surface (black curve). The dz2-projected\nDOS of the Co atom with largest \u0001 EI\nSOC(\u00190:48 meV)\nshows a rather sharp resonance at about 1 eV above\nthe Fermi energy in the minority spin channel [cyan\ncurve in Fig. 10(b)]. This feature drives the MCA\nswitch, in qualitative agreement with the model of Sec.\nII. In contrast, the other Co atoms, which are slightly\nless hybridized with the molecule, present a smaller13\n\u0001EI\nSOC. The center of the pz-dz2antibonding state\nin the minority channel is near the Fermi level, and\nthe magnitude of the MCA change is less pronounced.\nOverall, the results appear rather similar, even at the\nquantitative level, to those obtained in the case of\nC60/Co, thus con\frming the experimental observation\nof Ref. [29] that the e\u000bect of an Alq 3overlayer on Co\nultrathin \flms is rather similar to that of a C 60one.\nFinally we suggest that, instead of comparing Alq 3/Co\nand C 60/Co, experiments may consider the tris-(9-\noxidophenalenone)-aluminum(III) molecule, Al(OP) 3, a\nvariation of Alq 3with similar properties, but larger 9-\noxidophenalenone ligands99. These ligands will form a\nlarge number of pz-dz2chemical bonds with Co as already\ndemonstrated in spectroscopic measurements99, and this\nwill potentially enhance the MCA modi\fcations, as dis-\ncussed in Sec. IV C. Experimental studies along this di-\nrection will provide a further con\frmation of the phe-\nnomenology that we have described in this paper.\nV. CONCLUSION\nWe have investigated the modi\fcation of the MCA of\nCo slabs upon molecular adsorption, demonstrating that\nthe hybridization between the out-of-plane Co dz2or-\nbitals and molecular C pzorbitals favours perpendicular\nMCA.\nStarting from an elementary model, we have em-\nployed second-order perturbation theory to identify the\nvirtual electronic transitions between Co occupied and\nunoccupied states that contribute to the in-plane and\nout-of-plane MCA. We have then shown that the virtual\ntransitions leading to in-plane MCA are suppressed by\nthe formation of molecule-metal bonding and antibond-\ning states, while those leading to out-of-plane MCA are\nnot. This explains why perpendicular MCA is favoured\nat molecule-Co interfaces. The key parameter governing\nthe e\u000bect is the energy splitting between the in-plane\nCodx2\u0000y2orbital and the pz-dz2antibonding hybrid one.\nThe results obtained with the elementary model are\ncon\frmed by DFT calculations for several molecular\nsystems. Speci\fcally, we have shown that the electronic\nstructure of the surface layer of Co slabs is signi\fcantly\nmodi\fed by the hybridization with C pzorbitals leading\nto a suppression of the in-plane contributions to the\nMCA. This e\u000bect is rather general at chemisorbed\nmolecules/Co interfaces, and it can be further enhanced\nby selecting molecules that are very reactive with Co.\nAmong the various systems studied, there are also the\nprototypical spintronic interfaces C 60/Co and Alq 3/Co.\nOur results agree with previous studies for C 60/Co28\nand they support recent experimental observations re-\nporting a similar MCA modi\fcation in the two systems29.Finally and importantly, we have demonstrated that\nthe atomic MCA is correlated with the di\u000berence be-\ntween the atomic orbital magnetic moment for in-plane\nand out-of-plane spin magnetization, in accordance with\nBruno's model36. The orbital magnetic moment is a\nmicroscopic observable, which can be measured, for\nexample, via XMCD, and, therefore, experiments could\nprovide a de\fnite validation of the proposed mechanism\nbased on the pz-dz2hybridization.\nVI. ACKNOWLEDGEMENTS\nAD would like to thank V.A. Dediu, I. Bergenti, and\nM. Benini for stimulating discussions. AH and AD\nacknowledge funding from Science Foundation Ireland\n(SFI) and the Royal Society through the University Re-\nsearch Fellowship URF-R1-191769. AD and SS were\npartly supported by the H2020 program of the Euro-\npean Commission under the FET-Open project INTER-\nFAST (grant agreement No. 965046). SB, SS, and DO'R\nacknowledge support from Science Foundation Ireland\n(19/EPSRC/3605) and the Engineering and Physical Sci-\nences Research Council (EP/S030263/1). The computa-\ntional resources were provided by Trinity College Dublin\nResearch IT and the Irish Center for High End Comput-\ning (ICHEC).\nVII. SUPPLEMENTARY MATERIAL\nA. Spin-orbit matrix elements\nIn this section, we compute the SOC matrix elements\nused in Eq. (1) of the paper to obtain the MCA energy\nof the model Co(001) surface via perturbation theory.\nWe assume that the spin quantization axis is along the\nzCartesian axis, normal to the surface. Following Ref.\n39, we consider the atomic d\u000borbitals (\u000b=xy,yz,z2,\nxz,x2\u0000y2) with orbital momentum quantum number\nl= 2.\nTo begin with, we set the spin magnetic moment to\nbe out-of-plane, i.e. along the zdirection. The spin\nsand the spin magnetic quantum number sz=\u00061=2\nare good quantum numbers for the unperturbed system.\nThed\u000bstates,jd\u000b;szi, are then expressed in the basis\njm;sz=\u00061=2i=jl;m;i\njs;szi, wheremis the orbital\nmagnetic quantum number, m= 0;\u00061;\u00062. Speci\fcally,14\njdxy;\"i jdyz;\"i jdz2\"i jdxz\"i jdx2\u0000y2\"i jdxy#i jdyz#i jdz2#i jdxz#i jdx2\u0000y2;#i\njdxy\"i 0 0 0 0 i\u0018 0\u0018=2 0 i\u0018=2 0\njdyz\"i 0 0 0 i\u0018=2 0 - \u0018=2 0 ip\n3\u0018=2 0 \u0018=2\njdz2\"i 0 0 0 0 0 0 \u0000ip\n3\u0018=2 0\u0000p\n3\u0018=2 0\njdxz\"i 0\u0000i\u0018=2 0 0 0 - i\u0018=2 0p\n3\u0018=2 0 \u0000\u0018=2\njdx2\u0000y2\"i \u0000i\u0018 0 0 0 0 0 - \u0018=2 0 \u0018=2 0\njdxy#i 0\u0000\u0018=2 0 \u0000i\u0018=2 0 0 0 0 0 \u0000i\u0018\njdyz#i\u0018=2 0\u0000ip\n3\u0018=2 0\u0000\u0018=2 0 0 0 \u0000i\u0018=2 0\njdz2#i 0ip\n3\u0018=2 0p\n3\u0018=2 0 0 0 0 0 0\njdxz#ii\u0018=2 0\u0000p\n3\u0018=2 0 \u0018=2 0 i\u0018=2 0 0 0\njdx2\u0000y2#i 0\u0018=2 0 \u0000\u0018=2 0 i\u0018 0 0 0 0\nTABLE I. SOC matrix elements hd\u000b;szj^HSOCjd\f;s0\nzi. The spin magnetic quantum number sz= 1=2 (\u00001=2) is indicated as \"\n(#).\njdxy\"i jdyz\"i jdz2\"i jdxz\"i jdx2\u0000y2\"i jdxy#i jdyz#i jdz2#i jdxz#i jdx2\u0000y2#i\njdxy\"i 0 0 0 \u00182=4\u001820 0 0 \u0000\u00182=4\u00182\njdyz\"i 0 0 3 \u00182=4\u0000\u00182=4\u00182=4 0 0 \u00003\u00182=4\u00182=4\u0000\u00182=4\njdz2\"i 0 3\u00182=4 0 0 0 0 \u00003\u00182=4 0 0 0\njdxz\"i\u00182=4\u0000\u00182=4 0 0 0 \u0000\u00182=4\u00182=4 0 0 0\njdx2\u0000y2\"i \u0000\u00182\u00182=4 0 0 0 \u00182\u0000\u00182=4 0 0 0\njdxy#i 0 0 0 \u0000\u00182=4\u001820 0 0 \u00182=4\u0000\u00182\njdyz#i 0 0 \u00003\u00182=4\u00182=4\u0000\u00182=4 0 0 3 \u00182=4\u0000\u00182=4\u00182=4\njdz2#i 0\u00003\u00182=4 0 0 0 0 3 \u00182=4 0 0 0\njdxz#i \u0000\u00182=4\u00182=4 0 0 0 \u00182=4\u0000\u00182=4 0 0 0\njdx2\u0000y2#i\u00182\u0000\u00182=4 0 0 0 \u0000\u00182\u00182=4 0 0 0\nTABLE II. Di\u000berence between the SOC matrix elements computed with parallel and perpendicular magnetization jh^HSOCij2\nk\u0000\njh^HSOCij2\n?. The spin magnetic quantum number sz= 1=2 (\u00001=2) is indicated as \"(#).\nwe write\njdxy;szi=ip\n2(j\u00002;szi\u0000j2;szi); (7)\njdyz;szi=ip\n2(j\u00001;szi+j1;szi); (8)\njdz2;szi=j0;szi; (9)\njdxz;szi=1p\n2(j\u00001;szi\u0000j1;szi); (10)\njdx2\u0000y2;szi=1p\n2(j\u00002;szi+j2;szi): (11)\nWe then calculate the matrix elements\nhd\u000b;szj^HSOCjd\f;s0\nziof the SOC Hamiltonian\n^HSOC =\u0018^L\u0001^S=~2=\u0018\n2~2(^L\u0000^S++^L+^S\u0000) +\u0018\n~2^Lz^Sz,\nwhere ^L\u0006(^S\u0006), are the ladder orbital (spin) angular\nmomentum operators, and ^Lz(^Sz) is thezcomponents\nof the orbital (spin) angular momentum operator. The\nresults are listed in Table I.\nNext, we rotate the spin in such a way that it lays\nalong the in-plane direction, for example, x. Thus, we\nintroduce the states\njd\u000b;sx=\u00061=2i=1p\n2h\njd\u000b;sz= 1=2i\u0006jd\u000b;sz=\u00001=2ii\n;\n(12)\nand we easily calculate the matrix elements\njhd\u000b;sxj^HSOCjd\f;s0\nxij2.Finally, we compute the di\u000berence between the square\nmodulus of the matrix elements with spin parallel and\nperpendicular to the surface\njh^HSOCij2\nk\u0000jh^HSOCij2\n?\u0011\njhd\u000b;sxj^HSOCjd\f;s0\nxij2\u0000jhd\u000b;szj^HSOCjd\f;s0\nzij2;(13)\nwheresx=sz=\u00061=2 ands0\nx=s0\nz=\u00061=2. The re-\nsults are presented in Table II and can be used to de-\nrive the simple rules mentioned in Sec. II of the pa-\nper and discussed in Refs. 39{41. Negative (positive)\nterms correspond to virtual transitions, which promote\nin-plane (out-of-plane) MCA. The di\u000berence jh^HSOCij2\nk\u0000\njh^HSOCij2\n?enters the numerator of Eq. (1) of the paper\nand is used to obtain the MCA energy in Eqs. (2) and\n(3).\nB. DFT results for the Bz/hcp-Co interface\nWe present here the results of DFT calculations for\nthe Bz/hcp-Co interface showing that the physics driven\nby molecular adsorption is similar, at the atomic scale,\nto that described in the paper for Bz/fcc-Co.\nThe calculations are carried out for a 4-layer hcp(0001)\nslab with a 4\u00024 in-plane supercell. The molecule is ad-15\nsorbed on the top surface as displayed in Fig. 11 (inset),\nwhere the surface Co atoms bonded with the molecule\nare highlighted and labelled 53, 56, and 64. To compare\nthe electronic structure of the bare hcp-Co surface and\nof the Bz/hcp-Co interface, we plot in Fig. 11 the DOS\nprojected over the 3 dorbitals of the atom 64. The\nblack and the cyan lines are for the bare Co and for the\nBz/hcp-Co case, respectively. The DOS of the bare hcp\nsurface is qualitatively similar to that of the fcc surface\n(cf. Fig. 2 in the paper). The majority spin states are\ncompletely occupied, whereas the minority spin DOS\ncrosses the Fermi energy. However, the crystal \feld\nsplitting of the hcp surface is di\u000berent from that of the\nfcc one. In the hcp case, the dx2\u0000y2and thedxyorbitals\nform a \frst low energy doublet, while the dxzand thedyz\norbitals form a second doublet at higher energies. The\ndz2orbital is a singlet with the highest energy. In the\nminority spin channel, the DOS of the dz2splits into two\ndistinct peaks appearing on either side of the Fermi level.\nDespite the di\u000berence between hcp and fcc-Co, the\nchange of the electronic structure upon Bz adsorption\nis qualitatively similar in the two cases. Focusing in\nparticular on the minority spin channel, we see that the\npz-dz2antibonding state is located at about 1 eV above\nthe Fermi energy, whereas the bonding state is broad\nand centered at about 1 :5 eV below the Fermi energy.\nThe modi\fcation of the MCA of the hcp slab upon\nBz adsorption is summarized in Fig. 12(a), where we\nplot the total MCA energy EMCA (inset) and the atomic\nSOC energy di\u000berence \u0001 EI\nSOC between the in-plane\nand out-of-plane magnetization con\fgurations. EMCA\nis positive and equal to 2.55 meV for the bare hcp-Co\nslab, while it is enhanced to 3.91 meV for Bz/hcp-Co.\nThus, the easy magnetization direction of hcp-Co is\nout-of-plane even without Bz, unlike the case of fcc-Co\npresented in the paper. The e\u000bect of the molecule in\nhcp-Co is to further favour the perpendicular MCA.\nThis is in agreement with the experimental reports by\nBairagi et al.28for hcp(0001) thin \flms.\nThe atom resolved \u0001 EI\nSOC is displayed in Fig 12(a)\nas the red (black) dots for the Bz/hcp-Co (bare hcp-Co)\nslab. For the bare hcp-Co slab, the values are positive\nfor all the atoms in the bottom and top layers (i.e.,\nfor the atoms labelled 1-16 and 49-64, respectively).\nThe values are instead negative for the atoms in the\ncentral layers (i.e., for the atoms 17-48). In the case\nof the Bz/hcp-Co slab, \u0001 EI\nSOC increases for the atoms\nat the top surface, which forms now the interface with\nthe molecule, and switches to positive values for the\nlayer just underneath the interface. In particular, the\nmaximum positive \u0001 EI\nSOC, equal to about 0.43 meV, is\nfound for the Co atoms bonded with the Bz molecule.\nThis behaviour accounts for the overall increase of the\nout-of-plane MCA induced by the Bz absorption.\nFIG. 11. DOS projected over the 3 dorbitals of the Co atom\n64 at the Bz/hcp(0001)-Co interface. The Fermi energy is\nat 0 eV. The black and cyan lines are for the bare Co slab\nand the Bz/hcp-Co slab, respectively. The inset shows the\nside view of the Bz/hcp-Co 4-layer slab. Brown, white, and\nyellow spheres indicate C, H, and Co atoms, respectively.\nTo further understand the e\u000bect of the metal-molecule\nhybridization we present in Fig. 12(b) the SOC matrix\nelements \u0001 EI;mm0\nSOC for the various combinations of the\n3dorbitals of the Co atom 64. Positive (negative) val-\nues are in red (blue). The transitions with the largest\npositive values, promoting out-plane MCA, connect dxy\nwithdx2\u0000y2anddyzwithdxz. They are barely a\u000bected\nby the hybridization with the molecule. On the other\nhand, the main negative transition, which connects dyz\nwithdz2, is fully suppressed once the molecule is present.\nBesides, we note that there are also other two transitions\nwith negative values, namely those between dxyanddxz\nand between dyzanddx2\u0000y2. Although they do not van-\nish, their magnitude is reduced in the Bz/hcp-Co case\ncompared to the bare hcp-Co case. These results are\nqualitatively, and even quantitatively, similar those re-\nported for fcc-Co (cf. Fig. 3 in the paper). Therefore,\nwe conclude that the metal-molecule hybridization has\nthe same e\u000bect in promoting out-of-plane surface MCA\nin Bz/fcc-Co as well as in Bz/hcp-Co slabs.16\nFIG. 12. DFT results for the MCA of the hcp(0001)-Co and\nBz/hcp(0001)-Co slabs. (a) \u0001 EI\nSOCevaluated for all atoms.\nThe red, and black dots represent the values for the bare\nhcp Co slab and the Bz/hcp-Co slab respectively. Atoms 1-\n16 are at the slab bottom surface, atoms 49-64 are at the\ntop surface, and atoms 65-76 are the Bz atoms. The inset\ndisplays the MCA energy of the bare hcp-Co slab and of the\nBz/hcp-Co slab. (b) In-plane versus out-of-plane di\u000berence\nbetween the SOC matrix elements \u0001 EI;mm0\nSOC (in meV) for the\n3dorbitals of the surface Co atom 64, which is bonded with\nthe Bz molecule (see the inset of Fig. 11).\nC. Dependence of the MCA on the fcc-Co slab\nthickness\nThe MCA energies EMCA of ideal fcc-Co slabs and\nof Bz/fcc-Co slabs with thicknesses varying from 3 to\n10 layers are displayed in Fig. 13(a) (the 5-layer slab\nis the one extensively discussed in the paper). For\nthe bare Co slabs EMCA is negative (light blue bars),\nimplying an in-plane MCA, regardless of the number\nof layers, although the results oscillate as a function of\nthe slab thickness. A similar oscillating behaviour has\npreviously been observed in the literature and attributed\nto quantum well e\u000bects100. The adsorption of Bz on the\nslabs' top surfaces leads to a considerable reduction of\nthe magnitude of EMCA in almost all cases (see the pink\nbars) as we already observed in the paper for the speci\fc\ncase of a 5-layer slab. The removal of the molecule\nfrom the slabs' top surfaces without re-optimizing the\ngeometries [bare Co(-Bz) slabs] recovers a negative\nFIG. 13. MCA for fcc slabs of di\u000berent thicknesses. (a) Total\nMCA energy EMCA as a function of the number of layers;\nlight blue bars, pink bars, and dark blue bars respectively\nrepresent the results for bare fcc-Co slabs, Bz/fcc-Co slabs,\nand fcc-Co slabs, where the Bz molecule was removed without\nre-optimization of the atomic coordinates [bare Co (-Bz)]. (b)\n\u0001EI\nSOCfor all atoms of a 8-layer slab. The red, black, and\ngreen dots represent the results for the bare fcc-Co slab, the\nBz/fcc-Co slab, and the bare Co (-Bz) slab. Atoms 57-64 are\nat the slab's top surface.\nMCA energy comparable to that of the bare Co slabs\nthus demonstrating that the displacement of the surface\natoms has a minor e\u000bect on EMCA for all cases.\nThe mechanism leading to the reduction of the MCA\nenergy upon Bz adsorption is the same for thick slabs as\nfor the 5-layer slab discussed in the paper. It is indeed\ndue to a \\local\" e\u000bect, which involves only the surface\natoms hybridized with the molecules, and which is there-\nfore almost independent on the slab thickness. To see\nthis, we plot in Fig. 13(b) the values of \u0001 EI\nSOCfor all\natoms of a 8-layer slab, which is taken as a representative\ncase of all thick slabs. The trend is similar to that re-\nported in the paper [cf. Fig. 2(d)]. At the surfaces of the\nbare Co slab (atoms 1-8 and 57-64), \u0001 EI\nSOCis negative\nand equal to about 0 :31 eV. In contrast, in the Bz/Co\nslab, \u0001EI\nSOC switches sign and becomes approximately\nequal to 0:28 eV for those Co atoms that are bonded17\nwith Bz at the top surface, and for some of the Co atoms\nin the second layer underneath the interface. Thus, for\nall the considered slabs, there is a compensation betweenthe positive contributions of the interface and the nega-\ntive contributions of the rest of the slab to the total MCA\nenergy.\n∗andrea.droghetti@tcd.ie\n1P. Gambardella and S. Bl ugel, Magnetic Surfaces, Thin\nFilms and Nanostructures (Springer International Pub-\nlishing, Cham, 2020), pp. 625{698, URL https://doi.\norg/10.1007/978-3-030-46906-1_21 .\n2F. Hellman, A. Ho\u000bmann, Y. Tserkovnyak, G. S. D.\nBeach, E. E. Fullerton, C. Leighton, A. H. MacDonald,\nD. C. Ralph, D. A. Arena, H. A. D urr, et al., Rev. Mod.\nPhys. 89, 025006 (2017), URL https://link.aps.org/\ndoi/10.1103/RevModPhys.89.025006 .\n3G. Varvaro and F. Casoli, eds., Ultra-High-Density Mag-\nnetic Recording (Jenny Stanford Publishing, 2016), URL\nhttps://doi.org/10.1201/b20044 .\n4A. Hirohata, K. Yamada, Y. Nakatani, I.-L. Prejbeanu,\nB. Di\u0013 eny, P. Pirro, and B. Hillebrands, Journal of Mag-\nnetism and Magnetic Materials 509, 166711 (2020),\nISSN 0304-8853, URL https://www.sciencedirect.\ncom/science/article/pii/S0304885320302353 .\n5M. Cinchetti, V. A. Dediu, and L. E. Hueso, Nature\nMaterials 16, 507 (2017), ISSN 1476-4660, URL https:\n//doi.org/10.1038/nmat4902 .\n6N. Atodiresei, J. Brede, P. Lazi\u0013 c, V. Caciuc, G. Ho\u000b-\nmann, R. Wiesendanger, and S. Bl ugel, Phys. Rev. Lett.\n105, 066601 (2010), URL https://link.aps.org/doi/\n10.1103/PhysRevLett.105.066601 .\n7N. Atodiresei, V. Caciuc, P. Lazi\u0013 c, and S. Bl ugel, Phys.\nRev. B 84, 172402 (2011), URL https://link.aps.org/\ndoi/10.1103/PhysRevB.84.172402 .\n8N. Atodiresei and K. V. Raman, MRS Bulletin 39, 596-\n601 (2014).\n9T. L. A. Tran, D. Cakr, P. K. J. Wong, A. B. Preobrajen-\nski, G. Brocks, W. G. van der Wiel, and M. P. de Jong,\nACS Applied Materials & Interfaces 5, 837 (2013), pMID:\n23305202, URL https://doi.org/10.1021/am3024367 .\n10T. Moorsom, M. Wheeler, T. Mohd Khan, F. Al Ma'Mari,\nC. Kinane, S. Langridge, D. Ciudad, A. Bedoya-\nPinto, L. Hueso, G. Teobaldi, et al., Phys. Rev. B\n90, 125311 (2014), URL https://link.aps.org/doi/10.\n1103/PhysRevB.90.125311 .\n11N. M. Ca\u000brey, P. Ferriani, S. Marocchi, and S. Heinze,\nPhys. Rev. B 88, 155403 (2013), URL https://link.\naps.org/doi/10.1103/PhysRevB.88.155403 .\n12J. Brede, N. Atodiresei, S. Kuck, P. Lazi\u0013 c, V. Caciuc,\nY. Morikawa, G. Ho\u000bmann, S. Bl ugel, and R. Wiesendan-\nger, Phys. Rev. Lett. 105, 047204 (2010), URL https://\nlink.aps.org/doi/10.1103/PhysRevLett.105.047204 .\n13C. Barraud, P. Seneor, R. Mattana, S. Fusil, K. Bouze-\nhouane, C. Deranlot, P. Graziosi, L. Hueso, I. Bergenti,\nV. Dediu, et al., Nature Physics 6, 615 (2010), ISSN 1745-\n2481, URL https://doi.org/10.1038/nphys1688 .\n14S. Javaid, M. Bowen, S. Boukari, L. Joly, J.-B.\nBeaufrand, X. Chen, Y. J. Dappe, F. Scheurer, J.-\nP. Kappler, J. Arabski, et al., Phys. Rev. Lett. 105,\n077201 (2010), URL https://link.aps.org/doi/10.\n1103/PhysRevLett.105.077201 .\n15V. Hess, R. Friedrich, F. Matthes, V. Caciuc, N. Atodire-sei, D. E. B urgler, S. Bl ugel, and C. M. Schneider, New\nJournal of Physics 19, 053016 (2017), URL https://dx.\ndoi.org/10.1088/1367-2630/aa6ece .\n16D. M. Janas, A. Droghetti, S. Ponzoni, I. Cojocariu,\nM. Jugovac, V. Feyer, M. M. Radonji, I. Rungger,\nL. Chioncel, G. Zamborlini, et al., Advanced Materi-\nals35, 2205698 (2023), URL https://onlinelibrary.\nwiley.com/doi/abs/10.1002/adma.202205698 .\n17M. Callsen, V. Caciuc, N. Kiselev, N. Atodiresei, and\nS. Bl ugel, Phys. Rev. Lett. 111, 106805 (2013), URL\nhttps://link.aps.org/doi/10.1103/PhysRevLett.\n111.106805 .\n18R. Friedrich, V. Caciuc, N. S. Kiselev, N. Atodiresei, and\nS. Bl ugel, Phys. Rev. B 91, 115432 (2015), URL https:\n//link.aps.org/doi/10.1103/PhysRevB.91.115432 .\n19K. V. Raman, A. M. Kamerbeek, A. Mukherjee,\nN. Atodiresei, T. K. Sen, P. Lazi\u0013 c, V. Caciuc, R. Michel,\nD. Stalke, S. K. Mandal, et al., Nature 493, 509\n(2013), ISSN 1476-4687, URL https://doi.org/10.\n1038/nature11719 .\n20S. Boukari, H. Jabbar, F. Schleicher, M. Gruber,\nG. Avedissian, J. Arabski, V. Da Costa, G. Schmer-\nber, P. Rengasamy, B. Vileno, et al., Nano Letters 18,\n4659 (2018), pMID: 29991266, URL https://doi.org/\n10.1021/acs.nanolett.8b00570 .\n21T. Moorsom, S. Alghamdi, S. Stansill, E. Poli,\nG. Teobaldi, M. Beg, H. Fangohr, M. Rogers,\nZ. Aslam, M. Ali, et al., Phys. Rev. B 101,\n060408 (2020), URL https://link.aps.org/doi/10.\n1103/PhysRevB.101.060408 .\n22M. Benini, G. Allodi, A. Surpi, A. Riminucci, K.-\nW. Lin, S. Sanna, V. A. Dediu, and I. Bergenti,\nAdvanced Materials Interfaces 9, 2201394 (2022),\nURL https://onlinelibrary.wiley.com/doi/abs/10.\n1002/admi.202201394 .\n23F. A. Ma'Mari, T. Moorsom, G. Teobaldi, W. Deacon,\nT. Prokscha, H. Luetkens, S. Lee, G. E. Sterbinsky, D. A.\nArena, D. A. MacLaren, et al., Nature 524, 69 (2015).\n24R. Pang, X. Shi, and M. A. Van Hove, Journal\nof the American Chemical Society 138, 4029 (2016),\npMID: 26966934, URL https://doi.org/10.1021/jacs.\n5b10967 .\n25C. Fourmental, L. Le Laurent, V. Repain, C. Cha-\ncon, Y. Girard, J. Lagoute, S. Rousset, A. Coati,\nY. Garreau, A. Resta, et al., Phys. Rev. B 104,\n235413 (2021), URL https://link.aps.org/doi/10.\n1103/PhysRevB.104.235413 .\n26J. St ohr and H. Siegmann, Magnetism: from fundamen-\ntals to nanoscale dynamics. 2006 .\n27M. T. Johnson, P. J. H. Bloemen, F. J. A. den Broeder,\nand J. J. de Vries, Reports on Progress in Physics 59, 1409\n(1996), URL https://doi.org/10.1088/0034-4885/59/\n11/002 .\n28K. Bairagi, A. Bellec, V. Repain, C. Chacon, Y. Gi-\nrard, Y. Garreau, J. Lagoute, S. Rousset, R. Bre-\nitwieser, Y.-C. Hu, et al., Phys. Rev. Lett. 114,18\n247203 (2015), URL https://link.aps.org/doi/10.\n1103/PhysRevLett.114.247203 .\n29K. Bairagi, A. Bellec, V. Repain, C. Fourmen-\ntal, C. Chacon, Y. Girard, J. Lagoute, S. Rousset,\nL. Le Laurent, A. Smogunov, et al., Phys. Rev. B\n98, 085432 (2018), URL https://link.aps.org/doi/10.\n1103/PhysRevB.98.085432 .\n30Z.-H. Zhu, G. Levy, B. Ludbrook, C. N. Veen-\nstra, J. A. Rosen, R. Comin, D. Wong, P. Dosanjh,\nA. Ubaldini, P. Syers, et al., Phys. Rev. Lett. 107,\n186405 (2011), URL https://link.aps.org/doi/10.\n1103/PhysRevLett.107.186405 .\n31S. Jakobs, A. Narayan, B. Stadtm uller, A. Droghetti,\nI. Rungger, Y. S. Hor, S. Klyatskaya, D. Jungkenn,\nJ. St o]ckl, M. Laux, et al., Nano Letters 15, 6022 (2015),\npMID: 26262825, URL https://doi.org/10.1021/acs.\nnanolett.5b02213 .\n32B. Stadtm uller, J. Seidel, N. Haag, L. Grad,\nC. Tusche, G. van Straaten, M. Franke, J. Kirschner,\nC. Kumpf, M. Cinchetti, et al., Phys. Rev. Lett.\n117, 096805 (2016), URL https://link.aps.org/doi/\n10.1103/PhysRevLett.117.096805 .\n33S. Alotibi, B. J. Hickey, G. Teobaldi, M. Ali, J. Barker,\nE. Poli, D. D. O'Regan, Q. Ramasse, G. Burnell, J. Patch-\nett, et al., ACS Applied Materials & Interfaces 13,\n5228 (2021), pMID: 33470108, URL https://doi.org/\n10.1021/acsami.0c19403 .\n34B. Skubic, J. Hellsvik, L. Nordstr om, and O. Eriks-\nson, Journal of Physics: Condensed Matter 20, 315203\n(2008), URL https://dx.doi.org/10.1088/0953-8984/\n20/31/315203 .\n35R. F. L. Evans, W. J. Fan, P. Chureemart, T. A. Ostler,\nM. O. A. Ellis, and R. W. Chantrell, Journal of Physics:\nCondensed Matter 26, 103202 (2014), URL https://dx.\ndoi.org/10.1088/0953-8984/26/10/103202 .\n36P. Bruno, Phys. Rev. B 39, 865 (1989), URL https://\nlink.aps.org/doi/10.1103/PhysRevB.39.865 .\n37S. Bhandary, O. Gr\u0017 an as, L. Szunyogh, B. Sanyal,\nL. Nordstr om, and O. Eriksson, Phys. Rev. B 84,\n092401 (2011), URL https://link.aps.org/doi/10.\n1103/PhysRevB.84.092401 .\n38J. Zhang, P. V. Lukashev, S. S. Jaswal, and E. Y. Tsym-\nbal, Phys. Rev. B 96, 014435 (2017), URL https://link.\naps.org/doi/10.1103/PhysRevB.96.014435 .\n39F. Gimbert and L. Calmels, Phys. Rev. B 86,\n184407 (2012), URL https://link.aps.org/doi/10.\n1103/PhysRevB.86.184407 .\n40G. H. O. Daalderop, P. J. Kelly, and M. F. H. Schuurmans,\nPhys. Rev. B 50, 9989 (1994), URL https://link.aps.\norg/doi/10.1103/PhysRevB.50.9989 .\n41K. Kyuno, J.-G. Ha, R. Yamamoto, and S. Asano,\nJapanese Journal of Applied Physics 35, 2774 (1996),\nURL https://doi.org/10.1143/jjap.35.2774 .\n42R. Liz\u0013 arraga, F. Pan, L. Bergqvist, E. Holmstr om,\nZ. Gercsi, and L. Vitos, Sci. Rep. 7, 3778 (2017), URL\nhttps://doi.org/10.1038/s41598-017-03877-5 .\n43P. E. Bl ochl, Phys. Rev. B 50, 17953 (1994), URL https:\n//link.aps.org/doi/10.1103/PhysRevB.50.17953 .\n44G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993),\nURL https://link.aps.org/doi/10.1103/PhysRevB.\n47.558 .\n45J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.\nLett. 77, 3865 (1996), URL https://link.aps.org/doi/\n10.1103/PhysRevLett.77.3865 .46P. E. Bl ochl, O. Jepsen, and O. K. Andersen, Phys. Rev.\nB49, 16223 (1994), URL https://link.aps.org/doi/\n10.1103/PhysRevB.49.16223 .\n47E. A. Owen and D. M. Jones, Proceedings of the Physical\nSociety. Section B 67, 456 (1954), URL https://doi.\norg/10.1088/0370-1301/67/6/302 .\n48O. K. Andersen, H. L. Skriver, H. Nohl, and J. B., Pure\nAppl. Chem. 52, 93 (1979).\n49M. Weinert, R. E. Watson, and J. W. Davenport, Phys.\nRev. B 32, 2115 (1985), URL https://link.aps.org/\ndoi/10.1103/PhysRevB.32.2115 .\n50G. H. O. Daalderop, P. J. Kelly, and M. F. H. Schuurmans,\nPhys. Rev. B 41, 11919 (1990), URL https://link.aps.\norg/doi/10.1103/PhysRevB.41.11919 .\n51A. B. Klautau and O. Eriksson, Phys. Rev. B 72,\n014459 (2005), URL https://link.aps.org/doi/10.\n1103/PhysRevB.72.014459 .\n52S. Steiner, S. Khmelevskyi, M. Marsmann, and G. Kresse,\nPhys. Rev. B 93, 224425 (2016), URL https://link.\naps.org/doi/10.1103/PhysRevB.93.224425 .\n53C. J. Aas, P. J. Hasnip, R. Cuadrado, E. M. Plotnikova,\nL. Szunyogh, L. Udvardi, and R. W. Chantrell, Phys. Rev.\nB88, 174409 (2013), URL https://link.aps.org/doi/\n10.1103/PhysRevB.88.174409 .\n54J. Zhang, C. Franz, M. Czerner, and C. Heiliger, Phys.\nRev. B 90, 184409 (2014), URL https://link.aps.org/\ndoi/10.1103/PhysRevB.90.184409 .\n55P. V. Ong, N. Kioussis, D. Odkhuu, P. Khalili Amiri,\nK. L. Wang, and G. P. Carman, Phys. Rev. B 92,\n020407 (2015), URL https://link.aps.org/doi/10.\n1103/PhysRevB.92.020407 .\n56Q. Sun, S. Kwon, M. Stamenova, S. Sanvito, and\nN. Kioussis, Phys. Rev. B 101, 134419 (2020), URL\nhttps://link.aps.org/doi/10.1103/PhysRevB.101.\n134419 .\n57M. Blanco-Rey, J. I. Cerd\u0013 a, and A. Arnau, 21, 073054\n(2019), URL https://doi.org/10.1088/1367-2630/\nab3060 .\n58J. Trygg, B. Johansson, O. Eriksson, and J. M. Wills,\nPhys. Rev. Lett. 75, 2871 (1995), URL https://link.\naps.org/doi/10.1103/PhysRevLett.75.2871 .\n59I. Yang, S. Y. Savrasov, and G. Kotliar, Phys. Rev. Lett.\n87, 216405 (2001), URL https://link.aps.org/doi/10.\n1103/PhysRevLett.87.216405 .\n60T. Burkert, O. Eriksson, P. James, S. I. Simak,\nB. Johansson, and L. Nordstr om, Phys. Rev. B 69,\n104426 (2004), URL https://link.aps.org/doi/10.\n1103/PhysRevB.69.104426 .\n61R. Resta, Journal of Physics: Condensed Matter\n22, 123201 (2010), URL https://dx.doi.org/10.1088/\n0953-8984/22/12/123201 .\n62O. Eriksson, M. S. S. Brooks, and B. Johansson, Phys.\nRev. B 41, 7311 (1990), URL https://link.aps.org/\ndoi/10.1103/PhysRevB.41.7311 .\n63D. Ceresoli, U. Gerstmann, A. P. Seitsonen, and F. Mauri,\nPhys. Rev. B 81, 060409 (2010), URL https://link.\naps.org/doi/10.1103/PhysRevB.81.060409 .\n64Vasp wiki ,https://www.vasp.at/wiki/index.php/\nLSORBIT .\n65V. Antropov, L. Ke, and D. Aberg, Solid State Com-\nmunications 194, 35 (2014), ISSN 0038-1098, URL\nhttps://www.sciencedirect.com/science/article/\npii/S0038109814002476 .\n66M. Getzla\u000b, J. Bansmann, and G. Sch onhense, Sur-19\nface Science 323, 118 (1995), ISSN 0039-6028, URL\nhttps://www.sciencedirect.com/science/article/\npii/0039602894006415 .\n67S. J. Carey, W. Zhao, and C. T. Campbell, Sur-\nface Science 676, 9 (2018), ISSN 0039-6028, special\nIssue Dedicated to the 75 birthday of Peter R Nor-\nton, URL https://www.sciencedirect.com/science/\narticle/pii/S0039602817309779 .\n68M. Cinchetti, K. Heimer, J.-P. W ustenberg, O. Andreyev,\nM. Bauer, S. Lach, C. Ziegler, Y. Gao, and M. Aeschli-\nmann, Nature Materials 8, 115 (2009), ISSN 1476-4660,\nURL https://doi.org/10.1038/nmat2334 .\n69S. Lach, A. Altenhof, K. Tarafder, F. Schmitt, M. E.\nAli, M. Vogel, J. Sauther, P. M. Oppeneer, and\nC. Ziegler, Advanced Functional Materials 22, 989\n(2012), URL https://onlinelibrary.wiley.com/doi/\nabs/10.1002/adfm.201102297 .\n70F. Djeghloul, F. Ibrahim, M. Cantoni, M. Bowen, L. Joly,\nS. Boukari, P. Ohresser, F. Bertran, P. Le F\u0012 evre,\nP. Thakur, et al., Scienti\fc Reports 3, 1272 (2013), ISSN\n2045-2322, URL https://doi.org/10.1038/srep01272 .\n71A. Droghetti, S. Steil, N. Gro\u0019mann, N. Haag,\nH. Zhang, M. Willis, W. P. Gillin, A. J. Drew,\nM. Aeschlimann, S. Sanvito, et al., Phys. Rev. B\n89, 094412 (2014), URL https://link.aps.org/doi/10.\n1103/PhysRevB.89.094412 .\n72A. Droghetti, P. Thielen, I. Rungger, N. Haag, N. Gro\u0019-\nmann, J. St ockl, B. Stadtm uller, M. Aeschlimann, S. San-\nvito, and M. Cinchetti, Nature Communications 7,\n12668 (2016), ISSN 2041-1723, URL https://doi.org/\n10.1038/ncomms12668 .\n73S. Lach, A. Altenhof, S. Shi, M. Fahlman, and C. Ziegler,\nPhys. Chem. Chem. Phys. 21, 15833 (2019), URL http:\n//dx.doi.org/10.1039/C9CP02205H .\n74F. Mittendorfer and J. Hafner, Surface Sci-\nence 472, 133 (2001), ISSN 0039-6028, URL\nhttps://www.sciencedirect.com/science/article/\npii/S0039602800009298 .\n75Vasp wiki ,https://www.vasp.at/wiki/index.php/\nAngular_functions .\n76J. St ohr, Journal of Magnetism and Magnetic Ma-\nterials 200, 470 (1999), ISSN 0304-8853, URL\nhttps://www.sciencedirect.com/science/article/\npii/S0304885399004072 .\n77P. Carra, B. T. Thole, M. Altarelli, and X. Wang, Phys.\nRev. Lett. 70, 694 (1993), URL https://link.aps.org/\ndoi/10.1103/PhysRevLett.70.694 .\n78M. Tischer, O. Hjortstam, D. Arvanitis, J. Hunter Dunn,\nF. May, K. Baberschke, J. Trygg, J. M. Wills,\nB. Johansson, and O. Eriksson, Phys. Rev. Lett. 75,\n1602 (1995), URL https://link.aps.org/doi/10.1103/\nPhysRevLett.75.1602 .\n79J. St ohr, Journal of Electron Spectroscopy and Re-\nlated Phenomena 75, 253 (1995), ISSN 0368-2048, fu-\nture Perspectives for Electron Spectroscopy with Syn-\nchrotron Radiation, URL https://www.sciencedirect.\ncom/science/article/pii/0368204895025375 .\n80D. Li, A. Smogunov, C. Barreteau, F. m. c.\nDucastelle, and D. Spanjaard, Phys. Rev. B 88,\n214413 (2013), URL https://link.aps.org/doi/10.\n1103/PhysRevB.88.214413 .\n81H. Wende, M. Bernien, J. Luo, C. Sorg, N. Ponpandian,\nJ. Kurde, J. Miguel, M. Piantek, X. Xu, P. Eckhold, et al.,\nNature Materials 6, 516 (2007), ISSN 1476-4660, URLhttps://doi.org/10.1038/nmat1932 .\n82Y. Zhan, E. Holmstr om, R. Liz\u0013 arraga, O. Eriks-\nson, X. Liu, F. Li, E. Carlegrim, S. Stafstr om, and\nM. Fahlman, Advanced Materials 22, 1626 (2010),\nURL https://onlinelibrary.wiley.com/doi/abs/10.\n1002/adma.200903556 .\n83J. Miguel, C. F. Hermanns, M. Bernien, A. Kr uger,\nand W. Kuch, The Journal of Physical Chemistry Let-\nters2, 1455 (2011), URL https://doi.org/10.1021/\njz200489y .\n84C. F. Hermanns, K. Tarafder, M. Bernien, A. Kr uger, Y.-\nM. Chang, P. M. Oppeneer, and W. Kuch, Advanced Ma-\nterials 25, 3473 (2013), URL https://onlinelibrary.\nwiley.com/doi/abs/10.1002/adma.201205275 .\n85S. Shi, Z. Sun, A. Bedoya-Pinto, P. Graziosi, X. Li,\nX. Liu, L. Hueso, V. A. Dediu, Y. Luo, and\nM. Fahlman, Advanced Functional Materials 24, 4812\n(2014), URL https://onlinelibrary.wiley.com/doi/\nabs/10.1002/adfm.201400125 .\n86H.-J. Jang, J.-S. Lee, S. J. Pookpanratana, C. A. Hacker,\nI. C. Tran, and C. A. Richter, The Journal of Physical\nChemistry C 119, 12949 (2015), URL https://doi.org/\n10.1021/acs.jpcc.5b01222 .\n87R. Friedrich, V. Caciuc, N. Atodiresei, and S. Bl ugel,\nPhys. Rev. B 92, 195407 (2015), URL https://link.\naps.org/doi/10.1103/PhysRevB.92.195407 .\n88H. Yang, A. D. Vu, A. Hallal, N. Rougemaille, J. Coraux,\nG. Chen, A. K. Schmid, and M. Chshiev, Nano Letters\n16, 145 (2016), pMID: 26641927, URL https://doi.org/\n10.1021/acs.nanolett.5b03392 .\n89Z. H. Xiong, D. Wu, Z. Valy Vardeny, and J. Shi, Na-\nture427, 821 (2004), ISSN 1476-4687, URL https://\ndoi.org/10.1038/nature02325 .\n90V. Dediu, L. E. Hueso, I. Bergenti, A. Rimin-\nucci, F. Borgatti, P. Graziosi, C. Newby, F. Casoli,\nM. P. De Jong, C. Taliani, et al., Phys. Rev. B\n78, 115203 (2008), URL https://link.aps.org/doi/10.\n1103/PhysRevB.78.115203 .\n91M. Galbiati, State of the Art in Alq3-Based Spin-\ntronic Devices Springer International Publishing, Cham,\n2016), pp. 139{151, URL https://doi.org/10.1007/\n978-3-319-22611-8_7 .\n92V. A. Dediu, L. E. Hueso, I. Bergenti, and C. Taliani,\nNature Materials 8, 707 (2009), URL https://doi.org/\n10.1038/nmat2510 .\n93A. Curioni, M. Boero, and W. Andreoni, Chem-\nical Physics Letters 294, 263 (1998), ISSN 0009-\n2614, URL https://www.sciencedirect.com/science/\narticle/pii/S000926149800829X .\n94K. Tarafder, B. Sanyal, and P. M. Oppeneer, Phys. Rev.\nB82, 060413 (2010), URL https://link.aps.org/doi/\n10.1103/PhysRevB.82.060413 .\n95F. Bisti, A. Stroppa, M. Donarelli, S. Picozzi, and L. Ot-\ntaviano, Phys. Rev. B 84, 195112 (2011), URL https:\n//link.aps.org/doi/10.1103/PhysRevB.84.195112 .\n96A. Droghetti, M. Cinchetti, and S. Sanvito, Phys. Rev.\nB89, 245137 (2014), URL https://link.aps.org/doi/\n10.1103/PhysRevB.89.245137 .\n97A. Droghetti, Journal of Magnetism and Magnetic\nMaterials 502, 166578 (2020), ISSN 0304-8853, URL\nhttps://www.sciencedirect.com/science/article/\npii/S0304885319340156 .\n98Note1, the Co slab considered here is slightly di\u000berent\nfrom that used in the other sections. In the case of20\nAlq3/Co, the slab is constrained in-plane to have the same\nlattice constant of Cu, so to mimic the experiments of\nRefs.71,72. Nonetheless this has a minor e\u000bect on the re-\nsults and on the proposed mechanism for the MCA mod-\ni\fcation.\n99S. M uller, S. Steil, A. Droghetti, N. Grossmann,V. Meded, A. Magri, B. Sch afer, O. Fuhr, S. Sanvito,\nM. Ruben, et al., New Journal of Physics 15, 113054\n(2013), URL https://dx.doi.org/10.1088/1367-2630/\n15/11/113054 .\n100L. Le Laurent, C. Barreteau, T. Markussen, , Phys. Rev.\nB100, 174426 (2019), URL https://link.aps.org/doi/\n10.1103/PhysRevB.100.174426 ."    },    {        "title": "2302.11714v1.The_influence_of_crystalline_electric_field_on_the_magnetic_properties_of_CeCd3X3__X___P_and_As_.pdf",        "content": "arXiv:2302.11714v1  [cond-mat.str-el]  23 Feb 2023The influence of crystalline electric field on the magnetic pr operties of\nCeCd 3X3(X= P and As)\nObinna P. Uzoh1, Suyoung Kim1, and Eundeok Mun1,2\n1Department of Physics, Simon Fraser University,\n8888 University Drive, Burnaby, B.C. Canada and\n2Center for Quantum Materials and Superconductivity (CQMS) ,\nSungkyunkwan University, Suwon 16419, South Korea\nCeCd 3P3and CeCd 3As3compounds adopt the hexagonal ScAl 3C3-type structure, where mag-\nnetic Ce ions on a triangular lattice order antiferromagnet ically below TN∼0.42 K. Their crystalline\nelectric field (CEF) level scheme has been determined by fitti ng magnetic susceptibility curves, mag-\nnetization isotherms, and Schottky anomalies in specific he at. The calculated results, incorporating\nthe CEF excitation, Zeeman splitting, and molecular field, a re in good agreement with the experi-\nmental data. The CEF model, with Ce3+ions in a trigonal symmetry, explains the strong easy-plane\nmagnetic anisotropy that has been observed in this family of materials. A detailed examination of\nthe CEF parameters suggests that the fourth order CEF parame terB3\n4is responsible for the strong\nCEF induced magnetocrystalline anisotropy, with a large ab-plane moment and a small c-axis mo-\nment. The reliability of our CEF analysis is assessed by comp aring the current study with earlier\nreports of CeCd 3As3. For both CeCd 3X3(X= P and As) compounds, less than 40 % of Rln(2)\nmagnetic entropyis recovered by TNandfullRln(2) entropyis achieved at theWeiss temperature θp.\nAlthough the observed magnetic entropy is reminiscent of de localized 4 f-electron magnetism with\nsignificant Kondo screening, the electrical resistivity of these compounds follows a typical metallic\nbehavior. Measurements of thermoelectric power further va lidate the absence of Kondo contribution\nin CeCd 3X3.\nI. INTRODUCTION\nSeveralf-electron materials with triangular lattices\n(TL) have shown rich phenomena, where the spin-\norbit coupling enhances quantum fluctuations due to the\nhighly anisotropic interactions between 4 fmoments [1–\n6]. For example, a spin liquid state has been proposed in\ninsulating YbMgGaO 4[7–11] and NaYbS 2[12, 13]. The\nlow carrier density system YbAl 3C3shows a gap in the\nmagnetic excitation spectrum due to the dimerization\nof thefelectrons in Yb3+pairs [14–17]. Of interest is\nthe easy-plane antiferromagnets CeCd 3X3(X= P and\nAs), a new class of TL system, with a low antiferro-\nmagneticorderingtemperatureandextremelylowcarrier\ndensity [18–22].\nThe family of compounds Ce M3X3(M= Al, Cd, and\nZn,X= C, P, and As) have been investigated for their\nrobust ground state properties arising from the interplay\nbetween crystalline electric field (CEF) and magnetic\nexchange interaction on the triangular lattice [16, 18–\n24]. CeCd 3X3materials adopt the hexagonal ScAl 3C3-\ntype structure (space group P63/mmc) with the mag-\nnetic Ce3+ions having a trigonal ( D3d) point symme-\ntry as shown in Fig. 1(c) [20–23]. In this crystal struc-\nture, the Ce layers are well separated by the Cd and X\natoms, forming a layered 2D TL in the ab-plane, as de-\npicted in Figs. 1(a) and (b) [20–23]. Thermodynamic\nand transport property measurements have character-\nized CeCd 3X3as an low carrier density system, with\na strong easy plane magnetic anisotropy and antiferro-\nmagnetic ordering below TN∼0.42 K [19, 22]. The low\ncarrierdensitymeansthereareinsufficientchargecarriers\nto screen all local moments and to mediate Ruderman-\nFIG.1. (a)Crystal structureofthehexagonalCeCd 3X3(X=\nP and As). (b) Ce atoms forming two-dimensional triangular\nlattices. (c) Ce atom in a trigonal ( D3d) point symmetry\nsurrounded by P/As atoms.\nKittel-Kasuya-Yosida (RKKY) exchange interaction be-\ntween moments. In this regard, both RKKY and Kondo\ninteractions are expected to be weakened in these com-\npounds. We note that the electrical resistivity measure-\nment on CeCd 3As3grown by chemical vapor transport\nshows a semiconductor-like enhancement as decreasing\ntemperature [20], whiletheflux grownCeCd 3As3sample\nis metallic [19]. At the magnetic ordering temperature,\nroughly40% of Rln(2) entropyis recovered. This implies\nadoublet groundstate resultingfromCEFsplitting oflo-\ncalizedCeionenergylevels. Thehighlyenhancedspecific\nheat below 10 K and the reduced magnetic entropyat TN\nare reminiscent of Kondo lattice materials. However, the\nelectrical resistivity of CeCd 3X3shows no maximum or2\nTABLE I. CeCd 3As3: CEF parameters, energy eigenvalues,\neigenstates, and molecular field parameters.\nCEF and molecular field parameters\nB0\n2(K)B0\n4(K)B3\n4(K)λi(mole/emu)\n18.55 -0.08 23.02 λab=−5.7,\nλc= 1.0\nEnergy eigenvalues and eigenstates\nE (K) |−5\n2/angbracketright |−3\n2/angbracketright |−1\n2/angbracketright |1\n2/angbracketright |3\n2/angbracketright |5\n2/angbracketright\n0 0.0 0.0 −0.898 0.0 0.0 0.440\n0 0.440 0.0 0.0 0.898 0.0 0.0\n242 0.0 -1.0 0.0 0.0 0.0 0.0\n242 0.0 0.0 0.0 0.0 -1.0 0.0\n553 -0.898 0.0 0.0 0.440 0.0 0.0\n553 0.0 0.0 0.440 0.0 0.0 0.898\nlogarithmic upturns resulting from the Kondo scattering\nof conduction electrons from magnetic Ce ions. The re-\nsistivity of CeCd 3X3is the same as that of LaCd 3X3,\nimplying an absence of Kondo screening in these materi-\nals [19, 22].\nHere, we use the CEF scheme to clarify the anisotropic\nmagnetic properties of CeCd 3X3(X= P and As), where\nthe CEFanalysisisperformedusingthe PyCrystalField\npackage [26]. For CeCd 3As3, the CEF analyses of two\nprevious independent studies [20, 25], carried out only\nconsidering magnetic susceptibility and magnetization,\nhave shown discrepancies in their energy level splittings\nand first excited state wave functions. Thus, we extend\nthe CEF analysis to specific heat data to resolve these\ndiscrepancies. In contrast to CeCd 3As3, no CEF analy-\nsis has been carried out for CeCd 3P3. We show that the\neasy plane magnetic anisotropy observed in these com-\npounds can be explained by the strong CEF acting on\nCe3+ions. In addition, we provide evidence of the lack-\ning Kondo screening in these materials by way of ther-\nmoelectric power measurements.\nII. EXPERIMENTAL\nFor this work, single crystals of LaCd 3X3and\nCeCd3X3(X= P and As) were grown by high tem-\nperature ternary melt [19, 22, 27]. Magnetization mea-\nsurements as a function of temperature and magnetic\nfield were performed in a QD MPMS. Specific heat of\nthese compounds was measured in QD PPMS. The ob-\ntained results are consistent with earlier reports [19, 22].\nThermoelectricpowermeasurementswereperformed in a\nhome made two thermometer and one heater setup. De-\ntailedstudiesofthermodynamicandtransportproperties\nof these compounds are presented in Ref. [19, 22].\nIII. RESULTS & DISCUSSION\nFor CeCd 3X3systems, Ce atoms occupy a single site\nat the 2aWyckoff position that has a trigonal point sym-/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48 /s51/s53/s48/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48/s56/s48/s48\n/s84 /s32/s40/s75/s41/s72/s47/s77 /s32/s40/s109/s111/s108/s101/s47/s101/s109/s117/s41/s67/s101/s67/s100\n/s51/s65/s115\n/s51\n/s72/s32/s124/s124/s32/s99\n/s72/s32/s124/s124/s32/s97/s98/s72 /s32/s61/s32/s49/s32/s107/s79/s101/s40/s97/s41\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50\n/s72 /s32/s40/s107/s79/s101/s41/s40/s98/s41/s77 /s32/s40\n/s66/s47/s67/s101/s41/s67/s101/s67/s100\n/s51/s65/s115\n/s51\n/s72/s32/s124/s124/s32/s99/s72/s32/s124/s124/s32/s97/s98/s84 /s32/s61/s32/s49/s46/s56/s32/s75\n/s48/s46/s49 /s49 /s49/s48 /s49/s48/s48/s48/s50/s52/s54/s56\n/s84 /s32/s40/s75/s41/s32/s72 /s32/s61/s32/s48\n/s32/s57/s48/s32/s107/s79/s101/s32/s40/s72/s32/s124/s124/s32/s97/s98/s41/s40/s99/s41/s67\n/s109/s32/s40/s74/s47/s109/s111/s108/s101/s32/s75/s41/s67/s101/s67/s100\n/s51/s65/s115\n/s51\nFIG. 2. (a) Inverse magnetic susceptibility curves of\nCeCd 3As3, taken at H= 1 kOe for both H/bardblabandH/bardblc.\n(b) Magnetization isotherms at 1.8 K for both H/bardblaband\nH/bardblc. (c) Magnetic part of the specific heat at H= 0 and\n90 kOe along H/bardblab. Open symbols are experimental data.\nDashed lines represent CEF fits with no molecular field con-\ntribution. Solid lines are CEF fits with molecular field terms\nλab=−5.7 mole/emu and λc= 1.0 mole/emu.3\nmetry, shown in Fig. 1(c). In this point symmetry, the\nCEFHamiltonianrequiresonlythreeparameters[28,29]:\nHCEF=B0\n2O0\n2+B0\n4O0\n4+B3\n4O3\n4, where Om\nnare the\nStevens operators and Bm\nnare known as the CEF pa-\nrameters [30, 31]. It has to be noted here that thermo-\ndynamic and transport property measurements of these\ncompoundsindicateapossiblestructuralphasetransition\nbelow 132 K [16, 19, 22]. Additional CEF parametersbe-\nlow the transition temperature are probably required to\naccount for the change in the local environment of the Ce\natoms. However, the magnetic susceptibility curves show\na smooth evolution through the transition temperature.\nTherefore, we assume in our CEF analysis that the Ce\nions have the same trigonal symmetry below the transi-\ntion temperature. To account for the Zeeman effect and\nthe interaction between magnetic ions, the eigenvalues\nand eigenfunctions are determined by diagonalizing the\ntotal Hamiltonian: H=HCEF−gJµBJi(Hi+λiMi) (i\n=x,y, andz), where gJis the Land´ e g-factor,µBis the\nBohr magneton, Hiis the applied field, Jiis the angular\nmomentum operator, and Miis the magnetization. The\nsecond term is the Zeeman contribution and the third\nterm represents the effective molecular interactions λi.\nThe point charge CEF model assumes that the mag-\nnetic ions are well localized with a stable valence state.\nThe valence state of CeCd 3X3can be deduced from the\neffective moment of Ce ions. Figures 2(a) and 3(a) show\ninverse magnetic susceptibility, χ−1=H/M, curves of\nCeCd3As3and CeCd 3P3, respectively. At sufficiently\nhigh temperatures, magnetic susceptibility curves follow\nthe Curie-Weiss (CW) behavior: χ(T) =C/(T−θp),\nwhereCis the Curie constant and θpis the Weiss\ntemperature. The effective moments of CeCd 3P3and\nCeCd3As3are estimated by applying the CW law to\nthepolycrystallineaveragedmagneticsusceptibilitytobe\nµeff= 2.51µBandµeff= 2.54µB, respectively; which\nagreeverywellwith thetheoreticalvalue µeff= 2.54µB.\nFrom the effective moment values, it is reasonable to as-\nsume that Ce ions in these compounds are well local-\nized with 3+ valence state. As shown in Ref. [25, 32]\ntheB0\n2parameter mainly gives a measure of the magne-\ntocrystalline anisotropy and can be expressed in terms\nof the Weiss temperatures [33–35]. From the Curie-\nWeiss fit (not shown), anisotropic Weiss temperatures\nof CeCd 3As3are estimated to be θab∼9.3 K and θc=\n−283 K. For CeCd 3P3,θab= 9.3 K and θc=−248 K\nare obtained. The obtained θpvalues are consistent with\nearlier reports [19, 22]. These values are used to estimate\nthe leading CEF parameters: B0\n2∼30.5 K for CeCd 3As3\nandB0\n2∼26.7 K for CeCd 3P3.\nFigure 2 displays anisotropic magnetic susceptibility\ncurves, magnetization isotherms, and specific heat of\nCeCd3As3with the results ofCEF analysis. Table I sum-\nmarizes the obtained CEF parameters, including molec-\nular field parameters, and the energy eigenstates and\neigenvalues. The 2 J+1 degenerate levels for J= 5/2 of\nCe3+split into three Kramers doublets with energy level\nsplittings ∆ 1= 242K and ∆ 2= 553K. The ground stateand second excited state are in a mixture of |±5/2/angbracketrightand\n|∓1/2/angbracketrightstates. However, the first excited state is a pure\n|±3/2/angbracketrightstate.\nFirst, we discuss the CEF effects on thermodynamic\nproperties of CeCd 3As3without the molecular field con-\ntribution. In Fig. 2, blue dashed-lines represent the\nCEF evaluations with λi= 0. The CEF fit generally\nagrees with the inverse magnetic susceptibility as the\nlarge anisotropy between crystallographic directions is\ncaptured. For H/bardblcthe broad hump around 25 K is\nwell reproduced. The CEF model aligns very well with\nthe magnetization isotherm, M(H), forH/bardblcat 1.8 K,\nwhereasthereisaninconsistencybetweenthe CEFcalcu-\nlation and M(H) forH/bardblab, as shown in Fig. 2(b). The\nmagnetic contribution to the specific heat, Cm, atH=\n0 and 90 kOe is presented in Fig. 2(c), where Cmis es-\ntimated by subtracting the specific heat of non-magnetic\nanalog LaCd 3As3. In zero field, the broad maximum\nnear 130 K in Cmcan be reproduced by the CEF model.\nNote that the CEF calculation above the maximum can-\nnot capture the experimental data, which is caused by\nthe very large subtraction error as explained in Ref. [19].\nObviously, the sharpriseof Cmbelow 10Kand the sharp\npeak atTN= 0.42 K [19] are not captured by the CEF\nmodel. At H= 90 kOe for H/bardblab, the overall shape\nofCmis captured by the CEF model, but the maximum\ntemperature is higher than that of experimental data.\nThe CEF model without λidoes not adequately re-\nproduce M(H) andCmdata for H/bardblab. In order to\naccount for this mismatch, the molecular field interac-\ntions between Ce3+ions are incorporated. Red solid-\nlines in Fig. 2 represent the CEF model in the presence\nof the molecular field terms: λab=−5.7 mole/emu and\nλc= 1 mole/emu. The magnetic susceptibility curves\nwithλishow a slightly better agreement than that with\nλi= 0. However, as can be clearly seen in Fig. 2(b), the\nM(H) curve for H/bardblabis well captured by the combina-\ntion of the CEF and λi. Moreover, although the absolute\nvalue of the maximum in Cmat 90 kOe is slightly higher,\nthe position of the maximum temperature is well repro-\nduced by introducing λi. These results point to the im-\nportance of including the exchange interactions between\nCe3+magnetic ions.\nNow turning to the CEF analysis for CeCd 3P3, fol-\nlowing the same procedure applied to CeCd 3As3. The\nisostructural compounds CeCd 3P3and CeCd 3As3show\nremarkably similar magnetic properties, implying that\ntheir local CEF environments are in close resemblance.\nHence, it is expected that the CEF parameters will be\nquite similar for both compounds, giving rise to very\nsimilar CEF energy level splittings and eigenstates. Ta-\nble II shows a summary of CEF fit results of CeCd 3P3.\nAs expected, the obtained CEF parameters, eigenstates,\nand eigenvalues for CeCd 3P3are very similar to those of\nCeCd3As3(Table I). The positive B0\n2term indicates that\nthe magnetization lies in easy plane as seen in Fig. 3(b).\nThe large B3\n4term implies a mixed ground state with\n| ±5/2/angbracketrightand| ∓1/2/angbracketright, just like the case for CeCd 3As3.4\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48 /s51/s53/s48/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48/s56/s48/s48/s72/s47/s77 /s32/s40/s109/s111/s108/s101/s47/s101/s109/s117/s41\n/s84 /s32/s40/s75/s41/s67/s101/s67/s100\n/s51/s80\n/s51\n/s72/s32/s124/s124/s32/s99\n/s72/s32/s124/s124/s32/s97/s98/s72 /s32/s61/s32/s49/s32/s107/s79/s101/s40/s97/s41\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50/s77 /s32/s40\n/s66/s47/s67/s101/s41\n/s72 /s32/s40/s107/s79/s101/s41/s67/s101/s67/s100\n/s51/s80\n/s51\n/s72/s32/s124/s124/s32/s99/s72/s32/s124/s124/s32/s97/s98/s84 /s32/s61/s32/s49/s46/s56/s32/s75/s40/s98/s41\n/s48/s46/s49 /s49 /s49/s48 /s49/s48/s48/s48/s50/s52/s54/s56/s49/s48\n/s32/s48/s32/s107/s79/s101\n/s32/s57/s48/s32/s107/s79/s101/s32/s40/s72/s32/s124/s124/s32/s97/s98/s41/s67\n/s109/s32/s40/s74/s47/s109/s111/s108/s101/s32/s75/s41\n/s84 /s32/s40/s75/s41/s67/s101/s67/s100\n/s51/s80\n/s51/s40/s99/s41\nFIG. 3. (a) Inverse magnetic susceptibility curves of\nCeCd 3P3, taken at H= 1 kOe for both H/bardblabandH/bardblc.\n(b) Magnetization isotherms at 1.8 K for both H/bardblaband\nH/bardblc. (c) Magnetic part of the specific heat at H= 0 and\n90 kOe along H/bardblab. Open symbols are experimental data.\nDashed lines represent CEF fits with no molecular field con-\ntribution. Solid lines are CEF fits with molecular field terms\nλab=−6.2 mole/emu and λc= 0.3 mole/emu.TABLE II. CeCd 3P3: CEF parameters, energy eigenvalues,\neigenstates, and molecular field parameters.\nCEF and molecular field parameters\nB0\n2(K)B0\n4(K)B3\n4(K)λi(mole/emu)\n20.90 -0.03 26.00 λab=−6.2,\nλc= 0.3\nEnergy eigenvalues and eigenstates\nE (K) |−5\n2/angbracketright |−3\n2/angbracketright |−1\n2/angbracketright |1\n2/angbracketright |3\n2/angbracketright |5\n2/angbracketright\n0 0.0 0.0 −0.897 0.0 0.0 0.442\n0 0.442 0.0 0.0 0.897 0.0 0.0\n257 0.0 -1.0 0.0 0.0 0.0 0.0\n257 0.0 0.0 0.0 0.0 -1.0 0.0\n621 -0.897 0.0 0.0 0.442 0.0 0.0\n621 0.0 0.0 0.442 0.0 0.0 0.897\nThe first excited state is in a pure state of |±3/2/angbracketrightand\nthe second excited state is in an admixtureof |±5/2/angbracketrightand\n|∓1/2/angbracketrightstates. The energy level splittings correspond to\n257 K for the first excited state and 621 K for the second\nexcited state.\nFigure 3 shows magnetic susceptibility, magnetization,\nand specific heat curves of CeCd 3P3together with the\nCEF calculations. In the absence of molecular field\nterms, the CEF calculations (dashed-lines) agree very\nwell with the experimental H/Mcurves, as shown in\nFig. 3(a). Also, in zero field Cmagrees with the CEF fit\nat high temperatures, as shown in Fig. 3(c), implying the\nhightemperature broadmaximum (Schottky anomaly)is\ndue to the CEF effects. The inclusion of the molecular\nfield terms greatly alters the M(H) curve calculation for\nH/bardblaband adequately captures the experimental M(H)\ndata. In addition, the CEF calculation with molecular\nfield terms properly captures the low temperature maxi-\nmum inCmatH= 90 kOe. Therefore, the CEF model\nin the present of the molecular field interaction should\nbe used to describe the experimental data for CeCd 3P3.\nAs expected from the layered crystal structure, the ab-\nsolute value of λabis greater than λc, implying a strong\nexchange interaction in the ab-plane.\nThe obtained CEF parameters of CeCd 3P3are slightly\nlarger than those of CeCd 3As3. The slight difference\nin the values of their CEF parameters is quite surpris-\ning. When the distance between Ce and nearest As/P\natoms is considered, the difference between Ce-As dis-\ntance (3.05 ˚A) and Ce-P distance (2.96 ˚A) is about\n∼9 pm [19, 20, 22], which might not be large enough to\ncausedifferent CEF energylevel splittings. Note that the\nvalidity of the CEF Hamiltonian must be verified below\nthe (structural) phasetransition temperature Ts= 127K\nfor CeCd 3P3[22] and Ts= 136 K for CeCd 3As3[19].\nIn Fig. 4(a) we compare the energy level splittings and\neigenfunctions for three independent CEF analyses of\nCeCd3As3. For all three studies the CEF parameters are\nobtained from fits to the magnetic susceptibility curves.\nIn this study, the Hamiltonian includes the molecular\nfield term. In Ref. [20] (Study A), the CEF parame-\nters are obtained in the presence of antiferromagnetic5\n/s80/s114/s101/s115/s101/s110/s116/s32/s115/s116/s117/s100/s121/s83/s116/s117/s100/s121/s32/s65 /s83/s116/s117/s100/s121/s32/s66/s48/s32/s75/s32/s50/s52/s50/s32/s75/s53/s53/s51/s32/s75\n/s48/s32/s75/s32/s51/s55/s50/s32/s75/s53/s52/s53/s32/s75\n/s48/s32/s75/s32/s50/s52/s49/s32/s75/s50/s56/s50/s32/s75\n/s71\n/s49/s32/s61/s32 /s97 /s124/s43 /s53/s47/s50/s62/s32 /s43 /s32/s98 /s124/s43 /s49/s47/s50/s62/s71\n/s50/s32/s61/s32 /s103/s124/s43 /s51/s47/s50/s62/s71\n/s51/s32/s61/s32/s43 /s98 /s124/s43 /s53/s47/s50/s62/s32/s43/s32 /s97 /s124/s43 /s49/s47/s50/s62\n/s71\n/s49/s32/s61/s32 /s97 /s124/s43 /s53/s47/s50/s62/s32/s43/s32 /s98 /s124/s43 /s49/s47/s50/s62/s71\n/s50/s32/s61/s32 /s98 /s124/s43 /s53/s47/s50/s62/s32/s43/s32 /s97 /s124/s43 /s49/s47/s50/s62/s71\n/s51/s32/s61/s32 /s103/s124/s43 /s51/s47/s50/s62\n/s71\n/s49/s32/s61/s32 /s97 /s124/s43 /s53/s47/s50/s62/s32/s45/s32 /s98 /s124/s43 /s49/s47/s50/s62/s71\n/s50/s32/s61/s32 /s103/s124/s43 /s51/s47/s50/s62/s71\n/s51/s32/s61/s32 /s98 /s124/s43 /s53/s47/s50/s62/s32/s45/s32 /s97 /s124/s43 /s49/s47/s50/s62/s40/s97/s41\n/s49 /s49/s48 /s49/s48/s48/s48/s50/s52/s54/s56\n/s32/s72 /s32/s61/s32/s111\n/s32/s57/s48/s32/s107/s79/s101/s67\n/s109/s32/s40/s74/s47/s109/s111/s108/s101/s32/s75/s41\n/s84 /s32/s40/s75/s41/s32/s82/s101/s102/s46/s32/s65\n/s32/s82/s101/s102/s46/s32/s66\n/s32/s80/s114/s101/s115/s101/s110/s116/s32/s115/s116/s117/s100/s121/s67/s101/s67/s100\n/s51/s65/s115\n/s51\nFIG. 4. CEF profiles of CeCd 3As3for three independent\nstudies. Study A is from Ref. [20] and Study B is from\nRef. [25]. (a) CEF energy level splittings and eigenstates.\nPresent study: α= 0.44,β= 0.90, and γ=−1.\nStudy A: α= 0.32,β= 0.95, and γ= 1. Study B:\nα= 0.28,β= 0.96, andγ= 1. (b) Comparison between\nCmof present study and CEF calculations based on the three\nindependent studies, where λi= 0.\nexchange interaction based on a mean-field approach. In\nRef.[25](StudyB),theCEFparametersareevaluatedby\nfitting the magnetic susceptibility curves to only HCEF\n(without any interaction term).\nForallthreestudies, thegroundstate(Γ 1)isinamixed\nstate of|±5/2/angbracketrightand|∓1/2/angbracketrightwith a higher probability in\nthe|∓1/2/angbracketrightstate. This requires a mixing angle θin the\nwave function: cos( θ)| ±5/2/angbracketright+sin(θ)| ∓1/2/angbracketright. The ob-\ntained mixing angle is roughly similar for all three stud-\nies: 64◦inthisstudy, 72◦inRef.[20], and74◦inRef.[25].\nUnlike the ground state, the excited states Γ 2and Γ3do\nnot entirely agree for all three studies. In Ref. [20] there\nis a clear swap between Γ 2and Γ3wave functions (Study\nA). This is caused by the relatively high |B0\n4|value (=\n1.4K).In fact, weconfirmedthatany |B0\n4|>0.7Kwouldresult in the Γ 2being in a mixed state and the Γ 3being\nin a pure state, just as the case in Ref. [20]. Another\nimportant distinction among these three studies is in the\nenergylevel splittings. The energyeigenvalue for the sec-\nond excited state in Ref. [25] (Study B) is roughly two\ntimes smaller than that of the other two studies. How-\never, the energy level splitting from the ground to the\nfirst excited state ( ∼240 K) is comparable for all studies,\nclearly indicating that the ground state is well isolated\nfrom the excited states.\nFigure 4 (b) shows the magnetic part of the specific\nheat, together with the calculated specific heat curves\nby using CEF parameters. Solid lines, dashed lines, and\ndotted lines are CEF calculations from the present study,\nRef. [20], and Ref. [25], respectively. As shown in the\nfigure, when subtle differences are ignored, the high tem-\nperature maximum is captured by all three calculated\ncurves. This implies that the high temperature maxi-\nmum inCmis due to the CEFeffect and the groundstate\ndoublet is well isolated from the excited states. When\nmeasurement uncertainty and different sample quality\nare considered, the best CEF parameters among three\nparameter sets cannot be selected from the comparison\nwithCm. Therefore, further measurements such as in-\nelastic neutron scattering with Cd isotopes are necessary\nto distinctly specify the best CEF parameters in this sys-\ntem. In addition, the CEF scheme can be determined by\noptical spectroscopy techniques such as Raman scatter-\ning. Note that CEF parameters evaluated by the three\nindependent studies provide a qualitatively good descrip-\ntion of the experimental M/Hcurves.\nThe significance of B3\n4CEF parameter has been ob-\nserved in Ce-based antiferromagnets such as CeAuSn,\nCeIr3Ge7, and CeCd 3X3, where Ce ions are in a trig-\nonal environment [25, 28, 36]. These compounds indi-\ncate a large magnetic anisotropy with the ab-plane being\nthe magnetic easy plane, which can be qualitatively ex-\nplainedbytheCEFeffect. AdetailedCEFanalysisbased\non both the magnetic susceptibility and inelastic neutron\nscattering data of hexagonalCeAuSn indicates a mixture\nof|±5/2/angbracketrightand|∓1/2/angbracketrightCEF ground state doublet, a pure\n|±3/2/angbracketrightdoublet as the first excited state at ∼345 K, and\na mixture of |±5/2/angbracketrightand|∓1/2/angbracketrightdoublet as the second\nexcited state at ∼440 K [28, 36]. The anisotropy ratio\nof magnetic susceptibility between H/bardblabandH/bardblcis\nfound tobe χab/χc∼15nearTN. Bothmagneticsuscep-\ntibility [28] and neutron scattering [36] CEF evaluations\nclearly show a significant B3\n4contribution ( ∼19 K) with\na large mixing angle, which is consistent with the CEF\nanalysis of CeCd 3X3. The anisotropy ratio of CeCd 3X3\n(χab/χc∼36 at 1.8 K) is larger than that of CeAuSn\nbecause both B0\n2andB3\n4CEF parameters of CeCd 3X3\nare larger than those of CeAuSn. The CEF investiga-\ntion of the rhombohedral CeIr 3Ge7compound shows a\nvery similar CEF eigenstates and eigenvalues with those\nof other compounds. However, in CeIr 3Ge7, the reported\nCEF parameters (especially the term B3\n4= 67.3 K) are\nlarger than that of other compounds, thus inducing a6\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48/s51/s53/s48\n/s67/s101/s67/s100\n/s51/s65/s115\n/s51/s76/s97/s67/s100\n/s51/s65/s115\n/s51/s76/s97/s67/s100\n/s51/s80\n/s51\n/s67/s101/s67/s100\n/s51/s80\n/s51/s83 /s32/s40 /s86/s47/s75/s41\n/s84 /s32/s40/s75/s41\nFIG.5. Temperaturedependenceofthethermoelectric power ,\nS(T), ofRCd3X3(R= La and Ce, X= P and As). Vertical\narrows indicate the phase transition temperatures observe d in\nelectrical resistivity measurements [19, 22].\nhuge energy level splittings of 374 K and 1398 K. [25].\nIt has been suggested that the exceptionally large CEF\nsplitting can be related to the contribution of the 5 dlig-\nands of Ir atoms. [25]. In addition, the CEF analysis\non rhombohedral CePtAl 4Ge2antiferromagnet has also\nbeen conducted [29]. Unlike the above mentioned com-\npounds, the ground state and the second excited state of\nCePtAl 4Ge2arenotinamixedconfigurationof |5/2/angbracketrightand\n|1/2/angbracketrightstates. The sign and magnitude of B0\n2(= 13.26 K)\nandB0\n4(=−0.3 K) CEF parameters in CePtAl 4Ge2are\ncomparable to that of CeAuSn, CeIr 3Ge7, and CeCd 3X3\n(X= P and As). However, because the B3\n4term respon-\nsible for mixing is exceptionally small in CePtAl 4Ge2\nsystem ( B3\n4= 0±0.02 K), the ground state and sec-\nond excited state is in a pure | ±1/2/angbracketrightstate and a pure\n|∓5/2/angbracketrightstate, respectively. Wefoundthatthe groundand\nexcited states are without mixing for any |B3\n4|<1.1 K.\nThe small value of B0\n2andB3\n4implies a relatively smaller\nmagnetic anisotropy in CePtAl 4Ge2, which is clearly re-\nflected on its magnetic susceptibility data ( χab/χc∼4\nnearTN) [29].\nAlthough our CEF analysis on CeCd 3X3(X= P and\nAs) provides a comprehensive picture at high tempera-\ntures, a number of unanswered questions remains at low\ntemperatures. When the temperatureis much lowerthan\nthe CEF splitting, the lowest Kramers doublet is only\nrelevant to explain the observed magnetic ordering at\nTN= 0.42 K and upturn in Cmbelow 10 K. It is obvious\nthat the temperature dependence and absolute value of\nCmbelow 10 K cannot be explained by the CEF effects\nas shown in Figs. 3(c) and 4(c).\nSince magnetization isotherms at T= 1.8 K for both\ncompounds are reproduced well by CEF calculation with\nthe ground state wave functions, the reduced magnetiza-\ntion value cannot be associated with the Kondo screen-ing. This is consistent with the electrical resistivity re-\nsults of CeCd 3X3[19, 22]. The absence of Kondo effect\nin CeCd 3X3is also confirmed from thermoelectric power\n(TEP) measurements, as shown in Fig. 5. The observed\nTEP value of LaCd 3X3is an order of magnitude higher\nthan that of typical metals, implying low carrier density\nsystem. The temperature dependence of the TEP, S(T),\nof both La- and Ce-based compounds shows a hump at\nlow temperatures, which can be related to the phonon\ndrag. In general, S(T) of typical metals shows a max-\nimum, corresponding to the phonon-drag effect, where\nthe maximum is expected to be located between Θ D/5\nand Θ D/12 [37]. Many Ce- and Yb-based Kondo lat-\ntice systems have shown that S(T) indicates an extrema\nwith enhanced value, corresponding to the Kondo effect\nin conjunction with CEF effect. The TEP of CeCd 3X3\nexhibits behavior similar to that of LaCd 3X3, implying\nnegligible Kondo contributions. Hence, as suggested in\nRefs. [19, 22], the enhancement of the specific heat below\n10 K is likely related to either the magnetic frustration\nin triangular lattices [18–22] or simply the magnetic fluc-\ntuations observed in insulating antiferromagnets [38].\nThe electrical resistivity of CeCd 3X3compounds ex-\nhibits a metallic behavior, suggesting that Ruderman-\nKittel-Kasuya-Yosida (RKKY) interaction may be re-\nsponsible for the magnetic ordering. However, it would\nhave to be mediated by an extremely small number of\ncharge carriers [19, 22]. It has been qualitatively shown\nthat the magnetic ordering temperature of non-Kondo\nmaterials scale with the distance between Ce ions, where\nthe larger the Ce-Ce distance results in a smaller or-\ndering temperature [39]. For example, a non-Kondo\nmetal CeIr 3Ge7orders at an extremely low temperature\nTN= 0.63 K due to the large Ce-Ce distance ( ∼6˚A).\nWhen the Ce-Ce distance ( ∼4˚A) for CeCd 3X3is con-\nsidered, the magnetic ordering temperature is signifi-\ncantly suppressed compared to that of other Ce-based\nnon-Kondo systems. On the contrary, it has been sug-\ngested that the superexchange interaction in low car-\nrier density YbAl 3C3compound becomes dominant in-\nstead of the RKKY interaction, where the carrier con-\ncentration is estimated to be n∼0.01 per formula unit\n(f.u.) [14]. When the carrier concentrations of CeCd 3As3\n(n∼0.003/f.u.) [19] and CeCd 3P3(n∼0.002/f.u.) [22]\ncompounds are considered, it is reasonable to assume\nthat the superexchange interaction may be responsible\nfor the antiferromagnetic ordering below 0.42 K. In ad-\ndition, the partial H−Tphase diagram of these com-\npounds, especially the field-induced increase of TN, is\nsimilar to that of 2D insulating triangular lattice sys-\ntems with easy-plane anisotropy [40, 41]. Furthermore,\nthe low temperature specific heat of CeCd 3As3has been\nexplained by anisotropicexchangeHamiltonian for an in-\nsulating, layered triangular lattice [20].7\nIV. CONCLUSION\nAt high temperatures, the observed magnetic proper-\ntiesofCeCd 3X3triangularlatticecompoundscanbewell\nunderstood by considering the CEF effects with molecu-\nlar filed contributions. The large anisotropy in the mag-\nnetic susceptibility and magnetization and the electronic\nSchottky anomaly in the specific heat are explained by\nenergy level splittings of the J= 5/2 Hund’s rule ground\nstate of Ce3+ions into three doublets. The striking sim-\nilarity of the CEF profile of both compounds implies a\nvery close resemblance of their crystal field environment.\nThree independent CEF analyses on CeCd 3As3indicate\ninconsistent CEF profiles, requiring further studies such\nas inelastic neutron scattering. When the temperature is\nwell below the CEF splitting, the well-isolated Kramers’\ndoublet ground state is responsible for the antiferromag-netic ordering and the large enhancement of specific heat\nbelow 10 K. Further measurements such as magnetiza-\ntion, neutron scattering, and nuclear magnetic resonance\nare necessary to provide additional insight into the na-\nture of magnetism below TNand the role of anisotropic\nexchange interactions in the triangular motif.\nACKNOWLEDGMENTS\nThis work was supported by the Canada Research\nChairs, Natural Sciences and Engineering Research\nCouncil of Canada, and Canada Foundation for Inno-\nvation program. EM was supported by the Korean Min-\nistry of Science and ICT (No. 2021R1A2C2010925) and\nby BrainLink program funded by the Ministry of Science\nand ICT (2022H1D3A3A01077468)through the National\nResearch Foundation of Korea.\n[1] W.-J. Hu, S.-S. Gong, W. Zhu, and D. N. Sheng, Phys.\nRev. B92, 140403(R) (2015).\n[2] Y.-D. Li, X. Wang, and G. Chen, Phys. Rev. B 94,\n035107 (2016).\n[3] Y. Iqbal, W.-J. Hu, R. Thomale, D. Poilblanc, and F.\nBecca, Phys. Rev. B 93, 144411 (2016).\n[4] S.-S. Gong, W. Zhu, J.-X. Zhu, D. N. Sheng, and K.\nYang, Phys. Rev. B 96, 075116 (2017).\n[5] Z. Zhu, P. A. Maksimov, S. R. White, and A. L. Cherny-\nshev, Phys. Rev. Lett. 120, 207203 (2018).\n[6] J. G. Rau and M. J. P. Gingras, Phys. Rev. B 98, 054408\n(2018).\n[7] Y. Li, H. Liao, Z. Zhang, S. Li, F. Jin, L. Ling, L. Zhang,\nY. Zou, L. Pi, Z. Yang, J. Wang, Z. Wu, and Q. Zhang,\nSci. Rep. 5, 16419 (2015).\n[8] Y.Li, G. Chen, W.Tong, L.Pi, J. Liu, Z.Yang, X.Wang,\nand Q. Zhang, Phys. Rev. Lett. 115, 167203 (2015).\n[9] Y. Shen, Y.-D. Li, H. Wo, Y. Li, S. Shen, B. Pan, Q.\nWang, H. Walker, P. Steffens, M. Boehm, Y. Hao, D.\nL. Quintero-Castro, L. W. Harriger, M. D. Frontzek, L.\nHao, S. Meng, Q. Zhang, G. Chen, and J. Zhao, Nature\n540, 559 (2016).\n[10] Y. Li, D. Adroja, P. K. Biswas, P. J. Baker, Q. Zhang,\nJ. Liu, A. A. Tsirlin, P. Gegenwart, and Q. Zhang, Phys.\nRev. Lett. 117, 097201 (2016).\n[11] J. A. Paddison, M. Daum, Z. Dun, G. Ehlers, Y. Liu, M.\nB. Stone, H. Zhou, and M. Mourigal, Nat. Phys. 13, 117\n(2017).\n[12] M. Baenitz, P. Schlender, J. Sichelschmidt, Y. A. Onyki -\nienko, Z. Zangeneh, K. M. Ranjith, R. Sarkar, L. Hozoi,\nH. C. Walker, J.-C. Orain, H. Yasuoka, J. van den Brink,\nH. H. Klauss, D. S. Inosov, and T. Doert, Phys. Rev. B\n98, 220409(R) (2018).\n[13] J. Sichelschmidt, P. Schlender, B. Schmidt, M. Baenitz ,\nand T. Doert, J. Phys.: Condens. Matter 31, 205601\n(2019).\n[14] A. Ochiai, T. Inukai, T. Matsumura, A. Oyamada, and\nK. Katoh, J. Phys. Soc. Jpn. 76, 123703 (2007).[15] Y. Kato, M. Kosaka, H. Nowatari, Y. Saiga, A. Yamada,\nT. Kobiyama, S. Katano, K. Ohoyama, H. S. Suzuki, N.\nAso, and K. Iwasa, J. Phys. Soc. Jpn. 77, 053701 (2008).\n[16] A. Ochiai, K. Hara, F. Kikuchi, T. Inukai, E. Matsuoka,\nH. Onodera, S. Nakamura, T. Nojima, and K. Katoh, J.\nPhys.: Conf. Ser. 200, 022040 (2010).\n[17] K. Hara, S. Matsuda, E. Matsuoka, K. Tanigaki, A.\nOchiai, S. Nakamura, T. Nojima, and K. Katoh, Phys.\nRev. B85, 144416 (2012).\n[18] Y. Liu, S. Zhang, J. Lv, S. Su, T. Dong, G. Chen, and\nN. Wang, arXiv preprint arXiv:1612.03720 (2016).\n[19] S. R.Dunsiger, J. Lee, J. E. Sonier, andE. D.Mun, Phys.\nRev. B102, 064405 (2020).\n[20] K. E. Avers, P. A. Maksimov, P. F. S. Rosa, S.\nM. Thomas, J. D. Thompson, W. P. Halperin, R.\nMovshovich, and A. L. Chernyshev, Phys. Rev. B 103,\nL180406 (2021).\n[21] S. Higuchi, Y. Noshima, N. Shirakawa, M. Tsubota, and\nJ. Kitagawa, Mater. Res. Express 3, 056101 (2016).\n[22] J. Lee, A. Rabus, N. R. Lee-Hone, D. M. Broun, and E.\nMun, Phys. Rev. B 99, 245159 (2019).\n[23] S. S. Stoyko and A. Mar, Inorg. Chem. 50, 11152 (2011).\n[24] A. Ochiai, N. Kabeya, K. Maniwa, M. Saito, S. Naka-\nmura, and K. Katoh, Phys. Rev. B 104, 144420 (2021).\n[25] J. Banda, B. K. Rai, H. Rosner, E. Morosan, C. Geibel,\nand M. Brando, Phys. Rev. B 98, 195120 (2018).\n[26] A. Scheie, J. Appl. Crystallogr. 54, 356 (2021).\n[27] P. C. Canfield and Z. Fisk, Philos. Mag. B 65, 1117\n(1992).\n[28] C. L. Huang, V. Fritsch, B. Pilawa, C. C. Yang, M. Merz,\nand H. v. L¨ ohneysen, Phys. Rev. B 91, 144413 (2015).\n[29] S. Shin, V. Pomjakushin, L. Keller, P. F. S. Rosa, U.\nStuhr, C. Niedermayer, R. Sibille, S. Toth, J. Kim, H.\nJang, S.-K. Son, H.-O. Lee, T. Shang, M. Medarde, E.\nD. Bauer, M. Kenzelmann, and T. Park, Phys. Rev. B\n101, 224421 (2020).\n[30] K. W. H. Stevens, Proc. Phys. Soc. A 65, 209 (1952).\n[31] M. T. Hutchings, Solid State Phys. 16, 227 (1964).8\n[32] C. Klingner, C. Krellner, M. Brando, C. Geibel, and F.\nSteglich1, New J. Phys. 13083024 (2011).\n[33] Y.-L. Wang, Physics Letters A 35, 383 (1971).\n[34] G. J. Bowden, D. S. P. Bunbury, and M. A. H. McCaus-\nland, J. Phys. C: Solid State Phys. 41840 (1971).\n[35] N. Shohata, J. Phys. Soc. Jpn. 42, 1873 (1977).\n[36] D. T. Adroja, B. D. Rainford, and A. J. Neville, J. of\nPhys.: Condens. Matter 9, L391 (1997).\n[37] F. J. Blatt, P. A. Schroeder, C. L. Foiles, and D. Greig,\nSpringer US, Boston, MA (1976).\n[38] W. K. Robinson and S. A. Friedberg, Phys. Rev. 117,\n402 (1960).[39] Binod K. Rai, Jacintha Banda, Macy Stavinoha, R.\nBorth, D.-J. Jang, Katherine A. Benavides, D. A.\nSokolov, Julia Y. Chan, M. Nicklas, Manuel Brando, C.-\nL. Huang, and E. Morosan, Phys. Rev. B 98, 195119\n(2018).\n[40] L. Seabra, T. Momoi, P. Sindzingre, and N. Shannon,\nPhys. Rev. B 84, 214418 (2011).\n[41] D. H. Lee, J. D. Joannopoulos, J. W. Negele, and D. P.\nLandau, Phys. Rev. B 33, 450 (1986)."    },    {        "title": "2302.13810v1.Large_ordered_moment_with_strong_easy_plane_anisotropy_and_vortex_domain_pattern_in_the_kagome_ferromagnet_Fe__3_Sn.pdf",        "content": "Large ordered moment with strong easy-plane anisotropy and vortex-domain pattern\nin the kagome ferromagnet Fe 3Sn\nLilian Prodan1;\u0003, Donald M. Evans1, Sin\u0013 ead M. Gri\u000en2;3, Andreas Ostlin4, Markus Altthaler1, Erik Lysne5,\nIrina G. Filippova6, Serghei Shova7, Liviu Chioncel4, Vladimir Tsurkan1;6, and Istv\u0013 an K\u0013 ezsm\u0013 arki1\n1Experimentalphysik V, Center for Electronic Correlations and Magnetism,\nInstitute for Physics, Augsburg University, D-86135 Augsburg, Germany\n2Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA\n3Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, CA 94720\n4Theoretische Physik III, Center for Electronic Correlations and Magnetism,\nInstitute for Physics, Augsburg University, D-86135 Augsburg, Germany\nAugsburg Center for Innovative Technologies (ACIT),D-86135 Augsburg, Germany\n5Department of Materials Science and Engineering,\nNorwegian University of Science and Technology (NTNU), 7043 Trondheim, Norway\n6Institute of Applied Physics, MD 2028, Chis ,in\u0015 au, R. Moldova and\n7Department of Inorganic Polymers, \\Petru Poni\" Institute of\nMacromolecular Chemistry, Romanian Academy, 700487 Iasi, Romania\n(Dated: February 28, 2023)\nWe report the structural and magnetic properties of high-quality bulk single crystals of the kagome\nferromagnet Fe 3Sn. The dependence of magnetisation on the magnitude and orientation of the ex-\nternal \feld reveals strong easy-plane type uniaxial magnetic anisotropy, which shows a monotonous\nincrease from K1=\u00000:99\u0002106J=m3at 300 K to \u00001:23\u0002106J=m3at 2 K. Our ab initio elec-\ntronic structure calculations yield the value of total magnetic moment of about 6.9 \u0016B/f.u. and\na magnetocrystalline anisotropy energy density of 0.406 meV/f.u. (1 :16\u0002106J=m3) both being\nin good agreement with the experimental values. The self-consistent DFT computations for the\ncomponents of the spin/orbital moments indicate that the small di\u000berence between the saturation\nmagnetisations measured along and perpendicular to the kagome layers results from the subtle bal-\nance between the Fe and Sn spin/orbital moments on the di\u000berent sites. In zero \feld, magnetic\nforce microscopy reveals micrometer-scale magnetic vortices with weakly pinned cores that vanish\nat\u00183 T applied perpendicular to the kagome plane. Our micromagnetic simulations, using the ex-\nperimentally determined value of anisotropy, well reproduce the observed vortex-domain structure.\nThe present study, in comparison with the easy-axis ferromagnet Fe 3Sn2, shows that varying the\nstacking of kagome layers provides an e\u000ecient control over magnetic anisotropy in this family of\nFe-based kagome magnets.\nNotes. Version 2. This manuscript is under review in Physical Review B.\nI. INTRODUCTION\nMagnetic compounds with kagome-lattice arrangement\nof spins have recently attracted much attention due to\ntheir unusual magnetic and electronic properties related\nto the speci\fc topology of their electronic band struc-\ntures [1{5]. Recent theoretical and experimental stud-\nies have demonstrated that the existence of \rat bands,\nnodal points and nodal lines appearing close to the Fermi\nenergy signi\fcantly a\u000bect magnetic, magneto-transport\nand magneto-optical properties of kagome materials [6].\nIn particular, fascinating physical phenomena like giant\nanomalous and topological Hall e\u000bects, giant Nernst ef-\nfect, topological superconductivity, magnetic spin chiral-\nity and skyrmion bubbles have been reported for large\ngroup of kagome materials like van der Waals AV3Sb5\n(A= K, Cs, Rb) [7, 8], rare earth based ReT 6Sn6(Re\n= Gd, Tm, Tb, Y; T= Mn,V) [9, 10], magnetic Weyl-\nsemimetal Co 3Sn2S2[11, 12], and binary metals TxSny\n(T= Fe, Mn, Co; x:y= 1:1, 3:2, 3:1) [13{18]. The co-\nexistence of these e\u000bects provide an exceptional platform\nto study electronic band topology and its interplay withmagnetic spin and orbital e\u000bects.\nHere we focus on the magnetic properties of Fe 3Sn\nquasi-2D kagome magnet, investigated by anisotropy\nmeasurements, ab initio calculations and imaging of the\nmagnetic domain pattern. Fe 3Sn is an itinerant ferro-\nmagnet, a member of a family of iron stannides with\nthe general formula Fe xSny. Our particular interest\nis on compounds with the kagome-lattice arrangement,\nnamely compounds with x:y= 1:1; 3:1; 3:2. Depend-\ning on the ratio of x:y, i.e. the stacking of kagome\nlayers, these compounds realize various hexagonal mag-\nnetic space groups and di\u000berent magnetic ground state.\nFor example, FeSn crystallises in the P6=mmm sym-\nmetry and below 365 K it is an easy-plane antiferro-\nmagnet with ferromagnetic arrangement of spins within\neach kagome layer [19{21]. Fe 3Sn has a structure of\nP63=mmc symmetry and exhibits ferromagnetic order\nbelow 743 K [19, 22]. The crystal structure of Fe 3Sn\nis shown in Figs. 1(a) and (b). The ablayer consists\nof a breathing kagome lattice of two sizes of equilateral\nFe triangles with Sn atoms in the center of the kagome.\nThe unit cell of Fe 3Sn contains two adjacent, laterallyarXiv:2302.13810v1  [cond-mat.mtrl-sci]  6 Feb 20232\ndisplaced kagome layers, separated by half of the lat-\ntice constant along the caxis. The structure of Fe 3Sn\nis much simpler than of Fe 3Sn2, which realizes a third\ntype of stacking and crystallizes in the R3msymmetry.\nNamely, Fe 3Sn2contains two such basic blocks of Fe 3Sn\nbilayers stacked along the caxis and separated by a hon-\neycomb layer of Sn atoms [19, 22]. In contrast to Fe 3Sn,\nFe3Sn2is an easy-axis ferromagnet with a Curie tempera-\nture of 612 K [23], which shows skyrmion bubbles at room\ntemperature [18, 24]. However, this compound shows\na spin reorientation below room temperature, when the\neasy axis of magnetisation gets tilted towards the kagome\nplane [22, 25{27].\nPhysical properties of Fe 3Sn are investigated predomi-\nmantly on polycrystalline samples synthesized by solid\nstate reaction or by arc-melting of metalic Fe and Sn in\nargon atmosphere. Early M ossbauer studies of Trumpy\net al. [23] and Dj\u0013 ega-Mariadassou et al. [28] revealed\na ferromagnetic behavior of Fe 3Sn and concluded that\nthe spin moment is oriented along the caxis. In con-\ntrast, recent analysis of magnetic properties of Fe 3Sn by\nSales et al. [29], performed on \feld-oriented polycrys-\ntalline powder, implied predominantly in-plane orienta-\ntion of the magnetisation with the magnitude of easy-\nplane anisotropy K1=\u00001:8MJ=m3at 300 K. The easy-\nplane character of the anisotropy was further supported\nby magnetisation studies of Fayyazi et al. [30], that was\nperformed on an oriented plaque of very small crystals\nwith a mass of 10 \u0016g, obtained by reactive \rux technique.\nHowever, this work reported an unusual temperature de-\npendence of the anisotropy \feld, namely its decrease from\n200 K to 10 K.\nThe precise quanti\fcation of magnetic anisotropy is\nthe prerequisite to understand the diversity of complex\nspin textures, for which kagome magnets o\u000ber a fertile\nground [31]. Moreover, the magnetic anisotropy in in-\nterplay with the Dzyaloshinskii-Moriya interaction, com-\npeting and frustrated exchanges, or dipole-dipole inter-\nactions, can a\u000bect the properties of mesoscale spin tex-\ntures, including conventional domain walls, vortices, and\nskyrmions [32{37]. As a prime example, the competition\nbetween easy-axis magnetic anisotropy and dipole-dipole\ninteraction can stabilize magnetic bubbles and skyrmions\nin \fnite magnetic \felds [18, 24, 33, 38]. On the other\nhand, the in-plane magnetized systems, e.g., magnets\nwith easy-plane anisotropy, can develop magnetic vor-\ntices and anti-vortices [39{43].\nAlthough Fe xSnycompounds are known for long, the\nmicroscopic origin and the precise value of exchange in-\nteractions and magnetic anisotropy constants is still un-\nsettled in these compounds, even the spin arrangement\nis under debate in some of these materials. From the ex-\nperimental point of view, the situation is complicated by\nlarge variation of sample quality, leading to inconsisten-\ncies of the data available in literature. A large piece of\nthe data was obtained on polycrystalline samples, while\nthose reported for single crystals show large scattering\nevidently related to di\u000berent quality of samples, e.g., ho-mogeneity, deviation from the stoichiometry and impu-\nrity content. Therefore, precisely quatifying the magnetic\ninteractions in a single bilayer Fe 3Sn is a prerequisite\nfor understanding magnetism on the microscopic level in\nthese kagome magnets.\nHere we report detailed magnetometry studies per-\nformed on large stoichiometric single crystals grown by\nthe chemical transport reactions. The appropriate size\nof the crystals (several mg) allowed us to accurately\ndetermine their orientation prior to the magnetisation\nmeasurements along the di\u000berent crystallographic direc-\ntions. Moreover, our angular-dependent magnetisation\nmeasurements, performed in di\u000berent \felds and at vari-\nous temperatures, allowed a highly reliable quanti\fcation\nof the anisotropy constants of Fe 3Sn. The experimental\nstudies were complemented by ab initio electronic struc-\nture calculations to determine the magnetic properties\nin the ground state (magnetic moments on Fe and Sn as\nwell as anisotropy energies).\nThe spin/orbital moments were restricted to speci\fc\norientations and by including the spin-orbit coupling, cal-\nculations revealed easy-plane magnetic anisotropy. The\ncomputed uniaxial magnetocrystalline anisotropy energy,\nwhich arises from the collective e\u000bect of the crystal struc-\nture and spin-orbit coupling, is in excellent agreement\nwith the experimentally determined value. Furthermore,\na slight departure from the collinear magnetic con\fgu-\nration is predicted due to tilting of the orbital-magnetic\nmoment of Fe with respect to its spin moment. Finally,\nour MFM study reveals the formation of a complex do-\nmain pattern built of in-plane \rux enclosure domains, i.e.\nmagnetic vortices, on the micrometer scale. This vortex-\ndomain pattern is well reproduced by our micromagnetic\nsimulations using the magnetic interaction parameters\ndetermined experimentally and theoretically, further sup-\nporting the high reliability of these parameters.\nII. METHODS\nCrystal growth. Polycrystalline Fe 3Sn was prepared\nby solid state reactions of high purity elements of Fe\n(99.99%) and Sn (99.995%). A stoichiometric amount of\nFe and Sn has been mixed and sealed in quartz ampule\nevacuated to 10\u00003mbar. The mixture was annealed for\nfour days at 780\u000eC followed by quenching in ice water.\nIn order to achieve the full reaction and homogeneity two\nsintering cycles were performed. The phase purity after\neach sintering was checked by x-ray powder di\u000braction.\nSingle crystals of Fe 3Sn have been grown by the chemi-\ncal transport reactions method using the polycrystalline\npowder as starting material and I 2as the transport agent.\nThe growth was performed in exothermic conditions in a\ntwo-zone horizontal furnace with the temperature gradi-\nent of 50\u000eC in a temperature range of 850-800\u000eC. Crys-\ntals up to 1.5 mm size have been obtained after 6 weeks\nof transport. The chemical composition of single crystals\nwas analyzed by the ZEISS Crossbeam 550, using energy3\ndispersive x-ray spectroscopy (EDS) technique.\nSingle-crystal x-ray di\u000braction. The single-crystal\nx-ray di\u000braction measurement was carried out with a\nRigaku Oxford-Di\u000braction XCALIBUR E CCD di\u000brac-\ntometer equipped with graphite-monochromated MoK \u000b\nradiation. The data were corrected for the Lorentz and\npolarization e\u000bects and for the absorption by multi-scan\nempirical absorption correction methods. The structure\nwas re\fned by the full matrix least-squares method based\nonF2with anisotropic displacement parameters. The\nunit cell determination and data integration were car-\nried out using the CrysAlis package of Oxford Di\u000braction\n[44]. All calculations were carried out by the programs\nSHELXL2014 [45] and the WinGX software [46].\nMagnetic properties. Magnetic properties were\nmeasured by a SQUID magnetometer (MPMS 3, Quan-\ntum Design) in the temperature range of 1.8{900 K and\nmagnetic \felds up to 7 T. The angular-dependent magne-\ntization measurements have been performed both within\nabandacplanes on cylindrical sample with the acplane\nas the basal plane of the cylinder. The measurements in\n\felds up to 14 T were performed with a vibrating-sample\nmagnetometer using a physical properties measurement\nsystem (PPMS, Quantum Design).\nComputational Details. Density Functional Theory\n(DFT) calculations were performed with the Vienna Ab\ninitio Simulation Package (vasp) [47{49] using projec-\ntor augmented wave (PAW) pseudo-potentials [50, 51].\nWe treated the Fe(3d, 4s) and Sn(5s, 5p) electrons as\nvalence. Plane waves were expanded to an energy cut-\no\u000b of 800 eV, and a Monkhorst-Pack Gamma-centered k-\npoint grid of 10\u000210\u000210 for structural optimizations and\n20\u000220\u000220 for accurate total energy calculations to ob-\ntain the magnetic energetics. A smearing value of 0.02 eV\nwas used for all calculations. The PBEsol functional was\nused throughout [52]. Spin-orbit coupling was included\nself-consistently as implemented in VASP. The electronic\nconvergence criteria set to 10\u00008eV and the force conver-\ngence criteria set to 0.002 eV/ \u0017A. Our reference structure\nwas taken from the Materials Project (mp-20883) [53{\n56].\nELK [57] computation has been performed for the a\nequally large k-mesh (21\u000221\u000221), and the PBE-GGA\nfunctional was used with the experimental lattice param-\neters given in Table I. For the FPLAPW calculations the\nplane-wave cuto\u000b Kmaxwas set so that RmtKmax= 7:0,\nwhereRmtis the smallest mu\u000en-tin radius.\nMFM imaging. The magnetic domain pattern was\nimaged using a low-temperature attocube attoAFM I,\nin MFM mode, equipped with a superconducting mag-\nnet. The magnetic texture on an as-grown absurface was\nrecorded via a phase-sensitive feedback loop that records\nchanges in the resonance frequency, which are propor-\ntional to the gradient of the stray magnetic \feld of the\nsample along the magnetisation of the MFM tip, that\nis normal to the surface. The magnetic tips used were\nPPP-MFMR probes from NanoSensor with the magnetic\nmoment of 5 :09\u000210\u000013A=m2. All data was collectedat 100 K for experimental reasons, such as the thermal\nstability of the equipment.\nMicromagnetic simulations. Simulations were\ncarried out using MuMax3 v3.10 [58]. Here, the mi-\ncrostructure is found by minimizing a total energy made\nup of terms representing, Heisenberg exchange, \frst-\norder uniaxial anisotropy, Zeeman and demagnetisation\nenergy. Practically this is done by solving the equation\n\"=Z\nVS[Aex(\u0001m)2\u0000Kum2\nz+MsBextm\u00001\n2MsBdemm]dr\n(1)\nWhere the reduced magnetisation, m(x,y,z), is the\nmagnetisation M(x,y,z) divided by the saturation mag-\nnetization, Ms, within a sample volume VS, where\nthe magnetization makes up a continuous vector \feld.\nThe exchange sti\u000bness constant, the \frst order uniaxial\nanisotropy constant, the external magnetic \feld and the\ndemagnetisation \feld are represented by Aex,Ku,Bext,\nBdem, respectively. Periodic boundary conditions were\nnot necessary; the simulation was initiated from a ran-\ndom state, and easy-plane anisotropy (0, 0, 1) was used.\nThe mesh was 4\u00024\u00024 nm cells, and the geometry was\n2048\u00021024\u0002256 nm. Note, that MuMax3 is able to\ncalculate the MFM response as it knows the orientation\nof the magnetisation, and therefore the stray \felds, and\nthe MFM response comes from the interactions between\nthe MFM tip and the gradient of the stray \felds.\nIII. RESULTS AND DISCUSSIONS\nA. Structure, magnetic properties and anisotropy\nTypical as-grown Fe 3Sn crystals used in our study are\nshown in the inset of Fig. 2 (a). The energy-dispersive x-\nray analysis found an excess of Fe .1 %, indicating that\nthe stoichiometry Fe 3Sn is close to ideal. Single-crystal\nx-ray di\u000braction study revealed the hexagonal P63=mmc\n(#194) space group symmetry and showed no traces of\nimpurity phases. The calculated cell parameters aandc\nare close to those reported for polycrystalline Fe 3Sn [19,\n29]. The structural parameters obtained from the single-\ncrystal re\fnement are summarised in Table [ ?].\nThe magnetisation measurements were carried out on\na single crystal with a size of 0 :7\u00020:6\u00020:5mm3.\nThe temperature-dependent magnetisation in 1 T applied\nalong theaaxis exhibits a steep increase below 750\nK indicating the onset of ferromagnetic order, as shown\nin Fig. 2 (a). The Curie temperature of TC= 705 K was\nobtained as the location of the maximum in the temper-\nature derivative of the magnetisation, and is close to pre-\nviously reported data for polycrystalline samples [19, 22].\nFigure 2(b) shows the \feld-dependent magnetisation\ncurves,M(H), measured at 2 K for magnetic \felds ap-\nplied along three orthogonal directions. Two directions\nare in the basal abplane, [ 1210] and [10 10] being paral-\nlel and perpendicular to the aaxis, respectively, and the4\n030609012015018000.10.20.30.4In-plane (xy)O\nut-of-plane (xz)0\n30609012015018000.010.020.030.040.050.06l\nxl\nz\n(a)( b)(\nc)ba(\nd)(\ne)\ncb\na0\n3 0- 306 0- 609 0- 90/s102\n (degree)Energy (meV/f.u.)O rbital moment (µB)(f)\nFIG. 1. (a) Schematic representation of the crystal struc-\nture of layered Fe 3Sn. Red and green spheres represent Fe\nand Sn ions, respectively. Transparent and bold atoms are\nfrom di\u000berent planes. (b) Breathing kagome lattices of equi-\nlateral Fe triangles of two distinct sizes (marked by blue\nand red lines) in the abplane, as seen when viewed along\nthecaxis. (c)/(d) Magnetic structure with moment paral-\nlel/perpendicular to the caxis. Green and orange arrows\nshow Fe and Sn spin moments, respectively. (e) Calculated\nmagnetocrystalline anisotropy energy for rotation of Fe spin\nwithin the abplane (purple) and the acplane (green), as-\nsuming a ferromagnetic ordering. In the former no angular\ndependence is observed, while in the latter case the energy\nhas a maximum for spins pointing along the caxis (\u001e= 0).\n(f) Calculated orbital moment magnitude of Fe for in-plane\n(lx) and out-of-plane ( lz) projections as a function of Fe spin\ntilt angle.\nhexagonalcaxis (equivalently [0001]) is chosen as the\nthird direction. The magnetisation within the abplane\nreaches the saturation already at \u00181 T. We found that\nthe saturation is reached in a slightly higher \feld for\nHjj[1010] than for Hjj[1210], indicating that the aaxis\nis the easy axis of the magnetisation. For \felds along\nthecaxis, the saturation takes place at higher \felds,\nabove 3 T. The uniaxial magnetocrystalline anisotropy\nwithin the acplane was calculated from the hysteresisTABLE I. Structural re\fnement details and crystal data for\nFe3Sn determined through single-crystal x-ray di\u000braction.\nRe\fned empirical formula Fe 3Sn\nSpace group P63=mmc (No. 194)\na(\u0017A) 5.4604(3)\nc(\u0017A) 4.3458(3)\nVolume ( \u0017A3) 112.215(15)\nZ, D calc(g/cm3) 2, 8.472\n\u0016(mm\u00001) 29.551\n\u0002 range (\u000e) 4.310 - 29.014\nRe\rections collected/unique 577/68 [R int= 0.0467)]\nNo. variables 8\nGoodness-of-\ft on F21.001\nExtinction coe\u000ecient 0.174(9)\nR1, wR 2(all data) 0.0143, 0.0352\nAtom Position x y z Ueq\nFe 6h 0.1544(1) 0.3088(2) 0.2500 7(1)\nSn 2d 0.6667 0.3333 0.2500 5(1)\ncurves using an \\area method\" [59], considering the dif-\nference of the integralsRMs\n0HidMfor in-plane ( aaxis)\nand out-of-plane ( caxis) \feld orientations. Here Hiis\nthe internal \feld determined as the di\u000berence between\nthe applied magnetic \feld Hand the demagnetizing \feld\nDM, whereDis the demagnetisation coe\u000ecient. We\nobtained the value of Ku=\u00001:27\u0002106J=m3at 2 K\nfor the uniaxial anisotropy constant, which decreases to\nKu=-0.97\u0002106J=m3at room temperature. A similar ap-\nproach was used to calculate the sixth-order anisotropy\nin theabplane, that led to the value of 2 :3\u0002104J=m3\nat 2 K, which decreases to 1 :8\u0002104J=m3at 300 K. The\nanisotropy in the basal plane is nearly two orders of mag-\nnitude weaker than uniaxial anisotropy.\nThe inset in Fig.2(b) shows the temperature depen-\ndence of the magnetisation measured in 7 T. The value\nof the saturation moment is 2.27 \u0016B/Fe at 2 K for \feld\nin theabplane, which is\u00182 % larger than the saturation\nvalue along the caxis. Although this di\u000berence is rather\nsmall it cannot originate from demagnetisation e\u000bects,\nsince the sample has a circular shape in the acplane.\nAdditional measurements, not shown here, show this dif-\nference is present up to 14 T.\n[t]\n\u0003Ueqis de\fned as one third of the trace of the orthog-\nonalizedUijtensor.\nThe angular dependence of the magnetisation M(\u001e)\nwithin theac-plane, measured in di\u000berent \felds at 300 K,\nis shown in Figure 3(b). Here \u001eis the angle spanned by\nthe magnetic \feld and the caxis (see inset of Fig. 3(b)).\nAs a general trend, the modulation of M(\u001e) is suppressed\nwith increasing magnetic \feld. In addition, there is a\nchange in its functional form. While M(\u001e) exhibits a\nminimum and a maximum in low \felds parallel and per-\npendicular to the caxis, respectively, upon saturation a\nlocal maximum develops also for \felds applied along the c5\n02 4 6 02460\n3 006.06.604 008 000246M (µB/f.u.)µ\n0H (T)(a)(\nb)T\n = 2 KT (K)--- H || [1210] \nH || [1010] \nH || [0001]M (µB/f.u.)T\n (K)TC  µ0H = 1 T\nFIG. 2. (a) The temperature-dependent magnetisation of\nFe3Sn single crystal measured in 1 T applied along aaxis.\nThe inset shows as-grown Fe 3Sn single crystals on a mm-scale\npaper.(b) Magnetisation curves measured at 2 K in magnetic\n\felds applied in two directions in the abplane|parallel to\ntheaaxis or [ 1210] and perpendicular to the aaxis or [10 10],\nand with \felds along the caxis or [0001]. The inset shows\nthe temperature dependence of the magnetisation in 7 T for\nthese three directions.\naxis. To avoid complications due to the presence of mul-\ntiple magnetic domains, we determined the anisotropy\nbased onM(\u001e) curves measured in the fully \feld polar-\nized state, as shown in Fig. 3(b) for the 4 T curve. The\nangular dependence of the magnetisation was \ftted based\non commonly used phenomenological expression for the\nmagnetic anisotropy energy for a hexagonal ferromagnet\n[59]\nEA=K1sin2\u0012+K2sin4\u0012+K3sin6\u0012cos 6\t:(2)\nHere,K1andK2are the \frst- and second-order\nanisotropy constants, respectively. \u0012is the angle spanned\nby the magnetisation with the caxis, while \t denotes the\nangle between the projection of the magnetisation to the\nabplane and the aaxis. In our \ftting the K3term is ne-\n[a][ c]0\n9 01 802 70360\n4 59 01351806.26.31 T2 T4 TM|| (µB/f.u)0\n.5 T \nexp. \nfit 1 \nfit 2M|| (µB/f.u.)/s102\n (degree)(a)(\nb)FIG. 3. (a) Angular dependence of the magnetisation mea-\nsured in di\u000berent \felds at 300 K. The magnetisation compo-\nnent parallel to the \feld is detected. (b) Experimental data\n(open circles) and \ft to the experimental data (solid lines) in\n4 T at 300 K.\nglected, since our data show that the in-plane anisotropy\nis less than 2 % of the total magnetic anisotropy, as al-\nready discussed above.\nThe \ftting formula for calculation of the anisotropy\nconstantsK1andK2for angular-dependent magnetisa-\ntion measurements was derived by minimising the total\nenergy per unit volume described as E=EA+EZ+ED,\nwhereEZ=\u0000MsHcos (\u0012\u0000\u001e) corresponds to the Zee-\nman energy, and ED= 1=2(M2\njjD) is the magnetostatic\nenergy of a sample in its own \feld. The formula of \ftting\nof the experimental data is given by\nsin 2\u0012(K1+ 2K2sin2\u0012) =\u0000Hiq\nM2s\u0000M2\njj;(3)\nwhereMjjis the projection of Mson the applied \feld,\nwhich is measured in experiment (see inset of Fig. 3(b)).\nThe \ft to the angular-dependent magnetisation\n(marked as R method, \ft 1 in Fig. 3(b)) describes well\nthe angular dependence of M(\u001e) over the whole angu-\nlar range except for the immediate vicinity of the caxis,\nwhere it overshoots the experimental data. The exper-\nimentally observed di\u000berence in Msfor \felds parallel\nand perpendicular to the caxis indicates an anisotropy6\n-1.4-1.2-1.00.00.20\n1 002 003 00-1.4-1.2-1.0-0.8 K2 (R)   K2 (S-T) \nK1 (R)    K1 (S-T) \nK (MJ/m3) \n K1 + K2 (R)  \n K1 + K2 (S-T)  \n KU (Ar)K (MJ/m3)T\n (K)(a)(\nb)\nFIG. 4. (a) Temperature-dependent anisotropy constants K1\nandK2as determined from the angular dependence of the\nmagnetisation (circles, marked as R) and by the S-T method\n(triangles). (b) The uniaxial constant Kudetermined by the\n\\area method\" (Ar) is plotted together with K1+K2taken\nfrom panel (a). Lines are guides to the eye.\n(\u00182 %) of the g-tensor, i.e. imply peculiar orbital con-\ntributions to the magnetisation. (Please notice, if the\ng-tensor anisotropy is completely negligible, the mag-\nnetisation should point parallel to the applied \feld in\nthe high-\feld limit, hence \u0012in Eq. 2 and \u001ein Fig. 3(b)\nwould become identical.) From the \ft to the experimen-\ntal data we derived the value of the \frst anisotropy con-\nstantK1=\u00001:32\u0002106J=m3at 2 K, which is in good\nagreement with the K1=\u00001:34\u0002106J=m3determined\nby Sucksmith-Thompson (S-T) method (see below) [60].\nTo account for the angular dependence of the satu-\nration magnetisation anisotropy we used the approach\ndeveloped by J. Alameda etal: and A. S. Bolyachkin\netal: [61, 62]. This \ft (\ft 2 in Fig. 3(b)) improves\nthe description of the experimental data for angles close\nto thecaxis. Figure 4 (a) summarizes the tempera-\nture dependence of the anisotropy constants K1andK2,\nas determined by two independent methods: a) from\nthe \fts to the angular dependence of the magnetisa-\ntion measured in 4 T and b) calculated from the mag-\nnetisation curves along the caxis, following the mod-\ni\fed (S-T) method, which considers the anisotropy of\nthe saturation magnetisation [62]. At 300 K, both meth-\nods yield very close values K1=\u00000:99\u0002106J=m3and\n\u00001:01\u0002106J=m3, respectively. The magnitude of K1\nshows a monotonous increase with decreasing tempera-ture in contrast to non-monotonous behavior reported by\nFayyazietal: [30]. Moreover, the absolute values of K1\nreported in [30] are by about 20% lower than our ex-\nperimental values. These discrepancies likely arise due\nto limitations of proper sample orientation in the former\nstudy, which was not an issue in our case due to the larger\nsingle crystals. K2has opposite sign and is lower by ap-\nproximately one order magnitude than K1. At 300 K, we\nobtainedK2= 1:4\u0002105J=m3and 2:7\u0002104J=m3using\nthe R and S-T methods, respectively.\nThe reason of the di\u000berence in the values of K1andK2\nis unclear and may be related to di\u000berent data sets for \ft-\nting: high-\feld magnetisation data in the saturated state\nin R method and low-\feld data in S-T method. However,\nthe value of K2derived from the R method should be\ntaken with precaution because of larger error in determi-\nnation due to nonlinear extension of the rotator spring\nand change of its elasticity with lowering temperature.\nTo further validate our quanti\fcation of the uniaxial\nanisotropy in Fe 3Sn, in Fig. 4(b), we compare the sum\nofK1+K2, as determined using the angular dependent\nmagnetisation and the Sucksmith-Thompson methods,\nwith the overall Kuanisotropy, as obtained using the\n\\area method\". The values obtained by the three meth-\nods agree within less than 10% at all temperatures.\nB.Ab initio calculations of magnetic moments and\nanisotropy\nThe origins of magnetocrystalline anisotropy energy\n(MAE) is well-established for 3 d-transition metals as aris-\ning from a combination of the spin-orbit coupling and the\ncrystal-\feld environment of the magnetic species. Being\na ground state property the MAE can in principle be cal-\nculated using DFT, however, the small energy scales as-\nsociated with MAE in 3 dtransition-metal systems make\nthis challenging owing to the tight convergence and sam-\npling needed. Because of this, we compare two di\u000berent\nimplementations of DFT - a pseudo-potential code with\na plane-wave basis set (VASP), and an all-electron code\nwith LAPW basis (ELK).\nDFT computation of Fe 3Sn binary system has been\nrecently published [29, 30] and in particular it was found\nthat the easy magnetic axis lies in the hexagonal plane.\nHere, we \frst present our calculations of the magnetic\nproperties using VASP. We \fnd the calculated ground\nstate to be ferromagnetic order with a spin moment of\n2.20\u0016Bper Fe, an orbital moment of 0.06 \u0016Bper Fe, and\na small antiparallel moment of 0.18 \u0016Bon the Sn site,\ngiving a total magnetisation of 6.4 \u0016B/f.u., consistent\nwith our experimental value of 6.8 \u0016B/f.u.. We \fnd no\no\u000b-plane canting of moments aligned along the easy-plane\ndirection in our VASP calculations, as shown in Fig.1(f).\nTo calculate the magnetocrystalline anisotropy, we ro-\ntate the spin in the in-plane and out-of-plane directions,\n\fnding the corresponding energy for several angles, as de-\npicted in Figure 1(e). We \fnd the ground state magnetic7\norder to be ferromagnetic with an easy-plane anisotropy.\nIn fact, we \fnd the in-plane ( ab) spin directionality to\nbe fully degenerate (up to 2 \u0016eV), in agreement with\nthe negligible sixth-order ab-plane anisotropy found ex-\nperimentally is this study. Our calculated magnetocrys-\ntalline anisotropy energy by comparing the energy of the\nin-plane and out-of-plane spin axes is 0.406 meV/f.u.\n(1:16\u0002106J=m3), fully consistent with our experiment.\nNote that this value is by ca. 20 % lower/higher than the\ntheoretical estimate reported by Sales etal: [29]/Fayyazi\netal: [30]. In Fig.1(f) we also plot the variation of the\ncalculated orbital moment on Fe from varying the spin\neasy-axis in the ( ac) direction. For the ground-state spin\ndirection (fully in-plane), we \fnd no out-of-plane contri-\nbution to either the spin or orbital moment. However,\nas the spin axis is rotated in the out-of-plane direction,\nwe \fnd a gradual increase in the out-of-plane orbital mo-\nment which is compensated by a decrease in the in-plane\norbital moment.\nWe now discuss our results using ELK. For systems\ncontaining 3 d-electrons, the relatively weak SOC can be\ntreated in a Russel-Sounders ( LS)-coupling scheme, with\nwell de\fned spin and orbital quantum numbers. In this\ncase it is possible to de\fne a quantization direction (di-\nrection of magnetisation). In practice we have consid-\nered the quantization axis (spin moment ~S) aligned along\nthea- andc- crystallographic directions and performed\nDFT self-consistent computations for the components of\nthe spin/orbital moments. The values of Fe spin/orbital\nmoments are about 2.32/0.07 \u0016Bfor the considered ori-\nentations. We have noticed, however, that in the self-\nconsistent computation with the moment along the crys-\ntallographic a-direction the overall (unit cell) moment is\nslightly increased. The DFT self-consistent con\fguration\nfor spin moments of Fe and Sn atoms oriented along the\nc/a-directions respectively are shown in Figs. 1 (c)/(d).\nIn the ferromagnetic ground state the induced spin mo-\nment on Sn-sites is of about 0.12 \u0016B. Orbital moments\nof Sn although existing, have no signi\fcant magnitudes\n(\u001410\u00003\u0016B). The total magnetic moment per formula\nunit, as obtained from DFT is 6.9 \u0016B/f.u., in good agree-\nment with the experimental value of 6.8 \u0016B/f.u.\nThe DFT moments (on all atoms) computed for the\ncon\fguration aligned along the a-direction are somewhat\nsmaller, in particular the decrease in the magnitude of\nthe Sn atoms spin-moments (pointing opposite to the\nFe spins) are larger than the corresponding decrease for\nFe atoms. This is the cause for a slightly larger total\n(unit cell) moments along the a-direction. Thus, the\nanisotropy of the moments results from the subtle bal-\nance between spin and orbital moments on the di\u000berent\nsites of the unit cell.\nLet us comment further upon the distinction between\nthe anisotropy of magnetic moments and the macroscopic\nanisotropy energies. In systems (containing heavier ele-\nments) with lower than cubic symmetry the magnetic\nmoment per atom or the corresponding g-tensor is ex-\npected to be anisotropic and the spin-orbit interactionmight be important also in the absence of a long-range\norder. Below the ordering temperature, the cooperative\nalignment of the moments produces a net magnetisation.\nThe energy cost to rotate the magnetisation from the\neasy axis (lowest energy) into the hard direction is higher\nat lower temperatures. In the framework of second-order\nperturbation theory (at zero temperature) the anisotropy\nenergy is proportional to the anisotropy of the orbital\nmagnetic moment. Note also that for Fe 3Sn, for thecdi-\nrection con\fguration, the spin and orbital moments align\ncollinearly. At the same time, we found that the orbital\nmoments rotate slightly in the abplane away from the ini-\ntialadirection (angular departure of about 7 degrees).\nAlthough the bulk measurements presented here are not\nable to capture the small changes in the orientation and\nmagnitude, recent progress in XMCD allows for a precise\ndetermination of spin and orbital moments. Through the\nXMCD sum rules also the magnitude of the orbital mo-\nment and its anisotropy can be measured, note that the\nrelative change in the orbital moment magnitude and di-\nrection is important, not the absolute number, which is\nobviously small.\nC. Imaging and simulations of magnetic vortices\nTo move beyond the investigation of bulk properties,\nwe use MFM to image the local magnetic textures on an\nas-grownabplane. A contact-mode topography image\nis given in Fig. 5(a), to show that the surface is smooth\nenough for the magnetic signal to be collected in a single-\npass, constant height mode. While generally \rat, there is\na change in topography in the top-left of the image and\na few point defects which need to be considered when in-\nterpreting the MFM data. The magnetic microstructure\nof the sample is represented by the relative shift of the\ncantilever's resonance frequency, at a lift height of 150\nnm. Figure 5(b) shows a representative MFM image of\nthe virgin zero-\feld magnetic state. Strikingly this evi-\ndences the formation of smooth vortices across the sam-\nple. Such vortices are characteristic of systems where the\nmagnetisation is con\fned to a nearly isotropic plane, ei-\nther via easy-plane anisotropy or shape anisotropy, the\nlatter being the consequence of dipole-dipole interaction\nin thin samples.\nFollowing the procedure used for the bulk measure-\nments, we collect MFM images in a 3 T magnetic \feld,\nperpendicular to the sample surface. As expected from\nthe bulk measurements, 2(a), Figure 5(c) shows a nearly\nmono-domain, saturated, state. The remaing contrast\npredominantly originates at features that correlate with\nthe topography changes seen in Fig. 5 (a), suggesting\nthat they are not purely of magnetic origin. The \feld is\nremoved and a subsequent scan, Fig. 5(d), shows that the\nvortices have reformed but in di\u000berent positions and with\ndi\u000berent geometries. The di\u000berent positions of the vor-\ntices show that the vortex microstructure is not pinned.\nThe more angular geometry of vortex cores, see the high-8\n(d)\n0 T/s916\nf (Hz)0\n6/s916h(nm)0\n100\n3\n T0  T5  /s956m/s916\nf (Hz)0\n6/s916f (Hz)0\n16\n(a)( b)( c)( d)(\ne)( f)( g)\nFIG. 5. (a) Topographic image collected in contact mode on an as-grown Fe 3Snabsurface, also imaged in the MFM scans in\npanels (b), (c), and (d). (b)-(d) MFM images, recorded at 100 K, showing the shift in resonance frequency of the cantilever:\nLight/dark colors show an increase/decrease in the resonance frequency. (b) Extended domain walls meeting at vortex cores\nin the zero-\feld state. (c) Uniform MFM signal, observed except in the vicinity of topographic features, indicate uniform\nmagnetisation in 3 T. (d) The zero-\feld state after \feld treatment hosts a high density of vortices, some of which are much\nmore angular in structure. The di\u000berence between the fully relaxed vortex structure (b) and the metastable structure shortly\nafter the removal of a \feld (d) is highlighted by the dashed circle and square, respectively. (e) Micromagnetic simulations of\nthe local orientation of the magnetisation (gray arrows), viewed down the caxis. The false color code (bottom right) indicates\nthe in-plane orientation of the magnetisation vector. (f) Magni\fed view of one of the closure structures (vortices) in (e). (g)\nMFM response calculated for the domain pattern in panel (e).\nlighted square in Figure 5(d), are associated with larger\nstray \felds, this likely indicates that the microstructure,\nat 100 K, has not fully relaxed on the measurement time\nscale of tens of minutes.\nIn order to directly connect our images of vortex-\nantivortex microstructure to our bulk measurements, we\nuse micromagnetic simulation. The bulk data are used\nas input parameters to compute the expected orientation\nof the local magnetic moments, as shown in Fig. 5(e).\nHere, the small arrows represent the nanoscale orienta-\ntion of the magnetisation vector and the rainbow false\ncolors showing areas of the same orientation. The pattern\nis typical for an easy-plane magnet, with sets of vortex-\nantivortex pairs. Figure 5(f) shows one such example of\nthis, with a continuous in-plane rotation of magnetisa-\ntion.\nThe connection between the local moment and the\nstray \felds, that our MFM senses, can be counterintu-\nitive, and so we use MuMax3 to calculate the expected\nMFM response of the microstructure (shown in Fig. 5(e))\nwhich is presented in Fig. 5(g). This is possible once the\ntip material and lift height are given, as the force on the\nMFM tip goes as the gradient of the stray \feld { which is\nknown via the preceding magnetisation calculations. The\ncalculated MFM signal is in excellent qualitative agree-\nment with the observed signal of the zero-\feld state, e.g.\nFig. 5(b) and (d), showing a series of vortex points con-\nnected by domain walls. The simulations also show thatthe contrast at the walls depends on the exact nature\nof the magnetisation reorientation: some walls have very\nsharp bright-dark features, while others have smoother\ntransitions; again, as observed in the MFM data. Note,\nobserving in-plane closure domains with an MFM is long\nestablished for thin-\flms, for instance, Ref. [63].\nIV. CONCLUSIONS\nIn conclusion, we have grown high-quality bulk single\ncrystals of single kagome bilayer ferromagnet Fe 3Sn and\nperformed magnetisation and magnetic-force-microscopy\nstudies. The studies revealed a strong uniaxial easy-plane\nanisotropy characterized by the \frst-order anisotropy\nconstantK1=\u00000:99\u0002106J=m3at 300 K and\u00001:23\u0002\n106J=m3at 2 K. The three independent methods, ap-\nplied for the calculation of anisotropy constants in our\nwork, provide values that are in good agreement with\neach other as well as the anisotropy energy obtained from\nourab initio study. Moreover, the used angular depen-\ndent measurements allowed to evidence the anisotropy\nin the saturation magnetisation along the aand thec\naxes, which we ascribe to orbital contributions. Our DFT\ncalculations predict an induced spin moment on the Sn\nsites of a similar magnitude to the Fe's orbital moment,\nthe two pointing opposite to each other. All-electron\nDFT calculations suggest that the orbital-magnetic mo-9\nments of Fe tilt within the easy plane with respect to the\nmain crystallographic directions, thus a slight departure\nfrom the collinear magnetic con\fguration is obtained.\nAlthough the current experimental analysis of the bulk\nmagnetisation data can not distinguish between the or-\nbital collinear and non-collinear con\fguration within the\nab-plane, the neutron di\u000braction measurements might\ncon\frm the existence and the magnitude of the induce\nspin moment on Sn sites. Moreover, XMCD studies have\nthe potential to reveal the predicted non-collinearity of\nthe orbital moments.\nOur MFM study further con\frms the system is\nan easy-plane ferromagnet that contains a rich mi-\ncrostructure of the magnetisation pattern, dominated\nby (anti)vortices. The experimentally observed MFM\ncontrast is well reproduced by the micromagnetic sim-\nulations, using experimentally determined values of the\nmagnetic interaction parameters.\nOverall, our bulk and microscopic experimental studies\nin combination with ab initio and micromagnetic calcu-\nlations provide a multi-scale, highly reliable approach to\nquantify magnetic interactions, e.g., magnetic anisotropy\nand spin arrangement, and to reveal their origins in thekagome bilayer model system Fe 3Sn. This is an im-\nportant step towards understanding complex magnetism\nemerging in kagome magnets in presence of remarkable\nspin-orbit e\u000bects.\nACKNOWLEDGMENTS\nThis work was supported by the Deutsche Forschungs-\ngemeinschaft (DFG) through Transregional Research\nCollaboration TRR 80 (Augsburg, Munich, and\nStuttgart), and by the project ANCD 20.80009.5007.19\n(Moldova). DME wishes to thank and acknowledge\nfunding by the DFG individual fellowship, number (EV\n305/1-1). SMG acknowledge funding by the U.S. Depart-\nment of Energy, O\u000ece of Science, O\u000ece of Basic Energy\nSciences, Materials Sciences and Engineering Division\nunder Contract No. DEAC02-05-CH11231 (Nonequilib-\nrium magnetic materials program MSMAG). Computa-\ntional resources were provided by the National Energy\nResearch Scienti\fc Computing Center and the Molecular\nFoundry, DOE O\u000ece of Science User Facilities supported\nunder same contract.\n[1] L. Balents, Nature 464, 199 (2010).\n[2] I. Mazin, H. Jeschke, F. Lechermann, H. Lee, M. Fink,\nR. Thomale, and R. Valenti, Nat Commun 5, 4261\n(2014).\n[3] A. Bolens and N. Nagaosa, Phys. Rev. B 99, 165141\n(2019).\n[4] J. Yin, S. S. Zhang, G. Chang, Q. Wang, S. S. Tsirkin,\nZ. Guguchia, B. Lian, H. Zhou, K. Jiang, I. Belopolski,\nN. Shumiya, D. Multer, M. Litskevich, T. A. Cochran,\nH. Lin, Z. Wang, T. Neupert, S. Jia, H. Lei, and M. Z.\nHasan, Nat. Phys. 15, 443 (2019).\n[5] T. Chen, T. Tomita, S. Minami, M. Fu, T. Koretsune,\nM. Kitatani, I. Muhammad, D. Nishio-Hamane, R. Ishii,\nF. Ishii, R. Arita, and S. Nakatsuji, Nature Communi-\ncations 12(2021), 10.1038/s41467-020-20838-1.\n[6] H. Zhang, H. Feng, X. Xu, W. Hao, and Y. Du, Ad-\nvanced Quantum Technologies 4, 2100073 (2021).\n[7] S.-Y. Yang, Y. Wang, B. R. Ortiz, D. Liu, J. Gayles,\nE. Derunova, R. Gonzalez-Hernandez, L. \u0014Smejkal,\nY. Chen, S. S. P. Parkin, S. D. Wilson, E. S. Toberer,\nT. McQueen, and M. N. Ali, Science Advances 6(2020),\n10.1126/sciadv.abb6003.\n[8] B. R. Ortiz, S. M. Teicher, Y. Hu, J. L. Zuo, P. M.\nSarte, E. C. Schueller, A. M. Abeykoon, M. J. Krogstad,\nS. Rosenkranz, R. Osborn, R. Seshadri, L. Balents, J. He,\nand S. D. Wilson, Physical Review Letters 125, 247002\n(2020).\n[9] W. Ma, X. Xu, J.-X. Yin, H. Yang, H. Zhou, Z.-J. Cheng,\nY. Huang, Z. Qu, F. Wang, M. Z. Hasan, and S. Jia,\nPhysical Review Letters 126, 246602 (2021).\n[10] S. Peng, Y. Han, G. Pokharel, J. Shen, Z. Li,\nM. Hashimoto, D. Lu, B. R. Ortiz, Y. Luo, H. Li, M. Guo,\nB. Wang, S. Cui, Z. Sun, Z. Qiao, S. Wilson, and J. He,\nPhysical Review Letters 127, 266401 (2021).[11] E. Liu, Y. Sun, N. Kumar, L. Muechler, A. Sun, L. Jiao,\nS.-Y. Yang, D. Liu, A. Liang, Q. Xu, J. Kroder, V. S u\u0019,\nH. Borrmann, C. Shekhar, Z. Wang, C. Xi, W. Wang,\nW. Schnelle, S. Wirth, Y. Chen, S. T. B. Goennenwein,\nand C. Felser, Nat. Phys. 14, 1125 (2018).\n[12] S. N. Guin, P. Vir, Y. Zhang, N. Kumar, S. J. Watz-\nman, C. Fu, E. Liu, K. Manna, W. Schnelle, J. Gooth,\nC. Shekhar, Y. Sun, and C. Felser, Advanced Materials\n31, 1806622 (2019).\n[13] L. Ye, M. Kang, J. Liu, F. von Cube, C. R. Wicker,\nT. Suzuki, C. Jozwiak, A. Bostwick, E. Rotenberg, D. C.\nBell, L. Fu, R. Comin, and J. G. Checkelsky, Nature\n555, 638 (2018).\n[14] S. Nakatsuji, N. Kiyohara, and T. Higo, Nature 527,\n212 (2015).\n[15] M. Kang, L. Ye, S. Fang, J.-S. You, A. Levitan, M. Han,\nJ. I. Facio, C. Jozwiak, A. Bostwick, E. Rotenberg,\nM. K. Chan, R. D. McDonald, D. Graf, K. Kaznatcheev,\nE. Vescovo, D. C. Bell, E. Kaxiras, J. van den Brink,\nM. Richter, M. Prasad Ghimire, J. G. Checkelsky, and\nR. Comin, Nature Materials 19, 163 (2020).\n[16] H. Huang, L. Zheng, Z. Lin, X. Guo, S. Wang, S. Zhang,\nC. Zhang, Z. Sun, Z. Wang, H. Weng, L. Li, T. Wu,\nX. Chen, and C. Zeng, Phys. Rev. Lett. 128, 096601\n(2022).\n[17] T. Chen, S. Minami, A. Sakai, Y. Wang, Z. Feng,\nT. Nomoto, M. Hirayama, R. Ishii, T. Koretsune,\nR. Arita, and S. Nakatsuji, Science Advances 8,\neabk1480 (2022).\n[18] Z. Hou, W. Ren, B. Ding, G. Xu, Y. Wang, B. Yang,\nQ. Zhang, Y. Zhang, E. Liu, F. Xu, W. Wang, G. Wu,\nX. Zhang, B. Shen, and Z. Zhang, Advanced Materials\n29, 1701144 (2017).10\n[19] H. Giefers and M. Nicol, Journal of Alloys and Com-\npounds 422, 132 (2006).\n[20] L. H aggstr om, T. Ericsson, R. W appling, and K. Chan-\ndra, Physica Scripta 11, 47 (1975).\n[21] B. C. Sales, J. Yan, W. R. Meier, A. D. Christianson,\nS. Okamoto, and M. A. McGuire, Physical Review Ma-\nterials 3, 114203 (2019).\n[22] G. Le Ca er, B. Malaman, and B. Roques, Journal of\nPhysics F: Metal Physics 8, 323 (1978).\n[23] G. Trumpy, E. Both, C. Djega-Mariadassou, and\nP. Lecocq, Phys. Rev. B 2, 3477 (1970).\n[24] M. Altthaler, E. Lysne, E. Roede, L. Prodan, V. Tsurkan,\nM. A. Kassem, H. Nakamura, S. Krohns, I. Kezsmarki,\nand D. Meier, Phys. Rev. Research 3, 043191 (2021).\n[25] S. Kulshreshtha and P. Raj, Journal of Physics F: Metal\nPhysics 11, 281 (1981).\n[26] K. Heritage, B. Bryant, L. A. Fenner, A. S. Wills, G. Aep-\npli, and Y.-A. Soh, Advanced Functional Materials 30,\n1909163 (2020).\n[27] G. He, L. Peis, R. Stumberger, L. Prodan, V. Tsurkan,\nN. Unglert, L. Chioncel, I. K\u0013 ezsm\u0013 arki, and R. Hackl,\nPhysica Status Solidi (B) 259, 2100169 (2021).\n[28] C. Dj\u0013 ega-Mariadassou, P. Lecocq, G. Trumpy, J. Tr a\u000b,\nand P. Ostergaard, Il Nuovo Cimento XLVI B , 35\n(1966).\n[29] B. C. Sales, B. Saparov, M. A. McGuire, D. J. Singh,\nand D. S. Parker, Scienti\fc reports 4, 1 (2014).\n[30] B. Fayyazi, K. P. Skokov, T. Faske, I. Opahle,\nM. Duerrschnabel, T. Helbig, I. Soldatov, U. Rohrmann,\nL. Molina-Luna, K. Gueth, et al. , Acta Materialia 180,\n126 (2019).\n[31] J. Watanabe, Y. Araki, K. Kobayashi, A. Ozawa, and\nK. Nomura, Journal of the Physical Society of Japan 91\n(2022), 10.7566/jpsj.91.083702.\n[32] A. Bogdanov and D. Yablonskii, Sov Phys Solid State\n22, 399 (1980).\n[33] X. Yu, M. Mostovoy, Y. Tokunaga, W. Zhang, K. Ki-\nmoto, Y. Matsui, Y. Kaneko, N. Nagaosa, and\nY. Tokura, Proceedings of the National Academy of Sci-\nences 109, 8856 (2012).\n[34] T. Okubo, S. Chung, and H. Kawamura, Phys. Rev.\nLett. 108, 017206 (2012).\n[35] S. A. Montoya, S. Couture, J. J. Chess, J. C. T. Lee,\nN. Kent, D. Henze, S. K. Sinha, M.-Y. Im, S. D. Kevan,\nP. Fischer, B. J. McMorran, V. Lomakin, S. Roy, and\nE. E. Fullerton, Physical Review B 95, 024415 (2017).\n[36] J. Tang, L. Kong, Y. Wu, W. Wang, Y. Chen, Y. Wang,\nJ. Li, Y. Soh, Y. Xiong, M. Tian, and H. Du, ACS Nano\n14, 10986 (2020).\n[37] M. Prei\u0019inger, K. Karube, D. Ehlers, B. Szigeti, H.-\nA. K. von Nidda, J. S. White, V. Ukleev, H. M. R\u001cnnow,\nY. Tokunaga, A. Kikkawa, Y. Tokura, Y. Taguchi,\nand I. K\u0013 ezsm\u0013 arki, npj Quantum Materials 6(2021),\n10.1038/s41535-021-00369-8.\n[38] X. Yu, Y. Tokunaga, Y. Taguchi, and Y. Tokura, Ad-\nvanced Materials 29, 1603958 (2016).\n[39] M. E. Gouva, G. M. Wysin, A. R. Bishop, and F. G.\nMertens, Physical Review B 39, 11840 (1989).[40] K. Shigeto, T. Okuno, K. Mibu, T. Shinjo, and T. Ono,\nApplied Physics Letters 80, 4190 (2002).\n[41] F. P. Chmiel, N. W. Price, R. D. Johnson, A. D. Lami-\nrand, J. Schad, G. van der Laan, D. T. Harris, J. Irwin,\nM. S. Rzchowski, C.-B. Eom, and P. G. Radaelli, Nature\nMaterials 17, 581 (2018).\n[42] N. Gao, S. G. Je, M. Y. Im, J. W. Choi, M. Yang, Q. Li,\nT. Y. Wang, S. Lee, H. S. Han, K. S. Lee, W. Chao,\nC. Hwang, J. Li, and Z. Q. Qiu, Nature Communications\n10(2019), 10.1038/s41467-019-13642-z.\n[43] X. Zhang, Y. Zhou, K. M. Song, T.-E. Park, J. Xia,\nM. Ezawa, X. Liu, W. Zhao, G. Zhao, and S. Woo, Jour-\nnal of Physics: Condensed Matter 32, 143001 (2020).\n[44] CrysAlisPro Software system , software 1.171.41.118a\n(Rigaku Corporation: Oxford, UK, 2015).\n[45] G. M. Sheldrick, Acta Crystallographica Section C Struc-\ntural Chemistry 71, 3 (2015).\n[46] L. J. Farrugia, Journal of Applied Crystallography 32,\n837 (1999).\n[47] G. Kresse and J. Hafner, Physical Review B 47, 558\n(1993).\n[48] G. Kresse and J. Hafner, Physical Review B 49, 14251\n(1994).\n[49] G. Kresse and J. Furthm uller, Computational Materials\nScience 6, 15 (1996).\n[50] P. Bl ochl, Physical Review B 50, 17953 (1994).\n[51] G. Kresse and D. Joubert, Physical Review B 59, 1758\n(1999).\n[52] J. P. Perdew, K. Burke, and M. Ernzerhof, Physical\nReview Letters 77, 3865 (1996).\n[53] A. Jain, S. P. Ong, G. Hautier, W. Chen, W. D. Richards,\nS. Dacek, S. Cholia, D. Gunter, D. Skinner, G. Ceder,\nand K. a. Persson, APL Materials 1, 011002 (2013).\n[54] O. Nial, Svensk Kemisk Tidskrift 59, 165 (1947).\n[55] K. Buschow, P. van Engen, and R. Jongebreur, Journal\nof Magnetism and Magnetic Materials 38, 1 (1983).\n[56] F. Basile, P. Lecocq, and A. Michel, Annales de Chimie\n(Paris) 1969 , 297 (1969).\n[57] \\The Elk Code,\" http://elk.sourceforge.net/ .\n[58] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen,\nF. Garcia-Sanchez, and B. V. Waeyenberge, AIP Ad-\nvances 4, 107133 (2014).\n[59] B. D. Cullity and C. D. Graham, Introduction to Mag-\nnetic Materials (IEEE Press, John Wiley and Sons,\n2009).\n[60] W. Sucksmith and J. E. Thompson, Proceedings of the\nRoyal Society of London. Series A. Mathematical and\nPhysical Sciences 225, 362 (1954).\n[61] J. Alameda, J. Deportes, D. Givord, R. Lemaire, and\nQ. Lu, Journal of Magnetism and Magnetic Materials\n15-18 , 1257 (1980).\n[62] A. S. Bolyachkin, D. S. Neznakhin, and M. I. Bartashe-\nvich, Journal of Applied Physics 118, 213902 (2015).\n[63] H. J. Mamin, D. Rugar, J. E. Stern, R. E. Fontana, and\nP. Kasiraj, Applied Physics Letters 55, 318 (1989)."    },    {        "title": "2302.14667v1.Bistable_electric_field_control_of_single_atom_magnetocrystalline_anisotropy.pdf",        "content": " \n  \nBistable electric field control of single -atom magnetocrystalline anisotropy  \n \nAuthors  \nJose Martinez -Castro1,2,3 *, Cyrus F. Hirjibehedin1,2,4†, and David Serrate3,5* \nAffiliations   \n1 London Centre for Nanotechnology, University College London (UCL), London WC1H 0AH, \nUK. \n2 Department of Physics & Astronomy, UCL, London WC1E 6BT, UK.  \n3 Instituto de Nanociencia de Aragón and Laboratorio de Microscopías Avanzadas, Universidad \nde Zaragoza,  50018 Zaragoza, Spain.  \n4 Department of Chemistry, UCL, London WC1H 0AJ, UK.  \n5 Departamento de Física de la Materia Condensada, Universidad de Zaragoza, 50009 Zaragoza, \nSpain.  \n*Correspondence to: serrate@unizar.es, j.martinez@fz -juelich.de  \n†present address: M IT Lincoln Laboratory, Lexington, MA 02421 -6426, USA  \n \nAbstract  \nWe reversibly switch the polar environment of an individual magnetic atom with an electric \nfield to control the energy barrier for reversal of magnetization. By applying an electric field \nin the gap between the tip and sample of a scanning tunneling micros cope, we induce bistable \nchanges in the polarization of the region surrounding a chlorine vacancy in a monolayer of \nsodium chloride  on copper terminated by a monolayer of copper nitride. The displacement of \nthe sodium chloride ions alters the local electri c polarization and modifies the \nmagnetocrystalline anisotropy experienced by a single cobalt atom. When a cobalt atom is \nnear a chlorine vacancy, spin -sensitive inelastic electron tunneling spectroscopy \nmeasurements can reveal the change in anisotropy. The  demonstration of atomic -scale control \nof magnetic properties with electric fields opens new possibilities for probing the origins of \nmagnetoelectric coupling and will stimulate the development of model  artificial mutliferroic \nsystems.  \n \nMAIN  TEXT  \n \nIntroduc tion \nAchieving electric field control of magnetic properties is a major challenge in the development of \nnovel materials and devices, offering access to new material properties as well as potential \ntechnological improvements such as significantly reduced po wer consumption (1). A variety of \ndriving mechanisms to couple a material’s electronic and magnetic degrees of freedom have been \nproposed. For example, an electrostatic gate voltage can force electronic transport in quantum \nsystems to proceed through discrete spin states with a well -defined conductivity (2–5). In thin \nferromagnetic metals and semiconductors, the charge redistribution near a strong gating electric \nfield can significantly alter the magnetic coercivity (6) or ordering temperatures (7). In addition, \nspin-orbit coupling enables magnetization control by electrical currents through spin -torque  effects \n(8–10). Alternatively, direct coupling of electrostatic and magnetic degrees of freedom has been \nachieved in single phase multiferroic materials like BiFeO 3 (11) and hexagonal manganites (12, 13), or in heterostructures interfacing ferroelectric and magnetic thin films (14, 15). In spite of the \nintense research activity on multiferroic phenomena, a framework enabling fundamental studies on \nthe coupling of the electric polarization and the spin moment at interfaces is not yet fully developed \n(15).  \n \nMagnetocrystalline anisotropy energy (MAE) is one of the most relevant parameters for defining \nthe behavior of magnetic m aterials. It determines the susceptibility of a material’s magnetization to \nthermal activation, external magnetic fields, and electromagnetic radiation  (16). Different studies \nhave demonstrated that MAE in metallic thin films can be continuously tuned at t he nanoscale by \nthe application of an electric field (6, 17, 18). In these cases, the coupling mechanism is restricted \nto the response of the electronic density of states to the unscreened part of the electric field in the \nmetal. At the nanosc ale, the electric field can influence the MAE barrier and modify the \nmagnetization reversal attempt frequency in the superparamagnetic regime  (18).  \n \nIn the fundamental limit of an individual magnetic atom, the atom’s MAE is mainly controlled by \nthe structu re and charge distribution of the immediate environment (19–21). At the single molecule \nlevel, it is possible to modify the MAE by charging (22) as well as through controlled (reversible \nor non -reversible) deformation of the bond distance between the ion carrying the magnetic moment \nand its surrounding ligands (23–25). Analogously, an observable  change in  MAE can be achieved \nby local strain arising from deformation of the substrate supporting the magnetic moments (26). \nThis suggests that controlling the arrangement of the atoms surrounding a magnetic ion using an \nelectric field would also modify the single -atom MAE, resulting in an efficient way to implement \nexternal electric field control of magnetism.   \n \nIn this work, we show that the MAE of an individual magnetic atom can be manipulated through \nbistable atomic displacements controlled by an external electric field applied to a supporting dipolar \nsubstrate. By depositing a monolayer (ML) of NaCl on the ato mically thin polar insulator copper \nnitride (Cu 2N) capping bulk Cu(001), we induce a distortion in the NaCl that results in a net out of \nplane dipole similar to what has been observed for a bilayer of NaCl on Cu 2N (27). In the presence \nof a Cl vacancy, the NaCl can be bistably switched between two dipolar orientations using an \nelectric field applied from the tip of a scanning tunneling microscope (STM) used to study the \nsystem. By performing spin -sensitive inelastic ele ctron tunneling spectroscopy (IETS) on a Co \natom adsorbed on the NaCl -ML, we observe that the characteristic MAE measured for Co on bare \nCu2N (28, 29) is altered between two distinct value s following the bistable electric polarization of \nthe NaCl -ML that can be programmed by opposite electric fields in the tip -sample gap. These \nresults show  that electric field control of magnetic properties can be achieved in the limit of single \natoms on su rfaces. Extending this technique to other materials in which more detailed \ncharacterization can be performed would enable the development of model systems for \nunderstanding the interplay between MAE and polar order , shedding light on  the atomic -scale \norigi ns of multiferroic coupling. \nBistable  polarization switching  in a monolayer . Figure 1A shows a topographic STM image of \na NaCl ML covering Cu 2N nanoislands on Cu(001). As is seen for ultra -thin films of NaCl on many \nother substrates (30), the Na Cl ML on Cu 2N also contain Cl vacancies (Fig. 1A). Co atoms can also \nbe deposited on top of the NaCl ML, and as seen in Fig. 1B can be recognized as a bright protrusions \nsimilar to those observed for Co adsorbed on top of Cu 2N/Cu(001) (28). However, the appearance \nof Co adatoms on the NaCl ML strongly depends on the adsorption site. As seen in Fig. 1C, multiple \nadsorption sites can be identified for Co, for example on top of Na or Cl sites  (Fig S1A and S1B, \nrespectively) . In addition, the Co atom has an unusual appearance near a Cl vacancy (Fig. S1), \nwhere it does not have the round characteristic shape of a Co atom adsorbed on Cu 2N (Fig. 1C) or \nthe four -fold symmetry expected above Cl sites. This irregular shape is attributed to a Co atom \nbecause the appearance on high symmetry sites can be systematically and repeatedly recovered by \ndisplacing a Co atom using lateral manipulation near a Cl vacancy on the NaCl ML (Fig. S1). \nFurthermore, its identity can be confirmed because its spectroscopic signature is similar to that of \nCo on Cu 2N (Fig. 3). \n \nAs has been observed for the NaCl BL on Cu 2N (27), the polarization of the NaCl ML can be \nbistably switched in the presence of a Cl vacancy. When the t ip is positioned above a Cl vacancy, \nthe strongly self -poled polarization  of the NaCl layer,  which is induced by the polar orientation of \nthe underlying Cu 2N, can be switched by ramping the bias to positive values, thus applying a \npositive electric field ( i.e. pointing from the sample to the tip) (Fig. 2A). The polarization also can \nbe reversibly switched back to the original state by applying large enough negative electric fields. \nThe sharp change in the tunneling current is attributed to tunneling electro  resistance (TER), where \nthe reversal of the dipole orientation modifies the work function of the substrate and therefore the \nheight of the tunneling barrier (27, 31). \n \nSimultaneously acquired Kelvin probe measurements obtained using atomic force microscopy \n(AFM) further confirm the change in the work function of the substrate from dipolar reversal. As \nsee in Fig. 2B,  the shift of the resonance frequency as a fun ction of voltage  f(V) shows the expected \nparabolic behavior (32), with the contact potential difference Vcpd between the tip and substrate \nwork functions marked by the parabola’s maximum. The value of Vcpd clearly shifts for the two \nstates, indicating the difference in work function and electric polarization.   \n \nBistable switching of magnetic anisotropy.  Having seen that switching the Cl vacancy in the NaCl \nML results in a change of its electric polarization, we explore the impact of this change on the \nmagnetic properties of Co atoms adsorbed nearby. A Co atom on bare Cu 2N has a quantum spin \nS=3/2 (28). Because of the anisotropic arrangement of charge in the Cu 2N below the magnetic atom \n(21), the crystal field splits the states of spin projection along the z -axis Sz=1/2 and Sz=3/2 by an \nenergy Ean (19). These states further split in an applied magnetic field according t o the spin \nHamiltonian  \n \n𝐻̂=−𝑔𝜇𝐵𝑩̂∙𝑺̂+𝐷𝑆𝑧2 \n \nwhere g is the Landé factor, μ B is the Bohr magneton, and D the is the uniaxial anisotropy parameter. \nFor Co on Cu 2N, D > 0 so Sz=1/2 is the doubly degenerate ground state. Exchange coupling to the \nunderlying conduction electrons results in Kondo screening of this state (28), manifesting as a sharp \nresonance at  the Fermi energy EF (V=0). STM -based IETS induces transitions between the Sz states \n(21), resulting in conductance (d I/dV) steps when the sample bias matches Ean/e at ~ ± 5 mV .  \n As seen in Fig. 3, a similar spectrum is observed for a Co atom near a Cl vacancy in the NaCl ML: \nthe relative amplitude of the Kondo resonance is reduced and no additional conductance steps (i.e. \ninelastic spin excitations) are observed. This suggests that the additional NaCl ML above the Cu 2N \ndecreases the exchange coupling between the magnetic impurity and the nearby conducting \nelectrode without changing the total spin of the Co atom (28, 29). Spectroscopic measurements \nperformed on Co atoms adsorbed on Na or Cl sites on top of the NaCl ML but away from Cl \nvacancies did not show any characteristic IETS features in this energy range . This may be because  \nthe inelastic component of the tunneling current , which induces spin -flip excitations,  is too small \nto be resolved for experimental conditions in which Co atoms remains stable.  \n \nTo study the impact of the polarization switching on the  Co atom, we position the tip above a Co \natom located near a Cl vacancy and ramp the applied voltage. Current jumps at critical voltages \ncorresponding to critical electric fields (Fig. 4) confirm that polarization switching occurs even with \nthe Co atom pre sent (Fig. 5A). To ensure that no other process happened while setting the \nconditions for low bias spectroscopy, variations  in Vset and Iset before and after the polarization \nswitch were  continuously  monitored.  \n \nAs seen in Fig. 5B, two different spectrosco pic signatures can be distinguished for the two different \npolarization states (labeled A and B). Note that the electric field conditions during the spectroscopy \nin both states are identical. As demonstrated by the change in voltage of the IETS step, the MA E \nof the Co atom is decreased by a factor of two when the underlying NaCl ML is switched from state \nA to B, while the amplitude of the Kondo resonance is enhanced. The similarity of the spectra (i.e. \nsame number of inelastic excitations and the persistence  of a Kondo resonance) implies that the \nvalue of S and the sign of D remain the same for both states.  This excludes a charging process on \nthe Co atom as the origin of the bistable switching.  \n \nAdditional confirmation of the change in MAE experienced by the Co atom for the two different \nsurface polarization states is obtained by observing the evolution of  the spectra with magnetic field \n(Fig. 6), which is  illustrated in Fig. 5C. For the relatively small magnetic fields (up to B = 3 T) \naccessible in these studies, only a very small change in the energy of the inelastic tunneling step is \nexpected (28). The small shift that is observed is consistent with such changes. A much more \nprominent change, however, can be observed in the splitting of the Kondo resonance, which is much \nmore sensitive to changes in magnetic field (28). As seen in Fig. 6, the splitting of the Kondo \nresonance is different for the two different polarization states.  \n \nFigure S2 shows low energy spectroscopy for three other Co atoms near Cl vacancies as a functi on \nof the polarization. The general features are the same in all cases, though the MAE either increases \nor decreases from state A to B depending on the local environment. Variations are also observed \nfor changes in the Kondo resonance, which may be modifie d due to changes in the strength of the \ncoupling between the Co atom and the underlying metallic substrate. It is difficult to quantify these \nvariations because the adsorption site of the Co atom near the Cl vacancy as well as the underlying \nlattice struct ure nearby cannot be resolved ; this may be  due to the low symmetry around the Cl \nvacancy  and the relatively weak adsorption energy  of the Co atom , which precludes high -resolution \nimaging at small  tip-sample distances.   \n \nDiscussion  \n \nWe have demonstrated tha t electric field induced modification of the polarization of a substrate can \nbistably switch the magnetocrystalline anisotropy experienced by a single magnetic atom through \nthe rearrangement of the atomic positions of the neighboring ions. As is the case f or nanoscale magnetic data storage (33), bistable  modulation of an individual atom’s MAE could have applicates  \nfor classical and quantum information processing , potentially allowing for switching to an easily \nmodifiable state when writing data and then back a more stable state for longer -term storage. \nFurther development of atomic manipulation techniques on this or other switchable substrates \nwould also facilitate the construction of coupled spin systems, enabling construction of the smallest \npossible multiferroic systems in which a collective electric de gree of freedom could be used to \ncontrol the collective magnetic degree of freedom at the atomic scale.  If this can be performed on \na surface for which the switched polarization state can be fully characterized, then the system would \nprovide an ideal venue  for studying the coupling between collective polar and magneti c order  at the \nlevel of a single atomic spin . This would represent a model system for understanding the \nfundamentals of multiferroic behavior . \n \nMaterials and Methods  \n \nScanning Tunneling Microsc opy and Atomic Force Microscopy . Scanning tunneling \nmicroscopy (STM) experiments were performed using a Specs JT -STM, a commercial adaptation \nof the desig n described by Zhang et al. (34) as well as an Omicron Nanotechnology LT -STM with \na qPlus force sensor (35) installed for combined operation  of both STM and atomic force \nmicroscopy (AFM) . The qPlus sensor , with a resonance frequency f0 = 23379.5 Hz and a stiffness \nk ~ 1800 N/m (35), was operated in non-contact AFM mode with a phase -locked excitation at a \nconstant oscillation amplitude of A = 20 pm.  Both systems were operated in ultrahigh vacuum \nconditions, with typical chamber pressures below 2×10-10 mbar, and at base temperatures of 1.1 K \nand 4.5 K respectively. In the Specs JT -STM, a magnetic field up to 3 T can be applied \nperpendicular to the sample surface.  \n \nThe bias voltage V is quoted in sample bias convention. Topographic images were obtained in the \nconstant current imaging mode with V and tunn el current I set to Vset and Iset respectively.  \nDifferential conductance dI/dV measurements are obtained using a lock -in amplifier, with t ypical \nmodulation voltages of 150 V at ∼840 Hz added to V. Spectroscopy is acquired  by initially setting \nV=Vset and I=I set, disabling the feedback loop  to maintain position of the tip , and then sweeping V \nwhile recording I, dI/dV, and/or the shift of the resonance frequency f. \n \nCu(001) samples (MaTeck single crystal with 99.999% purity)  were prepared by repeated cycles \nof sputtering with Ar and annealing to 500 ˚C. Cu 2N is prepared on top of clean Cu(001) samples \nby sputtering with N 2 and annealing to 350 ˚C (36). Deposition  of NaCl was performed using a \nKnudsen effusion cell operated at 490 ˚C and with the Cu 2N/Cu(001) substrate at room temperature  \n(27). \n \nTopographic images obtained using STM were processed using WSxM (37). \n \nLateral atomic manipulation.  Co atoms on top of Na  sites on NaCl on Cu 2N/Cu(001) can be \nlaterally manipulated to the next Na atomic position by approaching the STM tip in closed feedback \nmode to maintain a constant Iset and (simultaneously) decreasing V. Typical s tarting conditions are \nVset=-1.3 V  and Iset=10 pA . V is then  decreas ed until a sharp jump in the tip position  is observed, \ntypically when V < 100 mV. The tip is then moved across  the NaCl ML island while the Co follows \nthe tip. The speed of the tip during lateral manipulation is below 10  nm/s.  \n  References and Notes:  \n \n1.  F. Matsukura, Y. Tokura, H. Ohno, Nat. Nanotechnol.  10, 209 –220 (2015).  \n2.  J. Paaske et al. , Nat. Phys.  2, 460 –464 (2006).  \n3.  N. Roch, S. Florens, V. Bouchiat, W. Wernsdorfer, F. Balestro, Nature . 453, 633 –637 \n(2008).  \n4.  C. Brüne et al. , Nat. Phys.  8, 485 –490 (2012).  \n5.  R. Vincent, S. Klyatskaya, M. Ruben, W. Wernsdorfer, F. Balestro, Nature . 488, 357 –360 \n(2012).  \n6.  M. Weishei t et al. , Science . 315, 349 –351 (2007).  \n7.  D. Chiba et al. , Phys. Rev. Lett.  104, 1–4 (2010).  \n8.  S. O. Valenzuela, M. Tinkham, Nature . 442, 176 –179 (2006).  \n9.  I. M. Miron et al. , Nat. Mater.  9, 230 –234 (2010).  \n10.  R. A. Buhrman, Science . 336, 555 (2012).  \n11.  J. Wang et al. , Science . 299, 1719 –1722 (2003).  \n12.  N. Fujimura, T. Ishida, T. Yoshimura, T. Ito, Appl. Phys. Lett.  69, 1011 –1013 (1996).  \n13.  T. Lottermoser, T. Lonkai, U. Amann, M. Fiebig, Nature . 430, 541 –544 (2004).  \n14.  C. A. F. Vaz, J. Phys. Condens. Matter . 24 (2012), doi:10.1088/0953 -8984/24/33/333201.  \n15.  J. M. Hu, L. Q. Chen, C. W. Nan, Adv. Mater.  28, 15–39 (2016).  \n16.  J. M. D. Coey, Magnetism and magnetic materials  (Cambridge University Press, 2010).  \n17.  A. Mardana, S. Ducharme,  S. Adenwalla, Nano Lett.  11, 3862 –3867 (2011).  \n18.  A. Sonntag et al. , Phys. Rev. Lett.  112, 017204 (2014).  \n19.  D. Gatteschi, R. Sessoli, J. Villain, Molecular Nanomagnets  (Oxford University Press, ed. \n2, 2006).  \n20.  I. G. Rau et al. , Science . 344, 988–92 (2014).  \n21.  C. F. Hirjibehedin et al. , Science . 317, 1199 –203 (2007).  \n22.  A. S. Zyazin et al. , Nano Lett.  10, 3307 –3311 (2010).  \n23.  B. W. Heinrich et al. , Nano Lett.  13, 4840 –4843 (2013).  \n24.  B. W. Heinrich, L. Braun, J. I. Pascual, K. J. Franke , Nano Lett.  15, 4024 –4028 (2015).  \n25.  J. J. Parks et al. , Science . 328, 1370 –1373 (2010).  \n26.  B. Bryant,  a Spinelli, J. J. T. Wagenaar, M. Gerrits,  a F. Otte, Phys. Rev. Lett.  111, \n127203 (2013).  \n27.  J. Martinez -Castro et al. , Nat. Nanotechnol.  13, 19–23 (2018).  \n28.  A. F. Otte et al. , Nat. Phys.  4, 847 –850 (2008).  \n29.  J. C. Oberg et al. , Nat. Nanotechnol.  9, 64–8 (2014).  \n30.  J. Repp, G. Meyer, S. Paavilainen, F. Olsson, M. Persson, Phys. Rev. Lett.  95, 225503 \n(2005).  31.  V. Garcia et al. , Nature . 460, 81–4 (2009).  \n32.  S. Kitamura, M. Iwatsuki, Appl. Phys. Lett.  72, 3154 –3156 (1998).  \n33.  M. R. Kryder et al. , Proc. IEEE . 96, 1810 –1835 (2008).  \n34.  L. Zhang, T. Miyamachi, T. Tomanić, R. Dehm, W. Wulfhekel, Rev. Sci. Instrum.  82, \n103702 (2011).  \n35.  F. J. Giessibl, Rev. Mod. Phys.  75 (2003).  \n36.  F. M. Leibsle, S. S. Dhesi, S. D. Barrett,  a. W. Robinson, Surf. Sci.  317, 309 –320 (1994).  \n37.  I. Horcas et al. , Rev. Sci. Instrum.  78, 013705 (2007).  \n \nAcknowledgements  \nWe thank Markus Ternes for stimulating discussions  and Marten Piantek  for experimental \ncontributions . J.M.C. and C.F.H. acknowledge financial support from Specs GmbH, EPSRC \n[EP/H002367/1  and EP/M009564/1 ], and the Leverhulme Trust [RPG - 2012 -754]; D.S. \nacknowledge s funding from MINECO [MAT2016 -78293 -C6-6-R], FEDER funds through the \nInterreg -POCTEFA program [TNSI/EFA194/16/], and the use of SAI -Universidad de Zaragoza.  \n \nAuthor contributions : J.M.C., D.S., and C.F.H. conceived of the project; J.M.C.  and D.S.  \nperformed the experiments and analy zed the resu lts; all authors discussed the results and \ncontributed to the writing of the paper.   Figures and Tables  \n \nFig. 1. Vacancies and Co atoms on NaCl ML.  (A) Topographic STM image of a Cu(001) surface \nmostly covered by Cu 2N islands with small stripes of Cu in between and NaCl ML on top (9 nm × \n9 nm; Vset=-1.3 V, Iset=50 pA). Red circles denote two of the Cl vacancies in the NaCl ML. ( B) \nTopographic STM image overview of a Cu(001) surface fully covered by Cu 2N with NaCl ML a nd \nBL on top with adsorbed single Co atoms (50 nm × 50 nm, Vset=-1.3 V , Iset=10 pA). ( C) Co atoms \nadsorbed on top of Cu 2N (green arc), Na site of NaCl ML (red arc), and a Cl vacancy (blue arc). A \nnearby Cl vacancy (red circle) is also highlighted (11 nm ×  11 nm). ( D) Side view of an illustration \nof an NaCl ML on top of Cu 2N with Cl vacancy as well as a Co atom on top.  \n(A) (B)\n(C) (D)\n \nFig. 2.  Hysteretic reversal of the polarization near a Cl vacancy in the NaCl ML.  (A) I(V) \nacquired above a Cl- anion near a Cl vacancy in the NaCl ML ( Vset=-1.3 V, Iset=50 pA). Two states \ncan be distinguished: a self -poled initial state (black) and a switched state (red). The sharp change \nin current indicates the switching of the polarization (dashed grey vertical  lines). Inset: topographic \nSTM image of the NaCl ML with the Cl vacancy. Red circle denotes the position where the \nspectroscopy was performed (5.8 nm × 5.8 nm, Vset=-1.3 V, Iset=50 pA). (B) Simultaneously \nacquired f(V). The change in the polarization of the NaCl ML is observed in a clear shift of Vcpd \nbetween the two states as indicated by the vertical black and red arrows, which mark the maximum \nof the respective parabolas.  \n-50050100150Current (pA)\n-1.0 -0.5 0.0 0.5 1.0-6-5-4-3Df (Hz)\nBias (V)(A)\n(B) \nFig. 3. Inelastic electron tunneling spectroscopy for Co atoms on Cu 2N and NaC l ML.  (A) \nSTM topographic image of a Cu 2N/Cu(001) surface and a partial coverage of NaCl ML. Co atoms \ncan be observed on top of both the Cu 2N (green arc) and NaCl ML (red arc). (40 nm × 20 nm, Vset=-\n1.3 V, Iset=10 pA). (B) Typical low energy spectroscopy of a Co atom on top of Cu 2N (28). A \ncharacteristic Kondo resonance peak centered at V=0 is observed along with conductance steps due \nto inelastic spin -flip excitations ( Vset=-15 mV, Iset=80 pA, Vm=150 V). (C) Low energy \nspectroscopy of a Co atom on NaCl ML close to a Cl vacancy obtained with the same time that was \nused to measure the spectrum in panel B. As for Co on Cu 2N, a Kondo resonance and inelastic \ntunneling steps can be observed ( Vset=-15 mV, Iset=30 pA , Vm= 150 V). \n(A)\n(B) (C)\n-15-10 -5 0 5 10 15468dI/dV (nA/V)\nBias (mV)-15-10 -5 0 5 10 152.02.22.42.6dI/dV (nA/V)\nBias (mV) \nFig. 4. Electric field induced polarization switching above a Co atom adsorbed near a Cl \nvacancy.  (A) I(V) spectroscopy of  the polarization  switching cycle for a Co atom near a Cl vacancy \nin the NaCl ML performed for different values of Iset to tune the change in tip -sample distance z. \nThe relative height variation was extracted from the change in the relative piezo movement. A \ncritical bias  is observed when the current step occurs, marking the transitions between the two \npolarization states. (B) Change in critical bias vs z with best -fit lines corresponding to critical \nelectric fields Ec=0.6 GV m-1 for positive applied electric fields and Ec=0.7 GV m-1 for negative \napplied electric fields. Vertical error bars represent the width of the step in  I(V). \n-0.4-0.2 0.0 0.2 0.4 0.6-1012345 1000 pA\n500 pA\n100 pACurrent (nA)\nBias (V)10 pA\n0.0 0.5 1.0 1.5 2.0-0.6-0.30.60.9\nBackwardCritical Bias (V)\nDz (Å)Forward(A) (B) \nFig. 5. Controlling MAE with polarization switching.  (A) Hysteretic polarization switching cycle \nperformed over an adsorbed Co atom near a Cl vacancy in the NaCl ML. Spectroscopy begins at \npoint A. As the voltage is increased along line 1, a polarization switch is induced. The voltage is \nthe decreased along line  2 back to the same voltage as point A, but now the current is different \n(point B). The voltage is then further decreased along line 3 until the polarization switches back to \nthe original state. Finally, as the voltage is increased along line 4 the spectru m returns to point A. \nPoints A and B also indicate the position in the cycle where IETS is performed. Inset: topographic \nSTM image of the measured Co atom on the NaCl ML system (7 nm × 5 nm, Vset=-1.3 V, Iset=10 \npA). (B) Low-bias d I/dV spectroscopy perform ed for the Co atom near a Cl vacancy in the NaCl \nML in the two different polarization states. The MAE, as measured from the voltage of the \nsymmetric steps in d I/dV, is significantly different in each case. ( Iset=200 pA for state A and 370 \npA for state B, Vset= -100 mV). (C) Energy level diagram for an S=3/2 system in the presence of \nnegative axial magnetic anisotropy as a function of out -of-plane magnetic field for two different \nMAE values, corresponding to the substrate polarization in state A (black) and B \n(red). (35)(36)(37)(38) \n-200 0 200 400 600-101234Current (nA)\nBias (mV)A\nB12\n34(A) (B) (C)\nBz(T)DE (A.U.)\n0 3Ean(B)Ean(A)\n-30-20-10 0 10 20 302.02.2\nBias (mV)dI/dV (nA/V)\n3.23.64.04.4\nState B\nState A\n \nFig. 6.  Magnetic field dependence of spin -excitat ion energy for the self -poled and switched \npolarization state.  (A) Low energy spectroscopy of a Co atom adsorbed on NaCl ML near a Cl \nvacancy measured at B=0 and B=3 T (shifted vertically for clarity) with the NaCl ML in the self -\npoled state (state A). A distinct Zeeman splitting of the Kondo resonance (vertical red arrows) near \nEF is observed at 3 T, as has been observed for Co on bare Cu 2N (28). A slight shift of the inelastic \nsteps with magnetic field, which is expected to be very small for these relatively small changes in \nB (28), can also be discerned.  (Vset=-100 mV, Iset=53 pA, Vm=125 V) (B) Same as panel A but for \nthe switched state (state B). ( Vset=-100 mV, Iset=61 pA, Vm=125 V). \n \nSupplementary Materials  \nFigures S1 -S2 \n \n  \n(B) \n-15 -10 -5 0 5 10 154567\nBias (mV)State B 0 TdI/dV (nA/V)State B 3 T\n-30 -20 -10 0 10 20 302,02,22,42,6\nBias (mV)State A 3 TdI/dV (nA/V)\nState A 0 T\n (A) Supplementary  Materials  \n \nFigures  S1-S2 \n \n \nFig. S1. Adsorption and manipulation of Co atoms on NaCl ML/Cu 2N. (A) Co atom brought to a \nCl site with lateral manipulation (marked with the green arrow). (7  nm × 4 nm). (B) Co atom from \npanel A brought to a Na site (marked with the green arrow) (7  nm × 4 nm). (C) Co atom sitting on \ntop of a Na site. A Cl vacancy is highli ghted with a red circle while the Co atom is marked with the \ngreen arrow (10  nm × 2.2 nm). (D) Co atom brought close to a Cl vacancy, resulting in a change in \nits apparent shape and size. (10  nm × 2.2 nm). (E) Co atom naturally adsorbed on top of a Cl \nvacancy. The adsorbed Co atom can be identified as the cloud -like feature (green arrow) while the \nCl vacancy has the characteristic four lobe structure (10 nm × 4 nm). All topographies were \nmeasured at Vset=-1.3 V, Iset=10 pA  except panel D with  Iset=50 pA . All manipulations were \ninitialized at Vset= -30mV, Iset=10 pA.  \n  \n(C)\n(D)\n(E)\n(A)\n(B) \nFig. S2.  Low energy spectroscopy of different on Co atoms adsorbed near a Cl vacancy in the NaCl \nML. (A-C) Low-bias d I/dV spectroscopy performed for three  different Co atom s near a Cl vacancy \nin the NaCl ML in the two different NaCl polarization states . The general features are the same in \nall cases, though the MAE either increases or decreases from state A to B depending on the details \nof the local environment, which cannot be resolved by STM imaging. We note that the main factor \ndetermining the single -ion MAE is the charge distribution around an atom, which fixes the crystal \nelectric field and influences the orbital configuration. Similar variations are also observed for \nchanges in the Kondo resonance. The setpoint conditions are (A) Vset=-20 mV, Iset=50 pA (State A) \nand Iset=20 pA (State B), Vm=125 V; (B) Vset=-20 mV, Iset=50 pA, Vm=125 V; (C)  Vset=-15 mV, \nIset=30 pA, Vm=125 V. \n \n \n(A) (B) (C)\n-15-10 -5 0 5 10 151.62.02.42.8\nState AdI/dV (nA/V)\nBias (mV)State B\n-15-10 -5 0 5 10 153.03.23.43.6\nBias (mV)dI/dV (nA/V)\n1.11.21.31.4State A\nState B\n-20 -10 0 10 201.52.02.53.03.5\nState AdI/dV (nA/V)\nBias (mV)State B"    },    {        "title": "2302.14745v1.Epitaxial_growth_and_characterization_of__001___NiFe_M____20____M___Cu__CuPt_and_Pt__superlattices.pdf",        "content": "arXiv:2302.14745v1  [cond-mat.mtrl-sci]  28 Feb 2023Epitaxial growth and characterization of (001) [NiFe/M] 20(M = Cu, CuPt and Pt)\nsuperlattices\nMovaffaq Kateb,1,2,∗Jon Tomas Gudmundsson,1,3,†and Snorri Ingvarsson1,‡\n1Science Institute, University of Iceland, Dunhaga 3, IS-10 7 Reykjavik, Iceland\n2Condensed Matter and Materials Theory Division, Departmen t of Physics,\nChalmers University of Technology, SE-412 96 Gothenburg, S weden\n3Space and Plasma Physics, School of Electrical Engineering and Computer Science,\nKTH Royal Institute of Technology, SE-100 44, Stockholm, Sw eden\n(Dated: March 1, 2023)\nWe present optimization of [(15 ˚A) Ni 80Fe20/(5˚A) M] 20single crystal multilayers on (001) MgO,\nwith M being Cu, Cu 50Pt50and Pt. These superlattices were characterized by high reso lution X-ray\nreflectivity (XRR) and diffraction (XRD) as well as polar mapp ing of important crystal planes. It\nis shown that cube on cube epitaxial relationship can be obta ined when depositing at substrate\ntemperature of 100◦C regardless of the lattice mismatch (5% and 14% for Cu and Pt, respectively).\nAt lower substrate temperatures poly-crystalline multila yers were obtained while at higher substrate\ntemperatures {111}planes appear at ∼10◦off normal to the film plane. It is also shown that as the\nepitaxial strain increases, the easy magnetization axis ro tates towards the direction that previously\nwas assumed to be harder, i.e. from [110] to [100], and eventu ally further increase in the strain makes\nthe magnetic hysteresis loops isotropic in the film plane. Hi gher epitaxial strain is also accompanied\nwith increased coercivity values. Thus, the effect of epitax ial strain on the magnetocrystalline\nanisotropy is much larger than what was observed previously in similar, but polycrystalline samples\nwith uniaxial anisotropy (Kateb et al.2021).\nPACS numbers: 75.30.Gw, 75.50.Bb, 73.50.Jt, 81.15.Cd\nKeywords: NiFe; Superlattice; Magnetic Anisotropy; Micro structure; Substrate Temperature\nI. INTRODUCTION\nSince the discovery of the giant magneto-resistance\n(GMR) effect by Fert [ 1] and Gr¨ unberg [ 2] in the late\n1980s, magnetic multilayers have been widely studied.\nIn many cases they present unique features that cannot\nbe achieved within the bulk state namely inter-layer ex-\nchange coupling [ 3], magnetic damping, due to the in-\nterface [4,5] rather than alloying [ 6], and perpendicular\nmagnetic anisotropy [ 7].\nThe GMR discovery, without a doubt, was an outcome\nof the advances in preparation methods such as molec-\nular beam epitaxy (MBE), that enabled deposition of\nmultilayer films with nanoscale thicknesses [ 8]. Thus, a\ngreat deal of effort has been devoted to enhancing the\npreparation methods over the years using both simula-\ntions [9–12] and experiments (cf. Ref. [ 13] and references\ntherein). Permalloy (Py) multilayers with non-magnetic\n(NM) Pt [ 13–15] or Cu [ 12,16–19] as spacers have been\nstudied extensively in recent years. Various deposition\nmethods have been utilized for preparing magnetic mul-\ntilayers such as MBE [ 16], pulsed laser deposition (PLD)\n[20], ion beam deposition [ 12,21], dc magnetron sputter-\ning (dcMS) [ 3,14,17,18], and more recently, high power\nimpulse magnetron sputtering (HiPIMS) [ 13].\nPermalloy (Py) is a unique material with regards to\n∗movaffaq.kateb@chalmers.se\n†tumi@hi.is\n‡sthi@hi.isstudying magnetic anisotropy, which has been shown to\nstrongly depend on the preparation method [ 22]. For\ninstance, uniaxial anisotropy can be induced in polycrys-\ntalline Py by several means [ 23]. However, it has been\nthought that the cubic symmetry of single crystal Py en-\ncourages magneto-crystalline anisotropy, while uniaxial\nanisotropy cannot be achieved. We have recently shown\nthatusingHiPIMS depositiononecandecreasetheNi 3Fe\n(L12) order, but maintain the single crystal form, to\nachieve uniaxial anisotropy. We attributed this to the\nhigh instantaneous deposition rate during the HiPIMS\npulse [24], which limits ordering compared to dcMS that\npresent cubic (biaxial) anisotropy. Regarding Py multi-\nlayers there has been a lot of focus on magneto-dynamic\nproperties recently while the effects of interface strain\non magnetic anisotropy has not received much attention.\nRooket al.[16]preparedpolycrystallinePy/Cumultilay-\ners by MBE and reported a weak anisotropy in them, i.e.\nhysteresis loops along both the hard and easy axes with\ncomplete saturation at higher fields. They compared the\ncoercivity values ( Hc) and saturation fields of their sam-\nples toHcand anisotropy field ( Hk) of sputter deposited\nmultilayers showing uniaxial anisotropy and concluded\nthat the latter gives more than twice harder properties.\nThey also reported an increase in Hcwith Py thickness\nand attributed this to the interface strain that relaxes\nwith increased thickness. Corrˆ ea et al.[14] prepared\nnanocrystalline Py/Pt multilayers on rigid and flexible\nsubstrates and in both cases obtained weak anisotropy\nbut two orders of magnitude larger Hc. Unfortunately,\nthey did not mention any change in magnetic anisotropy2\nupon straining the flexible substrate.\nRecently we showed that utilizing increased power to\nthe dcMS process, and in particular, by using HiPIMS\ndeposition that the interface sharpness in polycrystalline\n[Py/Pt] 20multilayers can be improved, due to increased\nionization of the sputtered species [ 13].Briefly,in dcMS\ndeposition the film forming material is composed mostly\nof neutral atoms [ 25], while in HiPIMS deposition a sig-\nnificant fraction of the film forming material consists of\nions [26,27]. In fact we have shown that higher ioniza-\ntion of the film-forming material leads to smoother film\nsurfaces and sharper interfaces using molecular dynam-\nics simulations [ 28,29]. We also showedthat by changing\nthe non-magnetic spacer material one can increase inter-\nface strain that is accompanied with higher Hc,Hkand\nlimited deterioration of uniaxial anisotropy [ 13].\nAnother aspectofpreparationisthat depositioncham-\nbers for multilayers mostly benefit from oblique deposi-\ntion geometry, which encourage uniaxial anisotropy in\nPy. The origin of uniaxial anisotropy induced by oblique\ndeposition has been proposed to be self-shadowing, but\nthis has not been systematically verified. We demon-\nstrated uniaxial anisotropy, even in atomically smooth\nfilms with normal texture, which indicates lack of self-\nshadowing [ 30,31]. We also showed that oblique deposi-\ntion is more decisive in definition of anisotropy direction\nthan application of an in-situmagnetic field for inducing\nuniaxial magnetic anisotropy. Also for polycrystallinePy\nfilms oblique deposition by HiPIMS presents a lower co-\nercivity and anisotropy field than when dcMS deposition\nis applied [ 32–34]. While none of the above mentioned\nresults verify self-shadowing they are consistent with our\ninterpretation of the order i.e. oblique deposition induces\nmore disorder than in-situmagnetic field and HiPIMS\nproduce more disorder than dcMS. Note that the level of\norder in polycrystals cannot be easily observed by X-ray\ndiffraction. Inthisregardweproposedamethodformap-\nping the resistivity tensor that is very sensitive to level of\norder in Py [ 23,35]. We reported much higher coercivity\nand deterioration of uniaxial anisotropy in (111)Py/Pt\nmultilayersobtainedbyHiPIMSdepositionofthePylay-\ners [13]. We attributed the latter effect to the interface\nsharpness and higher epitaxial strain when HiPIMS is\nutilized for Py deposition.\nHere, we study the properties of Py superlattices de-\nposited by dcMS with Pt, Cu and CuPt as non-magnetic\nspacers. Pt and Cu were chosen as spacer because they\nhavelatticeparametersof3.9and3.5 ˚A, respectively, and\ntherefore provide varying strain to the Py film which has\nlattice constant of 3.54 ˚A. In this regard, calibration of\nthe substratetemperatureduringdepositionwith respect\nto the desired thickness is of prime importance [ 36]. It\nis worth mentioning that dcMS deposition is expected to\ngive more ordered single crystal (001) Py layers in which\ncrystalline anisotropy is dominant [ 22]. This enables un-\nderstanding to what extent interface strain will affect\nmagnetocrystalline anisotropy of Py which we will show\nis much larger than the changes in uniaxial anisotropy inour latest study [ 13]. Section IIdiscusses the deposition\nmethod and process parameters for the fabrication of the\nsuperlattices and the characterization methods applied.\nIn Section IIIthe effects of substrate temperature on the\nproperties of the Py/Cu system are studied followed by\nexploring the influence of varying the lattice parameter\nof the non-magnetic layeron the structural and magnetic\nproperties of the superlattice. The findings are summa-\nrized in Section IV.\nII. EXPERIMENTAL APPARATUS AND\nMETHODS\nThe substrates were one side polished single crystal\n(001)MgO (Crystal GmbH) with surface roughness <5˚A\nand of dimensions 10mm ×10mm×0.5mm. The MgO\nsubstrates were used as received without any cleaning\nbut were baked for an hour at 600◦C in vacuum for\ndehydration, cooled down for about an hour, and then\nmaintained at the desired temperature ±0.4◦C during\nthedeposition. Thesuperlatticesweredepositedinacus-\ntom built UHV magnetron sputter chamber with a base\npressure below 5 ×10−7Pa. The chamber is designed\nto support 5 magnetron assemblies and targets, which\nare all located 22cm away from substrate holder with a\n35◦angle with respect to substrate normal. The shut-\nters were controlled by a LabVIEW program (National\nInstruments). The deposition was made with argon of\n99.999% purity as the working gas using a Ni 80Fe20at.%\nand Cu targets both of 75mm diameter and a Pt target\nof 50mm in diameter.\nThe Py depositions were performed at 150W dc power\n(MDX 500 power supply from Advanced Energy) at ar-\ngon working gas pressure of 0.25Pa which gives deposi-\ntion rate of 1.5 ˚A/s. Both pure Cu and Pt buffer layers\nweredeposited at dc powerof20W. For the deposition of\nCuPt alloy we calibrated Cu 50Pt50at.% at dc power at\n10 and 13W for Cu and Pt, respectively. This selection\nof powers provide a similar deposition rate of 0.45 ˚A/s in\nall cases. In order to ensure that the film thickness is as\nuniform as possible, we rotate the sample at ∼12.8rpm.\nThese deposition processes were repeated to fabricate su-\nperlattices consisting of 20 repetitions of 15 ˚A Py and 5 ˚A\nPt, Cu or Cu 50Pt50at.% (CuPt).\nX-ray diffraction measurements (XRD) were carried\nout using a X’pert PRO PANalitical diffractometer (Cu\nKα1and K α2lines, wavelength 0.15406 and 0.15444nm,\nrespectively) mounted with a hybrid monochroma-\ntor/mirror on the incident side and a 0.27◦collimator\non the diffracted side. We would like to remark that,\nKα2separation at 2 θ= 55◦is only 0.2◦and much less at\nthe smaller angles i.e. where our multilayer peaks are lo-\ncated. This is anorderofmagnitudesmallerthan the full\nwidth half maximum (FWHM) of our multiplayer and\nsatellite peaks. A line focus was used with a beam width\nof approximately 1mm. The film thickness, mass den-\nsity, and surface roughness, was determined by low-angle3\nX-ray reflectivity (XRR) measurements with an angular\nresolution of 0.005◦, obtained by fitting the XRR data\nusing the commercial X’pert reflectivity program, that is\nbased on the Parrat formalism [ 37] for reflectivity.\nThe magnetic hysteresis was recorded using a home-\nmadehigh sensitivitymagneto-opticKerreffect (MOKE)\nlooper. We use a linearly polarized He-Ne laser of wave-\nlength 632.8 nm as a light source, with Glan-Thompson\npolarizers to further polarize and to analyze the light\nafter Kerr rotation upon reflection off the sample sur-\nface. TheGlan-Thompsonpolarizerslinearlypolarizethe\nlight with a high extinction ratio. They are cross polar-\nized near extintion, i.e. their polarization states are near\nperpendicular and any change in polarization caused by\nthe the Kerr rotation at a sample’s surface is detected\nas a change in power of light passing through the ana-\nlyzer. The coercivitywas read directly from the easyaxis\nloops. The anisotropy field is obtained by extrapolating\nthe linear low field trace along the hard axis direction to\nthe saturation magnetization level, a method commonly\nused when dealing with effective easy axis anisotropy.\nIII. RESULTS AND DISCUSSION\nA. Effect of substrate temperature on structural\nand magnetic properties\nFigure1shows the XRR results from Py/Cu superlat-\ntices deposited at different substrate temperatures. The\nΛ andδindicated in the figure are inversely proportional\nto the superlattice period and the total thickness, re-\nspectively. It can be clearly seen that the fringes decay\nfaster for a Py/Cu superlattice deposited at substrate\ntemperature of 21◦C and 200◦C than when deposited at\n100◦C. This indicates lower surface roughness obtained\nin the Py/Cu superlattice deposited at 100◦C. When de-\nposited at room temperature, the large lattice mismatch\nbetween MgO and Py/Cu does not allow depositing a\nhigh quality superlattice. For substrate temperature of\n200◦C, however, it is difficult to grow a continuous Cu\nlayer with such a low thickness (5 ˚A). This is due to the\ndewetting phenomenon which causes the minimum Cu\nthickness that is required to maintain its continuity to\nbe 12˚A. Earlier, it has been shown that for substrate\ntemperature up to 100◦C Py/(1 ˚A) Cu showed a limited\nintermixing upon annealing [ 17]. The optimum substrate\ntemperature for deposition obtained here is very close to\n156◦C which has earlier been reported for the deposi-\ntion of (001)Fe/MgO [ 38] and (001)Fe 84Cu16/MgO [39]\nsuperlattices. We would like to remark that in our previ-\nous study we deposited 5nm Ta underlayer to reduce the\nsubstrate surface roughness [ 13]. However, Ta on MgO\nis non-trivial due to the large lattice mismatch (22%).\nBesides, Ta underlyer encourages polycrystalline /an}bracketle{t111/an}bracketri}ht\ntexture normal to substrate surface that does not serve\nour purpose here.\nFigure2shows the result of symmetric ( θ−2θ) XRD1 2 3 4 5 6 7 8 9 10\n2 (°)00.20.40.60.811.21.4Normalized log intensity (arb. units)Py/Cu 21 °C\nPy/Cu 100 °C\nPy/Cu 200 °C\nFIG. 1. Comparison of the XRR pattern from [Py/Cu] 20\nsuperlattices deposited on (001)MgO at different substrate\ntemperatures. The Λ and δare inversely proportional to the\nPy/Cu period and total thickness, respectively.\nscannormaltothefilmforPy/Cusuperlatticesdeposited\nat different substrate temperatures. It can be seen that\nno Cu and Py peak were detected in the superlattice de-\nposited at room temperature. Thus, epitaxial growth of\nPy and Cu were suppressed by the low substrate tem-\nperature. Furthermore, we studied room temperature\ndeposited Py/Cu using grazing incidence XRD which in-\ndicated a polycrystalline structure (not shown here). For\nsubstrate temperature of 100 – 200◦C there are clear\n(002)Py/Cu peaks indicating an epitaxial relationship\nin the (001)Py /bardbl(001)Cu /bardbl(001)MgO stack. However,\nthere is no sign of satellite peaks due to the Λ (Py/Cu)\nperiod. We explain this further when comparing the\nPy/Cu, Py/Pt, and Py/CuPt superlattices in Section\nIIIB.\nFigure3shows the pole figures from the {200}and\n{111}planes for Py/Cu superlattices deposited at differ-\nent substrate temperatures. For the Py/Cu superlattice\ndeposited at 21◦C, there is only a peak in the middle of\nthe{111}pole figure that indicates a weak /an}bracketle{t111/an}bracketri}htcontri-\nbution normal to the film plane. For a superlattice de-\nposited with substrate temperature of 100◦C the{200}\npole figure indicates an intense spot at ψ= 0 that is\ncorresponding to (002)Py/Cu planes parallel to the sub-\nstrate. There is also a weaker four-fold spot at ψ= 90◦\nandφ= 0,90,180 and 270◦from the {200}planes paral-\nlel to the substrate edges. In the {111}pole figure only\nfour-fold points appear at ψ= 54.7◦and with 45◦shifts\ninφwith respect to substrate edges. These are the char-\nacteristics of the so-called cube on cube epitaxy achieved\nat 100◦C. For deposition with substrate temperature of\n200◦C, however, there is a weak {111}ring atψ= 7.5◦.\nNotethatthese {111}planeswerenotdetectedbynormal\nXRD because the (111)Py/Cu peak appears at 2 θ≃42◦4\n30 40 50 60 70 80 90\n2 (°)00.20.40.60.811.21.4Normalized log intensity (arb. units)\nMgO\nCu\nPyPy-Cu 21 °C\nPy-Cu 100 °C\nPy-Cu 200 °C\nFIG. 2. Comparison of the XRD pattern from [Py/Cu] 20\nsuperlattices deposited on (001)MgO at different substrate\ntemperatures. The intense peak belongs to (002) planes of\nMgO and and the other peak is due to (002) planes of Py/Cu\nmultilayer.\nwhich is masked by the strong (002)MgO peak normal\nto the film plane.\nFIG. 3. Comparison of the {200}and{111}pole figures from\n[Py/Cu] 20superlattices deposited on (001)MgO at substrate\ntemperatures of 21, 100, and 200◦C. The background is re-\nmoved for better illustration.\nFigure4compares the MOKE response of Py/Cu su-\nperlattices deposited at different substrate temperatures.\nFor a superlattice deposited at room temperature, uniax-\nial anisotropy along the [100] direction is evident. This is\nexpected since the oblique deposition in a co-deposition\nchamber tends to induce uniaxial anisotropy in Py films\n[22,30–32]. However,the obliquedepositioncannot over-\ncome magnetocrystalline anisotropy due to symmetry in\nan orderedsingle crystalPy [ 22]. Thus, the lowsubstratetemperature must be accounted for the limiting order in\nthe Py layer and presence of uniaxial anisotropy.\n-10 1 (a) Py/Cu 21 °C  \n-10 1 (b) Py/Cu 100 °C Normilized intensity (arb. units) \n-6 -4 -2 0 2 4 6\nH (Oe)-10 1 (c) Py/Cu 200 °C  \nFIG. 4. MOKE response of different [Py/Cu] 20superlattices\ndeposited on (001)MgO at substrate temperatures (a) 21, (b)\n100 and (c) 200◦C. Each legend indicates probing orientation\nin the substrate plane.\nFor deposition at higher substrate temperatures, how-\never, biaxial anisotropy was obtained with the easy axes\nalong the [110] directions in plane. It is worth mention-\ning that the bulk crystal symmetry gives the easy axis\nalong the [111] direction which is forced into the film\nplane along the [110] direction due to shape anisotropy\n[22]. In the Py/Cu superlattice grown at 100◦C (Fig-\nure4(b)),/an}bracketle{t1¯10/an}bracketri}htis clearly an easy direction, with a very\nlowHcof 0.7Oe and double-hysteresis loops along the\n/an}bracketle{t110/an}bracketri}htdirection that saturates at 1.2Oe. For the Py/Cu\nsuperlattice deposited at 200◦C (Figure 4(c)), it seems\nthe double-hysteresis loops overlap and the other easy\naxis gives a step that in total present increased coerciv-\nity. With increasing substrate temperature not only do\nthe coercivities vary but also the shapes of the hystere-\nsis curves are different. When the substrate temperature\nduring deposition is 21◦C the magnetization, shown in\nFigure4(a), is much like we obtain for polycrystalline\nsingle layer films. When the substrate temperature is\nhigher, as shown in Figures 4(b) and (c), however, the\nanisotropy has rotated by 45 degrees and the hystere-\nsis loops have changed. The intermediate steps in the\nhysteresis curves are caused by antiferromagnetic align-\nment ofthe magneticlayers,that minimizes the exchange\nand dipolar magnetic interactions. In some cases this re-\nsultsinperfectlyzeromagneticremanence, whileinother\ncases the cancellation is not perfect. The non-magnetic\nCu spacer layer is only 5 ˚A in our case, just at the on-\nset of the first antiferromagnetic exchange coupling peak\nobserved by Parkin [ 40]. Double hysteresis curves have\nbeen observed in the Py/Cu system [ 18]. Note that Ni\nis miscible in Cu and during annealing a mixing of Ni5\nand Cu is possible. Such intermixing causes a decrease\nof magnetic homogenity and a reduction in the GMR\n[17,18].\nB. Effect of strain on structural and magnetic\nproperties\nIn order to explore the influence of strain on the\nmagnetic properties we deposited NM layers of Pt and\nCu50Pt50at. % alloy in addition to Cu discussed in Sec-\ntionIIIA. Pt has lattice constant of 3.9 ˚A which is larger\nthan of Py, which has lattice constant of 3.54 ˚A. There-\nfore, by going from Cu to Cu 50Pt50and then to Pt the\nstrainis graduallyincreased. Figure 5showsXRR results\nfrom different superlattices deposited on (001)MgO at\n100◦C. Note that the Λ peak is suppressed in the Py/Cu\nsuperlattice. One may think this arises from a diffused\nPy/Cu interface that leads to smooth density variation\nat the interface. This is not the case here and the Λ\npeaks intensity decreases due the similar density of Py\nand Cu. The latter has been shown to reduce the resolu-\ntion of the XRR measurement in Si/SiO 2by a few orders\nof magnitude [ 41].\n1 2 3 4 5 6 7 8 9 10\n2 (°)-0.200.20.40.60.811.21.4Normalized log intensity (arb. units)Py/Cu\nPy/Pt\nPy/CuPt\nFIG. 5. XRR measurements from the various superlat-\ntices, [Py/Cu] 20, [Py/Pt] 20, and [Py/CuPt] 20, deposited on\n(001)MgO at substrate temperature of 100◦C.\nThe layers thickness and their mass density as well as\nsurface and interface roughness obtained by fitting XRR\nresults for deposition at substrate temperature of 100◦C\nare summarized in table I. The period Λ is in all cases\nabout 19 ˚A withtPy∼16˚A andtNM∼3˚A. The film\nmass density of the Py layers is the highest (8.74 g/cm3)\nin the Py/Cu stack but is lowest (7.45 g/cm3) in the\nPy/CuPt stack.\nFigure6showsthe XRD results from Py/NM superlat-\nticesdeposited on(001)MgOatsubstratetemperatureof\n100◦C. The most intense peak, indicated by the verticalTABLE I. The Py and NM layer thicknesses ( t), roughness\n(Ra) and density ( ρ) extracted by fitting the XRR results\nof different superlattices deposited on (001)MgO at 100◦C\nsubstrate temperature.\nSample t(˚A) Ra (˚A)ρ(g/cm3)\nPy NM Λ Py NM Py NM\nPy/Cu 15.83.4619.37.626.258.749.8\nPy/Pt 15.92.9718.95.923.248.5527.2\nPy/CuPt 15.93.4319.32.254.947.4526.8\ndashed line, belong to the (002) planes of the MgO sub-\nstrate. Rather than exhibiting separate peaks for Py and\nNM, a single main (002)Py/NM peak is evident from all\nthe superlattices and indicated by 0 in the figure. This\npeak is closer to the Py side due to the higher thick-\nness of Py layers compared to the NM layers. The other\npeaks, indicated by ±are satellite peaks. The asymmet-\nric intensity of the satellite peaks is associated with the\nstrain in direction normal to the substrate. It is also\nclear that the main (002)Py/CuPt peak is located be-\ntween the Py/Cu and Py/Pt peaks. Due to the lack of\nany other peaks, we conclude that these superlattices are\nsingle crystalline with their /an}bracketle{t001/an}bracketri}htorientation normal to\nthe substrate surface. The satellite peaks suggest the pe-\n30 40 50 60 70 80 90\n2 (°)00.20.40.60.811.21.41.6Normalized log intensity (arb. units)0-3-2-10\n+1-3-2-10\n+1\n+2MgO\nPt\nCu\nPy Py/Cu\nPy/Pt\nPy/CuPt\nFIG. 6. Comparison of the XRD results from the various su-\nperlattices, [Py/Cu] 20, [Py/Pt] 20and [Py/CuPt] 20, deposited\nat substrate temperature of 100◦C. All the peaks are due to\nthe (002) plane and the vertical dashed line indicate the (00 2)\npeak position for the bulk state.\nriod Λ to be of 18.8 and 19.2 ˚A for Py/Pt and Py/CuPt,\nrespectively, which is slightly off compared to the values\ngiven in table I.\nFigure7shows the {002}and{111}pole figures from\ndifferent superlattices deposited on (001)MgO at sub-\nstrate temperature of 100◦C. Since the Py/Pt and\nPy/CuPt superlattices exhibit multiple (002) peaks the\npole figure were obtained for the main peak (indicated6\nby 0 in figure 6). It can be seen that all the pole figures\nare very similar. All these pole figures indicate a cube on\ncube epitaxial relationship.\nFIG. 7. Comparison of the {002}and{111}pole figures\nfrom the various superlattices, [Py/Cu] 20, [Py/Pt] 20and\n[Py/CuPt] 20, deposited on (001)MgO at substrate temper-\nature of 100◦C. The background is removed for better illus-\ntration.\nFigure8depicts the MOKE response from Py/Pt and\nPy/CuPtsuperlatticespreparedat100◦C.ForthePy/Pt\nsuperlattice we did not detect any clear easy direction\nin the film plane, the film appeared almost isotropic in\nthe film plane with Hcof 60 – 75Oe. The hysteresis in\nthe [100] and [110] directions are displayed in figure 8(a).\nAsidefrom Hcof3Oe,thePy/CuPtsuperlatticepresents\nbiaxial anisotropy similar to Py/Cu, cf. figure 4(b) and\n(c). However, a Py/Cu superlattice exhibits an easy\naxis along the [110] directions, while an easy axis ap-\npears along the [100] orientations for a Py/CuPt super-\nlattice. Note that the [100] directions are harder than\nboth the[110]and[111]directions. However,forcingeasy\naxes along the [100] direction in the single crystal Py on\n(001)MgO has been reported previously [ 42].\nForthepolycrystallinebuthighly(111)texturedPy/M\nmultilayers we observed limited change in coercivity and\nopening in hard axis with interface strain due to the\nchoice of M [ 13]. Here, for Py/Pt the coercivity increases\nan order of magnitude and cubic anisotropy is almost\ndestroyed.IV. SUMMARY\nIn summary it is shown that Py superlattices can be\nsuccessfullydepositedon(001)MgOwithinanarrowsub-\nstrate temperature window around 100◦C. For small lat-\ntice mismatch of 5% superlattice the easy axes detected\nalong the [110] directions is similar to the single crystal\n-100 -50 0 50 100\nH (Oe)-10 1 (a) Py/Pt\n-10 -5 0 5 10\nH (Oe)-10 1 Normilized intensity (arb. units)(b) Py/CuPt\nFIG. 8. MOKE response from (a) [Py/Pt] 20and (b)\n[Py/CuPt] 20superlattices. The legend indicate probing di-\nrection in the substrate plane.\nPy. It is also shown that the moderate lattice mismatch\n(7%) rotates the easy axes towards the [100] orientation\nandthecoercivityincreases. The higherlatticemismatch\nof 14% present nearly isotropic behaviour and a very\nhigh coercivity, simultaneously. Thus, the results indi-\ncate that the changes in magnetocrystalline anisotropy\ndue to epitaxial strain are much larger than the changes\nwe observed earlier in the case of uniaxial anisotropy.\nACKNOWLEDGMENTS\nThe authors would like to acknowledge helpful com-\nments and experimental help from Dr. Fridrik Mag-\nnus and Einar B. Thorsteinsson. This work was par-\ntially supported by the Icelandic Research Fund Grant\nNos. 228951, 196141, 130029 and 120002023.\n[1]M. N. Baibich, J. M. Broto, A. Fert, F. N. Van Dau,\nF. Petroff, P. Etienne, G. Creuzet, A. Friederich, and\nJ. Chazelas, Phys. Rev. Lett. 61, 2472 (1988) .\n[2]G. Binasch, P. Gr¨ unberg, F. Saurenbach, and W. Zinn,\nPhysical Review B 39, 4828 (1989) .[3]S. S. P. Parkin, N. More, and K. P. Roche,\nPhysical Review Letters 64, 2304 (1990) .\n[4]S. Mizukami, Y. Ando, and T. Miyazaki,\nJpn. J. Appl. Phys. 40, 580 (2001) .\n[5]S. Ingvarsson, L. Ritchie, X. Liu, G. Xiao,\nJ. Slonczewski, P. Trouilloud, and R. Koch,7\nPhysical Review B 66, 214416 (2002) .\n[6]S. Ingvarsson, G. Xiao, S. S. P. Parkin, and R. H. Koch,\nApplied Physics Letters 85, 4995 (2004) .\n[7]M. T. Johnson, P. J. H. Bloemen, F. J. A.\nDen Broeder, and J. J. de Vries,\nReports on Progress in Physics 59, 1409 (1996) .\n[8]A. Fert, Rev. Mod. Phys. 80, 1517 (2008) .\n[9]X. Zhou and H. Wadley,\nJournal of Applied Physics 84, 2301 (1998) .\n[10]X. Zhou and H. Wadley,\nPhilosophical Magazine 84, 193 (2004) .\n[11]X. Zhou, R. Johnson, and H. Wadley,\nPhysical Review B 69, 144113 (2004) .\n[12]C. B. Ene, G. Schmitz, R. Kirchheim, and A. H¨ utten,\nActa Materialia 53, 3383 (2005) .\n[13]M. Kateb, J. T. Gudmundsson, and S. Ingvarsson,\nJournal of Magnetism and Magnetic Materials 538, 168288 (2021) .\n[14]M. A. Corrˆ ea, R. Dutra, T. L. Marcondes,\nT. J. A. Moric, F. Bohn, and R. L. Sommer,\nMaterials Science and Engineering B 211, 115 (2016) .\n[15]D. Ley Dom´ ınguez, R. J. S´ aenz-Hern´ andez, A. F.\nArzate, A. I. Arteaga Duran, C. E. Ornelas\nGuti´ errez, O. Sol´ ıs Canto, M. E. Botello-Zubiate,\nF. J. Rivera-G´ omez, A. Azevedo, G. L. da Silva,\nS. M. Rezende, and J. A. Matutes-Aquino,\nJournal of Applied Physics 17, 17D910 (117) .\n[16]K. Rook, A. M. Zeltser, J. O. Artman, D. E.\nLaughlin, M. H. Kryder, and R. M. Chrenko,\nJournal of Applied Physics 69, 5670 (1991) .\n[17]L. van Loyen, D. Elefant, D. Tietjen, C. M.\nSchneider, M. Hecker, and J. Thomas,\nJournal of Applied Physics 87, 4852 (2000) .\n[18]M. Hecker, D. Tietjen, H. Wendrock, C. M.Schneider,\nN.Cramer, L. Malkinski, R. E. Camley, and Z. Celinski,\nJournal of Magnetism and Magnetic Materials 247, 62 (2002) .\n[19]S. Heitmann, A. H¨ utten, T. Hempel,\nW. Schepper, G. Reiss, and C. Alof,\nJournal of Applied Physics 87, 4849 (2000) .\n[20]J. Shen, Z. Gai, and J. Kirschner,\nSurface Science Reports 52, 163 (2004) .\n[21]M. Ueno and S. Tanoue,\nJournal of Vacuum Science and Technology A 13, 2194 (1995) .\n[22]M. Kateb, J. T. Gudmundsson, and S. Ingvarsson,\nAIP Advances 9, 035308 (2019) .\n[23]M. Kateb and S. Ingvarsson,\nJournal of Magnetism and Magnetic Materials 532, 167982 (2021) .\n[24]M. Kateb, J. T. Gudmundsson, P. Brault,\nA. Manolescu, and S. Ingvarsson,Surface and Coatings Technology 426, 127726 (2021) .\n[25]J.T.Gudmundsson, Plasma Sources Science and Technology 29, 113001 (2020) .\n[26]J. T. Gudmundsson, N. Brenning,\nD. Lundin, and U. Helmersson,\nJournal of Vacuum Science and Technology A 30, 030801 (2012) .\n[27]D. Lundin, T. Minea, and J. T. Gudmundsson, eds.,\nHigh Power Impulse Magnetron Sputtering: Fundamentals, Te chnologies, Challenges and Applications\n(Elsevier, Amsterdam, The Netherlands, 2020).\n[28]M. Kateb, H. Hajihoseini, J. T.\nGudmundsson, and S. Ingvarsson,\nJournal of Vacuum Science and Technology A 37, 031306 (2019) .\n[29]M. Kateb, J. T. Gudmundsson, and S. Ingvarsson,\nJournal of Vacuum Science and Technology A 38, 043006 (2020) .\n[30]M. Kateb and S. Ingvarsson, in\n2017 IEEE Sensors Applications Symposium (SAS)\n(IEEE, Glassboro, New Jersey, 13–15 March 2017) pp.\n1–5.\n[31]M. Kateb, E. Jacobsen, and S. Ingvarsson,\nJournal of Physics D: Applied Physics 52, 075002 (2018) .\n[32]M.Kateb, H.Hajihoseini, J.T.Gudmundsson, andS.In-\ngvarsson, J. Phys. D Appl. Phys. 51, 285005 (2018) .\n[33]H. Hajihoseini, M. Kateb, S. Ing-\nvarsson, and J. T. Gudmundsson,\nBeilstein Journal of Nanotechnology 10, 1914 (2019) .\n[34]M. Kateb, Induced uniaxial magnetic anisotropy in poly-\ncrystalline, single crystal and superlattices of permallo y,\nPh.D. thesis , University of Iceland (2019).\n[35]S. Ingvarsson, M. Kateb, and C. Scheuner, in\nBulletin of the American Physical Society ,Vol.62(APS,\n2017) pp. APS March Meeting 2017, abstract E47.012.\n[36]R. Loloee and M. A. Crimp,\nJournal of Applied Physics 92, 4541 (2002) .\n[37]L. G. Parratt, Physical Review 95, 359 (1954) .\n[38]R. Moubah, F. Magnus, T. Warnatz, G. K.\nP´ alsson, V. Kapaklis, V. Ukleev, A. Devishvili,\nJ. Palisaitis, P. Persson, and B. Hj¨ orvarsson,\nPhysical Review Applied 5, 044011 (2016) .\n[39]T. Warnatz, B. E. Skovdal, F. Mag-\nnus, H. Stopfel, D. Primetzhofer,\nA. Stein, R. Brucas, and B. Hj¨ orvarsson,\nJournal of Magnetism and Magnetic Materials 496, 165864 (2020) .\n[40]S. S. P. Parkin, Appl. Phys. Lett. 60, 512 (1992) .\n[41]J. Tiilikainen, J.-M. Tilli, V. Bosund, M. Mattila,\nT. Hakkarainen, J. Sormunen, and H. Lipsanen,\nJournal of Physics D: Applied Physics 40, 7497 (2007) .\n[42]F. Michelini, L. Ressier, J. Degauque, P. Baul´ es,\nA. R. Fert, J. P. Peyrade, and J. F. Bobo,\nJournal of Applied Physics 92, 7337 (2002) ."    },    {        "title": "2303.00347v1.Tailed_skyrmions____an_obscure_branch_of_magnetic_solitons.pdf",        "content": "Tailed skyrmions { an obscure branch of magnetic solitons\nVladyslav M. Kuchkin,1, 2, 3, \u0003Nikolai S. Kiselev,2and Pavel F. Bessarab1, 4\n1Science Institute, University of Iceland, 107 Reykjav\u0013 \u0010k, Iceland\n2Peter Gr unberg Institute and Institute for Advanced Simulation,\nForschungszentrum J ulich and JARA, 52425 J ulich, Germany\n3Department of Physics, RWTH Aachen University, 52056 Aachen, Germany\n4Department of Physics and Electrical Engineering, Linnaeus University, SE-39231 Kalmar, Sweden\n(Dated: March 2, 2023)\nWe report tailed skyrmions { a new class of stable soliton solutions of the 2D chiral magnet model.\nTailed skyrmions have elongated shapes and emerge in a narrow range of \felds near the transition\nbetween the spin spirals and the saturated state. We analyze the stability range of these solutions in\nterms of external magnetic \feld and magnetocrystalline anisotropy. Minimum energy paths and the\nhomotopies (continuous transitions) between tailed skyrmions of the same topological charge have\nbeen calculated using the geodesic nudged elastic bands method. The discovery of tailed skyrmions\nextends the diversity of already-known solutions illustrated by complex morphology solitons, such\nas tailed skyrmion bags with and without chiral kinks.\nI. INTRODUCTION\nChiral magnets represent a unique class of materi-\nals where the competition between local interaction,\nin particular, Heisenberg exchange and Dzyaloshinskii-\nMoriya interaction1,2(DMI), give rise to a vast variety\nof topological magnetic solitons such as k\u0019-skyrmions3{6,\nskyrmion bags7,8, and skyrmions with chiral kinks9,10.\nNote that the physical systems that exhibit such diver-\nsity of solitons possessing arbitrary topological charges\nand morphology are very rare in nature.\nThe primary parameter that distinguishes magnetic\nsolitons is the topological charge, which de\fnes the ho-\nmotopy class of a particular solution:\nQ=1\n4\u0019Z\nn\u0001(@xn\u0002@yn) dxdy; (1)\nwhere nis the magnetization unit vector \feld.\nThe solutions with identical Qcan be continuously\n(without the appearance of magnetic singularities) trans-\nformed into each other7. We refer to such transforma-\ntions as homotopies , which can be de\fned as follows.\nFor two magnetic textures n1(r) and n2(r) with iden-\nticalQ, there is a unity parametric vector \feld n(r;\u001c),\n\u001c2[0;1], which is continuously di\u000berentiable with re-\nspect of rand\u001cand obeys the boundary conditions\nn(r; 0) = n1(r),n(r; 1) = n1(r). Homotopy between\nany two con\fgurations n1(r) and n2(r) is not de\fned\nuniquely as there are in\fnitely many ways to introduce\nthe parameter \u001c. A constraint can be introduced by con-\nsidering only those homotopies that represent minimum\nenergy paths (MEPs), i.e. the paths lying lowermost on\nthe energy surface of the system. Homotopies satisfying\nthe MEP condition are also physically relevant as they\nde\fne thermal stability of the magnetic con\fgurations\nwithin harmonic rate theories. For solitons with di\u000ber-\nentQ, homotopies are, by de\fnition, impossible.\nThe representative and the most well-studied soliton in\nchiral magnets { \u0019-skyrmion with Q=\u00001 { is shown inFigure 1A-B. \u0019-skyrmions were theoretically predicted\nmore than thirty years ago by Bogdanov and Yablon-\nskii3{5. Nowadays, there is a long list of magnetic mate-\nrials where \u0019-skyrmions were observed experimentally11.\nAnother type of axially symmetric solitons called k\u0019-\nskyrmions ( k > 1) was presented by Bogdanov and\nHubert6. Despite the morphological diversity of k\u0019-\nskyrmions, they all belong to only two homotopy classes,\nwithQ= 0 andQ=\u00001 for even and odd k, respectively.\nThe non-axially symmetric solutions with arbitrary\ntopological charge, also known as skyrmion bags7,8and\nskyrmions with chiral kinks9,10were found only recently.\nThe skyrmion bags with positive topological charge have\nrecently been observed in FeGe plates using transmission\nelectron microscopy12. Co-existing skyrmion and its an-\ntiparticle (antiskyrmion) i.e. soliton with two chiral kinks\nwere observed experimentally in FeGe thin platelet13.\nIn this article, we introduce a new class of chiral\nmagnetic skyrmions we refer to as tailed skyrmions.\nWe demonstrate rich diversity of tailed skyrmions and\npresent a comprehensive study of their static proper-\nties, stability and \fnite-temperature dynamics. Tailed\nskyrmions can be thought of as various elongated\nskyrmions stabilized in the narrow range of external mag-\nnetic \felds close to the elliptical instability \feld of an or-\ndinary\u0019-skyrmion. Some examples of tailed skyrmions\nare shown in Fig. 1. To our knowledge, this type of soli-\ntons has not been reported in the literature.\nII. MODEL\nWe consider the two-dimensional (2D) micromagnetic\nmodel of a chiral magnet containing three energy terms:\nE=Z\nfwex(n) +wD(n) +wU(n)gldxdy; (2)\nwhere n=M=Msis the magnetization unit vector\n\feld which is uniform across the \flm thickness l,MsarXiv:2303.00347v1  [cond-mat.mes-hall]  1 Mar 20232\nis the saturation magnetization, wex(n) =Ajrnj2\nis the Heisenberg exchange interaction and wU(n) =\nMsBext(1\u0000nz) +K\u0000\n1\u0000n2\nz\u0001\nis the potential term con-\ntaining the Zeeman interaction and the easy-axis/easy-\nplane anisotropy. The external magnetic \feld is perpen-\ndicular to the plane of the \flm, Bextkez.\nThe DMI term wD(n) =Dw(n) is de\fned by combina-\ntions of Lifshitz invariants3, \u0003(k)\nij=ni@knj\u0000nj@kni. The\nresults presented below are valid for systems with Bloch-\ntype modulations14, N\u0013 eel-type modulations15{17as well\nas for crystals with D 2dor S 4point group symmetry3.\nWithout loss of generality, we assume Bloch-type DMI,\nw(n)=\u0003(x)\nzy+\u0003(y)\nxz, in our calculations. By introducing the\ncharacteristic size of chiral modulations LD= 4\u0019A=D\nand the characteristic magnetic \feld BD=D2=(2MsA),\nwe reduce the number of independent parameters to\ntwo, namely the dimensionless external magnetic \feld\nh=Bext=BDand anisotropy parameter u=K=(MsBD).\nIII. METHODS\nThis work involves several numerical methods. Stable\nmagnetic con\fgurations are obtained via direct energy\nminimization using the conjugate gradient method with\nstereographic projections of the n-\feld. For details, see\nSupplementary materials in Ref.18. MEPs) between the\nstable states are calculated using the geodesic nudged\nelastic band (GNEB) method19,20. Dynamical proper-\nties of tailed skyrmions are investigated via the numerical\nintegration of the Landau-Lifshitz-Gilbert (LLG) equa-\ntion21using the well-established semi-implicit method\nfrom Ref.22.\nThe calculations were performed and double-checked\nwith various software. In particular, we used Excalibur23\nand Mumax324for energy minimization and calculation\nof stability ranges. MEPs were calculated using the Spirit\ncode25. We used high-accuracy \fnite di\u000berence schemes\nin the energy minimization to remove numerical artifacts\nin the calculation of derivatives in Eq. (2). A typical\nmesh density in our simulations is 64 nodes per LD. The\ndetails can be found in Refs.7,9.\nFor the MEP calculations and stochastic LLG simula-\ntions, we used an e\u000bective spin-lattice model of nearest\nneighbors. All calculations were performed with periodic\nboundary conditions in the \flm plane. For visualization\nof the magnetization, we use the standard color code as\nexplained in Figure 1A and B.\nIV. INITIAL GUESSES\nTo obtain statically stable solutions for tailed\nskyrmions in Mumax3, we constructed initial guesses for\nthen-\feld by placing domains with magnetization an-\ntiparallel to the \feld, n= (0;0;\u00001), into the state with\nmagnetization pointing along the \feld, n= (0;0;1). In\nthe case of axially symmetric skyrmions, e.g., \u0019-skyrmionin Figure 1A and B, we use a circular shape domain with\nmagnetization down. For tailed skyrmions shown in Fig-\nure 1C-E, the domains with down magnetization should\napproximately mimic the shape of the skyrmion, which\ncan be achieved by placing circular domains in a chain.\nFor details, see the Supplementary materials with the\ncorresponding Mumax script.\nIn the simulation with Excalibur and Spirit, we used\nthe built-in options allowing one to drag and invert the\nmagnetization \feld interactively, as illustrated in Supple-\nmentary Movie 1. Notably, the interactive modi\fcation\nof the magnetization \feld with the drag tool permits for\nexamining whether the skyrmion has reached the optimal\nshape and size.\nV. RESULTS\nA. Stability diagram\nFigure 1C, D, and E show elementary tailed skyrmions.\nThe two solutions with two and three tails can be referred\nto as a stick-skyrmion and a spinner-skyrmion, respec-\ntively. We use these representative solutions to illustrate\nthe range of tailed skyrmion existence. In Figure 1F, the\ncritical \felds for stick-skyrmion and spinner-skyrmion are\nshown with blue and red curves, respectively. The \feld\nrange corresponding to stable tailed skyrmions is bound\nfrom above by the transition \feld into axially symmet-\nric\u0019-skyrmion, ht, and from below by elliptic instability\n\feld,he. Note that Figure 1F shows the deviation of ht\nandhefromh\u0003{ the critical \feld corresponding to the\ntransition between the spin-spiral state and the ferromag-\nnetic phase { so as to emphasize the di\u000berence between\nthe critical \felds which are quite close to each other. The\ndependence h\u0003(u) is the solution of the well-known equa-\ntion5:\n2\u0019\u00004p\nh+ 2u\u0000p\n2hpulnp\nh+ 2u+p\n2up\nh+ 2u\u0000p\n2u= 0;(3)\nwhich is derived from the criterion that the energy of\nthe isolated spiral (2 \u0019-domain wall) equals the energy of\nthe saturated ferromagnetic state. The functional depen-\ndenceh\u0003(u) is shown in the inset of Figure 1F.\nThe critical \felds for any tailed skyrmion solutions al-\nways meet at two points: i) the Bogomolnyi point h= 1,\nu=\u00000:5 and ii) the phase transition point between the\nspin spiral and ferromagnetic phases h= 0,u=\u00192=8.\nThus, the stability range of tailed skyrmions is quite sig-\nni\fcant and includes both in-plane and strong out-of-\nplane anisotropy systems.\nThe range of the external magnetic \feld correspond-\ning to stable spinner-skyrmion (see the domain bound\nby the red lines in Fig. 1F) is roughly eight times larger\nthan that for the stick-skyrmion (see the domain bound\nby the blue lines). The stability range for most of the\nother tailed-skyrmion solutions considered in this work3\nstick-skyrmion spinner-skyrmion π-skyrmion multi-tailed skyrmion\nA B C D E\nE\nD\nC\nBtailsz\nLD\nAnisotropy, u0 -0.5 0.5 1.0 1.504812\nhehtCritical fields, (he-h*)×103  ,(ht-h*)×10316\nhtF\nAnisotropy, u-0.5Magnetic filed,hSpin-spiralSaturated\n    state\nh*\n0\n0.5 0 11\n0.5\nReaction coordinate0 200 400 600 8000\n-1\n-2\n-3\n-4\nEnergy, E/(2A)1\n-5G\n90 110 130\nReaction coordinate0\n-4.0\n-8.0(E/2A+4.48)×1034.0\nFIG. 1. Tailed skyrmions . (A) The magnetization vector \feld of the \u0019-skyrmion. (B) Top view of a \u0019-skyrmion, where\ncolors encode the magnetization direction. (C)-(E) Color-coded spin texture of tailed skyrmions with Q=\u00001. (F) Stability\ndiagram of tailed skyrmions in terms of the magnetic \feld, h, and magnetocrystalline anisotropy, u. (G) Minimum energy path\nbetween skyrmions depicted in (B)-(E). All simulations were performed on the 4 LD\u00024LDdomain.\nwas found to coincide with high precision with the sta-\nbility range of the spinner-skyrmion. Therefore, the area\nbound by the red curves in Figure 1F can be considered\nas a good estimate for the stability range of all tailed-\nskyrmions.\nB. Homotopies and minimum energy paths\nFigure 1G shows the MEP connecting the \u0019-\nskyrmion, stick-skyrmion, spinner-skyrmion and four-\ntailed skyrmion (see Figure 1B, C, D, and E). The calcu-\nlated MEP turns out to be a homotopy as the topological\ncharge of the system remains the same, Q=\u00001, at ev-\nery point of the MEP. Some other examples of homotopy\nMEPs have already been presented in Ref.26, reporting\nthe method for \fnding exotic three-dimensional hybrid\nskyrmion tubes.\nStarting from the \u0019-skyrmion solution and following\nthe MEP we observe a sequential increase in the number\nof tails in the magnetic texture. The growth of a new tail\nrequires overcoming an energy barrier that turns out to\nbe signi\fcantly larger than that for the reverse process,\ni.e., removal of the tail. The latter de\fnes the stability ofthe tailed states. With an increasing external magnetic\n\feld, the barriers corresponding to the tail contraction\ndecrease and eventually disappear, leading to the tran-\nsition of the tailed skyrmions into the axially symmetric\n\u0019-skyrmion state.\nAs follows from the MEP calculations, the energy of\nthe tailed skyrmions increases with the number of tails,\nmeaning that tailed skyrmions represent higher energy\nmetastable states. The latter is consistent with the pre-\nvious study27where it was proven that the axially sym-\nmetric\u0019-skyrmion solution represents the global mini-\nmum among all solutions with Q=\u00001.\nThe question about the second lowest energy soliton\nwithQ=\u00001 remains open. For instance, 3 \u0019-skyrmion\nand some of the skyrmion bags seem to be promising\ncandidates. However, comparing the energies of tailed\nskyrmions, we found that the stick skyrmion may cer-\ntainly compete with these solutions. In particular, at\nh= 0:617 andu= 0, the energy of stick-skyrmion is only\n8:4% higher than that of the \u0019-skyrmion, while the en-\nergy of 3\u0019-skyrmion for these parameters is 11% higher.\nMoreover, the 3 \u0019-skyrmion is signi\fcantly bigger than\nthe\u0019-skyrmion and has a diameter of \u00188LD. The size\nof other skyrmion bags with Q=\u00001 is even larger. This4\nmeans that the stick-skyrmion is not only lower in energy\nthan other solutions but also has the shortest distance to\nthe\u0019-skyrmion in the con\fguration space.\nC. Thermally activated transition from \u0019-skyrmion\nto stick-skyrmion\nInitial state\n1×104 iterations 5×106\nHeating\nCooling Final stateA B C\nD E FLD\n1×1061×1051×107\nFIG. 2. Snapshots of the system during the heating-cooling\nprotocol of stochastic LLG simulations. (A) The initial con\fg-\nuration of\u0019-skyrmion. (B)-(D) Snapshots of the system after\n104, 5\u0001106and 107iterations of stochastic LLG simulations\nwith \fxed temperature of T= 0:2J=kB. (E), (F) Snapshots\nafter 105and 106LLG time steps with T= 0. Simulations\nare performed on the 8 LD\u00028LDcomputational domain for\nh= 0:617,u= 0.\nSince the tailed stick-skyrmion solution is the closest\nlocal minimum to the \u0019-skyrmion state, excitation of a\n\u0019-skyrmion by external stimuli such as thermal \ructua-\ntions can induce a nucleation of the stick-skyrmion. To\ndemonstrate this, we performed \fnite-temperature spin\ndynamics simulations for h= 0:617 andu= 0. We\nused the LLG time step \u0001 t= 0:01J\r\u0016\u00001\ns, and damping\nparameter \u000b= 0:1. The system was simulated at the\ntemperature of T= 0:2J=kB, which is signi\fcantly lower\nthan the Curie temperature, which was estimated to be\nTc= 0:7J=kBfor the given system28.\nThe initial magnetic texture of axially symmetric \u0019-\nskyrmion is shown in Figure 2A. The thermal \ructuations\nresult in skyrmion deformation, as seen in Figure 2B-D.\nAfter turning o\u000b the temperature and performing cool-\ning with the LLG simulation at T= 0, the deformed\nskyrmion quickly turned into a stick-skyrmion, as shown\nin Figure 2E-F and in Supplementary Movie 2. A longer\nheating of the system, in principle, can also give rise\nto the appearance of spinner-skyrmion and other multi-\ntailed skyrmions. However, such events have a much\nlower probability because of the large distance between\ncorresponding minima in the parameter space and higher\nenergy barriers, Figure 1G. Nevertheless, our results sug-\ngest a simple approach to the experimental observationof tailed skyrmions.\nD. Tailed skyrmions with arbitrary topological\ncharge\nSo far, we have only discussed solutions with Q=\u00001.\nHowever, the tails can in principle be added to any soli-\nton with an arbitrary topological charge. Adding a tail\ncreates a new soliton, but does not change the topologi-\ncal charge of the system.This section provides examples\nof such skyrmions. All solutions presented in the follow-\ning were obtained on a square simulation domain of size\nLx=Ly= 8LDfor parameters h= 0:617 andu= 0.\nIn Figure 3, we show topologically trivial solitons {\nskyrmionium with tails. Note that all solitons shown\nin Figure 3 are homotopically equivalent despite di\u000ber-\nent number of tails and their position on the inner or\nouter contour of the solitons. Noticeably, the skyrmion-\nium with more than three tails on the outer contour, see,\ne.g., Figure 3G-I, has rotational symmetry of order equal\nto the number of tails.\nLD\nA B C\nD E F\nG H I\nFIG. 3. Tailed skyrmions with Q= 0. Skyrmionium with\ntails on the inner contour only (A), (B), on both contours (C),\nand on the outer contour only (D) - (I). The scale bar is the\nsame for all images.\nFigure 4 provides examples of tailed skyrmion bags of\nvarious topological charges and demonstrates diversity of\ntailed skyrmions. For example, Figure 4C-E shows three\nskyrmion con\fgurations with the same topological charge\nQ=\u00001 but di\u000berent symmetry and di\u000berent number\nof tails. Figure 4G and H demonstrates two di\u000berent\nskyrmion bags with the same positive topological charge,\nQ= +3.\nAnother qualitatively new feature appears in the case\nof skyrmions with chiral kinks and tails. Figure 5 pro-5\n-1\n-2\n+5\n-1\n-1\n+2\n+3\nQ=-7\n+3A B C\nD E F\nG H ILD\nFIG. 4. Skyrmion bags with tails stabilized for u= 0 and\nh= 0:617.\n+1\nd\n+1\n+1\n+1\n+1\n+2Q=0\n+19A B C\nD E F\nG H ILD\nFIG. 5. Skyrmions with tails and chiral kinks stabilized for\nh= 0:617 andu= 0. (B) Detailed view of the spin texture\nnear the chiral kink of skyrmion shown in (A).\nvides several examples of such skyrmions having vari-\nous topological charges. Note that the chiral kink on\nthe outer contour of the skyrmion carries the topological\nchargeQ= +1. Because of this, the topological charge\nof skyrmions in Figure 5 A and B equals zero and does\nnot depend on the number of tails. The skyrmions with\nantikinks of topological charge Q=\u00001 are only stable\nat strong out-of-plane anisotropy9.\nFigures 5 C-E illustrate that the chiral kinks form cou-\n1.0\n-1.0\n-1.51.5\n0.5\n0\n-0.5ny 102\nC\nCB\nBA\nAF\nGE\nD\nLD\n0.0 2.0 4.0 -2.0 -4.0\nx/LDFIG. 6. (A) axially symmetric \u0019-skyrmion. (B), (C) stick-\nskyrmions of di\u000berent lengths. (D) chiral droplet soliton. (E)\nand (F) stick-skyrmion with one tail and one chiral kink.\nAll solutions co-exist for u= 1:194 andh= 0:01. The\nenergy minimization was performed on a large size domain,\nLy= 4LDandLx= 16LD, to avoid the in\ruence of peri-\nodic boundary conditions. All images have identical scale,\nas marked in (A). (G) y-component of magnetization along\nthe dashed symmetry line of the skyrmions (A)-(C). Hollow\ncircles denote the zeros of ny(x).\npled pairs in the presence of tails. A similar phenomenon\nwas observed for chiral kinks on the 2 \u0019-domain wall9.\nGenerally, the chiral kinks on a straight domain wall re-\npel each other at large distances and attract each other\nonly at small distances9. Similarly, a pair of kinks can\nform a coupled state on tailed skyrmions, with \fnite dis-\ntance between the kinks on the order of 0 :1LD. With\nincreasing \feld, the solutions with Q= +1 shown in\nFigures 5 C-E collapse to an antiskyrmion29. Figures 5\nF-I show skyrmion bags with positive topological charges\nhosting chiral kinks and tails simultaneously.\nE. E\u000bect of strong perpendicular anisotropy on\ntailed skyrmions\nSo far, the diversity of tailed skyrmions has been\ndemonstrated for the u= 0 case. However, all tailed\nskyrmion solutions can also be obtained for any value of\nuin the range from \u00001=2 to\u00192=8 (see the diagram in\nFig. 1F). Moreover, for strong perpendicular anisotropy,\nu.\u00192=8, we found distinct co-existing solutions rep-\nresenting tailed skyrmions of di\u000berent lengths. In Fig-\nure 6B, C, we show several examples of stick-skyrmions\nof di\u000berent lengths, which coexist for a given u,h, and\nare separated by small energy barriers. The energy of the6\nstick-skyrmion increases with its length. Note that such\ntailed skyrmions of di\u000berent lengths are observed only\nat strong anisotropy. As Figure 6G shows, the magnetic\nstructure of such stick-skyrmions is non-trivial. In par-\nticular, the ycomponent of magnetization is modulated\nalong the skyrmion symmetry axis giving an energy gain\nsu\u000ecient for the skyrmion stabilization. The number of\nzeros ofnyincreases with the length of a stick-skyrmion.\nInterestingly, the stick-skyrmion with one tail and one\nchiral kink (see Figure 6D-F) also exhibits multiple min-\nima corresponding to the solutions of di\u000berent lengths.\nVI. CONCLUSIONS\nIn conclusion, we presented in this work a new class\nof chiral skyrmions { tailed skyrmions. We identi\fed\nthe range of uniaxial anisotropy and perpendicular mag-\nnetic \feld where the tailed skyrmions remain statically\nstable. We show a wide diversity of tailed skyrmions\nwith arbitrary topological charges. The presence of tails\nchanges only the soliton's symmetry but does not change\nits topological charge. Thus, the transitions betweentailed and tail-free solutions are homotopies. We demon-\nstrated that such transitions can be identi\fed using the\nGNEB method. The energy barriers involved in the ho-\nmotopies tend to be smaller than what needs to be over-\ncome to change the topological charge of the system. As\na result, tailed skyrmions can be obtained from a reg-\nular\u0019-skyrmion by applying weak thermal \ructuations,\nas demonstrated by the stochastic LLG simulations.\nOur results contribute to the development of a com-\nplete picture of the diversity of statically stable magnetic\nsolitons in chiral magnets.\nACKNOWLEDGMENTS\nWe are grateful to F. Rybakov for fruitful discussions and\nproviding us with the Excalibur code. The authors ac-\nknowledge \fnancial support from the Icelandic Research\nFund (Grant No. 217750), the University of Iceland Re-\nsearch Fund (Grant No. 15673), the Swedish Re- search\nCouncil (Grant No. 2020-05110), and the European Re-\nsearch Council (ERC) under the European Union's Hori-\nzon 2020 research and innovation program (Grant No.\n856538, project \\3D MAGiC\").\n\u0003vkuchkin@hi.is\n1I. Dzyaloshinsky, A thermodynamic theory of \\weak\" fer-\nromagnetism of antiferromagnetics, J. Phys. Chem. Solids\n4, 241 (1958).\n2T. Moriya, Anisotropic superexchange interaction and\nweak ferromagnetism, Phys. Rev. 120, 91 (1960).\n3A. N. Bogdanov and D. A. Yablonskii, Thermodynami-\ncally stable \\vortices\" in magnetically ordered crystals.\nThe mixed state of magnets, Sov. Phys. JETP 68, 101\n(1989).\n4A. N. Bogdanov and A. Hubert, The Properties of Isolated\nMagnetic Vortices, Phys. Status Solidi B 186, 527 (1994).\n5A. N. Bogdanov and A. Hubert, Thermodynamically sta-\nble magnetic vortex states in magnetic crystals, J. Magn.\nMagn. Mater. 138, 255 (1994).\n6A. N. Bogdanov and A. Hubert, The stability of vortex-\nlike structures in uniaxial ferromagnets, J. Magn. Magn.\nMater. 195, 182-192 (1999).\n7F. N. Rybakov and N. S. Kiselev, Chiral magnetic\nskyrmions with arbitrary topological charge, Phys. Rev.\nB99, 064437 (2019).\n8D. Foster, C. Kind, P. J. Ackerman, J.-S. B. Tai,\nM. R. Dennis, and I. I. Smalyukh, Two-dimensional\nskyrmion bags in liquid crystals and ferromagnets, Nat.\nPhys. 15, 655 (2019).\n9V. M. Kuchkin, B. Barton-Singer, F. N. Rybakov, S.\nBl ugel, B. J. Schroers, and N. S. Kiselev, Magnetic\nskyrmions, chiral kinks and holomorphic functions, Phys.\nRev. B 102, 144422 (2020).\n10B. Barton-Singer, C. Ross, C. and B. J. Schroers, Magnetic\nskyrmions at critical coupling, Commun. Math. Phys. 375,\n2259 (2020).\n11Magnetic Skyrmion Materials Y. Tokura and N. Kanazawa\nChem. Rev. 121, 2857 (2021).12J. Tang, et al. Magnetic skyrmion bundles and their\ncurrent-driven dynamics. Nat. Nanotechnol. 16, 1086-1091\n(2021).\n13F. Zheng, N. S. Kiselev, L. Yang, V. M. Kuchkin,\nF. N. Rybakov, S. Bl ugel and R. E. Dunin-Borkowski,\nSkyrmion{antiskyrmion pair creation and annihilation in\na cubic chiral magnet, Nat. Phys. 18, 863{868 (2022).\n14E. Davoli, G. Di Fratta, D. Praetorius, and M.\nRuggeri, Micromagnetics of thin \flms in the pres-\nence of Dzyaloshinskii-Moriya interaction, Preprint at\narXiv:2010.15541.\n15N. Romming, C. Hanneken, M. Menzel, J. E. Bickel, B.\nWolter, K. Von Bergmann, A. Kubetzka, and R. Wiesen-\ndanger, Writing and Deleting Single Magnetic Skyrmions,\nScience 341, 636 (2013).\n16I. K\u0013 ezsm\u0013 arki, S. Bord\u0013 acs, P. Milde, E. Neuber, L. M.\nEng, J. S. White, H. M. R\u001cnnow, C. D. Dewhurst, M.\nMochizuki, K. Yanai, H. Nakamura, D. Ehlers, V. Tsurkan,\nand A. Loidl, N\u0013 eel-type skyrmion lattice with con\fned\norientation in the polar magnetic semiconductor GaV4S8,\nNat. Mater. 14, 1116 (2015).\n17N. Romming, A. Kubetzka, C. Hanneken, K.\nvon Bergmann, and R. Wiesendanger, Field-Dependent\nSize and Shape of Single Magnetic Skyrmions, Phys. Rev.\nLett. 114, 177203 (2015).\n18F.N. Rybakov, A.B. Borisov, S. Bl ugel, and N.S. Kiselev,\n\\New Type of Stable Particlelike States in Chiral Mag-\nnets,\" Phys. Rev. Lett. 115, 117201 (2015); Supplemen-\ntary material.\n19P.F. Bessarab, V.M. Uzdin, and H. J\u0013 onsson, Comput.\nPhys. Commun. 196, 335 (2015).\n20P.F. Bessarab, Phys. Rev. B 95, 136401 (2017).\n21L. D. Landau and E. M. Lifshitz, On the theory of the dis-\npersion of magnetic permeability in ferromagnetic bodies,7\nPhysik. Zeits. Sowjetunion 8, 153 (1935).\n22J. H. Mentink, M. V. Tretyakov, A. Fasolino, M. I. Kat-\nsnelson and Th, Rasing, J. Phys.: Condens. Matter 22,\n176001 (2010).\n23F. N. Rybakov and E. Babaev, Excalibur software ,\nhttp://quantumandclassical.com/excalibur/.\n24A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F.\nGarcia-Sanchez, and B. Van Waeyenberge, The design and\nveri\fcation of MuMax3, AIP Advances 4, 10 (2014).\n25G. P. M uller, M. Ho\u000bmann, C. Di\u0019elkamp, D. Sch urho\u000b,\nS. Mavros, M. Sallermann, N. S. Kiselev, H. J\u0013 onsson, and\nS. Bl ugel, Spirit: Multifunctional framework for atomisticspin simulations, Phys. Rev. B 99, 224414 (2019).\n26V. M. Kuchkin and N. S. Kiselev, Homotopy transitions\nand 3D magnetic solitons, APL Materials 10, 071102\n(2022).\n27C. Melcher, Chiral skyrmions in the plane, Proc. R. Soc.\nA470, 20140394 (2014).\n28V. M. Kuchkin, P. F. Bessarab, and N. S. Kiselev, Thermal\ngeneration of droplet soliton in chiral magnet, Phys. Rev.\nB105, 184403 (2022)\n29V. M. Kuchkin and N. S. Kiselev, Turning a chiral\nskyrmion inside out, Phys. Rev. B 101, 064408 (2020)."    },    {        "title": "2303.01823v2.Magneto_optical_sensing_of_the_pressure_driven_magnetic_ground_states_in_bulk_CrSBr.pdf",        "content": "Magneto-optical sensing of the pressure driven magnetic ground states in bulk CrSBr\nAmit Pawbake,1,∗Thomas Pelini,1Ivan Mohelsky,1Dipankar Jana,1Ivan Breslavetz,1Chang-Woo\nCho,1Milan Orlita,1Marek Potemski,1, 2Marie-Aude Measson,3Nathan Wilson,4Kseniia Mosina,5\nAljoscha Soll,5Zdenek Sofer,5Benjamin A. Piot,1M. E. Zhitomirsky,6, 7and Clement Faugeras1,†\n1LNCMI, UPR 3228, CNRS, EMFL, Universit´ e Grenoble Alpes, 38000 Grenoble, France\n2CENTERA Labs, Institute of High Pressure Physics, PAS, 01 - 142 Warsaw, Poland\n3Institut Neel, Universit´ e Grenoble Alpes, 38000 Grenoble, France\n4Walter Schottky Institut, Physics Department and MCQST,\nTechnische Universitat Munchen, 85748 Garching, Germany\n5Department of Inorganic Chemistry, University of Chemistry and\nTechnology Prague, Technicka 5, 166 28 Prague 6, Czech Republic\n6Universit´ e Grenoble Alpes, CEA, Grenoble INP, IRIG, Pheliqs, 38000 Grenoble, France\n7Institut Laue-Langevin, F-38042 Grenoble Cedex 9, France\n(Dated: October 25, 2023)\nCompetition between exchange interactions and magnetocrystalline anisotropy may bring new\nmagnetic states that are of great current interest. An applied hydrostatic pressure can further be\nused to tune their balance. In this work we investigate the magnetization process of a biaxial an-\ntiferromagnet in an external magnetic field applied along the easy axis. We find that the single\nmetamagnetic transition of the Ising type observed in this material under ambient pressure trans-\nforms under hydrostatic pressure into two transitions, a first-order spin flop transition followed by\na second order transition towards a polarized ferromagnetic state near saturation. This reversible\ntuning into a new magnetic phase is obtained in layered bulk CrSBr at low temperature by vary-\ning the interlayer distance using high hydrostatic pressure, which efficiently acts on the interlayer\nmagnetic exchange, and is probed by magneto-optical spectroscopy.\nThe generation of new magnetic phases by an exter-\nnal manipulation of the magnetic state is a requirement\nfor the implementation of new spintronic and of magnon-\ntronic functionalities. Successful manipulations of mag-\nnetic states have been achieved by using spin-polarized\nelectrical currents [1], electrostatic gating [2], or ultra fast\nlaser excitations [3, 4]. An alternative approach relies\non the direct modification of the magnetic interactions\nby changing the lattice parameters and/or the super-\nexchange paths by applying strain [5] or hydrostatic pres-\nsure [6, 7]. Such profound modifications affect both direct\nexchange magnetic interaction, and spin-orbit interac-\ntion related effects such as magnetocrystalline anisotropy,\nwhich determines the preferential directions of spins in a\nmagnetically ordered solid and the spin-waves energy.\nMagnetic van der Waals (vdW) materials are lay-\nered crystals with magnetic properties and that can be\nthinned down to the monolayer limit. They exhibit a\nbroad variety of ordered magnetic structures [8, 9]. Re-\ncently, vdW magnets have attracted a lot of interest\nand stimulated many experimental and theoretical stud-\nies aiming at understanding their magnetic properties\nand at using them within functional vdW heterostruc-\ntures [10, 11]. The observation of long range magnetic\norder in the 2D limit [12, 13] that is precluded by the\nMermin-Wagner theorem [14], is understood as a conse-\nquence of substantial magnetocrystalline anisotropy and\nof the associated energy gap in the spin-wave disper-\nsion [15]. vdW materials offer new exciting possibili-\nties to engineer magnetic properties of solid state sys-\ntems as they are characterized by a very strong struc-tural anisotropy: individual layers are bound together\nby weak vdW interactions. As a result, the interlayer\nspacing, the vdW gap, can be tuned efficiently through\nthe application of hydrostatic pressure [6, 7, 16, 17]. Such\nchanges strongly affect the short range interlayer inter-\nactions [18, 19], and the competition between magnetic\nexchange interactions and magnetic anisotropies. New\nmagnetic phases can be engineered in a magnetic system\nby changing the competition between anisotropies and\nexchange interactions.\nIn this Letter, we apply hydrostatic pressure to bulk\nCrSBr to tune the interlayer magnetic exchange interac-\ntion and the magnetocrystalline anisotropies. We sense\nthe induced changes by magneto-photoluminescence\nmeasurements with an external magnetic field applied\nalong one of the three crystallographic axis. The inter-\nlayer exchange parameter J ⊥responsible for the anti-\nferromagnetic (AFM) ordering is small in bulk CrSBr\nbecause of its lamellar structure. When decreasing the\nvdW gap, we observe an increase of J ⊥by 1700 % at\nP > 6 GPa. This large increase with respect to magne-\ntocrystalline anisotropies, stabilizes CrSBr into a partic-\nular magnetic ground state, characterized by an interme-\ndiate spin-flop phase appearing between the AFM ground\nstate at B= 0 and the ferromagnetic (FM) ground state\nwhen an external magnetic field, exceeding the satura-\ntion field, is applied (see Fig. 1b). This new magnetic\nground state is observed because of the possibility to\ntune the competition between the interlayer exchange\ninteraction and the magnetocrystaline anisotropies with-\nout changing the crystallographic stacking. IncreasingarXiv:2303.01823v2  [cond-mat.mes-hall]  24 Oct 20232\nfurther hydrostatic pressure, the in-plane magnetocrys-\ntalline anisotropy vanishes, transforming CrSBr into an\neasy-plane AFM.\nCrSBr is an air-insensitive layered direct band gap\nsemiconductor with E g= 1.5 eV and hosting tightly\nbound excitons that give rise to photoluminescence sig-\nnals close to 1 .35 eV [20] at low temperature. It crys-\ntalizes in the P mmn space group in an orthorhombic\nstructure in which monolayers are bound together by\nvdW interactions, see Fig. 1a. From the point of view\nof magnetism, CrSBr belongs to the class of easy-axis\nlayered AFM systems, with an in-plane anisotropy. Be-\nlow the Curie temperature of TC= 160 K, magnetic\nmoments in the layers orient ferromagnetically along the\neasy axis [21], the crystallographic baxis. Below the N´ eel\ntemperature of T N= 132 K, an AFM interaction couples\neach layer. Below T= 40 K, various experimental signa-\ntures of a change in the magnetic ground state have been\nreported [20, 22–25], even though the low temperature\nmagnetic behavior is the one of a layered antiferromag-\nnet [26]. This low temperature behaviour was recently\nattributed to the ferromagnetic ordering of defaults [27].\nWhen magnetically ordered, bulk CrSBr is an ensemble\nof anisotropic FM planes, with an AFM coupling of adja-\ncent planes [28]. In multilayer CrSBr, the interlayer hop-\nping is spin dependent which makes the energy of elec-\ntronic states different for a AFM or FM ground state [26].\nThis energy difference can be seen in the emission en-\nergy of the exciton which can then be used to deter-\nmine for instance the value of saturation magnetic fields,\nand to characterize the magnetic ground state [5]. The\nelectronic properties of CrSBr are strongly anisotropic\nand this is reflected in its optical [26, 29] and trans-\nport properties [30]. Low energy excitations in CrSBr\ninclude two magnon excitations at 25 and 35 GHz [31–\n33]. CrSBr exhibits two inequivalent magnetic ions in the\nmagnetic unit cell and hence, a higher energy magnon\nbranch [31]. Our infrared magneto-transmission exper-\niments presented in Fig. S1 of the Supplementary Ma-\nterials, show an absorption at 372 cm−1that disperses\nlinearly with the external magnetic field with a g-factor\nclose to 2, which we identify as the second magnon branch\nof bulk CrSBr.\nTo measure the low temperature magneto-optical\nproperties of bulk CrSBr under hydrostatic pressure, we\nuse the experimental set-up described in Ref. [34]. A dia-\nmond anvil cell (DAC) is placed on piezo motors to allow\nfor its displacement under the excitation laser spot pro-\nduced by a long working distance objective. The pressure\nvalue at low temperature is determined by the photolu-\nminescence of the ruby balls. Bulk CrSBr was grown\nby chemical vapor transport (see Supplementary Mate-\nrials). Crystals with typical sizes of few tens of microm-\neters were inserted in the pressure chamber of the DAC\nas shown in Fig. 1b and could be oriented taking advan-\ntage of the strong shape anisotropy of this material that\nFigure 1. a) Crystallographic structure of bulk CrSBr. b) Op-\ntical image of the pressure chamber of the diamond anvil cell\nshowing two pieces of CrSBr with different orientations and\nruby balls. c) (Solid lines) Schematics of the magnetic phase\ndiagram of easy-plane AFM for a fixed value of J⊥show-\ning the antiferromagnetic, the ferromagnetic and the spin-\nflop magnetic phases. The spin-flop phase appears when the\nanisotropy difference (D-E) becomes smaller than the inter-\nlayer exchange J⊥. We indicate the critical values of magnetic\nfield for spin-flip Bb\nc, for the abrupt Ising transition Bb\nISand\nfor the spin-flop Bb\nSF. The dashed lines represent a similar\nphase diagram with a higher value of J⊥. d) Typical photolu-\nminescence spectrum of bulk CrSBr at T= 5 K showing the\nexcitonic lines around 1 .35 eV and a broad emission at lower\nenergies.\nproduces crystals elongated along the acrystallographic\naxis, see Fig. 1b. The DAC can be oriented at room tem-\nperature to apply the external magnetic field along differ-\nent crystal directions (see Fig S2 of Supplementary Ma-\nterials). This allows to compare the pressure dependent\nmagneto-optical response giving access to the parameters\ndescribing the magnetic Hamiltonian. All measurements\npresented in this work have been performed at a temper-\nature of T= 5 K. Each value of the hydrostatic pressure\nwas first applied at room temperature, and the DAC was\nthen cooled down for optical spectroscopy measurements.\nA typical low temperature photoluminescence spec-\ntrum of bulk CrSBr is presented in Fig. 1d. It includes an\nensemble of sharp emission lines around 1 .35 eV related\nto excitons and exciton-polaritons [26, 35], while lower\nenergy broad emissions are related to defects [24].\nWe present in Fig. 2 false color maps of the evolution\nof the low temperature photoluminescence spectrum of\nCrSBr at ambient pressure and at P≈2 GPa with the3\nFigure 2. False color maps of the low temperature magneto-\nphotoluminescence response of bulk CrSBr for the different\nvalues of hydrostatic pressure when the magnetic field is ap-\nplied along the a axis (top row), the b axis (middle row) and\nthe c axis (bottom row). The white arrows indicate the criti-\ncal magnetic fields.\nmagnetic field applied along the a(top row), b(middle\nrow) and caxis (bottom row). When applying the mag-\nnetic field along the intermediate aor the hard caxis,\nthe two spin sublattices gradually cant towards the di-\nrection of the external field. The exciton energy slowly\nevolves from the B= 0 energy to a lower energy rep-\nresentative of the band structure of CrSBr with a FM\nmagnetic ordering [26]. When a hydrostatic pressure is\napplied, we observe a pronounced increase of the satura-\ntion magnetic fields Ba\ncandBc\nc(Bα\ncare the critical mag-\nnetic fields when the external magnetic field is applied\nalong the αaxis), indicating a change in the competi-\ntion between magnetic exchange and magnetocrystalline\nanisotropy. The complete evolution is presented in Fig.\nS3-S5 of the Supplementary materials. If the external\nBfield is applied along the baxis, the observed behav-\nior is different. At ambient pressure, the AFM to FM\ntransition is abrupt (Ising transition or strong anisotropy\nregime) at a saturation field Bb\nIS. When applying a hy-\ndrostatic pressure above P= 1 GPa, we observe a dif-\nferent behavior: the exciton energy remains constant up\nto a saturation field Bb\nSFabove which it initiates an ap-\nproximately linear red-shift, up to a second value of the\nmagnetic field labeled Bb\nc. For B > Bb\nc, the exciton emis-\nFigure 3. a) Critical magnetic fields along the hard-c axis\n(green symbols), the intermediate-a axis (orange symbols)\nand along the easy-b axis (black symbols). Open symbols\nare extrapolated from nearby measurements. b) Evolution of\nthe microscopic parameters describing the interlayer interac-\ntionJ⊥, the easy axis anisotropy Dand the intermediate-hard\naxis anisotropy E, as a function of the applied pressure. The\nspin-flop phase is observed when Bb\nSF< B < Bb\nc. The gray re-\ngion indicates the range of pressure for which 2 J⊥≈(D−E).\nsion energy remains constant, indicating that CrSBr is in\na FM ground state. The intermediate behavior observed\nwhen Bb\nSF< B < Bb\ncappears as a gradual decrease of\nthe emission energy when increasing the magnetic field.\nThis behavior is similar to that observed when the ex-\nternal magnetic field is applied along the aorcaxis and\ndescribes the gradual canting of the spins towards the\ndirection of the applied magnetic field.\nTo understand these results, we use the following mi-\ncroscopic spin Hamiltonian which is suitable for describ-\ning the two-sublattice orthorhombic AFM CrSBr:\nˆH=X\n⟨ij⟩JijSi·Sj+X\ni\b\n−D(Sb\ni)2+E\u0002\n(Sc\ni)2−(Sa\ni)2\u0003\n−gαµBBαSα\ni\t\n.\n(1)\nHereSiareS= 3/2 spins of chromium ions, Jijare ex-\nchange coupling constants, DandEare parameters of4\nbiaxial single-ion anisotropy with αbeing either b(easy\naxis), a(intermediate axis ( D > E )), and cthe hard\naxis. Stable magnetic configurations can be determined\nby treating spins as classical vectors and minimizing the\ncorresponding energy (1). In this way one can find that\nFM exchanges inside the ablayers play no role in the\nmagnetization process, which depends only on the AFM\ninterlayer coupling J⊥>0. The three nearest neigh-\nbours exchange integrals have been determined by neu-\ntron scattering and their values are found to be two or-\nders of magnitude larger than the interlayer one [36], so\nspins within the layers remain parallel to each other to\nminimize the intralayer exchange energy.\nMagnetic field applied perpendicular to the easy-axis\ncauses continuous spin tilting in the canted AFM state\n(CAF) till the full saturation is reached at the critical\n(second-order) field\ngaµBBa\nc= 2S(4J⊥+D−E), (2)\ngcµBBc\nc= 2S(4J⊥+D+E). (3)\nwhere gαare the g-factors when the external magnetic\nfield is applied along the αaxis.\nOnce an external field is applied along the easy axis\nof a collinear AFM, the magnetization remains zero till\nthe spin-flop transition. The corresponding expression\nfor the spin model of CrSBr\ngbµBBb\nSF= 2Sp\n(D−E)(4J⊥−D+E). (4)\nAbove Bb\nSF, the AFM sublattices rotate abruptly towards\nthe intermediate aaxis forming the CAF state. The full\nsaturation is reached at\ngbµBBb\nc= 2S(4J⊥−D+E). (5)\nThus, the conventional magnetization process for a\ncollinear AFM proceeds as ↑↓ → CAF → ↑↑ .\nThe above picture suggested by Louis N´ eel [37], needs\nto be modified for CrSBr. Because of a large D/J⊥, this\nAFM undergoes a direct first-order transition from the\ncollinear AFM state into the fully polarized state typical\nfor magnets with strong Ising anisotropy\ngbµBBb\nIS= 4J⊥S . (6)\nThe Ising-like transition for B∥bhas been experimen-\ntally observed in CrSBr [26, 33, 38]. It corresponds to\nan abrupt reversal of magnetization in every second Cr-\nlayer. Comparing the expressions (4)–(6) we find that\nthe conventional magnetization process is realized for\n(D−E)<2J⊥, whereas for ( D−E)>2J⊥, the di-\nrect transition ↑↓ → ↑↑ takes place. The corresponding\nT= 0 phase diagram is sketched in Fig. 1c.\nOur main experimental observation is the tuning of\nbulk CrSBr from the strong to the weak anisotropy\nregime for which a spin-flop phase is stabilized. Cross-\ning this boundary is a unique possibility offered by bulkCrSBr for which ( D−E) and 2 J⊥are comparable at am-\nbient pressure. The tuning is achieved by applying hy-\ndrostatic pressure which strongly enhances the interlayer\ncoupling J⊥such that ( D−E)/2J⊥becomes smaller.\nForP > 4 GPa, the measured saturation field Bb\ncalong\nthebaxis increases and becomes almost equal to the\none along the aaxis. This evolution describes a decrease\nof the in-plane anisotropy (( D−E)→0) such that the\n(a,b) plane becomes nearly isotropic.\nSpecifically, Fig. 3a shows the evolution of the mea-\nsured values of the saturation magnetic fields along the\nthree crystallographic axis. They strongly increase with\nthe applied hydrostatic pressure and, at each pressure\nvalue, we can determine from Eq. 2-6 the three micro-\nscopic parameters J⊥,DandE. The evolution with\npressure of these parameters is presented in Fig. 3b.\nAt ambient pressure we find that J⊥= 5.3µeV,D=\n33.0µeV and E= 16 .36µeV, in line with recent\nGHz investigations conducted on bulk material from the\nsame batch [33]. At ambient pressure, bulk CrSBr is\nin the strong anisotropy regime fulfilling the condition\n(D−E)>2J⊥. When applying hydrostatic pressure,\nthe lattice parameters and in particular the vdW gap,\nare changed and both the magnetic anisotropies and the\ninterlayer exchange interaction are modified accordingly.\nJ⊥is increased by 1700 % at P= 6 GPa, in line with re-\ncent results obtained at lower pressure values [39]. Both\nDandEalso increase significantly, but in a similar way,\nkeeping the difference D−Eat a constant value up to\nP∼3 GPa. The condition ( D−E)<2J⊥, indicative of\nthe weak anisotropy regime, is reached for P≃1 GPa.\nAbove this pressure value, we observe the intermediate\nspin-flop phase when applying an external magnetic field\nalong the baxis and two saturation fields as indicated in\nFig. 2 and 3. The range of magnetic field within which\nthe spin-flop phase is observed increases with the applied\npressure, as the vdW gap shrinks and J⊥is increased.\nThe variations of D or E are more likely related to defor-\nmations of the individual layers, along the c axis (bond\nangles) and within the plane of the layers, and also by a\npossible anisotropy of the Young modulus.\nAbove P= 3.5 GPa, we note two main changes in the\nevolution of the microscopic parameters with pressure :\nBoth D and E strongly decrease and become compara-\nble, and the evolution rate of J⊥with pressure increases\nsignificantly. These two points are the sign of change\nin the system eventhough the observed behavior of the\nmagneto-photoluminescence remains the same above and\nbelow this particular pressure value. As shown in Fig.\nS6 of the Supplementary Materials, applying hydrostatic\npressure to bulk CrSBr is a reversible process.\nTo conclude, by applying high hydrostatic pressure to\nbulk CrSBr, we have modified the competition between\nthe crystalline magnetic anisotropy and the interlayer ex-\nchange parameter. Based on low temperature magneto-\nphotoluminescence measurements under such extreme5\nexperimental conditions, we have identified a spin-flop\nphase stabilized when the interlayer exchange interac-\ntion strongly exceeds the magnetic anisotropy. These\nresults are understood in the frame of a spin Hamil-\ntonian including magneto-crystalline anisotropies which\ndescribes the appearance of the spin-flop magnetic phase\nwhen ( D−E)<2J⊥(weak anisotropy regime). Com-\npared to the experimental results, this model provides\nvalues for the microscopic parameters entering the spin-\nHamiltonian and their evolution can be traced up to\nP= 6 GPa. This work demonstrates the possibility of\ntailoring the competition between magnetic crystalline\nanisotropy and interlayer exchange interaction of vdW\nmagnets by external means and enlarges the technical\npossibilities for the design of two dimensional magnets.\nWe thank Nicolas Ubrig and Gerard Martinez for\nvaluable comments on the manuscript. This work has\nbeen partially supported by the EC Graphene Flag-\nship project. M.E.Z. was supported by ANR Grant\nNo. ANR-15-CE30-0004. M-A. M. acknowledges\nthe support from the ERC (H2020) (Grant agreement\nNo. 865826). N.P.W. acknowledges support from\nthe Deutsche Forschungsgemeinschaft (DFG,German\nResearch Foundation) under Germany’s Excellence\nStrategy—EXC-2111—390814868. We acknowledge the\nsupport of the LNCMI-EMFL, CNRS, Univ. Grenoble\nAlpes, INSA-T, UPS, Grenoble, France.\n∗amit.pawbake@lncmi.cnrs.fr\n†clement.faugeras@lncmi.cnrs.fr\n[1] J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers,\nand D. C. Ralph, Current-driven magnetization reversal\nand spin-wave excitations in co /cu/co pillars, Phys.\nRev. Lett. 84, 3149 (2000).\n[2] M. Weisheit, S. Fahler, A. Marty, Y. Souche,\nC. Poinsignon, and D. Givord, Electric field-induced\nmodification of magnetism in thin-film ferromagnets, Sci-\nence 315, 349 (2007).\n[3] E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y. Bigot,\nUltrafast spin dynamics in ferromagnetic nickel, Phys.\nRev. Lett. 76, 4250 (1996).\n[4] P. Nˇ emec, M. Fiebig, T. Kampfrath, and A. V. Kimel,\nAntiferromagnetic opto-spintronics, Nature Physics 14,\n229 (2018).\n[5] J. Cenker, S. Sivakumar, K. Xie, A. Miller, P. Thijssen,\nZ. Liu, A. Dismukes, J. Fonseca, E. Anderson, X. Zhu,\nX. Roy, D. Xiao, J.-H. Chu, T. Cao, and X. Xu, Re-\nversible strain-induced magnetic phase transition in a\nvan der waals magnet, Nature Nanotechnology 17, 256\n(2022).\n[6] T. Li, S. Jiang, N. Sivadas, Z. Wang, Y. Xu, D. We-\nber, J. E. Goldberger, K. Watanabe, T. Taniguchi, C. J.\nFennie, K. F. Mak, and J. Shan, Pressure-controlled in-\nterlayer magnetism in atomically thin CrI 3, Nature Ma-\nterials 18, 1303 (2019).\n[7] T. Song, Z. Fei, M. Yankowitz, Z. Lin, Q. Jiang,\nK. Hwangbo, Q. Zhang, B. Sun, T. Taniguchi, K. Watan-abe, M. A. McGuire, D. Graf, T. Cao, J.-H. Chu, D. H.\nCobden, C. R. Dean, D. Xiao, and X. Xu, Switching 2d\nmagnetic states via pressure tuning of layer stacking, Na-\nture Materials 18, 1298 (2019).\n[8] X. Jiang, Q. Liu, J. Xing, N. Liu, Y. Guo, Z. Liu, and\nJ. Zhao, Recent progress on 2d magnets: Fundamental\nmechanism, structural design and modification, Applied\nPhysics Reviews 8, 10.1063/5.0039979 (2021).\n[9] Q. H. Wang, A. Bedoya-Pinto, M. Blei, A. H. Dismukes,\nA. Hamo, S. Jenkins, M. Koperski, Y. Liu, Q.-C. Sun,\nE. J. Telford, H. H. Kim, M. Augustin, U. Vool, J.-\nX. Yin, L. H. Li, A. Falin, C. R. Dean, F. Casanova,\nR. F. L. Evans, M. Chshiev, A. Mishchenko, C. Petrovic,\nR. He, L. Zhao, A. W. Tsen, B. D. Gerardot, M. Brotons-\nGisbert, Z. Guguchia, X. Roy, S. Tongay, Z. Wang, M. Z.\nHasan, J. Wrachtrup, A. Yacoby, A. Fert, S. Parkin, K. S.\nNovoselov, P. Dai, L. Balicas, and E. J. G. Santos, The\nmagnetic genome of two-dimensional van der waals ma-\nterials, ACS Nano 16, 6960 (2022).\n[10] D. Zhong, K. L. Seyler, X. Linpeng, R. Cheng,\nN. Sivadas, B. Huang, E. Schmidgall, T. Taniguchi,\nK. Watanabe, M. A. McGuire, W. Yao, D. Xiao, K.-M. C.\nFu, and X. Xu, Van der waals engineering of ferromag-\nnetic semiconductor heterostructures for spin and val-\nleytronics, Science Advances 3, 10.1126/sciadv.1603113\n(2017).\n[11] L. Ciorciaro, M. Kroner, K. Watanabe, T. Taniguchi, and\nA. Imamoglu, Observation of magnetic proximity effect\nusing resonant optical spectroscopy of an electrically tun-\nable mose 2/crbr 3heterostructure, Phys. Rev. Lett. 124,\n197401 (2020).\n[12] B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein,\nR. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A.\nMcGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-\nHerrero, and X. Xu, Layer-dependent ferromagnetism in\na van der waals crystal down to the monolayer limit,\nNature 546, 270 (2017).\n[13] C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao,\nW. Bao, C. Wang, Y. Wang, Z. Q. Qiu, R. J. Cava, S. G.\nLouie, J. Xia, and X. Zhang, Discovery of intrinsic fer-\nromagnetism in two-dimensional van der waals crystals,\nNature 546, 265 (2017).\n[14] N. D. Mermin and H. Wagner, Absence of ferromag-\nnetism or antiferromagnetism in one- or two-dimensional\nisotropic heisenberg models, Phys. Rev. Lett. 17, 1133\n(1966).\n[15] S. Jenkins, L. R´ ozsa, U. Atxitia, R. F. L. Evans, K. S.\nNovoselov, and E. J. G. Santos, Breaking through the\nmermin-wagner limit in 2d van der waals magnets, Na-\nture Communications 13, 10.1038/s41467-022-34389-0\n(2022).\n[16] M. Yankowitz, S. Chen, H. Polshyn, Y. Zhang, K. Watan-\nabe, T. Taniguchi, D. Graf, A. F. Young, and C. R. Dean,\nTuning superconductivity in twisted bilayer graphene,\nScience 363, 1059 (2019).\n[17] W.-T. Hsu, J. Quan, C.-R. Pan, P.-J. Chen, M.-Y. Chou,\nW.-H. Chang, A. H. MacDonald, X. Li, J.-F. Lin, and\nC.-K. Shih, Quantitative determination of interlayer elec-\ntronic coupling at various critical points in bilayer Mos 2,\nPhys. Rev. B 106, 125302 (2022).\n[18] J. Xia, J. Yan, Z. Wang, Y. He, Y. Gong, W. Chen, T. C.\nSum, Z. Liu, P. M. Ajayan, and Z. Shen, Strong coupling\nand pressure engineering in WSe 2-MoSe 2heterobilayers,\nNat. Phys. 17, 10.1038/s41567-020-1005-7 (2021).6\n[19] A. Pawbake, T. Pelini, A. Delhomme, D. Romanin,\nD. Vaclavkova, G. Martinez, M. Calandra, M.-A. Meas-\nson, M. Veis, M. Potemski, M. Orlita, and C. Faugeras,\nHigh-pressure tuning of magnon-polarons in the layered\nantiferromagnet FePS 3, ACS Nano 16, 12656 (2022).\n[20] E. J. Telford, A. H. Dismukes, K. Lee, M. Cheng, A. Wi-\neteska, A. K. Bartholomew, Y.-S. Chen, X. Xu, A. N.\nPasupathy, X. Zhu, C. R. Dean, and X. Roy, Layered\nantiferromagnetism induces large negative magnetoresis-\ntance in the van der waals semiconductor CrSBr, Ad-\nvanced Materials 32, 2003240 (2020).\n[21] W. Liu, X. Guo, J. Schwartz, H. Xie, N. U. Dhale,\nS. H. Sung, A. L. N. Kondusamy, X. Wang, H. Zhao,\nD. Berman, R. Hovden, L. Zhao, and B. Lv, A three-stage\nmagnetic phase transition revealed in ultrahigh-quality\nvan der waals bulk magnet CrSBr, ACS Nano 16, 15917\n(2022).\n[22] E. J. Telford, A. H. Dismukes, R. L. Dudley, R. A. Wis-\ncons, K. Lee, J. Yu, S. Shabani, A. Scheie, K. Watan-\nabe, T. Taniguchi, D. Xiao, A. N. Pasupathy, C. Nuck-\nolls, X. Zhu, C. R. Dean, and X. Roy, Hidden low-\ntemperature magnetic order revealed through magneto-\ntransport in monolayer crsbr (2021), arXiv:2106.08471\n[cond-mat.mes-hall].\n[23] S. A. L´ opez-Paz, Z. Guguchia, V. Y. Pomjakushin,\nC. Witteveen, A. Cervellino, H. Luetkens, N. Casati,\nA. F. Morpurgo, and F. O. von Rohr, Dynamic mag-\nnetic crossover at the origin of the hidden-order in van\nder waals antiferromagnet CrSBr, Nature Communica-\ntions 13, 10.1038/s41467-022-32290-4 (2022).\n[24] J. Klein, Z. Song, B. Pingault, F. Dirnberger,\nH. Chi, J. B. Curtis, R. Dana, R. Bushati, J. Quan,\nL. Dekanovsky, Z. Sofer, A. Al` u, V. M. Menon, J. S.\nMoodera, M. Lonˇ car, P. Narang, and F. M. Ross, Sensing\nthe local magnetic environment through optically active\ndefects in a layered magnetic semiconductor, ACS Nano\n17, 288 (2022).\n[25] A. Pawbake, T. Pelini, N. P. Wilson, K. Mosina, Z. Sofer,\nR. Heid, and C. Faugeras, Raman scattering signatures of\nstrong spin-phonon coupling in the bulk magnetic van der\nwaals material crsbr, Phys. Rev. B 107, 075421 (2023).\n[26] N. P. Wilson, K. Lee, J. Cenker, K. Xie, A. H. Dismukes,\nE. J. Telford, J. Fonseca, S. Sivakumar, C. Dean, T. Cao,\nX. Roy, X. Xu, and X. Zhu, Interlayer electronic coupling\non demand in a 2d magnetic semiconductor, Nature Ma-\nterials 20, 1657 (2021).\n[27] F. Long, K. Mosina, R. H¨ ubner, Z. Sofer, J. Klein,\nS. Prucnal, M. Helm, F. Dirnberger, and S. Zhou, Intrin-\nsic magnetic properties of the layered antiferromagnet\ncrsbr (2023), arXiv:2309.04778 [cond-mat.mtrl-sci].\n[28] A. Avsar, Highly anisotropic van der waals magnetism,\nNature Materials 21, 731 (2022).\n[29] J. Klein, B. Pingault, M. Florian, M.-C. Heißenb¨ uttel,\nA. Steinhoff, Z. Song, K. Torres, F. Dirnberger, J. B.Curtis, M. Weile, A. Penn, T. Deilmann, R. Dana,\nR. Bushati, J. Quan, J. Luxa, Z. Sofer, A. Al` u,\nV. M. Menon, U. Wurstbauer, M. Rohlfing, P. Narang,\nM. Lonˇ car, and F. M. Ross, The bulk van der waals lay-\nered magnet CrSBr is a quasi-1d material, ACS Nano 17,\n5316 (2023).\n[30] F. Wu, I. Guti´ errez-Lezama, S. A. L´ opez-Paz, M. Gib-\nertini, K. Watanabe, T. Taniguchi, F. O. von Rohr,\nN. Ubrig, and A. F. Morpurgo, Quasi-1d electronic trans-\nport in a 2d magnetic semiconductor, Advanced Materi-\nals34, 2109759 (2022).\n[31] Y. J. Bae, J. Wang, A. Scheie, J. Xu, D. G. Chica, G. M.\nDiederich, J. Cenker, M. E. Ziebel, Y. Bai, H. Ren, C. R.\nDean, M. Delor, X. Xu, X. Roy, A. D. Kent, and X. Zhu,\nExciton-coupled coherent magnons in a 2d semiconduc-\ntor, Nature 609, 282 (2022).\n[32] T. M. J. Cham, S. Karimeddiny, A. H. Dismukes, X. Roy,\nD. C. Ralph, and Y. K. Luo, Anisotropic gigahertz anti-\nferromagnetic resonances of the easy-axis van der waals\nantiferromagnet CrSBr, Nano Letters 22, 6716 (2022).\n[33] C. W. Cho, A. Pawbake, N. Aubergier, A. L. Barra,\nK. Mosina, Z. Sofer, M. E. Zhitomirsky, C. Faugeras,\nand B. A. Piot, Microscopic parameters of the van der\nwaals crsbr antiferromagnet from microwave absorption\nexperiments, Phys. Rev. B 107, 094403 (2023).\n[34] I. Breslavetz, A. Delhomme, T. Pelini, A. Paw-\nbake, D. Vaclavkova, M. Orlita, M. Potemski, M.-\nA. Measson, and C. Faugeras, Spatially resolved opti-\ncal spectroscopy in extreme environment of low tem-\nperature, high magnetic fields and high pressure,\nReview of Scientific Instruments 92, 123909 (2021),\nhttps://doi.org/10.1063/5.0070934.\n[35] F. Dirnberger, J. Quan, R. Bushati, G. M. Diederich,\nM. Florian, J. Klein, K. Mosina, Z. Sofer, X. Xu,\nA. Kamra, F. J. Garc´ ıa-Vidal, A. Al` u, and V. M. Menon,\nMagneto-optics in a van der waals magnet tuned by self-\nhybridized polaritons, Nature 620, 533 (2023).\n[36] A. Scheie, M. Ziebel, D. G. Chica, Y. J. Bae,\nX. Wang, A. I. Kolesnikov, X. Zhu, and X. Roy,\nSpin waves and magnetic exchange hamiltonian\nin crsbr, Advanced Science 9, 2202467 (2022),\nhttps://onlinelibrary.wiley.com/doi/pdf/10.1002/advs.202202467.\n[37] L. N´ eel, Antiferromagnetism and ferrimagnetism, Pro-\nceedings of the Physical Society. Section A 65, 869\n(1952).\n[38] O. G¨ oser, W. Paul, and H. Kahle, Magnetic properties\nof CrSBr, Journal of Magnetism and Magnetic Materials\n92, 129 (1990).\n[39] E. J. Telford, D. G. Chica, M. E. Ziebel, K. Xie,\nN. S. Manganaro, C.-Y. Huang, J. Cox, A. H. Dis-\nmukes, X. Zhu, J. P. S. Walsh, T. Cao, C. R.\nDean, and X. Roy, Designing magnetic properties in\ncrsbr through hydrostatic pressure and ligand substi-\ntution, Advanced Physics Research , 2300036 (2023),\nhttps://onlinelibrary.wiley.com/doi/pdf/10.1002/apxr.202300036."    },    {        "title": "2303.11794v2.Robust_intralayer_antiferromagnetism_and_tricriticality_in_a_van_der_Waals_compound__VBr3_case.pdf",        "content": " \n 1 Robust intralayer antiferromagnetism and tricriticality in \na van der Waals compound: VBr 3 case \nDávid Hovančík1, Marie Kratochvílová1, Tetiana Haidamak1, Petr Doležal1, Karel Carva1, Anežka \nBendová1, Jan Prokleška1, Petr Proschek1, Martin Míšek2, Denis I. G orbunov3, Jan Kotek4, Vladimír \nSechovský1 and Jiří Pospíšil1* \n1 Charles University, Faculty of Mathematics and Physics, Department of Condensed Matter Physics, Ke Karlovu 5, 121 16 \nPrague 2, Czech Republic  \n2 Institute of Physics, Czech Academy of Sciences,  Na Slovance 2, 182 21 Prague 8, Czech Republic  \n3 Hochfeld -Magnetlabor Dresden (HLD -EMFL), Helmholtz -Zentrum Dresden -Rossendorf, 01328 Dresden, Germany  \n4 Department of Inorganic Chemistry, Faculty of Science, Charles University, Hlavova 8, 128 40 Prague 2,  Czech Republic.  \n \nKEYWORDS : VBr 3, van der Waals, antiferromagnetism, metamagnetism, tricritical  point   \nABSTRACT  \nWe studied magnetic states and phase transitions in the van der Waals antiferromagnet VBr 3 experimen-\ntally by specific heat and magnetization mea surements of single crystals in high magnetic fields  and the-\noretically by  the density functional theory calculations focused on exchange interactions. The magnetiza-\ntion behavior mimics  Ising antiferromagnets with magnetic moments pointing  out-of-plane due to strong \nuniaxial magnetocrystalline anisotropy. The out -of-plane magnetic field induces a spin -flip metamagnetic \ntransition of first -order type at low temperatures , while at higher temperatures the transition becomes \ncontinuous. The first -order and conti nuous transition segments in the field -temperature phase diagram \nmeet at a tricritical point . The magnetization response to the in -plane field manifests a continuous spin  \ncanting  which is completed  at the anisotropy field µ0HMA  27 T. At higher fields  the two magnetization \ncurves above saturate at the same value of magnetic moment µsat  1.2 µB/f.u., which is much smaller than \nthe spin -only ( S = 1) moment of the V3+ ion. The reduced moment can be explained by the existence of \nan unquenched  orbital magnetic  moment antiparallel to the spin. The orbital moment is a key ingredient \nof a mechanism responsible for the observed large anisotropy. The exact energy evaluation of possible \nmagnetic structures  shows that  the intralayer zigzag antiferromagnetic order  is preferred  which  renders \nthe antiferromagnetic ground state significantly more stable against the spin -flip transition than the other \noptions. The calculations also predict that a minimal distortion of the Br ion sublattice causes a radical \nchange of the orb ital occupation in the ground state, connected with the formation of the orbital moment \nand the stability of magnetic order.  \n \nI. INTRODUCTION  \nVan der Waals (vdW) magnets provide a natural platform for studying two -dimensional (2D) mag-\nnetism and its potential for advanced technologies like magnetooptics, spintronics, etc.1-6. Transition -\nmetal trihalides, TX3 (T = V , Cr ; X = Cl, Br, I), form an important group of vdW magnets. VI 37-9, CrBr 310, \n11, and CrI 312-15 become ferromagnetic (FM), whereas VBr 37, 16, 17, VCl 37, and CrCl 318, 19 are antiferromag-\nnetic (AFM) at low temperatures. Historically, FM materials were considered to be more interesting from \nthe point of view of applications, but recent discoveries show that AFM may hold even larger potential  \n 2 for magneti sm-based information storage and spintronics20-22. The TX3 compounds are dimorphic, adopt-\ning the rhombohedral BiI 3-type and the monoclinic AlCl 3-type (or related types) layered crystal struc-\ntures8, 9, 13, 14, 16, 23 -25. Both structure types are formed  by stacking the X-T-X triple layers (further referred \nto as monolayers). The T-ion layer in the BiI 3-type structure has a graphene like honeycomb form of a \nregular three -fold s ymmetry. The monolayers are equidistantly shifted, and the honeycomb network may \nbe distorted  in monoclinic structure8, 26. The weak  X-X vdW bond between neighboring monolayers allows \ntheir easy separation. This un ique property, combined with the magnetic ordering within a monolayer at \nfinite temperatures, provides a testing base for 2D magnetism and offers promising opportunities to fab-\nricate 2D nanoelectronic devices27, 28. \nThe V trihalides undergo a structural phase transition from a high -temperature trigonal structure to the \nlow-temperature monoclinic one at a temperature Ts ( 79, 90, 97 K for VI3, VBr 3, VCl3)16, 25. The distor-\ntions of the β angle in VBr 3 (90° 90.55° ) and VI 3 (90° 90.45° ) are only slightly different. Kong et al. \nreported VBr 3 antiferromagnetism below TN = 26.5 K with the magnetic moments perpendicular to the \nab-plane (out of plane direction) . We will mark this direction by  the symbol c* to distinguish it from the \nc-axis, which is not perpendicular to the ab plane in the monoclinic structure. The recent publication17 \nattributed the changes in Raman -scattering spectra around 90  K to the structural tra nsition associated with \na decrease in the crystal -structure symmetry from 𝑅3̅ to C2/m. The authors also reported minor hysteresis \nloops observed in low magnetic fields and ascribed them to a canted AFM order.  \nOur work is mainly devoted to aspects of VBr 3 physics not treated in previous studies presented in the \nliterature. The experiments were focused primarily on the influence of the magnetic field on the structural \nand magnetic phases in VBr 3, which can bring essential information on the mechanisms driving  the spe-\ncific phenomena. The structural -transition temperature Ts was found intact by magnetic fields contrary to \nthe reduction of Ts of the isostructural ferromagnet VI 3 in the out -of-plane field ( H || c*)24. The observed \ndramatically different responses of TN-related specific -heat anomalies and magnetization isotherms to the \nfields H || c* and H  c* reveal strong uniaxial magnetocrystalline anisotropy. VBr 3, similar to other an-\ntiferromagnets, undergoes a magnetic -field-induced AFM   PM metamagnetic phase transition  (MPT) \nat critical magnetic field Hc. We have observed a spin -flip transition in fi elds H || c* that is typical for an \nIsing -like antiferromagnet. In fields H  c*, i.e. in the a-b plane,  a continuous spin  canting  has been found \nrunning from the lowest fields and completed around 0Hc  27 T, which can be considered a reasonable \nestimate  of the anisotropy field. Such a high value of the anisotropy field indicates the presence of a \nsignificant V orbital magnetic moment. The existence of orbital moment can consistently explain also the \nsize of the saturated magnetic moment of 1.2  B/f.u. ob served in high magnetic fields ( 0Hc  T). This \nvalue is much smaller than the spin -only value (2 B) of the magnetic moment expected for the V3+ ion \nwith a quenched orbital moment.  \nA closer inspection of the evolution of specific heat and magnetization  isotherms  in H || c* reveal s that \nthe MPT  at lower temperatures has a first -order transition character  contrary to higher temperatures up to \nTN, where the MPT  becomes a continuous transition . This behavior is typical for antiferromagnets with a \ntricritica l point that separates the first -order and the continuous metamagnetic transition segments in the \nfield-temperature ( H-T) magnetic phase diagram29. The archetype of this unique phenomenon is FeCl 229-\n31 also a layered vdW compound32. Howe ver, the tricriticality has not been reported in any other vdW \nantiferromagnet.  \nPrevious first principles calculations for VBr 3 predicted a strong FM intralayer superexchange interac-\ntion like in  VI333. This result suggests that the AFM structure should be composed of FM monolayers that \nare AFM coupled in the out -of-plane direction (layered AFM order), e.g., in the case of CrPS 434. The \ncalculation results excluded only the Néel AFM order inside layers caused by the AFM nearest -neighbor \ninteractions. Nevertheless, sufficiently strong negative e xchange interactions between distant V next near-\nest neighbors within the monolayer could also lead to more complex intralayer orders such as a stripe or \na zigzag AFM order35, 36. Magnetism at a honeycomb lattice is  typically described by interactions up to \nthird nearest -neighbor ( J1-J2-J3 model ). The zigzag AFM order has been identified in other quasi -2D sys-\ntems on the honeycomb spin -lattice FePS 337, 38 and NiPS 339, 40. The interlayer interaction was predicted to  \n 3 be an order of magnitude smaller than the intralayer one. Other studies were devoted only to the theoretical \ncalculations of properties of VBr 3 monolayers41, 42. \nThe current state of understanding of the physics of the vdW AFM VBr 3 motivated us to focus the \npresent study on the role of crystal structure details on exchange interactions and the influence of the \napplied magnetic field on  the magnetic states. To study the impact of crystal -structure details in determin-\ning the character of the exchange interactions and the magnetic ground state, we performed density func-\ntional theory (DFT) calculations. Spin -orbit coupling (SOC), which play s a vital role in the formation of \nthe orbital moment, has been included. The calculations demonstrate that the ion network deformation is \ncrucial for the stability of the AFM ordering of VBr 3. Our findings contradict the previous assumption of \nlayered AFM  ordering since we have found that zig -zag AFM order is energetically more favorable than \nFM order in layers. The predicted energy difference between this state and FM order is in agreement with \nthe experimentally  measured field needed to reorient spins. W e have also shown that a particular displace-\nment of Br atoms has a powerful impact on the ground state orbital occupation and exchange interactions.  \n \nII. EXPERIMENTAL SECTION  \nA. Material synthesis and single -crystal growth  \nThe single crystals of VBr 3 have been g rown from pure elements (V 99.9 %, Br 2 99.5%) using the \nchemical vapor transport method. This approach prevented contamination of the final product by residuals \nfrom precursors readily used to produce Br 2, e.g. TeBr 4 used by Lyu et al.17, in the reaction and transport-\ning tube. First, a quartz tube with V metal powder kept at 300°C was evacuated overnight with simulta-\nneous baking down to a vacuum of 10-7 mbar for proper degassing. Then, the tube was filled with 6N \nargon gas and coole d to -60°C by dipping into an ethanol bath cooled by dry ice. Subsequently, the stoi-\nchiometric volume of Br 2 liquid was injected inside the frozen tube, where it instantly solidified. The \ncontinuously cooled tube with the mixture of V powder and solid Br 2 was then pre -evacuated by a Scroll \npump and evacuated by a turbomolecular pump for 5  minutes with no signatures of Br 2 evaporation. Fi-\nnally, the sealed quartz tube was inserted into a gradient furnace where a thermal gradient of 460/350°C \nwas kept for two weeks to transport all V metal from  the hot part. The single crystals of the black -reflective \ncolor of several millimet er square dimensions have been obtained. The single crystals seem stable for \nseveral hours with no significant degradation effect. The de sired 1:3 composition was confirmed by EDX \nanalysis. The crystallinity and orientation of the single crystals were confirmed by the Laue method show-\ning sharp reflections. The rhombohedral c -axis is perpendicular to the plane of plate -like crystals.  \nB. Magneti zation and specific heat study  \nSpecific heat was measured by the relaxation method and magnetization data with a vibrating sample \nmagnetometer in magnetic fields up to 14 T using a Quantum Design PPMS 14T (Quantum Design Inc.) \nand up to 18 T with a Cryogen ic cryomagnet (Cryogenic Ltd.). The Néel temperature TN was determined \nfrom the temperature dependence of specific heat as the point of the balance of the entropy released at the \nphase transition. The field dependence of specific heat was measured point by  point in a stable magnetic \nfield. To probe the angular dependence of magnetization in ac*-plane and ab-plane (7  T) we used a home-\nmade rotator for MPMS  7T with a rotation axis orthogonal to the applied magnetic field. The magnetiza-\ntion in pulsed magnetic f ields up to ∼58 T was measured at the Dresden High Magnetic Field Laboratory \nusing a high -field magnetometer43 with a coaxial pick -up coil system. Absolute values of the magnetiza-\ntion were calibrated using data obtained in steady fields. The measurements in magnetic fields were per-\nformed for two perpendicular directions of the field: in th e ab-plane and perpendicular to the a-b plane. \nThe c-axis in the monoclinic structure is not perpendicular to the ab plane. To avoid ambiguities, we use \nthe symbol c* for the direction perpendicular to the a-b plane ( c*  ab plane). In the trigonal structu re c* \nis parallel to the c-axis ( c* || c). We extra note, that the single crystals are very thin and fragile plates   \n 4 moreover unstable in air , and commonly used solvents or glues44. It complicated the preparation and fix-\nation of the samples for various experimental methods and  caused  slight uncertainty of their final masses \nwhich result ed in deviation of calculated absolute value s of the  physical quantities with 5 -10%-error  bar. \nC. Theoretical calculations  \nDensity fun ctional theory (DFT) calculations employed the full -potential linear augmented plane wave \n(FP-LAPW) method, as implemented in the band structure program ELK45. Spin -orbit coupling (SOC) \nplays a crucial role in V  trihalides. Therefore, it has been included in the calculations46, 47. The generalized \ngradient approximation (GGA) parametrized by Perdew -Burke - Ernzerhof48 has been used to perform \ngeometrical relaxation of the structure. We have used local density approximation (LDA) as the exchange -\ncorrelation potential to determine more s ubtle properties requiring high precision in energy, as are the \nexchange interactions and magnetic anisotropy energy since with GGA we have noticed numerical \ninstabilities and convergence problems. These may be related to the presence of multiple local ene rgy \nminima in the configurational space, or maybe of the same origin as those found in the pseudopotential -\nbased calculation of vdW trisulfides49. Since the material is known to be a Mott insulator, we have \nincluded the effect of electron -electron correlations in terms of the Hubbard correction term U = 4.3 eV  \n50-52 acting on V 3d electrons. Double counting was treated in the fully localized limit. Similar \nDFT+U+SOC calculatio ns have already successfully described the quasi -2D compound VI 352. The entire \nBrillouin zone has been sampled by 10 105 k-points and  the convergence w.r.t. k -mesh density has been \nverified. For total energy calculations, increased angular momentum cut -off of the expansion into \nspherical harmonics has been used with lmax = 14 for the APW functions and lmaxo = 8 for the muffin -tin \ndensit y and potential. To evaluate the interlayer interaction JL a unit cell doubled in the z -direction was \nused. Energies of calculated self -consistent ground states with forced FM and AFM interlayer alignment \n(LAFM)  then allow us to calculate JL = EFM  ELAFM for different possible geometries, similar to the \npressure dependence of JL calculated already for VI 351. To compar e the energies of the possible magnetic \norderings inside the layer we double the unit cell in one of the planar directions. The basis with 4 V atoms \nin the layer allows us to evaluate energies of AFM Néel, stripe, and zig -zag orders, as well as FM order53. \nIII. RESULTS AND DISCUSSION  \nTwo transitions were detected in specific heat data (see Fig. 1a) in agreement with Kong et al.16 and \nLyu et al.17. The sharp peak at Ts = 90 K corresponds to the structural transition between the monoclinic \nand trigonal phase s. The -shape anomaly at TN = 26.5 K indicates a  second -order phase  transition be-\ntween the low -temperature AFM state and the PM state (AFM   PM).  \n  \na) b)  \n 5 FIG.  1. a)The temperature dependence of specific heat of VBr 3 in zero magnetic field ( ) and the field \nof 13.5  T applied parallel ( ) and perpendicular ( ) to c*. The arrows mark the position of TN and Ts. \nThe inset shows the variation of the TN-related anomaly for the magnetic fields applied in the c* direction.  \nb) The specific -heat isotherms of VBr 3 at selected temperatures for the magnetic field parallel to c*. \nThe structural phase transition is intact by magnetic fields up to 18 T (13.5 T data are shown in Fig. 1a) \napplied in both principal directions. This result contrasts with the significant field dependence  of the cor-\nresponding structural phase transformation of isostructural vdW FM VI 3 in fields parallel to c*24, which \ncan be understood as a result of the ferromagnetic correlations detected in Raman spectra54, combined \nwith strong magnetoelastic interaction. On the other hand, the applied magnetic field pushes the TN-related \nanomaly in VBr 3 to lower temperatures and smears it out. These effects are much more pronounced for \nH || c*. No sign of magnetic phase transition is detected in magnetic fields  > 17.5 T. \nSpecific -heat isotherms measured in varying magnetic fields H || c* up to 18  T are shown in Fig. 1b. A \nbroad -peak anomaly on the Cp(H) isotherm manifests the second -order metamagnetic transition \n(AFMPM) at temperatures from  12 K and TN. For temperatures lower than  12 K, no anomaly is detected \non the Cp vs. H plot. That is consistent with Cp vs. T data for T < 12 K in the inset of Fig. 1a which barely \nchange with the applie d magnetic field. Considering Maxwell's relation  (eq 1) \n(𝜕𝑆\n𝜕𝐻)\n𝑇=(𝜕𝑀\n𝜕𝑇)\n𝐻      (1) \nthis behavior indicates a negligible  temperature dependence of magnetization in different magnetic fields \nT < 12 K. \nThe 2 -K M(H) isotherms H || c* and H  c* in static fields up to 18  T, are shown in Fig. 2a. The M(H) \nmeasured in H  c* remains linear up to 15 T and  then becomes convex . The convexity  is more pro-\nnounced with further increasing H. The increasing H || c* field induces a steep S -shape increa se in fields \nabove 15.5 T with an inflection point at 16.9  T. This anomaly resembles a spin -flip transition typical for \nantiferromagnets with a strong uniaxial magneto -crystalline anisotropy. The observed field hysteresis \n0H  0.3 T suggests a first -order metamagnetic transition.  \nTo reveal the limits of the AFM -phase stability in the H-T phase space, we employed pulsed magnetic \nfields up to 58 T. The 1.9 -K M(H) isotherms are displayed in Fig. 2a. For H || c*, the S -shape profile is \nbroader than in static fields, most likely due to a slower reaction of the magnetic moments to the fast field \npulse. The magnetization in fields above the transition gradually saturates to the final value of 1.2  B/f.u. \nat 58 T.  The M(H) curve for H  c* is linear, becoming convex above 20  T. The convexity increases with \nincreasing H up to the inflection point at  23 T, where the AFM   PM transition appears. Above 27  T, \nthe magnetization gradually saturates and a pproaches the H || c* curve. Both  M(H) curves  join above 40 T. \nThe observed saturated moment of 1.2  B/f.u. is much lower than  that expected for the V3+ spin-only \nmagnetic moment (2  B/f.u.)16, 41, 42, 47, 55 -57. It is worth noting that the experimentally observed saturated \nmoment for VBr 3 compares well with that measured on the FM VI 3 single crystals in which a large V \norbital magnetic moment has been recently confirmed by X -ray magnetic circular dichroism  experi-\nments58. This result, together with the observed strong magnetic anisotropy and l arge Hc, fields are clear \nindications that the V -ion in AFM VBr 3 also bears a significant orbital moment. The results of our anisot-\nropy study and H  c* magnetization isotherms are shown in Supplemental Material  (SM)59.  \n 6  \nFIG.  2. a)The magnetization isotherms of VBr 3 measured at 1.9 K in static fields (SF) up to 18 T and \npulsed fields (PF) H || c* and H  c* up to 58 T. Inset: The detail of the plots for H || c* between 13 and \n20 T. The arrow points to the critical field Hc. The absolute value of the calibrated magnetization isotherms \ncan vary with  an error  bar ± 10%.  b) Stoner -Wohlfart h simulation. Plott ed M along H per one site as a \nfunction of applied external field H, for H parallel to the easy axis (blue) or perpendicular to the easy axis \n(red). Solid lines are calculated according to the model assuming purely uniaxial anisotropy ( K2 = 0 meV), \nwhile c alculations depicted with dashed lines include  a higher -order term ( K2 = 0.2 meV).  \nMagnetization isotherms simulations based on the Stoner -Wohlfahrt model for the two sublattices have \nrecovered some characteristics of the observed M(H) curves at zero tempe rature. This model assumes two \nmacroscopic moments representing the two spin sublattices described by unit moments m1 and m2. These \nare coupled by an effective interaction J* and subjected to external field H and magnetic anisotropy60. \nThe corresponding free energy expressed per one V atom is then given by  (eq 2)  \nE = -J*m1  m2 + EAN(m1,m2) + H  (m1 + m2)µsat     (2) \nAssuming uniaxial anisotropy with an easy axis and magnit ude K1, we may rewrite the energies using \nthe angles ( ϑ1,ϑ2)  between the two “moments ” and the easy axis  (eq 3) . \nEAN(m1,m2) = K1  (sin2ϑ1 + sin2ϑ2)     (3) \nThe external magnetic field is applied in the direction given by unit vector h. We search for moments \ndirections minimizing the free energy. The observa ble quantity is the projection of total magnetization to \nthe direction of the field, denoted as M, per one V atom it is given as  (eq 4)  (value s = 1.3 B was used)  \nM = h  (m1 + m2)µs     (4) \nThe fact that the M(H) experimentally observed in the H || c* regime almost reaches saturation after \nthe sudden increase indicates that a spin -flip has occurred61. This behavior can be reproduced in this model \nif K1 ~ J*/2 or higher. Here we assumed J* = -1.2 meV , K1 = 0.6 meV, which provides good agreement \nwith the experimentally observed dependence for H || c*. But for the H  c* curve a slower approach to \nsaturation is predicted - see Fig. 2b. There can be many reasons for this, given the limitations of the model. \nAs the next step , we have extended the model by including a higher -order anisotropy term \nK2  (sin4ϑ1 + sin4ϑ2) whi ch is present in systems with hexagonal or rhombohedral symmetry62. These \nterms were found to strongly affect field -induced magnetization dynamics in antiferromagnets despite \ntheir small value63. A calculation assuming K2 = 0.2 meV leads to markedly improved agreement with the \nobserved slope for high H  c* (compare solid and dotted curves in Fig.  S5 in SM59). This indicates that \nthe spe cific behavior of M(H) curves can be explained by the difference in the presence of  anisotropy \nterms of fourth -power in sin ϑ. \nThe experimentally observed non -zero slope for small H || c* could be ascribed to an inclination of the \nfield from th e easy axis, or the high content of defects in samples. Since the angular dependence of mag-\nnetization  (Fig. S3 in SM ) shows that the c direction corresponds to the easy axis, the latter explanation \na) b)  \n 7 is more plausible. We have tested the model for various va lues of K1 and inclinations of field direction \nfrom the easy axis results are displayed  in SM59 Fig. S4. The employed semiclassical spin dynamics does \nnot capture the effect of quantum fluctuations. However, a recent study performed also on the J1-J2-J3 \nmodel for the honeycomb lattice finds that quantum fluctuations are strongly suppressed for the case of \nS = 1, so that the boundaries between different phases are only slightly changed as compared to the clas-\nsical solution64. Therefore we do not expect it to affect the main conclusions drawn from the semiclassical \napproach. Quantum fluc tuations in AFM V trihalides deserve further study.  \nSince there are indications of a significant orbital moment value in VBr 3, we have examined how it \ncould affect the M(H) curve . A crucial contribution to magnetic anisotropy originates from crystal field \neffects (magnetocrystalline anisotropy). This interacts with the orbital moment directly, but its effect is \ntransferred to spin via the spin -orbit interaction (SOI). Therefore, a sizable orbital moment can justify the \nhigh anisotropy that we used in our model, although the total moment is predominantly of spin origin. \nFurthermore, we have considered it as an extension  of our model where the spin and the orbital moment \nwould be treated separately. However, within the expected values of the magnitude of spin -orbit interac-\ntion, this does not lead to a change of properties that would be observable. A detailed description is pro-\nvided in SM . \nFig. 3 shows that the increasing temperature up to 12  K has a tiny effect on the first-order MPT \n(FOMPT) in  the M(H)  curves in the field H || c*. The critical field at 12 K is 0Hc = 16.4 T. This behavior \nreasonably  matches the Cp(H) dependence that is related via Maxwell's relation (see Eq.  1). The second -\norder  (continuous)  MPT  (SOMPT ) accompanied by magnetic fluctuations appe ars as a bump on the iso-\nthermal Cp vs. H plot while the anomaly disappear s at FOMPT  (Fig.  1b) because the  released  latent heat \nhas vanished  by the principle of the relaxation method  in PPMS apparatus (see as the  example in ref  65). \nWhen inspecting  Fig. 1b with decreasing temperature we observe a smeared bum p on the Cp vs. H (a \nsignature of SOMPT dependence ) still at 12 K whereas a FOMPT at this temperature is suggested by \nmagnetization behavior. We take this  as a si gn of  TCP proximity.  \n \nFIG.  3. The magnetization isotherms of VBr 3 measured at various temperatures in pulsed fields (PF) \nfor H || c* up to 27 T. \nThis evolution reflects the increasing influence of thermal fluctuations leading to the change of the \ncharacte r of the MPT to a continuous  SOMPT  at temperature  interval  Tcp < T < TN. It indicates that VBr 3 \nbelongs to the family of antiferromagnets in which the MPT is a FOMPT at low temperatures and in the \nhighest magnetic fields whereas, at higher temperatures and  lower fields, a continuous  (second -order) , i.e. \nSOMPT is observed.  The SOMPT  in the zero-field limit is corroborated by the -anomaly at TN in the \ntemperature dependence of  specific heat, in clear cont rast to rare systems with first -order AFM transition \nwith the peak -like anomaly66-69. The FOMPT is characterized by a sudden reversal of AFM -coupled FM \nsublattice(s) to the direction of the applied field. The high -field ( H > Hc) state is then characterized by \n \n 8 field-polarized magnetic moments. It resembles an FM alignment of magnetic moments, however,  it is a \nparamagnetic (not ferromagnetic) state29. It is used to  be called a polarized paramagnet (PPM) regime70. \n \nFIG.  4. The H-T phase diagram of VBr 3 for magnetic field applied along the c*. The Blue curve re p-\nresents the inflection points on MPT in pulsed field data, the red curve the position of the anomaly in \nspecific heat data ( Cp), the cyan curve represents the kink of MPT in steady field data (see Fig. S1), and \nmagenta point shows the position of anomalie s in field dependence of specific heat data. The yellow point \nshows the estimated position of TCP. The TCP is tentatively placed at TTCP = 12 K, 0HTCP = 16.4 T. \nM(S2) represe nts data measured on a sample of lower quality (less stable in air) for comparison.  \nThe Ising antiferromagnet FeCl 230, 31 with competing FM and AFM exchange interactions is considered \nthe archetype of this interesting family of materials.  To retain the FOMPT above 0  K and to permit the \noccurrence of hysteresis  at MT , a ferromagnetic intra -sublattice exchange is necessary  for a simple Ising \nsystem . The FM  exchange in VBr 3 is documented b y the Heisenberg exchange interactions in  the third \nnearest -neighbor model -see Table 2 where J1 > 0. In such a case a ratio * = TTCP/TN ≈ 0.45 for VBr 3 is \nestablished and  proportional to  (eq 5)  \n* = 1 - (A/3Γ)    (5) \nwhere A and Γ are molecular field coefficients30. \nThe H-T phase diagram of VBr 3 for H || c* is displayed in Fig. 4. The PM   AFM phase -transition \nline has two parts: a low -temperature part of FOMPTs and has high -temperature part of SOMPTs  sepa-\nrated by the tricritical point (TCP)29, 71 which we tentative ly place at [12 K, 16.4 T] in the H-T phase  \ndiagram . Unfortunately, we do have not more  M vs. H and Cp vs. H isotherm s data measured at steady \nfields enabling us to determin e TTCP and HTCP coordinates more precisely . The l ack of experimental facil-\nities pro viding steady fields within a reasonable T-H space prevents performing magnetization measure-\nments for  a reliable study of critical coefficient s in the interesting case of 2 D antiferromagnet with a  TCP.  \nWe emphasize that VBr 3 is not a single non -Ising antif erromagnet with FOMPTs and SOMPTs seg-\nments in the H-T phase space separated by TCP. Analogous behavior is found also in some antiferromag-\nnets characterized by strong uniaxial anisotropy71-73 as well as in one exhib iting strong orthorhombic an-\nisotropy74, 75. \nWe have used first -principles calculation methods to evaluate the energies of the FM -ordered system, \n3 plausible intralayer AFM orderings, and the layered AFM state. Firs t calculations were performed for \nthe lattice geometry determined by X -ray diffraction at 100 K, where the lattice parameters were found to \nbe a = 6.3711  Å, c = 18.3763  Å, and the Br planes are placed at hBr = 0.07928  c above (or below) the V \nplanes16. Our calculations predict that the magnetic ground state has a zigzag AFM  (Fig.  5a) order and not \n \n 9 the suggested layered AFM structure consi sting of ferromagnetically ordered layers with antiparallel ori-\nentation between neig hboring layers33 (Fig. 5b).  \n  \nFIG.  5. a) The predicted  AFM magnetic structure of VBr3. Only a single layer of V3+ ions is displaye d. \nThe AFM structure consists of FM zig -zag chains coupled antiferromagnetically within the plane. b) Lay-\nered AFM  structure (ferromagnetically ordered layers with antiparallel orientation between neighboring \nlayers)33. Two layers of V3+ ions are displayed, and lines in the vertical direction connect V ions stacked \non top of each other (differing only in the z coordinate).  c) Calculated eff ective exchange J* and anisot-\nropy K1 as a function of the distance r (in multiples of lattice parameter a). The occupation of the V3+ d-\nstates is calculated and schematically displayed in the diagrams.  \nAll other considered AFM orders, including the layered  AFM state, are energetically less favorable \n(Table 1). Using calculated energies of magnetic configurations one can map the problem to an e ffective \nHeisenberg Hamiltonian (eq 6) ,  \n𝐻=−1\n2∑𝐽𝑖𝑗𝑖,𝑗𝑠𝑖∙𝑠𝑗 (6) \nwhere si is the  unit vector corresponding to  the i -th spin in the system, Jij is the exchange energy \nbetween the i -th and j -th spins, and the sums run over magnetic atoms in the system. Magnetism at a \nhoneycomb lattice is efficiently described by the Heisenberg model with interact ions up to the third near-\nest neighbor ( J1-J2-J3 model, sche matically depicted in Fig.  6b). Individual  exchange interactions can be \ncalculated from the calculated four energies  of the four plausible magnetic order76, and are shown in \nTable  2. The energy d ifference between the AFM zig zag state and the FM state corresponds to the stability \nof the system w.r.t. a field -induced spin -flip transition. It can be denoted as 2. J*, an effective interaction \nbetween the two AF c oupled sublattices (in J1-J2-J3 model it equals –J1 - 4J2 - 3J3). We find J* = -1.9 meV \nper V atom. This finding appears to be consistent with the experimental knowledge about the presence of \nthe AFM state and its stability, as the calculated J* is rather close to the value we could use to simulate  \nthe experimentally observed spin reorientation transition ( Fig. 2) within the Stoner -Wohlfart h model . \n  \nb) c) \na)  \n 10 Table 1: The calculated energies of magnetic structure s relative  to the zigzag AFM.  \nType  of magnetic \nstructu re Relative  total ener gies E (meV/f.u.)  \n100 K structure  Relaxed structure  \nAFM zigzag  0 0 \nFM +1.91 +2.66  \nLayered AFM  +1.65  +2.09 \nAFM stripe  +3.26 +4.87  \nAFM Neél  +6.04 +4.81 \n \nTable 2: The calculated Heisenberg exchange interactions  Ji (in meV)  \nInteraction  J1 J2 J3 \n100 K structure  1.87 -0.57 -0.46 \nRelaxed structure  1.76 -0.33 -1.04 \nFIG.  6. a) VBr 3 lattice plane , predicted relaxation is shown . b) Heisenberg model with interactions up \nto the third nearest  neighbor ( J1-J2-J3). \nNote that the magnetic ally ordered state is present only in the low -temperature structure below Ts. The \nlow-temperature structure is only known to be slightly distorted so that its symmetry is reduced to the \nmonoclinic22, but the structure details about anion positions are unknown. Therefore, as the next step, we \nperformed geometrical relaxation of the  atomic positions, starting from the monoclinically distorted  struc-\nture. The optimiz ed structure exhibits the following significant changes compared to the original one: i) \na decrease of Br plane height hBr by only  0.002 c, ii) a slight planar shift of Br sites in the direction \ntowards the midpoint between its two nearest V atoms, so that its distance r from the hollow site in the \nhoneycomb lattice increases from the value of rm = 0.349 a (measured at 100  K) to ropt = 0.353a (see \nb) a)  \n 11 Fig. 6a). A similar distortion has been predicted by first principles computational optimization in BiI 3, but \nin the opposite direction, anions have moved towards the vacant point in the cation honeycomb lattice77. \nWe have studied the effect of these deformations on the difference between FM and zig -zag AFM order \nenergies, J*. Changing the Br height hBr does not introduce a significant change in the calculated exchan ge \ninteractions  to the extent suggested by the relaxation. The dependence of the effective exchange interac-\ntions J* on the planar Br distortion  (Fig.  6a) is shown in Fig. 5c. The  relaxed hBr was included in these \ncalculations. Note that the exchange varies  only slowly with r (within numerical precision), while at a \nspecific distance, a sudden change of the effective magnetic exchange occurs.  \nLike VI3, the electronic structure of VBr 3 may converge to two strikingly different solutions: a state \nwith a quenche d orbital moment, typical for 3d transition metals in a medium -strength crystal field, or a \nstate with a high orbital moment. This problem is connected with the position of different energy levels \nof trigonal symmetry electronic d orbitals in the ground st ate, in particular, with the question of whether \nthe a1g orbital or one of the 𝑒𝑔′ orbitals will be positioned above the Fermi level, as debated for VI 346, 47, 58, \n78. If the a1g orbital is unoccupied, 𝑒𝑔′ has to be fully occupied and the orbital moment would be sup-\npressed. For the spin moment, the calculations predict values close to 2 B/f.u. in agreement with Hund's \nrules, but the experimental magnetic moment is 1.2  B/f.u. This sharp transition is as sociated with a \nchange in the occupation of different electronic orbitals in the ground state. For r < rcrit states with fully \noccupied 𝑒𝑔′ are preferred, while a state with a1g occupied turns out to be favorable for r above this thresh-\nold value rcrit = 0.344 a. \nA correct Br position has to be used to obtain the correct ground state in calculations. For r = ropt our \ncalculations predict a sizeable mag netocrystalline anisotropy 0.89  meV , a value that is reasonably close \nto that we have used to describe  the field -induced spin -flip within the S toner-Wohlfahrt model . For ex-\nample, the assumption of the so -called ideal position (with r = a/3)77 would lead to a different magnetic \norder as well as an electronic structure. A compression of r by only ~3% from its optimal value is sufficient \nto overcome rcrit and reach this different state. Our calculations also predict easy plane preference for that \nsituation, together with significant o rbital reoccupation for moment orientation in -plane. The state  pre-\ndicted for  r < rcrit does not correspond to current observations, but it could be reached by the application \nof pressure or if specific phononic modes would be significantly occupied.  \nA zigz ag AFM magnetic structure was also predicted in vdW AFM FePS 3. The VBr 3 magnetization \nloops significantly differ from those of FePS 3 where specific magnetization plateaus (cascades) were de-\ntected53, 79. These occur due to a special combination of exchange interaction values that favor a situation \nwith partially flipped moments within a small range of applied fields, where for example six moments in \na unit cell point in one direction and two moments in the opposite direction79. Only a simple spin -flip \ntransition was found in VBr 3, therefore, based on the suggested models, we suppose the simple out -of-\nplane zigzag intralayer AFM structure as displayed in Fig. 5a. without preference for intermediate par-\ntially flipped states.   \n \nIV. CONCLUSIONS  \nWe have grown high -quality single crystals of  VBr 3 by chemical vapor transport directly from pure \nelements. The study addresses the character of antiferromagnetism and phase transitions by measuring \nspecific heat and magnetization in static magnetic fields up to 18.5 T and pulsed fields up to 58 T. T he \nstructural transition at Ts = 90 K remains intact by magnetic fields, which is in contrast with the decrease \nof Ts in the isostructural ferromagnet VI 3. TN was found to decrease with increasing magnetic field much \nfaster for the out -of-plane field  than for the in -plane -direction field. The magnetization response to the \nmagnetic field is strongly anisotropic. A first-order  metamagnetic spin-flip transition to PPM  occurs at \n0Hc = 16.9 T in the out -of-plane field at 2  K. This transition remains first -order at temperatures up to \n12 K. At higher temperatures up to TN the AFM   PM SOMPT  occurs at lower fields . These phenomena \nare observed in some Ising antiferromagnets with stro ng uniaxial anisotropy and competing AFM and FM \ninteractions.  Results of our stu dy suggest  VBr 3 to be a member of this group of materials  with TCP at  \n 12  [12 K, 16.4 T ]. Further experiments that we do have not available are needed to determine the coordi-\nnates of TCP precisely and determine the dimensionality  of the system  in terms of critical coefficients . In \nparticular, high steady field (at least up to 20 T) magnetization and electrical transport measurements are \ndesirable. To our best knowledge, VBr 3 seems  to be the only vdW antiferromagnet besides  the archetype \nFeCl 2 in which the tricriticality  has been reported until now . \nThe in -plane magnetization curve represents continuous spin  canting  towards  the PM state with spin \nmoments fully oriented along the app lied field in fields  above  27 T (at 2  K). The saturated magnetic mo-\nment observed in fields above 50  T µsat  1.2 µB/f.u. is much smaller than the spin -only moment (2  µB) of \na V3+ ion. This indicates the existence of a significant orbital magnetic moment, w hich partly compensates \nfor the spin moment similar to the VI 3 case.  \nOur calculations predict that the magnetic structure in VBr 3 is based on  the intral ayer antiferromagnetic \norder  in the form of a zigzag pattern . We have also found that the relaxation of Br atomic position plays \nan important role in the electronic structure calculations and the resulting magnetic properties of the sys-\ntem. A small change in the distances of Br atoms from the hollow site in the honeycomb lattice leads to a \nsudden redistribut ion of orbital occupation in the ground state. This suggests that phonon modes leading \nto a similar displacement would be strongly coupled to magnetic order here.  \nASSOCIATED CONTENT   \nSuplemental material . Results of magnetocrystalline anisotropy study of V Br3 compound by angular \ndependence of magnetization; extended results of  specific heat,  steady , and pulsed -field magnetization \nstudy; extra theoretical calculations supporting the conclusions concerning the magnetic structure  of VBr 3. \nACKNOWLEDGMENT  \nThis w ork is a part of the research project GAČR 21 -06083S which is financed by the Czech Science \nFoundation  and project GAUK 938220  financed by  the Charles University  Grant Agency . The single -\ncrystal growth and characterization, and experiments in steady magnet ic fields were carried out in the \nMaterials Growth and Measurement Laboratory MGML (see: http://mgml.eu ) which is supported within \nthe program of Czech Research Infrastructures (project no. LM2023065 ). This project was al so supported  \nby OP VVV project MATFUN under Grant No. CZ.02.1.01/0.0/0.0/15_003/0000487. We acknowledge \nthe support from HLD at HZDR, a member of the European Magnetic Field Laboratory (EMFL) where \nthe measurements in pulsed magnetic fields were done. This  work was supported by the Ministry of Ed-\nucation, Youth and Sports of the Czech Republic through the e -INFRA CZ (ID: 90140). We appreciate \nfruitful discussions with K. Výborný. The authors are also indebted to Dr. Ross H. Colman for critical \nreading and co rrecting of the manuscript.  \nREFERENCES  \n1 D. L. Duong, S. J. Yun, and Y. H. Lee, ACS Nano 11, 11803 (2017).  \n2 K. S. Burch, D. Mandrus, and J. G. Park, Nature 563, 47 (2018).  \n3 W. Zhang, P. K. J. Wong, R. Zhu, and A. T. S. Wee, InfoMat 1, 479 (2019).  \n4 M. Gibertini, M. Koperski, A. F. Morpurgo, and K. S. Novoselov, Nature Nanotechnology 14, \n408 (2019).  \n5 M. C. Wang, C. C. Huang, C. H. Cheung, C. Y. Chen, S. G. Tan, T. W. Huang, Y. Zhao, Y. F. \nZhao, G. Wu, Y. P. Feng, H. C. Wu, and C. R. Ch ang, Annalen Der Physik 532, 1900452 (2020).  \n6 S. Kumari, D. K. Pradhan, N. R. Pradhan, and P. D. Rack, Emergent Materials 4, 827 (2021).  \n7 J. A. Wilson, C. Maule, and J. N. Tothill, Journal of Physics C: Solid State Physics 20, 4159 \n(1987).  \n8 S. Son, M. J . Coak, N. Lee, J. Kim, T. Y. Kim, H. Hamidov, H. Cho, C. Liu, D. M. Jarvis, P. A. \nC. Brown, J. H. Kim, C. H. Park, D. I. Khomskii, S. S. Saxena, and J. G. Park, Physical Review \nB 99, 041402(R) (2019).  \n9 T. Kong, K. Stolze, E. I. Timmons, J. Tao, D. R. Ni,  S. Guo, Z. Yang, R. Prozorov, and R. J. Cava, \nAdvanced Materials 31, 1808074 (2019).   \n 13 10 H. Yoshida, J. Chiba, T. Kaneko, Y. Fujimori, and S. Abe, Physica B 237-238, 525 (1997).  \n11 L. D. Jennings and W. N. Hansen, Physical Review 139, A1694 (1965).  \n12 J. F. Dillon and C. E. Olson, Journal of Applied Physics 36, 1259 (1965).  \n13 M. A. McGuire, H. Dixit, V. R. Cooper, and B. C. Sales, Chemistry of Materials 27, 612 (2015).  \n14 H. Wang, V. Eyert, and U. Schwingenschlogl, Journal of Physics -Condensed Matter 23, 116003 \n(2011).  \n15 W. B. Zhang, Q. Qu, P. Zhu, and C. H. Lam, Journal of Materials Chemistry C 3, 12457 (2015).  \n16 T. Kong, S. Guo, D. Ni, and R. J. Cava, Physical Review Materials 3, 084419 (2019).  \n17 B. Lyu, L. Wang, Y. Gao, S. Guo, X. Zhou, Z. Hao, S. Wan g, Y. Zhao, L. Huang, J. Shao, and M. \nHuang, Physical Review B 106, 085430 (2022).  \n18 B. Kuhlow, Phys. Status Solidi A -Appl. Res. 72, 161 (1982).  \n19 C. Starr, F. Bitter, and A. R. Kaufmann, Physical Review 58, 977 (1940).  \n20 T. Jungwirth, X. Marti, P. Wadl ey, and J. Wunderlich, Nature Nanotechnology 11, 231 (2016).  \n21 P. Němec, M. Fiebig, T. Kampfrath, and A. V. Kimel, Nature Physics 14, 229 (2018).  \n22 L. Šmejkal, A. H. Macdonald, J. Sinova, S. Nakatsuji, and T. Jungwirth, Nature Reviews Materials \n7, 482 (2 022).  \n23 B. Morosin and A. Narath, Journal of Chemical Physics 40, 1958 (1964).  \n24 P. Doležal, M. Kratochvílová, V. Holý, P. Čermák, V. Sechovský, M. Dušek, M. Míšek, T. \nChakraborty, Y. Noda, S. Son, and J. G. Park, Physical Review Materials 3, 121401(R) ( 2019).  \n25 M. Kratochvílová, P. Doležal, D. Hovančík, J. Pospíšil, A. Bendová, M. Dušek, V. Holý, and V. \nSechovský, Journal of Physics: Condensed Matter 34, 294007 (2022).  \n26 T. Marchandier, N. Dubouis, F. Fauth, M. Avdeev, A. Grimaud, J. M. Tarascon, and G . Rousse, \nPhysical Review B 104, 014105 (2021).  \n27 P. V. Pham, S. C. Bodepudi, K. Shehzad, Y. Liu, Y. Xu, B. Yu, and X. Duan, Chemical Reviews \n122, 6514 (2022).  \n28 Y. Li, B. Yang, S. Xu, B. Huang, and W. Duan, ACS Applied Electronic Materials 4, 3278 (2022 ). \n29 E. Stryjewski and N. Giordano, Advances in Physics 26, 487 (1977).  \n30 I. S. Jacobs and P. E. Lawrence, Physical Review 164, 866 (1967).  \n31 D. P. Landau, Physical Review Letters 28, 449 (1972).  \n32 M. A. McGuire, Crystals 7, 121 (2017).  \n33 L. Liu, K. Y ang, G. Y. Wang, and H. Wu, Journal of Materials Chemistry C 8, 14782 (2020).  \n34 Y. Peng, S. Ding, M. Cheng, Q. Hu, J. Yang, F. Wang, M. Xue, Z. Liu, Z. Lin, M. Avdeev, Y. \nHou, W. Yang, Y. Zheng, and J. Yang, Advanced Materials 32, 2001200 (2020).  \n35 J. B. Fouet, P. Sindzingre, and C. Lhuillier, The European Physical Journal B 20, 241 (2001).  \n36 E. Rastelli, A. Tassi, and L. Reatto, Physica B+C 97, 1 (1979).  \n37 D. Lançon, H. C. Walker, E. Ressouche, B. Ouladdiaf, K. C. Rule, G. J. McIntyre, T. J. Hicks, H. \nM. Rønnow, and A. R. Wildes, Physical Review B 94, 214407 (2016).  \n38 K. Kurosawa, S. Saito, and Y. Yamaguchi, Journal of the Physical Society of Japan 52, 3919 \n(1983).  \n39 D. Lançon, R. A. Ewings, T. Guidi, F. Formisano, and A. R. Wildes, Physical Review B 98, \n134414 (2018).  \n40 A. R. Wildes, V. Simonet, E. Ressouche, G. J. McIntyre, M. Avdeev, E. Suard, S. A. J. Kimber, \nD. Lançon, G. Pepe, B. Moubaraki, and T. J. Hicks, Physical Review B 92, 224408 (2015).  \n41 S. Tomar, B. Ghosh, S. Mardanya, P. Rastogi, B. S . Bhadoria, Y. S. Chauhan, A. Agarwal, and S. \nBhowmick, J. Magn. Magn. Mater. 489, 165384 (2019).  \n42 J. X. Sun, X. Zhong, W. W. Cui, J. M. Shi, J. Hao, M. L. Xu, and Y. W. Li, Physical Chemistry \nChemical Physics 22, 2429 (2020).  \n43 Y. Skourski, M. D. Kuz’M in, K. P. Skokov, A. V. Andreev, and J. Wosnitza, Physical Review B \n83, 214420 (2011).  \n44 M. Kratochvílová, K. Uhlířová, M. Míšek, V. Holý, J. Zázvorka, M. Veis, J. Pospíšil, S. Son, J. \nG. Park, and V. Sechovský, Mater Chem Phys 278, 125590 (2022).   \n 14 45 http://elk.sourceforge.net/ . \n46 L. M. Sandratskii and K. Carva, Physical Review B 103, 214451 (2021).  \n47 K. Yang, F. Fan, H. Wang, D. I. Khomskii, and H. Wu, Physical Review B 101, 100402 (2020).  \n48 J. P. Perdew, K. Burke, and M. Ernzerhof, Physical Review Letters 77, 3865 (1996).  \n49 T. Y. Kim and C. -H. Park, Nano Letters 21, 10114 (2021).  \n50 V. I. Anisimov, F. Aryasetiawan, and A. I. Lichtenstein, J. Phys. -Condes. Matter 9, 767 (1997).  \n51 J. Valenta, M. Kratochvílová , M. Míšek, K. Carva, J. Kaštil, P. Doležal, P. Opletal, P. Čermák, P. \nProschek, K. Uhlířová, J. Prchal, M. J. Coak, S. Son, J. -G. Park, and V. Sechovský, Physical \nReview Materials 103, 054424 (2021).  \n52 V. I. Anisimov, J. Zaanen, and O. K. Andersen, Physi cal Review B 44, 943 (1991).  \n53 A. R. Wildes, D. Lançon, M. K. Chan, F. Weickert, N. Harrison, V. Simonet, M. E. Zhitomirsky, \nM. V. Gvozdikova, T. Ziman, and H. M. Rønnow, Physical Review B 101, 024415 (2020).  \n54 D. Hovančík, D. Repček, F. Borodavka, C. Ka dlec, K. Carva, P. Doležal, M. Kratochvílová, P. \nKužel, S. Kamba, V. Sechovský, and J. Pospíšil, The Journal of Physical Chemistry Letters 13, \n11095 (2022).  \n55 C. J. Ballhausen, Introduction to Ligand Field Theory  (McGraw -Hill Book Co., New York, 1962).  \n56 Y. Liu, M. Abeykoon, and C. Petrovic, Physical Review Research 2, 013013 (2020).  \n57 M. An, Y. Zhang, J. Chen, H. M. Zhang, Y. J. Guo, and S. Dong, Journal of Physical Chemistry \nC 123, 30545 (2019).  \n58 D. Hovančík, J. Pospíšil, K. Carva, V. Sechovský, and C. Piamonteze, Nano Letters 23, \n1175−1180 (2023).  \n59 Supplemental material.  \n60 R. Skomski, Simple Models of Magnetism  (Oxford University Press, 2008).  \n61 S. Dong, J. M. Liu, S. W. Cheong, and Z. F. Ren, Adv. Phys. 64, 519 (2015).  \n62 L. D. Landau, E. M. Lif shitz, and L. P. Pitaevskii, Electrodynamics of continuous media  (Oxford: \nButterworth -Heinemann, 1984).  \n63 A. N. Bogdanov, A. V. Zhuravlev, and U. K. Rößler, Physical Review B 75, 094425 (2007).  \n64 J. Merino and A. Ralko, Physical Review B 97, 205112 (2018 ). \n65 J. Pospíšil, M. Míšek, M. Diviš, M. Dušek, F. R. de Boer, L. Havela, and J. Custers, Journal of \nAlloys and Compounds 823, 153485 (2020).  \n66 D. Hovančík, M. Kratochvílová, P. Doležal, A. Bendová, J. Pospíšil, and V. Sechovský, Journal \nof Solid State C hemistry 316, 123580 (2022).  \n67 J. J. Huntzicker and E. F. Westrum, The Journal of Chemical Thermodynamics 3, 61 (1971).  \n68 A. Mori, Y. Miura, H. Tsutsumi, K. Mitamura, M. Hagiwara, K. Sugiyama, Y. Hirose, F. Honda, \nT. Takeuchi, A. Nakamura, Y. Hiranaka, M . Hedo, T. Nakama, and Y. Ōnuki, J. Phys. Soc. Jpn. \n83, 024008 (2014).  \n69 N. S. Sangeetha, S. Pakhira, Q. -P. Ding, L. Krause, H. -C. Lee, V. Smetana, A. -V. Mudring, B. B. \nIversen, Y. Furukawa, and David, Proceedings of the National Academy of Sciences 118, \ne2108724118 (2021).  \n70 W. Knafo, R. Settai, D. Braithwaite, S. Kurahashi, D. Aoki, and J. Flouquet, Physical Review B \n95, 014411 (2017).  \n71 J. Valenta, F. Honda, M. Vališka, P. Opletal, J. Kaštil, M. Míšek, M. Diviš, L. Sandratskii, J. \nPrchal, and V. Secho vský, Physical Review B 97, 144423 (2018).  \n72 T. N. Haidamak, J. Valenta, J. Prchal, M. Vališka, J. Pospíšil, V. Sechovský, J. Prokleška, A. A. \nZvyagin, and F. Honda, Physical Review B 105, 144428 (2022).  \n73 F. Honda, J. Valenta, J. Prokleška, J. Pospíšil,  P. Proschek, J. Prchal, and V. Sechovský, Physical \nReview B 100, 014401 (2019).  \n74 D. Hovančík, A. Koriki, A. Bendová, P. Doležal, P. Proschek, M. Míšek, M. Reiffers, J. Prokleška, \nJ. Pospíšil, and V. Sechovský, Physical Review B 105, 014436 (2022).   \n 15 75 J. Pospíšil, Y. Haga, Y. Kohama, A. Miyake, K. Shinsaku, T. Naoyuki, M. Vališka, P. Proschek, \nJ. Prokleška, V. Sechovsky, M. Tokunaga, K. Kindo, A. Matso, and E. Yamamoto, Physical \nReview B 98, 014430 (2018).  \n76 T. Olsen, J. Phys. D -Appl. Phys. 54, 314001 (2 021).  \n77 H. Yorikawa and S. Muramatsu, Journal of Physics: Condensed Matter 20, 325220 (2008).  \n78 H. Lane, E. Pachoud, J. A. Rodriguez -Rivera, M. Songvilay, G. Xu, P. M. Gehring, J. P. Attfield, \nR. A. Ewings, and C. Stock, Physical Review B 104, L020411 (2 021).  \n79 K. Okuda, K. Kurosawa, and S. Saito, High field magnetization process in FePS3  (North -Holland, \nNetherlands, 1983).  \n \n \n   \n 16 Supplemetal Material for : \nRobust intralayer antiferromagnetism and tricriticality  in a van der \nWaals compound: VBr 3 case \nDávid Hovančík1, Marie  Kratochvílová1, Tetiana Haidamak1, Petr Doležal1, Karel Carva1, Anežka \nBendová1, Jan Prokleška1, Petr Proschek1, Martin  Míšek2, Denis I. Gorbunov3, Jan Kotek4, Vladimír  \nSechovský1 and Jiří Pospíšil1* \n1 Charles University, Faculty of Mathematics and Physics, Department of Condensed Matter Physics,  Ke Karlovu 5, 121 16 \nPrague 2, Czech Republic  \n2 Institute of Physics, Czech Academy of Sciences, Na Slovance 2, 182 21 Prague 8, Czech Republic  \n3 Hochfeld -Magnetlabor Dresden (HLD -EMFL), Helmholtz -Zentrum Dresden -Rossendorf, 01328 Dresden, Germany  \n4 Department of Inorganic Chemistry, Faculty of Science, Charles University, Hlavova 8, 128 40 Prague 2, Czech Republic.  \nResults   \nThe steady field 14 T has shown only a tiny effect on linearly growing magnetization up to the temperature \nof 12 K (Fig.  S1). At higher temperatures , a signature of an upturn on magnetization was detected near -\nmaximal values of magnetic fields signalizing the presence of M PT which finally appears at a temperature \njust below TN. The effect of the magnetic field perpendicular to t he c*-axis is very weak in agreement \nwith the specific heat data. The  steady field 14 T  magnetization isotherms are linear with no sign of any \neffect  (inset of Fig. S2). Only a very weak deviation from the linear trend appears approaching TN. The \ndata demonstrates pronounced  magneto -crystalline anisotropy . These results agree with the work of Kong \net al.1. The pulsed field data up to 58 T for H  c* (Fig.  S2) shows the magnetic moment canting transi-\ntion, which is very  robust  to the increasing temperature . The first more distinguished effect on the shape \nof the isot herm is detect able at 20 K  in the vicinity of the TN. \n \n \n 17 FIG. S1. The magnetization isotherms of VBr 3 measured with PPMS14T  in the field applied  H || c*.The \nblack arrows mark the position of MPT, the values were used for the construction of the H-T phase dia-\ngram in Fig.  4. \n \nFIG. S2. The magnetization isotherms of VBr 3 measured at various temperatures in pulsed fields (PF)  \nand static fields (SF)  H  c* up to 47 T. Inset: The magnetization isotherms of VBr 3 measured at various \ntemperatures in static fields H  c* up to 14 T  without special orientation within the ab plane.  \n0 510 15 20 25 30 35 40 45 5000.10.20.30.40.50.60.70.80.911.11.21.3\n0 2 4 6 810 12 1400.10.20.30.40.5M (B/f.u.)\n0H (T) 1.9 K\n 3 K\n 10 K\n 20 KM (B/f.u.)\n0H (T) 1.4 K\n 3 K\n 10 K\n 20 K\n SF 1.8 KVBr3\nHc* \n 18  \nFIG. S3. Angular -dependent magnetization was taken in a) ac*-plane (the c*-axis corresponds to 0º \nand 180º) b) ab-plane for different temperatures measured in the magnetic field of 7 T. c) Polar plot of \nthe rotational magnetization for  ab-plane (blue) and ac*-plane (red) taken at 4 K. d) Low -field magneti-\nzation isotherms measur ed along the c*-axis (blue) and ab-plane (red).  \nTo probe the magneto -crystalline anisotropy in VBr 3, we measured the angular dependence of the mag-\nnetization in µ 0H = 7 T applied in the ac* and ab planes (see Fig.  S3). The diamagnetic contribution of \nthe ro tator was subtracted from the data for both rotation planes. Below TN, the magnetic signal in the ac* \nplane has two -fold symmetry with the minimum (maximum) along the c*-axis ( ab-plane), which corrob-\norates the AFM state with the Néel vector aligned along t he c*-axis. The ab-plane magnetic signal also \nexhibits a weak two -fold-like symmetry but with a much smaller maximum/minimum signal ratio (almost \nisotropic) than the result in the ac* plane. That can indicate the slightly tilted Néel vector from the c*-\naxis or the biaxiality of AFM anisotropy2. One can notice that a weak two -fold-like signal is already \npresent just above TN for both planes, however, it is more apparent for the ab-plane given the small scale \nof the relative change of magnetization. The difference between the relative change of magnetic signal in \nac-plane and ab-plane (almost isotropic) is clearly visible from  the polar plot in Fig.  S3c. The VBr 3 ani-\nsotropy and magnetic structure are in obvious contrast to the features of neighbor FM VI 33, 4. We have not \ntraced any hysteresis loop in the low magnetic field along th e c*-axis (see Fig. S3), contrary to results \nreported and ascribed to the canted AFM easy -axis by Lyu et al.5. \n \n 19  \nResults  – Stoner -Wohlfarth simulations  \nThe two -sublattice  Stoner -Wohlfarth model can provide an approximate picture of field -induced \nspin-flop and spin -flip transitions in antiferromagnets6. In Fig.  S4a we show how the type of tra nsition \ncan be controlled by magnetic anisotropy (other parameters are the same as used in the main article text). \nFor small fields applied along the easy axis, spins on the two sublattices are oriented antiparallel to each \nother due to the exchange intera ction. For the most common case of small anisotropy (here  we assume  \nK1 = 0.1 meV), at a sufficiently strong field, it is energetically favorable to rotate the moments into the \nspin-flop state so that there appears a projection into field direction, and the  exchange, Zeeman and ani-\nsotropy energy are partially satisfied. With increased anisotropy, this transition occurs at a higher field \nwith a smaller deviation from the easy axis, until a critical value of anisotropy is reached when it is more \nfavorable to c ompletely break the exchange interaction rather than the anisotropy energy, and a spin -flip \noccurs. We show the case with anisotropy slightly below this critical value ( K1 = 0.4 meV), so that small \ntilting from the easy axis occurs for a short range of H values and a higher value ( K1 = 0.6 meV) with a \ncomplete spin -flip. Fields applied perpendicular to the easy axis lead to tilting of moments towards the \nfield direction that is linearly dependent on the field magnitude. The field needed for saturation (co mplete \ntilt of moments) increases with increasing anisotropy.  \nWe also examine how the results would be affected by possible deviation of the field direction from \nthe orientation either along ( ϑH = 0˚) or perpendicular ( ϑH = 90˚) to the easy axis. The cases  with field \ncanted by 20˚ from these major directions (Fig.  S4b)) have some features common with the experimen-\ntally observed M(H) dependencies, but the deviation of 20˚ appears too large to be possible (?). On the \nother hand, for misalignment less than 10˚  the change is hardly visible within the given range of the graph.  \n \n \nFIG. S4. Stoner -Wohlfarth simulations. Magnetization M per f.u. as a function of the applied external \nfield H: a) for different values of anisotropy K1 with field applied either parallel to the easy axis (blue \nlines) or perpendicular to the easy axis (red lines). Dotted lines: K1 = 0.1 meV. Solid lines: K1 = 0.53 meV. \nDashed lines: K1 = 1 meV. b) for H applied at different angles from the easy axis (with  K1 = 0.53 meV).  \nDue to the large size orbital moment and the relatively modest spin -orbit coupling for 3d electrons , we \nnext consider a model where spin and orbital moments are treated separately, described by their own unit \nvectors s1,2 and l1,2. The moments are then given by the multiplication of these unit vectors with moment \nmagnitudes ( µs and µl), they interact with the external field, and are coupled by SOI . Exchange interaction \naffects only the spin component. Concerning anisotropy energy, we ass ume that the magnetocrystalline \npart is dominant, and therefore affects only the orbital component. The free energy is then modified as \nfollows (eq 1): \nb) a)  \n 20 E = -J*s1  s2 + EAN(l1,l2) + H  (s1 + s2)µs + H  (l1 + l2)µl + (l1s1  + l2s2)  (1) \nV ion is expected to be in the high spin state with spin moment of 2  B7 and ab initio calculations \npredi ct it to be rather close to this v alue, between 1.85 and 2.03 B1, 8-12. Therefore we set the spin moment \nsize to 2 B and the orbital moment size  to L = 0.7 B. SOI prefers antiparallel orientation to the spin so \nthat the total moment is again 1.3  B. In the limit of very strong SOI,  these moments remain rigidly cou-\npled and behave like one moment, leading to a similar result as in the previous case. Assuming a smaller \nSOI ( = 3.5 meV) we notice slight changes mainly in the H  c* curve (Fig.  S5), as the variation of the \nangle between  si and li provides an additional degree of freedom. That lowers total anisotropy energy \nconnected with spins pointing perpendicular to the easy axis. The increase of M with H is then faster in \nthis case. While the separation of spin and orbital moment wit hin the classical model is disputable, it \nseems to improve slightly the agreement with the experiment . \n \n \nFIG.  S5. Stoner -Wohlfahrt simulation. Plotted M along H per one site as a function of applied external \nfield H, for H parallel to the easy axis (blue) or perpendicular to the easy axis (red). Solid lines are calcu-\nlated according to the model in (eq  4), while dashed lines come from the model with a separate orbital \nmoment (eq  5). \nFig. S6 provides a detailed view of the dy namics in the model with sep arate spin and orbital contribu-\ntions. For small fields applied along the easy axis, spins on one of the sublattices are oriented along the \nfield, on the other one oppositely, similarly to the standard model with one moment per s ublattice (studied \nabove). The smaller orbital moments are oriented oppositely w.r.t the corresponding spins. Hence the total \nmoment on one subla ttice is ~1.3  μB, and the total moment projected  to the field direction is zero. A \nsufficiently strong field ov ercomes the effect of the exchange interaction and the second sublattice is \nflipped to be parallel to the field too. Orbital moments remain antiparallel to spins within the whole studied \nfield range. Therefore the dynamics is almost identical to the simple r model considering one moment per \nsublattice (Fig.  3b of the main text13). For the field applied perpendicular to th e easy axis, the behavior of \nspins is again similar to  the case study above. However, since the anisotropy is acting on the orbital mo-\nmentum, we see that it is advantageous for the system to tilt the spin momentum in the field direction \nslightly more than the orbital momentum. Some energy is thus gained from Zeeman energy on behalf of \nthe spin -orbit contribution. The system effectively behaves similarly to the total moment system, but with \nslightly enlarged momentum, as shown by the red dotted curve in Fig.  S6. We are not aware of any work \ndiscussing the possible separation of spin and orbital contributions in strong fields. It should be pointed \nout that this topic deserves further study.  \n \n 21  \nFIG. S6. Stoner -Wohlfarth simulations. Magnetic contributions of dif ferent  sublattices (per f.u.) as a \nfunction of the applied external field H, for H parallel to the easy axis (blue lines) or perpendicular to the \neasy axis (red lines). Solid lines: Total magnetization M. Dashed lines: spin contribution on sublattice 1 \n(projected to the direction of applied field h : μs h · s1 ). Dotted lines: total magnetic momentum magni-\ntude on sublattice 1 ( | μ s s1 + μ l  l1 | ). \n \nReferences  \n1 T. Kong, S. Guo, D. Ni, and R. J. Cava, Physical Review Materials 3, 084419 (2019).  \n2 P. Bloembergen and G. de Vries, Solid State Communications 13, 1109 (1973).  \n3 A. Koriki, M. Míšek, J. Pospíšil, M. Kratochvílová, K. Carva, J. Prokleška, P. Doležal, J. Kaštil, \nS. Son, J. -G. Park, and V. Sechovský, Physical Review B 103, 174401 ( 2021).  \n4 Y. Q. Hao, Y. Q. Gu, Y. M. Gu, E. X. Feng, H. B. Cao, S. X. Chi, H. Wu, and J. Zhao, Chinese \nPhysics Letters 38, 096101 (2021).  \n5 B. Lyu, L. Wang, Y. Gao, S. Guo, X. Zhou, Z. Hao, S. Wang, Y. Zhao, L. Huang, J. Shao, and M. \nHuang, Physical Review B 106, 085430 (2022).  \n6 R. Skomski, Simple Models of Magnetism  (Oxford University Press, 2008).  \n7 C. J. Ballhausen, Introduction to Ligand Field Theory  (McGraw -Hill Book Co., New York, 1962).  \n8 M. An, Y. Zhang, J. Chen, H. M. Zhang, Y. J. Guo, and S. Dong, Journal of Physical Chemistry \nC 123, 30545 (2019).  \n9 K. Yang, F. Fan, H. Wang, D. I. Khomskii, and H. Wu, Physical Review B 101, 100402 (2 020).  \n10 Y. Liu, M. Abeykoon, and C. Petrovic, Physical Review Research 2, 013013 (2020).  \n11 S. Tomar, B. Ghosh, S. Mardanya, P. Rastogi, B. S. Bhadoria, Y. S. Chauhan, A. Agarwal, and S. \nBhowmick, J. Magn. Magn. Mater. 489, 165384 (2019).  \n12 J. X. Sun, X.  Zhong, W. W. Cui, J. M. Shi, J. Hao, M. L. Xu, and Y. W. Li, Physical Chemistry \nChemical Physics 22, 2429 (2020).  \n13 Robust intralayer antiferromagnetism and tricriticality in a van der Waals compound: VBr 3 case \n(main text).  \n \n"    },    {        "title": "2304.12270v1.Fast_non_volatile_electric_control_of_antiferromagnetic_states.pdf",        "content": "Fast non-volatile electric control of antiferromagnetic states\nS. Ghara\u0003,1M. Winkler,1K. Geirhos,1L. Prodan,1V. Tsurkan,1, 2S. Krohns,1and I. K\u0013 ezsm\u0013 arki1\n1Experimental Physics V, Center for Electronic Correlations and Magnetism,\nUniversity of Augsburg, 86159 Augsburg, Germany\n2Institute of Applied Physics, Moldova\nElectrical manipulation of antiferromagnetic states, a cornerstone of antiferromag-\nnetic spintronics, is a great challenge, requiring novel material platforms. Here we\nreport the full control over antiferromagnetic states by voltage pulses in the insulating\nCo 3O4spinel. We show that the strong linear magnetoelectric e\u000bect emerging in its\nantiferromagnetic state is fully governed by the orientation of the N\u0013 eel vector. As\na unique feature of Co 3O4, the magnetoelectric energy can easily overcome the weak\nmagnetocrystalline anisotropy, thus, the N\u0013 eel vector can be manipulated on demand,\neither rotated smoothly or reversed suddenly, by combined electric and magnetic \felds.\nWe succeed with switching between antiferromagnetic states of opposite N\u0013 eel vectors\nby voltage pulses within a few microsecond in macroscopic volumes. These observa-\ntions render quasi-cubic antiferromagnets, like Co 3O4, an ideal platform for the ultra-\nfast (pico- to nanosecond) manipulation of microscopic antiferromagnetic domains and\nmay pave the way for the realization of antiferromagnetic spintronic devices.\nFerromagnets still represent an important material\nplatform for magnetic storage and spintronic devices1,2.\nHowever, they set restrictions on further miniaturization\nand speed-up of these devices due to several drawbacks,\nsuch as the cross-talking between the adjacent units via\ntheir stray \felds, high sensitivity to external magnetic\n\felds and relatively slow spin dynamics3,4.\nThough antiferromagnets are free from these limita-\ntions, their applicability has not been put into considera-\ntion for long. Even Louis N\u0013 eel, who discovered them,\nstated in his Nobel lecture that antiferromagnets had\nno practical importance5. Contradicting to his predic-\ntion, a plethora of recent studies imply that antiferro-\nmagnets can be viable replacements for ferromagnetic\ncomponents in spintronic devices3,4,6,7. In fact, emer-\ngent phenomena associated with antiferromagnets, such\nas anisotropic magnetoresistance8,9, spin Hall e\u000bect10{12,\nspin Seebeck e\u000bects13{15, N\u0013 eel spin-orbit torque16,17, ul-\ntrafast THz manipulation of antiferromagnetic (AFM)\nspin waves18and dynamics of AFM skyrmions19{22, open\nup new opportunities in spintronics. In fact, AFM spin-\ntronics, which is an impressively growing \feld at the fron-\ntiers of magnetism and electronics, is foreseen to trigger\na change of paradigm in information technology3,4,6,23.\nIn the last years, the electrical manipulation of AFM\ndomains and the orientation of the AFM N\u0013 eel vec-\ntor (L) have attracted much attention, as a possible\nroute to write-in and read-out information in AFM de-\nvices3,4,7,24{28. The spin-orbit coupling plays a key role\nin controlling the orientation of Lin itinerant and insu-\nlating antiferromagnets via electric currents and voltages,\nrespectively29. In non-centrosymmetric AFM metals, like\n\u0003email: somnath.ghara@physik.uni-augsburg.deCuMnAs17, Mn 2Au30and Fe 1=3NbS 231, the current in-\nduced rotation of Lis governed by the spin-orbit torque\nand has led to the realization of the \frst prototype of\nAFM random access memories. In non-centrosymmetric\nAFM insulators, such as Cr 2O332,33, LiCoPO 434{36and\nMnTiO 337,38, the linear magnetoelectric (ME) e\u000bect,\nthat is another manifestation of the spin-orbit coupling,\ncan be exploited for the electric \feld-driven switching\nof AFM domain states, as a working principle for AFM-\nME random access memories39. While the voltage-driven\nswitching of Lhas an advantage over current-driven\nswitching in terms of heat dissipation, especially for de-\nvices with high density of information and/or active el-\nements, such in situ voltage-driven switching has been\nobserved only in a few compounds so far, and in all cases\nonly in the vicinity of TN33,37,39,40.\nIn this proof-of-concept study, we demonstrate the in\nsitu isothermal switching of Ldriven by an electric \feld\ndeep in the AFM state of insulating Co 3O4. This com-\npound belongs to the family of only A-site magnetic\nspinels, where Co2+ions, occupying the diamond lattice\nformed by the Asites, have spins S=3/2 in tetrahedral\noxygen coordination41{43. The pyrochlore lattice of the\nBsites is occupied by non-magnetic Co3+ions. While\nCo3O4has a centrosymmetric crystal structure (space\ngroupFd\u00163m) in the paramagnetic state, the Co2+ions\nhave no inversion symmetry (site symmetry \u001643m)44and\nthe onset of N\u0013 eel-type AFM order at TN=30 K breaks the\nglobal inversion symmetry of the lattice, making Co 3O4\na non-centrosymmetric insulating antiferromagnet. Ear-\nlier reports showed that Co 3O4undergoes a transition to\na simple textbook-like collinear AFM state at TN41,44,\nin strong contrast to the other highly frustrated A-site\nspinels, such as MSc2S4(M=Mn, Fe)45,46. However, the\ndirection of the magnetic easy-axis, which can be either\nalong theh111i-type body-diagonals or along the h001i-\nTypeset by REVT EXarXiv:2304.12270v1  [cond-mat.str-el]  24 Apr 20232\ntype main cubic axes, has not been resolved yet due to\nthe scarce of single crystals.\nThis compound, along with a few other A-site mag-\nnetic spinels, such as CoAl 2O4and MnB2O4(B=Al,\nGa), has recently been reported to exhibit a linear ME\ne\u000bect44,47{49. Due to the non-centrosymmetric tetrahe-\ndral environment ( \u001643m) of the magnetic Co2+ions, the\nME free energy has the following form47:\nFME=Lx(MyPz+MzPy) +Ly(MzPx+MxPz)\n+Lz(MxPy+MyPx):(1)\nHere, thex,yandzaxes are the main cubic axes, whereas\nM,LandPstands for the magnetization, the AFM N\u0013 eel\nvector and the electric polarization, respectively. For\nweak magnetic \felds, when M\u001cLand the magnitude of\nLis nearly \feld independent, Eq. (1) yields a linear ME\nsusceptibility (^ \u000b), which solely depends on Laccording\nto,\n^\u000b\u00180\n@0LzLy\nLz0Lx\nLyLx01\nA: (2)\nThis speci\fc form of its ME coupling together with its\nweak magnetic anisotropy make Co 3O4an ideal bench-\nmarking material to investigate the electrical control of\nL. For example, when applying an electric \feld along\none of the main cubic axes together with a magnetic \feld\narbitrarily oriented in the plane perpendicular to it, the\nformation of an AFM mono-domain state with Lperpen-\ndicular to both \felds is expected, if the ME free energy\ncan overcome the magnetocrystalline anisotropy energy.\nHere, we demonstrate the full control over the orien-\ntation of L, achieved by simultaneous electric and mag-\nnetic \felds in the AFM state of Co 3O4. The electrical\nswitching of Lin macroscopic volumes is accomplished\nonly within a few microseconds, pointing to an ultrafast\nintrinsic switching mechanism for microscopic AFM do-\nmains.\nAFM domain selection via ME e\u000bect\nFigure 1 summarizes our results on the linear ME ef-\nfect obtained after poling with orthogonal electric and\nmagnetic \felds, namely Epkzaxis andHpkxaxis. As\nclear from Fig. 1a, the electric polarization emerges be-\nlowTNin proportion to the magnetic \feld, characteristic\nto the linear ME e\u000bect. The value of \u000bis found to be\n\u001811.3 ps/m at 4 K (inset of Fig. 1a), which is four times\nlarger compared to that of the prototypical ME material\nCr2O333. Note that \u000bstands for the xzcomponent of ^ \u000b\nthroughout this work. The linear behaviour of the ME ef-\nfect is directly evidenced in the magnetic \feld-dependent\nelectric polarization (Fig. 1b). The sign of \u000b, that is the\nsign of the slope of the P-Hcurve, depends on the rela-\ntive signs of the poling \felds: \u000bis positive/negative when\nthe product HpEpis positive/negative. These results, to-\ngether with the presence/absence of electric polarization\n-12-8-404 8 12-120-80-400408012001 02 03 04 004080120\nPz (µC/m2)T\n (K)14 T1\n21\n08\n6\n4\n2\n0\n Tc\nab\ny\nzL\n = (0, +Ly, 0)020400510T\n (K)/s97 (ps/m)+\nEp,−HpPz (µC/m2)/s109\n0H (T)+Ep,+Hp−\nEp,+Hp− Ep,/s45Hp+Ly− Lyx\nL = (0, -Ly, 0)4 KFIG. 1:jRealization of two distinct AFM mono-\ndomain states via ME poling. a , Temperature-dependent\nelectric polarization Pzmeasured after ME poling with\nEp=+4.2 kV/cm (kzaxis) and various \u00160Hp(kxaxis).\nThe inset shows the temperature-dependent ME coe\u000ecient\n\u000b.b, Magnetic \feld-dependent electric polarization at 4 K\nmeasured after ME poling with all four sign combinations of\nthe poling \felds, Ep=\u00064.2 kV/cm and \u00160Hp=\u000614 T. In\nall these cases, the electric polarization was measured with\nEpbeing switched o\u000b after ME poling. c, Schematics of + Ly\nand\u0000LyAFM mono-domain states in Co 3O4. The red dots\nbetween the Co2+ions of opposite magnetization indicate the\ncenters of inversion in the paramagnetic cubic state, which are\nbroken by the AFM order.\nfor perpendicular/parallel orientation of the electric and\nmagnetic \felds (shown in Fig. 2e and discussed below),\ncon\frm the speci\fc form of the linear ME e\u000bect, as given\nin Eqs. (1) & (2).\nFor electric and magnetic \felds, respectively, applied\nalong thezand thexaxes, the ME free energy term has\nthe simple form LyMxPzand, thus, the + LyAFM mono-\ndomain state is expected to emerge for HpEp>0 and\nthe\u0000Lymono-domain state for HpEp<0 (see Fig. 1c).3\nEiz\n0\n4590135180-100-50050100\nP\nz (µC/m2)/s102\n (deg)/s113 = 0°/s113\n = 180°zE\n || zfE || y-\n100-50050100P\ny (µC/m2)exy − xz /s113\n = −90°/s113 = +90°E || yx\ny-\n14-70 7 14-120-80-4004080120\n+ 8.7+\n4.5+\n1.7Pz (µC/m2)/s109\n0H (T)E (kV/cm) \n     +17.4 \n     +13.9−\n17.4 k\nV/cma +Ly+\nLy −Ly−Lybc d\nxyx\nyE || z4  K\nFIG. 2:jIn situ switching and rotation of L. a , Magnetic \feld ( Hkxaxis) dependence of the electric polarization at\n4 K measured in the presence of electric \felds ( Ekzaxis). b, Sketch showing the orthogonality of the AFM N\u0013 eel vector L\n(red arrow) to the magnetic \feld (blue arrow) and introducing the angles \u001eand\u0012.c, Schematic diagram showing the favoured\ndirection of L, either +Lzor\u0000Lz, depending on the orientation of the magnetic \feld rotated within the xyplane, when the\nelectric \feld is applied along the yaxis. d, Schematics of the in-plane rotation of Lwith either preceding (yellow shaded\nangular range) or following (green shaded angular range) the magnetic \feld rotating in the xyplane, when the electric \feld\nis applied along the zaxis. e/f, Dependence of the electric polarization component Py/Pzon the orientation of the magnetic\n\feld (14 T) rotating within the xyplane at 4 K in the presence of an electric \feld (+17.4 kV/cm) applied along y/zaxis. The\npurple curves in the two panels are the experimental data. The solid/dashed black curve in panel eindicates the calculated\nbehaviour of Py, according to Eq. (3), for the + Lz/\u0000Lzmono-domain state, and those in panel fshow the expected behaviour\nofPz, according to Eq. (4), when Lrotates with preceding/following the magnetic \feld within the xyplane. In all panels, the\nyellow and green background colours indicate \feld or angular regions, where Lis preserved (panels a,c&e) or continuously\nrotated (panels d&f). At the borders of these regions Lis switched suddenly.\nIndeed, such selection of the \u0006Lymono-domain states\nfor opposite signs of HpEpis implied by P-Hcurves in\nFig. 1b. Please note that in the present case, \u0006LyAFM\nstates are interchanged not only by time reversal but also\nby spatial inversion, that is why the sign of Lis linked\nto the sign of HpEp. Before measuring the electric polar-\nization atT= 4 K, shown in Fig. 1b, the poling electric\n\feld was switched o\u000b. Since the P-Hcurve is linear in\nthe whole magnetic \feld range, we can conclude that\nthe AFM mono-domain state, established by poling with\ncrossedEpandHp, was preserved upon the entire mag-\nnetic \feld sweep. We note that magnetic anisotropy with\neasy axes along the cubic h111i-type directions could pre-\nvent the formation of the \u0006LyAFM mono-domain states.\nHowever, the magnetization of Co 3O4is found to be in-\ndependent of the orientation of the magnetic \feld (see\nthe supplementary \fgure 1), which indicates the weak-\nness of magnetic anisotropy and the dominance of the\nexternal \felds in determining the orientation of Lvia\nthe ME coupling.\nMagnetic control and kinetics of L\nIn the following, we demonstrate that the AFM mono-\ndomain state can be achieved not only by ME poling but\ncan also be created in situ from a multi-domain state by\northogonal electric and magnetic \felds, even at temper-atures well below TN. In addition to the mono-domain\ncreation, we control the orientation of Lin two ways:\nWe reverse it via the reversal of either the electric or the\nmagnetic \feld, and rotate it by rotating the magnetic\n\feld, while the electric \feld is kept constant.\nFor these studies, the sample was cooled to 4 K with-\nout any poling \felds. In the absence of electric \feld, no\nmagnetically induced polarization was observed, indicat-\ning that the net Laverages to zero in the multi-domain\nstate. When starting from the virgin multi-domain state\nand applying a constant electric \feld of E=+17.4 kV/cm\nalong thezaxis, the sample is set to the + Lymono-\ndomain state by magnetic \felds exceeding 2 - 3 T, ap-\nplied along the xaxis. This is shown in Fig. 2a, where\nthe initial parabolic increase of the P-Hcurve turns to\nlinear above this \feld range, implying no further changes\nin domain population. Indeed, in +14 T the electric po-\nlarization reaches the same value as recorded after ME\npoling (cf. Figs. 1a and 1b), also evidencing that the\n+Lymono-domain state has been achieved. Intriguingly,\nwhen sweeping the magnetic \feld back with keeping the\nconstant electric \feld on, the slope of the P-Hcurve\nchanges sign for negative magnetic \felds, indicating the\nswitching to the \u0000Lymono-domain state. This is op-\nposite to the case with no electric \feld applied, when\nthe poled + Lymono-domain state is preserved upon the4\n-20-1001020-\n6-30361\n00102104106010200369121518-\n10-50510-\n20-100E (kV/cm)tr = 0.3 stimeI\n (nA)ab -20-1001 020-\n12-60612E (kV/cm)/s97 (ps/m)cd\n/s68/s97±\n17.4 kV/cm0\n1020 \n/s68/s97 (ps/m)fe 1\n0-510-310-11 0-610-410-2100t\nr (s)ts (s)1\n5 µstr = 30 µst\ns =×1t\nr = 1 ms×9t\ns =0\n.2 ms-\n14-70t\ns =3\n00 msI\n (µA)tr = 1 s×13860/s68\n/s97 (ps/m)p\nulse number/s97 (ps/m)/s1090H (T) / E (kV/cm)+\n14 TE (kV/cm)\nFIG. 3:jVoltage pulse-driven switching of L. a , The +, +,\u0000,\u0000, ... electric pulse sequence (dashed line/left scale)\nforEkzaxis and the measured switching current (solid line/right scale) at 4 K in the presence of +14 T ( kxaxis). b, ME\nhysteresis loop at 4 K in +14 T. The di\u000berence between the + and \u0000remanent value of the ME coe\u000ecient, \u0001 \u000b, is a measure\nof the switching e\u000eciency. The sequence of pulses is indicated by colours according to panel a.c, Dependence of the ME\ncoe\u000ecient\u000bon the strength of the pulsed electric \feld (purple), when +14 T is constantly applied, and on the strength of the\nstatic magnetic \feld (green), while exposing the sample to \u000617.4 kV/cm electric pulses. The +/ \u0000signs of\u000bcorresponds to\nthe +Ly/\u0000LyAFM mono-domain state. d, Time traces of the driving electric pulses (light blue line) for various rise times\n(tr) and the time traces of the corresponding switching currents (dark blue line). The latter are plotted on a common vertical\nscale after multiplication with numerical factors, as indicated. The estimated switching time of the AFM domains is indicated\nbyts.e, Dependence of the switching time ts(left scale) and the switching e\u000eciency \u0001 \u000b(right scale) on the rise time trof the\ndriving electric pulse. The data indicated by solid and open symbols were obtained on a thick and a thin crystal, respectively\n(see Methods). f, Dependence of the switching e\u000eciency on the number of +, \u0000, ... pulses sequences at 4 K in +14 T. The\nsolid and dashed lines in panel c,eandfare guides to the eyes.\nsign change of the magnetic \feld (Fig. 1b). Electric \felds\nas low as +1.7 kV/cm are still e\u000ecient in increasing the\npopulation of the + Lystate and switching the sign of\nthe excess + Lycomponent, but the mono-domain state\nis only reached for E\u0015+8.7 kV/cm, as seen in Fig. 2a.\nThe same domain selection and switching processes are\nalso demonstrated in Fig. 2a for the opposite sign of the\nelectric \feld.\nAfter demonstrating the in situ ME domain control, we\ninvestigate the kinetics of \u0006Lswitching and the rotation\nofL. This is done by analysing the dependence of electric\npolarization on the orientation of the magnetic \feld us-\ning Eqs. (1)-(2). Due to the negligibly small anisotropy\nenergy, Mcoaligns with magnetic \felds larger than 2-\n3 T, irrespective of the \feld direction, and Lcan point\nalong any direction perpendicular to M, as depicted in\nFig. 2b. Via the ME free energy, the application of an\nadditional electric \feld can select an unique L, which de-\ntermines the electric polarization, probed experimentally,\naccording to Eq. (2).\nFigure 2e and f respectively show the dependence of Py\nandPzon the orientation of the magnetic \feld, rotated\nin thexyplane. In the two cases, the electric \feld wasapplied along the measured electric polarization compo-\nnent, i.e., along the yorzaxis. Based on Eq. (1) and\nthe orthogonality of Lto the magnetic \feld, the angular\ndependences of the two electric polarization components\nhave the following forms:\nPy/MzLx+MxLz=\u0011cos(\u001e) sin(\u0012); (3)\nPz/MxLy+MyLx=\u0011cos(2\u001e) cos(\u0012): (4)\nwhere\u0011is a constant factor, \u001eis the angle between the\nmagnetic \feld and the xaxis, and\u0012is the angle of L\nwith respect to the xyplane, as shown in Fig. 2b. Since\nthe magnetic anisotropy is negligible, the angle \u0012is de-\ntermined by the ME free energy. When the electric \feld\nis applied along the yaxis, the ME free energy is mini-\nmized by the + Lzstate (\u0012= +90\u000e) and the\u0000Lzstate\n(\u0012=\u000090\u000e) for 0\u000e< \u001e < 90\u000eand 90\u000e< \u001e < 180\u000e, re-\nspectively, as depicted in Fig. 2c (For the details of the\ncalculation see the supplementary note 2 and supplemen-\ntary \fgure 2). Thus, upon the rotation of the magnetic\n\feld, the + Lz!\u0000Lzswitching is expected when the\nmagnetic and electric \felds become parallel, while the\nback-switching should take place when they are antipar-\nallel. The solid/dashed black curve in Fig. 2e indicates5\nthe calculated behaviour of Py, according to Eq. (3), for\nthe +Lz/\u0000Lzmono-domain state. The angular depen-\ndence of the measured Py(purple curve in Fig. 2e) is well\ncaptured by this model, con\frming the sudden switching\nbetween the\u0006Lzstates.\nWhen the electric \feld was applied along the zaxis, the\nME energy is minimized by \u0012= 0\u000efor\u000045\u000e< \u001e < 45\u000e\nand 135\u000e< \u001e < 225\u000e, and by\u0012= 180\u000efor 45\u000e< \u001e <\n135\u000eand 225\u000e<\u001e< 315\u000e, as depicted in Fig. 2d. This\nmeans Llies within in the xyplane during the entire ro-\ntation of H. More speci\fcally, the mutual orthogonality\namong H,EandL, according to Eq. (1), results in the\nsmooth rotation of L, within each of the four 90\u000eseg-\nments mentioned above. However, at the edges of these\nsegments, i.e. at \u001e= 45\u000e, 135\u000e, 225\u000eand 315\u000e,Lis\nsuddenly reversed. Therefore, if Lprecedes the magnetic\n\feld in one segment, then it follows the \feld in the next\nsegment, as sketched in Fig. 2d. (For more details see\nsupplementary note 2 and supplementary \fgure 3). The\nexperimentally observed behaviour of Pz, as shown by\npurple curve in Fig. 2f, is also con\frmed by our model.\nThese results speak for the full controllability of Lvia\nthe ME e\u000bect, and no signi\fcant in\ruence of magnetic\nanisotropy on the kinetics of L. Such an unconstrained\nin situ ME control of Leven atT\u001cTNmakes Co 3O4\nunique among antiferromagnets.\nPulsed voltage control of AFM domains\nNext, we demonstrate the in situ switching of Lby\npulsed electric \felds. Fig. 3a displays the e\u000bect of +, +,\n\u0000,\u0000, ... voltage pulse sequences on an initially multi-\ndomain crystal at 4 K. The pulsed electric \feld and the\nstatic magnetic \feld were oriented along the zandx\naxis, respectively. The current measured with the ap-\nplied electric \feld exhibits peaks only when the sign of\nthe voltage pulse is changed, i.e. at the \frst + and \frst\n\u0000pulses. This current peak originates from the electric\n\feld induced switching of the ME polarization. The lack\nof current peaks associated with the second + and the\nsecond\u0000pulses indicate that the \frst + and \u0000pulses\nestablished the + Lyand\u0000Lymono-domain states, re-\nspectively. During the switching process, the time de-\npendence of \u000bis obtained by dividing the electric polar-\nization (the time-integral of the current) by the magnetic\n\feld value. The switching process is more easily accessi-\nble in Fig. 3b, where we directly plot the magnetoelectric\ncoe\u000ecient versus the electric \feld, which resembles a typ-\nical ferroelectric hysteresis loop, and demonstrates that\n\u000bis reversed when the electric \feld is inverted. Note that\nthe AFM domain switching is non-volatile, as indicated\nby the high-remanent \u000bobserved after all electric pulses\nin Fig. 3b. This magnitude of remanent \u000bagrees with\nthe value in the inset of Fig. 1a, measured on a mono-\ndomain sample after switching o\u000b the electric poling \feld.\nThough a similar in situ ME control of AFM domains has\nbeen reported in Cr 2O3and MnTiO 3, the switching inthose cases are limited to the vicinity of TN33,37,39,40.\nIn Fig. 3c, we analyse the dependence of switching ef-\n\fciency on the strength of the pulsed electric \feld in\n+14 T at 4 K. While electric \felds >\u00189 kV/cm are re-\nquired to create\u0006Lymono-domain states, weaker elec-\ntric pulses still lead to considerably high \u0006\u000bvalues in\na reproducible fashion. This implies that a full control\noverLcan be achieved by lower electric \felds when mi-\ncroscopic AFM domains are subject to manipulation and\nnot macroscopic volumes, like in the present case. A sim-\nilar behaviour of \u000bon the strength of the static magnetic\n\feld indicates that magnetic \felds >\u00186 T are necessary\nto create in situ\u0006Lymono-domains by electric pulses of\n\u000617.4 kV/cm, as seen in Fig. 3c.\nNext, we investigate the dynamics of the switching\nwith particular interest on the switching speed, a pa-\nrameter highly relevant to device applications. The de-\npendence of the switching time ( ts) on the rise time of\nthe driving electric pulse ( tr) is shown in Fig. 3e. Time\ntraces of the switching current are displayed in Fig. 3d\nfor three di\u000berent values of tr, wheretsis obtained as\nthe full width of the current peak at half maximum. The\nswitching is found to be as fast as ts\u00195.5\u0016s fortr=\n10\u0016s. We have to stress that for all trvalues studied here,\nthe switching was completed during the rise of the driving\nvoltage, with the current peak located typically at t<\u0018tr\n2.\nThis observation, together with the linear dependence of\ntsontrwithout any sign of a saturation, suggests that\nthe values of the switching time, indicated in Figs. 3d &\n3e, are limited by the rise time of the drive pulse of our\nexperimental setup ( tr\u001510\u0016s), and the shortest intrinsic\nswitching time is less than 5.5 \u0016s.\nThe sound velocity associated with the AFM magnon\nband in Co 3O4,vm= 2.3 km/s50, imposes a fundamen-\ntal limit for the speed of the AFM domain walls. This\nmaximum domain wall speed leads to a switching time\nofts= 100 ps for 200 nm sized AFM domain in Co 3O4,\nwhich is considerably faster than ts= 100 ns observed\nfor heterostructure based on 200 nm thick Cr 2O3\flms51.\nWhether this fundamental speed limit can be reached\nor closely approached by AFM domain walls in Co 3O4\nrequires additional studies, investigating the switching\ndynamics on the nano- to picosecond timescale.\nFig. 3e also demonstrates that the change in \u000b(\u0001\u000b) as-\nsociated with the switching between \u0006Ly, i.e. the switch-\ning e\u000eciency, remains unchanged while the rise time of\nthe driving pulse is reduced to tr= 10\u0016s. Finally, we\nshow that Lcan be switched reversibly and repetitively in\nCo3O4. The switching e\u000eciency is not reduced upon sub-\nsequent switchings of Lup to million switching events,\nas seen in Fig. 3f. Instead a weak increase is observed,\nwhich is attributed to the training e\u000bect, often observed\nin ferroelectrics52,53.\nIn summary, we demonstrated the fast manipulation\nof AFM states in the isotropic collinear antiferromag-6\nnet Co 3O4by utilizing its strong linear ME e\u000bect. We\nshowed that the AFM N\u0013 eel vector can be either rotated\nsmoothly or reversed isothermally by electric \felds. Fur-\nthermore, our results revealed the microsecond dynam-\nics of the non-volatile and reversible switching process of\nmacroscopic AFM domains and indicate that the switch-\ning process can be even faster for microscopic domains.\nThe present study represents an important step towards\nthe on-demand electric control of AFM states, that may\neventually lead to the realization of fast-operation AFM\nspintronic device.\nData availability. The experimental data that sup-\nport the \fndings of this manuscript are available from\nthe corresponding author upon reasonable request.\n[1]\u0014Zuti\u0013 c, I., Fabian, J. & Sarma, S. D. Spintronics: Fun-\ndamentals and applications. Rev. Mod. Phys. 76, 323\n(2004).\n[2] Dietl, T. & Ohno, H. Dilute ferromagnetic semiconduc-\ntors: Physics and spintronic structures. Rev. Mod. Phys.\n86, 187 (2014).\n[3] Jungwirth, T., Marti, X., Wadley, P. & Wunderlich, J.\nAntiferromagnetic spintronics. Nat. Nanotechnol. 11,\n231{241 (2016).\n[4] Baltz, V. et al. Antiferromagnetic spintronics. Rev. Mod.\nPhys. 90, 015005 (2018).\n[5] N\u0013 eel, L. Magnetism and local molecular \feld. Science\n174, 985{992 (1971).\n[6] Jungwirth, T. et al. The multiple directions of antiferro-\nmagnetic spintronics. Nat. Phys. 14, 200{203 (2018).\n[7] Jung\reisch, M. B., Zhang, W. & Ho\u000bmann, A. Perspec-\ntives of antiferromagnetic spintronics. Phys. Lett. A 382,\n865{871 (2018).\n[8] Fina, I. et al. Anisotropic magnetoresistance in an an-\ntiferromagnetic semiconductor. Nat. Commun. 5, 1{7\n(2014).\n[9] Wang, C. et al. Anisotropic magnetoresistance in anti-\nferromagnetic Sr 2IrO 4.Phys. Rev. X 4, 041034 (2014).\n[10] Zhang, W. et al. Spin hall e\u000bects in metallic antiferro-\nmagnets. Phys. Rev. Lett. 113, 196602 (2014).\n[11] Kimata, M. et al. Magnetic and magnetic inverse spin\nhall e\u000bects in a non-collinear antiferromagnet. Nature\n565, 627{630 (2019).\n[12] Chen, X. et al. Observation of the antiferromagnetic spin\nhall e\u000bect. Nat. Mater. 1{5 (2021).\n[13] Seki, S. et al. Thermal generation of spin current in an\nantiferromagnet. Phys. Rev. Lett. 115, 266601 (2015).\n[14] Wu, S. M. et al. Antiferromagnetic spin seebeck e\u000bect.\nPhys. Rev. Lett. 116, 097204 (2016).\n[15] Rezende, S., Rodr\u0013 \u0010guez-Su\u0013 arez, R. & Azevedo, A. Theory\nof the spin seebeck e\u000bect in antiferromagnets. Phys. Rev.\nB93, 014425 (2016).\n[16]\u0014Zelezn\u0012 y, J. et al. Relativistic N\u0013 eel-order \felds induced by\nelectrical current in antiferromagnets. Phys. Rev. Lett.\n113, 157201 (2014).\n[17] Wadley, P. et al. Electrical switching of an antiferromag-\nnet. Science 351, 587{590 (2016).\n[18] Kampfrath, T. et al. Coherent terahertz control of an-tiferromagnetic spin waves. Nat. Photonics 5, 31{34\n(2011).\n[19] Gao, S. et al. Fractional antiferromagnetic skyrmion lat-\ntice induced by anisotropic couplings. Nature 1{5 (2020).\n[20] Legrand, W. et al. Room-temperature stabilization of an-\ntiferromagnetic skyrmions in synthetic antiferromagnets.\nNat. Mater. 19, 34{42 (2020).\n[21] Zhang, X., Zhou, Y. & Ezawa, M. Magnetic bilayer-\nskyrmions without skyrmion hall e\u000bect. Nat. Commun.\n7, 1{7 (2016).\n[22] Woo, S. et al. Current-driven dynamics and inhibition\nof the skyrmion hall e\u000bect of ferrimagnetic skyrmions in\nGdFeCo \flms. Nat. Commun. 9, 1{8 (2018).\n[23] Duine, R., Lee, K.-J., Parkin, S. S. & Stiles, M. D. Syn-\nthetic antiferromagnetic spintronics. Nat. Phys. 14, 217{\n219 (2018).\n[24] Song, C. et al. How to manipulate magnetic states of\nantiferromagnets. Nanotechnology 29, 112001 (2018).\n[25] Leo, N. et al. Magnetoelectric inversion of domain pat-\nterns. Nature 560, 466{470 (2018).\n[26] Parthasarathy, A. & Rakheja, S. Dynamics of magneto-\nelectric reversal of an antiferromagnetic domain. Phys.\nRev. Appl. 11, 034051 (2019).\n[27] Baldrati, L. et al. E\u000ecient spin torques in antiferromag-\nnetic CoO/Pt quanti\fed by comparing \feld-and current-\ninduced switching. Phys. Rev. Lett. 125, 077201 (2020).\n[28] Yan, H. et al. Electric-\feld-controlled antiferromagnetic\nspintronic devices. Adv. Mater. 32, 1905603 (2020).\n[29] Th ole, F., Keliri, A. & Spaldin, N. A. Concepts from the\nlinear magnetoelectric e\u000bect that might be useful for an-\ntiferromagnetic spintronics. J. Appl. Phys. 127, 213905\n(2020).\n[30] Bodnar, S. Y. et al. Writing and reading antiferro-\nmagnetic Mn 2Au by N\u0013 eel spin-orbit torques and large\nanisotropic magnetoresistance. Nat. Commun. 9, 1{7\n(2018).\n[31] Nair, N. L. et al. Electrical switching in a magnetically\nintercalated transition metal dichalcogenide. Nat. Mater.\n19, 153{157 (2020).\n[32] Borisov, P., Hochstrat, A., Chen, X., Kleemann, W. &\nBinek, C. Magnetoelectric switching of exchange bias.\nPhys. Rev. Lett. 94, 117203 (2005).\n[33] Iyama, A. & Kimura, T. Magnetoelectric hysteresis loops\nin Cr 2O3at room temperature. Phys. Rev. B 87, 180408\n(2013).\n[34] Kocsis, V. et al. Identi\fcation of antiferromagnetic do-\nmains via the optical magnetoelectric e\u000bect. Phys. Rev.\nLett.121, 057601 (2018).\n[35] Van Aken, B. B., Rivera, J.-P., Schmid, H. & Fiebig,\nM. Observation of ferrotoroidic domains. Nature 449,\n702{705 (2007).\n[36] Zimmermann, A. S., Meier, D. & Fiebig, M. Ferroic\nnature of magnetic toroidal order. Nat. Commun. 5, 1{6\n(2014).\n[37] Sato, T., Abe, N., Kimura, S., Tokunaga, Y. & Arima,\nT.-h. Magnetochiral dichroism in a collinear antiferro-\nmagnet with no magnetization. Phys. Rev. Lett. 124,\n217402 (2020).\n[38] Mufti, N. et al. Magnetoelectric coupling in MnTiO 3.\nPhys. Rev. B 83, 104416 (2011).\n[39] Kosub, T. et al. Purely antiferromagnetic magneto-\nelectric random access memory. Nat. Commun. 8, 1{7\n(2017).\n[40] He, X. et al. Robust isothermal electric control of ex-7\nchange bias at room temperature. Nat. Mater. 9, 579{585\n(2010).\n[41] Roth, W. The magnetic structure of Co 3O4.J. Phys.\nChem. Solids 25, 1{10 (1964).\n[42] Cova, F., Blanco, M. V., Han\rand, M. & Garbarino, G.\nStudy of the high pressure phase evolution of Co 3O4.\nPhys. Rev. B 100, 054111 (2019).\n[43] Sundaresan, A. & Ter-Oganessian, N. Magnetoelectric\nand multiferroic properties of spinels. J. Appl. Phys. 129,\n060901 (2021).\n[44] Saha, R. et al. Magnetoelectric e\u000bect in simple collinear\nantiferromagnetic spinels. Phys. Rev. B 94, 014428\n(2016).\n[45] Gao, S. et al. Spiral spin-liquid and the emergence of\na vortex-like state in MnSc 2S4.Nat. Phys. 13, 157{161\n(2017).\n[46] Fritsch, V. et al. Spin and orbital frustration in MnSc 2S4\nand FeSc 2S4.Phys. Rev. Lett. 92, 116401 (2004).\n[47] Ter-Oganessian, N. Cation-ordered A 1=2A0\n1=2B2X4mag-\nnetic spinels as magnetoelectrics. J. Magn. Magn. Mater.\n364, 47{54 (2014).\n[48] Ghara, S., Ter-Oganessian, N. & Sundaresan, A. Lin-\near magnetoelectric e\u000bect as a signature of long-range\ncollinear antiferromagnetic ordering in the frustrated\nspinel CoAl 2O4.Phys. Rev. B 95, 094404 (2017).\n[49] De, C. et al. Magnetoelectric e\u000bect in a single crystal of\nthe frustrated spinel CoAl 2O4.Phys. Rev. B 103, 094406\n(2021).\n[50] Zaharko, O. et al. Spin liquid in a single crystal of\nthe frustrated diamond lattice antiferromagnet CoAl 2O4.\nPhys. Rev. B 84, 094403 (2011).\n[51] Toyoki, K. et al. Magnetoelectric switching of perpendic-\nular exchange bias in Pt/Co/ \u000b-Cr 2O3/Pt stacked \flms.\nAppl. Phys. Lett. 106, 162404 (2015).\n[52] Gao, Y., Uchino, K. & Viehland, D. Domain wall release\nin \\hard\" piezoelectric under continuous large amplitude\nac excitation. J. Appl. Phys. 101, 114110 (2007).\n[53] Glaum, J., Genenko, Y. A., Kungl, H., Ana Schmitt,\nL. & Granzow, T. De-aging of Fe-doped lead-zirconate-\ntitanate ceramics by electric \feld cycling: 180\u000e-vs. non-\n180\u000edomain wall processes. J. Appl. Phys. 112, 034103\n(2012).\nMethods\nSample synthesis. The single crystals of Co 3O4\nwere grown by chemical transport reaction method us-\ning the preliminary treated high purity powder of Co 3O4\ntogether with anhydrous TeCl 4transport agent. The\ngrowth was performed in a horizontal two-zone furnace\nwith the source temperature of 830\u000eC and a temperature\ngradient of 40 to 25\u000eC. The crystallographic orientation\nof the crystals were determined by X-ray Laue di\u000brac-\ntion.\nME measurements. Crystals with \rat (001) sur-faces were prepared for the measurements. Contacts on\ntwo parallel (001) surfaces were made by silver paint.\nAll measurements were carried out in an Oxford helium-\n\row cryostat, where the electrical measurements were\nperformed up to the lowest temperature of 4 K and the\nhighest magnetic \feld of 14 T. Temperature- and mag-\nnetic \feld-dependent electric polarization were obtained\nvia the pyroelectric and ME current measurements, re-\nspectively, with a Keysight electrometer (B2987A). For\nangular-dependent electric polarization study, the ME\ncurrent was recorded under a rotating magnetic \feld of\n14 T at a constant electric \feld of +17.4 kV/cm by using\na home-designed probe, where the sample platform can\nbe rotated by a stepper motor. For the data shown in\nFig. 1, the sample was cooled with EpandHpfrom 45 K\nto 4 K and the pyroelectric/ME current was measured\nafter switching o\u000b the electric \feld. For the data shown\nin Fig. 2 & 3, the measurements were performed on a\nvirgin multi-domain sample without any ME cooling. In\nall these cases, the electric polarization were obtained by\nperforming integration of the measured pyroelectric/ME\ncurrent over time.\nPulsed electric \feld measurements. Electric po-\nlarization measurements in pulsed electric \felds were car-\nried out by a AixACCT TF Analyzer 2000 equipped with\nhigh voltage booster HVB 1000 and Krohn-Hite model\n7500 ampli\fer. In order to obtain the purely ME con-\ntribution, we subtracted the current arising from the di-\nelectric response in zero magnetic \feld from the current\nmeasured in applied magnetic \felds. The data indicated\nby the closed and opened symbols in Fig. 3e were ob-\ntained by the pulse \feld measurements on crystal with\nthickness of 250 \u0016m and 40\u0016m, respectively.\nAcknowledgements We thank Anton Jesche for as-\nsistance in X-ray Laue di\u000braction experiment and Pe-\nter Lunkenheimer, S\u0013 andor Bord\u0013 acs, Maxim Mostovoy, N.\nV. Ter-Oganessian and Sundaresan Athinarayanan for\nfruitful discussions. This work was supported by the\nDFG via the Transregional Research Collaboration TRR\n80 From Electronic Correlations to Functionality (Augs-\nburg/Munich/Stuttgart).\nAuthor Contributions: S.G. initiated this project;\nL.P. and V.T. synthesized the single crystals; M.W.,\nS.G., K.G. and S.K. performed the experiments and an-\nalyzed the data; S.G. and I.K. performed the theoreti-\ncal modeling and wrote the manuscript; I.K. supervised\nthis project. All authors discussed the content of the\nmanuscript.\nCompeting interests: The authors declare no com-\npeting interests."    },    {        "title": "2305.04744v1.Large_magnetocaloric_effect_in_the_kagome_ferromagnet_Li__9_Cr__3__P__2_O__7____3__PO__4____2_.pdf",        "content": "arXiv:2305.04744v1  [cond-mat.mtrl-sci]  8 May 2023Large magnetocaloric effect in the kagome ferromagnet Li 9Cr3(P2O7)3(PO4)2\nAkshata Magar,1Somesh K,1Vikram Singh,1J. J. Abraham,2,3Y. Senyk,2\nA. Alfonsov,2B. Büchner,2,4V. Kataev,2A. A. Tsirlin,5and R. Nath1,∗\n1School of Physics, Indian Institute of Science Education an d Research Thiruvananthapuram-695551, India\n2Leibniz IFW Dresden, D-01069 Dresden, Germany\n3Institute for Solid State and Materials Physics, TU Dresden , 01069 Dresden, Germany\n4Institute for Solid State and Materials Physics and Würzbur g-Dresden\nCluster of Excellence ct.qmat, TU Dresden, D-01062 Dresden , Germany\n5Felix Bloch Institute for Solid-State Physics, Leipzig Uni versity, 04103 Leipzig, Germany\nSingle-crystal growth, magnetic properties, and magnetoc aloric effect of the S= 3/2kagome\nferromagnet Li 9Cr3(P2O7)3(PO4)2(trigonal, space group: P¯3c1) are reported. Magnetization\ndata suggest dominant ferromagnetic intra-plane coupling with a weak anisotropy and the onset\nof ferromagnetic ordering at TC≃2.6K. Microscopic analysis reveals a very small ratio of inter-\nlayer to intralayer ferromagnetic couplings ( J⊥/J≃0.02). Electron spin resonance data suggest\nthe presence of short-range correlations above TCand confirms quasi-two-dimensional character\nof the spin system. A large magnetocaloric effect characteri zed by isothermal entropy change of\n−∆Sm≃31J kg−1K−1and adiabatic temperature change of −∆Tad≃9K upon a field sweep of 7 T\nis observed around TC. This leads to a large relative cooling power of RCP≃284J kg−1. The large\nmagnetocaloric effect, together with negligible hysteresi s render Li 9Cr3(P2O7)3(PO4)2a promising\nmaterial for magnetic refrigeration at low temperatures. T he magnetocrystalline anisotropy con-\nstantK≃ −7.42×104erg cm−3implies that the compound is an easy-plane type ferromagnet with\nthe hard axis normal to the ab-plane, consistent with the magnetization data.\nI. INTRODUCTION\nKagome lattice hosts a plethora of interesting phe-\nnomena. Its frustrated nature renders antiferromagnetic\nkagome insulators a natural playground for the experi-\nmental realization of quantum spin liquid [ 1]. Whereas\nkagome ferromagnets are not frustrated and develop\nmagnetic order, they are no less interesting because\nflat bands and Dirac fermions expected in this set-\nting have far-reaching implications for transport prop-\nerties. Recent work on ferromagnetic kagome metals\nexposed anomalous Hall and Nernst effects, as well as\nchiral edge states, in several intermetallic compounds,\nsuch as Fe 3Sn2[2], Co3Sn2S2[3], LiMn 6Sn6[4], and\nUCo0.8Ru0.2Al [5]. Concurrently, insulating kagome fer-\nromagnets were actively studied in the context of magnon\nHall effect and other exotic properties associated with\nDirac magnons [ 6–8].\nFerromagnets are further interesting as materials with\nthe large magnetocaloric effect (MCE) that can be in-\nstrumental in cooling via adiabatic demagnetization for\nreaching temperatures in the sub-Kelvin range [ 9–11].\nThis magnetic refrigeration technique is often consid-\nered as the most energy-efficient, cost-effective (as3He\nand4He are expensive), and environment-friendly re-\nplacement for the conventional refrigeration based on\ngas compression/expansion technique. For this purpose,\nmaterials with large magnetic moment, low magnetic\nanisotropy, low magnetic hysteresis, and extremely low\ntransition temperature are desirable [ 12,13]. The nature\n∗rnath@iisertvm.ac.inof the magnetic transition and the specific form of the\nmagnetic structure are also deciding factors for the per-\nformance of a MCE material. Ferromagnetic insulators\nwith second-order phase transition are proposed to be\nexcellent MCE materials, as only a small change in ap-\nplied magnetic field is sufficient to yield a large entropy\nchange and adiabatic temperature change, compared to\nany paramagnetic salt [ 10,13]. A very few ferromagnetic\ninsulators with low transition temperature are reported\nto satisfy the above prerequisites and qualify for low-\ntemperature applications [ 14–16].\nIn the following, we report the magnetic properties of\nLi9Cr3(P2O7)3(PO4)2(LCPP) which is a structural sib-\nling of the recently reported S= 5/2Heisenberg kagome\nantiferromagnet Li 9Fe3(P2O7)3(PO4)2(LFPP) with a\ntrigonal space group P¯3c1[17]. LFPP shows the onset of\nan antiferromagnetic (AFM) ordering below TN≃1.3K\nand a characteristic 1/3magnetization plateau below\nT∗≃5K. The NMR spectra along with the NMR spin-\nlattice relaxation time reveal the presence of an exotic\nsemiclassical nematic spin liquid regime between TNand\nT∗. On the contrary, LCPP is found to be a ferromag-\nnet and it undergoes a ferromagnetic (FM) ordering at\nTC≃2.5K. LCPP exhibits a large MCE around TCand\nappear to have strong potential for cryogenic applications\nsuch as low temperature sensors in space research, achiev-\ning sub-kelvin temperatures for basic research, hydrogen\nand helium gas liquefaction etc [ 9,18].\nII. METHODS\nPlatelet single crystals of LCPP with the lateral size\nof 0.5 mm to 1 mm were synthesized by a self-flux tech-26.9034 Å\nJ┴\nJ\nabc(a)\nFIG. 1. (a) Corner-sharing equilateral triangles of Cr3+form\na kagome lattice. The intraplane ( J) and interplanar ( J⊥)\ncoupling are shown. The kagome lattice layers are well sepa-\nrated from each other with interlayer distance of 6.9034 Å. ( b)\nInteractions between the magnetic Cr3+ions via PO 4tetra-\nhedra are shown.\nnique as reported in Ref. [ 19]. The mixture of starting\nmaterials, Li 3PO4, Cr2O3, and NH 4H2PO4in the molar\nratio 15:1:9 was kept in an alumina crucible and heated\ngradually to 900◦C. The cooling process involves three\nsteps. At first, the sample was cooled down to 850◦C at\na rate of 50◦C per hour and then to 600◦C at a slow rate\nof2◦C per hour. Finally, the sample was allowed to cool\nnaturally to room temperature. In order to dissolve the\nflux and separate the crystals, the sample was treated\nwith 1 M solution of acetic acid for five days followed by\nthe treatment with saturated NaCl solution and distilled\nwater. The final product after the treatment yields the\nmixture of mm-sized single crystals and polycrystalline\nsample. The large-sized crystals were hand-picked and\nthe remaining part is grinded to get the polycrystalline\nsample.\nRoom-temperature single-crystal x-ray diffraction\n(XRD) was performed on a good quality single crystal\nusing the Bruker KAPPA APEX-II CCD diffractome-\nter equipped with graphite monochromated Mo Kα1ra-\ndiation ( λ= 0.71073 Å). The APEX3 software was\nused to collect the data that were further reduced with\nSAINT/XPREP followed by an empirical absorption cor-\nrection using the SADABS program [ 20]. The phase pu-\nrity of the polycrystalline sample was confirmed from\npowder XRD (PANalytical Xpert-Pro, Cu Kαradia-\ntion with λav= 1.54182 Å). The temperature-dependent\npowder XRD measurement was performed in the temper-\nature range 15 K ≤T≤300 K with a low-temperature\n(Oxford Phenix) attachment to the diffractometer.\nMagnetization ( M) measurement was performed as a\nfunction of temperature ( T) and magnetic field ( H) using\na superconducting quantum interference device (SQUID)\n(MPMS-3, Quantum Design) magnetometer. The data\nwere collected in the temperature range 1.8 K ≤T≤\n350 K and in the magnetic field range 0 ≤H≤7 T. Heat\ncapacity ( Cp) as a function of T(0.5 K≤T≤300 K)\nandHwas measured on a small piece of sintered pellet\nusing the relaxation technique in the physical property\nmeasurement system (PPMS, Quantum Design). Mea-\nsurements below 2 K were carried out using an additional3He insert in the PPMS.\nHigh-field electron spin resonance (HF-ESR) spec-\ntroscopy was used to study the single crystals of LCPP.\nFor the measurements in a frequency range 75 - 330 GHz,\na vector network analyzer (PNA-X from Keysight Tech-\nnologies) was used and for frequencies up to 975 GHz\na modular Amplifier/Multiplier Chain (AMC from Vir-\nginia Diodes Inc.) was used for the generation of\nmicrowaves in combination with a hot electron InSb\nbolometer for detection. All measurements were per-\nformed at a given fixed frequency in the field-sweep mode\nup to 16 T, using a superconducting magnet system from\nOxford Inst. The sample was mounted onto a transmis-\nsion probe head which is then inserted in a4He variable\ntemperature insert (VTI) of the magnet cryostat to en-\nable measurements in a temperature range of 1.8−300K.\nDensity-functional (DFT) band-structure calculations\nwere performed in the FPLOcode [ 21] with the Perdew-\nBurke-Ernzerhof flavor of the exchange-correlation po-\ntential [ 22]. Correlation effects in the Cr 3dshell were\nincluded on the mean-field level within DFT+ Uusing the\non-site Coulomb repulsion parameter Ud= 2eV, Hund’s\ncoupling Jd= 1eV, and double-counting correction in\nthe atomic limit [ 23,24]. Exchange couplings Jijwere\nobtained by mapping [ 25] total energies of collinear mag-\nnetic configurations onto the spin Hamiltonian,\nH=/summationdisplay\n/angbracketleftij/angbracketrightJijSiSj (1)\nwhere the summation is over pairs, and S=3\n2. Energies\nwere converged on a kmesh with 64 points within the first\nBrillouin zone. Thermodynamic properties for the model\ndefined by Eq. ( 1) were obtained from quantum Monte-\nCarlo simulations performed with the loopalgorithm [ 26]\nof theALPSsimulation package [ 27]. Finite lattices with\nup to 752 sites and periodic boundary conditions were\nused.\nIII. RESULTS AND DISCUSSION\nA. X-ray Diffraction\nThe crystal structure of LCPP was solved from single-\ncrystal XRD data with direct methods using SHELXT-\n2018/2 [ 28] and refined by the full matrix least squares\nonF2using SHELXL-2018/3, respectively [ 29]. Details\nof the crystal structure and the refined parameters are\nsummarized in Table I. LCPP crystallizes in the trigonal\nspace group P¯3c1(No. 165). The refined atomic posi-\ntions at room temperature are listed in Table II. These\nstructural parameters are in good agreement with the\nprevious report [ 19].\nThe schematic view of the crystal structure of LCPP\nis presented in Fig. 1. It illustrates the corner sharing\nof CrO 6octahedra and PO 4tetrahedra forming equilat-\neral triangles with a geometrically deformed but regular3\nTABLE I. Crystallographic data for LCPP at room tempera-\nture, obtained from single-crystal XRD.\nEmpirical formula Cr 3Li9O29P8\nFormula weight(M r) 930.22 g mol−1\nTemperature 296(2) K\nCrystal system Trigonal\nSpace group P¯3c1\nLattice parameters a= 9.668(3) Åα= 90◦\nb= 9.668(3) Åβ= 90◦\nc= 13.610(6) Åγ= 120◦\nUnit cell volume 1101.7(8) Å3\nZ 2\nDensity (calculated) 2.804 g cm−3\nWavelength 0.71073 Å\nRadiation type MoK α1\nDiffractometer Bruker KAPPA APEX-II CCD\nCrystal size 0.049×0.035×0.027mm3\n2θrange 2.993 to 25.997◦\nIndex ranges −11≤h≤11\n−11≤k≤11\n−16≤l≤16\nF(000) 902\nReflections collected 6940\nIndependent reflections 735 [ Rint= 0.0454]\nData / restraints / parameters 735/0/76\nGoodness-of-fit on F21.098\nFinalRindices [I≥2σ(I)] R1= 0.0304,\nωR2= 0.0892\nRindices(all data) R1= 0.0351,\nωR2= 0.0919\nLargest diff. peak and hole +0.445/−0.970e.Å−3\nkagome lattice. Though all of the Cr3+– Cr3+distances\nin each hexagon are equal ( ∼4.949Å), the bond angles\nare different: three angles are about ∼146.3◦and the\nremaining three angles are about ∼93.7◦. The nearest-\nneighbour (NN) coupling between the Cr3+ions in the\nab-plane is denoted by J, while the shortest interplane\ndistance of ∼6.903Å leads to a weak coupling ( J⊥) be-\ntween the planes.\nIn order to confirm the phase purity and to scruti-\nnize the presence of any structural distortions, powder\nXRD data were collected at various temperatures. Le-\nBail analysis of the XRD patterns was performed us-\ningFullProf package [ 30] taking the initial structural\nparameters from the single crystal data (Table I). Fig-\nure2(a) and (b) present the powder XRD patterns at the\nhighest ( T= 300 K) and lowest ( T= 15 K) measured\ntemperatures, respectively, along with the Le-Beil fits.\nAll the peaks could be indexed based on the space group\nP¯3c1, suggesting phase purity of the polycrystalline sam-\nple. The obtained lattice parameters at room tempera-\nture are a=b= 9.6628(3) Å,c= 13.5769(3) Å, andTABLE II. Crystal structure of LCPP refined using single-\ncrystal XRD data. The atomic coordinates ( ×104) and the\nisotropic atomic displacement parameter Uiso(Å2×103), which\nis defined as one-third of the trace of the orthogonalized Uij\ntensor.\nAtomic\nsitesWyckoff\npositionsx y z U iso\nCr(1) 6f 5676(1) 0 2500 6(1)\nLi(1) 2b 0 0 5000 19(3)\nLi(2) 12g 3369(7) 2367(7) 4374(4) 16(1)\nLi(3) 4d 6667 3333 6178(7) 15(2)\nO(1) 12g 3758(2) −1053(2) 3331(1) 9(1)\nO(2) 6f 2120(3) 0 2500 9(1)\nO(3) 12g 792(2) −2535(2) 3440(2) 9(1)\nO(4) 12g 2299(3) 38(2) 4332(2) 11(1)\nO(5) 12g 6781(2) 1893(2) 3352(2) 10(1)\nO(6) 4d 6667 3333 4839(3) 23(1)\nP(1) 12g 2279(1) −894(1) 3440(1) 6(1)\nP(2) 4d 6667 3333 3736(1) 5(1)\nunit-cell volume Vcell≃1097.85(5) Å3, which are consis-\ntent with the single-crystal data. No extra peaks or fea-\ntures were observed in the XRD data corroborating the\nabsence of any structural transition or distortion down\nto 15 K.\nThe temperature variation of lattice parameters ( a,c,\nandVcell) is presented in Fig. 3. They are found to de-\ncrease monotonically upon cooling down to 15 K. Vcell(T)\nwas fitted by the equation [ 31]\nVcell(T) =γU(T)\nK0+V0, (2)\nwhereV0is the zero-temperature unit cell volume, K0\nis the bulk modulus, and γis the Grüneisen parameter.\nU(T)is the internal energy and it can be expressed in\nterms of the Debye approximation as\nU(T) = 9pkBT(T\nθD)3/integraldisplayθD/T\n0x3\nex−1dx. (3)\nHere,pis the number of atoms in the unit cell, kBis\nthe Boltzmann constant, and the Debye temperature is\nrepresented by θD. The fit (see Fig. 3) returns θD≃\n385K,γ/K0≃2.06×10−4Pa−1, andV0≃1094.2Å3.\nB. Magnetization\nMagnetic susceptibility χ(T) [≡M(T)/H]measured\nin an applied field of H= 0.5T perpendicular ( H⊥c)\nand parallel ( H∝bardblc) to the kagome plane is displayed\nin Fig. 4(a). With decreasing temperature, χ(T)in-\ncreases in a Curie-Weiss manner as expected in the high-\ntemperature paramagnetic (PM) regime, followed by a4\n/s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48/s32\n/s32/s32/s73\n/s111/s98/s115\n/s32/s32 /s73\n/s99/s97/s108\n/s32/s32 /s73\n/s111/s98/s115/s45/s32 /s73\n/s99/s97/s108/s32\n/s32/s32/s66/s114/s97/s103/s103/s32/s112/s111/s115/s105/s116/s105/s111/s110/s115/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s40/s97/s41\n/s84 /s32/s61/s32/s51/s48/s48/s32/s75\n/s32/s40/s100/s101/s103/s114/s101/s101/s41/s97\n/s32/s40/s98/s41\n/s84 /s32/s61/s32/s49/s53/s32/s75\n/s32/s32\n/s98\nFIG. 2. Powder XRD data measured at (a) T= 300 K and\n(b)T= 15K. The black solid line represents the Le-Bail fit\nof the data. Bragg positions are indicated by vertical bars\nand the solid green line at the bottom denotes the difference\nbetween experimental and calculated intensities. The inse t of\n(b) shows a representative single crystal.\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s57/s46/s54/s53/s50/s57/s46/s54/s53/s54/s57/s46/s54/s54/s48/s57/s46/s54/s54/s52/s49/s51/s46/s53/s53/s50/s49/s51/s46/s53/s54/s48/s49/s51/s46/s53/s54/s56/s49/s51/s46/s53/s55/s54\n/s32/s97\n/s32/s99/s99 /s32/s40 /s197 /s41/s32/s97 /s32/s40 /s197 /s41\n/s84 /s32/s40/s75/s41/s49/s48/s57/s51/s46/s53/s49/s48/s57/s53/s46/s48/s49/s48/s57/s54/s46/s53/s49/s48/s57/s56/s46/s48/s32/s86\n/s99/s101/s108/s108\n/s32/s102/s105/s116\n/s86\n/s99/s101/s108/s108/s32/s40 /s197/s51\n/s41\nFIG. 3. Variation of lattice parameters ( a,c, andVcell) with\ntemperature. The solid line denotes the fit of Vcell(T)by\nEq. (2).\nrapid enhancement at low temperatures. This rapid in-\ncrease suggests strong FM correlations below about 10 K.\nFurther, the susceptibility for H⊥candH∝bardblcshow only\na small difference even at low temperatures, which is an\nindication of weak magnetic anisotropy in the compound.\nFor a quantitative analysis, we have plotted the in-\nverse susceptibility ( 1/χ) as a function of temperature in/s49 /s49/s48 /s49/s48/s48/s48/s49/s50/s51\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s45/s52 /s45/s50 /s48 /s50 /s52/s45/s51/s45/s50/s45/s49/s48/s49/s50/s51\n/s50 /s52 /s54 /s56/s49/s49/s48/s49/s48/s48/s32\n/s72 /s32/s61/s32/s48/s46/s53/s32/s84/s32/s91/s99/s109/s51\n/s32/s40/s109/s111/s108/s32/s67/s114/s51/s43\n/s41/s45/s49\n/s93\n/s84 /s32/s40/s75/s41/s32/s72 /s32 /s32/s99\n/s32/s72 /s32/s99/s40/s97/s41\n/s40/s99/s41\n/s32/s32/s49/s47 /s32/s40/s99/s109/s45/s51\n/s32/s109/s111/s108/s32/s67/s114/s51/s43\n/s41\n/s84 /s32/s40/s75/s41/s32/s67/s87/s32/s102/s105/s116\n/s32/s72 /s32 /s32/s99/s40/s98/s41/s32\n/s32/s84 /s32/s61/s32/s49/s46/s56/s32/s75\n/s32/s72 /s32 /s32/s99\n/s32/s72 /s32/s99/s77 /s32/s91\n/s66/s32/s40/s67/s114/s51/s43\n/s41/s45/s49\n/s93\n/s72 /s32/s40/s84/s41\n/s40/s100/s41\n/s72 /s32 /s32/s32/s99/s32\n/s32/s48/s46/s48/s49/s32/s84\n/s32/s48/s46/s48/s50/s32/s84\n/s32/s48/s46/s48/s53/s32/s84\n/s32/s48/s46/s49/s32/s84\n/s32/s48/s46/s53/s32/s84\n/s32/s49/s32/s84/s84 /s32/s91/s99/s109/s51\n/s32/s75/s32/s40/s109/s111/s108/s32/s67/s114/s51/s43\n/s41/s45/s49\n/s93\n/s84 /s32/s40/s75/s41\nFIG. 4. (a) χ(T)measured in the field of H= 0.5T applied\nperpendicular ( H⊥c) and parallel ( H/bardblc) to the c-axis. (b)\nMagnetization at T= 1.8K as a function of applied field for\nbothH⊥candH/bardblcafter correcting for the demagnetiza-\ntion effect. (c) Inverse susceptibility ( 1/χ) as a function of\ntemperature for H⊥cand solid line is the CW fit. (d) χT\nvsTin different fields for H⊥cin low temperatures.\nFig.4(c) forH⊥c. ForT≥50 K, it exhibits a com-\npletely linear behaviour which was fitted by the modified\nCurie-Weiss (CW) law\nχ(T) =χ0+C\n(T−θCW). (4)\nHere,χ0is the temperature-independent susceptibility,\nCis the Curie constant, and θCWis the CW temper-\nature. The fit yields χ0≃ −3.6×10−4cm3mol−1,\nC≃1.92cm3K mol−1, andθCW≃6K. From the\nvalue of Cthe effective moment is calculated to be\nµeff≃3.92µBin agreement with the spin-only value\nof 3.87µBfor spin-3/2. The positive value of θCWsug-\ngests that the dominant exchange interactions between\nCr3+ions are FM in nature. Using the mean-field ex-\npression J/kB=−3|θCW|/zS(S+ 1)withz= 4neigh-\nbors on the kagome lattice, we estimate J/kB≃ −1.2K.\nMoreover, the small peak in χTin low magnetic fields\nrevealsTC≃2.6K [Fig. 4(d)].χ(T)measured in zero-\nfield cooled and field cooled conditions (not shown) in a\nsmall magnetic field of H= 0.01T forH⊥cshows no\ndifference, suggesting negligible hysteresis.\nThe magnetic isotherm ( MvsH) atT= 1.8K satu-\nrates in low fields of Hsat⊥c≃0.15T andHsat∝bardblc≃\n0.4T with the saturation magnetization of Msat∼\n3.2µB/Cr3+and2.8µB/Cr3+forH⊥candH∝bardblc,\nrespectively (not shown). The obtained Msatvalues are\nclose to the calculated Msat=gSµB≃2.952µBand\n2.937µB, taking the ESR values g≃1.968and 1.958 for\nH⊥candH∝bardblc, respectively. No visible hysteresis is\nobserved in any of the field directions. A slight difference5\nin the saturation field for H⊥candH∝bardblcmay be at-\ntributed to the anisotropic demagnetization field caused\nby the flat shape of the crystals [ 8]. The demagnetiza-\ntion factor is negligible when magnetic field is parallel\nto the crystal plates ( H⊥c). However, when field is\nperpendicular to crystal plate ( H∝bardblc), the demagne-\ntization effect is considerably amplified [ 32]. The data\nin Fig. 4(b) have been corrected for this demagnetizing\nfield asHeff=H0−4πNM whereHeffandH0are the\neffective and applied magnetic fields, respectively, Nis\nthe demagnetization factor, and Mis magnetic moment\nin emu cm−3. To calculate the demagnetization factor\nin the case of LCPP, we approximated the shape of the\nsample to a rectangular strip and did the calculation fol-\nlowing Ref. [ 33], which yields N= 0.84forH∝bardblc. The\nHsatin both directions after demagnetization correction\nis found to be almost same ( ∼0.15T).\nC. Heat Capacity\nTemperature-dependent heat capacity Cpmeasured on\nthe polycrystalline sample is shown in Fig. 5. TheCp\ndata exhibit a sharp λ-type anomaly at TC≃2.5K\ndemonstrating the transition to the magnetically ordered\nstate. Typically, in magnetic insulators, the major con-\ntributions to Cpare from magnetic ( Cmag) and phonon\n(Cph) parts. In high temperatures, Cphdominates over\nCmag, while at low temperatures it is reverse. One can\nestimate Cmagby subtracting Cphfrom the total heat\ncapacity. First, we approximate the phonon contribution\nby fitting the high- Tdata by a linear combination of one\nDebye and three Einstein terms as [ 34]\nCph(T) =fDCD(θD,T)+3/summationdisplay\ni=1giCEi(θEi,T).(5)\nThe first term in Eq. ( 5) is the Debye contribution to\nCph, which can be written as\nCD(θD,T) = 9nR/parenleftbiggT\nθD/parenrightbigg3/integraldisplayθD\nT\n0x4ex\n(ex−1)2dx. (6)\nHere,Ris the universal gas constant, θDis the character-\nistic Debye temperature, and nis the number of atoms\nin the formula unit. The second term in Eq. ( 5) gives the\nEinstein contribution to Cphthat has the form\nCE(θE,T) = 3nR/parenleftbiggθE\nT/parenrightbigg2eθE/T\n[eθE/T−1]2. (7)\nHere,θEis the characteristic Einstein temperature. The\ncoefficients fD,g1,g2, andg3represent the fraction of\natoms that contribute to their respective parts. These\nvalues are taken in such a way that their sum should be\nequal to 1 and are conditioned to satisfy the Dulong-\nPetit value ∼3nRat high temperatures. The high-\nTfit to the Cp(T)data was then extrapolated down/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s48/s53/s49/s48\n/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50/s48/s49/s50/s51/s52\n/s48 /s49 /s50 /s51/s48/s51/s54/s57/s49/s50\n/s32/s32\n/s32/s67\n/s112\n/s32/s67\n/s112/s104\n/s32/s67\n/s109/s97 /s103/s67\n/s112/s32/s91/s74/s32/s40/s109/s111/s108/s32/s75/s32/s67/s114/s51/s43\n/s41/s45/s49\n/s93/s40/s97/s41\n/s32/s32/s67\n/s112/s32/s91/s74/s32/s40/s109/s111/s108/s32/s75/s32/s67/s114/s51/s43\n/s41/s45/s49\n/s93\n/s84 /s32/s40/s75/s41/s32/s48/s32/s84\n/s32/s53/s32/s84\n/s32/s55/s32/s84\n/s40/s98/s41/s67\n/s109/s97/s103/s47/s84 /s32/s91/s74/s32/s40/s109/s111/s108/s32/s75/s50\n/s32/s67/s114/s51/s43\n/s41/s45/s49\n/s93/s32\n/s84/s32 /s40/s75/s41/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48\n/s83\n/s109/s97/s103/s47/s82 /s108/s110/s52/s32/s32\n/s32 /s32/s84 /s32/s67\n/s112/s32/s91/s74/s32/s40/s109/s111/s108/s32/s75/s32/s67/s114/s51/s43\n/s41/s45/s49\n/s93\n/s84/s32 /s40/s75/s41\nFIG. 5. (a) Heat capacity ( Cp) of LCPP measured in\nzero applied field. The solid line represents the simulated\nphonon contribution [ Cph(T)] and the dotted line represents\nthe magnetic contribution [ Cmag(T)]. Inset: Low tempera-\ntureCpmeasured in various applied fields. (b) Cmag/Tand\nSmag/Rln4in left and right y-axes, respectively, are plotted\nas a function of temperature. Inset: Cmag(T)vsT. Solid line\nis the power-law ( Cmag=aTα) fit in the low- Tregime.\nto low temperatures and subtracted from Cp(T). The\nobtained Cmag/Tis plotted as a function of tempera-\nture in the main panel of Fig. 5(b) and the correspond-\ning magnetic entropy is calculated to be Smag(T) =/integraltextT\n0KCmag(T′)\nT′dT′≃11.6J mol−1K−1at 12 K. This value\ncorresponds to the expected magnetic entropy for spin-3\n2:\nSmag=Rln4 = 11 .5J mol−1K−1. Unlike the conven-\ntional magnets which release the entire entropy near the\ntransition temperature, LCPP releases only ∼40% of\nthe total entropy at TCand the remaining entropy is re-\nleased only above 11 K, suggesting that TCis partially\nsuppressed as a result of low-dimensionality or magnetic\nfrustration [ 24].\nThe inset of Fig. 5(a) presents the Cp(T)data mea-\nsured in different applied fields. The influences of mag-\nnetic field is clearly reflected in the data. The zero-field\npeak broadens and shifts toward high temperatures with\nincreasing field, which is usual for ferromagnets. At low\ntemperatures, Cmag(T)in zero field could be well de-6\n/s48 /s50 /s52 /s54 /s56/s48/s49/s50/s51\n/s48 /s49 /s50 /s51/s48/s52/s56/s49/s50/s49/s48/s32/s75/s49/s46/s56/s32/s75\n/s32/s77 /s32/s91\n/s66/s32/s40/s67/s114/s51/s43\n/s41/s45/s49\n/s93\n/s72 /s32/s40/s84/s41/s72 /s32 /s32/s99/s40/s97/s41\n/s40/s98/s41\n/s32/s32/s77/s50\n/s32/s91\n/s66/s32/s40/s67/s114/s51/s43\n/s41/s45/s49\n/s93/s50\n/s72/s47/s77 /s32/s40/s84/s32/s67/s114/s51/s43\n/s66/s45/s49\n/s41/s49/s46/s56/s32/s75\n/s49/s48/s32/s75\nFIG. 6. (a) Isothermal magnetization ( MvsH) curves for\nH⊥cand (b) their corresponding Arrott plots ( M2vs\n(H/M)) for LCPP at different temperatures around TC.\nscribed using power-law ( Cmag∝Tα) behavior [inset of\nFig.5(b)] with an exponent α∼1.5that corresponds to\nFM spin-wave excitations [ 35].\nD. Magnetocaloric Effect\nMagnetocaloric effect (MCE) is an intrinsic property of\nmagnetic materials. Magnetic cooling is achieved by first\napplying magnetic field to the material isothermally and\nthen removing the field adiabatically. Therefore, MCE\nis generally quantified by the isothermal entropy change\n(∆Sm) and adiabatic temperature change ( ∆Tad) with\nrespect to the change in applied field. The ∆Smcan\nbe calculated from either magnetization isotherms ( M\nvsH) or heat capacity data measured in zero and non-\nzero magnetic fields. Figure 6(a) displays the magnetic\nisotherms measured in close temperature steps around/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s48/s45/s49/s48/s45/s50/s48/s45/s51/s48\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s48/s45/s49/s48/s45/s50/s48/s45/s51/s48/s49/s32/s84/s55/s32/s84/s32\n/s32/s83\n/s109/s32/s40/s74/s32/s107/s103/s45/s49\n/s32/s75/s45/s49\n/s41/s40/s97/s41\n/s72 /s32 /s32/s99\n/s40/s98/s41\n/s32/s32/s83\n/s109/s32/s40/s74/s32/s107/s103/s45/s49\n/s32/s75/s45/s49\n/s41\n/s84/s32 /s40/s75/s41/s32/s55/s32/s84\n/s32/s53/s32/s84\nFIG. 7. (a) Isothermal entropy change ( ∆Sm) vsTplotted\nupon sweeping the field to 0 T starting from different fields\nfrom 1 T to 7T, calculated using MvsHdata of single\ncrystals for H⊥cand employing Eq. ( 8). (b)∆SmvsTplot\nfor5T and7T, calculated using Cp(T)of polycrystalline\nsample in Eq. ( 9).\nTCforH⊥c. The first method utilizes Maxwell’s ther-\nmodynamic relation, (∂S/∂H)T= (∂M/∂T)H, and∆Sm\ncan be estimated using the MvsHdata as [ 36]\n∆Sm(H,T) =/integraldisplayHf\nHidM\ndTdH. (8)\nFigure 7(a) presents the plot of ∆Smas a function of\ntemperature ( T) in different values of ∆H=Hf−Hi.\n∆SmvsTexhibits a maximum entropy change around\n4.6 K, with a highest value of ∆Sm≃ −31J kg−1K−1\nfor the 7 T field change. As the magnetic anisotropy\nis negligibly small, no significant difference in ∆Smis\nexpected for H⊥candH∝bardblc[37].\nFurther, to cross check the large value of ∆Sm, we have\nalso estimated ∆Smfrom heat capacity data measured\nin zero field, 5 T, and 7 T. First, we calculated the total7\nentropy at a given field as\nS(T)H=/integraldisplayTf\nTiCp(T)H\nTdT, (9)\nwhereCp(T)His the heat capacity at a particular field\nHandTiandTfare the initial and final tempera-\ntures, respectively. We calculated ∆Smby taking the\ndifference of total entropy at non-zero and zero fields\nas∆Sm(T)∆H= [S(T)H−S(T)0]T. Here, S(T)Hand\nS(T)0are the total entropy in the presence of Hand\nin zero field, respectively. Figure 7(b) presents the esti-\nmated∆Smas a function of temperature in 5 T and 7 T\nmagnetic fields. The overall shape and peak position of\nthe∆Smcurves are identical with the curves [Fig. 7(a)]\nobtained from the magnetic isotherms but with a slight\nreduction in magnitude. This difference in magnitude\nat the peak position could be related to the polycrys-\ntalline sample used for heat capacity and single crystals\nfor magnetic measurements [ 38].\nSimilarly, the adiabatic temperature change ∆Tadcan\nbe estimated from either the combination of zero-field\nheat capacity and the magnetic entropy change obtained\nfrom magnetic isotherms or from the heat capacity alone\nmeasured in different magnetic fields. Using the heat\ncapacity in zero field and magnetization isotherm data,\nthe estimation of ∆Tadcan be done as [ 39]\n∆Tad=/integraldisplayHf\nHiT\nCpdM\ndTdH. (10)\nThe dependence of ∆TadonTfor different magnetic\nfields is shown in Fig. 8(a). The maximum value of ∆Tad\nis obtained to be ∼40K for∆H= 7T. However, as\nexplained in Ref. [ 40] the above expression overestimates\n∆TadsinceT/Cpis not constant over the range of applied\nfields as it was assumed. It is evident from the inset of\nFig.5(a) thatCpat low temperatures is changing drasti-\ncally as we apply magnetic field and this change should be\ntaken into account while calculating the entropy for that\nparticular field. Therefore, we tried to estimate ∆Tadby\ntaking the difference in temperatures corresponding to\ntwo different fields with constant (same) entropy as [ 40]\n∆Tad(T)∆H= [T(S)Hf−T(S)Hi]. (11)\n∆TadvsTfor∆H= 5T and7T calculated by this\nmethod is shown in Fig. 8(b). The maximum value of\n∆Tadat 7 T is around ∼9K which is significantly\nsmaller than the value obtained using the former method\n[Eq. (10)]. A similar difference has been reported earlier\nfor ErAl 2[40]. The latter method is considered to be\nmore reliable. It is expected to provide accurate results\nas the effect of magnetic field on Cpis accounted for.\nNote that the large values of ∆Smand∆Tadare not\nsufficient to characterize the potential of a material for\nthe magnetic refrigeration applications. Another im-\nportant parameter is the relative cooling power ( RCP)/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s48/s49/s48/s50/s48/s51/s48/s52/s48\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s48/s50/s52/s54/s56/s49/s48/s72 /s32 /s32/s99\n/s32\n/s32/s32/s45 /s84\n/s97/s100/s32/s40/s75/s41\n/s49/s32/s84/s55/s32/s84/s40/s97/s41\n/s40/s98/s41\n/s32/s32/s45 /s84\n/s97/s100/s32/s40/s75/s41\n/s84 /s32/s40/s75/s41/s32/s32/s55/s32/s84\n/s32/s32/s53/s32/s84\nFIG. 8. (a) ∆TadvsTplotted for different field changes of\n∆H= 1T to7T calculated using Eq. ( 10). (b)∆TadvsT\nplotted for ∆H= 5T and7T calculated using Eq. ( 11).\nwhich is a measure of the amount of heat transferred be-\ntween the cold and hot reservoirs in a refrigeration cycle.\nMathematically, it can be expressed as\nRCP=/integraldisplayThot\nTcold∆Sm(T,H)dT, (12)\nwhere,TcoldandThotcorrespond to temperatures of cold\nand hot reservoirs, respectively. The formula for RCP\ncan be approximated as\n|RCP|approx= ∆Speak\nm×δTFWHM, (13)\nwhere∆Speak\nmandδTFWHM are the maximum value of\nentropy change (or the peak value) and full width at\nhalf maximum of the ∆Smcurve, respectively. RCP\nas a function of Hcalculated using the ∆Smdata from\nFig.7(a) is plotted in Fig. 9(a). The maximum value of\nRCP is calculated to about ∼284J kg−1at 7 T.\nThe application of a MCE material is also decided by\nthe nature of its magnetic phase transition. In materi-8\n/s48 /s49 /s50 /s51 /s52 /s53 /s54 /s55 /s56/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48\n/s48 /s50 /s52 /s54 /s56/s49/s48/s50/s48/s51/s48\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s50 /s52 /s54/s45/s50/s52/s45/s49/s54/s45/s56/s48\n/s32/s82/s67/s80 /s32/s40/s74/s32/s107/s103/s45/s49\n/s41\n/s72 /s32/s40/s84/s41/s40/s97/s41\n/s32/s32/s83\n/s109/s112/s101/s97/s107\n/s40/s74/s32/s107/s103/s45/s49\n/s75/s45/s49\n/s41\n/s72 /s32/s40/s84/s41\n/s40/s98/s41\n/s32/s32/s110\n/s84 /s32/s40/s75/s41\n/s32/s32/s83\n/s109/s32/s40/s74/s32/s107/s103/s45/s49\n/s32/s75/s45/s49\n/s41\n/s72 /s32/s40/s84/s41/s50/s50/s46/s53/s32/s75\n/s49/s48/s32/s75\n/s51/s46/s54/s32/s75\nFIG. 9. (a) Relative cooling power ( RCP =∆Speak\nm×\nδTFWHM ) as a function of magnetic field and the inset shows\nthe value of entropy change at the peak position ∆Speak\nm. Solid\nlines are the fits as described in the text. (b) The exponent n\nplotted as a function of temperature which is obtained from\nfitting power law to ∆Smvs field isotherms. Inset shows the\n∆Smisotherms for temperatures near and well above TC.\nals with a first-order phase transition, though the peak\nheight of the ∆Smand∆TadvsTcurves is large but\nthe curve width is not very broad, which limits the rele-\nvance of these materials in a cyclic operation. The second\nproblem with the first-order transitions is the energy loss\ndue to magnetic and thermal hysteresis [ 41]. Further,\nrelatively large magnetic fields are required to perturb\nthe first-order magneto-structural transitions and induce\nlarge MCE which is another drawback of these mate-\nrials. On the other hand, materials with second-order\nphase transition do not show very large peaks, but their\nRCP values are large due to the increased curve widthand the absence of thermal hysteresis, both effects mak-\ning them promising for practical applications. In order to\nanalyze the nature of the phase transition, we construct\nthe Arrott plot [ 42] in Fig. 6(b) by using the isother-\nmal magnetization data presented in Fig. 6(a). Clearly,\nthe slope of M2vsH/M curves is positive in the entire\nmeasured temperature range, well below and above TC.\nAccording to the Banerjee criterion [ 43], positive slope\nimplies the second-order phase transition. This confirms\nthe continuous second-order nature of the PM to FM\nphase transition in LCPP.\nFurthermore, MCE is also utilized to characterize the\nnature of a phase transition [ 39,44,45]. According to\nthe scaling hypothesis [ 41], the∆Sm(T)curves for dif-\nferent values of ∆Hshould collapse on a single universal\ncurve when the ∆Sm(T)is normalized to its peak value\n∆Speak\nm. However, due to the low transition temperature,\nthe universal curve construction is implausible with the\npresent data. Therefore, we performed only the power-\nlaw analysis of ∆SmandRCP. In Fig. 9(a) main panel\nand inset, we fitted the RCP and∆Speak\nmdata by power\nlaws of the form RCP∝HNand|∆Speak\nm| ∝Hn, re-\nspectively. The exponents Nandn, which are related\nto the critical exponents ( β,γ, andδ), are estimated to\nbeN≃0.7andn≃0.5. In order to perceive the tem-\nperature dependence of n, we fitted the field-dependent\nisothermal magnetic entropy change ∆Sm(H)at various\ntemperatures across the transition using the power law\n∆Sm∝Hn[see inset of Fig. 9(b)] [44]. The obtained\nnvsTdata are plotted in Fig. 9(b) and provide in-\nformation concerning the nature of the transition. For\ninstance, for a second-order magnetic transition the ex-\nponent should have the value n≃2in the paramagnetic\nregion (T >> T C) andn(T)typically exhibits a mini-\nmum near TC[44]. Indeed, our n(T)demonstrates the\nexpected behavior, further confirming the second-order\nmagnetic transition in LCPP.\nIn Table III, we compare the main parameters of LCPP\nwith those of well-studied magnets having low transition\ntemperatures and large MCE. Though the ∆Speak\nmvalue\nof LCPP is comparable to the values for most of the\npotential low-temperature magnetic refrigerant materi-\nals, the width of ∆SmvsTcurves is not very broad.\nDue to which LCPP has a slightly reduced value of RCP\ncompared to others. Nevertheless, the obtained value of\nRCP≃284J kg−1is still significantly large and LCPP\nmay have strong prerequisites for cryogenic applications\nin sub-Kelvin temperatures [ 9,18].\nE. Electron Spin Resonance\n1. Frequency Dependence at T= 1.8K\nThe ferromagnetic resonance (FMR) measurements on\nLCPP were performed at T= 1.8K (< TC) for the two\norientations of the applied magnetic field, H∝bardblcand\nH⊥c. The frequency ( ν) vs resonance field ( Hres) de-9\nTABLE III. Comparison of the adiabatic temperature change ( ∆Tad), maximum entropy change ( ∆Speak\nm), and relative cooling\npower (RCP) of LCPP with some known magnets having low transition tempe ratures ( TCorTN) and large MCE in a field\nchange of ∆H= 5to 8 T. More compounds with the similar behaviour are listed i n Refs. [ 38,46,47].\nSystem TC/TN |∆Tad| | ∆Speak\nm| RCP ∆H Ref.\n(K) (K) (J kg−1K−1) (J kg−1) (T)\nLCPP 2.6 9 31 284 7 This work\nHoMnO 3 5 6.5 13.1 320 7 [ 48]\nErMn 2Si2 4.5 12.9 25.2 365 5 [ 16]\nEuTi0.9V0.3O3 4.5 17.4 41.4 577 7 [ 15]\nGdCrTiO 5 0.9 15.5 36 – 7 [ 49]\nEdDy 2O4 5 16 25 415 8 [ 50]\nEuHo 2O4 5 12.7 30 540 8 [ 50]\nMn32 0.32 6.7 18.2 – 7 [ 51]\nHoB2 15 12 40.1 - 5 [ 52]\nEuTiO 3 5.6 21 49 500 7 [ 46]\n/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48/s53/s48/s48/s32/s40/s71/s72/s122/s41\n/s72\n/s114/s101/s115/s32/s40/s84/s41/s32/s72 /s32/s124/s124/s32 /s99\n/s32/s72 /s32 /s32/s99\n/s32/s70/s77/s82/s32/s70/s105/s116/s44/s32 /s72 /s32/s124/s124/s32 /s99\n/s32/s70/s77/s82/s32/s70/s105/s116/s44/s32 /s72 /s32 /s32/s99/s84 /s32/s61/s32/s49/s46/s56/s32/s75\n/s84/s114/s97/s110/s115/s109/s105/s115/s115/s105/s111/s110/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\nFIG. 10. Left vertical scale: Frequency as a function of reso -\nnant field measured for both orientations at T= 1.8K. The\nhollow red and solid black squares represent the measured re s-\nonance fields for orientations H/bardblcandH⊥c, respectively.\nRight vertical scale: Selected HF-ESR spectra.\npendence of the FMR signal is shown in Fig. 10together\nwith few selected spectra (right axis). The spectra mea-\nsured at fields perpendicular to the c-axis are relatively\nbroader and more distorted when compared to the H∝bardblc\nmeasurements. The νvsHresdata were fitted with a\nspin-wave model for FMR, ν=h−1g/bardblµB(Hres−Ha)for\nH∝bardblcorientation and ν=h−1g⊥µB[Hres(Hres+Ha)]1/2\nforH⊥corientation to obtain the g-factors and the\nanisotropy field Ha. It was found that the g-factors in\nboth orientations are somewhat different amounting to\ng⊥= 1.968±0.003andg/bardbl= 1.958±0.003. Such a\nslightly anisotropic g-tensor is typical for a Cr3+ion (3d3,\nS= 3/2,L= 3) in a distorted octahedral ligand coordi-\nnation [ 53]. Further, the anisotropy field Hawas obtained\nas∼2864Oe from the fit. Generally, Haconsists of twocontributions [ 54]:\nHa= 4πM−2K/M. (14)\nThe first term accounts for the shape anisotropy and\nthe second term is the intrinsic magnetocrystalline\nanisotropy. In Eq. ( 14), the first term assumes the shape\nanisotropy of a thin plate with N= 1.\nUsing the saturation magnetization value, Ms=g/bardblS=\n2.94µB/Cr3+, the magnetocrystalline anisotropy con-\nstantK=−7.42×104erg cm−3was obtained. The neg-\native sign of Kimplies that LCPP is an easy-plane fer-\nromagnet with the hard magnetic axis normal to the ab-\nplane. We note that no other resonance excitations could\nbe found in a frequency range up to 975 GHz (4 meV).\n2. Temperature Dependence\nHF-ESR spectra of LCPP at various temperatures\nwere measured at a fixed excitation frequency of ν=\n141.310GHz for both field directions [see Figs. 11(a)\nand (b)]. The resonance fields Hreswere obtained from\nthe absorption minima of each spectrum and are plotted\nagainst temperature in Fig. 12(a). Interestingly, the Hres\nvsTdependence for both orientations does not converge\nto a paramagnetic line immediately at TC= 2.8K. Only\nabove 10 K, the resonance fields for H∝bardblcandH⊥cori-\nentations rapidly start to decrease and increase, respec-\ntively, toward the expected paramagnetic position and\nalmost merge around 100 K at a field corresponding to\ntheg-factor,g= 1.97. The lineshape distortions in mea-\nsurements at low temperatures for H⊥care accounted\nfor in the enlarged error bars.\nThe shift of the resonance position δH(T)from the\nparamagnetic one is shown in Fig. 12(b) (left y-axis).\nδH(T)is positive and larger when the external field is\nparallel to the magnetic hard axis, as compared to the\nsmaller negative shift for the in-plane field geometry, as10\n/s52/s46/s56 /s53/s46/s50 /s53/s46/s54 /s52/s46/s56 /s53/s46/s49 /s53/s46/s52/s40/s98/s41/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s51/s48/s48/s32/s75\n/s50/s53/s48/s32/s75\n/s53/s48/s32/s75/s55/s53/s32/s75/s49/s48/s48/s32/s75/s50/s48/s48/s32/s75\n/s49/s53/s48/s32/s75\n/s52/s32/s75/s55/s32/s75/s49/s48/s32/s75/s50/s48/s32/s75\n/s50/s32/s75/s51/s48/s32/s75\n/s50/s46/s50/s53/s32/s75/s51/s32/s75\n/s50/s46/s55/s53/s32/s75\n/s50/s46/s53/s32/s75/s72 /s32 /s32/s99\n/s32/s32\n/s72 /s32/s40/s84/s41/s49/s46/s56/s32/s75/s32/s61/s32/s49/s52/s49/s46/s51/s49/s48/s32/s71/s72/s122\n/s32/s32\n/s72 /s32/s40/s84/s41/s40/s97/s41\n/s72 /s32 /s32/s99\nFIG. 11. Temperature dependence of the HF-ESR spectra at\nan excitation frequency of 141.310 GHz for (a) H⊥cand (b)\nH/bardblc. Line shape distortions at T <20K in the H⊥cfield\ngeometry are instrumental artifacts arising due to the stro ng\nmagnetization of the sample.\nexpected for an easy-plane ferromagnet. Such a shift can-\nnot be ascribed entirely to the shape anisotropy, which\nshould play a role also in the paramagnetic state if the\nsample’s magnetization Mis large. The M(T)curve\nplotted in Fig. 12(b) (right y-axis) for comparison de-\ncreases right above TCmore rapidly than the δH(T)de-\npendence. Therefore, the line shift observed for both\norientations well above the Curie temperature may be in-\ndicative of short-range FM spin correlations on the fast\nESR time scale, typical for low-dimensional magnets such\nas, e.g., the quasi-two-dimensional van der Waals com-\npound Cr 2Ge2Te6[55]. This is also the case for LCPP\nowing to its layered crystal structure.\nF. Microscopic Analysis\nOur DFT calculations return nearest-neighbor ex-\nchange coupling J/kB≃ −0.5K within the kagome\nplanes. This value is somewhat dependent on the\nchoice of the DFT+ Uparameters, but the negative\nsign is robust and suggests ferromagnetic nature of the\nkagome network in LCPP. The origin of this ferro-\nmagnetic coupling deserves some attention, as the sib-\nling Fe3+compound is clearly antiferromagnetic [ 17].\nSuperexchange theory stipulates that antiferromagnetic\ncouplings are mediated by hoppings between half-filled\norbitals, whereas ferromagnetic couplings arise from hop-\npings between the half-filled and empty orbitals. In/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s32/s40/s98/s41/s32/s72 /s32/s124/s124/s32 /s99\n/s32/s72 /s32 /s32/s99/s72 /s32/s40/s84/s41/s61/s32/s49/s52/s49/s46/s51/s49/s48/s32/s71/s72/s122\n/s84 /s32/s40/s75/s41/s53/s46/s48/s53/s46/s50/s53/s46/s52/s72\n/s114/s101/s115/s32/s40/s84/s41/s103\n/s84 /s32/s61/s32/s51/s48/s48/s32/s75/s32/s61/s32/s49/s46/s57/s55/s40/s97/s41\n/s45/s48/s46/s48/s48/s53/s48/s46/s48/s48/s48/s48/s46/s48/s48/s53/s48/s46/s48/s49/s48/s48/s46/s48/s49/s53\n/s77\n/s72 /s32/s61/s32/s48/s46/s53/s32/s84/s32/s40/s101/s109/s117/s41\nFIG. 12. Temperature dependence of (a) the resonance field\nHresand (b) the resonance shift δH=Hres(T)−Hres(300 K)\n(lefty-axis) and the magnetization Min an applied field of\nH= 0.5T (right y-axis), for both H/bardblcandH⊥c.\nLCPP with the 3d3Cr3+magnetic ion, these orbitals\nhave the t2gandeσ\ngcharacters, respectively. Trigonal\nsymmetry of the crystal structure further splits the t2g\nlevels into a1gandeπ\ng.\n100\n0\n0 1 2a e1g g+π\n( )t2g\negσe eg gπ σ/c45\ne eg gπ π/c45200\n20 40a a1 1g g/c45a e1g g/c45σ\na e1g g/c45π\n60AFMFM\n80 0(b)(a)\nHopping amplitude (meV) Energy (eV)DOS (eV /f.u.)/c451\nFIG. 13. (a) PBE density of states for LCPP with the Fermi\nlevel placed at zero energy. (b) Nearest-neighbor Cr–Cr hop -\nping amplitudes within the kagome plane.\nDFT band structure of LCPP calculated on the PBE\nlevel features narrow t2g(a1g+eπ\ng) bands around the\nFermi level and almost equally narrow eσ\ngbands centered\nat around 2.0 eV (Fig. 13). This band structure is metal-\nlic because neither magnetism nor correlation effects have\nbeen taken into account. The small band width of 0.25 eV\nfor thet2gbands indicates that antiferromagnetic contri-11\n10/c45210/c451100\n10kagome planeexperiment\nkagome plane + interlayer\n100\nT(K)101102\nχ(cm mol Cr )3 3+( )/c451\n13\n069\n10Cp(J mol K )/c45 /c451 1\nT(K)\nFIG. 14. Magnetic susceptibility of LCPP measured in the\napplied field of 0.01 T upon field cooling and its fit using\nthe model of ferromagnetic kagome planes (dashed line) and\ncoupled ferromagnetic kagome planes ( J⊥/J= 0.01, solid\nline). The inset shows calculated specific heat for the cou-\npled kagome planes, with TC≃2.8K.\nbution to the exchange couplings should be minor. The\nt2g−eσ\nghoppings are small too, but somewhat larger than\nthet2g−t2ghoppings, as shown in Fig. 13(b). There-\nfore, ferromagnetic contribution to the exchange becomes\npredominant, and the overall coupling is ferromagnetic.\nMagnetic susceptibility of LCPP is well described by the\nmodel of nearest-neighbor ferromagnetic kagome planes\nwithJ/kB=−1.2K (g= 1.995). A minute interlayer\ncoupling J⊥/J= 0.01improves the fit below 3.5 K and\nleads to the Curie temperature TC= 2.8K in a perfect\nagreement with the experimental TC≃2.6K (Fig. 14).Our DFT calculations corroborate this result and reveal\na weakly ferromagnetic J⊥withJ⊥/J≃0.02.\nIV. SUMMARY\nWe synthesized single crystals of LCPP and confirmed\ntrigonalP¯3c1symmetry of this compound with the lat-\ntice constants a=b= 9.668(3) Å andc= 13.610(6) Å at\nroom temperature. Green-colored LCPP is a rare exam-\nple of an insulating kagome ferromagnet. Ferromagnetic\norder below TC≃2.6K is driven by the in-plane FM\ncoupling J/kB≃ −1.2K supplied with a minute inter-\nplane coupling J⊥/J= 0.02, which is also FM in nature.\nThe incomplete release of the magnetic entropy at TCand\nthe increased width of the ESR line above TCboth sug-\ngest quasi-2D magnetic behavior caused by the strong\nspatial anisotropy of FM couplings. The overall mag-\nnetic behavior of LCPP has striking resemblance with\nthat of other insulating kagome ferromagnets, such as α-\nMgCu 3(OD)6Cl2and Cu[1,3-bdc] [ 8,56]. Magnetization\nand ESR measurements on single crystals indicate a weak\neasy-plane anisotropy. The critical scaling of magnetiza-\ntion suggests a non-mean-field type second-order nature\nof the phase transition at TC. The low TCcombined with\nthe large values of ∆Sm,∆Tad, andRCP render LCPP\na promising magnetocaloric material for low-temperature\napplications.\nACKNOWLEDGMENTS\nAM, SK, VS, and RN would like to acknowledge\nSERB, India for financial support bearing sanction Grant\nNo. CRG/2019/000960.\n[1]P. Mendels and F. Bert, Quantum kagome frustrated\nantiferromagnets: One route to quantum spin liquids,\nC. R. Phys. 17, 455 (2016) ; T.-H. Han, J. S. Helton,\nS. Chu, D. G. Nocera, J. A. Rodriguez-Rivera, C. Bro-\nholm, and Y. S. Lee, Fractionalized excitations in the\nspin-liquid state of a kagome-lattice antiferromagnet,\nNature 492, 406 (2012) ; J. Carrasquilla, Z. Hao, and\nR. G. Melko, A two-dimensional spin liquid in quantum\nkagome ice, Nat. Commun. 6, 7421 (2015) .\n[2]L. Ye, M. Kang, J. Liu, F. von Cube, C. R. Wicker,\nT. Suzuki, C. Jozwiak, A. Bostwick, E. Rotenberg, D. C.\nBell, L. Fu, R. Comin, and J. G. Checkelsky, Mas-\nsive Dirac fermions in a ferromagnetic kagome metal,\nNature 555, 638 (2018) .\n[3]S. Howard, L. Jiao, Z. Wang, N. Morali, R. Batabyal,\nP. Kumar-Nag, N. Avraham, H. Beidenkopf, P. Vir,\nE. Liu, C. Shekhar, C. Felser, T. Hughes, and\nV. Madhavan, Evidence for one-dimensional chiral\nedge states in a magnetic Weyl semimetal Co3Sn2S2,\nNat. Commun. 12, 4269 (2021) .[4]D. Chen, C. Le, C. Fu, H. Lin, W. Schnelle, Y. Sun, and\nC. Felser, Large anomalous Hall effect in the kagome fer-\nromagnet LiMn6Sn6,Phys. Rev. B 103, 144410 (2021) .\n[5]T. Asaba and V. Ivanov and S. M. Thomas and S.\nY. Savrasov and J. D. Thompson and E. D. Bauer\nand F. Ronning , Colossal anomalous nernst effect in\na correlated noncentrosymmetric kagome ferromagnet,\nSci. Adv. 7, eabf1467 (2021) .\n[6]A. Mook, J. Henk, and I. Mertig, Magnon Hall effect and\ntopology in kagome lattices: A theoretical investigation,\nPhys. Rev. B 89, 134409 (2014) .\n[7]R. Chisnell, J. S. Helton, D. E. Freedman, D. K. Singh,\nR. I. Bewley, D. G. Nocera, and Y. S. Lee, Topologi-\ncal Magnon Bands in a Kagome Lattice Ferromagnet,\nPhys. Rev. Lett. 115, 147201 (2015) .\n[8]R. Chisnell, J. S. Helton, D. E. Freedman, D. K. Singh,\nF. Demmel, C. Stock, D. G. Nocera, and Y. S. Lee,\nMagnetic transitions in the topological magnon insula-\ntor Cu(1,3-bdc), Phys. Rev. B 93, 214403 (2016) .\n[9]K. A. GschneidnerJr, V. K. Pecharsky, and A. O.\nTsokol, Recent developments in magnetocaloric materi-12\nals,Rep. Prog. Phys. 68, 1479 (2005) .\n[10]V. K. Pecharsky and K. A. Gschneidner Jr,\nMagnetocaloric effect and magnetic refrigeration,\nJ. Magn. Magn. Mater. 200, 44 (1999) .\n[11]A. Kitanovski, Energy applications of magnetocaloric\nmaterials, Adv. Energy Mater. 10, 1903741 (2020) .\n[12]V. Franco, J. Blázquez, J. Ipus, J. Law, L. Moreno-\nRamírez, and A. Conde, Magnetocaloric effect:\nFrom materials research to refrigeration devices,\nProg. Mater. Sci. 93, 112 (2018) .\n[13]M.-H. Phan and S.-C. Yu, Review of the\nmagnetocaloric effect in manganite materials,\nJ. Magn. Magn. Mater. 308, 325 (2007) .\n[14]J. K. Murthy, K. D. Chandrasekhar, S. Mahana, D. Top-\nwal, and A. Venimadhav, Giant magnetocaloric effect in\nGd2NiMnO 6andGd2CoMnO 6ferromagnetic insulators,\nJ. Phys. D: Appl. Phys. 48, 355001 (2015) .\n[15]S. Roy, M. Das, and P. Mandal, Large low-field magnetic\nrefrigeration in ferromagnetic insulator EuTi0.9V0.1O3,\nPhys. Rev. Mater. 2, 064412 (2018) .\n[16]L. Li, K. Nishimura, W. D. Hutchison, Z. Qian, D. Huo,\nand T. NamiKi, Giant reversible magnetocaloric effect in\nErMn 2Si2compound with a second order magnetic phase\ntransition, Appl. Phys. Lett. 100, 152403 (2012) .\n[17]E. Kermarrec, R. Kumar, G. Bernard, R. Hé-\nnaff, P. Mendels, F. Bert, P. L. Paulose,\nB. K. Hazra, and B. Koteswararao, Classical\nSpin Liquid State in the S=5\n2Heisenberg\nKagome Antiferromagnet Li9Fe3(P2O7)3(PO4)2,\nPhys. Rev. Lett. 127, 157202 (2021) .\n[18]M.-J. Martínez-Pérez, O. Montero, M. Evangelisti,\nF. Luis, J. Sesé, S. Cardona-Serra, and E. Coronado,\nFragmenting gadolinium: Mononuclear polyoxometalate-\nbased magnetic coolers for ultra-low temperatures,\nAdv. Mater. 24, 4301 (2012) .\n[19]S. Poisson, F. d’Yvoire, NGuyen-Huy-Dung, E. Bretey,\nand P. Berthet, Crystal Structure and Cation\nTransport Properties of the Layered Monodiphos-\nphates: Li 9M3(P2O7)3(PO4)2(M=Al, Ga, Cr, Fe),\nJ. Solid State Chem. 138, 32 (1998) .\n[20]G. M. Sheldrick, Siemens area correction absorption cor-\nrection program, University of Göttingen, Göttingen,\nGermany (1994).\n[21]K. Koepernik and H. Eschrig, Full-potential nonorthogo-\nnal local-orbital minimum-basis band-structure scheme,\nPhys. Rev. B 59, 1743 (1999) .\n[22]J. P. Perdew, K. Burke, and M. Ernzerhof, Gen-\neralized Gradient Approximation Made Simple,\nPhys. Rev. Lett. 77, 3865 (1996) .\n[23]O. Janson, G. Nénert, M. Isobe, Y. Skourski,\nY. Ueda, H. Rosner, and A. A. Tsirlin, Magnetic\npyroxenes LiCrGe 2O6and LiCrSi 2O6: Dimensionality\ncrossover in a nonfrustrated S=3\n2Heisenberg model,\nPhys. Rev. B 90, 214424 (2014) .\n[24]K. Somesh, Y. Furukawa, G. Simutis, F. Bert,\nM. Prinz-Zwick, N. Büttgen, A. Zorko, A. A. Tsir-\nlin, P. Mendels, and R. Nath, Universal fluctuat-\ning regime in triangular chromate antiferromagnets,\nPhys. Rev. B 104, 104422 (2021) .\n[25]H. J. Xiang, E. J. Kan, S.-H. Wei, M.-H. Whangbo,\nand X. G. Gong, Predicting the spin-lattice or-\nder of frustrated systems from first principles,\nPhys. Rev. B 84, 224429 (2011) .[26]S. Todo and K. Kato, Cluster algo-\nrithms for general- Squantum spin systems,\nPhys. Rev. Lett. 87, 047203 (2001) .\n[27]A. F. Albuquerque, F. Alet, P. Corboz, P. Dayal,\nA. Feiguin, S. Fuchs, L. Gamper, E. Gull, S. Gürtler,\nA. Honecker, R. Igarashi, M. Körner, A. Kozhevnikov,\nA. Läuchli, S. R. Manmana, M. Matsumoto, I. P. Mc-\nCulloch, F. Michel, R. M. Noack, G. Pawłowski, L. Pol-\nlet, T. Pruschke, U. Schollwöck, S. Todo, S. Trebst,\nM. Troyer, P. Werner, and S. Wessel, The ALPS project\nrelease 1.3: Open-source software for strongly correlated\nsystems, J. Magn. Magn. Mater. 310, 1187 (2007) .\n[28]G. M. Sheldrick, Shelxt–integrated space-group and\ncrystal-structure determination, Acta Crystallogr. A:\nFoundations and Advances 71, 3 (2015).\n[29]G. M. Sheldrick, Shelxl-2018/3 software package, Univer-\nsity of Göttingen, Germany (2018).\n[30]J. Rodríguez-Carvajal, Recent advances in magnetic\nstructure determination by neutron powder diffraction,\nPhysica B 192, 55 (1993) .\n[31]S. J. Sebastian, S. S. Islam, A. Jain, S. M. Yusuf,\nM. Uhlarz, and R. Nath, Collinear order in the\nspin-5\n2triangular-lattice antiferromagnet Na3Fe(PO 4)2,\nPhys. Rev. B 105, 104425 (2022) .\n[32]E. Morosan, L. Li, N. P. Ong, and R. J. Cava, Anisotropic\nproperties of the layered superconductor Cu0.07TiSe2,\nPhys. Rev. B 75, 104505 (2007) .\n[33]R. Prozorov and V. G. Kogan, Effective demagnetiz-\ning factors of diamagnetic samples of various shapes,\nPhys. Rev. Applied 10, 014030 (2018) .\n[34]S. J. Sebastian, K. Somesh, M. Nandi, N. Ahmed, P. Bag,\nM. Baenitz, B. Koo, J. Sichelschmidt, A. A. Tsirlin,\nY. Furukawa, and R. Nath, Quasi-one-dimensional mag-\nnetism in the spin-1\n2antiferromagnet BaNa 2Cu(VO 4)2,\nPhys. Rev. B 103, 064413 (2021) .\n[35]E. Gopal, Specific heats at low temperatures (Springer,\nBoston, MA, 2012).\n[36]A. M. Tishin and Y. I. Spichkin, The magnetocaloric ef-\nfect and its applications , 1st ed. (CRC Press, 2016) p.\n476.\n[37]M. Balli, S. Jandl, P. Fournier, and M. M. Gospodi-\nnov, Anisotropy-enhanced giant reversible rotating\nmagnetocaloric effect in HoMn 2O5single crystals,\nAppl. Phys. Lett. 104, 232402 (2014) .\n[38]Y. Zhu, P. Zhou, T. Li, J. Xia, S. Wu, Y. Fu,\nK. Sun, Q. Zhao, Z. Li, Z. Tang, Y. Xiao, Z. Chen,\nand H.-F. Li, Enhanced magnetocaloric effect and\nmagnetic phase diagrams of single-crystal GdCrO 3,\nPhys. Rev. B 102, 144425 (2020) .\n[39]S. S. Islam, V. Singh, K. Somesh, P. K. Mukhar-\njee, A. Jain, S. M. Yusuf, and R. Nath, Un-\nconventional superparamagnetic behavior in the\nmodified cubic spinel compound LiNi0.5Mn1.5O4,\nPhys. Rev. B 102, 134433 (2020) .\n[40]V. K. Pecharsky and K. A. Gschneidner, Magnetocaloric\neffect from indirect measurements: Magnetization and\nheat capacity, J. Appl. Phys. 86, 565 (1999) .\n[41]V. Franco, J.S Blázquez, B. Ingale, and A. Conde,\nThe magnetocaloric effect and magnetic refrigera-\ntion near room temperature: Materials and models,\nAnnu. Rev. Mater. Res. 42, 305 (2012) .\n[42]R. Nath, V. O. Garlea, A. I. Goldman, and D. C. John-\nston, Synthesis, structure, and properties of tetragonal\nSr2M3As2O2(M3=Mn3, Mn2Cu, and MnZn 2) com-13\npounds containing alternating CuO 2-type and FeAs-type\nlayers, Phys. Rev. B 81, 224513 (2010) .\n[43]B. Banerjee, On a generalised approach to\nfirst and second order magnetic transitions,\nPhys. Lett. 12, 16 (1964) .\n[44]V. Singh, P. Bag, R. Rawat, and R. Nath, Critical behav-\nior and magnetocaloric effect across the magnetic tran-\nsition in Mn1+xFe4−xSi3,Sci. Rep. 10, 6981 (2020) .\n[45]V. Singh and R. Nath, Negative thermal expan-\nsion and itinerant ferromagnetism in Mn 1.4Fe3.6Si3,\nJ. Appl. Phys. 130, 033902 (2021) .\n[46]A. Midya, P. Mandal, K. Rubi, R. Chen, J.-S. Wang,\nR. Mahendiran, G. Lorusso, and M. Evangelisti, Large\nadiabatic temperature and magnetic entropy changes in\nEuTio 3,Phys. Rev. B 93, 094422 (2016) .\n[47]Y.-Z. Zheng, G.-J. Zhou, Z. Zheng, and R. E. P.\nWinpenny, Molecule-based magnetic coolers,\nChem. Soc. Rev. 43, 1462 (2014) .\n[48]A. Midya, P. Mandal, S. Das, S. Banerjee,\nL. S. S. Chandra, V. Ganesan, and S. R. Bar-\nman, Magnetocaloric effect in HoMnO 3crystal,\nAppl. Phys. Lett. 96, 142514 (2010) .\n[49]M. Das, S. Roy, N. Khan, and P. Mandal, Giant mag-\nnetocaloric effect in an exchange-frustrated GdCrTiO 5\nantiferromagnet, Phys. Rev. B 98, 104420 (2018) .[50]A. Midya, N. Khan, D. Bhoi, and P. Man-\ndal, Giant magnetocaloric effect in magnetically\nfrustrated EuHo 2O4and EuDy 2O4compounds,\nAppl. Phys. Lett. 101, 132415 (2012) .\n[51]M. Evangelisti, A. Candini, M. Affronte, E. Pasca, L. J.\nde Jongh, R. T. W. Scott, and E. K. Brechin, Magne-\ntocaloric effect in spin-degenerated molecular nanomag-\nnets,Phys. Rev. B 79, 104414 (2009) .\n[52]P. B. d. Castro, K. Terashima, T. D. Yamamoto,\nZ. Hou, S. Iwasaki, R. Matsumoto, S. Adachi, Y. Saito,\nP. Song, H. Takeya, and Y. Takano, Machine-learning-\nguided discovery of the gigantic magnetocaloric effect\nin HoB2 near the hydrogen liquefaction temperature,\nNPG Asia Materials 12, 35 (2020) .\n[53]A. Abragam and B. Bleaney, Electron paramagnetic res-\nonance of transition ions (Oxford University Press, Ox-\nford, 2012).\n[54]M. Farle, Ferromagnetic resonance of ultrathin metallic\nlayers, Rep. Prog. Phys. 61, 755 (1998) .\n[55]J. Zeisner, A. Alfonsov, S. Selter, S. Aswartham, M. P.\nGhimire, M. Richter, J. van den Brink, B. Büchner, and\nV. Kataev, Magnetic anisotropy and spin-polarized two-\ndimensional electron gas in the van der Waals ferromag-\nnetCr2Ge2Te6,Phys. Rev. B 99, 165109 (2019) .\n[56]D. Boldrin, B. Fåk, M. Enderle, S. Bieri, J. Ollivier,\nS. Rols, P. Manuel, and A. S. Wills, Haydeeite: A spin-1\n2\nkagome ferromagnet, Phys. Rev. B 91, 220408 (2015)"    },    {        "title": "2305.10876v2.Giant_coercivity_induced_by_perpendicular_anisotropy_in_Mn2_42Fe0_58Sn_single_crystals.pdf",        "content": "Giant coercivity induced by perpendicular anisotropy  in \nMn 2.42Fe0.58Sn single crystals \nWeihao Shen1, Yalei Huang1, Xinyu Yao1, Fangyi Qi1, Guixin Cao1, 2, * \n1Materials Genome Institute, Shanghai University, 200444 Shanghai, China \n2Zhejiang Laboratory, Hangzhou 311100, China \n \nAbstract \nWe report the discovery of a giant out-of- plane coercivity in the Fe -doped Mn 3Sn \nsingle crystals. The compound of Mn 2.42Fe0.58Sn exhibits a series of magnetic \ntransitions accompanying with large magnetic anisotropy and electric transport \nproperties. Compared with the ab- plane easy axis in Mn 3Sn, it switche s to the c -axis in \nMn 2.42Fe0.58Sn, producing a sufficiently large uniaxial anisotropy . At 2 K, a giant out -\nof-plane coercivity ( Hc) up to 3 T was observed , which originates from the large \nuniaxial magnetocrystalline anisotropy. The modified Sucksmith -Thompson method \nwas used to determine the values of the second -order and the fourth-order \nmagnetocrystalline anisotropy constant s K1 and K2, resulting in values of 6.0 × 104 J/m3 \nand 4.1 × 105 J/m3 at 2 K, respectively.  Even though the Curie temperature (TC) of 200 \nK for Mn 2.42Fe0.58Sn is not high enough for direct application, our research presents a \nvaluable case study of a typical  uniaxial anisotropy material . \nKeywords:  non-collinear  antiferromagnet ; coercivity; perpendicular magnetic \nanisotropy \n1. Introduction \nMagnetic materials possessing high Hc and perpendicular magnetic anisotropy \n(PMA) demonstrate great potential for a range of applications such as ultrahigh-density \n \n* Corresponding author, Email: guixincao@shu.edu.cn   \n \n perpendicular magnetic recording media, high- performance permanent magnets, and \nspintronic devices  [1–6]. Particularly,  large PMA  is advantageous for achieving low \nenergy consumption and high thermal stability for high- density information storage  [7–\n9]. On the other hand, rare earth permanent magnets are currently the highest -\nperforming permanent magnet materials  as known. However, the reserves of rare earth \nelements are extremely limited, and the costs of extraction and processing are high. \nTherefore, there is an urgent need to find high- performance permanent magnets without \nrare earth elements to replace the widely  used rare earth magnets .   \nIn recent years, Mn 3Sn, with hexagonal structure  (Fig. 1(a)) and a N éel \ntemperature of 420 K , has been widely studied for its excellent properties, such as the \nlarge anomalous Hall effect (AHE)  [10], topological Hall effect (THE)  [11], anomalous  \nNernst effect  [12], and exchange bias effect  [13]. The Mn atoms form a 2D Kagome \nlattice on the ab plane with Sn atoms sitting at the hexagon centers (Fig. 1( b)), and t wo \nneighboring Kagome Mn 3Sn layers are stacked along the c  axis (Fig. 1(a))  [14]. Mn 3Sn \nis a room temperature non -collinear antiferromagnet , while Fe3Sn, a compound with \nthe same crystalline structure as that of Mn 3Sn, is a ferromagnetic metal with small \ncoercivity  and an easy axis of magnetization in the ab plane that can be shifted to the \nc-axis with doping [15,16] . It was reported that substituted Fe atoms at Mn sites of  \nMn 3Sn disrupt potentially the symmetry of the Kagome lattice  [17], consequently \naffecting the nearest -neighbor Heisenberg exchange, and anisotropic energy et al . \nTherefore,  substitut ing Fe atoms at Mn sites of Mn 3Sn to enhance the magnetic \nproperties of Mn 3Sn is greatly expected as Fe substitution introduces ferromagnetism \nto the system . \nIn this letter , we investigated the magnetic phase transition s of Mn 2.42Fe0.58Sn \nsingle crystals  and observed a giant  out-of-plane Hc up to 3 T at 2 K. We find that doping \nFe into Mn 3Sn causes the easy magnetization axis of Mn 3Sn to switch from the ab-\nplane to the c -axis, producing a sufficiently large uniaxial anisotropy in Mn 2.42Fe0.58Sn. \nWe use the modified Sucksmith -Thompson method which takes into account \nmagnetization anisotropy and high- field susceptibilities to obtain magnetocrystalline \nanisotropy constants K 1 and K2, resulting in values of 6.0 × 104 J/m3 and 4.1 × 105 J/m3 at 2 K , respectively.  Both Hc and K2 increase with decreasing temperature (T), while K 1 \nis non- monotonically independent of T . Our result presents a typical uniaxial anisotropy \nsystem  with potential to obtain the large magnetocrystalline anisotropy through Fe -\ndoping. \n2. Experimental details  \nMn 2.42Fe0.58Sn single crystals were synthesized by the Sn- flux method with a molar \nratio of Mn:  Fe :Sn =7: 1.5:3 [18,19] . The obtained single crystals were shiny and \nhexagonal -rod-shape with typical dimensions of 2× 1×1mm3. Elemental compositions \nof the crystals were calculated to be Mn 2.42Fe0.58Sn by using an energy- dispersive x- ray \nspectroscopy (EDS , HGST FlexSEM -1000)  technique . The crystal structural  \ncharacterization was performed by x -ray diffraction (XRD)  using a Bruker D2 x- ray \ninstrument  with Cu K α radiation ( λ = 1.541 Å). Magnetization measurements  were \nperformed in a SQUID magnetometer (MPMS, Quantum Design). The x, y, and z axes \nthat we define correspond to the [ 21�1�0], [011�0], and [ 0001] directions, respectively , \nas shown  in the inset of Fig.2(d) . The longitudinal resistivity was measured by a \nstandard four -probe method using a physical property measurement system ( PPMS- 14, \nQuantum Design).   \n3. Results and discussion \nThe powder XRD pattern at  room temperature of the Mn 2.42Fe0.58Sn (Fig. 1( c)) \nshows that a ll the diffraction peaks of the sample can be indexed as the D0 19 type \nhexagonal structure with space group P6 3/mmc. The peak s of Mn 2.42Fe0.58Sn shift s to \nright compared to the reflection peaks of Mn 3Sn due to the smaller atomic radius of the \nFe substituting for Mn. Laue pattern of a polished (0 0 0 1) plane  displays  good \nsymmetry and sharpness of the diffraction spots  as shown Fig.1(d) , which prove s the \ngood orientation and high quality of our single crystals . \nFig. 2(a) show s the temperature dependence of the magnetization M(T) of \nMn 2.42Fe0.58Sn measured by applying a magnetic field of 𝜇𝜇0𝐻𝐻= 0.02 T  perpendicular \n(H⊥z) and parallel ( H∥z) to the z -axis, respectively . It can be seen that both the  M(T) curves for H ⊥z and H∥z under low field of  𝜇𝜇0𝐻𝐻= 0.02 T  displays  a bifurcation at \nalmost the same temperature at T p for ZFC and FC modes. Above  Tp, there appear two \ntransitions of T N ~ 297 K  (antiferromagnetic ) and TC ~ 200 K (ferromagnetic- like) for \nboth H⊥z and H∥z, respectively . However, when T  < TC, two additional transitions T p \n~ 187 K and T t ~163 K  emerge in the M(T) curve for H ∥z, with their  absence along  H\n⊥z. The Tt ~163 K  appeared only along H ∥z is regarded  as the rotation of magnetic \nmoments (spin reorientation) due to the enhancement of magnetocrystalline anisotropy \nwith decreasing temperature [20,21] .  \n This obviously different magnetic properti es for  H⊥z and H∥z indicate the large  \nmagnetic anisotropy in Mn 2.42Fe0.58Sn. The large anisotropy is also reflected in the \nM(T) curves with applied various magnetic field as displayed in Fig.2(b) and (c),  \nrespectively . For H⊥z, the M(T) under 0.02 T shows typical ferromagnetic (FC) \ncharacter with a clear FC  transition at T C and complete reproduces  for both ZFC and \nFC under 1T field. In comparison, the applied 1 T field for the M (T) curves along H ∥\nz causes a clear shift  of Tp to low temperature  (low -T) from ~187 K  of 𝜇𝜇0𝐻𝐻= 0.02 T to \n~50 K of 𝜇𝜇0𝐻𝐻= 1 T, with a antiferromagnetic  (AFM)  character . The ZFC and FC M (T) \ncurves were completely reproduced by applied 5 T field with a FM character as \ndisplayed in Fig.2(b). This phenomenon implies that the low -T phase of Mn 2.42Fe0.58Sn \nis not a completely single magnetic phase in nature  [22], it can be concluded that the \nground state of the sample is the coexistence of FM and AFM phases, consistent with \nprevious reports  [20,23] .  \nConsistent with the strong magnetic anisotropy in the Mn 2.42Fe0.58Sn, the large \nanisotropy is also reflected in the electrical transports as shown in Fig.2(d), which \ndepict zero -field in -plane  (ρxx) and out-of-plane resistivity ( ρzz) between 2 and 300 K , \nrespectively.  It can be seen that  ρxx increases with decreasing temperature, displaying a  \ntypical semiconducting behavior. However, ρzz exhibits a standard insulator -metal \ntransition at ~200 K, corresponding to the FM transition T C in the M (T) curves. Upon \nfurther cooling from 200 K, there emerges  a second transition with a typical upturn at \naround 40 K. These differences between ρ xx and ρzz in Mn 2.42Fe0.58Sn are apparently different from the metallic characteristics and isotropic electrical resistivity of Mn 3Sn \n[24]. This further displays the strong electrical anisotropy due to the Fe doping.  \nTo further understand the large a nisotropy in both the magnetic and transport \nproperties , we measured the  magnetization isotherms M (H) at various temperatures for \nboth H⊥z (Fig. 3(a)) and H∥z (Fig. 3( b)). Interestingly, corresponding to the magnetic \ntransitions TN and TC etc. observed in the M (T) curves, the M (H) curves also display \nvarying features  with temperatures . When the temperature is higher than TN, i.e., 300 \nK, a linear behavior of M  with H for both H ⊥z and H∥z was observed (see F ig. 3(a), \n3(b)), indicating the paramagnetic phase.  As the temperature is low er than T N but \nhigher than 200 K, i.e., 250 K, a significant hysteresis  was observed  for H⊥z as \nshown in inset of Fig. 3(a), suggest ing the weak ferromagnetic  property . The \nobservation of weak ferromagnetic is arising from  the uncompensated moment in \ntriangular antiferromagnetic structure , consistent with that in Mn 2.5Fe0.5Sn [25]. Below \nTC = 200 K, the hysteresis loops exhibit typical ferromagnetic characteristics, \nsuggesting that there is a magnetic transition from the antiferromagnetic to \nferromagnetic state.  In particular , an obvious hysteresis has  been found in the \nmagnetization curve at 2  K. The coercivity  increase from 0 T at 200 K to 3 T at 2 K, \nindicating the strong magnetocrystalline anisotropy at low temperature s as shown in \nFig. 3( b), consistent with the analysis of M(T) curves above . Meantime, it is noteworthy \nthat the M (H) curve s below 50 K appears the “collapse”  phenomenon at the small field \nregion,  which is related to the weak exchange -coupling between two magnetic phases  \n[26,27] . This evidence favors the coexistence of FM and AFM at low temperatures, in \naccord with the analysis in M (T) curves above. In addition, the large H c and a non-\nsaturation magnetization  until 7 T at 2 K also confirm the co existence of FM and AFM \nphases . To make a clear comparison, M (H) curves of Mn 2.42Fe0.58Sn for both 300 K and \n2 K under  H⊥z and H∥z are displayed in Figs . 3(c) and 3(d) , respectively. It can be \nseen that the easy axis of Mn 2.42Fe0.58Sn shifts from in -plane at 300 K to out -of-plane \nat 2 K, along the crystallographic  c-axis, which facilitates the usage in perpendicular \nmagnetic recording applications. At 2 K, t he out-of-plane Hc is up to 3 T, but the in-\nplane M(H) curve exhibits almost anhysteretic loop, zero remnant magnetization and high saturation field exceeding 7 T, which  further  realizes a large perpendicular \nmagnetic anisotropy . \nTo further investigate the large  out-of-plane Hc and perpendicular magnetic \nanisotropy , we plotted H c as a function of temperature for H∥z orientation in Fig. 4(a) . \nIt can be seen that Hc increases with decreasing temperature , and the m aximum value  \nreaches 3 T at 2 K . Considering the origin of the H c, we know that  the formation of \nhysteresis during the demagnetization process is related to the irreversibility of M  \ncaused by stress fluctuations, impurities, and general magnetic anisotropy present in \nferromagnetic materials  [28]. The demagnetization process is caused by both the \ndisplacement of magnetic domain walls  and domain wall rotation  [29]. The larger \nhysteresis means that the domain wall displacement and/or rotation are prevented \nduring the demagnetization process. To make this clear, we have fitted the Hc (T) \naccording to both weak domain wall pinning model 𝐻𝐻𝑐𝑐∝𝑇𝑇 and strong domain wall \npinning mechanism 𝐻𝐻𝑐𝑐1/2∝𝑇𝑇2/3, respectively  [30]. We found that our results don’t \nagree with both models , this indicates that basically our large H c is not through the \ndomain wall displacement . Whereas, the H c(T) curve agree well with log10𝐻𝐻𝑐𝑐∝𝑇𝑇 as \nshown in the inset of Fig.4(a). This display that the giant H c observed in our \nMn 2.42Fe0.58Sn single crystals is mainly related to the large magnetocrystalline \nanisotropy as it shown in Mn 2LiReO 6 compound [31]. In this case, the demagnetization \nprocess in our compound should be mainly  achieved through the rotation of the \nmagnetization vector. Therefore, we can conclude that the magnetic crystalline \nanisotropy is the main cause of the large coercivity  in our compound.  \nFurther, we estimate the magnetocrystalline anisotropy constants by using the \nmodified Sucksmith -Thompson (ST) m ethod. The anisotropy energy for hexagonal \ncrystals is represented by the phenomenological expression:  \n𝐸𝐸𝑎𝑎=𝐾𝐾1𝑆𝑆𝑆𝑆𝑆𝑆2𝜃𝜃+𝐾𝐾2𝑆𝑆𝑆𝑆𝑆𝑆4𝜃𝜃+𝐾𝐾3𝑆𝑆𝑆𝑆𝑆𝑆6𝜃𝜃… (1) \nwhere K1 and K2 are magnetocrystalline anisotropy constants and θ  is an angle between \nthe magnetization M and c  axis.  The values of K 1 and K2 generally can be  evaluated by \napplying the Sucksmith -Thompson (ST)  method [32]. However, our magnetization measurements revealed different magnetization values above the anisotropy field  as \nshown in Fig.3(d) . This phenomenon is called magnetization anisotropy and is \ndetermined by the parameter 𝑝𝑝=𝑀𝑀𝑆𝑆−𝐸𝐸𝐸𝐸−𝑀𝑀𝑆𝑆−𝐻𝐻𝐸𝐸\n𝑀𝑀𝐸𝐸𝐸𝐸, where MS-EA and MS-EH are values of \nthe saturation magnetization for field applied along the easy and hard magnetization \ndirections, respectively [33] . p affects values of magnetocrystalline anisotropy \nconstants calculated using the ST method. Therefore, we use the modified ST method \n[34,35]  which takes into account magnetization anisotropy and high- field \nsusceptibilities to obtain magnetocrystalline anisotropy constants K 1 and K2. According \nto this method,  the dependence of ℎ(1−4𝜇𝜇𝑝𝑝∗) on 𝜇𝜇 should be plotted instead of the \nclassical one 𝜇𝜇0𝐻𝐻𝑖𝑖/𝑀𝑀(𝑀𝑀2), where ℎ=𝜇𝜇0𝐻𝐻𝑖𝑖(𝑀𝑀𝑆𝑆−𝐸𝐸𝐸𝐸+𝜒𝜒𝐸𝐸𝐸𝐸𝜇𝜇0𝐻𝐻𝑖𝑖)2/2𝑀𝑀𝑆𝑆−𝐸𝐸𝐸𝐸,𝜇𝜇=[𝑀𝑀/\n(𝑀𝑀𝑆𝑆−𝐸𝐸𝐸𝐸+𝜒𝜒𝐸𝐸𝐸𝐸𝜇𝜇0𝐻𝐻𝑖𝑖)]2  and 𝑝𝑝∗=[𝑝𝑝𝑀𝑀𝑆𝑆−𝐸𝐸𝐸𝐸−(𝜒𝜒𝐻𝐻𝐸𝐸−𝜒𝜒𝐸𝐸𝐸𝐸)𝜇𝜇0𝐻𝐻𝑖𝑖]/[𝑀𝑀𝑆𝑆−𝐸𝐸𝐸𝐸+𝜒𝜒𝐸𝐸𝐸𝐸𝜇𝜇0𝐻𝐻𝑖𝑖] . \nWhere u 0Hi is the internal magnetic field defined as  𝜇𝜇0𝐻𝐻𝑖𝑖=𝜇𝜇0𝐻𝐻−4𝜋𝜋𝜋𝜋𝑀𝑀𝑣𝑣. Here, M v \nis the magnetization per unit volume and the multiplier 4 π is necessary for cgs units. \nSince the geometric shape of the measured samples is processed as a rectangle, the \ndemagnetization factors are estimated to be N x = 0.85( H⊥z) and N z = 0.8( H∥z). The \ncase of easy axis magnetic anisotropy satisfies the equation:  \n𝐾𝐾1+2𝐾𝐾2𝜇𝜇=ℎ(1−4𝜇𝜇𝑝𝑝∗)    (2) \nThe M(H) curve measured along the hard axis demonstrates nonlinear behavior in \nFig.3(a) , which indicates the significant role of the anisotropy constant K 2 in the \nmagnetization processes [36] . We obtained the saturation magnetization, M S-EA, MS-HA, \nfor these two axes by extrapolating the magnetization from high fields to zero field \nusing a linear function. The slope of this line corresponds to the high- field susceptibility, \nχEA, χHA. Field dependence of magnetization after high field susceptibility subtraction \ncorrection for Mn 2.42Fe0.58Sn single crystal at 150 K is presented in the inset of Fig. 4(b). \nThe anisotropy of magnetization was clearly observed as  MS-EA = 412 emu/cm3 and MS-\nHA = 291 emu/cm3 at 150 K. Using magnetization curves measured between 2 K and \n150 K and taking into account the effect of the demagnetizing field, we obtained \ntemperature dependenc e of the saturation magnetization M s (Fig. 4(b)), high- field \nsusceptibility χ (Fig. 4(c)) along EA and HA for Mn 2.42Fe0.58Sn, respectively . The solid lines in Fig. 4(b) is a fit of experimental M s(T) data for both H //Z and H ⊥z using the \nexpression [37] 𝑀𝑀𝑠𝑠(𝑇𝑇) =𝑀𝑀𝑠𝑠(0)[1−𝑠𝑠�𝑇𝑇\n𝑇𝑇𝐶𝐶�3\n2−(1−𝑠𝑠)�𝑇𝑇\n𝑇𝑇𝐶𝐶�52\n]1/3 . Here, Ms (0) is a \nsaturation magnetization at 0 K, T C is a Curie temperature and s  is a shape parameter, \nwhich relates to spin -wave  stiffness. According to the fitting, TC of the Mn 2.42Fe0.58Sn \nsingle crystal equals to  200 K independently of crystallographic direction, which is \nconsistent with that from M (T) data. The  other parameters  are listed in Table  1.  \nThe high- field susceptibility in both direction increases with increasing \ntemperature and the ratio  χHA > χEA is typical in Fig. 4(c) , which is associated with \nthermal fluctuations [34,35] . Using M s(T) values, we obtained  the p versus  T as \ndisplayed in the inset of Fig. 4(c). It can be seen that with increasing temperature, p( T) \nincreases monotonically. Based on above, the anisotropy constants, K 1 and K2 for \nMn 2.42Fe0.58Sn single crystal were determined by using the modified ST method. Thus, \na linear approximation of ℎ(1−4𝜇𝜇𝑝𝑝∗) versus 𝜇𝜇 provides K 1 and K 2 values at 150 K \nin the inset of Fig. 4(d). In this way, the K 1 and K2 at various temperatures are obtained \nand presented in Fig. 4(d). W ithin 2-150 K , both K1 and K2 are positive, which is \nconsistent with the fact that Mn 2.42Fe0.58Sn has an easy magnetization axis in this \ntemperature region. The temperature dependence of K 1 is non- monotonic, with a \nminimum at 2 K and a maximum of 0.9× 105 J/m3 at 100 K. It turns out that K 2 is an \norder of magnitude larger than K 1. K2 increases monotonically with decreasing \ntemperature, with similar trend as the temperature dependence of H c. Therefore, both \nHc and K2 reaches the maximum value at 2 K with H c = 3 T and K2 = 4.1× 105 J/m3. Our \nresults indicates that the uniaxial anisotropy of Mn 2.42Fe0.58Sn is compar able with the \nreported potential rare -earth  free permanent magnet materials , such as MnAl (1.67× 106 \nJ/m3) [38], MnBi (2.23× 106 J/m3) [39], CoPt (1.73× 106 J/m3) [40] and Fe 2P (2.32× 106 \nJ/m3) [41]. \n \n4. Conclusions  \nIn summary , we found a giant out -of-plane coercivity (3 T) in  high quality \nMn 2.42Fe0.58Sn single crystals  with a typical easy axis along z  direction due to the Fe doping. This unusually large coercivity originates from the perpendicular \nmagnetocrystalline anisotropy. The modified Sucksmith -Thompson method, which \ntakes into account magnetization anisotropy and high- field susceptibilities, has been \nused to obtain the magnetocrystalline anisotropy constants K 1 and K2. At 2 K, K1 and \nK2 result in values of 6.0 × 104 J/m3 and 4.1 × 105 J/m3, respectively. We found that \nboth the values of H c and K2 increase with decreasing temperature T , while K1 is non-\nmonotonically independent of T . Though the T C of 200 K for Mn 2.42Fe0.58Sn is not high \nenough for direct application, our study provides a model to obtain the perpendicular \nmagnetocrystalline anisotropy through Fe -doping in the Mn 3Sn compound.  \nACKNOWLEDGMENT  \nThis work was supported by Key Research Project of Zhejiang Lab (No. \n2021PE0AC02) , Department of Science and Technology of Zhejiang Province \n(2023C01182)  and Shanghai Engineering Research Center for Integrated Circuits and \nAdvanced Display Materials . A portion of this work was performed on the Steady High \nMagnetic Field Facilities, High Magnetic Field Laboratory, CAS.   \nFigure Captions:  \nFIG. 1. (a) Crystal structure of Mn 3Sn. (b) Triangular spin structure in the Kagome \nlayer ( ab plane). (c) Powder XRD data taken from the crushed single crystals of \nMn 2.42Fe0.58Sn. Inset shows photographic image of as -grown Mn 2.42Fe0.58Sn single \ncrystal. (d) Laue pattern of a polished surface  of Mn 2.42Fe0.58Sn single crystal  showing \nthe (0 0 0 1) plane.  \nFIG. 2. (a) M(T) curves under the field of 200 Oe  applied both in parallel and \nperpendicular to the z -axis for field -cooled (FC) and Zero- field-cooled (ZFC) modes , \nrespectively . (b) and (c) show M (T) curves  measured in the FC and ZFC modes under \nvarious  field in parallel and perpendicular to the z -axis, respectively.  The data in ( 𝜇𝜇0𝐻𝐻 \n= 0.02 T) is shifted along the y axis for better clarity. (d) ρ xx (T) (left) and  ρzz(T)  (right) \ncurves for Mn 2.42Fe0.58Sn.  \nFIG. 3. Magnet ization isotherms, M (H) of Mn 2.42Fe0.58Sn for H⊥z (a) and  H∥z (b) at various temperatures ; Inset shows enlarged curves of central region of the hysteresis \nloops for 250 K . (c) and (d) show M (H) isotherms of Mn 2.42Fe0.58Sn for H⊥z and H∥\nz at 300 K and 2 K, respectively. \nFIG. 4. (a) Coercivity as a function of temperature for H ∥z. Inset shows the fitted \nlogarithmic dependence of H c with temperature. (b)Temperature dependences of M s; \nInset shows the field dependence of magnetization after high field susceptibility \nsubtraction correction for Mn 2.42Fe0.58Sn single crystal at 150 K.  (c)Temperature \ndependences of the high -field susceptibility, χ; Inset shows the temperature dependence \nof magnetization anisotropy p.  (d) Temperature dependencies of magnetocrystalline \nanisotropy constant K 1 and K2 of the Mn 2.42Fe0.58Sn single crystal obtained by the \nmodified Sucksmith -Thompson method; Inset shows the  application of the method. \nTable 1  \nThe Curie temperature T C, saturation magnetizations M s at 0 K and shape parameters s  \nalong hard magnetization axis (HA) and easy magnetization axis (EA) of the Mn\n2.42Fe0.58Sn single crystal.  \nDirection  TC (K) Ms (0) (emu/cm3) s \nHA 200 463 1.53 \nEA 200 573 0.78 \n           REFERENCES  \n[1] A.T. McCallum, P. Krone, F. Springer, C. Brombacher, M. Albrecht, E. Dobisz, M. Grobis, D. \nWeller, O. Hellwig, L1 0 FePt based exchange coupled composite bit patterned films, Appl. \nPhys. Lett. 98 (2011) 242503. https://doi.org/10.1063/1.3599573.  \n[2] I.M. Miron, K. Garello, G. Gaudin, P.- J. Zermatten, M.V . Costache, S. Auffret, S. Bandiera, \nB. Rodmacq, A. Schuhl, P. Gambardella, Perpendicular switching of a single ferromagnetic \nlayer induced by in- plane current injection, Nature. 476 (2011) 189–193. \nhttps://doi.org/10.1038/nature10309.  \n[3] L. Zhu, S. Nie, K. Meng, D. Pan, J. Zhao, H. Zheng, Multifunctional L10-Mn 1.5Ga Films with \nUltrahigh Coercivity, Giant Perpendicular Magnetocrystalline Anisotropy and Large Magnetic Energy Product, Adv. Mater. 24 (2012) 4547–4551. https://doi.org/10.1002/adma.201200805.  \n[4] C.L. Zha, R.K. Dumas, J.W. Lau, S.M. Mohseni, S.R. Sani, I.V . Golosovsky, Å.F. Monsen, J. \nNogués, J. Åkerman, Nanostructured MnGa films on Si/SiO\n2 with 20.5 kOe room temperature \ncoercivity, J. Appl. Phys. 110 (2011) 093902. https://doi.org/10.1063/1.3656457.  \n[5] T.J. Nummy, S.P. Bennett, T. Cardinal, D. Heiman, Large coercivity in nanostructured rare -\nearth -free Mn xGa films, Appl. Phys. Lett. 99 (2011) 252506. \nhttps://doi.org/10.1063/1.3671329.  \n[6] O. Gutfleisch, M.A. Willard, E. Brück, C.H. Chen, S.G. Sankar, J.P. Liu, Magnetic Materials \nand Devices for the 21st Century: Stronger, Lighter, and More Energy Efficient, Adv. Mater. 23 (2011) 821 –842. https://doi.org/10.1002/adma.201002180.  \n[7] S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H.D. Gan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura, H. Ohno, A perpendicular -anisotropy CoFeB –MgO magnetic tunnel junction, \nNat. Mater. 9 (2010) 721–724. https://doi.org/10.1038/nmat2804.  \n[8] S. Mangin, D. Ravelosona, J.A. Katine, M.J. Carey, B.D. Terris, E.E. Fullerton, Current -\ninduced magnetization reversal in nanopillars with perpendicular anisotropy, Nat. Mater. 5 (2006) 210–215. https://doi.org/10.1038/nmat1595.  \n[9] K.C. Chun, H. Zhao, J.D. Harms, T.- H. Kim, J. -P. Wang, C.H. Kim, A Scaling Roadmap and \nPerformance Evaluation of In- Plane and Perpendicular MTJ Based STT -MRAMs for High -\nDensity Cache Memory, IEEE J. Solid -State Circuits. 48 (2013) 598–610. \nhttps://doi. org/10.1109/JSSC.2012.2224256.  \n[10] S. Nakatsuji, N. Kiyohara, T. Higo, Large anomalous Hall effect in a non- collinear \nantiferromagnet at room temperature, Nature. 527 (2015) 212–215. https://doi.org/10.1038/nature15723.  \n[11] J.M. Taylor, A. Markou, E. Lesne, P.K. Sivakumar, C. Luo, F. Radu, P. Werner, C. Felser, S.S.P. \nParkin, Anomalous and topological Hall effects in epitaxial thin films of the noncollinear antiferromagnet Mn\n3Sn, Phys. Rev. B. 101 (2020) 094404. \nhttps://doi.org/10.1103/PhysRevB.101.094404.  \n[12] X. Li, L. Xu, L. Ding, J. Wang, M. Shen, X. Lu, Z. Zhu, K. Behnia, Anomalous Nernst and Righi -Leduc Effects in Mn\n3Sn: Berry Curvature and Entropy Flow, Phys. Rev. Lett. 119 (2017) \n056601. https://doi.org/10.1103/PhysRevLett.119.056601.  \n[13] X.F. Zhou, X.Z. Chen, Y .F. You, L.Y . Liao, H. Bai, R.Q. Zhang, Y .J. Zhou, H.Q. Wu, C. Song, F. Pan, Exchange Bias in Antiferromagnetic Mn\n3Sn Monolayer Films, Phys. Rev. Appl. 14 \n(2020) 054037. https://doi.org/10.1103/PhysRevApplied.14.054037.  [14] D. Khadka, T.R. Thapaliya, S.H. Parra, X. Han, J. Wen, R.F. Need, P. Khanal, W. Wang, J. \nZang, J.M. Kikkawa, L. Wu, S.X. Huang, Kondo physics in antiferromagnetic Weyl \nsemimetal Mn 3+xSn1-x films, Sci. Adv. 6 (2020) eabc1977. \nhttps://doi.org/10.1126/sciadv.abc1977.  \n[15] B.C. Sales, B. Saparov, M.A. McGuire, D.J. Singh, D.S. Parker, Ferromagnetism of Fe 3Sn \nand Alloys, Sci. Rep. 4 (2014) 7024. https://doi.org/10.1038/srep07024.  \n[16] O.Yu. Vekilova, B. Fayyazi, K.P. Skokov, O. Gutfleisch, C. Echevarria -Bonet, J.M. \nBarandiarán, A. Kovacs, J. Fischbacher, T. Schrefl, O. Eriksson, H.C. Herper, Tuning the \nmagnetocrystalline anisotropy of Fe 3Sn by alloying, Phys. Rev. B. 99 (2019) 024421. \nhttps://doi.org/10.1103/PhysRevB.99.024421.  \n[17] Z.H. Liu, Q.Q. Zhang, Y .J. Zhang, H.G. Zhang, X.Q. Ma, E.K. Liu, Evolution of diverse Hall effects during the successive magnetic phase transitions in Mn\n2.5Fe0.6Sn0.9 Kagome -lattice \nalloy, J. Phys. Condens. Matter. 33 (2020) 115803. https://doi.org/10.1088/1361-648X/abd337.  \n[18] N.H. Sung, F. Ronning, J.D. Thompson, E.D. Bauer, Magnetic phase dependence of the anomalous Hall effect in Mn\n3Sn single crystals, Appl. Phys. Lett. 112 (2018) 132406. \nhttps://doi.org/10.1063/1.5021133.  \n[19] J. Yan, X. Luo, H.Y . Lv, Y . Sun, P. Tong, W.J. Lu, X.B. Zhu, W.H. Song, Y .P. Sun, Room -\ntemperature angular -dependent topological Hall effect in chiral antiferromagnetic Weyl \nsemimetal Mn 3Sn, Appl. Phys. Lett. 115 (2019) 102404. https://doi.org/10.1063/1.5119838.  \n[20] J. Liu, S. Zuo, H. Li, Y . Liu, X. Zheng, Y . Zhang, T. Zhao, F. Hu, J. Sun, B. Shen, Spontaneous \nmagnetic bubbles and large topological Hall effect in Mn 3-xFexSn compound, Scr. Mater. 187 \n(2020) 268–273. https://doi.org/10.1016/j.scriptamat.2020.06.034.  \n[21] A. Low, S. Ghosh, S. Changdar, S. Routh, S. Purwar, S. Thirupathaiah, Tuning of topological \nproperties in the strongly correlated antiferromagnet Mn 3Sn via Fe doping, Phys. Rev. B. 106 \n(2022) 144429. https://doi.org/10.1103/PhysRevB.106.144429.  \n[22] Q. Zhang, Z. Liu, X. Ma, Magnetic transition and the associated exchange bias, transport properties in Mn\n2.1FeSn 0.9 alloy, J. Phys. Appl. Phys. 54 (2021) 505001. \nhttps://doi.org/10.1088/1361 -6463/ac2641.  \n[23] T. Hori, H. Niida, Y . Yamaguchi, H. Kato, Y . Nakagawa, Antiferromagnetic to ferromagnetic transition of DO\n19 type (Mn 1-xFex)3Sn1-δ, J. Magn. Magn. Mater. 90 –91 (1990) 159–160. \nhttps://doi.org/10.1016/S0304- 8853(10)80053- 8. \n[24] S. Nakatsuji, N. Kiyohara, T. Higo, Large anomalous Hall effect in a non- collinear \nantiferromagnet at room temperature, Nature. 527 (2015) 212–215. https://doi.org/10.1038/nature15723.  \n[25] J.S. Kouvel, Exchange Anisotropy and Long‐Range Magnetic Order in the Mixed Intermetallic Compounds, (Mn, Fe)\n3Sn, J. Appl. Phys. 36 (1965) 980–981. \nhttps://doi.org/10.1063/1.1714287.  \n[26] D. Goll, M. Seeger, H. Kronmüller, Magnetic and microstructural properties of nanocrystalline exchange coupled PrFeB permanent magnets, J. Magn. Magn. Mater. 185 \n(1998) 49 –60. https://doi.org/10.1016/S0304- 8853(98)00030- 4. \n[27] J.P. Liu, R. Skomski, Y . Liu, D.J. Sellmyer, Temperature dependence of magnetic hysteresis \nof RCo\nx:Co nanocomposites (R=Pr and Sm), J. Appl. Phys. 87 (2000) 6740 –6742. \nhttps://doi.org/10.1063/1.372826.  [28] J.D. Livingston, A review of coercivity mechanisms (invited), J. Appl. Phys. 52 (1981) 2544–\n2548. https://doi.org/10.1063/1.328996.  \n[29] E.A. Gorbachev, E.S. Kozlyakova, L.A. Trusov, A.E. Sleptsova, M.A. Zykin, P.E. Kazin, Design of modern magnetic materials with giant coercivity, Russ. Chem. Rev. 90 (2021) 1287. https://doi.org/10.1070/RCR4989.  \n[30] P. Gaunt, Magnetic coercivity, Can. J. Phys. 65 (1987) 1194 –1199. \nhttps://doi.org/10.1139/p87- 195. \n[31] E. Solana -Madruga, C. Ritter, O. Mentré, J. Paul  Attfield, Á. M.  Arévalo -López, Giant \ncoercivity and spin clusters in high pressure polymorphs of Mn\n2LiReO 6, J. Mater. Chem. C. \n10 (2022) 4336 –4341. https://doi.org/10.1039/D2TC00451H.  \n[32] W. Sucksmith, J.E. Thompson, The magnetic anisotropy of cobalt, Proc. R. Soc. Lond. Ser. Math. Phys. Sci. 225 (1997) 362–375. https://doi.org/10.1098/rspa.1954.0209.  \n[33] A.S. Bolyachkin, D.S. Neznakhin, M.I. Bartashevich, The effect of magnetization anisotropy and paramagnetic susceptibility on the magnetization process, J. Appl. Phys. 118 (2015) 213902. https://doi.org/10.1063/1.4936604.  \n[34] A.S. Bolyachkin, D.S. Neznakhin, T.V . Garaeva, A.V . Andreev, M.I. Bartashevich, Magnetic \nanisotropy of YFe\n3 compound, J. Magn. Magn. Mater. 426 (2017) 740–743. \nhttps://doi.org/10.1016/j.jmmm.2016.10.133.  \n[35] D.S. Neznakhin, A.M. Bartashevich, A.S. V olegov, M.I. Bartashevich, A.V . Andreev, \nMagnetic anisotropy in RCo 3 (R = Lu and Y) single crystals, J. Magn. Magn. Mater. 539 (2021) \n168367. https://doi.org/10.1016/j.jmmm.2021.168367.  \n[36] D.Yu. Karpenkov, K.P. Skokov, M.B. Lyakhova, I.A. Radulov, T. Faske, Y . Skourski, O. Gutfleisch, Intrinsic magnetic properties of hydrided and non- hydrided Nd\n5Fe17 single \ncrystals, J. Alloys Compd. 741 (2018) 1012–1020. https://doi.org/10.1016/j.jallcom.2018.01.239.  \n[37] M.D. Kuz’min, Shape of Temperature Dependence of Spontaneous Magnetization of \nFerromagnets: Quantitative Analysis, Phys. Rev. Lett. 94 (2005) 107204. \nhttps://doi.org/10.1103/PhysRevLett.94.107204.  \n[38] S. Zhao, Y . Wu, C. Zhang, J. Wang, Z. Fu, R. Zhang, C. Jiang, Stabilization of τ -phase in \ncarbon -doped MnAl magnetic alloys, J. Alloys Compd. 755 (2018) 257–264. \nhttps://doi.org/10.1016/j.jallcom.2018.04.318.  \n[39] J.B. Yang, K. Kamaraju, W.B. Yelon, W.J. James, Q. Cai, A. Bollero, Magnetic properties of the MnBi intermetallic compound, Appl. Phys. Lett. 79 (2001) 1846–1848. \nhttps://doi.org/10.1063/1.1405434.  \n[40] X. Sun, Z.Y . Jia, Y .H. Huang, J.W. Harrell, D.E. Nikles, K. Sun, L.M. Wang, Synthesis and \nmagnetic properties of CoPt nanoparticles, J. Appl. Phys. 95 (2004) 6747 –6749. \nhttps://doi.org/10.1063/1.1667441.  \n[41] H. Fujii, T. Hokabe, T. Kamigaichi, T. Okamoto, Magnetic Properties of Fe\n2P Single Crystal, \nJ. Phys. Soc. Jpn. 43 (1977) 41–46. https://doi.org/10.1143/JPSJ.43.41.  \n \n \n  \n Fig.1 W.-H Shen et al.  \n \n \n \n  20 30 40 50 60 70 80 90\nab\n Mn\nSn a bc\nIntensity  (a.u.)\n2θ (deg.)(1011)\n(1120)\n(2020)\n(0002)(2021)\n(1122)(2131)\n(2021)\n(2240)\n(2023)\n(4041)\n(2133)\n(0004)2 mm(a) (b)\n(c) (d)(2130)Fig.2 W.- H Shen et al.  \n \n \n  0246\n0 100 200 3000123\n100 200 3000204060\n100 200 3000510152025H//z\nTNTCTpTt\nH⊥zM (emu/g)\nT (K)µ0H= 5 T\nµ0H = 0.02 TM (emu/g)\nT (K)H//z\nTpTp\nData × 4 µ0H = 1 T\nµ0H = 1T  \nµ0H = 0.02 TM (emu/g)\nT (K)TCH⊥z FC\n ZFC\nTN\n0 100 200 300240260280300\n I//x\nT (K)ρxx (µΩ cm)(c) (d)\nxyz[0110]\n[2110][0001]\n300310320330I//z ρzz (µΩ cm)(a) (b)Fig.3 W.-H Shen et al.  \n \n \n \n  -6 -4 -2 0 2 4 6-800-600-400-2000200400600800\n-6 -4 -2 0 2 4 6-800-600-400-2000200400600800\n-6 -4 -2 0 2 4 6-150-100-50050100150\n-6 -4 -2 0 2 4 6-800-600-400-2000200400600800H⊥z\nu0H (T) 2 K\n 50 K\n 150 K\n 200 K\n 250 K\n 300 KM (emu/cm3)(b)\n-0.1 0.0 0.1-10-50510M (emu/cm3)\nu0H (T)H//zM (emu/cm3)\nu0H (T) 2 K\n 50 K\n 150 K\n 200 K\n 250 K\n 300 KM (emu/cm3)\nu0H (T) H⊥z\n H//z\nT = 300 K(c) (d)\nM (emu/cm3)\nu0H (T) H⊥z\n H//z\nT = 2 K(a)Fig.4 W.-H Shen et al.  \n \n \n \n 020 40 60 80 100 120 14001234 0 40 80 120-2-101\n0 50 100 150300400500600\n0 50 100 15051015202530\n0 50 100 1500.10.20.30.4-6 -4 -2 0 2 4 6-400-2000200400\n0 40 80 120 1600.180.210.240.270.30\n0.0 0.2 0.4 0.6 0.8 1.00.00.51.01.5Hc (T)\nT (K)H//z(b)Log10 (Hc) (T)\nT (K)\nMs (emu/cm3)\nT (K) H⊥z\n H//z\n Fitted1\n Fitted2\n(c)\nχ (×10−4)\nT (K) H⊥z\n H//z(d)\nK (×106 J/m3)\nT (K) K1\n  K2(a)\nMs (emu/cm3\n)\nu0Hi (T)△M\nT = 150 KP\nT (K)\nh(1-4up*) ( ×106 J/m3)\nuT = 150 K"    },    {        "title": "2306.05681v1.Finite_temperature_second_order_perturbation_analysis_of_magnetocrystalline_anisotropy_energy_of_L10_type_ordered_alloys.pdf",        "content": "Finite-temperature second-order perturbation analysis of magnetocrystalline anisotropy energy of\n𝐿10-type ordered alloys\nShogo Yamashita∗and Akimasa Sakuma\nDepartment of Applied Physics, Tohoku University, Sendai 980-8579, Japan\n(Dated: June 12, 2023)\nWe present a novel finite-temperature second-order perturbation method incorporating spin-orbit coupling to\ninvestigate the temperature-dependent site-resolved contributions to the magnetocrystalline anisotropy energy\n(MAE), specifically 𝐾1(𝑇), in FePt, MnAl, and FeNi alloys. Our developed method successfully reproduces the\nresults obtained using the force theorem from our previous work. By employing this method, we identify the\nkey sites responsible for the distinctive behaviors of MAE in these alloys, shedding light on the inadequacy of\nthe spin model in capturing the temperature dependence of MAE in itinerant magnets. Moreover, we explore the\nlattice expansion effect on the temperature dependence of on-site contributions to 𝐾1(𝑇)in FeNi. Our results\nnot only provide insights into the limitations of the spin model in explaining the temperature dependence of\nMAE in itinerant ferromagnets but also highlight the need for further investigations. These findings contribute\nto a deeper understanding of the complex nature of MAE in itinerant magnetic systems.\nI. INTRODUCTION\nThe magnetocrystalline anisotropy energy (MAE) is an\nimportant characters of magnetic materials because it gov-\nerns the coercivity. Rare-earth permanent magnets are exam-\nples of magnets with the high coercivity and are used for\nmany modern applications. Recently, however, the develop-\nment of rare-earth-free magnets has been accelerating to avoid\nthe use of expensive rare-earth elements. 𝐿10-type transition\nmetal alloys such as FeNi are examples of rare-earth-free high-\nperformance permanent magnets[1]. Generally, coercivity has\nstrong temperature dependence; thus, we need to understand\nthe temperature dependence of the MAE for the development\nof high-performance transition-metal magnets at finite temper-\natures.\nThe MAE for the uniaxial crystal 𝐸MAE(𝑇,𝜃)is usually ex-\npressed as follows:\n𝐸MAE(𝑇,𝜃)=𝐾1(𝑇)sin2𝜃+𝐾2(𝑇)sin4𝜃+···, (1)\nwhere𝐾1(𝑇)and𝐾2(𝑇)are the anisotropy constants. How-\never, the theoretical description of the temperature dependence\nof the MAE remains controversial. In localized electron sys-\ntems, for instance, 4 𝑓electron systems such as permanent\nmagnets, theories based on the localized spin model combined\nwith the crystal field theory have been well-established [2–8].\nThe Callen–Callen power law[9–12] is a line of these theo-\nries that can describe the temperature dependence of 𝐾1(𝑇)\nand𝐾2(𝑇). In contrast, the cases of transition-metal magnets,\nwhich are treated as itinerant electron systems, are debatable.\nAt 0 K, the mechanism of the MAE in the itinerant elec-\ntrons systems, particularly 𝐾1(0), can be explained by the\nsecond-order perturbation formula in terms of the spin -orbit\ncoupling (SOC) according to the tight-binding model[13–18].\nHowever, finite-temperature expressions for the MAE based\non the band theory are not available yet. This is because\nthe band theory is based on the mean field theory and can-\nnot describe the spin-transverse fluctuations directly. One of\n∗shogo.yamashita.q1@dc.tohoku.ac.jpthe ways to describe the spin fluctuation based on the itiner-\nant electron theory is the functional integral method[19–24].\nThis method is usually combined with coherent potential ap-\nproximation (CPA). In this approach, the spin fluctuation can\nbe expressed by random spin states with respect to its direc-\ntion, which are called disordered local moment (DLM) states.\nFirst-principles calculations based on this scheme have been\nperformed by several authors[25–29] to investigate the finite-\ntemperature magnetic properties of magnetic materials as pio-\nneering works. Subsequently, the temperature dependences of\nthe MAE, transport properties, and Gilbert damping constants\nin the itinerant electron systems were investigated via the\nDLM-CPA method and the density functional theory[30–38],\nalong with model calculations[39–41]. In particular, the tem-\nperature dependence of the MAE for 𝐿10-type alloys has been\ncalculated by several authors[30–32, 35].\nRecently, we calculated the temperature dependence of the\nMAE for𝐿10-type FePt, MnAl, and FeNi using the DLM-\nCPA method[35]. The calculation results for FePt and MnAl\nindicated that the MAE decreases with an increase in the tem-\nperature. However, the calculated MAE for FeNi exhibited a\nunique behavior. It did not decrease monotonically with an\nincrease in the temperature; rather, it exhibited plateau-like be-\nhavior in the low-temperature region. This behavior is similar\nto that of Y 2Fe14B[34, 42, 43], for which the mechanism of the\ntemperature dependence of the MAE has been controversial.\nIn the present study, to analyze these behaviors, we decompose\nthe MAE of these alloys at finite temperatures into onsite and\npair contributions using the second-order perturbation (SOP)\nmethod in terms of the SOC. In this method, we can extend the\nformula to describe 𝐾1(0)in the tight-binding model[13–18]\nto the finite-temperature expression 𝐾1(𝑇).\nII. CALCULATION DETAILS\nTo develop the finite-temperature SOP formula, we use\nthe tight-binding linearized muffin-tin orbital (TB-LMTO)\nmethod[44–48] with the atomic sphere approximations com-\nbined with the DLM-CPA method. First, in the DLM-CPA\nmethod, we need to calculate a distribution function 𝜔({e},𝑇)arXiv:2306.05681v1  [cond-mat.mtrl-sci]  9 Jun 20232\nrepresenting the probability with which the spin vectors are\ndirected to{e}at temperature 𝑇. In this work, we adopt the\nsingle-site approximation; thus, 𝜔({e},𝑇)is decoupled into\nthe simple product of the probability at each site 𝜔𝑖(e𝑖,𝑇)as\nfollows:\n𝜔({e},𝑇)=Ö\n𝑖𝜔𝑖(e𝑖,𝑇). (2)\nIn a previous work, 𝜔𝑖(e𝑖,𝑇)was evaluated using the anal-\nogy of the Weiss field[28, 30–32]. However, in this work,\nwe determine 𝜔𝑖(e𝑖,𝑇)by evaluating the effective grand po-\ntentialΩeff({e},𝑇)of electrons. Here, we explain the calcu-\nlation ofΩeff({e},𝑇)and𝜔𝑖(e𝑖,𝑇). First, we introduce the\nGreen function including the spin-transverse fluctuation at fi-\nnite temperatures in the TB-LMTO method 𝐺(𝑧,{e})[35–38]\nas follows:\n𝐺𝑖 𝑗(𝑧;{e})=𝜆𝛽\n𝑖(𝑧;e𝑖)𝛿𝑖 𝑗+𝜇𝛽\n𝑖(𝑧;e𝑖)𝑔𝛽\n𝑖 𝑗(𝑧;{e})¯𝜇𝛽\n𝑗(𝑧;e𝑗),\n(3)\nwhere𝑧and𝑔𝛽\n𝑖 𝑗(𝑧;{e})are𝐸+𝑖𝛿and an auxiliary green\nfunction including spin-fluctuation, respectively. 𝜆𝛽\n𝑖(𝑧;e𝑖)and\n𝜇𝛽\n𝑖(𝑧;e𝑖)are given as follows:\n𝜆𝛽\n𝑖(𝑧;e𝑖)=(Δ𝑖(e𝑖))−1/2(1+(𝛾𝑖(e𝑖)−𝛽)𝑃𝛾\n𝑖(𝑧;e𝑖))(Δ𝑖(e𝑖))−1/2,\n(4)\n𝜇𝛽\n𝑖(𝑧;e𝑖)=(Δ𝑖(e𝑖))−1/2(𝑃𝛾\n𝑖(𝑧;e𝑖))−1𝑃𝛽\n𝑖(𝑧;e𝑖), (5)\n¯𝜇𝛽\n𝑖(𝑧;e𝑖)=𝑃𝛽\n𝑖(𝑧;e𝑖)(𝑃𝛾\n𝑖(𝑧;e𝑖))−1(Δ𝑖(e𝑖))−1/2, (6)\nwhere\nΔ−1/2\n𝑖(e𝑖)=𝑈†(e𝑖)(Δ𝑖)−1/2𝑈(e𝑖), (7)\n(𝑃𝛾\n𝑖(𝑧;e𝑖))−1=𝑈†(e𝑖)(𝑃𝛾\n𝑖(𝑧))−1𝑈(e𝑖), (8)\n𝛾𝑖(e𝑖)=𝑈†(e𝑖)(𝛾𝑖)𝑈(e𝑖), (9)\n𝑃𝛽\n𝑖(𝑧;e𝑖)=𝑈†(e𝑖)𝑃𝛽\n𝑖(𝑧)𝑈(e𝑖), (10)\n𝑃𝛽\n𝑖(𝑧)=𝑃𝛾\n𝑖(𝑧){1−[𝛽−𝛾𝑖]𝑃𝛾\n𝑖(𝑧)}−1, (11)\n𝑃𝛾\n𝑖(𝑧)=(Δ𝑖)−1/2[𝑧−𝐶𝑖](Δ𝑖)−1/2. (12)\nHere,𝛾𝑖,Δ𝑖, and𝐶𝑖are called potential parameters in the\nTB-LMTO method. The 𝛽values are summarized in several\npapers[37, 46, 47]. In this work, we neglect the SOC to\ncalculate the Ωeff({e},𝑇)and𝜔𝑖(e𝑖,𝑇). The effective grand\npotential of electronic part is expressed as\nΩeff({e},𝑇)∼1\n𝜋∫\nd𝜖𝑓(𝜖,𝑇,𝜇)∫𝜖\n−∞d𝐸ImTr𝐺(𝑧;{e})\n=−1\n𝜋∫\nd𝜖𝑓(𝜖,𝑇,𝜇)Im\u0002\nTr log𝜆𝛽(𝜖+;{e})+Tr log𝑔𝛽(𝜖+;{e})\u0003\n,\n(13)where𝑓and𝜇represent the Fermi-Dirac function and the\nchemical potential, respectively. The trace is taken over with\nrespect to sites 𝑖, orbitals𝐿, and spin indices 𝜎. From here, we\nexpand𝑔𝛽(𝜖+;{e})with the auxiliary coherent Green function\n¯𝑔𝛽(𝑧), which is defined as follows:\n¯𝑔𝛽(𝑧)=\u0010\n¯𝑃(𝑧)−𝑆𝛽\u0011−1\n, (14)\nwhere𝑆𝛽is given as\n𝑆𝛽=𝑆(1−𝛽𝑆)−1. (15)\n𝑆is a bare structure constant matrix[45]. The auxiliary Green\nfunction𝑔𝛽(𝜖+;{e})is expanded as follows:\n𝑔𝛽(𝑧;{e})=¯𝑔𝛽(𝑧)\u0010\n1+Δ𝑃(𝑧;{e})¯𝑔𝛽(𝑧)\u0011−1\n, (16)\nwhere\nΔ𝑃(𝑧;{e})=𝑃𝛽(𝑧;{e})− ¯𝑃(𝑧), (17)\nand ¯𝑃is a coherent potential function. We also need to obtain\n¯𝑃in a self-consistent manner (explained later). Using Eq.\n(16), Eq.(13) can be rewritten as follows:\nΩeff({e},𝑇)=−1\n𝜋∫\nd𝜖 𝑓(𝜖,𝑇,𝜇)Im\"\nTr log𝜆𝛽(𝜖+;{e})\n+Tr log ¯𝑔𝛽(𝑧)−Tr log\u0010\n1+Δ𝑃(𝑧;{e})¯𝑔𝛽(𝑧)\u0011#\n. (18)\nBy taking the trace with respect to site 𝑖, the grand potential\ncan be expressed as follows[37]:\nΩeff({e},𝑇)=Ω 0+∑︁\n𝑖ΔΩ𝑖(e𝑖,𝑇), (19)\nΔΩ𝑖(e𝑖,𝑇)=1\n𝜋Im∫\nd𝐸𝑓(𝐸,𝑇,𝜇)Tr𝐿𝜎log\u0010\n1+Δ𝑃𝑖(𝑧;e𝑖)¯𝑔𝛽\n𝑖𝑖(𝑧)\u0011\n.\n(20)\nHere, we used the fact that {e}dependence of 𝜆𝛽(𝜖+;{e})\nvanishes in our case. Therefore, 𝜔𝑖(e𝑖,𝑇)can be expressed as\nfollows:\n𝜔𝑖(e𝑖,𝑇)\n=exp(−ΔΩ𝑖(e𝑖,𝑇)/𝑘B𝑇)/∫\nde𝑖exp(−ΔΩ𝑖(e′𝑖,𝑇)/𝑘B𝑇),\n(21)\nwhere𝑘Bis the Boltzmann constant. Finally, we need to\ndetermine the converged 𝜔𝑖(e𝑖,𝑇)and ¯𝑃(𝑧)self-consistently.\nThe CPA condition to determine ¯𝑃(𝑧)is given as:\n∫\nde𝑖𝜔𝑖(e𝑖,𝑇)Δ𝑃𝑖(𝑧;e𝑖)h\n1+Δ𝑃𝑖(𝑧;e𝑖)¯𝑔𝛽\n𝑖𝑖(𝑧)i−1\n=0.\n(22)3\nWe use Eq. (14), Eq. (17), Eq. (20), Eq. (21), and Eq. (22)\nto obtain ¯𝑃𝑖and the converged 𝜔𝑖(e𝑖,𝑇)in a self-consistent\nmanner.\nOnce we obtain the converged 𝜔𝑖(e𝑖,𝑇), we can calculate the\nSOP formula at finite temperatures as follows:\n𝛿𝐸2nd(𝑇,n)=\n−1\n2𝜋∑︁\n𝑖 𝑗ImTr 𝐿𝜎∫∞\n−∞d𝐸 𝑓(𝐸,𝜇,𝑇)\n×⟨𝐺𝑖 𝑗(𝑧;{e},n)𝐻soc\n𝑗𝐺𝑗𝑖(𝑧;{e},n)𝐻soc\n𝑖⟩{𝜔𝑖(e𝑖,𝑇)},(23)\nwhere𝐻socandnrepresent the spin -orbit Hamiltonian and\nthe magnetization direction, respectively. Similar expressions\nwere used in several works[41, 49]. ⟨···⟩ denotes the average\nover{e}with a weight of 𝜔𝑖(e𝑖,𝑇), which is given as follows:\n⟨···⟩{𝜔𝑖(e𝑖,𝑇)}=Ö\n𝑖∫\nde𝑖𝜔𝑖(e𝑖,𝑇)(···). (24)\nThe rotation of the magnetization direction is expressed with\nthe SO(3) rotation matrices 𝑅(n)as follows[35, 50, 51]:\n𝑆𝑙1𝑙2𝑚1𝑚2(n)=∑︁\n𝑚3𝑚4𝑅𝑙1∗\n𝑚3𝑚1(n)𝑆𝑙1𝑙2𝑚3𝑚4𝑅𝑙2𝑚4𝑚2(n), (25)\nwhere∗denotes the complex conjugate. We substitute Eq.\n(25) into Eq. (14) to express the rotation of the direction of\nmagnetization.\nWe can decompose Eq. (23) into onsite 𝐸2nd\n𝑖𝑖and pair𝐸2nd\n𝑖 𝑗\ncontributions as follows:\n𝐸2nd\n𝑖𝑖(𝑇,n)\n=−1\n2𝜋ImTr 𝐿𝜎∫\nde𝑖𝜔𝑖(e𝑖,𝑇)∫∞\n−∞d𝐸𝑓(𝐸,𝑇,𝜇)\n×\u001a\n𝐻soc\n𝑖\u0010\n𝜆𝛽\n𝑖(𝑧;e𝑖)+𝜇𝛽\n𝑖(𝑧;e𝑖)¯𝑔𝛽\n𝑖𝑖(𝑧,n)𝜉𝑖(𝑧;e𝑖,n)¯𝜇𝛽\n𝑖(𝑧;e𝑖)\u0011\n×𝐻soc\n𝑖\u0010\n𝜆𝛽\n𝑖(𝑧;e𝑖)+𝜇𝛽\n𝑖(𝑧;e𝑖)¯𝑔𝛽\n𝑖𝑖(𝑧,n)𝜉𝑖(𝑧;e𝑖,n)¯𝜇𝛽\n𝑖(𝑧;e𝑖)\u0011\u001b\n,\n(26)\n𝐸2nd\n𝑖 𝑗(𝑇,n)\n=−1\n2𝜋∑︁\n𝑘ImTr 𝐿𝜎∫\nde𝑖𝜔𝑖(e𝑖,𝑇)∫\nde′𝑗𝜔𝑗(e′𝑗,𝑇)\n×∫∞\n−∞d𝐸 𝑓(𝐸,𝑇,𝜇){˜𝐻soc\n𝑖𝜒𝑖𝑘(1−Γ𝜒)−1\n𝑘 𝑗˜𝐻soc\n𝑗}. (27)\nHere, we introduce 𝜉𝑖(𝑧;e𝑖,n),˜𝜉𝑖(𝑧;e𝑖,n),˜𝐻soc\n𝑖,Γ, and𝜒,\nwhich are given as follows:\n𝜉𝑖(𝑧;e𝑖,n)=h\n1+Δ𝑃𝑖(𝑧;e𝑖)¯𝑔𝛽\n𝑖𝑖(𝑧,n)i−1\n, (28)\n˜𝜉𝑖(𝑧;e𝑖,n)=h\n1+¯𝑔𝛽\n𝑖𝑖(𝑧,n)Δ𝑃𝑖(𝑧;e𝑖)i−1\n. (29)˜𝐻soc\n𝑖=𝜉𝑖(𝑧;e𝑖,n)𝜇𝛽\n𝑖(𝑧;e𝑖)𝐻soc\n𝑖¯𝜇𝛽\n𝑖(𝑧;e𝑖)˜𝜉𝑖(𝑧;e𝑖,n),(30)\nΓ𝑖(𝑧,𝑇,n)\n=∫\nde𝑖𝜔𝑖(e𝑖,𝑇)h\nΔ𝑃𝛽\n𝑖(𝑧;e𝑖)˜𝜉𝑖(𝑧;e𝑖,n)i h\nΔ𝑃𝛽\n𝑖(𝑧;e𝑖)˜𝜉𝑖(𝑧;e𝑖,n)i\n,\n(31)\n𝜒𝑖 𝑗(𝑧,n)=¯𝑔𝛽\n𝑖 𝑗(𝑧,n)¯𝑔𝛽\n𝑗𝑖(𝑧,n)(1−𝛿𝑖 𝑗). (32)\nFor evaluating pair contributions, we expand the Green func-\ntion including the spin-fluctuation with the T-matrix to include\nthe vertex correction terms. Details regarding the deriva-\ntion of the vertex correction terms are provided in several\npapers[38, 52, 53].\nIn practical calculations, we neglect the Fermi–Dirac distri-\nbution function in Eqs. (26) and (27). This does not cause\nserious numerical errors. In the present study, the 𝐾1(𝑇)part\nof the MAE at finite temperatures is defined as follows:\n𝜖MAE(𝑇)\n=∑︁\n𝑖 𝑗\u0010\n𝐸2nd\n𝑖 𝑗(𝑇,𝜃=𝜋/2)−𝐸2nd\n𝑖 𝑗(𝑇,𝜃=0)\u0011\n∼𝐾1(𝑇).(33)\nUsing Eqs. (26) and (27), we can investigate the site-resolved\ncontributions in 𝐾1(𝑇)and its temperature dependences.\nFor calculation details, the lattice constants of each alloy are set\nto the same values used in a previous work[35]. The number\nof𝑘-points for each calculation is also the same as that in the\nprevious work.\nIII. RESULTS AND DISCUSSIONS\nTo examine the accuracy of the developed method, let us\nfirst investigate how the SOP method can reproduce the MAE\nobtained via the force theorem (FT) in our previous work[35].\nFigure 1 shows the MAE calculated using the FT and the SOP\nmethod for FePt, MnAl, and FeNi. Small differences between\nthe results of the two methods are observed for FePt and FeNi,\nwhereas little difference is observed for in MnAl. For MnAl,\nthe good agreement is reasonable, considering that the SOC of\nthis system is far weaker than those of FePt and FeNi. From\nthis view-point, the origin of the larger difference for FeNi\ncompared with FePt is not simple, because the SOC in FeNi is\nweaker than that in FePt. This may suggest the peculiar 𝐾1(𝑇)\nbehavior of FeNi, which will be discussed later. In total sum-\nmary, the qualitative behaviors of the temperature dependence\nof the MAE in the previous work can well be reproduced by\nthe SOP method focusing on 𝐾1(𝑇). The difference between\nthe FT and the SOP method may arise from the higher-order\nperturbation term, which contributes to 𝐾2(𝑇).\nThe total onsite and pair contributions to 𝐾1(𝑇)are also shown\nin Fig. 1. For FePt, onsite and pair contribution have opposite\nsigns for the whole temperature region. The onsite contri-\nbution is suppressed by the pair contribution, which leads to\nuniaxial anisotropy of 𝐾1(𝑇).4\nFor MnAl,𝐾1(𝑇)is mostly dominated by the onsite contri-\nbution. The pair contribution makes a small correction to the\n𝐾1(𝑇). In this case, both contributions have a positive sign.\nFor FeNi, the onsite and pair contributions have similar am-\nplitudes, whereas the signs are opposite, as in the case of FePt.\nIn addition, the peculiar behavior that 𝐾1(𝑇)exhibits a plateau\nin the low-temperature region is found to be due to the cancel-\nlation of the variations of the total onsite and total pair terms\nwith the temperature change.\nWe stress here that even though the crystal structures are the\nsame for these alloys, the alloys differ with regard to the break-\ndown of these SOP results into onsite and pair contributions.\nIn particular for FePt and MnAl, despite the similar behavior of\nthe temperature dependence of the total 𝐾1(𝑇), the onsite and\npair contributions differ significantly. Furthermore, compar-\ning FePt and FeNi reveals that the temperature dependences of\nonsite and pair contributions are differ significantly between\nthese two alloys. These characteristics of the 𝐾1(𝑇)of each\nalloy can be recognized via the present SOP theory at finite\ntemperatures, which we believe is one of the advantage of this\napproach.\nHere, to investigate the characteristics of the temperature de-\npendence of 𝐾1(𝑇)in the itinerant electron magnets, we com-\npare those results with the results expected from the spin\nmodel. The Hamiltonian is given by the XXZ model[35, 54–\n56] as follows:\n𝐻=−∑︁\n(𝑖, 𝑗)2𝐽𝑖 𝑗®𝑆𝑖·®𝑆𝑗−∑︁\n(𝑖, 𝑗)𝐷𝑖 𝑗𝑆𝑧\n𝑖𝑆𝑧\n𝑗−∑︁\n𝑖𝐷𝑖(𝑆𝑧\n𝑖)2,(34)\nwhere®𝑆𝑖,𝐽𝑖 𝑗,𝐷𝑖, and𝐷𝑖 𝑗are a classical spin vector, an\nexchange coupling constant, a single-site anisotropy coeffi-\ncient, and a two-site anisotropy coefficient, respectively. As\npresented in our previous work[35], analysis with this model\nrequires𝐷𝑖>0 andÍ\n𝑗𝐷𝑖 𝑗>0 for FePt and MnAl and\n𝐷𝑖<0 andÍ\n𝑗𝐷𝑖 𝑗>0 for FeNi to reproduce the temper-\nature dependence of the total MAE. However, if one naively\nassumes that the 𝐷𝑖and𝐷𝑖 𝑗terms correspond to onsite and\npair contributions, respectively, at first glance, the signs of the\nterms in the XXZ model used in the previous work are not con-\nsistent with the results in Fig. 1, except for the case of MnAl.\nThis mismatch originates from the fact that the temperature\ndependences of the MAE from the 𝐷𝑖and𝐷𝑖 𝑗terms in the\nXXZ model behave as approximately ∝𝑀3(𝑇)[11, 12] and\n∝𝑀2(𝑇)[54], respectively, regardless of the signs and am-\nplitudes of the parameters 𝐷𝑖and𝐷𝑖 𝑗, whereas those of the\nonsite and pair terms in the SOP method do not necessarily\nfollow such simple rules but exhibit various behaviors depend-\ning on the system. For this reason, to reproduce the peculiar\nbehavior of the total MAE of FeNi, the XXZ model has no\nother choice than to set 𝐷𝑖<0 andÍ\n𝑗𝐷𝑖 𝑗>0; however, the\nonsite and pair contributions can produce such behavior with\nopposite signs from the XXZ model. Thus, the results in Fig.\n1 imply that the spin model is too simple and is insufficient to\nexpress the temperature dependence of the MAE of itinerant\nmagnets.\n(a) FePt\n(b) MnAl\n(c) FeNiTotal-FTOnsitePairTotal-FTOnsitePair\nTotal-FTOnsitePairTotal-SOPTotal-SOPTotal-SOPFIG. 1. Temperature dependence of the total 𝐾1(𝑇)calculated via the\ndeveloped finite-temperature SOP method including vertex correction\nterms for (a) FePt, (b) MnAl, and (c) FeNi, which are indicated by\ngreen lines. For comparisons, the results calculated using the FT\nreported by Yamashita et al.[35] are shown as black lines. In addition,\nthe total onsite and total pair contributions are shown as red and blue\nlines, respectively.\nTo examine each contribution in detail, we decompose the\nSOP results into each onsite and pair contribution. The break-5\ndown of these contributions is shown in Fig. 2 for all the\nalloys. For FePt, the results are shown in Fig. 2 (a). The\ntotal onsite contribution in FePt is mostly from of Pt, and\nthe Fe contribution is far smaller. In addition, it is found that\nthe negative contribution of the total pair term in Fig. 1 (a)\nmainly comes from the Fe-Pt pair, and it is suppressed by the\npositive contribution from Pt-Pt pairs. This leads to a nega-\ntive total pair contribution. In this alloy, the onsite and pair\ncontributions related to Pt significantly affect the temperature\ndependence of 𝐾1(𝑇).\nThe results for MnAl in Fig. 2 (b) indicate that all the contri-\nbutions are positive and that the situation regarding the total\nonsite term is similar to that for FePt. It is mostly dominated by\nthe Mn onsite contribution. The second-largest contribution is\nthe Mn-Mn positive pair contribution, and the other contribu-\ntions are negligible. Thus, the onsite and pair contributions of\nMn determine the temperature dependence of 𝐾1(𝑇)in MnAl.\nFor FeNi, as shown in fig. 2 (c), the onsite contributions from\nFe and Ni are positive and have similar amplitudes, leading\nto a total positive onsite contribution. We also find that the\nnegative contribution of the total pair term mostly comes from\na pair of different atoms, i.e., Fe-Ni. While the Fe-Ni con-\ntribution is suppressed by other positive pair contributions, it\nfinally leads to negative finite total pair contributions.\nFrom these results, although most of the pair contributions are\nsuppressed by other pairs and the net contribution becomes\nsmall, the pair contributions play important roles over the\nwhole temperature region, particularly for FePt and FeNi. The\nimportance of the pair contributions was also investigated by\nKe[18] with the SOP method at 0 K. In this work, we con-\nfirmed that the pair contributions to 𝐾1(𝑇)significantly affect\nthe temperature dependence of 𝐾1(𝑇)at not only 0 K but also\nfinite temperatures.\nFinally, we virtually expand the lattice of FeNi to investi-\ngate the influence of electron itineracy on the onsite contribu-\ntion𝐾on\n𝑖(𝑇). Here, we fit 𝐾on\n𝑖(𝑇)by assuming the relation\n𝐾on\n𝑖(𝑇) ∝𝑀𝑖(𝑇)𝑛and investigate the temperature depen-\ndence of the exponent 𝑛at each site. As mentioned previ-\nously, the onsite contributions 𝐾on\n𝑖(𝑇)should follow the re-\nlation𝐾on\n𝑖(𝑇)∝𝑀𝑖(𝑇)3if the localized spin model is suit-\nable to explain the temperature dependence of 𝐾on\n𝑖(𝑇). We\nbriefly examine the validity of the single-site anisotropy term,\nwhich is the simplest term, for the temperature dependence\nof the MAE in the itinerant magnets. In Fig. 3, the tem-\nperature dependences of 𝑛of the onsite contributions for\nFeNi with various volumes are shown. If we expand the\nlattice, the𝑛value of𝐾on\nFe(𝑇) ∝𝑀Fe(𝑇)𝑛increases, and\nit reaches 3 in the low-temperature region. However, the 𝑛\nvalue of𝐾on\nNi(𝑇)∝𝑀Ni(𝑇)𝑛is always far from 3 and is not\nchanged drastically. If we use a spin model with the single-site\nanisotropy term to explain the temperature dependence of the\nonsite contributions, the 𝑛value must be fixed to 3 in the low-\ntemperature region regardless of the sign and amplitude of 𝐷𝑖.\nIn addition, if the lattice is expanded, the 𝐷𝑖and𝐷𝑖 𝑗values\nare expected to change, and these temperature dependences\nare not changed in the localized spin model. However, our\nresults imply that not only changing the value of 𝐷𝑖but also\n(a) FePt\n(b) MnAl(c) FeNiPtPt-PtFeFe-FeFe-Pt\nNiNi-NiFeFe-FeFe-NiAlAl-AlMnMn-MnMn-AlFIG. 2. Temperature dependence of the site-resolved contributions\nto𝐾1(𝑇)for (a) FePt, (b) MnAl, and (c) FeNi.\nchanging the temperature dependence of the onsite term itself\nwith expanding the lattice. Therefore, we can again conclude\nthat if we assume Eq. (34) to explain the temperature depen-\ndence of the MAE, even the single-site anisotropy term in Eq.\n(34) may not always be sufficient to describe the temperature\ndependence of 𝐾on\n𝑖(𝑇)in the itinerant ferromagnets.6\n(a) FeNiFe site\n(b) FeNiNi site\nFIG. 3. Temperature dependences of the exponent 𝑛of\n𝐾on\n𝑖(𝑇)/𝐾on\n𝑖(0)=(𝑀𝑖(𝑇)/𝑀𝑖(0))𝑛for Fe and Ni sites for various\nvolumes𝑉of FeNi. Black line represents the results for the original\nvolume. Other lines represent to the lattice-expanded results. 𝑉0is\nset to 22.63.IV. SUMMARY\nIn summary, we developed a finite-temperature SOP method\nto describe the temperature dependence of 𝐾1(𝑇)and applied\nit to𝐿10-FePt, MnAl, and FeNi. We confirmed that the devel-\noped method can reproduce the results of a previous work[35].\nWe also investigated the onsite and pair contributions to 𝐾1(𝑇)\nwith the developed method. We showed that not only the on-\nsite contributions but also the pair-contributions significantly\naffect the temperature dependence of 𝐾1(𝑇). In particular, the\nunique behavior of 𝐾1(𝑇)for FeNi is attributed to the com-\npetition of onsite and pair-site contributions. In addition, for\nsome results, it is found that the signs of onsite and pair contri-\nbutions do not agree with the conditions used in the previous\nwork[35]. Finally, we investigated the effect of electron itin-\neracy for the temperature dependence of the 𝐾on\n𝑖(𝑇)of FeNi\nwhile expanding the lattice parameters. We found that the\nexponent𝑛of the onsite contributions of both atoms depends\non the volume and is not fixed to 3, which is expected from a\nspin model. From the above, our results imply that the XXZ\nmodel, even the single-site anisotropy term in this model, is\ninsufficient for the itinerant ferromagnets.\nACKNOWLEDGMENTS\nS.Y. acknowledges Dr. Ryoya Hiramatsu, Dr. Yusuke\nMasaki, and Prof. Dr. Hiroaki Matsueda of Tohoku Uni-\nversity for fruitful discussions and support from GP-Spin at\nTohoku University, Japan. S.Y. also appreciates Dr. Juba\nBouaziz and Prof. Dr. Stefan Bl¨ ugel of Forschungszentrum\nJ¨ ulich for constructive comments on this work. This work was\nsupported by JSPS KAKENHI Grant Number JP19H05612 in\nJapan.\n[1] S. Goto, H. Kura, E. Watanabe, Y. Hayashi, H. Yanagihara, Y.\nShimada, M. Mizuguchi, K. Takanashi, and E. Kita, Sci. Rep.\n7, 13216 (2017).\n[2] J. F. Herbst, Rev. Mod. Phys. 63, 819 (1991).\n[3] M. Yamada, H. Kato, H. Yamamoto, and Y. Nakagawa, Phys.\nRev. B 38, 620 (1988).\n[4] R. Sasaki, D. Miura, and A. Sakuma, Appl. Phys. Express 8,\n043004 (2015).\n[5] T. Yoshioka and H. Tsuchiura, Appl. Phys. Lett. 112, 162405\n(2018).\n[6] T. Yoshioka, H. Tsuchiura, and P. Nov ´ak, Phys. Rev. B 102,\n184410 (2020).\n[7] S. Yamashita, D. Suzuki, T. Yoshioka, H. Tsuchiura, and P.\nNov´ak, Phys. Rev. B 102, 214439 (2020).\n[8] T. Yoshioka, H. Tsuchiura, and P. Nov ´ak, Phys. Rev. B 105,\n014402 (2022).\n[9] N. Akulov, Z. Phys. 100, 197 (1936).\n[10] C. Zener, Phys. Rev. 96, 1335 (1954).\n[11] E. R. Callen and H. B. Callen, Phys. Rev. 129, 578 (1963).[12] E. R. Callen and H. B. Callen, J. Phys. Chem. Solids 27, 1271\n(1966).\n[13] P. Bruno, Phys. Rev. B 39, 865 (1989).\n[14] G. van der Laan, J. Phys.: Condens. Matter 10, 3239 (1998).\n[15] Y. Kota and A. Sakuma, J. Phys. Soc. Jpn. 83, 034715 (2014).\n[16] I. V. Solovyev, P. H. Dederichs, and I. Mertig, Phys. Rev. B 52,\n13419 (1995).\n[17] L. Ke and M. van Schilfgaarde, Phys. Rev. B 92, 014423 (2015).\n[18] L. Ke, Phys, Rev. B. 99, 054418 (2019).\n[19] M. Cyrot, Phys. Rev. Lett. 25, 871 (1970).\n[20] J. Hubbard, Phys. Rev. B 19, 2626 (1979).\n[21] J. Hubbard, Phys. Rev. B 20, 4584 (1979).\n[22] H. Hasegawa, J. Phys. Soc. Jpn. 46, 1504 (1979).\n[23] H. Hasegawa, J. Phys. Soc. Jpn. 49, 178 (1980).\n[24] H. Hasegawa, J. Phys. Soc. Jpn. 49, 963 (1980).\n[25] T. Oguchi, K. Terakura, and N. Hamada, J. Phys. F 13, 145\n(1983).\n[26] A. J. Pindor, J. Staunton, G. M. Stocks, and H. Winter, J. Phys.\nF13, 979 (1983).7\n[27] J. Staunton, B. L. Gyorffy, A. J. Pindor, G. M. Stocks, and H.\nWinter, J. Magn. Magn. Mater. 45, 15 (1984).\n[28] B. L. Gyorffy, A. J. Pindor, J. Staunton, G. M. Stocks, and H.\nWinter, J. Phys. F 15, 1337 (1985).\n[29] J. B. Staunton and B. L. Gyorffy, Phys. Rev. Lett. 69, 371 (1992).\n[30] J. B. Staunton, S. Ostanin, S. S. A. Razee, B. L. Gyorffy, L.\nSzunyogh, B. Ginatempo, and E. Bruno, Phys. Rev. Lett. 93,\n257204 (2004).\n[31] J. B. Staunton, L. Szunyogh, A. Buruzs, B. L. Gyorffy, S. Os-\ntanin, and L. Udvardi, Phys. Rev. B 74, 144411 (2006).\n[32] A. De ´ak, E. Simon, L. Balogh, L. Szunyogh, M. Dos Santos\nDias, and J. B. Staunton, Phys. Rev. B 89, 224401 (2014).\n[33] M. Matsumoto, R. Banerjee, and J. B. Staunton, Phys. Rev. B\n90, 054421 (2014).\n[34] J. Bouaziz, C. E. Patrick, and J. B. Staunton, Phys. Rev. B 107,\nL020401 (2023).\n[35] S. Yamashita and A. Sakuma, J. Phys. Soc. Jpn. 91, 0934703\n(2022).\n[36] R. Hiramatsu, D. Miura, and A. Sakuma, Appl. Phys. Express\n15, 013003 (2021).\n[37] A. Sakuma and D. Miura, J. Phys. Soc. Jpn. 91, 084701 (2022).\n[38] R. Hiramatsu, D. Miura, and A. Sakuma, J. Phys. Soc. Jpn. 92,\n044704 (2023).\n[39] A. Sakuma, J. Phys. Soc. Jpn. 87, 034705 (2018).\n[40] D. Miura and A. Sakuma, J. Phys. Soc. Jpn. 90, 113601 (2021).\n[41] D. Miura and A. Sakuma, J. Phys. Soc. Jpn. 91, 023706 (2022).[42] M. Sagawa, S. Fujimura, H. Yamamoto, Y. Matsuura, and S.\nHirosawa, J. Appl. Phys. 57, 4094 (1985).\n[43] R. Grossinger, X. Sun, R. Eibler, K. Buschow, and H. Kirchmayr,\nJ. Magn. Magn. Mater. 58, 55 (1986).\n[44] O. K. Andersen, Z. Pawlowska, and O. Jepsen, Phys. Rev. B, 34,\n5253 (1986).\n[45] H. L. Skriver, The LMTO Method (Springer, Berlin, 1984).\n[46] I. Turek, V. Drchal, J. Kudrnovsk´ y, M. S ˇob, and P. Weinberger,\nElectronic Structure of Disordered Alloys, Surface and Inter-\nfaces (Kluwer Academic, Dordrecht, 1997).\n[47] J. Kudrnovsk´ y and V. Drchal, Phys. Rev. B, 41, 7515 (1990).\n[48] A. Sakuma, J. Phys. Soc. Jpn. 69, 3072 (2000).\n[49] N. Kobayashi, Y. Kota, and A. Sakuma, J. Phys. Soc. Jpn. 87,\n094714 (2018).\n[50] H. Ebert, Phys. Rev. B 38, 9390 (1988).\n[51] A. Sakuma, J. Phys. Soc. Jpn. 63, 1422 (1994).\n[52] W. H. Butler, Phys. Rev. B. 31, 3260, (1985).\n[53] K. Carve, I. Turek, J. Kudrnovsk´ y, and O. Bengone, Phys. Rev.\nB73, 144421 (2006).\n[54] O. N. Mryasov, U. Nowak, K. Y. Guslienko, and R. W. Chantrell,\nEurophys. Lett. 69, 805 (2005).\n[55] R. F. L. Evans, L. R ´ozsa, S. Jenkins, and U. Atxitia, Phys. Rev.\nB102, 020412(R) (2020).\n[56] R. Cuadrado, R. F. L. Evans, T. Shoji, M. Yano, M. Ito, G.\nHrkac, T. Schrefl, and R. W. Chantrell, J. Appl. Phys. 130,\n023901 (2021)."    },    {        "title": "2306.11952v1.First_principles_prediction_of_structural__magnetic_properties_of_Cr_substituted_strontium_hexaferrite__and_its_site_preference.pdf",        "content": "First-principles prediction of structural, magnetic properties of\nCr-substituted strontium hexaferrite, and its site preference\nBinod Regmi,1,2Dinesh Thapa,1,2Bipin Lamichhane,1,2Seong-Gon Kim,1,2,a)\n1Department of Physics and Astronomy, Mississippi State University, Mississippi State, Mississippi 39762, USA\n2Center for Computational Sciences, Mississippi State University, Mississippi State, Mississippi 39762, USAa)\nABSTRACT\nTo investigate the structural and magnetic properties of Cr-doped M-type strontium hexaferrite (SrFe 12O19)\nwith x = (0.0, 0.5, 1.0), we perform first-principles total-energy calculations relied on density functional\ntheory. Based on the calculation of the substitution energy of Cr in strontium hexaferrite and formation\nprobability analysis, we conclude that the doped Cr atoms prefer to occupy the 2a, 12k, and 4f 2sites which is\nin good agreement with the experimental findings. Due to Cr3+ion moment, 3 µB, smaller than that of Fe3+\nion, 5 µB, saturation magnetization (M s) reduce rapidly as the concentration of Cr increases in strontium\nhexaferrite. The magnetic anisotropic field (Ha)rises with an increasing fraction of Cr despite a significant\nreduction of magnetization and a slight increase of magnetocrystalline anisotropy (K1).The cause for the rise\nin magnetic anisotropy field (Ha)with an increasing fraction of Cr is further emphasized by our formation\nprobability study. Cr3+ions prefer to occupy the 2a sites at lower temperatures, but as the temperature\nrises, it is more likely that they will occupy the 12k site. Cr3+ions are more likely to occupy the 12k site\nthan the 2a site at a specific annealing temperature (>700 °C).\nI. INTRODUCTION\nHexaferrites, also known as hexagonal ferrites or\nhexagonal ferrimagnets, are a class of magnetic materials\nthat have been of great interest to researchers since their\ndiscovery in the 1950’s. These hexaferrites are found in\nnumerous types such as M, Y, Z, W, X, U-type com-\nmonlydopedwithzinc, strontium, nickel, aluminum, and\nmagnesium. The most common properties to all hexa-\nferrites include that all are ferrimagnetic, their proper-\nties of magnetism are based on the crystal structure, and\nthey take different amounts of energy to magnetize in a\nspecific direction within the crystal because of spin-orbit\ninteraction1.\nParticularly, we are interested in M-type strontium\nhexaferrite (SrFe 12O19, SFO) that falls to space group\nP63/mmcwhich has a crystal structure of hexagonal\nmagnetoplumbite. The unit cell of SFO having two for-\nmula units is presented in Fig(1). The iron ions in this\nstructure are coupled in a tetrahedral, trigonal bipyrami-\ndal, andoctahedralmannerbyoxygenions.The magnetic\nproperty is retained in SFO mainly due to the occupancy\nof Fe+3ions in five inequivalent sites (namely 2a, 2b, 4f 1,\n4f2, and 12k), three octahedral sites (namely 2a, 12k, and\n4f2), one trigonal bipyramidal site (2b), and one tetrahe-\ndral site (4f 1). However, the range (degree) of magnetic\nproperties will be influenced by the shape and size of\nmaterial particles mainly in the context of thin films,\nnanoparticles2–5.\nIn SFO, there is an involvement of interactions be-\ntween the moments or moments with lattice ions, tend\nto contribute anisotropic energy which is termed as mag-\nnetocrystalline anisotropy (MA). In a more explicit way,\na)Author to whom correspondence should be addressed. Electronic\nmail: sk162@msstate.eduit is the dependence of the magnetic properties on the ap-\npliedfielddirectionrelativetothecrystallattice. Magne-\ntocrystalline anisotropy energy (MAE), an integral prop-\nerty of a ferromagnetic crystal, is the energy difference\nto magnetize a crystal along easy and hard direction of\nmagnetization2,6. TheprimarysourceofMAisspin-orbit\ncoupling(SOC).Becauseofthiscoupling, orbitalsofelec-\ntronsarecoupledwiththespinofelectronsandfollowthe\nspin direction no matter how the magnetization changes\nFIG. 1. (a) One double formula unit cell of SrFe 12O19. Small\nmaroonspheresare Oatoms, andtwohugegrayspheresareSr\natoms. Fe3+ions are represented by colored spheres encircled\nby polyhedra made of Oatoms in a variety of inequivalent\nsites: 2a(red), 2b(blue), 4f1(green), 4f2(yellow), and 12k\n(purple).(b) A schematic representation of the Fe3+ions of\nSrFe 12O19in their lowest energy spin configuration. The local\nmagnetic moment at each atomic location is represented by\nthe arrows.arXiv:2306.11952v1  [cond-mat.mtrl-sci]  21 Jun 20232\nits direction in space6. The anisotropy that arises on a\ncrystal are mainly due to the shape of a magnetic parti-\ncle at the quantum scale, atomic diffusion at sufficiently\nhigh temperature, and the interaction between a ferro-\nmagnetic and an antiferromagnetic materials6–9.\nSFO is one of the best candidates among hexaferrite\ngroups owing to its industrial and electronic implica-\ntions. In the beginning years, Technophiles were moti-\nvated about SFO to make permanent magnets, record-\ning media, electric motors because of its high saturation\nmagnetization, large coercivity, optimum Curie temper-\nature, supreme magnetocrystalline anisotropy, and more\nchemical stability. Nowadays, due to technological ad-\nvancement, it is equally growing interest in the devel-\nopment of nano fibres, electronic components for mobile\nand wireless communications. Recently, researchers have\nbeen characterized Y, M, U, and Z ferrites as multifer-\nroicsevenatroomtemperature. Thesemultiferroicshave\na wide range of practical implications like multi-state\nmemory elements, memory media, and novel functional\nsensors2,10,11.\nSeveral successful investigations have been done on\nSFO to understand its electronic structure by com-\nputational and experimental approach. To uplift the\nstrength of magnetic and electric properties, researchers\nare substituted ions or pair of ions in a different con-\ncentration mainly on Fe sites of SFO. Majority of re-\nsearchers have followed the non-magnetic ions substitu-\ntion into Fe sites to enhance further the value of satura-\ntion magnetization (M s). In case of Zr-Cd substitution\n(SrFe 12−2x(ZrCd) xO19), the value of M saugmented in\nthe limit of concentration x= 0.2, whereas the value\nof coercivity declined with increasing concentration of\nZr-Cd12. Substitution of Er-Ni pair in SFO showed the\ncontinuous rise in the value of M sand coercivity in ac-\ncordance with the concentration13. However, the sub-\nstitution of certain pairs like Zn-Nb,14Zn-Sn,15–17and\nSn-Mg18,19showed the increasing trend of M sand de-\ncreasing pattern of coercivity.\nIn this study, we performed the first-principles total-\nenergy calculations to analyze the link between site occu-\npation and magnetic properties of substituted strontium\nhexaferrite, SrFe 12−xCrxO19with x= 0.5andx= 1.0.\nEvery configuration of substituted SFO appears with a\nparticular probability. To determine the formation prob-\nabilities of its various configurations at a typical anneal-\ning temperature ( 1000K), we used the Boltzmann dis-\ntribution function. We show that our calculation pre-\ndicts a decrease of saturation magnetization ( Ms) as well\nas a decrease in magnetic anisotropy energy (MAE) of\nSrFe 12−x(Cr) xO19atx= 0.5and1.0compared to the\npure M-type SFO. This result is also in good agree-\nment with the experimental observation as observed in\nGhasemi et al. (2009)19.II. COMPUTATIONAL DETAILS\nWe pursued the first-principles total-energy calcula-\ntions for configuration SrFe 12−x(Cr) xO19atx= 0.5and\n1.0. In our calculation, a unit cell of two formula units\nof SFO is used. The structural optimization calcula-\ntion along with total energies and forces were carried out\nusing density functional theory by projector augmented\nwave (PAW) potential as executed in VASP. Depending\non the ground state ferrimagnetic spin ordering of Fe,\nour calculations were solely focused on spin-polarized3,20.\nThe expansion of the wave function was in the form of\nplane waves with a 520eVenergy cut-off used for pristine\nSFO, Cr-substituted SFO. A 7×7×1Monkhorst-Pack\nk-mesh was used to sample a Brillouin zone with a Fermi-\nlevel smearing of 0.2eVapplied through the Methfessel-\nPaxton method21,22. The electronic relaxation was ac-\ncomplished till the change in free energy and the band\nstructure energy less than 10−7eV. In addition, we fully\noptimized the structure by relaxing the positions of ions\nand cell shape till the change in total energies between\ntwo ionic steps less than 10−4eV. We used the Perdew-\nBurke-Ernzerhof (PBE) generalized gradient approxima-\ntion (GGA) to describe fully the electron exchange-\ncorrelation effect23. Furthermore, we implemented the\nGGA +Umethod in the simplified rotationally invari-\nant approach described by Dudarev et al. to address\nthe localized 3delectrons in Fe24. The necessity of U eff\nfor Fe was fulfilled by setting 3.7eVbased on previous\nstudy. we set U effto be zero for all other elements25.\nToevaluatethemagnetocrystallineanisotropyenergy, we\nfirst carried out an accurate collinear calculation in the\nground state, and then we followed the spin-orbit cou-\nplingcalculationsintwodifferentspinorientationswithin\nnon-collinear setup. The substitution of foreign atoms\nin five crystallographic inequivalent Fesites can change\nthe magnetic characteristics of SFO. When foreign atoms\nare substituted in a SFO unit cell, there are a variety of\nenergetically distinct configurations. The magnetism of\nsubstituted SFO is highly dependent on the on-site pref-\nerences of the replaced atoms since SFO is ferrimagnetic.\nUnderstanding the site preference of substituted atoms\nis crucial in order to research how substitution affects\nmagnetic characteristics. The substitution energy can be\ncalculated to find the substituted atom’s preferred site.\nThe substitution energy Esub[i]for configuration iat0 K\nis given by\nEsub[i] =ECSFO [i]−ESFO−X\nβnβϵβ(1)\nwhere ECSFO [i]is the total energy per unit cell of Cr-\nsubstituted SFO in configuration i, whereas ESFOis the\ntotal energy per unit cell of pure SFOandϵβis the to-\ntal energy per atom for element β(β=CrandFe)in\nits most stable crystal structure. nβis the number of\natoms of type βadded or removed; if one atom is added,\nnβ= +1, and when one atom is withdrawn, nβ=−1.\nThe calculation of Magnetic Anisotropy Energy (MAE)3\nis important for understanding the preferred magnetiza-\ntion directions in a material. Mathematically, MAE is\ndefined as the difference between the two total energies\nwhere the spin quantization axes are oriented along two\ndistinct directions:26\nEa=E(100)−E(001) (2)\nwhere E(100)is the total energy with the spin quantiza-\ntion axis in the magnetically hard axis and E(001)is the\ntotal energy with the spin quantization axis in the mag-\nnetically easy axis. The total energies in Eq.(2) are com-\nputed by the non-self-consistent calculations, where the\nspin densities are kept constant. With the help of MAE,\nthe uniaxial magnetic anisotropy constant, K1, can be\ncomputed as27,28\nK1=Ea\nVsin2θ(3)\nwhere Vistheequilibriumvolumeoftheunitcelland θis\nthe angle between the two spin quantization axis orienta-\ntions (90◦in the present scenario). The anisotropy field,\nHa, which is related to the coercivity can be expressed\nas29\nHa=2K1\nMs(4)\nwhere K1is the magnetocrystalline anisotropy constant\nandMsis the saturation magnetization.\nWhen the difference in substitution energies Esubbe-\ntween different configurations is relatively small com-\npared to the thermal energy at high annealing temper-\natures ( ≳1000 K), the site preference of substituted\natoms in hexaferrite can change. This change in site oc-\ncupation preference can be described using the Maxwell-\nBoltzmann distribution, which relates to the formation\nprobability. The site occupation probability or the for-\nmation probability Pi(T)of configuration iat tempera-\ntureTis given by\nPi(T) =giexp (−∆Gi/kBT)P\njgjexp (−∆Gj/kBT), (5)\n∆Gi= ∆Ei+P∆Vi−T∆Si, (6)\n∆Si=kBln (gi)−kBln (g0), (7)\nwhere ∆Gi,∆Ei,∆Vi, and ∆Siare the change in free\nenergy, substitutionenergy, unitcellvolume, andentropy\nof the configuration irelative to the ground state con-\nfiguration. P,kB, and giare the pressure, Boltzmann\nconstant, and multiplicity of configuration i. g0is the\nmultiplicity of the ground state configuration. we con-\nsidered ∆Sito be the same for all configurations based\non prior literature2. Eq.(7) enhances the model through\nthe explicit computation of the entropy change concern-\ning the most stable configuration30,31.Hence, when the probability of higher energy configu-\nrationsbecomessignificantattheannealingtemperature,\nit can be inferred that in a substituted SFO sample, mul-\ntiple configurations exist rather than a single one. Con-\nsequently, any physical quantity of the SFO sample will\nbe a weighted average of the corresponding properties in\nthese different configurations.\n⟨Q⟩=X\niP1000 K (i)·Qi (8)\nwhere P1000 K (i)andQiare the formation probability at\n1000 Kand a physical quantity Qof the configuration\ni. The weighted average calculated by Eq.(8) is the ma-\nterial’s low-temperature property even though 1000 Kis\nused for computation because the crystalline configura-\ntions of CSFO will be distributed according to this value\nduring the annealing process.\nIII. RESULTS AND DISCUSSION\nIn order to visualize the doping effect of the Cr+3ion\non the structural and magnetic properties of SFO, we re-\nplaced the Fe+3from various lattice locations. We found\nthat it has a significant effect on the structural and mag-\nnetic properties of this system. We fully relaxed the vol-\nume, ionic positions, and shape of SFO and Cr-doped\nSFO. The crystal structure remains hexagonal under all\ncircumstances. We further carried out our calculations\nwhen we match the optimized lattice parameter of pure\nSFO (a = 5.928 Å, c = 23.195 Å) with experimental lat-\ntice constants (a = 5.890 Å, c = 23.182 Å); less than\n1%difference between the calculated and experimental\nvalues. For x = 1.0, the calculated lattice parameters (a\n= 5.930 Å, c = 23.076 Å) were found to be very con-\nsistent with experimental lattice parameters (a = 5.902\nÅ, c = 23.024 Å)32,33. Fig.2 shows the variation of lat-\ntice parameters from theory and experiment explicitly.\nTo compensate for the unavailability of the experimental\nlattice parameter at x = 0.5, we are comparing the ex-\nperimental values at x = 0.6 with our calculated values\nat x = 0.5 for the comparison in Fig. 2. The substitution\nof Cr in SFO has little effect on the lattice parameters\nor unit cell volume. Because the radius of Cr+3(0.630 Å)\nis similar to that of Fe+3(0.645 Å), this is an expected\noutcome.\nIn this paper, we measured the various physical quan-\ntities by varying the concentration of Cr in the unit cell\nof SFO. For the x = 0.5, one Cr atom was substituted at\none of the 24 Fe sites of the unit cell. Many of these Fe\nsites are equivalent when crystallographic symmetry op-\nerations are applied, leaving only five inequivalent struc-\ntures. We label these inequivalent configurations [2a],\n[2b], [ 4f1], [4f2], and [12k] using the crystallographic\nname of the Fe site. These structures were found by\nfully optimizing the unit cell shape, volume, and ionic\npositions. We estimated the substitution energy, Esub,4\nFIG. 2. Comparision of predicted and experimental lattice\nconstant of unit cell as a function of fraction of Cr (x).\nto comprehend the site preference of the substituted Cr\natom.\nTable I displays the results of our calculation for each\nof the five inequivalent configurations in ascending or-\nder of substitution energy( Esub). The configuration [2a]\nhas the lowest Esubfollowed by [12k], and [ 4f2] which\nis consistent with the experimental outcomes32,34. We\ncan conclude that the [2a] site is the most preferred site\nfor the Cr atom at 0 K. We used Eq.(5) to calculate the\nprobability of forming each configuration as a function\nof temperature. Because the change in volume between\ndifferent configurations is so small (less than 0.3 Å3), we\nmay discard the P∆Vterm as negligible (in the order of\n10−7eVat a standard pressure of 1 atm) compared to the\n∆Esub(i) term in Eq.(6). The entropy change ∆Shas\ntwo components: configurational, ∆Sc, and vibrational,\n∆Svib·31.∆Svibis around 0.1-0.2 kB/atom for binary\nsubstitutional alloys like the present system, and ∆Scis\n0.1732kB/atom. As a result, we assign ∆S= 0.3732kB/\natom.\nFig.(3) shows the formation probability of different\nconfigurations of SrFe 12−xCrxO19with x= 0.5at dif-\nferent temperatures. The foreign Cr3+ions prefer to oc-\ncupy host Fe3+ions from the 2aand the 12ksites. The\nformation probabilities of [2b],[4f1], and [4f2]are trivial\nand not displayed in Fig.(3). A Cr3+ion has a 100 %\nprobability of occupying the [ 2a] site at 0K and it’s value\nsharply declines as the temperature rises, while the prob-\nability of occupancy of Cr3+at the [ 12k] site is maximum\n(88%) at 1000K. At a typical annealing temperature of\n1000 KforCSFO, the site occupation probability of the\nsite[2a]and[12k]are7%and88.4%respectively. In\nCSFO, the doped Cr3+ions are more likely to occupy\nFe3+ions at the [ 12k] site than the [ 2a] site because of\nthe higher multiplicity of the [12k] site.\nFor the x = 1.0, two Cr atoms were substituted at\ntwo of the 24 Fe sites of the unit cell. Many of these\nFe sites are equivalent when crystallographic symmetryoperationsareapplied,leavingonly15inequivalentstruc-\ntures. These structures were found by fully optimizing\nthe unit cell shape, volume, and ionic positions. To un-\nderstand the site preference of a substituted Cr atom, we\nestimated the substitution energy, Esub. Table II pro-\nvides the results of our calculation for each of the fifteen\ninequivalent configurations in ascending order of substi-\ntution energy( Esub). The configuration [2a, 2a] has the\nlowest Esubfollowed by [12k, 2a], and [12k, 12k]. We also\nused Eq.(5) to calculate the probability of forming each\nconfiguration as a function of temperature. Because the\nchange in volume between different configurations is so\nsmall(lessthan0.7Å3), wemaydiscardthe P∆Vtermas\nnegligible (in the order of 10−7eVat a standard pressure\nof 1 atm) compared to the ∆Esub(i) term in Eq.(6).\nFig. (4) shows the formation probability of different\nconfigurations of SrFe 12−xCrxO19with x= 1.0at differ-\nenttemperatures. Theforeign Cr3+ionsprefertooccupy\nhost Fe3+ions from the [2a,2a],[12k,2a],and [12k,12k]\nsites. The formation probabilities of other sites are triv-\nial and not displayed in Fig.(4). A Cr3+ion has a 100 %\nprobability of occupying the [2a,2a]site at 0K and it’s\nvalue sharply declines as the temperature rises, while the\nprobability of occupancy of Cr3+at the [12k,12k]site\nis maximum (66.4 %) at 1000 K. At a typical annealing\ntemperature of 1000 KforCSFO, the site occupation\nprobability of the site [12k,2a]is17.8%. In CSFO, the\ndoped Cr3+ions are more likely to occupy Fe3+ions at\nthe[12k,12k]site than the [2a,2a]site because of it’s\nhigher multiplicity. We exclusively utilize the forma-\ntion probability at elevated temperatures for computing\nweighted averages. This is because the arrangement of\nCSFOconfigurationsduringtheannealingprocesswillbe\ndistributed according to these values. Table III displays\nthe weighted average of corresponding quantities as the\nconcentration of Cr3+increases. The volume of CSFO\ndecreases as we increase the concentration of Cr3+be-\nFIG.3. Temperaturedependenceoftheformationprobability\nof different configurations of SrFe 12−xCrxO19with x= 0.5.\nIn the plot, the configurations with trivial probabilities are\nnot displayed.5\nTABLE I. Physical properties of inequivalent configurations of SrFe 12−xCrxO19with x= 0and0.5: Doped amount(x),\nmultiplicity( g), substitution energy( Esub), total magnetic moment ( Mtot), volume of the unit cell ( V), saturation magnetization\n(Ms), magnetocrystalline anisotropy energy ( Ea), uniaxial magnetic anisotropy constant ( K1), anisotropy field ( Ha), and the\nformation probability at 1000K (P1000K). All values are for a double formula unit cell containing 64atoms.\nx config. g Esub(eV)mtot(µB)Volume(Å3)Ms(emu/g )Ea(meV) K1(KJ/m3)Ha(koe)P1000K\n0.0 SFO - - 39.99 706.18 105.19 0.87 197.12 7.50 1.000\n0.5 [2a]2 -3.41 38.00 704.70 100.00 1.20 272.45 10.89 0.076\n[12k]12 -3.33 38.00 705.04 100.00 0.90 204.25 8.17 0.884\n[4f2]4 -3.27 42.00 703.70 111.00 0.90 204.63 7.39 0.049\n[2b]2 -2.96 38.00 703.17 100.00 0.60 136.52 5.44 0.000\n[4f1]4 -2.33 42.00 705.49 111.00 0.20 45.36 1.64 0.000\nTABLE II. Physical properties of inequivalent configurations of SrFe 12−x(Cr) xO19with x= 1.0: Doped amount(x), substitu-\ntionenergy( Esub), totalmagneticmoment( mtot),volumeoftheunitcell( V), saturationmagnetization( Ms), magnetocrystalline\nanisotropy energy ( Ea), uniaxial magnetic anisotropy constant ( K1), anisotropy field ( Ha), and the formation probability at\n1000K (P1000K). All values are for a double formula unit cell containing 64atoms.\nx config. Esub(eV)mtot(µB)Volume(Å3)Ms(emu/g )Ea(meV) K1(KJ/m3)Ha(koe)P1000K\n1.0 [2a,2a]-6.84 36.00 703.73 95.00 1.50 341.39 14.36 0.001\n[12k,2a]-6.75 30.00 703.03 79.20 1.10 250.34 12.64 0.178\n[12k,12k]-6.69 36.00 703.24 95.00 0.80 182.01 7.66 0.664\n[2a,4f2]-6.68 40.00 701.63 106.00 1.30 296.45 11.20 0.009\n[12k,4f2]-6.62 34.00 701.89 89.70 0.90 205.16 9.13 0.145\n[4f2,4f2]-6.56 44.00 700.33 116.00 1.00 228.46 7.84 0.001\n[12k,2b]-6.30 30.00 701.93 79.20 0.50 113.97 5.75 0.001\n[2b,4f2]-6.15 34.00 698.58 89.70 0.70 160.33 7.10 0.000\n[2a,2b]-6.09 30.00 701.90 79.20 0.40 91.18 4.60 0.000\n[2b,2b]-5.99 36.00 699.37 95.00 2.00 457.55 19.15 0.000\n[12k,4f1]-5.66 40.00 704.40 106.00 0.20 45.43 1.72 0.000\n[4f1,4f2]-5.61 44.00 703.98 116.00 0.20 45.46 1.57 0.000\n[2a,4f1]-5.47 40.00 704.37 106.00 0.90 204.44 7.76 0.000\n[2b,4f1]-5.15 38.91 702.14 103.00 0.10 22.79 0.88 0.000\n[4f1,4f1]-4.67 44.00 705.71 116.00 0.60 136.03 4.70 0.000\nTABLE III. Weighted averages of physical properties of pure and Cr-doped strontium hexaferrite: volume of the unit cell (V),\ntotal magnetic moment (mtot), saturation magnetization (Ms), magnetocrystalline anisotropy energy ( Ea)), uniaxial magnetic\nanisotropy constant (K1), and anisotropy field (Ha). The column labeled (Ms)displays the calculated values at 0K along with\nthe bracketed experimental values from K. Praveena et al.35for the comparison.\nMaterial V(Å3)mtot(µB)Ms(emu/g) Ea(meV) K1(kJ/m3)Ha(kOe)\nSrFe 12O19 706.18 39 .99 105 .19(59 .33) 0 .87 197 .12 7 .50\nSrFe 11.5Cr0.5O19704.39 38 .16 100 .46(36 .01) 0 .91 208 .58 8 .30\nSrFe 11.0Cr1.0O19702.07 34 .63 91 .40(30 .04) 0 .87 198 .44 8 .79\ncause of the smaller atomic radius of Cr3+ion. The mag-\nnetic moment of CSFO is also in the decreasing trend as\nthe amount of doped Cr3+increases owing to its smaller\nmagnetic moment (3 µB) than the Fe3+(5µB). Similarly,\nthe saturation magnetization is decreasing monotonically\nas we increase the Cr3+concentration which is consis-\ntent with the K. Praveena et al.35. Although the value of\nmagnetocrystalline anisotropy (K1)is slightly increased,\nthe reduction in saturation magnetization (Ms)is much\nmore significant. So, their resultant effect causes the\nanisotropy field (Ha)to increase as the fraction of Cr is\nraised. In Table IV, we have provided the atomic contri-\nbution from each sublattice to the overall magnetic mo-mentofCSFO.Itcanbeobservedthatthetotalmagnetic\nmoment of the unit cell is slightly distinct from the sum\nof local magnetic moments. This disparity arises from\nthe contribution of the interstitial region to the overall\nmagnetic moment.\nIV. CONCLUSIONS\nFirst-principlestotal-energycalculationsbasedonden-\nsity functional theory were used to study Cr-substituted\nSFO( SrFe 12−xCrxO19)with x= 0.0,0.5,1.0. Theresults\nshowed that increasing the fraction of Cr atoms reduced6\nFIG.4. Temperaturedependenceoftheformationprobability\nof different configurations of SrFe 12−xCrxO19with x= 1.0.\nIn the plot, the configurations with trivial probabilities are\nnot displayed.\nTABLE IV. Contribution of atoms in each sublattice to the\ntotal magnetic moment of Cr-substituted strontium hexafer-\nrite structures [ 12k]and[12k,12k]compared with pure SFO.\nAll magnetic moments are in µB, and ∆mis measured rela-\ntive to the values for pure SFO.\nSFO x = 0.5, [12 k] x = 1.0, [12 k, 12 k]\nSite Atoms m Atoms m ∆mAtoms m ∆m\n2d 2 Sr -0.006 2 Sr -0.005 0.001 2 Sr -0.005 0.001\n2a 2 Fe 8.328 2 Fe 8.334 0.006 2 Fe 8.336 0.008\n2b 2 Fe 8.130 2 Fe 8.131 0.001 2 Fe 8.129 -0.001\n4f14 Fe -16.184 4 Fe -16.159 0.025 4 Fe -16.141 0.043\n4f24 Fe -16.416 4 Fe -16.410 0.006 4 Fe -16.415 0.001\n12k 1 Fe 4.180 1 Cr 2.680 0.319 1 Cr 2.689 0.311\n1 Fe 4.178 1 Fe 4.180 0.000 1 Cr 2.692 0.308\n10 Fe 41.794 10 Fe 41.792 -0.002 10 Fe 41.805 0.011\n4e 4 O 1.400 4 O 1.292 -0.108 4 O 1.165 -0.235\n4f 4 O 0.356 4 O 0.294 -0.062 4 O 0.239 -0.117\n6h 6 O 0.160 6 O 0.151 -0.009 6 O 0.142 -0.018\n12k 24 O 3.140 24 O 2.777 -0.363 24 O 2.422 -0.718\nΣm 39.053 37.056 35.057\nmtot 40 38 36\nthe total magnetic moment of the SFO unit cell. This re-\nduction in magnetization was obtained by low magnetic\nCr atoms replacing Fe3+ions at two of the majority spin\nsites, 2a and 12k, resulting negative contribution to the\nmagnetization. Our substitution energy and formation\nprobability analysis predicts that Cr atoms preferentially\noccupy the 2a, 12k, and 4f 2sites, consistent with experi-\nmentalobservations. IncreasingthefractionofCrinSFO\nleads to a rise in the magnetic anisotropic field (Ha)de-\nspite a decrease in magnetization and a slight increase\nin magnetocrystalline anisotropy (K1). This increase in\nanisotropic field (Ha)is supported by a formation prob-\nability study, which shows that at higher temperatures\n(>700 °C),Cr3+ions are more likely to occupy the 12k\nsite rather than the 2a site because of its higher multi-\nplicity.ACKNOWLEDGMENTS\nThis work was supported by the Center for Compu-\ntational Science (CCS) at Mississippi State University.\nComputertimeallocationhasbeenprovidedbytheHigh-\nPerformance Computing Collaboratory (HPC2)at Mis-\nsissippi State University.\n1R. C. Pullar, “Hexagonal ferrites: a review of the synthesis, prop-\nerties and applications of hexaferrite ceramics,” Progress in Ma-\nterials Science 57, 1191–1334 (2012).\n2V. Dixit, S.-G. Kim, J. Park, and Y.-K. Hong, “Effect of ionic\nsubstitutionsonthemagneticpropertiesofstrontiumhexaferrite:\nA first principles study,” AIP Advances 7, 115209 (2017).\n3C. Fang, F. Kools, R. Metselaar, R. De Groot, et al., “Mag-\nnetic and electronic properties of strontium hexaferrite srfe12o19\nfromfirst-principlescalculations,” JournalofPhysics: Condensed\nMatter 15, 6229 (2003).\n4G. D. Soria, P. Jenus, J. Marco, A. Mandziak, M. Sanchez-\nArenillas, F. Moutinho, J. E. Prieto, P. Prieto, J. Cerdá,\nC. Tejera-Centeno, et al., “Strontium hexaferrite platelets: a\ncomprehensive soft x-ray absorption and mössbauer spectroscopy\nstudy,” Scientific reports 9, 1–13 (2019).\n5X. Obradors, X. Solans, A. Collomb, D. Samaras, J. Rodriguez,\nM. Pernet, and M. Font-Altaba, “Crystal structure of strontium\nhexaferrite srfe12o19,” Journal of Solid State Chemistry 72, 218–\n224 (1988).\n6A. Aharoni, Introduction to the Theory of Ferromagnetism\n(Clarendon Press, New York, 1996).\n7T. Miyazaki and H. Jin, The physics of ferromagnetism , Vol. 158\n(Springer Science & Business Media, 2012).\n8W. H. Meiklejohn and C. P. Bean, “New magnetic anisotropy,”\nPhysical review 102, 1413 (1956).\n9Y. Hong, Z. Qiu, Z. Zheng, G. Wang, H. Yu, D. Chen, G. Han,\nD. Zeng, and J. Liu, “Influencing mechanisms of atomic diffu-\nsion and compositional distribution on the magnetic anisotropy\nof cr/smco/(cu)/cr thin films,” Acta Materialia 164, 627–635\n(2019).\n10Q. Li, J. Song, M. Saura-Múzquiz, F. Besenbacher, M. Chris-\ntensen, and M. Dong, “Magnetic properties of strontium hexa-\nferritenanostructuresmeasuredwithmagneticforcemicroscopy,”\nScientific reports 6, 1–7 (2016).\n11G. Tan and X. Chen, “Structure and multiferroic properties of\nbarium hexaferrite ceramics,” Journal of Magnetism and Mag-\nnetic Materials 327, 87–90 (2013).\n12M. N. Ashiq, M. J. Iqbal, and I. H. Gul, “Structural, magnetic\nand dielectric properties of zr–cd substituted strontium hexafer-\nrite(srfe12o19)nanoparticles,” JournalofAlloysandCompounds\n487, 341–345 (2009).\n13M. N. Ashiq, M. J. Iqbal, M. Najam-ul Haq, P. H. Gomez, and\nA. M. Qureshi, “Synthesis, magnetic and dielectric properties\nof er–ni doped sr-hexaferrite nanomaterials for applications in\nhigh density recording media and microwave devices,” Journal of\nmagnetism and magnetic materials 324, 15–19 (2012).\n14Q. Fang, H. Bao, D. Fang, and J. Wang, “Temperature depen-\ndence of magnetic properties of zinc and niobium doped stron-\ntium hexaferrite nanoparticles,” Journal of applied physics 95,\n6360–6363 (2004).\n15A. Ghasemi, V. Šepelák, X. Liu, and A. Morisako, “The role of\ncations distribution on magnetic and reflection loss properties of\nferrimagnetic srfe 12- x (sn 0.5 zn 0.5) x o 19,” Journal of Applied\nPhysics 107, 09A734 (2010).\n16A. Ghasemi and V. Šepelák, “Correlation between site prefer-\nenceandmagneticpropertiesofsubstitutedstrontiumferritethin\nfilms,” Journal of magnetism and magnetic materials 323, 1727–\n1733 (2011).\n17V. Dixit, D. Thapa, B. Lamichhane, C. N. Nandadasa, Y.-K.\nHong, and S.-G. Kim, “Site preference and magnetic properties7\nof zn-sn-substituted strontium hexaferrite,” Journal of Applied\nPhysics (2019).\n18A. Davoodi and B. Hashemi, “Magnetic properties of sn–mg sub-\nstituted strontium hexaferrite nanoparticles synthesized via co-\nprecipitation method,” Journal of Alloys and Compounds 509,\n5893–5896 (2011).\n19A. Ghasemi, V. Sepelak, X. Liu, and A. Morisako, “Microwave\nabsorptionpropertiesofmn–co–sndopedbariumferritenanopar-\nticles,” IEEE Transactions on Magnetics 45, 2456–2459 (2009).\n20E. Gorter, “Saturation magnetization of some ferrimagnetic ox-\nides with hexagonal crystal structures,” Proceedings of the IEE-\nPart B: Radio and Electronic Engineering 104, 255–260 (1957).\n21H.J.MonkhorstandJ.D.Pack,“Specialpointsforbrillouin-zone\nintegrations,” Physical review B 13, 5188 (1976).\n22M. Methfessel and A. Paxton, “High-precision sampling for\nbrillouin-zone integration in metals,” Physical Review B 40, 3616\n(1989).\n23J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient\napproximation made simple,” Physical review letters 77, 3865\n(1996).\n24S. Dudarev, G. Botton, S. Savrasov, C. Humphreys, and A. Sut-\nton, “Electron-energy-loss spectra and the structural stability of\nnickel oxide: An lsda+ u study,” Physical Review B 57, 1505\n(1998).\n25L. S. Liyanage, S. Kim, Y.-K. Hong, J.-H. Park, S. C. Erwin, and\nS.-G. Kim, “Theory of magnetic enhancement in strontium hex-\naferrite through zn–sn pair substitution,” Journal of magnetism\nand magnetic materials 348, 75–81 (2013).\n26P. Ravindran, A. Delin, P. James, B. Johansson, J. M. Wills,\nR. Ahuja, and O. Eriksson, “Magnetic, optical, and magneto-\noptical properties of mnx (x=as, sb, or bi) from full-potential\ncalculations,” Phys. Rev. B 59, 15680–15693 (1999).\n27F.Muñoz,A.H.Romero,J.Mejía-López,andJ.L.Morán-López,\n“Finitesizeeffectsonthemagnetocrystallineanisotropyenergyinfe magnetic nanowires from first principles,” Journal of Nanopar-\nticle Research 15(2013).\n28J.SmitandH.Wijn,“Physicalpropertiesofferrites,” in Advances\nin Electronics and Electron Physics , Vol. 6 (Elsevier, 1954) pp.\n69–136.\n29C.Kittel,“Physicaltheoryofferromagneticdomains,” Rev.Mod.\nPhys. 21, 541–583 (1949).\n30C. Gilmore, Materials science and engineering properties (Cen-\ngage Learning, 2014).\n31A. van de Walle and G. Ceder, “The effect of lattice vibrations\non substitutional alloy thermodynamics,” Rev. Mod. Phys. 74,\n11–45 (2002).\n32Q. Fang, H. Cheng, K. Huang, J. Wang, R. Li, and Y. Jiao,\n“Doping effect on crystal structure and magnetic properties\nof chromium-substituted strontium hexaferrite nanoparticles,”\nJournal of Magnetism and Magnetic Materials 294, 281–286\n(2005).\n33K. Kimura, M. Ohgaki, K. Tanaka, H. Morikawa, and\nF. Marumo, “Study of the bipyramidal site in magnetoplumbite-\nlike compounds, srm12o19 (m = al, fe, ga),” Journal of Solid\nState Chemistry 87, 186–194 (1990).\n34R. Parker, “Ferrite,” Proceedings of ICF-3 375(1980).\n35K. Praveena, M. Bououdina, M. P. Reddy, S. Srinath, R. Sand-\nhya, and S. Katlakunta, “Structural, magnetic, and electrical\nproperties of microwave-sintered cr3+-doped sr hexaferrites,”\nJournal of Electronic Materials 44, 524–531 (2015).\n36G. P. Nethala, R. Tadi, G. R. Gajula, K. N. C. Kumar, and\nV. K. Veeraiah, “Investigations on the structural, magnetic and\nmossbauer properties of cerium doped strontium ferrite,” Physica\nB: Condensed Matter (2018)."    },    {        "title": "2306.17592v2.Achiral_dipoles_on_a_ferromagnet_can_affect_its_magnetization_direction.pdf",        "content": "Achiral dipoles on a ferromagnet can affect its magnetization direction\nRagheed Alhyder,1Alberto Cappellaro,1Mikhail Lemeshko,1and Artem G. Volosniev1\nInstitute of Science and Technology Austria (ISTA),\nAm Campus 1, 3400 Klosterneuburg, Austria\n(*Electronic mail: alberto.cappellaro@ist.ac.at)\n(Dated: 15 September 2023)\nWe demonstrate the possibility of a coupling between the magnetization direction of a ferromagnet and the tilting angle\nof adsorbed achiral molecules. To illustrate the mechanism of the coupling, we analyze a minimal Stoner model that\nincludes Rashba spin-orbit coupling due to the electric field on the surface of the ferromagnet. The proposed mechanism\nallows us to study magnetic anisotropy of the system with an extended Stoner-Wohlfarth model, and argue that adsorbed\nachiral molecules can change magnetocrystalline anisotropy of the substrate. Our research’s aim is to motivate further\nexperimental studies of the current-free chirality induced spin selectivity effect involving both enantiomers.\nI. INTRODUCTION\nChirality induced spin selectivity (CISS) has become an\numbrella name for a number of seemingly related experimen-\ntal reports on the coupling between geometrical chirality and\nthe spin degree of freedom1–4. Initially, the effect was dis-\ncovered by analyzing photo-excited electrons after they pass\nthrough adsorbed biological molecules5,6and in currents of\nelectrons through organic chiral junctions7. Later, the inter-\nplay between chirality and the electron’s spin was observed in\nchemical reactions8,9, in enantioselective adsorption on mag-\nnetic substrates10–12, and even in fully inorganic systems13,14.\nBy now, there is a great collection of experimental set-ups and\nobservables that demonstrate CISS. In spite of this, detailed\ntheoretical understanding of the CISS effect, much needed for\na further development of the field, is still lacking15,16.\nOne problem in building theoretical models of CISS is that\nthe experimental platforms differ drastically from one another,\nand are traditionally studied using independent methods and\napproaches. This does not rule out the possibility of a unify-\ning mechanism behind the CISS effect; it suggests, however,\nto study the existing classes of experiments separately. From\nthe theoretical standpoint, it is natural to start by examining\nsystems where the interplay between spin and chirality is ob-\nserved without applied currents17–20, note also relevant theo-\nretical models in Refs. 40 ? ? . First, these systems are usu-\nally easier to analyze, as they can be studied using methods\ndeveloped for steady states. Second, looking into these sys-\ntems might help to answer fundamental questions about the\nCISS effect, in particular, about the origin of time-reversal-\nbreaking correlations necessary for spin-selective phenomena.\nIndeed, systems in equilibrium have vanishing currents, elim-\ninating the key ingredient of many CISS-related theoretical\nmodels21–37from the picture. In the absence of currents, time\nreversal breaking may occur, e.g., as a result of substrate mag-\nnetization38or of non-unitary effects (such as dephasing and\ndissipation) that are used in CISS models22,25,26,39,40.\nArguably, the most puzzling observation of current-free\nCISS experiments is the coupling between the direction of\nmagnetization in the substrate and the tilt of the organic\nmolecules17,19. Although this observation is typically ex-\nplained within the CISS framework, in this contribution we\nwant to highlight the fact that the magnetization of the sub-strate, M, can in principle couple directly to the polarization\nof the molecular layer, P, without the need to involve chi-\nrality. For example, a phenomenological term in the form\nβ(P·M)2can enter the energy functional since it is allowed\nby the time-reversal symmetry of the problem. Here, the pa-\nrameter βdetermines the strength of the effect. Some exper-\nimental results17,19can be explained by the term β(P·M)2if\nβis negative so that the magnetization prefers to be parallel\nto the direction of P, see Fig. 1. For example, this explains\nthe larger tilting angle for samples with the in-plane easy axis\nthan for the out-of-plane easy axis17, and also the correlation\nbetween the direction of the magnetization and the tilt angle\nof the molecular layer19.\nAs we shall illustrate in this paper, β(P·M)2is not the only\nterm allowed by symmetries, and a case-by-case study is prob-\nably needed to establish accurate phenomenological models\nfor every system of interest. For example, on the surface of a\nmetal only the normal component of the electric field should\nplay a role, so that the terms of the form β(P·n)2(M·n)2be-\ncome relevant, where nis the vector normal to the surface. In\nany case, the very possibility of a chirality-independent cou-\npling between the magnetization direction and a tilting angle\nmust motivate current-free experimental studies of the CISS\neffect involving both enantiomers. This might be especially\nimportant in light of the observation that, for certain sub-\nstrates, the effect of chirality on the strength of the coercive\nfield might be a next-to-leading-order effect18.\nII. MAGNETO-ELECTRIC COUPLING\nThe key message of our study is that the electric field at\nthe interface between a ferromagnet and a molecular layer\ncan couple to the vector M, see Fig. 1. This hypothesis falls\ninto the realm of magneto-electric coupling (MEC) effects in a\nbroad sense41. Many properties of MEC are well-established,\ntherefore, we find it appropriate to review them here only\nbriefly.\nIt is expected that the interface MEC is very sensitive to\nthe electronic and atomic structure of constituents42,43, in par-\nticular, because reactive metal surfaces of ferromagnetic ma-\nterials may lead to formation of chemical bonds between the\nmolecules and the substrate44. Therefore, a microscopic first-arXiv:2306.17592v2  [cond-mat.mes-hall]  14 Sep 20232\na\nb\nFIG. 1. Molecules adsorbed on a magnet modify surface electric\nfields, which, in turn, can change magnetocrystalline anisotropy\nof the substrate. Panel a (b) illustrates the preferred magnetiza-\ntion direction without (with) a molecular self-assembled layer. The\nmolecules are shown as dipoles, see panel b. Spin-polarized elec-\ntrons inside the substrate are presented as balls with arrows. Their\ndirection determines the direction of magnetization, M.\nprinciple calculation is required for every experimental set-up\nto accurately determine the resulting MEC. Here, we do not\nattempt such a calculation, as our goal is to provide a basic in-\ntuition for phenomenological terms that can couple the direc-\ntions of PandM. In particular, we illustrate the effect using\na toy model that explains a physical mechanism of coupling,\nand symmetries of the problem.\nThe mechanism connecting electric and magnetic effects in\nour case is Rashba spin-orbit coupling (SOC), which is known\nto be important at surfaces and interfaces45. Before introduc-\ning the toy model in the following section, we note that there\nare a number of other effects that can lead to the interplay be-\ntween magnetization of the surface and adsorbed molecules.\nFor example, the adsorption of molecules can increase the\nstrength of the magnetic exchange interaction on the surface46\nand even invert the magnetic anisotropy of thin films under\ncertain conditions47. These ‘magnetic hardening’ effects relyon chemical bonding between molecular adsorbates and the\nsubstrate, and cannot directly explain the experiments where\nthe ferromagnet is coated with gold19. Screening charge on\nthe surface of a ferromagnetic metal can also modify magne-\ntization of the surface42,48. However, in this effect the direc-\ntion of Mis not important, and therefore it cannot directly\nexplain the experimental data taken for different directions of\nmagnetization17. Finally, we note that density-functional the-\nory calculations on isolated ferromagnetic films indicate that\nexternal electric fields can be used to change the magnetiza-\ntion between in-plane and out-of-plane orientations49. This\nobservation is in line with our phenomenological results, see\nbelow.\nIII. ELECTROSTATICS OF THE MOLECULAR LAYER\nFormulation. The simplest approach to static properties of\nelectrons at interfaces is based upon electrostatics50,51. In this\napproach, the main effect of the organic layer is in the change\nof the electric field across the interface. For simplicity, we\nassume that the electric field on the substrate is determined\ncompletely by the molecules. Furthermore, as we are inter-\nested in surface effects, we restrict the motion of electrons to\nthexyplane. With these assumptions, we write the following\nStoner model (cf. Ref. 52),\nˆHtot=εkI2+PI·σσσ+ˆHSOC, (1)\nwhere I2is the 2 ×2 unity matrix, and εkis a dispersion rela-\ntion without SOC and Stoner terms. For simplicity, we as-\nsume that εkis independent of spin. PI·σσσis the Stoner\nterm, which we introduce to account for a ferromagnetic na-\nture of the substrate; here, σσσis the spin operator, I=|I|is\nthe Stoner parameter and Pis the polarization of the sub-\nstrate, see below. If ˆHSOCis vanishing, then the direction of I\ndefines the direction of magnetization, and a natural quantiza-\ntion axis for the spin with projections: ↑Iand↓I. The corre-\nsponding electronic densities are n↑I=N↑I/Sandn↓I=N↓I/S,\nwhere N↑I/↓Iis the number of particles and Sis the surface\narea. The corresponding polarization of the substrate reads\nP= (n↑I−n↓I)/(n↑I+n↓I). Without loss of generality, we\nassume that I=Ixex+Izez(thus Iy=0).\nThe last term in Eq. (1) describes a Rashba-like spin-orbit\ncoupling53–55\nˆHSOC=αRσσσ·\u0000\nE׈p\u0001\n, (2)\nwhich couples the electric field to electron’s spin, provid-\ning the basis for MEC. αRis a constant that determines the\nstrength of SOC; Eis a homogeneous (at least in first ap-\nproximation) electric field, originating from the charge re-\norganization concurrent to the chemical absorption process for\nthe molecular monolayer.\nSince the substrate is metallic, the electric field has to be\nperpendicular to its surface, E=Ezez. We assume that the\ncomponent Ezis proportional to Pz=Pcos(θP), and its ampli-\ntude is therefore determined by the tilt angle of the molecular\nlayer, θP, see Fig. 6 in the Appendix. The assumption Ez∼Pz3\nis not essential, and used only to simplify the discussion. It is\nonly important that by tilting the angle of the molecules, the\nvalue of Ezchanges. In general, the direction of Iis not per-\npendicular to the surface, and our goal here is to investigate\nthe energy as a function of Ezand the direction of I.\nFor the sake of completeness, it is worth remarking that one\nmust exercise care when working with Eq. (2), since hermitic-\nity of the resulting Hamiltonian is not guaranteed for at least\na couple of relevant situations. The first scenario, now well\nunderstood in solid-state physics, involves the dimensional re-\nduction, occuring when, for instance, a strong potential con-\nfines electrons on an effective one-dimensional ring (see Refs.\n56 and 57 for an extensive discussion about this issue). Sec-\nond, note that spin-orbit coupling is an inherently relativistic\neffect as its derivation might be based upon the Dirac theory?.\nThe lowest-order relativistic corrections to the Hamiltonian\nare given? ? ? ?by Eq. (2) and one more term, interpreted as\nthe coupling between the angular momentum of a radiation\nfield and the magnetic moment of the electron. For our pur-\npose, Eq. (2) is accurate because the electric field is assumed\nto be time- and position-independent and the relativistic cor-\nrections to the eigenstates are neglected.\nEigenstates of the Hamiltonian. To elucidate the interplay\nbetween Iand the Rashba SOC, we calculate the spectrum of\nˆHtotfrom Eq. (1). To this end, we work in the momentum\nrepresentation, where the total Hamiltonian is cast in the form\nˆHtot=εkI2−PIzσz−αREzσykx\n+σx\u0012\nαREzky−PIx\u0013\n.(3)\nNote that the Pauli matrices anticommute with each other, and\ntherefore, we can easily calculate the spectrum by considering\n(ˆHtot−εkI2)2. We derive\nEks=εk−ss\nα2\nRE2zk2x+PI2z+\u0012\nαREzky−PIx\u00132\n,(4)\nwhere s=±1 labels the spin manifold.\nIn Fig. 2 we plot the energy spectrum bands as provided\nby the equation above for εk=k2/2 (here, ¯h≡m≡1). As\na matter of convention, the lower bands have been labelled\nasE+(k). By construction, the Stoner term introduces a gap\nbetween the s=±1 bands, while the Rashba SOC (cf. Eq. (2))\nshifts the band minima.\nTotal energy of the system. The total energy of the system\nis obtained by summing over the lowest N↑I+N↓Istates|ks⟩,\nnamely\nE(tot)=∑Eks, (5)\nwhere the sum is most easily calculated in the continuum\nlimit: ∑k→S\n(2π)2Rd2k.\nTo illustrate this energy, we assume that the energy scale\nassociated with SOC is significantly smaller than all other en-\nergy scales, i.e., Ez→0. With this assumption, we write the\n-1-0.500.51-202468-1-0.500.51-101234a\nbFIG. 2. Energy spectrum of the Hamiltonian ˆHtotas given by Eq. (4).\nThe energy is plotted for kx=ky≡k. The parameter 1 /√nis used as\na unit of length ( nis the density of electrons in the substrate). a: the\nspectrum without the electric field, the parameters are PIz=0.25,\nPIx=0 and Ez=0. b: the spectrum for a non-vanishing electric\nfield, the parameters are PIz=0.25,PIx=0 and αREz=0.8 (with\nthe convention ¯h≡m≡n≡1).\nenergy up to E2\nz-terms as\nEks≃εk−s\u0012\nPI−αREzkysin(θI)+\nα2\nRE2\nzk2\n2PI−α2\nRE2\nzk2\nysin2(θI)\n2PI\u0013\n, (6)\nwhere sin (θI) =Ix/I. To demonstrate that the change in the\nenergy due to the presence of the organic layer\n∆E=E(tot)(Ez)−E(tot)(Ez=0), (7)\ncan depend on the angle θI, we assume that P=1 and I→∞,\ni.e., we work with a strong ferromagnet. In this case, only the\nfirst line in Eq. (6) is relevant because there is a high energy4\nMH/uni03D5H/uni03D5Mne/uni03D5e\n/uni03B8P\nFIG. 3. The geometry of the extended Stoner-Wohlfarth model. The\nangles are measured with respect to the normal to the substrate, n.\nThe figure shows the directions of magnetization, M; external mag-\nnetic field, H; easy axis, e; also shown is the tilting angle of the\ndipoles, θP. The direction of an arrow below each angle determines\nthe sign of the angle, e.g., in our convention φMis negative, whereas\nφHis positive.\nprice for flipping a spin of an electron. We derive\n1\nS∆E\nEF≃ − sin2(θI)m2α2\nRE2\nz\n4π, (8)\nwhereEFis the Fermi energy of the ‘spin-up’ particles. The\ndependence of ∆E/EFonθIshows that the system prefers\nto have in-plane magnetization. Note that Eq. (8) contradicts\nthe sketch in Fig. 3. However, this is not a point of concern\nfor two reasons. First, our goal here is to illustrate general\nprinciples of coupling, and the overall sign of the derived term\nmay change if other toy models are considered. Second, the\nsurface electric field Ezexists in real materials even without\nthe molecules. Therefore, the following representation of Ez\nmight be more accurate: Ez≃E0\nz+cos(θP)E1\nz. IfE0\nzandE1\nz\nhave different signs and |E0\nz| ≫ | E1\nz|, the situation in Fig. 3 is\nrestored. The most important conclusion of this section is that\nthe strength of the effect presented in Eq. (8) is determined by\nthe tilt angle of the molecules via Ez.\nIV. EXTENDED STONER-WOHLFARTH MODEL\nThe discussion in the previous section provides a ba-\nsis for studying magneto-electric coupling between organic\nmolecules and the ferromagnetic substrate. With it, we can ex-\ntend the simple Stoner-Wohlfarth model58,59and understand\nthe effect of the molecules on magnetic hysteresis and the in-\nterplay between the molecular tilting angle and the direction\nofM. Assuming that the ferromagnet has an uniaxial mag-\nnetic anisotropy, we write the energy, E, as\nE\nKuV=sin2(φM−φe)−µ0MsH\nKucos(φM−φH)\n−αcos2(θP)\nKudsin2(φM) (9)\nwhere the first line is the standard Stoner-Wohlfarth model,\nand the second line is our extension. Here, Vis the three-\ndimensional volume of the ferromagnet; Kuis the anisotropyparameter; Msis the saturation magnetization; His the mag-\nnitude of an external magnetic field; µ0is the vacuum per-\nmeability. The last term in Eq. (9) is derived in Eq. 8. It\nintroduces magnetic anisotropy due to the presence of organic\nmolecules. Its strength is determined by\nα=EFα2\nRE2\nzm2\n4πcos2(θP), (10)\nwhere the parameter dis the thickness of the ferromagnet. In\nwhat follows, we shall treat the amplitude and the sign of αas\nfree parameters, and estimate ‘realistic’ values of |α|only at\nthe end of this section. Finally, we note that we associate the\nangle of the magnetization with that of I, i.e., φM=θI.\nThe geometry of the system is presented in Fig. 3. We mea-\nsure all angles with respect to the normal to the surface. φM,\nφH, and φedefine the angles of the magnetization, external\nmagnetic field and the easy axis, respectively. We assume that\nH,e,nandMall lie in one plane. This assumption does not\nchange the qualitative nature of our discussion.\nWithout an external magnetic field. First, we consider\nthe system without an external field ( H=0)\nE\nVKu=sin2(φM−φe)−αcos2(θP)\nKudsin2(φM). (11)\nForφe=lπ/2 (lis any integer), the minimum of the energy is\nfor either in-plane or out-of-plane direction of M. For exam-\nple, for φe=0, we have\nE\nKuV=sin2(φM)(1−γ). (12)\nThe defining dimensionless parameter here is\nγ=αcos2(θP)\nKud. (13)\nNote that by changing the tilt of the molecules, we can change\nthe direction of the easy axis of the magnet, assuming that\nα/(Kud)>1, see Fig. 4.\nFor general values of φe, the equilibrium is reached when\nthe derivative of the energy with respect to the magnetization\ndirection is zero, ∂E/∂φM=0, so that\nsin(2φM−2φe)−γsin(2φM) =0. (14)\nWe see that the direction of the equilibrium direction of mag-\nnetization changes, which can explain the experimental obser-\nvation of Ref. 19. For example, for α→0 (γ→0), we can\nwrite\nφM≃φe+γsin(2φe). (15)\nFor general values of γ, we solve the equation numerically,\nsee Fig. 4. Note that the effect of small values of γis the most\npronounced for φe≃π/4.\nWith an external magnetic field. The extended Stoner-\nWohlfarth model provides insight into the modification of the\ncoercive field caused by the adsorbed molecules18. To illus-\ntrate this, let us assume an out-of-plane easy axis, φe=0,5\n0.20.40.60.811.21.400.20.40.60.811.21.41.6\nFIG. 4. Preferred direction of magnetization determined by φMas a\nfunction of the angle of the easy axis, φe. Different curves illustrate\ndifferent values of the parameter γfrom Eq. (13).\nand that the magnetic field is also perpendicular to the sur-\nfaceφH=0. In this case, the organic molecular layer simply\nchanges the value of the anisotropy parameter in the standard\nStoner-Wohlfarth model, i.e., Ku→Ku(α)where\nKu(α) =Ku(α=0)−αcos2(θP)\nd. (16)\nThe change of Kuaffects the magnetic hysteresis, in partic-\nular, the switching field (recall that φe=φH=0)\nHs(α) =2Ku(α)\nµ0Ms. (17)\nThe relative strength of the effect of the adsorbed molecules\non the switching field is (see Fig. 5 a)\nHs(α)\nHs(0)=1−αcos2(θP)\ndKu(α=0). (18)\nFor general values of φe, the Stoner-Wohlfarth model is not\nexactly solvable. Our numerical calculations show that the\neffect of the parameter γ=αcos2(θP)/(dKu(α=0))is qual-\nitatively similar to the case of φe=0, see Fig. 5 b.\nUsing Eq. (18) and the experimental data18, we can have an-\norder-of-magnitude estimate for the value of α, which is oth-\nerwise difficult to calculate because the strength of the SOC\neffect strongly depends on the electric field, whose accurate\nvalue is unknown.\nThe substrate of thickness d≃0.7 nm is made of iron for\nwhich60Ku≃3×10−4eV/nm3. Since the change in the\ncoercive field can be about 10% (so that γ≃0.1), we esti-\nmate α≃2×10−5eV/nm2. Here, we have assumed that the\nmolecules are perpendicular to the surface, θP=0. Using the\nvalue of αand the Fermi energy, EF∼11eV , we can also esti-\nmate the strength of the Rashba SOC: αREz∼10−4eV×nm –\n-1.5-1-0.500.511.5-1-0.500.51-1.5-1-0.500.511.5-1-0.500.51a\nbFIG. 5. Hysteresis loop in the ferromagnet, where MH= (φM−φH)\nis the magnetization projection on the magnetic field vector and h=\nµ0MsH/2Ku. a: Hysteresis for φH=0 and different values of the\nparameter γ. b: Same for φH=π/4.\na relatively small value55. Note that if there is an electric field\npresent without the molecules, i.e., Ez≃E0\nz+cos(θP)E1\nz, then\nwe actually estimate αRp\nE0zE1zin this way.\nV. CONCLUSIONS AND OUTLOOK\nWe have argued that the change of the surface electric field\ncaused by adsorbed molecules can affect the magnetocrys-\ntalline anisotropy of a ferromagnetic metallic substrate even if\nmolecules are achiral. To illustrate this claim, we have formu-\nlated a toy model in which the molecules change the surface\nelectric field, which is coupled to magnetization via a Rashba-\nlike SOC. The strength of coupling is given by the properties\nof the substrate and therefore no effect should be present for6\nsubstrates with weak SOC (cf. Refs. 24, 35, and 61).\nUsing the results of the toy model, we have introduced an\nextension of the Stoner-Wohlfarth framework, and calculated\nthe effect of the molecules on the preferred direction of mag-\nnetization. In particular, we have shown that the effect of the\nmolecules (for small values of γ) can be enhanced by working\nwith a ferromagnet whose easy axis has φe≃π/4, motivat-\ning experiments with such materials. Finally, we have illus-\ntrated the effect of the molecules on the magnetic hysteresis,\nand estimated phenomenological parameters using available\nexperimental data. Our findings show that isolating the effect\nof molecular chirality by changing magnetization of the sur-\nface is a daunting task, thus motivating further experimental\nresearch of chirality induced spin selectivity in current-free\nset-ups to use both enantiomers.\nLet us conclude by making a few remarks. First, note that\naccording to our results the direction of magnetization couples\nto the dipolar moment of both chiral and achiral molecules,\nand therefore, our results cannot unravel microscopic origin\nof the CISS effect. Second, our work does not intend to dis-\ncard the role of chirality in CISS experiments that explore the\nrealignment of magnetization upon the adsorption of a chi-\nral monolayer?. On the contrary, our work aims to moti-\nvate further theoretical studies that must contrast and com-\npare the roles of structural chirality and a magneto-electric\ncoupling driven by interface SOC. For instance, the term\nβ(P·n)2(M·n)2suggests that Mcan be either parallel or\nanti-parallel to n. In other words, these two directions are\nenergetically degenerate. Chirality then might be a key ingre-\ndient to break this degeneracy during the adsorption process\nleading to the results reported in Ref. ?. The symmetry-\nbreaking mechanism can be based upon an observation that\nthe helical structure of the molecule is coupled to the angu-\nlar momentum degree of freedom of the electron24,35,61. One\ncan speculate that this coupling in the presence of interface\nSOC can provide an energy barrier during molecular adsorp-\ntion that depends on chirality and the direction of magnetiza-\ntion, and hence can potentially drive the system into only one\nof the available energy minima, i.e., with Meither parallel or\nantiparallel to ndepending on chirality.\nAlthough the above mentioned frameworks35,61deal with\ntime-dependent setups, they might pave the way for a natural\nextension of our model. Indeed, if we assume that the role of\nchirality is to filter the orbital angular momentum, then we can\nimplement this filtering in SOC. Another possible extension\nof our work is to take into account a helical structure of the\nmolecule, which adds to the electric field Esome additional\nspatial chirality. To this end, one should study more general\nforms of the electric field, e.g., Ez=E0\nz+E1\nzcos(θP) +..., in-\ncluding the terms independent of the tilting angle. Such terms\ncan noticeably modify the overall dependence of observables\nonθP. Furthermore, if we consider higher-order electric mul-\ntipoles it should be possible to also study the effect of chiral-\nity. Finally, it is worth investigating the role of the electric\nfield generated by the molecules on the CISS effect with cur-\nrents, and the effect of other, in particular non-linear, types of\nSOC that can be present in the substrate.\n0+--+FIG. 6. Electric field lines (the curves at Z>0) created by two\ncharges (represented by the two balls at Z>0) above a metalic sur-\nface (the thick line at Z=0). The normal to the surface is along the\nZ-axis. The two charges placed at Z<0 are auxiliary mirror charges.\nThe lines at Z<0 do not represent the physical electric field, and are\nshown only for illustrative purposes.\nACKNOWLEDGMENTS\nWe thank Zhanybek Alpichshev, Mohammad Reza Safari,\nBinghai Yan and Yossi Paltiel for enlightning discussions.\nM.L. acknowledges support by the European Research\nCouncil (ERC) Starting Grant No.801770 (ANGULON).\nA. C. received funding from the European Union’s Horizon\nEurope research and innovation program under the Marie\nSkłodowska-Curie grant agreement No. 101062862 - Neq-\nMolRot.\nAppendix A: Electric field of a single dipole\nHere, we discuss the electric field on the surface of a metal\ncreated by a single dipole. The geometry is shown in Fig. 6.\nThe corresponding electric field is (in cgs units):\nEz=2qz1\nr3\n1−2qz2\nr3\n2, (A1)\nwhere ziis the position of the charge above the ferromagnet,\nandriis the distance from the charge to the point on the sur-\nface, see Fig. 6. For simplicity, we assume a point dipole,\nd, so that q(z1−z2) =dz, and Ez=2dz/r3\n1. This expression\nshows that the electric field that enters our calculations indeed\ndepends on the direction of the dipoles.\nWe note that the realistic electric field generated by the\nmolecules is unlikely to be that of a point dipole. It is also\nnot homogeneous, in particular, because the molecules are\nextended entities. However, the qualitative properties of our\nmain findings should be insensitive to these assumptions.7\n1R. Naaman, Y . Paltiel, and D. H. Waldeck, “Chiral molecules and the elec-\ntron spin,” Nature Reviews Chemistry 3, 250–260 (2019).\n2D. H. Waldeck, R. Naaman, and Y . Paltiel, “The spin selectivity effect in\nchiral materials,” APL Materials 9(2021), 10.1063/5.0049150.\n3R. Naaman, Y . Paltiel, and D. H. Waldeck, “Chiral induced spin selectivity\nand its implications for biological functions,” Annual Review of Biophysics\n51, 99–114 (2022).\n4Y . Xu and W. Mi, “Chiral-induced spin selectivity in biomolecules, hybrid\norganic–inorganic perovskites and inorganic materials: a comprehensive\nreview on recent progress,” Mater. Horiz. , – (2023).\n5K. Ray, S. P. Ananthavel, D. H. Waldeck, and R. Naaman, “Asymmet-\nric scattering of polarized electrons by organized organic films of chiral\nmolecules,” Science 283, 814–816 (1999).\n6B. Göhler, V . Hamelbeck, T. Z. Markus, M. Kettner, G. F. Hanne, Z. Vager,\nR. Naaman, and H. Zacharias, “Spin selectivity in electron transmis-\nsion through self-assembled monolayers of double-stranded DNA,” Science\n331, 894–897 (2011).\n7Z. Xie, T. Z. Markus, S. R. Cohen, Z. Vager, R. Gutierrez, and R. Naaman,\n“Spin specific electron conduction through DNA oligomers,” Nano Letters\n11, 4652–4655 (2011).\n8W. Zhang, K. Banerjee-Ghosh, F. Tassinari, and R. Naaman, “Enhanced\nelectrochemical water splitting with chiral molecule-coated fe3o4 nanopar-\nticles,” ACS Energy Letters 3, 2308–2313 (2018).\n9T. S. Metzger, S. Mishra, B. P. Bloom, N. Goren, A. Neubauer, G. Shmul,\nJ. Wei, S. Yochelis, F. Tassinari, C. Fontanesi, D. H. Waldeck, Y . Paltiel,\nand R. Naaman, “The electron spin as a chiral reagent,” Angewandte\nChemie International Edition 59, 1653–1658 (2020).\n10K. Banerjee-Ghosh, O. B. Dor, F. Tassinari, E. Capua, S. Yochelis, A. Ca-\npua, S.-H. Yang, S. S. P. Parkin, S. Sarkar, L. Kronik, L. T. Baczewski,\nR. Naaman, and Y . Paltiel, “Separation of enantiomers by their enantiospe-\ncific interaction with achiral magnetic substrates,” Science 360, 1331–1334\n(2018).\n11M. R. Safari, F. Matthes, K.-H. Ernst, D. E. Bürgler, and C. M. Schnei-\nder, “Enantiospecific adsorption on a ferromagnetic surface at the single-\nmolecule scale,” (2022), arXiv:2211.12976 [cond-mat.mtrl-sci].\n12K. Santra, Y . Lu, D. H. Waldeck, and R. Naaman, “Spin selectivity dam-\nage dependence of adsorption of dsdna on ferromagnets,” The Journal of\nPhysical Chemistry B 127, 2344–2350 (2023), pMID: 36888909.\n13K. B. Ghosh, W. Zhang, F. Tassinari, Y . Mastai, O. Lidor-Shalev, R. Naa-\nman, P. Möllers, D. Nürenberg, H. Zacharias, J. Wei, E. Wierzbinski, and\nD. H. Waldeck, “Controlling chemical selectivity in electrocatalysis with\nchiral cuo-coated electrodes,” The Journal of Physical Chemistry C 123,\n3024–3031 (2019).\n14A. Inui, R. Aoki, Y . Nishiue, K. Shiota, Y . Kousaka, H. Shishido, D. Hi-\nrobe, M. Suda, J.-i. Ohe, J.-i. Kishine, H. M. Yamamoto, and Y . Togawa,\n“Chirality-Induced Spin-Polarized State of a Chiral Crystal CrNb 3S6,”\nPhys. Rev. Lett. 124, 166602 (2020).\n15F. Evers, A. Aharony, N. Bar-Gill, O. Entin-Wohlman, P. Hedegård, O. Hod,\nP. Jelinek, G. Kamieniarz, M. Lemeshko, K. Michaeli, V . Mujica, R. Naa-\nman, Y . Paltiel, S. Refaely-Abramson, O. Tal, J. Thijssen, M. Thoss, J. M.\nvan Ruitenbeek, L. Venkataraman, D. H. Waldeck, B. Yan, and L. Kronik,\n“Theory of chirality induced spin selectivity: Progress and challenges,” Ad-\nvanced Materials 34, 2106629 (2022).\n16J. Fransson, “The chiral induced spin selectivity effect what it is, what it\nis not, and why it matters,” Israel Journal of Chemistry 62, e202200046\n(2022).\n17N. Sukenik, F. Tassinari, S. Yochelis, O. Millo, L. T. Baczewski, and\nY . Paltiel, “Correlation between ferromagnetic layer easy axis and the\ntilt angle of self assembled chiral molecules,” Molecules 25(2020),\n10.3390/molecules25246036.\n18S. Miwa, K. Kondou, S. Sakamoto, A. Nihonyanagi, F. Araoka, Y . Otani,\nand D. Miyajima, “Chirality-induced effective magnetic field in a phthalo-\ncyanine molecule,” Applied Physics Express 13, 113001 (2020).\n19I. Meirzada, N. Sukenik, G. Haim, S. Yochelis, L. T. Baczewski, Y . Paltiel,\nand N. Bar-Gill, “Long-time-scale magnetization ordering induced by an\nadsorbed chiral monolayer on ferromagnets,” ACS Nano 15, 5574–5579\n(2021), pMID: 33591720.\n20H. Alpern, M. Amundsen, R. Hartmann, N. Sukenik, A. Spuri, S. Yoche-\nlis, T. Prokscha, V . Gutkin, Y . Anahory, E. Scheer, J. Linder, Z. Salman,\nO. Millo, Y . Paltiel, and A. Di Bernardo, “Unconventional meissner screen-ing induced by chiral molecules in a conventional superconductor,” Phys.\nRev. Mater. 5, 114801 (2021).\n21R. Gutierrez, E. Díaz, R. Naaman, and G. Cuniberti, “Spin-selective trans-\nport through helical molecular systems,” Phys. Rev. B 85, 081404 (2012).\n22A.-M. Guo and Q.-f. Sun, “Spin-selective transport of electrons in dna dou-\nble helix,” Phys. Rev. Lett. 108, 218102 (2012).\n23E. Medina, F. López, M. A. Ratner, and V . Mujica, “Chiral molecular films\nas electron polarizers and polarization modulators,” EPL (Europhysics Let-\nters) 99, 17006 (2012).\n24J. Gersten, K. Kaasbjerg, and A. Nitzan, “Induced spin filtering in elec-\ntron transmission through chiral molecular layers adsorbed on metals with\nstrong spin-orbit coupling,” The Journal of Chemical Physics 139, 114111\n(2013).\n25A.-M. Guo and Q.-F. Sun, “Spin-dependent electron transport in protein-\nlike single-helical molecules,” Proceedings of the National Academy of\nSciences 111, 11658–11662 (2014).\n26S. Matityahu, Y . Utsumi, A. Aharony, O. Entin-Wohlman, and C. A. Bal-\nseiro, “Spin-dependent transport through a chiral molecule in the presence\nof spin-orbit interaction and nonunitary effects,” Phys. Rev. B 93, 075407\n(2016).\n27E. Diaz, P. Albares, P. G. Estevez, J. M. Cerveró, C. Gaul, E. Diez, and\nF. Domínguez-Adame, “Spin dynamics in helical molecules with nonlinear\ninteractions,” New Journal of Physics 20, 043055 (2018).\n28J. Fransson, “Chirality-induced spin selectivity: The role of electron corre-\nlations,” The Journal of Physical Chemistry Letters 10, 7126–7132 (2019).\n29K. Michaeli and R. Naaman, “Origin of spin-dependent tunneling through\nchiral molecules,” The Journal of Physical Chemistry C 123, 17043–17048\n(2019).\n30S. Dalum and P. Hedegård, “Theory of chiral induced spin selectivity,”\nNano Letters 19, 5253–5259 (2019), pMID: 31265313.\n31J. Fransson, “Vibrational origin of exchange splitting and ”chiral-induced\nspin selectivity,” Phys. Rev. B 102, 235416 (2020).\n32A. Ghazaryan, M. Lemeshko, and A. G. V olosniev, “Filtering spins by scat-\ntering from a lattice of point magnets,” Communications Physics 3(2020),\n10.1038/s42005-020-00445-8.\n33A. Ghazaryan, Y . Paltiel, and M. Lemeshko, “Analytic model of chiral-\ninduced spin selectivity,” The Journal of Physical Chemistry C 124, 11716–\n11721 (2020), pMID: 32499842.\n34S. Alwan and Y . Dubi, “Spinterface origin for the chirality-induced spin-\nselectivity effect,” Journal of the American Chemical Society 143, 14235–\n14241 (2021), pMID: 34460242.\n35Y . Liu, J. Xiao, J. Koo, and B. Yan, “Chirality-driven topological electronic\nstructure of DNA-like materials,” Nature Materials 20, 638–644 (2021).\n36Y . Wolf, Y . Liu, J. Xiao, N. Park, and B. Yan, “Unusual spin polarization in\nthe chirality-induced spin selectivity,” ACS Nano 16, 18601–18607 (2022),\npMID: 36282509.\n37C. Vittmann, J. Lim, D. Tamascelli, S. F. Huelga, and M. B. Plenio,\n“Spin-dependent momentum conservation of electron–phonon scattering in\nchirality-induced spin selectivity,” The Journal of Physical Chemistry Let-\nters14, 340–346 (2023), pMID: 36625481.\n38S. Sanvito, “The rise of spinterface science,” Nature Physics 6, 562–564\n(2010).\n39M. Geyer, R. Gutierrez, V . Mujica, and G. Cuniberti, “Chirality-induced\nspin selectivity in a coarse-grained tight-binding model for helicene,” The\nJournal of Physical Chemistry C 123, 27230–27241 (2019).\n40A. G. V olosniev, H. Alpern, Y . Paltiel, O. Millo, M. Lemeshko, and\nA. Ghazaryan, “Interplay between friction and spin-orbit coupling as a\nsource of spin polarization,” Phys. Rev. B 104, 024430 (2021).\n41W. Eerenstein, N. D. Mathur, and J. F. Scott, “Multiferroic and magneto-\nelectric materials,” Nature 442, 759–765 (2006).\n42M. K. Niranjan, C.-G. Duan, S. S. Jaswal, and E. Y . Tsymbal, “Elec-\ntric field effect on magnetization at the fe/MgO(001) interface,” Applied\nPhysics Letters 96, 222504 (2010).\n43P. K. Kumari and M. K. Niranjan, “Interface magnetoelectric effect and\nits sensitivity on interface structures in fe/agnbo3 and srruo3/agnbo3 het-\nerostructures: A first-principles investigation,” Journal of Magnetism and\nMagnetic Materials 517, 167372 (2021).\n44M. Cinchetti, V . A. Dediu, and L. E. Hueso, “Activating the molecular\nspinterface,” Nature Materials 16, 507–515 (2017).\n45A. Soumyanarayanan, N. Reyren, A. Fert, and C. Panagopoulos, “Emer-8\ngent phenomena induced by spin–orbit coupling at surfaces and interfaces,”\nNature 539, 509–517 (2016).\n46M. Callsen, V . Caciuc, N. Kiselev, N. Atodiresei, and S. Blügel, “Magnetic\nhardening induced by nonmagnetic organic molecules,” Phys. Rev. Lett.\n111, 106805 (2013).\n47K. Bairagi, A. Bellec, V . Repain, C. Chacon, Y . Girard, Y . Garreau,\nJ. Lagoute, S. Rousset, R. Breitwieser, Y .-C. Hu, Y . C. Chao, W. W. Pai,\nD. Li, A. Smogunov, and C. Barreteau, “Tuning the magnetic anisotropy at\na molecule-metal interface,” Phys. Rev. Lett. 114, 247203 (2015).\n48S. Zhang, “Spin-dependent surface screening in ferromagnets and magnetic\ntunnel junctions,” Phys. Rev. Lett. 83, 640–643 (1999).\n49C.-G. Duan, J. P. Velev, R. F. Sabirianov, Z. Zhu, J. Chu, S. S. Jaswal,\nand E. Y . Tsymbal, “Surface magnetoelectric effect in ferromagnetic metal\nfilms,” Phys. Rev. Lett. 101, 137201 (2008).\n50A. Natan, L. Kronik, H. Haick, and R. Tung, “Electrostatic properties of\nideal and non-ideal polar organic monolayers: Implications for electronic\ndevices,” Advanced Materials 19, 4103–4117 (2007).\n51O. L. A. Monti, “Understanding interfacial electronic structure and charge\ntransfer: An electrostatic perspective,” The Journal of Physical Chemistry\nLetters 3, 2342–2351 (2012).\n52A. V . Gold, “Review paper: Fermi surfaces of the ferromagnetic transition\nmetals,” Journal of Low Temperature Physics 16, 3–42 (1974).\n53E. Rashba, Sov. Phys.-Solid State 2, 1109 (1960).\n54Y . A. Bychkov and É. I. Rashba, “Properties of a 2D electron gas withlifted spectral degeneracy,” Soviet Journal of Experimental and Theoretical\nPhysics Letters 39, 78 (1984).\n55A. Manchon, H. C. Koo, J. Nitta, S. M. Frolov, and R. A. Duine, “New\nperspectives for rashba spin–orbit coupling,” Nature Materials 14, 871–882\n(2015).\n56F. E. Meijer, A. F. Morpurgo, and T. M. Klapwijk, “One-dimensional ring\nin the presence of rashba spin-orbit interaction: Derivation of the correct\nhamiltonian,” Phys. Rev. B 66, 033107 (2002).\n57B. Berche, C. Chatelain, and E. Medina, “Mesoscopic rings with spin-orbit\ninteractions,” European Journal of Physics 31, 1267 (2010).\n58E. C. Stoner and E. P. Wohlfarth, “A mechanism of magnetic hysteresis\nin heterogeneous alloys,” Philosophical Transactions of the Royal Society\nof London. Series A, Mathematical and Physical Sciences 240, 599–642\n(1948).\n59C. Tannous and J. Gieraltowski, “The stoner–wohlfarth model of ferromag-\nnetism,” European Journal of Physics 29, 475 (2008).\n60B. D. Cullity and C. D. Graham, Introduction to Magnetic Materials (2nd\ned.)(Wiley-IEEE Press., 2008).\n61Y . Adhikari, T. Liu, H. Wang, Z. Hua, H. Liu, E. Lochner, P. Schlottmann,\nB. Yan, J. Zhao, and P. Xiong, “Interplay of structural chirality, elec-\ntron spin and topological orbital in chiral molecular spin valves,” (2022),\narXiv:2209.08117 [cond-mat.mtrl-sci]."    },    {        "title": "2307.02213v1.Multi_level_recording_in_dual_layer_FePt_C_granular_film_for_heat_assisted_magnetic_recording.pdf",        "content": " \n Multi-level recording in dual -layer FePt -C granular film for heat-assisted magnetic \nrecording  \n \nP. Tozman1*, S. Isogami1, I. Suzuki1, A. Bolyachkin1, H. Sepehri -Amin1, S.J. Greaves2 \nH. Suto1, Y . Sasaki1, H.T. Y . Chang3, Y Kubota3, P. Steiner3, P.-W. Huang3, K. Hono1 \nand Y .K. Takahashi1* \n \n1 National Institute for Materials Science, Tsukuba, Ibaraki 305 -0047,  Japan  \n2Research Institute of Electrical Communications Tohoku University, Sendai, Japan  \n3 Seagate Technology, Recording Media Organization, Fremont, CA. 94538 USA  \n \n \nAbstract  \n  \nMulti -level magnetic recording is a new concept for increasing the data storage capacity \nof hard disk drives.  However, its implementation has been limited by a lack of suitable \nmedia capable of  storing information at multiple levels. Here in, we  overcome  this \nproblem by developing dual FePt-C nanogranular films separated by a Ru-C breaking \nlayer  with a cubic crystal structure . The FePt grains in the bottom and top layers  of the \ndeveloped media exhibited different effective magnetocrystalline anisotropies and Curie \ntemperatures . The  former is realized  by different degree s of ordering in the L1 0-FePt \ngrains , whereas the latter was attributed to the  diffusion of Ru , thereby enabling separate \nmagnetic recordings at each layer under different magnetic fields and temperatures . \nFurthermore, t he magnetic measurements and heat-assisted  magnetic recording \nsimu lations showed that these  media enable d 3-level recording and could potentially  be \nextended  to 4-level recording , as the ↑↓ and ↓↑ states exhibit ed non-zero magnetization.  \n \n \n \n \n \n \n \n \nCorresponding authors:  \npelin.tozman @tu-darmstadt.de , TAKAHASHI .Yukiko@nims.go.jp   \n Evolution of magnetic recording  \nThe digital transformation from Industry 4.0  to 5.0 has resulted in  a significant \ngrowth in big data, increasing the demand for data storage [1 -3]. However,  with the \nexpected increase in energy consumption owing to the construction of more data centers, \nan environmental ly friendly solution is required to satisfy this demand . One solution \ninvolves  increas ing the storage capacity of hard disk drive s (HDD s), whic h currently \nserve as the primary storage devices in data centers. HDD s comprise  a spinning disk \ncoated with a nanogranular ferro magnetic layer  (media)  and read/write head that \nmagnetically reads and write s data to the disk. The storage capacity  of HDDs is \ndetermined by the bit size, which refers to the physical dimensions of the region on the \nmedia  where several ferromagnetic nanograins are magnetized i n the same direction.  \nReducing the bit size facilitate s increased  data storage in the same physical space [ 4], \nthereby  increas ing the areal density of data bits from the current value of 1.5 Tbit/in2 to 4 \nTbit/in2 and beyond [3-8]. This is achieved by reducing the grain size of the granular \nmedia while maintaining the long -term thermal stability of the data by satisfying the \ncondition KuV/kBT > 60, where Ku is the uniaxial magnetocrystalline anisotropy constant, \nV is the volume of a grain, kB is the Boltzman n constant, and T is the temperature [ 9,10]. \nThis condition is fulfilled  by L10-FePt granular media , where FePt grains with a very high \nKu ≈7 MJ/m3 are uniformly dispersed in a nonmagnetic segregant  [7,9,11,12]. However , \nthis medium  cannot be used in conventional recording technologies because it requires  a \nhigh magnetic field , more than 3 T,  for the writing process . This problem is overcome via \nheat-assisted magnetic recording (HAMR) , which facilitates  the writing of  data on high-\nKu and high-coercivity  (Hc) media material s [11,13]. Unlike conventional recording, \nHAMR equips a write head with an optical transducer  to heat a small area on the recording  \n media around its Curie temperature Tc, thereby lowering the high-Ku locally and enabling  \nthe switch ing of the magnetization direction by a write field generated from the write \nhead  (Fig. 1(a)) .  \nAccording to the Advanced Storage Research Consortium (ASRC) , the primary \nfocus for increasing the areal density in HAMR is to improve the nanostructure of the \ngranular FePt -X media [ 7,9], where X indicates nonmagnetic segregants. By refining the \ngrain size  to 4.3 nm and obtaining a high degree of L10-order, the areal density is expected \nto reach 4 Tbit/in2 [7,9,10,14,15 ]. To achieve such a fine microstructure, various  X, \nincluding  metal oxides , B, h-BN, B 4C, and C , have been added to FePt -based granular \nfilms, with C and h -BN provid ing a small average grain size , <D>= 5-8 nm while \nmaintaining uniaxial anisotropy [ 11, 14-20]. Thus, a maximum areal density of 2.77 \nTbit/in2 was achieved by Seagate for a demo HAMR [3].  \nHowever, FePt grain s smaller than  4 nm (<D> < 4 nm) cannot maintain their \ncrystallographic L1 0 order , resulting in  thermal fluctuation s in magnetization owing to \ntheir low anisotropy energy  [18]. Therefore, increasing the areal density of future \ngeneration s of HDD recording media  requires a new recording concept  that is not rel iant \non grain size miniaturization  [8,21]. One solution is bit -pattern media (1 bit = 1 grain) \ninstead of granular media (1 bit = multiple grains). Numerical calculation s predicted that \nbit-patterned media can achieve an areal density of 10 Tbit/in2 [21-23]. However, bit -\npattern ed media cannot be  mass produc ed because of the ir costly nanofabrication process. \nAnother solution involves the  use of multi -level recording [ 24-26], as explain ed in the \nfollowing Section.  \nConcept of multi -level recording in HAMR  \nA multi -level recording concept has been proposed for HAMR technologies,  \n which uses media comprising  multiple recording layers with different Tc and controls t he \nmagnetization reversal in each layer by tuning the temperature  of the heated spot [27-29]. \nFig. 1 shows  a schematic of the dual-layer version of a multi -level HAMR  in comparison \nwith the  traditional single -layer version  of a 2 -level HAMR . The single -layer  version  \nemploys a  constant laser power to heat a small region of the magnetic layer  around Tc, \nand the write  field is used to switch the magnetization direction between  the up  ↑ (“1”) \nand down  ↓ (“-1”) magnetization state s. This facilitates the  storage of a single bit of data \n(Fig 1 (a))  [7,13]. Following the magnetization switch , the heated region is rapidly cooled \nunder an applied field to store the magnetic polarity of the region. However , in the dual -\nlayer version, the Curie temperature of the bottom layer , Tc2, is higher  than that of the top \nlayer  Tc1 (Tc1 < Tc2) [29,30]. In the first write , the laser power is set  as high to align the \nmagnetization direction of the bottom layer with the polarity of the write field . Moreover, \nduring the first write, the top layer is inevitably written because of its low Tc, resulting  in \nthe same magnetization pattern  in both layers  containing  only the up ↑↑  (“1”) and down \n↓↓  (“-1”) magnetization states  similar to the single -layer version . Thereafter, i n the \nsecond write, the laser power is lowered to facilitate writing in  only the upper layer \nwithout disturbing the magnetization configuration of the bottom layer. This process \nproduces four magnetic states corresponding  to ↑↑ (“1”), ↓↓ ( “-1”), ↑↓ (“0”), and ↓↑ (“0”), \nas shown in Fig. 1 (b) . These additional magnetization st ates extend the recording from \nthe conventional 2 -level s to 3-level s, and they can potentially be extended to 4-level s if \nthe read head can distinguish the ↑↓ or ↓↑ states . \nExperimentally, Tc1 can be differentiated from Tc2 either by varying the Fe \ncontent in FePt or by using various additives in FePt, such as Cu,  Ni, Cr, Mn, B, and Ag  \n[11,12,15,17,29,31 -34]. These layers can be  magnetically decoupled by introducing a  \n breaking layer between them . In this study, we developed  the first experimental synthesis \nof a multi -level recording medium . Its nanostructure , magnetization -switching \nmechanism , and read/write performance were investigated using experimental and \nnumerical tools.  \n \nFig. 1  Schematic of the concept of multi -level  magnetic recording compared to current  2-level  \nmagnetic recording technology in hard disk drives. In both cases, a small area  on the recording \nmedi um is heated to approximately its Tc to switch the magnetization . The a rrow indicates the writing \ndirection. An optical transducer provides a Gaussian heat sp ot (red color) utilized to heat the area . The \nmagnetic state is stored  by cooling the heated area to ambient temperature. a In the con ventional 2 -\nlevel recording, the laser power is set as a constant, and magnetization switches by changing  the \npolarity of the write field  to obtain a magnetization pattern with (↑ (“1”) and ↓(“ -1”)). b For multi -\nlevel recording , the laser power is tuned  along with the polarity of the write field  to manipulate the \nmagnetization switch ing of each layer. During the first write  operation , a high  laser power  is set to \nheat a small area above  Tc2 of the bottom layer ( T>Tc2) and thus write both layers . During the second \nwrite, the laser power is lowered below Tc2 (T>Tc1) to write only the top layer . This delivers four \ndifferent  magnetic states  (↑↑ (“1”), ↓↓ ( “-1”), ↑↓ (“0”), and ↓↑ (“0”))  which correspond to  a 4-level \nrecording . The recording medi um is only a few nm thick, while the breaking layer is half that thickness.  \n \n \n Experimental  \nA. Breaking layer   \nWe experimentally investigated various nonmagnetic metallic breaking layer s (BL) \nto magnetically decouple the top and bottom recording la yers to realize multi -level \nrecording. An ideal BL  should facilitate  the growth of the top FePt layer with  a smooth \nand flat surface , high Ku, and (001) epitaxy . To select the most appropriate BL, we first \nmodeled the idealized interface between the FePt layers using a continuous film , not a \ngranular  one. Consequently,  the effect s of mismatch strain  and surface roughness were \nminimized , and only the effect of the BL on the magnetic properties, particularly on the \nswitching separation,  was investigated . Ru, Cu , Pt, and W  were considered for use as a \nBL owing to their small lattice mismatch  (0.5–1.7%) with FeP t (Supplementary Fig. S1) . \nHigh -magnification , cross -sectional high -angle annular dark field (HAADF) and \nscanning transmission electron microscopy (STEM) images showed that, among these \nelements, only Ru and Pt exhibited a flat interface with (001) epitaxial growth (Fig. 2 \n(a,b) and Supplementary Fig. S2). Currently , Pt is not consider ed because both layers \nexhibit the same composition, resulting in similar Tc1 and Tc2 values in their M-T curve s. \nIn contrast, composition al differentiation occurs in the case of face-centered  cubic (fcc) \nRu [35]. The e nergy -dispersive X -ray spectroscopy (EDS) line scan profile shows the \ndifference  in the composition of  the top ( Fe39Pt61) and bottom (Fe 42Pt58) layers and the \ndiffusion of trace amount s into the bottom  within a depth of approximately  1 nm from the \nRu laye r (Fig. 2(a,c)). This is expected to induce a differen ce in  the Tc values,  which is \nessential for multi -level recording [36]. In addition , despite the predominant (001) epitaxy, \nthe top FePt layer also  displayed in -plane variants  and a lack of L10 order during the initial \nstage of growth (~1 nm) (Fig. 2(a,b)) . This affect s the effective anisotropy , thus varying  \n the switching field  and resulting  in a step-like behavior in the Hall bar resistance \nmeasurement corresponding to the two coercivity values (Fig. 2(d)) . The W and Cu \nbreaking layers form ed unexpected (111) epitax ies and island -like microstructure s, \nrespectively,  as shown in Supplementary Fig. S3.   \nAlthough  Ru magnetically decouple d the top and bottom layers , it diffused  locally  \ninto the bottom layer and  created  local  variants  and disordered  regions  in the top layer . \nTherefore,  to understand  the comprehensive  effect of Ru  on the magnetic properties and \nchemical order, a continuous  FePt layer was grown on MgO substrates with Ru content s \nof x = 0, 0.5, 2, 4, and 8 at.% . Ru substitution decreased the L1 0 order while increasing \nthe lattice parameter c (Supplementary Fig S4). Therefore, one reason for the partially \ndisordered region could  be the diffusion of Ru in to the FePt layers , which results in  a \ndifferent strain state owing to the expansion of the lattice parameter.  \nThe effective  anisotropy constant of Ku for x = 0 in Fe42Pt58 was 3.06 MJ/m3 at 300 \nK, which is consistent  with the literature [ 37]. Increasing the Ru content to  x = 8 in \nFe42(Pt58-xRux) decreased Ku to 0.8 MJ/m3 (Supplementary Fig. S5) [34]. Therefore, Ku as \nwell as Hc, can be modif ied in each layer via Ru substitution or diffusion.  Further,  Tc \ndecreased from 610 K for Fe 42Pt58 to 540 K via the substitution of  a small amount of Ru \nin Fe 42Pt57.5Ru0.5 (Supplementary Fig. S6 (a,b)) [34,36,38]. The modification of Ku and \nTc via Ru introduction benefi ts multi -level recording  by controlling the magnetization \nreversal in each layer if the Ru substitution remain s at x ≤ 0.5 at.% .   \n  \nFig. 2. Microstructural features of FePt/Ru/FePt continuous film.  a, b HAADF and STEM -\nEDS maps of Fe,  Pt, and Ru with c line scan for FePt (17  nm)/Ru (2 nm)/FePt (17 nm). c EDS \nline scan reveals the difference in the composition of the top and bottom layers, with a trace \namount of Ru diffusion into the bottom. d Transverse normalized Hall -bar resistance (R xy) \nmeasurement as a function of the external magnetic field , where R xy corresponds to the out -of-\nplane of the magnetization, showing  that each layer has a different switching field . \nB. Dual nanogranular magnetic layer s \nDual nanogranular magnetic layer s as a multi -level recording medi um was obtained by \ngrowing  a (001) -textured FePt-20 vol. % C granular  bottom  layer  (4.5 nm)  on a MgO  \n(001) substrate , followed by a Ru-C breaking layer  (2.9 nm)  and FePt-C granular top layer . \nThis was confirmed via the cross -sectional HAADF -STEM image and EDS map in Fig. \n3 (a,b).  Carbon is preferred as a segrega nt to reduc e the average grain size and maximize \nthe grain density  while maintaining the L1 0 order  [14,39,40]. The FePt grains primarily \nexhibit ed a columnar shape with a relatively flat surface and certain curved grains . In Fig. \n3(c), the high-magnification HADDF image reveals  that the FePt grains in the bottom \n \n layer exhibit  mainly well L10-ordered (001) texture with some in -plane variants. In \ncontrast , the FePt grains in  the top layer exhibit  more in -plane variants and partially \ndisordered  regions . The EDS maps and line scans in Fig. 3(e-g) show that Ru diffused to \nboth layer s and to the in tergranular region . To investigate t he influence of the Ru -C BL \non the degree of the L1 0 order , the XRD pattern s of the single - and dual-layer  media were \nevaluated.  The large chemical order parameter, S = 0.8, of the single FePt -C layer  \ndecrease d to 0.63 with the growth of the BL and top layer ( Supplementary Fig. S7). \nAnother factor that disturb ed the (001) texture was the shape of FePt grains. Interfacial \nstrain is required to suppress the formation of in -plane variants , as is the case for the \nFePt/MgO interface in the bottom layer  [41,42]. However, such strain may lack at the \ncurved Ru -FePt interface , leading to an increase in the number of in-plane variants in the \ntop layer.  \n \nFig. 3 Overall nanostructure of the developed media for multi -level magnetic recording.  a \nHADDF image  b EDS map for FePt-20 vol.%  C/Ru - 20 vol% C/FePt -20 vol% C  of granular dual \nlayer media shows that FePt -C layers are separated by using a Ru-C breaking layer. c, d High \nmagnifi cation  HADDF images indicate that the additional in -plane variants are present on the top \nlayer, in addition to the (001) texture . e High magnifi cation  EDS map with f, g its line scan  \nreveal ing that Ru diffuses to both the layers and to the intergranular region . \n \n \n The average grain size of the media was obtained as  12.7±5.7nm  from  the top-view \nbright -field transmission electron microscopy ( TEM ) images in Fig. 4(a). Fig. 4(b, c) \nshow the experimental  out-of-plane  (OOP) and in -plane (IP) magnetization curves , along \nwith the simulated curves . The IP curve indicates strong perpendicular magnetic \nanisotropy . Based on half of the mid -distance between the saturation and plateau region s \nof the OOP curve, t he coercivities  of the top ( Hc1) and bottom ( Hc2) layer s were \ndetermined to be μ0Hc1 = 0.24 T and μ0Hc2 = 3.82 T . The large μ0Hc2 value confirms good \nepitaxial growth of the highly ordered FePt (001). To interpret the magnetic properties of \nthe obtained two -layer FePt films and distinguish them from the individual layers, the \nmeasured OOP hysteresis loops were analyzed using micromagnetic simulations. We \nreproduc ed the microstructural features of the fi lms using a finite element model  to \nperform micromagnetic approximations  of the demagnetization curves , where  the mean \nmagnetic anisotropy constants  (Ktop and Kbot), their standard deviations, and the volume \nfraction s of the in -plane variants  were  used as approximation parameters  (Supplementary \nFig. S8) . The simulated M-H curve  for Kbot = 2.70 ± 0.45 MJ/m3 (Ktop = 0.6 ± 0.1 MJ/m3) \nand for 8 vol. % (30 vol.%) of in-plane variants  in the bottom (top) layer is consistent  with \nthe experimental ones. An increase in the number of in -plane variants on the top layer \nwas observed using high -resolution cross -sectional TEM , resulting  in a decrease in Ktop. \nThis was attributed to the distribution  of the L1 0 order and Ru diffusion (Fig.  3 (c)).  Based \non the micromagnetic simulation , the in-plane variants of the top layer had minimal  \nimpact on the shape of the demagnetization curve. However , increasing Ku while \ndecreasing the variants enhance d remanence  (µ0Mr) and eliminate d the plateau region  \n(Fig. 4 (c)) . Nevertheless, the high slope d M/dH clearly indicates a two -layer  recording \nsystem with distinct Ku of top and bottom layers.    \n   \nFig. 4. Microstructure of  granular  FePt -C/Ru -C/FePt -C films and corresponding \nmagnetization curves.  a In-plane bright -field TEM images , with an inset  showing the grain size \ndistribution , reveal  that the ave rage grain  size is 12.7 ±5.7nm . b Corresponding out -of-plane (⊥) \nand in -plane (//) magnetization curves show  that μ0Hc1= 0.24 T and μ0Hc2 = 3.82 T . c The \nsimulated  (line)  and experimental  (circle)  M-H curves are shown together , along with the \nvariation of the simulated M-H curve with an anisotropy  constant K top and a variant V top.  \nTemperature -dependent m agnetic measurement  and writing simulation  \nThe Tc of each layer in the dual nanogranular magnetic layers was estimated from \nM-T measurement at 0.1 T in Fig. 5( a), and it was found to be Tc1 = 526 K and  Tc2 = 620 \nK. Tc can be affected by various factors such as Ru diffusion, deviation in the  Fe and Pt \ncontent s of the FePt grains , and the L10 order parameter . Therefore, Tc1 = 526 K may \nbelong  to the top layer , which contains  most of these factors .  \nThe writing performance of the HAMR media was investigated using \ntemperature -dependent  magnetic measurement s (Fig. 5(b)).  Unlike in the HAMR writing \nprocedure, in this magnetic  writing test, the medium was heated as a whole , rather than \nlocally.  First, the medi um was saturated , corresponding  to the magnetic state of ↑↑ (“1”),  \nunder 7 T at 300 K . To switch the top layer  with a low Ktop (0.6 MJ/m3) and Tc1 (526 K) , \na small reverse field (µ0H = -0.1 T ) was applied while  increasing the  temperature from \n300 K to Tan = 400, 500, 600 , and 700  K. The medium was then cooled to 300 K to store \nthe obtained magnetic state. Consequently, the magnetization of the top layer was \nreversed , and M = 0 was obtained by cooling the medium from Tan = 600 to 300 K , a \n \n temperature higher than Tc1 = 526 K , Tan>Tc1. This process deliver ed a magnetic state of \n↓↑ (“0”). For switching the bottom layer with high Kbot (2.7 ±0.45 MJ/m3), the reverse \nfield was increase d to µ0H = -1 T, and the same process was followed . The magnetization \nof the bottom layer was reversed  by cooling from 700 to 300 K , a temperature higher than \nTc2 = 610 K, Tan>Tc2. This delivered the magnetic state of  ↓↓ (“ -1”).  \n \nFig. 5. Temperature -dependent magnetic measurement of granular FePt -C/Ru -C/FePt -C \nmedia . a M-T measurement is performed under 0.1 T wherein  the top and bottom layer s exhibit \nTc1 of 526 K and Tc2 of 620 K. b Schematic representation of magnetic measurement to mimic the \nwriting procedure . c M-T measurement  under an applied field of µ0H = - 0.1 T to switch  the \nmagnetization of the top layer  at 600 K corresponding to the ↓↑ state (“0”) . d Increasing µ 0H = -\n1 T to switch the magnetization of the bottom layer corresponding to ↓↓ (“ -1”). \nIn addition , a multi-level HAMR recording simulation was performed based on \nthe experimentally  measured parameters . A heat spot with a Gaussian intensity \ndistribution and globally uni form write field (0.2 T was used to heat the media and switch \n \n the magnetization in the desired direction as the media cooled  down ). Fig. 6 (a , b) shows \nimages of the top and bottom layers at the end of a two -pass writing , which is described \nin “Concept of multi -level recording in HAMR.” Writing started with an AC-erased \nmedium  corresponding  to each grain in a magnetically randomly oriented state . The first \npass was written at 680 K, which is higher than the T c of both laye rs, and a bit length of \n50 nm . This  resul ted in the successful writing of both layers  but differing track width s, \nwhich were narrower (≈50 nm) and wider (≈100 nm) for the  bottom and  top layer s, \nrespectively,  owing to the different  Tc (Tc1<Tc2). For the second pass, the laser power was \nlowered to 540 K , below Tc2, and the bit length was increased to 100 nm  to write only the \ntop layer. However, r emnants  of the first track remained  at the track edge of the  top layer , \nand the magneti zation within the written bits was not fully saturated . This is attributed to \ncertain grains  switching in the direction  opposite to that of the write field during cooling, \nwhich is used to store the magnetic state.  This problem can be solved by either increasing \nthe write field or increasing Ku of the top layer.  Fig. 6 ( c) shows the magnetization along \nthe down -track direction in the top and bottom layers, averaged over a 40 nm track width,  \nalong with  the sum of the magnetization of the two layers. Herein,  four di fferent \nmagnetization states were observed. The magnetization correspond ing to ↑↓ and ↓↑ was \nM≠0 , which was easily distinguish ed. Further,  the recording can be expanded from 3 -\nlevel s to 4-level s.   \n  \nFig. 6. Multi -level HAMR recording simulation for  granular FePt -C/Ru -C/FePt -C media \nAverage magnetization patterns of 50 tracks after a two -pass write. a Pass 1: Writing on the \nbottom layer with Tmax = 680 K, bit length = 50 nm. b Pass 2: Writing on the top layer with Tmax \n= 540  K c Average magnetization along the down -track direction for the top and bottom layers . \nThe magnetization of the top and bottom layers  and the sum of  the magnetization of  both layers  \nare indicated by a blue dash line, red solid line , and green circle, respectively.   \n \nConclusion  \nProof  of concept in the multi -level recording has been developed for the first time \nsuccessfully  by separating FePt-C (20 vol. %) nanogranular layers with a cubic Ru -C \nbreaking layer . Both layers exhibited mainly a L1 0 ordered [001] texture along with \ncertain in -plane variants based on micromagnetic simulation with a finite element model \nreproduced from experimental microstructural features. The higher variants in the top \nlayer are att ributed to the Ru diffusion and the spherical shape of FePt grains, resulting in \na low µ 0Hc1 and Ktop compared to the bottom layer . The Tc of each layer differed owing to \nRu diffusion, deviation s in the Fe and Pt content of the composition, and the order \nparameters, which facilitates multi -level recording.  \nThe writing performance of the media was evaluated using experimental magnetic \nmeasurements and a HAMR recording simulation. Both layers were written successfully \nfollowing the first pass ; however, durin g the second pass, certain remnants of the first \n \n track remain ed in the top layer. This can be overcome by increasing Ktop by improving the \ndegree of the L10 order  and by searching for a suitable spacer layer.  After the two -pass \nwriting, the magnetic states of ↑↑, ↓↓, and ↓↑ were obtained, with ↑↓ and ↓↑ having M≠0, \nthus, allowing them to be distinguished and enabling an expansion from 3 -level to 4 -level \nrecording.  For this concept, d ue to the magnetic field st rength of the magnetic head and \nthe Curie temperature requirements, the number of layers is estimated to be three at most.  \nFurther optimization of microstructure and magnetic properties would pave the way to 10 \nTbit/in2. \nMethods  \nExperimental methods  \nAll the films were prepa red using an ultrahigh -vacuum co -sputtering system with a base \npressure of approximately 10-7 Pa. To investigate  the breaking layer , FePt (17  nm)/X (with \na gradual thickness (2 –7 nm)/FePt (17 nm )) was deposited on an STO (001) subst rate at \n500 ºC for X =  Ru, Cu , and Pt . The dual -layer  films were patterned into  Hall bars for \ntransverse  Hall bar resistance measurement s. An alternating MgO/FePt -C film stack  (20 \nvol.%) (4.5 nm)/FePt(0.26 nm)/Ru -C (20 vol.%) (2.9 nm)/Ru (0.11  nm)/FePt -C (20 \nvol.%) (4.5 nm)/C -cap was grown at 500  ºC as a granular dual medi um. The average \nheights of the layers (FePt -C (5.5±0.7 nm )/Ru-C (3.0±0.6 nm )/ FePt -C (5.1±0.6 nm )/) \nobtained from the EDS maps were consistent  with the design.  Thin FePt  and Ru layers \nwere deposited to prevent C migration. A C capping layer was deposited at 300 K to \nprevent surface oxidation. MgO single -crystal substrates are preferred for promoting  a \nstrong [001] texture an d minimizing misorientation in (001) -textured Fe Pt grains [ 9,42]. \nThe film stack of MgO/FePt -Ru (30 nm)/C -cap (4.5 nm) was grow n at a substrate \ntemperature of 500 ºC to perform the  comparison s. X-ray diffraction (XRD) Rigaku  \n SmartLab with a Cu X -ray source was used for crystallographic and degree of L1 0 order \nanalyses. Electron -transparent thin specimens were prepared for transmission electron \nmicroscopy (TEM) using a focused ion beam lift -out technique (FEI Helios Nanolab 650 ). \nTo prevent damage during the ion  beam milling, the film surfaces  were coated with Ni. \nTEM was performed using a Titan G2 80 –200 with a probe aberration corrector. E nergy -\ndispersive  X-ray spectroscopy (EDS) was conducted using an FEI Super -X EDX detector. \nA 7T Quantu m Design superconducting quantum interference device magnetometer \n(SQUID) was used for the magnetic measurements. The films were embedded in high -\ntemperature cement for the measurement s at temperatures above 380 K. The Curie \ntemperature ( Tc) was determined  by the least square s fitting using Kuz’min’s equation \n[43]. The magnetization direction (θM) dependences of torque s were measured using the \nanomalous Hall effect  to deduce the first - and second -order uniaxial anisotropy constants \nK1 and K2 as described by Ono et al. [ 41].  \nMicromagnetic Approximation of Hysteresis Loops :  \nA finite element model of a two -layer FePt granular medi um was developed using \nV oronoi -based tessellation . The bottom layer was represented by 5 nm thick columnar \ngrains with a  top-view grain size of 12.5 (2.0) nm , which  reproduc ed the experimental \ngrain size distribution. The t op layer with a thickness of ttop was fabricated by the out -of-\nplane t ranslation of the bottom layer through a 3 -nm-thick nonmagnetic gap (breaking \nlayer). Following the TEM observations of  the real samples, in-plane variants were \nintroduced in to the model by randomly splitting  certain  grains into  several  regions  with \nmutually orthogonal easy magnetization axes. The total volume fraction s of the in -plane \nvariants were controlled in both layers using Vtop and Vbot. The magnetic anisotropy \nconstant was distributed among the grains , with mean values of Ktop and Kbot as well as  \n standard deviations that were individually prescribed for the top and bottom layers , \nrespectively . The saturation magnetization and exchange stiffness values were assumed \nto be the same for all grains,  that is,  μ0Ms = 1.43 T and A = 10 pJ/m, respectively [ 39]. \nMicromagnetic simulations  were performed  using the FastMag software by solving the \nLandau –Lifshitz –Gilbert equation with a damping constant α = 1  [42]. The parameters \nttop, Vtop, Ktop, Vbot, and Kbot were varied in a grid -search manner to approximate the \nexperimental hysteresis loops.  \nMultilevel HAMR recording simulation:  \nThe model uses the Landau –Lifshitz –Bloch equation, which is suitable for modeling  the \nbehavior of magnetic materials at temperature s up to and beyond Tc [43]. The temperature \ndependence of Ms was calculated by assuming a Brillouin function with J = 1, and Ku was \nderived from Ms as Ku(T) µ Ms(T)2 [44]. Each image depicts the average magnetization of \n50 tracks, where a magnetic head was used to read or write data.  The simulation \nparameters used were the experimental Tc and predicted Ku based on the experimental M-\nH. The recording medium had an average grain size , average grain pitch, and grain size \ndistribution  of 13 nm, 14 nm, and 9% , respectively . The top and bottom layers were both \n6 nm thick, separated by a 3 nm nonmagnetic layer.  The heat spot was moved along the \nmedium at a velocity of 4 m/s. The ambient temperature was set at 300 K. The heat spot \nhad a two-dimensional Gaussian t emperature distribution in the media with a half width \nat half maximum ( HWHM ) of 50 nm, that is, 50 nm from the center of the heat spot , and \nthe temperature was 300 + (0.5 ×(Tmax - 300)) K  where Tmax is maximum temperature . The \ntracks showed a signal -to-noise ratio closest to the average of the 50 written tracks. Thus, \nreversed grains within written bits are more likely to occur in the top layer.   \n Acknowledgments : This work was supported in part by the JST -CREST (JPMJC22C3) \nand MEXT program: Data Creation and Utilization -Type Material Research and \nDevelopment Project (JPMXP1122715503).  \n \nReferences:  \n[1] Reinsel,  D., et al.  The digitization of the world from edge to core . Framingham: \nInternational Data Corporation  16, 1–26 (2018).  \n[2] Özdemir , V . & Hekim,  N. Birth of industry 5.0: Making sense of big data with artificial \nintelligence, “the internet of things” and next -generation technology policy . Omics J. \nIntegr. Biol.  22, 65–76 (2018).   \n[3] Albuquerque,  G., et al.  HDD reader technology roadmap to an areal density of 4 tbpsi \nand beyond . IEEE Trans . on Magn . 58, 1–10 (2021) . \n[4] Weller,  D., et al.  Review Article: FePt heat assisted magnetic recording media . J. Vac. \nSci. Technol. B  34, 060801 (2016 ).  \n[5] Wang,  X., et al.  HAMR recording limitations and extendibility . IEEE Trans. on Magn.  \n49 686–692 (2013) . \n[6] Wu,  A. Q. , et al.  HAMR areal density demonstration of 1+ Tbpsi on spinstand . IEEE \nTrans. on Magn.  49, 779–782 (2013) . \n[7] Weller, D., et al. A HAMR media technology roadmap to an areal density of 4 Tb/in2. \nIEEE Trans. on Magn.  50, 1–8 (2013).  \n[8] Roddick, E. , et al. A new Advanced Storage Research Consortium HDD Technology \nRoadmap 2022 . IEEE 33rd Magnetic Recording Conference (TMRC) , Milpitas, CA, USA, \n1–2, (2022) DOI: 10.1109/TMRC56419.2022.9918580  \n[9] Hono,  K., et al. Heat -assisted magnetic recording media m aterials . MRS Bull . 43, 93 –\n99 (2018).  \n[10] Varaprasad,  B. S. D. , et al. Microstructure control of L1 0-ordered FePt granular film \nfor heat -assisted magnetic recording (HAMR) media . JOM  65, 853 –861 (2013) . \n[11] Weller,  D., et al.  L10 FePtX –Y media for heat‐assisted magnetic recording . Phys. \nStatus Solidi (a)  210, 1245 –1260  (2013) .  \n[12] Tozman,  P., et al.  Preparation of Fe ‐Pt powders from ingot by a simple and cost‐\neffective spark erosion device . Cryst. Res. Technol.  57, 2200117  (2022) . \n[13] Rottmayer,  R. E. , et al.  Heat -assisted magnetic recording . IEEE Trans. on Magn.  42, \n2417 –2421  (2006) . \n[14] Suzuki,  I. Control of grain density in FePt -C granular thin films during initial  \n growth . J. Magn. Magn. Mater.  500, 166418  (2020) . \n[15] Bolyachkin,  A., et al. Transmission electron microscopy image based micromagnetic \nsimulations for optimizing nanostructure  of FePt -X heat -assisted magnetic recording \nmedia . Acta Mater.  227, 117744  (2022) . \n[16] Pandey, H., et al. Structure optimization of F ePt–C nanogranular films for heat -\nassisted magnetic recording media . IEEE Trans. on Magn.  52, 1 –8 (2015) . \n[17] Zhang, L., et al. L10-ordered high coercivity (FePt) Ag –C granular thin films for \nperpendicular recording . J. Magn. Magn. Mater. 322, 2658 –2664  (2010) . \n[18] Takahashi, Y . K. , et al. Size dependence of ordering in FePt nanoparticles . J. Appl. \nPhys.  95, 2690 –2696  (2004) . \n[19] Varaprasad, B .S.D., et al. FePt–BN granular HAMR media with high grain aspect \nratio and high L1 0 ordering on corning LotusTM NXT glass . AIP Adv. 13, 035002  (2023).  \n[20] Xu, C. , et al, Bias sputtering of granular L10 -FePt films with hexagonal boron \nnitride grain boundaries . arXiv preprint arXiv:2303.11411  (2023).  \n[21] V ogler, C. , et al.  Heat -assisted magnetic recording of bit -patterned media beyond 10 \nTb/in2. Appl. Phys. Lett . 108, 102406 (2016).  \n[22] Albrecht, T. R. , et al. Bit-patterned magnetic recording: Theory, media fabrication, \nand recording performance . IEEE Trans. on Magn.  51, 1–42 (2015) .  \n[23] Albrecht,  M., et al. Magnetic dot arrays with multiple storage layers . J. Appl. \nPhys.  97, 103910  (2005).  \n[24] Suto,  H., et al. Layer -selective switching of a double -layer perpendicular magnetic \nnanodot using microwave assistance . Phys. Rev. Appl.  5, 014003  (2016) . \n[25] Winkler,  G., et al.  Microwave -assisted three -dimensional multilayer magnetic \nrecording . Appl. Phys. Lett.  94, 232501  (2009).  \n[26] Greaves, S. J., et al.  Areal density capability of dual -structure media for microwave -\nassisted magnetic recording . IEEE Trans. on Magn.  55, 1–9 (2019) . \n[27] Kobayashi , T., et al. Media design for three -dimensional heat -assisted magnetic \nrecording . J. Magn Soc. Jpn.  44, 122–128 (2020) .  \n[28] Kobayashi,  T., et al. Information stability in three -dimensional heat -assisted \nmagnetic recording . J. Magn Soc. Jpn.  44, 34–39 (2020).  \n[29] Mehta, V . V . , et al. Heat -assisted magnetic recording (HAMR) disk drive with disk \nhaving multiple continuous magnetic recording layers . U.S. Patent  No: 9,601,144, (2017 ). \n[30] Chang, T. Y ., et al. Zero -state insert dual layer HAMR media recording . IEEE Trans. \non Magn.  59, 3200107  (2022).  \n[31] Park,  J., et al.  Magnetic properties of Fe -Mn-Pt for heat assisted magnetic recording \napplications . J. Appl. Phys.  117, 053911  (2015) .  \n [32] Ono,  T., et al. Experimental investigation of off -stoichiometry and 3d transition metal \n(Mn, Ni, Cu) -substitution in single -crystall ine FePt thin films . AIP Adv. 6, 056011  (2016) . \n[33] Granz, S. D. , et al. Granular L1 0 FePt-B and FePt -B-Ag (001) thin films for heat \nassisted magnetic recording . J. Appl. Phys. 111, 07B709  (2012) . \n[34] Kota,  Y. First-principles calculation of Curie temperature tuning in L1 0-type FePt by \nelement substitution of Mn, Cu, Ru, and Rh.  J. Magn Soc. Jpn.  44, 1 –4 (2020) . \n[35] Zhang , D.-L., et al. L10 FePd synthetic antiferromagnet through an fcc Ru spacer \nutilized for perpendicular magnetic tunnel junctions . Phys . Rev. Appl . 9, 044028 (2018) . \n[36] Ono, T., et al. Addition of Ru to L1 0-FePt thin film to lower Curie temperature . Appl. \nPhys. Express . 9, 123002  (2016) . \n[37] Okamoto, S., et al. Chemical -order -dependent magnetic anisotropy and exchange \nstiffness constant of FePt (001) epitaxial films . Phys. Rev. B 66, 024413  (2002) . \n[38] Watanabe, M. & Masumoto, T. Extraordinary Hall effect in Fe –Pt alloy thin films \nand fabrication of micro Hall devices . Thin Solid Films  405, 92–97 (2002) . \n[39] Wang,  J., et al.  Impact of carbon segregant on microstructure and magnetic properties \nof FePt -C nanogranular films on MgO (001) substrate . Acta Mater.  166, 413–423 (2019) .  \n[40] Perumal,  A., et al.  Perpendicular thin films of carbon -doped FePt for ultrahigh -\ndensity magnetic recording media . IEEE Trans. on Magn. 39, 2320 –2322  (2003) . \n[41] Feng, C., et al.  Nonvolatile modulation of electronic structure and correlative \nmagnetism of L1 0-FePt films using significant strain induced by shape memory \nsubstrates . Sci. Rep.  6, 1–10 (2016) . \n[42] Wang, J., et al.  Effect of MgO underlayer misorientation on the texture and magnetic \nproperty of FePt –C granular film . Acta Mater . 91, 41–49 (2015).  \n[43] Kuz'min,  M. D.  Shape of temperature dependence of spontaneous magnetization of \nferromagnets: quantitative analysis . Phys. Rev. Lett.  94, 107204  (2005) .  \n[41] Ono, T., et al.  Novel torque magnetometry for uniaxial anisotropy constants of thin \nfilms and its application to FePt granular thin films . Appl. Phys. Exp . 11, 033002  (2018) .  \n[42] Chang, R., et al. FastMag: Fast micromagnetic simulator for complex magnetic \nstructures . J. Appl . Phys . 109, 07D358  (2011) . \n[43] Evans, R. F. L. , et al. Stochastic form of the Landau -Lifshitz -Bloch equation . Phys. \nRev. B  85, 014433 (2012 ). \n[44] Greaves, S. J., et al. Modelling of thermally assisted magnetic recording with the \nLLB equation and Brillouin functions . J. Appl. Phys. 117, 17C505, (2015).  \n \n \n  \n Supplementary Information  \n \nP. Tozman1*, S. Isogami1, I. Suzuki1, A. Bolyachkin1, H. Sepehri -Amin1, S.J. Greaves2 \nH. Suto1, Y . Sasaki1, H.T. Y . Chang3, Y Kubota3, P. Steiner3, P.-W. Huang3, K. Hono1 \nand Y .K. Takahashi1* \n \n1 National Institute for Materials Science, Tsukuba, Ibaraki 305 -0047,  Japan  \n2Research Institute of Electrical Communications Tohoku University, Sendai, Japan  \n3 Seagate Technology, Recording Media Organization, Fremont, CA. 94538 USA  \n \nA. Magnetic properties influenced by various breaking layers  \nFig. S1(a) shows the film stack of FePt (17  nm)/ breaking layer ( X) (2–7 nm)/FePt (17 \nnm), which was used to investigate the X-dependent magnetic properties . This stack was \ndeposited on an STO (001) substrate at 500  ºC for X = Ru, Cu , Pt, and W. The lattice \nparameter s of these elements are presented in Fig . S1(b) [1,2].  Fig. S1(c) shows the \ntransverse resistance (Rxy) of the Hall bar devices as a function of the magnetic field \napplied along the film normal  (0H), where R xy was normalized by the value for 0H = 2 \nT.  \nFig. S1. (a) Stacking structure with highly textured FePt films grown on an STO (001) substrate. \n(b) Lattice parameters of various breaking layers  (X), FePt , and substrates . Note that the * symbol \nindicates Ru with the fcc structure  (c) Normalized transverse resistance (R xy) of the  Hall bar \ndevices with the 2 nm -thick X = Pt, W, Cu , and Ru.  \n \n B. Structural analysis using transmission electron microscopy  \nFig. S2  shows  the nanobeam  electron diffraction patterns along with high-magnification , \ncross -sectional high -angle annular dark -field (HAADF) scanning transmission electron \nmicroscopy (STEM) image s and energy -dispersive X -ray spectroscopy (EDS) line for  \nFePt (17  nm)/ Pt (2 nm)/FePt (17 nm). The bottom and top  FePt layer s exhibit  (001) \nepitaxy , that is,  fine crystallographic orientation along the [001]  direction  (Fig. S2(a,b)) . \nBoth layers were composed of Fe40Pt60, which was slightly richer in Pt than the single \nFePt layer with Fe 42Pt58 (Fig. S2(c)). Because both layers exhibit the same composition, \nthe M-T curve,  measured under 0H = 0.1 T, shows a single Tc resulting in similar Tc1 and \nTc2 values (Fig. S2(e)). Therefore, Pt can be considered a spacer for granular media if Tc1 \nand Tc2 are differentiated  in the future.  \n \nFig S2 . (a) Nanobeam electron diffraction patterns with FePt -[100] zone axis for the bottom - and \ntop-FePt layer s. (b) High-resolution HAADF -STEM images . (c) STEM -EDS maps of Fe and Pt \nalong with (d) line scan . (e) Temperature -dependent normalized magnetization, M-T curve, for \nthe FePt/Pt/FePt structure.  \n \n \n \n In Fig. S3, the nanobeam  electron diffraction patterns along with high-magnification \ncross -sectional HAADF -STEM image s and EDS maps are shown for FePt (17  nm)/ W (2 \nnm)/FePt (17 nm). The bottom  FePt and W -breaking layer s exhibit (001) epitaxy . \nAlthough the bottom interface between the bottom  FePt and W is smooth, that between \nthe top  FePt and W is rough,  which causes (111) growth in the top FePt.   \n \nFig. S3. (a) Nanobeam electron diffraction patterns of bottom -FePt, W breaking layer  with [ 110] \nzone axis , and top -FePt with zone axis [110]. (b) High-resolution HAADF -STEM images . (c) \nSTEM -EDS maps for  the FePt/W/FePt structure.   \n \nC. Pt substitution by Ru  \nFig. S 4(a–c), show the XRD patterns, atomic order parameter S, and lattice parameter c, \nrespectively, as  function s of the Ru content  in Fe 42(Pt58-xRux). The peaks observed at \napproximately 23 and 48 originate from the superlattice 001 and fundamental 002 \nplanes, respectively. The S parameter of FePt  was consistent with the value reported in \nthe literature for a given substrate temperature of 500 ºC [3]. Furth er, the Ru substitution \ndecreased  the L10 order while increasing the lattice parameter c. \n \n Fig. S 4. (a) Out-of-plane X-ray diffraction (XRD) patterns for the Fe42(Pt58-xRux) films with 0 ≤ x \n≤8 along with the Ru content dependence of (b) the atomic order parameter S and (c)  the lattice \nparameter c.  \n \nThe magnetocrystalline anisotropy constants K1 and K2 were deduced using a  torque \nmeasurement method based on the anomalous Hall effect (AHE) in the  temperature range  \nof 100 –370 K [ 4] (Fig. S 5 (a)). Although the temperature dependence of K1 follows the \nsame trend as in the literature, K2 has an opposite sign to the reported value [ 5]. This \ndiscrepancy may be because of  the different analysis techniques : the Sucksmith \nThompson (ST) and AHE torque methods used for determining K1 and K2, wherei n the \nST method induces a large error, particularly at K2 [6]. Increasing the Ru content in \nFe42(Pt58-xRux) decreased K1 and changed the sign of K2 from negative to positive . \n \n Fig. S 5. (a) Temperature dependence of magnet ocrystalline  anisotropy constants  of K1 and K2 in \nthe Fe42(Pt58-xRux) films with 0 ≤ x ≤ 8 at.%. Filled  and half-filled marks correspond to K1 and K2, \nrespectively. Ru content dependence of (b) the anisotropy field (μ0Ha) and (c) the effective \nanisotropy constant, Ku = K1 + K2  \n \nFig. S6(a) shows the saturation magnetization versus temperature for Fe 42(Pt58-xRux), \nwhere the circle and line indicate  the experimental and fitted data, respectively. The Curie \ntemperature ( Tc) was determined by the least square s fitting using Kuz’min’s equation [7].  \nTc of Fe42Pt58 decreased from 610 to 540 K upon substituting only a small  amount of Ru \nin Fe 42Pt57.5Ru0.5. A further increase in Ru content was expected to further decrease Tc, \nfollowing a linear trend with a smaller decrement rate of ≈15%/at.% , which is close to the \nrate reported in  the literature,  i.e., ≈17%/at.% (Fig. S 6 (b)) [8]. The decrease in Tc was \nfollowed by a decrease in Ms, as shown in Fig. S 6(c), which has been  confirmed via DFT \ncalculation [9].  \n \n Fig. S 6. (a) Temperature -dependent saturation magnetization  (μ0Ms) for the  Fe42(Pt58-xRux) with 0 \n≤ x ≤ 8 at.% , where the dots and  lines indicate  experimental and fitted data, respectively . Ru \ncontent dependence of (b) the Curie temperature  (Tc) and ( c) the saturation magnetization ( μ0Ms). \n \nThe chemical order of Fe and Pt in the FePt grains of the FePt-C granular layer was \nestimated using S = 0.85×√𝐼001𝑒𝑥𝑝/𝐼002𝑒𝑥𝑝 [10]. A large chemical ordering parameter, S = \n0.8, was observed, even at a thickness as small as 4.5 nm (Fig. S 7(a)). This value is \ncomparable to that reported in the literature for a single FePt -C granular layer with a \nthickness of 8 nm [ 11,12]. However,  S decrease d to approximately 0.63 in the  stacking \nwith dual FePt-C layer s, as shown in Fig. S7(b), which can be attributed to the lower S in \nthe top layer.  \n \n  \nFig. S 7. Out-of-plane  XRD patterns and estimated chemical order parameters ( S) for the stacking \nwith (a) single FePt-C nano granular layer and (b)  dual layer s. \n \n \nD. Micromagnetic simulation for the dual nanogranular films  \n \nIn Fig. S8(a), the microstructural features of the  dual-nanogranular  film were  reproduce d \nusing a finite element model  based on the experimental microstructure . The table in Fig. \nS8(b) shows the estimated magnetic anisotropy constant ( Ku), volume fraction of the in -\nplane variants ( V), and height of the layer  (t) for the top and bottom layers  from the \nmicromag netic approximation. As the actual thickness of the top layer may deviate from \nthe nominal thickness during deposition on the curved interface of the bottom layer,  this \nstudy introduced  the thickness of the top layer as an approximation parameter.  \n \n Fig. S8. (a) The micromagnetic model along with (b) estimated parameters for the FePt -20 \nvol.%C/Ru - 20 vol% C/FePt -20 vol% C.  \n \nReferences:  \n[1] https://mits.nims.go.jp/  \n[2] Zhang,  D.-L., et al.  L10 Fe−Pd synthetic antiferromagnet through an fcc Ru spacer \nutilized for perpendicular magnetic tunnel junctions . Phys . Rev. Appl . 9, 044028  (2018) . \n[3] Takahashi, Y . K. , et al. Microstructure and magnetic properties of FePt  thin films \nepitaxially grown on MgO (001) substrates . J. Magn. Magn. Mater.  267 248–255 (2003) . \n[4] Ono, T., et al. Novel torque magnetometry for uniaxial anisotropy constants of thin \nfilms and its application to FePt granular thin films . Appl. Phys. Exp.  11 033002  (2018) .  \n[5] Okamoto, S ., et al.  Chemical -order -dependent magnetic anisotropy and exchange \nstiffness constant of FePt (001) epitaxial films . Phys . Rev. B 66, 024413  (2002) . \n[6] Ogawa, D., et al. Magnetic anisotropy constants of ThMn 12-type Sm(Fe 1–xCox)12 \ncompounds and their temperature dependence . J. Magn. Magn. Mater.  497, 165965  \n(2020) . \n[7] Kuz'min, M. D.  Shape of temperature dependence of spontaneous magnetization of \nferromagnets: quantitative analysis . Phys. Rev. Lett.  94, 107204 (2005).  \n[8] Ono, T., et al.  Addition of Ru to L1 0-FePt thin film to lower Curie temperature . Appl. \nPhys. Express , 9, 123002  (2016) . \n[9] Kota , Y . First-principles calculation of Curie temperature tuning in L1 0-type FePt by \nelement substitution of Mn, Cu, Ru, and Rh.  J. Magn Soc. Jpn.  44, 1 -4 (2020) . \n[10] Christodoulides, J.A. , et al. Magnetic, structural and microstructural properties of \nFePt/M (M= C, BN) granular films . IEEE Trans. Magn.  37, 1292 -1294  (2001) . \n[11] Wang,  J., et al.  Impact of carbon segregant on microstructure and magnetic \nproperties of FePt -C nanogranular films on MgO (001) substrate . Acta Mater . 166, 413-\n \n 423 (2019) . \n[12] Varaprasad,  B. C. S. , et al. MgO -C interlayer for grain size control in FePt -C media \nfor heat assisted magnetic recording.  AIP Adv . 7, 056503  (2016) . \n \n "    },    {        "title": "2307.03458v1.Anomalous_Nernst_effect_in_perpendicularly_magnetised_τ_MnAl_thin_films.pdf",        "content": "arXiv:2307.03458v1  [cond-mat.mtrl-sci]  7 Jul 2023Anomalous Nernst effect in perpendicularly magnetised τ-MnAl thin films\nD. Scheffler,1S.Beckert,1H.Reichlova,1,2T.G. Woodcock,3S.T.B. Goennenwein,4and A. Thomas1,3\n1)Institute for Solid State and Materials Physics, Technisch e Universit¨ at Dresden, 01062 Dresden,\nGermany\n2)Institute of Physics ASCR, v.v.i., Cukrovarnick´ a 10, 162 5 3, Praha 6, Czech Republic\n3)Leibniz Institute for Solid State and Materials Research Dr esden (IFW Dresden), 01069 Dresden,\nGermany\n4)Department of Physics, University of Konstanz, 78457 Konst anz, Germany\n(Dated: 10 July 2023)\nτ-MnAl is interesting for spintronic applications as a ferro magnet with perpendicular magnetic anisotropy due to its\nhigh uniaxial magnetocrystalline anisotropy. Here we repo rt on the anomalous Nernst effect of sputter deposited τ-\nMnAl thin films. We demonstrate a robust anomalous Nernst eff ect at temperatures of 200 K and 300 K with a hysteresis\nsimilar to the anomalous Hall effect and the magnetisation o f the material. The anomalous Nernst coefficient of (0.6±\n0.24)µV/K at 300 K is comparable to other perpendicular magnetic an isotropy thin films. Therefore τ-MnAl is a\npromising candidate for spin-caloritronic research.\nI. INTRODUCTION\nτ-MnAl is a L10ordered ferromagnetic compound with\na high Curie temperature of approx. 650 K, a saturation\nmagnetisation of 600 kAm−1and a high uniaxial magnetic\nanisotropy of 1.7 MJm−3making it an interesting material\nfor the application as a rare earth free permanent magnet1–3.\nThe material also displays perpendicular magnetic anisotr opy\n(PMA), which is crucial for many spintronic applications, e .g.\nmagnetic storage or spin transfer torque switching4–7. Conse-\nquently, the magnetotransport properties of τ-MnAl thin films\nwere studied in terms of the anomalous Hall effect (AHE)\nand tunnel magnetoresistance8–16. However, the magneto-\nthermal properties of τ-MnAl are still unknown, despite re-\nnewed interest in large anomalous Nernst (ANE) effect val-\nues in emerging materials17–19, which are often discussed in\nthe context of intrinsic Berry phase related contributions20.\nThe large ANE values also renewed interest in its applicatio n\npotential as, e.g., a heat flux sensor21or energy harvesting\nelements22, in particular, in PMA materials23–25.\nIn this manuscript, we report on the anomalous Nernst ef-\nfect (ANE) in epitaxial thin films of τ-MnAl grown via mag-\nnetron sputtering at various temperatures. To measure the\nANE (the thermal counterpart of the anomalous Hall effect),\na thermal gradient ∇Txapplied to the sample drives a volt-\nage perpendicular to both the thermal gradient ∇Txand the\nmagnetic order vector. We show that the ANE is sponta-\nneous (present at zero magnetic field) and that it exhibits a\nrobust hysteresis with a coercivity of approximately µ0Hc≈\n1 T, comparable the Hcvalues obtained from magnetome-\ntry measurements with a superconducting quantum interfer-\nence device vibrating sample magnetometer (SQUID-VSM)\nand the AHE. We compare the anomalous Nernst coefficient\nSA\nxyto other PMA materials and find that τ-MnAl thin films\nare an interesting candidate material for spintronic and sp in-\ncaloritronic research and devices.II. FILM GROWTH, STRUCTURE AND MAGNETIC\nPROPERTIES\nMnAl thin films were deposited by magnetron sputtering\nonto MgO(001) substrates using a Bestec UHV sputter tool.\nThe films investigated in this manuscript had a thickness of\ntMnAl=78nm and a Cr(001) buffer layer ( tCr=19 nm) was\nutilised to reduce the lattice mismatch between the MgO sub-\nstrate and the thin films12,26,27.\nPrior to deposition, the substrates were subsequently\ncleaned in acetone and ethanol in a ultrasonic bath for 10 min -\nutes (each). After transferring to the chamber, the substra tes\nwere annealed at 800°C for 1 h to eliminate surface contami-\nnations. The Cr buffer layer was deposited at room tempera-\nture with a power of 30 W (DC) using a sputtering pressure of\npdep,Cr=3×10−3mbar (Ar). A subsequent annealing step at\n600°C for 1 h in vacuum was necessary for the crystallisation\nof Cr. The MnAl film was deposited by cosputtering, apply-\ning 60 W rf power to a MnAl(50/50 at.%) target and 10 W dc\npower to an aluminium target under a sputtering pressure of\npdep,Al=5×10−3mbar (Ar). A temperature of 300 °C during\nthe deposition and an in situ post-annealing at 500 °C for 1 h\nunder vacuum is needed for the crystallisation and the chem-\nical ordering of τ-MnAl. While several samples were fab-\nricated and measured, in this work we focus on the results\nof one representative τ-MnAl thin film. Similar results were\nahieved for the other τ-MnAl thin films.\nThe crystalline structure was investigated by X-ray diffra c-\ntion (XRD) using Cu- Kα1radiation ( λ=0.15406nm).\nThe symmetrical 2 θ−ωscan is shown in Fig 1a.\nThe epitaxial relation of the system is given by\nMgO(001)[100]//Cr(001)[110]// τ-MnAl(001)[110] and\naccording reflexes are expected in the XRD data: Besides\nthe MgO(002) peak of the substrate and the Cr(002) peak of\nthe buffer layer, only τ-MnAl(001) and τ-MnAl(002) can be\nobserved, corroborating the epitaxial growth of τ-MnAl and\nindicating a predominant (001)-orientation of the film. In a d-\ndition, the presence of the superlattice peak of (001) τ-MnAl\nindicates a high degree of chemical ordering.\nThe magnetic properties were studied via SQUID-VSM2\n20 40 60\n2θ (°)10−510−410−310−210−1100norm. Intensit \nMgO(002)(a)\nτ - MnAl(001) τ - MnAl(002)\nCr(002)\n−5 0 5\nμ0H(T)−300−200−1000100200300M (kAm−1)(b)\n300 K\n200 K\n100 K\nFIG. 1. (a) Symmetrical 2 θ−ωXRD scan of 78 nm thick τ-\nMnAl thin film. (b) SQUID-VSM measurement of the same thin film\nat different temperatures. The magnetic field was applied pe rpendic-\nular to the film plane (solid lines) and parallel to it (dotted lines).\nat different field orientations and temperatures, as shown i n\nFig.1b. The film is ferromagnetic and does show the ex-\npected perpendicular magnetic anisotropy with a saturatio n\nmagnetisation of Ms=272 kA/m and a high perpendicular co-\nercive field of µ0Hc=1.15 T at 100 K. Similar values were\nreported for other τ-MnAl thin films26,27. At higher temper-\natures, the saturation magnetisation as well as the coerciv e\nfield decrease which is expected closer to the Curie tempera-\nture of the system1. Compared to bulk samples of MnAl-C3,\nthe film grown here has a much lower saturation magnetisa-\ntion (cf. film: Ms=272 kA/m, bulk: Ms=620 kA/m) but\na much higher coercivity (cf. film: µ0Hc=1.15 T, extruded\nbulk, typically µ0Hc=0.35 T). Please note that the maximum\nfield of µ0H=±7 T is not sufficient for the saturation of the\nfilm when the magnetic field is applied in the magnetically\nhard film plane.\nIII. ANOMALOUS HALL EFFECT\nFor the magneto-transport measurements, the τ-MnAl film\nwas then patterned into Hall bars by a combination of optical\nlithography and ion beam etching. Subsequently, the on-chi p\nheaters were defined by a lift-off process with 10 nm of sput-\ntered Pt. The complete structure is shown in Figure 2a. The\nmagneto-transport response was recorded in a cryostat with\nvariable temperature insert (VTI) to control the sample bas e\ntemperature. An external magnetic field of up to µ0Hz=±5 T\nwas applied perpendicular to the film surface (out-of-plane ).\nThe heat gradient for the anomalous Nernst measurements\nwas generated by the on-chip Pt heater.\nFirst, we evaluate the magneto-transport properties of our\nτ-MnAl thin film by measuring the AHE: A current Ix=\n100µA is applied along the Hall bar, i.e. between contacts\n#5−#8 as shown in Fig. 2a. The transversal voltage Vy\n(#4−#6) generated by the Hall effect and the longitudinal\nvoltage Vx(#6−#7) are measured simultaneously as a function\nof the applied perpendicular magnetic field. We determine\nthe longitudinal resistivity of the sample as ρxx,s=Vxwts/Ixl\nand transversal Hall resistivity as ρxy,s=Vyts/Ix, where wandlare the width and contact separation of the measured Hall\nbar, respectively. tsdenotes the combined thickness of the τ-\nMnAl film and the buffer layer.\nIt is important to note that the applied current as well as the\ngenerated Hall voltage of the τ-MnAl thin film are affected\n(shorted) by the highly conductive Cr buffer layer. Therefo re,\nthe longitudinal resistivity ρxxand transversal Hall resistivity\nρxyof theτ-MnAl thin film are calculated as\nρxx=tMnAlρxx,Crρxx,s\ntρxx,Cr−tCrρxx,s(1)\nρxy=ρxy,s(ρxx,CrtMnAl+ρxxtCr)2\nρ2\nxx,CrtMnAlts(2)\nwhere ρxx,Cris the longitudinal resistivity of the Cr buffer\nlayer. ρxx,Crwas measured on a separate sample with a single\nCr layer. Please note, that we exclude any possible (anoma-\nlous) Hall voltages generated within the Cr layer. For a de-\ntailed derivation of equations 1and2, please refer to the sup-\nplementary material.\nThe anomalous Hall resistivity of τ-MnAl ρAHE\nxy was ex-\ntracted by subtracting the ordinary ρOHE\nxy determined by a lin-\near fit to the high field slope of ρxy. Ultimately, we calculate\nthe longitudinal conductivity σxxand the anomalous Hall con-\nductivity σAHE\nxy ofτ-MnAl as28\nσAHE\nxy=−ρAHE\nxy\nρAHE\nxy2+ρ2\nxx,σxx=1/ρxx (3)\nThe magnetic field dependence of σAHE\nxy at different tem-\nperatures is given in Fig. 2b. In accordance to previous\nstudies8,11,15and to the magnetisation hysteresis of the τ-\nMnAl thin film, the anomalous Hall conductivity is scaling\nwith the magnetisation of τ-MnAl and is showing a hysteresis\nwith a high coercive field of approximately 1 T. Consequen-\ntially, a decrease of the switching field and a decrease of the\nmagnitude towards higher temperatures can be observed.\nThe AHE is often discussed in terms of different intrinsic\nand extrinsic contributions33,34. Here, we follow the typical\nprocedure to investigate the origin of the AHE based on the\nlongitudinal and transversal conductivity, in which three dis-\ntinct regions are distinguished33. Ourτ-MnAl thin film can\nbe classified into the intermediate regime (see Fig. 2c) ac-\ncording to the conductivity, suggesting that the anomalous\nHall effect is dominated by the scattering-independent int rin-\nsic and side-jump contributions33. This is in accordance to\na previous study of τ-MnAl thin films on GaAs by Zhu et\nal.11although our measured σAHE\nxy and longitudinal conduc-\ntivity σxxare significantly higher. We attribute this to the dif-\nferent preparation procedure of the τ-MnAl thin films. As\nalready pointed out by Zhu et al.11, the intrinsic Hall effect\nis dependent on the electronic structure of τ-MnAl which is\neasily modified by changes of the chemical composition and\nchemical disorder11,35.3\n(a)\n−5 0 5\nμ\n0H\nz(T)−200−1000100200σAHE\nx− (Ω−1\ncm−1\n)(b)\n300 K\n200 K\n100 K\n104\n106\nσ\nxx (Ω−1\ncm−1\n)100101102103|σAHE\nx−| (Ω−1\ncm−1\n)intrin)ic x -\ntrinsic(c)\nthis work\nMnAl[9]\nFePt[27]\nFePt[29]Ni[28]\nCo[28]\nMnSi[30]\nFIG. 2. (a) Schematics of the patterned MnAl film and the Pt hea ter.\nThe contacts of the Hall bar are numbered to enable a simple de scrip-\ntion of the measurement scheme. (b) Magnetic field dependenc e of\nthe anomalous Hall conductivity σAHE\nxy for different temperatures.\n(c) Summary of absolute values of the anomalous Hall conduct ivity\nσAHE\nxy as a function of the longitudinal conductivity σxxin various\nthin film materials11,29–32, adapted from Oneda et al.33.\nIV. ANOMALOUS NERNST EFFECT\nFor the measurement of the ANE, we apply a current of Ih=\n8 mA to one of the Pt heaters to generate a heat gradient ∇Tx\nalong the longitudinal direction and measured the transver sal\nvoltage Vy. To determine the heat gradient ∇Txwe adopted\nmethods used previously for ANE measurements36,37: two\ntransversal arms of the Hall bar serve as resistive local tem -\nperature sensors. The heat gradient is then calculated as\n∇Tx=∆Tx/Lwhere ∆TxandLare the temperature difference\nand distance between the two transversal legs. For further d e-\ntails on the thermometry of the ANE measurement, please see\nthe supplementary material.\nSimilar to the Hall effect, the electrically conductive Cr\nbuffer layer also impacts the magnitude of the anomalous\nNernst effect. Thus, the Nernst effect signal is calculated as\nSxy=Vy(ρxx,CrtMnAl+ρxxtCr)\nρxx,CrtMnAl1\n∇Txd(4)\nwhere d=1400µm is the contact distance of the transver-\nsal contacts. A derivation of equation 4is given in the supple-\nmentary material.\nThe field-dependent Nernst signal Sxyfor different temper-\natures thus obtained is given in Fig. 3a.Sxyconsists of a neg-\native linear function superimposed by a hysteresis at 200 K\nand 300 K. No hysteresis can be resolved at a sample temper-\nature of 100 K. Similar to the Hall effect, the Nernst effect\ncan phenomenologically be understood as the combination ofFIG. 3. (a) Magnetic field dependence of Sxyat different tempera-\ntures. Vywas measured between #4 and #6 at 100 K and #3 and #7\nat 200 K and 300 K. (b) Magnetic field dependence of SANE\nxy for dif-\nferent temperatures. The error of SA\nxywas calculated based on the\nerror of the heat gradient and the noise of the measured Nerns t volt-\nage and is exemplary given for 300 K. (c) Comparison of the abs olute\nAnomalous Nernst coefficient |SA\nxy|for different PMA materials and\nthe results of this study, for a sample temperature of 300 K20,23,38.\n(d) Evolution of the anomalous Nernst coefficient with tempe rature.\nthe ordinary Nernst contribution SONE\nxyscaling linear with the\nexternal magnetic field Hand the magnetisation dependent\nSANE\nxy23\nSxy=SONE\nxy+SANE\nxy=µ0HNONE+SA\nxyM\nMs(5)\nwhere NONEandSA\nxyare the ordinary and anomalous Nernst\ncoefficients, respectively. The hysteretic contribution c ontri-\nbution to Sxythus corresponds to the ANE where as the gen-\neral negative slope in each measurement can be assigned to\nthe ONE.\nThe anomalous Nernst contribution SANE\nxy is calculated by\nsubstracting the ONE from the Nernst effect and is given in\nFig. 3b as a function of the external field. SANE\nxy switches\nin sign at around ±1 T, in close analogy to the magnetization\ncurve and the AHE. This supports the notion that the thermal\nvoltage is generated by the ANE of the τ-MnAl thin film. For\na sample temperature of 100 K, no clear ANE signal can be\nobserved.\nFrom the data in Fig. 3b, we obtain an anomalous Nernst\ncoefficient SA\nxy= (0.60±0.24)µV/K at 300 K and (0.43±\n0.16)µV/K at 200 K. For the error calculation we consider an\nerror of the heat gradient of ±0.1Kmm−1as well as the noise4\nTABLE I. Measured transport coefficients of the (isolated) τ-MnAl thin film at different temperatures. (*) These values a re not measured but\nestimated by using the Mott relation28.\nT(K) σxx(Ω−1cm−1)×103σxy(Ω−1cm−1) Sxx(µV/K) SA\nxy(µV/K) αxy(A/Km)\n100 37.174 193.05 8.01 0.1 ±0.17 * 0.155*\n200 17.088 176.69 9.64 0.43 ±0.17 0.904\n300 11.494 134.61 22.84 0.6 ±0.24 0.998\nlevel of the measured anomalous Nernst voltage which yields\na relative high error of SA\nxy. Given this high error, we expect\nthe ANE at 100 K to be below the resolution limit of our ANE\nmeasurement as visible in Fig. 3b.\nComparing the ANE at 200 K and 300 K, we observe a de-\ncrease of the ANE towards lower temperatures, which is in\ncontrast to the trend observed in AHE and magnetometry (see\nFig. 3d). Similar correlations are reported for other ferro-\nmagnetic materials at temperatures well below TC23,37,39–41,\nincluding L10-ordered alloys23. There, the ANE vanishes to-\nwards T=0 K as the entropy change of the magnet reaches\nzero23.\nTo gain further insights into the temperature dependence\nof the ANE, we also measured the Seebeck effect of the τ-\nMnAl thin film. For these experiments, a heating current of\n8 mA was applied and the voltage Vxwas measured between\ncontact #2 and #4 (see Fig. 2a). Again, the electrical conduc-\ntive Cr layer impacts the magnitude of the Seebeck effect and\nthe Seebeck coefficient Sxxis thereby calculated by\nSxx=−Vx(ρxxtCr+ρxx,CrtMnAl)−Vx,CrρxxtCr\nρxx,CrtMnAl∆Tx(6)\nwhere Vx,Cris the voltage generated by the Seebeck effect\nof the Cr layer42. More details on the Seebeck measurement\nas well as a derivation of Eq. 6are given in the supplementary\nmaterial. Using Sxx, we can calculate the anomalous Nernst\nconductivity αxyfor the temperatures of 200 K and 300 K,\ngiven by the general relation of the transport coefficients28:\nαxy=Sxxσxy+Sxyσxx (7)\nAll transport coefficients are summarised in Table I.\nBy fitting of both the measured Sxyandαxyat 200 K and\n300 K to the Mott relation28:\nSxy=ρxy\nρxx/parenleftBigg\nπ2k2\nB\n3eλ′\nλ−(n−1)Sxx/parenrightBigg\n(8)\nαxy=ρxy\nρ2\nxx/parenleftBigg\nπ2k2\nB\n3eλ′\nλ−(n−2)Sxx/parenrightBigg\n(9)\nwhere kBandedenote the Boltzmann constant and the el-\nementary charge, we can determine a pair ofλ′\nλ=6.43×\n10−19J−1andn=2 for the τ-MnAl thin film. Given thesevalues, we can finally estimate the anomalous Nernst coeffi-\ncient at 100 K to be SA\nxy=0.1±0.17µV/K. The high error\nexceeds the estimated value of SA\nxyand hence explains why\nthe ANE is not resolvable at 100 K.\nThe magnitudes of SA\nxyare well in the range reported for\nother ferromagnetic thin film materials20,23,38,43as compiled\nin Fig. 3c. As shown for other L10-ordered thin films, the\nmagnitude of SA\nxyis scaling with the magnetisation and mag-\nnetic anisotropy of the material20,23,43. However, as SA\nxyis\nstrongly dependent on film thickness44,45, a direct compari-\nson of different materials is challenging. In addition, the mag-\nnetic properties of L10-ordered materials are highly dependent\non the chemical composition and structural ordering20,27,46,47.\nThe variance of the AHE in τ-MnAl which we discussed ear-\nlier can also be extended to the ANE, given the similar physi-\ncal origin of both effects. Therefore further anomalous Ner nst\nmeasurement in dependence of film thickness, chemical com-\nposition and structural disorder will be important to gain a bet-\nter understanding of the magneto-thermal transport proper ties\nofτ-MnAl.\nV. CONCLUSION\nWe prepared τ-MnAl thin films with perpendicular mag-\nnetic anisotropy via sputter deposition, and measured the\nanomalous Nernst effect at various temperatures. The ANE\nfeatures a robust hysteresis and spontaneous signal in zero\nmagnetic field, in close qualitative agreement with anoma-\nlous Hall effect and SQUID magnetometry data. At 300\nK the anomalous Nernst coefficient reaches SA\nxy= (0.60±\n0.24)µV/K, a value well within the range of ANE coefficients\nreported in other PMA materials with similar magnetic prop-\nerties. Our results show that τ-MnAl in thin film form is an\ninteresting material for spin-caloritronic research and d evices.\nVI. ACKNOWLEDGEMENTS\nWe thank Torsten Mix for helpful discussions. HR ac-\nknowledges Czech Science Foundation GACR 22-17899K.\nThis work was funded by the Deutsche Forschungsgemein-\nschaft (DFG, German Research Foundation) – Project num-\nber: 380033763.\n1J. M. D. Coey, Journal of Physics: Condensed Matter 26, 064211 (2014) .\n2S. Kontos et al. ,Energies 13(2020), 10.3390/en13215549 .\n3L. Feng et al. ,Acta Materialia 199, 155 (2020) .5\n4D. Apalkov et al. ,Proceedings of the IEEE 104, 1796 (2016) .\n5S. Ikeda et al. ,Nature Materials 9, 721 (2010) .\n6M. Cubukcu et al. ,Applied Physics Letters 104, 042406 (2014) .\n7C. Onur Avci et al. ,Applied Physics Letters 100, 212404 (2012) .\n8Y . Takeuchi et al. ,Applied Physics Letters 120, 052404 (2022) .\n9X. Zhang et al. ,Applied Physics Letters 110, 252403 (2017) .\n10L. J. Zhu et al. ,Nature Communications 7, 10817 () .\n11L. J. Zhu et al. ,Phys. Rev. B 93, 195112 () .\n12H. Saruyama et al. ,Japanese Journal of Applied Physics 52, 063003 (2013) .\n13K. K. Meng et al. ,Applied Physics Letters 110, 142401 (2017) .\n14K. Meng et al. ,Solid State Communications 255-256 , 15 (2017) .\n15L. Luo et al. ,Journal of Applied Physics 119, 103902 (2016) .\n16N. Anuniwat et al. ,Applied Physics Letters 102, 102406 (2013) .\n17T. Asaba et al. ,Science Advances 7, eabf1467 (2021) .\n18A. Sakai et al. ,Nature 14, 1119 (2018) .\n19B. He et al. ,Joule 5, 3057 (2021) .\n20Z. Shi et al. ,Phys. Rev. Applied 13, 054044 (2020) .\n21W. Zhou and Y . Sakuraba, Applied Physics Express 13, 043001 (2020) .\n22M. Mizuguchi and S. Nakatsuji, Sci. Technol. Adv. Mater. 20, 262 (2019) .\n23K. Hasegawa et al. ,Applied Physics Letters 106, 252405 (2015) .\n24R. Ando et al. ,Journal of Electronic Materials 45, 3570 (2016) .\n25J. Xu et al. ,Nano Letters 19, 8250 (2019) , pMID: 31658813.26D. Oshima et al. ,AIP Advances 10, 025012 (2020) .\n27M. Hosoda et al. ,Journal of Applied Physics 111, 07A324 (2012) .\n28Y . Pu et al. ,Phys. Rev. Lett. 101, 117208 (2008) .\n29Y . M. Lu et al. ,Phys. Rev. B 87, 094405 (2013) .\n30T. Miyasato et al. ,Phys. Rev. Lett. 99, 086602 (2007) .\n31P. He et al. ,Phys. Rev. Lett. 109, 066402 (2012) .\n32M. Lee et al. ,Phys. Rev. B 75, 172403 (2007) .\n33S. Onoda et al. ,Phys. Rev. B 77, 165103 (2008) .\n34N. Nagaosa et al. ,Rev. Mod. Phys. 82, 1539 (2010) .\n35S. Sato et al. ,Scientific Reports 10, 12489 (2020) .\n36H. Reichlova et al. ,Applied Physics Letters 113, 212405 (2018) .\n37G.-H. Park et al. ,Phys. Rev. B 101, 060406 (2020) .\n38S. Isogami et al. ,Applied Physics Letters 118, 092407 (2021) .\n39W.-L. Lee et al. ,Phys. Rev. Lett. 93, 226601 (2004) .\n40A. Sakai et al. ,Nature (2020), 10.1038/s41586-020-2230-zDO .\n41A. Ghosh et al. ,Journal of Applied Physics 125, 153902 (2019) .\n42S. Arajs et al. ,Journal of the Less Common Metals 26, 157 (1972) .\n43M. Ikhlas et al. ,Nature Physics 13, 1085 (2017) .\n44T. C. Chuang et al. ,Phys. Rev. B 96, 174406 (2017) .\n45H. Kannan et al. ,Scientific Reports 7, 6175 (2017) .\n46M. Oogane et al. ,Japanese Journal of Applied Physics 56, 0802A2 (2017) .\n47W. Zhou et al. ,Applied Physics Letters 118, 152406 (2021) .arXiv:2307.03458v1  [cond-mat.mtrl-sci]  7 Jul 2023Anomalous Nernst effect in perpendicularly magnetised τ-MnAl\nthin films - Supplementary Material\nD. Scheffler and S.Beckert\nInstitute for Solid State and Materials Physics,\nTechnische Universit¨ at Dresden, 01062 Dresden, Germany\nH.Reichlova\nInstitute for Solid State and Materials Physics,\nTechnische Universit¨ at Dresden, 01062 Dresden, Germany a nd\nInstitute of Physics ASCR, v.v.i., Cukrovarnick´ a 10, 162 53, P raha 6, Czech Republic\nT.G. Woodcock\nLeibniz Institute for Solid State and Materials Research\nDresden (IFW Dresden), 01069 Dresden, Germany\nS.T.B. Goennenwein\nDepartment of Physics, University of Konstanz, 78457 Konst anz, Germany\nA. Thomas\nInstitute for Solid State and Materials Physics,\nTechnische Universit¨ at Dresden, 01062 Dresden, Germany a nd\nLeibniz Institute for Solid State and Materials Research\nDresden (IFW Dresden), 01069 Dresden, Germany\n(Dated: July 10, 2023)\nI. DERIVATION OF ANOMALOUS HALL AND NERNST EFFECT\nCONSIDERING A CONDUCTIVE BUFFER LAYER\nIn general, the longitudinal resistivity ρxx, Hall resistivity ρxy, Hall conductivity σxyand\nNernst effect Sxyof a film stack are caluculated as:\nρxx=Vxwt\nIxl(1)2\nρxy=Vyt\nIx(2)\nσxy=−ρxy\nρ2xy+ρ2xx(3)\nSxy=Vy\n∇Txd(4)\nwherew,l,dandtare the bar width, the longitudinal contact separation, transver sal\ncontact separation and film thickness of the Hall bar, respectively .Ixis the applied current\nalong the x-direction, ∇Txthe temperature gradient along the x-direction. The voltage is\nrecorded in x-direction ( Vx) andy-direction ( Vy). However, if the film consist of a ferromag-\nnetic film layer with thickness tFand a buffer layer with thickness tB, the applied current\nand the transversal Hall/Nernst voltage of the ferromagnet are shorted by the buffer layer.\nA. Influence on longitudinal resistivity.\nIn the longitudinal direction, the conductive buffer layer carries a p art of the current. We\nmodel this by two individual resistors representing the buffer layer (RB) and the ferromag-\nnetic film layer ( RF) which are connected parallel (see electrical circuit Fig. S1). Then the\ntotal resistance/resistivity Ris defined by:\n1\nR=1\nRF+1\nRB(5)\nt\nρxx=tF\nρxx,F+tB\nρxx,B(6)\nThen, the resistivity of the ferromagnetic film ρxx,Fis calculated as:\nρxx,F=tF\nt\nρxx−tB\nρxx,B=tFρxxρxx,B\ntρxx,B−tBρxx(7)\nII. INFLUENCE ON HALL/NERNST EFFECT\nHere we assume an electrical circuit as in Fig.S2. The voltage generat ed by the Hall or\nNernst effect of the functional film layer ( Vy,F) is treated as a battery which has an internal3\nIx\nRB\nRF\nVVx\nFig. S1: Electrical circuit for the longitudinal direction measuremen t\nresistance equal to the resistance of this layer ( RF), which is shorted by the parallel buffer\nlayer resistance RB.\nVVy\nVy,FRB\nRF\nFig. S2: Electrical circuit for the anomalous Hall/Nernst effect\nVF\nyin this electrical circuit is thereby calculated by:\nVy,F=Vy(RF+RB\nRB) (8)\nVy,F=Vy(ρxx,BtF+ρxx,FtB)\nρxx,BtF(9)\nFor the anomalous Hall effect, using this equation and equation 2 for the ferromagnetic\nthin film yields:4\nρxy,F=Vy(ρxx,BtF+ρxx,FtB)\nρxx,BIx,F(10)\nIx,Fis the current flowing through the ferromagnetic film and is not equa l to the applied\ncurrentIx. It can be calculated assuming the parallel resistor electrical circu it in Fig. S1:\nIx,F=IxxRF\nRB+RF=Ixρxx,BtF\nρxx,BtF+ρxx,FtB(11)\nthen using 10 and 11:\nρxy,F=Vy(ρxx,BtF+ρxx,FtB)2\nIxρ2\nxx,BtF=ρxy(ρxx,BtF+ρxx,FtB)2\nρ2\nxx,BtFt(12)\nFor the anomalous Nernst coefficient, we assume a constant heat g radient in both buffer\nand ferromagnetic thin film layer, hence the anomalous Nernst coeffi cient is calculated using\nequation 4 and 9:\nSxy,F=Vy,F\n∇Txd=Vy(ρxx,BtF+ρxx,FtB)\nρxx,BtF1\n∇Txd(13)\nIII. THERMOMETRY FOR ANOMALOUS NERNST MEASUREMENT\nFor the measurement of the heat gradient generated by the on-c hip Platinum heaters, a\ncurrent of 50 µA was applied longitudinal between two contacts of the hall bar, i.e. c ontact\n#2 and #4. The voltage was measured between contacts #1 −#2 and #4 −#5. The two\nmeasured voltages are representative for the transversal con tact legs at contact #2 and #4.\nFor calibration as resistive temperature sensors, the voltages wh ere monitored as a function\nofthebasetemperatureofthecryostat, showninFig. S5. Thecu rveofeachsensorwasfitted\nto a polynomial function which was utilised as calibration function for t he temperature.\nThen, the base temperature of the cryostat was set to a consta nt temperature and the\nheating current was increased in 2mA steps up to Ih= 8mA, as shown in Fig. S5 exemplary\nforT= 200K. For each step, the temperature of both sensors was calc ulated based on\nthe calibration function. The temperature gradient was then calcu lated as∇Tx= ∆Tx/L\nwhere ∆TxandLare the temperature difference and distance between the two mea sured5Heater\nCr/MnAl 400 µm#1\n#6 #7 #9 #8#10#2 #4 #3\n#5Ix /\n∇Txx\nyz\nAHE/\nANEµ0Hz\nFig. S3: Schematics of the structured MnAl thin film. For better co mprehension, all\ncontacts of the Hall bar are numbered.\n100 200 300\nT  (K)246V\nth (mV)Th_01\nTh_02\n0 5000\nt (s)4.554.604.654.70V\nth (mV)V\nth-Th_01\n050100150\nP  (mW)P\nh\nFig. S4: (left) Voltage as a function of the cryostat base tempera ture (right) Voltage\nmeasured for the stepwise increase of the heating current at 200 K.\ncontact pairs. The measured temperature gradient as a function of the heating power is\nshown in Fig. S5. The temperature gradient is increasing linearly with t he applied heating\npower supporting that the presented measurement routine is suit able for the calculation\nof the temperature gradient. For the maximum heating current of Ih= 8mA which was\nalso used for the ANE measurements, the temperature gradient w as measured as ∇Tx=\n(0.15;0.33;0.41)Kmm−1for the temperatures of T= (100;200;300)K, respectively.\nHowever, it must be noted that the voltages were measured in thre e point configuration.6\n0 25 50 75 100 125 150\nP\nh (mW)0.00.10.20.30.4∇ T  (Kmm−1\n)300K\n200K\n100K\nFig. S5: Temperature gradient as a function of the heating power f or different cryostat\nbase temperatures.\nTherefore we assume a high error of the measured temperature g radient of ±0.1Kmm−1.7\nIV. SEEBECK EFFECT MEASUREMENT AND INFLUENCE OF CHROMIUM\nBUFFER\nIn the Seebeck measurement, the measured voltage is generated by both the Seebeck\neffect of the τ-MnAl and the buffer layer. For that reason, we assume an electric al circuit\nas in Fig.S6. The voltage generated by the Seebeck effect of the fun ctional film layer ( Vx,F)\nand the buffer layer ( Vx,B) are treated as a battery which have an internal resistance equa l\nto the resistance of layers ( RF,RB) which are then connected in parallel.\nVVx\nVx,F\nVx,FRB\nRF\nFig. S6: Electrical circuit for the Seebeck effect.\nVxis thereby calculated as:\nVx=Vx,F(RB\nRB+RF)+Vx,B(RF\nRB+RF) (14)\nand hence Vy,Fis calculated as:\nVx,F=Vx(ρxx,FtB+ρxx,BtF)−Vx,Bρxx,FtB\nρxx,BtF(15)\nUltimately the Seebeck coefficient is calculated by\nSxx,F=−Vx,F\n∆T(16)\nThe measured voltages for the different base temperatures as we ll as the calculated See-\nbeck coefficient are summarised in table I. The temperature differen ce ∆Twas measured as8\ndescribed in the previous section. The Seebeck coefficient of Cr was taken from reference1.\nT(K)Vx(µV)Vx,MnAl(µV)Sxx,Cr(µVK−1)Sxx,MnAl(µVK−1)\n100-0.587 -0.718 5 8.015\n200-2.516 -2.187 12.5 9.64\n300-7.012 -7.147 22 22.84\nTABLE I: Results of the Seebeck effect measurement9\n[1] Arajs, S. et al., Seebeck coefficient of binary chromium alloys containing mol yb-\ndenum or tungsten , Journal of the Less Common Metals, volume 26, 1 (1972),\nhttps://doi.org/10.1016/0022-5088(72)90018-5\n[2] Pu,Y. et al., Mott Relation for Anomalous Hall and Nernst Effects in Ga1−xMnxAs\nFerromagnetic Semiconductors , Phys. Rev. Lett., volume 101, 11 (2008)\nhttps://link.aps.org/doi/10.1103/PhysRevLett.101.11 7208"    },    {        "title": "2307.11304v1.First_principles_prediction_of_phase_transition_of_YCo__5__from_self_consistent_phonon_calculations.pdf",        "content": "First-principles prediction of phase transition of YCo 5from self-consistent phonon calculations\nGuangzong Xing,1,∗Yoshio Miura,1and Terumasa Tadano1,†\n1Research Center for Magnetic and Spintronic Materials,\nNational Institute for Materials Science, Tsukuba, Ibaraki, 305-0047, Japan\n(Dated: July 24, 2023)\nRecent theoretical study has shown that the hexagonal YCo 5is dynamically unstable and distorts into a stable\northorhombic structure. In this study, we show theoretically that the orthorhombic phase is energetically more\nstable than the hexagonal phase in the low-temperature region, while the phonon entropy stabilizes the hexagonal\nphase thermodynamically in the high-temperature region. The orthorhombic-to-hexagonal phase transition\ntemperature is ∼165 K, which is determined using the self-consistent phonon calculations. We investigate the\nmagnetocrystalline anisotropy energy (MAE) using the self-consistent and non-self-consistent (force theorem)\ncalculations with the spin-orbit interaction (SOI) along with the Hubbard Ucorrection. Then, we find that the\northorhombic phase has similar MAE, orbital moment, and its anisotropy to the hexagonal phase when the self-\nconsistent calculation with the SOI is performed. Since the orthorhombic phase still gives magnetic properties\ncomparable to the experiments, the orthorhombic distortion is potentially realized in the low-temperature region,\nwhich awaits experimental exploration.\nI. INTRODUCTION\nCaCu 5-type RECo 5(RE = rare earth) intermetallic com-\npound has emerged as promising permanent magnets owing\nto its high Curie temperature, saturation magnetization, and\nstrong coercivity [1–5]. The coercivity is related to the in-\ntrinsic properties of a material, namely magnetocrystalline\nanisotropy energy (MAE), which can be obtained by measur-\ning the energy difference along different directions of mag-\nnetization with respect to the crystal axes [6–9]. Among the\nRECo 5family, magnetic properties of YCo 5have been exten-\nsively studied both from experiments and by first-principles\nmethods based on density functional theory (DFT) [4, 10].\nThe saturated magnetization of ∼910 emu/cm3, the mag-\nnetocrystalline anisotropy constant of 7.38 ×107erg/cm3at\n4.2 K, and the Curie temperature of 977 K were obtained in\nRef. [8]. Previous studies indicate that DFT calculations at\nthe level of local density approximation (LDA) or the gener-\nalized gradient approximation (GGA) significantly underesti-\nmate the orbital magnetic moments of Co atoms and the MAE\nin comparison with the experimental values. These underes-\ntimation problems have been tackled by including the orbital\npolarization scheme [11, 12] or by using DFT+ U[13, 14] or\nLDA+DMFT [15] approaches.\nAll of the previous theoretical calculations have been per-\nformed using the CaCu 5-type structure [Fig. 1(a)], which dis-\nplays a layered hexagonal lattice (space group: P6/mmm )\nwith two different kinds of Co atoms labeled as Co 2cand\nCo3g[16]. However, a recent first-principles study has shown\nthat a phonon at the L point of the first Brillouin zone is dy-\nnamically unstable at 0 K [17], indicating that the hexagonal\nphase distorts into a lower-energy phase, as schematically il-\nlustrated in Fig. 1(d). Indeed, the previous study has reported\nthat the distorted orthorhombic phase (space group: Imma )\nis energetically more stable than the hexagonal phase. Given\n∗XING.Guangzong@nims.go.jp\n†TADANO.Terumasa@nims.go.jpthe presence of the double-well potential energy surface of the\nunstable phonon mode [Fig. 1(d)], there should be a structural\nphase transition induced by the soft phonon excited at finite\ntemperatures. Since the magnetic properties, such as MAE,\nis sensitive to the crystal structure, understanding the temper-\nature evolution of the crystal structure is crucial for reliably\ncomparing the magnetic properties of YCo 5obtained from\ntheory and experiments.\nHere, by using the state-of-the-art anharmonic phonon cal-\nculation method, we demonstrate that the orthorhombic-to-\nhexagonal phase transition of YCo 5can take place with heat-\ning. We first show the orthorhombic phase predicted to be\nenergetically more stable than the hexagonal phase at 0 K\nwith various exchange-correlation (XC) functionals, including\nGGA, GGA+ U, meta-GGA SCAN, and HSE06. As the tem-\nperature rises, the hexagonal phase becomes thermodynam-\nically more stable than the orthorhombic phase at ∼165 K,\nwhich is estimated from the Helmholtz free energies com-\nputed using the self-consistent phonon method [18, 19]. The\nMAE values of both phases computed using the force theorem,\nwhich is a non-self-consistent calculation with the spin-orbit\ninteraction (SOI), within GGA underestimate the experimen-\ntal values, whereas a self-consistent calculation with the SOI\nunder the GGA +Uscheme yields larger MAE values com-\nparable to the experiments. By contrast, the MAE values in\nthe orthorhombic phase obtained by the force theorem within\nGGA+Ustill fall short of experimental values for all the ap-\nplied Uvalues. We attribute this to the fact that both the\norbital moment of ∼0.1µBand orbital moment anisotropy\n(OMA) with the value of ∼0.03µB, which is proportional\nto the MAE, at the Co site were significantly underestimated\nby the force theorem compared to the corresponding experi-\nmentally measured values of ∼0.2µB[15] and ∼0.06µB[8],\nrespectively.\nThe structure of this paper is organized as follows. The the-\noretical approach to calculate MAE and computational details\nare described in Secs. II A and II B, respectively. The main\nresults are shown in Sec. III. The orthorhombic-to-hexagonal\nphase transition induced by lattice anharmonicity is discussed\nin Sec. III A. In Sec. III B, we evaluate the MAE for both phasesarXiv:2307.11304v1  [cond-mat.mtrl-sci]  21 Jul 20232\nusing both force theorem and self-consistent methods within\nthe GGA +U. Finally, we summarize the study in Sec. IV.\nII. METHODS\nA. MAE from DFT calculation\nThe MAE at 0 K was evaluated based on DFT as\nKDFT\nu=E0,⊥−E0,∥, (1)\nwhere E0,⊥andE0,∥are the ground state total energies with\nthe magnetic moment ( m) being aligned along the hard and\neasy axes, respectively. These energies were computed by\nperforming DFT calculations with the SOI using the force the-\norem [20] (a non-self-consistent approach) and self-consistent\napproaches. The easy axis for both phases was found along\nthe [001] direction ( m∥c). To determine the hard axis, we\ncalculated the total energy by varying the azimuthal angle ϕ\nin the basal plane ( m⊥c) from 0 to 2 πwith a step ofπ\n6. The\nenergy difference between different ϕin the hexagonal phase\nwas negligible. In the orthorhombic phase, the maximum en-\nergy difference was ∼0.4 MJ/m3with the lowest energy at\nϕ=π\n2([010] direction). Therefore, we employed the hard\naxis along the [100] direction for the hexagonal phase and\nalong the [010] direction for the orthorhombic phase to obtain\nKDFT\nu. Although the choice of the hard axis may slightly af-\nfect the MAE, the small energy difference does not affect the\noverall conclusion.\nAccording to the Bruno relation [21], MAE is proportional\nto the OMA defined as\n∆mo=mCo\no,∥−mCo\no,⊥, (2)\nwhere mCo\no,∥andmCo\no,⊥are the orbital magnetic moment at the\nCo site along the easy and hard axes, respectively.\nB. Computational details\nThe DFT calculations in this work were performed mainly\nby using the projector augmented wave (PAW) method [22],\nas implemented in VASP code (version 6.2.1) [23]. The rec-\nommended set of the PAW potentials, which accounts for\nthe scalar relativistic effect, was used. A kinetic-energy\ncutoff of 400 eV and a k-point mesh density of ∼450 ˚A3\nwere employed. We adopted the second-order Methfessel-\nPaxton [24] (MP) smearing method with a width of 0.2 eV\nfor structural optimization. A much denser k-mesh density\nof∼6000 ˚A3and the tetrahedron method with the Bl ¨ochl\ncorrection [25] were employed to calculate the static energy\ndifference ∆E=Eorth\n0−Ehex\n0between the orthorhombic and\nhexagonal phase, the MAE, the density-of-state (DOS), and the\ncrystal orbital Hamilton populations (COHP). For these calcu-\nlations, the initial local moments of 3 µBand−0.3µBwere\nused for the Co and Y atoms, respectively. The SOI was also\nincluded in the MAE calculation. The subsequent phonon cal-\nculations were performed within the GGA by Perdew, Burke,and Ernzerhof (GGA-PBE) [26] without the Hubbard Ucor-\nrection for the following reasons. First, as we elaborate be-\nlow, we confirmed that the orthorhombic phase is energetically\nmore stable at 0 K, irrespective of the XC functionals. Sec-\nond, phonon frequencies are sensitive to the employed lattice\nparameters, and GGA-PBE yields lattice parameters of the\nhexagonal phase that are in accord with the experimental val-\nues [27] within the error of ∼1%. Lastly, we encountered\ntechnical challenges in achieving convergences when applying\nGGA+Ufunctionals for displaced supercells. The calculated\npotential energy surface was not smooth enough to obtain re-\nliable harmonic force constants, despite repeated adjustments\nto the mixing parameters eventually led to convergence.\nIt has been reported that MAE obtained using GGA-PBE\nsignificantly underestimated the experimental values. To ad-\ndress the underestimation problem, we employ the GGA +U\nscheme [28] to account for the correlation effect of the Co- 3d\nelectrons. To compute the total energies under the GGA +U\nscheme with the SOI, we employed two different approaches:\nself-consistent (SC) and the force theorem (FT) calculations.\nThe main difference is whether the charge density from\nthe self-consistent spin-polarized calculation is updated self-\nconsistently or not; in the SC method, the charge density is\noptimized again, whereas it is not so in the FT. Additionally,\nwe employed the full-potential linearized augmented plane-\nwave (FLAPW) method [29], as implemented in the WIEN2k\ncode [30], to investigate the effect of the core electrons on the\nMAE and orbital moment of YCo 5. The crystal structure opti-\nmized using V ASP within GGA-PBE was adopted to calculate\nthe MAE, the OMA, and the orbital moment. We used LAPW\nsphere radii of 2.015 and 2.115 for Co and Y, respectively, and\na basis set cut-off parameter of RminKmax= 9.\nTo compute the vibrational Helmholtz free energy Fvib\nwithin the first-order self-consistent phonon (SCP) theory,\nsecond-order and fourth-order interatomic force constants\n(IFCs) are required. A 2×2×2supercell containing 48\n(96) atoms was used for calculating the second-order IFCs\nof the hexagonal (orthorhombic) phase. Here, each atom in\nthe supercell was displaced from its equilibrium position by\n0.03 ˚A, and the atomic forces were calculated using V ASP.\nFrom the generated displacement-force dataset, we estimated\nthe second-order IFCs by ordinary least squares. For anhar-\nmonic IFCs, a supercell containing 48 atoms was adopted\nfor both phases. We employed the compressive sensing lat-\nTable I. Calculated the static electronic energy difference ( ∆E, see\ntext) between the hexagonal and orthorhombic phases and the squared\nfrequency of the soft mode at the L point ( ω2\nL) for hexagonal YCo 5\nusing different XC functionals.\nXC-functional ∆E(meV/f.u.) ω2\nL(cm−2)\nLDA −44.08 −7826.4\nPBE −23.24 −4301.2\nPBEsol −23.78 −4425.9\nGGA + U(U= 1eV) −52.39 ···\nSCAN −97.31 ···\nHSE06 −134.48 ···3\n¢EorthorhombichexagonalQLzyx\nY2aCo2cCo3g1Co3g2Y4eCo8hCo8fCo4c(a)(b)(d)hexagonalorthorhombic\nxyabc(c)\nFigure 1. Crystal structure of the hexagonal (a) and orthorhombic (b) phase of YCo 5. The Y and Co atoms are represented by the large and\nsmall spheres, respectively. The atomic Wyckoff position for different atoms is marked with the corresponding colors. (c) Top view of the\ncrystal structures of hexagonal (top panel) and orthorhombic (lower panel) YCo 5. (d) Schematic diagram of the double-well potential obtained\nby displacing atoms along the direction of the L-point soft phonon mode of the hexagonal phase. QLis the normal coordinate amplitude, and\n∆Erepresents the static energy difference between the two phases.\ntice dynamics to extract the anharmonic IFCs [31] from the\ndisplacement-force training datasets. Here, we uniformly sam-\npled 200 training structures from the last 2500 steps of an ab\ninitio molecular dynamics calculation ( 5000 steps in total) at\n10 K for both phases. For each sampled structure, we further\ndisplaced all atoms by 0.1 ˚A in random directions. We have\nconfirmed that 200 sample structures were sufficient and the\nchange in ∆Fvib=Forth\nvib−Fhex\nvibbetween the orthorhom-\nbic and hexagonal phase after adding 50 more structures (250\nstructures in total) was smaller than ∼0.5 meV/f.u. at 500 K.\nIn our phonon calculations based on GGA-PBE, we found\nthat the harmonic phonon dispersion of the hexagonal phase\nis sensitive to the broadening parameter σof the MP method\nused for the IFCs calculations, which is particularly notice-\nable when σ≳0.1eV. Thus, we carefully investigated the σ\ndependence of the predicted structural transition temperature\n(Tc)and found that σas small as 0.085 eV was desirable for\nobtaining a converged Tcvalue. We discuss this point in Sec.\nIII A. All the phonon calculations in this study were performed\nusing the alamode code [19, 32].\nIII. RESULTS AND DISCUSSION\nA. Phase transition induced by lattice anharmonicity\nFirst, to assess the stability of the hexagonal and orthorhom-\nbic phases at 0 K, we calculated the ground state energies\nusing various XC functionals: LDA [33], PBE, its variantfor solids (PBEsol) [34], GGA+ U[28] with U= 1 eV, the\nstrongly constrained and appropriately normed (SCAN) meta-\nGGA [35], and the Heyd-Scuseria-Ernzerhof (HSE)06 hybrid\nfunctional [36]. For each XC functional other than HSE06,\nthe crystal structures of the two phases were fully relaxed with\nthe collinear ferromagnetic spin configuration. The detailed\nstructural parameters are listed in Tables S1 and S2 of the\nsupplemental material (SM) [37]. For HSE06, the crystal\nstructures from the PBE calculation were used. The difference\nof the ground state energy ∆E=Eorth\n0−Ehex\n0, where EX\n0\nbeing the energy of the phase Xper formula unit (f.u.), is\nsummarized in Table I. It is clear that the orthorhombic phase\nis predicted to be energetically more stable at 0 K irrespective\nof the employed XC functionals. In the following, the crystal\nstructures for both phases optimized by GGA-PBE are used\nto perform further phonon and magnetic property calculations\nexcept for the squared frequency ω2\nLof the soft-mode tabulated\nin Table I, for which the crystal structures optimized by each\nXC functional are used.\nNext, we discuss the thermodynamic stability of the two\nphases at finite temperatures by comparing the Helmholtz free\nenergies defined as\nF(V, T) =E0(V0) +Fvib(V, T) +Fel(V, T), (3)\nwhere E0(V)is the static electronic energy obtained from a\nconventional DFT calculation, and Fel(V, T)andFvib(V, T)\nare the electronic and vibrational free energy at temperature\nTand cell volume V. Since the lattice constants reportedly\nincrease only by ∼0.04 ˚A at 600 K [27], we neglect the ther-\nmal expansion effect and use the optimized cell volume V0at4\n100200300400500Temperature (K)0102030¢F(meV/f.u.)(c)TcF(SCP)vib+Fel˜F(QHA)vib¢Fcorrvib°MK°ALHA0100200300Frequency (cm°1)(a)\n100150200250300Temperature (K)\nX°RWS°TW0100200300Frequency (cm°1)(b)\nFigure 2. Temperature-dependent anharmonic phonon dispersion of\nhexagonal (a) and orthorhombic (b) phases of YCo 5. The colormap\nshows the self-consistent phonon solutions at the temperature range\nof 100–300 K, and the dashed lines are harmonic lattice dynamics\nresults. Imaginary frequency is shown as negative. (c) Calculated\ndifference of the free energies, ∆F(T)(see text). The horizontal\ndashed line is the difference of static energy −∆E= 23.24meV/f.u.\nHere, the phonon calculation are performed with σ= 0.075eV.\n.\n0 K for computing the Helmholtz free energy. Also, since the\nmagnetic phase transition, e.g., anomaly in the heat capacity,\nhas never been observed in the low-temperature region, the\nmagnetic entropy is also omitted in this study assuming that\nits difference between the two phases is negligible. Then, the\nelectronic free energy Fel(V0, T)and vibrational free energyFvib(V0, T)play a central role in determining the thermody-\nnamic stability. Felis obtained based on the fixed density-of-\nstates approximation [38, 39]. When all phonon modes are\ndynamically stable within the harmonic approximation (HA),\nit is straightforward to estimate Fvib(V0, T). However, for the\nhexagonal YCo 5, the method based on the HA breaks down\ndue to the presence of the soft mode at the L point. Hence, in\nthis study, we employ the SCP scheme. In this approach, we\nfirst compute finite-temperature phonon frequencies by solving\nthe following SCP equation [40, 41]\nΩ2\nq=ω2\nq+1\n2X\nq1Φ(q;−q;q1;−q1)αq1. (4)\nHere, ωqis the angular frequency of the phonon mode qob-\ntained within the HA, Ωqis the frequency at finite temperature\nrenormalized by the anharmonic interaction Φ(q;−q;q1;−q1)\nassociated with the fourth-order anharmonicity of the potential\nenergy surface, and αq=ℏ\n2Ωq[1+2n(Ωq)]withn(ω)being the\nBose-Einstein distribution function. Once the above equation\nis solved for Ωq, the vibrational free energy can be evaluated\nasF(SCP)\nvib(V0, T) =˜F(QHA)\nvib(V0, T) + ∆ Fcorr\nvib(V0, T), which\nis the sum of the quasi-harmonic (QH) term computed using\nΩqand the correction term necessary to satisfy the correct\nthermodynamic relationship of −dFvib/dT=Svib[42, 43].\nMore details of the mathematical expressions for F(SCP)\nvibcan\nbe found in the SM [37].\nFigure 2(a) and (b) show the phonon dispersion curves of\nhexagonal and orthorhombic YCo 5within the HA (dashed\nlines) and SCP theory (solid lines). The SCP frequencies from\n100 K to 300 K in the step of 50 K are shown with different col-\nors. It is clear that the harmonic phonon of the hexagonal YCo 5\nis dynamically unstable with the most unstable mode occurring\nat the L points. This instability was also observed with other\nXC functionals, as indicated by the negative ω2\nLvalues in Ta-\nble I. By contrast, the harmonic phonon is dynamically stable\nfor the distorted orthorhombic phase [see Fig. 2(b)]. The soft\nmodes of the hexagonal phase at L and M points mainly involve\nthe in-plane [ xy-plane in Fig. 1(c)] displacements of the Co 2c\natoms together with relatively small displacements of the other\natoms. Recently, it has been shown that the phonon instability\nat the L point originates from the strong antibonding nature\nof the Co 2c-Co2cbond [17], as evidenced by a sharp peak in\nthe crystal orbital Hamilton populations (COHP) [44] near the\nFermi level. To see how the antibonding nature changes with\nthe structural distortion, we compare the calculated −COHP\nvalues in Fig. 3(a). The originally equidistant Co 2c-Co2cbonds\nsplit into three different Co 8h-Co8hbonds labeled as a,b, and\nc[see bottom panel of Fig. 1(c)], with the corresponding bond\nlength of 2.75 ˚A, 2.57 ˚A, and 3.21 ˚A, respectively. Hence, the\n−COHP values for the three inequivalent Co 8h-Co8hbonds\nare averaged here. Compared with the large population of\nantibonding states [red curve in Fig. 3(a)] at Fermi energy in\nhexagonal YCo 5, the magnitude of −COHP in the orthorhom-\nbic phase is reduced significantly, which can be explained by\nthe smaller projected DOS [Fig. 3(b)] of the dx2−y2orbital at\nthe Co 8hsites. Such a large reduction in the −COHP value by\nstructural distortion is consistent with the trend of phonon sta-5\n−1.0−0.50.0 0.5 1.0−0.4−0.20.00.20.4−COHP(a)\nCo2c-Co 2c\nCo8h-Co 8h\n−1.0−0.50.0 0.5 1.0\n/epsilon1−/epsilon1F(eV)−101PDOS (states/eV)(b)\ndxy,dx2−y2(Co 2c)\ndxy(Co 8h)\ndx2−y2(Co 8h)\nFigure 3. (a) Crystal orbital Hamilton populations (COHP) calcula-\ntion of the Co 2c-Co2cbond in the hexagonal phase and Co 8h-Co8h\nbond in the orthorhombic phase. (b) Projected density of states (DOS)\nfor3dstates at Co 2c(Co8h) site. The solid and dashed lines represent\nthe minority and majority spin states, respectively. The data of the\nhexagonal YCo 5is adapted from Ref. [17].\nbility we observed. Interestingly, the projected DOS in Fig. S1\nof the SM [37] shows that the peak of the Co 2cdx2−y2orbital\nat the Fermi energy does not change appreciably even when\nwe used the SCAN functional and GGA +Uwith various U\nparameters. This suggests the presence of phonon instability\nin the hexagonal YCo 5even with the XC functionals other than\nGGA-PBE, in accord with the trend of ∆Eshown in Table I.\nAs can be seen in Figs. 2(a) and 2(b), the phonon fre-\nquencies significantly increase by the quartic anharmonic-\nity at finite temperatures, particularly notable in the low-\nenergy optical modes. Since SCP theory postulates the\nrenormalized phonons to be stable in Eq. (4), we observe\nthat the finite-temperature phonons are stable irrespective\nof the temperature for both phases. Hence, the informa-\ntion of the self-consistent phonon dispersion alone cannot\ndetermine the structural phase transition temperature Tcre-\nliably [45, 46]. In this study, we employ another approach\nbased on Eq. (3), where Tcis estimated from the differ-\nence in the electronic and vibrational free energy defined as\n∆F(T) =Forth\nvib(T)+Forth\nel(T)−[Fhex\nvib(T)+Fhex\nel(T)]. The\ncalculated ∆F(T)is shown in Fig. 2(c). The increase of ∆F\nwith heating occurs because the electronic and vibrational en-\ntropy gradually enhances the stability of the hexagonal phase.Eventually, the hexagonal phase becomes more stable when\nFhex(T)≤Forth(T), or equivalently −∆E≤∆F(T), is\nsatisfied. From this condition, Tcof∼165 K was obtained\nwhen evaluating ∆F(T)using the MP method with the broad-\nening parameter σof 0.075 eV, as shown in Fig. 2(c). While a\nmagneto-elastic study showed that the hexagonal phase to be\nstable down to ∼100 K under hydrostatic pressure [47, 48], a\ndetailed structure analysis at ambient pressure below ∼165 K\nis necessary to test the present prediction.\nHere, we discuss the dependency of Tcon the employed\nσvalue. As shown in Fig. S2 of the SM [37], the harmonic\nphonon dispersion of the hexagonal phase was sensitive to the\nσvalue, particularly noticeable near the A (0,0,1\n2)point of\nthe Brillouin zone. By contrast, we did not observe such a\nsignificant σdependence for the phonon dispersion of the or-\nthorhombic phase. These different behaviors are consistent\nwith the large difference in the projected DOS at the Fermi\nlevel, shown in Fig. 3(b). Interestingly, in the case of an-\nharmonic phonon calculations, the phonon frequencies were\nnot so sensitive to σeven for the hexagonal phase. This oc-\ncurred presumably because the relatively large and random\ndisplacements we used for computing anharmonic IFCs lifted\nthe degeneracy and thereby reduced the DOS at the Fermi level,\nmaking the anharmonic IFCs less sensitive to σ. Because of\nthe observed σdependency of the phonon frequencies, the\n∆F(T)value also changed slightly with σ, leading to a size-\nable shift in Tc, as shown in Fig. S3(b) of the SM [37]. With\ndecreasing σfrom 0.2 eV to 0.085 eV, Tcgradually decreased\nfrom∼240 K to ∼165 K. In the smaller σregion of 0.060–\n0.085 eV, the calculated Tcvalue remained nearly constant at\n∼165 K, indicating convergence of the calculation with respect\ntoσ. We also confirmed that a σvalue as small as 0.085 eV\ncan give a potential energy surface of the third mode at the A\npoint which agrees almost perfectly with that computed using\nthe tetrahedron method with the Bl ¨ochl correction, whereas a\nlarger σoverestimated the Hessian, as shown in Fig. S2(b) of\nthe SM [37].\nB. MAE for two phases\nIn the following, we compare the MAE values of the two\nphases obtained from two different approaches: FT [20], i.e.,\na non-self-consistent calculation with the SOI, and SC cal-\nculations with the SOI. For the hexagonal phase, the MAE\nvalue, defined as KDFT\nu =E0,⊥−E0,∥, was 0.75 MJ/m3\nwithin GGA-PBE, which significantly underestimates the ex-\nperimental values of ∼7.4 MJ/m3[8] and ∼6.0 MJ/m3[9].\nThis underestimation problem could be resolved by adding\nthe Hubbard Ucorrection. Zhou et al. [49] reported that\nthe GGA +Umethod does not adequately describe the orbital\nground state for strongly correlated f-electron systems due to\nthe significant anisotropy in the self-interaction error (SIE) of\ntheforbitals. However, in YCo 5without felectrons, the\n3delectrons of Co are more itinerant and less correlated, and\nthe SIE anisotropy should be less significant in comparison\nwith the typical energy scale of the crystal field and band-\nwidth [11, 50]. Therefore, the GGA +Umethod is still valid6\n0 1 2\nU(eV)0510KDFT\nu(MJ/m3)\nExp.1(a)\north\nhexNguyen et al.\n0 1 2\nU(eV)0.00.10.20.3∆mo(µB/f.u.)\nExp.1(b)\north\nhex\n0 1 2\nU(eV)0.500.751.001.25mo(µB/f.u.)\n Exp.2(c)\north\nhex\nFigure 4. Dependence of magnetocrystalline anisotropy energy, KDFT\nu, (a) and orbital moment anisotropy, ∆mo, (b) and orbital moment (c)\nfor the hexagonal and orthorhombic phase of YCo 5under the GGA +Uscheme. The solid and translucent symbols represent the quantities\nmentioned above obtained using the self-consistent and force theorem approaches, respectively. The unfilled diamond is the theoretical results\nfrom Ref. [13] based on the force theorem, and the horizontal dash-dotted lines, labeled as Exp.1, and Exp.2 are experimental results from\nRef. [8], and Ref. [15], respectively. Note that the ∆moandmoof Y atom are not included in (b) and (c) for comparison with the experimental\ndata.\nfor calculating the MAE and orbital moment, which has been\ndemonstrated in the previous GGA +Ustudies on some Co\ncompounds [51, 52].\nFor hexagonal YCo 5, our FT calculation in Fig. 4(a) nicely\nreproduced the Udependence of MAE from a previous FT re-\nsult based on GGA +U[13], and the KDFT\nu value became com-\nparable with the experimental values when U∼1.75 eV. How-\never, at this Uvalue, the FT still underestimated the total orbital\nmoments and their anisotropy (OMA), ∆mo=mCo\no,∥−mCo\no,⊥,\nfor all Co sites, as shown in Figs. 4(c) and (b), respectively. We\nobserved that the SC approach can yield the MAE and orbital\nmoment results much more consistent with the experimental\nvalues, as we elaborate below. The SC scheme enhanced the\nKDFT\nu values at all the Uvalues, as shown in Fig. 4(a). Conse-\nquently, an agreement with experimental values was obtained\nfor MAE at U∼0.75 eV, which is smaller than the Uof∼1.75\neV for the FT approach. At U=0.75 eV, the SC approach gave\nthe orbital moment of ∼0.8µB/f.u., in better agreements with\nthe experimental value of ∼1µB/f.u. [15] [see Fig. 4(c)]. Al-\nthough perfect agreements in the MAE, orbital moment, and\nOMA were not reached simultaneously with a single Uvalue,\nit is clear that the SC scheme gives more reliable results than\nthe FT approach.\nFor the orthorhombic phase, we obtained the U-dependent\nMAE, OMA, and the orbital moment which were qualitatively\nsimilar to those of the hexagonal phase when the SC approach\nwas used, as shown by filled markers in Figs. 4(a), (b), and\n(c). One minor difference can be found in the Uvalue to\nreproduce experimental MAE; for the orthorhombic phase, a\nslightly larger Uvalue of ∼1.0 eV appears to give essentially\nthe same magnetic properties as the hexagonal phase. Even at\nU= 0.75eV, which was applied to the hexagonal phase, the\nMAE of 6.12 MJ/m3can still reach the range of experimental\nobservations (6.0-7.4 MJ/m3). Interestingly, we found that the\nFT failed completely for the orthorhombic phase, as inferred\nfrom the consistently lower MAE values in Fig. 4(a), which canbe attributed to the underestimation of the OMA [Fig. 4(b)].\nIn the context of the GGA +Uscheme with the SOI, the FT\napproach only modifies the GGA energy due to the Coulomb\ninteraction between the Co 3delectrons occupying the same\nion. However, the change in occupancy due to the on-site\nelectron-electron Coulomb interaction is neglected because\nthe charge density is not self-consistently updated. Therefore,\nit is reasonable that the FT calculations underestimate the or-\nbital moment, the OMA, and consequently the MAE. We note\nthat SC approach could reproduce the experimentally-observed\nlarger orbital moments and OMA at the Co 2c(Co8h) than the\nother Co sites for both of the two phases. The details of the\nsite dependency are summarized in the SM [37]. In addition,\nthe total spin and orbital magnetic moment obtained by the SC\ncalculation with U= 0.75eV (U= 1.0eV) for the hexago-\nnal (orthorhombic) phase is 8.44 µB/f.u. (8.52 µB/f.u.), and\nagrees well with the experimental value of ∼8.3µB/f.u. [8] and\n∼8.4µB/f.u. [4], while the total moment of 7.80 µB/f.u. (7.73\nµB/f.u.) determined without Uunderestimates the experimen-\ntal results. We have confirmed that other GGA +Ucorrection\nschemes [53–55] as well as a full-potential all-electron calcu-\nlation [29, 30] give quantitatively similar MAE values. (Please\nsee details in the SM [37]).\nIV. SUMMARY\nTo summarize, we have theoretically demonstrated a possi-\nble orthorhombic-to-hexagonal structural phase transition of\nYCo 5induced by heating. We found the orthorhombic phase\nis energetically more stable than the hexagonal phase at 0 K ir-\nrespective of the employed exchange-correlation functionals.\nThe stable, temperature-dependent phonons of these phases\nwere obtained by incorporating the anharmonic renormaliza-\ntion using the self-consistent phonon approach, and the theo-\nretical phase transition temperature of ∼165 K was obtained by7\ncomparing the calculated Helmholtz free energies. We com-\npared the MAE, the associated orbital moment anisotropy, and\nthe orbital moments of the two phases computed using the force\ntheorem and self-consistent calculations with the SOI based on\nthe GGA +Uscheme. We showed that the magnetic properties\nof the orthorhombic phase were similar to those of the hexag-\nonal phase when the self-consistent approach was employed.\nSince the calculated magnetic properties also agreed well with\nthe available experimental data, we expect the predicted or-\nthorhombic phase to be existent in the low-temperature region,\nwhich awaits experimental verification.ACKNOWLEDGMENTS\nThis work was partly supported by MEXT Program: Data\nCreation and Utilization-Type Material Research and Devel-\nopment Project (Digital Transformation Initiative Center for\nMagnetic Materials) Grant Number JPMXP1122715503. The\nfigures of crystal structures are created by the vesta soft-\nware [56].\n[1] C. E. Patrick and J. B. Staunton, Temperature-dependent magne-\ntocrystalline anisotropy of rare earth/transition metal permanent\nmagnets from first principles: the light RCo 5(R= Y, La-Gd) in-\ntermetallics, Phys. Rev. Mater. 3, 101401(R) (2019).\n[2] S. Kumar, C. E. Patrick, R. S. Edwards, G. Balakrishnan, M. R.\nLees, and J. B. Staunton, Torque magnetometry study of the\nspin reorientation transition and temperature-dependent mag-\nnetocrystalline anisotropy in NdCo 5, J. Phys.: Condens. Matter\n32, 255802 (2020).\n[3] H. Ucar, R. Choudhary, and D. Paudyal, An overview of the first\nprinciples studies of doped RE-TM 5systems for the develop-\nment of hard magnetic properties, J. Magn. Magn. Mater. 496,\n165902 (2020).\n[4] C. E. Patrick, S. Kumar, G. Balakrishnan, R. S. Edwards,\nM. R. Lees, E. Mendive-Tapia, L. Petit, and J. B. Staunton,\nRare-earth/transition-metal magnetic interactions in pristine and\n(Ni,Fe)-doped YCo 5and GdCo 5, Phys. Rev. Mater. 1, 024411\n(2017).\n[5] O. Gutfleisch, M. A. Willard, E. Br¨ uck, C. H. Chen, S. G.\nSankar, and J. P. Liu, Magnetic materials and devices for the\n21st century: Stronger, lighter, and more energy efficient, Adv.\nMater. 23, 821 (2010).\n[6] J. Trygg, B. Johansson, O. Eriksson, and J. M. Wills, Total\nenergy calculation of the magnetocrystalline anisotropy energy\nin the ferromagnetic 3dmetals, Phys. Rev. Lett. 75, 2871 (1995).\n[7] P. B lo ´nski, A. Lehnert, S. Dennler, S. Rusponi, M. Etzkorn,\nG. Moulas, P. Bencok, P. Gambardella, H. Brune, and J. Hafner,\nMagnetocrystalline anisotropy energy of Co and Fe adatoms on\nthe (111) surfaces of Pd and Rh, Phys. Rev. B 81, 104426 (2010).\n[8] J. M. Alameda, D. Givord, R. Lemaire, and Q. Lu, Co energy\nand magnetization anisotropies in RCo 5intermetallics between\n4.2 K and 300 K, J. Appl. Phys. 52, 2079 (1981).\n[9] J. Franse, N. Thuy, and N. Hong, Individual site magnetic\nanisotropy of the iron and cobalt ions in rare earth-iron and rare\nearth-cobalt intermetallic compounds, J. Magn. Magn. Mater.\n72, 361 (1988).\n[10] P. Larson and I. Mazin, Calculation of magnetic anisotropy en-\nergy in YCo 5, J. Magn. Magn. Mater. 264, 7 (2003).\n[11] L. Steinbeck, M. Richter, and H. Eschrig, Itinerant-electron\nmagnetocrystalline anisotropy energy of YCo 5and related com-\npounds, Phys. Rev. B 63, 184431 (2001).\n[12] G. H. O. Daalderop, P. J. Kelly, and M. F. H. Schuurmans, Mag-\nnetocrystalline anisotropy of YCo 5and related RECo 5com-\npounds, Phys. Rev. B 53, 14415 (1996).\n[13] M. C. Nguyen, Y. Yao, C.-Z. Wang, K.-M. Ho, and V. P.\nAntropov, Magnetocrystalline anisotropy in cobalt based mag-\nnets: a choice of correlation parameters and the relativisticeffects, J. Phys.: Condens. Matter 30, 195801 (2018).\n[14] A. Asali, J. Fidler, and D. Suess, Influence of changes in elec-\ntronic structure on magnetocrystalline anisotropy of YCo 5and\nrelated compounds, J. Magn. Magn. Mater. 485, 61 (2019).\n[15] J.-X. Zhu, M. Janoschek, R. Rosenberg, F. Ronning, J. D.\nThompson, M. A. Torrez, E. D. Bauer, and C. D. Batista,\nLDA+DMFT approach to magnetocrystalline anisotropy of\nstrong magnets, Phys. Rev. X 4, 021027 (2014).\n[16] The threefold degenerate Co 3gatoms will be separated into two\ninequivalent sites with multiplicity of 1 (denoted as Co 3g1) and\n2 (denoted as Co 3g2) when the magnetization is along the hard\naxis ([100] direction).\n[17] G. Xing, Y. Miura, and T. Tadano, Lattice dynamics and its\neffects on magnetocrystalline anisotropy energy of pristine and\nhole-doped YCo 5from first principles, Phys. Rev. B 105, 104427\n(2022).\n[18] D. Hooton, The use of a model in anharmonic lattice dynamics,\nPhilos. Mag. 3, 49 (1958).\n[19] T. Tadano and S. Tsuneyuki, Self-consistent phonon calcula-\ntions of lattice dynamical properties in cubic SrTiO 3with first-\nprinciples anharmonic force constants, Phys. Rev. B 92, 054301\n(2015).\n[20] G. H. O. Daalderop, P. J. Kelly, and M. F. H. Schuurmans,\nFirst-principles calculation of the magnetocrystalline anisotropy\nenergy of iron, cobalt, and nickel, Phys. Rev. B 41, 11919 (1990).\n[21] P. Bruno, Tight-binding approach to the orbital magnetic mo-\nment and magnetocrystalline anisotropy of transition-metal\nmonolayers, Phys. Rev. B 39, 865 (1989).\n[22] G. Kresse and D. Joubert, From ultrasoft pseudopotentials to\nthe projector augmented-wave method, Phys. Rev. B 59, 1758\n(1999).\n[23] G. Kresse and J. Furthm¨ uller, Efficient iterative schemes for\nab initio total-energy calculations using a plane-wave basis set,\nPhys. Rev. B 54, 11169 (1996).\n[24] M. Methfessel and A. T. Paxton, High-precision sampling for\nBrillouin-zone integration in metals, Phys. Rev. B 40, 3616\n(1989).\n[25] P. E. Bl ¨ochl, O. Jepsen, and O. K. Andersen, Improved tetra-\nhedron method for brillouin-zone integrations, Phys. Rev. B 49,\n16223 (1994).\n[26] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient\napproximation made simple, Phys. Rev. Lett. 77, 3865 (1996).\n[27] A. Andreev, Chapter 2 thermal expansion anomalies and sponta-\nneous magnetostriction in rare-earth intermetallics with cobalt\nand iron, Handb. Magn. Mater. 8, 59 (1995).\n[28] S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys,\nand A. P. Sutton, Electron-energy-loss spectra and the structural8\nstability of nickel oxide: An LSDA+U study, Phys. Rev. B 57,\n1505 (1998).\n[29] D. J. Singh and L. Nordstrom, Planewaves, Pseudopotentials,\nand the LAPW method , 2nd ed. (Springer, Berlin, 2006).\n[30] K. Schwarz, P. Blaha, and G. K. Madsen, Electronic structure\ncalculations of solids using the WIEN2k package for material\nsciences, Comput. Phys. Commun. 147, 71 (2002).\n[31] F. Zhou, W. Nielson, Y. Xia, and V. Ozolin ¸ ˇs, Lattice anhar-\nmonicity and thermal conductivity from compressive sensing\nof first-principles calculations, Phys. Rev. Lett. 113, 185501\n(2014).\n[32] T. Tadano, Y. Gohda, and S. Tsuneyuki, Anharmonic force con-\nstants extracted from first-principles molecular dynamics: ap-\nplications to heat transfer simulations, J. Phys.: Condens. Matter\n26, 225402 (2014).\n[33] W. Kohn and L. J. Sham, Self-consistent equations including\nexchange and correlation effects, Phys. Rev. 140, A1133 (1965).\n[34] J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov, G. E.\nScuseria, L. A. Constantin, X. Zhou, and K. Burke, Restoring the\ndensity-gradient expansion for exchange in solids and surfaces,\nPhys. Rev. Lett. 100, 136406 (2008).\n[35] J. Sun, A. Ruzsinszky, and J. P. Perdew, Strongly constrained\nand appropriately normed semilocal density functional, Phys.\nRev. Lett. 115, 036402 (2015).\n[36] A. V. Krukau, O. A. Vydrov, A. F. Izmaylov, and G. E. Scuse-\nria, Influence of the exchange screening parameter on the per-\nformance of screened hybrid functionals, J. Chem. Phys. 125,\n224106 (2006).\n[37] See Supplemental Material at [URL] for details of SCP vibra-\ntional free energy, density of states at Co 2csite computed with\nvarious exchange-correlation (XC) functionals, smearing width\ndependency of harmonic frequency and transition temperature,\ninfluence of using different GGA +Uscheme on magnetic prop-\nerties, Co site dependence of orbital moment and its anisotropy\nunder the GGA +Uscheme, and lattice information obtained\nwith different XC functionals for hexagonal and orthorhombic\nYCo 5, which includes Ref. [57, 58].\n[38] G. Xing, T. Ishikawa, Y. Miura, T. Miyake, and T. Tadano, Lat-\ntice dynamics effects on finite-temperature stability of R 1−xFex\n(R= Y, Ce, Nd, Sm, and Dy) alloys from first principles, J. Alloys\nand Compd. 874, 159754 (2021).\n[39] A. Togo and I. Tanaka, First principles phonon calculations in\nmaterials science, Scr. Mater. 108, 1 (2015).\n[40] T. Tadano and S. Tsuneyuki, First-principles lattice dynamics\nmethod for strongly anharmonic crystals, J. Phys. Soc. Jpn. 87,\n041015 (2018).\n[41] T. Tadano and S. Tsuneyuki, Quartic anharmonicity of rattlers\nand its effect on lattice thermal conductivity of clathrates from\nfirst principles, Phys. Rev. Lett. 120, 105901 (2018).\n[42] Y. Oba, T. Tadano, R. Akashi, and S. Tsuneyuki, First-principles\nstudy of phonon anharmonicity and negative thermal expansion\nin ScF 3, Phys. Rev. Mater. 3, 033601 (2019).[43] T. Tadano and S. Tsuneyuki, Ab initio prediction of structural\nphase-transition temperature of SrTiO 3from finite-temperature\nphonon calculation, J. Ceram. Soc. Jpn. 127, 404 (2019).\n[44] R. Dronskowski and P. E. Bl ¨ochl, Crystal orbital hamilton\npopulations (COHP): energy-resolved visualization of chemi-\ncal bonding in solids based on density-functional calculations,\nJ. Phys. Chem. 97, 8617 (1993).\n[45] R. Bianco, I. Errea, L. Paulatto, M. Calandra, and F. Mauri,\nSecond-order structural phase transitions, free energy curva-\nture, and temperature-dependent anharmonic phonons in the\nself-consistent harmonic approximation: Theory and stochastic\nimplementation, Phys. Rev. B 96, 014111 (2017).\n[46] T. Tadano and W. A. Saidi, First-principles phonon quasiparticle\ntheory applied to a strongly anharmonic halide perovskite, Phys.\nRev. Lett. 129, 185901 (2022).\n[47] H. Rosner, D. Koudela, U. Schwarz, A. Handstein, M. Hanfland,\nI. Opahle, K. Koepernik, M. Kuz’min, K.-H. M¨ uller, J. Mydosh,\net al. , Magneto-elastic lattice collapse in YCo 5, Nat. Phys. 2, 469\n(2006).\n[48] D. Koudela, U. Schwarz, H. Rosner, U. Burkhardt, A. Handstein,\nM. Hanfland, M. D. Kuz’min, I. Opahle, K. Koepernik, K.-H.\nM¨ uller, and M. Richter, Magnetic and elastic properties of YCo 5\nand LaCo 5under pressure, Phys. Rev. B 77, 024411 (2008).\n[49] F. Zhou and V. Ozolin ¸ ˇs, Obtaining correct orbital ground states\nin f-electron systems using a nonspherical self-interaction-\ncorrected LDA+U method, Phys. Rev. B 80, 125127 (2009).\n[50] M. Richter, Band structure theory of magnetism in 3d-4f com-\npounds, J. Phys. D: Appl. Phys. 31, 1017 (1998).\n[51] F. Donati, Q. Dubout, G. Aut `es, F. Patthey, F. Calleja, P. Gam-\nbardella, O. V. Yazyev, and H. Brune, Magnetic moment and\nanisotropy of individual Co atoms on graphene, Phys. Rev. Lett.\n111, 236801 (2013).\n[52] J.-C. Tung, H.-R. Fuh, and G.-Y. Guo, Anomalous and spin hall\neffects in hcp cobalt from GGA+U calculations, Phys. Rev. B\n86, 024435 (2012).\n[53] A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen, Density-\nfunctional theory and strong interactions: Orbital ordering in\nMott-Hubbard insulators, Phys. Rev. B 52, R5467 (1995).\n[54] M. T. Czy˙ zyk and G. A. Sawatzky, Local-density functional and\non-site correlations: The electronic structure of La 2CuO 4and\nLaCuO 3, Phys. Rev. B 49, 14211 (1994).\n[55] E. R. Ylvisaker, W. E. Pickett, and K. Koepernik, Anisotropy and\nmagnetism in the LSDA+U method, Phys. Rev. B 79, 035103\n(2009).\n[56] K. Momma and F. Izumi, VESTA 3 for three-dimensional vi-\nsualization of crystal, volumetric and morphology data, J Appl\nCryst 44, 1272 (2011).\n[57] J. Schweizer and F. Tasset, Polarised neutron study of the RCo 5\nintermetallic compounds. I. the cobalt magnetisation in YCo 5,\nJ. Phys. F 10, 2799 (1980).\n[58] A. Heidemann, D. Richter, and K. H. J. Buschow, Investigation\nof the hyperfine fields in the compounds LaCo 13, LaCo 5, YCo 5\nand ThCo5 by means of inelastic neutron scattering, Z. Phys. B\n22, 367 (1975)."    },    {        "title": "2307.12692v1.In_plane_magnetocrystalline_anisotropy_in_the_van_der_Waals_antiferromagnet_FePSe__3__probed_by_magneto_Raman_scattering.pdf",        "content": "In-plane magnetocrystalline anisotropy in the van der Waals antiferromagnet\nFePSe 3probed by magneto-Raman scattering\nDipankar Jana,1,∗Piotr Kapuscinski,1Amit Pawbake,1Anastasios Papavasileiou,2Zdenek\nSofer,2Ivan Breslavetz,1Milan Orlita,1, 3Marek Potemski,1, 4,†and Clement Faugeras1,‡\n1Laboratoire National des Champs Magn´ etiques Intenses, LNCMI-EMFL,\nCNRS UPR3228,Univ. Grenoble Alpes, Univ. Toulouse,\nUniv. Toulouse 3, INSA-T, Grenoble and Toulouse, France\n2Chemistry Department, University of Chemistry and Technology Prague, 16628 Prague, Czech Republic\n3Institute of Physics, Charles University, Ke Karlovu 5, Prague, 121 16, Czech Republic\n4CENTERA Labs, Institute of High Pressure Physics, PAS, 01 - 142 Warsaw, Poland\nMagnon gap excitations selectively coupled to phonon modes have been studied in FePSe 3layered\nantiferromagnet with magneto-Raman scattering experiments performed at different temperatures.\nThe bare magnon excitation in this material has been found to be split (by ≈1.2 cm−1) into two\ncomponents each being selectively coupled to one of the two degenerated, nearby phonon modes.\nLifting the degeneracy of the fundamental magnon mode points out toward the biaxial character of\nthe FePS3 antiferromagnet, with an additional in-plane anisotropy complementing much stronger,\nout-of-plane anisotropy. Moreover, the tunability, with temperature, of the phonon- versus the\nmagnon-like character of the observed coupled modes has been demonstrated.\nI. INTRODUCTION\nvan der Waals (vdW) magnets, layered materials in-\ncluding magnetic ions, are today at the heart of an in-\ntense effort to investigate and characterize their magnetic\nproperties. This family of materials includes a broad va-\nriety of magnetic ground states such as ferromagnetism\nin CrI 3[1] or Cr 2Ge2Te6[2] and antiferromagnetism in\nthe MPX 3family [3], where M=Fe, Cr, Co, Mn, Ni and\nX=S or Se, together with a rich physics of interlayer\nferro- or antiferromagnetic interaction across the vdW\ngap. Beyond the fundamental aspects related to mag-\nnetic properties of purely two-dimensional (2D) systems,\nthe physics of 2D vdW magnets is also foreseen to play\nan important role in future spintronic devices [4, 5] or\nto impose new properties on neighboring materials via\nproximity effects [6]. Magnons are among the possible\nelementary excitations of magnetic systems and investi-\ngating their properties provides a unique knowledge of\nthe magnetic ground state [7–10].\nThe magnon-phonon interaction in magnetic materials\nhas recently attracted a lot of interest as it determines\nthe magnon dynamics, a property crucial for the field of\nmagnontronics [11–18]. In the case of iron-based antifer-\nromagnets, a strong single-ion anisotropy of Fe-ion boosts\nthe magnon energy in the range of meV (with respect to\nµeV for other compounds). This allows one to inves-\ntigate the physics of magnon excitations using Raman\nscattering techniques with a typical low energy cut-off\naround 1 meV and with a spatial resolution of ∼1µm.\nSuch high magnon energy coincides with energies of op-\ntical phonons in covalent solids, indicating a possible\n∗dipankar.jana@lncmi.cnrs.fr\n†marek.potemski@lncmi.cnrs.fr\n‡clement.faugeras@lncmi.cnrs.frinfluence of the magnon-phonon interaction on the ob-\nserved magnon or phonon energies. Similar phenomena\nhave been evidenced in recent works where the effects of\nmagnon-phonon interaction are monitored by adjusting\nthe energy detuning between the magnon and the phonon\nexcitations by external means [14, 19–21]. In the case of\neasy-axis antiferromagnets, up and down spins of the two\nsublattices precess in the exchange field of the other one\nwith a circular precession in opposite directions. Col-\nlective oscillations of this spin system lead to a doubly\ndegenerate magnon excitation [22]. In the presence of\nmagnon-phonon interaction, as in the case of FePS 3, the\ndegeneracy of the magnon gap excitation is lifted by the\ninteraction [20, 21]. Note that this degeneracy can also be\nintrinsically lifted even in the absence of magnon-phonon\ninteraction, as it is observed in bi-axial antiferromagnets\nsuch as NiPS 3, MnPSe 3, and in CrSBr, by an additional\nin-plane magnetocrystalline anisotropy [9, 13, 23, 24].\nIdentifying the source of magnon degeneracy is hence\ncrucial as it brings a detailed knowledge of the magnetic\nsystem and its interacting state.\nAs recently reported, the magnon gap in bulk FePSe 3\nis nearly degenerate with a pair of optical phonon modes\nwith chiral character [25], and the two magnon com-\nponents coupled independently to the phonon with the\ncorresponding chirality. The magnon-phonon interac-\ntion in bulk FePSe 3is hence chirality selective, very dis-\ntinct from the one observed in the analogous compound\nFePS 3[14, 20] where both magnons couple independently\nto the same phonon mode. This situation makes it diffi-\ncult to disentangle the effects of magnon-phonon interac-\ntion from those of magnetocrystalline anisotropy on the\nmagnon excitation spectrum, and in particular on the\ndegeneracy lifting at B= 0.\nIn this work, we combine Raman scattering tech-\nniques with different environments of variable temper-\nature, from T= 4.2 K up to T= 140 K, and of higharXiv:2307.12692v1  [cond-mat.mtrl-sci]  24 Jul 20232\nFIG. 1. Magnetic structure of FePSe 3in antiferromagnetic\nphase. The grey sphere and red/blue arrows represent Fe2+\nion and spin direction respectively. Vertical lines guide the\nFe2+ion staking along the c-axis. The figure is created using\nthe VESTA software package [26].\nmagnetic fields up to B= 30 T, to disentangle all cou-\npled magnon-phonon excitations in bulk FePSe 3and to\npinpoint the origin of the zero-field splitting of the cou-\npled modes in the case of selective magnon-phonon in-\nteraction. We show the presence of two non-degenerate\nmagnon gap excitations that are selectively coupled with\ntwo degenerate phonons, leading to the apparent split-\nting in the phonon modes. Our experiments indicate an\nintrinsic lifting of the magnon degeneracy, most probably\nrepresentative of a weak magneto-crystalline anisotropy.\nII. MATERIALS AND EXPERIMENTAL SETUP\nFePSe 3crystallizes in the trigonal R ¯3 space group [27,\n28]. Within each layer, Fe atoms are surrounded by six\nSe atoms in octahedral coordination, while P atoms are\ntetrahedrally coordinated by three Se atoms and one P\natom, forming a [P 2Se6]4−unit. The Fe cations are orga-\nnized in a honeycomb structure within the ab-plane while\nthe layers are stacked along the c-axis. The hexagonal\nlattice arrangement of Fe ions of FePSe 3in the antifer-\nromagnetic phase is schematically presented in Fig. 1 to-\ngether with the orientation of their spins. Ferromagnetic\nzig-zag spin chains along the b-axis are coupled antifer-\nromagnetically with the adjacent chains along the a-axis.\nThe spin direction is perpendicular to the layers’ plane\n(along c-axis). The interlayer coupling is ferromagnetic.\nThe investigated samples were grown by chemical va-\npor transport. A stoichiometric amount of Fe (99.9%,\n-100 mesh, STREM, France), phosphorus (99.9999%,\ngranules, Wuhan Xinrong New Materials), and selenium\n(99.9999%, granules, Wuhan Xinrong New Materials)\nwere placed in quartz glass ampoule (30 ×180 mm, wall\nthickness 3 mm) in amount corresponding to 12 g of\nFePSe 3together with iodine (2mg/mL) and melt sealed\nunder high vacuum. The ampoule was first placed in\na horizontal furnace and heated at 500◦C for 50 hoursand on 600◦C for 50 hours. The heating and cooling\nrate was 1◦C/min. For the crystal growth, the ampoule\nwas placed in a horizontal two-zone furnace and the first\ngrowth zone was heated on 750◦C and the source zone\nwas kept on 500◦C for 2 days, then the gradient was re-\nversed on 700◦C for source zone and 650◦C for growth\nzone. The growth zone temperature was gradually de-\ncreased from 650◦C to 600◦C over a period of 10 days.\nAmpoule was cooled at room temperature and opened in\nan argon-filled glovebox.\nLarge bulk crystals were then inserted either on the\ncold finger of a helium flow cryostat or in a homemade\nsetup for magneto-optical investigations. In both set-\nups, the temperature can be varied from liquid helium\ntemperature up to T= 140 K. Magneto-optical experi-\nments were performed in the Faraday configuration with\nthe magnetic field applied perpendicular to the plane\nof the layers. The excitation laser was produced by a\nsemiconductor-based laser ( λ= 515 nm) or a He-Ne gas\nlaser ( λ= 633 nm), focused on the sample with a mi-\ncroscope objective of NA= 0 .5 (helium flow cryostat) or\nNA= 0 .83 (magneto-optical measurements). Scattered\nsignals were collected using the same objective and were\nthen analyzed by a grating spectrometer equipped with\na liquid nitrogen-cooled charge-coupled device (CCD). A\nset of reflection-based Bragg filters are used in both the\nexcitation and collection paths to clean the laser line and\nto reject the backscattered laser.\nIII. MAGNON-POLARONS IN BULK FePSe 3\nTo unambiguously identify the magnon contribution to\nthe global Raman scattering response, one has to change\na thermodynamic parameter, such as temperature or to\napply an external magnetic field, and to analyze how\nthe different contributions to the Raman scattering spec-\ntrum evolve. The magnon energy gap at Γ, observable in\nRaman scattering [29], is expected to close due to ther-\nmal fluctuations when increasing temperature [30] and to\nvanish at TN. Magnons also couple to an external mag-\nnetic field and, when increasing the magnetic field, the\nindividual magnon components evolve in energy in a way\nspecific to the magnetic system and to the geometry used\nfor the magneto-Raman experiment [22]. In the follow-\ning, we will apply both approaches to identify magnon ex-\ncitations, disentangle magnon-polarons and identify the\nsource of magnon degeneracy lifting in bulk FePSe 3.\nWe present in Fig. 2a the unpolarized low-temperature\nRaman scattering response of bulk FePSe 3atB= 0 T\n(red curve). It is composed of several phonon modes la-\nbeled P 1at 66.4 cm−1, P2at 71.1 cm−1, P3at 77.5 cm−1,\nP7at 147 .7 cm−1, and magnon-phonon coupled modes la-\nbeled MP 1,2at 112 .2 cm−1, and a doublet labeled MP 3\nand MP 4at 116 .4 and 117 .3 cm−1, respectively. The Ra-\nman scattering response of phonons of bulk FePSe 3has\nbeen described previously [28]. Cui et al. [25] have re-\ncently shown that MP 1−4are coupled phonon-magnon3\nFIG. 2. a) B= 0 T (red solid line) and B= 30 T (black solid line) Raman scattering spectra of bulk FePSe 3at low-temperature\n(T= 4.2 K). Uncoupled phonon modes are labeled as P iand the coupled magnon-phonon modes are labeled as MP i. At\nB= 0 T, MP 1and MP 2are not resolved (see Fig. S2 of Supplementary Materials for polarization-resolved measurements)\nwhile there is a visible splitting between MP 3and MP 4. (b) False color map of low-temperature Raman scattering of FePSe 3as\na function of the magnetic field applied along the crystal c-axis. A few representative scattering spectra, measured at magnetic\nfield strengths of 0 T, 5 T, and 10 T are also plotted. The weak MP 5,6peaks correspond to two additional coupled modes\n(not shown in Fig. 2a) which get prominent when in resonance with the high energy magnon mode. Dashed lines display\nmagnon-polaron modes (MP i) as calculated using Eq. 1.\nmodes, arising from a pair of quasi-degenerate chiral\nphonons and antiferromagnetic magnons. Interestingly\nthe chiral phonons in bulk FePSe 3are coupled selectively\nto the magnon component of the same chirality. In con-\ntrast to the strong coupling case, the selective coupling\ndoes not lift degeneracies either of the degenerate chiral\nphonons or of the degenerate AFM magnons (see also\nSupplementary Materials Fig. S1a and Fig. S1b-d for\nother possibilities). The B= 0 T Raman scattering re-\nsponse of bulk FePSe 3(red solid line in Fig. 2a) clearly\nshows a splitting of the MP 3,4peaks, while MP 1,2appear\nmainly as a single peak. Using polarization-resolved low-\ntemperature Raman scattering techniques, one can in-\ndeed resolve a splitting for both pair of modes MP 1/MP 2\n(0.2 cm−1) and of MP 3/MP 4(1.0 cm−1), see Fig. S2 of\nthe Supplementary Material [31]. The observation of an\nenergy splitting ∆MP 1,2and ∆MP 3,4of different ampli-\ntude for both pairs of coupled modes is not consistent\nwith the model of selective coupling among degenerate\nmagnon and phonon, and indicates that the degeneracy\nof either the phonon or the magnon excitations is intrin-\nsically lifted by a mechanism other than magnon-phonon\ninteraction.\nThe application of an external magnetic field along the\neasy axis (c-axis) modifies the magnon excitation spec-\ntrum by inducing a linear Zeeman effect which further\nmodifies the magnon-phonon energy detuning. The evo-\nlution of the Raman scattering response with respect to\nthe magnetic field is presented in Fig. 2b in the form of\na false color plot up to B= 14 T and the spectrum at\nB= 30 T is represented in Fig. 2a (black curve). While\nthe splitting of MP 3,4is clearly visible even at B= 0 T,the splitting of MP 1,2into two components with increas-\ning magnetic field confirms the presence of two quasi-\ndegenerate modes at B= 0 T. The bare phonon and\nmagnon modes are recovered with increasing magnetic\nfield. MP 1and MP 4disperse linearly with the mag-\nnetic field, indicating a pronounced magnon-like char-\nacter while the energy of MP 2and MP 3changes slightly\nand saturates at a constant value for B > 10 T, reveal-\ning a phonon-like character at the high magnetic field.\nMP 4further interacts with two more phonon modes MP 5\nand MP 6with very weak intensity, showing an avoided\ncrossing close to B= 8 T. Above this field, MP 6dis-\nplays a magnon-like behavior. As can be seen in the\nB= 30 T spectrum of Fig. 2a, the MP 1mode is interact-\ning with three P 1−3low energy phonons as their energy\nat high field is different from that at B= 0. Despite the\nclose proximity with MP 6at high magnetic fields, the\nP7phonon energy remains unaffected essentially due to\nthe fact that this high-energy phonon does not involve\nthe vibration in the magnetic lattice [28, 32]. An inter-\nesting observation in Fig. 2a is that, at B= 30 T (black\nspectrum), when the two magnon-like components MP 1,6\nare well separated from each other and from the phonon\nmodes, MP 2,3have merged into a single peak at the bare\nphonon energy (See Fig. S3 of Supplementary Materi-\nals [31]). This is strong evidence that the bare phonon\nmodes are degenerate. As we will demonstrate in the fol-\nlowing, the apparent splitting between each pair of cou-\npled modes(MP 1,2and MP 3,4) is intrinsic to magnons in\nbulk FePSe 3and we attribute it to a magnetocrystalline\nanisotropy.\nThe energy of the coupled modes observed in Raman4\nFIG. 3. False color map of Raman scattering of FePSe 3as a\nfunction of temperature using λ= 515 nm excitation laser.\nThree characteristic spectra at T= 100 K, T= 50 K, and\nT= 4.2 K are displayed. Solid lines are the bare mean\nmagnon energy (cyan solid line) and the bare phonon energy\n(green solid and dotted line) as deduced from our modeling of\nmagneto-Raman scattering experiments at different temper-\natures (see Fig. 4a-d).\nscattering experiments is analyzed in the frame of selec-\ntive coupling between magnon and phonon as presented\nin Ref. [25] and considering zero-field splitting of the\nmagnon gap excitation using the following Hamiltonian:\nH6×6=\nM10 0 δ1δ2δ3\n0M2δ10δ2δ3\n0δ1P40 0 0\nδ10 0 P40 0\nδ2δ20 0 P50\nδ3δ30 0 0 P6\n(1)\nwhere M1,2are the bare energies of the two antiferro-\nmagnetic magnon branches. The magnetic field depen-\ndence of these modes is defined as M1,2(B) = M0±p\n∆2+ (gµbB)2where M 0is the mean magnon en-\nergy and 2∆ the energy splitting between the magnon\nbranches. P iare the bare phonon energies, δ1,2,3are\nthe magnon-phonon coupling parameters for P 4, P5, and\nP6. Each component of the doubly degenerate P 4modes\nis coupled selectively to the two magnon components\nM1and M 2while P 5and P 6couple identically to both\nmagnon branches. We omit in this description the po-\ntential influence of P1−3phonons as their energy dif-\nference with the magnon is large at B= 0 T. Using\nthe Bogolyubov transformations[33], one can obtain the\neigenstates of the magnon-phonon coupled modes and\nthe corresponding coupling constants using this Hamilto-\nnian. These results are presented in Fig. 2b with dashed\nlines and represent the experimental evolutions accu-\nrately using the following parameters: M0= 115 .4 cm−1,\n∆ = 0 .6 cm−1,g= 2.14 cm−1,P4= 113 .8 cm−1,P5=\n123.9 cm−1,P6= 126 .7 cm−1and coupling constantsδ1= 2.36 cm−1,δ2= 1.36 cm−1andδ3= 0.68 cm−1.\nWhen introducing the coupling constants, these bare\nmodes transform into magnon-polarons MP i, as labeled\nin Fig. 2 and the 2∆ energy gap is distributed among\nthe MP imodes. The observation of larger energy split-\nting for MP 3,4than for MP 1,2(see Fig. 2a) is an in-\ndication that MP 3,4are more magnon-like while MP 1,2\nare more phonon-like at B= 0 T. This approach shows\nthat when bulk FePSe 3is magnetically ordered, the AFM\nmagnon excitations are nearly degenerate with the pair\nof phonons P4. This situation is seldom found in nature\nand, to create the magnon-polaron resonance, one usu-\nally has to tune the magnon or the phonon energies by\nexternal means.\nAnother way to control the magnon-phonon energy de-\ntuning is to change temperature. We have performed\ntemperature-dependent Raman scattering measurements\nto evidence the effect of the closing of the magnon gap on\nmagnon-polarons, see Fig. 3. Starting from low temper-\nature, magnon-polarons are first weakly affected by the\nchange of temperature, but both MP 1,2and MP 3,4evolve\nsignificantly above T= 30 K. The energy of MP 3,4grad-\nually changes from 117 cm−1at low temperature down\nto 114 cm−1close to T= 90 K and remains constant\nup to T= 140 K. Simultaneously, the energy of MP 1,2\nat 112 cm−1also decreases along with a decrease of its\nintensity. Above T= 70 K, the intensity of MP 1,2van-\nishes. MP 1,2and MP 3,4have an avoided crossing behav-\nior which indicates a coupling between these two pairs\nof modes. Though magnon excitations in this material\ncannot be directly observed at T= 100 K, the emergence\nof the magnetic order below TNcan be traced with the\nactivation of P 1−3phonons related to the folding of zone\nedges onto the Γ-point of the phonon Brillouin zone (see\nFig. S4 of Supplementary Materials [31]).\nTo be able to determine accurately the parameters de-\nscribing the magnetic and phonon excitations using Eq. 1\nfor different temperatures, and in particular the value of\nthe magnon splitting energy ∆, we have applied a mag-\nnetic field to separate the magnon and phonon excitations\nand suppress the effect of their interaction. These results\nare presented as false color plots of the magnetic field\nevolution of the Raman scattering intensity in Fig. 4a-d\nforT= 30 ,40,50 and 60 K, respectively. The dashed\nlines in this figure are the results of the calculation using\nEq. 1. We do not observe significant changes in the val-\nues of the magnon g-factor, of the magnon-phonon cou-\npling constants and the main effect of increasing temper-\nature lies in the decrease of the bare magnon energy M0.\nMost important is that the magnon splitting energy ∆,\nnecessary to describe Raman scattering data, appears to\nbe independent of temperature and also of the magnon-\nphonon energy detuning. Because the magnon energy is\ndecreasing when increasing temperature, the pair of MPs\nshowing the maximum splitting at B= 0 changes from\nMP 3,4atT= 30 K to MP 1,2atT= 60 K when the bare\nmagnon energy is smaller than the bare phonon energy.\nThis increase of the splitting of MP 1,2with temperature5\nFIG. 4. (a-d) False color map of FePSe 3Raman scattering as a function of the magnetic field applied along the crystal c-axis\nat temperatures 30 K, 40 K, 50 K, and 60 K respectively. Two representative scattering spectra, measured at B= 0 T and at\nB= 5 T are also plotted. Dashed lines correspond to the simulation of the magnon-polaron modes (MP i) following Eq. 1.\nis not observed in the experiment due to the thermal\nbroadening of these Raman modes, however, MP 3,4have\nmerged into a single peak as shown in Fig. S2 of the\nSupplementary Material [31], which is reproduced by our\nsimulation. This is a strong indication that this energy\nsplitting is intrinsic to the magnon excitation and is not\nimposed by the magnon-phonon interaction.\nSolving Eq. 1 brings the coupled mode energies as\neigenvalues and the eigenvectors which components de-\nscribe the bare-mode content of the coupled modes to-\ngether with their evolution when applying a magnetic\nfield and changing the magnon-phonon energy detuning.\nThe evolution of the magnon content of the MPs as a\nfunction of the magnetic field is presented in Fig. 5a and\nb for T= 5 K and T= 60 K, respectively. At T= 5 K,\nMP 3,4are mainly of magnon-like character while MP 1,2\nare mainly phonon-like. This character changes when the\ntemperature is increased, as presented in Fig. 5c.\nWhen applying an external magnetic field at T= 5 K,\nthe magnon-like character of MP 4increases significantly\nas the Zeeman effect increases the magnon-phonon en-ergy detuning. Simultaneously, the magnon-like charac-\nter of MP 3increases. At high enough magnetic field,\nMP 1,4are mostly of magnon-like character while MP 2,3\nhave become mostly phonon-like. At T= 60 K, the\nbare magnon energy is smaller than the phonon en-\nergy and the lower-in-energy pair of MPs now shows the\nlargest energy splitting. These MP 1,2show the largest\nmagnon-like character, see Fig; 5b, while MP 3,4are now\nmostly phonon-like. When applying a magnetic field, the\nmagnon content of MP 1,4increases while MP 2,3evolves\ntowards a phonon-like character. Extending this analy-\nsis to different temperatures, we can gain some insights\ninto the change of the MP is content when changing the\nmagnon-phonon energy detuning with temperature, see\nFig. 5c. When increasing temperature, the magnon con-\ntent of MP 3,4decreases, and they both become mainly\nphonon-like above T= 40 K. MP 1,2follows the inverse\nevolution, from a mainly phonon-like character at low\ntemperature to a mainly magnon-like character above\nT= 40 K. This analysis allows tracing the evolution of\nthe bare-modes composition entering the coupled modes6\nFIG. 5. (a) and (b) Simulation of the magnon-like behavior of the magnon-polaron modes (MP i) driven by a magnetic field\nat 5 K and 60 K respectively. The behavior of the MP 4mode is affected due to further coupling with other high-energy\nphonon modes (see Fig. 2a at the high field), which are not shown here. (c) Simulation of the magnon-like behavior of the\nmagnon-polaron modes (MP i) as a function of temperature at zero-field.\nfor different temperatures and with an applied magnetic\nfield.\nWhile earlier reports have considered the existence\nof two distinct phonon modes interacting with magnon\ngap excitations [25], our experimental findings provide\nnew evidence showing that these phonons are degener-\nate. Consequently, we attribute the observed zero-field\nsplitting of both coupled modes of different amplitude,\nto the splitting of the underlying magnon gap excita-\ntion. One possible cause for this splitting is an addi-\ntional anisotropy, induced by the substitution of S by Se\natoms. Indeed, FePS 3has a monoclinic structure and is\nan easy-axis antiferromagnet with no or very little mag-\nnetocrystalline anisotropy [30] leading to a degenerate\nmagnon gap excitation, while FePSe 3has an orthorhom-\nbic structure and shows all signatures of an easy-axis an-\ntiferromagnet with an anisotropy that lifts the magnon\ngap excitation degeneracy. This situation is similar to\nthe case MnPX 3where MnPS 3is an easy-axis antiferro-\nmagnet with a monoclinic structure while MnPSe 3has\nan orthorhombic structure and shows an additional mag-\nnetocrystalline anisotropy that lifts the degeneracy of\nmagnon gap excitations [13].IV. CONCLUSION\nBased on the Raman scattering measurements com-\nbined with an environment of variable temperature\nand high magnetic fields, we have evidenced selectively\ncoupled magnon-polarons in the bulk antiferromagnet\nFePSe 3. We have used a magnon-phonon coupling ma-\ntrix formalism to reproduce our experimental data and\nextract the bare modes energies, type of coupling, the\nmagnon-phonon coupling constants, and the coupled\nmode composition in terms of bare modes. Our results\nclearly indicate the lifting of the degeneracy of the anti-\nferromagnetic magnon modes (of 1 .2 cm−1at low tem-\nperature) in bulk FePSe 3which can appear due to an\nadditional in-plane magneto-crystalline anisotropy.\nACKNOWLEDGMENTS\nThe work has been supported by the EC Graphene\nFlagship project. M.P. also acknowledges support from\nthe Foundation for Polish Science (MAB/2018/9 Grant\nwithin the IRA Program financed by EU within SG OP\nProgram). Z.S. was supported by ERC-CZ program\n(project LL2101) from Ministry of Education Youth and\nSports (MEYS).\n[1] B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein,\nR. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A.\nMcGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-\nHerrero, and X. Xu, Layer-dependent ferromagnetism in\na van der Waals crystal down to the monolayer limit,\nNature 546, 270 (2017).\n[2] C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao,\nW. Bao, C. Wang, Y. Wang, Z. Q. Qiu, R. J. Cava, S. G.\nLouie, J. Xia, and X. Zhang, Discovery of intrinsic fer-\nromagnetism in two-dimensional van der Waals crystals,\nNature 546, 265 (2017).\n[3] X. Jiang, Q. Liu, J. Xing, N. Liu, Y. Guo, Z. Liu, and\nJ. Zhao, Recent progress on 2D magnets: Fundamentalmechanism, structural design and modification, Applied\nPhysics Reviews 8, 031305 (2021).\n[4] T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich,\nAntiferromagnetic spintronics, Nature Nanotechnology\n11, 231 (2016).\n[5] V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono,\nand Y. Tserkovnyak, Antiferromagnetic spintronics, Rev.\nMod. Phys. 90, 015005 (2018).\n[6] V. Voroshnin, A. V. Tarasov, K. A. Bokai, A. Chikina,\nB. V. Senkovskiy, N. Ehlen, D. Y. Usachov, A. Gruneis,\nM. Krivenkov, J. Sanchez-Barriga, et al. , Direct spec-\ntroscopic evidence of magnetic proximity effect in MoS 2\nmonolayer on graphene/Co, ACS nano 16, 7448 (2022).[7] D. Lan¸ con, H. C. Walker, E. Ressouche, B. Oulad-\ndiaf, K. C. Rule, G. J. McIntyre, T. J. Hicks, H. M.\nRønnow, and A. R. Wildes, Magnetic structure and\nmagnon dynamics of the quasi-two-dimensional antifer-\nromagnet FePS 3, Phys. Rev. B 94, 214407 (2016).\n[8] S. Calder, A. Haglund, A. I. Kolesnikov, and D. Mandrus,\nMagnetic exchange interactions in the van der waals lay-\nered antiferromagnet MnPSe 3, Physical Review B 103,\n024414 (2021).\n[9] A. R. Wildes, J. R. Stewart, M. D. Le, R. A. Ewings,\nK. C. Rule, G. Deng, and K. Anand, Magnetic dynamics\nof NiPS 3, Phys. Rev. B 106, 174422 (2022).\n[10] A. R. Wildes, B. F˚ ak, U. B. Hansen, M. Enderle, J. R.\nStewart, L. Testa, H. M. Rønnow, C. Kim, and J.-G.\nPark, Spin wave spectra of single crystal CoPS 3, Phys.\nRev. B 107, 054438 (2023).\n[11] D. Bozhko, V. Vasyuchka, A. Chumak, and A. Serga,\nMagnon-phonon interactions in magnon spintronics, Low\nTemperature Physics 46, 383 (2020).\n[12] F. Godejohann, A. V. Scherbakov, S. M. Kukhtaruk,\nA. N. Poddubny, D. D. Yaremkevich, M. Wang,\nA. Nadzeyka, D. R. Yakovlev, A. W. Rushforth, A. V.\nAkimov, et al. , Magnon polaron formed by selectively\ncoupled coherent magnon and phonon modes of a surface\npatterned ferromagnet, Physical Review B 102, 144438\n(2020).\n[13] T. T. Mai, K. F. Garrity, A. McCreary, J. Argo, J. R.\nSimpson, V. Doan-Nguyen, R. V. Aguilar, and A. R. H.\nWalker, Magnon-phonon hybridization in 2d antiferro-\nmagnet MnPSe 3, Science advances 7, eabj3106 (2021).\n[14] S. Liu, A. G. Del ´Aguila, D. Bhowmick, C. K. Gan,\nT. T. H. Do, M. Prosnikov, D. Sedmidubsk` y, Z. Sofer,\nP. C. Christianen, P. Sengupta, et al. , Direct observation\nof magnon-phonon strong coupling in two-dimensional\nantiferromagnet at high magnetic fields, Physical Review\nLetters 127, 097401 (2021).\n[15] K. Park, J. Oh, J. C. Leiner, J. Jeong, K. C. Rule, M. D.\nLe, and J.-G. Park, Magnon-phonon coupling and two-\nmagnon continuum in the two-dimensional triangular an-\ntiferromagnet CuCrO 2, Physical Review B 94, 104421\n(2016).\n[16] S. Bao, Z. Cai, W. Si, W. Wang, X. Wang, Y. Shang-\nguan, Z. Ma, Z.-Y. Dong, R. Kajimoto, K. Ikeuchi, et al. ,\nEvidence for magnon-phonon coupling in the topologi-\ncal magnet Cu 3TeO 6, Physical Review B 101, 214419\n(2020).\n[17] S. Streib, N. Vidal-Silva, K. Shen, and G. E. Bauer,\nMagnon-phonon interactions in magnetic insulators,\nPhysical Review B 99, 184442 (2019).\n[18] K. Wang, J. He, M. Zhang, H. Wang, and G. Zhang,\nMagnon–phonon interaction in antiferromagnetic two-\ndimensional mxenes, Nanotechnology 31, 435705 (2020).\n[19] K. A. Hay and J. B. Torrance, Magnon-phonon interac-\ntion observed in far infrared studies of FeCl 2−2H20 and\nCoCl 2−2H20, Journal of Applied Physics 40, 999 (1969).\n[20] D. Vaclavkova, M. Palit, J. Wyzula, S. Ghosh, A. Del-\nhomme, S. Maity, P. Kapuscinski, A. Ghosh, M. Veis,\nM. Grzeszczyk, et al. , Magnon polarons in the van der\nwaals antiferromagnet FePS 3, Physical Review B 104,\n134437 (2021).\n[21] A. Pawbake, T. Pelini, A. Delhomme, D. Romanin,\nD. Vaclavkova, G. Martinez, M. Calandra, M.-A. Meas-\nson, M. Veis, M. Potemski, et al. , High-pressure tuning ofmagnon-polarons in the layered antiferromagnet FePS 3,\nACS nano 16, 12656 (2022).\n[22] S. M. Rezende, Fundamentals of Magnonics , 1st ed., Lec-\nture notes in physics (Springer Nature, Cham, Switzer-\nland, 2020).\n[23] T. M. J. Cham, S. Karimeddiny, A. H. Dismukes, X. Roy,\nD. C. Ralph, and Y. K. Luo, Anisotropic gigahertz anti-\nferromagnetic resonances of the easy-axis van der Waals\nantiferromagnet CrSBr, Nano Letters 22, 6716 (2022).\n[24] C. Cho, A. Pawbake, N. Aubergier, A. Barra, K. Mosina,\nZ. Sofer, M. Zhitomirsky, C. Faugeras, and B. Piot, Mi-\ncroscopic parameters of the van der waals CrSBr antifer-\nromagnet from microwave absorption experiments, Phys-\nical Review B 107, 094403 (2023).\n[25] J. Cui, E. V. Bostroem, M. Ozerov, F. Wu, Q. Jiang, J.-\nH. Chu, C. Li, F. Liu, X. Xu, A. Rubio, and Q. Zhang,\nChirality selective magnon-phonon hybridization and\nmagnon-induced chiral phonons in an atomically thin an-\ntiferromagnet, Nat. Comm. 14, 3396 (2023).\n[26] K. Momma and F. Izumi, Vesta 3 for three-dimensional\nvisualization of crystal, volumetric and morphology data,\nJournal of applied crystallography 44, 1272 (2011).\n[27] A. Wiedenmann, J. Rossat-Mignod, A. Louisy, R. Brec,\nand J. Rouxel, Neutron diffraction study of the layered\ncompounds MnPSe 3and FePSe 3, Solid State Communi-\ncations 40, 1067 (1981).\n[28] M. Scagliotti, M. Jouanne, M. Balkanski, G. Ouvrard,\nand G. Benedek, Raman scattering in antiferromagnetic\nFePS 3and FePSe 3crystals, Physical Review B 35, 7097\n(1987).\n[29] P. A. Fleury and R. Loudon, Scattering of light by one-\nand two-magnon excitations, Phys. Rev. 166, 514 (1968).\n[30] T. Sekine, M. Jouanne, C. Julien, and M. Balkanski,\nLight-scattering study of dynamical behavior of antifer-\nromagnetic spins in the layered magnetic semiconductor\nFePS 3, Phys. Rev. B 42, 8382 (1990).\n[31] See Supplemental Material at [URL will be inserted by\npublisher] for details on coupled magnon-phonon mode\nsimulation, linear polarization-resolved Raman scatter-\ning, 30 T coupled mode simulation, and temperature-\ndependent Raman scattering of FePSe 3,.\n[32] A. Hashemi, H.-P. Komsa, M. Puska, and A. V.\nKrasheninnikov, Vibrational properties of metal phos-\nphorus trichalcogenides from first-principles calculations,\nThe Journal of Physical Chemistry C 121, 27207 (2017).\n[33] R. M. White, M. Sparks, and I. Ortenburger, Diagonal-\nization of the antiferromagnetic magnon-phonon interac-\ntion, Phys. Rev. 139, A450 (1965).Supplementary Materials for ”In-plane magnetocrystalline anisotropy in the van der\nWaals antiferromagnet FePSe 3probed by magneto-Raman scattering”\nDipankar Jana,1,∗Piotr Kapuscinski,1Amit Pawbake,1Anastasios Papavasileiou,2Zdenek\nSofer,2Ivan Breslavetz,1Milan Orlita,1, 3Marek Potemski,1, 4,†and Clement Faugeras1,‡\n1Laboratoire National des Champs Magn´ etiques Intenses, LNCMI-EMFL,\nCNRS UPR3228,Univ. Grenoble Alpes, Univ. Toulouse,\nUniv. Toulouse 3, INSA-T, Grenoble and Toulouse, France\n2Chemistry Department, University of Chemistry and Technology Prague, 16628 Prague, Czech Republic\n3Institute of Physics, Charles University, Ke Karlovu 5, Prague, 121 16, Czech Republic\n4CENTERA Labs, Institute of High Pressure Physics, PAS, 01 - 142 Warsaw, Poland\nI. SIMULATION FOR THE MAGNETIC FIELD DEPENDENCE OF MAGNON-PHONON COUPLED\nMODES: OTHER POSSIBILITIES\nFIG. S1. Simulations to explore the magnetic field dependence of magnon-phonon coupled modes under different coupling\nscenarios: (a) selective coupling among degenerate P 4modes and degenerate M 1,2modes, excluding the coupling with P 5and\nP6modes, (b) selective coupling among degenerate P 4modes and degenerate M 1,2modes, including the coupling with P 5and\nP6modes, (c) selective coupling among non-degenerate P 4modes and degenerate M 1,2modes, and (d) Nonselective coupling\namong all the degenerate P 4modes and degenerate M 1,2modes. In Fig.S1a, we observe that the selective coupling among\nthe degenerate coupled modes does not create any apparent splitting among the modes. The inclusion of coupling with P 5\nand P 6phonon modes (Fig.S1b) generates an apparent splitting that is small. Simulation with non-degenerate phonon modes\n(Fig.S1c) to match the apparent zero-field gap deviates largely from the experimental data at higher fields. Simulation with\nnon-selective coupling (Fig.S1d) exhibited even greater deviation from the experimental outcome. Please note that in Fig.Sb-d,\nwe considered the coupling with the P 5and P 6phonon modes, even though it was not explicitly shown.\n∗dipankar.jana@lncmi.cnrs.fr\n†marek.potemski@lncmi.cnrs.fr\n‡clement.faugeras@lncmi.cnrs.fr2\nII. LINEAR POLARIZATION-RESOLVED RAMAN SCATTERING\nFIG. S2. Linear polarization-resolved Raman scattering spectra at two different temperatures, 4.5 K and 100 K. In Ref. 1 of\nthe main script, the coupled modes were reported to exhibit circular polarization. However, our findings indicate that these\nmodes at B=0 T also exhibit opposite linear polarization when excited with a linearly polarized laser. At 4.2 K, we estimate\nan apparent splitting of 0.15 cm−1between MP 1,2and 0.9 cm−1between the MP 3,4modes. The spectra at 100 K are scaled\nfour times for better clarity. At this temperature, only the phonon modes (MP 3,4) are present among the coupled modes.\nInterestingly, we do not observe any splitting between MP 3,4modes, which confirms the degeneracy of the bare phonons (P 4).\nIII. SIMULATED SPLITTING BETWEEN THE PHONON-LIKE MODES (MP 2,3) AT 30 T\nFIG. S3. Low-temperature Raman scattering spectra at 6 T and 30 T showing only the coupled modes MP 2,3. The well-\nseparated peaks at 6 T are combined into a single peak at 30 T. We employed two Lorenzian peaks with equal widths to\nfit the modes at 6 T. To simulate these merged modes at 30 T, we constrain the width (0.3 cm−1) of the peaks to be the\nsame as that obtained at 6 T and the intensity to be the same for both modes. The peak energy of the two modes is treated\nas an adjustable parameter. Remarkably, a gap of 0.3 cm−1is estimated between the two components, aligning closely with\nthe simulated field-dependent splitting (0.35 cm−1) of MP 2,3modes at 30 T. This confirms that the relatively larger splitting\namong the coupled modes at B=0 T is not due to the non-degeneracy of the phonon modes P 4.3\nIV. TEMPERATURE DEPENDENT RAMAN SCATTERING IN WIDE ENERGY RANGE\nFIG. S4. False color map of the Raman scattering response of FePSe 3as a function of temperature. The lowest energy peaks,\nlabeled P 1,3, disappear when the temperature reaches Neel temperature (T N) and are considered as phonons due to the folding\nof the Brillouin zone boundary onto the Γ point."    },    {        "title": "2307.15470v2.Revisiting_Néel_60_years_on__the_magnetic_anisotropy_of___mathrm_L_1_0__FeNi__tetrataenite_.pdf",        "content": "Revisiting Néel 60 years on: the magnetic anisotropy of L1 0FeNi\n(tetrataenite)\nChristopher D. Woodgate,1,a)Christopher E. Patrick,2Laura H. Lewis,3, 4and Julie B. Staunton1,b)\n1)Department of Physics, University of Warwick, Coventry, CV4 7AL, United Kingdom\n2)Department of Materials, University of Oxford, Oxford, OX1 3PH, United Kingdom\n3)Department of Chemical Engineering, Northeastern University, Boston, MA 02115, USA\n4)Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA 02115,\nUSA\n(Dated: 13 October 2023)\nThe magnetocrystalline anisotropy energy of atomically ordered L1 0FeNi (the meteoritic mineral tetrataenite) is studied\nwithin a first-principles electronic structure framework. Two compositions are examined: equiatomic Fe 0.5Ni0.5and an\nFe-rich composition, Fe 0.56Ni0.44. It is confirmed that, for the single crystals modelled in this work, the leading-order\nanisotropy coefficient K1dominates the higher-order coefficients K2andK3. To enable comparison with experiment, the\neffects of both imperfect atomic long-range order and finite temperature are included. While our computational results\ninitially appear to undershoot the measured experimental values for this system, careful scrutiny of the original analysis\ndue to Néel et al. [J. Appl. Phys. 35, 873 (1964)] suggests that our computed value of K1is, in fact, consistent with\nexperimental values, and that the noted discrepancy has its origins in the nanoscale polycrystalline, multivariant nature\nof experimental samples, that yields much larger values of K2andK3than expected a priori . These results provide fresh\ninsight into the existing discrepancies in the literature regarding the value of tetrataenite’s uniaxial magnetocrystalline\nanisotropy in both natural and synthetic samples.\nI. INTRODUCTION\nPermanent magnets are of critical importance in many tech-\nnological applications, prime examples being in electrical\npower generation and electric motors that are essential in the\ntransition to a low-carbon future. At present, the vast ma-\njority of permanent magnets used for advanced applications\ncontain significant amounts of rare-earth elements such as\nsamarium and neodymium1, with examples being SmCo 52\nand Nd 2Fe14B3,4; the latter compound also requires incorpo-\nration of rare and expensive dysprosium and/or terbium for\nrobust high-temperature performance. These rare-earth ele-\nments are a constrained resource accompanied by concerns\naround price volatility and the long-term stability of the global\nsupply chain5,6. There are also concerns over the environmen-\ntal impact of their extraction7and processing. Overall these\nconditions motivate a globally concerted effort to develop\nrare-earth-free permanent magnets for use in advanced appli-\ncations. One such candidate under consideration is atomically\nordered Fe-Ni that crystallises in the tetragonal L1 0structure8.\n(The L1 0structure is visualised in Fig. 1.) This compound,\nwhich is naturally found in extremely slowly cooled metallic\nmeteorites, is also referred to as tetrataenite9.\nAlthough there are significant challenges associated with\nits formation that arise from a low atomic ordering temper-\nature and sluggish kinetics10, the L1 0phase is thermody-\nnamically stable relative to its atomically disordered, face-\ncentered-cubic (fcc) counterpart, and a number of attempts\nhave been made to synthesise it in the lab10–23. Tetrataenite is\nknown, both from experimental measurements10–24and from\na)Christopher.Woodgate@warwick.ac.uk\nb)J.B.Staunton@warwick.ac.uktheoretical calculations25–31, to possess a sufficiently large\nuniaxial magnetocrystalline anisotropy energy (MAE) to be\nuseful as a ‘gap magnet’32, with anticipated performance be-\nlow that of the rare-earth magnets but above that of oxide fer-\nrite magnets.\nTo date, computational modelling of tetrataenite’s magne-\ntocrystalline anisotropy has focused on evaluating its uniax-\nial anisotropy energy, KU, which is the difference in energy\n1a1d\n1a1d\nFIG. 1. Visualisation of the L1 0ordered structure imposed on a face-\ncentred tetragonal lattice. The conventional simple tetragonal cell,\nwith 1a sites at the corners and a 1d site at the centre, is indicated\nby dashed black lines. The simple tetragonal unit cell has lattice pa-\nrameters c′=c,a′=b′=a/√\n2, where aandcare the face-centred\ntetragonal lattice parameters. Perfectly ordered L1 0Fe0.5Ni0.5has\nall 1a (blue) sites occupied by Fe atoms, and all 1d (red) sites occu-\npied by Ni atoms. The disordered Fe 0.5Ni0.5alloy in which every\nsite is occupied with a 50% probability of either Fe or Ni has a face-\ncentred cubic lattice with c=a.arXiv:2307.15470v2  [cond-mat.mtrl-sci]  12 Oct 20232\nReference Sample Type Composition S T (K) Anisotropy Constants (MJm−3)\nKUK1 K2 K3\nNéel et al. (1964)10Bulk Sample (Disc) Fe 0.5Ni0.5 0.41–0.45 (Estimate) 293 0.55 0.32 0.23\nPaulevé et al. (1968)11Bulk Sample (Sphere) Fe 0.5Ni0.5 0.55 0.3 0.17 0.08\nShima et al. (2007)12Film Fe 0.5Ni0.5 0.6±0.2 Room T. 0.63\nMizugichi et al. (2011)13Thin Film Fe 0.5Ni0.5 0.33 Room T. 0.58\nKojima et al. (2011)14Film Fe 0.5Ni0.5 0.5±0.1 Room T. 0.50 ±0.01\nKojima et al. (2012)15Film Fe 0.5Ni0.5 0.48±0.05 Room T. 0.70 ±0.02\nKojima et al. (2014)16Film Fe 0.5Ni0.5 0.4 Room T. 0.7\nFilm Fe 0.6Ni0.4 0.2 Room T. 0.93\nPoirier et al. (2015)24Meteorite Fe 0.57Ni0.43 0.84\nFrisk et al. (2016)18Film Fe 0.5Ni0.5 35 <0.2\nFrisk et al. (2017)19Film Fe 0.5Ni0.5 300 ∼0.35\nItoet al. (2023)22Film Fe 0.5Ni0.5 0.60 Room T. 0.55\nTABLE I. Experimental measurements of the magnetocrystalline anisotropy of FeNi, with sample description and atomic order parameter, S,\nwhere provided. Blank fields in the table indicate values not provided in the associated reference. Uncertainties are quoted where provided\nin the associated reference. KUdenotes the conventional measurement of the uniaxial anisotropy, the difference in energy between a system\nmagnetised in the ˆzdirection and one magnetised in the ˆxdirection. We note that the order parameter provided by Néel et al. in Ref. 10 is an\nestimate, rather than a direct measurement.\nbetween the system magnetised in the ˆzdirection compared\nto that along the ˆxdirection, rather than resolving it into its\nseparate coefficients, i.e. K U=K1+K2+K3. In addition,\nthese calculations have primarily focused on the equiatomic\nbinary composition, Fe 0.5Ni0.5, atT=0 K and with perfect\natomic order25–31. However, experimental measurements of\nthis material’s magnetocrystalline anisotropy are invariably\nperformed at finite temperature (typically room temperature)\non samples with imperfect atomic order10–19,22,24,33. (A sum-\nmary of the experimental data can be found in Table I.) These\nfactors are significant, because both decreasing atomic long-\nrange order and increasing temperature reduce the magne-\ntocrystalline anisotropy of L1 0materials34–36.\nComputational predictions25–31give an MAE for L1 0\nFe0.5Ni0.5in the range 0.23–0.78 MJm−3, while experimental\nobservations10–16,18,19,22,24fall in the range 0.2–0.93 MJm−3.\nAlthough at first glance there appears to be consistency be-\ntween experiment and theory, we again emphasise that the-\noretical calculations are performed at T=0 K on systems\nwith perfect atomic order, while experimental measurements\nare performed at room temperature on samples with imperfect\natomic order. Further analysis of the experimental data8,10,27\nhas suggested that the MAE of this material could reach\nthe range 1.0–1.3 MJm−3. Typically, ab initio estimates\nof MAE values for L1 0transition-metal/noble-metal systems\nhave been in good agreement with the corresponding ex-\nperimental data, for example for FePd37, FePt34,35,37,38, and\nCoPt38, where the ab initio values for idealised systems tend\nto, if anything, slightly overestimate MAE values. The dis-\ncrepancy between experimental and computational results for\nFeNi is therefore striking, and we suggest that it highlights\na gap in understanding of the underlying physical origin of\nmagnetocrystalline anisotropy in this material.\nIn addition, we note that almost all computational studies,and most experiments, do not resolve the anisotropy energy\ninto its separate coefficients, K1,K2, and K3. These coeffi-\ncients have their origins in the symmetry of the underlying\ncrystal structure. For a tetragonal crystal structure, K1is the\nleading (second-order) term in an expansion of the magne-\ntocrystalline anisotropy energy, while K2andK3are higher-\norder (fourth order) terms39. Significantly, the two seminal\nworks on this material by Néel et al.10and Paulevé et al.11\ndo indeed provide values for these separate K2andK3coef-\nficients but they are unusually large for a uniaxial material.\nThese large values of K2andK3are noted to make a signif-\nicant contribution to the total uniaxial anisotropy coefficient,\nKU.\nIn this work, we address the issues outlined above. We use\na fully relativistic, first-principles-based framework to study\nthe magnetocrystalline anisotropy of this system, providing\nresults for both the equiatomic composition Fe 0.5Ni0.5as well\nas for an Fe-rich composition that is more consistent with ex-\nperimentally determined composition of tetrataenite in some\nmeteorite samples8,9,40, Fe 0.56Ni0.44. We use our modelling\nframework to study the dependence of the magnetic torque on\nthe direction of magnetisation and confirm that the leading-\norder anisotropy coefficient K1dominates higher-order co-\nefficients K2andK3for the univariant single crystals mod-\nelled in this work. We go on to examine the magnetocrys-\ntalline anisotropy energy of this compound at finite temper-\nature, and find that its value remains high up to and well\nbeyond room temperature. We also consider the effect of\nvarying long-range atomic order and find that, for both com-\npositions, decreasing atomic order consistently decreases the\ncomputationally predicted MAE. Finally, we perform calcu-\nlations in which the effects of both finite temperature and im-\nperfect atomic ordering are included. From a careful read-\ning of the original analysis due to Néel et al.10, we propose3\nthat our predicted MAE values for ideal single crystals are\nconsistent with the experimental data, and that it is instead\nthe nanocrystalline, multivariant nature of experimental sam-\nples41that yield larger values of K2andK3than expected a\npriori .\nThese results have significant implications. First, they pro-\nvide insight into the discrepancies between measured and pre-\ndicted anisotropies of this compound in the existing literature,\nemphasising that the higher-order anisotropy coefficients K2\nandK3appear to make a significant contribution to this ma-\nterial’s experimentally observed total uniaxial anisotropy en-\nergy. Second, these results demonstrate the impact of vary-\ning Fe:Ni ratio on the predicted value of K1, and conclude\nthat an equiatomic composition is expected to exhibit superior\nmagnetic performance compared to those of off-stoichiometry\ncompositions. Finally, it is shown that the finite-temperature\nmagnetocrystalline anisotropy of this material is remarkably\nrobust, lending support for its potential use in elevated tem-\nperature applications. This work therefore supports renewed\ncontemplation of this compound as a suitable material for ad-\nvanced magnets.\nThis paper is structured as follows. In Section II, we briefly\noutline the underlying theory and our methodology for obtain-\ning the MAE from first principles using the torque method as\nwell as provide relevant computational details. Then, in Sec-\ntion III, we provide results of our calculations and carefully\nset them in an experimental context. Finally, in Section IV,\nwe summarise our results, venture an outlook on their impli-\ncations, and suggest possible further work.\nII. METHODOLOGY\nOur technique for modelling the magnetocrystalline\nanisotropy energy of a material is based on evaluations of\ntorque when a system is magnetised in a given direction.\nThese torques are obtained by evaluating derivatives of an en-\nergy, where the energy is obtained ab initio via density func-\ntional theory (DFT) calculations in which relativistic effects,\nnotably spin-orbit coupling, are included42. The method is a\nnumerically robust way of calculating MAE values which are\non the scale of 1 meV/atom or smaller. Further, by consider-\ning torque as a function of angle, the method enables direct\nevaluation of the separate anisotropy coefficients K1,K2, and\nK3, as well as the usual uniaxial anisotropy constant KU.\nWe use the Korringa-Kohn-Rostoker (KKR) formulation of\nDFT and the coherent potential approximation (CPA)43to de-\nscribe the effects of partial atomic order37. In addition, the\neffects of magnetic fluctuations at finite temperature are in-\ncluded via the disordered local moment (DLM) picture44.\nA. Magnetocrystalline anisotropy\nThe magnetocrystalline anisotropy energy describes the\nvariation in energy of a ferromagnetic system due to a rota-\ntion of the magnetisation direction. For a ferromagnet withmagnetisation direction ˆn, the MAE can be expressed as37\nK=∑\nγκγgγ(ˆn), (1)\nwhere κγdenote MAE coefficients and gγdenote spherical\nharmonics fixing the orientation of ˆnwith respect to the crys-\ntal axes. It is convenient to represent ˆnin spherical polar co-\nordinates by ˆn= (sinθcosφ,sinθsinφ,cosθ), where θand\nφdenote polar and azimuthal angles, respectively, in a coor-\ndinate frame specified by the crystal axes.\nIn a tetragonal uniaxial ferromagnet, the MAE is known to\nbe well-described by39\nK=K1sin2θ+K2sin4θ+K3sin4θcos4φ, (2)\nwhere K1,K2, and K3are the conventional anisotropy coeffi-\ncients. The total energy of the system is then written as\nE(ˆn) =Eiso+K1sin2θ+K2sin4θ+K3sin4θcos4φ,(3)\nwhere Eisorepresents the isotropic portion of the energy. The\nMAE coefficients K1,K2, and K3can be accessed from deriva-\ntives of this energy with respect to magnetisation directions.\nGiven an energy of the form in Eq. 3, we have that\n∂E\n∂θ=K1sin2θ+2K2sin2θsin2θ\n+2K3sin2θsin2θcos4φ, (4)\n∂E\n∂φ=−4K3sin4θsin4φ, (5)\nand evaluation of these quantities at particular values of θand\nφleads directly to the coefficients.\nMoreover, the magnetocrystalline anisotropy energy differ-\nence between the system magnetised in the ˆzandˆxdirections,\nKU:=E((1,0,0))−E((0,0,1)) = K1+K2+K3,(6)\nis given by the evaluation of the derivative,∂E\n∂θatθ=π\n4and\nφ=0 directly, enabling us to obtain the MAE in a one-shot\ncalculation. In practice, for a uniaxial ferromagnet, one ex-\npects K1to dominate K2andK339, and the anisotropy is typi-\ncally written as\nK=K1sin2θ. (7)\nB. Evaluating magnetic torques ab initio\nTo evaluate these derivatives and, therefore, the MAE coef-\nficients, we use an expression from the KKR multiple scatter-\ning formalism for the magnetic torque34,37,45,46,\nT(ˆn)=−∂E(ˆn)\n∂ˆn, (8)\nin which the effects of finite-temperature local moment fluctu-\nations44can also be included. Using the stationary properties\nof the relativistic DFT energy E(ˆn)with respect to the charge4\nand magnetisation densities, the derivation of the torque be-\ngins with the single-electron energy-sum part of this energy.\nIn terms of the integrated electronic density of states, N(ˆn)(E),\nfor a system magnetised along a direction ˆn, this energy is\nwritten as\nE(ˆn)=−ZE(ˆn)\nFN(ˆn)(E)dE. (9)\nIn multiple scattering theory the integrated density of states is\nexpressed particularly succinctly using the Lloyd formula43,47\nN(ˆn)(E) =N0(E)−1\nπImlndet\u0010\nt(ˆn;E)−1−G0(E))\u0011\n,(10)\nwhere t(ˆn;E)describes an array of single-site-scattering t-\nmatrices (combined into a super matrix in both site and angu-\nlar momentum space) and G0(E)specifies the structure con-stants which contain all the information on the spatial location\nof the scatterers43.\nAt a site i, the t-matrix describing the scattering of an elec-\ntron from an effective scalar potential and local magnetic field,\naligned with the spin-polarisation of the electron density in\nthat region and located in the unit cell surrounding the site,\nis obtained from the solution of the Dirac equation. This so-\nlution is found in a coordinate frame in which the z-axis of\nthe local coordinate frame is aligned with ˆn48,49. A simple\ntransformation produces a t-matrix for the general coordinate\nframe,\nti(ˆn;E) =R(ˆn)ti(ˆz;E)R(ˆn)+(11)\nwhere R(ˆn) =expiαˆm(ˆm·J),αˆmis the angle of rotation about\nan axis ˆm=ˆz∧ˆn\n|ˆz∧ˆn|, and Jis the total angular momentum. The\ntorque quantity T(ˆn)\nαˆu=−∂E(ˆn)\n∂αˆu, describing the variation of the\ntotal energy with respect to a rotation of the magnetisation\nabout a general axis ˆu, is\nT(ˆn)\nαˆu=−1\nπZE(ˆn)\nFIm∂\n∂αˆuh\nlndet\u0010\nt(ˆn;E)−1−G0(E)\u0011i\ndE (12)\nwhich can be written as37\nT(ˆn)\nαˆu=−1\nπZE(ˆn)\nFIm∑\niTr\u0012\nτ(ˆn)\nii(E)∂\n∂αˆu\u0000\nR(ˆn)t(ˆn;E)−1R(ˆn)+\u0001\u0013\ndE (13)\nwhere the KKR scattering path operator43,50is\nτ(ˆn)=\u0012\u0010\nt(ˆn)\u0011−1\n−G0\u0013−1\n. (14)\nSince∂R(ˆn)\n∂αˆu=i(J·ˆu)R(ˆn)and∂R(ˆn)+\n∂αˆu=−i(J·ˆu)R(ˆn), we obtain\nT(ˆn)\nαˆu=1\nπZE(ˆn)\nFImi∑\niTr\u0010\nτ(ˆn)\nii(E)\u0002\n(J·ˆu)t(ˆn;E)−1−t(ˆn;E)−1(J·ˆu)\u0003\u0011\ndE. (15)\nThis closed-form expression enables us to evaluate the torque\nfor a given magnetisation direction ˆn. By sampling a range\nof values of θandφ, it is therefore possible to extract values\nof the coefficients K1,K2, and K3separately along with the\nconventional uniaxial anisotropy, KU. We note that this tech-\nnique for evaluating the MAE has previously exhibited excep-\ntional fidelity when applied to other alloys crystallising in the\nL10structure, giving values in excellent agreement with ex-\nperimental literature for the FePd37and FePt34systems, along\nwith explaining the dependence of the MAE on the long-range\natomic order parameter35. In addition, the method has also\nbeen used with success to study a number of rare-earth-based\nsystems51,52and other magnetic materials53.C. Computational Details\nWe use the all-electron HUTSEPOT code54to generate\nself-consistent, one-electron potentials within the KKR for-\nmulation of density functional theory (DFT)43,55. We per-\nform spin-polarised, scalar-relativistic calculations within the\natomic sphere approximation (ASA)56with an angular mo-\nmentum cutoff of lmax=3 for basis set expansions, a 20 ×\n20×20k-point mesh for integrals over the Brillouin zone,\nand employ a 24-point semi-circular Gauss-Legendre grid in\nthe complex plane to integrate over valence energies. We\nuse the local density approximation (LDA) and the exchange-\ncorrelation functional of Perdew-Wang57. For all calculations,\nwe use lattice parameters of a=b=3.560 Å, and c=3.5775\nÅ, obtained in an earlier first-principles study29. The c/ara-\ntio is 1.0048, consistent with a very low tetragonality. These\nlattice parameters are broadly consistent with both earlier ex-\nperimental17,33,58,59and computational8,25,26,28studies. The\nconventional simple tetragonal (A6) representation of the L1 0\nunit cell with a two atom basis is used, as visualised in Fig. 1.\nA comment should be made about the suitability of the\nASA as used in this work to study the MAE via calculations\nof the magnetic torque of the FeNi system. The ASA descrip-\ntion of the self-consistent potentials of the KKR-CPA formu-\nlation of DFT is known to be most suitable for description\nof close-packed systems49,60. L1 0FeNi, with its underlying\nface-centred tetragonal lattice and a c/aratio close to unity,\ncan be classified as close-packed and we therefore believe the\nASA to be appropriate. In addition, the relativistic DLM pic-\nture required to study finite-temperature effects has only been\nimplemented within a spherical potential approximation, such\nas the ASA employed in this work. So-called ‘full potential’\nschemes, such as those used in Ref. 28, are expected to be\nnecessary to accurately describe more complex crystal struc-\ntures, especially if the MAE is calculated from direct energy\ndifferences rather than the magnetic torque.\nUtilization of the inhomogeneous CPA61,62allows obser-\nvation of the effects of partial atomic order on the predicted\nMAE. An order parameter Sis defined to describe partial\nFe-Ni atomic order in our configurations, where S=1 de-\nscribes a maximally ordered state, and S=0 corresponds to\na state which is completely disordered and both lattice sites\nshown in Fig. 1 are equivalent. For the equiatomic compo-\nsition, Fe 0.5Ni0.5, the concentration of Fe on the 1a (1d) site\nis given by 0 .5+0.5S(0.5−0.5S). For the Fe-rich composi-\ntion, Fe 0.56Ni0.44, the concentration of Fe on the 1a (1d) site is\ngiven by 0 .56+0.44S(0.56−0.44S). The Ni concentrations\ncan be inferred as the concentrations of Fe and Ni must sum\nto unity on all lattice sites.\nTo compute the MAE of our considered structures, we use\nthe MARMOT code45, which solves the single-site scatter-\ning problem on a fully relativistic footing, naturally including\nspin-orbit effects. The crystal structure for the MARMOT cal-\nculations is the same as that used for the HUTSEPOT calcula-\ntions, and the angular momentum expansion was again trun-\ncated at lmax=3, while the remaining input parameters were\nleft at the MARMOT default values. This includes a 240 ×40\nmesh for angular sampling of the CPA integral and an adaptive\nscheme for Brilluoin zone integrations. MARMOT is used for\nboth the zero and finite temperature calculations.\nIII. RESULTS\nA. Functional Form of the Anisotropy: K1,K2, and K3\nTo explore the functional form of the magnetocrystalline\nanisotropy we evaluate the magnetic torque over a range of\nvalues of θandφand fit the ab initio data to the form given\nin Eqs. 4 and 5. Displayed in Table II are our obtained val-\nues of K1,K2, and K3for Fe 0.5Ni0.5and Fe 0.56Ni0.44atT=0\nK with the perfect atomic order parameter S=1.0. NotablyComposition Anisotropy Constants (MJm−3)\nK1 K2 K3 KU\nFe0.5Ni0.5 0.9582 −0.0008 0.0001 0.96\nFe0.56Ni0.44 0.6122 −0.0003 0.0006 0.61\nTABLE II. Computed values of K1,K2,K3, and KUfor Fe 0.5Ni0.5\nand Fe 0.56Ni0.44with atomic order parameter S=1 and temperature\nT=0 K. It can be seen that K1is orders of magnitude larger than K2\norK3for both compositions.\nK1is almost four orders of magnitude larger than K2andK3.\nWe find similar results for calculations performed on systems\nwith imperfect atomic order and at finite temperature; K1is al-\nways at least two orders of magnitude greater than K2andK3.\nWe note that Ref 28 found a similar outcome in their compu-\ntational study for perfectly ordered ( S=1) L1 0Fe0.5Ni0.5at\nT=0 K; the leading-order anisotropy constant K1dominated\nhigher-order constants K2andK3. We are able to extend this\nresult and find that the dominance of K1over K2andK3holds\nfor systems modelled with imperfect atomic order and at fi-\nnite temperature. This outcome confirms that the MAE values\nof ideal single crystals modelled in our calculations are well-\ndescribed by the standard formula for a uniaxial ferromagnet\nand, for the remainder of this paper, the torque will always be\nevaluated at the positions θ=π/4,φ=0 to directly obtain\nthe uniaxial anisotropy constant K1=KU.\nWe calculate K1for maximally ordered (S=1) L1 0\nFe0.5Ni0.5atT=0 K to be 0.96 MJm−3, while for Fe 0.56Ni0.44\nwe calculate it to be 0.61 MJm−3. Our calculated MAE for\nthe equiatomic Fe 0.5Ni0.5composition is broadly consistent\nwith values of 0.23–0.78 MJm−3that were previously ob-\ntained using a variety of other computational techniques25–31.\nHowever, we note that these values appear to be low in com-\nparison with experimentally determined ones10–19,22,24,33of\n0.2–0.93 MJm−3, especially given that these measurements\nwere performed on samples with imperfect atomic order and\nat room temperature. As noted in the introduction, analysis\nof the experimental data8,10,27has suggested that the magne-\ntocrystalline anisotropy energy of L1 0-type FeNi could reach\nthe range 1.0-1.3 MJm−3. Since ab initio computational esti-\nmates of the MAE for L1 0transition metal systems are usually\nin good agreement with experimental data and, if anything,\ntend to overestimate the anisotropy constants, this computa-\ntional underestimate stands out.\nB. Effects of finite temperature and imperfect atomic order\nTo make the comparison between theory and experiment\nmore direct we incorporate the effects of temperature and im-\nperfect atomic order into the computational modelling. We\ncan thus simulate the MAE of a single-variant FeNi crystal\nat room temperature with the atomic order parameter as fur-\nnished in the experimental reports (Table I).\nFor these finite-temperature calculations, we use the disor-\ndered local moment (DLM) description of the magnetocrys-6\n0 250 500 750 1000\nT(K)0.00.20.40.60.81.0K1(MJm−3)\nL10Fe0.5Ni0.5\nK1\n0 250 500 750 1000\nT(K)0.00.20.40.60.81.0K1(MJm−3)\nL10Fe0.56Ni0.44\nK1\n0123\nM (µB)\nM\n0123\nM (µB)\nM\nFIG. 2. Anisotropy constant K1and magnetisation Mcalculated as a function of temperature for a perfectly ordered crystal of L1 0Fe0.5Ni0.5\n(left) and for a maximally ordered crystal of L1 0Fe0.56Ni0.44(right). It can be seen that, in both cases, the predicted finite-temperature\nanisotropy remains high up to and beyond room temperature. Neither case obeys a simple power law of K1∝Mα. Notably, for the Fe-rich\ncomposition, the excess Fe on the Ni sublattice magnetically disorders very rapidly, indicating that the magnetocrystalline anisotropy of this\ncomposition at T=300 K is actually higher than its zero-temperature value.\ntalline anisotropy at finite temperature, as outlined in Sec.II.\nThis method has been used with fidelity in the past to study\nother ferromagnetic L1 0-structured materials34,37as well as\nrare-earth-based permanent magnets51,52.\nShown in in the left panel of Fig. 2 is a plot of K1for max-\nimally ordered L1 0Fe0.5Ni0.5as a function of temperature,\nalong with a plot of the total magnetisation, M, of the two-\natom unit cell. It can be seen that, as expected, the anisotropy\ndecreases as temperature increases, but that the decrease is\nvery gradual, with the anisotropy energy at T=300 K com-\nputed as 0.889 MJm−3, around 90% of the T=0 value. This\nrobust finite-temperature performance is intriguing; it is clear\nthat the anisotropy energy does not obey a simple power law\n(K1∝Mα) as it does in other L1 0materials such as FePt34.\nThis aspect was studied by Yamashita and Sakuma63,64, who\nused an atomistic spin model to find similar unusual finite\ntemperature behaviour in L1 0FeNi.\nProceeding, in the right panel of Fig. 2 is a plot of the\nsame quantities for the Fe-rich composition in its maximally\nordered state, with the 1a site having an occupancy of pure Fe,\nwhile the 1d site has occupancy Ni 0.88Fe0.12. As we increase\nthe temperature away from zero, there is a sharp jump in KU\nand an associated decrease in M, resulting in a predicted KU\nvalue at finite temperature which is only marginally lower than\nfor the equiatomic composition. This sharp increase in KU\nfor the Fe-rich composition is associated with a rapid mag-\nnetic disordering of the excess Fe on the 1d lattice site in\nour calculations. This excess Fe, which makes a significant\nnegative contribution to the total magnetic torque in the zero-\ntemperature calculations, makes a much smaller contribution\nto the torque once it has magnetically disordered, resulting in\nan increase in the predicted KUvalue for this composition.\nThe rapid magnetic disordering is understood by considering\nthe underlying face-centered tetragonal lattice. Pure Fe on an\nfcc lattice with this lattice spacing exhibits competition be-tween ferromagnetic and antiferromagnetic ordering65, and in\nthis system this competition is manifest by a rapid disordering\nof the excess Fe on the Ni sites.\nOur computed Curie temperature of L1 0Fe0.5Ni0.5is 1025\nK, in reasonable agreement with experimental measurements\non the material that measured a Curie temperature of at least\n830 K8. We note that our Curie temperature determination is\nobtained within a mean-field theory and is therefore expected\nto be an overestimate of its true value. The Curie temper-\nature of 1220 K obtained by Yamashita and Sakuma63,64,\nusing a perturbative approach to treat spin-orbit coupling, is\nhigher than the value we obtain. For the Fe-rich composition,\na marginally reduced Curie temperature of 935 K is predicted\nwithin our model. For both compositions, these Curie temper-\natures are theoretical estimates, as the material is expected to\natomically disorder below this temperature10,66.\nA comment should also be made about magnetic moments\npredicted in our modelling. For reference, we provide the\nmagnetic moments at T=0 K for an atomic order parame-\nter of S=1 for both of the studied compositions in Table III.\nThe total magnetisation of the system compares well with ex-\nperimental data, for example with the magnetisation of just\nCompositionMagnetic Moment ( µB)\nTotal Fe (1 a) Ni (1 d) Fe(1 d)\nFe0.5Ni0.5 3.24 2.62 0.62\nFe0.56Ni0.44 3.43 2.60 0.63 2.26\nTABLE III. Magnetic moments associated with Fe and Ni L1 0sys-\ntems considered in this work, at T=0 K and with an atomic order\nparameter of S=1. We also give the total magnetisation of the two-\natom unit cell. The Fe-rich composition has excess Fe sitting on the\n1dsite.7\n0.00 0.25 0.50 0.75 1.00\nAtomic Order Parameter, S0.00.20.40.60.81.0K1(MJm−3)\nT= 0 K\nFe0.5Ni0.5\nFe0.56Ni0.44\nFIG. 3. MAE as a function of atomic long-range order parameter, S,\nfor L1 0Fe-Ni for both both the equatomic and Fe-rich compositions,\nevaluated at T=0 K. For both cases, decreasing long-range order\ndecreases the predicted MAE, although the drop is sharper for the\nequiatomic composition than for the Fe-rich case.\nover 1.6 µB/atom for an equiatomic composition measured in\nRef. 19.\nWe now move on to considering the effects of imperfect\natomic ordering on the predicted MAE. In Fig. 3, we show\nthe zero temperature MAE of both compositions as a function\nof atomic long-range order parameter, S. Decreasing atomic\nlong-range order consistently decreases the calculated MAE\nfor both compositions. This behavior is expected based on\nresults obtained for other L1 0compounds, such as FePt and\nFePd, from both experimental36and computational35studies.\nWe note that recent work by Izardar et al.29, which used su-\npercell calculations to represent partially ordered L1 0FeNi\nstructures, concluded that the MAE for S=0.75 was com-\nparable to that obtained from perfect S=1.0, with a value\naround 0.54 MJm−3. This outcome is contrary to our results\nwhich show a sizeable MAE decrease with decreased atomic\norder. However, Izardar et al. utilized only 32 atoms in each\nsupercell—a relatively small number—and obtained a large\nspread in calculated MAE values for the specific supercell\nconfigurations used to represent a particular order parameter.\nC. Comparison with experimental data\nTo make as close as possible a comparison with the exist-\ning experimental data, we now provide results for calculations\nincluding the effects of both imperfect atomic long-range or-\nder and finite temperature. Visualised in Fig. 4 is a plot of\nK1against Sat a simulated temperature of T=300 K. When\nthe results of the equiatomic composition, Fe 0.5Ni0.5, are con-\nsidered, it can be seen that K1is only fractionally decreased\nat 300 K compared to its value at 0 K, lending support for\nthis material’s potential use in applications with high oper-\nating temperatures. For the Fe-rich composition, there is a\nmarked increase in K1across the range of Svalues comparedto the data obtained at T=0 K, particularly at larger values of\nS, where the system is close to being fully atomically ordered.\nThis increase is associated with the unusual finite temperature\nbehaviour visualised in the right-hand panel of Fig. 2, associ-\nated with the excess Fe on the 1d lattice site rapidly disorder-\ning with increasing temperature.\nWe now use the calculated temperature and atomic long-\nrange order dependence of the MAE to make a direct com-\nparison with the experimental values for FeNi summarised in\nTable I. Considering first the total uniaxial anisotropy, KU, it\nappears that the calculated values are significantly lower than\nthe experimental ones. For example, the experimental deter-\nmination of Ito et al.22ofKU=0.55 MJm−3for a single-\nvariant thin film of L1 0Fe0.5Ni0.5with S=0.6 atT=293 K\nmay be contrasted with with our calculated anisotropy coeffi-\ncients, for the same conditions, of K1=0.3616, K2=0.0005,\nK3=−0.0001 MJm−3that provide a uniaxial anisotropy con-\nstant of KU=0.36 MJm−3, a markedly smaller value.\nTable I’s summary of experimental data is obtained from\na variety of sources. We note that not all references disclose\nthe atomic order parameter Sfor the sample, making a direct\ncomparison with our results challenging. In addition, only two\nreferences by Neél and co-workers10,11resolve the measured\nanisotropy, KUinto its separate components, K1,K2,K3, and\nof these two references only one10provides an atomic long-\nrange order parameter in addition to the measured anisotropy\ncoefficients.\nWhen we consider the resolution of the total uniaxial\nanisotropy into its separate coefficients, K1,K2, and K3, the\norigin of the discrepancy becomes clear. For example, the\nexperimental value of K1= 0.32 MJm−3reported by Néel and\n0.00 0.25 0.50 0.75 1.00\nAtomic Order Parameter, S0.00.20.40.60.81.0K1(MJm−3)\nT= 300 K\nFe0.5Ni0.5\nFe0.56Ni0.44\nFIG. 4. MAE as a function of atomic long-range order parameter, S,\nfor L1 0Fe-Ni for both both the equatomic and Fe-rich compositions,\nevaluated at T=300 K. As for the calculations at T=0 K, decreas-\ning atomic order impacts the predicted anisotropy of the system. The\npredicted values of K1for Fe 0.5Ni0.5are only slightly decreased com-\npared to their value at T=0 K, while for the Fe-rich composition the\nanisotropy at T=300 K is increased compared to its zero tempera-\nture value, due to the rapid magnetic disordering of the excess Fe on\nthe Ni sublattice.8\ncoworkers in Ref. 10 for a specimen of estimated order param-\neterS=0.41–0.45 at T=293 K is in close alignment with our\nresults, as shown in Fig. 4. However, Ref. 10 also reports ex-\nperimental determinations of large values of K2andK3, which\nwe do not find. These large values are unexpected for a uniax-\nial ferromagnetic material, an assessment noted by the authors\nof the original study. Indeed one of the authors, Néel, devel-\noped a theory to account for the unexpectedly large K2andK3\nvalues in these nanocrystalline samples10,11.\nThe picture described by Néel is of a polycrystalline sam-\nple consisting of individual nanocrystals with L1 0atomic lay-\nering directions (determining the magnetic easy axis) that are\nnot all aligned along a common direction. (Note that there are\nthree possible directional variants of the L1 0structure, with\nlayering perpendicular to ˆx,ˆy, orˆzdirections.) Néel proposes\nthat these nanocrystals are smaller than the size of magnetic\ndomains within the sample, with a suggested nanocrystal size\nof approximately 150 Å. It is concluded that exchange cou-\npling between nanocrystals can lead to finite macroscopic val-\nues of K2andK3, even though individual nanocrystals them-\nselves are assumed to have a negligible value of these coef-\nficients. It is known that meteoritic tetrataenite samples have\nnanoscale structure41. In addition, laboratory-grown films of-\nten contain step edges, dislocations and other structural imper-\nfections which may lead to similar exchange-coupling mecha-\nnisms that could impact the measured anisotropy. We empha-\nsise that these large K2andK3values, which are not captured\nwithin first-principles calculations, appear to make a signifi-\ncant contribution to the total uniaxial anisotropy, KU, of this\nsystem.\nNéel’s argument is predicated on his assumption that the\napplication of a magnetic field along the ˆxdirection during\nthe annealing process will favor L1 0layering along that same\ndirection. To quantify this effect, we use our computational\nformalism for modelling atomic arrangements in alloys67–69\nin which relativistic effects and, therefore spin-orbit coupling,\ncan be included34. We find that, for a sample magnetised\nalong the ˆxdirection, the predicted ordering is indeed an L1 0-\ntype layering along the ˆxdirection, with an ordering temper-\nature of 508 K. However, the difference in energy between\nlayering along ˆxdirection and layering along ˆyorˆzdirec-\ntions is only 0.7 MJm−3, comparable to our calculated mag-\nnetocrystalline anisotropy energy constant K1. This small but\ndetectable difference supports Néel’s hypothesis that samples\nannealed with an applied field parallel to the ˆxdirection will\nproduce crystallites with L1 0layering along all three crystal\naxes, but layering along the ˆxdirection will be favoured over\nthat along the other two directions. The hypothesis that an in-\nhomogeneous, nanostructured sample can yield unexpectedly\nlarge higher-order anisotropy coefficients is also supported by\nother authors, for example, by recent computational work on\ninhomogeneous FePt using atomistic spin modelling70.\nWe suggest, therefore, that more data from both experimen-\ntal measurements and computational modelling are required\nto resolve this intriguing apparent discrepancy between the-\nory and experiment. It may be that a mechanism, similar to\nthat outlined by Néel, is unrecognised but at play in a num-\nber of the experimental datasets. Further experimental workthat resolves the magnetocrystalline anisotropy into its sepa-\nrateK1,K2, and K3components will enable definitive conclu-\nsions to be drawn about this disparity. In addition, atomistic\nand micromagnetic modelling are anticipated to provide fur-\nther insight into the proposed exchange coupling mechanism\nand attendant magnetic properties of the multivariant geome-\ntry described by Néel.\nIV. CONCLUSIONS\nIn summary, we have carried out an ab initio computational\nstudy of the magnetocrystalline anisotropy energy of two L1 0-\ntype FeNi compositions, equiatomic Fe 0.5Ni0.5and Fe-rich\nFe0.56Ni0.44. This rare-earth-free compound is of interest as\na sustainable advanced permanent magnet material. We con-\nfirm that the modeled magnetic torque of ideal single crystals\nis well-described by the standard formula for a uniaxial ferro-\nmagnet, with the leading-order magnetocrystalline anisotropy\nenergy coefficient K1dominating the higher-order coefficients\nK2andK3. This dominance persists to finite temperature\nand is also predicted in samples with imperfect atomic order.\nMoreover we find that a significant MAE persists to high tem-\nperatures, affirming this material’s potential as a useful ‘gap’\nmagnet.\nTo compare computational outcomes as faithfully as possi-\nble with experimental ones, we have included the effects of\nboth finite temperature and imperfect atomic order on the pre-\ndicted magnetocrystalline anisotropy. It is confirmed that both\nincreasing temperature and decreasing atomic order decrease\nthe MAE of this compound. We have made a detailed compar-\nison of our computed anisotropy coefficients with those of the\nexperimental literature and find that our calculated values of\nK1appear to be consistent with the experimental data. How-\never, the experimental reports from Néel et al.10and Paulevé\net al.11detail significant K2andK3values, which we do not\nfind.\nWe suggest that the noted discrepancy between computa-\ntional and experimental literature values of the uniaxial mag-\nnetocrystalline anistropy constant KUhas its origins in these\nlarge K2andK3values observed in the early experiments.\nThese values significantly contribute to the experimentally ob-\nserved uniaxial magnetocrystalline anisotropy constant, KU. It\nis proposed that these large K2andK3values have their ori-\ngins in the polycrystalline, multivariant nature of experimental\nsamples.\nFurther experimental work should seek to resolve separate\nvalues for K1,K2, and K3at both very low and at finite temper-\natures to better understand the magnetocrystalline anisotropy\nof this material. Further computational/theoretical work could\nfocus on micromagnetic or atomistic spin modelling to bet-\nter understand the model proposed by Néel10and its conse-\nquences.9\nACKNOWLEDGMENTS\nWe gratefully acknowledge the support of the UK Engi-\nneering and Physical Sciences Research Council, Grant No.\nEP/W021331/1. C.D.W. is supported by a studentship within\nthe UK Engineering and Physical Sciences Research Council-\nsupported Centre for Doctoral Training in Modelling of Het-\nerogeneous Systems, Grant No. EP/S022848/1. This work\nwas also supported in by the U.S. Department of Energy,\nOffice of Basic Energy Sciences under Award Number DE\nSC0022168 (for atomic insight) and by the U.S. National\nScience Foundation under Award ID 2118164 (for advanced\nmanufacturing aspects).\nAUTHOR DECLARATIONS\nJournal Submission\nThis article has been submitted to the Journal of Applied\nPhysics.\nConflict of Interest\nThe authors have no conflicts to disclose.\nAuthor Contributions\nChristopher D. Woodgate: Conceptualization (equal);\nData curation (lead); Formal analysis (lead); Investigation\n(lead); Methodology (equal); Validation (lead); Visualisa-\ntion (lead); Writing – original draft (lead); Writing – re-\nview & editing (lead). Christopher E. Patrick: Soft-\nware (supporting); Writing – review & editing (supporting).\nLaura H. Lewis: Conceptualization (equal); Funding Ac-\nquisition (equal); Project administration (equal); Supervision\n(equal); Writing – review & editing (supporting). Julie B.\nStaunton: Conceptualization (equal); Funding Acquisition\n(equal); Methodology (equal); Project administration (equal);\nResources (lead); Supervision (equal); Writing – review &\nediting (supporting).\nDATA AVAILABILITY STATEMENT\nThe data that support the findings of this study are available\nfrom the corresponding author upon reasonable request.\n1J. Coey, “Perspective and Prospects for Rare Earth Permanent Magnets,”\nEngineering 6, 119–131 (2020).\n2K. Strnat, G. Hoffer, J. Olson, W. Ostertag, and J. J. Becker, “A Fam-\nily of New Cobalt-Base Permanent Magnet Materials,” Journal of Applied\nPhysics 38, 1001–1002 (1967).\n3M. Sagawa, S. Fujimura, N. Togawa, H. Yamamoto, and Y . Matsuura,\n“New material for permanent magnets on a base of Nd and Fe (invited),”\nJournal of Applied Physics 55, 2083–2087 (1984).4J. J. Croat, J. F. Herbst, R. W. Lee, and F. E. Pinkerton, “Pr-Fe and Nd-\nFe-based materials: A new class of high-performance permanent magnets\n(invited),” Journal of Applied Physics 55, 2078–2082 (1984).\n5K. Smith Stegen, “Heavy rare earths, permanent magnets, and renewable\nenergies: An imminent crisis,” Energy Policy 79, 1–8 (2015).\n6R. McCallum, L. Lewis, R. Skomski, M. Kramer, and I. Anderson, “Prac-\ntical Aspects of Modern and Future Permanent Magnets,” Annual Review\nof Materials Research 44, 451–477 (2014).\n7J. Bai, X. Xu, Y . Duan, G. Zhang, Z. Wang, L. Wang, and C. Zheng,\n“Evaluation of resource and environmental carrying capacity in rare earth\nmining areas in China,” Scientific Reports 12, 6105 (2022).\n8L. H. Lewis, A. Mubarok, E. Poirier, N. Bordeaux, P. Manchanda,\nA. Kashyap, R. Skomski, J. Goldstein, F. E. Pinkerton, R. K. Mishra, R. C.\nKubic Jr, and K. Barmak, “Inspired by nature: investigating tetrataenite for\npermanent magnet applications,” Journal of Physics: Condensed Matter 26,\n064213 (2014).\n9R. S. Clarke and E. R. D. Scott, “Tetrataenite—ordered FeNi, a new mineral\nin meteorites,” American Mineralogist 65, 624–630 (1980).\n10L. Néel, J. Pauleve, R. Pauthenet, J. Laugier, and D. Dautreppe, “Magnetic\nProperties of an Iron—Nickel Single Crystal Ordered by Neutron Bom-\nbardment,” Journal of Applied Physics 35, 873–876 (1964).\n11J. Paulevé, A. Chamberod, K. Krebs, and A. Bourret, “Magnetization\nCurves of Fe–Ni (50–50) Single Crystals Ordered by Neutron Irradiation\nwith an Applied Magnetic Field,” Journal of Applied Physics 39, 989–990\n(1968).\n12T. Shima, M. Okamura, S. Mitani, and K. Takanashi, “Structure and\nmagnetic properties for L10-ordered FeNi films prepared by alternate\nmonatomic layer deposition,” Journal of Magnetism and Magnetic Mate-\nrials310, 2213–2214 (2007).\n13M. Mizuguchi, T. Kojima, M. Kotsugi, T. Koganezawa, K. Osaka, and\nK. Takanashi, “Artificial Fabrication and Order Parameter Estimation of\nL10-ordered FeNi Thin Film Grown on a AuNi Buffer Layer,” Journal of\nthe Magnetics Society of Japan 35, 370–373 (2011).\n14T. Kojima, M. Mizuguchi, and K. Takanashi, “L1 0-ordered FeNi film\ngrown on Cu-Ni binary buffer layer,” Journal of Physics: Conference Series\n266, 012119 (2011).\n15T. Kojima, M. Mizuguchi, T. Koganezawa, K. Osaka, M. Kotsugi, and\nK. Takanashi, “Magnetic Anisotropy and Chemical Order of Artificially\nSynthesized L1 0-Ordered FeNi Films on Au–Cu–Ni Buffer Layers,”\nJapanese Journal of Applied Physics 51, 010204 (2012).\n16T. Kojima, M. Ogiwara, M. Mizuguchi, M. Kotsugi, T. Koganezawa,\nT. Ohtsuki, T.-Y . Tashiro, and K. Takanashi, “Fe–Ni composition depen-\ndence of magnetic anisotropy in artificially fabricated L1 0-ordered FeNi\nfilms,” Journal of Physics: Condensed Matter 26, 064207 (2014).\n17A. Makino, P. Sharma, K. Sato, A. Takeuchi, Y . Zhang, and K. Takenaka,\n“Artificially produced rare-earth free cosmic magnet,” Scientific Reports 5,\n16627 (2015).\n18A. Frisk, B. Lindgren, S. D. Pappas, E. Johansson, and G. Andersson, “Res-\nonant x-ray diffraction revealing chemical disorder in sputtered L1 0FeNi\non Si(0 0 1),” Journal of Physics: Condensed Matter 28, 406002 (2016).\n19A. Frisk, T. P. A. Hase, P. Svedlindh, E. Johansson, and G. Andersson,\n“Strain engineering for controlled growth of thin-film FeNi L1 0,” Journal\nof Physics D: Applied Physics 50, 085009 (2017).\n20S. Goto, H. Kura, E. Watanabe, Y . Hayashi, H. Yanagihara, Y . Shimada,\nM. Mizuguchi, K. Takanashi, and E. Kita, “Synthesis of single-phase L10-\nFeNi magnet powder by nitrogen insertion and topotactic extraction,” Sci-\nentific Reports 7, 13216 (2017).\n21N. Maât, I. McDonald, R. Barua, B. Lejeune, X. Zhang, G. Stephen,\nA. Fisher, D. Heiman, I. Soldatov, R. Schäfer, and L. Lewis, “Creating,\nprobing and confirming tetragonality in bulk FeNi alloys,” Acta Materialia\n196, 776–789 (2020).\n22K. Ito, T. Ichimura, M. Hayashida, T. Nishio, S. Goto, H. Kura, R. Sasaki,\nM. Tsujikawa, M. Shirai, T. Koganezawa, M. Mizuguchi, Y . Shimada, T. J.\nKonno, H. Yanagihara, and K. Takanashi, “Fabrication of L10-ordered\nFeNi films by denitriding FeNiN(001) and FeNiN(110) films,” Journal of\nAlloys and Compounds 946, 169450 (2023).\n23Y . P. Ivanov, B. Sarac, S. V . Ketov, J. Eckert, and A. L. Greer, “Direct For-\nmation of Hard-Magnetic Tetrataenite in Bulk Alloy Castings,” Advanced\nScience 10, 2204315 (2023).10\n24E. Poirier, F. E. Pinkerton, R. Kubic, R. K. Mishra, N. Bordeaux,\nA. Mubarok, L. H. Lewis, J. I. Goldstein, R. Skomski, and K. Barmak,\n“Intrinsic magnetic properties of L1 0FeNi obtained from meteorite NWA\n6259,” Journal of Applied Physics 117, 17E318 (2015).\n25Y . Miura, S. Ozaki, Y . Kuwahara, M. Tsujikawa, K. Abe, and M. Shi-\nrai, “The origin of perpendicular magneto-crystalline anisotropy in L1 0\n–FeNi under tetragonal distortion,” Journal of Physics: Condensed Matter\n25, 106005 (2013).\n26A. Edström, J. Chico, A. Jakobsson, A. Bergman, and J. Rusz, “Electronic\nstructure and magnetic properties of L 1 0 binary alloys,” Physical Review\nB90, 014402 (2014).\n27L. H. Lewis, F. E. Pinkerton, N. Bordeaux, A. Mubarok, E. Poirier, J. I.\nGoldstein, R. Skomski, and K. Barmak, “De Magnete et Meteorite: Cos-\nmically Motivated Materials,” IEEE Magnetics Letters 5, 1–4 (2014).\n28M. Werwi ´nski and W. Marciniak, “ Ab initio study of magnetocrystalline\nanisotropy, magnetostriction, and Fermi surface of L1 0FeNi (tetrataenite),”\nJournal of Physics D: Applied Physics 50, 495008 (2017).\n29A. Izardar and C. Ederer, “Interplay between chemical order and magnetic\nproperties in L 1 0 FeNi (tetrataenite): A first-principles study,” Physical\nReview Materials 4, 054418 (2020).\n30D. Tuvshin, T. Ochirkhuyag, S. C. Hong, and D. Odkhuu, “First-principles\nprediction of rare-earth free permanent magnet: FeNi with enhanced mag-\nnetic anisotropy and stability through interstitial boron,” AIP Advances 11,\n015138 (2021).\n31M. Si, A. Izardar, and C. Ederer, “Effect of chemical disorder on the mag-\nnetic anisotropy in L 1 0 FeNi from first-principles calculations,” Physical\nReview Research 4, 033161 (2022).\n32J. Coey, “Permanent magnets: Plugging the gap,” Scripta Materialia 67,\n524–529 (2012).\n33M. Kotsugi, H. Maruyama, N. Ishimatsu, N. Kawamura, M. Suzuki,\nM. Mizumaki, K. Osaka, T. Matsumoto, T. Ohkochi, T. Ohtsuki, T. Ko-\njima, M. Mizuguchi, K. Takanashi, and Y . Watanabe, “Structural, mag-\nnetic and electronic state characterization of L1 0-type ordered FeNi alloy\nextracted from a natural meteorite,” Journal of Physics: Condensed Matter\n26, 064206 (2014).\n34J. B. Staunton, S. Ostanin, S. S. A. Razee, B. L. Gyorffy, L. Szun-\nyogh, B. Ginatempo, and E. Bruno, “Temperature Dependent Magnetic\nAnisotropy in Metallic Magnets from an Ab Initio Electronic Structure The-\nory: L 1 0 -Ordered FePt,” Physical Review Letters 93, 257204 (2004).\n35J. B. Staunton, S. Ostanin, S. S. A. Razee, B. Gyorffy, L. Szunyogh, B. Gi-\nnatempo, and E. Bruno, “Long-range chemical order effects upon the mag-\nnetic anisotropy of FePt alloys from an ab initio electronic structure theory,”\nJournal of Physics: Condensed Matter 16, S5623–S5631 (2004).\n36S. Okamoto, N. Kikuchi, O. Kitakami, T. Miyazaki, Y . Shimada, and\nK. Fukamichi, “Chemical-order-dependent magnetic anisotropy and ex-\nchange stiffness constant of FePt (001) epitaxial films,” Physical Review\nB66, 024413 (2002).\n37J. B. Staunton, L. Szunyogh, A. Buruzs, B. L. Gyorffy, S. Ostanin, and\nL. Udvardi, “Temperature dependence of magnetic anisotropy: An ab initio\napproach,” Physical Review B 74, 144411 (2006).\n38A. Sakuma, “First Principle Calculation of the Magnetocrystalline\nAnisotropy Energy of FePt and CoPt Ordered Alloys,” Journal of the Phys-\nical Society of Japan 63, 3053–3058 (1994).\n39L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of\nContinuous Media , 2nd ed., Course of Theoretical Physics No. 8 (Elsevier\nButterworth-Heinemann, Amsterdam Heidelberg, 2009).\n40P. Wasilewski, “Magnetic characterization of the new magnetic mineral\ntetrataenite and its contrast with isochemical taenite,” Physics of the Earth\nand Planetary Interiors 52, 150–158 (1988).\n41A. Kovács, L. H. Lewis, D. Palanisamy, T. Denneulin, A. Schwedt, E. R.\nScott, B. Gault, D. Raabe, R. E. Dunin-Borkowski, and M. Charilaou,\n“Discovery and Implications of Hidden Atomic-Scale Structure in a Metal-\nlic Meteorite,” Nano Letters 21, 8135–8142 (2021).\n42X. Wang, R. Wu, D.-s. Wang, and A. J. Freeman, “Torque method for\nthe theoretical determination of magnetocrystalline anisotropy,” Physical\nReview B 54, 61–64 (1996).\n43J. S. Faulkner, G. M. Stocks, and Y . Wang, Multiple scattering theory elec-\ntronic structure of solids , 1st ed. (IOP Publishing, Bristol, UK, 2018).\n44B. L. Gyorffy, A. J. Pindor, J. Staunton, G. M. Stocks, and H. Winter, “A\nfirst-principles theory of ferromagnetic phase transitions in metals,” Journalof Physics F: Metal Physics 15, 1337–1386 (1985).\n45C. E. Patrick and J. B. Staunton, “MARMOT: magnetism, anisotropy, and\nmore, using the relativistic disordered local moment picture at finite tem-\nperature,” Electronic Structure 4, 017001 (2022).\n46S. Ouazi, S. Vlaic, S. Rusponi, G. Moulas, P. Buluschek, K. Halleux,\nS. Bornemann, S. Mankovsky, J. Minár, J. Staunton, H. Ebert, and\nH. Brune, “Atomic-scale engineering of magnetic anisotropy of nanostruc-\ntures through interfaces and interlines,” Nature Communications 3, 1313\n(2012).\n47P. Lloyd, “Wave propagation through an assembly of spheres: II. The den-\nsity of single-particle eigenstates,” Proceedings of the Physical Society 90,\n207–216 (1967).\n48P. Strange, J. Staunton, and B. L. Gyorffy, “Relativistic spin-polarised scat-\ntering theory-solution of the single-site problem,” Journal of Physics C:\nSolid State Physics 17, 3355–3371 (1984).\n49H. Ebert, D. Ködderitzsch, and J. Minár, “Calculating condensed matter\nproperties using the KKR-Green’s function method—recent developments\nand applications,” Reports on Progress in Physics 74, 096501 (2011).\n50B. L. Gyorffy and M. J. Stott, Band Structure Spectroscopy of Metals and\nAlloys (Academic Press, London; New York, 1973).\n51C. E. Patrick and J. B. Staunton, “Temperature-dependent magnetocrys-\ntalline anisotropy of rare earth/transition metal permanent magnets from\nfirst principles: The light R Co 5 ( R = Y , La-Gd ) intermetallics,” Physical\nReview Materials 3, 101401 (2019).\n52J. Bouaziz, C. E. Patrick, and J. B. Staunton, “Crucial role of Fe in deter-\nmining the hard magnetic properties of Nd 2 Fe 14 B,” Physical Review B\n107, L020401 (2023).\n53C. E. Patrick, “Quantifying temperature-induced magnetic disorder effects\non the anisotropy of MnBi,” Physical Review B 106, 014425 (2022).\n54M. Hoffmann, A. Ernst, W. Hergert, V . N. Antonov, W. A. Adeagbo,\nR. M. Geilhufe, and H. Ben Hamed, “Magnetic and Electronic Properties\nof Complex Oxides from First-Principles,” Physica Status Solidi (b) 257,\n1900671 (2020).\n55R. M. Martin, Electronic Structure: Basic Theory and Practical Methods\n(Cambridge University Press, Cambridge, UK, 2004).\n56G. M. Stocks, W. M. Temmerman, and B. L. Gyorffy, “Complete Solution\nof the Korringa-Kohn-Rostoker Coherent-Potential-Approximation Equa-\ntions: Cu-Ni Alloys,” Physical Review Letters 41, 339–343 (1978).\n57J. P. Perdew and Y . Wang, “Accurate and simple analytic representation of\nthe electron-gas correlation energy,” Physical Review B 45, 13244–13249\n(1992).\n58J. F. Albertsen, “Tetragonal Lattice of Tetrataenite (Ordered Fe-Ni, 50-50)\nfrom 4 Meteorites,” Physica Scripta 23, 301–306 (1981).\n59A. Montes-Arango, L. Marshall, A. Fortes, N. Bordeaux, S. Langridge,\nK. Barmak, and L. Lewis, “Discovery of process-induced tetragonality in\nequiatomic ferromagnetic FeNi,” Acta Materialia 116, 263–269 (2016).\n60T. Hoshino, M. Asato, and T. Mizuno, “Accuracy of atomic-sphere approx-\nimation in electronic-structure calculations for transition-metal systems,”\nJournal of Magnetism and Magnetic Materials 226-230 , 384–385 (2001).\n61J. S. Faulkner and G. M. Stocks, “Calculating properties with the coherent-\npotential approximation,” Physical Review B 21, 3222–3244 (1980).\n62D. D. Johnson, D. M. Nicholson, F. J. Pinski, B. L. Györffy, and G. M.\nStocks, “Total-energy and pressure calculations for random substitutional\nalloys,” Physical Review B 41, 9701–9716 (1990).\n63S. Yamashita and A. Sakuma, “First-Principles Study for Finite Tempera-\nture Magnetrocrystaline Anisotropy of L10-Type Ordered Alloys,” Journal\nof the Physical Society of Japan 91, 093703 (2022).\n64S. Yamashita and A. Sakuma, “Finite-temperature second-order perturba-\ntion analysis of magnetocrystalline anisotropy energy of L 1 0 -type ordered\nalloys,” Physical Review B 108, 054411 (2023).\n65F. J. Pinski, J. Staunton, B. L. Gyorffy, D. D. Johnson, and G. M.\nStocks, “Ferromagnetism versus Antiferromagnetism in Face-Centered-\nCubic Iron,” Physical Review Letters 56, 2096–2099 (1986).\n66E. Dos Santos, J. Gattacceca, P. Rochette, G. Fillion, and R. Scorzelli,\n“Kinetics of tetrataenite disordering,” Journal of Magnetism and Magnetic\nMaterials 375, 234–241 (2015).\n67C. D. Woodgate and J. B. Staunton, “Compositional phase stability in\nmedium-entropy and high-entropy Cantor-Wu alloys from an ab initio all-\nelectron Landau-type theory and atomistic modeling,” Physical Review B\n105, 115124 (2022).11\n68C. D. Woodgate and J. B. Staunton, “Short-range order and compositional\nphase stability in refractory high-entropy alloys via first-principles theory\nand atomistic modeling: NbMoTa, NbMoTaW, and VNbMoTaW,” Physical\nReview Materials 7, 013801 (2023).69C. D. Woodgate, D. Hedlund, L. H. Lewis, and J. B. Staunton, “Interplay\nbetween magnetism and short-range order in medium- and high-entropy\nalloys: CrCoNi, CrFeCoNi, and CrMnFeCoNi,” Physical Review Materials\n7, 053801 (2023).\n70N. T. Binh, S. Jenkins, S. Ruta, R. F. L. Evans, and R. W. Chantrell,\n“Higher-order magnetic anisotropy in soft-hard magnetic materials,” Phys-\nical Review B 107, L041410 (2023)."    },    {        "title": "2308.04594v2.Giant_magnetic_and_optical_anisotropy_in_cerium_substituted_M_type_strontium_hexaferrite_driven_by_4_f__electrons.pdf",        "content": "arXiv:2308.04594v2  [cond-mat.mtrl-sci]  17 Aug 2023Giant magnetic and optical anisotropy in cerium-substitut ed M-type strontium\nhexaferrite driven by 4 felectrons\nChurna Bhandari1∗and Durga Paudyal1,2\n1The Critical Material Institute, Ames National Laboratory , Iowa State University, Ames, IA 50011, USA\n2Department of Electrical and Computer Engineering, Iowa St ate University, Ames, Iowa 50011, USA\nBy performing density functional calculations, we find a gia nt magnetocrystalline anisotropy\n(MCA) constant in abundant element cerium (Ce) substituted M-type hexaferrite, in the energeti-\ncally favorable strontium site, assisted by a quantum confin ed electron transfer from Ce to specific\niron (2a) site. Remarkably, the calculated electronic stru cture shows that the electron transfer\nleads to the formation of Ce3+and Fe2+at the2asite producing an occupied Ce( 4f1) state be-\nlow the Fermi level that adds a significant contribution to MC A and magnetic moment. A half\nCe-substitution forms a metallic state, while a full substi tution retains the semiconducting state\nof the strontium-hexaferrite (host). In the latter, the ban d gap is reduced due to the formation\nof charge transferred states in the gap region of the host. Th e optical absorption coefficient shows\nan enhanced anisotropy between light polarization in paral lel and perpendicular directions. Calcu-\nlated formation energies, including the analysis of probab le competing phases, and elastic constants\nconfirm that both compositions are chemically and mechanica lly stable. With successful synthesis,\nthe Ce-hexaferrite can be a new high-performing critical-e lement-free permanent magnet material\nadapted for use in devices such as automotive traction drive motors.\nI. INTRODUCTION\nThe increasing demand for high-performance perma-\nnent magnets for higher energy efficiency miniaturized\ndevices with the limited global supply of critical rare-\nearth elements has urged the magnetism community to\ndiscover an alternative to critical rare-earth-based mag-\nnets. Although the hexaferrites are useful critical rare-\nelement-free permanent magnets, the low value of max-\nimum energy product (BHmax)[1] limits their usage for\nhigh-performance magnetic devices. Continuous theo-\nretical and experimental efforts have been made to im-\nprove their intrinsic magnetic properties. In particular,\nan electron doping via 5d-element La substitution is a\npromising approach in uplifting the magnetocrystalline\nanisotropy (MCA) constant ( K1) ofSrFe12O19[2, 3], al-\nbeit it reduces the net magnetic moment. The magnetic\nrare-earth elements are desirable over La because a large\nspin-orbit coupling (SOC) in 4f-states changes the MCA\ndrastically, as seen in other rare-earth-based permanent\nmagnets[4–9] including Sm-substituted hexaferrites[10] .\nCe substitution is preferred in hexaferrite as Ce is natu-\nrally abundant and is a low-cost 4f-element. Although\na few experiments have been performed with the Ce-\nsubstitution [4–6, 11], its site preference and effect on\nstability and the electronic, magnetic, and optical prop-\nerties remain elusive.\nExperimentally, Ce-substitution has been made in\nthe Ba-hexaferrites, e.g., 20% in Ba-site and ∼5%\nin Fe-site. Although these experiments adopt a sol-\ngel approach, sample preparations differ. Mosleh et\nal.[6] used Fe(NO 3)3, Ba(NO 3)2, CeO 2, ethylenedi-\namine tetraacetic acid (EDTA(C 10H16N2O8), citric acid\n∗cbb@ameslab.gov (corresponding author)(C6H8O7), and ammonia solution as precursor to make\nBa1−xCexFe12O19(0≤x≤0.2), while Pawar et al.[11]\nused Ba(NO 3)2, Ce(NO 3)36H2O and Fe(NO 3)39H2O,\nand C 6H8O7in forming BaCe xFe12−xO19(0≤x≤0.3).\nChang et al. [5] used slightly different precursor to syn-\nthesize BaCe 0.05Fe11.95O19. These reports point to two\ncontrasting site preferences, for instance, the Sr-site in\nRef. [6] and the Fe-site in Ref. [5 and 11].\nMosleh et al.[6] found no specific trend in the magne-\ntization and coercivity by increasing Ce concentration;\nfor instance, they found an increase of magnetization for\nx= 0.1and then a decrease for x= 0.2. Pawar et al.[11]\nfound a decrease in the magnetization and Curie temper-\nature (T C) as well as an increase of coercivity with an in-\ncrease of Ce where the site occupancy is different. Chang\net al.[5] report that the tendency increases in anisotropy\nfield with Ce doping. In a separate experiment, Ban-\nihashemi et al. [12] also synthesized Sr 1−xCexFe12O19\n(x=0.0, 0.05, 0.10) utilizing hydrothermal process and\nconfirmed the formation of the nano-crystalline phase of\nthe M-type hexaferrite from x-ray diffraction (XRD) and\nFourier transform infrared spectroscopy (FTIR). They\nidentified the band gap dependency (1.37–1.51 eV) on\nthe Ce content. However, to the best of our knowledge,\nno theoretical studies are available in the literature on\nCe-substituted hexaferrite.\nIn this paper, we predict the stability and electronic,\nmagnetic, and optical properties of Ce-substituted\nstrontium hexaferrites by performing ab initio cal-\nculations with density functional theory (DFT). We\nfind the following key results. i) Both full and half\nCe-substituted compositions have negative formation\nenergies and are mechanically stable. ii) Ce donates\nan electron to Fe( 2a), leading to significant changes in\nthe electronic structure resulting in a metal for x= 0.5\nand a semiconductor for x= 1. iii) The net magnetic\nmoment and K1are improved with Ce, especially K1is2\nenhanced by one order of magnitude. iv) The computed\noptical spectra for x= 1show significant anisotropy\nbetween light polarization in parallel and perpendicular\ndirections.\nII. METHODOLOGY AND CRYSTAL\nSTRUCTURE\nWe used Vienna Simulation Package (VASP)[13, 14]\nto solve Kohn-Sham eigenvalue equations within the\nprojector augmented wave (PAW) method and Perdew-\nBurke-Ernzerhof (PBE) generalized gradient approxi-\nmation (GGA) exchange-correlation functional includ-\ning the on-site electron-correlation[15, 16]. Fully self-\nconsistent charge density was obtained using plane wave\ntotal energy cut-off 500 eV for all elements and 7×7×1\nk-mesh for the Brillouin zone integration. For GGA +\nUcalculations, including structural relaxation, we used\nDudarev’s[17] Ueff=U−J= 3and4.5eV for Ce( 4f)\nand Fe(3d) orbitals following the previous theoretical\nreports[18, 19], and Heyd–Scuseria–Ernzerhof (HSE06)\ncalculations. We employed HSE06 with Hartree-Fock\n(HF) mixing parameter, α= 0.25and the screening pa-\nrameter,w= 0.2Å−1in the electronic and optical prop-\nerties calculations[20, 21]. For the magnetic anisotropy\nconstant, we used fully self-consistent noncollinear GGA\n+U+ SOC calculations in standard 64 atom hexaferrite\nunit cell. The unit cell consists of Wyckoff’s positions:\nSr at2d, Fe at2a,2b,12k,4f1, and4f2, and O at 4f,4e,\n6h, and two 12k[19]. The full Ce-substitution at Sr 2d-\nsite retains the original crystal symmetry to P63/mmc ,\nspace group no. 194, while a half-Ce slightly lowers the\ncrystal symmetry P6m2, space group no. 187 (hexago-\nnal),including the atomic site-symmetries . The chosen\nk-mesh produces well converged MCA in M-type hexa-\nferrite structures[19, 22]\nThe optimized structural parameters are given in\nTable I. The GGA + Uoverestimates volume (Table I)\nfor experimentally known SrFe12O19[23], and a similar\ntrend is seen in the full and partial Ce-substitutions. The\nCe-substitution leads to a reduction in volume as the\natomic size of Ce3+(1.01 Å) is slightly smaller than Sr2+\n(1.12 Å). Specifically, the reduction is due to a slight\ndecrease in the out-of-plane lattice constant ( c) (Table I).\nTABLE I. DFT optimized lattice constants in Å and volume\nper unit cell, V, in Å3of Sr1−xCexFe12O19with GGA + U\ncalculations.\nx a b c c/a V\n0 5.927 5.927 23.17 3.909 704.91\n0.5 5.933 5.933 23.016 3.879 701.69\n1 5.927 5.927 22.919 3.867 697.45III. FORMATION ENERGY AND\nMECHANICAL PROPERTIES\n1. Formation energy\nNow we discuss the formation energy of CeFe12O19\n(x= 1) to examine chemical stability. The magnetic\norder of Fe sublattices remains the same as the original\nSr-hexaferrite. This is confirmed by total energy calcu-\nlations of all possible spin configurations in all Ce-doped\ncompositions. All the calculations hereafter correspond\nto the Sr-hexaferrite type magnetic state unless explicitl y\nstated. It is well-known that CeO and Fe 2O3both exist\nin crystalline form[24, 25]. We compute the formation\nenergy of CeFe12O19, using the relation\nEf(CeFe12O19) =E(CeFe12O19)−E(CeO)−6E(Fe2O3),\n(1)\nwhereE(CeFe12O19),E(CeO), and E(Fe2O3) are total\nenergies of CeFe12O19, CeO, and Fe 2O3per formula unit\n(f.u.), which yields Ef=−3.31eV per f.u. Alternatively,\nwe compute the formation energy with respect to Ce 2O3,\nFe2O3, and FeO using the expression\nEf(CeFe12O19) =1\n2[2E(CeFe12O19)−E(Ce2O3)\n−11E(Fe2O3)−2E(FeO)],(2)\nand find -2.08 eV per f.u. In both cases, CeFe12O19is\nchemically stable. Further, we compute Efrelative to\nCeO2phase as\nEf= 2E(CeFe12O19)−E(CeO2)\n−12E(Fe2O3)−E(Ce), (3)\nwhich is -3.88 eV /f.u.\nWe also examine the formation energy for x= 0.5with\nrespect to SrO, CeO, and Fe 2O3as\nEf(Sr0.5Ce0.5Fe12O19) =E(Sr0.5Ce0.5Fe12O19)\n−0.5E(CeO)−0.5E(SrO)−6E(Fe2O3)(4)\nand findEf=−2.94eV /f.u. It is higher than for\nx= 1obtained with a similar expression, meaning a half\nCe-substituted composition is chemically less stable.\nTo clarify Ce site-preference in Fe-sites, we compute\nthe self-consistent total energy for (Sr2Ce1Fe23O38,x∼\n0.04)one Ce substituted for one Fe atom in 4f2,4f1,12k,\n2b, and2asites in GGA + U, in fully relaxed geometries.\nThe computed energies are 0, 0.51, 1.03, 1.06, and 1.66\nin eV relative to the 4f2site, which indicates 4f2as the\nmost favorable Fe site. Interestingly Ce enhances the\nunit cell volume by ∼2%as compared to Sr-hexaferrite,\npresumably due to its larger ionic radius. In contrast,\nthe volume shrinks by ∼1%if Ce replaces Sr. As noted\nearlier, the Sr has a larger ionic size, in which Ce fits rela-\ntively easily, with minimal distortion of crystal structur e,\nwhile in Fe-sites, Ce creates more spacing that leads to3\nlattice expansion.\nTo further scrutinize the site preference, we compute\nthe formation energy of one Ce substituted Sr-hexaferrite\nin the lowest energy Fe site, i.e., 4f2, by using the relation\nEf(Sr2CeFe23O38) =E(Sr2CeFe23O38)−E(CeO2)\n−2E(SrO)−11E(Fe2O3)−E(FeO).(5)\nIt gives -1.88 eV/f.u., significantly larger than Ce at Sr-\nsite. The formation energy of other compositions for Ce\nat other remaining Fe-sites is even larger as the total\nenergy is larger than for Ce at the 4 f2site. The formation\nenergy results, therefore, confirm that Ce has a strong\nSr-site preference over the Fe-sites.\nExperimentally Ce is doped in Ba-hexaferrites where\nthe site preference is much debated. The experimental\nreports show that site occupancy depends on how the\nsample is synthesized. Indeed, there are some differences\nin sample preparation. For example, Mosleh et al.[6] used\nslightly different experimental precursor than Pawar et\nal.[11]. Both experiments report lattice parameters ob-\ntained with x-ray diffraction (XRD), which behave oppo-\nsitely to the pristine system. Ce at the Ba site is shown\nto reduce the cell volume in Ref. [6]. In contrast, in\nRefs. [11] and 5], the authors show an increase in volume\nwith Ce at the Fe site. These trends are consistent with\nour results.\n2. Mechanical and dynamical stability\nMechanical stability is described by calculating the\nstiffness tensor using the energy-strain relationship,\nwhich, in Voigt notation, is\nE=V0\n26/summationdisplay\ni=16/summationdisplay\nj=1Cijǫiǫj, (6)\nwhereV0,Cij, andǫiare the volume of unit cell, the\nelastic constants, and the components of strain tensor,\nrespectively. As the crystal structure of Ce-substituted\nhexaferrites is hexagonal (space group P63/mmc ), its\nstability can be described by five independent non-zero\nelastic constants C11,C12,C13,C33, andC44. The elastic\nstiffness tensor (not given here) is the definitive positive\nand has all six eigenvalues positive for all compositions.\nAll three elastic stability criteria for hexagonal crystal\n[26],e.g.,C11>|C12|,2C2\n13< C33(C11+C12), and\nC44>0are satisfied by the computed elastic constants\n(Table II) for substituted as well as pure compositions\nconfirming their mechanical stability. Interestingly, bul k\nmodulus (K) and shear modulus (G) are similar for x= 0\nandx= 1compositions while smaller for x= 0.5. These\nresults suggest that a half-Ce-substitution makes it less\nstiff, likely due to the peak of the density of states at\nthe Fermi level, which we discuss later. The zone-center\n(Γ-point) phonon frequencies are all positive, indicating\nthat these are likely dynamically stable. Currently, weTABLE II. Elastic constants ( Cij), bulk modulus (K), and\nshear modulus (G) in Voigt notation (in units of GPa) of\nSr1−xCexFe12O19obtained with GGA +Ucalculations.\nCij x= 1 x= 0.5x= 0\nC11 297.932 226.311 283.214\nC12 130.157 109.500 143.012\nC13 110.131 84.067 102.011\nC33 267.509 203.541 248.651\nC44 71.192 43.796 65.769\nK 173.801 134.603 167.461\nG 79.422 54.435 71.698\nFIG. 1. Phonon dispersion in Sr-hexaferrites computed alon g\nthe high symmetry k-points with GGA + U.\nsucceed in the full Brillouin zone phonon dispersion\ncalculations, as shown in Fig. 1, for pure strontium\nhexaferrite, and we find no soft-mode frequencies that\nconfirm the dynamical stability. We believe Ce and\nLa substituted-hexaferrite, where an extra electron is\nadded to the system , presumably exhibit similar phonon\nbehaviors at finite k-points. However, a thorough study\nof phonon behaviors and thermodynamic stability with\nelement substitutions in hexaferrites requires compu-\ntationally time-consuming calculations, which is worth\ninvestigating in the future.\nIV. ELECTRONIC STRUCTURE\nThe band structure for x= 1.0is shown in Fig. 2(a).\nThe GGA + U+ SOC predicts an indirect band gap of\n0.5 eV analogous to La-substitution[19]. The reduction in\nthe band gap is due to the appearance of a localized band\nin the gap region of the parent compound. The narrow\nband corresponds to the 3d2\nzorbital of Fe( 2a) as shown\nin Fig. 2 (c) ( in red ). Besides, there are extra new fea-\ntures in band structure with varying Ce. First, four occu-\npied3dbands show up slightly above the highly localized4\nAΓM K Γ\nk - path-2-1.6-1.2-0.8-0.400.40.81.2E - EF (eV)\nFe(2a)-3dz2\nCe (4f)(a)\nAΓM K Γ\nk - pathFe(2a)-3dz2\nCe (4f)(b)\n-10 -8 -6 -4 -2 0 2 4 6\nE - EF (eV)-303charge\ntransfer-404-303x = 0.5\nx = 1.0-404PDOS (states / eV /atom /spin)Fe (12k)Fe (4f2)Fe (2b)Ce (2d) O (ave.)\nFe (4f1)\nFe (2a)(c)\n-404-505-0.300.3\nFIG. 2. GGA + U+ SOC computed band structure of\nSr1−xCexFe12O19for (a)x= 1and (b)x= 0.5. (c) Compar-\nison of partial density of states (PDOS) of individual atoms\nforx= 0.5(in black ) andx= 1(in red )obtained with GGA\n+U.(d) Electron charge density contour occupying the va-\nlence bands near the Fermi level (in e/Å3) forx= 1. The\noccupied region of PDOS corresponds to the net charge (1 e)\noccupying a single 4f-orbital of Ce, confirming the formation\nof Ce3+. The Fe( 2a)-PDOS occupied by transferred electrons\nis shown in bottom panel (c). Consistent with the electronic\nstructure of x= 1, the charge contours indicate the localiza-\ntion of Ce-substituted extra electrons near the Fe( 2a) and the\nlocalization of one electron left in Ce( 4f)-orbitals. The shape\nof the charge contour of electrons at the Ce atom is polarized\nalong the c-axis, which is responsible for enhancing MCA. For\nx= 0.5, the same is true except Fe( 2a)-3d2\nzis a half-filled that\nleads to a metal (b). The 3d2\nzbands split into two manifolds\nas the net charge transfer slightly differs between two Fe( 2a)\ndue to the proximity effect.\nCe(4f) band. Interestingly, these 3dbands are pushed\ndown to the 4fband forx= 0.5. Second, the band struc-\nture [Fig. 2(b)] shows a metallic state for x= 0.5as the\nFermi level lies in the middle of the spin-polarized (spin-\n↓) band and PDOS (c) in black . The half-filled bands are\nsplit into two, likely due to a disproportionate amount of\ncharge transfer from Ce, as the two Fe( 2a) are not at\nthe equal distance from Ce. Interestingly, the half-filled\nbands are more dispersive along the planar distance, indi-\ncating asymmetry in the electron’s effective mass, which\nmay lead to interesting transport properties.\nTo better understand electron-electron correlation in\n3dand4fstates, we perform more advanced HSE06-8-6-4 -2 0 2 4\nE - EF (eV)-30-20-100102030PDOS (states /eV) Ce\nFe(2a)\nTotal\nFIG. 3. Partial density of states (PDOS) calculated using\nHSE06 for x= 1showing a band gap and electron localization.\ncalculations for x= 1. We show HSE06 calculated to-\ntal and partial DOS of Ce and Fe( 2a) in Fig. 3. For\ncompleteness, we compute the band gap of the pure sys-\ntem: HSE06 yields 2.4 eV, which is slightly higher than\nthe experimental (1.9 eV) and GGA + U(1.74 eV) val-\nues. We note that Ufor Fe(3d) is well established in the\nhexaferrites[19, 22], while for Ce (4f), it is unknown pri-\nori. With the help of HSE06 results, we repeat the GGA\n+Ucalculations by fine-tuning Ueffvalues for both 4f\nand3dorbitals. The GGA + Uresults correspond to\nthe optimal Ueffvalue, as stated in the method section\n(II). Qualitatively, both the GGA + Uand the HSE06\nyield a similar band gap. Not surprisingly, the theoret-\nical indirect band gap is smaller than the experiment.\nThe experimental band gap of (1.31-1.51 eV) is a direct\noptical band gap, as it was obtained from a linear fit of\nα2(hν)vs.hν. The calculated GGA + U+ SOC direct\nband gap of ∼0.8eV atΓis also underestimated from\nthe experiment, as discussed in Ref. [12], probably due to\nthe presence of impurities in the low Ce-doped sample.\nTo get a more quantitative picture of the localization\nof transferred electrons, we compute the net electrons in\nindividual atoms by integrating the occupied PDOS in\nGGA +U. First, we analyze the valence state of Ce,\nwhich tends to fluctuate between Ce3+and Ce4+[27–29].\nCe-atom consists of four valence electrons with two 6s,\none5d, and one 4f, meaning it can donate two more\nelectrons than Sr: 5s2. We find the net charge ∼1e\nresiding in Ce confirming Ce3+which is in agreement\nwith Mosleh et al.’s [6] report with Fourier-transform\ninfrared spectroscopy (FTIR) data in Ce-substituted\nhexaferrites. Next, we analyze the net charge transfer\nto the remaining atoms. The extra electron occupies\nsymmetry allowed[19] Fe at 2a-site as given in Table\nIII. Forx= 0.5, the Fe( 2a)-atoms share an electron\nleading to a half-filled 3d2\nz-orbital. On the other hand,\nthe amount of net electron transfer in the other Fe-sites\nis negligible. The sharp localization of electron at 2a-site\nis also confirmed by a charge density contour [Fig.2(c)]\ncomputed for x= 1within the 1 eV energy below the\nFermi level.5\nTABLE III. Spin and orbital magnetic moment (inµB)of\nindividual atom in Sr 1−xCexFe12O19(x= 0.5,1), the total\nmagnetic moment per unit cell including the interstitial co n-\ntribution, and the net amount of electron transfer ( qt)(in e)\nto different atomic sites in units of electrons. For oxygen, o nly\nthe average value is given.\nx= 1 x= 0.5\nAtomµsµl qtµsµl qt\nFe(2a) 3.64 0.01 1.679 4.02 0.05 0.800\nFe(2b) 4.14 0.01 0.000 4.12 0.03 0.000\nFe(12k) 4.24 0.02 0.031 4.25 0.02 0.020\nFe(4f1) -4.19 -0.02 0.004 -4.19 -0.02 0.002\nFe(4f2) -4.11 -0.02 0.059 -4.13 -0.02 0.020\nO 0.10 0.0 0.227 0.11 0.00 0.100\nCe(2d) -0.27 2.17 -0.92 0.85 -\nTotal 37.41 4.48 2.0 38.01 1.09 1.0\nV. MAGNETIC PROPERTIES\nNow, we discuss the magnetic moments as presented in\nTable III. The spin moments of all Fe-atoms are, to some\nextent, the same as in SrFe12O19except for Fe( 2a) in\nwhich the moment is reduced by about 0.5 µBfor Fe2+\n(S= 2). As the 4f-states are less than half-filled, the\nsigns of orbital and spin magnetic moments follow Hund’s\nrule. Forx= 0.5, the Ce orbital moment is significantly\nquenched, likely due to a stronger effect of the crystalline\nelectric field in the metallic state. Interestingly, the spi n\nand orbital moments nearly cancel, leading to no gain in\nthe net magnetic moment. Indeed, experimentally, the\nmagnetic moment is also shown to decrease for x= 0.2[6],\nwhich agrees with our results for x= 0.5. On the other\nhand, forx= 1, Ce orbital moment (2.17 µB) is much\nlarger than the spin moment (-0.27 µB), which leads to an\nincrease of the net magnetic moment about 1 µBrelative\ntoSrFe12O19(20µB). These results suggest that Ce\ncontrols the net magnetic moment of hexaferrites.\nNext, we compute T Cby taking the energy difference\nbetween the ferromagnetic and ferrimagnetic structure of\nthe nearest neighbor Fe-sublattices, as T C=S2¯Jij/kB\nwhereS= 5/2is spin of each Fe and ¯Jijis average near-\nest neighbor exchange interaction. The calculated T C\nare 825.53 K and 777.74 K for x= 1and 0.5, which\nare slightly smaller than for SrFe12O19(893 K). The pre-\ndicted T Cfor CeFe 12O19(CeM) is slightly smaller than\nforLaFe12O19(851 K), possibly due to the neglect of\nthe exchange interaction between Ce3+and Fe3+inour\noversimplified mean-field approximation calculations.\nWe further calculated the T Cusing the static linear-\nresponse approach for CeM and LaFe 12O19(LaM), as im-\nplemented in a Green’s function (GF), the linearized\nmuffin-tin orbital (LMTO) method[30, 31] in conjunction\nwith the atomic sphere approximation (ASA) and the lo-\ncal density approximation (LDA) and U(Ueff= 3eV for\nCe and 4.5 eV for Fe). The effective exchange interaction\nJ0(i)for magnetic ion at site iis obtained by using the re-TABLE IV. Exchange coupling (in meV) between the nearest\nneighbor magnetic ions in Ce-hexaferrites computed from GF -\nASA in the LDA + U. Fe atoms in different sites are labeled\nby corresponding Wyckoff positions.\nCe2a2b12k4f14f2\nCe 0.00 0.01 0.97 0.36 0.29 0.00\n2a - -0.12 0.00 -3.50 19.43 0.00\n2b - - -0.98 -9.36 0.80 24.08\n12k - - - -10.34 18.23 15.24\n4f1 - - - - -2.91 -0.57\n4f2 - - - - - -0.52\nTABLE V. GGA + U+ SOC calculated magnetic anisotropy\nconstant, K1, in Sr 1−xCexFe12O19(xis a variable 0to1)\nand its comparison with experiment and previous theory for\nSr/LaFe 12O19. The values are presented in units of MJ/m3\nforUeff= 2−4for Ce and 4.5eV for Fe.\nK1 x= 0x= 1x= 0.5 LaFe 12O19\nThis worka0.35 10.31 2.70 0.66\nThis workb0.35 8.14 1.71 0.66\nThis workc0.35 6.34 1.14 0.66\nExpt.[32, 33] 0.35−0.36- - 0.5−0.8\nlationJ0(i) =/summationtext\njJij, wherejis the number of the near-\nest neighbor atoms and Jijis the pairwise exchange cou-\npling for Heisenberg Hamiltonian H=−/summationtext\ni,jJijˆei.ˆej,ˆei\nandˆejare unit vectors along the direction of local mag-\nnetic moments. Then, T Cis computed using the mean-\nfield approximation, i.e., TC=2\n3/summationtext\niJ0(i)/kB. The com-\nputed nearest neighbor Jijare given in Table IV, which\ncorrectly show the ordering of magnetic ions, i.e., pos-\nitive and negative values for antiferromagnetically and\nferromagnetically aligned spins. As expected, the lead-\ning magnetic exchange-couplings are found between the\nnearest neighbor Fe3+ions at2a−4f1,2a−12k,2b−4f2,\n2b−12k,12k−4f1,12k−4f2, and12k−12ksites, while\nthe exchange couplings with Ce3+are negligible.\nWe found T C= 905 K for LaM and 914 K for CeM,\nwhich agree qualitatively with respective values obtained\nwith an oversimplified approach. For CeM, T Cobtained\nby including the exchange interactions of Ce3+with\nFe3+is slightly larger. Overall there are no significant\ndifferences in T Cwith La and Ce substitutions.\nVI. MAGNETOCRYSTALLINE ANISOTROPY\nNow we turn our discussion to magnetic anisotropy.\nTo predict MCA, we employ a fully self-consistent rela-\ntivistic non-collinear scheme to compute the total energy\nin two different crystalline directions, viz., E001(magne-\ntocrystalline easy) and E100(magnetocrystalline hard)6\nSr/Ce 2a 2b 12k 4f14f2\nAtom -1001020304050Eaniso(meV)Sr1-xCexFe12O19\nx = 0x = 1\nx = 0.5\nFIG. 4. On-site anisotropy energy ( Eaniso) contributions.\nThe Ce has the highest Eanisocontribution. Fe atoms are\nlabeled by their corresponding Wyckoff position. The onsite\nFe contribution is positive at 12kand2band negative at 2a.\nThe contribution is negligible at 4f1and4f2-sites.\nalong the crystalline c, i.e.,[0,0,1]anda, i.e,[1,0,0]\ndirections, including SOC in GGA + U. The mag-\nnetic anisotropy energy ( MAE ) is then obtained as\nMAE=E100−E001from which we evaluate K1as\nK1=MAE/V , whereVis the unit cell volume.\nK1is computed for different values of Uefffor Ce(4f)\n[Table V]. For Ueff∼3eV, the computed value of K1\nin 100% Ce-doped hexaferrite is nearly 8 times larger\nthan in strontium-hexaferrite. These results clearly show\na big influence of 4f-electrons. Qualitatively, the effect\nof4f-electrons can be understood by the charge den-\nsity contours of occupying states near the Fermi level.\nAs seen from Ce’s prolate type of charge density con-\ntour, it costs huge energy to rotate it in other direc-\ntions. The most direct quantitative evidence for the\nsite-contribution to MAE can be obtained from the en-\nergy difference between easy and hard axes (anisotropy\nenergy) viz., Eaniso=Esoc[100]−Esoc[001], where\nEsoc[001]andEsoc[100]are atomic SOC energies for mag-\nnetization directions [0,0,1]and[1,0,0]. The net MAE\nis obtained from the expression[34]1\n2/summationtext\niEi\naniso, where\nidenotes the atomic site. Figure 4 shows the Eanisoto\nMAE for different Ce substitutions. For x= 0, the MAE\nis primarily governed by Fe( 2b) and Fe( 12k) with some\ncontribution from Fe at 4f1and4f2. Although small,\nFe(2a) gives a planar contribution. Interestingly, elec-\ntron doping to Fe( 2a) results in uniaxial or planar MAE\ndepending on the substituted elements. For instance, it\nproduces a uniaxial contribution and is a leading term\nin La substituted hexaferrite. In CeM, the shape of the\n3d2\nzorbital is significantly distorted [Fig. 1(d)] from the\nc-direction, unlike in LaM, probably due to the 4felec-\ntrons. The Fe( 2b) site contribution is uniaxial and re-\nmains unchanged after substitutions. The 4f1or4f2\ncontribution changes sign; however, it does not affect the\nMAE since it is much smaller. Of note, as expected in\n4f-electron systems, the Ce-site contribution is larger byTABLE VI. Calculated magnetic properties and compari-\nson with available experiments (forx= 0) at room tem-\nperature otherwise stated for Ce-substituted compositions\n(Sr1−xCexFe12O19) withUeff= 3eVfor Ce and 4.5eV for\nFe viz., magnetic anisotropy constant ( K1), saturation mag-\nnetization ( MS), anisotropy field ( Ha), BHmax, and magnetic\nhardness parameter ( κ) (units are given inside the parenthe-\nsis).\nparameter x= 0x= 1x= 0.5Expt[35]\nK1(MJ/m3) 0.35 8.14 1.71 0.35\nMS(T) 0.60 0.64 0.59 0.31a, 0.48 - 0.59b\nHa(KOe) 14.58 320.04 72.56 14.88a-20.01\nBHmax(kJ/m3) 72.42 81.12 69.82 45- 69.53a\nκ 1.135.01 2.47 1.35\norders of magnitude than other sites.\nA similar 4f-electrons effect in MAE is found for\nx= 0.5, while the increase is not proportional to Ce.\nIn the metallic system, the nonlinear trend in MAE is\nexpected; because the itinerant electrons can perturb the\n4fcharge distribution and affect the crystalline electric\nfield. Indeed, the Ce-site contribution to MAE [Fig.\n4] is reduced, consistent with the reduced Ce-orbital\nmagnetic moment. Lastly, the Fe-sites contribution to\nMAE is less significant.\nTable VI summarizes the computed magnetic proper-\nties of Ce-substituted compositions and compares them\nwith experimental values of pure Sr-hexaferrite. The\nsaturation magnetization ( MS) is enhanced in full Ce-\nhexaferrite. However, 50% Ce substitution does not\nincrease the MS.Our calculated MSlies within the\nrange of experimental values, which vary from 0.31 to\n0.59 T depending on the temperature, sample prepara-\ntion techniques, and micromagnetic structures. Magnetic\nanisotropy field ( Ha) can be estimated using the relation\nHa=2K1\nMS. (7)\nFor Ce-substituted compositions, Haenhances by about\na factor of 10, consistent with the predicted value of K1.\nBHmaxwhich is estimated from MSas\nBHmax=µ0M2\nS\n4. (8)\nBHmaxis the largest for full Ce. Magnetic hardness pa-\nrameterκis given by\nκ=/radicalBig\nK1/µ0M2\nS, (9)\nand it satisfies the criterion[35] ( κ >1) of hard magnet\nfor all compositions. The full Ce substitution of Sr\nproduces the hardest permanent magnet with the largest\nvalue ofκ.7\nVII. OPTICAL ANISOTROPY\nTo understand the effect of Ce( 4f)-states in optical\nproperties of CeFe 12O19, we compute the optical ab-\nsorption coefficient α(ω) =ωε2(ω)\ncn(ω),wherec,ω, and\nn(ω) =1√\n2[ε1(ω) +/radicalbig\nε1(ω)2+ε2(ω)2]1\n2, andε2(ω)are\nspeed of light, photon energy, refractive index, and the\nimaginary part of the macroscopic dielectric constant\nfrom which the real part of the macroscopic dielectric\nconstantε1(ω)is obtained using Kramers-Kronig trans-\nformation. The ε2(ω)is calculated from the expression\nε2(ω) =4π2e2\nm2ω2/summationdisplay\nv,c/integraldisplay\nBZd3k\n(2π)3|ˆe·Mcv(k)|2δ(/planckover2pi1ω−ǫck+ǫvk),\n(10)\nwhereˆeandMcv(k) =/angbracketleftψck|p|ψvk/angbracketrightare the light polariza-\ntion vector and the momentum matrix element between\nthe conduction and the valence states. We find signifi-\ncant changes in the α(ω)as compared with LaFe12O19,\nwhere vertical transitions occur between Fe( 3d) states as\ndiscussed in our previous paper[19]. Overall the inten-\nsity ofα(ω)is enhanced considerably with Ce, including\nthe appearance of two main peaks labeled by βandγ\nin Fig. 5. It is expected as the 4f-states are located\nnear the Fermi-level, for which the strength of matrix ele-\nments of dipole moments corresponding to the transition\nfrom the occupied 4fto unoccupied 4f-states is much\nstronger. The unoccupied 4f-states are split into two\nmanifolds because of the combined effects of crystal-field,\nexchange, and SOC. β-peak corresponds to the transition\nfrom the occupied 4fto lower unoccupied at ∼2eV,\nandγcorresponds to upper unoccupied 4f-states at ∼4\neV. The optical absorption is spatially localized to Ce-\natom, which could be interesting to examine experimen-\ntally. The other interesting result is enhanced anisotropy\nin optical absorption. Indeed, the real parts of dielec-\ntric constant ε1are 6.94 for E/bardblaand 6.18 for E/bardblc\nconsistent with α(ω).\nHSE06 calculations reveal qualitatively similar opti-\ncal spectra, except now that the absorption peaks are\nshifted. In particular, β-peak is shifted by about 1 eV,\nwhileγ-peak shifts by about 0.5 eV. These are consis-\ntent with the alignment of Ce( 4f)-bands relative to the\nFermi level. The height of β-peak is similar, while γ-peak\nis reduced by about 2. Interestingly, HSE06 + SOC pro-\nduces another peak of about 1.3 eV corresponding to the\ninterband transition between the Fe( 3d)-states for E/bardblc,\nwhich is much weaker with the GGA + U+ SOC cal-\nculations. The magnitudes of ε1are 3.94 for E/bardblaand\n3.65E/bardblc. The difference in the dielectric constants\nbetween the two approaches is expected, partly due to\nthe use of different numbers of virtual states in the cal-\nculations. Unfortunately, the convergence of εin HSE06\ncalculations is always problematic due to the huge com-\nputational cost. However, both approaches yield a simi-lar trend in the anisotropy.\nVIII. METASTABLE SITES AND MAGNETISM\nBy performing the GGA + U+ SOC calculations, we\nexamine the effect of Fe-site substitutions by Ce in mag-\nnetic properties and local spin quantization. Depend-\ning on the Ce-site occupancy, the net magnetic moment\nchanges from 34 µBto 44µB, as given in Table VII. To\nillustrate it, we discuss the PDOS of Ce and Fe (nearest\nto Ce) while replacing one Fe at 4f2(the most favorable\nFe site) in Fig. 6. The 4f-states are delocalized below\nthe Fermi level. Quite interestingly, one electron of Ce\ntransfers only to the nearest Fe( 2a), forming a localized\n3dz2state. It reduces Fe3+to Fe2+, as found in Ce at Sr-\nsite. Ce remains to be Ce3+, which helps to increase the\nnet magnetic moment to 44 µBfor the following reasons:\n(i) the negative spin magnetic moment contribution of\nabout 5µBat4f2of one Fe-atom vanishes and (ii) the\nmagnetic moment of Ce itself is much smaller ∼1µB.\nA similar situation holds in 4f1-site. At the 2bsite, Ce\ngives an extra electron to Fe( 2a), producing Fe2.5+and\nfurther reducing the net moment to 34 µB. Interestingly,\nat the12ksite, our calculations show that the Ce donates\nan extra electron to the nearest Fe( 12k), resulting in a\nsimilar net moment.\nNext, we calculate total energies by choosing the spin\nquantization directions along [0,0,1], [1,0,0], and [1,1, 1]\nusing self-consistent GGA + U+ SOC method and in-\nvestigate the Ce-induced spin-canting by analyzing the\nEaniso andMAE .Fortunately, we did not have to\nconstrain the spins because they remained aligned with\nthe initialized directions even after completing the full\nself-consistent calculations. The computed Eaniso(not\nshown here) suggest that the local spin magnetic mo-\nment directions of Ce( 4f2) and Fe( 2a) are not along\nthe [0,0,1] (Gorter’s type[36]), rather they are along the\n[1,1,1], while the directions of other remaining Fe-spin\nmagnetic moments remain the same. As Ce( 4f) with a\nlargeEanisoprimarily controls the magnetic anisotropy,\nthe whole spin-system becomes uni-axial along the crys-\ntalline [1,1,1] direction with MAE=E001−E111= 0.45\nmeV. This local spin-canting could arise due to the fol-\nlowing reasons: (i) a complex spin structure around the\ndefect and (ii) a significant change in the crystalline elec-\ntric field. Experimentally Sm is already known to replace\nSr[10]. In a separate experimental work[7], authors re-\nport that Sm causes a spin-canting at Fe-sites. Pawar\net al. also discussed the possibility of spin canting if Ce\nis doped at Fe-sites[11]. Although our DFT calculations\nsuggest Fe-spins canting near the impurity, it does not\ndramatically change the net magnetic moment (see Ta-\nble VII) as reported in experiment[7], in which MSis\nshown to decrease significantly by increasing the Ce con-\ncentration.8\n0 1 2 3 4 56\nEnergy (eV)0246810 α (105 cm-1)\nE || a\nE || cβγ(a) \n0 1 2 3 4 56\nEnergy (eV)0246810\nβγ(b)\nFIG. 5. Optical absorption coefficient, α(ω), of full Ce-substituted hexaferrite ( x= 1), calculated with GGA + U+ SOC (a)\nand HSE06 + SOC (b). The strength of dipole-transition betwe en4fstates is rather strong compared to 3d-states, resulting\nin two peaks βandγat∼2and∼4eV in GGA + U+ SOC. These peaks shift by ∼1eV with HSE06. In the HSE06 results,\nan additional peak appears in the low energy region around 1. 3 eV for E/bardblc.\n-6-4 -2 0 2 4 6\nE - EF (eV)05101520PDOS (states /eV)Ce (4f)\nFe (3d)    extra\nelectron\nFIG. 6. PDOS of Ce( 4f) and Fe( 3d) at2a(the nearest to\nthe Ce in the unit cell) computed with GGA + U+ SOC for\nCe-substituted at Fe- 4f2site. The extra Ce electron transfers\nto the Fe( 3d) states localized below the Fermi level.\nTABLE VII. GGA + Ucalculated the net magnetic moments\n(m) (inµB) for SrCeFe 23O19by replacing a Fe atom with Ce\nin different sites and the optimized lattice constants aandc\n(in Å).\nparameter 2a2b12k4f14f2\na 5.95 5.99 5.98 5.96 5.96\nc 23.48 23.15 23.36 23.41 23.68\nm 36 34 34 44 44\nIX. CONCLUSION\nIn summary, by performing DFT, including hybrid\nfunctional calculations and analyzing formation energies\nand elastic constants, we uncover the lattice stability ofCe-substituted M-type strontium hexaferrites. Both full-\nand half-Ce-substituted compositions are chemically\nand mechanically stable. The full Ce results in fully\noccupied narrow 3d2\nz-derived bands of Fe( 2a), yielding\na reduced band gap. On the other hand, the half-Ce\nleads to a half-filled 3d2\nz-band producing a metallic state.\nThese results, including the computed net amount of 4 f\nelectron in Ce, confirm the formation of Ce3+. The Ce\nincreases the MCA of the parent compound by at least\nan order of magnitude, which is a direct consequence\nof Ce(4 f)-electrons. The net magnetic moment also\nincreases with the full Ce as the net Ce-contribution\nresults from the orbital moment following Hund’s rule.\nIn addition, due to Ce( 4f), the computed optical spectra\nfor CeFe 12O19show significant anisotropy between light\npolarization in parallel and perpendicular directions\nwith enhanced transitions between Ce( 4f) and con-\nduction bands. If grown successfully, we believe the\nCe-substituted M-type hexaferrite can be a potential\ncandidate material for a high-performance permanent\nmagnet.\nACKNOWLEDGMENTS\nThis work is supported by the Critical Materials In-\nstitute, an Energy Innovation Hub funded by the U.S.\nDepartment of Energy, Office of Energy Efficiency and\nRenewable Energy, Advanced Manufacturing Office. The\nAmes National Laboratory is operated for the U.S. De-\npartment of Energy by Iowa State University of Sci-\nence and Technology under Contract No. DE-AC02-\n07CH11358.\n[1] H. Warlimont, Hard magnetic materials: Alnico: Datashe et from landolt-börnstein - group viii advanced materials a nd technologies · volume 2a1: “powder metallurgy data” in sp ringermaterials (https://doi.org/10.1007/10689123_31 )\n(2003).[2] F. Lotgering, Journal of Physics and Chemistry of Solids 35, 1633 (1974).9\n[3] D. Seifert, J. Tópfer, F. Langenhorst, J.-\nM. L. Breton, H. Chiron, and L. Lechevallier,\nJournal of Magnetism and Magnetic Materials 321, 4045 (2009).\n[4] K. M. U. Rehman, M. Riaz, X. Liu, M. W. Khan,\nY. Yang, K. M. Batoo, S. F. Adil, and M. Khan,\nJournal of Magnetism and Magnetic Materials 474, 83 (2019).\n[5] S. Chang, S. Kangning, and C. Pengfei,\nJournal of Magnetism and Magnetic Materials 324, 802 (2012).\n[6] Z. Mosleh, P. Kameli, A. Poor-\nbaferani, M. Ranjbar, and H. Salamati,\nJournal of Magnetism and Magnetic Materials 397, 101 (2016).\n[7] N. Yasmin, M. Mirza, S. Muhammad, M. Za-\nhid, M. Ahmad, M. Awan, and A. Muhammad,\nJournal of Magnetism and Magnetic Materials 446, 276 (2018).\n[8] S. Ounnunkad, Solid State Communications 138, 472 (2006).\n[9] B. Rai, S. Mishra, V. Nguyen, and J. Liu,\nJournal of Alloys and Compounds 550, 198 (2013).\n[10] J. Jalli, Y.-K. Hong, S.-H. Gee, S. Bae, J. Lee, J. C. Sur,\nG. S. Abo, A. Lyle, S.-I. Lee, H. Lee, and T. Mewes,\nIEEE Transactions on Magnetics 44, 2978 (2008).\n[11] R. Pawar, S. Desai, Q. Tamboli,\nS. E. Shirsath, and S. Patange,\nJournal of Magnetism and Magnetic Materials 378, 59 (2015).\n[12] V. Banihashemi, M. E. Ghazi, and M. Izadifard,\nJournal of Materials Science: Materials in Electronics 30, 17374 (2019).\n[13] G. Kresse and J. Furthmüller,\nPhys. Rev. B 54, 11169 (1996).\n[14] G. Kresse and J. Furthmüller,\nComputational Materials Science 6, 15 (1996).\n[15] G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).\n[16] D. Hobbs, G. Kresse, and J. Hafner,\nPhys. Rev. B 62, 11556 (2000).\n[17] S. L. Dudarev, G. A. Botton, S. Y.\nSavrasov, C. J. Humphreys, and A. P. Sutton,\nPhys. Rev. B 57, 1505 (1998).\n[18] I. L. M. Locht, Y. O. Kvashnin, D. C. M. Rodrigues,\nM. Pereiro, A. Bergman, L. Bergqvist, A. I. Licht-\nenstein, M. I. Katsnelson, A. Delin, A. B. Klau-\ntau, B. Johansson, I. Di Marco, and O. Eriksson,Phys. Rev. B 94, 085137 (2016).\n[19] C. Bhandari, M. E. Flatté, and D. Paudyal,\nPhys. Rev. Materials 5, 094415 (2021).\n[20] J. Heyd, G. E. Scuseria, and M. Ernzerhof,\nThe Journal of Chemical Physics 118, 8207 (2003).\n[21] J. Heyd, G. E. Scuseria, and M. Ernzerhof,\nThe Journal of Chemical Physics 124, 219906 (2006).\n[22] V. c. v. Chlan, K. Kouřil, K. c. v. Uličná,\nH. Štěpánková, J. Töpfer, and D. Seifert,\nPhys. Rev. B 92, 125125 (2015).\n[23] G. K. Thompson and B. J. Evans,\nJournal of Applied Physics 75, 6643 (1994).\n[24] J. Leger, N. Yacoubi, and J. Loriers,\nJournal of Solid State Chemistry 36, 261 (1981).\n[25] G. Shirane, D. E. Cox, W. J. Takei, and S. L. Ruby,\nJournal of the Physical Society of Japan 17, 1598 (1962).\n[26] F. Mouhat and F. m. c.-X. Coudert,\nPhys. Rev. B 90, 224104 (2014).\n[27] O. Eriksson, L. Nordström, M. S. S. Brooks, and B. Jo-\nhansson, Phys. Rev. Lett. 60, 2523 (1988).\n[28] O. Isnard, S. Miraglia, and F. Villain,\nJournal de Physique IV Proceedings 7, C2 (1997).\n[29] E. Shoko, M. F. Smith, and R. H. McKenzie,\nPhys. Rev. B 79, 134108 (2009).\n[30] M. van Schilfgaarde and V. P. Antropov,\nJournal of Applied Physics 85, 4827 (1999).\n[31] D. Pashov, S. Acharya, W. R. Lambrecht,\nJ. Jackson, K. D. Belashchenko, A. Chan-\ntis, F. Jamet, and M. van Schilfgaarde,\nComputer Physics Communications 249, 107065 (2020).\n[32] L. Jahn and H. G. Müller,\nphysica status solidi (b) 35, 723 (1969).\n[33] B. T. Shirk and W. R. Buessem,\nJournal of Applied Physics 40, 1294 (1969).\n[34] Y. Sun, Y.-X. Yao, M. C. Nguyen, C.-Z. Wang, K.-M.\nHo, and V. Antropov, Phys. Rev. B 102, 134429 (2020).\n[35] J. M. D. Coey, IEEE Transactions on Magnetics 47, 4671 (2011).\n[36] E. W. Gorter, Proceedings of the IEE - Part B: Radio and El ectronic Engineering 104, 255 (1957)."    },    {        "title": "2308.11177v1.Effect_of_Electron_Phonon_Scattering_on_the_Anomalous_Hall_Conductivity_of_Fe__3_Sn__A_Kagome_Ferromagnetic_Metal.pdf",        "content": "Effect of Electron-Phonon Scattering on the Anomalous Hall Conductivity of Fe 3Sn:\nA Kagome Ferromagnetic Metal\nAchintya Low, Susanta Ghosh, Soumya Ghorai, and Setti Thirupathaiah∗\nDepartment of Condensed Matter and Materials Physics,\nS. N. Bose National Centre for Basic Sciences, JD Block,\nSalt Lake, Sector III, Kolkata, West Bengal, India, 700106.\nWe report on magnetic and magnetotransport studies of a Kagome ferromagnetic metal, Fe 3Sn.\nOur studies reveal a large anomalous Hall conductivity ( σzx) in this system, mainly contributed\nby temperature independent intrinsic Hall conductivity ( σint\nzx=485±60 S/cm) and temperature de-\npendent extrinsic Hall conductivity ( σext\nzx) due to skew-scattering. Although σext\nzxvalue is large and\nalmost equivalent to the intrinsic Hall conductivity at low temperatures, it drastically decreases\nwith increasing temperature, following the relation σext\nzx=σext\nzx0\n(aT+1)2, under the influence of electron-\nphonon scattering. The presence of electron-phonon scattering in this system is also confirmed by\nthe linear dependence of longitudinal electrical resistivity at higher temperatures [ ρ(T)∝T]. We\nfurther find that Fe 3Sn is a soft ferromagnet with an easy-axis of magnetization lying in the ab\nplane of the crystal with magnetocrystalline anisotropy energy density as large as 1.02 ×106J/m3.\nI. INTRODUCTION\nHall effect due to which the fast-moving charge carri-\ners get deflected transversely under the external magnetic\nfields in metals and semiconductors [1] has found lately\nmany real-life technological applications [2, 3]. While an\nordinary Hall effect has been noticed in the nonmagnetic\nmetals, an anomalous Hall effect (AHE) was found in\nthe collinear ferromagnetic metals [4, 5] and in the non-\ncollinear antiferromagnetic metals [6, 7]. Moreover, the\nanomalous Hall effect produces substantially higher Hall\nresistivity than the ordinary Hall effect, and the field\ndependent Hall resistivity perfectly scales with magneti-\nzation [4, 5]. Though the origin of AHE in non-collinear\nAFM metals is widely understood by the presence of non-\nzero Berry phase in momentum space, several mecha-\nnisms were proposed to understand the AHE in collinear\nferromagnets.\nForemost, Karplus and Luttinger (KL) predicted that\nthe AHE in ferromagnetic metals originates from the in-\nterband scattering of the charge carriers under spin-orbit\ncoupling [8], which is recently connected to the Berry cur-\nvature of the electronic state of solids [9]. Since the KL\ntheory does not take into account the impurity scatter-\ning effects and mainly talks on the intrinsic band struc-\nture, considered as the intrinsic theory of AHE. Later\non, Smit et al.,proposed an extrinsic theory of AHE\nby incorporating the impurity scattering [10]. The ex-\ntrinsic AHE happens by two scattering mechanisms, (i)\nthe skew-scattering: whereby the charge carriers scatter\nasymmetrically by the localized magnetic impurities [10]\nand (ii) the side-jump: whereby the charge carrier takes\na small side jump up on scattering with impurity un-\nder spin-orbit coupling [11, 12]. Although many ferro-\nmagnetic metals are known to show the AHE [5], the\n∗setti@bose.res.inKagome ferromagnets are quite fascinating systems as\nthey show an intrinsic geometrical frustration, leading\nto several exotic electronic properties such as the flat-\nbands near the Fermi level [13–16], quantum spin-liquid\nground state [17, 18], Chern insulating state [19, 20],\nWeyl fermions [21–23], Dirac fermions [21, 24, 25], and\nmagnetic topological Skyrmions [26, 27], manifesting the\nanomalous and topological Hall effects.\nThus, the ferromagnetic metal Co 3Sn2S2is found to\nexhibit giant intrinsic anomalous Hall conductivity ( σxy\n≈505 S/cm) [28] due to the presence of Weyl nodes\nnear the Fermi level [29] and Fe 3Sn2is found to show ex-\ntremely large anomalous Hall conductivity ( σxy≈1150\nS/cm) [30] below 2K and large topological Hall conduc-\ntivity (-0.875 µΩ cm) above room temperature [31, 32].\nOn the other hand, recently, a few reports on polycrys-\ntalline Fe 3Sn suggested it to be a ferromagnetic metal in\nwhich the Fe atoms form a Kagome network in the ab\nplane. Further, it was also shown that Fe 3Sn exhibits a\nlarge magnetocrystalline anisotropy energy [33, 34] in ad-\ndition to the anomalous Nernst effect [35]. Although the\nprevious report shows temperature-dependent anomalous\nHall conductivity to some extent, a thorough understand-\ning of the anomalous Hall effect in Fe 3Sn is still missing,\nespecially the influence of electron-phonon scattering on\nthe AHC.\nFe3Sn belongs to the Ni 3Sn-type family of crystal\nstructure with an in-plane Kagome network. Unlike its\nsister compound Mn 3Sn which is a non-collinear antifer-\nromagnet metal, Fe 3Sn is a collinear in-plane ( ab-plane)\nferromagnetic metal. In this paper, we mainly focus on\nthe anomalous Hall effect of Fe 3Sn as a function temper-\nature. For this, we have grown high-quality single crys-\ntals of Fe 3Sn and performed magnetic and magnetotrans-\nport studies. Our results unravel two important contri-\nbutions to the total Hall conductivity of Fe 3Sn. One\nof them is the temperature independent intrinsic Hall\nconductivity originated from the electronic band struc-\nture and the other one is the temperature dependent ex-arXiv:2308.11177v1  [cond-mat.mtrl-sci]  22 Aug 20232\n10\n5\n0\n0.8 0.6 0.4 0.2 0.0\nabc\ncb\na\nIobs\nIcal\nIobs-Ical\nBragg positionIntensity(a.u.)\nEnergy (keV)SnFeSnFe\nSn\nFe\nIntensity(a.u.)\n2θ(°)\nIntensity(a.u.)\n2θ(°)102103\n(020)\nFe\nSn\nAtomic\n(%)Weight\n(%)Element\n74.9 58.4 Fe\n25.1 41.6 Sn\nSn\nSn\nSnSnFe\n1 mm(a)\n(b)\n(c)\n(030)\n(040)\n10 μm\n10 μm\n31 21 14 7 0 50 36 24 12 0\nFIG. 1. (a) Schematic crystal structure and Kagome lattice\nof Fe 3Sn. (b) Powder XRD pattern of crushed Fe 3Sn single\ncrystals. Inset of (b) shows intensity reflections correspond to\n(02¯20) Brag planes. Left panel in (c) presents EDS spectra of\nFe3Sn along with tabulated elemental ratios and photographic\nimage showing typical Fe 3Sn single crystals. Right panels in\n(c) show the elemental mapping of measured single crystal for\nFe and Sn.\ntrinsic Hall conductivity originated from the asymmet-\nric skew-scattering. Most importantly, we observe that\nthe extrinsic skew-scattering Hall conductivity strongly\ndepends on the inelastic electron-phonon scattering rate\n(γ),σext\nzx=σext\nzx0\n(γ/γ 0+1)2. In addition, the linear depen-\ndence of longitudinal electrical resistivity confirms the\npresence of electron-phonon scattering at higher temper-\natures. We further show that Fe 3Sn is a soft ferromagnet\nwith an easy-axis of magnetization lying parallel to the\nabplane. We derive a magnetocrystalline anisotropy en-\nergy density as large as 1.02 ×106J/m3.II. EXPERIMENTAL DETAILS\nHigh quality single crystals of Fe 3Sn were grown by the\nsolid-state crystal growth (SSCG) technique. In SSCG\nmethod, the crystals are grown out of polycrystalline\nmatrix. Initially, Fe powder (99.99%, Stern chemicals)\nand Sn powder (99.995%, Alfa Aesar) were taken in sto-\nichiometric ratio, grounded thoroughly, and heated at\n810oC for 7 days. As prepared Fe 3Sn powder was again\ngrounded and pressed into a pellet which was then an-\nnealed at 810oC for another 45 days. Several small rod-\nshaped shiny crystals with a typical size of 1 mm ×0.2\nmm×0.2 mm were grown on the surface of the pellet.\nX-ray diffraction (XRD) technique was performed on a\nrod-shaped single crystal and on the crushed crystals us-\ning Rigaku SmartLab 9kW Cu K αX-ray source. Ele-\nmental analysis was done using the Energy Dispersive\nX-ray Spectroscopy (EDS of EDAX) suggests an actual\nchemical composition of Fe 2.98Sn, which is very close\nto the nominal composition of Fe 3Sn. For the electri-\ncal transport and magnetotransport measurements lin-\near four-probe and Hall probe connections were made,\nrespectively, using the copper wire and silver paint. Mag-\nnetic and magnetotransport measurements were carried\nout on the 9T Physical Properties Measurement Sys-\ntems (PPMS, Quantum Design-DynaCool) using VSM\nand ETO options. To eliminate the longitudinal voltage\ncontribution due to any misalignment of the connections,\nthe Hall resistivity was measured by applying both posi-\ntive and negative magnetic fields and average Hall resis-\ntivity was calculated by ρH=ρH(H)−ρH(−H)\n2.\nIII. RESULTS AND DISCUSSIONS\nFig 1(a) depicts schematic crystal structure of Fe 3Sn\nwhere the Fe atoms form a Kagome structure with Sn sit-\nting at the center of the Kagome lattice. Fig. 1(b) shows\nthe powder XRD pattern of crushed Fe 3Sn single crys-\ntals, confirming the hexagonal Ni 3Sn type crystal struc-\nture with the space group of P6 3/mmc (No. 194). Inset\nof Fig. 1(b) depicts the XRD performed on a rod-shaped\nsingle crystal, showing intensity of reflections from the\n(0 2¯2 0) Bragg plane, suggesting that the length of rod-\nshaped crystals is parallel to the c-axis. Rietveld refine-\nment performed on the powder XRD of crushed single\ncrystals using the Fullprof software [36] derives the lat-\ntice parameters a=b=5.4631(4) ˚Aand c=4.3552(4) ˚A, in\ngood agreement with previous reports [33]. Left panel of\nFig. 1(c) shows the EDS data from which the atomic and\nweight percentage of the elements are tabulated on the\ntop-left inset of Fig. 1(c). Photographic image of typical\nFe3Sn single crystals is shown in Fig. 1(c). Elemental\nmapping performed for Fe and Sn using the EDAX is\nshown in the right-side panel of Fig. 1(c), implies good\nhomogeneity of single crystals.\nTemperature dependent magnetization [ M(T)] be-\ntween 2 and 300 K performed on Fe 3Sn single crystal with3\n3.0\n2.5\n2.0\n1.5\n1.0\n0.5\n0.0\n800 700 600 500 400 30020\n15\n10\n5\n0\n800 750 700 6506.8\n6.6\n6.4\n6.2\n250200150100500\n6\n4\n2\n0\n-2\n-4\n-6\n-5 -4 -3 -2 -1 0 1 2 3 4 56.8\n6.6\n6.4\n6.2\n250 200 150 100500-6-4-20246\n-2-1012\n-400-2000200400\n-5 -4 -3 -2 -1 0 1 2 3 4 5M( )(b)\nCurie-Weiss\nfitting-1(emu-1molOe)\nT(K)ߤ0H = 0.3 T5 K\n50 K\n100 K\n150 K\n200 K\n250 K5 K\n100 KM( )\nT(K)MS( )\ny[011ത0]\nx[21ത1ത0]H || y\n5 K\n50 K\n100 K\n150 K\n200 K\n250 KM( )\nμ0H (T)T(K)MS( )\nz[0001]H || z(d)5 K\n25 K\n50K75 K\n100 K\n125 K\n150 K\n175 K\n200 K\n250 K( )\nH || y(e)\n5 K\n25 K\n50 K75 K\n100 K\n125 K\n150 K\n175 K\n200 K\n250 K(Scm-1)\nμ0H (T)\n-Vx\n+Vx-Iz +Iz\nH || y(f)(c)\nT(K)M( )(a)\nH || y\nH || zߤ0H = 0.05 T FC\nZFC450\n400\n350\n300\n250\n5 4 3 2 1\nFIG. 2. (a) and (b) Magnetization measured as a function of temperature on Fe 3Sn single crystals and pellets, respectively.\nInset of (b) is Curie-Weiss fitting of inverse susceptibility plotted as a function of temperature. (c) and (d) are the magnetization\nisotherms [ M(H)] for H∥yandH∥z, respectively. Insets in (c) and (d) show the saturation magnetization. (e) and (f) Hall\nresistivity and Hall conductivity, respectively, plotted as a function of field at various sample temperatures for H∥y.\nthe field ( H) applied parallel to y-axis ( H∥y) and paral-\nlel to z-axis ( H∥z) in the field-cooled mode is shown in\nFig. 2(a). From Fig. 2(a) it is evident that below 300 K,\nM(T) is completely temperature independent. Fig. 2(b)\nshows M(T) performed between 300 and 800 K on Fe 3Sn\npellet in the zero-field-cooled mode in order to identify\nthe ferromagnetic transition. From Curie-Weiss fitting\nof the inverse susceptibility ( χ−1(T)), using the formula\nχ=C\nT−θ, we derive a Curie constant of C= 7.1±0.1\nemu.mol−1Oe−1K−1and a Curie-Weiss temperature of\nθ= 689 ±2 K, which are consistent with a previous re-\nport [33]. The effective magnetic moment of Fe atom in\nFe3Sn is found to be µeff(Fe)=2 .51µBusing the relation\nµeff=2.828√\nC[37].\nFigs. 2(c) and 2(d) depict the magnetization isotherms\n[M(H)] measured at various sample temperatures for\nH∥yandH∥z, respectively. The saturation field\nforH∥yis around 0.5 T whereas it is 2 T for H∥z,\nclearly suggesting that Fe 3Sn has an easy-axis of mag-\nnetization parallel to the ab-plane. Also, the absence of\nhysteresis in the M(H) data for the filed applied parallel\nboth directions makes this system a good soft ferromag-\nnet. The saturation magnetic moment per Fe atom is\nfound to be Ms=2.2µBwhich is close to value of effec-\ntive magnetic moment µeff=2.51µB. From the mag-\nnetization isotherms shown in Figs. 2(c) and 2(d), we\nestimated the magnetocrystalline anisotropic energy den-\nsityKu= 1.02×106J/m3using the relation Ku=\nµ0RMs\n0[Hy(M)−Hz(M)]dMafter excluding the geo-\nmetrical demagnetization factor [38]. See supplemental\nmaterial for more details on the calculations of demag-\nnetization factor [39]. Here, Msrepresents saturationmagnetization, HzandHyrepresent H∥zandH∥y,\nrespectively.\nHall resistivity ( ρzx) as a function of field is shown\nin Fig. 2(e) measured at various sample temperatures.\nHere, the current is applied along the z-axis and the mag-\nnetic field is applied along the y-axis to measure the Hall\nvoltage along the x-axis of the crystal as depicted in the\ninset of Fig. 2(e). It is evident from Fig. 2(e) that Fe 3Sn\nshows anomalous Hall effect with a little normal Hall ef-\nfect that is visible at very high temperatures. Next, the\nHall conductivity ( σzx) is calculated using the formula\nσzx=−ρzx\nρ2zx+ρ2zz, where ρzzis the longitudinal resistivity\nmeasured along the z-axis of the crystal. Fig. 2(f) shows\nσzxplotted as a function of field in which we observe a\nlarge anomalous Hall conductivity in the range of 410-425\nS-cm−1between 75 and 250 K when measured at 5 T,\nand a sudden drop to 370 S-cm−1is observed upon low-\nering the sample temperature below 75 K. Though high\ntemperature Hall conductivity obtained in this study is\nconsistent with previous report made on polycrystalline\nFe3Sn [35], the low temperature Hall conductivity of 200\nS-cm−1at 2 K observed in Ref. [35] is significantly smaller\nthan our findings ( ≈370 S-cm−1at 5 K).\nNext, Fig. 3(a) depicts temperature dependent longi-\ntudinal resistivity ( ρzz) measured with current applied\nalong the z-axis of the crystal. We notice that ρzz\nquadratically depends on the temperature ( ρzz∝T2)\nup to ≈75 K, demonstrating a Fermi-liquid type nature\nof the resistivity at low temperatures [40]. Nevertheless,\nabove 75 K, ρzzshows linear depends on the tempera-\nture ( ρzz∝T) due to a strong electron-phonon interac-\ntion [41]. The temperature dependent resistivity profile4\n400\n350\n300\n250\n250 200 150 100 505\n4\n3\n2\n1\n250 200 150 100 505\n4\n3\n2\n1\n250 200 150 100 502.0\n1.5\n1.0\n0.5\n250 200 150 100 50\n250\n200\n150\n100\n50\n250 200 150 100 50\n80\n70\n60\n50\n40\n250 200 150 100 502.8\n2.4\n2.0\n1.6\n1.2\nT(K)( )\nT(K)T(K)\nT(K)R0(10-4cm3/C)\nn(1022cm-3)T(K)RS(10-2cm3/C)\nSH(10-1V‒1)\nT(K)( )(a) (c) (e)\n(b) (d) (f)ρzzβTαT2\n(108S2/cm2)( )\nExp. data\n−ߪ௭௫=ߩ௭௫௫௧ߪ௭௭ଶ+ܾ\nT(K)( )\nߪ௭௫௫௧=ߪ௭௫௫௧\n(ܶܽ+ 1)ଶExp. dataAHA(%)\nT (K)ߪzz\n(104)\nFIG. 3. (a) Longitudinal electrical resistivity ( ρzz) and electrical conductivity ( σzz) plotted as a function of temperature. (b)\nand (c) Normal Hall coefficient ( R0) and Anomalous Hall coefficient ( RS) plotted as a function of temperature, respectively.\nCharge carrier density ( n) is shown in the inset of (b). Top inset in (c) shows anomalous Hall scale factor ( SH) and bottom inset\nin (c) shows anomalous Hall angle percentage (AHA%). (d) Anomalous Hall conductivity (- σA\nzx) as a function of temperature.\n(e) -σA\nzxvs. σ2\nzzplot. The dashed line in (e) is a linear fitting using the relation - σA\nzx=ρext\nzx0σ2\nzz+b. (f) Extrinsic anomalous Hall\nconductivity σext\nzxplotted as a function of temperature. The solid green curve in (f) is a fit with the equation σext\nzx=σext\nzx0\n(aT+1)2.\nof our Fe 3Sn single crystal is consistent with previous re-\nport on the polycrystalline Fe 3Sn [35]. Particularly, the\nlinear dependence of ρzzabove 75 K is in very good agree-\nment with Ref. [35]. Temperature dependent longitudinal\nconductivity ( σzz) also is shown in Fig. 3(a). In general,\nthe total Hall resistivity presented in Fig. 2(e) can be\nexpressed by the empirical formula, ρH=ρN\nH+ρA\nH[5].\nWhere, the first term represents normal Hall contribution\nρN\nH=µ0R0Hand the second term represents the anoma-\nlous Hall contribution which in turn depends on the mag-\nnetization ( M) asρA\nH=µ0RSM. Here, R0andRSare\nnormal and the anomalous Hall coefficients, respectively.\nFurther, with the help of normal Hall coefficient ( R0) one\ncan calculate the charge carrier ( q) density using the re-\nlation, n=1\nR0|q|. Since the field dependent anomalous\nHall effect is a replica of magnetization, anomalous Hall\nresistivity saturates beyond a critical field and becomes\nalmost field independent while the normal Hall resistivity\nlinearly depends with field.\nThus, in order to separate the anomalous Hall resistiv-\nity from the normal Hall contribution, we fitted the high\nfield region of the total Hall resistivity with a linear func-\ntion of field and subtracted the normal Hall contribution\nfrom the total Hall resistivity. In this way, we extracted\nvarious important parameters such as the normal ( R0)\nand anomalous ( RS) Hall coefficients. Fig. 3(b) depictsR0plotted as a function of temperature. The positive\nR0values throughout the measured temperature range as\nobserved in Fig. 3(b) suggest hole-carrier dominant elec-\ntrical transport in Fe 3Sn. Inset of Fig. 3(b) demonstrates\nalmost temperature independent hole carrier density ( nh)\nwhich is of the order of ∼1022cm−3, suggesting Fe 3Sn\nto be a good metal. Fig. 3(c) shows the anomalous Hall\ncoefficient (R S) plotted a function of temperature. Top-\nleft inset of Fig. 3(c) presents the anomalous Hall scaling\nfactor ( SH), defined as SH=−σA\nzx\nM=µ0RS\nρzz2(∼=ρA\nzx\nMρzz2),\nplotted as a function of temperature. We observe that\nSHis almost temperature independent within the error-\nbars. Moreover, the value of SH=0.03±0.01V−1derived\nin this study is within the range of 0.01-0.2 V−1, for any\ntypical ferromagnetic metal [42, 43]. Bottom-right in-\nset of Fig. 3(c) shows temperature dependent anomalous\nHall angle (AHA), defined as the deviation of electron\nflow from the current direction, is calculated using the\nformula AHA(%)=σzx\nσzz×100(%). We clearly notice that\nAHA(%) ≈3% at 250 K which decreases with tempera-\nture. The value of AHA(%) ≈3% is close to the AHA\nvalues reported on a similar Kagome ferromagnetic sys-\ntem Fe 3Sn2(≈1.1%) [43], Kagome antiferromagnetic sys-\ntems such as Mn 3Sn (≈3.2%) [6] and Mn 3Ge (≈5%) [7],\nbut much smaller than the Shandite Kagome ferromag-\nnet Co 3Sn2S2(≈20%) [28].5\nSeveral mechanisms were proposed for understanding\nthe anomalous Hall effect in magnetic and nonmagnetic\nmetals. In most of the proposals, the anomalous Hall\nresistivity ( ρA\nzx) is represented mainly by the function\nof longitudinal resistivity ( ρzz),ρA\nzx=f(ρzz). More\nexplicitly, (i) The intrinsic Karplus-Luttinger mecha-\nnism of AHE describes the Hall resistivity ∝ρ2\nzzdue\nto the interband scattering in presence of strong spin-\norbit coupling [8], (ii) The extrinsic side-jump mecha-\nnism of AHE describes the Hall resistivity ∝ρ2\nzzdue\nto side-jump scattering of charge carriers at the impu-\nrities [11], (iii) The extrinsic skew-scattering mechanism\nof AHE describes the Hall resistivity ∝ρzzdue to skew-\nscattering of charge carrier at the impurities [10], (iv) a\nρα\nzz(1< α < 2) dependency of ρA\nzxwas proposed in the\ncase of bad metals [44]. For our case, the mechanism\n(iv) can be ignored as the longitudinal conductivity of\nFe3Sn is found to be σzz=31.4e2\nhc(where cis the lat-\ntice constant) which is in the metallic regime [45]. Re-\ncently, a new mechanism (TYJ) was proposed by Tian\net. al. [46] in order to understand the AHE in ferro-\nmagnetic metals. According to TYJ theory, the anoma-\nlous Hall resistivity is described by ρA\nzx=f(ρzz0, ρzz)\nwhich includes the residual resistivity ( ρzz0). Thus, fol-\nlowing the TYJ theory [46, 47], the anomalous Hall re-\nsistivity takes the form ρA\nzx= (αρzz0+βρ2\nzz0) + bρ2\nzz\nand the anomalous Hall conductivity is represented by\n−σA\nzx= (ασ−1\nzz0+βσ−2\nzz0)σ2\nzz+b=ρext\nzx0σ2\nzz+b. Here, αand\nβare real constants, σzz0= 1/ρzz0is the residual conduc-\ntivity, and bis the intrinsic Hall conductivity originated\nfrom the momentum space Berry curvature. It is well\nknown that the intrinsic Hall conductivity ( b) is usually\ntemperature independent [5, 28] except for the systems\nshowing electronic or magnetic phase transitions [9, 48].\nSince Fe 3Sn shows neither electronic nor magnetic tran-\nsition within the measured temperature range of 2-250\nK, change in the Berry phase is not expected. From\nFig. 3(d), we notice that the anomalous Hall conductivity\nis almost constant at higher temperatures, but decreases\nbelow 150 K. To understand this phenomenon, we em-\nployed TYJ mechanism to fit the data of - σA\nzxvs. σ2\nzzas\nshown in Fig. 3(e), demonstrating a good fitting within\nthe error-bars. From the fitting , we extracted intrinsic\nHall conductivity b=485±60 S/cm that is close to the\npreviously reported values on polycrystalline Fe 3Sn (≈\n500 S/cm) [35] and predicted by a theoretical calculation\n(≈600 S/cm at EF) [49].\nNext, Fig. 3(f) shows the extrinsic Hall conductivity\n(σext\nzx) extracted from the total conductivity by subtract-\ning the intrinsic Hall contribution ( b). From Fig. 3(f), it\nis evident that the extrinsic Hall conductivity decreases\nwith increasing sample temperature. Further, the sign\nof extrinsic Hall conductivity is opposite to the intrin-\nsic Hall conductivity, resulting into the reduced total\nHall conductivity at lower temperatures. As we know,\nin the clean limit, for ℏ/τ→0 (τis the relaxation time)\nthe extrinsic skew-scattering contribution diverges [50].\nOn the other hand, from Fig. 3(f), we can see that asthe sample temperature decreases the extrinsic Hall con-\nductivity rapidly increases. This observation indicates\nthat skew-scattering playing a major role to generate the\nanomalous Hall conductivity in Fe 3Sn. The longitudi-\nnal resistivity ( ρzz) suggests electron-phonon scattering\nat higher temperatures in this system. In order to un-\nderstand the phonon influence on the anomalous Hall\nconductivity, we employed the mechanism proposed by\nShitade and Nagaosa [45] which involves the electron-\nphonon inelastic scattering rates ( γ). In this mecha-\nnism, the extrinsic Hall conductivity decays with γfol-\nlowing the relation σext\nzx=σext\nzx0\n(γ/γ 0+1)2. Since the inelas-\ntic scattering rate is proportional to the longitudinal re-\nsistivity [ γ∼(ρzz−ρzz0)] and ρzz−ρzz0linearly de-\npends on temperature [( ρzz−ρzz0)∝T] above 75 K [see\nFig. 3(a)], we can rewrite the equation as σext\nzx=σext\nzx0\n(aT+1)2.\nHere, a= 0.011±0.003 K−1represents the measure\nof inelastic electron-phonon scattering strength. In this\nway, we could fit the σext\nzxdata very well as shown in\nFig. 3(f). Thus, our results clearly demonstrate the influ-\nence of electron-phonon interaction on the extrinsic skew-\nscattering anomalous Hall conductivity which decreases\nwith increasing temperature. Finally, we would like to\nmention here that during the course of our manuscript\npreparation a preprint on the magnetic studies of Fe 3Sn\nsingle crystals has appeared [51]. The magnetic prop-\nerties presented in their study are consistent with our\nfindings. Particularly, the magnetocrystalline anisotropy\nenergy density of 1.23 ×106J/m3reported in Ref. [51]\nis in good agreement with the value of 1.02 ×106J/m3\nfound in this study.\nIV. SUMMARY\nIn summary, we have successfully grown high quality\nsingle crystals of Fe 3Sn. In this study, we mainly fo-\ncussed on understanding the anomalous Hall effect as\na function temperature. Our results unravel two main\ncontributions to the total Hall conductivity of Fe 3Sn.\nOne of them is the temperature independent intrinsic\nHall conductivity originated from the electronic band\nstructure and the other one is the temperature depen-\ndent extrinsic Hall conductivity due to the asymmetric\nskew-scattering. Most importantly, we find that the ex-\ntrinsic skew-scattering Hall conductivity is significantly\ninfluenced by the electron-phonon scattering at higher\ntemperatures and obeys the relation σext\nzx=σext\nzx0\n(aT+1)2. In\naddition, the longitudinal electrical resistivity ( ρzz) con-\nfirm the presence of electron-phonon scattering as the re-\nsistivity linearly depends on the temperature. We show\nthat Fe 3Sn is a soft ferromagnet with easy-axis magne-\ntization lying parallel to the abplane. We estimate a\nmagnetocrystalline anisotropic energy density as large as\n1.02×106J/m3in Fe 3Sn.6\nACKNOWLEDGMENTS\nThe authors thank the Science and Engineering Re-\nsearch Board (SERB), Department of Science and Tech-\nnology (DST), India for the financial support (Grant No.SRG/2020/000393). This research has made use of the\nTechnical Research Centre (TRC) Instrument Facilities\nof S. N. Bose National Centre for Basic Sciences, estab-\nlished under the TRC project of Department of Science\nand Technology, Govt. of India.\n[1] E. H. Hall, On a new action of the magnet on electric cur-\nrents, American Journal of Mathematics 2, 287 (1879).\n[2] R. S. Popovic, Hall effect devices (CRC Press, 2003).\n[3] E. Ramsden, Hall-effect sensors: theory and application\n(Elsevier, 2011).\n[4] E. Hall, On the new action of magnetism on a permanent\nelectric current, The London, Edinburgh, and Dublin\nPhilosophical Magazine and Journal of Science 10, 301\n(1880).\n[5] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and\nN. P. Ong, Anomalous Hall effect, Rev. Mod. Phys. 82,\n1539 (2010).\n[6] S. Nakatsuji, N. Kiyohara, and T. Higo, Large anomalous\nHall effect in a non-collinear antiferromagnet at room\ntemperature, Nature 527, 212 (2015).\n[7] A. K. Nayak, J. E. Fischer, Y. Sun, B. Yan, J. Karel,\nA. C. Komarek, C. Shekhar, N. Kumar, W. Schnelle,\nJ. K¨ ubler, C. Felser, and S. S. P. Parkin, Large anoma-\nlous Hall effect driven by a nonvanishing Berry curvature\nin the noncolinear antiferromagnet Mn 3Ge, Sci. Adv. 2,\ne1501870 (2016).\n[8] R. Karplus and J. M. Luttinger, Hall Effect in Ferromag-\nnetics, Phys. Rev. 95, 1154 (1954).\n[9] K. Manna, L. Muechler, T.-H. Kao, R. Stin-\nshoff, Y. Zhang, J. Gooth, N. Kumar, G. Kreiner,\nK. Koepernik, R. Car, J. K¨ ubler, G. H. Fecher,\nC. Shekhar, Y. Sun, and C. Felser, From Colossal to\nZero: Controlling the Anomalous Hall Effect in Magnetic\nHeusler Compounds via Berry Curvature Design, Phys.\nRev. X 8, 041045 (2018).\n[10] J. Smit, The spontaneous hall effect in ferromagnetics II,\nPhysica 24, 39 (1958).\n[11] L. Berger, Side-Jump Mechanism for the Hall Effect of\nFerromagnets, Phys. Rev. B 2, 4559 (1970).\n[12] S. K. Lyo and T. Holstein, Side-Jump Mechanism for Fer-\nromagnetic Hall Effect, Phys. Rev. Lett. 29, 423 (1972).\n[13] B. Sutherland, Localization of electronic wave functions\ndue to local topology, Phys. Rev. B 34, 5208 (1986).\n[14] E. H. Lieb, Two theorems on the Hubbard model, Phys.\nRev. Lett. 62, 1201 (1989).\n[15] D. Leykam, A. Andreanov, and S. Flach, Artificial flat\nband systems: from lattice models to experiments, Ad-\nvances in Physics: X 3, 1473052 (2018).\n[16] C. Wu, D. Bergman, L. Balents, and S. Das Sarma, Flat\nBands and Wigner Crystallization in the Honeycomb Op-\ntical Lattice, Phys. Rev. Lett. 99, 070401 (2007).\n[17] L. Savary and L. Balents, Quantum spin liquids: a re-\nview, Rep. Prog. Phys. 80, 016502 (2016).\n[18] C. Broholm, R. J. Cava, S. A. Kivelson, D. G. Nocera,\nM. R. Norman, and T. Senthil, Quantum spin liquids,\nScience 367, eaay0668 (2020).\n[19] C. L. Kane and E. J. Mele, Quantum Spin Hall Effect in\nGraphene, Phys. Rev. Lett. 95, 226801 (2005).[20] H. Huang, Z. Liu, H. Zhang, W. Duan, and D. Vander-\nbilt, Emergence of a Chern-insulating state from a semi-\nDirac dispersion, Phys. Rev. B 92, 161115 (2015).\n[21] P. B. Pal, Dirac, Majorana, and Weyl fermions, Amer. J.\nPhys. 79, 485 (2011).\n[22] G. Xu, H. Weng, Z. Wang, X. Dai, and Z. Fang, Chern\nSemimetal and the Quantized Anomalous Hall Effect in\nHgCr 2Se4, Phys. Rev. Lett. 107, 186806 (2011).\n[23] K. Landsteiner, Anomalous transport of Weyl fermions\nin Weyl semimetals, Phys. Rev. B 89, 075124 (2014).\n[24] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang,\nM. I. Katsnelson, I. V. Grigorieva, S. Dubonos,\nand a. Firsov, Two-dimensional gas of massless Dirac\nfermions in graphene, Nature 438, 197 (2005).\n[25] K. Nomura and A. H. MacDonald, Quantum Transport\nof Massless Dirac Fermions, Phys. Rev. Lett. 98, 076602\n(2007).\n[26] N. Nagaosa and Y. Tokura, Topological properties and\ndynamics of magnetic skyrmions, Nat. Nanotechnol. 8,\n899 (2013).\n[27] A. Fert, N. Reyren, and V. Cros, Magnetic skyrmions:\nadvances in physics and potential applications, Nat. Rev.\nMater. 2, 1 (2017).\n[28] E. Liu, Y. Sun, N. Kumar, L. Muechler, A. Sun, L. Jiao,\nS.-Y. Yang, D. Liu, A. Liang, Q. Xu, et al. , Giant\nanomalous Hall effect in a ferromagnetic kagome-lattice\nsemimetal, Nat. Phys. 14, 1125 (2018).\n[29] N. Morali, R. Batabyal, P. K. Nag, E. Liu, Q. Xu,\nY. Sun, B. Yan, C. Felser, N. Avraham, and H. Bei-\ndenkopf, Fermi-arc diversity on surface terminations of\nthe magnetic Weyl semimetal Co 3Sn2S2, Science 365,\n1286 (2019).\n[30] L. Ye, M. Kang, J. Liu, F. von Cube, C. R. Wicker,\nT. Suzuki, C. Jozwiak, A. Bostwick, E. Rotenberg, D. C.\nBell, L. Fu, R. Comin, and J. G. Checkelsky, Massive\nDirac fermions in a ferromagnetic kagome metal, Nature\n555, 638 (2018).\n[31] Z. Hou, W. Ren, B. Ding, G. Xu, Y. Wang, B. Yang,\nQ. Zhang, Y. Zhang, E. Liu, F. Xu, W. Wang, G. Wu,\nX. Zhang, B. Shen, and Z. Zhang, Observation of Var-\nious and Spontaneous Magnetic Skyrmionic Bubbles at\nRoom Temperature in a Frustrated Kagome Magnet with\nUniaxial Magnetic Anisotropy, Advanced Materials 29,\n1701144 (2017).\n[32] H. Li, B. Ding, J. Chen, Z. Li, Z. Hou, E. Liu, H. Zhang,\nX. Xi, G. Wu, and W. Wang, Large topological Hall ef-\nfect in a geometrically frustrated kagome magnet Fe 3Sn2,\nAppl. Phys. Lett. 114, 192408 (2019).\n[33] B. C. Sales, B. Saparov, M. A. McGuire, D. J. Singh, and\nD. S. Parker, Ferromagnetism of Fe 3Sn and alloys, Sci.\nRep.4, 1 (2014).\n[34] O. Y. Vekilova, B. Fayyazi, K. P. Skokov, O. Gutfleisch,\nC. Echevarria-Bonet, J. M. Barandiar´ an, A. Kovacs,\nJ. Fischbacher, T. Schrefl, O. Eriksson, and H. C. Her-7\nper, Tuning the magnetocrystalline anisotropy of Fe 3Sn\nby alloying, Phys. Rev. B 99, 024421 (2019).\n[35] T. Chen, S. Minami, A. Sakai, Y. Wang, Z. Feng,\nT. Nomoto, M. Hirayama, R. Ishii, T. Koretsune,\nR. Arita, and S. Nakatsuji, Large anomalous Nernst effect\nand nodal plane in an iron-based kagome ferromagnet,\nSci. Adv. 8, eabk1480 (2022).\n[36] J. Rodr´ ıguez-Carvajal, FullProf, CEA/Saclay, France\n1045 , 132 (2001).\n[37] S. Mugiraneza and A. M. Hallas, Tutorial: a begin-\nner’s guide to interpreting magnetic susceptibility data\nwith the Curie-Weiss law, Communications Physics 5,\n95 (2022).\n[38] R. Prozorov and V. G. Kogan, Effective Demagnetizing\nFactors of Diamagnetic Samples of Various Shapes, Phys.\nRev. Appl. 10, 014030 (2018).\n[39] See supplemental information for the calculations on ge-\nometrical demagnetizaiton factor.\n[40] K. Behnia, On the Origin and the Amplitude of T-Square\nResistivity in Fermi Liquids, Ann. Phys. 534, 2100588\n(2022).\n[41] G. Varelogiannis and E. N. Economou, Small-q electron-\nphonon scattering and linear dc resistivity in High-T c\noxides, Europhys. Lett. 42, 313 (1998).\n[42] C. Zeng, Y. Yao, Q. Niu, and H. H. Weitering, Linear\nMagnetization Dependence of the Intrinsic Anomalous\nHall Effect, Phys. Rev. Lett. 96, 037204 (2006).\n[43] Q. Wang, S. Sun, X. Zhang, F. Pang, and H. Lei, Anoma-\nlous Hall effect in a ferromagnetic Fe 3Sn2single crystal\nwith a geometrically frustrated Fe bilayer kagome lattice,\nPhys. Rev. B 94, 075135 (2016).\n[44] J. M. Lavine, Extraordinary Hall-Effect Measurements\non Ni, Some Ni Alloys, and Ferrites, Phys. Rev. 123,\n1273 (1961).\n[45] A. Shitade and N. Nagaosa, Anomalous Hall Effect in\nFerromagnetic Metals: Role of Phonons at Finite Tem-\nperature, J. Phys. Soc. Jpn. 81, 083704 (2012).\n[46] Y. Tian, L. Ye, and X. Jin, Proper Scaling of the Anoma-\nlous Hall Effect, Phys. Rev. Lett. 103, 087206 (2009).\n[47] D. Hou, G. Su, Y. Tian, X. Jin, S. A. Yang, and Q. Niu,\nMultivariable Scaling for the Anomalous Hall Effect,\nPhys. Rev. Lett. 114, 217203 (2015).\n[48] N. H. Sung, F. Ronning, J. D. Thompson, and E. D.\nBauer, Magnetic phase dependence of the anomalous Hall\neffect in Mn 3Sn single crystals, Appl. Phys. Lett. 112,\n132406 (2018).\n[49] C. Shen, I. Samathrakis, K. Hu, H. K. Singh, N. Fortu-\nnato, H. Liu, O. Gutfleisch, and H. Zhang, Thermody-\nnamical and topological properties of metastable Fe 3Sn,\nnpj Comput. Mater. 8, 248 (2022).\n[50] S. Onoda, N. Sugimoto, and N. Nagaosa, Intrinsic Versus\nExtrinsic Anomalous Hall Effect in Ferromagnets, Phys.\nRev. Lett. 97, 126602 (2006).\n[51] L. Prodan, D. M. Evans, S. M. Griffin, A. ¨Ostlin, M. Alt-\nthaler, E. Lysne, I. G. Filippova, S. Shova, L. Chioncel,\nV. Tsurkan, and I. K´ ezsm´ arki, Large ordered moment\nwith strong easy-plane anisotropy and vortex-domain\npattern in the kagome ferromagnet Fe 3Sn (2023).\n[52] M. Kang, L. Ye, S. Fang, J.-S. You, A. Levitan, M. Han,\nJ. I. Facio, C. Jozwiak, A. Bostwick, E. Rotenberg, et al. ,\nDirac fermions and flat bands in the ideal kagome metal\nFeSn, Nat. Mater. 19, 163 (2020).[53] M. Han, H. Inoue, S. Fang, C. John, L. Ye, M. K. Chan,\nD. Graf, T. Suzuki, M. P. Ghimire, and W. J. Cho, Evi-\ndence of two-dimensional flat band at the surface of an-\ntiferromagnetic kagome metal FeSn, Nat. Commun. 12,\n5345 (2021).\n[54] K. Kuroda, T. Tomita, M.-T. Suzuki, C. Bareille, A. Nu-\ngroho, P. Goswami, M. Ochi, M. Ikhlas, M. Nakayama,\nS. Akebi, et al. , Evidence for magnetic Weyl fermions in\na correlated metal, Nat. Mater. 16, 1090 (2017).\n[55] H. Yang, Y. Sun, Y. Zhang, W.-J. Shi, S. S. P. Parkin,\nand B. Yan, Topological Weyl semimetals in the chiral\nantiferromagnetic materials Mn 3Ge and Mn 3Sn, New J.\nPhys. 19, 015008 (2017).\n[56] T. McGuire and R. Potter, Anisotropic magnetoresis-\ntance in ferromagnetic 3d alloys, IEEE Trans. Magn. 11,\n1018 (1975).\n[57] M. Kataoka, Resistivity and magnetoresistance of ferro-\nmagnetic metals with localized spins, Phys. Rev. B 63,\n134435 (2001).\n[58] D. A. Goodings, Electrical Resistivity of Ferromagnetic\nMetals at Low Temperatures, Phys. Rev. 132, 542\n(1963).\n[59] J. S. Moodera and D. M. Mootoo, Nature of half-metallic\nferromagnets: Transport studies, J. Appl. Phys. 76, 6101\n(1994).\nV. SUPPLEMENTARY INFORMATION8\n-6-4-20246\n-5 -4 -3 -2 -1 0 1 2 3 4 56\n4\n2\n0\n-2\n-4\n-6\n-5 -4 -3 -2 -1 0 1 2 3 4 55 K\n50 K\n100 K\n150 K\n200 K\n250 K5 K\n100 KM( )\ny[011ത0]\nx[21ത1ത0]H || y 5 K\n50 K\n100 K\n150 K\n200 K\n250 KM( )\nμ0Heff(T)\nz[0001]H || z(a) (b)\nμ0Heff(T)\nFIG. 4. Magnetization isotherms [ M(Heff)] measured at various sample temperatures for H∥y(a) and H∥z(b). Here, Heff\nis the effective magnetization field after excluding the shape demagnetizing factor of the approximately rod-shaped sample\nusing the formula, Heff=H−NdM, where Ndis the demagnetization factor. For our rod-shaped sample, the calculated\nN−1\nd= 2 +1√\n(2)a\ncforH∥yis 0.45 and N−1\nd= 1 + 1 .6c\naforH∥zis 0.15 [38]. Here, a=0.2 mm and c=0.7 mm are the sample\ndiameter and length, respectively."    },    {        "title": "2308.11183v1.Tuning_of_Electrical__Magnetic__and_Topological_Properties_of_Magnetic_Weyl_Semimetal_Mn___3_x__Ge_by_Fe_doping.pdf",        "content": "Tuning of Electrical, Magnetic, and Topological\nProperties of Magnetic Weyl Semimetal Mn 3+xGe\nby Fe doping\nSusanta Ghosh, Achintya Low, Soumya Ghorai, Kalyan\nMandal, and Setti Thirupathaiah∗\nDepartment of Condensed Matter and Materials Physics, S. N. Bose National Centre\nfor Basic Sciences, Kolkata, West Bengal-700106, India\nE-mail: setti@bose.res.in\n23 August 2023\nAbstract. We report on the tuning of electrical, magnetic, and topological\nproperties of the magnetic Weyl semimetal (Mn 3+xGe) by Fe doping at the Mn site,\nMn (3+x)−δFeδGe (δ=0, 0.30, and 0.62). Fe doping significantly changes the electrical\nand magnetic properties of Mn 3+xGe. The resistivity of the parent compound displays\nmetallic behavior, the system with δ=0.30 of Fe doping exhibits semiconducting or\nbad-metallic behavior, and the system with δ=0.62 of Fe doping demonstrates a metal-\ninsulator transition at around 100 K. Further, we observe that the Fe doping increases\nin-plane ferromagnetism, magnetocrystalline anisotropy, and induces a spin-glass state\nat low temperatures. Surprisingly, topological Hall state has been noticed at a Fe\ndoping of δ=0.30 that is not found in the parent compound or with δ=0.62 of Fe\ndoping. In addition, spontaneous anomalous Hall effect observed in the parent system\nis significantly reduced with increasing Fe doping concentration.\n1. Introduction\nTopological materials with kagome lattice structure offer many intriguing quantum\nphenomena such as the Weyl [1, 2] and Dirac fermions [3, 4] in solids, quantum spin\nliquid state [5], superconductivity [6], anomalous Hall effect (AHE) [7], topological Hall\neffect (THE) [8], and skyrmion lattice [9, 10]. Though there exist many topological\nkagome systems, the topological kagome magnets have gained much attention due to\ntheir potential technological applications in spintronics as well as from a fundamental\nscience point of view due to strong electronic correlations present in these systems\n[11, 12]. For instance, Co 3Sn2S2is a ferromagnetic Weyl semimetal showing giant AHE\nand zero-field Nernst effect [7, 13]. The noncollinear and coplanar antiferromagnetic\n(AFM) Weyl semimetals, Mn 3X(X = Sn and Ge), show large AHE originating from\nthe nonzero Berry phase in momentum space [14, 15, 16, 17, 18]. On the other hand,arXiv:2308.11183v1  [cond-mat.mtrl-sci]  22 Aug 20232\nferromagnetic Weyl semimetal Fe 3Sn2shows both THE and AHE [19]. A nonzero Berry\nphase in the momentum space leads to an intrinsic anomalous Hall effect. In contrast,\nthe nonzero total scalar-spin chirality [ χijk=Si.(Sj×Sk)] summed over the lattice,\nproducing the nonzero real-space Berry curvature, generates the topological Hall effect\n[20, 21, 17].\nIn noncentrosymmetric magnets such as MnSi [22, 23], FeGe [24, 25], MnGe\n[9], and FeCoSi [26], THE originates from the chiral-spin structure stabilized by\nthe Dzyaloshinskii-Moriya interaction (DMI).On the other hand, in centrosymmetric\nmagnets such as La 1−xSrxMnO 3[27], Fe 3Sn2[28, 29], and Mn 3−δFeδSn[30], THE\noriginates from the chiral-spin structure stabilized by strong magnetic anisotropy. To\ndate, many systems of various magnetic orderings have been found to show THE,\nincluding the skyrmionic crystals [22, 31, 32], the antiferromagnets (AFM) [33, 34],\nthe spin glass systems [35, 36], the frustrated magnets [37, 38], the double-exchanged\nferromagnets [39, 40], and the magnetic skyrmion arrays [41, 42, 43, 44]. Among them,\nthe frustrated kagome magnets such as PdCrO 2[38], Gd2PdSi 3[45], and Mn 3+xSn\nandFe3Sn2[8, 17] are more interesting as they are rich in physics due to the\ntriangular spin-lattice in the kagome network. Mn 3Ge, like its sister compound Mn 3Sn,\nis a noncollinear antiferromagnet with a hexagonal crystal structure of space group\nP63/mmc [46, 14, 16, 15]. Here, the Mn atoms form the kagome network with triangles\nand hexagons in the ab-plane of the crystal, with the Ge atom sitting at the centre\nof hexagon [47, 48, 15, 16]. Mn 3+xSnshows a spin-reorientation transition at around\n260–270 K, while no such spin-reorientation transition is found in Mn 3+xGedown to\nthe lowest possible temperature. Further, Mn 3+xSnshows a low-temperature spin-glass\ntransition at around 50 K but is not found in Mn 3+xGe. Earlier, we found that Fe doping\nat the Mn site can significantly tune the electrical, magnetic, and magnetotransport\nproperties of Mn 3−δFeδSn[30].\nIn this contribution, we systematically studied the electrical, magnetic, and Hall\neffect properties of Mn (3+x)−δFeδGe(δ=0, 0.30, and 0.62) by varying the Fe doping\nconcentration. Fe doping significantly changes the electrical resistivity and magnetic\nproperties of Mn 3+xGe. Most importantly, we observe a topological Hall state only in the\ncompound with δ=0.30 of Fe doping but not in the parent system and the system with\nδ=0.62 of Fe doping. In addition, increasing the Fe doping concentration significantly\nreduces the spontaneous anomalous Hall effect observed in the parent system. In the\nmanuscript, we thoroughly explain our experimental observations.\n2. Experimental details\nSingle crystals of Mn (3+x)−δFeδGe (δ=0, 0.30, and 0.62) were prepared by the melt-\ngrowth method using a high-temperature muffle furnace. In this method, Manganese\n(Alfa Aesar 99.95%), Iron (Alfa Aesar 99.99%), and Germanium (Alfa Aesar 99.999%)\npowders were taken in stoichiometric ratio, mixed thoroughly in the glove-box under an\nargon environment before sealing them in a preheated quartz ampoule under a vacuum3\nTable 1: Lattice parameters obtained from the single crystal XRD (SCXRD) measurements.\nComposition a ( ˚A) b ( ˚A) c ( ˚A) α(o) β(o) γ(o)\nMn3.48Ge (δ=0) 5.350(5) 5.352(5) 4.308(9) 90.06 (17) 90.18 (16) 120.00 (2)\nMn2.97Fe0.30Ge (δ=0.30)5.313(2) 5.314(2) 4.302(18) 90.06 (3) 89.91 (3) 119.93 (4)\nMn2.69Fe0.62Ge (δ=0.62)5.294(6) 5.282(8) 4.281(11) 90.5 (2) 89.7 (2) 120.00 (3)\nof 10–4mbar. The sealed quartz tube was then heated in a muffle furnace up to 1050oC\nand kept at that temperature for 24 hours. The tube was slowly cooled to 760oC for δ=0,\n780oC for δ=0.30, and 800oC for δ=0.62 at a rate of 2 K/h. After prolonged annealing\nat the respective growth temperatures for five more days, the ampules were quenched\nin ice water to avoid the impurity phases. As-grown single crystals of Mn (3+x)−δFeδGe\nwere looking shiny and had a typical size of 2 ×1.5 mm2.\nPhase purity and crystal structure of the single crystals were examined by using\nsingle crystal X-ray diffraction (SCXRD, SuperNova, Rigaku) and powder X-ray\ndiffraction (XRD, Rigaku SmartLab) with Cu-K αradiation of wavelength λ=1.5406 ˚A.\nUsing energy dispersive X-ray spectroscopy (EDS of EDAX), we find the actual chemical\ncomposition of the as-prepared samples to be Mn 3.48±0.02Ge, Mn 2.97±0.05Fe0.30±0.02Ge, and\nMn2.69±0.04Fe0.62±0.02Ge. Usually, Mn 3+xGe forms with excess Mn without any control\non the Mn concentration [47, 16, 49, 46]. Thus, we also got the crystals with excess Mn.\nFor convinience, we represent the systems Mn 3.48Ge, Mn 2.97Fe0.30Ge, and Mn 2.69Fe0.62Ge\nin the manuscript by δ=0, 0.30, and 0.62, respectively, wherever applicable. Electrical\ntransport, Hall effect, and magnetic property studies were performed using a 9-Tesla\nphysical property measurement system (PPMS, Dynacool, Quantum Design) within\nthe temperature range of 2–380 K. Electrical transport and Hall measurements were\nperformed using the four-probe technique. Copper leads were attached to the sample\nusing EPO-TEK H20E silver epoxy.\n3. Results\n3.1. Structural Properties\nMn3Ge crystalizes into the Ni 3Sn-type hexagonal structure with a space group of\nP63/mmc (194), with the Mn atoms sitting on the ab-plane to form a breathing kagome\nlattice, while the Ge atoms sit at the center of the hexagons [15, 16]. A couple of\nkagome lattice planes are stacked along the c-axis per unit cell. XRD patterns taken4\non the single crystal of Mn (3+x)−δFeδGe is shown in Fig. 1, suggesting that the crystal\ngrowth is parallel to the c-axis. The bottom right inset in Fig. 1 shows zoomed-in\n(0002) peaks of the parent, δ=0.30, and δ=0.62 compounds. Here we observe that\nwith increased Fe doping, (0002) peaks are shifted to higher 2 θvalues. This indicates\na decrease in the lattice parameter c. Optical images of the as-grown single crystals\nof different compositions are shown in the middle insets of Fig. 1. Lattice parameters\nobtained from the single crystal XRD of δ=0, 0.30, and 0.62 compounds are tabulated\nin Tab. 1.\n(0002)\n(0004)\n(a)\n2ϴ(°)Intensity(a.u.)\n(0002)\nδ= 0δ= 0.30δ= 0.62\nFigure 1: Powder XRD patterns of Mn 3.48Ge (δ=0), Mn 2.97Fe0.30Ge (δ=0.30), and Mn 2.69Fe0.62Ge\n(δ=0.62) single crystals. Bottom-right inset shows zoomed-in (0002) peaks of δ=0, 0.30, and 0.62\ncompounds, demonstrating the shift in peak position with doping. Photographic image of the respective\ncompositions are shown.5\nFC\nZFC\nFC\nZFCH Ʇz\nH ‖ zFC\nZFC\nFC\nZFCH Ʇz\nH ‖ zFC\nZFC\nFC\nZFCH Ʇz\nH ‖ zM(µB/f.u.)\nµ0H (Tesla) µ0H (Tesla) µ0H (Tesla)T (K) T (K) T (K)\nH Ʇz\nH ‖ zH Ʇz\nH ‖ zH Ʇz\nH ‖ zT = 2K T = 2K T = 2K\nT = 200 K T = 200 KT = 200 Kµ0H = 500 Oeµ0H = 500 Oe\nµ0H = 500 Oe\nδ = 0.30 δ = 0.62 δ = 0(a) (c) (b)\n(e) (d) (f)M(µB/f.u.)\nM(µB/f.u.)M(µB/f.u.)\nM(µB/f.u.)\nM(µB/f.u.)TSG=15 KTt=100 K\nTN=298 K\nTSG=20 KTN=230 KTN=365 Kδ = 0.30 δ = 0.62 δ = 0\nFigure 2: Temperature-dependent magnetization [ M(T)] of Mn (3+x)−δFeδGe measured in zero-field-\ncooled (ZFC) and field-cooled (FC) modes with H∥zandH⊥zforδ=0 (a), δ=0.30 (b), and δ=0.62\n(c). Similarly, magnetization isotherms [ M(H)] measured at 2 K with H∥zandH⊥zforδ=0 (d),\nδ=0.30 (e), and δ=0.62 (f). Insets of (d), (e), and (f) are M(H) data measured at 200 K from δ=0,\n0.30, and 0.62, respectively. See Supplemental Material for the M(H) data measured at several other\nsample temperatures.\n3.2. Magnetic Properties\nTo explore the magnetic properties, magnetization as a function of temperature [ M(T)]\nwas measured in the field-cooled (FC) and zero-field-cooled (ZFC) modes under a\nmagnetic field ( H) of 500 Oe applied parallel ( H∥z) and perpendicular ( H⊥z)\nto the z-axis as shown in Figs. 2(a), 2(b), and 2(c) from the δ=0, 0.30, and 0.62\ncompounds, respectively. From Fig. 2(a), we identify an antiferromagnetic (AFM)\ntransition at a N´ eel temperature of T N=365 K in Mn 3.48Ge, consistent with previous\nreports on Mn 3+xGe where TNwas found between 365 K and 400 K depending on the\nxvalue [47, 15, 16, 46, 50]. Below TN, the magnetization increases with decreasing\ntemperature and saturates at very low temperatures for both H∥zandH⊥z\ndirections. From Fig. 2(b), we observe a similar behavior of magnetization from the\nδ=0.30 sample, except that TNis reduced to 298 K. A down-turn in the ZFC M(T)\ndata is noticed at a very low sample temperature of 15 K, which is clearly visible with\nH⊥z. This hints at a spin-glass-like transition induced by the Fe doping [51, 52, 53, 54].\nNext, from Fig. 2(c), we observe a further decrease in TNto 230 K with a Fe doping\nofδ=0.62 in addition to a sudden drop in magnetization at 100 K from both H∥z\nandH⊥zdirections. This kind of magnetization drop earlier reported on Mn 3Sn was\nunderstood as an AFM-to-AFM magnetic transition [55, 56]. Moreover, at 20 K, we6\nobserve a downturn in the ZFC M(T) data, clearly visible from the H⊥zdirection,\nwhich is sharper than δ=0.30 compound.\nInterestingly, from the M(T) of Mn 3.48Ge measured in the FC mode, we notice\nalmost two times higher in-plane magnetization ( H⊥z) than the out-of-plane\nmagnetization ( H∥z) at 2 K, indicating a strong magnetic anisotropy in this systems.\nSuch strong magnetic anisotropy in Mn 3.48Ge, despite being an AFM metal, mainly\noriginates from the geometrical frustration of the Mn magnetic moments within the\nkagome lattice plane, producing a finite net magnetic moment [47, 57]. Further, from\nFigs. 2(b) and 2(c) we can notice that the in-plane magnetization increases nearly three\ntimes and four times compared to the out-of-plane magnetization for the Fe doping\nconcentration of δ=0.30 and 0.62, respectively. Thus, the magnetic anisotropy rapidly\nincreases with Fe doping concentration.\nFigs. 2(d), 2(e), and 2(f) show magnetization isotherms [ M(H)] measured at 2 K\nwith H∥zandH⊥zdirections from the δ=0, 0.30, and 0.62 compounds, respectively.\nFrom Fig. 2(d), we observe a spontaneous magnetization at lower fields, which then\nlinearly increases with the field. Further, consistent with M(T) data [see Fig. 2(a)],\nthe in-plane spontaneous magnetization is nearly two times higher than that of out-\nof-plane magnetization at lower fields. Inset in Fig. 2(d) shows M(H) data taken\nat 200 K. Nevertheless, we do not find any significant difference in the M(H) data\nbetween 2 and 200 K. On the other hand, more interestingly, from the M(H) data of\ntheδ=0.30 compound [see Fig. 2(e)], we find a quite complex magnetization isotherm.\nThat means the out-of-plane ( H∥z) magnetization increases more rapidly than the\nin-plane ( H⊥z) magnetization with increasing applied field and eventually dominates\nthe in-plane magnetization beyond 0.3 T of applied field when measured at 2 K. We\nfurther notice that at 4 T of the applied field, the out-of-plane magnetization is nearly\nthree times higher than that of the in-plane. This observation is in stark contrast to\nthe parent system, where we observe dominating in-plane magnetization throughout the\napplied field up to 4 T. The inset in Fig. 2(e) shows the isotherm measured at 200 K,\ndemonstrating that the in-plane magnetization dominates the out-of-plane throughout\nthe applied field up to 4 T, which is similar to the parent system. Further, Fe doping\nofδ=0.62 converts the system more into an in-plane ferromagnet (with a sigmoid-like\nM(H) loop) and an out-of-plane antiferromagnet (with a linear M(H) loop), as shown\nin Fig. 2(f).\n3.3. Electrical and Magnetotransport Properties\nFig. 3(a) depicts the zero-field in-plane longitudinal electrical resistivity ( ρxx) of the\nδ=0, 0.30, and 0.62 compounds measured between 2 and 300 K. We notice from the\nresistivity that the parent compound, Mn 3.48Ge, exhibits overall a metallic behavior\nwith temperature that is in agreement with earlier reports on Mn 3+xGe [16]. On the\nother hand, with the Fe doping of δ=0.30, the system exhibits a semiconducting or\na bad-metallic behavior as ρxxincreases with decreasing temperature. With further7\nδ= 0.30\nδ= 0.62δ= 0ρxx(mΩ-cm) ρxy(µΩ-cm)\nµ0H (Tesla)ρxy(µΩ-cm)\nT (K)(a)\n(b)(c)\nδ= 0\nδ= 0.30\nδ= 0.62\nδ= 0\nδ= 0.30\nδ= 0.62(d)\n(e)\nTMI= 100 K\nμ0H = 1 T\nρxy(µΩ-cm) ρxy(µΩ-cm)\nFigure 3: Temperature-dependent (a) longitudinal resistivity ( ρxx) and (b) Hall resistivity ( ρxy)\nplotted for δ=0, 0.30, and 0.62. Inset in (b) is the zoomed-in image taken to demonstrate the increase\nin total Hall resistivity of δ=0.62 below 230 K. Field-dependent Hall resistivity ( ρxy) measured at\nvarious temperatures for δ=0 (c), δ=0.30 (d), and δ=0.62 (e) compounds. Inset in (d) and (e) are the\nzoomed-in images taken around zero field.\nincreasing the Fe doping to δ=0.62, we observe that the ρxxincreases with decreasing\ntemperature before a sharp drop in the resistivity takes place at 100 K [see Fig. 3(a)],\nbelow 100 K the system behaves like a metal. Such a sudden drop in the resistivity at\n100 K is possibly due to the in-plane AFM-to-AFM magnetic transition [see Fig. 2(c)]\nleading to the metal-insulator (MI) transition as reported earlier on Mn 3−δFeδSn [30]\nand on Mn 2.34Fe0.66Ge [58].\nFig. 3(b) depicts the Hall resistivity [ ρxy(T)]δ=0, 0.30, and 0.62 compounds\nmeasured as a function of temperature. Here, ρxystands for the in-plane Hall8\n-2-1012\n4 2 0 -2 -4-4-2024-4-2024\n-4-2024\nT (K)R0(10-2cm3/C) RS(cm3/C)(d)(c)(b)\nn(1022cm-3)2 K\n50 K\nµ0H (Tesla) µ0H (Tesla)ρH(µΩ.cm)5 K20 K(a)\n10 K 100 Kߩ௫௬\nߩே+ߩ\nߩ௫௬்\nߩ௫௬\nߩே+ߩ\nߩ௫௬்ߩ௫௬\nߩே+ߩ\nߩ௫௬்ߩ௫௬\nߩே+ߩ\nߩ௫௬்\nߩ௫௬\nߩே+ߩ\nߩ௫௬்\nߩ௫௬\nߩே+ߩ\nߩ௫௬்\nFigure 4: (a) Field-dependent in-plane Hall resistivity ( ρxy) measured at different temperature of\nδ=0.30. The red curves are experimental data and the solid-black curves are the fits using the Eqn. 1\nand green curves are the topological Hall resistivity ( ρT\nxy). (b) Normal Hall coefficient ( R0). (c) Carrier\ndensity plotted as a function of temperature. (d) Anomalous Hall coefficient ( RS) plotted as a function\nof temperature.\nresistivity ( ab-plane) measured with the current applied parallel to the x-axis and\nfield applied parallel to the z-axis, while the Hall voltage is measured along the y-\naxis. Hall resistivity was measured using both positive (+H) and negative (-H) fields\nto exclude the magnetoresistance contribution due to possible misalignment of the four-\nprobe connection. The resultant Hall resistivity was calculated using the formula,\nρH(H,T)−ρH(−H,T)\n2. Thus, from Fig. 3(b), we can observe that the ρxyof the parent\ncompound monotonically increases (negative value) with decreasing temperature which\nthen saturates at low temperatures. This observation agrees with previous reports on\nthis system [15]. ρxy(T) of the parent system bears a resemblance to the M(T) data\n[see Fig. 2(a)]. On the other hand, the ρxyofδ=0.30 compound rapidly increases\n(positive) with decreasing temperature, possibly due to a rapid increase in the out-of-9\nplane magnetization [see Fig. 5(d)]. Next, from the ρxyofδ=0.62, we can find significant\nHall signal only within the temperature range of 100–230 K, which is consistent with\ntheM(T) data as shown in Fig. 2(c) where we notice one antiferromagnetic transition\n(AFM) at TN=230 K and the other one (also AFM type) at Tt=100 K.\nField-dependent Hall resistivity [ ρxy(H)] measured at various temperatures is\nplotted in Figs. 3(c), 3(d), and 3(e) from the δ=0, 0.30, and 0.62 compounds,\nrespectively. From the parent compound, we observe a sudden jump in ρxynear the\nzero fields, which saturates with further increasing the field as shown in Fig. 3(c), which\nis a signature of the anomalous Hall effect (AHE). This observation is consistent with\nprevious reports on similar systems [15, 16]. Also, in agreement with ρxy(T) as shown in\nFig. 3(b), the saturated ρxy(H) value increases with decreasing temperature and becomes\nnearly constant below 100 K. On the other hand, with a Fe doping of δ=0.30, the AHE\nis significantly decreased, as we do not observe a sharp jump near the zero fields but only\na gradual increase in ρxywith the field. Particularly at higher temperatures ( >100 K),\nwe observe a linear dependence of ρxyon the applied field due to the dominating normal\nHall contribution. However, at lower temperatures ( <100 K), we still find a non-linear\nand hysteretic ρxy(H) behavior. Further increasing the Fe doping to δ=0.62, as shown\nin Fig. 3(e), the AHE signal is hardly visible at most of the measured temperatures\nexcept a small AHE is noticed at 110 and 200 K. Thus, the total Hall resistivity of\nδ=0.62 shown in Fig. 3(b) is mainly contributed by the normal Hall resistivity except\nfor the temperature range of 100-230 K where a small AHE contribution also present.\nRed-colored data in Fig. 4(a) depict the total Hall resistivity of δ=0.30 fitted using\nEqn. 1 measured at various sample temperatures. Black-colored data in Fig. 4(a) are\nthe fits using the Eqn. 1. We can notice that ρxy(H) is not fitted well at lower fields\nbecause of the topological Hall contribution. Therefore, to extract the topological Hall\ncontribution, one needs to subtract anomalous and normal Hall contributions from the\ntotal Hall resistivity such as ρT\nxy(H) =ρxy(H)-[ρN\nxy(H) +ρA\nxy(H)]. Green-colored data\nin Fig. 4(a) depict the obtained topological Hall resistivity plotted as a function of field\nat various temperatures. From Fig. 4(a), we notice that the topological Hall effect is\nmore significant at low temperatures with a maximum of ρT\nxy= 0.69 µΩ cm for a critical\nfield of 0.6 T at 2 K, which disappears at higher temperatures ( >50 K).\nρH=ρN\nH+ρA\nH=R0µ0H+RSµ0M (1)\nHere, R 0is the normal Hall coefficient and R Sis the anomalous Hall coefficient.\nNext, in addition to ρxy(H) ofδ=0.30, ρxy(H) ofδ=0 and 0.62 were also fitted by\nthe Eqn. 1 to estimate the values of normal (R 0) and anomalous (R S) Hall coefficients.\nFig. 4(b) shows temperature-dependent normal Hall coefficient. Since the normal Hall\ncoefficient R 0is related to carrier density ( n) and charge ( q) asR0=1\nn|q|, we can obtain\nthe carrier concentration as a function of temperature, shown in Fig. 4(c). From Fig.\n4(c), we can notice that the carrier concentration of the parent system is almost constant\nwith temperature. For δ=0.30, it hardly changes with temperature down to 50 K and\nwith further decreasing the temperature the carrier concentration rapidly increases. On10\nthe other hand, the carrier concentration of δ=0.62 is nearly the same at both 2 and 300\nK but it significantly decreases as the sample approaches 100 K. Such a change in carrier\ndensity is possibly related to the magnetic transition around this temperature. Further,\nthe anomalous Hall effect is quantified by the anomalous Hall coefficient (R S). From\nFig. 4(d), we can find large R Svalues for the parent system at 2 K, which decrease with\nincreasing temperature, consistent with the total Hall resistivity [see Fig. 3(b)]. On the\nother hand, fluctuating R Svalues are noticed in the δ=0.30 compound. To be precise,\npositive and negative RSvalues are found below and above the sample temperature of 50\nK, respectively. Such a crossover from positive to negative RSvalues indicates the sign\nchange in the anomalous Hall effect as observed in the inset of Fig. 3(d) [59]. Negligible\nRSvalues found from the δ=0.62 suggest negligible AHE which is in agreement with\nFig. 3(e). We further notice a change in the sign of AHE between the parent system\n(−ve) and the Fe doped system of δ=0.62 (+ ve), possibly due to the change in carrier\nconcentration with Fe doping as a sign of the Hall conductivity or Berry curvature is\nsensitive to the chemical potential [49, 60]. The same can be valid for the δ=0.30 system\nas well, in which we observe a change in the sign of AHE below (+ ve) and above ( −ve)\n50 K [see Figs. 4(c) and 4(d)].\n4. Discussions\nOverall, the electrical transport, magnetic, and magnetotransport properties of\nMn3.48Ge are consistent with previous reports [47, 15, 16, 49, 46]. During this\nmanuscript preparation, a report on Mn 2.34Fe0.66Ge has appeared in the literature [58],\nthoroughly discussing the magnetic and magnetotransport properties. Our magnetic\nand magnetotransport results on δ=0.62 compound agree with Ref. [58]. That means\nthe antiferromagnetic transitions observed at 230 K and 100 K in our studied system are\nvery close to the reported values of 240 K and 120 K, respectively. Further, the absence\nof in-plane anomalous Hall effect ( ρxy) is consistent with the report. However, a small\nout-of-plane AHE ( ρzy) is reported in Ref. [58] with a maximum of ρzy=0.7 µΩ cm at\n130 K, which we could not study on our skinny samples. Nevertheless, the dominating\nnormal Hall resistivity at all measured temperatures is in excellent agreement with\nRef. [58]. In addition, the electrical transport and magnetic properties of our studied\nsystem Mn 2.69Fe0.62Ge are consistent with several other previous reports on Mn 3−δFeδGe\n(δ≥0.45) [61, 62, 63]. Specifically, the metal-insulator transition observed at 100 K in\nourδ=0.62 system is in agrement with the previous report on (Mn 0.83Fe0.2)3.25Ge [63].\nNext, the new results of this manuscript, not yet reported so far, are\nthe electrical transport, magnetic, and Hall effect on Mn 2.97Fe0.30Ge compound.\nInterestingly, although both the parent and δ=0.62 compounds show an increase in\nlongitudinal resistivity ( ρxx) with temperature, at least in the low-temperature regime,\nMn2.97Fe0.30Ge does not display any metallic nature of the resistivity. Means the\nhighest resistivity observed at 2 K gradually decreases with increasing temperature,\nlike in magnetic semiconductors [64, 65, 66]. On the other hand, we could observe11\nthe topological Hall effect in δ=0.30 which was not found in the parent and δ=0.62\nsystems. There exist several mechanisms to understand the origin of the topological\nHall effect, such as the Dzyaloshinskii-Moria (DM) interaction in noncentrosymmetric\nsystems [44, 67, 68], uniaxial magnetocrystalline anisotropy in centrosymmetric\nsystems [27, 69, 30], and chiral domain-wall-induced skyrmion lattice [70, 71, 72, 73].\nFurthermore, the recent theoretical and experimental studies suggest the stabilization of\nskyrmion lattice in centrosymmetric kagome systems [74, 75, 17] such as Gd2PdSi 3[45],\nGd3Ru4Al12[76] due to geometrical magnetic frustration. Recently, we have also shown\nan enhancement of magnetocrystalline anisotropy in addition to the formation of the\nnoncoplanar spin structure generating the topological Hall effect in Mn 3−δFeδSn [30].\nThe same could be true in the present case of δ=0.30 compound as well.\nFig. 5 schematically demonstrates the magnetic structure observed in the studied\ncompositions of δ=0, 0.30, and 0.62. Fig. 5(a) depicts the Mn 3Ge magnetic structure\nwith an inverse-triangular arrangement of the Mn magnetic moments. Here, the Mn\nmagnetic moments are in a noncollinear but coplanar antiferromagnetic order, leading\nto zero scalar spin chirality χijk=Si.(Sj×Sk) = 0. Thus, no topological Hall effect\nhas been observed but only the anomalous Hall effect due to nonzero Berry curvature\nin the momentum space [46, 14, 16, 15]. With a Fe doping of δ=0.62, Mn 2.69Fe0.62Ge\nhas a collinear out-of-plane AFM ordering as demonstrated in Ref. [58] by the neutron\ndiffraction study as well in this study [see Fig. 2(f)], is schematically shown in Fig. 5(c).\nSince in this case also the scalar spin chirality is zero, no topological Hall effect is\nanticipated.\nFig. 5(b) schematically shows the noncoplanar arrangement of the Mn magnetic\nmoments in Mn 2.97Fe0.30Ge, where the magnetic moments align neither ferromagnetically\nnor antiferromagnetically in any particular direction under the external applied field. In\nthis case, the scalar spin chirality will be nonzero, χijk=Si.(Sj×Sk)̸= 0. As a result,\nthe topological Hall resistivity is warranted, as shown in Fig. 4(a). Further, to confirm\nthat the system has a noncoplanar spin structure, we plotted M(T) of Mn 2.97Fe0.30Ge\nunder the applied field of 1 T in Fig. 5(d). From Fig. 5(d), we can notice that the in-plane\nmagnetization gradually increases with decreasing temperature and nearly saturates\nbelow 150 K. On the other hand, the out-of-plane magnetization rapidly increases with\ndecreasing temperature and dominates the in-plane magnetization below 50 K. Such\na crossover from dominating in-plane to out-of-plane magnetization indicates that the\nspins are canted towards out-of-plane directions, producing field-induced noncoplanar\nspin structure. This argument is further supported by Fig. 2(e), in which the in-plane\nmagnetization dominates the out-of-plane magnetization at higher temperatures (T =\n200 K). In contrast, the out-of-plane magnetization dominates the in-plane at 2 K, above\n0.3 T of applied fields. Therefore, our results suggest that the noncoplanar spin structure\nstabilized by the strong magnetocrystalline anisotropy originates the topological Hall\neffect in Mn 2.97Fe0.30Ge [30].12\nδ= 0\nδ= 0.62δ= 0.30 (a) (b)\n(c) (d)\nM(µB/f.u.)\nT (K)µ0H = 1 TH Ʇz\nH ‖ zδ= 0.30χijk= Si.(Sjx Sk) = 0\nχijk= Si.(Sjx Sk) = 0χijk= Si.(Sjx Sk) ≠0Mn\nGe\nMn\nGe\nFeMn\nGe\nFe\nFigure 5: Schematic representation of the Mn magnetic moment arrangement in the parent (a), δ=0.30\n(T<50 K) (b), and δ=0.62 (T <100 K) (c) compounds. (d) Magnetization plotted as a function of\ntemperature measured at 1 T of applied field from δ=0.30 sample.\n5. Summary\nWe have systematically studied the electrical, magnetic, and magnetotransport\nproperties of Mn (3+x)−δFeδGe(δ=0, 0.30, and 0.62). We find that the electrical\nresistivity of the parent compound displays metallic behavior, while the system with\nδ=0.30 of Fe doping exhibits resistivity similar to a dilute magnetic semiconductor. With\nfurther Fe doping of δ=0.62, the system demonstrates a metal-insulator transition at 100\nK. Fe doping increases ferromagnetism and magnetocrystalline anisotropy. It induces a\nspin-glass-like state at low temperatures. In addition, the spontaneous anomalous Hall\neffect observed in the parent system is significantly reduced with increasing Fe doping\nconcentration. Importantly, the topological Hall effect observed in Mn 2.97Fe0.30Ge\n(δ=0.30), is not found from the parent system Mn 3.48Ge or Mn 2.69Fe0.62Ge (δ=0.62).\n6. Acknowledgement\nS.G. acknowledges the University Grants Commission (UGC), India, for the Ph.D.\nfellowship. The authors thank SERB (DST), India, for the financial support (Grant13\nNo. SRG/2020/00393). This research has used the Technical Research Centre (TRC)\nInstrument facilities of the S. N. Bose National Centre for Basic Sciences, established\nunder the TRC project of the Department of Science and Technology (DST), Govt.\nof India. The authors thank Prof. Kamaraju Natarajan and Siddhanta Sahu, IISER\nKolkata, for the SCXRD measurements.\nReferences\n[1] Xu G, Weng H, Wang Z, Dai X and Fang Z 2011 Phys. Rev. Lett. 107(18) 186806 URL\nhttps://link.aps.org/doi/10.1103/PhysRevLett.107.186806\n[2] Liu D F, Liang A J, Liu E K, Xu Q N, Li Y W, Chen C, Pei D, Shi W J, Mo S K, Dudin P,\nKim T, Cacho C, Li G, Sun Y, Yang L X, Liu Z K, Parkin S S P, Felser C and Chen Y L 2019\nScience 3651282–1285 URL https://doi.org/10.1126%2Fscience.aav2873\n[3] Liu Z, Li M, Wang Q, Wang G, Wen C, Jiang K, Lu X, Yan S, Huang Y, Shen D, Yin J X,\nWang Z, Yin Z, Lei H and Wang S 2020 Nat. Commun. 11URL https://doi.org/10.1038%\n2Fs41467-020-17462-4\n[4] Ye L, Kang M, Liu J, von Cube F, Wicker C R, Suzuki T, Jozwiak C, Bostwick A, Rotenberg\nE, Bell D C, Fu L, Comin R and Checkelsky J G 2018 Nature 555 638–642 URL https:\n//doi.org/10.1038%2Fnature25987\n[5] Nakano H and Sakai T 2017 J Phys : Conf Ser 868 012006 URL https://doi.org/10.1088%\n2F1742-6596%2F868%2F1%2F012006\n[6] Wu X, Schwemmer T, M¨ uller T, Consiglio A, Sangiovanni G, Di Sante D, Iqbal Y, Hanke W,\nSchnyder A P, Denner M M, Fischer M H, Neupert T and Thomale R 2021 Phys. Rev. Lett.\n127(17) 177001 URL https://link.aps.org/doi/10.1103/PhysRevLett.127.177001\n[7] Liu E, Sun Y, Kumar N, Muechler L, Sun A, Jiao L, Yang S Y, Liu D, Liang A, Xu Q et al. 2018\nNature physics 141125–1131 URL https://doi.org/10.1038/s41567-018-0234-5\n[8] Li X, Collignon C, Xu L, Zuo H, Cavanna A, Gennser U, Mailly D, Fauqu´ e B, Balents L, Zhu Z\nand Behnia K 2019 Nat. Commun. 103021\n[9] Kanazawa N, Onose Y, Arima T, Okuyama D, Ohoyama K, Wakimoto S, Kakurai K, Ishiwata\nS and Tokura Y 2011 Phys. Rev. Lett. 106(15) 156603 URL https://link.aps.org/doi/10.\n1103/PhysRevLett.106.156603\n[10] M¨ uhlbauer S, Binz B, Jonietz F, Pfleiderer C, Rosch A, Neubauer A, Georgii R and B¨ oni P 2009\nScience 323915–919 URL https://doi.org/10.1126/science.1166767\n[11] Zhang H, Feng H, Xu X, Hao W and Du Y 2021 Advanced Quantum Technologies 42100073 URL\nhttps://doi.org/10.1002%2Fqute.202100073\n[12] Chowdhury N, Khan K I A, Bangar H, Gupta P, Yadav R S, Agarwal R, Kumar A and Muduli\nP K 2023 Proceedings of the National Academy of Sciences, India Section A: Physical Sciences\nURL https://doi.org/10.1007%2Fs40010-023-00823-1\n[13] Morali N, Batabyal R, Nag P K, Liu E, Xu Q, Sun Y, Yan B, Felser C, Avraham N and Beidenkopf\nH 2019 Science 3651286–1291 URL https://doi.org/10.1126/science.aav2334\n[14] Nakatsuji S, Kiyohara N and Higo T 2015 Nature 527212–215 URL https://doi.org/10.1038/\nnature15723\n[15] Nayak A K, Fischer J E, Sun Y, Yan B, Karel J, Komarek A C, Shekhar C, Kumar N, Schnelle\nW, K¨ ubler J et al. 2016 Sci. Adv. 2e1501870 URL https://www.science.org/doi/10.1126/\nsciadv.1501870\n[16] Kiyohara N, Tomita T and Nakatsuji S 2016 Phys. Rev. Applied 5(6) 064009 URL https:\n//link.aps.org/doi/10.1103/PhysRevApplied.5.064009\n[17] Rout P K, Madduri P V P, Manna S K and Nayak A K 2019 Phys. Rev. B 99(9) 094430 URL\nhttps://link.aps.org/doi/10.1103/PhysRevB.99.094430\n[18] Taylor J M, Markou A, Lesne E, Sivakumar P K, Luo C, Radu F, Werner P, Felser C and Parkin14\nS S P 2020 Phys. Rev. B 101(9) 094404 URL https://link.aps.org/doi/10.1103/PhysRevB.\n101.094404\n[19] Li H, Ding B, Chen J, Li Z, Hou Z, Liu E, Zhang H, Xi X, Wu G and Wang W 2019 Appl. Phys.\nLett.114192408 URL https://doi.org/10.1063%2F1.5088173\n[20] Shiomi Y, Mochizuki M, Kaneko Y and Tokura Y 2012 Phys. Rev. Lett. 108(5) 056601 URL\nhttps://link.aps.org/doi/10.1103/PhysRevLett.108.056601\n[21] Denisov K S, Rozhansky I V, Averkiev N S and L¨ ahderanta E 2018 Phys. Rev. B 98(19) 195439\nURL https://link.aps.org/doi/10.1103/PhysRevB.98.195439\n[22] Neubauer A, Pfleiderer C, Binz B, Rosch A, Ritz R, Niklowitz P G and B¨ oni P 2009 Phys. Rev.\nLett.102(18) 186602 URL https://link.aps.org/doi/10.1103/PhysRevLett.102.186602\n[23] Nagaosa N and Tokura Y 2013 Nat. Nanotechnol. 8899–911 URL https://doi.org/10.1038%\n2Fnnano.2013.243\n[24] Yu X Z, Kanazawa N, Onose Y, Kimoto K, Zhang W Z, Ishiwata S, Matsui Y and Tokura Y 2010\nNat. Mater. 10106–109 URL https://doi.org/10.1038%2Fnmat2916\n[25] Huang S X and Chien C L 2012 Phys. Rev. Lett. 108(26) 267201 URL https://link.aps.org/\ndoi/10.1103/PhysRevLett.108.267201\n[26] Yu X Z, Onose Y, Kanazawa N, Park J H, Han J H, Matsui Y, Nagaosa N and Tokura Y 2010\nNature 465901–904 URL https://doi.org/10.1038%2Fnature09124\n[27] Yu X Z, Tokunaga Y, Kaneko Y, Zhang W Z, Kimoto K, Matsui Y, Taguchi Y and Tokura Y 2014\nNat. Commun. 5URL https://doi.org/10.1038%2Fncomms4198\n[28] Hou Z, Ren W, Ding B, Xu G, Wang Y, Yang B, Zhang Q, Zhang Y, Liu E, Xu F, Wang\nW, Wu G, Zhang X, Shen B and Zhang Z 2018 Adv. Mater. 301706306 URL https:\n//doi.org/10.1002%2Fadma.201706306\n[29] Du Q, Hu Z, Han M G, Camino F, Zhu Y and Petrovic C 2022 Phys. Rev. Lett. 129(23) 236601\nURL https://link.aps.org/doi/10.1103/PhysRevLett.129.236601\n[30] Low A, Ghosh S, Changdar S, Routh S, Purwar S and Thirupathaiah S 2022 Phys. Rev. B 106(14)\n144429 URL https://link.aps.org/doi/10.1103/PhysRevB.106.144429\n[31] Spencer C S, Gayles J, Porter N A, Sugimoto S, Aslam Z, Kinane C J, Charlton T R, Freimuth F,\nChadov S, Langridge S, Sinova J, Felser C, Bl¨ ugel S, Mokrousov Y and Marrows C H 2018 Phys.\nRev. B 97(21) 214406 URL https://link.aps.org/doi/10.1103/PhysRevB.97.214406\n[32] Li Y, Kanazawa N, Yu X Z, Tsukazaki A, Kawasaki M, Ichikawa M, Jin X F, Kagawa F and\nTokura Y 2013 Phys. Rev. Lett. 110(11) 117202 URL https://link.aps.org/doi/10.1103/\nPhysRevLett.110.117202\n[33] S¨ urgers C, Fischer G, Winkel P and v L¨ ohneysen H 2014 Nature Communications 5URL\nhttps://doi.org/10.1038%2Fncomms4400\n[34] Ueland B G, Miclea C F, Kato Y, Ayala–Valenzuela O, McDonald R D, Okazaki R, Tobash P H,\nTorrez M, Ronning F, Movshovich R, Fisk Z, Bauer E, Martin I and Thompson J 2012 Nat.\nCommun. 3URL https://doi.org/10.1038%2Fncomms2075\n[35] Taniguchi T, Yamanaka K, Sumioka H, Yamazaki T, Tabata Y and Kawarazaki S 2004 Phys. Rev.\nLett.93(24) 246605 URL https://link.aps.org/doi/10.1103/PhysRevLett.93.246605\n[36] Fabris F W, Pureur P, Schaf J, Vieira V N and Campbell I A 2006 Phys. Rev. B 74(21) 214201\nURL https://link.aps.org/doi/10.1103/PhysRevB.74.214201\n[37] Machida Y, Nakatsuji S, Maeno Y, Tayama T, Sakakibara T and Onoda S 2007 Phys. Rev. Lett.\n98(5) 057203 URL https://link.aps.org/doi/10.1103/PhysRevLett.98.057203\n[38] Takatsu H, Yonezawa S, Fujimoto S and Maeno Y 2010 Phys. Rev. Lett. 105(13) 137201 URL\nhttps://link.aps.org/doi/10.1103/PhysRevLett.105.137201\n[39] Wang L M 2006 Phys. Rev. Lett. 96(7) 077203 URL https://link.aps.org/doi/10.1103/\nPhysRevLett.96.077203\n[40] Vistoli L, Wang W, Sander A, Zhu Q, Casals B, Cichelero R, Barth´ el´ emy A, Fusil S, Herranz G,\nValencia S, Abrudan R, Weschke E, Nakazawa K, Kohno H, Santamaria J, Wu W, Garcia V and\nBibes M 2018 Nat. Phys. 1567–72 URL https://doi.org/10.1038%2Fs41567-018-0307-515\n[41] Ohuchi Y, Kozuka Y, Uchida M, Ueno K, Tsukazaki A and Kawasaki M 2015 Phys. Rev. B 91(24)\n245115 URL https://link.aps.org/doi/10.1103/PhysRevB.91.245115\n[42] Soumyanarayanan A, Raju M, Oyarce A L G, Tan A K C, Im M Y, Petrovi´ c A P, Ho P, Khoo\nK H, Tran M, Gan C K, Ernult F and Panagopoulos C 2017 Nat. Mater. 16898–904 URL\nhttps://doi.org/10.1038%2Fnmat4934\n[43] Liu C, Zang Y, Ruan W, Gong Y, He K, Ma X, Xue Q K and Wang Y 2017 Phys. Rev. Lett.\n119(17) 176809 URL https://link.aps.org/doi/10.1103/PhysRevLett.119.176809\n[44] Kanazawa N, Kubota M, Tsukazaki A, Kozuka Y, Takahashi K S, Kawasaki M, Ichikawa M,\nKagawa F and Tokura Y 2015 Phys. Rev. B 91(4) 041122 URL https://link.aps.org/doi/\n10.1103/PhysRevB.91.041122\n[45] Kurumaji T, Nakajima T, Hirschberger M, Kikkawa A, Yamasaki Y, Sagayama H, Nakao H,\nTaguchi Y, hisa Arima T and Tokura Y 2019 Science 365914–918 URL https://doi.org/10.\n1126%2Fscience.aau0968\n[46] Chen T, Tomita T, Minami S, Fu M, Koretsune T, Kitatani M, Muhammad I, Nishio-Hamane D,\nIshii R, Ishii F et al. 2021 Nature communications 121–14 URL https://doi.org/10.1038/\ns41467-020-20838-1\n[47] Yamada N, Sakai H, Mori H and Ohoyama T 1988 Physica B+ C 149 311–315 URL https:\n//doi.org/10.1016/0378-4363(88)90258-6\n[48] Arras E, Caliste D, Deutsch T, Lan¸ con F and Pochet P 2011 Phys. Rev. B 83(17) 174103 URL\nhttps://link.aps.org/doi/10.1103/PhysRevB.83.174103\n[49] Xu L, Li X, Lu X, Collignon C, Fu H, Koo J, Fauqu´ e B, Yan B, Zhu Z and Behnia K 2020 Sci.\nAdv.6eaaz3522 URL https://doi.org/10.1126/sciadv.aaz3522\n[50] Wuttke C, Caglieris F, Sykora S, Scaravaggi F, Wolter A U B, Manna K, S¨ uss V, Shekhar C,\nFelser C, B¨ uchner B and Hess C 2019 Phys. Rev. B 100(8) 085111 URL https://link.aps.\norg/doi/10.1103/PhysRevB.100.085111\n[51] Dho J, Kim W S and Hur N H 2002 Phys. Rev. Lett. 89(2) 027202 URL https://link.aps.org/\ndoi/10.1103/PhysRevLett.89.027202\n[52] Feng W J, Li D, Ren W J, Li Y B, Li W F, Li J, Zhang Y Q and Zhang Z D 2006 Phys. Rev. B\n73(20) 205105 URL https://link.aps.org/doi/10.1103/PhysRevB.73.205105\n[53] Wang B S, Tong P, Sun Y P, Zhu X B, Yang Z R, Song W H and Dai J M 2010 Appl. Phys. Lett.\n97042508 URL https://doi.org/10.1063%2F1.3469924\n[54] Bag P, Baral P R and Nath R 2018 Physical Review B 98(14) 144436 URL https://link.aps.\norg/doi/10.1103/PhysRevB.98.144436\n[55] Duan T F, Ren W J, Liu W L, Li S J, Liu W and Zhang Z D 2015 Appl. Phys. Lett. 107082403\nURL https://doi.org/10.1063%2F1.4929447\n[56] Sung N H, Ronning F, Thompson J D and Bauer E D 2018 Appl. Phys. Lett. 112 132406 URL\nhttps://doi.org/10.1063%2F1.5021133\n[57] Tomiyoshi S, Yamaguchi Y and Nagamiya T 1983 J. Magn. Magn. Mater. 31629–630 URL\nhttps://doi.org/10.1016/0304-8853(83)90610-8\n[58] Rai V, Stunault A, Schmidt W, Jana S, Perßon J, Soh J R, Br¨ uckel T and Nandi S 2023 Phys.\nRev. B 107(18) 184413 URL https://link.aps.org/doi/10.1103/PhysRevB.107.184413\n[59] Gu H, Tian J, Kang C, Wang L, Pang R, Shen M, Liu K, She L, Song Y, Liu X and Zhang W\n2022 Appl. Phys. Lett. 121191906 URL https://doi.org/10.1063%2F5.0108940\n[60] Wang X, Pan D, Zeng Q, Chen X, Wang H, Zhao D, Xu Z, Yang Q, Deng J, Zhai T, Wu G, Liu E\nand Zhao J 2021 Nanoscale 13(4) 2601–2608 URL http://dx.doi.org/10.1039/D0NR07946D\n[61] Hori T, Yamaguchi Y and Nakagawa Y 1992 J. Magn. Magn. Mater. 104-107 2045–2046 ISSN\n0304-8853 URL https://www.sciencedirect.com/science/article/pii/030488539291661C\n[62] Hori T, Niida H, Kato H, Yamaguchi Y and Nakagawa Y 1995 Physica B: Condensed Matter\n211 93 ISSN 0921-4526 research in High Magnetic Fields URL https://doi.org/10.1016/\n0921-4526(94)00952-R\n[63] Du J, Li D, Li Y B, Sun N K, Li J and Zhang Z D 2007 Phys. Rev. B 76(9) 094401 URL16\nhttps://link.aps.org/doi/10.1103/PhysRevB.76.094401\n[64] Dietl T 2010 Nat. Mater. 9965 URL https://doi.org/10.1038%2Fnmat2898\n[65] Deng Z, Jin C, Liu Q, Wang X, Zhu J, Feng S, Chen L, Yu R, Arguello C, Goko T, Ning F,\nZhang J, Wang Y, Aczel A, Munsie T, Williams T, Luke G, Kakeshita T, Uchida S, Higemoto\nW, Ito T, Gu B, Maekawa S, Morris G and Uemura Y 2011 Nat. Commun. 2422 URL\nhttps://doi.org/10.1038%2Fncomms1425\n[66] Yu S, Zhao G, Peng Y, Wang X, Liu Q, Yu R, Zhang S, Zhao J, Li W, Deng Z, Uemura Y J and\nJin C 2020 Crystals 10690 URL https://www.mdpi.com/2073-4352/10/8/690\n[67] Liu J and Balents L 2017 Phys. Rev. Lett. 119(8) 087202 URL https://link.aps.org/doi/10.\n1103/PhysRevLett.119.087202\n[68] Sukhanov A S, Heinemann A, Kautzsch L, Bocarsly J D, Wilson S D, Felser C and Inosov D S\n2020 Phys. Rev. B 102(14) 140409 URL https://link.aps.org/doi/10.1103/PhysRevB.102.\n140409\n[69] Preißinger M, Karube K, Ehlers D, Szigeti B, von Nidda H A K, White J S, Ukleev V, Rønnow\nH M, Tokunaga Y, Kikkawa A, Tokura Y, Taguchi Y and K´ ezsm´ arki I 2021 npj Quantum\nMaterials 6URL https://doi.org/10.1038%2Fs41535-021-00365-y\n[70] Gudnason S B and Nitta M 2014 Phys. Rev. D 89(8) 085022 URL https://link.aps.org/doi/\n10.1103/PhysRevD.89.085022\n[71] Cheng R, Li M, Sapkota A, Rai A, Pokhrel A, Mewes T, Mewes C, Xiao D, De Graef M and Sokalski\nV 2019 Phys. Rev. B 99(18) 184412 URL https://link.aps.org/doi/10.1103/PhysRevB.99.\n184412\n[72] Nagase T, So Y G, Yasui H, Ishida T, Yoshida H K, Tanaka Y, Saitoh K, Ikarashi N, Kawaguchi\nY, Kuwahara M and Nagao M 2021 Nat. Commun. 12URL https://doi.org/10.1038%\n2Fs41467-021-23845-y\n[73] Yang K, Nagase K, Hirayama Y, Mishima T D, Santos M B and Liu H 2021 Nat. Commun. 12\nURL https://doi.org/10.1038%2Fs41467-021-26306-8\n[74] Okubo T, Chung S and Kawamura H 2012 Phys. Rev. Lett. 108(1) 017206 URL https://link.\naps.org/doi/10.1103/PhysRevLett.108.017206\n[75] Hayami S, Ozawa R and Motome Y 2017 Phys. Rev. B 95(22) 224424 URL https://link.aps.\norg/doi/10.1103/PhysRevB.95.224424\n[76] Hirschberger M, Nakajima T, Gao S, Peng L, Kikkawa A, Kurumaji T, Kriener M, Yamasaki Y,\nSagayama H, Nakao H, Ohishi K, Kakurai K, Taguchi Y, Yu X, hisa Arima T and Tokura Y\n2019 Nat. Commun. 10URL https://doi.org/10.1038%2Fs41467-019-13675-4"    },    {        "title": "2308.13124v1.Thermal_effect_on_microwave_pulse_driven_magnetization_switching_of_Stoner_particle.pdf",        "content": "Thermal effect on microwave pulse driven magnetization switching of Stoner particle\nS. Chowdhury and M. A. S. Akanda\nPhysics Discipline, Khulna University, Khulna 9208, Bangladesh\nM. A. J. Pikul\nDepartment of Physics, Colorado State University, Fort Collins, Colorado 80523, USA\nM. T. Islam∗\nPhysics Discipline, Khulna University, Khulna 9208, Bangladesh and\nCenter for Spintronics and Quantum Syetems, State Key Laboratory for Mechanical Behavior of Materials,\nXi’an Jiaotong University, No. 28 Xianning West Road Xi’an, Shanxi 710049, China\nTai Min\nCenter for Spintronics and Quantum Syetems, State Key Laboratory for Mechanical Behavior of Materials,\nXi’an Jiaotong University, No. 28 Xianning West Road Xi’an, Shanxi 710049, China\nRecently it has been demonstrated that the cosine chirp microwave pulse (CCMP) is capable of\nachieving fast and energy-efficient magnetization-reversal of a nanoparticle with zero-Temperature.\nHowever, we investigate the finite temperature, Teffect on the CCMP-driven magnetization rever-\nsal using the framework of the stochastic Landau Lifshitz Gilbert equation. At finite Temperature,\nwe obtain the CCMP-driven fast and energy-efficient reversal and hence estimate the maximal\ntemperature, Tmax at which the magnetization reversal is valid. Tmax increases with increasing\nthe nanoparticle cross-sectional area/shape anisotropy up to a certain value, and afterward Tmax\ndecreases with the further increment of nanoparticle cross-sectional area/shape anisotropy. This\nis because of demagnetization/shape anisotropy field opposes the magnetocrystalline anisotropy,\ni.e., reduces the energy barrier which separates the two stable states. For smaller cross-sectional\narea/shape anisotropy, the controlling parameters of CCMP show decreasing trend with tempera-\nture. We also find that with the increment easy-plane shape-anisotropy, the required initial frequency\nof CCMP significantly reduces. For the larger volume of nanoparticles, the parameters of CCMP\nremains constant for a wide range of temperature which are desired for the device application.\nTherefore, The above findings might be useful to realize the CCMP-driven fast and energy-efficient\nmagnetization reversal in realistic conditions.\nI. INTRODUCTION\nObtaining swift and efficient magnetization switching\nof a single nanoparticle has attracted much attention be-\ncause of its non-volatility [1–3] and speedy data process-\ning properties [4]. In the last decades, several control-\nling parameters, for instance, magnetic fields, microwaves\nfields [5, 6], spin-polarized electric current and spin-orbit\ntorque [7–21] are employed to reverse magnetization re-\nversal. However, these methods are facing specific chal-\nlenging issues in memory-device applications. Particu-\nlarly, in the case of the magnetic field, it requires a large\nfield and a longer switching time [5]; also field localiza-\ntion to a bit-cell is a bottleneck. On the other hand, the\nspin-polarized-current can induce magnetization switch-\ning by the mechanism of spin transfer torque (STT) and /\nor spin-orbit torque (SOT) [22–27]. However, for STT-\nMRAM-based devices, the threshold current density is\nlarge and it creates Joule heat which, in turn, causes\nthe device lifetime and reliability issues [28–34]. For\nSOT-MRAM-based device applications, the main hin-\ndrance is that the requirement of two transistors for\n∗Correspondence email address:torikul@phy.ku.ac.bdeach bit-cell enhances the area of the bit-cell [35]. Later\non, people employ the microwave field with constant or\ntime-dependent frequency profile to drive magnetization\nswitching at zero temperature [36–49]. However, without\nconsidering the thermal effect, the recent study [50] re-\nported that the swift and energy-efficient magnetization\nswitching is obtained by a cosine chirped microwave pulse\n(cosine CMP). This is because the frequency changing (of\nCosine CMP) is closely matched with the magnetization\nprecession frequency which leads to the stimulated en-\nergy absorption (emission) by magnetization efficiently\nbefore (after) crossing the energy barrier. However, in\npractice, the temperature is prevailing everywhere, and\ndevices are operated at room temperature. So, from\nthe practical point of view, it is interesting to verify\nwhether the Cosine CMP still efficiently drives magne-\ntization switching at finite temperature. In addition,\nthe study [50] reported that the increase of easy-plane\nshape anisotropy (i.e., the increase of demagnetization\nfield) makes the magnetization switching faster which is\nexpected for device application. But, the increase of the\ndemagnetization field (which opposes the magnetocrys-\ntalline anisotropy) reduces the height of the energy bar-\nrier, originated by anisotropy, which may cause thermal\ninstability issues at room /operating temperature. ThusarXiv:2308.13124v1  [cond-mat.mes-hall]  25 Aug 20232\nat operating temperature, there is a possibility of spon-\ntaneous magnetization switching which may increase the\nerror rate, which is actually undesired. Therefore, in\nthis study, we include the finite/room temperature in the\nsystem and relax it, afterward we investigate the cosine\nCMP-driven magnetization switching to check whether\nthe switching is robust at operating (room) tempera-\nture and how the parameters (i.e., the optimal initial\nfrequency and field amplitude) of cosine CMP are altered\nwith temperature.\nThis study interestingly finds that the cosine CMP-\ndriven swift and energy-efficient switching is robust even\nwith a finite /room temperature. For the considered\nnanoparticles /stoner particle, we also estimate the max-\nimal temperature, Tmat which the magnetization is\nvalid. Tmincreases with increasing the nanoparticle vol-\nume (by increasing cross-sectional area, A=xy) or\nshape-anisotropy coefficient up to a certain value, and\nTmsignificantly decreases with the further increment of\nnanoparticle cross-sectional area /shape-anisotropy coef-\nficient. Here the demagnetization /shape anisotropy re-\nduces the effective uniaxial anisotropy i.e., reduces the\nenergy barrier which separates the two stable states.\nStill, the cosine CMP-driven magnetization switching is\nvalid for a wide operating temperature above room tem-\nperature. For the smaller volume of nanoparticles /stoner\nparticle, the controlling parameters of cosine CMP, i.e.,\nthe minimal amplitude Hmw(T), the optimal rate R\n(GHz), and the minimal initial frequency f0(GHz) show\na decreasing trend with temperature. We also find that\nwith the increment of easy-plane shape-anisotropy, the\nrequired initial frequency of cosine CMP significantly\nreduces. For the larger volume of nanoparticles, the\nparameters of cosine CMP remain constant for a wide\nrange of temperature that is desired for the device ap-\nplication. Note that this strategy might be employed to\nswitch the magnetization of other materials, for instance,\nsynthetic antiferromagnetic/ ferrimagnetic nanoparticles\nby in-plane cosine chirped current pulse via spin-orbit\ntorque. Therefore, the above findings might be useful to\nrealize the cosine CMP-driven fast and energy-efficient\nmagnetization switching in practical spintronics device\napplications with considering realistic conditions.\nII. MODEL AND METHOD\nWe assume a single-domain magnetic nanoparti-\ncle/Stoner particle of volume V=Ah, where Ais the\ncross-sectional area and his thickness, and its uniaxial\nanisotropy is directed along zaxis at temperature T, as\nshown in Figure 1(a). The nanoparticle’s size is chosen\nin such a way that the magnetization of can be treated\nas a macrospin which is indicated by the unit-vector m\nwith saturation magnetization Ms. An easy-plane shape\nanisotropy is approximated the demagnetization field\nwhich is opposite to the magnetocrystalline anisotropy\nfield. The shape anisotropy field is indicated as Hshape =\nmf0\n0-f(b) (a)\nτ t\nzy\nxhmw(x)hmw(y)\nTCosine CMPFIG. 1. (a) Schematic figure shows a single domain\nnanoparticle /stoner particle with in-plane orthogonal mi-\ncrowave field components at finite temperature, where mde-\nnotes the direction of magnetization. (b) Time-depended fre-\nquency outline for cosine CMP from + f0to−f0.\nKshapemzˆz, where Kshape =−µ0(Nz−Nx)Msis the\nshape anisotropy coefficient, NzandNxare demagne-\ntization factors [51, 52] and µ0= 4π×10−7N/A2is the\nvacuum magnetic permeability. The total anisotropy is\ndominated by the magnetocrystalline anisotropy Hani=\nKanimzˆzand hence the magnetic particle possesses two\nstable states (i.e., the maligns parallel to ˆzand−ˆz).\nAt finite temperatures, the dynamics of magnetization\nare governed by the stochastic Landau Lifshitz Gilbert\n(sLLG) with the application of a circularly polarized co-\nsine CMP [53].\ndm\ndt=−γm×[Htot+Hth] +αm×dm\ndt, (1)\nwhere αandγare the Gilbert damping constant and the\ngyromagnetic ratio respectively. Additionally, the total\nmagnetic field ( Htot) consists of the effective field, which\ncomes from the exchange field2Aex\nMs∇2m, and effective\nthe easy-axis anisotropy field along zdirection, and the\nexternal microwave field ( Hmw).\nThe stochastic thermal field is denoted by Hth, which\ncomes from a finite temperature. The thermal field ex-\nhibits the following relations which are described by the\nGaussian process [54, 55].\n⟨Hth,ip(t)⟩= 0,\n⟨Hth,ip(t)Hth,jq(t+ ∆t)⟩=2αkBT\nγMs∆Vδijδpqδ(∆t),(2)\nwhere kBis the Boltzman constant, pandqare the\nCartesian thermal field components, ∆ Vis the volume\nof a single micromagnetic cell, and iandjdesignate the\nmicromagnetic cells. Depending on the temperature, the\nthermal /random field is generated as\nHth, i,p =⃗ ηs\n2αkBT\nγMs∆V∆t(3)\nwhere ∆ tis the time step and ⃗ ηis a random vector which\nchanges with time step and provides the normal distri-\nbution with zero average.3\nWithout microwave field and with zero-temperature,\nthere are two ground states (or energy minima) m∥\nˆzandm∥ −ˆzin which the magnetization likes to\nstay due to uniaxial anisotropy. The main task is to\nswitch the magnetization from one energy minima to\nanother energy minima, purposely the study [50] re-\nported that, at zero temperature, the cosine CMP is\nefficient to achieve fast switching. The cosine CMP\nis constructed as Hmw=Hmw[cosϕ(t)ˆx+ sin ϕ(t)ˆy],\nwhere Hmwis microwave amplitude and ϕ(t) is the\nphase. ϕ(t) is denoted as 2 πf0cos (2 πRt)t,R(in GHz)\nis the controlling parameter, and f(t) is the instanta-\nneous frequency of cosine CMP, is defined as f(t) =\n1\n2πdϕ\ndt=f0[cos (2 πRt)−(2πRt) sin (2 πRt)] which sweeps\nfrom + f0to final −f0as shown in Figure 1(b). The\nphysical picture of obtaining the fast and energy-efficient\nmagnetization switching is that the cosine CMP induces\nstimulated microwave absorption (emissions) by (from)\nthe macro-spin before (after) crossing over the energy\nbarrier at zero temperature and, for detail, the formula-\ntion of the rate of energy change is given in the appendix\n[50]. In this study, we use the material parameters re-\nported in Ref. [56], Ms= 106A/m,Hk= 0.75 T or\n3.75×105J/m3,γ= 1.76×1011rad/(T·s), exchange\nconstant Aex= 13×10−12J/m, and α= 0.01 to mimic\nPermalloy. This study shows a strategy that would work\nfor other materials also since Permalloy does not pos-\nsess such a high anisotropy [57, 58]. MuMax3 Package\n[59] has been used to solve the stochastic-LLG equation\nusing the adaptive-Heun solver.\nWe study the nanoparticles by increasing the cross-\nsectional area A=xywhile the thickness h= 8 nm\nis kept constant. Particularly, the nanoparticles with the\nareas are A1= 8×8 nm2,A2= 12×12 nm2,A3= 16×16\nnm2, and A4= 22×22 nm2so that the demagnetiza-\ntion field is induced in the opposite direction of uniaxial\nanisotropy. We discretize the sample by the unit-cell size\n2×2×2 nm3. For efficient and stable calculation [60, 61],\nwe employ the fixed time step of 10−14s. According to\nthe practical requirement, we consider the switched state\nis obtained if the magnetization reaches at mz=−0.7.\nIII. NUMERICAL RESULTS\nFirstly we investigate the magnetization switching of\nthe nanoparticle 12 ×12×8 nm3driven by cosine CMP\nwith the initial frequency f0= 17 .70 GHz and the mi-\ncrowave field Hmw= 0.045 T, which are similar parame-\nters to the study [50] for T= 0 K. By keeping f0= 17.70\nGHz and Hmw= 0.045 T fixed, then we try to deter-\nmine the optimal R(at which the switching is fastest)\nwhich is obtained 0 .42 GHz for T= 0. Then we in-\nclude the temperature and relax the system. Afterward,\nwe apply the cosine CMP to the system and study the\nswitching at finite temperatures. In principle, we sim-\nulate 30 independent switchings by varying the random\nnumbers /thermseeds for the same parameters of mate-rial and of cosine CMP ( f0= 17.70 GHz, Hmw= 0.045\nT and R= 0.39 GHz) and take the ensemble average\nas shown in figure 2 (b). In this way, we estimate each\nobservations /data points of this study. Now for T= 300\nK, we study the system 12 ×12×8 nm3by cosine CMP\nwith the parameters ( f0= 17.70 GHz, Hmw= 0.045 T\nandR= 0.42 GHz), but the switching is not obtained as\nexpected which shown by the red line in the 2(a). Then R\nis optimized (for fixed f0= 17.70 GHz and Hmw= 0.045\nT), and for the optimal R= 0.39 GHz, we find the fastest\nmagnetization switching as shown by the blue line in Fig-\nure 2 (a). Therefore, it is noted that optimal Rdepends\non temperature, which is a crucial issue that needs to be\naddressed for realistic applications.\nIt is reported that, with enlarging the volume of\nthe samples /nanoparticles sizes, the thermal stability\n(Ea/Et) increases since the anisotropy energy, ( Ea=\nKV,Kis effective anisotropy coefficient) is proportional\nto volume, which increases the energy barrier [62–66]\nbut thermal energy Et=kBTdoes not depend on vol-\nume. In this study, we focus on the samples with in-\ncreasing cross-sectional area A=xywhile the thick-\nness h= 8 nm is kept constant so that the volume\n(V=Ah) is enlarged. Specifically, our considered sam-\nples are with A1= 8×8,A2= 12×12,A3= 16×16\nandA4= 22×22 nm2while the thickness z=h= 8\nnm is fixed. Therefore, in these sample shapes, the de-\nmagnetization field /shape anisotropy (which is the op-\nposite of the crystalline anisotropy) would be induced\nbecause of the magnetization of the nanoparticle. For\ndifferent A, by finding the demagnetization-factors Nz\nandNx[51, 67] analytically, the shape anisotropy of the\nsamples Kshape =µ0(Nz−Nx)Msmzˆzhas been calcu-\nlated. So, shape anisotropy Kshape are 0 T, 0 .17718\nT, 0.3064 T and 0 .4459 T, for A1,A2,A3andA4re-\nspectively. The Hshape actually opposes the anisotropy\nfieldHaniand reduces the stability as well as resonance\nfrequency f0=γ\n2π[Hani−µ0(Nz−Nx)Ms]. Therefore,\nthere is an issue to study how increment of the shape\nanisotropy, as well as the volume of the sample, affect\n(a) (b)\nFIG. 2. (a) The time evolutions of mzof nanoparticle 12 ×\n12×8 nm3induced by cosine CMP with f0= 17 .7 GHz,\nHmw= 0.045 T, and optimal Rfor different temperature T\n(0 and 300 K). (b) At temperature T= 300 K, the average\nmagnetization switching (bold line) of nanoparticle 12 ×12×8\nnm3is shown from the 30 independent switchings induced by\ncosine CMP with Hmw= 0.045,f0= 17.7 GHz, and R= 0.39\nGHz.4\nTABLE I. Optimal rate R, Minimal initial frequency f0, and Minimal amplitude Hmwin different temperature.\nCross-sectional Shape Temperature, Optimal RMinimal initial Minimal\narea, anisotropy, T(K) rate, frequency, Amplitude,\nA(nm2) Kshape (T) R(GHz) f0(GHz) Hmw(T)\nA1 0 T= 0 17.22 18.8 0.0450\nT= 300 15.12 18.8 0.0445\nT= 1000 16.38 18.7 0.0439\nA2 0.17718 T= 0 17.64 17.7 0.0450\nT= 300 16.38 17.7 0.0450\nT= 1400 14.28 17.6 0.0447\nA3 0.3064 T= 0 17.22 13.5 0.0450\nT= 300 16.8 13.1 0.0450\nT= 1500 15.96 13.5 0.0447\nA4 0.4459 T= 0 27.3 7.7 0.0450\nT= 300 27.3 7.1 0.0450\nT= 800 27.3 6.3 0.0450\n0200 400 600 800 1000 1200 1400 16000.300.350.400.450.500.550.600.650.70 R(GHz )\nT(K)A1\nA2\nA3\nA4\n0.00 .1 0.20 .3 0.40 .50.300.350.400.450.500.550.600.650.70\nKshape(T)R(GHz )\n7008009001000110012001300140015001600\nTm(K)(a) (b)\nFIG. 3. (a) The optimal Ras a function of Tfor different\nKshape (A1= 8×8 nm2,A2= 12×12 nm2,A3= 16×16\nnm2, and A4= 22×22 nm2). (b) Tm(red line) and optimal\nRcorresponding to Tm(Black line) are the function of shape\nanisotropy Kshape.\nthe thermal stability and the parameters of cosine CMP.\nPurposely, we investigate the cosine CMP-driven magne-\ntization switching of the patterned samples of A1,A2A3,\nandA4by varying Rand estimate optimal Rfor differ-\nentT. Fig 3(a) demonstrates the change of optimal R\nas a function of thermal effect, Tfor different samples.\nOptimal Rshows a decreasing trend for lower (smaller\ncross-sectional area) Kshape, albeit with some fluctua-\ntion. However, for higher Kshape, optimal Ris larger (the\nmagnetization switching is faster) and remains constant\nwith T. For each sample, there is a maximal temperature\nTmfor which the magnetization switching is valid, which\nis indicated by the vertical dashed line in Figure 3(a).\nTm’s are higher than the room temperature. The above\nfindings are useful for device applications. In Fig 3(b),\nwe explicitly plot Tmand optimal RatTmas a function\nKshape, it is found that Rdecreases till a certain value of\nKshape and for further increment of Kshape,Rincreases\nsignificantly, i.e., the switching time becomes smallest.\nThe reason can be attributed as (using basic knowledge)\nthe effective saturation magnetization Msdecreases with\ntemperature because of the spin-wave generation [68].\nForA1,A2andA3, i.e., smaller volume, Msdecreases\nfaster with Trefers to the study [69], so optimal Rde-\ncreases as a function of Twith some fluctuations, whichmight be absent if one can take a large number of ensem-\nble average, and the change of optimal Rwith Twould\nbe more consistent. However, for larger A4, i.e., larger\nvolume, the decrement of Mswith Tis not significant,\nso optimal Rremains constant. For A1,A2andA3,Tm\nincreases because of the increment of the sample volume,\nrather than Kshape as it is not dominant, which plays a\nrole to increase the thermal stability. But for the larger\nA4,Kshape becomes dominant and it reduces the uniaxial\nanisotropy as well as the energy barrier which separates\nthe two stable states. This is why thermal stability de-\ncreases, i.e., Tmdecreases significantly.\nSubsequently, we determine the optimal f0of cosine\nCMP, with the investigation of magnetization switching,\nwith Tby keeping amplitude Hmw= 0.045 T and the\noptimal Rfor corresponding Tfixed. Figure 4(a) shows\nthe magnetization switching (black line) induced by co-\nsine CMP with Hmw= 0.045 T, f0= 13 .5 GHz, and\nR= 0.37 GHz for T= 0 K, but at T= 300 K, the magne-\ntization switching (red line) is no longer valid with these\nparameters. Then we tune f0by keeping Hmw= 0.045T\nand optimal R(atT= 300 K) fixed and we obtain mag-\nnetization switching (blue line) with f0= 13 .10 GHz.\nThen for different Kshape or samples, we investigate the\nvariation of f0with Tand demonstrate in Figure 4(b).\nWith increasing the Kshape, the initial frequency f0de-\ncreases significantly, which is expected as the Hshape ac-\ntually acts in the opposite direction of the Haniand thus\nf0decreases as shown in the Figure 4 (a). For the sam-\nples of Kshape = 0 T, 0.17718 T and 0.3064 T, minimal\nf0= remains almost constant up to a maximal temper-\nature Tmwhich is useful for practical device realization.\nBut, for the sample of Kshape = 0.4459 T, optimal shows\na decreasing trend. This is because of the reduction of\neffective Ms(since the demagnetization field is strong),\nwhich leads to a decrement of intrinsic frequency γMsas\nwell. As a result, all dynamics become slow, as though\nthe time scale of the magnetization dynamics has been\nexpanded. Explicitly, Tmand optimal f0atTmas a func-\ntion of Kshape are plotted in the Figure 4(b), it is observed\nthatTminitially increases up to certain temperature and5\n(a) (b) (c)\nFIG. 4. (a) The switching of mzof nanoparticle with A3=\n16×16 nm2driven by the cosine CMP (using Hmw= 0.045 T\nandR= 0.37 GHz) with optimal frequency f0in GHz for dif-\nferent finite temperature, T. (b) Temperature Tdependence\noff0while Hmw= 0.045 T and the optimal Rat correspond-\ning finite temperature, T. (c) The maximal temperature Tm\n(red line) and f0(black line) at Tmas the function of the\nshape anisotropy Kshape.\nthen Tmdecreases rapidly for further increment of Kshape\n. This is because the Kshape opposes and reduces the\neffective anisotropy Hkand thus reduces the energy bar-\nrier.\nLastly, we study how the minimally required Hmwof\ncosine CMP varies with temperature Tby keeping the op-\ntimal f0andRat corresponding Tfixed. For Kshape = 0\norA1= 8×8 nm2, the variation of Hmw(black line)\npresented in the Figure 5(a) and it is found that the\nrequired Hmwremains almost constant up to a maxi-\nmum temperature Tmwhich is indicated by the vertical\ndashed line. After Tm, the magnetization switching is\nnot obtained. Similarly, for other Kshape or samples, we\nstudy the magnetization switching to find minimal Hmw\nof cosine CMP varies with temperature Tby keeping the\noptimal f0andRat corresponding Tfixed and presented\nin Figure 5(a). It is noted that for higher Tand lower\nKshape, the required Hmware smaller. To be more ex-\nplicit, the Tmand the minimal HmwatTm[Hmw(Tm)]\nare plotted in Figure 5(b) which shows that thermal sta-\nbility Tmincreases with Kshape up to a certain value and\nthen decreases with the further increment of Kshape. It\nis because of a similar reason as the increment of the\nsample volume, rather than Kshape which is not domi-\nnant, increases the thermal stability. But for the larger\nA4,Kshape becomes dominant and it reduces the uniaxial\nanisotropy and thus the thermal stability decreases, i.e.,\nTmdecreases significantly. From the above study, we es-\ntimate the minimal Hmw,f0, and optimal Rat different\nTfor different Kshape or the cross-sectional areas. the\nminimal Hmw,f0and optimal Rat three different T(=\n0 K , 300 K and Tm) are summarized in the Table I.\nIV. DISCUSSIONS AND CONCLUSIONS\nThe recent study [50] has demonstrated that the cosine\nCMP is capable of driving fast and energy-efficient mag-\nnetization switching of a nanoparticle with T= 0. How-\never, this study investigates the cosine CMP-driven mag-\nnetization switching of the nanoparticle by including fi-\nniteTsince the temperature is ubiquitous in nature. We\n0200 400 6008001000 1200 1400 16000.0360.0390.0420.0450.0480.0510.054Hmw(T)\nT(K)A1\nA2\nA3\nA4(a) (b)\n0.00 .1 0.20 .3 0.40 .50.010.020.030.040.050.06\nKshape(T)Hmw(T)\n7008009001000110012001300140015001600\nTm(K)FIG. 5. (a) Temperature Tdependence of Hmwwhile minimal\nf0andRat corresponding Tare fixed. (b) Tm(red line) and\nthe minimal Hmw(black line) at Tmas a function of the shape\nanisotropy Kshape.\nfound that cosine CMP-driven fast and energy-efficient\nswitching is still valid in finite Tor even at room tem-\nperature, which is important in practice. For the lower\nvolume of samples, the required minimal amplitude, the\noptimal R, and minimal frequency decrease with temper-\nature because of the decrement of magnetization with T.\nWe also find that with the increment of easy-plane shape-\nanisotropy, the required initial frequency of cosine CMP\nsignificantly reduces. For the larger volume of nanoparti-\ncles, the parameters of cosine CMP remains constant for\na wide range of temperature which is useful practically.\nIt is mentioned that there is a very recent study which\ndemonstrates how to generate such cosine CMP in prac-\ntice [70] and furthermore, several technologies [71, 72] are\navailable to generate such cosine CMP. It is suggested to\nsimulate magnetization switching for a real nanoparticle\nand thus obtain the optimal parameters; later these pa-\nrameters can be employed for all the same nano-particles\nof the device. In addition, it is mentioned that this strat-\negy might be applicable to switch the magnetization of\nsynthetic antiferromagnetic /ferrimagnetic nanoparticles.\nTherefore, the above findings may pave the way to realize\nthe future-generation highly dense and speedy processing\nmemory devices.\nACKNOWLEDGMENTS\nM T Islam acknowledges the National Key R&D\nProgram of China (Grant No. 2021YFA1202200), the\nKhulna University Research Cell (Grant No. KU/RC-\n04/2000-158), Khulna, Bangladesh, and the Ministry of\nEducation (BANBEIS, Grant No. SD2019972).\nAppendix A: Calculation of the rate of change of\nenergy\nThe energy is a function of magnetization and mi-\ncrowave field, i.e., E(m,Htot). The rate of change of\nenergy is then,\ndE\ndt=∂E\n∂mdm\ndt+∂E\n∂HtotdHtot\ndt6\nwhere, Htot=Heff+Hmw, the Hamilton’s equation of\nmotion for the system is,\n∂E\n∂Htot=−m\n∂E\n∂m=−Htot\nNow, after substituting the values ofdm\ndt,∂E\n∂Htotand∂E\n∂m\nin the equation, we get\ndE\ndt=−Htot·(−γm×Htot−αm×(γm×Htot))\n−m·dHmw\ndt\n⇒dE\ndt=γm·(Htot×Htot) +αγHtot(m×(m×Htot))\n−m·dHmw\ndt\n⇒dE\ndt=αγHtot(m×(m×Htot))−m·dHmw\ndt\n⇒dE\ndt=αγ(m×Htot)·(Htot×m)−m·dHmw\ndt\n⇒dE\ndt=−αγ(m×Htot)·(m×Htot)−m·dHmw\ndt\n⇒dE\ndt=−αγ|m×Htot|2−m·dHmw\ndt\nWe represent the second term of right-hand side of the\nabove equation as ˙E. We can write it in the explicit form\nas the following. The rate of change of microwave fieldHmwis\n˙Hmw=dHmw\ndt\n=d\ndt(Hmw[cosϕ(t)ˆx+ sin ϕ(t)ˆy])\n=Hmw[−sinϕ(t)ˆx+ cos ϕ(t)ˆy]dϕ\ndt\n=Hmw[−sinϕ(t)ˆx+ cos ϕ(t)ˆy]\u0014ϕ(t)\nt−d\ndt\u0012ϕ(t)\nt\u0013\nt\u0015\nAnd, the magnetization mis recast by\nm=mxˆx+myˆy\n= sin θ(t) cosϕm(t)ˆx+ sin θ(t) sinϕm(t)ˆy\nwhere θ(t) and ϕm(t) is the polar and azimuthal angle of\nthe magnetization mrespectively.\nNow, substituting the expression of mand ˙Hmwin the\nexpression of ˙E, we get,\n˙E=−m·˙Hmw\n=Hmwsinθ(t)[−sinϕ(t) cosϕm(t)\n+ cos ϕ(t) sinϕm(t)]·\u0014ϕ(t)\nt−d\ndt\u0012ϕ(t)\nt\u0013\nt\u0015\n=Hmwsinθ(t) sin ( ϕ(t)−ϕm(t))\u0014ϕ(t)\nt−d\ndt\u0012ϕ(t)\nt\u0013\nt\u0015\nWe can define Φ( t) =ϕm(t)−ϕ(t), and then we get,\n˙E=Hmwsinθ(t) sin Φ( t)\u0014ϕ(t)\nt−d\ndt\u0012ϕ(t)\nt\u0013\nt\u0015\nwhereh\nϕ(t)\nt−d\ndt\u0010\nϕ(t)\nt\u0011\nti\ndefines the angular frequency\nω(t).\n[1] S. Sun, C. B. Murray, D. Weller, L. Folks, and\nA. Moser, Monodisperse FePt nanoparticles and ferro-\nmagnetic FePt nanocrystal superlattices, science 287,\n1989 (2000).\n[2] S. Woods, J. Kirtley, S. Sun, and R. Koch, Direct inves-\ntigation of superparamagnetism in Co nanoparticle films,\nPhysical review letters 87, 137205 (2001).\n[3] D. Zitoun, M. Respaud, M.-C. Fromen, M. J. Casanove,\nP. Lecante, C. Amiens, and B. Chaudret, Magnetic en-\nhancement in nanoscale CoRh particles, Physical review\nletters 89, 037203 (2002).\n[4] B. Hillebrands and K. Ounadjela, Spin dynamics in con-\nfined magnetic structures I & II , Vol. 83 (Springer Science\n& Business Media, 2003).\n[5] A. Hubert and R. Sch¨ afer, Magnetic domains: the anal-ysis of magnetic microstructures (1998).\n[6] Z. Z. Sun and X. R. Wang, Fast magnetization switching\nof stoner particles: A nonlinear dynamics picture, Phys-\nical Review B 71, 174430 (2005).\n[7] J. C. Slonczewski et al. , Current-driven excitation of\nmagnetic multilayers, Journal of Magnetism and Mag-\nnetic Materials 159, L1 (1996).\n[8] L. Berger, Emission of spin waves by a magnetic multi-\nlayer traversed by a current, Physical Review B 54, 9353\n(1996).\n[9] M. Tsoi, A. Jansen, J. Bass, W.-C. Chiang, M. Seck,\nV. Tsoi, and P. Wyder, Excitation of a magnetic multi-\nlayer by an electric current, Physical Review Letters 80,\n4281 (1998).\n[10] J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers,7\nand D. C. Ralph, Current-driven magnetization reversal\nand spin-wave excitations in Co /Cu/Co pillars, Phys.\nRev. Lett. 84, 3149 (2000).\n[11] X. Waintal, E. B. Myers, P. W. Brouwer, and D. C.\nRalph, Role of spin-dependent interface scattering in gen-\nerating current-induced torques in magnetic multilayers,\nPhys. Rev. B 62, 12317 (2000).\n[12] J. Z. Sun, Spin-current interaction with a monodomain\nmagnetic body: A model study, Physical Review B 62,\n570 (2000).\n[13] J. Sun, Spintronics gets a magnetic flute, Nature 425,\n359 (2003).\n[14] M. D. Stiles and A. Zangwill, Anatomy of spin-transfer\ntorque, Physical Review B 66, 014407 (2002).\n[15] Y. B. Bazaliy, B. Jones, and S.-C. Zhang, Current-\ninduced magnetization switching in small domains of dif-\nferent anisotropies, Physical Review B 69, 094421 (2004).\n[16] R. Koch, J. Katine, and J. Sun, Time-resolved reversal of\nspin-transfer switching in a nanomagnet, Physical review\nletters 92, 088302 (2004).\n[17] W. Wetzels, G. E. Bauer, and O. N. Jouravlev, Efficient\nmagnetization reversal with noisy currents, Physical re-\nview letters 96, 127203 (2006).\n[18] A. Manchon and S. Zhang, Theory of nonequilibrium in-\ntrinsic spin torque in a single nanomagnet, Physical Re-\nview B 78, 212405 (2008).\n[19] I. M. Miron, G. Gaudin, S. Auffret, B. Rodmacq,\nA. Schuhl, S. Pizzini, J. Vogel, and P. Gambardella,\nCurrent-driven spin torque induced by the rashba effect\nin a ferromagnetic metal layer, Nature materials 9, 230\n(2010).\n[20] I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten,\nM. V. Costache, S. Auffret, S. Bandiera, B. Rodmacq,\nA. Schuhl, and P. Gambardella, Perpendicular switching\nof a single ferromagnetic layer induced by in-plane cur-\nrent injection, Nature 476, 189 (2011).\n[21] L. Liu, C.-F. Pai, Y. Li, H. Tseng, D. Ralph, and\nR. Buhrman, Spin-torque switching with the giant spin\nhall effect of tantalum, Science 336, 555 (2012).\n[22] Y. Zhang, H. Yuan, X. Wang, and X. Wang, Breaking the\ncurrent density threshold in spin-orbit-torque magnetic\nrandom access memory, Physical Review B 97, 144416\n(2018).\n[23] S. M. Vlasov, G. J. Kwiatkowski, I. S. Lobanov, V. M.\nUzdin, and P. F. Bessarab, Optimal protocol for spin-\norbit torque switching of a perpendicular nanomagnet,\nPhysical Review B 105, 134404 (2022).\n[24] M. Aryal, B. Choi, and T. Speliotis, Magnetic-field-\nfree spin–orbit torque-driven magnetization dynamics\nin cofeb/ β-w-based nanoelements, AIP Advances 12,\n015123 (2022).\n[25] M. Wang, Z. Wang, C. Wang, and W. Zhao, Field-free\ndeterministic magnetization switching induced by inter-\nlaced spin–orbit torques, ACS Applied Materials & In-\nterfaces 13, 20763 (2021).\n[26] V. Krizakova, M. Perumkunnil, S. Couet, P. Gam-\nbardella, and K. Garello, Spin-orbit torque switching of\nmagnetic tunnel junctions for memory applications, Jour-\nnal of Magnetism and Magnetic Materials 562, 169692\n(2022).\n[27] D. G. Ovalle, A. Pezo, and A. Manchon, Spin-orbit\ntorque for eld-free switching in c {3v}crystals, arXiv\npreprint arXiv:2301.01133 (2023).\n[28] J. Grollier, V. Cros, H. Jaffres, A. Hamzic, J.-M. George,G. Faini, J. B. Youssef, H. Le Gall, and A. Fert, Field\ndependence of magnetization reversal by spin transfer,\nPhysical Review B 67, 174402 (2003).\n[29] H. Morise and S. Nakamura, Stable magnetization states\nunder a spin-polarized current and a magnetic field,\nPhysical Review B 71, 014439 (2005).\n[30] T. Taniguchi and H. Imamura, Critical current of spin-\ntransfer-torque-driven magnetization dynamics in mag-\nnetic multilayers, Physical Review B 78, 224421 (2008).\n[31] Y. Suzuki, A. A. Tulapurkar, and C. Chappert, Spin-\ninjection phenomena and applications, in Nanomag-\nnetism and Spintronics (Elsevier, 2009) pp. 93–153.\n[32] Z. Z. Sun and X. R. Wang, Theoretical limit of the\nminimal magnetization switching field and the optimal\nfield pulse for stoner particles, Physical review letters 97,\n077205 (2006).\n[33] X. R. Wang and Z. Z. Sun, Theoretical limit in the mag-\nnetization reversal of stoner particles, Physical review let-\nters98, 077201 (2007).\n[34] X. R. Wang, P. Yan, J. Lu, and C. He, Euler equation\nof the optimal trajectory for the fastest magnetization\nreversal of nano-magnetic structures, EPL (Europhysics\nLetters) 84, 27008 (2008).\n[35] Y. Kim, X. Fong, K.-W. Kwon, M.-C. Chen, and K. Roy,\nMultilevel spin-orbit torque mrams, IEEE Transactions\non Electron Devices 62, 561 (2014).\n[36] G. Bertotti, C. Serpico, and I. D. Mayergoyz, Nonlinear\nmagnetization dynamics under circularly polarized field,\nPhysical Review Letters 86, 724 (2001).\n[37] Z. Sun and X. R. Wang, Strategy to reduce minimal mag-\nnetization switching field for stoner particles, Physical\nReview B 73, 092416 (2006).\n[38] S. I. Denisov, T. V. Lyutyy, P. H¨ anggi, and K. N. Trohi-\ndou, Dynamical and thermal effects in nanoparticle sys-\ntems driven by a rotating magnetic field, Physical Review\nB74, 104406 (2006).\n[39] S. Okamoto, N. Kikuchi, and O. Kitakami, Magnetization\nswitching behavior with microwave assistance, Applied\nPhysics Letters 93, 102506 (2008).\n[40] J.-G. Zhu and Y. Wang, Microwave assisted magnetic\nrecording utilizing perpendicular spin torque oscillator\nwith switchable perpendicular electrodes, IEEE Trans-\nactions on Magnetics 46, 751 (2010).\n[41] C. Thirion, W. Wernsdorfer, and D. Mailly, Switching of\nmagnetization by nonlinear resonance studied in single\nnanoparticles, Nature materials 2, 524 (2003).\n[42] K. Rivkin and J. B. Ketterson, Magnetization rever-\nsal in the anisotropy-dominated regime using time-\ndependent magnetic fields, Applied physics letters 89,\n252507 (2006).\n[43] Z. Wang and M. Wu, Chirped-microwave assisted magne-\ntization reversal, Journal of Applied Physics 105, 093903\n(2009).\n[44] N. Barros, M. Rassam, H. Jirari, and H. Kachkachi, Op-\ntimal switching of a nanomagnet assisted by microwaves,\nPhysical Review B 83, 144418 (2011).\n[45] N. Barros, H. Rassam, and H. Kachkachi, Microwave-\nassisted switching of a nanomagnet: Analytical determi-\nnation of the optimal microwave field, Physical Review\nB88, 014421 (2013).\n[46] T. Tanaka, Y. Otsuka, Y. Furomoto, K. Matsuyama,\nand Y. Nozaki, Selective magnetization switching with\nmicrowave assistance for three-dimensional magnetic\nrecording, Journal of Applied Physics 113, 1439088\n(2013).\n[47] G. Klughertz, P.-A. Hervieux, and G. Manfredi, Autores-\nonant control of the magnetization switching in single-\ndomain nanoparticles, Journal of Physics D: Applied\nPhysics 47, 345004 (2014).\n[48] Z. Juthy, M. Pikul, M. Akanda, and M. Islam, Shape\nanisotropy effect on magnetization reversal induced by\nlinear down chirp pulse, Physica B: Condensed Matter\n630, 413671 (2022).\n[49] M. T. Islam, X. Wang, Y. Zhang, and X. Wang,\nSubnanosecond magnetization reversal of a magnetic\nnanoparticle driven by a chirp microwave field pulse,\nPhysical Review B 97, 224412 (2018).\n[50] M. T. Islam, M. A. S. Akanda, M. A. J. Pikul, and\nX. Wang, Fast magnetization reversal of a magnetic\nnanoparticle induced by cosine chirp microwave field\npulse, Journal of Physics: Condensed Matter 34, 105802\n(2021).\n[51] J. Dubowik, Shape anisotropy of magnetic heterostruc-\ntures, Physical Review B 54, 1088 (1996).\n[52] A. Aharoni, Demagnetizing factors for rectangular fer-\nromagnetic prisms, Journal of Applied Physics 83, 3432\n(1998).\n[53] T. L. Gilbert, A phenomenological theory of damping in\nferromagnetic materials, IEEE transactions on magnetics\n40, 3443 (2004).\n[54] W. Fuller Brown Jr, Thermal fluctuations of a single-\ndomain particle, Journal of Applied Physics 34, 1319\n(1963).\n[55] D. Hinzke, N. Kazantseva, U. Nowak, O. N. Mryasov,\nP. Asselin, and R. W. Chantrell, Domain wall properties\nof FePt: From bloch to linear walls, Phys. Rev. B 77,\n094407 (2008).\n[56] J. Lu and X. R. Wang, Magnetization reversal of single\ndomain permalloy nanowires, Journal of magnetism and\nmagnetic materials 321, 2916 (2009).\n[57] T. Hasegawa, S. Kanatani, M. Kazaana, K. Takahashi,\nK. Kumagai, M. Hirao, and S. Ishio, Conversion of FeCo\nfrom soft to hard magnetic material by lattice engineering\nand nanopatterning, Scientific Reports 7, 13215 (2017).\n[58] B. Dieny and M. Chshiev, Perpendicular magnetic\nanisotropy at transition metal/oxide interfaces and appli-\ncations, Reviews of Modern Physics 89, 025008 (2017).\n[59] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen,\nF. Garcia-Sanchez, and B. Van Waeyenberge, The de-\nsign and verification of mumax3, AIP advances 4, 107133\n(2014).\n[60] M. T. Islam, X. S. Wang, and X. R. Wang, Thermal gra-dient driven domain wall dynamics, Journal of Physics:\nCondensed Matter 31, 455701 (2019).\n[61] M. Islam, M. Akanda, F. Yesmin, M. Pikul, and J. Is-\nlam, Role of shape anisotropy on thermal gradient-driven\ndomain wall dynamics in magnetic nanowires, Modern\nPhysics Letters B 37, 2350013 (2023).\n[62] R. H. Koch, G. Grinstein, G. Keefe, Y. Lu, P. Trouilloud,\nW. Gallagher, and S. Parkin, Thermally assisted magne-\ntization reversal in submicron-sized magnetic thin films,\nPhysical review letters 84, 5419 (2000).\n[63] Z. Li and S. Zhang, Thermally assisted magnetization\nreversal in the presence of a spin-transfer torque, Physical\nReview B 69, 134416 (2004).\n[64] J. De Vries, T. Bolhuis, and L. Abelmann, Temper-\nature dependence of the energy barrier and switching\nfield of sub-micron magnetic islands with perpendicular\nanisotropy, New journal of physics 19, 093019 (2017).\n[65] J. M. Iwata-Harms, G. Jan, H. Liu, S. Serrano-Guisan,\nJ. Zhu, L. Thomas, R.-Y. Tong, V. Sundar, and P.-\nK. Wang, High-temperature thermal stability driven by\nmagnetization dilution in CoFeB free layers for spin-\ntransfer-torque magnetic random access memory, Scien-\ntific reports 8, 1 (2018).\n[66] W. Liu, B. Cheng, S. Ren, W. Huang, J. Xie, G. Zhou,\nH. Qin, and J. Hu, Thermally assisted magnetization con-\ntrol and switching of Dy3Fe5O12and Tb 3Fe5O12ferri-\nmagnetic garnet by low density current, Journal of Mag-\nnetism and Magnetic Materials 507, 166804 (2020).\n[67] A. Aharoni, Demagnetizing factors for rectangular fer-\nromagnetic prisms, Journal of Applied Physics 83, 3432\n(1998).\n[68] C. Kittel, Introduction to Solid State Physics , 8th ed.\n(Wiley, 2004).\n[69] M. Islam, M. Pikul, and X. Wang, Thermally assisted\nmagnetization reversal of a magnetic nanoparticle driven\nby a down-chirp microwave field pulse, Journal of Mag-\nnetism and Magnetic Materials 537, 168174 (2021).\n[70] M. Sameh, Y. Shukrinov, A. Y. Ellithi, T. El Sherbini,\nand M. Nashaat, Josephson current-assisted reversal of\na single-domain nanoscale ferromagnet driven by co-\nsine chirp pulse, Journal of Physics: Condensed Matter\n(2023).\n[71] L. Cai, D. A. Garanin, and E. M. Chudnovsky, Reversal\nof magnetization of a single-domain magnetic particle by\nthe ac field of time-dependent frequency, Physical Review\nB87, 024418 (2013).\n[72] L. Cai and E. M. Chudnovsky, Interaction of a nanomag-\nnet with a weak superconducting link, Physical Review\nB82, 104429 (2010)."    },    {        "title": "2309.06110v1.Predicting_synthesizable_manganese_nitride_with_unprecedentedly_giant_magnetocrystalline_anisotropy_energy.pdf",        "content": "1Predictingsynthesizablemanganesenitridewithunprecedentedlygiant\nmagnetocrystallineanisotropyenergy\nZe-JinYang1一\n1SchoolofScience,ZhejiangUniversityofTechnology,Hangzhou,310023,China\nUsingmoderncrystalstructurepredictionprogram(CALYPSO),wesearched\nmanyexperimentallysynthesizablelow-energystructureswithperfectornearly\nperfecteasy-axismagnetocrystallineanisotropyenergy(MAE)inmanganesenitride,\nincludingMnN,Mn2N,Mn3N2,Mn5N2,Mn4N,respectively,whicharethemore\nfrequentlystudiedstoichiometriesbyexperimentalresearchers.MnN(dI24)shows\ngiantMAEwithvaluesofE001=1006,E010=0,E100=920μeV/atom(samehereinafter),\nrespectively.Oneperfecteasy-axisMAEinMn3N2(P42/mmc)withcorrespondent\nvaluesofE010=E100=12isobserved,theothernearlyperfecteasy-axisMAEone(Ibam)\nwithrespectivevaluesofE001=324andE010=345isobserved.Fouralmosttotally\nperfecteasy-axisMAEstructuresareobtainedinMn2N,includingP4/mmmwith\nindividualE001=249andE100=250,PccmwithE001=E100=62,P4/nmmwithE001=58\nandE100=60,ImmawithE001=108andE100=109,respectively.Threestructures\nincludingoneperfectcandidatesarefoundinMn4N,includingFmmmwithindividual\nE001=126andE010=121,I4/mmmwithE010=127andE100=133,I4/mmmwithE001=\nE100=169,respectively.Toomanyvaluablestructuresaredeservedtobefurther\nstudiedbyboththeoreticalandexperimentalscientists.Thepresentstudymight\nattractcloseattentiontotheseseveralcompounds.\n一zejinyang@zjut.edu.cn2I.Introduction\nPermanentmagnetswithextraordinarymagneticpropertiesareextensivelyusedin\nmodernindustrysuchasincomputerharddiskdrivesandwindturbinegeneratorsas\nwellaselectricorhybridcarmotors,thesepropertiesmainlyincludeslargeCurie\ntemperature(Tc)andsaturationmagnetization(µ0Ms)aswellasthe\nmagnetocrystallineanisotropyenergy(MAE),particularlyforthoseuniaxial(Ku)or\neasy-magnetization-axisMAEcompounds.MAEdefinestheenergiesormagnetic\nworkrequiredforbringingthedirectionofthemagnetizationfromtheeasydirection\ntothatimposedbytheperformedmagneticfield(viceversa,itistheenergythat\nforcesthemomentstoalignalongtheeasyaxis,ortheenergyrequiredtoreorientthe\nelectronspinsinaferromagnetfromeasy-tohard-axisdirections).SmCo5and\nNdFe14Barethetwomostprominentsubstancesincommercialproduction,while\nbothofthemcontainrare-earthelementswhichareveryexpensiveinpricedueto\ntheirverylimitedresourceinourearth.Forthemoreextensivemodernindustry\napplications,theresearchershavetosearchorsynthesizemorecheapmaterialswith\nsameproperties.\nMn-Ncompoundsexistrichmagneticphases,suchasuntilnow,theε-Mn4N,\nζ-Mn2N(frequentlycalledMn5N2),θ-MnN,andη-Mn3N2phaseshavebeen\nsynthesized[1-5],alsoincludingθ-Mn6N5.26etc[6],whereastheirMAEislessstudied\ndueprobablytothepracticaldifficultieswhetherintheoreticalcomputationor\nexperimentalmeasure,thusitisimportanttoexplorewhetherthereexiststhe\nexcellentMAEinMn-Ncompounds.Moreover,consideringthepossibledifficultyto3synthesizetheabovealloyscompounds,itisalsoverydifficulttosynthesizethe\nperfectstoichiometry,itisnecessarytotheoreticalpredictionthemtoguide\nexperimentalexplorationinordertoreducethecost.Mnisverycheaprelativetothat\nofCo,studyonMnisthusabsolutelynecessary.Basedonthesemotivations,we\ncarefullysearchedandanalysizedtheobtainedstructuresforMnN,Mn2N,Mn3N2,\nMn5N2,Mn4N,respectively,manypapershavediscussedthestructuresandthe\npropertiesoftheabovealloys,wethusignoretorepeatthemforsimplicity.Foreasier\nreadingpurpose,wealsoignorethedecimaldataforMAEatthediscussionsection.\nII.Methods\nThestructuresaresearchedbyCALYPSOcode[7,8],andfurtheroptimizedby\nprojector-augmentedwavemethod[9].Thespin\rpolarizedgeneralizedgradient\napproximationandthePerdew-Burke-Ernzerhoffunctionalareusedtomodelthe\nexchange-correlationenergy[10].Theelectronicandforceconvergenceisbelow10-7\neVand0.01eV/Å,thek-pointdensityisbelow2π×0.02Å−1,andplan-wavebasisis\n350eV,whicharesufficientforthemagneticatomsbasedonthecalculationsoniron\nsilicides[11].Theon-sitecoulombinteractiontermU(andJ)isignoredbasedonthe\npreviousconclusionofironsilicides[11].\nIII.Resultsanddiscussions\nMnN\nAllofthelowest-energyaxesofthestructuresaresettotheenergyreferenceand\nignoredforsimplicity.Thelowest-energyMnNstructure(ID_1,2,3,Table1)hasa\nsymmetryofmF34(No.216),consistingwithpreviouscalculations[12].4Furthermore,I4/mmm(No.139)ormF3m(No.225)isthemetastablestructure,also\nagreeingwithprevioussynthesization[12].ThemF34structurepresentsE010=454,\nE100=624μeV/atom,respectively,whichisverylargeMAEdespiteitsnon-perfect\neasy-axisdistributions.SeveralhundredsofμeV/atomisagiantvaluethatisnot\nusuallyoccurredinthematerials.Owingtothepracticalapplicationonlyconcernsthe\ntotalMAEvalue,wethusignoretodiscussthespinandorbitalmomentcomponents.\nSuchlargevaluesstillhavecertainapplicationsinsomespecialtechniquefields\nrequiringmulti-axisstep-likeenergydistributionfeatures.ThismF34mightbe\neasilysynthesizedduetoitsperfectcubicstructure,inwhicheachMnhasfour\nfirst-nearestneighborsN,andeachNalsohasfourfirst-nearestneighborsMn,similar\ntothatofmethane.Inaddition,thelatticeenergywillbelowered(fromEf=-0.2523to\n-0.2639eV/atom)ifthelatticeparameterelongatesslightly(from4.2074to4.2376Å),\nasaresulttheaxialenergyorderandvaluealsoredistribute,withvaluesofE001=642,\nE010=347μeV/atom,respectively.Thisphenomenonillustratesthestrongatomic\ndistancedependenceoftheMAEunderthelatticeparameterelongationoflessthan\nabout0.7%.NegligiblelatticeparameterelongationinID_3relativetothatofID_1\nwillalternateaxialenergyorder,revealingthestrongatomicsitedependenceof\nspin-orbitalcoupling.\nAslightlydistortedstructure(ID_4,Cm,No.8,Ef=-0.2582eV/atom),whose\nMAEisreducedcorrespondinglytoE001=262,E010=200μeV/atom.Itsatomic\narrangementisgenerallysamewiththoseofID_1-3,justaswhosesymmetryis\nmF34whendeterminingitbyatoleranceof0.1Å.Toobtainthesitedependenceof5theMAE,themoreusuallyusedmethodistoshifttheatomgraduallywhilekeeping\ntheotheratomsunchanged,whichwillcausethelatticeinstableandinduce\nsubstantialstresswithinthelatticemoreorless,correspondingtoahypotheticaland\ntransientstatestructure,andsuchstrainedstructureisimpossibletobesynthesized\nexperimentally.However,thepresentsimilarstructuresshowmultipleand\nsimultaneousvariationsinthewholelatticeparametersandcorrespondtoastable\nstate,comprehensivelyillustratedthecomplexityofthespin-orbitalcoupling.Allof\nthestudiedsimilarstructuresmightbesynthesizedwithnon-equilibriumtechnique.\nTherefore,thepresentstudyismorepracticableanduseful.\nTheexperimental[4]NaCl-fct(face-centeredtetragonal)MnNhasalmostfcc\n(facecentercubic)structureandrelativelyhigherenergy(ID_5,Ef=-0.1737eV/atom)\nthanthatofmF34(No.216,Ef≈-0.25eV/atom),whoseMAEareE001=48,E100=87\nμeV/atom,respectively.However,aperfectfccstructurewillproducegiantMAEin\nID_6,itisE001=2388,E100=124μeV/atom,respectively.\nThesearchedlowest-energystructure(ID_7),withasymmetryofF222(No.22)\nandEf=-0.265eV/atom,haszeromagneticmomentandMAE.Theotherseveral\nstructures,withenergyrangesofabout-0.21~-0.25eV/atom,havegenerallysame\natomicdistributionsandlargeMAEwiththeonlyoneexception(ID_8,Ef=-0.2399\neV/atom,Pmmm,No.59)thatshowsthenegligibleMAE,suchnegligibleMAEisdue\nprobablytoitsspecialatomicfirst-nearestneighborcoordinatenumber(CN),namely,\ntheCNofMnandNare6.Inotherwords,allofthefirstnearestneighborsofMnare\nN,viceversa,whichreducedthespin-orbitalcouplingstrengthandthuscanceledthe6magnetism.Manystructureshavethemethane-likebuildingblock,inwhichtheMn\nandNhavethesamefourCNvalue,suchatomicarrangementwillleavesufficient\nspacefortheneighboringMninteraction.Inotherwords,theaverageanglebetween\nthetwoneighborMn-NisfarlargerthanthatofCN=6(about90ﾟ).Suchasthe\naveragevaluesofthefouranglesoftheN-Mn-NwithinMnN4buildingblockisabout\n110ﾟ.\nAstructure(ID_9),withEf=-0.212eV/atomandsymmetrydI24(No.122),\nabout40meV/atomhigherthanthatoflowest-energystructure,shows\nunprecedentedlygiantnearlyperfecteasy-axisMAE,withvaluesofE001=1006,\nE100=920μeV/atom,respectively,whichisverypossibletobesynthesized\nexperimentallyandhasverypromisingapplications.Thisgiantvalueindicatesthat\natomicarrangementplaysakeyroleduringthespin-orbitalcouplingdespitethatthe\nchemicalstoichiometrybetweenmagneticandnon-magneticatomsis1:1.Inaddition,\naslightdistortionstructure(ID_10)includinganelongationof0.1Åalongcaxisand\nshrinkage0.005Åalongaandbaxeswilldecrease16meV/atominenergyand\nproduceE010=1030,E100=131μeV/atom,respectively,suchsmallvariationdoesn’t\nchangethestructuresymmetry,whichfurtherdemonstratesthattheMAEisquite\nsensitiveontheatomicsite.Inaword,itisclearthatthelatticedistortionunavoidably\nenhancedthelatticeenergy,whereasitalsoenhancedtheMAEinmanycases.Infact,\nmanymetastablephasesusuallypresentlargeMAEthanthatofgroundstate.\nMeanwhile,theavailableID_11presentssimilarMAEwiththatofID_10.\nConclusively,MnNisanimportantcompoundthatconsistsofmanylow-energy7synthesizablestructureswithlargenearlyperfecteasy-axisMAE.Usually,magnetic\ntransitiontemperatureislinearlydependentonthemagneticmomentmagnitude[13]\ninbinarymanganesenitrides,largemomentusuallyinduceslargeMAE,whereasthe\npresentMnNdoesnotobeythegeneralempiricalrule.\nTable1Experimentalandsearchedlow-energyMnNstructuresinformation,\nformationofenergy(Ef,eV/atom),magnetocrystallineanisotropyenergy(MAE,\nμeV/atom),identificationnumber(ID),symmetrydeterminationunderdifferent\ntolerances(Å),latticeparametersandangles,a,b,c,(Å),α,β,γ,(ﾟ),thewords\n“Experimental”and“Searched”havebeenabbreviatedas“Exp.”and“Sea”for\nsimplicity,respectively.The‘available’meansthatthecoordinatedataisdownloaded\nfromwww.materialproject.org.Thebig(purple)andsmall(blue)ballsareMnandN\natoms,respectively,thesameisapplicableinthefollowingtables.\nID\nEf\nExp./Sea.MAE\n001\n010\n100Sym.\n0.001\n0.01\n0.1Unitcell\nMn\nNa,b,c,\nα,β,γ,\n1\n-0.2523\navailable0\n454.006\n624.335mF34216\nmF34216\nmF34216\n4.20745\n4.20745\n4.20745\n90,90,90\n2\n-0.2639\nSea.642.9137\n347.0275\n0mF34216\nmF34216\nmF34216\n4.23763\n4.23763\n4.23763\n90,90,9083\n-0.252\nSea.579.391\n206.887\n0mF34216\nmF34216\nmF34216\n4.20785\n4.20785\n4.20785\n90,90,90\n4\n-0.2582\nSea.262.877\n200.112\n0Cm,8\nCm,8\nmF34216\n5.23877\n2.95821\n4.19565\n90\n145.0479\n90\n5\n-0.1737\nNaCl-fct\nExp.48.0325\n0\n87.335I4/mmm139\nmF3m225\nmF3m225\n4.05778\n4.05778\n4.05002\n90,90,90\n6\n-0.1976\navailable2388.3\n0\n124.68mF3m225\nmF3m225\nmF3m225\n4.06495\n4.06495\n4.06495\n90,90,90\n7\n-0.265\nSea.0\n0\n0F222,22\n24mI,119\n24mI,119\n4.12575\n4.38498\n4.12459\n90,90,90\n8\n-0.2399\nSea.7.85\n0\n2.265Pmmm,59\nPmmm,59\nPmmm,59\n2.89174\n2.80830\n3.94227\n90,90,90\n9\n-0.2123\nSea.1006.19\n0\n920.2125dI24,122\ndI24,122\ndI24,122\n4.32858\n4.32858\n7.86636\n90,90,90910\n-0.2382\nSea.0\n1030.9\n131.762dI24,122\ndI24,122\ndI24,122\n4.32373\n4.32373\n7.95674\n90,90,90\n11\n-0.1987\navailable185.495\n1313.07\n0P63mc186\nP63mc186\nP63mc186\n2.91368\n2.91368\n5.21045\n90,90,120\nMn3N2\nManysearchedMn3N2structureshavecomparableenergieswithintheEfrange\nof-0.2~-0.21eV/atom,inwhichthelowest-energystructureofMn3N2hasaEfof\nabout-0.21eV/atom.Infact,almostallofthemhavenearlyidenticalstructuresor\nlatticeshapes(α=β=γ=90ﾟ)exceptforsomeminordifferenceinlatticeparametersor\natomiccoordinates,whichmightbeoneofthereasonableoriginsoftheircomparable\nMAEalongthea(100),b(010),c(001)axes,respectively.Indetail,theMAEare\naboutE001≈110,E100≈80μeV/atom,respectively.Thecommonsymmetrymightbe\nI4/mmm(No.139)whenatoleranceorcriterionof0.01Åisused,whereasP42/mcm\n(No.132)orP42/mnm(No.136)mightbeobtainedunder0.001Åoriginatingfromthe\nsubtlelatticedistortion,asisclearlyseenfromthevariationsofthelatticeparameters\naandbwithafluctuationof≥0.01Å,seestructuresID_1andID_2(showninTable2)\nandID_S1-3showninTableS1ofsupplementalinformation(SI)fordetails.\nTherefore,itispossibletosynthesizethisstructureastherealwaysexistsastateat\nthisenergyrange,inadditiontoitsmediummagnetismisalsousefulinsomespecial\nfields.10Alowest-energyMn3N2structure(ID_2,Table2)issearchedandshows\ndecreasedeasy-planeMAE,E001≈61,E100≈0.6μeV/atom,respectively.Itssymmetryis\nCmcm(No.63)atatoleranceof0.01Å,orI4/mmm(No.139)atatoleranceof0.001\nÅ,respectively.Thespin-orbitalcouplingstrengthistotallyreducedatallthethree\naxeswhencomparingitscorrespondentaxialenergytothoseofID_1.Therefore,this\nisaconvincingexampletojustifythefactthatthelatticedistortionindeedcould\nsubstantiallyenhanceorreducetheMAEevenunderthesamelatticeshapewithonly\ncertainlengtheningandshorteningalongthelatticeaxis.\nAMn3N2structure(ID_3)showsEf≈-0.2eV/atom,E001≈92,E100≈192μeV/atom,\nrespectively,whosesymmetryisPnma(No.62)atatoleranceof0.01Å,orPbam\n(No.55)atatoleranceof0.001Å,respectively.Clearly,thisstructurealsohas\nrelativelylargedistortionwiththoseofID_1andID_2,whoseenergyisabout10\nmeV/atomhigherthanthatofID_2.ThecurrentlysearcheddiverseMAEcould\nenrichourunderstandingtothecouplingmechanismandguidetheexperimental\nsynthesization.\nTheEfofthesearchedMn3N2structure(ID_4,Cmcm,No.63)isabout-0.2\neV/atom,whereasitsmagneticmomentiszero,presentinganisotropyenergy\ndistributionandthecancellationofthespinandorbitalmoment,clearly,thiskindof\nstructurehasdirectconnectionwiththoseofID_1andID_2accordingtothe\nhexagonalstructurefeature.\nTheotherenergeticallycomparableMn3N2structuresshowaEfrangeofabout\n-0.17~-0.18eV/atom,about30meV/atomhigherthanthoseof-0.2~-0.21eV/atom11discussedabove.StructureswithenergydifferencesbelowaboutseveralmeVusually\nbelongtothesamephaseoriginatingfromtheverysimilaratomicarrangement.Thus,\nthesestructuresprovidevaluablereferencesforcomparisonthestructure-MAE\nrelationshipascandidatepopulationslocatingatdifferentpotentialenergyvalleys,\nsimilarly,itisalsoagoodideatocatalogueanddiscussthemtogether.\nOneofthestoichiometry(thecoordinatefilescouldbedownloadedfrom\nwww.materialproject.org)Mn3N2phase(ID_5)hasaEfof-0.156(I4mmm,No.139),\nwhichpresentslargeMAE,includingE001≈360,E010≈206μeV/atom,respectively.\nID_6showsacontractionalongthethreeaxescomparedwiththatofID_5,whereas\nitsMAEremainsalmostunchanged,indicatinganisotropiccontraction.Meanwhile,a\nsmalllatticeshorteninginID_6didn’tchangetheenergyorderandvaluesamongthe\nthreeaxesrelativetothatofID_5,differingwiththecaseofID_1ofMnNinTable1.\nID_5hasasimilarunitcellwiththoseofstructureshavingEf≈-0.2~-0.21eV/atom,\nsuchasID_1andID_2.AsfarasthelargeMAEvaluesareconcerned,ID_5might\nalsohassomepotentialapplicationdespiteitsnon-perfecteasy-axisMAEnature.The\notherexperimentalone(ID_7,13mP,No.164)isahigh-energystate,whose\nEf≈-0.125eV/atomandMAEissmall,indetail,E001≈21,E010≈10μeV/atom,\nrespectively.TheothersimilarstructureswiththatofID_5isshowninTableS1-2.\nStructuraltensionorcompressionindeedcouldadjustMAE,suchasoneMn3N2\nstructure(ID_8)showsEf=-0.17eV/atom,withasymmetryofP63/mcm(No.193),\nwhoseMAEisabout7,10,0μeV/atomalong001(c)and010(b)directionrelativeto\nthatof100(a)direction,respectively.Slightelongationalonga,b,cdirections12(a=b=4.6214(4.6432),c=9.1971(9.2236)Å)withavalueofabout0.02Åwillenlarge\nthecorrespondingMAEtoabout29,74,0μeV/atomwiththecostofenhancementof\nlatticeenergyabout2meV/atom,asisshowninTableS3(ID_S4).Similarly,the\nMAEofID_S5(TableS4)willbeabout98,0,87μeV/atombutthelatticeenergy\nremainsalmostunchangedwhenthelatticeparameterschangetoa=b=4.6393,\nc=9.1192Å,α=β=90ﾟ,γ=120.0001ﾟ,respectively.Roughlyspeaking,thesestructures\nshowgenerallysimilaratomicarrangements.Strictlyspeaking,thespacegroupofthe\nthreestructuresaredifferent,suchasID_8hasmostsymmetryoperationandthus\nshowsagroupofP63/mcm(No.193)atanythreedifferentkindsoftolerances\nincluding0.001,0.01,0.1Å,respectively.ItstillisP63/mcmataprecisionof0.1Åin\nID_S4(TableS3),whereasitisP63cm(No.185)attolerancesofand0.001and0.01Å,\nmeaningthattheatomicrelativeshiftrangeislargerthan0.1Åandcorrespondsto\nabout5%elongationofthechemicalbonds.ThisistrueinthecaseofID_S5(Table\nS4),whosesymmetryisP63(No.173)atatoleranceof0.001ÅorP6322(No.182)at\ntolerancesof0.1and0.01Å,respectively.Thenegligiblevariationisdeservedtobe\ncarefullyanalysizedastheelectricdeviceoftenworksathightemperatureconditions,\noriginatingprobablyfromthelargecurrentintensityorthelongworkingtimeorthe\nslowheatdissipationandsoon,whichwillinevitablyenlargethelatticesizeandthus\nchangetheMAE.TheothersimilarcasesareshowninTablesS3-4forreference\npurpose.\nAnearlyperfecteasy-axisMAEstructure(ID_9)issearched,asisshownin\nTable2,whoseEf=-0.137eV/atom,E001≈325,E010≈345μeV/atom,respectively.Its13energyisabout20meV/atom(Ibam,No.72)higherthanthatofexperimentalI4/mmm\n(No.139,Ef=-0.156eV/atom),thusitisstillpossibletobesynthesizedexperimentally.\nSimilarly,alargeMAEstructure(ID_10)isalsosearchedwithasymmetryofP21\n(No.4),whoseEf=-0.12eV/atom,E001≈381,E010≈307μeV/atom,respectively.A\nperfecteasy-axisMAEstructure(ID_11)issearched,whoseEf=-0.129eV/atom,\nE010=E100=12μeV/atom,respectively,correspondingtoasymmetryofP42/mmc\n(No.131)atatolerancesof0.1and0.01Å,whereasitisCccm(No.66)atatolerance\nof0.001Å,respectively,presentingsomeapplicationsinthelow-MAErequirements\nfields.Inaword,Mn3N2deservedtobesynthersizedastherealwaysexistsuseful\nMAEatdifferentenergystates.\nTable2Experimentalandsearchedlow-energyMn3N2structuresinformation,\nID\nEf\nExp./Sea.MAE\n001\n010\n100Sym.\n0.001\n0.01\n0.1Unitcell\nMn\nNa,b,c,\nα,β,γ,\n1\n-0.2121\nSea.102.86\n0\n77.645P42/mnm,136\nI4/mmm,139\nI4/mmm,139\n3.98585\n3.98585\n11.61646\n90,90,90\n2\n-0.2148\nSea.61.74\n0\n0.635Cmcm,63\nI4/mmm,139\nI4/mmm,139\n3.98358\n3.98358\n11.62871\n90,90,90\n3\n-0.2033\nSea.92.5\n0\n192.445Pnma,62\nPbam,55\nPbam,55\n6.87030\n5.71719\n4.90334\n90,90,90144\n-0.2019\nSea.0\n0\n0Cmcm,63\nCmcm,63\nCmcm,63\n2.58051\n6.69912\n9.86924\n90,90,90\n5\n-0.1566\nSea.360.789\n206.156\n0I4/mmm,139\nI4/mmm,139\nI4/mmm,139\n2.86260\n2.86260\n12.05071\n90,90,90\n6\n-0.1572\nExp.361.671\n206.488\n0I4/mmm,139\nI4/mmm,139\nI4/mmm,139\n2.828973\n2.828973\n11.82875\n90,90,90\n7\n-0.1254\nExp.21.442\n10.378\n013mP,164\n13mP,164\n13mP,164\n3.05375\n3.05375\n5.88195\n90,90,120\n8\n-0.1701\nSea.7.095\n10.82\n0P63/mcm,193\nP63/mcm,193\nP63/mcm,193\n4.62140\n4.62140\n9.19712\n90,90,120\n9\n-0.1369\nSea.324.075\n345.485\n0Ibam,72\nIbam,72\nIbam,72\n4.43632\n5.12811\n7.66036\n90,90,90\n10\n-0.1202\nSea.381.455\n307.855\n0P21,4\nP21,4\nCmc21,36\n3.80226\n11.81158\n3.80017\n90，\n92.3100，\n901511\n-0.1293\nSea.0\n12.61\n12.61Cccm,66\nP42/mmc,131\nP42/mmc,131\n3.59118\n3.59411\n12.99978\n90,90,90\nMn2N\nAsisshowninTable3,experimentalMn2NPbcn(No.60)haslowestenergy\namongallofthestudiedstructures,whereasdifferentsynthesizationtechniques\nusuallyobtaindifferentstructuresduemainlytothecomplicatedchemicalreaction,as\nisseenfromID_S1-6(TableS5).Forclearpurpose,weplotallofthesecandidate\nstatesinFigure1.Unfortunately,thesestructureshaven’tverylargemagnetismand\nperfecteasy-axisMAE.Theaxialenergyorderdependsonthethreelattice\nparametersamongID_S1-6buttheaxialenergydifferencesarealmostunchanged,\nmeaningthattheorbitalandspinmomentsarenotsignificantlychanged.Weshow\nthesestructuresinordertoemphasizethatthelow-energyMn2Npresentsnotlarge\nMAE.\nAsisshowninTable3,theEfandsymmetryofID_2are-0.2451eV/atomand\nP63/mmc(No.194),respectively,withMnN3-typebuildingblock(tetrahedron).The\ncorrespondentcoordinatenumber(CN)ofMn(orMnN3)andN(orMn6Noctahedron)\nis3and6,thedistancebetweenMnandNisabout1.95Å,inaddition,thedistance\nbetweentwonearestneighboringMnisabout2.43Å.Thisstructureshowsthe\nlayered-likestructurewithrelativeweakinteractionbetweenthetwonearest\nneighboringMn.TheanglesofthethreeN-Mn-Nconnectionare83.7ﾟ.Projections16towardsaandbaxesareidenticalandthustheiraxialenergiesarealsoidentical,Mn\ndistributesatthecenterofanequilateraltriangleformedbythethreeNatomsifthe\nprojectionsofMnistowardscaxisora,bplane,thusthemagnetismalongcaxisis\nalsozero,theprojectionorientationofthenearest-neighborsinglelayertowardsaorb\naxisisjustcontrary,thusthemagnetismalsocanceledaccordingtothestructure\nfeatureofP63/mmc.\nID_3(Ef=-0.2457eV/atom,P212121,No.19)showsmediumMAEwithvaluesof\nE001=36,E100=119μeV/atom,respectively.Thislow-symmetrystructurecausesthe\ndiversedistancebetweentheMn5-N(withinoneunitcell),namely,1.9404,1.8421,\n1.9907,1.8686,1.9279Å,respectively.\nEasy-axisMAEisthemostidealcandidatematerialandcouldpresentextremely\nimportantindustrialapplications.ThesearchedstructureID_4(Fmmm,No.69)isa\npromisingstructure,asisshowninTable3,whoseEfis-0.203eV/atomandMAEare\nE001=249,E100=250μeV/atom,respectively,inaddition,itsMnN3buildingblockis\nbendotherthanplanarstructure(thefouratomslayinaplane).Asimilarstructure\n(Table3,ID_5)withthatofID_4showsEf=-0.2295eV/atom(P42/mnm,No.136)and\nmediumMAE,includingE001=10,E100=37μeV/atom,respectively.Usually,electric\ndeviceoperatesathightemperatureandthusthelatticemightbeexpandedinevitably\noranisotropically,thusID_4mightproducetotallyperfecteasy-axisMAEathigh\ntemperatureconditions.\nStrictlyspeaking,manystructuresshowEfwithin-0.19~-0.2eV/atomwith\ndiversesymmetries,whereastheyareessentialthesamestructureduemainlytothe17moreorlessdistortions.Roughlyspeaking,theyhavethesymmetryP42/mnm\n(No.136)withatoleranceof0.01Å.Thesestructuresshowdiversebutmediumeven\nsmallMAE,whereasnoperfecteasy-axisMAEisformed,asareshowninTables\nS6-S8,ID_S7-19.\nOnestructure(Table3,ID_6)havingEf=-0.173eV/atomandsymmetryPccm\n(No.49)showsalmostperfecteasy-axisMAEwithmediumvaluesofE001=62,\nE100=62μeV/atom,respectively,whichmightbesynthesizedexperimentally.The\nsameprojectionsofID_6towardsaandcaxesconfirmtheirnearlysameaxial\nenergies.TheaveragedistanceofthefirstnearestneighborsoftheNwithitssixMn\natomsis1.97Å.TheaverageMnatomicdistancebetweenthetwoneighboringlayer\nis2.36Å.Thislayerstructureissimilarwiththatofnon-magnetismstructure(ID_2in\nTable3).TheirdifferenceisthattheangleoftheMn6Nbuildingblocks,itisabout90\nﾟinID_6butitis83.7ﾟinID_2.DuetothenottotallysameMn-Ndistancein\nMn6Nbuildingblock,theMAEisthusformedinID_6.Meanwhile,onestructure\nID_3(P212121,No.19)presentingsimilarenergywiththatofID_2issearched,whose\nE001=36,E100=119μeV/atom,respectively.\nAperfecteasy-planeMAEstructureID_7(Table3)issearched,presentinglarge\nMAEvalues,E001=270μeV/atom,respectively,whichshowsdiversesymmetriesand\nhasaEfof-0.1634eV/atom.AsimilarstructureisshowninID_S21inTableS9.\nUsuallyeasy-axisMAEstructureshavemoreextensiveapplicationsthanthoseof\neasy-planeones,whereaseasy-planestructuresmightalsohavepotentialapplications\ninthefuturemagneticfields.18OnestructureID_8(Table3)withEf=-0.134eV/atomandnearlyperfect\neasy-axisMAEispredicted,whoseE001=60,E100=58μeV/atom,respectively.Its\nsymmetryisdependentontheusedtolerance,suchasitisP2/c(No.13)at0.001Å,\nPmmn(No.59)at0.01Å,P4/nmm(No.129)at0.1Å,respectively.Averysimilar\ncasecouldalsobeenseeninID_9,whichshowsevenlowerenergythanthatofID_8.\nID_9structureshowscertainsimilaritywiththestructureID_S20(TableS9,\nEf=-0.1399eV/atom),whereastheMAEoflatterisdecreased,suchas,E001=5,\nE100=38μeV/atom,respectively,whereasbothofwhichhavesameCN,namely,theN\nhassixfirst-nearestMnneighbors,andoneoftheMnhasfivefirst-nearestN\nneighbors,whereastheotherMnhasonlyonefirst-nearestNneighbors.TableS10is\nthesimilarstructurewithinenergyrangesofabout-0.1～0.11eV/atom.Agenerally\nsamestructure(TableS11,ID_S22)withcertaindistortionissearched,presenting\nsimilarenergybutsmallerMAEcomparedwiththatofID_8,indetail,whose\nEf=-0.137eV/atomandE001=38,E100=32μeV/atom,respectively.Moreover,both\nstructureshavenearlysamelatticeparameters,indicatingMAEisverysensitiveon\ntheatomiccoordinates.Projectionstowardsa,corientationsarenearlysame,which\ncausesthenearlysameaxialanisotropyenergy.\nThelistedTableS12-14isforreferencepurpose.Ahigh-energystructure(Table\nS14,ID_S25)withEf=-0.099eV/atompresentsnearlysameMAEwithrespective\nvaluesofE010=72,E100=74μeV/atom,theseMAEwilltransformtoabout0,117,80\nundersmalldistortion(TableS14,ID_S26).\nAnotherstructure(Table3,ID_10)showsnearlyperfecteasy-axisMAEwith19respectivevaluesofE001=108,E100=109μeV/atom,inaddition,itsEf=-0.112eV/atom.\nItsNhassixfirst-nearestMnneighborsanditsMnhasthreefirst-nearestNneighbors.\nID_11isasimplelatticewithmediumMAE,differingwiththatofID_2whichshows\nzeromoment.\nOnestructure(Table3,ID_12)showsgiantMAE,itisE001=254,E010=1376\nμeV/atom,respectively,withsymmetryPmc2(No.26)andEf=-0.151eV/atom,in\nwhichthemagneticmomentare-0.6and1.36μmforthetwodifferentkindsofMn\natoms,respectively.ThevaluesoftheNare-0.002μB.Usuallylargemomentinduces\nlargeMAE,whereasitisnotthecaseinthisstructure.\nTwostructuresshowlargeMAE,themaximumaxialenergyapproachestoabout\n300μeV/atom,itsEfandsymmetry(TableS12,ID_S23)are-0.154eV/atomandI41\n(No.80)atatoleranceof0.001Å,whoseMAEareE001=281,E100=26μeV/atom,\nrespectively.ItsNatomhassixMnneighbors,itsMnatomshasfourortwoNatoms,\nrespectively.Theotherstructure(TableS12,ID_S24)showsanearlyperfect\neasy-planeMAEwithvaluesofE001=302,E010=0.9μeV/atom,respectively,which\nEf=-0.106eV/atomandsymmetryisPmma(No.51)atatoleranceof0.001Å.\nGenerallyspeaking,thesetwoarethesamestructure.\nAperfecteasy-axisMAEstructure(TableS16,ID_S27)ispredicted,with\nEf=-0.1eV/atom,E010=E100=53.69μeV/atom,respectively.Itssymmetryis\n13mP(No.164)andhastwoneighboringMnlayers,whichisverysimilarwiththatof\nnon-magnetismstructure(Table3,ID_2),whereasthepresentE010=E100=53μeV/atom\niscausedbythelatticedistortionasalloftheα,β,γhavenegligiblevariations,such20asα=90.0005ﾟ，β=89.9995ﾟ，γ=119.9995ﾟ,respectively.It’sstructureismoresimilar\nwiththatofID_11inTable3,whereasitsenergyishigherthanthoseofID_2and\nID_11inTable3.\nTable3Experimentalandsearchedlow-energyMn2Nstructuresinformation\nID\nEf\nExp./Sea.MAE\n001\n010\n100Sym.\n0.001Å\n0.01\n0.1Unitcell\nMn\nNa,b,c,\nα,β,γ,\n1\n-0.3323\nExp.59.9158\n18.585\n0Pbcn,60\nPbcn,60\nPbcn,60\n4.37186\n4.80842\n5.53472\n90,90,90\n2\n-0.2451\nSea.0\n0\n0P63/mmc,194\nP63/mmc,194\nP63/mmc,194\n2.59805\n2.59805\n8.78183\n90,90,120\n3\n-0.2457\nSea.36.2166\n0\n119.1833P212121,19\nP212121,19\nP212121,19\n4.55135\n5.81844\n4.10135\n90,90,90\n4\n-0.203\nSea.249.3958\n0\n250.0416Fmmm,69\nP4/mmm,123\nP4/mmm,123\n5.56460\n7.67059\n5.56905\n90,90,90\n5\n-0.2295\nSea.10.2033\n0\n37.5916P42/mnm,136\nP42/mnm,136\nP42/mnm,136\n4.55210\n4.55210\n2.77969\n90,90,90\n6\n-0.1736\nSea.62.4833\n0\n62.0333Pccm,49\nPccm,49\nPccm,49\n3.96094\n7.05265\n3.92921\n90,90,90217\n-0.1634\nSea.270.0666\n0\n0Cmmm,65\nPmmm,47\nP4/mmm,123\n5.51706\n5.54639\n7.58013\n90,90,90\n8\n-0.1345\nSea.60.2833\n0\n58.05P2/c,13\nPmmn,59\nP4/nmm,129\n3.95580\n7.01459\n3.94784\n90，\n89.6052，\n90\n9\n-0.1425\nSea.58.875\n0\n60.6333Pccm,49\nP4/nmm,129\n3.94971\n7.02324\n3.95589\n90,90,90\n10\n-0.1125\nSea.108.7583\n0\n109.075Pna21,33\nPna21,33\nImma,74\n3.77101\n7.44347\n3.73550\n90,90,90\n11\n-0.1979\navailable0\n83.38\n13.576613mP,164\n13mP,164\n13mP,164\n2.78588\n2.78588\n4.32834\n90,90,120\n12\n-0.1516\nSea.254.2166\n1,376.02\n0Pmc2,26\nPmc2,26\nPmc2,26\n2.55864\n4.34378\n4.79211\n90,90,90\nMn5N2\nTheenergydifferencebetweenthesearchedlowest-energystructure(～-0.11\neV/atom)andthevalueatconvexhull(～-0.24eV/atom)isabout0.13eV/atom,we\nthusonlygiveabriefdiscussionforMn5N2,asisshowninTableS17.TheEfofthe\nlowest-energystructureofMn5N2isabout-0.11eV/atom,whichmightbesynthesized\nexperimentallywhenitsdecomposedproductsarehigh-energymetastableMn2Nand22Mn3N,asisshowninFigure1.Wedidn’tsearchMn3Nowingtotheabsenceofthe\nexperimentaldata,wethusignoredthedetailedanalysistoMn5N2despiteitslarge\nMAE,suchasonestructureID_S1withEf=-0.113eV/atomshowsverylargeMAE,it\nisE010=252,E100=215μeV/atom,respectively,presentinganearlyperfecteasy-axis\nMAE.TheNissurroundedbysixMn,whereasMnhasthreeNneighborswiththe\naveragedistanceofabout2Å.\nMn4N\nAsisshowninTable4(ID_1),thesymmetryandEfoftheexperimentalMn4N\naremPm3(No.221)and-0.136eV/atom,respectively,withMAEofE001=97,\nE010=12μeV/atom,respectively.SeveralstructuresID_S1-S5showevenlowerenergy\nthanthatofexperimentalone,whereastheypresentnegligiblemagnetism,thuswe\nignoretodiscussthem,asisshowninTableS18(TableS19-S22arealsothe\ninformationforMn4N).Infact,ID_S1,ID_S3,ID_S4arealmostthesamestructure\nwithdifferentdegreeofdistortion,ID_S2andID_S5arealsoalmostthesamewith\nthatofexperimentalstructureundersomedistortion.Despitethatthechemical\nstoichiometryofMn:Nis4:1,noneofthesearchedlow-energystructuresshowsvery\nlargeMAE,themaximumvalueisabout200μeV/atom,farsmallerthanthoseof\nMnN,indicatingthatthemagnetismisnotproportionaltothemagneticatomcontent.\nTherefore,consideringthemoreexpensivepriceofMnthanthatofNelement,\nsynthesizationofcheapMnNorMn2Nshouldbethefirstchoice.\nOneMn4NstructureID_2showsslightlyhigherenergy(Ef=-0.129eV/atom)\nthanthatofexperimentalone(Ef=-0.136eV/atom),asisshowninTable4,which23showsasymmetryofFmmm(No.69)andpresentsnearlyperfecteasy-axisMAEand\nmediumvalues,E001=126,E010=121μeV/atom,respectively,thusitisavery\npromisingmagneticcandidatematerial.\nOneMn4NstructureID_3alsopresentsnearlyperfecteasy-axisMAEwith\nmediumvaluesandisonlyabout20meV/atomhigher(Ef=-0.1128eV/atom)thanthat\nofexperimentalone,asisshowninTable4,whosesymmetryisImm2(No.44)under\natoleranceof0.001ÅorI4/mmm(No.139)underatoleranceof0.1Å,the\ncorrespondentMAEvaluesareE010=127andE100=133μeV/atom.Aniso-energy\nstructureID_S1-2listedinTableS22showsevenlargerMAE,withvaluesofE010≈\n205,E100≈130μeV/atom,respectively,behavingasymmetryofI4/mmm(No.139)\nunderalldifferentkindsoftolerance0.001,0.01,0.1Å.Verysmallvolumechange\n(a=b=3.72534/3.72484,c=7.44036/7.43791Å)doesn’tchangethelatticeenergyand\nMAEbetweenID_S1andID_S2.However,whenthelatticeparameterschangeto\na=3.72588,b=3.72621,c=7.43791Å,theE010reducedevidently,from205to127\nμeV/atom(showninTableS22andID_S3),meaningthatthevariationrangehas\nexceededthespin-orbitalcouplinginteractionzone.\nOneMn4NstructureID_4presentstotallyperfecteasy-axisMAEandrelatively\nlargevalues,asisshowninTable4,withvaluesofE001=E100=169μeV/atom,\nrespectively,thusitisoneoftheidealcandidatematerialsandabsolutelydeservedto\nbesynthesizedexperimentally.Itisabout30meV/atomhigherthanthatof\nexperimentaldatawithEf=-0.1eV/atom.ThesimilarstructuresandMAEinducedby\ntheslightdistortionareshowninTableS19ID_S6-7,inparticular,ID_S7hasalmost24identicalenergywiththatofexperimentaldataandnearlyperfectandlargeeasy-axis\nMAE.DespitethatID_S8showsslightlyhigherenergythanthatofID_S7,italso\nshowsnearlyperfectandlargeeasy-axisMAE,similartothatofID_S7.Inaword,\nmanyimportantmetastablephasesofMn4Ncouldbesynthesizedanddeservedtobe\ncarefullystudiedexperimentally.\nTable4Experimentalandsearchedlow-energyMn4Nstructuresinformation,\nID\nEf\nExp./Sea.MAE\n001\n010\n100Sym.\n0.001\n0.01\n0.1Unitcell\nMn\nNa,b,c,\nα,β,γ,\n1\n-0.1363\nExp.97.982\n12.654\n0mPm3,221\nmPm3,221\nmPm3,221\n3.72357\n3.72357\n3.72357\n90,90,90\n2\n-0.1292\nSea.126.325\n121.475\n0Fmmm,69\nFmmm,69\nFmmm,69\n5.28113\n10.38140\n7.32419\n90,90,90\n3\n-0.1128\nSea.0\n127.408\n133.934Imm2,44\nImm2,44\nI4/mmm,139\n3.72588\n3.72621\n7.43791\n90,90,90\n4\n-0.1004\nSea.169.19\n0\n169.19I4/mmm,139\nI4/mmm,139\nI4/mmm,139\n5.27033\n7.40067\n5.27023\n90,90,9025\nFigure1.Theconvexhullofthemanganesenitride,inwhichthealpha[4]MnmI34\n(No.217)andN23Pa(205)isusedtoplotthisfigure,P213(198)ofN2hasidentical\nenergywiththatof3Pa.Twoconvexhullforthesecompoundsunderdifferent\ntemperatures[14]andpressures[15]areplotted,respectively.Consideringthatalpha\nMnmI34(No.217)has58atomsintheunitcell,thusweignoreitsMAE\ncalculations,wealsoignoretheMAEcalculationforthesecondlowest-energy\nstructureasitisabout64meV/atomhigherthanthatofmI34.\nIV.Commonfeaturesofthesecompoundswithdifferentstoichiometries\nThereare58atomsinthegroundstatealphaMnmI34(No.217)and20atoms\ninP4132(No.213),theenergydifferencebetweenabovetwoelementisabout0.06\nmeV/atom,whicharethetwolowest-energystructures.Ourextensivesearchfinds\nthatthealloysofthebinarymanganesenitridesgenerallyshowsimpleunitcell,and26theknownandpartialavailableexperimentaldataaremetastablestatescompared\nwiththesearchedstructures,originatingprobablyfromthecomplicatedinteractionof\nthemagneticMnatomsduringexperimentalsynthesization.Therefore,the\nlow-energystructuresinherittheirparentmanganesestructuresduringthedopingof\nthenitrogen.\nV.Conclusions\nOurextensivesearchfoundmanyperfectornearlyperfecteasy-axisMAE\nstructures,whicharesynthesizableexperimentallyandpresentsveryuseful\napplicationsinmodernindustry.Itisverynecessaryforexperimentalresearchersto\nsynthesizethemaspresentmanganesenitridesarenotveryexpensiveinthemarket.\nThemainmotivationofthisstudyistofindpotentialapplicationsmaterialsin\nmanganesenitridesinthefieldofspin-electronics.Themagneticpropertiesthuscould\nbedeeplystudiedoncethestructureissynthesizedexperimentally.Thegeneral\ndependenceoftheMAEonthedistortedstructureiscarefullypresented.Manylarge\nMAEstructuresaresearchedinMnNdespiteitslessMnconent,deviatingthebasic\nknowledgethatthemoremagneticatomshave,thelargeMAEwillbe.Anearly\nperfecteasy-axisgiantMAEstructureinMnNissearchedandalsoissynthesizable\nexperimentally,presentingunprecedentedlygiantMAEinthisnewenergymaterials\nfield.Severallargenearlyperfecteasy-axisMAEstructuresinMn3N2metastable\nstatesaresearched,whicharealsosynthesizableexperimentally,presentingimportant\npotentialapplicationsinthemagneticfields.Wealsosearchedseveralalmosttotally\nperfecteasy-axisMAEstructuresintheMn2Nmetastablestates,thesephaseshave27greatpromisingtobesynthesizedbythenon-equilibriumtechnology.Wesearched\nonenearlyperfecteasy-axisMAEstructureinMn5N2metastablestatesthatis\npossibletobesynthesizedwhenconsideringitsdecomposedproductsaremetastable\nMn2NandMn3N.Finally,threeperfectornearlyperfecteasy-axisMAEstructuresin\ntheMn4Nmetastablestatesaresearched.Inaword,theseabouttenimportant\nstructureshaveextensiveapplicationpromisinginthespin-electronicsfields.\n1 Suzuki,K.,Kaneko,T.,Yoshida,H.etal.Crystalstructureandmagneticpropertiesofthe\ncompoundMnN.J.AlloysComp.2000,306:66-71.\n2 Aoki,M.,Yamane,H.,Shimada,M.etal.SinglecrystalgrowthofMn2NusinganIn-Naflux.\nMater.Res.Bull.2004,39:827-832.\n3 Jacobs,H.,Stiive,C.Hochdrucksynthesederη-phaseimsystemMn-N:Mn3N2.J.\nLess-CommonMet.1984,96:323-329.\n4 Gokcen,N.A.TheMn-N(manganese-nitrogen)system.Bull.AlloyPhaseDiagrams1990,11:\n33-42.\n5 Kreiner,G.,Jacobs,H.MagnetischeStrukturvonMn3N2.J.AlloysComp.1992,183:\n345-362.\n6 Leineweber,A.,Niewa,R.,Jacobs,H.etal.Themanganesenitridesg-Mn3N2andh-Mn6N5+x:\nnuclearandmagneticstructures.J.Mater.Chem.2000,10:2827-2834.\n7 Wang,Y.C.,Lv,J.,Zhu,L.etal.CrystalStructurePredictionviaParticleSwarmOptimization.\nPhys.Rev.B2010,82:094116-094123.\n8 Wang,Y.C.,Lv,J.,Zhu,L.etal.CALYPSO:AMethodforCrystalStructurePrediction.Comput.\nPhys.Commun.2012,183:2063-2070.\n9 Kresse,G.,Joubert,D.Fromultrasoftpseudopotentialstotheprojectoraugmented-wave\nmethod.Phys.Rev.B1999,59:1758-1775.\n10 Perdew,J.P.,Burke,K.,Ernzerhof,M.Generalizedgradientapproximationmadesimple.Phys.\nRev.Lett.1996,77:3865-3868.\n11 Yang,Z.J.,Wu,S.Q.,Zhao,X.etal.Structuresandmagneticpropertiesofironsilicidefrom\nadaptivegeneticalgorithmandfirst-principlescalculations.J.Appl.Phys.2018,124:\n073901-073907.\n12 Huang,D.J.,Niu,C.P.,Yan,B.M.etal.SynthesisofManganeseMononitridewithTetragonal\nStructureunderPressure.Crystals2019,9:511-518.\n13 Tabuchi,M.,Takahashi,M.,Kanamaru,F.Relationbetweenthemagnetictransition\ntemperatureandmagneticmomentformanganesenitridesMnNγ(O<γ<1).J.AlloysComp.\n1994,210:143-148.\n14 Rognerud,E.G.,Rom,C.L.,Todd,P.K.etal.KineticallyControlledLow-Temperature\nSolid-StateMetathesisofManganeseNitrideMn3N2.Chem.Mater.2019,31.\n15 Li,L.,Bao,K.,Zhao,X.B.etal.BondingPropertiesofManganeseNitridesatHighPressure\nandtheDiscoveryofMnN4withPlanarN4Rings.J.Phys.Chem.C2021,125:24605-24612.28"    },    {        "title": "2309.06279v1.Probing_spatial_variation_of_magnetic_order_in_strained_SrMnO__3__thin_films_using_Spin_Hall_Magnetoresistance.pdf",        "content": "Probing spatial variation of magnetic order in strained SrMnO 3\nthin films using Spin Hall Magnetoresistance\nJ. J. L. van Rijn* T. Banerjee*\nJ.J.L. van Rijn, Prof. T. Banerjee\nUniversity of Groningen, Zernike Institute for Advanced Materials, 9747 AG Groningen, The Nether-\nlands\nEmail Address: j.j.l.van.rijn@rug.nl, t.banerjee@rug.nl\nKeywords: Complex oxides, Antiferromagnets, Multiferroic, Spin Hall magnetoresistance\nSrMnO 3(SMO) is a magnetic insulator and predicted to exhibit a multiferroic phase upon straining. Strained films of SMO display\na wide range of magnetic orders, ranging from G-type to C-and A-type, indicative of competing magnetic interactions. The potential\nof spin Hall magnetoresistance (SMR) is exploited as an electrical probe for detecting surface magnetic order, to read surface mag-\nnetic moments in SMO and its spatial variation, by designing and positioning electrodes of different sizes on the film. The findings\ndemonstrate antiferromagnetic domains with different magnetocrystalline anisotropies along with a ferromagnetic order, where the\nmagnetization arises from double exchange mediated ferromagnetic order and canted antiferromagnetic moments. Further, from a\ncomplete analysis of the SMR, a predominance of antiferromagnetic domain sizes of 3.5 µm2is extracted. This work enhances the\napplicability of SMR in unraveling the richness of correlation effects in complex oxides, as manifested by the detection of coexisting\nand competing ground states and lays the foundation for the study of magnon transport for different magnetoelectric based comput-\ning applications.\n1 Introduction\nThe emergence of coexisting magnetic orders that couples with other ferroic orders is a thriving play-\nground for the exploration of topological textures in materials. Multiferroic materials, due to their in-\ntrinsic magnetoelectric coupling are natural platforms for this and serve as building blocks for low power\ncomputing applications [1, 2]. Strain control in thin films is an established method that enriches and en-\nhances this coupling and has recently invigorated research in this direction [3, 4].\nComplex oxide materials are a versatile material class where strong correlation between structural, mag-\nnetic and electronic properties can be exploited to create emergent phases that can be tailored by vary-\ning the strain quotient. Our material of choice, SrMnO 3(SMO), is an emerging magnetic insulator in\nthis class and predicted to exhibit multiferroicity upon straining [5, 6, 7, 8]. Interestingly, significant\nmagnetoelectric coupling is predicted in SMO, since both its antiferromagnetic order and the induced\nferroelectric order originate from the body-centered manganese atom in its perovskite structure. This is\nin contrast to the well-studied material Bismuth Ferrite, in which the magnetic and ferroelectric proper-\nties originate from the iron and bismuth atoms respectively. In SMO, the strong co-dependence of struc-\ntural and magnetic properties induces a wide variety of antiferromagnetic orders, ranging from G-type\nin bulk to C-type and A-type for strained SMO [5], indicative of competing magnetic interactions. Es-\ntablishing the magnetic order is an essential ingredient to the understanding of the magnetoelectric cou-\npling. Possessing staggered arrangement of magnetic moments in the sublattices that cancels out macro-\nscopically, makes it challenging for common probes to read or manipulate their states in such antifer-\nromagnets. Although scanning probe techniques such as spin polarized scanning tunneling microscopy,\nmagnetic exchange force microscopy and NV centered microscopy are promising [9, 10, 11], they prove to\nbe technically challenging for the study of such orders at low temperatures.\nHere, we study the magnetic order in strained SMO thin films, providing new insights into the magnetic\ndomains, combining magnetic measurements using bulk magnetization studies in addition to the surface\nprobing techniques, mediated by the Spin Hall effect, termed Spin Hall magnetoresistance (SMR). SMR\nhas proven to be an effective tool for studying magnetic order, initially demonstrated for ferromagnetic\norder [12, 13, 14], and thereafter shown to reveal paramagnetic, antiferromagnetic order and even com-\nplex magnetic order in spin-spiral and skyrmionic systems [15, 16, 17, 18]. The dependence of SMR on\nsurface magnetic moments, rather than the net magnetization in the material, makes it particularly po-\ntent for investigating surface antiferromagnetic order. We show how this surface magnetic order and its\n1arXiv:2309.06279v1  [cond-mat.mtrl-sci]  12 Sep 2023spatial variation can be read by designing devices of different sizes, revealing the nature of the underly-\ning magnetic domains. This is accomplished by modifying the magnetic structure, yielding a sinusoidal\ndependence of the SMR induced resistivity, both in its amplitude and phase, by rotating an external\nmagnetic field.\n2 Results and discussion\n2.1 Strained SrMnO 3surface magnetization\nSrMnO 3(SMO) films are grown on SrTiO 3substrates using pulsed laser deposition. Three different thick-\nnesses of 2 nm, 6 nm and 12 nm are chosen for the study of bulk magnetic properties in these films. The\nstructural and magnetization measurements are presented in Figure 1 . The atomic force micrograph\ndisplays the surface structure of the 12 nm thick film displaying a smooth film with a root mean square\nroughness of 162 nm. A four-axis cradle PANalytical x-ray diffractometer (Cu k αradiation, λ= 1.54 ˚A)\nis utilized to establish the structural properties of the films by 2 θscans and reciprocal space mapping.\nNo x-ray diffraction peak was observed for the 2 nm thin film. The Q h00peak positions for the 6 nm and\n12 nm films in the reciprocal space maps are situated on the substrate peak as indicated by the dotted\nline verifying epitaxial growth. From the bulk lattice parameters of STO (3.905 ˚A) and SMO (3.805 ˚A)\n[19], an in-plane tensile strain of 2.6% is determined. A -0.6% out-of-plane strain is extracted from the\n2θscans. This contraction is relatively small with respect to the in-plane strain, implying that strain re-\nlaxation mechanisms play a role. For tensile strained SMO in particular, oxygen vacancy formation is\nknown to be an important factor. Although the critical thickness for strain relaxation of SMO films on\nSTO substrates have been determined to be approximately 20 nm [20], oxygen vacancies effectively relax\nstrain by modifying the valence state of the manganese (Mn) cations at these thicknesses as well. The\nionic radius of Mn4+ions is smaller than the oxygen vacancy mediated Mn3+valence state which accom-\nmodates for the volume expansion in tensile strained films [20, 21, 22].\nThe bulk magnetization of the strained SMO films are studied using a Quantum Design superconduct-\ning quantum interference device (SQUID) magnetometer using in-plane magnetic fields up to 7 T at a\ntemperature of 10 K (5 K) for the 2 nm and 12 nm (6 nm) films, shown in figure 1d-e. Evidently, all\nfilms display a hysteresis curve indicative of a net magnetization originating from ferro-, or ferrimag-\nnetic order. Upon normalizing the total measured magnetization to the respective film volume, figure\n1e, it is noticeable that the saturation magnetization is strongly dependent on the thickness such that\nthe net magnetization is considerably larger for thinner films. Surprisingly, the curves for the 2 nm and\n12 nm films are strikingly similar. The hysteresis curve for the 6 nm film is also comparable, however it\ndisplays a larger saturation magnetization (M s) and field which we attribute to the lower measurement\ntemperature of 5 K. The hystereses in figure 1d display independence on the film thickness suggesting\nthat the magnetization arises from the film surface or at the interface. We note that any linear field de-\npendence in the bulk magnetization measurements are subtracted, effectively masking any contribution\nfrom canted antiferromagnetic moments. The temperature dependent M sis displayed in the inset.\nFrom the structural data, we infer the presence of strain relaxation, most probably due to oxygen de-\nfects. Oxygen deficiency drastically affects the magnetic order in strained SMO films, possibly resulting\nin ferrimagnetic surface magnetization [23, 14]. Two reasons are put forth to explain the ferromagnetic\nhysteresis loops. Firstly, oxygen vacancies create mixed valency of the Mn cations promoting double ex-\nchange interaction, arising from the reduction in kinetic energy by electron hopping. Double exchange\nfavouring ferromagnetic interaction, effectively weakens the superexchange and antiferromagnetic ex-\nchange interaction. Secondly, oxygen defects modify locally the structure such that the bond angles be-\ntween the Mn and oxygen can alter up to 30◦, further weakening the antiferromagnetic interaction in ac-\ncordance with the Goodenough-Kanamori-Anderson rules [24]. From this we infer that the ferromagnetic\nhysteresis loops are well explained by the formation of oxygen vacancies. From first principle calcula-\ntions, the formation energy of oxygen vacancies is lowest in structural domain walls [25] and additionally\nat the surface [23, 24]. Since the amount of oxygen deficiency induced magnetization is independent of\nthe thickness, we argue that a deficient layer forms directly after growth rather than during growth and\n22.2 Direct probing of magnetically anisotropic antiferromagnetic domains using spin Hall Magnetoresistance\n-6 -4 -2 0 2 4 6-40-30-20-10010203040\n  M ( emu )\nField (T) 2 nm (10 K)\n 6 nm (5 K)\n 12 nm (10 K)\n0 400030  Ms (emu/cc)\n  \nTemperature (K)\n-6 -4 -2 0 2 4 6-450-300-1500150300450\n 2 nm (10 K)\n 6 nm ( 5 K)\n 12 nm (10 K)\n  M (emu/cc)\nMagnetic field (T)\n0 3000400\n  Ms (emu/cc) \nTemperature (K)\n44 45 46 47 48 49 50 51 5210-1100101102103104105106107108\n  Intensity (Arb. U.)\n2 (o) 2 nm\n 6 nm\n 12 nm\n0.740.760.780.800.82\n  \n Q00l (1/Å)\n0.22 0.24 0.26 0.28 0.300.740.760.780.800.82\n \nQh00 (1/Å)Q00l (1/Å)\n0.760.780.80\n \n Q00l (1/Å)2 nm\n6 nm\n12  nm\nSTO (103)SMO \n(103)\nSTO (103)\nSMO \n(103)\n(a)\n(b)(c) (d)\n(e)STO (103)\nSTO \n(002)\nSMO \n(002)\nFigure 1: Structural and bulk magnetization results for three SrMnO 3films with thicknesses 2 nm, 6 nm and 12 nm. In\n(a), an atomic force microscopy scan of the 12 nm film shows the surface structure with a roughness of 162 pm indicating\nan atomically flat surface. In (b) and (c), X-ray diffraction techniques are utilized to confirm the structure of the three\nfilms. In (b), 2 θscans show the (002) planes and in (c) reciprocal space maps around the SrTiO 3(103) peak are displayed.\nField dependent bulk magnetization measurements are displayed in (d) and (e), where (d) shows the total SMO magne-\ntization which is normalized to the film volume in (e). All M-H hystereses are taken with the field along the [100] crystal\ndirection and are corrected for the linear diamagnetic background from the SrTiO 3substrate.\nis inevitable in uncapped strained SMO films. Since the surface magnetization is similar for the three\nfilms, we focus on the 6 nm film for probing the local magnetic structure further.\n2.2 Direct probing of magnetically anisotropic antiferromagnetic domains using spin Hall\nMagnetoresistance\nAlthough the bulk magnetization reveals the presence of ferromagnetic interaction, the local surface mag-\nnetic order is more effectively studied by utilizing a technique that is sensitive to this. In this work, spin\nHall Magnetoresistance (SMR) is chosen, which is a result of the interaction at the interface of a heavy\nmetal, in our case Pt, and a magnetic insulator, SMO. Upon application of a current in Pt, the spin ac-\ncumulation generated by the spin Hall effect is reflected (absorbed) if the surface magnetic moments are\nordered parallel (perpendicular) to the spin accumulation, resulting in a relatively low (high) Pt resistiv-\nity. The dependence of the Pt resistivity by manipulation of magnetic moments with an externally ap-\nplied magnetic field is effectively a probe of the surface magnetic order. Important to note here is that\nwith this technique, only the local magnetic order directly in proximity of the Pt structure is measured.\nIn this work, we explore the dependence of the position and the size of the Pt electrodes in revealing\nthe local magnetic order. Figure 2 shows the angle dependent magneto resistance response (ADMR)\nof two small identical Pt devices (7 x 0.5 µm2) separated by a distance of 1 mm deposited on the 6 nm\nthick SMO sample. The Pt resistivity is recorded upon applying an AC current of 0.3 mA, as shown in\nfigure 2a. The angular dependence of the Pt resistivity when rotating a magnetic field in-plane is dis-\nplayed in figure 2b. Strikingly, the two devices have similar oscillation amplitudes, yet a distinctly differ-\nent phase. When rotating a field in-plane, the SMR is expected to show a A ·sin2(x-ϕ0) dependence. For\n32.2 Direct probing of magnetically anisotropic antiferromagnetic domains using spin Hall Magnetoresistance\n-270 -180 -90 0 90 180 2700.00.51.01.52.02.53.0\n  0 (10-4)\n (o) D1\n D2\n sin2(x) fit(a) (b) (c)\n(d) (e) (f)\n-1 0 1 2 3 4 5 6 7 80.00.51.01.52.02.5\n    (10-4)\nMagnetic field (T) D1\n D2\n Fit\n0 50 100 150 200 250 3000.00.51.01.52.02.5\n  0 (10-4)\nTemperature (K) D1\n D2\n0 50 100 150 200 250 30045607590105120135\n  0 (o)\nTemperature (K) D1\n D2\nxyz\nBα\nJc\n[110]\n[-110]\nDomain \nwallPt\nFigure 2: Spin Hall magnetoresistance measurements for two devices on a 6 nm film is displayed, revealing local probing of\nthe antiferromagnetic domains. In (a) the SMR measurement geometry is illustrated. In (b), the angle dependence of both\ndevices at 5 K and an externally applied magnetic field strength of 7 T shows different phase shifts. The quadratic SMR\namplitude dependence on magnetic field strength at 5 K is shown in (c) originating from the magnetic domain manipula-\ntion. Panels (d) and (e) compare the temperature dependent SMR amplitude and phase respectively for both device. The\nillustration in (f) visualizes the domain dependent probing of the different devices.\nferromagnetic insulators, a phase of 90◦is expected when a sufficiently large magnetic field aligns the\nmagnetic moments with the field. However, for antiferromagnets, the SMR generally displays a phase\nof 0◦, since the magnetic moments in the sublattices tend to align perpendicular to the applied field to\nreduce the Zeeman energy, by canting slightly towards the magnetic field. As is clear from the fit, both\ndevices show a sin2(x) dependence, with a distinct phase of around 135 and 45◦for device 1 (D1) and\ndevice (D2) respectively. The ADMR is further analyzed by plotting the amplitude as a function of the\nmagnetic field strength and suggests a quadratic dependence [26, 27]. The 135◦phase shift with a quadratic\ndependence is explained by taking into account preferential magnetic anisotropy axes for oxygen defi-\ncient SMO and domain formation [24]. Here, we note that no spin-flop and spin-flip effects are observed\nsince there is no abrupt change of the SMR signal with field nor a saturation of the SMR amplitude.\nThe [110] magnetic easy axis is stabilized by the symmetry breaking introduced by the oxygen vacancy\nformation and the resulting Pt resistivity for an in-plane magnetic field rotation is given by\nρSMR(α)∝ρ12H2\nH2\nMDcos2\u00121\n4π−α\u0013\n, (1)\nwhere ρ1is a SMR coefficient related to the Spin Hall Effect in Pt [13], His the magnetic field strength,\nHMDis the monodomainization field and αis the angle of the magnetic field with current direction. The\nphase shift in D2 located at a different position on the film is then explained by symmetry, since both\n[110] and [-110] anisotropy axes can be favoured upon formation of an oxygen vacancy. The observation\nof distinctly different phases in both devices suggest that the small Pt devices directly probe local mag-\nnetic moments at the interface with the SMO film, and reveals the direction of their dominant magnetic\nanisotropy axis, as indicated in Figure 2f. This shows that SMR proves to be effective in mapping spa-\n42.3 Device size dependence of Spin Hall Magnetoresistance\ntial differences in the local magnetic order in antiferromagnetic films that exhibit magnetic domains.\nThe temperature dependence of the extracted amplitude and phase from the SMR measurements is dis-\nplayed in Figure 2d and 2e respectively. With temperature, both devices show a gradual decrease in am-\nplitude which signifies the reduction in magnetic moment with temperature, saturating between 100 K\nand 150 K, in line with the predicted N´ eel temperature of 121 K, for strained oxygen deficient SMO films\n[24]. The phase of both devices is temperature dependent and transitions towards a phase of 90◦at tem-\nperatures above 100 K. We attribute the shift of the phase to temperature dependent magnetocrystalline\nanisotropy. Above the N´ eel temperature, the phase of both devices stabilizes. Due to the absence of mag-\nnetic order, we attribute this temperature independent ADMR signal to Hanle Magnetoresistance, previ-\nously observed in Pt nanostructures [28]. From the above discussion, it is evident that the positioning of\nthe Pt devices is crucial to map the spatial differences in the magnetic anisotropy in the different antifer-\nromagnetic domains.\n2.3 Device size dependence of Spin Hall Magnetoresistance\nBesides the positioning of the Pt devices on the SMO films, the relevance of the Pt device size is inves-\ntigated, shown in Figure 3 . In Figure 3a-c the SMR response of two devices, the small device (7 x 0.5\nµm2) D1 and a significantly larger device (50 x 10 µm2) (D3) are studied. Remarkably, the resistivity\nmodulation of the large device, D3, is reduced compared to the response of D1. Secondly, the phase of\nD3 is extracted to be 90◦, associated with ferromagnetic order, suggesting that this is the dominant con-\ntributor for the SMR in the larger device. We propose that the surface magnetization induced by the\noxygen vacancies play an important role in the SMR response for large devices. From the ferromagnetic\nSMR signal, we infer that the contributions from the antiferromagnetic domains with different magnetic\nanisotropies average out such that the resulting response originates from the remaining net magnetiza-\ntion. The large device encompasses a sufficient amount of differently aligned domains, yielding the non-\nzero contribution from the aligned magnetic moments.\nThe temperature dependent phase of D3, in Figure 3c displays a deviation of around 12◦with the peak\nposition at 75 K before stabilizing between 100 K and 150 K at 90◦. This phase change is attributed to\nthe disappearance of the double exchange mediated ferromagnetic magnetization (Figure 1e inset) at\naround 50 K. The temperature dependent amplitude in Figure 3b for D3 displays an increase of the am-\nplitude with temperature. This increase is attributed to the decrease of the magnetocrystalline anisotropy\nwith temperature, observed also from the gradual phase shift of D1. A decrease of magnetocrystalline\nanisotropy increases canting of antiferromagnetic moments towards the magnetic field, effectively in-\ncreasing the net magnetization that contributes to the ferromagnetic SMR response. The discontinuity\nobserved from 100 K agrees well with the N´ eel temperature of 121 K, above which the SMO film is para-\nmagnetic. Considering this, we do not expect the increasing amplitude above 150 K to be related with\nSMR, and hence the exact nature of the sinusoidal response is unclear. Summarizing, we observe that\nwhile the small devices probe the individual antiferromagnetic domains, large devices primarily pick up\nthe net magnetization from double exchange mediated ferromagnetic order and canted antiferromagnetic\nmoments, visualized in Figure 3d. From the differences of the measured SMR signals in the small and\nlarge device, and the discussions in Section 2.2 we conclude that the domain size is on the order of 3.5\nµm2from a dominant contribution of a single magnetic anisotropy axis.\n3 Conclusion\nIn this work, we demonstrate how we decipher the spatial differences in the magnetic order in strained\nSMO thin films by designing the size and positioning of the Pt devices, from the analysis of the SMR.\nSurface magnetization contribution in the films can be deduced from the analysis of the observed bulk\nmagnetization studies and arises from the formation of oxygen vacancies due to strain relaxation. Spin\nHall magnetoresistance measurements on Pt devices at different locations on the sample reveal local an-\ntiferromagnetic domains as proven by the direct observation of both the [110] and [-110] magnetic do-\n5(a) (b)\n(c) (d)\n0 50 100 150 200 250 3000.00.51.01.52.02.5\n  /0 (10-4)\nTemperature (K) D1\n D3\n0 50 100 150 200 250 30090105120135\n  0 (o)\nTemperature (K) D1\n D3\n90 180 270 360 4500.00.40.81.21.62.02.42.8\n  /0 (10-4)\n (0) D1\n D3\n Sin2(x) fit\nDomain \nwall[110]\n[-110]\nFigure 3: Dependence of SMR on the size of the Pt device on the 6 nm film showing the surface ferromagnetic interaction.\nIn (a), the angular dependence at 5 K and 7 T clearly reveals the difference between the smallest device (7 x 0.5 µm2) D1\nand the largest device (50 x 10 µm2) D3. The temperature dependent SMR amplitude and phase shift of the two devices\nare displayed in (b) and (c), taken with a magnetic field strength of 7 T. The illustration in (d) visualizes how the ferro-\nmagnetic interaction is dominant for larger devices.\nmains. Interestingly a size dependence of the Pt devices i) reveal a ferromagnetic order for larger device\ndimension, that arises due to the averaging of the antiferromagnetic order as indicated from the SMR\nphase and ii) from a complete analysis of the SMR on devices of different dimensions we infer a predomi-\nnance of antiferromagnetic domain sizes of 3.5 µm2.\n4 Experimental section\nSrMnO 3(SMO) thin films are grown on 5x5 mm SrTiO 3(STO) substrates using pulsed laser deposition.\nThe STO substrates are TiO 2terminated using buffered hydrofluoric acid before an annealing process\nin oxygen at 960oC. The KrF excimer laser with a wavelength of 248 nm pulses at 1 Hz with a fluence\nof 2 J/cm2in an oxygen pressure of 0.05 mbar and temperature of 800oC. The growth is monitored us-\ning in-situ reflective high electron energy diffraction, from which layer-by-layer growth is confirmed by\nanalyzing the diffraction pattern and its intensity oscillations. After growth, the films are annealed for\n30 minutes in a 100 mbar oxygen environment at 600oC to optimize the stoichiometry of the magnetic\nfilms.\nThe Pt devices are fabricated using electron beam lithography and deposited using DC sputtering. All\nPt devices are 7 nm thick and have Ti/Au contacts pads. The Pt resistivity is recorded by applying an\nAC current with a frequency of 7.777 Hz for all measurements. The current used for the small devices\n6REFERENCES\nD1 and D2 is 0.3 mA, whereas for all measurements of the larger devices a current of 1 mA is used.\nAcknowledgements\nJ. J. L. vR acknowledges financial support from Dieptestrategie grant (2019), Zernike Institute for Ad-\nvanced Materials. J. J. L. vR and T.B. acknowledges technical support from J. Baas, J. G. Holstein, H.\nH. de Vries, F. van der Velde and A. Joshua and acknowledges A. Das, S. Chen, A. S. Goossens and A.\nJaman. This work was realized using NanoLab-NL facilities.\nReferences\n[1] S. Manipatruni, D. E. Nikonov, C.-C. Lin, T. A. Gosavi, H. Liu, B. Prasad, Y.-L. Huang, E. Bon-\nturim, R. Ramesh, I. A. Young, Nature 2019 ,565, 7737 35.\n[2] J. Scott, Nature materials 2007 ,6, 4 256.\n[3] N. A. Spaldin, R. Ramesh, Nature materials 2019 ,18, 3 203.\n[4] R. Gupta, R. Kotnala, Journal of Materials Science 2022 ,57, 27 12710.\n[5] J. H. Lee, K. M. Rabe, Physical review letters 2010 ,104, 20 207204.\n[6] A. Edstr¨ om, C. Ederer, Physical Review Materials 2018 ,2, 10 104409.\n[7] A. Edstr¨ om, C. Ederer, Physical Review Letters 2020 ,124, 16 167201.\n[8] X. Zhu, A. Edstr¨ om, C. Ederer, Physical Review B 2020 ,101, 6 064401.\n[9] A. Haykal, J. Fischer, W. Akhtar, J.-Y. Chauleau, D. Sando, A. Finco, F. Godel, Y. Birkh¨ olzer,\nC. Carr´ et´ ero, N. Jaouen, et al., Nature communications 2020 ,11, 1 1704.\n[10] N. Hauptmann, J. W. Gerritsen, D. Wegner, A. A. Khajetoorians, Nano letters 2017 ,17, 9 5660.\n[11] T. Kosub, M. Kopte, R. H¨ uhne, P. Appel, B. Shields, P. Maletinsky, R. H¨ ubner, M. O. Liedke,\nJ. Fassbender, O. G. Schmidt, et al., Nature communications 2017 ,8, 1 13985.\n[12] M. Althammer, S. Meyer, H. Nakayama, M. Schreier, S. Altmannshofer, M. Weiler, H. Huebl,\nS. Gepr¨ ags, M. Opel, R. Gross, et al., Physical Review B 2013 ,87, 22 224401.\n[13] N. Vlietstra, J. Shan, V. Castel, B. Van Wees, J. B. Youssef, Physical Review B 2013 ,87, 18\n184421.\n[14] A. Das, V. E. Phanindra, A. Watson, T. Banerjee, Applied Physics Letters 2021 ,118, 5.\n[15] G. R. Hoogeboom, A. Aqeel, T. Kuschel, T. Palstra, B. J. van Wees, Applied Physics Letters 2017 ,\n111, 5.\n[16] F. Feringa, G. Bauer, B. Van Wees, Physical Review B 2022 ,105, 21 214408.\n[17] A. Aqeel, N. Vlietstra, J. A. Heuver, G. E. Bauer, B. Noheda, B. J. van Wees, T. Palstra, Physical\nReview B 2015 ,92, 22 224410.\n[18] V. E. Phanindra, A. Das, J. van Rijn, S. Chen, B. van Wees, T. Banerjee, arXiv preprint\narXiv:2201.08685 2022 .\n[19] O. Chmaissem, B. Dabrowski, S. Kolesnik, J. Mais, D. Brown, R. Kruk, P. Prior, B. Pyles, J. Jor-\ngensen, Physical Review B 2001 ,64, 13 134412.\n[20] E. Langenberg, L. Maurel, G. Antorrena, P. A. Algarabel, C. Mag´ en, J. A. Pardo, ACS Omega\n2021 ,6, 20 13144.\n7REFERENCES\n[21] A. Marthinsen, C. Faber, U. Aschauer, N. A. Spaldin, S. M. Selbach, MRS communications 2016 ,\n6, 3 182.\n[22] P. Agrawal, J. Guo, P. Yu, C. H´ ebert, D. Passerone, R. Erni, M. D. Rossell, Physical Review B\n2016 ,94, 10 1.\n[23] M. Kaviani, U. Aschauer, Physical Chemistry Chemical Physics 2022 ,24, 6 3951.\n[24] J. van Rijn, D. Wang, B. Sanyal, T. Banerjee, Physical Review B 2022 ,106, 21 214415.\n[25] C. Becher, L. Maurel, U. Aschauer, M. Lilienblum, C. Mag´ en, D. Meier, E. Langenberg, M. Trassin,\nJ. Blasco, I. P. Krug, et al., Nature nanotechnology 2015 ,10, 8 661.\n[26] J. Fischer, O. Gomonay, R. Schlitz, K. Ganzhorn, N. Vlietstra, M. Althammer, H. Huebl, M. Opel,\nR. Gross, S. T. Goennenwein, et al., Physical Review B 2018 ,97, 1 014417.\n[27] S. Gepr¨ ags, M. Opel, J. Fischer, O. Gomonay, P. Schwenke, M. Althammer, H. Huebl, R. Gross,\nJournal of Applied Physics 2020 ,127, 24.\n[28] S. V´ elez, V. N. Golovach, A. Bedoya-Pinto, M. Isasa, E. Sagasta, M. Abadia, C. Rogero, L. E.\nHueso, F. S. Bergeret, F. Casanova, Physical review letters 2016 ,116, 1 016603.\n8"    },    {        "title": "2309.08898v1.Investigation_of_the_Anomalous_and_Topological_Hall_Effects_in_Layered_Monoclinic_Ferromagnet_Cr___2_76__Te__4_.pdf",        "content": "Investigation of the Anomalous and Topological Hall Effects in Layered Monoclinic Ferromagnet\nCr2.76Te4\nShubham Purwar,1Achintya Low,1Anumita Bose,2Awadhesh Narayan,2and S. Thirupathaiah1,∗\n1Department of Condensed Matter and Materials Physics,\nS. N. Bose National Centre for Basic Sciences, Kolkata, West Bengal, India, 700106.\n2Solid State and Structural Chemistry Unit, Indian Institute of Science, Bengaluru 560012, India.\n(Dated: September 19, 2023)\nWe studied the electrical transport, Hall effect, and magnetic properties of monoclinic layered ferromagnet\nCr2.76Te4. Our studies demonstrate Cr 2.76Te4to be a soft ferromagnet with strong magnetocrystalline\nanisotropy. Below 50 K, the system shows an antiferromagnetic-like transition. Interestingly, between 50 and\n150 K, we observe fluctuating magnetic moments between in-plane and out-of-plane orientations, leading to\nnon-coplanar spin structure. On the other hand, the electrical resistivity data suggest it to be metallic throughout\nthe measured temperature range, except a kink at around 50 K due to AFM ordering. The Rhodes-Wohlfarth\nratioµeff\nµs= 1.89( >1)calculated from our magnetic studies confirms that Cr 2.76Te4is an itinerant ferromagnet.\nLarge anomalous Hall effect has been observed due to the skew-scattering of impurities and the topological\nHall effect has been observed due to non-coplanar spin-structure in the presence of strong magnetocrystalline\nanisotropy. We examined the mechanism of anomalous Hall effect by employing the first principles calculations.\nI. INTRODUCTION\nTwo-dimensional (2D) magnetic materials with topological\nproperties [1–3] have sparked significant research attention\nrecently due to their potential applications in spintronics and\nmagnetic storage devices [4–6]. Importantly, these are the\nvan der Waals (vdW) magnets possessing peculiar magnetic\nproperties with strong magnetocrystalline anisotropy\n(MCA) [7–9]. In general, the Heisenberg-type ferromagnet\ndoes not exist with long-range magnetic ordering at a\nfinite temperature in the 2D limit due to dominant thermal\nfluctuations [10]. However, the strong magnetic anisotropy\nthat usually present in the low-dimensional materials can\nstabilize the long-range magnetic ordering to become a 2D\nIsing-type ferromagnet [11]. Till date many 2D ferromagnets\nhave been discovered experimentally [12, 13], but only a\nfew of them can show the topological signatures such as\nthe topological Hall effect (THE) or skyrmion lattice. For\ninstance, the recent microscopic studies on Cr 2Ge2Te6[14],\nFe3GeTe 2[15], and Fe 5GeTe 2[16, 17] demonstrated\ntopological magnetic structure in the form of skymion\nbubbles in their low-dimensional form.\nOn the other hand, soon after predicting the layered\nCrxTey-type systems as potential candidates to realize the\n2D ferromagnetism in their bulk form [18, 19], a variety\nof Cr xTeycompounds were grown including CrTe [20],\nCr2Te3[21], Cr 3Te4[22], and Cr 5Te8[23]. Interestingly,\nall these systems are formed by the alternative stacking of\nCr-full (CrTe 2-layer) and Cr-vacant (intercalated Cr-layer)\nlayers along either a-axis or c-axis [24]. Thus, the Cr\nconcentration plays a critical role in forming the crystal\nstructure, magnetic, and transport properties. The compounds\nlike Cr 5Te8, Cr2Te3, and Cr 3Te4are reported to crystallize in\nmonoclinic or trigonal structures, whereas Cr 1–xTe(x<0.1)\ncrystallizes in the hexagonal NiAs-type structures [25]. The\n∗setti@bose.res.inelectronic band structure calculations performed on CrTe,\nCr2Te3, and Cr 3Te4suggest a strong out-of-plane Cr 3d\negorbital, dz2– dz2, overlapping along the c-axis to have\nrelatively smaller nearest neighbor Cr – Cr distance [24]. In\naddition, Cr 5Te8[23], Cr 1.2Te2[26], Cr 0.87Te [27] are known\nto show topological properties in the hexagonal phase.\nIn this work, we systematically investigate the electrical\ntransport, Hall effect, and magnetic properties of monoclinic\nCr2.76Te4which is very close to the stoichiometric\ncomposition of Cr 3Te4. We observe that the easy-axis of\nmagnetization is parallel to the bc-plane, leading to strong\nmagnetocrystalline anisotropy. Below 50 K, the system\nshows antiferromagnetic-like transition. In addition, we find\nfluctuating Cr magnetic moments between in-plane and out-\nof-plane directions within the temperature range of 50 and\n150 K. Electrical resistivity data suggest Cr 2.76Te4to metallic\nthroughout the measured temperature range with a kink at\naround 50 K due to AFM ordering. Our studies clearly\npoint Cr 2.76Te4to an itinerant ferromagnet. Magnetotransport\nmeasurements demonstrate large anomalous Hall effect\n(AHE) and topological Hall effect (THE) in this system.\nFirst-principles calculations point to an intrinsic AHE due\nto non-zero Berry curvature near the Fermi level, while\nexperimentally it is found to be an extrinsic AHE due to the\nskew-scattering [28].\nII. EXPERIMENTAL DETAILS\nHigh quality single crystals of Cr 2.76Te4were grown by\nthe chemical vapor transport (CVT) technique with iodine as\na transport agent as per the procedure described earlier [29].\nExcess Iodine present on the crystals was removed by washing\nwith ethanol several times and dried under vacuum. The\nas-grown single crystals were large in size ( 3×2mm2),\nwere looking shiny, and easily cleavable in the bc-plane.\nPhotographic image of typical single crystals is shown in the\ninset of Fig. 1(a). Crystal structural and phase purity of thearXiv:2309.08898v1  [cond-mat.mtrl-sci]  16 Sep 20232\n(c)\n2θ(degree)70 60 50 40 30 20 10Counts(a.u.)\n2θ(degree)(200)\n(400)\n(600)\n(800)(a)\nCr3Te4\n90 80 70 60 50 40 30 20\n(11-1)\n(400)\n(111) (31-1)\n(311)\n(402)\n(022)\n(222)(800)(b)Iexp\nIcal\nIexp-Ical\nBragg planeCounts(a.u.)\n(422)\n(71-5)\n(820)\n(82-4)\n(33-3)\n(332)(31-3)Cr2Cr1\nTe\nFIG. 1. (a) XRD pattern from Cr 2.76Te4single crystals. Inset in\n(a) shows the photographic image of the single crystals. (b) X-\nray diffraction pattern from the crushed Cr 2.76Te4single crystals,\noverlapped with Rietveld refinement. (c) Schematic crystal structure\nof Cr 2.76Te4obtained from the Rietveld refinement.\nsingle crystals were identified by the X-ray diffraction (XRD)\ntechnique using Rigaku X-ray diffractometer (SmartLab,\n9kW) with Cu K αradiation of wavelength 1.54059 ˚A.\nCompositional analysis of the single crystals was done\nusing the energy dispersive X-ray spectroscopy (EDS of\nEDAX). Magnetic and transport studies were carried out on\nthe physical property measurement system (9 Tesla-PPMS,\nDynaCool, Quantum Design). See Supplemental Material for\nmore discussion on the chemical composition of the studied\nsystem [30]. Electrical resistivity and Hall measurements\nwere performed in the standard four-probe method. To\neliminate the longitudinal magnetoresistance contribution due\nto voltage probe misalignment, the Hall resistance was\ncalculated as ρyz(H)=[ρyz(+H) – ρyz(–H)]/2 .\nIII. DENSITY FUNCTIONAL THEORY CALCULATIONS\nWe have performed density functional theory (DFT)\ncalculations using the Quantum Espresso package [31,\n32]. We used fully relativistic pseudopotentials in\norder to include the spin-orbit interaction. Generalized\ngradient approximation was considered based on Perdew-\nBurke-Ernzerhof implementation [33] within the projector\naugmented wave (PAW) basis [34]. For wave function\nand charge density expansions, cutoff values of 50 Ry and\n300 Ry were chosen, respectively. For the self-consistent\ncalculation, a 7 ×7×7 Monkhorst-Pack grid was used [35]. In\norder to consider the van der Waals forces, Semi-empirical\nGrimme’s DFT-D2 correction [36] was included. We further\nconstructed tight-binding model based on the maximally\nlocalized Wannier functions using the wannier90 code [37],\nwith Cr 3d, Cr3s, Te5p, and Te 5sorbitals as the basis. Then\nutilizing the as obtained tight-binding model, we calculated\nTcH ǁ bc\nH ǁ aTc≈ 310K(a)\n(b)(c)\n(d)ZFCZFC\nFCFCM(μB/Cr)\nT (K)\nH ǁ bc\nH ǁ aΔM(μB/Cr)\nT (K)ΔM = M FC-MZFCH = 1000 Oe\nM(μB/Cr)H ǁ a\nH (Tesla)3K\n110K\n200K\n300K\n-5 0 5\nH (kOe)\nH (Tesla)M(μB/Cr)H ǁ bc\n3K\n300K150K90K\n5 0 -5\nH (kOe)50 K 150 K\n125 KdM/dTZFC\nFCH ǁ a\nT (K)FIG. 2. (a) Temperature dependent magnetization M(T) measured\nunder ZFC and FC modes with a magnetic field H=1000 Oe for H∥a\nandH∥bc. (b) Variation of magnetization ∆M=(M FC-MFC) plotted\nas a function of temperature. Inset in (b) shows first derivative of\nmagnetization with respect to the temperature (dM/dT) of the data\nshown in (a) for H∥a. (c) and (d) Field dependent magnetization\nM(H) measured at different temperatures for H∥aandH∥bc,\nrespectively.\nBerry curvature along the high symmetry directions using\nKubo formula [38] encoded in wannier90 code [37]. We have\ncalculated intrinsic anomalous Hall conductivity (AHC) by\nintegrating the x-component of Berry curvature over the entire\nBZ using WannierTools code [39].\nIV . RESULTS AND DISCUSSION\nFig. 1(a) shows the XRD pattern of Cr 2.76Te4single crystal\nwith intensity peaks of (l 0 0) Bragg plane, indicating that the\ncrystal growth plane is along the a-axis. Inset in Fig. 1(a)\nshows the photographic image of Cr 2.76Te4single crystal.\nFig. 1(b) shows XRD pattern of crushed Cr 2.76Te4single\ncrystals measured at room temperature. All peaks in the XRD\npattern can be attributed to the monoclinic crystal structure of\nC12/m1 space group (No.12) without any impurity phases,\nconsistent with the crystal phase of Cr 3Te4[40]. Rietveld\nrefinement confirms the monoclinic structure with lattice\nparameters a=13.9655(2) ˚A,b=3.9354(4) ˚A,c=6.8651(7) ˚A,\nα=β=90◦, and γ=118.326(7)◦. These values are in good\nagreement with previous reports on similar systems [41, 42].\nFig. 1(c) shows schematic crystal structure of Cr 2.76Te4\nprojected onto the ac-plane (top panel) and ab-plane (bottom\npanel). Cr1 atoms are located in the Cr-vacant layer with an\noccupancy of 0.189/ u.c, whereas Cr2 atoms are located in the3\n(a) (c)\nVz\nIy(b)\nHǁx50K\nVz\nIz\nρyz(μΩ-cm)\nH (Tesla)\n3K\n15K\n30K50K\n70K\n110K150K\n200K\n250K࣋ࢀࢠ࢟ρyz\nρN+ρA\n0.28\n0.24\n80 60 40200 ρ0+a*T2ρzz(mΩ-cm)\nT (K)\nFIG. 3. (a) Temperature dependent longitudinal resistivity ρzz(T). Top-inset in (a) shows low temperature resistivity fitted by ρ(T) = ρ0+ bT2\nand bottom-inset in (a) shows schematic diagram of linear-four-probe contacts. (b) Hall measuring geometry is shown schematically. (c) Hall\nresistivity ( ρyz) plotted as a function of magnetic field measured at various temperatures. In (c), Red curves represent the experimental data\nof total Hall resistivity ( ρyz), black curves represent the contributions from the normal and anomalous Hall resistivities ( ρN+ρA), and Blue\ncurves represent the topological Hall resistivity ( ρTyz). See the text for more details.\nCr-full layer with an occupancy of 0.5/ u.c. The intercalated\nCr atoms (Cr1) are sandwiched within the van der Waals gap\ncreated by the two CrTe 2layers, as shown in Fig. 1(c). The\nΩcm)2.5\n2.0\n1.5\n1.0\n0.5\n280 210 140 70 040\n30\n20\n10\n0\nR0(Ωcm/Oe)x10-12n(cm-3)x1021\nT (K)Ω-1cm-1)\nT (K)(a)\n(d) (c)μΩcm)\nT (K)SH(V-1)(b)\nρzz(μΩcm)110K\n(m=1.03) ρ௬௭=aρ௭௭\nFIG. 4. Temperature dependence of (a) Normal Hall coefficient R0\n(left axis) and charge carrier density n(right axis). (b) Anomalous\nHall scaling coefficient SHplotted as a function of temperature. (c)\nAnomalous Hall resistivity ( ρAyz) and Hall conductivity ( σAyz). (d)\nPlot of ρAyzvs.ρzz. Dashed line in (d) is linear fitting with equation\nshown on the figure.EDS measurements suggest an actual chemical composition\nof Cr 2.76Te4.\nTo explore the magnetic properties of Cr 2.76Te4,\nmagnetization as a function of temperature [ M(T) ] was\nmeasured as shown in Fig. 2(a) at a field of 1000 Oe applied\nparallel to the bc-plane ( H∥bc) and a-axis ( H∥a) for\nboth zero-field-cooled (ZFC) and field-cooled (FC) modes.\nWe observe that Cr 2.76Te4exhibits paramagnetic (PM) to\nferromagnetic (FM) transition at a Curie temperature ( TC)\nof 310 K, which is close to the Curie temperature of Cr 3Te4\n(TC=316 K). Decrease in the sample temperature results into\na decrease in the magnetization for both H∥bcandH∥a\nat around 50 K, possibly due to spin-canting emerged from\nthe coupling between in-plane ( bc-plane) AFM order and\nout-of-plane ( a-axis) FM orders [43, 44]. Also, the in-plane\nsaturated magnetic moment of 1.78 µB/Cr is almost 4 times\nhigher than the out-of-plane saturated magnetic moment\nof 0.43 µB/Cr at 2 K with an applied field of 1000 Oe,\nclearly demonstrating strong magnetocrystalline anisotropy\nin Cr 2.76Te4. From the magnetization difference between\nZFC and FC, ∆M = M FC– MZFC, shown in Fig. 2(b), we\nnotice significant magnetization fluctuations as the maximum\nof∆Mvaries between in-plane and out-of-plane directions in\ngoing from 50 K to 150 K [45].\nThe magnetic state of Cr 2.76Te4is further explored by\nmeasuring the magnetization isotherms [ M(H) ] for H∥a\nandH∥bcat various sample temperatures as shown in\nFigs. 2(c) and 2(d), respectively. Consistent with M(T) data,\nthe magnetization saturation occurs at an applied field of 0.74\nT and 1.4 T for H∥bcandH∥a, respectively, suggesting\nbc-plane to be the easy-magnetization plane. Also, Cr 2.76Te4\nis a soft ferromagnet as it has a negligible coercivity [see\nFig. S1(a) in the Supplemental Material [30]]. The observed\nsaturation magnetisation ( Ms) 2.586 µB/Cr and 2.55 µB/Cr\nforH∥aandH∥bc, respectively, are smaller than the stand\nalone Cr atom (3 µB), indicating correlated magnetic states\nin Cr 2.76Te4. These observations are in good agreement with\nreport on Cr 2.76Te4[46].\nNext, coming to the main results of this contribution,\nFig. 3(a) exhibits temperature dependent longitudinal\nelectrical resistivity ( ρzz) of Cr 2.76Te4.ρzz(T) suggests\nmetallic nature throughout the measured temperature\nrange [47]. However, a kink at around 50 K is noticed in\nthe resistivity, related to the AFM ordering [see Fig. 2(a)].\nBottom inset of Fig. 3(a) depicts schematic diagram of\nlinear-four-probe measuring geometry and the top inset\nelucidates the quadratic nature of low temperature resistivity\nup to 50 K as it can be explained well by the Fermi liquid\n(FL) theory, ρ(T) = ρ0+ aT2where ρ0is the temperature\nindependent residual resistivity. Schematic diagram of\nHall measuring geometry is shown in Fig. 3(b). The Hall\nresistivity, ρyz, is measured with current along the y-axis\nand magnetic field applied along the x-axis to get the\nHall voltage along the z-axis. Thus, Fig. 3(c) shows field\ndependent Hall resistivity ρyz(black curve) measured at\nvarious sample temperatures. The total Hall resistivity\n(ρyz) may have contributions from the normal Hall effect\n(ρN) and the anomalous Hall effect ( ρA). Thus, the\ntotal Hall resistivity can be expressed by the empirical\nformula, ρyz(H) = ρN(H) + ρA(H) = µ0R0H + µ0RSM,\nwhere R0and RSare the normal and anomalous Hall\ncoefficients, respectively. These coefficients can be obtained\nby performing a linear fit using the relationρyz\nµ0H= R 0+RSM\nH\nas shown in Fig. S1(c) of the Supplemental Material [30].\nHaving obtained the normal and anomalous Hall coefficients,\nwe can now fit the total Hall resistivity (red curves)\nusing the equation, ρyz(H) = µ0R0H + µ0RsM. The\nfitting should be nearly perfect if there is no topological\nHall contribution. However, from Fig. 3(c) we can\nclearly notice that the fitting (red curve) is not perfect.\nTherefore, the topological Hall resistivity also contributes\nto the total Hall resistivity which can be expressed by\nρyz(H) = ρN(H) + ρA(H) + ρT(H). Thus, the topological\nHall contribution (blue curve) is extracted using the relation,\nρT(H) = ρyz(H) – [ ρN(H) + ρA(H)] [48–50]. See the\nSupplemental Material for more details [30].\nFig. 4(a) depicts the normal Hall coefficient ( R0) and\nthe charge carrier density ( n) derived using the formula,\n(R0= 1/n|e|), plotted as a function of temperature. We\nclearly notice from Fig. 4(a) that as the temperature decreases\nthe carrier density increases up to 110 K. However, below\n110 K, the carrier density decreases with temperature and\ngets saturated below 50 K. Fig. 4(b) (left axis) presents\nthe anomalous Hall resistivity ( ρA\nyz) at zero field obtained\nby linearly intersecting the field axis [see Fig. 3(c)].\nMaximum anomalous Hall resistivity is noticed at around\n110 K. Anomalous Hall conductivity, σA\nyz, derived usingthe formula σA\nyz=ρyz\nρ2yz+ρ2zzis shown in the right axis of\nFig. 4(c), again to find the maximum Hall conductivity\nofσA\nyz=27Ω–1cm–1at around 110 K. Fig. 4(b) shows\nanomalous scaling coefficient SH=ρAyz\nMρ2zz, plotted as a\nfunction of temperature. The values of SHare inline with\nthe itinerant ferromagnetic systems of SH=0.01-0.2 V–1[51].\nIn general, anomalous Hall effect can occur in solids either\nintrinsically originated from the nonzero-Berry curvature in\nthe momentum space [52, 53] or extrinsically due to side-\njump/scew-scattering mechanisms [28, 52, 54]. Therefore, to\nelucidate the mechanism of AHE in Cr 2.76Te4, we plotted ρA\nyz\nvs.ρzzas shown in Fig. 4(d). From Fig. 4(d) it is evident that\nρA\nyzlinearly changes with ρzz[ρyz=αρm\nzzform = 1.03 ±\n0.02] up to 110 K and then deviates on further increasing\nthe sample temperature [55]. In the case of itinerant\nferromagnetic system, the Hall resistivity can be expressed\nby the relation ρyz=αρzz+βρ2\nzzwhere αandβare the\nscrew-scattering and side-jump terms, respectively [47, 56,\n57]. That means, in the case of skew-scattering ρA\nyzlinearly\ndepends on ρzz, while ρA\nyzquadratically depends on ρzzin\nthe case of side-jump. Since the Hall resistivity ( ρyz) linearly\ndepends on ρzz, the skew-scattering could be the most suitable\nmechanism of AHE observed in this system [58, 59]. Note\nthat the intrinsic Berry curvature contribution to the AHR also\nquadratically dependents on ρzz[55]. To make a point, it\nlooks that 110 K is the critical temperature of the properties\ndiscussed in Figs. 4(a)-(d). However, as per the dM/dT data\nshown in the inset of Fig. 2(b), we think that the critical\ntemperature could be ≈125 K instead of 110 K as we can\nsee significant change in magnetization across ≈125 K.\nOur experimental findings on the AHE have been examined\nusing the density functional theory calculations as presented\nin Fig. 5. For the calculations, we considered primitive unit\ncell of Cr 3Te4, consisting of three Cr (one Cr1 and two Cr2\ntype) and four Te atoms. The magnetic spins of Cr atoms were\nconsidered along the x-direction. Our calculations suggest a\nferromagnetic ground state with average magnetic moments\nof 3.23 µBand 3.15 µBper Cr1 and Cr2 atoms, respectively,\nslightly higher than the experimental average value of 2.568\nµB/Cr. In Fig. 5(b), we present bulk electronic band structure\nof Cr 3Te4. The system is found to be metallic with several\nbands crossing the Fermi level ( EF). Several band crossing\npoints are found near EFalong C2– Y2,D2– A,A –Γand\nL2–Γk-paths, but slightly away from the high symmetry\npoints. Next, we explore the Berry curvature ( Ω) calculated\nusing the formula Ω(k) = ∇(k)×A(k) (where A(k) is the\nBerry connection). Variation of the x-component of Berry\ncurvature ( Ωx) at EFalong the high symmetry k-path is\nshown in Fig. 5(c). We can notice from Fig. 5(c) that Ωx\nis strongly enhanced along the A –Γ(kz)direction. Further,\nFig. 5(d) shows the anomalous Hall conductivity (AHC), σyz,\nof Cr 3Te4calculated as a function of energy using the Eqn. 1.\nσyz=e2\n¯hZ\nf(εn(k))Ωx\nn(k)dk\n(2π)3(1)\nWhere, f(εn(k))is the Fermi-Dirac distribution function.5\na\nbc(a) (b)\n(c)(d)\n(e)\nFIG. 5. (a) Schematic monoclinic unit cell of Cr 2.76Te4. (b) Electronic band structure of Cr 2.76Te4calculated with inclusion of spin-orbit\ncoupling (SOC). In (b) the Weyl points near the Fermi level are encircled. (c) x-component of the Berry curvature, Ωx, calculated at the Fermi\nlevel. Inset in (c) shows zoomed-in view of Ωxfor the A –Γsegment. (d) Anomalous Hall conductivity, σyz, plotted as a function of energy.\n(e) High symmetry points defined on the Brillouin zone of monoclinic primitive unit cell.\nOur calculations suggest an intrinsic AHC of σyz≈260\nΩ–1cm–1nearEF, much larger than the experimental value\nofσA\nyz=27Ω–1cm–1. Such a small AHC suggests dominant\nimpurity scattering contribution to the total anomalous Hall\neffect of Cr 3Te4[60], i.e., the skew-scattering contribution in\nthe dirty limit [61]. However, despite the system is in the dirty\nlimit, the total AHC should be at least comparable to the value\nof intrinsic AHC ( ≈260Ω–1cm–1) as it has contributions\nfrom both intrinsic Berry-curvature and extrinsic skew-\nscattering. In contrast, experimentally, we find a much\nsmaller AHC compared to theoretical calculations. Note\nhere that the calculations performed on the stoichiometric\nCr3Te4predict a large intrinsic AHC near EFdue to the\npresence of several band crossing points (Weyl points). But\nthe intrinsic AHC rapidly decreases as we move away from\nEF. This is particularly true when EFis shifted to lower\nbinding energies [see Fig. 5(d)]. Therefore, a genuine\nreason behind the discrepancy of AHC between experiment\nand theory could be that the experiments are performed on\na slightly off-stoichiometric composition of Cr 2.76Te4with\n8% Cr deficiency per formula unit, shifting the Fermi level\ntowards the lower binding energy. Maximum topological Hall\nresistivity ( ρT\nmax) is plotted as a function of temperature in\nFig. 6(a) by the black-colored data points. Also, green-colored\ndata points in Fig. 6(a) represent the magnetocrystallineanisotropy constant Kucalculated using the Eqn. 2.\nKu=µ0ZMs\n0[Hbc(M) – H a(M)] dM (2)\nHere, M srepresents saturation magnetization. H bcand H a\nrepresent H∥bcandH∥a, respectively.\nFrom Fig. 6(a), we can notice that the topological Hall\nresistivity is highest within the temperature range of 50 and\n150 K. Also, most importantly, the temperature dependance\nofKuresembles ρT\nmax. In Fig. 6(b), the black-colored data\npoints illustrate temperature dependent saturation field ( Hs)\nbeyond which the system becomes ferromagnetic [extracted\nfrom Fig. 2(c)] and the red-colored data points illustrate\ntemperature dependent field ( HT) beyond which ρT\nyzbecomes\nzero [extracted from Fig. 3(c)]. Interestingly, both fields HT\nandHsperfectly overlap at all measured temperatures. This\ncan be understood using the schematics shown in Fig. 6(c).\nMeans, at a given temperature up to the saturation field ( H<\nHs), the system possess non-coplanar spin structure and hence\nshows the topological Hall effect [62]. However, for the\napplied field beyond magnetic saturation ( H>Hs), the system\nbecomes ferromagnetic and thus THE disappears.\nFurther, the maximum topological Hall resistivity ρT\nmax\n≈1.1µΩ-cm over a broad temperature range of (50K<T<\n150K) observed in this vdW ferromagnetic Cr 2.76Te4is in the6\nHs\nNon coplanar,\nTopologicalFerromagnetic,\nNon topologicalH(T)\nT (K)\nH < ࡴ࢙ H > (c)(b) (a)\nHǁaߩ௫்\nT (K)Ku(kJ/m3)\nࡴ࢙HTHǁa\nFIG. 6. (a) Maximum value of ρTmax and magnetocrystalline\nanisotropy ( Ku) plotted as a function of temperature. (b) Saturation\nmagnetic field ( Hs) and magnetic field ( HT) above which the\ntopological Hall effect disappears for H∥a. (c) Schematic\nrepresentation of Crspin configuration for H<Hs(left) and H>Hs.\nSee the text for more details.\nsame order of magnitude found in the chiral semimetals such\nas Mn 2PtSn [63], Gd 3PdSi 3[64], LaMn 2Ge2[65], and in\nother Cr xTeybased systems [23, 26]. Interestingly, so far THE\nis observed in Cr xTeysystems in their hexagonal (trigonal)\ncrystal structure [23, 26, 66] but not in the monoclinic\nstructure that we found in the present study. As discussed\nabove, the easy-axis of magnetization in Cr 2.76Te4is found to\nbe in the bc-plane and thus for small applied fields the Cr spins\nare canted out of the bc-plane towards the a-axis for H∥a. In\nthis way, the non-coplanar spin-structure has been generated\nfor sufficiently smaller fields [23, 26, 65–68].\nSeveral mechanisms are proposed to understand the\ntopological Hall effect. Such as the antisymmetric\nexchange or Dzyaloshinskii–Moriya (DM) interaction in\nthe noncentrosymmetric systems [69–71] and the uniaxial\nmagnetocrystalline anisotropy in the centrosymmetric\nsystems [72–75]. In the case of monoclinic Cr 3Te4\n(C21/m1 ) which is a centrosymmetric crystal, the chiral-spin\nstructure could be stabilized by the strong MCA. This analogyis completely supported by our experimentally estimated\nMCA values at various temperatures as shown in Fig. 6(a).\nMost importantly, highest ρT\nmax value has been obtained at\nthe highest magnetocrystalline anisotropy of Ku=165 kJ/m3.\nThis is because, in presence of the chiral-spin structure,\nthe itinerant electrons acquire real-space Berry curvature\nassociated with finite scalar-spin chirality χijk= Si.(Sj×Sk)\nwhich serves as fictitious magnetic field to generate the\ntopological Hall signal [76–78].\nV . CONCLUSIONS\nTo summarize, we have grown high quality single\ncrystal of layered ferromagnetic Cr 2.76Te4in the monoclinic\nphase to study the electrical transport, Hall effect, and\nmagnetic properties. Our studies suggest Cr 2.76Te4to\nbe a soft ferromagnet with a negligible coercivity. The\neasy-axis of magnetization is found to be parallel to the\nbc-plane, leading to strong magnetocrystalline anisotropy.\nBelow 50 K, an antiferromagnetic-like transition is noticed.\nInterestingly, in going from 50 K to 150 K the strength of\nmagnetic moments switches between out-of-plane to in-plane,\nsuggests fluctuating Cr spins. From the electrical resistivity\nmeasurements the system is found to be metallic throughout\nthe measured temperature range. Also, a kink at around\n50 K due to AFM ordering is noticed. Magnetotransport\nmeasurements demonstrate large anomalous Hall effect\n(AHE) and topological Hall effect (THE) in this systems.\nFirst-principles calculations point to an intrinsic AHE due\nto non-zero Berry curvature near the Fermi level, while\nexperimentally it is found to be an extrinsic AHE due to skew-\nscattering. Topological Hall effect has been observed due\nto the non-coplanar spin-structure in the presence of strong\nmagnetocrystalline anisotropy.\nVI. ACKNOWLEDGEMENTS\nA.B. acknowledges support from Prime Minister’s\nResearch Fellowship (PMRF). A.N. thanks startup grant of\nthe Indian Institute of Science (SG/MHRD-19-0001) and\nDST-SERB (SRG/2020/000153). S.T. thanks the Science and\nEngineering Research Board (SERB), Department of Science\nand Technology (DST), India for the financial support (Grant\nNo.SRG/2020/000393). This research has made use of the\nTechnical Research Centre (TRC) Instrument Facilities of S.\nN. Bose National Centre for Basic Sciences, established under\nthe TRC project of Department of Science and Technology,\nGovt. of India.\n[1] M. Augustin, S. Jenkins, R. F. Evans, K. S. Novoselov, and\nE. J. Santos, Properties and dynamics of meron topological\nspin textures in the two-dimensional magnet CrCl 3, Nature\nCommunications 12, 185 (2021).\n[2] L. Chen, J.-H. Chung, M. B. Stone, A. I. Kolesnikov, B. Winn,\nV . O. Garlea, D. L. Abernathy, B. Gao, M. Augustin, E. J. G.Santos, and P. Dai, Magnetic field effect on topological spin\nexcitations in CrI3, Phys. Rev. X 11, 031047 (2021).\n[3] K. Choudhary, K. F. Garrity, J. Jiang, R. Pachter, and\nF. Tavazza, Computational search for magnetic and non-\nmagnetic 2d topological materials using unified spin–orbit\nspillage screening, npj Computational Materials 6, 49 (2020).7\n[4] N. Nagaosa and Y . Tokura, Topological properties and\ndynamics of magnetic skyrmions, Nature nanotechnology 8,\n899 (2013).\n[5] J. R. Schaibley, H. Yu, G. Clark, P. Rivera, J. S. Ross, K. L.\nSeyler, W. Yao, and X. Xu, Valleytronics in 2d materials, Nature\nReviews Materials 1, 1 (2016).\n[6] A. Fert, N. Reyren, and V . Cros, Magnetic skyrmions: advances\nin physics and potential applications, Nature Reviews Materials\n2, 1 (2017).\n[7] C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y . Xia, T. Cao, W. Bao,\nC. Wang, Y . Wang, et al. , Discovery of intrinsic ferromagnetism\nin two-dimensional van der waals crystals, Nature 546, 265\n(2017).\n[8] K. L. Seyler, D. Zhong, D. R. Klein, S. Gao, X. Zhang,\nB. Huang, E. Navarro-Moratalla, L. Yang, D. H. Cobden, M. A.\nMcGuire, et al. , Ligand-field helical luminescence in a 2D\nferromagnetic insulator, Nature Physics 14, 277 (2018).\n[9] Z. Fei, B. Huang, P. Malinowski, W. Wang, T. Song, J. Sanchez,\nW. Yao, D. Xiao, X. Zhu, A. F. May, et al. , Two-dimensional\nitinerant ferromagnetism in atomically thin Fe3GeTe 2, Nature\nmaterials 17, 778 (2018).\n[10] N. D. Mermin and H. Wagner, Absence of ferromagnetism\nor antiferromagnetism in one- or two-dimensional isotropic\nheisenberg models, Phys. Rev. Lett. 17, 1133 (1966).\n[11] B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein,\nR. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A.\nMcGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-Herrero,\nand X. Xu, Layer-dependent ferromagnetism in a van der Waals\ncrystal down to the monolayer limit, Nature 546, 270 (2017).\n[12] K. S. Burch, D. Mandrus, and J.-G. Park, Magnetism in two-\ndimensional van der Waals materials, Nature 563, 47 (2018).\n[13] C. Gong and X. Zhang, Two-dimensional magnetic crystals\nand emergent heterostructure devices, Science 363, eaav4450\n(2019).\n[14] M.-G. Han, J. A. Garlow, Y . Liu, H. Zhang, J. Li, D. DiMarzio,\nM. W. Knight, C. Petrovic, D. Jariwala, and Y . Zhu, Topological\nmagnetic-spin textures in two-dimensional van der waals\nCr2Ge2Te6, Nano Letters 19, 7859 (2019).\n[15] Y . You, Y . Gong, H. Li, Z. Li, M. Zhu, J. Tang, E. Liu,\nY . Yao, G. Xu, F. Xu, and W. Wang, Angular dependence of the\ntopological hall effect in the uniaxial van der waals ferromagnet\nFe3GeTe 2, Phys. Rev. B 100, 134441 (2019).\n[16] Y . Gao, S. Yan, Q. Yin, H. Huang, Z. Li, Z. Zhu, J. Cai,\nB. Shen, H. Lei, Y . Zhang, and S. Wang, Manipulation of\ntopological spin configuration via tailoring thickness in van der\nwaals ferromagnetic Fe5–xGeTe 2, Phys. Rev. B 105, 014426\n(2022).\n[17] M. Schmitt, T. Denneulin, A. Kov ´acs, T. G. Saunderson,\nP. R¨ußmann, A. Shahee, T. Scholz, A. H. Tavabi, M. Gradhand,\nP. Mavropoulos, et al. , Skyrmionic spin structures in layered\nFe5GeTe 2up to room temperature, Communications Physics\n5, 254 (2022).\n[18] Y . Zhu, X. Kong, T. D. Rhone, and H. Guo, Systematic\nsearch for two-dimensional ferromagnetic materials, Phys. Rev.\nMaterials 2, 081001 (2018).\n[19] Y . Liu and C. Petrovic, Critical behavior of the quasi-two-\ndimensional weak itinerant ferromagnet trigonal chromium\ntelluride Cr0.62Te, Phys. Rev. B 96, 134410 (2017).\n[20] T. Eto, M. Ishizuka, S. Endo, T. Kanomata, and T. Kikegawa,\nPressure-induced structural phase transition in a ferromagnet\nCrTe , Journal of Alloys and Compounds 315, 16 (2001).\n[21] F. Wang, J. Du, F. Sun, R. F. Sabirianov, N. Al-Aqtash,\nD. Sengupta, H. Zeng, and X. Xu, Ferromagnetic Cr2Te3\nnanorods with ultrahigh coercivity, Nanoscale 10, 11028(2018).\n[22] B. Hessen, T. Siegrist, T. Palstra, S. Tanzler,\nand M. Steigerwald, Hexakis (triethylphosphine)\noctatelluridohexachromium and a molecule-based synthesis\nof chromium telluride, Cr3Te4, Inorganic chemistry 32, 5165\n(1993).\n[23] Y . Wang, J. Yan, J. Li, S. Wang, M. Song, J. Song, Z. Li,\nK. Chen, Y . Qin, L. Ling, H. Du, L. Cao, X. Luo, Y . Xiong,\nand Y . Sun, Magnetic anisotropy and topological hall effect in\nthe trigonal chromium tellurides Cr5Te8, Phys. Rev. B 100,\n024434 (2019).\n[24] J. Dijkstra, H. H. Weitering, C. F. van Bruggen, C. Haas,\nand R. A. de Groot, Band-structure calculations, and magnetic\nand transport properties of ferromagnetic chromium tellurides\n(CrTe,Cr 3Te4,Cr2Te3), Journal of Physics: Condensed\nMatter 1, 9141 (1989).\n[25] H. Ipser, K. L. Komarek, and K. O. Klepp, Transition metal-\nchalcogen systems viii: The CrTe phase diagram, Journal of\nthe Less Common Metals 92, 265 (1983).\n[26] M. Huang, L. Gao, Y . Zhang, X. Lei, G. Hu, J. Xiang, H. Zeng,\nX. Fu, Z. Zhang, G. Chai, et al. , Possible topological hall effect\nabove room temperature in layered Cr1.2Te2ferromagnet,\nNano Letters 21, 4280 (2021).\n[27] J. Liu, B. Ding, J. Liang, X. Li, Y . Yao, and W. Wang, Magnetic\nskyrmionic bubbles at room temperature and sign reversal of\nthe topological hall effect in a layered ferromagnet Cr0.87Te,\nACS nano 16, 13911 (2022).\n[28] J. Smit, The spontaneous hall effect in ferromagnetics II,\nPhysica 24, 39 (1958).\n[29] T. Hashimoto, K. Hoya, M. Yamaguchi, and I. Ichitsubo,\nMagnetic properties of single crystals Cr2–δTe3, Journal of the\nPhysical Society of Japan 31, 679 (1971).\n[30] See Supplemental Material at [URL will be inserted by\npublisher] for topological Hall resistivity analysis.\n[31] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car,\nC. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni,\nI. Dabo, et al. , QUANTUM ESPRESSO: a modular and open-\nsource software project for quantum simulations of materials, J.\nPhys. Condens. Matter 21, 395502 (2009).\n[32] P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. B.\nNardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli,\nM. Cococcioni, et al. , Advanced capabilities for materials\nmodelling with Quantum ESPRESSO, J. Phys. Condens. Matter\n29, 465901 (2017).\n[33] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized Gradient\nApproximation Made Simple, Phys. Rev. Lett. 77, 3865 (1996).\n[34] P. E. Bl ¨ochl, Projector augmented-wave method, Phys. Rev. B\n50, 17953 (1994).\n[35] H. J. Monkhorst and J. D. Pack, Special points for Brillouin-\nzone integrations, Phys. Rev. B 13, 5188 (1976).\n[36] S. Grimme, Semiempirical GGA-type density functional\nconstructed with a long-range dispersion correction, J. Comput.\nChem. 27, 1787 (2006).\n[37] A. A. Mostofi, J. R. Yates, Y .-S. Lee, I. Souza, D. Vanderbilt,\nand N. Marzari, wannier90: A tool for obtaining maximally-\nlocalised Wannier functions, Comput. Phys. Commun. 178, 685\n(2008).\n[38] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den\nNijs, Quantized Hall Conductance in a Two-Dimensional\nPeriodic Potential, Phys. Rev. Lett. 49, 405 (1982).\n[39] Q. Wu, S. Zhang, H.-F. Song, M. Troyer, and A. A. Soluyanov,\nWannierTools: An open-source software package for novel\ntopological materials, Comput. Phys. Commun. 224, 405\n(2018).8\n[40] M. Yamaguchi and T. Hashimoto, Magnetic Properties of\nCr3Te4in Ferromagnetic Region, Journal of the Physical\nSociety of Japan 32, 635 (1972).\n[41] S. Ohta, Magnetic properties of (Cr1–uVu)3Te4, Journal of\nthe Physical Society of Japan 54, 1076 (1985).\n[42] D. Babot, M. Wintenberger, B. Lambert-Andron, and\nM. Chevreton, Propri ´et´es magn ´etiques et conductibilit ´e\nelectrique des compos ´es ternaires Cr3Se4–xTex, Journal of\nSolid State Chemistry 8, 175 (1973).\n[43] C. Li, K. Liu, D. Jiang, C. Jin, T. Pei, T. Wen, B. Yue,\nand Y . Wang, Diverse Thermal Expansion Behaviors in\nFerromagnetic Cr1–δTewith NiAs-Type, Defective Structures,\nInorganic Chemistry 61, 14641 (2022).\n[44] L.-Z. Zhang, A.-L. Zhang, X.-D. He, X.-W. Ben, Q.-L. Xiao,\nW.-L. Lu, F. Chen, Z. Feng, S. Cao, J. Zhang, and J.-Y . Ge,\nCritical behavior and magnetocaloric effect of the quasi-two-\ndimensional room-temperature ferromagnet Cr4Te5, Phys.\nRev. B 101, 214413 (2020).\n[45] P. Chen, C. Zhang, Y . Wang, B. Lv, P. Liu, Y . Tian, J. Ran,\nC. Gao, Q. Liu, and D. Xue, Topological Hall effect in frustrated\nB2-ordered Mn0.74Co0.57Al0.69 films, Phys. Rev. B 104,\n064409 (2021).\n[46] K. Shimada, T. Saitoh, H. Namatame, A. Fujimori, S. Ishida,\nS. Asano, M. Matoba, and S. Anzai, Photoemission study of\nitinerant ferromagnet Cr1–δTe, Phys. Rev. B 53, 7673 (1996).\n[47] Y . Liu and C. Petrovic, Anomalous hall effect in the trigonal\nCr5Te8single crystal, Phys. Rev. B 98, 195122 (2018).\n[48] S. Nakatsuji, N. Kiyohara, and T. Higo, Large anomalous Hall\neffect in a non-collinear antiferromagnet at room temperature,\nNature 527, 212 (2015).\n[49] K. Kuroda, T. Tomita, M.-T. Suzuki, C. Bareille, A. Nugroho,\nP. Goswami, M. Ochi, M. Ikhlas, M. Nakayama, S. Akebi,\nR. Noguchi, R. Ishii, N. Inami, K. Ono, H. Kumigashira,\nA. Varykhalov, T. Muro, T. Koretsune, R. Arita, S. Shin,\nT. Kondo, and S. Nakatsuji, Evidence for magnetic Weyl\nfermions in a correlated metal, Nat. Mater. 16, 1090 (2017).\n[50] N. Kanazawa, Y . Onose, T. Arima, D. Okuyama, K. Ohoyama,\nS. Wakimoto, K. Kakurai, S. Ishiwata, and Y . Tokura, Large\nTopological Hall Effect in a Short-Period Helimagnet MnGe,\nPhys. Rev. Lett. 106, 156603 (2011).\n[51] S. N. Kaul, Anomalous Hall effect in nickel and nickel-rich\nnickel-copper alloys, Phys. Rev. B 20, 5122 (1979).\n[52] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P.\nOng, Anomalous hall effect, Rev. Mod. Phys. 82, 1539 (2010).\n[53] C. Zeng, Y . Yao, Q. Niu, and H. H. Weitering, Linear\nmagnetization dependence of the intrinsic anomalous hall\neffect, Phys. Rev. Lett. 96, 037204 (2006).\n[54] L. Berger, Side-jump mechanism for the hall effect of\nferromagnets, Phys. Rev. B 2, 4559 (1970).\n[55] J. Shen, S. Zhang, T. Liang, J. Wang, Q. Zeng, Y . Wang,\nH. Wei, E. Liu, and X. Xu, Intrinsically enhanced anomalous\nHall conductivity and Hall angle in Sb-doped magnetic Weyl\nsemimetal Co3Sn2S2, APL Mater. 10, 090705 (2022).\n[56] Y . Tian, L. Ye, and X. Jin, Proper scaling of the anomalous hall\neffect, Phys. Rev. Lett. 103, 087206 (2009).\n[57] Y . Wang, C. Xian, J. Wang, B. Liu, L. Ling, L. Zhang, L. Cao,\nZ. Qu, and Y . Xiong, Anisotropic anomalous hall effect in\ntriangular itinerant ferromagnet Fe3GeTe 2, Phys. Rev. B 96,\n134428 (2017).\n[58] Y . Liu and C. Petrovic, Anomalous Hall effect in the trigonal\nCr5Te8single crystal, Phys. Rev. B 98, 195122 (2018).\n[59] Z. Z. Jiang, X. Luo, J. Yan, J. J. Gao, W. Wang, G. C. Zhao,\nY . Sun, J. G. Si, W. J. Lu, P. Tong, X. B. Zhu, W. H. Song,\nand Y . P. Sun, Magnetic anisotropy and anomalous Hall effectin monoclinic single crystal Cr5Te8, Phys. Rev. B 102, 144433\n(2020).\n[60] Y . Liu, H. Tan, Z. Hu, B. Yan, and C. Petrovic, Anomalous Hall\neffect in the weak-itinerant ferrimagnet FeCr 2Te4, Phys. Rev.\nB103, 045106 (2021).\n[61] S. Onoda, N. Sugimoto, and N. Nagaosa, Intrinsic Versus\nExtrinsic Anomalous Hall Effect in Ferromagnets, Phys. Rev.\nLett. 97, 126602 (2006).\n[62] X. Zheng, X. Zhao, J. Qi, X. Luo, S. Ma, C. Chen, H. Zeng,\nG. Yu, N. Fang, S. U. Rehman, et al. , Giant topological hall\neffect around room temperature in noncollinear ferromagnet\nNdMn 2Ge2single crystal, Applied Physics Letters 118,\n072402 (2021).\n[63] Z. Liu, A. Burigu, Y . Zhang, H. M. Jafri, X. Ma, E. Liu,\nW. Wang, and G. Wu, Giant topological hall effect in tetragonal\nheusler alloy Mn2PtSn , Scripta Materialia 143, 122 (2018).\n[64] T. Kurumaji, T. Nakajima, M. Hirschberger, A. Kikkawa,\nY . Yamasaki, H. Sagayama, H. Nakao, Y . Taguchi, T.-h. Arima,\nand Y . Tokura, Skyrmion lattice with a giant topological hall\neffect in a frustrated triangular-lattice magnet, Science 365, 914\n(2019).\n[65] G. Gong, L. Xu, Y . Bai, Y . Wang, S. Yuan, Y . Liu, and\nZ. Tian, Large topological hall effect near room temperature\nin noncollinear ferromagnet LaMn 2Ge2single crystal, Phys.\nRev. Materials 5, 034405 (2021).\n[66] Y . He, J. Kroder, J. Gayles, C. Fu, Y . Pan, W. Schnelle,\nC. Felser, and G. H. Fecher, Large topological hall effect in an\neasy-cone ferromagnet Cr0.9B0.1Te, Applied Physics Letters\n117, 052409 (2020).\n[67] H. Li, B. Ding, J. Chen, Z. Li, E. Liu, X. Xi, G. Wu,\nand W. Wang, Large anisotropic topological hall effect in\na hexagonal non-collinear magnet Fe5Sn3, Applied Physics\nLetters 116, 182405 (2020).\n[68] Q. Wang, Q. Yin, and H. Lei, Giant topological hall effect of\nferromagnetic kagome metal Fe3Sn2, Chinese Physics B 29,\n017101 (2020).\n[69] X. Yu, Y . Onose, N. Kanazawa, J. H. Park, J. Han, Y . Matsui,\nN. Nagaosa, and Y . Tokura, Real-space observation of a two-\ndimensional skyrmion crystal, Nature 465, 901 (2010).\n[70] W. Jiang, G. Chen, K. Liu, J. Zang, S. G. Te Velthuis, and\nA. Hoffmann, Skyrmions in magnetic multilayers, Physics\nReports 704, 1 (2017).\n[71] O. Meshcheriakova, S. Chadov, A. K. Nayak, U. K. R ¨oßler,\nJ. K ¨ubler, G. Andr ´e, A. A. Tsirlin, J. Kiss, S. Hausdorf,\nA. Kalache, W. Schnelle, M. Nicklas, and C. Felser,\nLarge Noncollinearity and Spin Reorientation in the Novel\nMn2RhSn Heusler Magnet, Phys. Rev. Lett. 113, 087203\n(2014).\n[72] P. K. Rout, P. V . P. Madduri, S. K. Manna, and A. K.\nNayak, Field-induced topological hall effect in the noncoplanar\ntriangular antiferromagnetic geometry of Mn3Sn, Phys. Rev. B\n99, 094430 (2019).\n[73] B. Ding, Z. Li, G. Xu, H. Li, Z. Hou, E. Liu, X. Xi, F. Xu,\nY . Yao, and W. Wang, Observation of magnetic skyrmion\nbubbles in a van der waals ferromagnet Fe3GeTe 2, Nano\nLetters 20, 868 (2019).\n[74] D. A. Gilbert, B. B. Maranville, A. L. Balk, B. J. Kirby,\nP. Fischer, D. T. Pierce, J. Unguris, J. A. Borchers, and K. Liu,\nRealization of ground-state artificial skyrmion lattices at room\ntemperature, Nature communications 6, 1 (2015).\n[75] M. Nakamura, D. Morikawa, X. Yu, F. Kagawa, T.-h. Arima,\nY . Tokura, and M. Kawasaki, Emergence of topological\nhall effect in half-metallic manganite thin films by tuning\nperpendicular magnetic anisotropy, Journal of the Physical9\nSociety of Japan 87, 074704 (2018).\n[76] Y . Machida, S. Nakatsuji, Y . Maeno, T. Tayama, T. Sakakibara,\nand S. Onoda, Unconventional anomalous hall effect enhanced\nby a noncoplanar spin texture in the frustrated kondo lattice\nPr2Ir2O7, Phys. Rev. Lett. 98, 057203 (2007).[77] Y . Taguchi, Y . Oohara, H. Yoshizawa, N. Nagaosa, and\nY . Tokura, Spin chirality, berry phase, and anomalous hall effect\nin a frustrated ferromagnet, Science 291, 2573 (2001).\n[78] H. Wang, Y . Dai, G. M. Chow, and J. Chen, Topological hall\ntransport: materials, mechanisms and potential applications,\nProgress in Materials Science , 100971 (2022)."    },    {        "title": "2309.08974v1.Colossal_linear_magnetoelectricity_in_polar_magnet_Fe2Mo3O8.pdf",        "content": " \nColossal linear magnetoelectricity in polar magnet Fe 2Mo 3O8 \n \nYuting Chang1#, Yakui Weng2#, Yunlong Xie3#, Bin You1, Junfeng Wang1, Liang Li1, Jun-Ming \nLiu4, Shuai Dong5* & Chengliang Lu1* \n1 School of Physics & Wuhan National High Magne tic Field Centre, Huazhong University of \nScience and Technology, Wuhan 430074, China \n2 School of Science, Nanjing University of  Posts and Telecommunications, Nanjing 210023, \nChina  \n3 Institute for Advanced Materials, Hube i Normal University, Huangshi 435001, China  \n4 Laboratory of Solid State Microstructure s, Nanjing University, Nanjing 210093, China  \n5 Key Laboratory of Quantum Materials and Devi ces of Ministry of Education, School of \nPhysics, Southeast Unive rsity, Nanjing 211189, China  \n   \nAbstract:  Linear magnetoelectric effect is an attractive phenomenon in condensed matters \nand provides indispensable te chnological functionalities. Here a colossal linear \nmagnetoelectric effect with diagonal component α\n33 reaching up to ~480 ps/m is reported in a \npolar magnet Fe 2Mo 3O8, and this effect can persist in a broad range of magnetic field (~20 T) \nand is orders of magnitude larger than repor ted values in literature. Such an exceptional \nexperimental observation can be well reprod uced by a theoretical model affirmatively \nunveiling the vital contributions from the exch ange striction, while the sign difference of \nmagnetocrystalline anisotropy can al so be reasonably figured out.  \n  \n\n# These authors contributed equally to this work. \n* Corresponding authors: C.L.L. (cllu@ hust.edu.cn), S.D.  (sdong@seu.edu.cn). Magnetoelectric (ME) effects refer to the control of magnetization ( M) by electric field E \nor control of polarization ( P) by magnetic field H, which provide fascinating functionalities to \ndevelop next-generation electron ic devices, e.g. ultra-sensi tive sensors and energy-saving \nmemories [1-3] . There are various manifestations of ME  effects, which can be realized in \nmany kinds of materials including but not limite d to ME composites, multiferroics, as well as \naxion insulators [4]. Of particular interest is the mo st canonical linear ME effect, i.e. H \ninduces electric polarization  Pi=αijHj, or E causes a variation of magnetization μ0Mj=αijEi, \nwith αij the ME tensor and  i/j the component of spatial coordinates [5,6] . In those linear \nmagnetoelectrics, the interplay of broken space-inve rsion and time-reversal symmetries can \nyield a bunch of emergent physics and functi ons, including ferrotoroidicity, ME monopole, \nnon-reciprocal transport of quasi-particles, etc [7-9] . Recently, it was found that the writing \nthreshold could be reduced drastically in memory  device made of the paradigmatic linear ME \nCr2O3, implying the technological im portance of linear ME in e nhancing energy efficiency \n[10]. Although the studies on linear ME can be tr aced back to 1960s, generally the intrinsic \nlinear magnetoelectricity to date  remains weak, with typical α values of 1~10 ps/m, which has \nbeen a long-term issue of ME activities [5,6] . \nThermodynamically it is well known that αij is just bounded by the permittivity εii and \npermeability μjj of solids, i.e. αij2<εiiμjj [11]. To obtain large αij, nonlinear regions of εii and μjj \nare usually employed, especially near critical point of a ME system where the permittivity and/or permeability may be significantly am plified. For example, in those type-II \nmultiferroics where spontaneous P’s are generated by some special magnetic orders, magnetic \ntransitions or metamagnetic transition s can trigger remarkable changes of P’s and result in \nvery large ME coefficients [12]. However, such a scenario can only work in the proximate \nregion near the transition point. In this sense, the operating window of pot ential ME devices \nbased on these materials would be narrow. \nIn fact, the integration upon H is the total change of P (i.e. P). With fixed P, the larger \nα, the narrower operating H-window. Such a trade-off between ME sensitivity and operating \nwindow makes it difficult to obtain an optimal system with superior performances in both sides (large α & wide H-window), once the total P generated by magnetism is small. \nTherefore, alternative ro utes should be developed. It has been well recognized th at the symmetric exchange st riction allows much larger P’s \n(e.g. on the order of ~1 μC/cm2 in E-type antiferromagnetic (AFM) manganites) [13,14] , \nwhile other mechanisms for magnetism-driven polarization (e.g. the inverse Dzyaloshinskii-Moriya interaction (DMI) and the spin-dependent p-d hybridization) rely on \nthe spin-orbit coupling (SOC) a nd typically lead to small P’s (~0.001-0.1 μC/cm\n2) [15-17] . \nNevertheless, only large P’s from exchange striction is not sufficient. In fact, some magnetic \ntextures (e.g. E-type phase in o-LuMnO 3 [18] and o-YMnO 3 [19]) are robust against H, \nleading to weak ME response. In other cases, la rge nonlinear ME effects, instead of linear one, \noccur in relative narrow H-window, triggered by the firs t-order phase transitions [20-22] . \nThese facts indicate that additional physical i ngredient has to be involved to obtain large \nlinear ME.  \nIn this Letter, the ME response of a polar magnet Fe 2Mo 3O8 is studied. Our experimental \nstudy finds a colossal linear ME effect in a broad H-window, far beyond the consequence near \nthe first-order transition point. Specifically, the diagonal linear ME component in its \nferrimagnetic (FRM) phase is unprecedentedly large: α33~480 ps/m at 20 K, orders of \nmagnitude larger than reported values in literature [5]. Our theoretical calculations suggest \nthat the exchange striction tuned polarity and sign difference  of magnetocrysta lline anisotropy \nare essential in physics.   \nFigure 1.  (a) Schematic of crystal and magnetic structures of Fe 2Mo 3O8. (b-d) show \nT-dependence of magne tization along the c-axis and the ab-plane, c-axis dielectric constant ε, \nand ΔPc along the c-axis, respectively. (e-g) H-dependences of Mc, magneto-current I, and \nintegrated ΔPc along the c-axis at T=50 K, respectively.  \n As shown in Fig. 1(a), Fe\n2Mo 3O8 consists of Fe-O and Mo-O  blocks stacking along the \nc-axis, and the polar arrangement of FeO 4 tetrahedra manifests the crystallographic polarity \n[23,24] . The corner-shared FeO 4 tetrahedra and FeO 6 octahedra form a buckled honeycomb \nstructure within the ab-plane. Fe 2Mo 3O8 has an AFM ground state (magnetic point group \n6'mm' ) below TN~60 K with all Fe2+-spins pointing along the c-axis [23], which can be \nevidenced by the anomalies in the temperature ( T) dependence of M as shown in Fig. 1(b). \nThe remarkable difference between Mc(T) and Mab(T) persists up to room temperature, \nimplying its strong magnetic anisotropy. Corresponding to the AFM transition, a sharp peak arises in the T-dependence of dielectric constant ε(T) (Fig. 1(c)), as well as an obvious change of Pc(T) (Fig. 1(d)), revealing the ons et of magnetically driven P based on the polar structure. \nThe exchange striction mechanism is  responsible for this emergent P [23,24] . \nBy applying H//c-axis above a critical field Hcri, a FRM phase is stabilized where the Fe \ntetrahedral and octahedral irons (Fe T vs Fe O) form individual ferromagnetic sublattices and \nalign in opposite directions (Fig. 1(a )). As shown in Fig. 1(e), a jump of ΔMc~0.6 μB/f.u. at \nHcri indicates that Fe T and Fe O sites have different magnetic moments. Concomitantly, the \nAFM-FRM transition causes st riking magneto-current at Hcri, and the integrated ΔPc(H) curve \nexhibits a step-like anomaly with ΔPc~0.06 μC/cm2, as shown in Figs. 1(f-g), consistent with \nthe previous report [24]. The H-stabilized FRM phase belongs to the magnetic point group \n6m'm' , which allows diagonal lin ear ME components. Previ ous works studied the ME \nresponse in a relatively narrow H-T window, and indeed observed  approximatively linear ME \nwith α33~16 ps/m in the FRM phase of Fe 2Mo 3O8, although there were also nonlinear terms \nmixed-in  [24,25] . A single-site mechanism, i.e. g-factor effect, was proposed to explain this \nlinear ME, instead of the symmetr ic exchange responsible for P, i.e. a two-site mechanism.  \n \nFigure 2. (a) Out-of-plane ( Mc) and in-plane magnetization ( Ma) as a function of H measured \nat T=10 K. (b) H-dependence of ΔPc measured with H//c-axis and H⊥c-axis at T=10 K. For \nbetter view, the black curves are vert ically shifted from the zero point.  \n In the following, we performed extensive m easurements of magnetization and electric \npolarization over a large H range up to ~58 T using a pulse d high magnetic field apparatus, \nwhich allows in-depth exploration of the linea r ME of the FRM phase. As shown in Fig. 2(a), \nM\nc(H) measured with H//c-axis at T=10 K also displays a clear  AFM-FRM transition with a very high critical field Hcri~32 T, and the related jump remains ΔMc~0.6 μB/f.u.. Slightly \nabove Hcri, Mc(H) shows another anomaly at H2 ~39 T, and then a subsequent linear increase \nas H>H2. The anomaly at H2 will be discussed later. Mc reaches ~7.5 μB/f.u. at H~58 T, close \nto the fully polarized M~8.5 μB/f.u. for Fe 2Mo 3O8. Different from Mc(H), the in-plane \nmagnetization Mab continuously increases with H, without any anomaly up to H~58 T, \nconfirming the strong magnetic anisotropy of Fe 2Mo 3O8.    \nIn accordance with Mc(H), ΔPc(H) measured with  H//c-axis at 10 K also shows an abrupt \nincrease at Hcri, and then a crossover anomaly at H~H2. A fascinating behavior is the \nemergence of remarkable and perfect linear e nhancement of ΔPc(H) above H2. The estimated \nlinear ME coefficient in a wide range is as  large as 460 ps/m, far beyond other known linear \nME materials. For instance, α~4 ps/m for Cr 2O3 [26], α~2 ps/m for LiNiPO 4 [27], α~10 ps/m \nfor GaFeO 3 [28], α~30 ps/m for TbPO 4 [29], and α~40 ps/m for Co 4Nb2O9 [30]. With the \nmeasured dielectric constant ε~15 at T<T N for Fe 2Mo 3O8 (Fig. 1(c)), one may translate \ndP/dH~460 ps/m to d E/dH~2800 mV/cm-Oe. This value is already comparable with many \ncomposite systems, such as commercially used composite Pb(Zr,Ti)O 3/terfrnol-D (4800 \nmV/cm-Oe) [31]. For H //ab-plane, ΔPc(H) remains zero till the maximum field 58 T, \nindicating the absence of non- diagonal ME terms.    \nAn interesting phenomenon is that both ΔPc(H) and  Mc(H) show similar linear behavior \nabove H2. In particular, Mc(H) exhibits considerable enhancement when H>H2, suggesting \ndrastic modification of the spin  arrangement. This is distinctly different from the case of \nlinear magnetoelectrics ba sed on the single-site mechanisms such as the g-factor mechanism \nand single-ion anisotropy mechanism,  which exhibit nearly saturated M(H) behavior as \nappeared in (Fe 1-xZnx)2Mo 3O8 and Ga 2-xFexO3 [24,28] . Here the estimated linear ΔPc(H) \nreaches ~0.45 μC/cm2, which is much larger than the induced- P due to the inverse DMI \n(~0.01 μC/cm2) [2,15] . Therefore, this remarkable linear variation ΔPc(H) is more like to be \ndue to the exchange striction effect (i.e. two-site mechanis m). As mentioned above, complex \nME response was reporte d at relatively low H-range (i.e.  Hcri<H<H2) in Fe 2Mo 3O8 [24]. \nTherefore, the crossover at H2 could be ascribed to a transi tion from complex ME response to \npure linear ME, namely from individual magne tic-site effect to two-site mechanism.  \nComprehensive characterizations of Mc(H) and ME response have been performed over a broad T-range from 1.6 K to 50 K. As shown in Fig. 3(a), the AFM-FRM transition is well \nvisible at all measured temperatures, and Hcri shifts toward low field with increasing T. H2 is \nnot as well-identified as Hcri especially in high- T range (i.e. above 30 K), and just shows slight \nshift with T, implying different underlying physics. Corre sponding behaviors are also seen in \nΔPc(H) curves measured at various T’s, as shown in Fig. 3(b). Importantly, both Mc(H) and \nΔPc(H) display almost perfect  linear behavior above H2 at all T’s, and the slope shows clear \nT-dependence.  \nAlthough here pulsed magnetic field is used, a ll above ME results should be considered \nas static properties instead of dynamic ones, as confirmed by a comparison study of some key \nparameters obtained using steady and pulsed ma gnetic fields (see Supplementary Materials \nFig. S3 [32]). The duration time t~10.5 ms of the H-pulse remains far longer than the time \nscale of spin dynamics which should be in the THz level for Fe 2Mo 3O8 [33]. Based on the \nlinear behaviors of both Mc(H) and ΔPc(H) at high- H, it is able to precisely estimate the \nrelationship between Mc and Pc, i.e. d P/dM. As shown in Fig. 3(c), d P/dM distributes roughly \nat 0.12-0.14 below TN~60 K. The obtained linear ME coefficients α33 (H//c) are plotted as a \nfunction of T in Fig. 3(d), in which α33 reaches a plateau of ~480 ps/m below 20 K. The \ncolossal linear magnetoelectricity can al so be obtained in Zn-substituted Fe 2Mo 3O8 with \nreduced Hcri but robust H2 (see Supplementary Materials [32]). \n A previous theoretical work has studied th e ME related to phase transitions of Fe\n2Mo 3O8 \n[25]. To understand the colossal linear ME in the FRM phase, we performed density \nfunctional theory (DFT) calc ulations. The AFM ground state and electronic band gap (~1.37 \neV) can be obtained when a proper Ueff and SOC are incorporated simultaneously, in \nagreement with previous theoretical works [34,35] . The FRM state is slightly higher in energy \n(0.14  meV/f.u.) than the AFM one. More details of the DFT calculations can be found in \nSupplementary Materials [32].  \nFigure 3. (a) and (b) show Mc(H) and ΔPc(H) at various temperatures, respectively. For better \nview, the curves have been shifted vertically. (c) d P/dM as a function of temperature. (d) \nT-dependence of linear magnetoelectricity α33. \n \nIn DFT calculations, the preferred global spin direction is indeed along the c-aixs, while \nother orientations lead to higher en ergy. The local spin moments at Fe O and Fe T are -3.74 μB \nand 3.68 μB respectively, implying the high-spin states. The orbital moment at the Fe O site is \nalmost quenched, but the SOC effect yiel ds a significant orbital moment of 0.34 μB for Fe T, \nwhich is parallel to its spin moment. Simila r result was also predicted by earlier studies \n[36,37] . Noting that here the DFT moments, no matte r from spin or orbital, are integrated \nvalues within the default Wigner-Seitz sphe re assigned in pseudopotentials, while the \ncorresponding integrity valu es of local spin and orbital moments should be 4 μB and 0.5 μB \nrespectively [36]. Therefore, the large orbital moment of Fe T is the major contribution to the \nuncompensated M of FRM phase, leading to the jump ~0.5 μB/f.u. in Mc(H) curve at Hcri (Fig. \n1(e)).  \nBecause of SOC, Fe T has a much higher magnetocryst alline anisotropy energy (MAE) \nthan Fe O. According to our DFT study, the easy axis of Fe T is along the c-axis, with a large MAE coefficient AT =-5.11 meV/Fe. However, unexpectedly, the c-axis is a hard axis for Fe O, \nwith a weaker MAE AO=1.97 meV/Fe. Since AT is much larger than AO, the overall MAE \nprefers the easy axis along the c-axis, consistent with the neutron experiments [38].  However, \nthis sign difference of MAE between Fe T and Fe O is beyond previous scenario [23,24] , which \nplays a vital role to determine the ME beha vior under field (to be explained below). \nUnder a magnetic field along the c-axis, the magnetic state first transits from the AFM to \nFRM phase at Hcri, with a net magnetization ~0.5 μB/f.u. along the c-axis. In other words, the \nmagnetic moment of Fe T (Fe O) is parallel (anti-parallel ) to the magnetic field (i.e. c-axis). \nFurther increasing of magnetic field can only lead to the magnetic canting of Fe O. Then the \nfree energy during this spin rotati on process can be expressed as: \nF=F0+[3Jab+Jc]ꞏcosθ +AOꞏcos2θ - HꞏmOꞏcosθ,                         ( 1 )  \nwhere Jab (Jc) is the Heisenberg-type effective exch ange between in-plane (out-of-plane) \nnearest-neighboring Fe O and Fe T; θ is the moment angle of Fe O as indicated in the inset of Fig. \n4(a). The third item is the MAE of Fe O; and the last item is the Zeeman energy of Fe O. Here \nthe MAE and Zeeman energy of Fe T is not explicitly expressed since they are not changing \nduring the process and thus can be absorbed into the free energy base F0.  \nAccording to Eq. 1, the starting magnetic field to rotate the Fe O moment is \n[3Jab+Jc-2AO]/mO, corresponding to the experimental H2. For comparison, Hcri at T=0 K to \ndrive the AFM to FMR transition can be expressed as -2 Jc/[mT-mO], which are not identical to \nH2 in physics. Based on the experimental value Hcri=32 T at 1.6 K and mT-mO=0.5 μB, Jc can \nbe derived as -0.46 meV . And our model based on Eq. 1 (with 3 Jab+Jc=11.5 meV and AO=1.45 \nmeV) can well mimic the M-H  behavior in the whole range, as shown in Fig. 4(a). First, \nbelow Hcri, the ground state is AFM, with M=0; Then between Hcri and H2, the magnetic state \ncan be understood as a collinear FM R state without spin canting at T=0 K. Thus the Mc-H \ncurve shows a terrace behavior between Hcri and H2 at low temperature, and thus the ME \ncoefficient in this region is very li mited, as revealed in previous studies [24]. After H2, the \nmagnetic moment of Fe O starts to rotate, resulting in the increasing of Mc. According to Eq. 1, \nthe Mc-H curve is indeed linear with a slope mO2/2AO, reaching ~7.5 μB/f.u. at 58 T. Note the \nthree independent coefficients ( Jc, 3Jab+Jc, AO) used in our model are very close to the DFT \nextracted ones (-0.07 meV , 10.55 meV , 1.97 meV). Considering the simplified spin model and tolerance of DFT values, such a consistent between model and DFT values are rather well, at \nleast in a semiquantitative level. In particular, the sign difference of AT and AO is crucial to \nobtain the linear ME behavior  after the te rrace. Or if AO<0, the magnetic transition will be in a \nsudden jump manner at a critical  field (like what occurs at Hcri), instead of a continuous \nrotation of Fe O spin in a broad region. \n \nFigure 4.  (a) The simulated M-H evolution based on the simplified spin model (Eq. 1). Model \ncoefficients: mO=4 μB, Jc=-0.46 meV , 3 Jab+Jc=11.5 meV , and AO=1.45 meV . (b) DFT \ncalculated ΔPc as a function Mc during the spin rotation of mO. For comparison, both the \nnon-relaxed case and relaxed one are calculate d. In the non-relaxed case, the crystalline \nstructure is fixed as the optim ized FRM phase without spin canting. Thus, only the bias of \nelectron cloud is considered, namely pure el ectronic contribution. In  the relaxed one, the \nstructure is optimized with SOC enabled to mi mic the full contributions from both electronic \nand lattice. For both cas es, the ME behaviors ( ΔPc-Mc) are linear. (c) The ME phase diagram \nof Fe 2Mo 3O8 constructed based on experimental a nd theoretical data. PM: paramagnetic.  \n \nThe broad-range linear ME beha vior suggests that here the spin-related polarization is \nmainly contributed by the exchange striction, namely in the form of ΔPc ~ S FeꞏSO ~ cosθ ~ΔMc, \ninstead of the inverse DMI (in the form of S Fe×SO ~ sinθ ~ΔMab). Since exchange striction is a \nnon-relativistic effect which is relative strong, the ME coefficient here can be so large. Then \nthe concrete value of ΔPc can be obtained in our DFT calcu lation by rotating the spin of Fe O, \nas shown in Fig. 4(b). Impressively, the ΔPc-Mc curve due to the Fe O spin rotation is indeed \nlinear. The DFT estimated slop of ΔPc(Mc) is d P/dM~0.086 at T=0 K, close to the \nexperimental value of d P/dM~0.12 shown in Fig. 3(c).  \nBased on above experimental results and theoretical calculations, a ( H, T) phase diagram \nis summarized in Fig. 4(c), where various magnetic phase transitions and crossover behavior \nof the linear ME are presented. In low- H region (i.e. below Hcri), the AFM state with spins \naligned along the c-axis is stabilized, and no li near ME is identified. At Hcri<H<H2, the AFM \nphase is transformed to the FRM one, and thus  exhibits linear ME. Nevertheless, the linear \nME appearing at Hcri<H<H2 hosts an individual-site mechanism [24], different from the origin \nof the magnetically driven P. Above H2, H is sufficiently large to tune the exchange striction \nof neighboring Fe2+, leading to another linear ME behavi or with colossal coefficient, i.e. \nα33=480 ps/m at 20 K.    \n  Acknowledgements:  This work is supported by the Na tional Nature Science Foundation of \nChina (Grant Nos. 12174128, 11834002, 11704139, 92163210, and 12074135), Hubei \nProvince Natural Science F oundation of China (Grant Nos. 2020CFA083 and 2021CFB574), \nthe Interdisciplinary program of Wuhan National High Magnetic Field Center (WHMFC) at \nHuazhong University of Science and Technology (Grant No. WHMFC202205). The high \nmagnetic field measurements were performed at WHMFC.    References  \n[1] S.-W. Cheong and M. Mostovoy, Nat. Mater. 6, 13 (2007). \n[2] C. L. Lu, M. Wu, L. Lin, and J.-M. Liu, Natl. Sci. Rev. 6, 653 (2019). \n[3] S. Dong, H. Xiang, and E. Dagotto, Natl. Sci. Rev. 6, 629 (2019). \n[4] T. Morimoto, A. Furusaki, and N. Nagaosa, Phys. Rev. B 92, 085113 (2015). \n[5] M. Fiebig, J Phys. D: Appl. Phys. 38, R123 (2005). \n[6] W. Eerenstein, N. D. Mathur, and J. F. Scott, Nature 442, 759 (2006). \n[7] N. A. Spaldin and R. Ramesh, Nat. Mater. 18, 203 (2019). \n[8] Y . Tokura and N. Nagaosa, Nat. Commun. 9, 3740 (2018). \n[9] S.-W. Cheong, D. Talbayev, V . Kiryukhin, and A. Saxena, npj Quantum Mater. 3, 19 \n(2018). \n[10] T. Kosub  et al. , Nat. Commun. 8, 13985 (2017). \n[11] W. F. Brown, R. M. Hornreich, and S. Shtrikman, Phys. Rev. 168, 574 (1968). \n[12] K. Zhai  et al. , Nat. Commun. 8, 519 (2017). \n[13] S. Picozzi, K. Yamauchi, B. Sanyal, I. A. Sergienko, and E. Dagotto, Phys. Rev. Lett. 99, \n227201 (2007). \n[14] I. A. Sergienko, C. Sen, and E. Dagotto, Phys. Rev. Lett. 97, 227204 (2006). \n[15] S. Dong, J.-M. Liu, S.-W. Cheong, and Z. Ren, Adv. Phys. 64, 519 (2015). \n[16] T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima, and Y . Tokura, Nature 426, 55 \n(2003). \n[17] H. Murakawa, Y . Onose, S. Miyahara, N.  Furukawa, and Y . Tokura, Phys. Rev. Lett. 105, \n137202 (2010). \n[18] S. Ishiwata, Y . Kaneko, Y . Tokunaga, Y . Ta guchi, T.-h. Arima, and Y . Tokura, Phys. Rev. \nB 81, 100411 (2010). \n[19] S. Ishiwata, Y . Tokunaga, Y . Taguc hi, and Y . Tokura, J Am. Chem. Soc. 133, 13818 \n(2011). \n[20] N. Lee, Y . J. Choi, M. Ramazanoglu, W. Ratcliff, V . Kiryukhin, and S. W. Cheong, Phys. \nRev. B 84, 020101 (2011). \n[21] J. W. Kim  et al. , Phys. Rev. B 89, 060404 (2014). \n[22] L. Ponet  et al. , Nature 607, 81 (2022). [23] Y . Wang, G. L. Pascut, B.  Gao, T. A. Tyson, K. Haule, V . Kiryukhin, and S. W. Cheong, \nSci. Rep. 5, 12268 (2015). \n[24] T. Kurumaji, S. Ishiwata, and Y . Tokura, Phys. Rev. X 5, 031034 (2015). \n[25] M. Eremin, K. Vasin, and A. Nurmukhametov, Materials 15, 8229 (2022). \n[26] G. T. Rado and V . J. Folen, J Appl. Phys. 33, 1126 (1962). \n[27] T. B. S. Jensen  et al. , Phys. Rev. B 79, 092413 (2009). \n[28] T. Arima  et al. , Phys. Rev. B 70, 064426 (2004). \n[29] G. T. Rado, J. M. Ferrari, and W. G. Maisch, Phys. Rev. B 29, 4041 (1984). \n[30] Y . Chang  et al. , Phys. Rev. B 107, 014412 (2023). \n[31] A. V . C. J. Ryu, K. Uchino, and H-E. Kim, Jpn. J. Appl. Phys. 40, 4948 (2001). \n[32] See Supplemental Material at  for more e xperimental results and th eoretical calculations. \n[33] T. Kurumaji, Y . Takahashi, J. Fujioka, R.  Masuda, H. Shishikura, S. Ishiwata, and Y . \nTokura, Phys. Rev. B 95, 020405 (2017). \n[34] K. Park  et al. , Phys. Rev. B 104, 195143 (2021). \n[35] T. N. Stanislavchuk  et al. , Phys. Rev. B 102, 115139 (2020). \n[36] S. Reschke, A. A. Tsirlin, N. Khan, L.  Prodan, V . Tsurkan, I. Kézsmárki, and J. \nDeisenhofer, Phys. Rev. B 102, 094307 (2020). \n[37] I. V . Solovyev and S. V . Streltsov, Phys. Rev. Mater. 3, 114402 (2019). \n[38] D. Bertrand and H. Kerner-Czeskleba, J. Phys. 36, 379 (1975). \n \n "    },    {        "title": "2310.13688v2.Exchange_driven_spin_Hall_effect_in_anisotropic_ferromagnets.pdf",        "content": "arXiv:2310.13688v2  [cond-mat.mtrl-sci]  8 Feb 2024Exchange-driven spin Hall effect in anisotropic ferromagne ts\nK. D. Belashchenko\nDepartment of Physics and Astronomy and Nebraska Center for Materials and Nanoscience,\nUniversity of Nebraska-Lincoln, Lincoln, Nebraska 68588, USA\n(Dated: February 9, 2024)\nCrystallographic anisotropyofthespin-dependentconduc tivitytensorcanbeexploitedtogenerate\ntransverse spin-polarized current in a ferromagnetic film. This ferromagnetic spin Hall effect is\nanalogous to the spin-splitting effect in altermagnets and d oes not require spin-orbit coupling.\nFirst-principles screening of 41 non-cubic ferromagnets r evealed that many of them, when grown as\na single crystal with tilted crystallographic axes, can exh ibit large spin Hall angles comparable with\nthe best available spin-orbit-driven spin Hall sources. Ma croscopic spin Hall effect is possible for\nuniformly magnetized ferromagnetic films grown on some low- symmetry substrates with epitaxial\nrelations that prevent cancellation of contributions from different orientation domains. Macroscopic\nresponse is also possible for any substrate if magnetocryst alline anisotropy is strong enough to lock\nthe magnetization to the crystallographic axes in different orientation domains.\nI. INTRODUCTION\nThe spin Hall effect [ 1,2] is utilized for generating\ntransverse spin currents which can be injected across\ninterfaces into adjacent layers in heterostructures. If\nthat spin current is injected into a magnetic layer, it\ncan be converted into magnetization torque. For heavy-\nmetal spin Hall layers, this is called spin-orbit torque [ 3]\nand is used as an alternative to spin-transfer torque for\napplications including magnetic random-access memory\n(MRAM), high-frequency oscillators, and neuromorphic\ncomputing [ 4,5].\nThe spin Hall effect also exists in ferromagnets [ 6–18],\nwhere it has both time-reversal even and odd [ 18,19]\ncomponents. Theoddcomponent, referredtoasthemag-\nnetic spin Hall effect, was also observed in noncollinear\nantiferromagnets [ 20–25]. The structure of the spin cur-\nrent response tensor in the presence of spin-orbit cou-\npling may be deduced from the magnetic point group of\nthe material [ 8]. Some magnets can exhibit spin Hall re-\nsponse even without spin-orbit coupling, in which case\nthe symmetry of the response tensor may be determined\nusing spin groups treating spatial and spin degrees of\nfreedom separately from each other [ 26]. This was first\nnoted for noncollinear antiferromagnets [ 20,21].\nTransverse spin currents without spin-orbit coupling\narealsopossible[ 27–30]inso-calledaltermagnets[ 31,32],\ni.e., collinear antiferromagnets with k-dependent ex-\nchange splitting allowed by symmetry. In this case, spin-\norbit coupling usually plays a minor role and can be ne-\nglected, but the crystallographicorientation of the film is\nimportant. The exchange-driven spin Hall current in al-\ntermagnets has been referred to as the spin-splitting cur-\nrent[27–29]. For collinear two-sublattice antiferromag-\nnets without spin-orbit coupling, the symmetry of the\nmacroscopic response properties, including the spin Hall\ntensor, is fully described by the black-and-white(Heesch-\nShubnikov) point group, while the polarization of the\nspin current is decoupled from the crystal axes and is\nsimply parallel to the antiferromagnetic order parameter\n[33].In contrast to the spin Hall effect in heavy metals,\nwhich is typically dominated by the intrinsic mecha-\nnism [2], the spin-splitting current in altermagnets scales\nwith the longitudinal conductivity, leading to a disorder-\nindependent spin Hall angle. This feature is shared with\nthe magnetic spin Hall effect, which is also time-reversal-\nodd [18]. The predicted spin Hall angle for altermagnetic\nRuO2is about 0.27 [ 27], which compares favorably with\nspin-orbit-driven sources like Pt or β-W. However, ob-\nservation of this large response may be hindered by the\ndifficulty in achieving a single-domain state in RuO 2due\nto the absence of uncompensated magnetization [ 29].\nIn this paper, I show that exchange-driven spin Hall\neffect can also be observed in ferromagnets and ferrimag-\nnets thanks to the anisotropy of the spin-dependent con-\nductivity tensor. As in the case of altermagnets, gener-\nation of the transverse spin current imposes restrictions\non the bulk symmetry and the orientation of the film. In\nparticular,theferromagnetcannotbecubic, anditneeds\nto be grown such that the surface normal does not coin-\ncide with a principal axis of the conductivity tensor. Us-\ning first-principles calculations, I then estimate the spin\nHall angles for a number of tetragonal, hexagonal, and\northorhombic ferromagnets and ferrimagnets, identifying\na few with spin Hall angles as large as 0.3–0.4. Where\npossible, suitable substrates are suggested for growing\nfilms with the tilted high-symmetry axes, as required for\nthe observation of the exchange-driven spin Hall current.\nIf the film is uniformly magnetized, observation of the\nspin Hall current over a macroscopically large surface\narea requires that the bilayer has sufficiently low sym-\nmetry so that the spin Hall current is not cancelled out\nby averagingoverdegenerateorientationdomains. These\nsymmetry requirements may be fully relaxed if the ferro-\nmagnet has strong magnetocrystalline anisotropy which\nlocks the magnetization to the crystallographic axes in\ndifferent orientation domains.\nThe paper is organized as follows. Section IIpresents\nthe analysis of the exchange-driven spin Hall response in\na film with an anisotropic spin-dependent conductivity\ntensor. Section IIIdescribes computational details, and2\nthe results are displayed in Section IV. Section Vdis-\ncusses crystallographic orientation domains and the sta-\ntistical average of the spin Hall conductivity over these\ndomains. The conclusions are summarized in Section VI.\nII. TRANSVERSE SPIN CURRENT IN A\nMAGNETIC FILM WITH OPEN BOUNDARY\nCONDITIONS\nConsider a metallic film with a collinear magnetiza-\ntion whose spin-dependent conductivity tensor is σλ\nαβ,\nwhere the lower indices denote Cartesian components,\nandλ∈ {↑,↓}is the spin projection. I will also use\nλ=±1 in equations. The conductivity tensor is sym-\nmetric because spin-orbit coupling is neglected, which\nimposes time-reversal invariance independently in each\nspin channel, and the spin projection is defined with re-\nspect to the orientation of the magnetic order parameter.\nLet thezaxis point along the film normal direction and\nthexaxis along the externally applied electric field Ex.\nBecause σλ\nαβis anisotropic, the charge Hall response\nwill generally lead to the build-up of Hall voltage across\nthe thickness of the film. The resulting electric field\ncomponent Ezis determined under the assumption of\nopen-circuit boundary conditions, i.e., vanishing charge\ncurrent density perpendicular to the film plane, jz= 0.\nWe then find the steady-state bulk spin current flowing\nperpendicular to the film plane. This spin current corre-\nsponds to the bulk spin Hall source term. To determine\nthe spin current and accumulation profiles in a specific\nmultilayer, this source term needs to be included in the\nspin-diffusion equations [ 34,35]. A somewhat similar\nproblem was considered for the transverse spin current\ndriven by relativistic sources in a ferromagnetic film [ 9].\nExplicitly, we have\njx=/summationdisplay\nλ/bracketleftbig\nσλ\nxxEx+σλ\nxzEz/bracketrightbig\n(1)\njz=/summationdisplay\nλ/bracketleftbig\nσλ\nzxEx+σλ\nzzEz/bracketrightbig\n(2)\njs\nz=/summationdisplay\nλ/bracketleftbig\nσλ\nzxEx+σλ\nzzEz/bracketrightbig\nλ (3)\nwherejs\nzis the spin current flowing in the zdirection.\nSettingjz= 0 and eliminating Ez, we find the effec-\ntive spin Hall conductivity σSH=js\nz/Exand the effec-\ntive longitudinal conductivity ˜ σxx=jx/Exunder open\nboundary conditions:\nσSH=σzzσs\nzx−σs\nzzσzx\nσzz(4)\n˜σxx=σxxσzz−σ2\nxz\nσzz(5)\nwhereσαβ=/summationtext\nλσλ\nαβandσs\nαβ=/summationtext\nλλσλ\nαβ. We are\ninterested in the spin Hall angle under open boundary\nconditions defined as θSH=σSH/˜σxx.For the given crystallographic orientation of the film,\nthe charge and spin conductivity tensors in Eqs. ( 4)-(5)\nmay be expressed through their principal components.\nFor simplicity, let us assume the principal axes of σαβ\nandσs\nαβare the same, which is true if they are fixed\nby symmetry; this is the case for tetragonal, hexagonal,\nand orthorhombic lattices considered below. Further, let\nus also assume that one of the principal axes is oriented\nin the plane of the film along the yaxis. I denote the\nother two principal axes as 1 and 2 and the angle made\nby principal axis 1 with the zaxis asθ. Applying the\nrotation by angle θwith respect to the yaxis to the\ntensors expressed in their principal axes and substituting\nin Eqs. ( 4)-(5), we find, after some algebra:\nθSH=1\n2(β1−β2)sin2θ (6)\nwhereβi=σs\ni/σiis the conventionally defined [ 35,36]\nspin polarization of the conductivity along its principal\naxisi.\nEquation ( 6) shows that |θSH| ≤1, and the maximal\nvalue of 1 is reached if θ=π/4 andβ1=−β2= 1.\nThis maximal condition requires that the current along\nprincipal axes 1 and 2 is carried exclusively by spin-up\nand spin-down electrons, respectively, i.e., the ferromag-\nnet behaves as a half-metal with opposite spin polariza-\ntions along the two principal axes. While this condition\nis unlikely to be satisfied by any material, we will see\nbelow that values of |θSH|exceeding 0.1 are quite com-\nmon, which compares favorablywith the spin Hall angles\ntypical of heavy metals such as Pt [ 2].\nIn the relaxation-time approximation [ 37], the conduc-\ntivity tensor is given by\nσλ\nαβ=e2/summationdisplay\nn/integraldisplay\nvλ\nnαvλ\nnβτλ\nn∂f(Eλ\nn)\n∂µd3k\n(2π)3(7)\nwherevλ\nn(k),τλ\nn(k), andEλ\nn(k) are the group veloc-\nity, relaxation time, and energy of the Bloch band nof\nspinλ, andf(E) is the Fermi-Dirac distribution func-\ntion at chemical potential µfor the given temperature.\nThe relaxation-time approximation ( 7) is appropriate if\ndisorder-induced broadening does not lead to significant\noverlap between electronic bands [ 38].\nThe relaxation times generally depend on the kind of\nimpurities or other disorder present in the system, but,\nfor a typical good metal with an extended Fermi surface\nand localized disorder that is typical for sputtered mul-\ntilayers, it is reasonable to assume that τλ\nn(k) depends\nonly on spin λ. Thenτλcan be pulled out of the integral\nin (7), and we have σλ\nαβ=e2τλKλ\nαβ, where the tensor\nKλ\nαβdepends only on the electronic structure and can be\ncalculated using standard techniques.\nIntroducing the spin polarizations of the relaxation\ntime,Pτ= (τ↑−τ↓)/(τ↑+τ↓), and of the principal value\niof tensor K,Pi= (K↑\ni−K↓\ni)/(K↑\ni+K↓\ni), we find\nβi=Pτ+Pi\n1+PτPi(8)3\nand\nβ1−β2=(1−P2\nτ)(P1−P2)\n(1+P1Pτ)(1+P2Pτ). (9)\nEquation( 9)showsthatthe difference P1−P2, whichis\nan easily calculable band structure property, can be used\nasa descriptorto screenfor materialswith large |β1−β2|.\nWe should, however, bear in mind that this descriptor is\nonly adequate as long as Pτis not too large, in which\ncase|β1−β2|can be significantly reduced compared to\n|P1−P2|. On the other hand, if P1/ne}ationslash=−P2, there is a\nrange of Pτof a certain sign that makes |β1−β2|greater\nthan|P1−P2|.\nThe (isotropic) transport spin polarization βin ele-\nmental ferromagnets Fe, Co, and Ni alloyed with a small\namount of a second magnetic element varies drastically,\nincludingin sign, dependingonthe choiceofthatelement\n[36,39]. This suggests that magnetic impurities can be\nused to tune Pτin a wide range as long as scattering is\ndominated by these impurities.\nThespinpolarization Poftheisotropictensor Kinbcc\nFe, fcc Co, and fcc Ni, calculated as described in Section\nIII, is, respectively, 0.34, 0.05, and 0.02. The values for\nCo and Ni are small compared to the typical measured\nvalues of βin multilayer structures, such as 0.3–0.5 in\nCo or 0.5-0.8 in permalloy [ 36]. A theoretical calculation\nwith explicit treatment of substitutional disorder in fcc\nNi1−xFexalloys resulted in βexceeding 0.5 in the whole\ncompositional range [ 40]. This comparison suggests that\nβin these ferromagnets is often dominated by Pτ. How-\never, in searching for materials with a large anisotropy of\nβ, which, according to ( 6), is required for efficient trans-\nverse spin current generation, the anisotropy of Pis of\ncentral importance, according to ( 9). Therefore, in the\nfollowing I use the quantity ˜θij=1\n2(Pi−Pj) as a proxy\nfor the spin Hall angle of a film with geometry described\nbefore Eq. ( 6).\nIII. COMPUTATIONAL DETAILS\nUsing the Materials Project database [ 41], a set of\nexperimentally reported ferro- and ferrimagnetic mate-\nrials was selected using the following criteria: no more\nthan three constituent elements, no rare earths (with the\nexception of GdNi) or actinides, metallic and magnetic\nDFT ground state, at least one 3 datom from V to Ni,\nwith the tetragonal, hexagonal, or orthorhombic crystal\nstructure. This set was further pruned, removing materi-\nalswithnoexperimentallyreportedmagneticphasetran-\nsition, in part using the database supplied with Ref. 42.\nHalf-metallic, quasi-two-dimensional, and weakly ferro-\nmagnetic materials were also excluded, along with those\nwith a large number of atoms per unit cell. The result-\ning set is by no means complete, but it provides a broad,\nrepresentative selection.\nThe lattice structures were taken from the Materials\nProject database [ 41] and standardized using the spgliblibrary [43] with a large tolerance of 0.5% to remove the\noccasionalspurioussymmetryreductions. Thestructures\nwerethen optimized using the projector-augmentedwave\n(PAW)method[ 44], implementedintheViennaAbInitio\nSimulation Package (VASP) [ 45–47], using the general-\nized gradient approximation of Perdew-Burke-Ernzerhof\n(PBE) [48] for the exchange-correlation functional. The\noptimized lattice constants and magnetizations are listed\nin Table I.\nFor ferrimagnetic Mn 2Sb, it was found that the\nLDA+Umethod with U= 2.0 eV applied to the 3 dshell\noftheMnatomsinoctahedralsitessignificantlyimproves\nagreement with experimental data ( a= 4.03˚A,c= 6.53\n˚A,M= 1.66µB/f.u.) [49] for the lattice constants and\nthe magnetization. This calculation is labeled Mn 2Sb+U\nin Table I.\nThe tensor Kλ\nαβwas calculated using the smooth\nFourier interpolation technique [ 51] implemented in the\nBoltzTrap2 package [ 52], which was slightly modified to\nobtain spin-resolved information. After structural op-\ntimization, the band eigenvalues were calculated on a\nΓ-centered k-point mesh with the distance between the\nmeshpoints inthe directionofeachreciprocallatticevec-\ntorset to approximately0 .1˚A−1. The smoothFourierin-\nterpolation included approximately 20 times more stars\nof real-space lattice vectors than irreducible k-points. It\nwas checked that these parameters provided results for\n˜θijconverged to about 0.01.\nIV. SPIN HALL ANGLES\nTableIIlists the calculated values of ˜θij=1\n2(Pi−Pj)\nfor selected materials. Note that ˜θxy= 0 by symmetry\nin tetragonal and hexagonal magnets.\nSome of the largest values of |˜θij|are found in L1 0-\nordered MnGa (0.29), hexagonal MnSb (0.29), and or-\nthorhombic FeB (0.37) and Mn 3Sn2(0.35). These values\ncomparefavorablewith commonlyused heavy-metalspin\nHall sources, such as Pt and β-W [2,3].\nThe last two columns of Table IIinclude possible sub-\nstrates, selected with the help of the Materials Project\ndatabase [ 41], which can form a lattice-matched inter-\nface with the given material, and the corresponding min-\nimal co-incident area (MCIA) [ 53] for that epitaxy. All\nof these epitaxial relations have the ferromagnetic film\noriented so that there is no principal axis perpendicu-\nlar to the film plane, thereby allowing spin Hall effect\nfor a given orientation domain. Of course, there is no\nguarantee that these epitaxial relations can be realized\nin practice, and they should be regarded as illustrative\nexamples.4\nTABLE I. Optimized ( a,b,c) and experimental [ 50] (aexp,bexp,cexp) lattice constants ( ˚A) and calculated magnetization M\n(µB/f.u.) for selected materials. Tetragonal, hexagonal, and orthorhombic magnets are separated by horizontal lines. MP ID is\nthe Materials Project identifier [ 41]. Mn 2Sb+Uis Mn 2Sb treated with LDA+ U(see text).\nMaterial MPID abcaexpbexpcexpM\nFePt mp-2260 2.723 3.764 2.729 3.7133.28\nFePd mp-2831 2.708 3.770 2.722 3.7143.29\nCoPt mp-949 2.690 3.715 2.691 3.6842.27\nMnAl mp-771 2.755 3.476 2.772 3.572.32\nMnGa mp-1001836 2.705 3.660 2.748 3.6762.51\nFeNi mp-2213 2.514 3.577 2.533 3.5823.24\nFe2Bmp-1915 5.046 4.228 5.11 4.2493.67\nCo2Bmp-493 4.947 4.245 5.015 4.222.00\nMnAu 4mp-12565 6.520 4.045 6.45 4.044.16\nVAu4mp-1069697 6.507 4.035 6.4 3.981.95\nMn2Sbmp-20664 3.921 6.419 4.085 6.5341.28\nMn2Sb+Ump-20664 4.026 6.527 4.085 6.5341.60\nFe3Bmp-1181327 8.535 4.227 8.647 4.2825.82\nFe3Pmp-18708 9.042 4.374 9.107 4.4605.68\nFe8Nmp-555 5.681 6.223 5.718 6.28819.3\nMn2Ga5mp-607225 8.809 2.692 8.803 2.6944.41\nBe12Crmp-1590 7.205 4.113 7.238 4.1741.44\nMnAlGe mp-20757 3.882 5.922 3.89 5.9252.02\nFe5B2Pmp-9913 5.441 10.31 5.477 10.3458.52\nCrTe mp-794 4.128 6.273 4.005 6.2423.88\nMnBi mp-568382 4.307 5.738 4.305 6.1183.52\nFe2Pmp-778 5.800 3.432 5.877 3.4372.96\nFe3Nmp-1804 4.652 4.317 4.708 4.3886.17\nYCo5mp-2827 4.912 3.941 4.951 3.9757.03\nMnAs mp-610 3.665 5.498 3.710 5.6912.90\nMnSb mp-786 4.094 5.602 4.12 5.793.19\nZrFe2mp-1190681 4.970 8.076 4.952 16.123.07\nHfFe2mp-956096 4.944 8.027 4.968 8.0983.05\nFe3Gemp-21078 5.129 4.210 5.178 4.2266.42\nFe3Snmp-20883 5.468 4.302 5.458 4.3616.99\nYFe3mp-1192321 5.087 16.20 5.133 16.395.57\nFe5Si3mp-449 6.697 4.677 6.541 4.5587.62\nMn5Ge3mp-617291 7.121 4.948 7.184 5.05313.3\nMnP mp-2662 3.1505.1865.8423.1725.2585.9181.38\nFeB mp-20787 2.9423.9905.4052.9464.0535.4951.16\nMn3Sn2mp-600428 5.3987.4578.4205.5107.5658.5987.74\nFe3Bmp-973682 4.3655.4036.6434.3875.4376.716.08\nFe3Cmp-510623 4.4785.0336.7224.5125.0826.7425.58\nCo3Bmp-20373 4.3965.1456.6034.4185.2326.6363.53\nCo3Cmp-20925 4.4264.936.6864.4835.0336.7313.05\nGdNi mp-542632 3.78410.404.2113.77810.364.2217.39\nAlFe2B2mp-3805 2.91311.022.8472.92611.022.8712.63\nV. SPIN HALL DOMAINS\nAs will be discussed shortly below, many epitaxial re-\nlations allow more than one energetically degenerate ori-\nentation domain of the ferromagnetic film on the same\nsubstrate, characterized by different spin Hall conduc-\ntivity tensors. In a large-area film, degenerate orienta-\ntion domains should occur with equal probabilities, and\nmacroscopic physical responses correspond to statistical\naverages over these domains. Of course, several nonde-\ngenerate interface terminations may coexist in one sam-\nple, resulting in more than one set of degenerate orienta-\ntion domains occurring with different probabilities. ThissectiondiscussesmacroscopicspinHallresponseofamul-\ntidomain ferromagnetic film from the symmetry point of\nview. If this response survives after statistical averaging\nover orientation domains, it can be observed either us-\ning macroscopic probes over length scales exceeding the\norientation domain size or as an ensemble average over\nsingle-domain nanoscale devices where orientations do-\nmains occur at random.\nI start with general considerations and then consider\nthe case of uniform magnetization in Section VB, which\nimposes stringent symmetry requirements on the sub-\nstrate and the epitaxial relation. Then Section VC\nturnstothe caseofstrongmagnetocrystallineanisotropy,5\nTABLE II. Spin polarizations Piof the principal values of tensor Kλ\nαβand maximal exchange-driven spin Hall angles ˜θij\ncalculated assuming Pτ= 0 and θ=π/4 for selected magnetic materials. Tc(K) is the experimental Curie temperature.\nSubstrate/Magnet: a possible substrate, obtained using th e Materials Project [ 41], and its epitaxial relation, such that the\ninterface normal of the magnetic material is not oriented al ong any of the principal axes of its Kλ\nαβtensor. MCIA: minimal\nco-incident area ( ˚A2) for that epitaxy [ 53].Gsub: crystallographic point group of the substrate. Gbil(listed when Gsub=C1v):\ncrystallographic point group of the bilayer. Substrates al lowing a finite macroscopic spin Hall effect for a uniformly ma gnetized\nferromagnetic film ( Gsub=C1orGsub=Gbil=C1v) are highlighted in bold. All cases with Gsub=C1vandGbil=C1belong\nto case (3) of Sec. VB. a-TiO 2stands for anatase and TiO 2for rutile.\nMaterial TcPxPyPz˜θzx˜θzy˜θxySubstrate/Magnet MCIAGsubGbil\nFePt 753 0.34 0.24 −0.05 0NdGaO 3(011)/(101) 51C1vC1v\nFePd 1193 0.37 0.31 −0.03 0NdGaO 3(011)/(101) 51C1vC1v\nCoPt 846 0.23 0.07 −0.08 0YAlO 3(011)/(101) 51C1vC1v\nMnAl 650 0.05 0.55 0.25 0NdGaO 3(011)/(101) 49C1vC1v\nMnGa 629 0.07 0.64 0.29 0NdGaO 3(011)/(101) 49C1vC1v\nFeNi 842 0.27 0.36 0.04 0h-BN (0001)/(101) 11C6v\nFe2B1013 0.37 0.06 −0.16 0LiGaO 2(010)/(101) 33C1vC1\nCo2B 433 0.34 0.63 0.14 0LiGaO 2(010)/(101) 32C1vC1\nMnAu 4385 0.45 0.49 0.02 0GaSe (0001)/(101) 51C6v\nVAu4 60−0.54 −0.58 −0.02 0\nMn2Sb 580 0.50 0.15 −0.17 0MgF2(101)/(111) 78C1vC1\nMn2Sb+U580 0.35 −0.09 −0.22 0MgF2(101)/(111) 78C1vC1\nFe3B 828 0.47 0.70 0.12 0\nFe3P 693 0.70 0.77 0.03 0Fe2O3(0001)/(101) 91C3v\nFe8N1500 0.03 0.26 0.12 0BaTiO 3(110)/(101) 48C1vC1\nMn2Ga5450 0.47 0.20 −0.14 0TiO2(100)/(101) 83C2v\nBe12Cr 50−0.13 −0.54 −0.21 0\nMnAlGe 520 0.42 0.88 0.23 0a-TiO 2(101)/(101) 83C1vC1v\nFe5B2P628 0.23 0.17 −0.03 0GdScO 3(001)/(101) 64C2v\nCrTe 340 0.30 0.68 0.19 0LiAlO 2(110)/(10 ¯11) 90C2\nMnBi 633 0.86 0.42 −0.22 0o-WTe 2(001)/(10 ¯11) 89C2v\nFe2P 217 0.28 0.43 0.07 0LiF (110)/(10 ¯11) 71C2v\nFe3N 858 −0.14 −0.01 0.07 0C (0001)/(11 ¯21) 79C6v\nYCo5980 0.51 0.63 0.06 0GaN (10 ¯11)/(10¯11) 57C1vC1\nMnAs 318 0.75 0.29 −0.23 0GaN (10 ¯11)/(10¯11) 116C1vC1\nMnSb 851 0.91 0.33 −0.29 0MgF2(110)/(10 ¯11) 82C2v\nZrFe2630 −0.38 −0.41 −0.02 0\nHfFe2600 −0.26 −0.05 0.10 0MgO (100)/(11 ¯21) 72C4v\nFe3Ge 670 0.16 0.46 0.15 0Cu(100)/(10 ¯11) 94C4v\nFe3Sn 748 0.72 0.52 −0.10 0LaAlO 3(10¯10)/(10¯11)70C1vC1\nYFe3 552 −0.08 0.21 0.15 0\nFe5Si3383 0.51 0.20 −0.15 0GaAs (100)/(11 ¯21) 67C2v\nMn5Ge3296 0.54 0.66 0.06 0BaF2(100)/(11 ¯21) 76C4v\nMnP 2930.27−0.230.24−0.010.240.25MgO (100)/(110) 36C4v\nFeB 598−0.53−0.360.210.370.28−0.09SiO2(10¯10)/(110) 27C1\nMn3Sn2262−0.250.440.270.26−0.09−0.35CaF2(100)/(011) 61C4v\nFe3B 8280.660.400.690.020.140.13MgO (100)/(011) 37C4v\nFe3C 4800.25−0.010.460.110.240.13SiO2(10¯10)/(101) 81C1\nCo3B 7470.660.520.730.040.110.07ZrO2(101)/(101) 41C1vC1v\nCo3C 4980.380.380.490.050.060.00SiO2(10¯10)/(101) 79C1\nGdNi 780.10−0.11−0.19−0.14−0.040.10BaTiO 3(111)/(101) 58C1vC1\nAlFe2B22850.730.550.730.000.090.09CdS (0001)/(111) 46C6v\nwhich enables macroscopic spin Hall response even for\nhigh-symmetry substrates.\nA. General considerations\nLetGsubandGFMbe the (nonmagnetic) surface point\ngroups of the substrate alone and the ferromagneticlayeralone, respectively, and Gbilthe point group of the epi-\ntaxial bilayer. Gbilis a common subgroup of Gsuband\nGFM, and I assume Gbil=Gsub∩GFMwhen viewed as\na set. Of course, Gbildepends on the mutual orientation\nof the symmetry elements of GsubandGFM.\nNeglecting the non-symmorphic components of the\nsymmetry operations automatically incorporates statis-\ntical averaging over interface roughness [ 54]. This is the6\nreasonto considerpoint groupsratherthan spacegroups.\nBecause the ferromagnetic layer is assumed to have\ntiltedprincipalaxesoftheconductivitytensor,werequire\nthatGFMis at most C1v, which implies that Gbilis also\nat mostC1v.\nIfGbilis apropersubgroup of Gsub, the same sub-\nstrate supports several energetically degenerate crystal-\nlographic orientations of the ferromagnetic layer grown\non top of that substrate. The set of these degenerate\norientations is analogous to the orientation states of a\nferroic crystal [ 55]. Different orientation domains are\nmapped onto each other by the symmetry operations in\nGsub. The number of inequivalent orientation domains is\nequal to the index of GbilinGsub(see Theorem 3 of Ref.\n[55]), and they can be classified by the left cosets of Gbil\ninGsub.\nB. Uniform magnetization\nHereIdiscussstatisticalaveragingofthe spinHallcon-\nductivity over one set of degenerate orientation domains\nthat are all magnetized in the same direction, allowing\nthe spin degree of freedom to be treated as a scalar quan-\ntity. This case is realized when the magnetocrystalline\nanisotropy is weak, and the entire film is uniformly mag-\nnetized by an application of an external magnetic field.\nIn the most generalcase, the effective longitudinalcon-\nductivities ( 5) and the effective Hall conductivities of dif-\nferent orientation domains from the same degenerate set\nmay be different, leading to an inhomogeneous current\ndensity and electric field distribution. The inhomogene-\nity of the electric field over different orientation domains\ncan, in principle, leadtoafinite spinHallresponseevenif\nthe average spin Hall conductivity vanishes. I will disre-\ngard this possibility and assume that the effective charge\nconductivity is approximately isotropic.\nGeneralizing Eq. ( 4) to an arbitrary direction of the\nin-plane electric field, the spin Hall current flowing per-\npendicular to the film can be written as js\nz=pSHE,\nwhere (pSH)α=σs\nzα−σ−1\nzzσs\nzzσzαandα∈ {x,y}. The\nin-plane polar vector pSHdescribes the out-of-plane spin\nHall current of a given orientation domain and is invari-\nant under GFM. In the case considered in the derivation\nofEq. (6), when one principal axisof ˆ σλlies in the plane,\npSHis perpendicular to that axis.\nThe star of vectors p(i)\nSHdescribing different orienta-\ntion domains ibelonging to the same degenerate set is\ngenerated by the symmetry operations from Gsub. The\nmacroscopic spin Hall response is given by an average\nover this star. For it to be finite, it is necessary that\nGsuballows an in-plane polar vector, which is only the\ncase forGsub=C1orC1v. ForGsub=C1there is only\none orientation domain, and it is sufficient that pSH/ne}ationslash= 0\nfor it.\nForGsub=C1v, letσvbe the mirror plane of the Gsub\ngroup. There are three possibilities:1.Gbil=Gsub=C1v. There is only one orientation\ndomain because Gbilis not a proper subgroup of\nGsub. Because we have already assumed that GFM\nis at most C1v, andGbil=C1vis its subgroup,\nit must be that GFM=Gsubwith the same σv.\nTherefore, pSH/bardblσvand finite.\n2.Gbil=C1andpSHhas a finite projection on σv.\nThis means that either GFM=C1orGFM=C1v\nbut its symmetry plane is not orthogonal to σvof\nGsub. Therearetwoorientationdomainsbut p(1)\nSH+\np(2)\nSH/ne}ationslash= 0, and the microscopic average is finite.\n3.Gbil=C1andpSH⊥σv. This happens if\nGFM=C1vwith the symmetry plane orthogonal\ntoσvofGsub. There are two orientation domains\nwithp(1)\nSH=−p(2)\nSH, and the microscopic average\nvanishes.\nThus, a finite macroscopic spin Hall current is allowed\nforGsub=C1and only in cases (1) and (2) for Gsub=\nC1v. Tomaximizethiscurrent,theelectricfieldshouldbe\nalignedparallelto the averageof p(i)\nSHoverthe orientation\ndomains.\n1. Substrate examples\nThe point groups Gsubare listed in Table IIfor all\nsubstrates. Many of them are higher than C1v, which\nguarantees that the macroscopic spin Hall conductivity\nvanishes after averaging over uniformly magnetized de-\ngenerate orientation domains.\nA generic orientation of the surface plane would re-\nsult inGsub=C1for any crystalline substrate and allow\nmacroscopic spin Hall effect, but this case is uncommon\nfor readily available substrates. In Table IIit is realized\nby SiO 2(10¯10), which may be matched with orthorhom-\nbic FeB (110), Fe 3C (101), or Co 3C (101). The D3bulk\npoint group of SiO 2has no mirror planes, and the (10 ¯10)\nplane (also known as the Y-cut) is not orthogonal to a\nsymmetry axis.\nFor substrates with Gsub=C1v, we need to dis-\ntinguish among the three cases listed above, and Ta-\nbleIIalso includes Gbil. Substrates corresponding to\nCase (1), which allows macroscopic spin Hall effect, have\nGsub=Gbil=C1vand are highlighted in bold in Ta-\nbleII. The largest values of ˜θαβestimated for these sys-\ntems are in L1 0-ordered MnAl (101) and MnGa (101) on\nNdGaO 3(011) and for MnAlGe (101) on anatase TiO 2\n(101) [56]. Note that the L1 0-ordered τ-phase of MnAl\nis metastable but can be stabilized by allowing with Ga\n[57]. NdGaO 3(011) and YAlO 3(011) substrates [ 58]\npresent an interesting case where the C3vpoint group\nof the (111) surface of the parent perovskite structure is\nbroken by the bulk orthorhombic distortion, resulting in\nGsub=C1v.\nCase (2) for Gsub=C1vrequires that the symmetry\nplanes of the substrate and the ferromagnet are neither7\nparallel nor orthogonal to each other. I have not found\nexamples of this case.\nCase (3) for Gsub=C1v, which results in zero macro-\nscopic spin Hall conductivity, is rather common and is\nrealized, for example, in (101)-oriented FePt or MnGa\ngrown on rutile TiO 2(101) and in all other cases in Ta-\nbleIIwithGsub=C1vthat are not highlighted.\nC. Case of strong magnetocrystalline anisotropy\nIn this case, the magnetizations in different orientation\ndomains are generally not collinear with each other. As I\nshow below, this allows one to obtain a finite macroscop-\nically averaged spin Hall current even if Gsubincludes a\nhigh-order symmetry axis, and the spin polarization of\nthat current can be switched both in sign and in orienta-\ntion by changing the direction of the initializing external\nmagnetic field and the direction of the current flow.\nBecause there is no longer a globally defined direction\nof magnetization, we need to restore the vector character\nof the spin polarization when averaging over the orien-\ntation domains. I will assume that the magnetization is\nuniform inside each orientation domain, which is reason-\nable if the domain size is large compared to the magnetic\ndomain wall width. The out-of-plane spin currentfor ori-\nentation domain ican be written as j(i)\ns=ˆ m(i)(p(i)\nSH·E),\nwhere the unit vector ˆ m(i)denotes the direction of the\nmagnetization in orientation domain i. The subscript z\nfor the direction of spin current flow has been omitted\nbut is implied throughout. The macroscopic average of\nthe 3×2 spin conductivity tensor is given by the ten-\nsor ˆσSH=/summationtext\niˆ m(i)⊗p(i)\nSH/Ni, whereNiis the number of\norientation domains.\nTo be specific, let us consider the case of dominant\nbulk easy-axis anisotropy in a tetragonal or hexagonal\nferromagnet with GFM=C1v. In this case the magneti-\nzation lies in the plane containing the surface normal and\nthe bulk fourfold or sixfold (or threefold) axis, which, as\nabove, is tilted by angle θwith respect to the zaxis. The\nangle made by ˆ m(i)with the zaxis will be denoted as\nθm, which may differ from θdue to the demagnetizing\nfield but is the same for all i, at least once the external\nfield has been switched off.\nAlong with the p(i)\nSHvector, the operations in Gsubalso\ntransform the orientation of the easy magnetization axis\nwhile keeping θunchanged. First, assume the sample is\nmagnetized by an out-of-plane magnetic field B=Bzˆz,\nso thatm(i)\nzis the same in all orientation domains. Be-\ncausem(i)\nzbehavesasascalarunder Gsub, thez-polarized\nspin current behaves exactly as described in Section VB\nfor the case of uniform magnetization.\nOn the other hand, the in-plane projection m(i)\n/bardbltrans-\nforms like a polar vector under Gsub, similar to p(i)\nSH.\n(Thatm(i)\n/bardblformally behaves here as a polar rather than\naxial vector is enforced by the external field which selectsthe same sign of m(i)\nzfor all orientation domains.) Our\nassumption of GFM=C1vleads to m(i)\n/bardbl/bardblp(i)\nSH. Thus,\nthe spin Hall conductivity ˆ σ/bardbl\nSHgiving the out-of-plane\nflow of in-plane-polarized spins is equal to the average\nof±pSHˆp(i)\nSH⊗ˆp(i)\nSHsinθmover all orientation domains,\nwhere ˆp(i)\nSH=p(i)\nSH/pSHand the overall sign is that of\nm(i)\n/bardbl·p(i)\nSH, which in turn depends on the sign of Bz. The\naverage over isurvives under anypoint group Gsub. If\nGsubincludes a threefold or higher rotation axis, ˆ σ/bardbl\nSH\nreduces to a scalar σ/bardbl\nSH=±1\n2pSHsinθm. Thus, we find\naninterestingsituationwherethe spin polarizationofthe\nmacroscopicallyaveragedspin currentflowingin the zdi-\nrection is locked parallel to the direction of the in-plane\ncharge current flow. This is in contrast to the conven-\ntional spin-orbit-driven spin Hall effect in nonmagnetic\nmaterials where the spin current would be polarized per-\npendicular to the current flow direction.\nThe situation is more complicated if the system is ini-\ntialized by an in-plane magnetic field B, rather than\nout-of-plane as above. The spin Hall current depends\nonGsub, on the orientation of the initializing magnetic\nfield, and on the orientation of the charge current flow,\nwith respect to the symmetry elements of GsubandGFM.\nFor example, consider Gsub=C4vandGFM=C1v, and\nalign the xaxis parallel to the coincident mirror planes\nofGsubandGFMfor one of the orientation domains.\nThere are four orientation domains with p(i)\nSHoriented\nalong the positive and negative directions of the xandy\naxes. For each domain, it is assumed the magnetization\npicks one of the two directions along its easy axis such\nthatm(i)B>0. Then the averaged spin Hall current\nis strictly z-polarized: js=1\n2ˆzpSHcosθm(±1,±1)·E,\nwhere the signs for the vector components in brackets\nare such that this vector lies in the same quadrant with\nB.\nAnother interesting example is when Bis applied\nin thexdirection with a small tilt toward z. In\nthis case, the average spin Hall current is js=\n1\n2pSH(0,Eysinθm,Excosθm), and its polarization can be\nswitched between in-plane and out-of-plane by changing\nthe direction of the current flow.\nAs a function of the orientation of the initializing mag-\nnetic field B, the spin Hall conductivity tensor has dis-\ncontinuities along the contours on the sphere where one\nor more orientation domains switch their magnetization.\nAfter initialization, the measurement may be performed\nin the remanent state.\nVI. SUMMARY\nA crystalline ferromagnetic (or ferrimagnetic) film\nwith tilted principal axes of the conductivity tensor\ncan exhibit exchange-driven spin Hall effect, generating\nspin-polarized current perpendicular to the film surface,\nthanks to the anisotropyof the conductivity tensor. This8\neffect is analogous to the spin-splitting current in alter-\nmagnets and does not require spin-orbit coupling. First-\nprinciples screening of 41 tetragonal, hexagonal, and or-\nthorhombic ferromagnets revealed that, for certain crys-\ntallographic orientations, the spin Hall angle is greater\nthan 0.1 in 26 of them, and greater than 0.2 in 12.\nTo allow the use of this exchange-driven spin Hall ef-\nfect on a macroscopic scale, the response should not van-\nish after averaging over the crystallographic orientation\ndomains of the ferromagnetic film. For a uniformly mag-\nnetized sample, this is possible only if the substrate has\nlow enough symmetry ( C1orC1v); moreover, for a C1v\nsubstrate, the ferromagnetic film should not be aligned\nin such a way that two degenerate orientation domains\nhave opposite spin Hall angles. If these conditions are\nnot met, the macroscopic spin Hall conductivity of a\nuniformly magnetized film vanishes after averaging over\ndegenerate orientation domains. Several cases listed in\nTableIIdo meet these conditions, most notably MnAl\n(101) and MnGa (101) on NdGaO 3(011) and MnAlGe\n(101) on anatase TiO 2(101), with rather large estimated\nspin Hall angles. In all systems listed in Table IIthe spin\nHall effect may be observed locally for each orientation\ndomain, for example, with a spatially resolved measure-\nment of the magneto-optic Kerr effect.\nThe spin Hall effect can alsobe observedon the macro-\nscopic scale, regardless of the substrate symmetry, if\nthe ferromagneticfilm hasmagnetocrystallinganisotropy\nstrong enough to lock the magnetization to the crystal-\nlographic easy axis in each orientation domain. Multi-\nple possibilities exist depending on the crystallographic\nsymmetryand the orientationofthe initializing magnetic\nfield, including the averaged spin Hall current being po-larized parallel to the current flow or perpendicular to\nthe film plane.\nExchange-driven spin Hall effect in ferromagnets ex-\npands the range of available spin Hall sources for appli-\ncations in magnetic heterostructures. In particular, one\ncan envision a trilayer heterostructure in which the fer-\nromagnetic source layer is separated from the detector\nlayer by a nonmagnetic spacer, such as Cu, which is suf-\nficientlythicktoeliminatetheexchangecouplingbetween\nthe layers. If the source ferromagnet has strong magne-\ntocrystalline anisotropy, the spin Hall current can have\na large out-of-plane component, facilitating the switch-\ning of the detector layer with perpendicular magnetiza-\ntion; this opportunity is advantageous for applications\nin magnetic random-access memory. In contrast to an\naltermagnet, the source ferromagnetic layer can be de-\nterministically initialized by an external magnetic field.\nIntermsofitsdevicestructure, theferromagnetictrilayer\nspin Hall device is similar to a spin-orbit torque bilayer\n[3], where, in contrast with spin-transfer torque devices,\nthe read and write current paths are separated.\nACKNOWLEDGMENTS\nI thank Ilya Krivorotov for fruitful discussions and\nMark Stiles, Evgeny Tsymbal, Igor Mazin, Vivek Amin,\nAlexey Kovalev, and Xin Fan for useful comments about\nthe manuscript. This work was supported by the Na-\ntional Science Foundation through Grants No. DMR-\n1916275 and DMR-2324203. Calculations were per-\nformed utilizing the Holland Computing Center of the\nUniversity of Nebraska, which receives support from the\nNebraska Research Initiative.\n[1] M. Dyakonov and V. Perel, Current-induced\nspin orientation of electrons in semiconductors,\nPhys. Lett. A 35, 459 (1971) .\n[2] J. Sinova, S. O. Valenzuela, J. Wunderlich,\nC. H. Back, and T. Jungwirth, Spin Hall effects,\nRev. Mod. Phys. 87, 1213 (2015) .\n[3] A. Manchon, J. ˇZelezn´ y, I. M. Miron, T. Jung-\nwirth, J. Sinova, A. Thiaville, K. Garello, and\nP. Gambardella, Current-induced spin-orbit torques\nin ferromagnetic and antiferromagnetic systems,\nRev. Mod. Phys. 91, 035004 (2019) .\n[4] Q. Shao, P. Li, L. Liu, H. Yang, S. Fukami, A. Razavi,\nH. Wu, K. Wang, F. Freimuth, Y. Mokrousov, M. D.\nStiles, S. Emori, A. Hoffmann, J. ˚Akerman, K. Roy, J.-P.\nWang, S.-H. Yang, K. Garello, and W. Zhang, Roadmap\nof spin–orbit torques, IEEE Trans. Magn. 57, 1 (2021) .\n[5] C. Song, R. Zhang, L. Liao, Y. Zhou, X. Zhou, R. Chen,\nY. You, X. Chen, and F. Pan, Spin-orbit torques: Mate-\nrials, mechanisms, performances, and potential applica-\ntions,Prog. Mater. Sci. 118, 100761 (2021) .\n[6] B. F. Miao, S. Y. Huang, D. Qu, and C. L. Chien,\nInverse spin Hall effect in a ferromagnetic metal,\nPhys. Rev. Lett. 111, 066602 (2013) .[7] H. Wang, C. Du, P. Chris Hammel, and F. Yang,\nSpin current and inverse spin Hall effect in ferromag-\nnetic metals probed by Y 3Fe5O12-based spin pumping,\nAppl. Phys. Lett. 104, 202405 (2014) .\n[8] M. Seemann, D.K¨ odderitzsch, S.Wimmer,andH.Ebert,\nSymmetry-imposed shape of linear response tensors,\nPhys. Rev. B 92, 155138 (2015) .\n[9] T. Taniguchi, J. Grollier, and M. D. Stiles,\nSpin-transfer torques generated by the anoma-\nlous Hall effect and anisotropic magnetoresistance,\nPhys. Rev. Appl. 3, 044001 (2015) .\n[10] D. Tian, Y. Li, D. Qu, S. Y. Huang, X. Jin, and\nC. L. Chien, Manipulation of pure spin current in\nferromagnetic metals independent of magnetization,\nPhys. Rev. B 94, 020403 (2016) .\n[11] H. Wu, X. Wang, L. Huang, J. Qin, C. Fang, X. Zhang,\nC. Wan, and X. Han, Separation of inverse spin Hall ef-\nfect andanomalous Nernsteffectinferromagnetic metals,\nJ. Magn. Magn. Mater. 441, 149 (2017) .\n[12] K. S. Das, W. Y. Schoemaker, B. J. van Wees, and\nI. J. Vera-Marun, Spin injection and detection via the\nanomalous spin Hall effect of a ferromagnetic metal,\nPhys. Rev. B 96, 220408 (2017) .9\n[13] J. D. Gibbons, D. MacNeill, R. A. Buhrman,\nand D. C. Ralph, Reorientable spin direction for\nspin current produced by the anomalous Hall effect,\nPhys. Rev. Appl. 9, 064033 (2018) .\n[14] A. Bose, D. D. Lam, S. Bhuktare, S. Dutta, H. Singh,\nY. Jibiki, M. Goto, S. Miwa, and A. A. Tulapurkar, Ob-\nservation of anomalous spin torque generated by a ferro-\nmagnet, Phys. Rev. Appl. 9, 064026 (2018) .\n[15] Y. Omori, E. Sagasta, Y. Niimi, M. Gradhand, L. E.\nHueso, F. Casanova, andY. Otani, Relation betweenspin\nHall effect and anomalous Hall effect in 3 dferromagnetic\nmetals,Phys. Rev. B 99, 014403 (2019) .\n[16] V. P. Amin, J. Li, M. D. Stiles, and P. M.\nHaney, Intrinsic spin currents in ferromagnets,\nPhys. Rev. B 99, 220405 (2019) .\n[17] A. Davidson, V. P. Amin, W. S. Aljuaid, P. M.\nHaney, and X. Fan, Perspectives of electrically\ngenerated spin currents in ferromagnetic materials,\nPhys. Lett. A 384, 126228 (2020) .\n[18] L. Salemi and P. M. Oppeneer, Theory of magnetic spin\nand orbital Hall and Nernst effects in bulk ferromagnets,\nPhys. Rev. B 106, 024410 (2022) .\n[19] A. Mook, R. R. Neumann, A. Johansson, J. Henk,\nand I. Mertig, Origin of the magnetic spin Hall\neffect: Spin current vorticity in the Fermi sea,\nPhys. Rev. Res. 2, 023065 (2020) .\n[20] J. ˇZelezn´ y, Y. Zhang, C. Felser, and B. Yan,\nSpin-polarized current in noncollinear antiferromagnets ,\nPhys. Rev. Lett. 119, 187204 (2017) .\n[21] M. Kimata, H. Chen, K. Kondou, S. Sugimoto, P. K.\nMuduli, M. Ikhlas, Y. Omori, T. Tomita, A. H. MacDon-\nald, S. Nakatsuji, and Y. Otani, Magnetic and magnetic\ninverse spin Hall effects in a non-collinear antiferromag-\nnet,Nature565, 627 (2019) .\n[22] J. Holanda, H. Saglam, V. Karakas, Z. Zang, Y. Li,\nR. Divan, Y. Liu, O. Ozatay, V. Novosad, J. E. Pear-\nson, and A. Hoffmann, Magnetic damping modulation\nin IrMn 3/Ni80Fe20via the magnetic spin Hall effect,\nPhys. Rev. Lett. 124, 087204 (2020) .\n[23] K. Kondou, H. Chen, T. Tomita, M. Ikhlas, T. Higo,\nA. H. MacDonald, S. Nakatsuji, and Y. Otani, Gi-\nant field-like torque by the out-of-plane magnetic\nspin Hall effect in a topological antiferromagnet,\nNat. Commun. 12, 6491 (2021) .\n[24] S. Hu, D.-F. Shao, H. Yang, C. Pan, Z. Fu, M. Tang,\nY. Yang, W. Fan, S. Zhou, E. Y. Tsymbal, and X. Qiu,\nEfficient perpendicular magnetization switching by a\nmagnetic spin Hall effect in a noncollinear antiferromag-\nnet,Nat. Commun. 13, 4447 (2022) .\n[25] R. K. Han, X. P. Zhao, H. R. Qin, H. L. Sun, H. L.\nWang, D. H. Wei, and J. H. Zhao, Field-free magnetiza-\ntion switching in CoPt induced by noncollinear antifer-\nromagnetic Mn 3Ga,Phys. Rev. B 107, 134422 (2023) .\n[26] D. Litvin and W. Opechowski, Spin groups,\nPhysica 76, 538 (1974) .\n[27] R. Gonz´ alez-Hern´ andez, L. ˇSmejkal, K. V´ yborn´ y,\nY. Yahagi, J. Sinova, T. Jungwirth, and\nJ.ˇZelezn´ y, Efficient electrical spin splitter based\non nonrelativistic collinear antiferromagnetism,\nPhys. Rev. Lett. 126, 127701 (2021) .\n[28] H. Bai, L. Han, X. Y. Feng, Y. J. Zhou, R. X.\nSu, Q. Wang, L. Y. Liao, W. X. Zhu, X. Z. Chen,\nF. Pan, X. L. Fan, and C. Song, Observation of spin\nsplitting torque in a collinear antiferromagnet RuO 2,Phys. Rev. Lett. 128, 197202 (2022) .\n[29] S. Karube, T. Tanaka, D. Sugawara, N. Kadoguchi,\nM. Kohda, and J. Nitta, Observation of spin-\nsplitter torque in collinear antiferromagnetic RuO 2,\nPhys. Rev. Lett. 129, 137201 (2022) .\n[30] A. Bose, N. J. Schreiber, R. Jain, D.-F. Shao, H. P.\nNair, J. Sun, X. S. Zhang, D. A. Muller, E. Y. Tsymbal,\nD.G.Schlom,andD.C.Ralph,Tiltedspincurrentgener-\nated by the collinear antiferromagnet ruthenium dioxide,\nNat. Electron. 5, 267 (2022) .\n[31] L.ˇSmejkal, J. Sinova, and T. Jungwirth, Beyond conven-\ntional ferromagnetism and antiferromagnetism: A phase\nwith nonrelativistic spin and crystal rotation symmetry,\nPhys. Rev. X 12, 031042 (2022) .\n[32] L. ˇSmejkal, J. Sinova, and T. Jungwirth,\nEmerging research landscape of altermagnetism,\nPhys. Rev. X 12, 040501 (2022) .\n[33] I. Turek, Altermagnetism and magnetic\ngroups with pseudoscalar electron spin,\nPhys. Rev. B 106, 094432 (2022) .\n[34] T. Valet and A. Fert, Theory of the perpen-\ndicular magnetoresistance in magnetic multilayers,\nPhys. Rev. B 48, 7099 (1993) .\n[35] I. ˇZuti´ c, J. Fabian, and S. Das Sarma,\nSpintronics: Fundamentals and applications,\nRev. Mod. Phys. 76, 323 (2004) .\n[36] J. Bass, CPP magnetoresistance of mag-\nnetic multilayers: A critical review,\nJ. Magn. Magn. Mater. 408, 244 (2016) .\n[37] J. M. Ziman, Electrons and Phonons: The Theory\nof Transport Phenomena in Solids (Oxford University\nPress, 1960) Chap. VII.\n[38] E. Y. Tsymbal and D. G. Pettifor, Effects of band struc-\nture and spin-independent disorder on conductivity and\ngiant magnetoresistance in Co/Cu and Fe/Cr multilay-\ners,Phys. Rev. B 54, 15314 (1996) .\n[39] I. Campbell and A. Fert, Transport properties of ferro-\nmagnets, in Handbook of Ferromagnetic Materials ,Vol.3\n(Elsevier, 1982) pp. 747–804.\n[40] A. A. Starikov, P. J. Kelly, A. Brataas,\nY. Tserkovnyak, and G. E. W. Bauer, Unified\nfirst-principles study of Gilbert damping, spin-flip\ndiffusion, and resistivity in transition metal alloys,\nPhys. Rev. Lett. 105, 236601 (2010) .\n[41] A.Jain, S.P. Ong, G. Hautier, W. Chen, W. D.Richards,\nS. Dacek, S. Cholia, D. Gunter, D. Skinner, G. Ceder,\nand K. A. Persson, Commentary: The Materials Project:\nA materials genome approach to accelerating materials\ninnovation, APL Mater. 1, 011002 (2013) .\n[42] C. J. Court and J. M. Cole, Data descriptor: Auto-\ngenerated materials database of Curie and N´ eel tem-\nperatures via semi-supervised relationship extraction,\nSci. Data 5, 180111 (2018) .\n[43] A. Togo and I. Tanaka, Spglib: a software library\nfor crystal symmetry search (2018), version 2.0.2,\narXiv:1808.01590 .\n[44] P. E. Bl¨ ochl, Projector augmented-wave method,\nPhys. Rev. B 50, 17953 (1994) .\n[45] G. Kresse and J. Furthm¨ uller, Efficiency of ab-\ninitio total energy calculations for metals and\nsemiconductors using a plane-wave basis set,\nComput. Mater. Sci. 6, 15 (1996) .\n[46] G. Kresse and J. Furthm¨ uller, Efficient iterative schem es\nfor ab initio total-energy calculations using a plane-wave10\nbasis set, Phys. Rev. B 54, 11169 (1996) .\n[47] G. Kresse and D. Joubert, From ultrasoft pseu-\ndopotentials to the projector augmented-wave method,\nPhys. Rev. B 59, 1758 (1999) .\n[48] J. P. Perdew, K. Burke, and M. Ernzerhof, Gen-\neralized gradient approximation made simple,\nPhys. Rev. Lett. 77, 3865 (1996) .\n[49] F. J. Darnell, W. H. Cloud, and H. S. Jarrett, X-\nray and magnetization studies of Cr-modified Mn 2Sb,\nPhys. Rev. 130, 647 (1963) .\n[50] C. R. Groom, I. J. Bruno, M. P. Lightfoot, and\nS. C. Ward, The Cambridge Structural Database,\nActa Crystallogr. Sect. B 72, 171 (2016) .\n[51] W. E. Pickett, H. Krakauer, and P. B. Allen,\nSmooth Fourier interpolation of periodic functions,\nPhys. Rev. B 38, 2721 (1988) .\n[52] G. K. H. Madsen, J. Carrete, and M. J. Verstraete,\nBoltzTraP2, a program for interpolating band struc-\nturesandcalculating semi-classical transport coefficient s,Comput. Phys. Commun. 231, 140 (2018) .\n[53] A. Zur and T. C. McGill, Lattice match: An application\nto heteroepitaxy, J. Appl. Phys. 55, 378 (1984) .\n[54] K. D. Belashchenko, Equilibrium magnetization at\nthe boundary of a magnetoelectric antiferromagnet,\nPhys. Rev. Lett. 105, 147204 (2010) .\n[55] K. Aizu, Possible species of ferromagnetic, ferroelec tric,\nand ferroelastic crystals, Phys. Rev. B 2, 754 (1970) .\n[56] The θangles corresponding to the epitaxial relations\nlisted in Table IIare notπ/4, but the effect of this dif-\nference is usually minor. For example, for MnAl (101) we\nhaveθ≈63◦, and the estimate of ˜θzxis reduced by a\nfactor sin2 θ≈0.81.\n[57] T. Mix, F. Bittner, K.-H. M¨ uller, L. Schultz, and\nT. Woodcock, Alloying with a few atomic per-\ncent of Ga makes MnAl thermodynamically stable,\nActa Mater. 128, 160 (2017) .\n[58] The notation for NdGaO 3, YAlO 3, LiGaO 2, and GdSc O3\nassumes the orthorhombic axes are ordered by increasing\nlattice constant."    },    {        "title": "2311.04099v1.Magnetism_in_AV3Sb5__A___Cs__Rb__K___Complex_Landscape_of_the_Dynamical_Magnetic_Textures.pdf",        "content": "1 \n Magnetism in AV 3Sb5 (A = Cs, Rb, K) : Complex Landscape  of the \nDynamical Magnetic Textures  \n \nDebjani Karmakar1,2,#,*, Manuel Pereiro1,#, Md. Nur Hasan3, Ritadip Bharati4, Johan \nHellsvik5, Anna Delin6,7, Samir Kumar Pal3, Anders Bergman1, Shivalika Sharma8, Igor Di \nMarco1,8,9, Patrik Thunström1, Peter M. Oppeneer1 and Olle Eriksson1,*  \n \n1 Department of Physics and Astronomy,  \nUppsala University, Box 516, SE -751 20, Uppsala, Sweden  \n \n2 Technical Physics Division  \nBhabha Atomic Research Centre , Mumbai 400085, India  \n \n3 Department of Chemical and Biological Sciences,  \nS. N. Bose National Centre for Basic Sciences,  \nBlock JD, Sector -III, SaltLake, Kolkata 700 106, India  \n \n4 School of Physical Sciences  \nNational Institute of Science Education and Research  \nHBNI, Jatn i - 752050, Odisha, India  \n \n5 PDC Center for High Performance Computing,  \nKTH Royal Institute of Technology, SE -100 44 Stockholm, Sweden  \n \n6 Department of Applied Physics  \nKTH Royal Institute of Technology , SE-106 91 Stockholm, Sweden  \n \n7 Swedish e -Science Re search Center (SeRC),  \nKTH Royal Institute of Technology, SE -10044 Stockholm, Sweden  \n \n8Asia Pacific Center for Theoretical Physics,  \nPohang, 37673, Republic of Korea  \n \n9 Department of Physics, Pohang University of Science and Technology,  \nPohang, 37673, Republic of Korea  \n \n \n \n \n# D.K. and M.P. equally contributed  \n \n*Corresponding Authors:  \nDebjani Karmakar  \nEmail: debjani.karmakar@physics.uu.se  \n \nOlle Eriksson  \nEmail: olle.eriksson@physics.uu.se  \n 2 \n Abstract  \n \nWe have investigate d the dynamical magnetic properties of the V -based kagome stibnite \ncompounds by combining the ab-initio  calculated magnetic parameters of a spin Hamiltonian  \nlike inter-site exchange  parameters , magnetocrystalline anisotropy and site projected magnetic \nmoments , with full-fledged  simulations of atomistic spin -dynamic s. Our calculations reveal \nthat in addition to  a ferromagnetic order along the [001] direction, the system hosts a complex \nlandscape of magnetic configuration s comprised of commensurate and incommensurate spin-\nspiral s along the [010] direction. The p resence of such  chiral magnetic textures  may be the key \nto solve the mystery  about the orig in of the experimentally observed  inherent  breaking of the \nC6 rotational symmetry - and the time-reversal symmetry .  \n \n  3 \n For the V-based kagome stib nites , AV 3Sb5 (A=Cs, Rb, K), a complete picture  of the origin  of \nthe experimental ly observe d spontaneously broken  time-reversal as well as  C6 rotational \nsymmetries  and their respective interplay with the underlying magnetic  structure is missing . \nThis lacuna  has proliferated  into wide variations of inconclusive explanations towards the \nexperimental results like muon spin -rotation (SR) or anomalous Hall Effect  (AHE) [1,2]. In \nthe zero -field SR spectra, the depolarization function of single crystals of these  materials \ndeviate from the standard  Gaussian Kubo -Toyabe behavior , for which the possible reaso ns may \nbe (a) an inhomogeneous distribution of the nuclear moments, (b) the presence of an electric \nfield gradient due to a chiral charge -order (CO) or (c) a contribution of purely electronic origin  \nemanating  from the chiral distribution of the spin moments  [1]. Below the CO transition and at \nhigh-field, the significant increase of the rate of spin-relaxation  of muons  and the enhanced \nspread of the internal field s are suggested to be the experimental signatures of a broken time -\nreversal symmetry  enrooted to the ir electronic attribute s [1]. In the same vein , the \nexperimentally observed giant AHE for this series exhibits an anomalous Hall ratio, which is \nan order of magnitude higher than that of intrinsic magnetic system s like bcc Fe [3]. The nature \nof AHE is proposed  to be of extrinsic type  related to  the spin-fluctuation s across  the triangular \nV-clusters  leading to  an enhanced skew scattering [2]. Currently proposed microscopic \nmechanis ms behind the nature of the SR spectra, the giant AHE and their interconnection \nwith the broken C 6 rotational - and time -reversal symmetries  are far from conclusive, suggesting  \na wide varieties of reasons, viz., the presence of orbital current, electron nematicity or a 2×2 \ncharge modulation  producing  chiral charge density wave s [1-3]. The effect s of spin -density \nwave, complex spin -texture and dynamical magnetic ground state s on the CO and the broken \nsymmetries therein are yet to get analyzed . \nIn recent literatures, there are controversies regarding the presence of broken time -reversal \nsymmetry in  the first member of this kagome superconducting series, viz., CsV 3Sb5. Saykin et 4 \n al. [4] have  used high resolution polar Kerr effect measurements to demonstrate an absence of \nany observable Kerr effect in CsV 3Sb5, which poses a question -mark against the presence of \nspontaneous breaking of time -reversal symmetry in this compound. On the other hand, a lbeit \nbeing centrosymmetric, this system reveals a chiral transport behavior via second -harmonic \ngeneration under an in -plane applied magnetic field  [5] implyi ng an inherently broken mirror \nsymmetry . Such occurrence of electronic magnetochiral anisotropy is prominent below 35K \nand th ereby  indicates interplay of time -reversal symmetry breaking and electronic chirality  in \nCsV 3Sb5 below this transition temperature .  \nThe t wo-dimen sional (2D) kagome lattice with nearest neighbor (NN) antiferromagnetic \n(AFM) exchange interactions ( J) is a well-known archetype of inherently frustrated systems , \nwhere, it is impossible to optimize all exchange interactions . Here, after including  diverse  NN \ninteractions , panoply  of degenerate magnetic ground states  may arise , where the out -of-plane \nAFM  or ferrimagnetic  order competes with the co -planar 1200 structures  with uniform or \nstaggered chirality [ 6, 7]. Depending on the temperature  (T), their magnetic ground states \nencompass  three different regimes, viz., i) a spin-diffusive paramagnetic regime ( T > |J|), ii) an \nintermediate cooperative regime leading to a spin-liquid arrangement  (0.001| J| < T < |J|) and \niii) an ultralow temperature coplanar regime ( T < 0.001| J|). The first crossover at T ~ J may \nhave a finite local spin-correlation without any long -range order (LRO) and for the second one \nat T ~ 10-3J, the entropic selection favors a coplanar order with anisotropic dynamics [8-11]. \nFor the coplanar configuration, thermal or quantum fluctuations resolve the  ground -state \ndegeneracy after stabilizing  a LRO  in the order by disorder  process [ 11], the most common of \nwhich is the 1200 spin-texture.  With increasing temperature, such chiral LRO is interven ed \nafter generation of weathervane defects leading to the rotational staggering of spins around the \nlocal spin -axis [ 6]. 5 \n The series AV 3Sb5, as suggested from their complex magnetic spectral functions [12], is an \nitinerant -electron system  with a correlated electronic structure . The low -energy magnetic \nexcitations of such systems can be evaluated  via a multi -scale approach  upon satisfaction of \nthree conditions, viz., 1) the magnetic configurational energy dependence can be mapped onto \na generalized Heisenberg model -Hamiltonian , 𝐻=−∑ 𝑒𝑖𝛼𝐽̂𝑖𝑗𝛼𝛽𝑒𝑗𝛽\n𝑖≠𝑗 ,   𝛼,𝛽=𝑥,𝑦,𝑧, with the \nunit vector ei designating the local spin direction  at the i-th site , 2) the exchange tensor 𝐽̂𝑖𝑗𝛼𝛽 can \nbe calculated from the first -principles and 3) the ground state many -electron systems relies on \nthe adiabatic approximations, with decoupled  faster process of inter-site electronic hopping \nfrom the slower moving magn etic excitations or “frozen” magnons [13]. In our prior study \n[12], the fully relativistic exchange tensor was reported using the  Lichtenstein -Katnelson -\nAntropov -Gubanov (LKAG)  formalism  coupled to  the dynamical mean field theory (DMFT)  \n[14-20]. The tensor  possesse s terms  like the symmetric -isotropic Heisenberg (𝐽𝑖𝑗), anti -\nsymmetric and anisotropic  Dzyaloshinskii -Moriya (DM) and symmetric anisotropic \nHeisenberg interaction s (Γ).  The  DM and Γ  interactions are calculated by using the relations \nlike; 𝐷𝑖𝑗𝑧=1\n2(𝐽̂𝑖𝑗𝑥𝑦−𝐽̂𝑖𝑗𝑦𝑥) , Γ𝑖𝑗𝑧=1\n2(𝐽̂𝑖𝑗𝑥𝑦+𝐽̂𝑖𝑗𝑦𝑥) and analogous ones for the x- and y-\ncomponent s.  Fig. 1 presents the plots of the inter -site exchange interactions for various shells \nof NN of V-based stibnites  extracted with an FM interlayer interacti on. The negative values of \n𝐽𝑖𝑗 correspond to an AFM coupling . A closer scrutiny of this figure reveal s that the magnetic \ninteractions in this system are composed of various compe ting mechanisms. For the first NN \nshell , the 𝐽𝑖𝑗 values are  roughly two order s of magnitude larger than the 𝐷𝑖𝑗 and Γ𝑖𝑗 values.  \nHowever, f rom the next NN -shell,  the anisotropic interactions 𝐷𝑖𝑗 and Γ𝑖𝑗 becomes comparable \nto the 𝐽𝑖𝑗′s. Such a comp eting  environment of complex magnetic interactions can be a fertile  \nground for obtaining  chiral magnetic ground -states . It may be mentioned in passing that for \nsystems like metallic and Mott insulating kagome systems , chiral  magnetism may appear from 6 \n very simpl e correlated Hamiltoniian [21-22]. Moreover , at the same NN distance s, the flipping \nsign of the 𝐷𝑖𝑗′s implies  a signature of  the chiral magnetic textures  as per the Moriya rule .  \n \nFIG. 1: The inter -site exchange parameters as a function of NN distances in local coordinate \nsystem: 𝐽𝑖𝑗, Dij and Γ𝑖𝑗 for (a) -(c) CsV 3Sb5;  (d) -(f)  RbV 3Sb5 and (g)-(k) KV 3Sb5. Note that, f or \nsome NN distances, we have different strength of the interactions  in differe nt directions , \nimplying  anisotropy in the directional distribution of the interactions.  The difference in colour \nat the bar plot signifies the difference of the strength of various magnetic exchange interactions  \nat a particular value of NN distance.  \n \nTo obtain inversion asymmetric chiral environments  with significant  𝐷𝑖𝑗 and Γ𝑖𝑗, systems were  \nartificially designed  using  metallic multilayers like Mn/Cr/W[110] or Pd -Fe/Ir[111] involving  \ncomponents of high spin -orbit coupling (SOC) [ 23-26]. Recently, such spiral magnetic textures \nhave gained attention by virtue of their additional  capability  to explore the topological spin-\n7 \n transport via skyrmions [25,26] for spin -torque devices [27-29]. Here , the limitations \nintroduced by the instabilities like W alker breakdown on the mobility of the domain walls can \nbe escaped by reaching the super -magnonic regime  [28-32]. A sustained search for such chiral \nmagnetic systems  constitutes an active  area of research . In this le tter, we demonstrate that the \nV-based stibnite kagome system , with the  presence of local moments and a balanced \ncompetition between the inter-site exchange s and SOC, constitutes a breeding ground for  long-\nperiod, inhomogeneous chiral ma gnetic superstructures.  \n \nFIG. 2: The dynamical structure factors 𝑆(𝑞,𝜔) plotted along the high -symmetry paths, (a), (e)  \nand (i) the resultant total , (b), (f) and (j) 𝑆𝑥-projected, (c), (g) and (l) 𝑆𝑦-projected, (d), (h) and \n(l) 𝑆𝑧 -projected 𝑆(𝑞,𝜔) for CsV 3Sb5, RbV 3Sb5 and KV 3Sb5, respectively. The dynamical spin -\ncorrelations shown in the figure are those around the magnetic excitation energy minimum at \nthe K -point.   \n \nThe ground -state magnetic properties of complex  magnetic systems are deri ved after \nconstruc ting a classical atomistic spin -model, where t he bilinear  effective Hamiltonian can be \nwritten as : ℋ𝑚𝑜𝑑𝑖,𝑗=−𝐽𝑖𝑗𝒆𝑖.𝒆𝑗−𝑫𝑖𝑗𝒆𝑖×𝒆𝑗−𝒆𝑖Γ𝑖𝑗𝒆𝑗−𝜅∑ (𝒆𝑘.𝒆𝑘𝑟)2\n𝑘=𝑖,𝑗  [27,33,34]. Here 𝒆𝑖 \nand 𝒆𝑗 are normalized atomic spin moments  at the atomic sites i and j and 𝒆𝑘𝑟 is the easy axis, \n8 \n along the arbitrary unit vector r. The four different terms of th is Hamiltonian represent  the \nenergies corresponding to the isotropic Heisenberg interaction (J), the DM interaction ( DM), \nthe symmetric anisotropic exchange interaction () and the magnetocrystalline anisotropy (K)  \n[24].  \nIn a periodic lattice, the most general solution of th is model generates a homogeneous spin -\nspiral, where the angle between the resultant spin -moments of two adjacent neighbours are \nconstant throughout the spir al [35]. The atomic -site (i) spin -moments of a such spin-spiral can \nbe mapped onto a unit circle with 1 ≤𝑖≤𝑁, with N being the number of atomic spins in a \nspiral, the pitch of which can be defined as 𝜆=𝑁𝑎, with a being the lattice constant. The \nchirality of the spiral is dependent on the clockwise (negative) or anti -clockwise (positive) \ndirectionality of the mapping. Naturally , the DM interaction has a significant role in \ndetermining the chirality of the spiral  [23-26]. \nThe presence of  SOC and a large number of magnetic atoms render the first -principles \ntreatment  of a long-pitched spin-spiral computationally taxing [7,29]. Therefore , for chiral \nmagnetic systems [24,36], a continuum micro -magnet ic model is defined , where  the energy of \nthe spin -spiral  is written in terms of the local magnetization vector m as 𝐸[𝑚]=\n1\n𝜆∫𝑑𝑥[𝐴\n4𝜋2(𝒎)̇2+𝑫\n2𝜋.(𝒎×𝒎)̇+𝒎𝑇𝜅𝒎]𝜆\n0. Here A, D and 𝜅 are the spin -stiffness constant, \nthe effective DM vector and the anisotropy tensor respectively , while  𝒎̇ is the gradient of the \nmagneti zation.  For a generalized spin -spiral with a wave vector q, this magnetic moment m at \na given  atomic position 𝑹𝑛𝛼 can be written in the tensor format as 𝒎𝑛𝛼=\n𝑚𝛼(𝑠𝑖𝑛𝜃𝛼𝑐𝑜𝑠(𝜑𝑛𝛼+𝛾𝛼)\n𝑠𝑖𝑛𝜃𝛼𝑠𝑖𝑛(𝜑𝑛𝛼+𝛾𝛼)\n𝑐𝑜𝑠𝜃𝛼) [5,24]. Here 𝜃𝛼 and 𝛾𝛼 are the cone and the phase angle s \nrespectively. The azimuthal angle corresponding to the local magnetic moment can be \ncalculated as,  𝜑𝑛𝛼=𝒒.𝑹𝑛𝛼. Such  spiral magnetic ground states can be either commensurate 9 \n or incommensurate  with the underlying lattice , the solution of which  can be  obtain ed by \nfollowing the Petit’s method [37], using a series of  effective coordinate transformation s of the \nlocal spins into a ferromagnetic (FM) order  [37,38].  \n \nFIG. 3: (a)-(c) The 3D view of the dynamical ground -state spin -textures obtained from the  J + \nK + Γ + DM  Hamiltonian ; (d)-(f) spin -configurations on the star -of-David of the kagome  \nlattice ; (g)-(i) the corresponding mag netization densities for CVS, RVS and KVS respectively.  \nThe colour bars associated with the spin -textures indicate the projection of the magnetization \nalong z-direction. The magnetic textures correspond to the minimum of magnetic excitation \nenergy at the z one-centre (Γ -point).   \nFor the present kagome -series , the atomistic description s of the spin -moments can be used to  \nconstruct the spin -model because of its sufficiently localized electronic densities around the \natomic sites and the invariance of the resultant atomic spin-moments with respect to their \norientations. The chiral ground states can be obtained  by dynamically solving the atomistic  \nmodel, where, at every magnetic site, around the direction of the classical spin, the dynamics \nof the small fluctuations of the spins are calculated, following the implementations in the \nUppASD software  [34].  \n10 \n The spin-spin correlation functions , and their Fourier transform,  alias the dynamical structure \nfactor, also accessible  experimentally by inelastic neutron scattering, can be calculated as \n𝑆(𝑞,𝜔)=1\n2𝜋𝑁∑ 𝑒i𝑞(𝑟𝑖−𝑟𝑗)\n𝑖,𝑗 ∫𝑑𝜏𝑒−𝜄𝜔𝜏∝\n−∝⟨𝒆𝑖𝒆𝑗(𝜏)⟩, where, 𝒓𝑖 is the position vectors of the \nmagnetic atoms and 𝒆𝑖 is the local spin-vectors. For the incommensurate spin structures, in \naddition to 𝜔(𝑞), the magnon dispersion also constitutes modes like 𝜔(𝑞±𝑄) and the \ncorresponding correlations 𝑆(𝑞,𝜔) and 𝑆(𝑞±𝑄,𝜔) describes rigid rotations on the ordering \nplane and the canting with respect to the same plane respectively [38], where Q is the ordering \nwave -vector.  \nFigure 2(a) -(i) represen t the spin -component projected dynamical structure factors S(𝑞,𝜔) \ncalculated for the q-values along the high -symmetry path s for CsV 3Sb5 (CVS), RbV 3Sb5 (RVS) \nand KV 3Sb5 (KVS)  respectively after considering  the full Hamiltonian . The highest values of \nthe spin-correlations are obtained along the adiabatic magnon dispersion lines, which in Fig.2 \nare represented by the transition in colo ur from green to blue  in the adjacent colour -scale . The \nkagome system  in its unit cell possesses  3 V atom s and in the full non -collinear manner, the \nthree possible magnetic orientations should give rise  to nine modes. However, along -A \ndirection , the out -of-plane magnon dispersions are triply degenerate.  In addition, t he x and y \ncomponents of the structure fac tor show the minimum of the energy dispersion at the K point \nrather than the  point , indicating  that the ir magnetic ground -state collapse s into a spin -spiral \nstate. This state is  shown in Fig. 3 ( a-c), and is analysed in more detail below . The x and y \ncomponents of 𝑆(𝑞,𝜔) are seen to have significant values of the magnon gap at the Γ -point. \nThe z-component and thus the total dynamical structure factor are seen to display a much \nsmaller value of the gap. The reason behind this gap is the high value o f the magnetocrystalline \nanisotropy, amount ing to 0.29 mRy , 0.22 mRy and 0.28 mRy per V atom for CVS, RVS and \nKVS respectively , as per the DFT + DMFT + SOC calculations . The derived anisotropy 11 \n parameters indicate that a ll three systems display  a uniaxial, out-of-plane  easy-axis pattern  of \nspin-orientation s. \nThe spin-textures of the dynamically obtained magnetic ground state s after solving the full \nHamiltonian with all J, DM,  and K  terms are plotted in Fig . 3(a)-(c). The corresponding spin -\narrangements in the star -of-David kagome pattern are plotted in Fig. 3(d) -(f) with the \nmagnetization densities in the Fig . 3(g)-(i).  The colour scale of the Fig 3 represents the values \nof the normalized out -of-plane ( z) projection s of the spins . This figure evi dences that the \ndynamical magnetic ground state s of CVS and RVS are similar , manifesting  a superposition of \nthree types of underlying magnetic order s, viz. a FM arrangement along the [001] direction  and \ntwo spin-spiral configuration s, viz. one commensurate and the other incommensurate spin \nspiral arrangements along [010]. T he pitches corresponding to the commensurate and the \nincommensurate spin -spirals are calculated to be 3 a and 1.5 a respectively , with a being the \nlattice parameter . The ground -state spin -texture of KVS, as seen in Figure 3(c) , is consisted of \nadditional complexities , viz. a superposition of i) FM order along [001], ii) one commensurate \nspin-spiral along [010] and iii) two more inter -penetrating commensurate spirals along [010]. \nThe pitches of all these spirals are 3 a. In Fig S1 and S2 of supplementary materials (SM), the \ncalculation s of the pitches are explained  [39]. In Fig.  4(a)-(c), the ground -state commensurate \nand incommensurate spin-spiral textures  are displayed after dynamically solving the full  \nHamiltonian.  Here, t he adjacent non-collinear atomistic spins are oriented at an angle of 1200 \nwith respect to each other and thus the magnetic ground -state constitutes  a three -dimensional  \n(3D)  1200 spin-configuration . In Fig S3 of SM, the 3D 1200 spin-configurations are illustrated . \nTo demonstrate the importance of the complete Hamiltonian  in achieving the ground -state, we \nhave presented the spin-spiral textures by solving  the spin -Hamiltonian with several \ncombinations, viz., only J (Fig 4(d) -(f)), J + K (Fig 4(g) -(i)) and J +  (Fig 4(j) -(l)). The 12 \n corresponding dynamical structure factors 𝑆(𝑞,𝜔) with the detailed spin -texture s are plotted \nin Fig S 4 – S9 of SM [3 9].  \n \nFIG. 4: (a)-(c) The 3D view of the commensurate  and incommensurate  spirals with J + K + Γ \n+ DM  Hamiltonian , (d)-(f) spin -spirals with Hamiltonian having only J, (g) -(i) with J + K , (j)-\n(l) with J +  for CVS, RVS and KVS respectively. For KVS, all  the spirals are commensurate. \nThe colo ur bar indicates the projection of the magnetization along the z-direction.   \n \nA detailed observation  of Fig . 4 evinces that albeit the pitches and the overall types of the spin-\nspiral s remains the same for all three syst ems for all combinations,  the 3D 1200 spin-\nconfigurations are stabilized only with the complete Hamiltonian. All of these spin -dynamical \nground states are simulated at an ultralow temperature of 0.001K. With increasing temperature , \nwe have investigated t he evolution of the spin -textures in Fig 5, where the temperature -induced \nincrease of configurational entropy leads to a prominent presence of  weathervane defects  [39]. \n13 \n However, the spin -spiral textures remain intact up  to 5K , which is higher  than the \nsuperconducting critical temperature, Tc of this series .  \n \nFIG. 5: (a) -(c) The 3D view of the magnetic ground -state spin -textures with at 5 K; (d) -(f) The \n3D view of the magnetic ground -state spin -textures at 10 K;  (g) -(i) The 3D view of the \nmagnetic ground -state spin -textures at 20 K;  for CVS, RVS and KVS respectively. All \ncalculations include Heisenberg exchange and symmetric - and antisymmetric anisotropic \nexchange interactions as well as magneto crystalline anisotropy.   \n \nPresence of such chiral magnetic states at finite temperatures may impart an important impact \ntowards the analysis of the experimental observation of electronic magneto -chiral anisotropic \neffects  in CVS, where, there is a sudden onset of unusually large ch iral transport with lowering \nof temperature below 35K  [5]. For such centrosymmetric system, the mirror symmetries are \nexperimentally observed to be inherently broken by the itinerant carriers in the correlated \nphases. Although, for CVS, the presence of time-reversal symmetry breaking is under scrutiny \nafter the absence of any polar Kerr signal, as observed by Saykin et al.  [4], the presence of \nother spontaneously broken symmetries cannot be ruled out. Occurrence of c hiral magnetism \nand spin -spiral ground st ates in this series  can be a potential reason behind the experimentally \nobserved broken symmetries. It should also be noted that the structural instability and the \nrespective distortion is quite weak and there are disputes and debates accompanied with the yet \n14 \n undetermined  low-temperature crystal structure [40-43].   Thus, the  presence of such inherent \nchiral dynamical magnetic structure in the kagome system may provide an answer to the \nsimultaneous and spontaneous occurr ence of time -reversal , mirror  and C6 rotational symmetry \nbreaking in the CO and SC phases.     \nIn conclusion, for the first part of the comprehensive study  of the V-based kagome stibnites  \n[10], we addressed the effects of the formation of the local moments at the  V-sites in the \nkagome series AV 3Sb5 (A = Cs, Rb, K) on their charge -order and superconductivity  [10]. In \nthis second part, we have investigated the dynamical magnetic ground state of the system. The \nunderlying arrangement of the spin -magnetic moments reveals that , in addition to the FM long -\nrange order along  the [001]  direction , there are complex landscapes of spin -spiral \nconfigurations  in the [010] direction, consisting of commensurate and incommensurate chiral \nmagnetic textures . The p resence of such chiral magnetic textures has the capability of \nspontaneously break ing the time-reversal , mirror or  the C 6 rotational symmetry , and thus may \nprovide a valid  explanation to the experimental findings . In addition, the theoretically derived \ndynamical str ucture factors support  the presence of complex, collective magnetic excitations \nwithin the system, which may be validated by future experimental studies. Thus, our thorough \nanalysis of the dynamical aspects of the magnetism for the V-based kag ome stibnites  can be \nhelpful in providing an explanation to the existing experimental queries and may also provide \na motivation for  the future experimental studies.  \nAcknowledgements  \nFinancial support from Vetenskapsrådet (grant numbers VR 2016 -05980 and V R 2019 -05304), \nand the Knut and Alice Wallenberg foundation (grant numbers 2018.0060, 2021.0246, \n2022.0108 and 2022.0079) is acknowledged. The computations were enabled by resources \nprovided by the National Academic Infrastructure for Supercomputing in Swe den (NAISS) and \nthe Swedish National Infrastructure for Computing (SNIC) at NSC and PDC, partially funded \nby the Swedish Research Council through grant agreements no. 2022 -06725 and no. 2018 -\n05973. OE also acknowledges support from STandUPP and eSSENCE. DK acknowledges \nBARC supercomputing facility. MNH acknowledges CSIR (India) for fellowship. IDM \nacknowledges support from the JRG Program at APCTP through the Science and Technology 15 \n Promotion Fund and Lottery Fund of the Korean Government, as well as from th e Korean \nLocal Governments -Gyeongsangbuk -do Province and Pohang City. IDM and SS also \nacknowledge financial support from the National Research Foundation of Korea (NRF), funded \nby the Ministry of Science and ICT (MSIT), through the Mid -Career Grant No. \n2020R1A2C101217411.  \n \n \nReferences  \n[1] C. Mielke, D. Das, J. X. Yin, H. Liu, R. Gupta, Y. X. Jiang, M. Medarde, X. Wu, H. C. \nLei, J. Chang, P. Dai, Q. Si, H. Miao, R. Thomale, T. Neupert, Y. Shi, R. Khasanov, M. Z. \nHasan, H. Luetkens, and Z. Guguchia, Nature 602, 245 (2022).  \n [2] Y.-X. Jiang, J .-X. Yin, M. M. Denner, N. Shumiya, B. R. Ortiz, G. Xu, Z. Guguchia, J. \nHe, M. S. Hossain, X. Liu, J. Ruff, L. Kautzsch, S. S. Zhang, G. Chang, I. Belopolski, Q. \nZhang, T. A. Cochran, D. Multer, M. Litskevich, Z. -J. Cheng, X. P. Yang, Z. Wang, R. \nThomale, T. Neupert, S. D. Wilson, and M. Z. Hasan, Nat. Mater. 20, 1353 (2021).  \n \n[3] S.-Y. Yang, Y. Wang, B. R. Ortiz, D. Liu, J. Gayles, E. Derunova, R. Gonzalez -\nHernandez, L. Šmejkal, Y. Chen, S. S. P. Parkin, S. D. Wilson, E. S. Toberer, T. McQueen, \nand M. N. A li, Sci. Adv. 6, eabb6003 (2020).  \n[4] D. R. Saykin, C . Farhang, E . D. Kountz, D . Chen, B . R. Ortiz, C . Shekhar, C . Felser, S . \nD. Wilson, R . Thomale, J . Xia, and A . Kapitulnik, Phys. Rev. Lett. 131, 016901 (2023).  \n \n[5] C. Guo, C . Putzke, S . Konyzheva, X . Huang, M . Gutierrez -Amigo, I . Errea, D . Chen, \nM. G. Vergniory, C . Felser, M . H. Fischer, T . Neupert and P. J. W. Moll, Nature 611, 461 –466 \n(2022).  \n \n[6]       W. Schweika, M. Valldor, and P. Lemmens, Phys. Rev. Lett. 98, 067201 (2007).  \n \n[7]     Daniel Boyko, Avadh Saxena and Jason T Haraldsen, Ann. Phys. (Berlin) 1900350 \n(2019) . \n \n[8]       Mathieu Taillefumier, Julien Robert, Christopher L. Henley, Roderich Moessner, and \nBenjamin Canals, Phys. Rev. B 90, 064419 (2014).  \n \n[9]       Masahide Nishiya ma, Satoru Maegawa, Toshiya Inami, and Yoshio Oka Phys. Rev. B \n67, 224435 (2003).  \n \n[10]       Jonas Becker and Stefan Wessel Phys. Rev. B 100, 241113(R) (2019) . \n[11]      Tao Xie, Qiangwei Yin, Qi Wang, A. I. Kolesnikov, G. E. Granroth, D. L. Abernathy, \nDongliang Gong, Zhiping Yin, Hechang Lei, and A. Podlesnyak, Phys. Rev. B 106, 214436 \n(2022) . \n \n[12] M. N. Hasan, R. Bharati, J. Hellsvik, A. Delin, S. K. Pal, A. Bergman, S. Sharma, I. D. \nMarco, M. Pereiro, P. Thunström, P. M. Oppeneer, O. Eriksson, and D. Karmakar, Phys. Rev. \nLett. (accepted), https://journals.aps.org/prl/accepted/ea077Y18R6f1c990d0859378e0 \n12ab2662a4490ae  16 \n [13] M. Ležaić, P. Mavropoulos, G. Bihlmayer, and S. Blügel, Phy s. Rev. B 88, 134403 \n(2013).  \n[14] A. I. Liechtenstein, M. I. Katsnelson, V. P. Antropov, and V. A. Gubanov, J. Magn. \nMagn. Mater. 67, 65 (1987).  \n[15] Y. O. Kvashnin, O. Grånäs, I. Di Marco, M. I. Katsnelson, A. I. Lichtenstein, and O. \nEriksson, Phys. Rev. B 91, 125133 (2015).  \n[16] Y. O. Kvashnin, A. Bergman, A. I. Lichtenstein, and M. I. Katsnelson, Phys. Rev. B \n102, 115162 (2020).  \n[17] A. Grechnev, I. Di Marco, M. I. Katsnelson, A. I. Lichtenstein, J. Wills, and O. \nEriksson, Phys. Rev. B 76, 035107 (2007).  \n[18] P. Thunström, I. Di Marco, and O. Eriksson, Phys. Rev. Lett. 109, 186401 (2012).  \n[19] J. M. Wills, M. Alouani, P. Andersson, A. Delin, O. Eriksson, and O. Grechnyev, Full-\npotential electronic structure method: Energy and force calculations wit h density functional \nand dynamical mean field theory (Springer Science & Business Media, 2010), Vol. 167.  \n[20] A. Szilva, Y. Kvashnin, E . A. Stepanov, L . Nordström, O . Eriksson, A . I. Lichtenstein, \nand M . I. Katsnelson, Rev. Mod. Phys. 95, 035004 (2023) . \n[21]  M. Udagawa and Y . Motome, Phys. Rev. Lett. 104, 106409 (2010).  \n[22] L. Messio, B . Bernu, and C . Lhuillier, Phys. Rev. Lett. 108, 207204 (2012).  \n[23] M. Bode, M. Heide, K. von Bergmann, P. Ferriani, S. Heinze, G. Bihlmayer, A. Kubetzka, \nO. Pietzsch, S. Blügel, and R. Wiesendanger, Nature 447, 190 (2007).  \n[24] B. Zimmermann, M. Heide, G. Bihlmayer, and S. Blügel, Phys. Rev. B 90, 115427 \n(2014).  \n[25] S. Heinze, K. von Bergmann, M. Menzel, J. Brede, A. Kubetzka, R. Wiesendanger, G. \nBihlmayer, and S. Blügel, Nature Phys 7, 713 (2011).  \n[26] N. Romming, C. Hanneken, M. Menzel, J. E. Bickel, B. Wolter, K. von Bergmann, A. \nKubetzka, and R. Wiesendanger, Science 341, 636 (2013).  \n[27] A. Neubauer, C. Pfleiderer, B. Binz, A. Rosch, R. Ritz, P. G. Niklowitz, and P. Böni, \nPhys. Rev. Lett. 102, 186602 (2009).  \n[28] C. Franz, F. Freimuth, A. Bauer, R. Ritz, C. Schnarr, C. Duvinage, T. Adams, S. Blügel, \nA. Rosch, Y. Mokro usov, and C. Pfleiderer, Phys. Rev. Lett. 112, 186601 (2014).  \n[29] T. Schulz, R. Ritz, A. Bauer, M. Halder, M. Wagner, C. Franz, C. Pfleiderer, K. \nEverschor, M. Garst, and A. Rosch, Nature Phys 8, 301 (2012).  \n[31] R. M. Otxoa, P. E. Roy, R. Rama -Eiroa, J. Godinho, K. Y. Guslienko, and J. \nWunderlich, Commun Phys 3, 190 (2020).  \n[32] K.-S. Ryu, L. Thomas, S. -H. Yang, and S. Parkin, Nature Nanotech. 8, 527 (2013).  \n[28] S. Emori, U. Bauer, S. -M. Ahn, E. Martinez, and G. S. D. Beach, Nature Mater 12, 611 \n(2013).  17 \n [33] C. Etz, L. Bergqvist, A. Bergman, A. Taroni, and O. Eriksson, J. Phys.: Condens. Matter \n27, 243202 (2015).  \n[34] B. Skubic, J. Hellsvik, L. Nordström, and O. Eriksson, J. Phys.: Condens. Matter 20, \n315203 (2008).  \n[35] P. Kurz, F. Förster, L. Nordström, G. Bihlmayer, and S. Blügel, Phys. Rev. B 69, \n024415 (2004).  \n[36] J. C. Leiner, T. Kim, K. Park, J. Oh, T. G. Perring, H. C. Walker, X. Xu, Y. Wang, S. \nW. Cheong, and J. -G. Park, Phys. Rev. B 98, 134412 (2018).  \n[37] S. Pe tit, JDN 12, 105 (2011).  \n[38] S. Toth and B. Lake, J. Phys.: Condens. Matter 27, 166002 (2015).  \n[39] See Supplemental materials for the description of the commensurate and    \nincommensurate spirals, the 1200 structures, the spin -textures with different ter ms of the full \nspin-Hamiltonian and the corresponding dynamical structure factor plots  and the temperature \ndependent spin -textures .  \n[40]  H. Zhao, H . Li, B . R. Ortiz, S . M. L. Teicher, T . Park, M . Ye, Z . Wang, L . Balents, S . \nD. Wilson and I. Zeljkovic, Nature  599, 216 –221 (2021) . \n[41] B. R. Ortiz, S . M.L. Teicher, Y . Hu, J . L. Zuo, P . M. Sarte, E . C. Schueller, A.M. M . \nAbeykoon, M . J. Krogstad, S . Rosenkranz, R . Osborn, R . Seshadri, L . Balents, J . He, and S . D. \nWilson, Phys. Rev. Lett. 125, 247002 (202 0). \n[42] J. Luo, Z. Zhao, Y. Z. Zhou, J. Yang, A. F. Fang, H. T. Yang, H. J. Gao, R. Zhou and \nG-Q Zheng, npj Quantum Mater. 7, 30 (2022).  \n[43] Q. Stahl, D. Chen, T. Ritschel, C. Shekhar, E. Sadrollahi, M. C. Rahn, O. Ivashko, M. \nv. Zimmermann, C. Felser , and J. Geck, Phys. Rev. B 105, 195136 (2022).  1 \n Supplementary Materials for  \nMagnetism in AV 3Sb5 (A = Cs, Rb, K): Complex Landscape of the \nDynamical Magnetic Textures  \nDebjani Karmakar1,2,#,*, Manuel Pereiro1,#, Md. Nur Hasan3, Ritadip Bharati4, Johan \nHellsvik5, Anna Delin6,7, Samir Kumar Pal3, Anders Bergman1, Shivalika Sharma8, Igor Di \nMarco1,8,9, Patrik Thunström1, Peter M. Oppeneer1 and Olle Eriksson1,*  \n \n1 Department of Physics and Astronomy,  \nUppsala University, Box 516, SE -751 20, Uppsala, Sweden  \n \n2 Technical Physics Division  \nBhabha Atomic Research Centre , Mumbai 400085, India  \n \n3 Department of Chemical and Biological Sciences,  \nS. N. Bose National Centre for Basic Sciences,  \nBlock JD, Sector -III, SaltLake, Kolkata 700 106, India  \n \n4 School of Physical Sciences  \nNational Institute of Science Education and Research  \nHBNI, Jatni - 752050, Odisha, India  \n \n5 PDC Center for High Performance Computing,  \nKTH Royal Institute of Technology, SE -100 44 Stockholm, Sweden  \n \n6 Department of Applied Physics  \nKTH Royal Institute of  Technology , SE-106 91 Stockholm, Sweden  \n \n7 Swedish e -Science Research Center (SeRC),  \nKTH Royal Institute of Technology, SE -10044 Stockholm, Sweden  \n \n8Asia Pacific Center for Theoretical Physics,  \nPohang, 37673, Republic of Korea  \n \n9 Department of Physics, Pohang University of Science and Technology,  \nPohang, 37673, Republic of Korea  \n \n \n \n# D.K. and M.P. contributed  equally  \n \n*Corresponding Authors:  \nDebjani Karmakar  \nEmail: debjani.karmakar@physics.uu.se  \n \nOlle Eriksson  \nEmail: olle.eriksson@physics.uu.se  \n 2 \n  \nFIG. S1: The commensurate and incommensurate spin -spirals for CsV 3Sb5 and RbV 3Sb5. The \nhighlighted green and yellow boxes correspond to the pitches of the commensurate and incommensurate \nspirals respectively. On the right hand panel, we have shown the corresponding pitches on the 2D \nkagome lattice plane. The parallelogram at the left c orner of the 2D -lattice shows the unit cell having \nthe dimension of the lattice parameter a. The pitches of the commensurate and incommensurate spirals \nare 3 a and 1.5 a respectively, as can be seen from the figure.  \n  \n3 \n  \nFIG. S2: The middle panel shows the 3 D view of the three commensurate spin -spirals for KV 3Sb5. The \nhighlighted green, yellow and cyan boxes correspond to the pitches of the three commensurate spirals \nrespectively. On the left and right hand panel, we have shown the corresponding pitches on th e 2D \nkagome lattice planes. The parallelogram at the left corner of the 2D -lattice shows the unit cell having \nthe dimension of the lattice parameter a. The pitches of all the three commensurate spirals are 3 a, as \ncan be seen from the figure.  \n \n \n \n \n \n \n \n \n \n \n \n \n4 \n  \nFIG. S3:  The three -dimensional (3D) -1200 magnetic textures for CVS, RVS and KVS.  \n \n \n \n \n \n \n \n \n \n5 \n  \nFIG. S4: The spin -component projected  and total  dynamical structure factors 𝑆(𝑞,𝜔) with Heisenberg \nexchange  of (a-d) CsV 3Sb5, (e-h) RbV 3Sb5 and (i -l) KV 3Sb5. \n  \n6 \n  \nFIG. S5: The spin -component projected and total dynamical structure factors 𝑆(𝑞,𝜔) with Heisenberg \nexchange  and magneto crystalline  anisotropy  of (a-d) CsV 3Sb5, (e-h) RbV 3Sb5 and (i -l) KV 3Sb5. \n  \n7 \n  \nFIG. S 6: The spin -component projected and total dynamical structure factors 𝑆(𝑞,𝜔) with Heisenberg \nexchange  and symmetric an isotropic exchange interactions of (a -d) CsV 3Sb5, (e-h) RbV 3Sb5 and (i -l) \nKV 3Sb5. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n8 \n  \nFIG. S7: (a)-(c) The 3D view of the dynamical ground -state spin -textures with only Heisenberg \nexchange ; (d)-(f) spin -configurations on the star -of-David of  the kagome  lattice ; (g)-(i) the \ncorresponding magnetization densities for CVS, RVS and KVS respectively.  \n \n \n \n \n \n \n \n \n \n \n \n \n9 \n  \n \n \nFIG. S8: (a)-(c) The 3D view of the dynamical groun d-state spin -textures with Heisenberg \nexchange and magneto crystalline anisotropy ; (d)-(f) spin -configurations on the star -of-David of \nthe kagome  lattice ; (g)-(i) the corresponding magnetization densities for CVS, RVS and KVS \nrespectively.  \n \n \n \n \n \n \n \n \n \n10 \n  \nFIG. S9: (a)-(c) The 3D view of the dynamical ground -state spin -textures with Heisenberg \nexchange and symmetric anisotropic exchange interactions ; (d)-(f) spin -configurations on the star -\nof-David of the kagome  lattice ; (g)-(i) the corresponding magnetizati on densities for CVS, \nRVS and KVS respectively.  \n \n \n \n \n \n"    },    {        "title": "2311.13215v1.Engineering_magnetic_domain_wall_energies_in_multiferroic_BiFeO__3__via_epitaxial_strain.pdf",        "content": "Engineering magnetic domain wall energies in multiferroic BiFeO 3via epitaxial strain\nSebastian Meyer,1, 2, 3, ∗Bin Xu,4, 5,†Laurent Bellaiche,5and Bertrand Dup´ e1, 2, 3\n1Nanomat/Q-mat/CESAM, Universit´ e de Li` ege, B-4000 Sart Tilman, Belgium\n2TOM/Q-mat/CESAM, Universit´ e de Li` ege, B-4000 Sart Tilman, Belgium\n3Fonds de la Recherche Scientifique (FRS-FNRS), Bruxelles, Belgium\n4Jiangsu Key Laboratory of Thin Films, School of Physical Science and Technology, Soochow University, Suzhou 215006, China\n5Physics Department and Institute for Nanoscience and Engineering,\nUniversity of Arkansas, Fayetteville, Arkansas 72701, USA\n(Dated: November 23, 2023)\nEpitaxial strain has emerged as a powerful tool to tune magnetic and ferroelectric properties\nin functional materials such as in multiferroic perovskite oxides. Here, we use first-principles cal-\nculations to explore the evolution of magnetic interactions in the antiferromagnetic multiferroic\nBiFeO 3(BFO), one of the most promising multiferroics for future technology. The epitaxial strain\nin BFO(001) oriented film is varied between εxx,yy∈[−2 %,+2 %]. We find that both strengths\nof the exchange interaction and Dzyaloshinskii-Moriya interaction (DMI) decrease linearly from\ncompressive to tensile strain whereas the uniaxial magnetocrystalline anisotropy follows a parabolic\nbehavior which lifts the energy degeneracy of the (111) easy plane of bulk BFO. From the trends of\nthe magnetic interactions we can explain the destruction of cycloidal order in compressive strain as\nobserved in experiments due to the increasing anisotropy energy. For tensile strain, we predict that\nthe ground state remains unchanged as a function of strain. By using the domain wall (DW) energy,\nwe envision the region where isolated chiral magnetic texture might occur as function of strain i.e.\nwhere the DW and the spin spiral energy are equal. This transition between −1.5 % and −0.5 %\nof strain should allow topologically stable magnetic states such as antiferromagnetic skyrmions and\nmerons to occur. Hence, our work should trigger experimental and theoretical investigations in this\nrange of strain.\nI. INTRODUCTION\nFor over a decade, topologically stable magnetic struc-\ntures, such as domain walls (DWs) and skyrmions have\nbeen studied extensively based on their theoretical pre-\ndiction in the early 90’s [1, 2] and their first experimen-\ntal observation 20 years later [3, 4]. Their potential in\nlow-power, high-density information storage, spintronic\napplications and their unique stability, resistance to ex-\nternal magnetic fields, and efficient dynamics make them\na fascinating area of study [5–8]. The formation and\ncontrol of these skyrmions have primarily been explored\nin ferromagnetic materials, but their antiferromagnetic\ncounterparts are predicted to exceed their potential [9–\n12]. While the stabilization of skyrmions in FMs can be\nachieved e.g. by the application of an external magnetic\nfields [13–18], antiferromagnets are unaffected by those.\nTherefore, the stabilization of chiral magnetic textures\nin antiferromagnets remains a challenge and depends on\nthe precise control of magnetic interactions.\nEpitaxial strain allows for precise modification of a ma-\nterial’s lattice structure through heteroepitaxial growth\nand has emerged as a powerful tool to engineering mag-\nnetic properties in materials. For example, it offers the\nmeans to tailor crystal symmetries [19, 20], magnetic\nanisotropies, and exchange interactions and even to con-\ntrol the magnetic ground state of a material [21]. In\n∗These two authors contributed equally; E-mail:\nsmeyer@uliege.be\n†These two authors contributed equallythis way, skyrmions could be stabilized in ferromagnetic\n(FM) systems [22–25]. Such a technique can pave the\nway to the stabilization and control of magnetic struc-\ntures such as skyrmions in antiferromagnets. Strain en-\ngineering has already been used in multiferroic oxides,\nwhere several ferroic orders, such as the polarization and\nthe magnetization, can be tuned [26–29]. More interest-\ningly, their coupling called the magneto-electric coupling\nis affected by strain. Since they do not only posses a\nmagneto-elastic, but also a magneto-electric and piezo-\nelectric coupling, the magnetism can be tuned in various\nways. To that end, one of the most promising multifer-\nroic material is BiFeO 3(BFO) where ferroelectricity and\nantiferromagnetism can coexist far beyond room temper-\nature [30–34].\nBesides being extensively studied in the bulk, the in-\nterest in strained BFO has also risen up due to the variety\nof non-collinear states that it can host. Bulk BFO has\nan antiferromagnetic (AFM) spin spiral ground state -\nthe so-called type-I cycloid of 62 nm pitch [35, 36]. This\nspin spiral can be destroyed via biaxial strain when BFO\nthin film is grown on various substrates [21, 37–40] or by\nuniaxial strain [28] which suggests a high magneto-elastic\nresponse. This response can affect all magnetic interac-\ntions in BFO e.g. the magnetic exchange, the anisotropy\nand the Dzyaloshinskii-Moriya interaction (DMI) whom\nratios have yet to be quantified via density functional\ntheory (DFT). A region of particular importance is the\ntransition between collinear and non-collinear magnetic\nground states as function of strain because more com-\nplex isolated magnetic textures, such as skyrmions, anti-\nskyrmions or merons/anti-merons can form.arXiv:2311.13215v1  [cond-mat.mtrl-sci]  22 Nov 20232\nIn this paper, we explore the different magnetic inter-\nactions of BiFeO 3under the influence of bi-axial strain\nbased on DFT. Varying the nominal strain εxx,yy ∈\n[−2 %,+2 %] from compression to elongation, we deter-\nmine the magnetic exchange interaction, Dzyaloshinskii-\nMoriya interaction and the anisotropy energy. We study\nthe influence of epitaxial strain on the magnetism in\ntheCcphase of BFO to identify the region of strain,\nwhere the transition from non-collinear to collinear or-\nder takes place. We find that both exchange and DMI\ndecrease linearly with strain whereas the anisotropy fol-\nlows a parabolic behavior, preferring the [11 2] direction\nfor compressive strain and the [1 10] direction for tensile\nstrain. Following the ratios of magnetic interactions, we\nconclude that with compressive strain, collinear G-type\nantiferromagnetic order is enhanced due to an increase of\nthe anisotropy and magnetic exchange over the DMI. In\nthe tensile regime, the results are ambiguous. From our\ncalculations, we expect that the ground state spin spi-\nral of the bulk R3cphase in BFO is preserved; however,\nsince the energies are extremely close, small distortions\nor impurities can also favor collinear order. At last we\nanalyze the evolution of the domain wall energy to lo-\ncate the region of strain where chiral non-collinear mag-\nnetic textures such as antiferromagnetic skyrmions and\nmerons could be stabilized in BiFeO 3. We identify the\ntransition between a non-collinear and a collinear ground\nstate in BFO in between −1.5 % and −0.5 % of compres-\nsive strain where such states should occur. The presented\nwork should therefore trigger experimental and theoreti-\ncal investigations in that range of strain.\nThis paper is structured as follows: In Sec. II, we\ndescribe our procedure for the DFT calculations. In\nSec. III, the results of the DFT calculations are presented\nwhich are discussed in Sec. IV. A conclusive statement is\ngiven in Sec. V.\nII. METHODS\nA. Structural Relaxation\nWe have relaxed all strained structures of BiFeO 3using\nthe Abinit package [41, 42] and the projector augmented\nwave (PAW) method [43]. Therefore, strain ε=a′−aBFO\naBFO\nis applied from −2% to 2% for the in-plane axes of the\nCcphase of BFO. Here, a′is the varied in-plane lattice\nconstant and aBFOdenotes the BFO bulk lattice constant\nof the R3cphase. Note that the Ccphase is imposed at\nε= 0 % with the lattice constant of R3cBFO. While εis\nvaried from compressive strain of −2 % to tensile strain\nof +2 %, the other structural parameters are relaxed [in-\ndicated by blurred colors of the atoms and bonds, cf.\nsketch in Fig. 1 (a)].\nWe have used a√\n2×√\n2×2 unit cell hosting 20 atoms.\nThe magnetic configuration is chosen as the collinear G-\ntype antiferromagnetic order. We have converged theforces on each atom down to 1 ×10−5hartree/bohr. The\nexchange and correlation functional is treated with lo-\ncal spin density approximation + U(LSDA+ U), with a\nHubbard U= 4.0 eV parameter and J= 0.4 eV on the\nFe atoms, which are typical values for first-principles cal-\nculations on BFO [33, 44, 45]. The wave-functions are\nexpanded with a plane-wave basis set using a kinetic en-\nergy cutoff of Ecut= 30 hartree. By using a k-point-mesh\nfor the structural relaxation of 8 ×8×6, we obtain a mag-\nnetic moment of the Fe atoms of |mFe|= 4.0µBfor all\nrelaxed structures. The results of the relaxation are pre-\nsented in Fig. 1 (b,c) and are in accordance with previous\nstudies [19, 20].\nB. Determination of Magnetic Interactions\nWe use the relaxed structures as described above to de-\ntermine the magnetic interactions, such as the Heisenberg\nexchange interaction, the Dzyaloshinskii-Moriya (DM)\ninteraction and the uniaxial anisotropy energy for each\nvalue of strain. The first two interactions are determined\nby calculating energy dispersions of flat homogeneous\nspin spiral states without (for the exchange) and with\n(for the DMI) spin-orbit coupling. For both interactions,\nwe apply the pseudo-cubic approximation to compare the\nresults we have obtained for the relaxed R3cstructure\n[45, 46].\nSpin spirals are the general solution of the Heisenberg\nmodel on a periodic lattice and can be characterized by\nthe spin spiral vector q. This vector determines the prop-\nagation direction of the spin spiral as well as the canting\nangle between two neighboring spins. A magnetic mo-\nment miat an atom position riis given by\nmi=m\u0010\ncos(q·ri),sin(q·ri),0\u0011\n(1)\nwhere mis the magnitude of the magnetic moment.\nThe energy dispersions E(q) are obtained apply-\ning the full-potential linearized augmented plane wave\n(FLAPW) approach [47–49], as implemented in the\nfleur code [50]. For all these calculations, we have used\nthe LSDA+ U[51] with a Hubbard U= 4.0 eV param-\neter and J= 0.4 eV on the Fe atoms. Muffin-tin radii\nwere set to 2.80 bohrs, 2.29 bohrs, and 1.29 bohrs for\nBi, Fe, and O atoms, respectively and a large plane-wave\ncutoff kmaxof 4.6 bohr−1. These parameters result in a\nmagnetic moment of m= 4.0µBfor all phases in agree-\nment with experiments [36]. Calculations along the full\npaths of the Brillouin zone (BZ) without SOC have been\nperformed self-consistently using the generalized Bloch\ntheorem [52] and a k-point mesh of 10 ×10×10. To\naccurately determine the energies around the magnetic\nground state (R points of the BZ) at |q| →R, the mag-\nnetic force theorem [53, 54] has been applied using a\ndense k-point set of 64000 kpoints (i.e., 40 ×40×40).\nThe energy dispersion without SOC is interpreted using\nthe Heisenberg exchange interaction. For the energy con-\ntribution due to SOC, ∆ ESOC, we add SOC in first-order3\nstrain εxx,yy (%)0 1 -1 -2 2monoclinic angle (˚)90\n89.75\n89.5\n89.25\n89 -10-9\nOxygen Octahedral tilt\nω (˚)-8strain εxx,yy (%)\n0 1 -1 -2 2\n10lattice constant of\n√2×√2×2 unit cell\n(bohr)\n1112131415\n157516001625\nVolume of √2×√2×2\nunit cell (bohr3)1650\nmonoclinic\nangleωc\nωaωbcaxis\naaxisbaxisVolume(a)\n(c)(b)\nBi\nOFe\n[110][110][001]\nbi-axial strain ± εxx,yystructural\nrelaxation\na\nb\nFigure 1. Structural properties of strained BiFeO 3. (a) Sketch\nof strained CcBFO in cubic representation: Bi-axial strain ε\nis applied in the xyplane with an angle of 90◦between the\naandbaxis, whereupon the other parameters ( caxis, mono-\nclinic angle and all atomic positions) are relaxed - indicated by\nblurred colors (for more details, see text). (b) Relaxed lattice\nconstant ( a,bandcaxes, squares on left ordinate) vs. value of\nstrain. Right ordinate: Unit cell volume (blue filled circles).\nNote that the relaxation has been done in a√\n2×√\n2×2\nunit cell - the values are presented within this notation. (c)\nLeft ordinate: Monoclinic angle of the caxis (black squares)\nand components of the oxygen octahedral tilts ωiaround the\niaxis (red circles for the second ordinate). Note that due to\nthe imposed Ccphase at 0 % strain, the tilt angles around the\nthree axes are not exactly on top of each other due to shear\nstrain in the xyplane.\nperturbation theory [55, 56] for every previously calcu-\nlated point. The resulting curve has been interpreted\nwith the Dzyaloshinskii-Moriya interaction [57, 58].\nWe determine the Heisenberg exchange interaction\nconstants Jijbeyond nearest neighbors by mapping the\nHeisenberg Hamiltonian\nHex=−X\nijJij(mi·mj) (2)onto the energy dispersion E(q) of flat spin spiral states\nneglecting spin-orbit coupling (SOC). We include seven\nneighbors for the exchange interaction using the pseudo-\ncubic approximation for all values of strain. To sim-\nplify our model, we approximate the curvature of the\ncombined exchange interactions J1,...,7atq→R,|q| ∈\n[0,0.05] with an effective nearest neighbor exchange in-\nteraction Jeff.\nFor the Dzyaloshinskii-Moriya (DM) interaction, we\nuse the calculated data of the energy contribution due\nto SOC ∆ ESOC. Therefore, we use the converse spin-\ncurrent (sc) model,\nHSC=−X\nijCij(u×eij)·(mi×mj) (3)\nwhere Cijdetermines the strength of the DMI, uis the\nunit vector along the polarization direction and eijis the\nunit vector between magnetic sites i, j. We include three\nneighbors for the sc DMI in pseudo-cubic approximation.\nFor the effective DMI, Ceff, we linearly map Eq (3) to the\nSOC contribution at q→R.\nWe also have calculated the magnetic anisotropy in\nBiFeO 3. For that, we have assumed the collinear G-type\nAFM state for each structure, where we have applied\nSOC in different directions of the respective BZ. We have\nused 20 ×20×20k-point-mesh in the whole Brillouin\nzone and applied the second quantization with the Force\nTheorem [53] to determine the energy differences between\nthe hard axis ([111] axis) and other directions as shown\nin the results. For the description of the domain wall\nenergy, we will use an effective anisotropy Keffthat is\nthe lowest energy of the respective direction with strain.\nThis method for calculating magnetic interaction in\nBFO has also been described in detail in Refs. 45 and 46.\nC. Error Estimation\nIn the following, we will analyze small energy differ-\nences that qualitatively affect the ground state of BiFeO 3.\nEspecially for the DFT calculations, we reach the limit of\naccuracy below ∼0.1 meV. It is impossible to estimate\nthe error in DFT calculations itself stemming mostly\nfrom the exchange correlation functional simply because\nthe amount of necessary calculations for that is beyond\nthe scope of any DFT study that is not addressed to\nthat specific problem. Hence, the absolute values in our\ncalculations may vary depending on the functional, we\ncan only conclude on the trends. The LSDA+ Uapprox-\nimation with the functional of Ref. 51 has been used\nprevioulsy in Refs. 45 and 46.\nFor the effective magnetic interactions including a fit-\nting procedure in the first step, namely the exchange and\nthe DMI, we will address the error based on the error of\nthe fits. For the DMI, the DFT data is perfectly linear\nand the value Ceffis determined with a negligible error of\nless than 0.01 %. The values for the exchange interactions4\nJion the other hand are fitted with an error ∆ Ji. This\nerror affects our estimate on Jeffbecause it describes this\nenergy dispersion curve within q→R. Since the energy\nis the sum for all Ji, we apply the sum rule to estimate\nthe error on Jeffas\n∆Jeff=1\n6vuut7X\ni=1(∆Ji)2Ni (4)\nwhere Niare the number of neighbors in the i-th shell,\nwhich for J1are 6 neighbors in the cubic lattice. The\nfraction 1/6in front of the square root accounts for the\nnumber of the nearest neighbors.\nUsing multiplication or division, we will use the error\nestimate according to\n∆Q=|Q|s\u0012∆Jeff\nJeff\u00132\n=|Q|∆Jeff\nJeff(5)\nwhere Qis the quantity we determine.\nIII. RESULTS\nFigure 2 shows the trends of magnetic interactions in\nBiFeO 3(BFO) under the application of bi-axial strain.\nA. Exchange Interaction\nFor all applied values of strain, the nearest neigh-\nbor Heisenberg exchange interaction J1[black points\nin Fig. 2 (a)] contains the dominant exchange con-\ntribution and prefers collinear antiferromagnetic or-\nder. Its strength decreases linearly about 10 % from\n−28 meV/Fe atom at ε=−2 % to −26 meV/Fe atom\natε= +2 %. The second largest contribution stems\nfrom the second neighbor J2≈ −1.5 meV/Fe atom (light\ngreen squares). Its strength does not vary drastically\nwith strain, but it leads to exchange frustration with\nstrain as the ratio J2/J1increases. The largest vari-\nation can be seen for the third neighbor exchange in-\nteraction J3[blue triangles in Fig. 2 (a)]. For com-\npressive strain, it prefers ferromagnetic (FM) alignment\nwhereas for tensile strain, it prefers AFM alignment. It\nlinearly decreases from J3(ε=−2 %) = 1 meV/Fe atom\ntoJ3(ε= +2 %) = −1 meV/Fe atom. Neighbors beyond\nthe third shell in the system hold a minor contribution\n(|J4,5,6,7|/|J1| ≤0.03) and do not change significantly\nwith strain. Even though J5undergoes a sign change it\nonly appears as a slight correction in our method.\nTo simplify the model, we evaluate the effective near-\nest neighbor exchange Jeffas shown in black circles of\nFig. 2 (a). It reduces its strength linearly with strain\n(from ∼ − 20 meV/Fe atom to ∼ − 15 meV/Fe atom),\nwith a rate of ∆ Jeff(ε) = 1 .33meV/Fe atom\n%strain. That means\nthat starting from BFO with no strain, collinear order ispreferred by the exchange interaction with compressive\nstrain, whereas it is weakened with tensile strain. This\ntrend is in agreement with Ref. [59], where the authors\nalso find a linear trend in the largest exchange contribu-\ntion in our range of strain [60].\nFrom Fig. 2 (a) it is evident Jeffdeviating from J1.\nThe ratio F=Jeff/J1is a good indicator for exchange\nfrustration in magnetic systems [61], where F ∼ 1 de-\nscribes a slightly exchange frustrated system and ratios\nfor 0 <F<<1 a strongly frustrated system. For BFO,\nFdecreases with strain from F(ε=−2%) = 0 .75 to\nF(ε= +2%) = 0 .60. That means, that a slight ex-\nchange frustration is induced from compressive to ten-\nsile strain. Note that for F ∼ 0.2 [62] and F ∼ 0.3\n[61], ferromagnetic skyrmions in the absence of mag-\nnetic field have been observed and predicted, respec-\ntively. These ratios are much smaller than in BiFeO 3\nand hence, magnetic skyrmions driven by exchange frus-\ntration are not expected to occur in BFO. In order to\nbe stabilized, skyrmions and other topological magnetic\nstructures rely on the interplay of magnetic exchange,\nDMI and anisotropy.\nB. Dzyaloshinskii-Moriya Interaction\nWe investigate the behavior of the DMI under strain\nin Fig. 2 (b). As we have shown for bulk BFO [46], we\nhave used the converse spin-current model [63, 64] to de-\nscribe our DFT results for the DMI [Eq. (3)]. C1(black\npoints) is decreasing almost linearly from compressive to\ntensile strain, the variation of C2(green squares) and C3\n(blue triangles) is rather small; however, they have oppo-\nsite signs at almost the same magnitude for each value of\nstrain. Effectively, the DMI (black circles) decreases with\nstrain from Ceff= 0.33 meV/Fe atom at ε=−2 % to\nCeff= 0.25 meV/Fe atom at ε= +2 %. This is a change\nrate of about ∆ Ceff(ε) =−0.02meV/Fe atom\n%strain. That means\nthat compared to R3cBFO, the DMI increases with com-\npression preferring non-collinear order and weakens with\ntensile strain which lowers the energy for non-collinear\norder. This is opposite to the trend of the exchange in-\nteraction, hence it is important to compare their ratios.\nC. Magnetocrystalline Anisotropy\nAs third magnetic interaction, we focus on the mag-\nnetocrystalline anisotropy [Fig. 2 (c)]. We determine\nthe energy of different crystallographic directions with\nrespect to the hard [111] axis. In Ref. 46, we have\nshown that in R3cbulk BFO, the magnetocrystalline\nanisotropy energy is lowest and degenerate within the\nfull (111) plane. Here, in the absence of strain ( ε= 0 %),\nwe see the same phenomenon: The calculated directions\n[110] (black points), [11 2] (light green squares), 10 1 (pur-\nple pentagons) and [1 21] (orange upside-down triangles)\nhave the same energy at −0.07 meV/Fe atom (Note that5\n0 0 1 1 -1 -1 -2 -2 2 2\nstrain εxx,yy(%) strain εxx,yy(%)0 0 1 1 -1 -1 -2 -2 2 2strain εxx,yy(%) strain εxx,yy(%)Exchange energy constants Jij (meV/Fe atom)0\n-1\n-30-25-20-1512\nE[uvw] – E[111] (meV/Fe atom)DM energy constants Cij(meV/Fe atom)00.2\n-0.2\n-0.4\n0\n-0.05\n-0.10.4\nexchange anisotropyDM interaction(a)\nJ1C1Ceff\nJ2C2J3\nC3J4J5 J6\nJ7\nJeff\nJeff (ε)Ceff (ε)\nK[110] (ε)K[112] (ε)(b)\n(c)[001] [111]\n[110]\n[112][100][121][101]\n[112] [101][110][121][001]\n[111]\nFigure 2. Trend of magnetic interactions in BiFeO 3under bi-axial strain. (a) Magnetic Heisenberg exchange interaction\nparameters J1...7from pseudo-cubic approximation for different values of strain. Note that the nearest neighbor exchange J1is\ndominant and the effective nearest neighbor exchange Jeffmatches the curvature of a energy dispersion at q→R. The error on\nJeffis stemming from ∆ Jias described in the text. (b) Evaluated spin-current driven Dzyaloshinskii-Moriya interaction ( C1...3\nfor the first three nearest neighbors and in effective nearest-neighbor approximation Ceff), (c) Energy differences for uniaxial\nanisotropy calculations of different magnetization directions with respect to the [111] direction - the inset shows a sketch with\nall directions in pseudo-cubic approximation. Note that in the absence of strain, magnetization directions within the (111)\nplane are energetically degenerate. In all graphs, dashed lines serve as guide to the eye, bold solid lines Jeff(ε),Ceff(ε) and\nK[uvw](ε) are linear fits for Jeff,Ceffand quadratic fits for the preferred anisotropy directions respectively.\na tiny deviation is caused by the imposed Ccphase at 0%\nof strain). The [100] and the [001] directions (dark green\nrhombus and lighter green x, respectively) are slightly\nhigher in energy.\nApplying strain changes this behavior and the energy\ndegeneracy of the different directions is lifted. For ten-\nsile strain ε >0, the [1 10] direction is almost constant\nin energy while the other directions become unfavorable,\nmost of all the [11 2] direction. For compressive strain,\nthe effect is reversed. The [1 10] direction gains the least\nenergy, whereas the [11 2] direction becomes the preferred\nmagnetization direction by the anisotropy. The value in-\ncreases from −0.07 meV/Fe atom without strain by 40 %\nto−0.1 meV/Fe atom at ε=−2 %. We focus on the\ntwo directions with lowest energies, [11 2] and [1 10]. The\nbehavior of their energies with respect to applied strain\ndiffers qualitatively from those of the exchange and DMI\nof panel (a,b). Here, we can apply a quadratic fit to de-\nscribe the trend of the anisotropy with strain K[uvw](ε).\nFollowing the trend for the two interactions, for larger\ncompressive strains, the [1 10] direction will be lower in\nenergy than the [11 2] direction - in accordance with DFTcalculations of Ref. [65] where the authors also find an\neasy (111) plane for 0 % of strain and the easy axis for\nlarge compression. However, in the low strain regime,\nour results slightly differ.\nWith respect to the R3cphase of BFO, compression\nleads to an increased MAE, which favors the collinear or-\nder of the G-type AFM state. For tensile strain, the MAE\nremains similar to that of bulk BFO, so a change of the\nmagnetic ground state is not expected from the MAE. In\nthe following, we will discuss the magnetic ground states\nwe derive from our calculations, compare them to cur-\nrent experimental results and show which regime of strain\ncould stabilize potential topological magnetic structures.\nIV. DISCUSSION\nFor over a decade, the different magnetic textures in\nstrained BiFeO 3have been intensively investigated in ex-\nperiments. However, while several studies show a de-\nstruction of the cycloidal order with large compressive6\nstrain of BFO on SrTiO 3substrates (estimated strain\nof−1.5 %< ε < −0.5 % depending on the thickness of\nBFO) [21, 37–40, 66, 67], others report the persistence of\ncycloidal order in the same system [68, 69]. For tensile\nstrain, the literature is not equivalently vast, but the de-\nstruction of the cycloid with large tensile strain has also\nbeen observed [21, 69]. There, the amount of applied\nstrain to stabilize collinear order differs by ∼1 %. These\nresults show that the magnetic ground state in strained\nBFO is very sensitive to the thickness and growth condi-\ntions. Different cycloids, such as the type-I and type-II\ncycloids in [1 10] and [11 2] directions, respectively as well\nas the collinear G-type AFM states are extremely close\nin energy. Hence, small distortions have a large impact\non the magnetic ground state. As we will show in the fol-\nlowing, this behaviour is in agreement with our calcula-\ntions because different magnetic states are within energy\nranges down to 10 µeV/atom.\nIn general, the magnetic ground state is driven by the\ninterplay between the magnetic exchange J, the DMI C\nand the anisotropy K. For example, when Kis large\nas compared to JandCa FM or AFM ground state is\nfavored depending on the sign of J. When Cis large, a\nnon-collinear ground state is favored since the DM favors\n90◦spin spirals. The ratio between the different contri-\nbutions is therefore important. We compare these inter-\nactions in Figure 3. In panel (a), the ratio of |Jeff|/|Ceff|\nis shown in grey (squares show the ratios of DFT deter-\nmined magnetic interactions - the bold line is the ratio of\nthe fitted interactions with strain). Increasing the ratios\nmeans that collinear order is enhanced, decreasing ratios\ndestabilize collinear magnetic ground states.\nApplying compressive strain, the exchange interaction\nslightly increases with respect to the DMI - collinear G-\ntype AFM order is more favorable than in the absence\nof strain. With tensile strain, the ratio lowers - hence,\nnon-collinear order gains energy. The anisotropy [panel\n(b)] increases for compressive strain from about 24 % to\nalmost 29 % of the DMI due to the increasing preference\nof the [11 2] direction shown in Fig. 2 (c). This enhances\nthe stability of collinear order with a quantization axis\nalong [11 2]. With tensile strain, the preferred magneti-\nzation direction changes to [1 10] with a constant value\nof the anisotropy. As the DMI decreases, the ratio K/C\nincreases and will also stabilize collinear magnetic order.\nNote that the increase of anisotropy with respect to DMI\nstarting from 0 % of strain differs for compressive and\ntensile strain - e.g. a ratio of 0.26 is reached at ∼+1.5 %\nwithin the tensile regime, but at around ∼ −0.75 % in\nthe compressive regime. Due to both increasing J/Cand\nK/Cwith compression, we expect collinear order to be\nfavored even in low strain regimes compared to the R3c\nbulk BFO phase. For tensile strain, the conclusion is not\nthat unambiguous: while the exchange loses energy with\nrespect to the DMI, the anisotropy gains energy. The\nmagnetic behavior crucially depends on both trends. In\nFig. 2, the effective exchange and DMI depend linearly on\nthe nominal strain, but the anisotropy can be described\n556065|Jeff| / |Ceff|from DFT|Keff| / |Ceff|\n0.240.250.260.270.28from fits\nfrom DFT\nfrom DFTfrom fits\nfrom fits\n0 1 -1 -2 2\nstrain εxx,yy(%)0 1 -1 -2 2strain εxx,yy(%)DW energy\nEDW(meV / Å2) \n00.050.1(a)\n(b)\n(c)PRB 103,\n214423 (2021)\nRenormalizationpossible \ntopological structuresFigure 3. Interplay between the magnetic interactions for\ndifferent values of strain. (a) Ratio of magnetic exchange\nJeffwith respect to the Dzyaloshinskii-Moriya interaction Ceff.\nNote that the large error estimate stems solely from the small\nerror ∆ Jeff. The squares show the values, determined from\nDFT whereas the solid line shows the ratioJeff(ε)/Ceff(ε)from\nthe fits of panel (a) and (b) of Fig. 2. (b) Ratio between\nanisotropy energy KeffandCeff. The circles show the values,\ndetermined from DFT whereas the solid line shows the ratio\nKeff(ε)/Ceff(ε)from the fits of panel (c) and (b) of Fig. 2, where\nKeffdenotes the direction of lowest energy. (c) Domain wall\nenergy EDW[Eq. (6)] as a function of applied strain. The blue\npoints are derived from the DFT values and the blue solid line\nshows the trend from the fits of Jeff(ε),Ceff(ε) and Keff(ε).\nThe red point shows the determined value for the R3cphase\nof BFO according to Ref. [45] where the red line denotes the\ntrend which is renormalized to that point.\nwith a quadratic behavior.\nTo quantify where isolated chiral magnetic textures\nmay be stable, we have used the domain wall energy,\ndefined as [70, 71]\nEDW(ε) =4\na2(ε)p\nJeff(ε)·Keff(ε)−2π\na2(ε)Ceff(ε),(6)\nwitha(ε) being the pseudo-cubic lattice constant for each\nvalue of strain, taken from the diagonal [111] direction\nto address for the changes in the caxis due to relax-\nation. We also renormalized ainto the [1 10] direction to\naccount for the known type-I cycloid. As compared to7\nother quantities such as the isolated skyrmion energies\nor vortex energies, the DW energy is independent of the\nsize of the magnetic textures and therefore does not re-\nquire the expensive spin dynamics energy minimization\nin a magnetic supercell of at least 100 ×100 nm at least\n[72]. The DW energy is directly quantifying the energy\ncost or gain of non collinear textures based on the effec-\ntive exchange, the anisotropy and the DMI. Furthermore,\na domain wall can be approximated as half of a cycloid,\nwhere for positive values of energy, collinear magnetic or-\nder is preferred and for negative values of EDW, a spin\nspiral ground state is expected [73].\nIn Fig. 3 (c), the DW energy with respect to the ap-\nplied strain is shown. For all values of strain, the DWs\nare higher in energy than the collinear state (blue points\nwith error bars); however, the energies are extremely\nsmall (below 0.1 meV). As described above, compressing\nBFO prefers collinear order, i.e. an increased DW energy.\nFor tensile strain, the energy differences are almost zero,\nmeaning that we predict a similar ground state compared\nto the magnetic state at 0 % strain. The trend coincides\nwith experimental observations: For compressive strain,\nthe collinear G-type AFM state is preferred, driven by\nthe large increase of anisotropy in the [11 2] direction and\nthe increase in exchange with respect to the DMI. This\nresult is robust against small errors within our calculated\ninteractions. For tensile strain, our results show the same\ndifficulties as for experimental observations: The energies\nof the states are so close that slightly differing strains, the\nmagnetic ground state can differ (cf. Refs. [21] and [69]).\nThis is indicated by the error bars from our calculations.\nAs a reference point in red, we plot the estimated\nDW energy from Ref. [45], where we investigated the\nR3cphase of BFO and slightly adapted the DFT pa-\nrameters to obtain a spin cycloid with a pitch of about\n∼62 nm. With these parameters (corresponding to\nCeff= 0.66 meV/Fe atom, Keff= 0.069 meV/Fe atom\nandJeff=−11.78 meV/Fe atom), the DW energy is\nslightly negative ( −0.017 meV/˚A). Following our esti-\nmated trend for the magnetic interactions by renormal-\nizing the DW energy from fits (red curve), we can expect\na collinear G-type AFM from ∼ −1.5 % of compressive\nstrain and a cycloidal ground state for low compressive\nstrains. In between these regimes, at compressive strain\n−1.5 %< εxx,yy <−0.5 %, we expect the transition be-\ntween non-collinear and collinear order for BFO. This\nis the region where topologically stable magnetic struc-\ntures, such as isolated skyrmions or merons and anti-\nmerons could be stabilized in BFO. An investigation of\nthose state is beyond the scope of this paper and part of\na different study.\nV. CONCLUSION\nWe have investigated the evolution of magnetic inter-\nactions in BiFeO 3under bi-axial strain between −2 %<εxx,yy <+2 % using first-principles calculations. Go-\ning from compressive to tensile strain, both exchange\nand Dzyaloshinskii-Moriya interaction decrease linearly\nwhereas the uniaxial anisotropy prefers a [11 2] magneti-\nzation direction for the calculated range of compressive\nstrain and a [1 10] axis for tensile strain. Comparing the\nratios of the three interactions, collinear G-type antifer-\nromagnetic order in BiFeO 3is strengthened for compres-\nsive strain due to a large increase of anisotropy by about\n40 % in accordance with experimental observations. For\ntensile strain our results show a similar magnetic ground\nstate as for bulk BFO, where for a slight change in the\nratios, also the collinear ground state can be favored. At\nthe transition between the cycloid and the collinear AFM\nstate for medium compressive strain more complex mag-\nnetic structures such as skyrmions or merons might occur\nwhich is why we believe this regime to be interesting for\nfurther investigation.\nACKNOWLEDGMENTS\nB.D., S.M. thank Prof. Philippe Ghosez, Louis Bas-\ntogne, Dr. Subhadeep Bandyopadhyay, Dr. He Xu\nand Prof. Matthieu J. Verstraete for helpful discus-\nsions. This work is supported by the National Nat-\nural Science Foundation of China under Grant No.\n12074277, the startup fund from Soochow University\nand the support from Priority Academic Program De-\nvelopment (PAPD) of Jiangsu Higher Education In-\nstitutions. S.M., B.D., and L.B. acknowledge the\nDARPA Grant No. HR0011727183-D18AP00010 (TEE\nProgram) and the European Union’s Horizon 2020 re-\nsearch and innovation programme under Grant Agree-\nments No. 964931 (TSAR). L.B. also thanks ARO\nGrant No. W911NF-21-1-0113 and ARO Grant No.\nW911NF-21-2-0162 (ETHOS). Computing time was pro-\nvided by ARCHER and ARCHER2 based in the United\nKingdom at National Supercomputing Service with sup-\nport from the PRACE aisbl, from the Consortium\nd’´Equipements de Calcul Intensif (FRS-FNRS Belgium\nGA 2.5020.11) and the LUMI CECI/Belgium for award-\ning this project access to the LUMI supercomputer,\nowned by the EuroHPC Joint Undertaking, hosted by\nCSC (Finland) and the LUMI consortium through LUMI\nCECI/Belgium, ULiege-NANOMAT-SKYRM-1. Sebas-\ntian Meyer is a Postdoctoral Researcher [CR] of the\nFonds de la Recherche Scientifique – FNRS. Bertrand\nDup´ e is a Research Associate [CQ] of the Fonds de la\nRecherche Scientifique – FNRS.8\n[1] A. Bogdanov and D. Yablonskii, Zh. Eksp. Teor. Fiz 95,\n178 (1989).\n[2] A. Bogdanov and A. Hubert, Journal of Magnetism and\nMagnetic Materials 138, 255 (1994).\n[3] S. M¨ uhlbauer, B. Binz, F. Jonietz, C. Pfleiderer,\nA. Rosch, A. Neubauer, R. Georgii, and P. Boni, Sci-\nence323, 915 (2009).\n[4] X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han,\nY. Matsui, N. Nagaosa, and Y. Tokura, Nature 465, 901\n(2010).\n[5] N. S. Kiselev, a. N. Bogdanov, R. Sch¨ afer, and U. K.\nR¨ oßler, Journal of Physics D: Applied Physics 44, 392001\n(2011).\n[6] A. Fert, V. Cros, and J. Sampaio, Nature Nanotechnol-\nogy8, 152 (2013).\n[7] S. Parkin and S.-H. Yang, Nature Nanotechnology 10,\n195 (2015).\n[8] C. Back, V. Cros, H. Ebert, K. Everschor-Sitte, A. Fert,\nM. Garst, T. Ma, S. Mankovsky, T. L. Monchesky,\nM. Mostovoy, N. Nagaosa, S. S. P. Parkin, C. Pfleiderer,\nN. Reyren, A. Rosch, Y. Taguchi, Y. Tokura, K. von\nBergmann, and J. Zang, Journal of Physics D: Applied\nPhysics 53, 363001 (2020).\n[9] T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich,\nNature Nanotechnology 11, 231 (2016).\n[10] X. Zhang, Y. Zhou, and M. Ezawa, Scientific Reports 6,\n24795 (2016).\n[11] J. Barker and O. A. Tretiakov, Physical Review Letters\n116, 147203 (2016).\n[12] T. Jungwirth, J. Sinova, A. Manchon, X. Marti, J. Wun-\nderlich, and C. Felser, Nature Physics 14, 200 (2018).\n[13] N. Romming, C. Hanneken, M. Menzel, J. E. Bickel,\nB. Wolter, K. von Bergmann, A. Kubetzka, and\nR. Wiesendanger, Science 341, 636 (2013).\n[14] C. Moreau-Luchaire, C. Moutafis, N. Reyren, J. Sampaio,\nC. A. F. Vaz, N. Van Horne, K. Bouzehouane, K. Garcia,\nC. Deranlot, P. Warnicke, P. Wohlh¨ uter, J.-M. George,\nM. Weigand, J. Raabe, V. Cros, and A. Fert, Nature\nNanotechnology 11, 444 (2016).\n[15] A. Soumyanarayanan, M. Raju, A. L. Gonzalez Oyarce,\nA. K. C. Tan, M.-Y. Im, A. P. Petrovi´ c, P. Ho,\nK. H. Khoo, M. Tran, C. K. Gan, F. Ernult, and\nC. Panagopoulos, Nature Materials 16, 898 (2017).\n[16] O. Boulle, J. Vogel, H. Yang, S. Pizzini, D. de Souza\nChaves, A. Locatelli, T. O. Mente¸ s, A. Sala, L. D. Buda-\nPrejbeanu, O. Klein, M. Belmeguenai, Y. Roussign´ e,\nA. Stashkevich, S. M. Ch´ erif, L. Aballe, M. Foerster,\nM. Chshiev, S. Auffret, I. M. Miron, and G. Gaudin,\nNature Nanotechnology 11, 449 (2016).\n[17] M. Herv´ e, B. Dup´ e, R. Lopes, M. B¨ ottcher, M. D.\nMartins, T. Balashov, L. Gerhard, J. Sinova, and\nW. Wulfhekel, Nature Communications 9, 1015 (2018).\n[18] S. D. Pollard, J. A. Garlow, J. Yu, Z. Wang, Y. Zhu, and\nH. Yang, Nature Communications 8, 14761 (2017).\n[19] A. J. Hatt, N. A. Spaldin, and C. Ederer, Physical Re-\nview B: Condensed Matter and Materials Physics 81,\n54109 (2010).\n[20] B. Dup´ e, S. Prosandeev, G. Geneste, B. Dkhil, and\nL. Bellaiche, Physical review letters 106, 237601 (2011).\n[21] D. Sando, A. Agbelele, D. Rahmedov, J. Liu, P. Rovil-\nlain, C. Toulouse, I. C. Infante, A. P. Pyatakov, S. Fusil,E. Jacquet, C. Carr´ et´ ero, C. Deranlot, S. Lisenkov,\nD. Wang, J.-M. L. Breton, M. Cazayous, A. Sacuto,\nJ. Juraszek, A. K. Zvezdin, L. Bellaiche, B. Dkhil,\nA. Barth´ el´ emy, and M. Bibes, Nature Materials 12, 641\n(2013).\n[22] J. M. Hu, T. Yang, and L. Q. Chen, npj Computational\nMaterials 2018 4:1 4, 1 (2018).\n[23] J. Wang, Y. Shi, and M. Kamlah, Phys. Rev. B 97,\n024429 (2018).\n[24] S. Budhathoki, A. Sapkota, K. M. Law, S. Ranjit,\nB. Nepal, B. D. Hoskins, A. S. Thind, A. Y. Borisevich,\nM. E. Jamer, T. J. Anderson, A. D. Koehler, K. D. Ho-\nbart, G. M. Stephen, D. Heiman, T. Mewes, R. Mishra,\nJ. C. Gallagher, and A. J. Hauser, Phys. Rev. B 101,\n220405(R) (2020).\n[25] Y. Zhang, J. Liu, Y. Dong, S. Wu, J. Zhang, J. Wang,\nJ. Lu, A. R¨ uckriegel, H. Wang, R. Duine, H. Yu, Z. Luo,\nK. Shen, and J. Zhang, Phys. Rev. Lett. 127, 117204\n(2021).\n[26] V. Govinden, P. Tong, X. Guo, Q. Zhang, S. Mantri,\nM. M. Seyfouri, S. Prokhorenko, Y. Nahas, Y. Wu,\nL. Bellaiche, T. Sun, H. Tian, Z. Hong, N. Valanoor,\nand D. Sando, Nature Communications 14, 4178 (2023).\n[27] B. W. Qiang, N. Togashi, S. Momose, T. Wada, T. Hajiri,\nM. Kuwahara, and H. Asano, Applied Physics Letters\n117, 142401 (2020).\n[28] P. Hemme, J.-C. Philippe, A. Medeiros, A. Alekhin,\nS. Houver, Y. Gallais, A. Sacuto, A. Forget, D. Colson,\nS. Mantri, B. Xu, L. Bellaiche, and M. Cazayous, Phys.\nRev. Lett. 131, 116801 (2023).\n[29] P. Dufour, A. Abdelsamie, J. Fischer, A. Finco,\nA. Haykal, M. F. Sarott, S. Varotto, C. Carr´ et´ ero,\nS. Collin, F. Godel, N. Jaouen, M. Viret, M. Trassin,\nK. Bouzehouane, V. Jacques, J.-Y. Chauleau, S. Fusil,\nand V. Garcia, Nano Letters 23, 9073 (2023).\n[30] Y. E. Roginskaya, Y. Y. Tomashpol’Ski ˇi, Y. N. Venevt-\nsev, V. M. Petrov, and G. S. Zhdanov, Soviet Physics\nJETP 23, 47 (1966).\n[31] J. B. Neaton, C. Ederer, U. V. Waghmare, N. A. Spaldin,\nand K. M. Rabe, Physical Review B: Condensed Matter\nand Materials Physics 71, 014113 (2005).\n[32] P. Ravindran, R. Vidya, A. Kjekshus, H. Fjellv˚ ag, and\nO. Eriksson, Physical Review B: Condensed Matter and\nMaterials Physics 74, 224412 (2006).\n[33] I. A. Kornev, S. Lisenkov, R. Haumont, B. Dkhil, and\nL. Bellaiche, Phys. Rev. Lett. 99, 227602 (2007).\n[34] R. Haumont, I. Kornev, S. Lisenkov, L. Bellaiche,\nJ. Kreisel, and B. Dkhil, Physical Review B: Condensed\nMatter and Materials Physics 78, 134108 (2008).\n[35] I. Sosnowska, T. P. Neumaier, and E. Steichele, J. Phys.\nC Solid State Phys. 15, 4835 (1982).\n[36] I. Sosnowska, R. Przenios lo, P. Fischer, and\nV. Murashov, Journal of Magnetism and Magnetic Ma-\nterials 160, 384 (1996).\n[37] F. Bai, J. Wang, M. Wuttig, J. Li, N. Wang, A. P. Py-\natakov, A. K. Zvezdin, L. E. Cross, and D. Viehland,\nApplied Physics Letters 86, 032511 (2005).\n[38] H. B´ ea, M. Bibes, S. Petit, J. Kreisel, and A. Barth´ el´ emy,\nPhilosophical Magazine Letters 87, 165 (2007).\n[39] T. Zhao, A. Scholl, F. Zavaliche, K. Lee, M. Barry, A. Do-\nran, M. P. Cruz, Y. H. Chu, C. Ederer, N. A. Spaldin,9\nR. R. Das, D. M. Kim, S. H. Baek, C. B. Eom, and\nR. Ramesh, Nature Materials 5, 823 (2006).\n[40] W. Ratcliff, D. Kan, W. Chen, S. Watson, S. Chi, R. Er-\nwin, G. J. McIntyre, S. C. Capelli, and I. Takeuchi,\nAdvanced Functional Materials 21, 1567 (2011).\n[41] www.abinit.org.\n[42] X. Gonze, B. Amadon, G. Antonius, F. Arnardi,\nL. Baguet, J.-M. Beuken, J. Bieder, F. Bottin,\nJ. Bouchet, E. Bousquet, N. Brouwer, F. Bruneval,\nG. Brunin, T. Cavignac, J.-B. Charraud, W. Chen,\nM. Cˆ ot´ e, S. Cottenier, J. Denier, G. Geneste, P. Ghosez,\nM. Giantomassi, Y. Gillet, O. Gingras, D. R. Hamann,\nG. Hautier, X. He, N. Helbig, N. Holzwarth, Y. Jia,\nF. Jollet, W. Lafargue-Dit-Hauret, K. Lejaeghere, M. A.\nMarques, A. Martin, C. Martins, H. P. Miranda, F. Nac-\ncarato, K. Persson, G. Petretto, V. Planes, Y. Pouil-\nlon, S. Prokhorenko, F. Ricci, G.-M. Rignanese, A. H.\nRomero, M. M. Schmitt, M. Torrent, M. J. van Set-\nten, B. Van Troeye, M. J. Verstraete, G. Z´ erah, and\nJ. W. Zwanziger, Computer Physics Communications\n248, 107042 (2020).\n[43] P. E. Bl¨ ochl, Phys. Rev. B 50, 17953 (1994).\n[44] C. Paillard, B. Xu, B. Dkhil, G. Geneste, and L. Bel-\nlaiche, Phys. Rev. Lett. 116, 247401 (2016).\n[45] B. Xu, S. Meyer, M. J. Verstraete, L. Bellaiche, and\nB. Dup´ e, Phys. Rev. B 103, 214423 (2021).\n[46] S. Meyer, B. Xu, M. J. Verstraete, L. Bellaiche, and\nB. Dup´ e, Phys. Rev. B 108, 024403 (2023).\n[47] H. Krakauer, M. Posternak, and A. J. Freeman, Physical\nReview B 19, 1706 (1979).\n[48] E. Wimmer, H. Krakauer, M. Weinert, and A. J. Free-\nman, Physical Review B 24, 864 (1981).\n[49] M. Weinert, E. Wimmer, and A. J. Freeman, Physical\nReview B 26, 4571 (1982).\n[50] www.flapw.de.\n[51] S. H. Vosko, L. Wilk, and M. Nusair, Canadian Journal\nof Physics 58, 1200 (1980).\n[52] L. M. Sandratskii, Journal of Physics: Condensed Matter\n3, 8565 (1991).\n[53] A. Mackintosh and O. Andersen, ed. M. Springford, Cam-\nbridge Univ. Press, London , 149 (1980).\n[54] A. Oswald, R. Zeller, P. J. Braspenning, and P. H.\nDederichs, Journal of Physics F: Metal Physics 15, 193\n(1985).\n[55] M. Heide, G. Bihlmayer, and S. Bl¨ ugel, Physica B: Con-\ndensed Matter 404, 2678 (2009).\n[56] S. Meyer, B. Dup´ e, P. Ferriani, and S. Heinze, Phys.\nRev. B 96, 094408 (2017).\n[57] I. Dzyaloshinsky, Journal of Physics and Chemistry of\nSolids 4, 241 (1958).\n[58] T. Moriya, Physical Review 120, 91 (1960).\n[59] M. R. Walden, C. V. Ciobanu, and G. L. Brennecka,\nJournal of Applied Physics 130, 104102 (2021).\n[60] Note that in [59], the authors distinguish between in-\nplane and out-of-plane components of the exchange,whereas we use a pseudo-cubic approximation.\n[61] S. Meyer, Complex spin structures in frustrated ultrathin\nfilms, Ph.D. thesis (2020).\n[62] S. Meyer, M. Perini, S. von Malottki, A. Kubetzka,\nR. Wiesendanger, K. von Bergmann, and S. Heinze, Nat.\nCommun. 10, 3823 (2019).\n[63] H. Katsura, N. Nagaosa, and A. V. Balatsky, Phys. Rev.\nLett. 95, 057205 (2005).\n[64] D. Rahmedov, D. Wang, J. ´I˜ niguez, and L. Bellaiche,\nPhys. Rev. Lett. 109, 037207 (2012).\n[65] Z. Chen, Z. Chen, C.-Y. Kuo, Y. Tang, L. R. Dedon,\nQ. Li, L. Zhang, C. Klewe, Y.-L. Huang, B. Prasad,\nA. Farhan, M. Yang, J. D. Clarkson, S. Das, S. Mani-\npatruni, A. Tanaka, P. Shafer, E. Arenholz, A. Scholl,\nY.-H. Chu, Z. Q. Qiu, Z. Hu, L.-H. Tjeng, R. Ramesh,\nL.-W. Wang, and L. W. Martin, Nature Communica-\ntions 9, 3764 (2018).\n[66] I. C. Infante, S. Lisenkov, B. Dup´ e, M. Bibes, S. Fusil,\nE. Jacquet, G. Geneste, S. Petit, A. Courtial, J. Juraszek,\nL. Bellaiche, A. Barth´ el´ emy, and B. Dkhil, Phys. Rev.\nLett. 105, 057601 (2010).\n[67] D. Sando, A. Agbelele, C. Daumont, D. Rahmedov,\nW. Ren, I. C. Infante, S. Lisenkov, S. Prosandeev,\nS. Fusil, E. Jacquet, C. Carr´ et´ ero, S. Petit, M. Cazay-\nous, J. Juraszek, J.-M. L. Breton, L. Bellaiche, B. Dkhil,\nA. Barth´ el´ emy, and M. Bibes, Philosophical Transac-\ntions of the Royal Society A: Mathematical, Physical and\nEngineering Sciences 372, 20120438 (2014).\n[68] X. Ke, P. P. Zhang, S. H. Baek, J. Zarestky, W. Tian,\nand C. B. Eom, Physical Review B 82, 134448 (2010).\n[69] A. Haykal, J. Fischer, W. Akhtar, J.-Y. Chauleau,\nD. Sando, A. Finco, F. Godel, Y. A. Birkh¨ olzer,\nC. Carr´ et´ ero, N. Jaouen, M. Bibes, M. Viret, S. Fusil,\nV. Jacques, and V. Garcia, Nature Communications 11,\n1704 (2020).\n[70] D. J. Craik, Magnetism: principles and applications\n(2003).\n[71] E. D. T. de Lacheisserie, D. Gignoux, and M. Schlenker,\nMagnetism , Vol. 1 (Springer Science & Business Media,\n2005).\n[72] Note that compared to this manuscript, in [74], the\nauthors assume a [111] directed anisotropy which ex-\nceeds the bulk value by a factor of 3 to stabilize mag-\nnetic skyrmions in BFO. Here, we have calculated the\nanisotropy based on DFT and only find an easy axis\nwithin (111) plane.\n[73] Note that contrary to the spin spiral pitch λ, the DW\nwidth ∆ ∝p\nJ/K is independent of the DMI, hence for\nspin spirals with a certain pitch length the quantitative\ndescription differs as well as its energy. However, due to\nthe simplicity of Eq. (6), the trends for the magnetic\nground state energies can be recognized directly and give\nqualitatively the same results as spin spiral energies.\n[74] Z. Li, T. Chirac, J. Tranchida, V. Garcia, S. Fusil,\nV. Jacques, J.-Y. Chauleau, and M. Viret, Phys. Rev.\nRes.5, 043109 (2023)."    },    {        "title": "2311.18360v1.Growth_of_high_quality_CrI3_single_crystals_and_engineering_of_its_magnetic_properties_via_V_and_Mn_doping.pdf",        "content": "1 \n Growth of high -quality CrI 3 single crystals and engineering of its \nmagnetic properties  via V and Mn doping  \nShuang Pan , Yuqing Bai, Jiaxuan Tang, Peihao Wang, Yurong You, Guizhou Xu*, Feng Xu*  \nMIIT Key Laboratory of Advanced Metallic and Intermetallic Materials Technology, School \nof Materials Science and Engineering, Nanjing University of Science and Technology, Nanjing \n210094, China  \n \nAbstract  \nCrI 3, as a soft van der Waals layered magnetic material, has been widely concerned and explored \nfor its magnetic complexity and tunability. In this work , high quality and large size thin CrI 3, V and \nMn doped single crystals  were prepared by chemical vapor trans fer method. A remarkable \nirreversible  Barkhausen effect was observed in CrI 3 and CrMn 0.06I3, which can be attributed to the \nlow dislocation density that facilitat es move ment of  the domain walls . In addition, the introduction \nof the doping element Mn allows higher satura tion magnetization intensity . Cr0.5V0.5I3 exhibit s \nsubstantially  increased coercivity force  and larger magnetocrystalline anisotropy  compared to CrI 3, \nwhile kept similar Curie temperature and good  environmental stability. The first principle s \ncalculation s suggest direct and narrowed band gaps in  Cr0.5V0.5I3 and VI3 comparing to CrI 3. The \nsmaller band gaps and good hard magnetic  property make Cr0.5V0.5I3 an alternative choice  to future \nresearch of spintronic devices.  \n \n \nKeywords : 2D magnetic materials; doped CrI 3 single crystal; Barkhausen effect  \n \n \n \n \n \n \n \n \n \n                                                             \n* Corresponding Author: E -mail: gzxu@njust.edu.cn  \n* Corresponding Author:  E-mail: xufeng@njust.edu.cn  2 \n 1. Introduction  \nTwo-dimensional (2D)  layered ferromagnetic (FM) materials  have been the subject of extensive \nresearch due to their highly tunable physical properties that show great potential in fundamental \nphysics and next generation spintronic s [1-4]. Typical examples of 2D layered FM materials  are \nCr2Ge2Te6 and CrI 3 [5-7], where the Curie temperatures (Tc) of bulk and monolayer CrI 3 are around \n61 K and 45 K, respectively  [8, 9] . As a layered material, CrI 3 exhibits 3D -Ising magnetism  [10, 11] , \nlow dissociation energy  [12], high in -plane stiffness  and easy exfoliation  [9, 13] , maintaining good \nand stable ferromagnetism even after exfoliation into monolayers  [3]. It is interesting to note that \nthe magnetic properties of CrI 3 depend on the number of layers : with mono -layer  and bulk being \nferromagnetic (FM) , while double -layers exhibit ing antiferromagnetic (AFM) behavior  and \nmultilayers being the coexistence of  FM and AFM state [14-16]. The unique magnetic transition  \nprovid es and increases the opportunity and possibility to explore and design low -dimensional \nintrinsic ferromagnetic semiconductors.  \nIn recent years, in addition to the studies on layer control  of CrI 3, great  efforts have been made \nto modify its magnetic behavior, such as  hydrostatic pressure  [17, 18]  and electrostatic doping  [19, \n20]. The application of hydrostatic pressure can alter the interlayer magnetis m by influenc ing the \nform and strength of the interlayer magnetic coupling . The electrostatic doping is achieved by  \napplying voltage  on CrI 3-based heterostructures. Both  method s require elaborate fabrication  of \nmicro devices, which is complex and not easy to operate . There have also been  theoretical \npredictions based on first principles calcul ations and Monte Carlo simulations that the electron or \nhole doping can enhance the  FM stability of CrI 3 as well as  its Curie temperature  [21]. However , \nthere is still rare experimental investigation about the introduction of foreign ions in CrI 3 to tune its \nmagnetic behavior.  Also in most of the current studies of CrI 3, the preparation method still follows \nthe one proposed in ref.  [22], which is modified here to increase the quality of the single crystals \nwhile greatly reducing the preparation time . \nIn this work, a high-quality and large -size CrI 3 single crystal was prepared in a relative short \ntime with the method of chemical vapor transport  (CVT ). By comprehensive study of its structure \nand physical properties, a significant Barkhausen effect  was observed during a period of increasing \nmagnetic field. Mn and V ions  are successfully introduced to the matrix CrI 3 phase by similar \nmethod. It is found that  V doped sample ( Cr0.5V0.5I3) reveals larger magnetocrystalline anisotropy  3 \n and higher coercivity  force ( Hc) compared to CrI 3, while Mn doping can possibly  affect the structure \nof CrI 3 and enhance the average magnetic moment of Cr ion , leading to the higher saturation \nmagnetization.  Our results could possibly provide better choices when constructing spintronic \ndevices based on CrI 3. \n2. Experiment al section  \n2.1 Sample preparation  \nThin  CrI 3 were synthesized by chemical vapor transfer reaction , using chromium powder and \nanhydrous iodine beads as raw materials in a molar ratio of 1:3 , sealed in a vacuumed quartz tube . \nThe length, inner diameter and wall thickness of the quartz tubes were 19.5  cm, 16  mm and 2  mm, \nrespectively.  The quartz tube was placed in a dual -temperature zone tube f urnace with  one side was \nset at 923 K , where  a total mass of 0.5  g of raw material was put , and the other side  where  the crystal \nwould grow  was kept at a temperature of 623 K . This temperature gradient was maintained for 3 \ndays and then the tube was cooled to room temperature . Finally, a large -size of CrI 3 single crystal \n(about 9  mm in dimension ) was formed, as seen in  Fig. 1, with the thickness in the range of 1 -5 m. \nThin  CrI 3 as a layered van der Waals (vdW ) material exhibits a monoclinic phase  with space \ngroup C2/m  at room temperature and a rhombohedral phase with space group  R3_\n at low temperature , \nand the corresponding phase  transiti on temperature is reported to be about 210 -220 K [14, 22, 23] . \nIn the monoclinic phase, each layer of Cr3+ is coordinated by six nonmagnetic I- to form an \noctahedron, which further shares edges to build a honeycomb network. In the rhombohedral phase, \nthe structure consists of three layers of I -Cr-I stacked along the c*-axis, and the three adjacent layers \nare b onded by weak vdW  forces to form a honeycomb lattice  [24-26].  \nThe V and Mn doped samples were both obtained by similar method , only the atomic  ratio and \nlow temperature end changes  to Cr:V /Mn:I of 0. 5:0.5:4 and 723 K , respectively . After  scanning the ir \ncomposition  by EDS, it is noted that despite the obtained V doped CrI 3 has the designed proportion , \nthe atomic rat io of Mn is only 0.06  (generally below 0.1) . By repeated measurements, the existence \nof Mn atom is confirmed, and the ratio varies between 0.04 -0.08. The low solubility of Mn in CrI 3 \nmay be originated from that there is no isostructural phase of MnI 3 to CrI 3, hence the extra Mn  is \nassumed  to occupy the interstitial sites, like the vdW gaps. But for V doping, as VI3 has the same \nstructure to CrI 3, they are supposed to substitute each other more easily. The size of the doped 4 \n samples was reduced to a maximum lateral size of about 4 -8 mm or even smaller, but the sample \nthickness was almost unchanged.  For VI 3, the growing temperatures in the high and low temperature \nregions were changed to 673 K  and 493 K , respectively . The maximum dimension of VI 3 single \ncrystal can reach to 1 cm, but it  has lower stability and easier to dissolve in air . \n \nFig. 1 Schematic illustration of the general synthetic process  for CrI 3, including VI 3, Cr0.5V0.5I3 and \nCrMn 0.06I3, along with their typical crystal structure and general size  \n2.2 Measurement and calculation details  \nThe surface morphology  of the samples w as observed using a C arl Zeiss optical microscope \n(Axio  Vert. A1 ). The X-ray powder diffraction (Bruker -AXS D8 Advance) was applied to analyze \nthe phase purity and structure of the samples.  The composition and elemental composition were \nexamined using energy dispersive spectroscopy (EDS) in a scanning electron  microscope (SEM, \nFEI Quanta  250F).  The magnetic properties are characterized using the Physical Property \nMeasurement System  (PPMS, Quantum Design), which measures temperature and magnetic field \nin the range of 10 -300 K and -4-+4 T, respectively . \nFirst principles calculations were performed with the projector augmented wave (PAW) method, \nas implemented in the Vienna ab initio simulation package (VASP) [27, 28] . The exchange \ncorrelation effect was treated wit h generalized gradient approximation (GGA) function in the form \n5 \n of Perdew -Burke -Ernzerhof (PBE) parametrization. A Hubbard U energy was added to the d \nelectron of V and Cr atom. The static self -consistency calculations were carried out on a k grid of \n9×9×9 , with the cutoff energy of 600 eV for the plane wave basis set .  \n3. Results and discussion  \nFig. 2(a) displays the XRD pattern of CrI 3 single crystal collected at 300 K. It is supposed to be \nthe monoclinic structure based on the literature [22], where the peaks can be well indexed to be ( 00l) \ndirections, indicating that the crystal surface is parallel to the ab plane. For si mplicity, c* is applied \nto represent the direct ion normal of ab plane in the following text. The distinct layered structure of \nCrI 3 is clearly shown by optical microsc opy in the two insets of Fig. 2 (a). The magnetization loops \nfor both H//c* and H//ab directions  are obtained at 10 K, as shown in Fig. 2 (b), where the initial \nmagnetization curve of H//c* saturates faster than that of H//ab, mean ing that the easy magnetization \naxis is the c*-axis. The coerciv e field along the easy axis is ~200 Oe, implying soft magnetic \nbehavior of CrI 3. In contrast to the continuous variation of the magnetic field in H//ab, there is a \nsignificant magnetization jump at μ0H≈±2.0 T for H//c*. The calculated jump degree value δ = [M \n(2.2 T)-M (2.0 T)] /M (2.0 T) is 2%, similar to that  reported in ref. [23, 29] . It is considered as a \nfield-driven AFM to FM transition , where  two types of magnetic orders  may coexist due to the \nstacking difference at low temperature , i.e. FM rhombohedral stacking and AFM monoclinic \nstacking  [23, 30] . As the crystals in the present study  were grown in a shorter time,  the stacking \ndisorder may increase, resulting the obvious jump in the  M-H curve . This metamagnetic transition  \ncan produce high magnetoresistance effects and be applied to spintronic devices, a phenomenon that \nalso appears in other literature and is thought to be common in few -layer CrI 3 [30, 31] . However, \nthe presence of a series of jumps around 0.5-1.0 T clearly distinguishes from the experiments of \nothers, which wi ll be explained in detail below. The inset in Fig. 2(b) depicts the temperature \ndependent magnetization response at μ0H=0.1 T in both H//ab  and H//c* directions, where a clear \nanisotropic magnetic response can be  observed. The data clearly indicate the FM of CrI 3 with a \nspecific Tc close to 57.9 K, which is consistent with the Tc previously reported for thin CrI 3 [23]. \nFig. 2(c) displays the c*-axis magnetization measured at various temperatures ( T= 10, 30 and 50 K) \nby sweeping the field between -4 and +4 T. It can be seen that the sub -magnetic jump at  μ0H≈±2.0T \ngradually moves to a lower field and eventually disappears at T=50 K as the temperature increases .  \nFig. 2 (d) was obtained by magnifying the M-H curve of H//c* from Fig. 2 (b). The ascending 6 \n and descending field curves do not overlap at low temperatures, indicating hysteresis behavior.  In \naddition , it is noted that a series of significant discontinuous jumps emerged  in the increase of the \nexternal magnetic field . Based on the random nature, it  is supposed to be  the Barkhausen effect, \nwhich accompanies the pinning and depinning of the domain walls . The Barkhausen signal  in a \nnormal bulk ferromagn etic material is gener ally smal l and it was first  detected by a voltage pulse \non a coil . In the polycrystalline or  ferromagnet with high hardness, the dislocation  or defect density  \nis relatively high , leading to strong  pining of the domain wall, thus Barkhausen  activity is not easy \nto generate and the signal is low  [32]. It is assumed the same reason that the Barkhausen effect is \nabsent  in previous reported single crystalline  CrI 3. However, due to the high quality  of our sample , \nthe large production of dislocations can be avoided . In addition , the obtained sample is naturally \nthin, so that the number of the vdW  gaps is smaller, facilitat ing the movement of the domain walls. \nA simple schematic  diagram about the domain wall displacement is  shown in Fig. 2 (e). The drawn \nslice is parallel to the ab plane, and the magnetization direction is along the easy magnetization axis \nof CrI 3, that is , out of ab-plane. The hollow  pining sites are  sparsely and  randomly distributed . In \ncase of applying the magnetic field parallel to the c*-axis, the volume of the domains in the same \ndirection  gradually increases, and each time the domain wall passes a line of pining sites, it can \ncause an abrupt jump in the magnetization curve  (region from  i-ii), until the domain walls  finally  \ndisappear.  The inset in Fig. 2 (d) shows similar behavior during sweeping from the negative magnetic \nfield. This behavior of Barkhausen jump , have rarely been observed in previous reports of CrI 3 and \nother similar 2D systems , reflect the high quality  and low defect of our sample s. 7 \n  \nFig. 2 (a) X-ray power diffraction (XRD)  of CrI 3. (b) M(H) curves of CrI 3 at 10 K for H//ab and \nH//c*. Inset:  M-T curves of CrI 3 at μ0H =0.1 T  for H//ab and H//c*. (c) M-H curves of CrI 3 at 10, 30 \nand 50 K for H//c*. (d) M-H curves of CrI 3 at 10 K for H//c* with a scan field between 0.3 and 1.8 \nT. Inset: partial enlarged view of M-H curves  at 10 K.  (e) Schematic diagram of CrI 3 domain wall \ndisplacement .  \nThen V and Mn element are introduced to the pristine CrI 3, aiming to tune the \nmagnetocrystalline anisotropy  or Curie temperature of CrI 3. The X-ray diffraction  patterns  of the \nobtained Cr 0.5V0.5I3 and CrMn 0.06I3 samples , along with VI 3, are shown in Fig.  3(a). All the peak \npositions highly coincide with those of CrI 3, indicating of their similar structures. The interlayer \nspacing  of VI 3, Cr 0.5V0.5I3 and CrMn 0.06I3 also approach to each other, which are around  6.68, 6.47 \n8 \n and 6.43 Å, respectively , within the error of 0.04 Å. The (002) peaks of them are enlarged to clearly  \nreveal  the evolution , where  the position of doped samples slightly shifted to right overall,  indicating  \nlittle shrink of the lattice after doping . Fig. 3(b) shows the temperature dependent magnetization \ncurves of CrI 3, VI3, Cr 0.5V0.5I3 and CrMn 0.06I3 at an applied magnetic field  μ0H=0.1 T, parallel to the \nc*-axis, where their Curie temperatures  can be  clear ly determined . By repeated measurements, the  \nTc for CrMn 0.06I3 is determined to be  57.8±0.1  K, close to that of CrI 3, while for Cr0.5V0.5I3, the Tc \nmoves to a lower temperature of 5 5.0±0.2  K. The Tc of VI 3 is lower (~50 .2±0.1  K), agree ing with \nthat reported in the literature  [33, 34] . Fig. 3 (c) compares the H//c* magnetization of all \ncompositions  at 10  K. It is found that the saturation magnetic moment of CrI 3 is about 32 A m2 kg-\n1 (2.5 μB), which is lower than its theoretical value (~3 .1 μB) [35]. The possible reason is the \ndegradation  of thin layer CrI 3 in the air, causing the increase of its real mass,  thus resulting in \ndecrease  of the magnetic moment.  It is interesting to note that despite the low doping efficiency, \nCrMn 0.06I3 exhibit a much larger saturation magnetic moment  about 44  A m2 kg-1 (3.4 μB) comparing \nto CrI 3. Considering the largest magnetic moment of Mn ion, that is 4 μB, the in crement in CrMn 0.06I3 \ncan only reach 0.24 μB. Therefore, it is assumed  that the Mn doping may also affect the structure of \nCrI 3 and enhance the  average  magnetic moment of Cr  ion. Further, the isothermal magnetization \ncurves of CrMn 0.06I3 at different temperat ures in Fig. 3 (d) exhibit  similar reversible jump at ~ ±2.0 \nT. The Barkhausen signal can also be observed at 10 K, but with weaker amplitude.  9 \n Fig.3 (a) XRD  of VI3, Cr 0.5V0.5I3 and CrMn 0.06I3 single crystals  and (002) enlarged diffraction peaks \nfor all peaks . (b) M-T curves along  H//c* at μ0H =0.1 T  for all samples . Inset: dM/dT vs T for all \nsamples . (c) M-H curves for them  at 10 K  for all samples . (d) Isothermal magnetization curves of \nCrMn 0.06I3 for T=10, 30 and 50 K measured in the out -of-plane field . \nFor V doped sample, the saturation magnetization lowers , due to the smaller moment  of V ions, \nas seen in the following part of calculation.  However, the remnant magnetization grows, as with \nincreasing V doping, the coercive field grows substantially from ~200 to ~4600 Oe, as shown in \nFig. 4(a).  Fig. 4(b) shows the in- and out -of-plane M-H curves of Cr 0.5V0.5I3 at 10  K. The anisotropic \nfield is expected to be greater than 4 T and much higher than CrI 3. Therefore, the increasing \ncoercivity can be related with growing magnetocrystalline anisotropy  in V doped CrI 3. For the \nmiddle compos ition of Cr 0.5V0.5I3, the Hc is about 1880 Oe, and it gradually decreases with \nincreasing temperature, as seen in  the inset of  Fig. 4(b). Moreover, after V doping, the AFM -FM \nsub-magn etic transit ion is disappeared. It may be caused by the stable FM exchange interaction in \nVI3, whether in few layer s or bulk [36].The Barkhausen jumps also vanish, accompanying with the \nenlargement of Hc. The hard mag netic properties of Cr0.5V0.5I3, while similar Tc and saturation \n10 \n magnetization, make it  a better  choic e comparing to CrI 3 when constructing  spintronic devices.  \n \nFig. 4 (a) M-H curves  of CrI 3, VI3 and Cr 0.5V0.5I3 at T =10 K for H//c*. (b) M-H curves  of Cr 0.5V0.5I3 \nfor H//c*and H//ab. Inset: M-H curves of Cr 0.5V0.5I3 at 10, 30 and 50 K for H//c*. \nFinally,  the electronic  structure s of the V doped sample are also investigated by first principles  \ncalculations. It is found that the  band structure  of VI 3 presents  an incorrect  metallic state without \nconsidering the electron -electron  correlation Hubbard U energy, suggesting that VI 3 is a FM Mott \ninsulator [37]. Based on the literature [38], a Hubbard U energy  of 3.68  eV was applied to the  d-\nelectron of  V atom . For CrI 3, we find the inclusion of U barely altered  the band structure. For \nconsistency, a common U value of 2.5  eV was still added to  Cr. As shown in Fig.  5, they all exhibit \ninsulating state , with a rather flat d -band lying across the Fermi level.  It is noted that  the indirect \nband gap in CrI 3 turns to direct one s in Cr 0.5V0.5I3 and VI 3 samples, accompanying with a decreasing \nbandgap magnitude.  Obvious  stronger  spin-splitting  is observed  for CrI 3 across the Fermi level, \ncorresponding to  a lager magnetic moment comparing with VI 3.  \nFig. 5 Spin resolved band structure of CrI 3 (a), Cr 0.5V0.5I3 (b) and VI 3 (c). Their bandgap values are \nalso indicated in the figure.  \n11 \n 4. Conclusion  \nIn conclusion, we have successfully synthesized large -size and high -quality CrI 3, VI3, Cr 0.5V0.5I3 \nand CrMn 0.06I3 single crystals by optimization  of the  preparation process and investigated their \nstructural and magnetic properties. Remarkable  irreversible Barkhausen jump was observed in the \nM-H curves of  thin CrI 3 crystal. The low dislocation density could possibly account for it . The large  \nBarkhausen  signal  in layered CrI 3 can offer  a good platform to investigate the mechanism  of this \neffect . Besides , the specific domain motion could be interesting due to the layered structure and is \nworth to  be investigated in more detail by future in -situ domain observation or micro -magnet ic \nsimulations.  By small doping of Mn into CrI 3, the saturation magnetization can substantially \nenhance, while the magnetization process is almost maintained.  For V doping, bo th the \nmagnetocrystalline anisotropy  and coercivity significantly grows, making  Cr0.5V0.5I3 a good hard \nmagnet , while keep ing similar  Tc and good environmental -stabil ity as CrI 3. By analyz ing their \nelectronic structure s with first pri nciples  calculations,  it is found that the indirect band gap in CrI 3 \nturns to direct ones in Cr 0.5V0.5I3 and VI 3, which also exhibit smaller gap magnitude. Moreover, the \nV contained sample exhibit large  electron correlation effect, indicating great  modulation possibility \nof their transport properties.   \nAcknowledgement s \nThis work is supported by National Natural Science Foundation of China (Grant No. \n11604148), and the Open Fund of Large Facilities in Nanjing University of Science and Technology . \nReferences  \n[1] Y . Wei, X. Tang, J. Shang, L. Ju, L. Kou, Two -dimensional functional materials: from properties to \npotential applications, Int. J. Smart Nano Mat. 11 (2020) 247 -264.  \n[2] K.L. Seyler, D. Zhong, D.R. Klein, S. Gao, X. Zhang, B. Huang, E. Navarro -Moratalla, L. Yang, D.H. \nCobden, M.A. McGuire, W. Yao, D. Xiao, P . Jarillo -Herrero, X. Xu, Ligand -field helical luminescence in a \n2D ferromagnetic insulator, Nat. Phys. 14 (2017) 277 -281.  \n[3] C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y . Xia, T. Cao, W. Bao, C. Wang, Y .  Wang, Z.Q. Qiu, R.J. Cava, S.G. \nLouie, J. Xia, X. Zhang, Discovery of intrinsic ferromagnetism in two -dimensional van der Waals crystals, \nNature 546 (2017) 265 -269.  \n[4] Y . Huang, Y .H. Pan, R. Yang, L.H. Bao, L. Meng, H.L. Luo, Y .Q. Cai, G.D. Liu, W.J. Zha o, Z. Zhou, L.M. Wu, \nZ.L. Zhu, M. Huang, L.W. Liu, L. Liu, P . Cheng, K.H. Wu, S.B. Tian, C.Z. Gu, Y .G. Shi, Y .F. Guo, Z.G. Cheng, \nJ.P . Hu, L. Zhao, G.H. Yang, E. Sutter, P . Sutter, Y .L. Wang, W. Ji, X.J. Zhou, H.J. Gao, Universal mechanical \nexfoliation of large -area 2D crystals, Nat. Commun. 11 (2020) 2453.  \n[5] S. Wei, X. Liao, C. Wang, J. Li, H. Zhang, Y . Zeng, J. Linghu, H. Jin, Y . Wei, Emerging intrinsic magnetism \nin two -dimensional materials: theory and applications, 2D Mater. 8 (2020) 012005.  12 \n [6] Z. Sh u, T. Kong, Spin stiffness of chromium -based van der Waals ferromagnets, J. Phys -Condens. Mat. \n33 (2021) 195803.  \n[7] C. Xu, J. Feng, H. Xiang, L. Bellaiche, Interplay between Kitaev interaction and single ion anisotropy \nin ferromagnetic CrI 3 and CrGeTe 3 monolayers, npj Comput. Mater. 4 (2018) 57.  \n[8] S. Ghosh, N. Stojic, N. Binggeli, Overcoming the asymmetry of the electron and hole doping for \nmagnetic transitions in bilayer CrI 3, Nanoscale 13 (2021) 9391 -9401.  \n[9] B. Huang, G. Clark, E. Navarro -Moratalla, D.R. Klein, R. Cheng, K.L. Seyler, D. Zhong, E. Schmidgall, \nM.A. McGuire, D.H. Cobden, W. Yao, D. Xiao, P . Jarillo -Herrero, X. Xu, Layer -dependent ferromagnetism \nin a van der Waals crystal down to the monolayer limit, Nature 546 (2017) 270 -273.  \n[10] G.T. L in, X. Luo, F.C. Chen, J. Yan, J.J. Gao, Y . Sun, W. Tong, P . Tong, W.J. Lu, Z.G. Sheng, W.H. Song, \nX.B. Zhu, Y .P . Sun, Critical behavior of two -dimensional intrinsically ferromagnetic semiconductor CrI 3, \nAppl. Phy. Lett. 112 (2018) 072405  \n[11] Y . Liu, C. Petrovic, Three -dimensional magnetic critical behavior in CrI 3, Phys. Rev. B 97 (2018) \n014420.  \n[12] W.B. Zhang, Q. Qu, P . Zhu, C.H. Lam, Robust intrinsic ferromagnetism and half semiconductivity in \nstable two -dimensional single -layer chromium trihalides, J . Mater. Chem. C 3 (2015) 12457 -12468.  \n[13] J. Liu, Q. Sun, Y . Kawazoe, P . Jena, Exfoliating biocompatible ferromagnetic Cr -trihalide monolayers, \nPhys. Chem. Chem. Phys. 18 (2016) 8777 -8784.  \n[14] B. Huang, J. Cenker, X. Zhang, E.L. Ray, T. Song, T. Taniguc hi, K. Watanabe, M.A. McGuire, D. Xiao, X. \nXu, Tuning inelastic light scattering via symmetry control in the two -dimensional magnet CrI 3, Nat. \nNanotechnol. 15 (2020) 212 -216.  \n[15] K. Guo, B. Deng, Z. Liu, C. Gao, Z. Shi, L. Bi, L. Zhang, H. Lu, P . Zhou, L.  Zhang, Y . Cheng, B. Peng, Layer \ndependence of stacking order in nonencapsulated few -layer CrI 3, Sci. China Mater. 63 (2019) 413 -420.  \n[16] W. Chen, Z. Sun, Z. Wang, L. Gu, X. Xu, S. Wu, C.J.S. Gao, Direct observation of van der Waals \nstacking -dependent interlayer magnetism, Science 366 (2019) 983 -987.  \n[17] S. Mondal, M. Kannan, M. Das, L. Govindaraj, R. Singha, B. Satpati, S. Arumugam, P . Mandal, Effect \nof hydrostatic pressure on ferromagnetism in two -dimensional CrI 3, Phys. Rev. B 99 (2019) 180407.  \n[18] T. Li, S. Jiang, N. Sivadas, Z. Wang, Y . Xu, D. Weber, J.E. Goldberger, K. Watanabe, T. Taniguchi, C.J. \nFennie, K. Fai Mak, J. Shan, Pressure -controlled interlayer magnetism in atomically thin CrI 3, Nat. Mater. \n18 (2019) 1303 -1308.  \n[19] S. Jiang, J. Shan,  K.F. Mak, Electric -field switching of two -dimensional van der Waals magnets, Nat. \nMater. 17 (2018) 406 -410.  \n[20] S. Jiang, L. Li, Z. Wang, K.F. Mak, J. Shan, Controlling magnetism in 2D CrI 3 by electrostatic doping, \nNat. Nanotechnol. 13 (2018) 549 -553.  \n[21] H. Wang, F. Fan, S. Zhu, H. Wu, Doping enhanced ferromagnetism and induced half -metallicity in \nCrI 3 monolayer, EPL. 114 (2016).  \n[22] M.A. McGuire, H. Dixit, V.R. Cooper, B.C. Sales, Coupling of Crystal Structure and Magnetism in the \nLayered, Ferromagnet ic Insulator CrI 3, Chem. Mater. 27 (2015) 612 -620.  \n[23] Y . Liu, L. Wu, X. Tong, J. Li, J. Tao, Y . Zhu, C. Petrovic, Thickness -dependent magnetic order in CrI 3 \nsingle crystals, Sci. Rep. 9 (2019) 13599.  \n[24] T. Song, Z. Fei, M. Yankowitz, Z. Lin, Q. Jiang, K. Hwangbo, Q. Zhang, B. Sun, T. Taniguchi, K. Watanabe, \nM.A. McGuire, D. Graf, T. Cao, J.H. Chu, D.H. Cobden, C.R. Dean, D. Xiao, X. Xu, Switching 2D magnetic \nstates via pressure tuning of layer stacking, Nat. Mater. 18 (2019) 1298 -1302.  \n[25] Y . Choi, P .J . Ryan, D. Haskel, J.L. McChesney, G. Fabbris, M.A. McGuire, J.W. Kim, Iodine orbital 13 \n moment and chromium anisotropy contributions to CrI 3 magnetism, Appl. Phys. Lett. 117 (2020) 022411.  \n[26] A. Frisk, L.B. Duffy, S. Zhang, G. van der Laan, T. Hesjedal, Ma gnetic X -ray spectroscopy of two -\ndimensional CrI 3 layers, Mater. Lett. 232 (2018) 5 -7. \n[27] G. Kresse, J. Furthmiiller, Efficiency of ab -initio total energy calculations for metals and \nsemiconductors using a plane -wave basis set, Comput. Mater. Sci. 6 (199 6) 15 -50. \n[28] P .E. Blochl, Projector augmented -wave method, Phys. Rev. B 50 (1994) 17953 -17979.  \n[29] V. Bayzi Isfahani, J. Filipe Horta Belo da Silva, L. Boddapati, A.G. Rolo, R.M.F. Baptista, F.L. Deepak, \nJ.P .E. de Araújo, E. de Matos Gomes, B.G. Almeida , Functionalized magnetic composite nano/microfibres \nwith highly oriented van der Waals CrI 3 inclusions by electrospinning, Nanotechnology 32 (2021) 145703.  \n[30] B. Niu, T. Su, B.A. Francisco, S. Ghosh, F. Kargar, X. Huang, M. Lohmann, J. Li, Y . Xu, T. Tan iguchi, K. \nWatanabe, D. Wu, A. Balandin, J. Shi, Y .T. Cui, Coexistence of Magnetic Orders in Two -Dimensional \nMagnet CrI 3, Nano Lett. 20 (2020) 553 -558.  \n[31] H.H. Kim, S. Jiang, B. Yang, S. Zhong, S. Tian, C. Li, H. Lei, J. Shan, K.F. Mak, A.W. Tsen, Magnet o-\nMemristive Switching in a 2D Layer Antiferromagnet, Adv. Mater. 32 (2020) e1905433.  \n[32] J.D. C, Dynamics of domain magnetization and the Barkhausen effect, Czech. J. Physi. 50 (2000) \n893-924.  \n[33] A. Koriki, M. Míšek, J. Pospíšil, M. Kratochvílová, K. C arva, J. Prokleška, P . Doležal, J. Kaštil, S. Son, \nJ.G. Park, V. Sechovský, Magnetic anisotropy in the van der Waals ferromagnet VI 3, Phys. Rev. B 103 \n(2021) 174401.  \n[34] Y . Liu, M. Abeykoon, C. Petrovic, Critical behavior and magnetocaloric effect in VI 3, Phys. Rev. Res. \n2 (2020) 013013.  \n[35] J.F. Dillon, C.E. Olson, Magnetization, Resonance, and Optical Properties of the Ferromagnet CrI 3, J. \nAppl. Phys. 36 (1965) 1259 -1260.  \n[36] S. Tian, J.F. Zhang, C. Li, T. Ying, S. Li, X. Zhang, K. Liu, H. Lei, Ferroma gnetic van der Waals Crystal \nVI3, J. Am. Chem. Soc. 141 (2019) 5326 -5333.  \n[37] P . Doležal, M. Kratochvílová, V. Holý, P . Čermák, V. Sechovský, M. Dušek, M. Míšek, T. Chakraborty, \nY . Noda, S. Son, J. -G. Park, Crystal structures and phase transitions of the van der Waals ferromagnet VI 3, \nPhysical Review Materials 3 (2019).  \n[38] Y .P . Wang, M.Q. Long, Electronic and magnetic properties of van der Waals ferromagnetic \nsemiconductor VI 3, Phys. Rev.B 101 (2020) 024411.  \n "    },    {        "title": "2312.02701v1.Iron_and_gold_thin_films__first_principles_study.pdf",        "content": "arXiv:2312.02701v1  [cond-mat.mtrl-sci]  5 Dec 2023Iron and gold thin films: first-principles study\nJustyna Rychły-Gruszecka, Hubert Głowi´ nski, Justyn Snar ski-Adamski, Piotr Ku´ swik, Mirosław Werwi´ nski\nInstitute of Molecular Physics, Polish Academy of Sciences , Smoluchowskiego 17, 60-179 Pozna´ n, Poland\nUsing density functional theory, we carried out systematic calculations for a series of ultrathin iron layers with thic knesses\nranging from one atomic monolayer to eleven monolayers (up t o about 1.5 nm). We considered three cases: (1) iron layers bo th on\na gold substrate and coated with gold, (2) iron layers on a gol d substrate but without coverage and (3) free-standing iron layers\nadjacent to a vacuum. For our models we chose initial bcc Fe(0 01) surfaces and fcc Au(001) substrates. Based on the calcul ations, we\ndetermined the details of the geometry and magnetic propert ies of the systems. We calculate lattice parameters, magnet ic moments,\nCurie temperatures and magnetocrystalline anisotropy ene rgies. From the thickness dependence we determined the volu me and\nsurface contributions to the magnetic anisotropy constant . The further analysis allowed us to determine the thickness ranges of the\noccurrence of perpendicular magnetic anisotropy, as well a s the effect of thickness and the presence of a substrate and c ap layer\non the direction of the magnetization easy axis.\nIndex Terms —iron, gold, thin films, magnetic anisotropy, magnetocryst alline anisotropy, density functional theory\nOne of the most important properties of a magnetic thin\nfilm is its magnetocrystalline anisotropy energy (MAE) sinc e\nit is one of the key factors determining the orientation of\nmagnetization at remanence. Sufficiently strong anisotrop y\nwith the axis of easy magnetization perpendicular to the\nsurface of the layers allows perpendicular magnetization o f\nthe sample in remanence, which, for example, is required\nin magnetic tunnel junctions to maximize resistance output .\nThin film systems with perpendicular anisotropy, due to the\npossibility of obtaining magnetic textures with perpendic ularly\nmagnetized very narrow domains, are also important for data\nstorage applications. In this context, layered systems are\nparticularly interesting because of their ability to tune e ffective\nmaterial parameters, such as magnetic anisotropy. Among su ch\nsystems, iron-based layered heterostructures are of signi ficant\ninterest.\nFirst-principles calculations of iron layers on gold have\nbeen going on since the 1980s [1]. It was predicted, for\nexample, that in Fe/Au multilayers above the thickness of\nthree atomic monolayers of iron, there is a change in the\nsign of the MAE interpreted as a change in the direction of\nmagnetization of the layer from perpendicular (perpendicu lar\nmagnetic anisotropy - PMA) to lying in the plane of the\nlayer [2]. The calculations showed a significant increase in the\nmagnetic moment values of several atomic layers at both the\nFe surface and Fe/Au interfaces. Earlier calculations, how ever\nsophisticated, usually failed to reproduce accurate magne tic\nresults due to the potential shape approximations used, as\nwell as the most commonly assumed bulk values of lattice\nparameters and the lack of optimization of geometries of lay ers\nand heterostructures.\nIn this work we performed first-principles calculations for\ndifferent thicknesses of iron layers deposited on fcc-gold\nsubstrates. We have considered a range of iron layers from\nsingle atomic monolayer up to eleven monolayers. In all the\nheterostructures considered, the thickness of the gold lay ers\nequals eleven atomic monolayers. We consider three types\nof structures. The first type is sandwich-type structures – f or\nwhich layers of iron and gold are alternated with each other,\nFig. 1. Selected structural models of Au 11Fenmultilayers. In each case,\nseveral unit cells are shown in the layer plain. In contrast, only one unit cell\nis shown in the direction perpendicular to the layer plain.\nsee Fig. 1. The second type is iron layers with two different\nsurfaces - surrounded by a gold substrate on one side and a\nvacuum on the other. The third type is free-standing iron lay ers\nadjacent to a vacuum.\nWe performed density functional theory (DFT) calculations\nusing the full-potential local orbital (FPLO) code [3] in th e\nFPLO18.00-52 version. Using Hellmann–Feynman forces, we\noptimized the lattice parameter aand the Wyckoff positions z\nwhich allowed us to determine the lattice parameter c.\nFigure 2 shows the dependence of lattice parameters on\nFe layer thickness. Assuming perfect epitaxial growth of Fe\nlayers on Au substrate in the model, the heterostructure mod el\nis described by a single unit cell with a lattice parameter a\ncommon to Fe and Au layers. In the case of gold layers, the\npresented lattice parameter adefines an fcc Au unit cell in\na body centered tetragonal (bct) representation. Due to the\ninfluence of the Au substrate, the obtained a(c) values are\nhigher (lower) than the values for the bulk Fe structure. TheFig. 2. Lattice parameters calculated for Fe layers in Au 11Fenmultilayers\nand for Fe layers in Au 11Fenheterostructure with vacuum (uncovered Fe\nsurface). Calculations were performed with the FPLO18 code using the PBE\nfunctional.\nFig. 3. Magnetocrystalline anisotropy energy (MAE) as a fun ction of Fe\nlayer thickness (expressed as the number of atomic monolaye rsn) for Fe\nlayers in Au 11Fenmultilayers and for Fe layers in Au 11Fenheterostructure\nwith vacuum (uncovered Fe surface). Results for free-stand ing Fe layers are\nincluded as a reference. Calculations were performed with t he FPLO18 code\nusing the PBE functional.\nhighest (lowest) values of obtained a(c) occur for the thinnest\none-atom-thick bcc-Fe layer. (In addition, the figure shows the\nvalues of lattice parameters afor bulk fcc Au and bcc Fe.) The\nthinner the Fe layer, the more its structure deviates from cu bic\ntoward a flattened tetragonal cell with a lattice parameter r atio\nc/a of about 0.9. As is known from previous studies, such\na relatively strong deformation due to the influence of the\nsubstrate and overlay layer can have a large impact on the\nvalue and direction of magnetic anisotropy.\nFigure 3 shows the dependence of magnetocrystalline\nanisotropy energy (MAE) on the thickness of the Fe layer\nfor both Au/Fe multilayers, as well as for Fe layers on Au\nsubstrate without a cap Au layer. Multilayers tend to magnet ize\nout-of-plane for thin Fe layers. However, for thicker Fe lay ers\nthey magnetize in-plane. This is similar behavior to that\nfound for free-standing Fe layers. However, this relations hip is0.0 0.5 1.0 1.5 2.0 2.5 3.001234\nKu = -0.97 MJ m-3 + 2 * 0.86 mJ m-2 * 1/tEffective uniaxial anisotropy Ku ( MJ m-3 )\nInverse of thickness 1/ t (109 m-1)fitting formula:  Ku = Kv + 2Ks/tn = 2\n3\n4\n5\n6\n78\nFig. 4. Effective uniaxial anisotropy ( Ku) as a function of inverse thickness\n(1/t) for free-standing Fe layers with thickness of two to eight m onolayers\n(bct phase). Fit allows determination of surface magnetic a nisotropy ( Ks).\nreversed for Fe layers on Au substrate surrounded by vacuum.\nThe above result points to the presence of a cap layer as\nanother important factor affecting the direction and value of\nmagnetic anisotropy.\nFigure 4 shows the effective uniaxial anisotropy ( Ku) of\nthin Fe films as a function of the inverse of thickness (1/ t).\nFitting with the equation Ku=Kv+ 2Ks/t[4] allows us\nto determine the surface magnetic anisotropy ( Ks), which\nis equal to 0.86 mJ m−2and is in good agreement with\nexperimental results. It is known that the value of Ksstrongly\ndepends on the type of interface. Heinrich et al. [5] observe d\nthe strongest surface anisotropy Ksfor the Fe(001)/vacuum\ninterface (0.96 mJ m−2(0.96 ergs/cm2)), followed by the\nFe/Ag (0.81 mJ m−2) and the Fe/Au (0.47 mJ m−2) interfaces,\nall at room temperature. At lower temperatures (77 K), the\nlatter two results are slightly higher (no low Tdata for the\nFe/vacuum interface).\nACKNOWLEDGEMENT\nWe acknowledge the financial support of the Na-\ntional Science Center Poland under the decision DEC-\n2018/30/E/ST3/00267 (SONATA-BIS 8). Part of the compu-\ntations were performed on resources provided by the Poznan\nSupercomputing and Networking Center (PSNC).\nREFERENCES\n[1] C. Li, A. J. Freeman, and C. L. Fu, “Monolayer magnetism: E lectronic\nand magnetic properties of Fe/Au(001),” J. Magn. Magn. Mater. , vol. 75,\nno. 3, pp. 201–208, Dec. 1988.\n[2] L. Szunyogh, B. ´Ujfalussy, and P. Weinberger, “Magnetic anisotropy of\niron multilayers on Au(001): First-principles calculatio ns in terms of the\nfully relativistic spin-polarized screened KKR method,” Phys. Rev. B ,\nvol. 51, no. 15, pp. 9552–9559, Apr. 1995.\n[3] K. Koepernik and H. Eschrig, “Full-potential nonorthog onal local-orbital\nminimum-basis band-structure scheme,” Phys. Rev. B , vol. 59, no. 3, pp.\n1743–1757, 1999.\n[4] K. Kyuno, J.-G. Ha, R. Yamamoto, and S. Asano, “First-Pri nciples\nCalculation of the Magnetic Anisotropy Energies of Ag/Fe(0 01) and\nAu/Fe(001) Multilayers,” J. Phys. Soc. Jpn. , vol. 65, no. 5, pp. 1334–\n1339, May 1996.\n[5] B. Heinrich, Z. Celinski, J. F. Cochran, A. S. Arrott, and K. Myrtle,\n“Magnetic anisotropies in single and multilayered structu res (invited),” J.\nAppl. Phys. , vol. 70, no. 10, pp. 5769–5774, Nov. 1991."    },    {        "title": "2401.02275v1.Anisotropy_of_the_anomalous_Hall_effect_in_the_altermagnet_candidate_Mn__5_Si__3__films.pdf",        "content": "Anisotropy of the anomalous Hall effect in the altermagnet candidate Mn 5Si3films\nMiina Leivisk ¨a,1Javier Rial,1Anton ´ın Badura,2, 3Rafael Lopes Seeger,1Isma ¨ıla Kounta,4Sebastian Beckert,5\nDominik Kriegner,5, 3Isabelle Joumard,1Eva Schmoranzerov ´a,2Jairo Sinova,6Olena Gomonay,6Andy Thomas,5, 7\nSebastian T. B. Goennenwein,8Helena Reichlov ´a,5Libor ˇSmejkal,6, 3Lisa Michez,4Tom´aˇs Jungwirth,3and Vincent Baltz1\n1Univ. Grenoble Alpes, CNRS, CEA, Grenoble INP , IRIG-SPINTEC, F-38000 Grenoble\n2Department of Chemical Physics and Optics, Faculty of Mathematics and Physics, Charles University, Prague, Czechia\n3Institute of Physics, Czech Academy of Sciences, Prague, Czechia\n4Aix-Marseille University, CNRS, CINaM, Marseille, France\n5Institute of Solid State and Materials Physics, TU Dresden, Dresden, Germany\n6Institute for Physics, Johannes Gutenberg University Mainz, Mainz, Germany\n7Leibniz Institute of Solid State and Materials Science (IFW Dresden), Helmholtzstr. 20, 01069 Dresden, Germany\n8Department of Physics, University of Konstanz, Konstanz, Germany\nAltermagnets are compensated magnets belonging to spin symmetry groups that allow alternating spin polar-\nizations both in the coordinate space of the crystal and in the momentum space of the electronic structure. In\nthese materials the anisotropic local crystal environment of the different sublattices lowers the symmetry of the\nsystem so that the opposite-spin sublattices are connected only by rotations, which results in an unconventional\nspin-polarized band structure in the momentum space. This low symmetry of the crystal structure is expected to\nbe reflected in the anisotropy of the anomalous Hall effect. In this work, we study the anisotropy of the anoma-\nlous Hall effect in epitaxial thin films of Mn 5Si3, an altermagnetic candidate material. We first demonstrate\na change in the relative N ´eel vector orientation when rotating the external field orientation through systematic\nchanges in both the anomalous Hall effect and the anisotropic longitudinal magnetoresistance. We then show\nthat the anomalous Hall effect in this material is anisotropic with the N ´eel vector orientation relative to the\ncrystal structure and that this anisotropy requires high crystal quality and unlikely correlates with the magne-\ntocrystalline anisotropy. Our results provide further systematic support to the case for considering epitaxial thin\nfilms of Mn 5Si3as an altermagnetic candidate material.\nI. INTRODUCTION\nThe physical properties of magnetic materials are governed\nby the intrinsic symmetries of their crystal and spin structures,\nwhich are represented by symmetry groups. A good example\nin the realm of magnetotransport properties is the anomalous\nHall effect (AHE) [1], where a longitudinal electric field E\ngenerates a transversal Hall current jHthat is perpendicular\nto the Hall vector ( h= (σzy,σxz,σyx)T, where σi jare the off-\ndiagonal terms of the conductivity matrix) i.e. jH=h×E.\nAshis an axial vector and odd upon time-reversal ( T), non-\nvanishing AHE is only observed in systems where the sym-\nmetries allow for such a vector. In ferromagnetic materials\nthe net magnetization is also a T-odd axial vector and thus\nhis allowed [1] while in magnetic materials with compen-\nsated order, his allowed upon sufficient lowering of the crys-\ntal symmetries [2–7]. In antiferromagnetic materials [8, 9]\nwhere the opposite-spin sublattices are connected by transla-\ntion or inversion, his forbidden by symmetry, while in alter-\nmagnetic materials [10] this connecting symmetry is rotation,\nwhich allows for h[11–17]. This lowered symmetry in alter-\nmagnets stems from the two sublattices being inequivalent in\nterms of the local crystal environment. Altermagnets are char-\nacterised by alternating spin-splitting of the energy bands [18–\n22] and they show various core spintronic properties such as\nthe AHE presented above, the Nernst effect (thermal counter-\npart of the AHE) [23], spin-polarized current generation [24–\n26], as well as giant and tunnel magnetoresistances [27–29] in\nheterostructures.\nIn polycrystalline soft ferromagnets, his allowed by sym-\nmetry for any magnetization orientation. Therefore, the AHEis proportional to the component of the magnetization that is\nparallel to h, i.e. to Mscosθ, where MSis the saturation mag-\nnetization and θis the angle between the external field and\nthe film normal, assuming that the field is strong enough to\nsaturate the magnetization. Exceptions to this behavior can\narise in crystalline ferromagnets [30, 31] or ferromagnets with\nstrong magnetocrystalline anisotropy, which causes a lag be-\ntween the external field and the magnetization. In altermag-\nnets, specific N ´eel vector ( L) orientations can either allow or\nexclude the Hall vector h(L)by symmetry [7, 11, 13] so that\nanisotropic AHE beyond Mscosθcan arise.\nAnisotropic AHE beyond Mscosθin compensated magnets\nconstitutes a hallmark of altermagnetism, as has been recently\ndemonstrated in RuO 2[11] (intrinsic d-wave altermagnet, i.e.\nthe electronic structure displays four lobes with alternating\nspin-polarization) and MnTe showing spontaneous AHE [13]\n(intrinsic g-wave, which acquires a d-wave character in the\npresence of spin-orbit coupling). Both RuO 2and MnTe owe\ntheir altermagnetism to the presence of non-magnetic atoms\nthat lower the symmetry of the system such that the opposite-\nspin sublattices are not connected by either inversion or trans-\nlation [10]. The saturation of Lin RuO 2and MnTe requires\nhigh fields of 68 [11, 12] and 4 T [13], respectively, so for fur-\nther studies on the anisotropy of AHE in altermagnets a new\nmaterial candidate where L-saturation is realized at smaller\nfields would be ideal.\nRecently, epitaxial thin films of Mn 5Si3have emerged as\nsuch candidate [15] as the collinear compensated phase of\nMn 5Si3with a hexagonal unit cell and four magnetic Mn\natoms on the Mn2 sites in a checkerboard arrangement of the\nsublattices shown in Figure 1(a) fulfills the symmetry crite-arXiv:2401.02275v1  [cond-mat.mes-hall]  4 Jan 20242\na)\nαxy\nz\nVxxVxyH\nxyz\nθ[0001]\nIx\nIxd) c)\nMn2Mn1\nSi\nα\n[1010]_\n[1120]_\n[0110]_\n[1100]_[2110]__\n[1210]__b)\n100 µm\nI I\nVxx Vxx VxyVxy\n+_\n+__\n+\n[2110]_ _\n[0110]_\nFIG. 1. (Color Online) (a) The hexagonal unit cell of epitaxial thin\nfilms of Mn 5Si3. The two magnetically ordered opposite-spin sub-\nlattices of Mn-atoms are highlighted in red and blue. Note, that there\nare two other options (’quadrupole domains’) for the checkerboard\ndistribution of the magnetic Mn atoms among the Mn2 sites. (b) Op-\ntical image of a typical Hall bar device. (c) Illustration of a Hall\nbar with the angle θdefining the external field Horientation relative\nto the film normal z, which coincides with the [0001] crystal axis\nof Mn 5Si3. (d) Top view of a Hall bar showing the angle α, which\nrefers to the orientation of the current channel Ixrelative to the crys-\ntal axis [0¯110]. Varying αis achieved through the fabrication of a set\nof Hall bars oriented along several crystal directions.\nria for altermagnetism [15]. Note, that in the bulk phase of\nMn 5Si3the hexagonal unit cell is not preserved during the\nmagnetic ordering and the opposite-spin sublattices become\nconnected by translation making bulk Mn 5Si3antiferromag-\nnetic [32–35]. However, when grown as epitaxial thin films\n[16] the substrate stabilises the hexagonal unit cell of Mn 5Si3\nover the entire temperature range [15], fulfilling the symme-\ntry requirements for making epitaxial Mn 5Si3a potential can-\ndidate for the altermagnetic phase with a d-wave symmetry.\nIn good accordance with the theoretical predictions and the\nstructural properties epitaxial Mn 5Si3exhibits a sizable spon-\ntaneous AHE despite a vanishingly small net magnetization\n[15].\nIn this work we investigate the anisotropy of AHE in epi-\ntaxial Mn 5Si3in terms of the external field orientation rela-\ntive to the crystal structure. We show that a systematic rota-\ntion of the field in different crystal planes inflicts systematic\nchanges on the magnetotransport properties, indicating that\nwe are able to control the relative L-orientation with experi-\nmentally reasonable external field strengths of the order of 2\nT. Moreover, we show that changing the relative L-orientation\nyields anisotropic behavior of the AHE, beyond cos θ, that\nrequires a high crystal quality of the epitaxial films and is\nunlikely related to the magnetocrystalline anisotropy, which\nfurther demonstrates that the properties of epitaxial Mn 5Si3\nclosely match those of altermagnetic materials.II. METHODS\nThe epitaxial thin films of Mn 5Si3are grown on Si(111)\nsubstrates by molecular beam epitaxy using optimized growth\nparameters for realizing films with high crystalline qual-\nity and maximum Mn 5Si3content [16]. A few nm-thick\nMnSi phase acts as a thin seed layer aiding the nucleation\nof Mn 5Si3on Si(111) [16]. Unless otherwise specified, the\nfilms used in this work contain 96 % Mn 5Si3and 4 % MnSi\nand they have a thickness of ∼17 nm. The high crys-\ntallinity and the crystal orientation of the films have been\nverified using transmission electron microscopy (TEM) in\nour previous works [15, 16]. The epitaxial relationship be-\ntween the Si substrate, MnSi seed layer and Mn 5Si3film\nis Si(111) [1¯10]//MnSi(111) [¯211]//Mn 5Si3(0001) [01¯10]. The\nfilms were patterned into microscopic Hall bars shown in Fig-\nure 1(b) using optical UV lithography and subsequent ion\nbeam etching. The orientations of the Hall bar current chan-\nnels relative to the crystal axis [2¯1¯10]are defined by the angle\nαas illustrated in Figure 1(d). We measure simultaneously\nthe longitudinal ( Vxx) and transversal ( Vxy) voltages (denoted\nin Figure 1(b,c)) as a function of the external field strength\nand orientation to obtain the corresponding field- and angular-\ndependences of the longitudinal ( ρxx) and transversal ( ρxy) re-\nsistivities. We apply the external field always in the (yz) plane,\ni.e. the plane perpendicular to the current channel. Its exact\norientation relative to the [0001] crystal axis (film normal) is\ngiven by the angle θ, as illustrated in Figure 1(c). Note that we\nwill demonstrate later on in the text that the important param-\neter causing the anisotropic AHE is the crystal plane in which\nthe external field rotates rather than the exact crystal orienta-\ntion of the current channel - we alter the Hall bar orientation\nsimply to ensure that the external field and the current chan-\nnel are always perpendicular to each other unless otherwise\nstated. The current density is 3 ×105A cm−2throughout this\nwork. For all the measurements presented in this work, the\ntemperature was kept at 110 K, which is between the two crit-\nical temperatures of our films [15, 16]. At this temperature the\nfilms exhibit characteristic features of an altermagnetic phase,\ni.e. a sizable spontaneous AHE despite a vanishingly small\nnet magnetization [15].\nIII. RESULTS AND DISCUSSIONS\nFirst, we study how the AHE resistivity ( ρAHE) evolves\nwhen the external field orientation, parameterized by θ, is\nchanged. We consider a Hall bar where the external field\nrotates in the (0 ¯110) plane ( α= 90 deg). We have first\nmeasured the hysteresis loops of ρxyfor selected θ, from\nwhich a linear ρOHE background (slope ∼0.045 µΩcm\nT−1), originating from the ordinary Hall effect has been re-\nmoved and only the antisymmetric-in-field part of the signal\nis used: ρAHE=ρxy,asym= (ρxy(H)−ρxy(−H))/2. The sym-\nmetric part ρxy,sym= (ρxy(H) +ρxy(−H))/2, is negligible in\nour sample at the specific measurement temperature support-\ning the collinear spin configuration [17]. The different contri-\nbutions to ρxyare summarized in Figure 2(a). For the remain-3\nb)\nAsym.Sym.OHEa) θ = 0 deg 30 deg 90 deg\n150 deg 180 deg[0001] [2110]\n[0001]___\nFIG. 2. (Color Online) (a) Disentangling the different contributions - symmetric-in-field ρxy,sym, antisymmetric-in-field ρxy,asym, and ordinary\nHall effect ρOHE - to the field-dependence of transverse resistivity ρxy. The signal of interest here is the antisymmetric contribution, relating\nto the anomalous Hall effect (AHE) resistivity ρAHE. (b) The hysteresis loops of the corresponding AHE conductivity σAHE=ρAHE/ρ2xxat\nselected external field orientations θat 110 K. The sweep direction is indicated by the small arrows. The σAHE is at maximum at θ=0 deg\n(field out-of-plane) and decreases when the field rotates away from it, almost vanishing at θ=0 deg (field in-plane). This is in contrast with\nthe weak magnetization M, which is isotropic for θ=0 and 90 deg. Note that we have removed the linear diamagnetic background in the\nhysteresis loops of M.\ning of the paper, we will convert ρAHE to AHE conductivity\nσAHE=ρAHE/ρ2\nxx, which is used to characterize the intrinsic\neffects ( ρxxat 110 K is ∼180µΩcm).\nThe hysteresis loops of σAHE for various θare shown\nin Figure 2(b). We observe that the σAHE at saturation\n(σAHE ,max) is maximum when the field is close to the out-of-\nplane [0001] axis ( θ=0 deg) and vanishes when the field\nis close to the in-plane [2 ¯1¯10] axis ( θ=90 deg). As ex-\npected from the previous AHE study on the epitaxial Mn 5Si3\n[15], 180 degree rotation of this field changes the polarity of\nσAHE. The evolution of σAHE for the field orientations be-\ntween [0001] and [000 ¯1] is better visible in Figure 3 where\nwe directly show σAHE as a function of θfor a 4 T exter-\nnal field. Between [0001] and [000 ¯1]σAHE shows step-like\nbehavior: when the field is oriented close to either [0001] or\n[2¯1¯10] (in-plane) crystal axes the slope is considerably more\nflat than in-between these directions. We note that the back-\nward sweep does not trace the forward sweep because the sat-\nuration is not reached at 4 T at all field orientations as seen in\nFigure 2(b).\nTo unravel the possible origin of this anisotropic behavior\nof the AHE, we will first study the option of weak magne-\ntization giving a contribution. Analogously to the AHE ex-\nperiments, we have measured the sample magnetization M\nalong the external field for two field orientations θ=0 and\n90 deg as shown in Figures 2b and 3. After removing a linear\ndiamagnetic background, we observe that the magnetization\nloops contain two components: a non-hysteretic part that re-\nlates to the substrate and the sample holder, and a hysteretic\npart that we attribute to the Mn 5Si3film as the magnitude of\nits coercivity agrees with that of the AHE-signal. This hys-\nteretic component suggests a slight spontaneous canting of\nthe magnetic sublattice moments and can play a role in thereversal mechanism of the N ´eel vector. We observe that the\nweak magnetization of the hysteretic component is small, i.e.\n∼0.1µB/u.c., and this value for θ=0 and 90 deg is similar.\n[0001] [2110]_ _\nV\nx\nyHθ[0001]\nIx[2110]_ _\n[0001]_[2110][0001]\n__\nMn2Mn1\nSi\nVxy\nFIG. 3. (Color Online) The AHE conductivity σAHE as a function\nof the orientation θof a 4 T external field, at 110 K (full line), for a\ncurrent applied along the [01¯10]direction ( α= 90 deg). The forward\nsweep is indicated by the full line and the backward sweep by the dot-\nted line. The AHE signal does not correlate with the θ-dependence\nof the weak saturation magnetization Msof the hysteretic component\n(circles), which is isotropic at θ=0 and 90 deg. Inset: measurement\ngeometry recalled and crystal structure of the (01 ¯10) plane in which\nthe external field His rotated.4\nMn5Si3:MnSiMn5Si3:MnSia)\nb)\nFIG. 4. (Color Online) The field strength (a) and θ-orientation (b)\ndependence (at 2 T) of σAHE at 110 K for epitaxial films contain-\ning 96:4 and 75:25 ratios of Mn 5Si3:MnSi. The former is an alter-\nmagnetic candidate and shows sizable AHE with large coercivity and\nunconventional step-like θ-dependence. The latter is expected not al-\ntermagnetic and it shows smaller AHE with vanishing coercivity and\nconventional isotropic θ-dependence.\nWe note that the behavior shown in Figure 2(b) would also be\nexpected for a ferromagnet as Mz∼0 atθ=90 deg, while\nthe wide plateau in Figure 3 could be a consequence of the\nmagnetocrystalline anisotropy. We will now discuss further\nexperiments to show that the anisotropic AHE is more likely\nto originate from the altermagnetic phase of our films.\nFirst, we have measured the AHE in Mn 5Si3films with\na larger proportion of MnSi, namely 75 % Mn 5Si3and 25\n% MnSi. Increasing the MnSi content deteriorates the crys-\ntallinity of the Mn 5Si3[16], leading to a more textured film.\nWe expect this to either prohibit altermagnetism as the strain\nnecessary for stabilizing the hexagonal unit cell can relax at\nthe grain boundaries, or even if the hexagonal unit cell is re-\ntained in the individual grains their different crystal orienta-\ntions result in overall cancelling out of the altermagnetic ef-\nfects. Figure 4 confirms that the magnetotransport properties\nof this film are different from those of the 96:4 Mn 5Si3:MnSi)\nMn 5Si3epitaxial films with higher crystal quality. The field\ndependence of σAHEof the 75:25 control sample has a reduced\namplitude and a vanishing coercivity, while the θ-dependence\nof the σAHE shows roughly a cos θbehavior, pointing to a\nferromagnetic character. These results are in stark contrast\nwith the behavior of the 96:4 sample and demonstrate that the\nAHE with anisotropic θ-dependence and non-zero coercivity\nrequires a high crystal quality of the Mn 5Si3epitaxial films,\nwhich is consistent with the altermagnetic phase being at the\norigin of the AHE requiring the system to have a specific sym-\nmetry. Moreover, this shows that the spurious MnSi phase\nx10-3\n120 deg\nα = 0 deg\n60 deg\n30 deg\n90 deg\n150 deg\n[0001] [0001]\n[0001] [0001] [2110]__\n[2110]__\n[0110]_α\nHIx\nH[0001]\nIx\nVxyθ\n[0110]_ _\n_\nMn2Mn1\nSi\n[0001][0001]\n[2110][0110]\n___x10-3FIG. 5. (Color Online) The normalized AHE conductivity\nσAHE/σAHE ,maxand the longitudinal anisotropic magnetoresistance\n(AMR ∥) when the external field is rotated in different crystal planes.\nIn both cases, two distinct behaviors can be observed depending on\nwhether the crystal plane is equivalent to (0 ¯110) or (2 ¯1¯10). Here, the\nfield strength is 4 T and the temperature is 110 K. The inserts recall\nthe geometry used and the planes in which the field is rotated.\nwith Tc∼40 K is also unlikely origin of the AHE signal as\ntheσAHEshould in that case be stronger for the 75:25 sample.\nNext, to further study the anisotropy of AHE, we vary sys-\ntematically the crystal plane in which the external field and in\neffect the N ´eel vector L(confirmed later using the anisotropic\nmagnetoresistance measurements) is rotated and again ob-\nserve how the σAHE is affected. This was realized by pat-\nterning multiple Hall bars with their current channels oriented\nalong different crystal axes. We recall that these orientations\nare denoted by α, which is the angle between the current chan-\nnel and the crystal axis [2¯1¯10]as was shown in Figure 1(d).\nFor each Hall bar we rotate the field in a plane perpendicular\nto the current channel and thus effectively we are changing the\ncrystal plane in which the field rotates. We have selected α’s\nthat reflect the hexagonal crystal structure of our material and\ncan be divided into two sets: for α=0,60,120 deg the current\nchannel is along [2 ¯1¯10] or equivalent so the field rotates in the\ncrystal plane (2 ¯1¯10) or equivalent; for α=30,90,150 deg the\ncurrent channel is along [0 ¯110] or equivalent and the field ro-\ntates in the crystal plane (0 ¯110) or equivalent. In Figure 5 we\nshow the θ-dependence of the σAHE (with 4 T field) for the\ntwo sets of αs. For each set, σAHEis maximum when the field\nis close to [0001] axis but significant differences arise when\nthe field orients towards the sample plane: for α=30,90,1505\n[0110]_[2110]_ _\nα = 0 deg\nα = 90 degField in (0110)_\n_\nField in (2110)_\nFIG. 6. (Color Online) Two perpendicular Hall bars measured simul-\ntaneously at 110 K with 2 T field. The anisotropic θ-dependence of\nσAHE relates to the crystal plane in which the field rotates instead of\nthe crystal axis along which the current channel orients.\ndeg, when the field orients in the [2 ¯1¯10] direction or equiva-\nlent, we observe a wide plateau around θ= 90 deg, familiar\nfrom Figure 3, while for α= 0, 60, 120 deg when the field ori-\nents in the [0 ¯110] direction or equivalent we observe a steeper\nlinear slope instead. We have verified that this anisotropy per-\nsists also at 2 and 7 T fields. The hexagonal symmetry of the\nAHE signals for the different αsuggests that the three possi-\nbilities for the distribution of the magnetic Mn2 atoms (Figure\n1) are likely present in equal populations: if each Mn2 site is\non average magnetic, the α-dependence of σAHE should re-\nflect the hexagonal symmetry of the crystal structure. If one of\nthese ’quadrupole domains’ dominated, lower symmetry for\ntheα-dependence of σAHEwould be expected. We have veri-\nfied that the key factor responsible for the change of behavior\nbetween the different αs is the crystal plane in which the field\nrotates rather than the crystal axis along which the current\nchannel is oriented, as observed in Figure 6. We note that for\nFigure 6 we have used a different sample of the same material,\nwhich explains some of the differences in the θ-dependence\nof the σAHEcompared to the other figures. However, the over-\nall behavior is the same, which confirms the reproducibility\nof the results discussed here. In summary, the results in Fig-\nures 5 and 6 allow us to conclude that the anisotropic behavior\ndepends on the L-orientation relative to the hexagonal crystal\nsymmetry of the Mn 5Si3epitaxial films.\nWe also consider magnetocrystalline anisotropy as the ori-\ngin of the anisotropic AHE unlikely: if we interpret the results\nshown in Figures 3 and 5 considering that the different crys-\ntal planes have strongly different magnetocrystalline energy\nlandscapes, then for α= 30, 90, and 150 deg it is more ener-\ngetically favorable for Lto remain aligned along a crystal axis\nyielding σAHE=0 atθ=90 deg compared to α= 0, 60, and120 deg. However, the hysteresis loops measured at θ=90\ndeg for both α=0 (not shown) and 90 deg (Figure 2(b)) su-\nperimpose at 4 T, indicating that it is unlikely that at this field\nLis lagging behind the external field around θ=90 deg for\nα= 30, 90, and 150 deg more than for α= 0, 60, and 120 deg.\nMoreover, we observe the anisotropy in the α-dependence of\nσAHE persisting up to 7 T, which is well above the satura-\ntion field of σAHE at any θ, as shown in Figure 2(b). Along\nwith our demonstration that the anisotropic AHE is crystal\nquality-dependent and depends on the L-orientation relative\nto the hexagonal crystal symmetry, these observations provide\nfurther support to the main AHE contribution in our Mn 5Si3\nfilms being from altermagnetism. These results constitute the\nmain conclusion of our work and they systematically support\nthe previous calculations and experiments on the same system\n[15].\nFinally, we confirm the ability to manipulate Lwith the ex-\nternal field by measuring the anisotropic magnetoresistance\n(AMR), which is known to depend on the L-orientation.\nAs shown in Figure 5, we have measured the α- and θ-\ndependence of the longitudinal AMR (AMR ||), which relates\ntoρxxas AMR ||(θ) = (ρxx(θ)−⟨ρxx⟩)/⟨ρxx⟩, where ⟨ρxx⟩in-\ndicates ρxxaveraged over all angles. AMR is an even func-\ntion of the field and known to have two components: a non-\ncrystalline one scaling with the angle between the order pa-\nrameter and the current directions, and a crystalline one re-\nlating to the order parameter orientation relative to the crystal\naxes [36]. We show the θ-dependence for α= 0, 60, 120 deg\nandα= 30, 90, 150 deg Hall bars in Figure 5 at conditions\nidentical to the AHE measurements. Again we observe dif-\nferent behaviors for the two sets of α, as expected from the\ncrystalline AMR depending on the crystal plane in which the\norder parameter rotates: for α= 0, 60, 120 deg the depen-\ndence follows a cos2 θbehavior while for α= 30, 90, 150 deg\nhigher order even cosines appear in the signal.\nAlthough the exact N ´eel vector orientation and rotation un-\nder the action of the external magnetic field cannot be deduced\nfrom the results presented, in this final paragraph we wish to\nopen a discussion on the interpretation of the anisotropic AHE\nin the light of the symmetries of the epitaxial Mn 5Si3consid-\nering this material to be a strong candidate for the altermag-\nnetic phase. First, it should be noted that in our measure-\nment geometry (Figure 1), the changes in σAHE correspond\nto changes in the component of halong the [0001] direction\n(out-of-plane). In altermagnets, the presence and orientation\nofhdepend on the L-vector orientation because the latter in-\nfluences the symmetries of the system. While accurate cal-\nculation of the entire θ-dependence of h(L)is inhibited by\nthe unknown relevant micromagnetic parameters, the value\nofσAHE for some specific orientations can be inferred from\nsymmetry-considerations. The symmetry analysis on the ex-\npected spin structure of the epitaxial Mn 5Si3shown in Fig-\nure 1(a) indicates that for non-zero σAHE theL-vector can-\nnot point along the [0001] axis ( hexcluded), along the main\nin-plane axes (only hin-plane allowed), or in the planes of\nthe high-symmetry in-plane axes highlighted in Figure 1(a)\n(only hin-plane allowed). Here, the crystal symmetries of the\nplanes corresponding to α=0, 60, and 120 deg and α=30,6\n90, and 150 deg differ, as seen in Figure 5 and therefore the\nσAHElikely differ, too.\nIV . CONCLUSION\nIn conclusion, the implications of our measurements on\nthe external field orientation dependence of σAHE in epitax-\nial Mn 5Si3films are two-fold. First, the systematic changes\ninσAHEsupported by the systematic changes in AMR ||upon\nthe field rotation underlines our ability to manipulate the L-\norientation with an external field of the order of 2 T or more.\nThis is an important premise for further experimentally feasi-\nble spin transport studies in altermagnets where control over\ntheL-orientation is critical. Second, our key result is that the\nσAHEin epitaxial Mn 5Si3films shows anisotropic behavior as\na function of the external field-induced orientation of Lrel-ative to the underlying crystal structure, and this anisotropic\nbehavior depends on the crystal quality and is unlikely to re-\nlate to the magnetocrystalline anisotropy. This observation of\nthe anisotropic σAHE adds to the growing array of properties\nof epitaxial Mn 5Si3advocating the altermagnetic phase in this\nmaterial.\nACKNOWLEDGMENTS\nThis work was supported by the French national research\nagency (ANR) and the Deutsche Forschungsgemeinschaft\n(DFG) (Project MATHEEIAS- Grant No. ANR-20-CE92-\n0049-01 / DFG-445976410). H. R., E. S., S. T. B. G. and\nA. T. are supported by the DFG-GACR grant no. 490730630.\nJ. R. acknowledges MINECO for the Margaritas Salas pro-\ngram. D. K. acknowledges the academy of sciences of the\nCzech Republic for the Lumina quaeruntur program.\n[1] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P.\nOng, Anomalous Hall effect, Rev. Mod. Phys. 82, 1539 (2010).\n[2] J. K ¨ubler and C. Felser, Non-collinear antiferromagnets and the\nanomalous Hall effect, EPL (Europhysics Letters) 108, 67001\n(2014).\n[3] H. Chen, Q. Niu, and A. H. MacDonald, Anomalous Hall effect\narising from noncollinear antiferromagnetism, Phys. Rev. Lett.\n112, 017205 (2014).\n[4] S. Nakatsuji, N. Kiyohara, and T. Higo, Large anomalous Hall\neffect in a non-collinear antiferromagnet at room temperature,\nNature 527, 212 (2015).\n[5] A. K. Nayak, J. E. Fischer, Y . Sun, B. Yan, J. Karel, A. C. Ko-\nmarek, C. Shekhar, N. Kumar, W. Schnelle, J. K ¨ubler, C. Felser,\nand S. S. P. Parkin, Large anomalous Hall effect driven by a\nnonvanishing Berry curvature in the noncolinear antiferromag-\nnet M 3Ge, Science Advances 2, e1501870 (2016).\n[6] L. ˇSmejkal, R. Gonz ´alez-Hern ´andez, T. Jungwirth, and\nJ. Sinova, Crystal time-reversal symmetry breaking and spon-\ntaneous Hall effect in collinear antiferromagnets, Science Ad-\nvances 6, eaaz8809 (2020).\n[7] L. ˇSmejkal, A. H. MacDonald, J. Sinova, S. Nakatsuji, and\nT. Jungwirth, Anomalous Hall antiferromagnets, Nature Re-\nviews Materials 7, 482 (2022).\n[8] T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, Antifer-\nromagnetic spintronics, Nature Nanotechnology 11, 231–241\n(2016).\n[9] V . Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and\nY . Tserkovnyak, Antiferromagnetic spintronics, Rev. Mod.\nPhys. 90, 015005 (2018).\n[10] L. ˇSmejkal, J. Sinova, and T. Jungwirth, Beyond conventional\nferromagnetism and antiferromagnetism: A phase with nonrel-\nativistic spin and crystal rotation symmetry, Phys. Rev. X 12,\n031042 (2022).\n[11] Z. Feng, X. Zhou, L. ˇSmejkal, L. Wu, Z. Zhu, H. Guo,\nR. Gonz ´alez-Hern ´andez, X. Wang, H. Yan, P. Qin, X. Zhang,\nH. Wu, H. Chen, Z. Meng, L. Liu, Z. Xia, J. Sinova, T. Jung-\nwirth, and Z. Liu, An anomalous Hall effect in altermagnetic\nruthenium dioxide, Nature Electronics 5, 735 (2022).[12] T. Tschirner, P. Keßler, R. D. Gonzalez Betancourt, T. Kotte,\nD. Kriegner, B. B ¨uchner, J. Dufouleur, M. Kamp, V . Jovic,\nL. Smejkal, J. Sinova, R. Claessen, T. Jungwirth, S. Moser,\nH. Reichlova, and L. Veyrat, Saturation of the anomalous hall\neffect at high magnetic fields in altermagnetic ruo2, APL Mate-\nrials 11, 10.1063/5.0160335 (2023).\n[13] R. D. Gonzalez Betancourt, J. Zub ´aˇc, R. Gonzalez-Hernandez,\nK. Geishendorf, Z. ˇSob´aˇn, G. Springholz, K. Olejn ´ık,\nL.ˇSmejkal, J. Sinova, T. Jungwirth, S. T. B. Goennenwein,\nA. Thomas, H. Reichlov ´a, J. ˇZelezn ´y, and D. Kriegner, Spon-\ntaneous anomalous Hall effect arising from an unconventional\ncompensated magnetic phase in a semiconductor, Phys. Rev.\nLett. 130, 036702 (2023).\n[14] K. P. Kluczyk, K. Gas, M. J. Grzybowski, P. Skupi ´nski, M. A.\nBorysiewicz, T. Fas, J. Suffczy ´nski, J. Z. Domagala, K. Grasza,\nA. Mycielski, M. Baj, K. H. Ahn, K. V ´yborn ´y, M. Sawicki, and\nM. Gryglas-Borysiewicz, Coexistence of anomalous Hall effect\nand weak net magnetization in collinear antiferromagnet MnTe\n(2023), arXiv:2310.09134.\n[15] H. Reichlov ´a, R. L. Seeger, R. Gonz ´alez-Hern ´andez, I. Kounta,\nR. Schlitz, D. Kriegner, P. Ritzinger, M. Lammel, M. Leivisk ¨a,\nV . Pet ˇr´ıˇcek, P. Dole ˇzal, E. Schmoranzerov ´a, A. Bad’ura,\nA. Thomas, V . Baltz, L. Michez, J. Sinova, S. T. B. Goennen-\nwein, T. Jungwirth, and L. ˇSmejkal, Macroscopic time rever-\nsal symmetry breaking by staggered spin-momentum interac-\ntion (2021), arXiv:2012.15651 [cond-mat.mes-hall].\n[16] I. Kounta, H. Reichlova, D. Kriegner, R. Lopes Seeger,\nA. Bad’ura, M. Leiviska, A. Boussadi, V . Heresanu, S. Bertaina,\nM. Petit, E. Schmoranzerova, L. Smejkal, J. Sinova, T. Jung-\nwirth, V . Baltz, S. T. B. Goennenwein, and L. Michez, Com-\npetitive actions of MnSi in the epitaxial growth of Mn 5Si3thin\nfilms on Si(111), Phys. Rev. Mater. 7, 024416 (2023).\n[17] A. Badura, D. Kriegner, E. Schmoranzerov ´a, K. V ´yborn ´y,\nM. Leivisk ¨a, R. L. Seeger, V . Baltz, D. Scheffler, S. Beckert,\nI. Kounta, L. Michez, L. ˇSmejkal, J. Sinova, S. T. B. Goennen-\nwein, J. ˇZelezn ´y, and H. Reichlov ´a, Even-in-magnetic-field part\nof transverse resistivity as a probe of magnetic order (2023),\narXiv:2311.14498.7\n[18] O. Fedchenko, J. Minar, A. Akashdeep, S. W. D’Souza, D. Vasi-\nlyev, O. Tkach, L. Odenbreit, Q. L. Nguyen, D. Kutnyakhov,\nN. Wind, L. Wenthaus, M. Scholz, K. Rossnagel, M. Hoesch,\nM. Aeschlimann, B. Stadtmueller, M. Klaeui, G. Schoenhense,\nG. Jakob, T. Jungwirth, L. Smejkal, J. Sinova, and H. J. Elmers,\nObservation of time-reversal symmetry breaking in the band\nstructure of altermagnetic RuO 2(2023), arXiv:2306.02170.\n[19] S. Lee, S. Lee, S. Jung, J. Jung, D. Kim, Y . Lee, B. Seok,\nJ. Kim, B. G. Park, L. ˇSmejkal, C.-J. Kang, and C. Kim,\nBroken Kramers’ degeneracy in altermagnetic MnTe (2023),\narXiv:2308.11180.\n[20] T. Osumi, S. Souma, T. Aoyama, K. Yamauchi, A. Honma,\nK. Nakayama, T. Takahashi, K. Ohgushi, and T. Sato, Obser-\nvation of giant band splitting in altermagnetic MnTe (2023),\narXiv:2308.10117.\n[21] S. Reimers, L. Odenbreit, L. Smejkal, V . N. Strocov, P. Con-\nstantinou, A. B. Hellenes, R. J. Ubiergo, W. H. Campos, V . K.\nBharadwaj, A. Chakraborty, T. Denneulin, W. Shi, R. E. Dunin-\nBorkowski, S. Das, M. Kl ¨aui, J. Sinova, and M. Jourdan, Direct\nobservation of altermagnetic band splitting in CrSb thin films\n(2023), arXiv:2310.17280.\n[22] R. M. Sattigeri, G. Cuono, and C. Autieri, Altermagnetic sur-\nface states: towards the observation and utilization of alter-\nmagnetism in thin films, interfaces and topological materials,\nNanoscale 15, 16998–17005 (2023).\n[23] X. Zhou, W. Feng, R.-W. Zhang, L. Smejkal, J. Sinova,\nY . Mokrousov, and Y . Yao, Crystal thermal transport in alter-\nmagnetic RuO 2(2023), arXiv:2305.01410.\n[24] R. Gonz ´alez-Hern ´andez, L. ˇSmejkal, K. V ´yborn ´y, Y . Yahagi,\nJ. Sinova, T. c. v. Jungwirth, and J. ˇZelezn ´y, Efficient electri-\ncal spin splitter based on nonrelativistic collinear antiferromag-\nnetism, Phys. Rev. Lett. 126, 127701 (2021).\n[25] A. Bose, N. J. Schreiber, R. Jain, D.-F. Shao, H. P. Nair, J. Sun,\nX. S. Zhang, D. A. Muller, E. Y . Tsymbal, D. G. Schlom, and\nD. C. Ralph, Tilted spin current generated by the collinear anti-\nferromagnet ruthenium dioxide, Nature Electronics 5, 267–274\n(2022).\n[26] H. Bai, L. Han, X. Y . Feng, Y . J. Zhou, R. X. Su, Q. Wang, L. Y .\nLiao, W. X. Zhu, X. Z. Chen, F. Pan, X. L. Fan, and C. Song,\nObservation of spin splitting torque in a collinear antiferromag-net RuO 2, Phys. Rev. Lett. 128, 197202 (2022).\n[27] B. Chi, L. Jiang, Y . Zhu, G. Yu, C. Wan, J. Zhang, and X. Han,\nCrystal facet orientated altermagnets for detecting ferromag-\nnetic and antiferromagnetic states by giant tunneling magne-\ntoresistance effect (2023), arXiv:2309.09561.\n[28] L. ˇSmejkal, A. B. Hellenes, R. Gonz ´alez-Hern ´andez, J. Sinova,\nand T. Jungwirth, Giant and Tunneling magnetoresistance in\nunconventional collinear antiferromagnets with nonrelativistic\nspin-momentum coupling, Phys. Rev. X 12, 011028 (2022).\n[29] S. Xu, Z. Zhang, F. Mahfouzi, Y . Huang, H. Cheng, B. Dai,\nW. Cai, K. Shi, D. Zhu, Z. Guo, C. Cao, Y . Liu, A. Fert,\nN. Kioussis, K. L. Wang, Y . Zhang., and W. Zhao, Spin-flop\nmagnetoresistance in a collinear antiferromagnetic tunnel junc-\ntion (2023), arXiv:2311.02458.\n[30] W. Limmer, M. Glunk, J. Daeubler, T. Hummel, W. Schoch,\nR. Sauer, C. Bihler, H. Huebl, M. S. Brandt, and S. T. B.\nGoennenwein, Angle-dependent magnetotransport in cubic and\ntetragonal ferromagnets: Application to (001)- and (113)a-\noriented (Ga,Mn)As, Phys. Rev. B 74, 205205 (2006).\n[31] E. Roman, Y . Mokrousov, and I. Souza, Orientation dependence\nof the intrinsic anomalous Hall effect in hcp cobalt, Phys. Rev.\nLett. 103, 097203 (2009).\n[32] G. H. Lander, P. J. Brown, and J. B. Forsyth, The antiferromag-\nnetic structure of Mn 5Si3, Proceedings of the Physical Society\n91, 332 (1967).\n[33] A. Z. Menshikov, A. P. V okhmyanin, and Y . A. Dorofeev, Mag-\nnetic structure and phase transformations in Mn 5Si3, physica\nstatus solidi (b) 158, 319 (1990).\n[34] P. J. Brown, J. B. Forsyth, V . Nunez, and F. Tasset, The low-\ntemperature antiferromagnetic structure of Mn 5Si3revised in\nthe light of neutron polarimetry, Journal of Physics: Condensed\nMatter 4, 10025 (1992).\n[35] P. J. Brown and J. B. Forsyth, Antiferromagnetism in Mn 5Si3:\nthe magnetic structure of the AF2 phase at 70 K, Journal of\nPhysics: Condensed Matter 7, 7619 (1995).\n[36] A. W. Rushforth, K. V ´yborn ´y, C. S. King, K. W. Edmonds,\nR. P. Campion, C. T. Foxon, J. Wunderlich, A. C. Irvine,\nP. Va ˇsek, V . Nov ´ak, K. Olejn ´ık, J. Sinova, T. Jungwirth, and\nB. L. Gallagher, Anisotropic magnetoresistance components in\n(Ga,Mn)As, Phys. Rev. Lett. 99, 147207 (2007)."    },    {        "title": "2401.02809v1.Integrated_ab_initio_modelling_of_atomic_order_and_magnetic_anisotropy_for_rare_earth_free_magnet_design__effects_of_alloying_additions_in___mathrm_L_1_0__FeNi.pdf",        "content": "Integrated ab initio modelling of atomic order and magnetic anisotropy for\nrare-earth-free magnet design: effects of alloying additions in L1 0FeNi\nChristopher D. Woodgate,1,∗Laura H. Lewis,2, 3and Julie B. Staunton1,†\n1Department of Physics, University of Warwick, Coventry, CV4 7AL, United Kingdom\n2Department of Chemical Engineering, Northeastern University, Boston, MA 02115, USA\n3Department of Mechanical and Industrial Engineering,\nNortheastern University, Boston, MA 02115, USA\n(Dated: January 5, 2024)\nWe describe an integrated modelling approach to accelerate the search for novel, single-phase,\nmulticomponent materials with high magnetocrystalline anisotropy (MCA). For a given system we\npredict the nature of atomic ordering, its dependence on the magnetic state, and then proceed\nto describe the consequent MCA. Crucially, within our modelling framework, the same ab initio\ndescription of the material’s electronic structure determines both aspects. We demonstrate this\nholistic method by studying the effects of alloying additions in FeNi, examining systems with the\ngeneral stoichiometry Fe 4Ni3X, including X=Pt, Pd, Al, and Co. The atomic ordering behaviour\npredicted on adding these elements, fundamental for determining a material’s MCA, is rich and\nvaried. Equiatomic FeNi has been reported to require ferromagnetic order to establish the tetragonal\nL10order suited for significant MCA. Our results show that when alloying additions are included in\nthis material, annealing in an applied magnetic field and/or below a material’s Curie temperature\nmay also promote tetragonal order, along with an appreciable effect on the predicted MCA.\nI. INTRODUCTION\nPermanent magnets play a crucial role in modern soci-\nety, with applications in electrical power generation and\nconversion, essential in the global transition to clean en-\nergy. The ‘gold-standard’ of current permanent magnetic\nmaterials are those based on the rare-earth elements,\nsuch as Nd 2Fe14B [1, 2] and SmCo 5[3], which have large\nmagnetic energy products and are widely used in appli-\ncations [4]. However, the rare-earth elements are a con-\nstrained resource, with issues around the stability of the\nsupply chain, price volatility, and the environmental im-\npact of their extraction [5–7]. There is therefore a desire\nto develop new materials which use reduced concentra-\ntions of rare-earth elements, or no rare-earth elements\nat all. In addition, there is currently a ‘gap’ in perma-\nnent magnet performance between the rare-earth-based\nsupermagnets and the significantly weaker oxide ferrite\nmagnets [8].\nComputational modelling approaches have an impor-\ntant role to play in the process of materials discovery. Not\nonly can they be used to screen potential compositions\nto suggest which are stable and have the desired physi-\ncal properties for a particular application, but via an ab\ninitio description of materials’ electronic structure, they\nalso have the potential to advance fundamental physical\nunderstanding of why existing materials perform the way\nthey do. This insight can facilitate the optimisation of\nmaterials for applications, as well as guide the search for\nimproved compositions and processing techniques.\nIn this work, we demonstrate a new computational\n∗Christopher.Woodgate@warwick.ac.uk\n†J.B.Staunton@warwick.ac.uktechnique which enables holistic assessment of a mag-\nnetic alloy’s phase behaviour and subsequent magne-\ntocrystalline anisotropy, and is therefore suitable for the\ndiscovery of new magnetic materials. As a case study, we\napply the approach to suggest new alloy compositions for\nuse as advanced permanent magnets. We begin by de-\nscribing in detail the binary FeNi system, which is cur-\nrently under consideration as a potential ‘gap’ magnet [9].\nWe then proceed to consider the addition of a third al-\nloying element, studying systems with the composition\nFe4Ni3X. The approach is based on complementary tech-\nniques for studying the phase behaviour of multicompo-\nnent alloys [10–13] and the magnetocrystalline anisotropy\nof (partially) ordered intermetallic phases [14–16]. Cru-\ncially, both the phase behaviour of a material and its\nmagnetocrystalline anisotropy are examined on the same\nfirst-principles footing, enabling us to examine how the\nelectronic ‘glue’ of an alloy drives atomic ordering and\nproduces subsequent magnetic properties. This holistic\napproach is important; it will always be possible to com-\nputationally design metastable structures with extraordi-\nnary magnetic properties, but for any proposed material\nto be useful, it must be able to be synthesised in the\nlaboratory.\nOur workflow for holistically studying the phase sta-\nbility and subsequent magnetic properties of a given al-\nloy composition, illustrated in Fig. 1, proceeds as follows.\nFirst, using the Korringa-Kohn-Rostoker (KKR) [17] for-\nmulation of density functional theory (DFT) [18], we gen-\nerate the self-consistent potentials and associated elec-\ntron density of the disordered alloy—the solid solution—\nwhere averaging over substitutional atomic disorder is\nperformed using the coherent potential approximation\n(CPA) [19]. Then, using the S(2)theory for multicompo-\nnent alloys [10–13], we perform a linear response analysis\nto rigorously calculate the two-point correlation function,arXiv:2401.02809v1  [cond-mat.mtrl-sci]  5 Jan 20242\nDFT calculation for\ndisordered alloy\n, , ...Linear response calculationObtain ordered structures\nvia concentration wavesPredict MAE at both zero\nand finite temperature\nParameters for atomistic\nmodelling\n Hutsepot, JuKKR, ...  Theory Atomistic Methods MARMOT\nMonte Carlo simulations to\nobtain ordered supercells\nFIG. 1. Visualisation of the workflow used in this paper for holistically assessing the phase behaviour and subsequent magnetic\nproperties of a given alloy composition.\nan atomic short-range order (ASRO) parameter, ab ini-\ntio. From this linear-response calculation, we can infer\n(partial) chemical orderings directly using a concentra-\ntion wave analysis [20], or instead fit to a pairwise Hamil-\ntonian describing the internal energy of the alloy that is\nsuitable for atomistic modelling. Once the (partially) or-\ndered phase of interest has been established, we are able\nto use a DFT-based description of the magnetic torque at\nboth zero and finite temperature to assess the system’s\nmagnetocrystalline anisotropy [14, 16].\nA. Recipe for a Good Permanent Magnet\nBefore presenting the case study of ordered FeNi, it is\nworth discussing the desirable intrinsic physical proper-\nties a material must possess to be suitable for advanced\npermanent magnetic applications. Putting aside the con-\nsideration of shape anisotropy, used to produce magnets\nsuch as those of the Alnico family [21], there are three in-\ntrinsic physical quantities which determine a material’s\nsuitability for use as an advanced permanent magnet.\nFirst is the Curie temperature, TC, the temperature be-\nlow which spontaneous magnetisation occurs even in the\nabsence of an applied magnetic field [22]. Second is the\nsaturation magnetisation, MS, which is the material’s\nmagnetic polarization when fully magnetised [23]. Third\nis its magnetocrystalline anisotropy (MCA), which mea-\nsures the energetic cost of rotating the magnetisation vec-\ntor,M, away from the material’s energetically favorable\n‘easy axis’ established at the atomic level by spin-orbit\ncoupling and crystal symmetry considerations [24]. The\nMCA can be related to a material’s coercivity, an ex-\ntrinsic, structure-sensitive property that is a measure of\nthe energetic cost of demagnetising the material [25]. In\nturn, the coercivity leads to a figure of merit known as the\n‘maximum energy product’ which can be used to quan-\ntitatively compare the performance different permanentmagnets at a given temperature [25].\nIdeally, a permanent magnet will have a high Curie\ntemperature, well above its desired operating tempera-\nture, to ensure that its magnetisation remains large dur-\ning operation. Inevitably, this means that 3 dtransition\nmetals must be incorporated, such as Fe or Co which have\nhigh elemental Curie temperatures of 1043 and 1388 K\nrespectively [26]. While it is the highly localised felec-\ntrons of elements such as Nd and Sm which give rise to\nthe extraordinary magnetic energy products of the rare-\nearth supermagnets [27], all use significant quantities of\nFe or Co in their compositions. This is because these\ntransition metals often stabilise magnetic order at high\ntemperatures and can therefore play a crucial role in de-\ntermining materials’ hard magnetic properties during op-\neration [28]. Similarly, a good permanent magnet will\nhave a large saturation magnetisation, which again re-\nquires the use of 3 delements such as Fe and Co in any\ncandidate composition.\nThe MCA of a material is determined by a mate-\nrial’s crystal structure and the associated local crystallo-\ngraphic environment around magnetic moments. Perma-\nnent magnets generally possess a uniaxial crystal struc-\nture [25], typically tetragonal or hexagonal. These uni-\naxial crystal structures give rise to uniaxial magnetocrys-\ntalline anisotropy, K, to leading order, of the form [24]\nK=K1sin2θ (1)\nwhere θdescribes the angle of rotation of the magneti-\nsation away from the easy crystal axis, and K1is the\nmagnetocrystalline anisotropy constant. In conjunction\nwith a large saturation magnetisation, MS, a value of\nK1larger than approximately 1 MJm−3, is considered to\nbe necessary for a material to be of use as a permanent\nmagnet for advanced applications [8].\nIn summary, in the search for novel permanent mag-\nnets for applications it is necessary to search for a stable\nintermetallic compound with a uniaxial crystal structure3\n(and associated uniaxial magnetocrystalline anisotropy),\na high Curie temperature, and high saturation mag-\nnetisation. In addition, with regard to the aforemen-\ntioned economic and environmental considerations, com-\npositions should be pursued that use reduced concentra-\ntions of constrained elements such as Nd, Sm, and Co, as\nwell as noble metals.\nB. L1 0Materials\nA class of materials which fulfill all of the above re-\nquirements are those intermetallic compounds forming on\nthe face-centred cubic (fcc) lattice and crystallising in the\nL10structure, visualised in Fig. 2. This family includes\nmaterials already experimentally verified to have superb\nmagnetic properties, such as FePt [14, 29], FePd [16, 30],\nand CoPt [31, 32], but also includes materials consisting\nentirely of comparatively cheap, abundant elements such\nas MnAl [33, 34] and FeNi [9, 35, 36]. In the present\nwork, as a demonstration of our modelling approach, we\nwill focus on the last of these materials, L1 0FeNi.\nAlthough L1 0FeNi (the meteoritic mineral tetrataen-\nite) does not have a theoretically predicted maximum\nenergy product as large as those of the rare-earth super-\nmagnets [9], the relative abundance and availability of its\nconstituent elements, in conjunction with its anticipated\nproduction based on principles used in steel processing,\nmean that its environmental footprint could be signifi-\ncantly lower than for rare-earth-based magnets. In addi-\n1a1d\n1a1d\nFIG. 2. Visualisation of L1 0ordered structure imposed on the\nface-centred cubic lattice. The conventional simple tetragonal\n(st) cell is indicated by dashed black lines and contains two\nnon-equivalent sites with Wyckoff labels 1a and 1d. The 1a\nsites are at the corners of the cell, while the 1d site is at the\ncentre. This st unit cell has lattice parameters c′=a,a′=\nb′=a/√\n2, where ais the conventional fcc lattice parameter.\nThe simple tetragonal representation, with its minimal two-\natom basis and a cell elongated in the ˆzdirection than in\nmakes the origin of a uniaxial anisotropy in L1 0materials\nclear.tion, while there remain many open questions concerning\ntetrataenite’s predicted maximum energy product, it is\nprojected to be higher than those of weak magnetic mate-\nrials such as Alnico and the oxide ferrite magnets [9, 36].\nAs such, tetrataenite may be poised to fill the current gap\nin permanent magnetic performance and accessibility.\nHowever, L1 0FeNi proves to be extremely challenging\nto synthesise in a laboratory setting. Although the L1 0\nphase is reported to be stable at typical operating tem-\nperatures for applications [37], its formation is restricted\nby a low atomic ordering temperature and consequently\nsluggish kinetics [35, 38]. As a result, the fcc solid solu-\ntion (Strukturbericht designation A1) is retained to room\ntemperature, only ordering very slowly under conven-\ntional processing techniques. Typically, bulk samples ob-\nserved experimentally either come from meteorites [39],\nwhich have cooled slowly over a period of millions of\nyears; have had their atomic mobility improved during\nthe annealing process e.g. by bombardment with high-\nenergy neutrons [35, 38]; or have had their structure mod-\nified to foster faster diffusion, e.g.through severe plastic\ndeformation [40].\nII. RESULTS\nA. FeNi\nWe begin our case study by examining the phase\nbehaviour and magnetocrystalline anisotropy of the\nequiatomic binary system, FeNi. Using the all-electron\nHUTSEPOT code [41], we perform self-consistent calcu-\nlations within the KKR-CPA formulation of DFT [17] to\nmodel the electronic structure of the disordered fcc (A1)\nphase in both its paramagnetic and ferromagnetic states.\nFig. 3 shows the total and species-resolved density of\nstates for the A1 phase in both its paramagnetic and fer-\nromagnetic states, where the paramagnetic phase is mod-\nelling within the disordered local moment (DLM) pic-\nture [42–44]. The broken magnetic symmetry in the fer-\nromagnetic state can most clearly be seen to differentiate\nthe bonding contributions from majority and minority\nspin electrons, owing to the different exchange splitting\nofd-states associated with Fe and Ni. This response is ex-\npected to significantly alter both the nature of predicted\natomic ordering in the system, and the temperature at\nwhich this ordering emerges.\nWe then use our linear response theory to assess the\nphase behaviour of this system in both its paramagnetic\nand ferromagnetic states. The S(2)theory for multicom-\nponent alloys [10] has previously been used with success\nto study atomic arrangements in the Cantor alloy (CrM-\nnFeCoNi) and its derivatives [11], the refractory high-\nentropy alloys [12], and to examine the impact of mag-\nnetic ordering on atomic arrangements [13]. The theory\nworks with partial atomic occupancies of lattice sites,\n{ciα}. For example, for the binary FeNi solid solution,\nfor each site iin the underlying fcc lattice, ciα= 0.54\n0123DoS (eV−1)A1 FeNi, PM\nFe\nNi\nTotal\n−8−6−4−2 0 2\nE−EF(eV)−101DoS (eV−1)Spin↑\nSpin↓A1 FeNi, FM\nFIG. 3. Species-resolved density of states for disordered\n(A1) FeNi in its paramagnetic (left) and ferromagnetic (right)\nstates. The paramagnetic state is modelling within the disor-\ndered local moment picture. The total DoS curve is given by\nthe average of the species-resolved curves, weighted by their\nconcentrations. For the ferromagnetic state, the upper half\nof the plot represents the majority-spin DoS, while the bot-\ntom half represents the DoS in the minority spin channel.\nMagnetic order can be seen to clearly alter the nature of the\ncontributions to the total DoS from both Fe and Ni, owing\nto the significant difference in exchange splitting of Fe and Ni\nd-states.\nfor both α= 1 (Fe) and α= 2 (Ni). The theory then\nuses the language of concentration waves [20] to describe\nordered structures, writing\nciα=cα+X\nkeik·Ri∆cα(k). (2)\nThe overall (average) concentration is represented by\ncα, while an applied chemical fluctuation (concentra-\ntion wave) is denoted ∆ cα(k). For example, the L1 0\nstructure is specified by k= (0 ,0,1), in units of2π\na,\nand ∆ c1(2)= +(−)0.5 describing alternating layers of\nFe and Ni atoms. (For clarity, when tabulating results,\nwe choose to normalise the chemical fluctuation, writ-\ning ∆ α= ∆cα/∥∆cα∥, as this allows the relative size\nof fluctuations to be objectively compared.) The anal-\nysis of the alloy’s phase stability then proceeds by per-\nforming a Landau-type series expansion of the free en-\nergy of the system, allowing the energetic cost of vari-\nous imposed chemical fluctuations to be assessed. This\nLandau-type theory allows us to infer the highest tem-\nperature Tordat which, for some kord, there is a chemicalfluctuation ∆ cαthat renders the solid solution unstable,\nresulting in a chemical ordering. Crucially, the chemi-\ncal polarisation [11], ∆ cα, informs us how the chemical\nspecies partition themselves onto the various sublattices\nof the predicted ordered structure. The S(2)theory is\nfully first-principled, and leads to a linear response cal-\nculation based on the self-consistent one-electron poten-\ntials generated using HUTSEPOT to infer chemical or-\ndering. These potentials can be non-magnetic [12], fer-\nromagnetic [13], or paramagnetic [11], as appropriate to\nthe system considered.\nPredicted ordering temperatures, wavevectors, and\nchemical polarisations for FeNi are tabulated in I. No-\ntably, L1 0ordering, characterised by the wavevector\nkord= (0,0,1), is only predicted when we model the\nmaterial when it is in its ferromagnetic state, reflective\nof the electronic structure of the material below its Curie\ntemperature. The predicted ordering temperature is 507\nK, which compares reasonably well with other theoreti-\ncal [45] estimates and experimental [35] determinations.\nThese results are highly significant; they suggest that any\nmaterials processing aiming to synthesise the L1 0form of\nFeNi must be carried out below its Curie temperature. It\nmay also be possible that heat treatment of the sample\nin an applied magnetic field will promote chemical order-\ning, as applying a magnetic field could induce a (small)\nspin polarisation to the system [13].\nIn addition to the above linear response analysis of\nthe phase stability, the S(2)theory also enables extrac-\ntion of lattice-based atom-atom interaction energies for\nfurther atomistic modelling of the phase behaviour of a\nsystem. This approach is based on discrete site occupan-\ncies,{ξiα}, where ξiα=1 if site iis occupied by an atom\nof species α, and ξiα=0 otherwise. The internal energy\nof the system is described by the conventional Bragg-\nWilliams Hamiltonian [46, 47], which takes the form\nH({ξiα}) =1\n2X\niα;jα′Viα;jα′ξiαξjα′. (3)\nThe phase behaviour of the system can then be simulated\nMagnetic State Tord(K) kord ∆ Fe ∆ Ni\nParamagnetic 175 (0 ,0.675,0.675) 0 .707−0.707\nFerromagnetic 507 (0 ,0,1) 0 .707−0.707\nTABLE I. Comparison of the predicted atomic ordering tem-\nperatures and concentration wave modes for FeNi in both\nits paramagnetic and ferromagnetic states. Note that we\nchoose to normalise the chemical fluctuation, writing we\nchoose to normalise the chemical fluctuation, writing ∆ α=\n∆cα/∥∆cα∥. The paramagnetic state is modelled within the\ndisordered local moment picture. It is only when the ferro-\nmagnetic state is modelled that an L1 0ordering (indicated by\nkord= (0,0,1) is predicted. This suggests that any materials\nprocessing aiming to achieve L1 0atomic ordering needs to be\nperformed when the material is below its Curie temperature.5\n0 250 500 750 1000 1250 1500\nT(K)0246C(kB/atom)\nC\nError\n0.00.20.40.60.81.0\nPpq\n(n)FM FeNi\nNi-Fe 1st\nNi-Fe 2nd\nFIG. 4. Atomic order parameters ( Ppq\nn) and specific heat\ncapacity (C) as a function of temperature from lattice-based\nMonte Carlo simulations for the FeNi system modelled in its\nferromagnetic (FM) state. The phase transition just below\n500 K is a transition from the atomically disordered fcc (A1)\nphase to the atomically ordered L1 0phase.\nusing sampling methods such as the Metropolis Monte\nCarlo algorithm [48].\nFor FeNi, our computed atom-atom interaction ener-\ngies are tabulated in the Supplemental Material [66]. Us-\ning these interactions, we perform simulated annealing on\nan ensemble of 10 simulations of systems of 2048 atoms\nwith periodic boundary conditions applied, and using\natomic-swaps rather than substitutions to conserve the\noverall concentration of each species [48]. To assess the\nnature of atomic ordering in the system, we use condi-\ntional probabilities averaged over configurations, denoted\nPpq\nn, representing the probability of species pbeing an nth\nnearest-neighbour of species q. These parameters there-\nfore represent atomic short-range order (ASRO) param-\neters. Using the fluctuation-dissipation theorem, it is\nalso possible to estimate the specific heat capacity of the\nsimulated system [48], which helps to identify the tem-\nperature at which phase transitions occur.\nVisualised in Fig. 4 are our ASRO parameters and spe-\ncific heat capacity determinations as a function of tem-\nperature for FeNi in its ferromagnetic state. The A1-L1 0\nphase transition is characterised by PFe-Ni\n1 = 1/3, and\nPFe-Ni\n2 = 0, which occurs a little below 500 K in our\nsimulations, consistent with the result obtained from the\nlinear response analysis, and confirms the L1 0structure\nas the ground-state of the system.\nGiven a predicted atomically ordered structure, such\nas L1 0FeNi, we use the same first-principles-based frame-\nwork to describe its magnetic characteristics and, there-\nfore, suitability for applications. Within our framework,\nwe generate new HUTSEPOT [41] potentials for the pre-\ndicted (partially) ordered phases, and then use the MAR-\nMOT [49] code to calculate the phases’ magnetic prop-\nerties at both zero and finite temperature [14]. Our S(2)\ntheory is based on the same underlying formulation of\nDFT as is used in MARMOT, and therefore we have anintegrated description of the electrons driving both chem-\nical ordering and producing the magnetic properties.\nWe calculate the MCA for perfect L1 0FeNi at zero-\ntemperature using the same lattice parameters as was\nemployed for the linear response calculation ( c/a= 1)\nto be K1= 0.94 MJm−3. Experimental estimates [35,\n38, 39, 50–57] have put the MCA of this material in\nthe region 0.2–0.93 MJm−3, while previous theoretical\ncalculations [36, 37, 58–65] have obtained values in the\nrange 0.34–0.96 MJm−3. Our magnetisation of M=\n1.66µB/atom is also consistent with earlier theoreti-\ncal [11, 64, 65] and experimental [56] studies. We there-\nfore take these results as verification that our treatment\nmodels the L1 0FeNi system successfully.\nB. Fe 4Ni3Pt\nTo promote atomic ordering and enhance magne-\ntocrystalline anisotropy in this class of materials, one\napproach to consider is the addition of a third alloying\nelement [67]. As an illustrative example of the effects of\nsuch alloying additions on atomic arrangements in this\nmaterial, we consider the addition of a small amount of\nPt to the (Fe,Ni) solid solution. Although Pt is unlikely\nto ever find its way into a mass-produced magnet on ac-\ncount of its high cost and low abundance, it is known\nthat the binary FePt system readily crystallises into the\nL10-structure with an atomic ordering temperature of\naround 1600 K [68]. In addition, the L1 0phase of FePt\nis known to have a large MCA energy ( K1≃15 MJm−3)\nattributed to the heavy 5 dPt atoms driving large spin-\norbit coupling [14]. Our expectation is that the addition\nof Pt to FeNi to form Fe(Ni,Pt) could yield an energet-\nically stable intermetallic phase with magnetocrystalline\nanisotropy between that of FeNi and FePt. As a demon-\nstration, we consider the substitution of Pt at a concen-\ntration of 12.5%, i.e.studying a system with the overall\ncomposition Fe 4Ni3Pt.\nFirst, in a naive calculation, we calculate the magne-\ntocrystalline anisotropy of this system assuming that Pt\nexclusively occupies sites on the Ni sublattice in the L1 0\nstructure. That is, the 1a site is occupied by pure Fe,\nwhile the 1d site has chemical occupancy Ni 0.75Pt0.25.\nUnder this assumption, a MCA of K1= 3.44 MJm−3\nis computed, a value significantly larger than that for\nobtained pure L1 0FeNi. However, this calculation does\nnot account for the phase stability of the compound. To\nassess the likelihood of formation of the ordered phase,\nwe again apply the S(2)theory for multicomponent al-\nloys [10–13] to this composition.\nTabulated in II are predicted atomic ordering temper-\natures, concentration wavevectors, and chemical polari-\nsations for Fe 4Ni3Pt in both its paramagnetic and fer-\nromagnetic states. In contrast to results obtained for\npure FeNi, the wavevector associated with ordering of the\nPt-modified composition is found to be kord= (0,0,1)\nin both magnetic states. The predicted atomic order-6\ning temperatures are likewise similar, initially suggesting\nthat the magnetic state of the disordered phase might\nnot be as critical to the onset of atomic ordering in the\nPt-modified material as in the binary FeNi. However,\nwhen the chemical polarisation of the concentration wave\nis inspected, it can be seen that both the paramagnetic\nand ferromagnetic orderings are characterised by the ar-\nrangements of the Pt atoms, with Fe and Ni distribu-\ntions largely unaffected and remaining disordered. We\nassociate this dominance with the relative atomic size\ndifference between the 3 delements Fe and Ni, and the\nlarger 5 delement Pt, an effect which has been noted in\nalloys such as the Ni-Pt system.\nFrom this linear response calculation, we infer an L1 0\nordering in both ferromagnetic and paramagnetic states,\nand at higher temperatures than for the FeNi binary. To\npredict the (partial) atomic site occupancies, we take the\nchemical polarisation of the concentration wave and al-\nlow this concentration wave to ‘grow’ until (at least) one\nchemical species has an associated lattice site occupancy\nof zero, this being the most atomically ordered structure\nconsistent with the concentration wave. For the atomic\nordering predicted for the paramagnetic solid solution,\nthe 1 asite occupancy is given by Fe 0.614Ni0.386, while\nthe 1 dsite occupancy is given by Fe 0.386Ni0.364Pt0.25.\n(See Fig. 2 for reference.) The computed MCA for this\nordered state when the alloy is quenched to low tempera-\ntures is K1= 0.96 MJm−3, a value that is only fraction-\nally larger than that of pure FeNi. However, for the solid\nsolution in a ferromagnetic state, the predicted atomic\nordering better separates Fe and Ni, with the 1 asite\noccupancy for this condition as Fe 0.678Ni0.322and the\n1dsite occupancy as Fe 0.322Ni0.428Pt0.25. The computed\nMCA is now higher, at K1= 1.22 MJm−3, a value that\nislarger than the predicted MCA for maximally ordered,\nunmodified L1 0FeNi.\nIn a multicomponent alloy system such as the Fe 4Ni3Pt\ncomposition, the linear response calculation may not con-\nvey a full picture of the nature of atomic order in the sys-\ntem. However, we are able to recover a simple pairwise\natomistic model from it and perform further Monte Carlo\nsimulations to better understand the nature of the atomic\narrangements. (See Fig. 1 for an overview of the work-\nMagnetic State Tord(K) kord ∆ Fe ∆ Ni ∆ Pt\nParamagnetic 794 (0 ,0,1) 0 .672 0 .065−0.737\nFerromagnetic 1163 (0 ,0,1) 0 .795−0.237−0.558\nTABLE II. Comparison of the predicted atomic ordering tem-\nperatures and concentration wave modes for Fe 4Ni3Pt in both\nits paramagnetic and ferromagnetic states. The paramagnetic\nstate is modelled within the disordered local moment picture.\nBoth orderings are dominated by Pt, on account of the relative\natomic size difference between it and the smaller 3 delements\nFe and Ni, however there is better distinguishment between\nFe and Ni in the ferromagnetic state.\n0 250 500 750 1000 1250 1500\nT(K)0246C(kB/atom)\nC\nError\n0.00.20.40.60.81.0\nPpq\n(n)FM Fe 4Ni3Pt\nNi-Fe 1st\nNi-Fe 2nd\nPt-Fe 1st\nPt-Fe 2nd\nPt-Ni 1st\nPt-Ni 2ndFIG. 5. Atomic order parameters ( Ppq\nn) and specific heat\ncapacity (C) as a function of temperature from lattice-based\nMonte Carlo simulations for the Fe 4Ni3Pt system modelled\nin its ferromagnetic (FM) state. There are three predicted\nphase transitions, indicated by the three peaks in the specific\nheat capacity, with the tetragonal Fe-Ni ordering occuring at\nthe lowest temperature.\nflow.) Visualised in Fig. 5 are atomic radial distribution\nprobabilities and a measure of the configurational contri-\nbution to the specific heat capacity (SHC) of the system.\nIt can be seen that there are in fact three distinct crystal-\nlographic orderings which can occur. The first ordering,\nat relatively high temperature, is cubic and Pt-driven—\nit represents Pt occupying the corners of the parent fcc\nlattice, with Ni and Fe arranged in a disordered manner.\nThe second ordering is also Pt-driven, and represents Pt\nexpelling other Pt atoms to third-nearest neighbour oc-\ncupancies. Finally, the last ordering is an tetragonal or-\ndering of the Fe and Ni atoms, with Pt atoms unaffected.\nThe structure of lowest energy for this atomistic model\nof Fe 4Ni3Pt is visualised in Fig. 6. It has a clear, uni-\naxial crystal structure, and consists of alternating layers\nof Fe and ordered Ni-Pt. The computed MCA of this\nordered intermetallic phase is K1= 2.77 MJm−3, ap-\nproximately three times that of equiatomic FeNi. How-\never, within our modelling framework this ground-state\nstructure only becomes thermodynamically stable at low\ntemperatures, suggesting it will be challenging to synthe-\nsise it in the laboratory. This low atomic ordering tem-\nperature is associated with an increased configurational\nentropy contribution for a three-component system over\nthat donated by a binary composition, an aspect which\nwill tend to drive down atomic ordering temperatures.\nIn addition, in our simulations, ordering between Pt and\nother elements at higher temperature ‘lock’ the Pt atoms\nto particular lattice sites at relatively high temperatures,\nlimiting the availability of lattice sites that allow Fe and\nNi to freely interchange positions and therefore inhibiting\natomic ordering.\nThese results, therefore, present a mixed picture. On\none hand, adding Pt to the FeNi base composition clearly\nhas the potential to produce a phase with a ground7\nPtFeNi\nPtFeNi\nFIG. 6. The predicted ground state of Fe 4Ni3Pt—a tetrag-\nonal, atomically ordered, intermetallic phase. The conven-\ntional, cubic unit cell of the underlying fcc lattice is marked\nwith black, dashed lines. The MCA of this ordered phase is\npredicted to be 2.766 MJm−3, approximately three times that\nof L1 0FeNi.\nstate configuration possessing a high magnetocrystalline\nanisotropy that is significantly higher than that of bi-\nnary FeNi. However, the increased configurational en-\ntropy and inhibited transformational pathways of the Pt-\nmodified compositon mean that this ground state config-\nuration is only stable at low temperatures with concomi-\ntant synthesis challenges.\nC. X= Al, Co, Pd, Mo, Cr\nBuilding off of the previous example of (FeNi)Pt, to\ndemonstrate the efficacy of our approach for searching\nfor ordered, intermetallic phases with large predicted\nmagnetocrystalline anisotropy energies, and to examine\nwhether alloying provides a viable, cost-effective route\nfor producing atomically ordered L1 0FeNi in the labo-\nratory, we consider five more alloying additions, added\nindividually to the FeNi lattice: X= Al, Co, Pd, Mo,\nCr. Al is chosen as a light abundant element known to\nform intermetallic phases with both Fe and Ni [69]. Co\nis chosen as it is known to provide a higher Curie tem-\nperature than Fe [26], potentially leading to improved\nmagnetic properties. Pd, with an isoelectronic valence\nstructure to Pt [70], is chosen for comparison with the\nearlier results. Finally, Cr and Mo are chosen as early\ntransition metals, to examine whether the early 3 dtran-\nsition metals will have an impact on atomic ordering.\nOnce again, we first calculate an ‘ideal’ MCA, assum-\ning that the elemental addition substitutes only on the\nNi sites. Results of this calculation are given in the right-\nhand column of Table III. Intriguingly, the addition of Co\nresults in a notable decrease in MCA compared to that ofbinary FeNi. This outcome is associated with the Co on\nthe Ni sublattice making a negative contribution to the\ntotal magnetic torque in our calculations. The additives\nCr, Al, and Mo look promising, as the addition of any\nof these elements to the Ni sublattice increases the MCA\nthat is predicted by our calculations. It therefore appears\nthat the addition of these elements alters the electronic\nstructure of the material in such a way as to enhance the\npredicted MCA. This phenomenon has been noted before\nin other computational studies, e.g. in a study of gen-\neral Fe-Ni-Al systems [69]. Once again, though, we stress\nthat these calculations represent an idealised picture, and\nthe thermodynamic stability of a given phase must be as-\nsessed before drawing conclusions as to its suitability as\na gap permanent magnet.\nTo this end, results of the chemical stability analysis for\nthe system Fe 4Ni3X, for the considered additions, X=\nAl, Co, Pd, Mo, Cr, are tabulated in III. The chemical\nstability analysis is always performed for the system mod-\nelled in both its paramagnetic and ferromagnetic states,\nto assess the degree to which chemical order is coupled to\nthe magnetic state. Where an L1 0ordering is inferred,\nindicated by kord= (0,0,1), the maximally ordered L1 0\nstructure that is consistent with this type of ordering is\nderived, and subsequently its MCA is calculated. Our\nresults for predicted chemical ordering demonstrate that\nthe assumption that the additive will substitute for Ni\nis flawed; our inferred chemical ordering never leads to a\nperfect L1 0-like structure and, therefore, the anisotropy\nis always reduced compared to that obtained from the\n‘ideal’ lattice structure. This result demonstrates that it\nis crucial to simultaneously examine the phase behaviour\nof a candidate material and quantify its magnetic prop-\nerties, to ensure that any proposed intermetallic phase\nwill be thermodynamically stable, and thus realizable.\nThe computed outcomes provided in Table III are ex-\namined one by one. Considering first the results for the\naddition of Al, we see that, for both the paramagnetic\nand ferromagnetic cases, the concentration wavevector\ncharacterising predicted ordering is not commensurate\nwith a simple ordered structure. This outcome is at-\ntributed to competing Ni-Al and Fe-Al interactions. This\nreasoning is supported by experimental finding: the Ni-\nAl system has a strong tendency to form an intermetallic\nphase on the fcc lattice [71], while the Fe-Al system forms\non a bcc lattice [72]. In our Monte Carlo simulations, the\nresults of which are visualised in the Supplemental Ma-\nterial [66], the (FeNi)Al system tends to phase segregate\ninto Al-rich and Al-deficient regions. Next considering\nthe results for Cr and Co additions, in both cases an\nL10ordering is predicted only in the case of the system\nbeing in the magnetically ordered state while, in con-\ntrast, L1 0ordering is not predicted for the paramagnetic\nstate. For Mo additions, L1 0ordering is not predicted in\neither magnetic state. Finally, the results for Pd addi-\ntions are comparable with those for the addition of Pt,\nalthough the ordering temperatures and the MCA values\nare notably reduced. The decreased ordering temper-8\nComposition Magnetic State Tord(K) kord(2π/a) ∆Fe ∆Ni ∆XMCA (MJm−3)M(µB/atom)\nHolistic Ideal Holistic Ideal\nFeNi PM 175 (0, 0.675, 0.675) 0 .707−0.707 — —\nFM 507 (0, 0, 1) 0 .707−0.707 0.94 0.94 1.66 1.66\nFe4Ni3Al PM 686 (0, 0.3, 1) 0 .115 0 .643−0.758 — —\nFM 765 (0, 0.3, 1) 0 .462 0 .352−0.814 — 1.66 — 1.46\nFe4Ni3Cr PM 136 (0, 0.6, 0.6) 0 .285−0.805 0 .520 — —\nFM 853 (0, 0, 1) 0 .790−0.220−0.570 0.26 1.79 1.09 1.13\nFe4Ni3Co PM 151 (0, 0, 0) 0 .099−0.751 0 .652 — —\nFM 433 (0, 0, 1) 0 .749−0.656−0.092 0.42 0.31 1.75 1.75\nFe4Ni3Mo PM 401 (0.4, 0.5, 0.6) 0 .169 0 .607−0.776 — —\nFM 607 (0, 0.675, 0.675) 0 .344 0 .470−0.814 — 1.37 — 1.26\nFe4Ni3Pd PM 361 (0, 0, 1) 0 .460 0 .354−0.814 0.17 1.63\nFM 696 (0, 0, 1) 0 .813−0.338−0.475 0.20 1.05 1.64 1.66\nFe4Ni3Pt PM 794 (0, 0, 1) 0 .672 0 .065−0.737 0.96 1.63\nFM 1163 (0, 0, 1) 0 .795−0.237−0.558 1.22 3.44 1.63 1.66\nTABLE III. Computed atomic ordering temperatures ( Tord) and concentration wave modes ( kord, ∆cα) for holistic modelling\nof the preferred ordering induced by an alloying addition, followed by their consequent predicted magnetisations ( M), and\nmagnetocrystalline anisotropy energies. Note that we choose to normalise the chemical fluctuation, writing we choose to\nnormalise the chemical fluctuation, writing ∆ α= ∆cα/∥∆cα∥.\nature is associated with a reduced difference in atomic\nradius between chemical species, in turn reducing chemi-\ncal ordering tendencies. The reduced MCA is associated\nwith a reduced spin-orbit coupling in the system, associ-\nated with the smaller mass of the Pd ions compared to\nPt ions.\nIII. DISCUSSION\nThere are three key conclusions that can be drawn from\nthis study. The first is that, when considering the ther-\nmodynamic stability of a candidate hard magnetic phase,\nit is crucial to consider its magnetic state as well. Our\nresults for FeNi suggest that L1 0ordering is only pre-\ndicted once the parent system has ferromagnetically or-\ndered, suggesting that processing of this phase should\ntake place below its Curie temperature and/or in an ap-\nplied magnetic field, thereby promoting desired atomic\nordering [73]. A similar story holds true when consider-\ning alloying additions to this system; different treatments\nof the magnetic state within our modelling can result in\ndifferent predicted atomic ordering tendencies.\nThe second key conclusion is that it is vital to assess\nthe thermodynamic stability of any proposed hard mag-\nnetic compound. It will always be possible to imagine\nmetastable crystal structures with extraordinary mag-\nnetic properties, but it is only by considering phase equi-\nlibria, as is conducted in this study, that a holistic as-\nsessment can be made about a candidate material’s suit-\nability for applications.\nFinally, although these results, and in particular re-\nsults presented for Pt additives in FeNi, suggest thatsome alloying additions may yield increased ordering\ntemperatures and crystal structures with improved mag-\nnetocrystalline anisotropy compared to the base FeNi bi-\nnary, it is not the case that all additives promote L1 0\nordering. It is necessary, therefore, to perform a com-\nplete stability analysis of a given specified composition,\nrather than assume attainment of an L1 0ordered phase\nfrom the outset.\nFurther work should seek to perform high-throughput\nsearches across a wider range of potential compositions,\nto search for novel, multi-component alloy phases with\ndesirable magnetic properties. Results of these compu-\ntational studies can be implemented into experimental\nactivities, to more rapidly realize new types of advanced\nmagnetic materials. We are in the process of adapting\nour codes for such studies.\nIV. METHODS\nA. The S(2)theory for multicomponent alloys\nOur approach for modelling atomic order in multicom-\nponent alloys uses a Landau-type expansion of the free\nenergy of the system about a homogeneous (disordered)\nreference state to obtain the two-point correlation func-\ntion, an ASRO parameter, ab initio . The effects of the\nresponse of the electronic structure and the rearrange-\nment of charge due to the applied inhomogeneous chem-\nical perturbation are fully included. The theory is the\nnatural multicomponent extension of the S(2)theory de-\nveloped for binary alloys [74, 75], and it has its ground-9\nings in statistical physics and the in seminal papers on\nconcentration waves authored by Khachaturyan [20] and\nGyorffy and Stocks [74]. Full details of the theory, its\nimplementation, and extensive discussion can be found\nin earlier works [10–13].\nThe approach assumes a fixed, ideal lattice, face-\ncentred cubic (fcc) for the alloys studied in this paper,\nwhich represents the averaged atomic positions in the\nsolid solution. A description of the atomic arrangements\nin a substitutional alloy with this fixed underlying lattice\nis given by the site occupation numbers, {ξiα}, where\nξiα=1 if site iis occupied by an atom of species α, and\nξiα=0 otherwise. Each lattice site must be constrained\nto have one (and only one) atom sitting on it, expressed\nasP\nαξiα=1 for all lattice sites i. The overall concentra-\ntion of a chemical species, cαis given by cα=1\nNP\niξiα,\nwhere Nis the total number of lattice sites in the sys-\ntem. A natural choice of atomic long-range order pa-\nrameter is given by the ensemble average of the site oc-\ncupancies, ciα=⟨ξiα⟩, where the {ciα}are referred to\nas the site-wise concentrations. In the atomically disor-\ndered limit, corresponding to the solid solution at high\ntemperatures, these occupancies will be spatially homo-\ngeneous and take the values of the overall concentration\nof each chemical species, i.e.limT→∞ciα=cα. Below\nany atomic disorder-order transition, however, the site\noccupancies acquire a spatial dependence. These spa-\ntially dependent occupancies (or concentrations) can be\nwritten as a fluctuation to the concentration distribution\nof the homogeneous system, ciα=cα+ ∆ciα. As the\nunderlying system is lattice-based and possesses transla-\ntional symmetry, it is natural to write these fluctuations\nin reciprocal space using a concentration wave formalism,\nas pioneered by Khachaturayan[20]. In this manner we\nwrite\nciα=cα+X\nkeik·Ri∆cα(k) (4)\nto describe a chemical fluctuation, where Riis the\nlattice vector with corresponding occupancy ciα. As\nan illustrative example, Fig. 2 shows L1 0-type crys-\ntallographic order imposed on the fcc lattice for an\nequiatomic, ABbinary system, cα= (1\n2,1\n2). The com-\npletely L1 0-ordered structure, represented by alternate\nlayers of atoms of species AandB, is described by\nk={(0,0,1),(0,0,−1)}, with the (normalised) change\nin concentration ∆ cα=1√\n2(−1,1). This formalism is\nextensible to multicomponent alloys.\nCorrelations between atomic species above any\ndisorder-order transition temperature are quantified by\nthe two-point correlation function, written as\nΨiαjα′=⟨ξiαξjα′⟩ − ⟨ξiα⟩⟨ξjα′⟩, (5)\nwhich is an atomic short-range order (ASRO) parameter.\nThis quantity is intrinsically related to the energetic cost\nof chemical fluctuations [10], as ASRO will be dominated\nby those chemical fluctuations which are least costly en-\nergetically.To find the energy cost we approximate the free energy,\nΩ, of an alloy with an inhomogeneous site-wise concen-\ntration distribution, {ciα}, by\nΩ(1)[{ciα}] =−1\nβX\niαciαlnciα\n−X\niανiαciα+⟨Ωel⟩0[{ciα}],(6)\nwhere the three terms on the right-hand side of Eq. (2)\ndescribe entropic contributions, site-wise chemical poten-\ntials, and an average of the electronic contribution to the\nfree energy of the system, respectively. The free energy\nof the system is then expanded about the homogeneous\nreference state ( i.e.the solid solution) in terms of the\n{ciα}. This Landau-type series expansion is written\nΩ(1)[{ciα}] = Ω(1)[{cα}] +X\niα∂Ω(1)\n∂ciα\f\f\f\n{cα}∆ciα\n+1\n2X\niα;jα′∂2Ω(1)\n∂ciα∂cjα′\f\f\f\n{cα}∆ciα∆cjα′+. . . .\n(7)\nThe site-wise chemical potentials present in Eq. 6 serve\nas Lagrange multipliers in the linear response the-\nory, but their variation is not relevant to the under-\nlying physics so terms involving these derivatives are\ndropped [10–12]. The symmetry of the solid solution\nat high temperature—and the requirement that any im-\nposed fluctuation conserves the overall concentrations of\neach chemical species—ensures that the first-order term\nvanishes. The change in free energy, δΩ(1)as a result of\na fluctuation is therefore written (to second order) as\nδΩ(1)=1\n2X\niα;jα′∆ciα[β−1C−1\nαα′−S(2)\niα,jα′]∆cjα′,(8)\nwhere C−1\nαα′=δαα ′\ncαis associated with the entropic contri-\nbutions, and the term −∂2⟨Ωel⟩0\n∂ciα∂cjα′≡S(2)\niα;jα′is the second-\norder concentration derivative of the average energy of\nthe disordered alloy. The evaluation of this term has\nbeen covered in depth in earlier works [10, 11] and for\nbrevity we will omit discussion of it here.\nIt should be noted that S(2)\niα;jα′is evaluated in recipro-\ncal space in our codes, and therefore the change in free\nenergy of Eq. 8 is written accordingly as:\nδΩ(1)=1\n2X\nkX\nα,α′∆cα(k)[β−1C−1\nαα′−S(2)\nαα′(k)]∆cα′(k).\n(9)\nThe matrix in square brackets [ β−1C−1\nαα′−S(2)\nαα′(k)], re-\nferred to as the chemical stability matrix, is related to\nan estimate of the ASRO, Ψ iα;jα′. (Eigenvalues of the\nchemical stability matrices for the compounds considered\nin this study are visualised in the Supplemental Mate-\nrial [66].) When searching for an disorder-order transi-\ntion, we start with the high temperature solid solution,10\nlower the temperature and look for the temperature at\nwhich the lowest lying eigenvalue of this matrix passes\nthrough zero for any k-vector in the irreducible Brillouin\nZone. When this eigenvalue passes through zero at some\ntemperature Tordand wavevector kord, we infer the pres-\nence of an disorder-order transition with chemical polar-\nisation ∆ cαgiven by the associated eigenvector. In this\nfashion we can predict both dominant ASRO and also\nthe temperature at which the solid solution becomes un-\nstable and a crystallographically ordered phase emerges.\nB. Pairwise Atomistic Model\nTo explore the alloy phase space further, we map\nthe concentration derivatives of the internal energy of\nthe alloy, S(2)\nαα′(k), to a pairwise real-space interaction.\nThis real-space model, referred to as the Bragg-Williams\nmodel [46, 47], is lattice-based and has a Hamiltonian of\nthe form\nH=1\n2X\niα;jα′Viα;jα′ξiαξjα′. (10)\nThe effective pairwise interactions, Viα;jα′, are recovered\nfrom S(2)\nαα′(k) by means of a inverse Fourier transform.\nThe mapping from reciprocal-space to real-space and fix-\ning of the gauge degree of freedom on the Viα;jα′is spec-\nified in earlier works [10–12]. These interactions are as-\nsumed to be isotropic and the sum in Eq. 3 is then taken\nas a sum over coordination shells, i.e.first-nearest neigh-\nbours, second-nearest neighbours, etc.\nC. Monte Carlo Simulations\nTo investigate the phase behaviour of these systems\nwith this atomistic model, we perform simulated anneal-\ning using the Metropolis Monte-Carlo algorithm with\nKawasaki dynamics [48]. These dynamics naturally con-\nserve overall concentrations of each chemical species by\npermitting only swaps of pairs of atoms.\nThe algorithm starts by initialising the occupancies of\neach lattice site randomly, with the only constraint be-\ning the overall number of atoms of each chemical species,\nspecified by the overall concentrations. A pair of atomic\nsites (not necessarily nearest-neighbours) is selected at\nrandom, and the change in internal energy ∆ Hresult-\ning from swapping the site occupancies is calculated. If\nthe change in energy is negative (∆ H < 0) the swap is\naccepted unconditionally, while if the change is positive\n(∆H > 0) the swap is accepted with acceptance proba-\nbility e−β∆H. To collect the configurational contribution\nto the specific heat capacity (SHC) of the system, we use\nthe fluctuation-dissipation theorem [76] and the expres-\nsion\nC=1\nkbT2\u0000\n⟨E2⟩ − ⟨E⟩2\u0001\n. (11)To quantify the emergent ASRO in these simulations,\nwe use the conditional probability of finding one species\nneighbouring another, writing Ppq\nnto denote the con-\nditional probability of an atom of type qneighbour-\ning an atom of type pon coordination shell n. In the\nhigh-temperature limit, these tend to the value cq,i.e.\nthe overall concentration of species q. Divergence from\nthis homogeneous value is indicative of emergent atomic\nshort- and long-range order.\nD. Evaluating magnetocrystalline anisotropy ab\ninitio\nThe magnetocrystalline anisotropy of a system de-\nscribes the change in energy of the system due a rotation\nof the magnetisation direction. For a ferromagnet with\nmagnetisation direction ˆn, the MCA can be expressed as\nK=X\nγKγgγ(ˆn), (12)\nwhere Kγdenote MCA coefficients, and gγdenote spher-\nical harmonics fixing the orientation of ˆnwith respect to\nthe crystal axes. These spherical harmonics must respect\nthe symmetry of the underlying crystal structure. It is\nconvenient to represent ˆnin spherical polar coordinates\nbyˆn= (sin θcosϕ,sinθsinϕ,cosθ), where θandϕde-\nnote polar and azimuthal angles respectively in a coordi-\nnate frame specified by the crystal axes.\nIn a uniaxial ferromagnet, the MCA is well-\napproximated by\nK=K1sin2θ (13)\nand the total energy of the system can be written as\nE(ˆn) =Eiso+K1sin2θ, (14)\nwhere Eisorepresents the isotropic portion of the energy.\nThe MCA coefficient K1can be calculated from deriva-\ntives of this energy with respect to magnetisation direc-\ntions. Given an energy of the form in Eq.14, we have\nthat\n∂E\n∂θ=K1sin 2θ. (15)\nThe magnetocrystalline anisotropy difference between\nthe system magnetised in the ˆzandˆxdirections is the\nsum,\n∆E=E([1,0,0])− E([0,0,1]) = K1. (16)\nEvaluation of the derivative,∂E\n∂θatθ=π\n4gives K1di-\nrectly, enabling us to obtain the MCA in a one-shot cal-\nculation. Evaluation at other angles enables extraction\nof coefficients of higher-order anisotropy coefficients, if\ndesired.\nUsing the MARMOT code [49], which works with a\nfully relativistic formulation of density functional theory11\n(DFT), these torques can be evaluated ab initio . A dis-\ncussion of the relevant formulae for the torque in terms of\nkey quantities of multiple scattering theory can be found\nin Refs. [16] and [36].\nE. Computational Details\nThroughout, the all-electron HUTSEPOT code [41] is\nused to generate self-consistent, one electron potentials\nwithin the KKR formulation of density functional theory\n(DFT) [17, 18]. Chemical disorder is described within\nthe coherent potential approximation (CPA) [77, 78],\nwhile the magnetic disorder of the paramegnetic state\nis modelled within the disordered local moment (DLM)\npicture [42–44]. We perform spin-polarised, scalar-\nrelativistic calculations within the atomic sphere approx-\nimation (ASA) [79] with an angular momentum cutoff of\nlmax= 3 for basis set expansions, a 20 ×20×20k-point\nmesh for integrals over the Brillouin zone, and a 24 point\nsemi-circular Gauss-Legendre grid in the complex plane\nto integrate over valence energies. We use the local den-\nsity approximation (LDA) and the exchange-correlation\nfunctional is that of Perdew-Wang [80]. The linear re-\nsponse results are obtained from an computational im-\nplementation of the theory described in Section IV A and\ndiscussed in detail in earlier works [10–12].\nMagnetocrystalline anisotropy energies are evaluated\nusing the MARMOT code [49], which takes in HUTSE-\nPOT potentials as a starting point for its evaluation of\nthe magnetic torques described in Section IV D. Rather\nthan include the effects of spin-orbit coupling in an ap-\nproximate way, by adding another term to the scalar-\nrelativistic Hamiltonian, MARMOT works in a fully rel-\nativistic setting in which spin-orbit effects are included\nnaturally. We use parameters consistent with the HUT-\nSEPOT calculations, and other relevant parameters are\nleft at their MARMOT default values. This includes an-\ngular sampling of the CPA integral of 240 ×40 and an\nadaptive meshing scheme for Brillouin zone integrations.\nDATA AVAILABILITY\nThe data produced for this study are available through\nZenodo via the DOI 10.5281/zenodo.10425682. Specific\nquestions should be directed to the corresponding au-\nthor(s).CODE AVAILABILITY\nThe all-electron HUTSEPOT code [41] used for con-\nstructing the self-consistent one-electron potentials of\nDFT is available at hutsepot.jku.at. The code im-\nplementing the S(2)theory for multicomponent al-\nloys [10] is available from the authors on reasonable\nrequest. The code for performing lattice-based Monte\nCarlo simulations to study alloy phase behaviour using\nthe obtained atom-atom interchange parameters [11] is\navailable via the DOI 10.5281/zenodo.10379949. The\nMARMOT code [49], used for evaluating magnetocrys-\ntalline anisotropies in this work, is available at war-\nwick.ac.uk/marmotcode.\nACKNOWLEDGMENTS\nThe authors would like to express their gratitude\nto Dr Christopher E. Patrick (Department of Materi-\nals, University of Oxford) for fruitful discussions, soft-\nware support, and feedback on the draft version of this\nmanuscript.\nThis work was supported by the UK Engineering\nand Physical Sciences Research Council, Grant No.\nEP/W021331/1, by the U.S. Department of Energy, Of-\nfice of Basic Energy Sciences under Award Number DE\nSC0022168 (for atomic insight) and by the U.S. National\nScience Foundation under Award ID 2118164 (for ad-\nvanced manufacturing aspects). C.D.W. was also sup-\nported by a studentship within the UK Engineering and\nPhysical Sciences Research Council-supported Centre for\nDoctoral Training in Modelling of Heterogeneous Sys-\ntems, Grant No. EP/S022848/1.\nAUTHOR CONTRIBUTIONS\nC.D.W., L.H.L., and J.B.S. conceived the approach.\nJ.B.S. led development of the MARMOT package used\nfor the magnetocrystalline anisotropy calculations and\nalso developed the code for the ASRO linear response\ncalculations. The electronic structure calculations, lin-\near response analysis, and evaluation of magnetocrys-\ntalline anisotropy energies were performed by C.D.W.\nand J.B.S.. Atomistic simulations and subsequent anal-\nysis were performed by C.D.W.. The first draft of the\nmanuscript was written by C.D.W. and subsequently in-\nput was received from all authors. J.B.S. and L.H.L.\nsupervised the project and led funding acquisition.\nCOMPETING INTERESTS\nThe authors declare no competing interests.12\n[1] M. Sagawa, S. Fujimura, N. Togawa, H. Yamamoto, and\nY. Matsuura, Journal of Applied Physics 55, 2083 (1984).\n[2] J. J. Croat, J. F. Herbst, R. W. Lee, and F. E. Pinkerton,\nJournal of Applied Physics 55, 2078 (1984).\n[3] K. Strnat, G. Hoffer, J. Olson, W. Ostertag, and J. J.\nBecker, Journal of Applied Physics 38, 1001 (1967).\n[4] L. H. Lewis and F. Jim´ enez-Villacorta, Metallurgical and\nMaterials Transactions A 44, 2 (2013).\n[5] R. McCallum, L. Lewis, R. Skomski, M. Kramer, and\nI. Anderson, Annual Review of Materials Research 44,\n451 (2014).\n[6] K. Smith Stegen, Energy Policy 79, 1 (2015).\n[7] J. Bai, X. Xu, Y. Duan, G. Zhang, Z. Wang, L. Wang,\nand C. Zheng, Scientific Reports 12, 6105 (2022).\n[8] J. Coey, Scripta Materialia 67, 524 (2012).\n[9] L. H. Lewis, A. Mubarok, E. Poirier, N. Bordeaux,\nP. Manchanda, A. Kashyap, R. Skomski, J. Goldstein,\nF. E. Pinkerton, R. K. Mishra, R. C. Kubic Jr, and\nK. Barmak, Journal of Physics: Condensed Matter 26,\n064213 (2014).\n[10] S. N. Khan, J. B. Staunton, and G. M. Stocks, Phys.\nRev. B 93, 054206 (2016).\n[11] C. D. Woodgate and J. B. Staunton, Physical Review B\n105, 115124 (2022).\n[12] C. D. Woodgate and J. B. Staunton, Physical Review\nMaterials 7, 013801 (2023).\n[13] C. D. Woodgate, D. Hedlund, L. H. Lewis, and J. B.\nStaunton, Physical Review Materials 7, 053801 (2023).\n[14] J. B. Staunton, S. Ostanin, S. S. A. Razee, B. L. Gyorffy,\nL. Szunyogh, B. Ginatempo, and E. Bruno, Physical Re-\nview Letters 93, 257204 (2004).\n[15] J. B. Staunton, S. Ostanin, S. S. A. Razee, B. Gyorffy,\nL. Szunyogh, B. Ginatempo, and E. Bruno, Journal of\nPhysics: Condensed Matter 16, S5623 (2004).\n[16] J. B. Staunton, L. Szunyogh, A. Buruzs, B. L. Gy-\norffy, S. Ostanin, and L. Udvardi, Physical Review B 74,\n144411 (2006).\n[17] J. S. Faulkner, G. M. Stocks, and Y. Wang, Multiple scat-\ntering theory electronic structure of solids , 1st ed. (IOP\nPublishing, Bristol, UK, 2018).\n[18] R. M. Martin, Electronic Structure: Basic Theory and\nPractical Methods (Cambridge University Press, Cam-\nbridge, UK, 2004).\n[19] P. Soven, Physical Review 156, 809 (1967).\n[20] A. G. Khachaturyan, Progress in Materials Science 22, 1\n(1978).\n[21] R. Skomski and J. Coey, Scripta Materialia 112, 3 (2016).\n[22] S. Blundell, Magnetism in condensed matter , reprint ed.,\nOxford master series in condensed matter physics No. 4\n(Oxford Univ. Press, Oxford, 2014).\n[23] J. M. D. Coey, Magnetism and Magnetic Materials , 1st\ned. (Cambridge University Press, 2001).\n[24] L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii,\nElectrodynamics of Continuous Media , 2nd ed., Course\nof Theoretical Physics No. 8 (Elsevier Butterworth-\nHeinemann, Amsterdam Heidelberg, 2009).\n[25] J. Coey, Engineering 6, 119 (2020).\n[26] C. Kittel, Introduction to Solid State Physics , 8th ed.\n(Wiley, Hoboken, NJ, 2005).\n[27] C. E. Patrick and J. B. Staunton, Physical Review Ma-\nterials 3, 101401 (2019).[28] J. Bouaziz, C. E. Patrick, and J. B. Staunton, Physical\nReview B 107, L020401 (2023).\n[29] S. Okamoto, N. Kikuchi, O. Kitakami, T. Miyazaki,\nY. Shimada, and K. Fukamichi, Physical Review B 66,\n024413 (2002).\n[30] H. Shima, K. Oikawa, A. Fujita, K. Fukamichi, K. Ishida,\nand A. Sakuma, Physical Review B 70, 224408 (2004).\n[31] H. Kanazawa, G. Lauhoff, and T. Suzuki, Journal of Ap-\nplied Physics 87, 6143 (2000).\n[32] A. Sakuma, Journal of the Physical Society of Japan 63,\n3053 (1994).\n[33] T. Mix, F. Bittner, K.-H. M¨ uller, L. Schultz, and\nT. Woodcock, Acta Materialia 128, 160 (2017).\n[34] A. Sakuma, Journal of the Physical Society of Japan 63,\n1422 (1994).\n[35] L. N´ eel, J. Pauleve, R. Pauthenet, J. Laugier, and\nD. Dautreppe, Journal of Applied Physics 35, 873 (1964).\n[36] C. D. Woodgate, C. E. Patrick, L. H. Lewis, and\nJ. B. Staunton, Journal of Applied Physics 134, 163905\n(2023).\n[37] L. H. Lewis, F. E. Pinkerton, N. Bordeaux, A. Mubarok,\nE. Poirier, J. I. Goldstein, R. Skomski, and K. Barmak,\nIEEE Magnetics Letters 5, 1 (2014).\n[38] J. Paulev´ e, A. Chamberod, K. Krebs, and A. Bourret,\nJournal of Applied Physics 39, 989 (1968).\n[39] E. Poirier, F. E. Pinkerton, R. Kubic, R. K. Mishra,\nN. Bordeaux, A. Mubarok, L. H. Lewis, J. I. Goldstein,\nR. Skomski, and K. Barmak, Journal of Applied Physics\n117, 17E318 (2015).\n[40] A. Montes-Arango, L. Marshall, A. Fortes, N. Bordeaux,\nS. Langridge, K. Barmak, and L. Lewis, Acta Materialia\n116, 263 (2016).\n[41] M. Hoffmann, A. Ernst, W. Hergert, V. N. Antonov,\nW. A. Adeagbo, R. M. Geilhufe, and H. Ben Hamed,\nPhysica Status Solidi (b) 257, 1900671 (2020).\n[42] A. J. Pindor, J. Staunton, G. M. Stocks, and H. Winter,\nJournal of Physics F: Metal Physics 13, 979 (1983).\n[43] J. Staunton, B. Gyorffy, A. Pindor, G. Stocks, and\nH. Winter, Journal of Magnetism and Magnetic Mate-\nrials45, 15 (1984).\n[44] B. L. Gyorffy, A. J. Pindor, J. Staunton, G. M. Stocks,\nand H. Winter, Journal of Physics F: Metal Physics 15,\n1337 (1985).\n[45] L.-Y. Tian, O. Eriksson, and L. Vitos, Scientific Reports\n10, 14766 (2020).\n[46] W. L. Bragg and E. J. Williams, Proceedings of the\nRoyal Society of London. Series A, Containing Papers of\na Mathematical and Physical Character 145, 699 (1934).\n[47] W. L. Bragg and E. J. Williams, Proceedings of the Royal\nSociety of London. Series A - Mathematical and Physical\nSciences 151, 540 (1935).\n[48] D. P. Landau and K. Binder, A Guide to Monte Carlo\nSimulations in Statistical Physics , 4th ed. (Cambridge\nUniversity Press, Cambridge, UK, 2014).\n[49] C. E. Patrick and J. B. Staunton, Electronic Structure\n4, 017001 (2022).\n[50] T. Shima, M. Okamura, S. Mitani, and K. Takanashi,\nJournal of Magnetism and Magnetic Materials 310, 2213\n(2007).\n[51] M. Mizuguchi, T. Kojima, M. Kotsugi, T. Koganezawa,\nK. Osaka, and K. Takanashi, Journal of the Magnetics13\nSociety of Japan 35, 370 (2011).\n[52] T. Kojima, M. Mizuguchi, and K. Takanashi, Journal of\nPhysics: Conference Series 266, 012119 (2011).\n[53] T. Kojima, M. Mizuguchi, T. Koganezawa, K. Osaka,\nM. Kotsugi, and K. Takanashi, Japanese Journal of Ap-\nplied Physics 51, 010204 (2012).\n[54] T. Kojima, M. Ogiwara, M. Mizuguchi, M. Kot-\nsugi, T. Koganezawa, T. Ohtsuki, T.-Y. Tashiro, and\nK. Takanashi, Journal of Physics: Condensed Matter 26,\n064207 (2014).\n[55] A. Frisk, B. Lindgren, S. D. Pappas, E. Johansson, and\nG. Andersson, Journal of Physics: Condensed Matter 28,\n406002 (2016).\n[56] A. Frisk, T. P. A. Hase, P. Svedlindh, E. Johansson, and\nG. Andersson, Journal of Physics D: Applied Physics 50,\n085009 (2017).\n[57] K. Ito, T. Ichimura, M. Hayashida, T. Nishio, S. Goto,\nH. Kura, R. Sasaki, M. Tsujikawa, M. Shirai, T. Ko-\nganezawa, M. Mizuguchi, Y. Shimada, T. J. Konno,\nH. Yanagihara, and K. Takanashi, Journal of Alloys and\nCompounds 946, 169450 (2023).\n[58] A. Izardar and C. Ederer, Physical Review Materials 4,\n054418 (2020).\n[59] M. Si, A. Izardar, and C. Ederer, Physical Review Re-\nsearch 4, 033161 (2022).\n[60] M. Werwi´ nski and W. Marciniak, Journal of Physics D:\nApplied Physics 50, 495008 (2017).\n[61] Y. Miura, S. Ozaki, Y. Kuwahara, M. Tsujikawa, K. Abe,\nand M. Shirai, Journal of Physics: Condensed Matter 25,\n106005 (2013).\n[62] D. Tuvshin, T. Ochirkhuyag, S. C. Hong, and D. Odkhuu,\nAIP Advances 11, 015138 (2021).\n[63] A. Edstr¨ om, J. Chico, A. Jakobsson, A. Bergman, and\nJ. Rusz, Physical Review B 90, 014402 (2014).\n[64] S. Yamashita and A. Sakuma, Journal of the Physical\nSociety of Japan 91, 093703 (2022).\n[65] S. Yamashita and A. Sakuma, Physical Review B 108,\n054411 (2023).[66] See supplemental material at URL_will_be_inserted_\nby_publisher for visualisation of chemical stability ma-\ntrices, atom-atom interchange parameters, and results for\nadditional Monte Carlo simulations for the alloys studied\nin this work.\n[67] L.-Y. Tian, O. Gutfleisch, O. Eriksson, and L. Vitos, Sci-\nentific Reports 11, 5253 (2021).\n[68] O. K. Von Goldbeck, in IRON—Binary Phase Diagrams\n(Springer Berlin Heidelberg, Berlin, Heidelberg, 1982)\npp. 91–94.\n[69] V. V. Sokolovskiy, Physical Review Materials 6, 025402\n(2022).\n[70] A. Montes-Arango, N. Bordeaux, J. Liu, K. Barmak, and\nL. Lewis, Journal of Alloys and Compounds 648, 845\n(2015).\n[71] H. Okamoto, Journal of Phase Equilibria & Diffusion 25,\n394 (2004).\n[72] O. K. Von Goldbeck, in IRON—Binary Phase Diagrams\n(Springer Berlin Heidelberg, Berlin, Heidelberg, 1982)\npp. 5–9.\n[73] L. H. Lewis and P. S. Stamenov, Advanced Science ,\n2302696 (2023).\n[74] B. L. Gyorffy and G. M. Stocks, Phys. Rev. Lett. 50, 374\n(1983).\n[75] J. B. Staunton, D. D. Johnson, and F. J. Pinski, Physical\nReview B 50, 1450 (1994).\n[76] M. P. Allen and D. J. Tildesley, Computer Simulation\nof Liquids , second edition ed. (Oxford University Press,\nOxford, United Kingdom, 2017).\n[77] J. S. Faulkner and G. M. Stocks, Physical Review B 21,\n3222 (1980).\n[78] D. D. Johnson, D. M. Nicholson, F. J. Pinski, B. L.\nGy¨ orffy, and G. M. Stocks, Physical Review B 41, 9701\n(1990).\n[79] G. M. Stocks, W. M. Temmerman, and B. L. Gyorffy,\nPhysical Review Letters 41, 339 (1978).\n[80] J. P. Perdew and Y. Wang, Physical Review B 45, 13244\n(1992)."    },    {        "title": "2401.15907v1.Magnetic__thermodynamic__and_magnetotransport_properties_of_CeGaGe_and_PrGaGe_single_crystals.pdf",        "content": "The magnetic, thermodynamic, and magnetotransport properties of CeGaGe and\nPrGaGe single crystals\nDaloo Ram,1Sudip Malick,1Zakir Hossain,1,∗and Dariusz Kaczorowski2,†\n1Department of Physics, Indian Institute of Technology, Kanpur 208016, India\n2Institute of Low Temperature and Structure Research, Polish Academy of Sciences,ulica Ok´ olna 2, 50-422 Wroclaw, Poland\nWe investigate the physical properties of high-quality single crystals CeGaGe and PrGaGe using\nmagnetization, heat capacity, and magnetotransport measurements. Gallium-indium binary flux\nwas used to grow these single crystals that crystallize in a body-centered tetragonal structure.\nMagnetic susceptibility data reveal a magnetic phase transition around 6.0 and 19.4 K in CeGaGe\nand PrGaGe, respectively, which is further confirmed by heat capacity and electrical resistivity\ndata. A number of additional anomalies have been observed below the ordering temperature in the\nmagnetic susceptibility data, indicating a complex magnetic structure. The magnetic measurements\nalso reveal a strong magnetocrystalline anisotropy in both compounds. Our detailed analysis of the\ncrystalline electric field (CEF) effect as observed in magnetic susceptibility and heat capacity data\nsuggests that J= 5/2 multiplet of CeGaGe splits into three doublets, while J= 4 degenerate ground\nstate of PrGaGe splits into five singlets and two doublets. The estimated energy levels from the\nCEF analysis are consistent with the magnetic entropy.\nI. INTRODUCTION\nThe competing interactions resulting from the hy-\nbridization of 4 fand conduction electrons in the strongly\ncorrelated electron system yields remarkable physical fea-\ntures such as the Kondo effect, heavy fermion behavior,\nintermediate valency, crystalline electric field (CEF) ef-\nfect, magnetic ordering, superconductivity, and so on [1–\n8]. Therefore, rare-earth-based intermetallic compounds\nhave always been appealing because they provide a plat-\nform for investigating such exotic physical properties.\nThe introduction of strong correlation into a nontriv-\nial topological system results in more intriguing ground\nstates. For instance, Ce 3Bi4Pd3is reported to be a\nWeyl-Kondo heavy-fermion semimetal [9, 10], whereas\nCe3Bi4Pt3displays a topological Kondo insulating state\n[10, 11]. On the other hand, the coexistence of relativistic\nfermions and magnetism in these compounds manifests\nanomalous transport [12–15].\nThe ternary tetragonal compounds RAlX (R = La −Nd\nand Sm; X = Si and Ge) have gained attention recently\nsince this class of compounds displays a Weyl semimetal\n(WSM) state [16–26]. Interestingly, RAlX crystallizes\nin two space groups: LaPtSi-type noncentrosymmetric\nspace group I41md(No. 109) and α-ThSi 2-type cen-\ntrosymmetric space group I41/amd (No. 141). LaAlGe\ncrystallizes in a noncentrosymmetric structure and ex-\nhibits a type-II WSM state as determined by angle-\nresolved photoemission spectroscopy [17]. Moreover,\nmagnetic compounds are more intriguing because they\nprovide a framework for investigating the interaction be-\ntween relativistic Weyl fermions and magnetism. For\ninstance, CeAlSi is a noncollinear ferromagnetic (FM)\nWSM that exhibits an anisotropic anomalous Hall effect\n∗zakir@iitk.ac.in\n†d.kaczorowski@intibs.pl[23] and PrAlSi is a centrosymmetric ferromagnet mani-\nfesting an anomalous Hall effect for magnetic field applied\nalong caxis [25]. In addition, the substitution of Ge with\nSi in PrAlGe results in a crossover from intrinsic to ex-\ntrinsic anomalous Hall conductivity [27]. Further, this\nfamily of compounds also exhibits strong magnetocrys-\ntalline anisotropy and the CEF effect [22–26, 28]. More-\nover, a number of recent reports reveal the presence of a\nWSM state exhibiting complex magnetic structure and\nanomalous transport in CeAlSi [23], CeAlGe [29], PrAlSi\n[25], NdAlSi [30], and NdAlGe [28, 31].\nIn this report, we present a thorough investigation of\nthe physical properties of CeGaGe and PrGaGe single\ncrystals. CeGaGe is a FM Kondo lattice system with an\nordering temperature of 5.5 K as determined in poly-\ncrystalline samples [32, 33]. However, to the best of\nour knowledge there is no report on PrGaGe. Here, we\nhave grown single crystals of RGaGe (R = Ce and Pr)\nas well as its nonmagnetic counterpart, LaGaGe, using\nthe flux method. Powder x-ray diffraction (XRD) pat-\nterns of crushed single crystals revealed that RGaGe (R\n= La−Pr) crystallizes in a noncentrosymmetric structure\nwith space group I41md. Magnetic susceptibility data\nalong with the heat capacity and electrical resistivity\ndata suggest a complex magnetic ordering and CEF ef-\nfect in CeGaGe and PrGaGe. Further, both compounds\nexhibits negative magnetoresistance near the magnetic\nordering temperature.\nII. EXPERIMENTAL DETAILS\nHigh-quality single crystals of RGaGe (R = La −Pr)\nwere grown using gallium-indium binary flux. Con-\nstituent elements La (99.9%, Alfa Aesar), Ce (99.9%,\nAlfa Aesar), Pr (99.9%, Alfa Aesar), Ga (99.9999%, Alfa\nAesar), Ge (99.999%, Alfa Aesar), and In (99.99%, Alfa\nAesar) are taken in the molar ratio R:Ga:Ge:In = 1:2:1:8.arXiv:2401.15907v1  [cond-mat.str-el]  29 Jan 20242\nTABLE I. The obtained lattice parameters of RGaGe from\npowder XRD data recorded at room temperature.\nLattice parameters LaGaGe CeGaGe PrGaGe\na(˚A) 4.3542(2) 4.2911(5) 4.2599(4)\nb(˚A) 4.3542(2) 4.2911(5) 4.2599(4)\nc(˚A) 14.581(1) 14.576(2) 14.521(2)\nV(˚A3) 276.43(3) 268.39(6) 263.51(5)\nThe elements were mixed up properly and placed in an\nalumina crucible. Next, the crucible was sealed inside\nquartz tube with partial argon pressure. The sealed\nquartz tube was put into a muffle furnace and heated\nto 1050◦C for 24 h. Then the furnace was slowly cooled\ndown to 500◦C at a rate of 3◦C/h. At this point the\nexcess flux was removed by centrifuging. Platelike shiny\nsingle crystals with a typical dimension of 3 ×3×0.4\nmm3were obtained as shown in lower insets of bottom\npanel of Fig 1.\nThe structural characterization of obtained crystals\nwas carried out using XRD method in a PANalytical\nX’Pert PRO diffractometer with Cu K α1radiation. The\nchemical compositions of the crystals were checked by\nenergy-dispersive spectroscopy (EDS) in a JEOL JSM-\n6010LA electron microscope. Electrical resistivity and\nmagnetotransport measurements were performed in a\nQuantum Design physical property measurement system\n(PPMS) utilizing the usual four-probe method. Heat\ncapacity measurements were performed using the relax-\nation method in the heat capacity option of PPMS. The\nmagnetic properties were measured using a Quantum De-\nsign magnetic property measurement system.\nIII. RESULTS AND DISCUSSION\nA. Crystal structure\nPowder XRD patterns of crushed crystals, recorded\nat room temperature, are shown in Figs. 1(a) −1(c).\nThe analysis of the powder XRD patterns using the Ri-\netveld refinement method suggest all three RGaGe (R =\nLa−Pr) compounds crystallize in a body-centered tetrag-\nonal structure with space group I41md(No. 109) [34].\nThe estimated lattice parameters as presented in Table\nI agree well with the previous report [32]. A schematic\ndiagram of the crystal structure is displayed in the inset\nof Fig. 1(b). The structure consists of four formula unit\nper unit cell, in which all three atoms (R, Ga, and Ge)\nhave 4 aWyckoff site. There are stacking of R, Ga, and\nGe layers along (001) direction. This structure has two\nvertical mirror planes ( σv) but lacks a horizontal mir-\nror plane ( σh), resulting in inversion symmetry breaking.The observed single crystal XRD pattern of RGaGe can\nbe indexed by (00 l) Miller indices, as depicted in Figs.\n1(d)−1(f), indicating that the crystallographic caxis is\nperpendicular to the flat plane of single crystals. More-\nover, the peaks are extremely sharp, suggesting the high\nquality of the single crystals. Further, the full width at\nhalf maximum (FWHM) of the rocking curve [see upper\ninsets of Figs. 1(d) −1(f)] for the peak (0012) is about\n0.04◦, which is very small, confirming that all single crys-\ntals are of excellent quality. EDS data collected from\nmultiple points and areas on the surface of crystals re-\nveal expected stoichiometry.\nB. Magnetic properties\nThe temperature-dependent magnetic susceptibilities\nχc(H∥c) and χab(H∥ab) for zero-field cooled (ZFC)\nand field cooled (FC) configurations at 0.1 T for both\ncompounds are shown in Figs. 2(a), 2(b), 2(d), and\n2(e). The sudden upturn in χcaround TC= 6.0 K in\nCeGaGe and TC= 19.4 K in PrGaGe suggest a FM\nphase transition. However, χabslowly increases with de-\ncreasing temperature down to TCin both compounds.\nInterestingly, the ZFC and FC data exhibit a bifurcation\nbelow TCalong both crystallographic directions, more\nprominent along H∥ab. Moreover, χabfor ZFC drops\nbelow TCin an AFM fashion. The value of χc/χabis\nabout 10 above T Cand reaches a value of ∼105 at 2 K\nin both compounds, which suggests strong magnetocrys-\ntalline anisotropy in these compounds. A similar tem-\nperature dependence with strong magnetic anisotropy\nwas observed in same family of compounds RAlX (R =\nCe−Nd; X = Ge and Si) [23–26]. In the magnetically\nordered state, χ(T) shows an anomaly around 3.7 K in\nCeGaGe, and two anomalies around 11.0 and 16.5 K in\nPrGaGe, as is evident from the dχ/dT data, presented in\nthe insets of Figs. 2(a) and 2(b). The observed anomalies\nmay be associated with the transition from incommen-\nsurate to commensurate order, which has recently been\nobserved in WSM NdAlSi [26, 30] and NdAlGe [28, 31].\nTherefore, a comprehensive investigation utilizing neu-\ntron diffraction is required to reveal the complex mag-\nnetic structure of CeGaGe and PrGaGe. The inverse\nmagnetic susceptibility data are plotted as a function of\ntemperature and fitted well above TCwith the modified\nCurie-Weiss law, χ(T) =χ0+C/(T−ΘP), where Cis\nthe Curie constant, Θ Pis the paramagnetic Curie tem-\nperature, and χ0is temperature-independent magnetic\nsusceptibility. The insets of Figs. 2(b) and 2(e) illustrate\nthe fitting of Curie-Weiss law for CeGaGe and PrGaGe,\nrespectively. The obtained fitting parameters are listed\nin Table II. The negative values of χ0may appear due\nto the diamagnetic contribution of sample holder. The\npositive Θ Palong the caxis indicates the dominant FM\nexchange interactions whereas negative Θ Palong the ab\nplane suggests an AFM coupling. Similar values of Θ P\nwere also observed in isostructural compounds such as3\n103 05 07 09 0 \n (\n0012)(008)\nIntensity (arb. unit)2\nθ (degree)(004)PrGaGe(f)3\n9.439.539.6FWHM =\n 0.04 °θ\n (degree)\n103 05 07 09 0(e) \n (\n0012)(008)(004)\nIntensity (arb. unit)2\nθ (degree)CeGaGe3\n9.339.439.5θ\n (degree)FWHM =\n 0.03 °\n103 05 07 09 0\n (\n0012)(008)(004)Intensity (arb. unit) \n 2\nθ (degree)LaGaGe5\n mm(d)3\n9.339.439.5FWHM =\n 0.04 °θ\n (degree)\n103 05 07 09 0Intensity (arb. unit)2\nθ (degree)PrGaGe(c)\n103 05 07 09 0(b)\n \n Intensity (arb. unit)2\nθ (degree)CeGaGe\n103 05 07 09 0(a)  \n Yobs \nYcalc \nYobs-Ycalc \nBragg positionIntensity (arb. unit)2\nθ (degree)LaGaGe\nFIG. 1. Powder XRD patterns along with Rietveld refinement of crushed single crystals (a) LaGaGe, (b) CeGaGe, and (c)\nPrGaGe recorded at room temperature. The solid black line and red circles represent the calculated pattern and experimental\ndata. The blue line shows difference between experimental and calculated intensities. The olive bars mark the Bragg positions.\nInset of (b) displays the crystal structure of RGaGe. Single crystal XRD patterns are presented (d) LaGaGe, (e) CeGaGe, and\n(f) PrGaGe. Insets of (d), (e), and (f) show optical images of single crystals (lower) and the rocking curves (upper).\nTABLE II. The estimated value of χ0,µeff, and Θ Pfrom\nmodified Curie-Weiss fit of CeGaGe and PrGaGe.\nCeGaGe PrGaGe\nH∥c H ∥ab H ∥c H ∥ab\nχ0(10−4emu/mol) -3.97 -0.16 -1.23 -3.71\nµeff(µB) 2.71 2.59 3.70 3.78\nΘP(K) 28.7 -41.9 32.3 -21.9\nCeAlGe [22], CeAlSi [23], and PrAlGe [21, 24]. The cal-\nculated effective moments µeffof CeGaGe and PrGaGe\nare very close to Ce3+(2.54 µB) and Pr3+(3.58 µB)\nions, respectively. Below 100 K, the inverse suscepti-\nbility deviates from the Curie-Weiss law and exhibits a\nhump around 50 K, as depicted in the insets of Figs. 2(b)\nand 2(e). Such a hump appears due to the CEF effect\nas observed in numerous rare-earth compounds such as\nCeAgAs 2[4], PrSi [35], and Pr 2Re3Si5[36].\nNext, we have analyzed the magnetic susceptibility\ndata using CEF schemes. For a tetragonal system, the\n(2J+1) levels of J= 5/2 (Ce3+) split into three dou-\nblets and J= 4 (Pr3+) multiplet splits into two doublets\nalong with five singlets [37]. The CEF Hamiltonian cor-TABLE III. CEF parameters, energy levels and the corre-\nsponding wave functions for CeGaGe\nCEF parameters\nB0\n2(K) B0\n4(K) B4\n4(K) B0\n6(K)B4\n6(K)\n-6.40±0.20 -0.418 ±0.035 3.32 ±0.270 0 0\nλz= 10.0 ±1.7 (mol/emu) λx,y= 18.7 ±1.8 (mol/emu)\nEnergy levels and wave functions\nE (K) |+5/2⟩ |+3/2⟩ |+1/2⟩ |− 1/2⟩ |− 3/2⟩ |− 5/2⟩\n251.4±12.6 -0.38 0 0 0 -0.92 0\n251.4±12.6 0 -0.92 0 0 0 -0.38\n126.9±7.4 0 0 1 0 0 0\n126.9±7.4 0 0 0 1 0 0\n0 0.92 0 0 0 -0.38 0\n0 0 -0.38 0 0 0 0.924\n510152025302468 \n /s61539ab  (10-2 emu/mol)T\n (K)H || ab CeGaGe(b)0\n1 00200300024C\nurie-Weissχ-1 (102 mol/emu)T\n (K)H || c H || ab\n102 03 00246 \n Z\nFCF\nC/s61539c (emu/mol)T\n (K)H || cCeGaGe(a)2\n468-10T\nc/s61539abd/s61539/dT (arb. unit)T\n (K)/s61539cF\nC\n01 002 003 00012H\n || cPrGaGe(χ-χ0)-1 (102 mol/emu)T\n (K)H || ab(f)C\nEF Fit\n01 002 003 0001234H\n || cH || ab(χ-χ0)-1 (102 mol/emu)T\n (K)CeGaGe( c)C\nEF Fit\n204 06 08 0234567 \n /s61539ab  (10-2 emu/mol)T\n (K)(e)   H || ab PrGaGe0\n1 00200300012χ-1 (102 mol/emu)T\n (K)Curie-Weiss  H || cH || ab\n204 06 08 001234 \n /s61539c (emu/mol)T\n (K)H || c( d) P rGaGe8\n12162024-0.2-0.10.0FC/s61539\nab d/s61539/dT (arb. unit)T\n (K)/s61539c T\nc\nFIG. 2. The temperature-dependent magnetic susceptibility ( χ) measured at 0.1 T along (a) H∥cand (b) H∥abof CeGaGe.\nThe open and filled circles represent magnetic susceptibility measured in ZFC and FC configurations, respectively. The χ(T)\nof PrGaGe for ZFC and FC configurations was measured at 0.1 T for (d) H∥cand (e) H∥ab, respectively. The insets (a) and\n(d) display the temperature-dependent d χ/dTfor FC configuration for CeGaGe and PrGaGe, respectively. The insets of (b)\nand (e) depict the inverse magnetic susceptibilities of CeGaGe and PrGaGe at 0.1 T, respectively, as a function of temperature.\nThe orange lines represent Curie-Weiss fitting. The temperature-dependent inverse magnetic susceptibility of (c) CeGaGe and\n(f) PrGaGe at field 0.1 T. The black solid lines in (c) and (f) represent the calculated inverse susceptibility using CEF model.\nresponding to tetragonal site symmetry and C4vpoint\nsymmetry is as follows\nHCEF =B0\n2O0\n2+B0\n4O0\n4+B4\n4O4\n4+B0\n6O0\n6+B4\n6O4\n6(1)\nwhere Bq\npand Oq\npare the CEF parameters and the\nStevens operators, respectively [38, 39]. The magnetic\nsusceptibility based on the CEF scheme, χi\nCEF (i=x,\ny,z), is given by\nχi\nCEF =NA(gJµB)2\nZ\u0014X\nnβ|⟨n|Ji|n⟩|2e−βEn\n+X\nn̸=m|⟨m|Ji|n⟩|2e−βEn−e−βEm\nEm−En\u0015\n,(2)\nwhere gJis the Land´ e factor, Z=P\nne−βEn, and β=\n1/kBT.|n⟩is the nth eigenfunction and Enis the cor-\nresponding eigenvalue. Jiis the component of angular\nmomentum [40]. To analyze the CEF, we have plotted\ninverse susceptibility as a function of temperature, sub-\ntracting temperature-independent term ( χ0) as shown in\nFigs. 2(c) and 2(f). The magnetic susceptibility ( χi-χi\n0),\nincluding the molecular field contribution λiand the CEF\ncontribution, can be expressed as(χi−χi\n0)−1= (χi\nCEF)−1−λi (3)\nThe calculated temperature-dependent magnetic suscep-\ntibilities along different crystallographic directions using\nthe above equation, presented in Figs. 2(c) and 2(f),\nagree well with the experimental data. The obtained val-\nues of CEF parameters and the energy levels of CeGaGe\nand PrGaGe are given in Tables III and IV, respectively.\nThe estimated parameters and energy levels are com-\nparable to those of other Ce- and Pr-based compounds\n[41, 42]. We have also recalculated the CEF parameter\nB0\n2directly from the paramagnetic Curie-Weiss temper-\natures to check the consistency of our analysis using fol-\nlowing expression [43],\nΘc\nP−Θab\nP=3\n10B0\n2(2J−1)(2J+ 3) (4)\nThe calculated value of B0\n2is -7.3 and -2.4 K for CeGaGe\nand PrGaGe, respectively, which are in good agreement\nwith estimated value from the CEF model fit.\nFurthermore, we measured the isothermal magnetiza-\ntions M(H) of CeGaGe and PrGaGe, as shown in Figs.\n3(a) and 3(b), respectively. Magnetization along H∥c5\nTABLE IV. CEF parameters, energy levels and the corresponding wave functions for PrGaGe\nCEF parameters\nB0\n2(K) B0\n4(K) B4\n4(K) B0\n6(K) B4\n6(K) λi(mol/emu)\n-2.752 ±0.338 0.050 ±0.012 0.106 ±0.019 0 ±0.0002 0.006 ±0.0018 λz= 8.9 ±2.1;λx,y= 7.5 ±1.9\nEnergy levels and wave functions\nE (K) |+4⟩ | +3⟩ | +2⟩ | +1⟩ | 0⟩ |− 1⟩ |− 2⟩ |− 3⟩ |− 4⟩\n202.8±29.2 -0.177 0 0 0 -0.968 0 0 0 -0.177\n158.9±23.0 0 0 0 0.996 0 0 0 0.089 0\n158.9±23.0 0 0.089 0 0 0 0.996 0 0 0\n77.8±10.5 0 0 -1/√\n2 0 0 0 -1/√\n2 0 0\n68.9±10.3 0 0 -1/√\n2 0 0 0 1/√\n2 0 0\n48.9±26.1 -1/√\n2 0 0 0 0 0 0 0 1/√\n2\n38.6±26.2 0.685 0 0 0 -0.250 0 0 0 0.685\n0 0 0 0 0.089 0 0 0 -0.996 0\n0 0 -0.996 0 0 0 0.089 0 0 0\nincreases sharply and saturates at 0.15 and 0.35 T for\nCeGaGe and PrGaGe, respectively. The saturated value\nof magnetization is 1.55 µBfor CeGaGe and 2.95 µBfor\nPrGaGe, which is less than the value of free Ce3+(gJµ B\n= 2.16 µB) and Pr3+(gJµ B= 3.20 µB) ions. Such low\nvalue of saturation magnetization may be due to the pres-\nence of CEF effect [21, 23, 24]. Whereas, magnetization\nalong H∥abis substantially lower and does not satu-\nrate even at 7 T. Such a variation in magnetization in\ndifferent directions indicates strong magnetocrystalline\nanisotropy. Moreover, M(H) data also indicate that the c\naxis is the magnetic easy axis as magnetization saturates\neasily at low applied field. A small hysteresis is observed\nin both measured directions in the magnetization data,\nsupporting the FM nature in CeGaGe and PrGaGe. We\nhave also determined the isothermal magnetization con-\nsidering the CEF scheme using the expression\nMi=gJµB\nZX\nn|⟨n|Ji|n⟩|e−βEn(5)\nThe eigenvalue and associated eigenfunction of the above\nexpression can be calculated by diagonalizing the given\ntotal Hamiltonian\nH=HCEF−gjµBJi(H+λiMi), (6)\nwhere the first term of the Hamiltonian is defined in Eq.\n(1). The second and third terms of the above Hamilto-nian are the Zeeman and the molecular field term, respec-\ntively. The calculated magnetizations of CeGaGe and\nPrGaGe are represented in Figs. 3(a) and 3(b) by solid\nblack lines obtained by solving Eqs. (5) and (6). The\nabove calculation roughly reproduces the magnetization\nas we neglected the exchange interaction in the Hamilto-\nnian. However, this calculation perfectly recognizes the\neasy and hard axes of magnetization. Furthermore, we\nhave estimated the saturation magnetization ( Msat) us-\ning the Zeeman term [40, 44]\nHx,z=gJµB⟨Γmix,1|Jx,z|Γmix,1⟩Bz (7)\nwhere Γ mix,1is the ground state wave function. The es-\ntimated and observed value of saturation magnetization\nis presented in Table V. A slight discrepancy between the\ncomputed and observed numbers is due to the use of the\nsimplified CEF model [36].\nC. Specific heat\nThe temperature-dependent heat capacity ( Cp) of sin-\ngle crystalline RGaGe (R = La −Pr) at a constant pres-\nsure is presented in Figs. 4(a) −4(c), respectively. A\nsharp anomaly at 6.0 and 19.4 K for CeGaGe and\nPrGaGe, respectively in the Cpdata confirms the bulk\nmagnetic ordering as observed in magnetic susceptibil-\nity data. The heat capacity reaches an expected value6\nTABLE V. The calculated saturation magnetizations using\nEq. (7) and observed value at 1.7 K and 7 T.\nCeGaGe PrGaGe\nEstimated Observed Estimated Observed\nMsat\n∥c(µB) 1.64 1.63 2.38 2.96\nMsat\n⊥c(µB) 0.68 0.72 0 0.74\n02 4 6 0.00.51.01.5 \n M (µB /f.u.)/s61549\n0H (T)CeGaGe(a)H\n || cH\n || ab0.00.10.20.30.00.81.6M (µB /f.u.)/s61549\n0H (T)\n02 4 6 0.01.02.03.0  \nM (µB /f.u.)/s61549\n0H (T)PrGaGe(b)H\n || cH\n || ab0.00.20.40.60123M (µB /f.u.)/s61549\n0H (T)\nFIG. 3. The isothermal magnetization of (a) CeGaGe and (b)\nPrGaGe was measured at 1.7 K as a function of the applied\nmagnetic fields. Insets of (a) and (b) show a zoomed view\nof low fields magnetization for the H∥cof CeGaGe and\nPrGaGe, respectively. The black lines are the result of the\nCEF calculations.\nof Dulong-Petit limits Cp= 3nR= 74.83 J/mol K for\nall three compounds, where nis the number of atoms\nin formula unit and Ris the universal gas constant. The\nCp(T) of the nonmagnetic LaGaGe can be expressed well\nby considering the Debye ( CD) and Einstein ( CE) modes\nof heat capacity as presented by the formula\nCp(T) =γT+mCD(T) + (1 −m)CE(T) (8)where mis the weight factor and γis the Sommerfeld\ncoefficient. CD(T) and CE(T) are defined as\nCD(T) = 9 nR\u0012T\nΘD\u00133ZΘD/T\n0x4ex\n(ex−1)2dx, (9)\nand\nCE(T) = 3 nR\u0012ΘE\nT\u00132eΘE/T\n(eΘE/T−1)2, (10)\nwhere Θ Dand Θ Eare the Debye and Einstein tempera-\ntures, respectively [45, 46]. The obtained fitting param-\neters are Θ D= 295 K, Θ E= 100 K and m= 0.76. We\nhave estimated the γfrom Cpdata of LaGaGe by ap-\nplying the formula Cp/T=γ+βT2in low-temperature\nregime (2 K ≤T≤7 K). The linear fit of Cp/TvsT2\ncurve as shown in the inset of Fig. 4(a) gives γ= 2(2)\nmJ/mol K2andβ= 0.44(1) mJ/mol K4. The estimated\nvalue γis comparable to the previous report [32]. Fur-\nthermore, to estimate γfor CeGaGe and PrGaGe, we\nhave fitted Cpdata in the magnetically ordered state as\npresented in the inset of Figs. 4(b) and 4(c) using the\nfollowing formula\nCp(T) =γT+βT3+δT3/2e−∆/T. (11)\nThe last term of the above equation is the contribution\nto the Cpdue to the spin-wave for a ferromagnet with\nan energy gap ∆ in the magnon spectrum [47]. To per-\nform the above fitting we have used same βobtained\nfrom LaGaGe, which reduces the fitting parameter. So-\nobtained fitting parameters are listed in Table VI. The\nestimated value of γfor CeGaGe and PrGaGe are com-\nparable to other FM compounds such as CeIr 2B2[48],\nPr2Rh2Ga [49] and Ce 11Pd4In9[50]. However, it is less\nthan typical Kondo/heavy fermion compounds like CeIn 3\n[51] and PrV 2Al20[52].\nThe magnetic contribution to the Cpfor CeGaGe and\nPrGaGe was calculated by subtracting the Cpof the non-\nmagnetic reference LaGaGe, assuming that the phonon\ncontribution in CeGaGe and PrGaGe is approximately\nequal to that of LaGaGe. The temperature dependence\nof magnetic heat capacity [ Cm(T)] is presented in Figs.\n4(d) and 4(e). In addition to the prominent peak at\nlow temperatures due to the magnetic phase transition,\na broad hump around 60 and 30 K for CeGaGe and\nPrGaGe is seen in Cm(T), respectively. The broad peak\ncan be attributed to a Schottky-type anomaly, which\narises from the CEF-induced splitting of energy levels\nCe3+and Pr3+. The generalized formula of Schottky\ncontribution to the heat capacity for multi-level CEF\nscheme is given by7\n101 000481216 \n R\nln9Sm (J/mol K)T\n (K)Rln2Rln4Rln6 \n CeGaGe \n PrGaGe(f)\n101 0005101520 \n Cm (J/mol K)T\n (K)(e)  \n           Seven-level CEFPrGaGe\n101 0002468 \n Cm (J/mol K)T\n (K)(d) C eGaGe \n             Three-level CEF\n01 002 003 0001530456075 \n Cp (J/mol K)T\n (K)(c) P\nrGaGeT\nC 1\n02 03 008162432Cp (J/mol K)T\n (K)\n01 002 003 0001530456075 \n Cp (J/mol K)T\n (K)(b) C\neGaGeT\nC3\n6 9 04812Cp (J/mol K)T\n (K)\n01 002 003 0001530456075 \n Cp (J/mol K)T\n (K)3nR \n Debye + Einstein(a)L\naGaGe0\n4 08 00.020.04Cp/T (J/mol K2)T\n 2 (K2)\nFIG. 4. The temperature-dependent heat capacity of single crystals (a) LaGaGe, (b) CeGaGe, and (c) PrGaGe. The red solid\nline in (a) represents the fitting of the Debye plus Einstein model. The inset of (a) shows linear fitting to Cp/TvsT2at\nlow temperatures. The purple solid line in the insets of (b) and (c) is the fitting of Eq. 11 to the Cpdata below TC. The\ntemperature dependence of magnetic contribution to Cpof (d) CeGaGe and (e) PrGaGe. The solid black line shows the fitting\nofCm(T) data using different levels CEF scheme. (f) The magnetic entropy of CeGaGe and PrGaGe calculated as a function\nof temperature.\nCSch(T) =\u0012R\nT2\u0013\u0014X\nigie−∆i/TX\nigi∆i2e−∆i/T\n− X\nigi∆ie−∆i/T!2\u0015 X\nigie−∆i/T!−2\n,\n(12)\nwhere giis the degeneracy of the ith state with ∆ ienergy\ngap splitting [47]. Here, we have used the same CEF pa-\nrameters obtained from CEF analysis of magnetic suscep-\ntibility data to reproduce the Schottky anomaly observed\ninCm(T). Interestingly, the calculated Schottky anomaly\nmatches well with the experimental data for both com-\npounds, as presented in Figs. 4(d) and 4(e). This justifies\nthe validity of the CEF parameters and the energy level\nsplittings of both compounds.\nNext, we have calculated magnetic entropy using the\nexpression Sm(T) =RCm\nTdT. The magnetic entropy as\na function of temperature for CeGaGe and PrGaGe is\npresented in Fig. 4(f). According to CEF analysis, six\nenergy levels in CeGaGe split into three doublets, re-\nsulting in three plateaus in Sm(T) near Rln2,Rln4, and\nRln6. In case of PrGaGe, the Sm(T) reaches a value of\n7.9 J/mol K at TC, which lies between a value of mag-\nnetic entropy of ground state Rln2 and Rln4. Therefore,TABLE VI. Obtained fitting parameters of low temperature\nCpdata of CeGaGe and PrGaGe using Eq. (11).\nγ(mJ/mol K2)δ(J/molK5/2) ∆ (K)\nCeGaGe 13 1.54 3.90\nPrGaGe 35 0.58 14.42\na strong thermal population of the excited levels is in-\ndicated, and the first excited level is probably slightly\nhigher in energy than TC. The total magnetic entropy\nRln6 for CeGaGe and Rln9 for PrGaGe is released at 252\nand 210 K, respectively, indicating the overall splitting\nof CEF levels within this energy window. The observed\nSm(T) is consistent with our CEF analysis.\nD. Magnetotransport\nThe electrical resistivity ρ(T) measured within the\nabplane in CaGaGe and PrGaGe single crystals in\nthe temperature range of 2 −300 K is shown in Figs.\n5(a) and 5(b), respectively. The ρ(T) of CeGaGe and8\n02 4 6 8 1 0-6-4-202 \n MR (%)/s61549\n0H (T)(d) CeGaGe2  K4\n K1\n5 K6\n K10 K5 K30 K\n02 4 6 8 1 0-3-2-10(e) PrGaGe5\n0 K4\n0 K3\n0 K2\n0 K10 KMR (%)/s61549\n0H (T)2 K\n28 1 42 02 6859095100 \n /s61554 (µΩ cm)T\n (K) 0 T \n1 T \n3 T \n5 T \n7 T(c) CeGaGe\n01 002 003 00140160180200(\nb) PrGaGe0\n T9\n T/s61554 (µΩ cm)T\n (K)102 0135140145/s61554 (µΩ cm)T\n (K)Eq. 13\n01 002 003 0080100120140160 \n /s61554 (µΩ cm)T\n (K)(a) CeGaGe \n0 T3\n6912808590/s61554 (µΩ cm)T\n (K)Eq. 13\nFIG. 5. Temperature-dependent electrical resistivity of (a) CeGaGe and (b) PrGaGe measured in between 2 −300 K. The\ninsets show the fitting of Eq. 13 in the magnetically ordered state. (c) The temperature dependence of electrical resistivity of\nCeGaGe measured under external magnetic fields. The MR of (d) CeGaGe and (e) PrGaGe as a function of the magnetic field\nfor different temperatures.\nPrGaGe show metallic behavior down to 2 K with a sharp\nanomaly around 6.0 and 19.4 K, respectively. The low-\ntemperature anomaly is associated with the magnetic\nphase transition where resistivity drops rapidly due to\nreduction of spin scattering as systems order magneti-\ncally. The residual resistivity ratio (RRR) of CeGaGe\nand PrGaGe is ∼2.1 and ∼1.5, respectively, which is\nsmall compared to typical metals. Such value of RRR\nmay appear due to site disorder between Ga and Ge.\nSimilar RRR has also been observed in RAlX (R =\nLa−Nd and X = Ge, Si) family for site disorder be-\ntween Al and Ge/Si [22–24]. On the other hand, ρ(T)\nof CeGaGe exhibits a broad hump from 220 to 50 K,\nmay arise from the CEF splitting, which is also evident\nfrom thermodynamic measurement. Similarly a concave\ncurvature is observed in the ρ(T) data of PrGaGe. Such\nbehavior of ρ(T) is commonly seen in rare-earth com-\npounds like CeAuGa 3[53], Ce 2NiSi 3[7] and Pr 7Ru3[54].\nFor both compounds, the ρ(T) in the magnetically order\nstate can be described using the following expression\nρ(T) =ρ0+AT2exp\u0012\n−∆\nT\u0013\n, (13)\nwhere ρ0is the residual resistivity, and the second term\nis the contribution to resistivity due to FM magnon. A\nis related to the strength of electron-magnon scattering,\nand ∆ is the energy gap in the magnon spectrum [55].\nThe so-obtained value of fitting parameters from Eq. 13TABLE VII. The estimated fitting parameters ρ0,Aand ∆\nfrom the resistivity data using Eq. (13).\nρ0(µΩ cm) A(µΩ cm/K2) ∆ (K)\nCeGaGe 81.3 0.45 3.33\nPrGaGe 134.0 0.05 13.36\nare listed in Table VII. The estimated value of ∆ is very\nclose to that obtained from the heat capacity data, con-\nfirming consistency of our analysis.\nTheρ(T) of CeGaGe and PrGaGe under various ex-\nternal magnetic fields is presented in Figs. 5(b) and 5(c).\nThe low temperature anomaly in ρ(T) due to magnetic\nordering for both compounds is getting smeared out by\nthe applied field, and the magnitude of electrical resistiv-\nity is reduced near TC. The field dependence of MR of\nCeGaGe and PrGaGe at various temperatures for H∥c\nare represented in Figs. 5(d) and 5(e), respectively. The\nMR is defined as [ ρ(H)−ρ(0)]/ρ(0), where ρ(H) and\nρ(0) are the resistivities in the presence and absence of a\nmagnetic field, respectively. The MR of both compounds\nbecomes negative close to the magnetic ordering temper-\nature and remains negative throughout the paramagnetic\nregion. The maximum value of MR reaches about -6.5 %9\nand -3.8 % for CeGaGe and PrGaGe, respectively. The\nnegative MR for both samples near TCresults from the\nsuppression of spin scattering by the applied magnetic\nfield, which is the typical MR behavior for an FM [56, 57].\nInterestingly, at low temperatures (T < TC), the MR for\nboth samples in a weak field regime is negative, but as\nthe field intensity increases, the MR rises and becomes\npositive. The magnetic moments become polarized at\nlow applied fields ( ∼0.4 T), and MR becomes negative\ndue to suppression of spin disorder; however, as the field\nstrength increases, the Lorentz force becomes dominant,\nand MR starts to increase, as observed in PrAlGe [24]\nand CeIr 2B2[48].\nIV. SUMMARY\nWe have studied the magnetic, thermodynamic, and\nmagnetotransport properties of RGaGe (R = Ce and\nPr) single crystals synthesized using gallium-indium flux.\nThe powder XRD patterns indicated that these crys-\ntals crystallize in a body-centered tetragonal structure\nwith space group I41md(No. 109). The temperature-\ndependent magnetic susceptibility and resistivity mea-\nsurements show a magnetic phase transition in CeGaGeand PrGaGe at TC= 6.0 and 19.4 K, respectively. Fur-\nther, heat capacity data shows a sharp peak at the mag-\nnetic transition, which confirms the bulk nature of the\nmagnetic ordering. The temperature and field-dependent\nmagnetization data reveal strong magnetic anisotropy\nand the presence of CEF in both compounds. The CEF\nanalysis for inverse magnetic susceptibility and heat ca-\npacity data of CeGaGe and PrGaGe reveal that J= 5/2\ndegenerate ground state of Ce3+ion splits into three dou-\nblets, whereas J= 4 multiplet of Pr3+ion splits into two\ndoublets and five singlets. The field-dependent MR for\nboth compounds is positive at low temperatures and high\nfields, it switches to negative near the magnetic transition\ntemperature and remains negative in the paramagnetic\nregion.\nV. ACKNOWLEDGMENT\nWe acknowledge IIT Kanpur and the Depart-\nment of Science and Technology, India, [Order No.\nDST/NM/TUE/QM-06/2019 (G)] for financial sup-\nport. We thank Suman Sanki for valuable discus-\nsion. D.K. acknowledges financial support from the Na-\ntional Science Centre (Poland) under Research Grant No.\n2021/41/B/ST3/01141.\n[1] L. Wang, C. Wang, Z. Liu, J. Cheng, S. Miao, Y. Song,\nY. Shi, and Y.-f. Yang, Phys. Rev. B 100, 085122 (2019).\n[2] D. Das, D. Gnida,  L. Bochenek, A. Rudenko,\nM. Daszkiewicz, and D. Kaczorowski, Scientific Reports\n8, 16703 (2018).\n[3] F. Steglich, J. Aarts, C. D. Bredl, W. Lieke, D. Meschede,\nW. Franz, and H. Sch¨ afer, Phys. Rev. Lett. 43, 1892\n(1979).\n[4] R. Mondal, R. Bapat, S. K. Dhar, and A. Thamizhavel,\nPhys. Rev. B 98, 115160 (2018).\n[5] V. K. Anand, Z. Hossain, G. Behr, G. Chen, M. Nicklas,\nand C. Geibel, Journal of Physics: Condensed Matter\n19, 506205 (2007).\n[6] N. D. Mathur, F. M. Grosche, S. R. Julian, I. R. Walker,\nD. M. Freye, R. K. W. Haselwimmer, and G. G. Lon-\nzarich, Nature 394, 39 (1998).\n[7] S. Malick, D. Das, and Z. Hossain, Journal of Magnetism\nand Magnetic Materials 482, 108 (2019).\n[8] P. Opletal, E. Duverger–N´ edellec, K. Miliyanchuk,\nS. Malick, Z. Hossain, and J. Custers, Journal of Alloys\nand Compounds 927, 166941 (2022).\n[9] H.-H. Lai, S. E. Grefe, S. Paschen, and Q. Si, Proceedings\nof the National Academy of Sciences 115, 93 (2018).\n[10] S. Dzsaber, L. Prochaska, A. Sidorenko, G. Eguchi,\nR. Svagera, M. Waas, A. Prokofiev, Q. Si, and S. Paschen,\nPhys. Rev. Lett. 118, 246601 (2017).\n[11] D. J. Campbell, Z. E. Brubaker, C. Roncaioli, P. Saraf,\nY. Xiao, P. Chow, C. Kenney-Benson, D. Popov, R. J.\nZieve, J. R. Jeffries, and J. Paglione, Phys. Rev. B 100,\n235133 (2019).[12] M. Hirschberger, S. Kushwaha, Z. Wang, Q. Gibson,\nS. Liang, C. A. Belvin, B. A. Bernevig, R. J. Cava, and\nN. P. Ong, Nature materials 15, 1161 (2016).\n[13] R. Singha, S. Roy, A. Pariari, B. Satpati, and P. Mandal,\nPhys. Rev. B 99, 035110 (2019).\n[14] S. Wolgast, C. Kurdak, K. Sun, J. W. Allen, D.-J. Kim,\nand Z. Fisk, Phys. Rev. B 88, 180405(R) (2013).\n[15] E. Cheng, W. Xia, X. Shi, H. Fang, C. Wang, C. Xi,\nS. Xu, D. C. Peets, L. Wang, H. Su, L. Pi, W. Ren,\nX. Wang, N. Yu, Y. Chen, W. Zhao, Z. Liu, Y. Guo, and\nS. Li, Nature Communications 12, 6970 (2021).\n[16] H. Su, X. Shi, J. Yuan, Y. Wan, E. Cheng, C. Xi, L. Pi,\nX. Wang, Z. Zou, N. Yu, W. Zhao, S. Li, and Y. Guo,\nPhys. Rev. B 103, 165128 (2021).\n[17] S.-Y. Xu, N. Alidoust, G. Chang, H. Lu, B. Singh, I. Be-\nlopolski, D. S. Sanchez, X. Zhang, G. Bian, H. Zheng,\nM.-A. Husanu, Y. Bian, S.-M. Huang, C.-H. Hsu, T.-R.\nChang, H.-T. Jeng, A. Bansil, T. Neupert, V. N. Stro-\ncov, H. Lin, S. Jia, and M. Z. Hasan, Science Advances\n3, e1603266 (2017).\n[18] W. Cao, Y. Su, Q. Wang, C. Pei, L. Gao, Y. Zhao, C. Li,\nN. Yu, J. Wang, Z. Liu, Y. Chen, G. Li, J. Li, and Y. Qi,\nChinese Physics Letters 39, 047501 (2022).\n[19] J. Zhao, W. Liu, A. Rahman, F. Meng, L. Ling, C. Xi,\nW. Tong, Y. Bai, Z. Tian, Y. Zhong, Y. Hu, L. Pi,\nL. Zhang, and Y. Zhang, New Journal of Physics 24,\n013010 (2022).\n[20] P. Puphal, V. Pomjakushin, N. Kanazawa, V. Ukleev,\nD. J. Gawryluk, J. Ma, M. Naamneh, N. C. Plumb,\nL. Keller, R. Cubitt, E. Pomjakushina, and J. S. White,\nPhys. Rev. Lett. 124, 017202 (2020).10\n[21] D. Destraz, L. Das, S. S. Tsirkin, Y. Xu, T. Neupert,\nJ. Chang, A. Schilling, A. G. Grushin, J. Kohlbrecher,\nL. Keller, P. Puphal, E. Pomjakushina, and J. S. White,\nnpj Quantum Materials 5, 5 (2020).\n[22] H. Hodovanets, C. J. Eckberg, P. Y. Zavalij, H. Kim,\nW.-C. Lin, M. Zic, D. J. Campbell, J. S. Higgins, and\nJ. Paglione, Phys. Rev. B 98, 245132 (2018).\n[23] H.-Y. Yang, B. Singh, J. Gaudet, B. Lu, C.-Y. Huang,\nW.-C. Chiu, S.-M. Huang, B. Wang, F. Bahrami, B. Xu,\nJ. Franklin, I. Sochnikov, D. E. Graf, G. Xu, Y. Zhao,\nC. M. Hoffman, H. Lin, D. H. Torchinsky, C. L. Broholm,\nA. Bansil, and F. Tafti, Phys. Rev. B 103, 115143 (2021).\n[24] B. Meng, H. Wu, Y. Qiu, C. Wang, Y. Liu, Z. Xia,\nS. Yuan, H. Chang, and Z. Tian, APL Materials 7, 051110\n(2019).\n[25] M. Lyu, J. Xiang, Z. Mi, H. Zhao, Z. Wang, E. Liu,\nG. Chen, Z. Ren, G. Li, and P. Sun, Phys. Rev. B 102,\n085143 (2020).\n[26] J.-F. Wang, Q.-X. Dong, Z.-P. Guo, M. Lv, Y.-F. Huang,\nJ.-S. Xiang, Z.-A. Ren, Z.-J. Wang, P.-J. Sun, G. Li, and\nG.-F. Chen, Phys. Rev. B 105, 144435 (2022).\n[27] H.-Y. Yang, B. Singh, B. Lu, C.-Y. Huang, F. Bahrami,\nW.-C. Chiu, D. Graf, S.-M. Huang, B. Wang, H. Lin,\nD. Torchinsky, A. Bansil, and F. Tafti, APL Materials 8,\n011111 (2020).\n[28] H.-Y. Yang, J. Gaudet, R. Verma, S. Baidya, F. Bahrami,\nX. Yao, C.-Y. Huang, L. DeBeer-Schmitt, A. A. Aczel,\nG. Xu, H. Lin, A. Bansil, B. Singh, and F. Tafti, Phys.\nRev. Mater. 7, 034202 (2023).\n[29] X. He, Y. Li, H. Zeng, Z. Zhu, S. Tan, Y. Zhang, C. Cao,\nand Y. Luo, Science China Physics, Mechanics & Astron-\nomy66, 237011 (2023).\n[30] J. Gaudet, H.-Y. Yang, S. Baidya, B. Lu, G. Xu, Y. Zhao,\nJ. A. Rodriguez-Rivera, C. M. Hoffmann, D. E. Graf,\nD. H. Torchinsky, P. Nikoli´ c, D. Vanderbilt, F. Tafti, and\nC. L. Broholm, Nature Materials 20, 1650 (2021).\n[31] C. Dhital, R. L. Dally, R. Ruvalcaba, R. Gonzalez-\nHernandez, J. Guerrero-Sanchez, H. B. Cao, Q. Zhang,\nW. Tian, Y. Wu, M. D. Frontzek, S. K. Karna, A. Meads,\nB. Wilson, R. Chapai, D. Graf, J. Bacsa, R. Jin, and J. F.\nDiTusa, arXiv preprint arXiv:2302.05596 (2023).\n[32] S. Dhar, S. Pattalwar, and R. Vijayaraghavan, Physica\nB: Condensed Matter 186-188 , 491 (1993).\n[33] N. Patil and S. Dhar, Physica B: Condensed Matter 223-\n224, 359 (1996).\n[34] P. Puphal, C. Mielke, N. Kumar, Y. Soh, T. Shang,\nM. Medarde, J. S. White, and E. Pomjakushina, Phys.\nRev. Materials 3, 024204 (2019).\n[35] P. K. Das, A. Bhattacharyya, R. Kulkarni, S. K. Dhar,\nand A. Thamizhavel, Phys. Rev. B 89, 134418 (2014).[36] S. Sanki, V. Sharma, S. Sasmal, V. Saini, G. Dwari, B. B.\nMaity, R. Kulkarni, R. Prakash Pandeya, R. Mondal,\nA. Lakshan, S. Ramakrishnan, P. Pratim Jana, K. Maiti,\nand A. Thamizhavel, Phys. Rev. B 105, 165134 (2022).\n[37] U. Walter, Journal of Physics and Chemistry of Solids\n45, 401 (1984).\n[38] K. W. H. Stevens, Proceedings of the Physical Society.\nSection A 65, 209 (1952).\n[39] M. Hutchings, Solid State Physics, 16, 227 (1964).\n[40] J. Banda, B. K. Rai, H. Rosner, E. Morosan, C. Geibel,\nand M. Brando, Phys. Rev. B 98, 195120 (2018).\n[41] N. Van Hieu, T. Takeuchi, H. Shishido, C. Tonohiro,\nT. Yamada, H. Nakashima, K. Sugiyama, R. Settai,\nT. D. Matsuda, Y. Haga, M. Hagiwara, K. Kindo,\nS. Araki, Y. Nozue, and Y. ¯Onuki, Journal of the Physical\nSociety of Japan 76, 064702 (2007).\n[42] Y. Nakano, F. Honda, T. Takeuchi, K. Sugiyama,\nM. Hagiwara, K. Kindo, E. Yamamoto, Y. Haga, R. Set-\ntai, H. Yamagami, and Y. ¯Onuki, Journal of the Physical\nSociety of Japan 79, 024702 (2010).\n[43] Y.-L. Wang, Physics Letters A 35, 383 (1971).\n[44] H. Jin, W. Cai, J. Coles, J. R. Badger, P. Klavins, S. Dee-\nmyad, and V. Taufour, Phys. Rev. B 106, 075131 (2022).\n[45] S. Malick, J. Singh, A. Laha, V. Kanchana, Z. Hossain,\nand D. Kaczorowski, Phys. Rev. B 105, 045103 (2022).\n[46] D. Ram, J. Singh, M. K. Hooda, O. Pavlosiuk, V. Kan-\nchana, Z. Hossain, and D. Kaczorowski, Phys. Rev. B\n107, 085137 (2023).\n[47] E. S. R. Gopal, Specific heats at low temperatures\n(Plenum, New York, 1966).\n[48] A. Prasad, V. K. Anand, U. B. Paramanik, Z. Hossain,\nR. Sarkar, N. Oeschler, M. Baenitz, and C. Geibel, Phys.\nRev. B 86, 014414 (2012).\n[49] B. Sahu, arXiv preprint arXiv:2102.04106 (2021).\n[50] D. Das and D. Kaczorowski, Journal of Magnetism and\nMagnetic Materials 471, 315 (2019).\n[51] H. Shishido, T. Shibauchi, K. Yasu, T. Kato, H. Kontani,\nT. Terashima, and Y. Matsuda, Science 327, 980 (2010).\n[52] M. Tsujimoto, Y. Matsumoto, T. Tomita, A. Sakai, and\nS. Nakatsuji, Phys. Rev. Lett. 113, 267001 (2014).\n[53] B. Lv and B. Liu, Nuclear Analysis 1, 100002 (2022).\n[54] S. Krolak, H. Swiatek, K. Gornicka, M. Winiarski,\nW. Xie, R. Cava, and T. Klimczuk, Journal of Alloys\nand Compounds 929, 167279 (2022).\n[55] D. Kaczorowski and A. Szytu la, Journal of Alloys and\nCompounds 614, 186 (2014).\n[56] V. K. Anand, D. T. Adroja, and A. D. Hillier, Phys. Rev.\nB85, 014418 (2012).\n[57] S. Nallamuthu, A. Dzubinska, M. Reiffers, J. R. Fernan-\ndez, and R. Nagalakshmi, Physica B: Condensed Matter\n521, 128 (2017)."    },    {        "title": "2401.18000v1.Prediction_of_stable_nanoscale_skyrmions_in_monolayer_Fe__5_GeTe__2_.pdf",        "content": "Prediction of stable nanoscale skyrmions in monolayer Fe 5GeTe 2\nDongzhe Li,1,∗Soumyajyoti Haldar,2and Stefan Heinze2, 3\n1CEMES, Universit ´e de Toulouse, CNRS, 29 rue Jeanne Marvig, F-31055 Toulouse, France\n2Institute of Theoretical Physics and Astrophysics, University of Kiel, Leibnizstrasse 15, 24098 Kiel, Germany\n3Kiel Nano, Surface, and Interface Science (KiNSIS), University of Kiel, 24118 Kiel, Germany\n(Dated: February 1, 2024)\nUsing first-principles calculations and atomistic spin simulations, we predict stable isolated skyrmions with a\ndiameter below 10 nm in a monolayer of the two-dimensional van der Waals ferromagnet Fe 5GeTe 2, a material\nof significant experimental interest. A very large Dzyaloshinskii-Moriya interaction (DMI) is observed due to\nthe intrinsic broken inversion symmetry and strong spin-orbit coupling for monolayer Fe 5GeTe 2. We show that\nthe nearest-neighbor approximation, often used in literature, fails to describe the DMI. The strong DMI together\nwith moderate in-plane magnetocrystalline anisotropy energy allows to stabilize nanoscale skyrmions in out-of-\nplane magnetic fields above ≈2T. The energy barriers of skyrmions in monolayer Fe 5GeTe 2are comparable\nto those of state-of-the-art transition-metal ultra-thin films. We further predict that these nanoscale skyrmions\ncan be stable for hours at temperatures up to 20 K.\nMagnetic skyrmions – localized, stable spin textures\nwith intriguing topological and dynamical properties – have\nemerged as a promising avenue to realize next-generation\nspintronics devices [1–6]. During the last ten years, the\nmain focus of the community so far has been on skyrmions\nin bulk systems [7–9] and interface-based systems of ul-\ntrathin films [10–13], and multilayers [14–16]. Recently,\nmagnetic skyrmions were discovered in atomically thin two-\ndimensional (2D) van der Waals (vdW) materials, provid-\ning an ideal playground to push skyrmion technology to the\nsingle-layer limit [17, 18]. Stabilizing skyrmions in 2D mag-\nnets can avoid pinning by defects due to high-quality vdW\ninterfaces and the possibility of easy control of magnetism via\nexternal stimuli.\nThe Dzyaloshinskii-Moriya interaction (DMI), which\nprefers a canting of the spins of adjacent magnetic atoms,\nis often recognized as the key ingredient in forming mag-\nnetic skyrmions. The DMI originates from spin-orbit coupling\n(SOC) and relies on broken inversion symmetry. However,\nmost 2D magnets exhibit inversion symmetry, therefore, the\nDMI is suppressed. Several strategies have been proposed\nto achieve DMI by breaking inversion symmetry. These in-\nclude the family of Janus vdW magnets [19–21], electric field\n[22], and 2D vdW heterostructures [23–25]. In particular, the\nFenGeTe 2family ( n= 3,4,5) with high Curie temperature\n(Tc) near room temperature has been proposed as a promis-\ning candidate for magnetic skyrmions. N ´eel-type magnetic\nskyrmions are reported in Fe 3GeTe 2heterostructures by ex-\nperiments [23–26] and explained by ab initio theory [27, 28]\nin terms of the emergence of DMI at the interface. All-\nelectrical skyrmion detection has also recently been proposed\nin tunnel junctions based on Fe 3GeTe 2[29]. More recently,\nseveral experimental groups reported the observation of topo-\nlogical spin structures (i.e., skyrmions or merons) in 2D vdW\nFe5GeTe 2[30–34]. Additionally, Fe 5GeTe 2exhibits high Tc\nabove room temperature [35, 36], which makes it promis-\ning for spintronics device applications. However, the quan-\ntification of individual skyrmions’ stability and lifetime in\n∗Corresponding author: dongzhe.li@cemes.fr\nFe\nGe\nTe\nxy(a)\n(b)\nFe1Fe2\nFe3\nFe4\nFe5\nyzFIG. 1. (a) Top and (b) side views of the atomic structure of the\nFe5GeTe 2monolayer. The black dashed lines draw up the 2D primi-\ntive cell.\nFe5GeTe 2, crucial for device applications, has been reported\nneither in experiments nor in theory.\nIn this Letter, we predict the formation of nanoscale\nskyrmions in the atom-thick vdW Fe 5GeTe 2monolayer based\non first-principles calculations and atomistic spin simulations.\nThe diameters of these skyrmions are below 10 nm, which\nis technologically desirable for improving the controllability\nand integrability of skyrmion-based functional devices. How-\never, such small skyrmions have not yet been observed in\n2D vdW magnets. The origin of these nanoscale skyrmions\nis attributed to strong DMI together with moderate in-plane\nmagnetocrystalline anisotropy energy (MAE) and weak ex-\nchange frustration. Furthermore, the calculated energy bar-\nriers of skyrmion collapse for Fe 5GeTe 2monolayer are ∼80\nmeV at a moderate magnetic field of about 2 T. This substan-arXiv:2401.18000v1  [cond-mat.mtrl-sci]  31 Jan 20242\n0.5 0.0 0.5 1.00306090120150Ess(q)E FM(meV/uc)\nK M\nM\nDFT\nFit up to 1stNN\nFit up to 7thNN\n0.5 0.0 0.5 1.05.0\n2.5\n0.02.5ESOC(q) (meV/uc)\n0.08 0.04 0.0 0.04 0.08\n|q| (2 /a)\n024E(q)E FM(meV/uc)\nM\nK\n(a)\n(b)\n(c)without SOC\nwith SOCCCWCW\nwith SOC\nFIG. 2. (a) Energy dispersion of flat cycloidal spin spirals ( ESS) for\nFe5GeTe 2along the high symmetry path M-Γ-K-M without SOC.\nThe symbols represent the DFT calculations in scalar-relativistic ap-\nproximation, while the dashed and solid lines are the fits to the\nHeisenberg exchange interaction up to the first nearest neighbor\n(NN) and up to the seventh NN, respectively. (b) Energy contri-\nbution of cycloidal spin spirals due to SOC ( ∆ESOC), also known\nas DMI contribution. All energies are measured with respect to the\nFM state ( EFM) at the Γpoint. Note that positive and negative en-\nergies represent a preference for clockwise (CW) and counterclock-\nwise (CCW) spin configurations. (c) Zoom around the FM state ( Γ\npoint), including the Heisenberg exchange, the DMI, and the MAE,\ni.e.E(q) =ESS(q) + ∆ ESOC(q) +K/2. The DMI leads to CCW\ncharity, and the MAE is responsible for the constant energy shift\n(K/2) of the spin spirals with respect to the FM state.\ntial energy barrier is comparable to that of ultrathin films [37–\n39], which serve as prototype systems for hosting nanoscale\nskyrmions. Finally, we also calculate explicitly skyrmion life-\ntime as a function of magnetic field and temperature. Our\nresults demonstrate that Fe 5GeTe 2monolayer is an excellent\ncandidate to experimentally observe nanoscale skyrmions in a\n2D vdW magnet.\nOur first-principles calculations were performed using the\nFLEUR code [40] based on the full-potential linearized aug-mented plane wave method (see Supplemental Material for\ncomputational details [41]). We have calculated the en-\nergy dispersions E(q)of flat spin spiral states [42, 43]\nfor the Fe 5GeTe 2monolayer. A magnetic moment miat\natom position Rifor a flat spin spiral is given by mi=\nM[cos(q·Ri),sin(q·Ri),0], with Mdenoting the size of\nthe magnetic moment.\nTo determine the interactions of magnetic moments for\nthe Fe 5GeTe 2monolayer, we adopt the following atomistic\nspin Hamiltonian, which is fitted from spin spiral calculations\nwithout and with SOC\nH=−X\nijJij(mi·mj)−X\nijDij·(mi×mj)\n+X\niK(mz\ni)2−X\niM(mi·Bext)(1)\nwhere miandmjare normalized magnetic moments at po-\nsitionRiandRirespectively. The four magnetic interaction\nterms correspond to the Heisenberg isotropic exchange, the\nDMI, the MAE, and the external magnetic field, and they are\ncharacterized by the interaction constants Jij,Dij, and K,\nandBext, respectively. Note that our spin model is adapted to\na collective 2D model by treating five Fe layers of the FGT as\na whole system, similar to a monolayer system. All magnetic\ninteraction parameters are measured in meV/unit cell (uc).\nWe consider a Fe 5GeTe 2monolayer where one Fe 1is situ-\nated above Ge as shown in Fig. 1(a), the so-called UUU con-\nfiguration. The calculated lattice constant is about 3.96 ˚A,\nwhich agrees well with previous results [44]. We are aware\nthat there are two possible configurations for monolayer\nFe5GeTe 2, namely UUU and UDU configurations [45, 46].\nIn this work, we focus on the UUU configuration since\nskyrmionic spin structures are experimentally observed in this\nconfiguration [30, 31]. For both spin channels, Fe 5GeTe 2ex-\nhibits a metallic property (See Fig. S1 in Supplemental Mate-\nrial [41]). The calculated spin moments are −0.21µB,2.21µB,\n1.70µB,1.28µB, and 2.41µBfor the Fe 1, Fe2, Fe3, Fe4, and\nFe5atoms, respectively.\nWe focus first on spin spiral calculations without SOC\nalong the M-Γ-K-M high-symmetry direction (Fig. 2(a)).\nFrom the fitted parameters shown in Table I, we find the\nHeisenberg exchange to be largely dominated by the nearest\nneighbor contribution. This is also clearly seen from the fitted\ncurve using only the nearest neighbor (NN) term (see dotted\nline in Fig. 2(a)). On the other hand, exchange constants up to\nseventh neighbors (Table I) are necessary to fit the DFT results\naccurately (solid line in Fig. 2(a)). Without SOC, the energy\ndispersion shows a minimum at the Γpoint, which represents\nthe ferromagnetic (FM) state. The energy difference between\ntheΓand the M point (row-wise antiferromagnetic (AFM)\nstate) is found to be about 137 meV for five Fe atoms. This\nenergy difference is much smaller than that in the Fe 3GTe2\nor Fe 4GeTe 2monolayers (see Fig. S2 in Supplemental Mate-\nrial [41]), leading to much smaller J1values (see Table I in\nSupplemental Material [41]). We also note that the spin mo-\nment variation is mainly from Fe 1and Fe 4atoms (Fig. S3 in\nSupplemental Material [41]).3\nJ1/Jeff J2J3J4J5 J6 J7 D1/D eff D2 D3 D4 D5 D6 D7 K m s\nFull model 14.803 0.898 0.466 0.916 0.231 −0.443−0.599 −0.659 −0.523−0.083 0.020 −0.008−0.021−0.008 0.380 7.306\nEffective model 16.457 – – – – – – −0.880 – – – – – – 0.380 7.306\nTABLE I. Full model: Shell-resolved Heisenberg exchange constants ( Ji) and DMI constants ( Di) obtained by fitting the energy contribution\nto spin spirals without and with SOC, and spin moments from DFT calculations as presented in Fig. 2 for the Fe 5GeTe 2monolayer. Effective\nmodel: Parameters of the effective NN exchange and DMI model obtained by fitting the DFT results. A positive (negative) sign represents\nan FM (AFM) coupling. A positive (negative) sign of Didenotes a preference for CW (CCW) rotating cycloidal spin spirals. The Fe 5GeTe 2\nmonolayer favors in-plane MAE (i.e., K > 0). Note that all parameters are treated as a collective 2D spin model in a hexagon symmetry (i.e.,\nfive Fe atoms in the supercell are treated as a whole). The magnetic moments are given in µB/unit cell (UC), and other parameters are given in\nmeV/uc.\nWhen SOC is taken into account, the DMI arises due to\nthe broken inversion symmetry. The Fe 5GeTe 2favors cy-\ncloidal spin spirals with a counter-clockwise (CCW) rota-\ntional sense, as seen from the calculated energy contribution\nto the dispersion due to SOC, ∆ESOC(q)(Fig. 2(b)). If we\napply the NN approximation, we obtain the effective DMI\nconstant Deff=−0.88meV , and the corresponding micro-\nmagnetic DMI is given by D=3√\n2Deff\nNFa2(aandNFare the lat-\ntice constant and the number of ferromagnetic layers). For the\nFe5GeTe 2monolayer, the value is approximately 0.76 mJ/m2,\nwhich is in reasonable agreement with the one calculated by\nthe supercell approach ( ∼0.48−0.67 mJ/m2) [30] based on\nthe NN approximation. However, as clearly seen in Fig. 2(b),\nthe dashed lines with an effective NN DMI fail to capture our\nDFT results beyond the regime of small |q|. The inclusion of\ninteractions up to the seventh NN is needed to reproduce the\nDFT data accurately, and this is also reflected in the fitted pa-\nrameters presented in Table I. The second NN interaction is\non the same order of magnitude compared to the first NN one.\nAdditionally, SOC also introduces MAE. We find the easy\nmagnetization axis of the Fe 5GeTe 2monolayer to be in-plane\nwith a MAE of 0.38 meV/uc. As observed in Fig. 2(c), a spin\nspiral energy minimum of −0.1 meV/uc compared to the FM\nstate occurs in the ΓK direction, which corresponds to a spin\nspiral period of λ= 2π/|q|= 19.9nm.\nIncluding all interactions, i.e. Heisenberg exchange, DMI,\nand MAE, results in the spin spiral energy dispersion shown\nin Fig. 2(c). The energy contribution from the MAE leads to\nan energy offset of K/2for spin spirals with a long period,\ni.e. small value of |q|, with respect to the FM state, as can be\nseen in a zoom of E(q)around Γ. Interestingly, the ground\nstate within the NN approximation is the FM state, while it\nis the spin spiral state when we include interactions up to the\nseventh NN. It is worth emphasizing that the spin spiral curve\nbecomes extremely flat near the Γpoint. It has been demon-\nstrated that a flat energy dispersion around Γis beneficial for\nstabilizing nanoscale skyrmions in ultrathin films [13].\nTo check the possibility of stabilizing nanoscale magnetic\nskyrmions in Fe 5GeTe 2, we performed atomistic spin simula-\ntions using the spin model described by Eq. (1) with the full\nset of DFT parameters. We apply atomistic spin-dynamics\nvia the Landau–Lifshitz equation to obtain isolated mag-\nnetic skyrmions. Minimum energy paths between the ini-\ntial skyrmion and final FM state were calculated using the\ngeodesic nudged elastic band (GNEB) method [47]. The en-\nergy barriers stabilizing skyrmions against collapse were ob-tained from the saddle point along this path. Finally, we used\nharmonic transition-state theory to quantify skyrmion stabil-\nity by calculating their lifetime [38, 48–50] (see Supplemental\nMaterial for computational details [41]).\nWe create isolated skyrmions in the field-polarized back-\nground with an out-of-plane magnetization direction due to\nan applied magnetic field and fully relax these spin struc-\ntures by solving the damped Landau–Lifshitz equation self-\nconsistently. The in-plane FM state (denoted as FM ∥) ex-\nhibits slightly lower energy compared to the out-of-plane FM\nstate, owing to the in-plane MAE of Fe 5GeTe 2(see Fig. S4 in\nSupplemental Material [41]). At zero magnetic field, we do\nnot observe the emergence of skyrmions. Instead, we observe\nlabyrinth domains with chiral N ´eel domain walls, which hold\nthe lowest energy state. As we gradually increase the mag-\nnetic field Bperpendicular to the monolayer, the labyrinthine\ndomain shrinks and eventually vanishes, giving rise to isolated\nmagnetic skyrmions at a critical field of Bc= 2.2 T. Note, that\nfor fields above B= 2K/M ≈1.8T applied perpendicular\nto the film the Zeeman energy exceeds the MAE [51].\nOur atomistic spin simulations predict N ´eel-type magnetic\nskyrmions stabilized in the FM background (see inset of\nFig. 3(a) for skyrmion profile and spin texture at B= 2.2T).\nAs expected, the skyrmion size decreases in Fe 5GeTe 2with\nincreasing magnetic field (Fig. 3(a)). Interestingly, nanoscale\nskyrmions occur with a radius below 6.4 nm at B >2.2T, and\nisolated skyrmions can be obtained. Note that the skyrmion\nradius is estimated using the Bocdanov- Θprofile [52] within\na wide range of B= 2.2∼7.6T. At B > 7.6T, skyrmions\ncollapse into the FM state.\nTo provide insight into skyrmion stability in monolayer\nFe5GeTe 2, we show in Fig. 3(b) the calculated energy bar-\nriers protecting skyrmions from collapsing into the FM state\nwith respect to B−Bc. Remarkably, we obtain an energy\nbarrier of more than 80 meV at the critical field Bc= 2.2T.\nThis value is comparable to state-of-the-art ultra-thin films\nthat serve as prototype systems to host nanoscale skyrmions\n[37–39]. Note, that previous work on skyrmion stability in 2D\nmagnets [27] reported smaller values than those found here for\nthe Fe 5GeTe 2monolayer.\nFrom the energy decomposition of the barrier, we con-\nclude that the DMI and MAE are mainly responsible for the\nskyrmion stability (Fig. 3(b)). We also note that the energy\nbarrier can be enhanced if we increase the in-plane MAE, a\nquantity that can be easily tuned in experiments by doping or\ntemperature [53–55]. However, if the in-plane MAE is in-4\nSP-1(d)\nSP\nSP+1\nTemperature (T)B - Bc (T)Time (s)f0/T (s-1K-1) B (T)02468Skyrmion radius (nm)\n0 1 2 3 4 5\nB B c(T)\n160\n80\n080160Energy barrier (meV)\nExchange\nDMI\nMAEZeeman\nTotal\n0 20 40 60\nReaction coordinate (rad)200\n150\n100\n50\n050100150200Energy barrier (meV)\nB= 2.2 TSkSP\nFM\n1.00\n0.75\n0.50\n0.25\n0.00\nTopological chargeQ\n10 2 3 41019\n1017\n1015\n4 8 12 16 20 246.0\n2.05.0\n4.0\n3.01000 years\n1 week\n1 hour1025\n1015\n105\n10-5\n5(a)\n(b)(c)\nB = 2.2 T\n-1-0.500.51\n-16-12-8-40481216\ndistance (nm)mz\nFIG. 3. Skyrmion radius and energy barrier for skyrmion collapse in the Fe 5GeTe 2monolayer evaluated using magnetic interaction parameters\nfrom DFT. (a) Skyrmion radius as a function of B−Bcwhere Bcis the critical field for stabilizing metastable magnetic skyrmions. The\ncalculated value is Bc= 2.2T for Fe 5GeTe 2. The skyrmion profile and spin texture at B= 2.2T are shown as insets. (b) Total energy\nbarriers and their decomposition (see legend) of isolated skyrmions versus B. (c) Energy contributions from the different interactions (the\nsame legend as in panel b, left axis) are shown versus the reaction coordinate along the minimum energy path from the initial (skyrmion) state\nto the final (FM) state through the saddle point (SP). The energies are summed over all atoms of the simulation box and are given relative to\nthe energies of the initial isolated skyrmion state. The topological charge (open circles, right axis) is plotted versus the reaction coordinate.\n(c) Corresponding spin structures before (SP-1), after (SP+1), and at the saddle point (SP) are shown. Note that the skyrmion collapse occurs\nvia radially symmetrical shrinking. (d) Calculated attempt frequencies f0on a logarithmic scale with respect to Bz(top panel). Skyrmion\nlifetimes of Fe 5GeTe 2obtained in harmonic transition state theory based on the spin model with DFT parameters as a function of magnetic\nfield and temperature (bottom panel).\ncreased larger magnetic fields are needed to obtain the field-\npolarized phase in which skyrmions can be stabilized.\nTo obtain information about transition mechanisms from\nthe skyrmion state to the FM state via the saddle point (SP),\nwhich determines the barrier, we show in Fig. 3(c) the de-\ncomposition of the energy along the minimum energy path for\nskyrmion collapse at B= 2.2T. The topological charge, cal-\nculated by Q=R\nm·(∂m\n∂x×∂m\n∂y)dxdy , changes from −1to0\nat the SP. The skyrmion is annihilated via the radial symmetric\ncollapse mechanism in which the skyrmion shrinks symmetri-\ncally to SP and then collapses into the FM state [50]. Again, it\nis clear that the DMI and MAE prefer the skyrmion (Sk) state\nand decrease the total barrier, while the Zeeman term strongly\nfavors the FM state. Due to frustration, the Heisenberg ex-\nchange energy gives a small positive annihilation barrier.\nThe stability of metastable magnetic skyrmions can be\nquantified by their mean lifetime, τ, which is given by the\nArrhenius law τ=f−1\n0exp\u0010\n∆E\nkBT\u0011\n, where ∆E,f0, and T\nare energy barrier, attempt frequency, and temperature, re-\nspectively. The calculated f0within harmonic transition state\ntheory [48] is shown in Fig. 3(d). As expected, f0depends\nstrongly on the magnetic field. This effect is similar to that ob-\nserved in ultrathin transition-metal films, which can be traced\nback to a change of entropy with skyrmion radius and pro-\nfile [49, 56]. From the temperature and field dependence\nof the skyrmion lifetime (Fig. 3(d)), we predict that isolatedskyrmions in the Fe 5GeTe 2are stable up to hours at a temper-\nature at about 20 K and B= 2.2T. Therefore, these nanoscale\nskyrmions can be probed by experiments using current state-\nof-the-art techniques e.g., spin-polarized scanning tunneling\nmicroscopy or Lorentz transmission electron microscopy.\nFinally, we plot in Fig. 4 the comparison of atomistic spin\nsimulation results from parameters with the full and effective\nmodels. The critical magnetic field is found to be about 1.4 T\nin the case of the effective model, which is 0.8 T smaller\nthan the one for the full model. In both models, we have\nsub-10 nm radius skyrmions (Fig. 4(a)) and energy barriers\nof a few 10 meV (Fig. 4(b)). From the GNEB calculation\natB−Bc= 0.6T, the energy barriers for the full and ef-\nfective models are rather different. We find more than 3 times\nlarger energy barrier with the full model, indicating the impor-\ntance of including full magnetic parameters for the Fe 5GeTe 2\nmonolayer.\nIn summary, we propose, based on first-principles cal-\nculations and atomistic spin simulations, that monolayer\nFe5GeTe 2is a compelling 2D vdW magnet with skyrmionic\nphysics. Due to the large DMI together with moderate in-\nplane MAE, Fe 5GeTe 2monolayer can exhibit metastable\nnanoscale (sub-10 nm) skyrmions in an out-of-plane mag-\nnetic field above 2.2 T. We also highlight the importance of\nincluding beyond nearest neighbor DMI terms in the atom-\nistic spin model. The energy barriers protecting skyrmions\nagainst collapse are up to 80 meV , which are comparable to5\n0246810Skyrmion radius (nm)\nFull model\nEffective model\n0 1 2 3 4 5\nB B c(T)\n0306090Energy barrier (meV)\n0 10 20 30 40 50 60\nReaction coordiate80\n40\n04080Energy barrier (meV)\n(a)\n(b)(c)\nB - Bc = 0.6 T\nFIG. 4. Full model versus effective model comparison. (a) Skyrmion radius vs. magnetic field. (b) Skyrmion energy barriers with respect to\nB. The estimated critical magnetic fields Bcare 2.2 T and 1.4 T for the full model and effective model, respectively. (c) Minimum energy paths\nof skyrmion collapse: Energies of the spin configurations during a skyrmion collapse are shown over the reaction coordinate corresponding\nto the progress of the collapse at B−Bc= 0.6T. Our simulations were conducted using both the effective exchange model (in red) and\nconsidering all magnetic interactions from the DFT calculation (in black).\nthose of state-of-the-art transition-metal ultrathin films. Us-\ning harmonic transition-state theory, we predict that nanoscale\nskyrmions are stable in monolayer Fe 5GeTe 2with lifetimes of\nhours up to 20 K.Acknowledgments\nThis study has been supported through the ANR Grant No.\nANR-22-CE24-0019. This study has been (partially) sup-\nported through the grant NanoX no. ANR-17-EURE-0009\nin the framework of the “Programme des Investissements\nd’Avenir”. S. Ha and S. He. gratefully acknowledge financial\nsupport from the Deutsche Forschungsgemeinschaft (DFG,\nGerman Research Foundation) through SPP2137 “Skyrmion-\nics” (project no. 462602351). This work was performed using\nHPC resources from CALMIP (Grant 2021/2023-[P21023])\nand the North- German Supercomputing Alliance (HLRN).\n[1] A. Fert, N. Reyren, and V . Cros, Magnetic skyrmions: ad-\nvances in physics and potential applications, Nat. Rev. Mater.\n2, 1 (2017).\n[2] K. Everschor-Sitte, J. Masell, R. Reeve, and M. Kl ¨aui, Perspec-\ntive: Magnetic skyrmions – overview of recent progress in an\nactive research field, J. Appl. Phys. 124, 240901 (2018).\n[3] S. Luo, M. Song, X. Li, Y . Zhang, J. Hong, X. Yang, X. Zou,\nN. Xu, and L. You, Reconfigurable skyrmion logic gates, Nano\nLett.18, 1180 (2018).\n[4] B. G ¨obel, I. Mertig, and O. Tretiakov, Beyond skyrmions: Re-\nview and perspectives of alternative magnetic quasiparticles,\nPhys. Rep. 895, 1 (2021).\n[5] S. Li, W. Kang, X. Zhang, T. Nie, Y . Zhou, K. Wang, and\nW. Zhao, Magnetic skyrmions for unconventional computing,\nMater. Horiz. 8, 854 (2021).\n[6] C. Psaroudaki and C. Panagopoulos, Skyrmion qubits: A new\nclass of quantum logic elements based on nanoscale magneti-\nzation, Phys. Rev. Lett. 127, 067201 (2021).[7] S. M ¨uhlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch,\nA. Neubauer, R. Georgii, and P. Boni, Skyrmion lattice in a\nchiral magnet, Science 323, 915 (2009).\n[8] X. Yu, N. Kanazawa, Y . Onose, K. Kimoto, W. Zhang, S. Ishi-\nwata, Y . Matsui, and Y . Tokura, Near room-temperature forma-\ntion of a skyrmion crystal in thin-films of the helimagnet FeGe,\nNat. Mater. 10, 106 (2011).\n[9] J. H. Yang, Z. L. Li, X. Z. Lu, M.-H. Whangbo, S.-H. Wei, X. G.\nGong, and H. J. Xiang, Strong Dzyaloshinskii-Moriya Interac-\ntion and origin of ferroelectricity in Cu 2OSe 3, Phys. Rev. Lett.\n109, 107203 (2012).\n[10] N. Romming, C. Hanneken, M. Menzel, J. Bickel, B. Wolter,\nK. von Bergmann, A. Kubetzka, and R. Wiesendanger, Writ-\ning and deleting single magnetic skyrmions, Science 341, 636\n(2013).\n[11] B. Dup ´e, M. Hoffmann, C. Paillard, and S. Heinze, Tailoring\nmagnetic skyrmions in ultra-thin transition metal films, Nat.\nCommun. 5, 4030 (2014).6\n[12] O. Boulle, J. V ogel, H. Yang, S. Pizzini, D. de Souza Chaves,\nA. Locatelli, T. Mentes ¸, A. Sala, L. Buda-Prejbeanu, O. Klein,\net al. , Room-temperature chiral magnetic skyrmions in ultrathin\nmagnetic nanostructures, Nat. Nanotechnol. 11, 449 (2016).\n[13] S. Meyer, M. Perini, S. von Malottki, A. Kubetzka, R. Wiesen-\ndanger, K. von Bergmann, and S. Heinze, Isolated zero field\nsub-10 nm skyrmions in ultrathin Co films, Nat. Commun. 10,\n3823 (2019).\n[14] C. Moreau-Luchaire, C. Moutafis, N. Reyren, J. Sampaio,\nC. Vaz, N. Van Horne, K. Bouzehouane, K. Garcia, C. Der-\nanlot, P. Warnicke, et al. , Additive interfacial chiral interaction\nin multilayers for stabilization of small individual skyrmions at\nroom temperature, Nat. Nanotechnol. 11, 444 (2016).\n[15] A. Soumyanarayanan, M. Raju, A. Gonzalez Oyarce, A. Tan,\nM. Im, A. Petrovic, P. Ho, K. Khoo, M. Tran, C. Gan, et al. ,\nTunable room-temperature magnetic skyrmions in Ir/Fe/Co/Pt\nmultilayers, Nat. Mater. 16, 898 (2017).\n[16] M. Raju, A. Yagil, A. Soumyanarayanan, A. K. Tan, A. Al-\nmoalem, F. Ma, O. Auslaender, and C. Panagopoulos, The evo-\nlution of skyrmions in Ir/Fe/Co/Pt multilayers and their topo-\nlogical hall signature, Nat. Commun. 10, 696 (2019).\n[17] M. Han, J. Garlow, Y . Liu, H. Zhang, J. Li, D. DiMarzio,\nM. W. Knight, C. Petrovic, D. Jariwala, and Y . Zhu, Topolog-\nical magnetic-spin textures in two-dimensional van der waals\nCr2Ge2Te6, Nano Lett. 19, 7859 (2019).\n[18] B. Ding, Z. Li, G. Xu, H. Li, Z. Hou, E. Liu, X. Xi, F. Xu,\nY . Yao, and W. Wang, Observation of magnetic skyrmion bub-\nbles in a van der waals ferromagnet Fe 3GeTe 2, Nano Lett. 20,\n868 (2020).\n[19] J. Liang, W. Wang, H. Du, A. Hallal, K. Garcia, M. Chshiev,\nA. Fert, and H. Yang, Very large Dzyaloshinskii-Moriya in-\nteraction in two-dimensional janus manganese dichalcogenides\nand its application to realize skyrmion states, Phys. Rev. B 101,\n184401 (2020).\n[20] C. Xu, J. Feng, S. Prokhorenko, Y . Nahas, H. Xiang, and L. Bel-\nlaiche, Topological spin texture in janus monolayers of the\nchromium trihalides Cr(I, X)3, Phys. Rev. B 101, 060404(R)\n(2020).\n[21] W. Du, K. Dou, Z. He, Y . Dai, B. Huang, and Y . Ma, Sponta-\nneous magnetic skyrmions in single-layer CrInX 3(X= Te, Se),\nNano Lett. 22, 3440 (2022).\n[22] J. Liu, M. Shi, J. Lu, and M. P. Anantram, Analysis of electrical-\nfield-dependent Dzyaloshinskii-Moriya interaction and mag-\nnetocrystalline anisotropy in a two-dimensional ferromagnetic\nmonolayer, Phys. Rev. B 97, 054416 (2018).\n[23] Y . Wu, S. Zhang, J. Zhang, W. Wang, Y . Zhu, J. Hu, G. Yin,\nK. Wong, C. Fang, C. Wan, et al. , N´eel-type skyrmion in\nWTe 2/Fe3GeTe 2van der waals heterostructure, Nat. Commun.\n11, 3860 (2020).\n[24] T. Park, L. Peng, J. Liang, A. Hallal, F. S. Yasin, X. Zhang,\nK. M. Song, S. J. Kim, K. Kim, M. Weigand, G. Sch ¨utz,\nS. Finizio, J. Raabe, K. Garcia, J. Xia, Y . Zhou, M. Ezawa,\nX. Liu, J. Chang, H. C. Koo, Y . D. Kim, M. Chshiev, A. Fert,\nH. Yang, X. Yu, and S. Woo, N ´eel-type skyrmions and their\ncurrent-induced motion in van der waals ferromagnet-based\nheterostructures, Phys. Rev. B 103, 104410 (2021).\n[25] Y . Wu, B. Francisco, Z. Chen, W. Wang, Y . Zhang, C. Wan,\nX. Han, H. Chi, Y . Hou, A. Lodesani, G. Yin, K. Liu, Y . Cui,\nK. Wang, and J. Moodera, A van der waals interface hosting\ntwo groups of magnetic skyrmions, Adv. Mater. 34, 2110583\n(2022).\n[26] M. Yang, Q. Li, R. Chopdekar, R. Dhall, J. Turner, J. Carlstr ¨om,\nC. Ophus, C. Klewe, P. Shafer, A. N’Diaye, et al. , Creation of\nskyrmions in van der waals ferromagnet Fe 3GeTe 2on (Co/Pd) nsuperlattice, Sci. Adv. 6, eabb5157 (2020).\n[27] D. Li, S. Haldar, and S. Heinze, Strain-driven zero-field near-10\nnm skyrmions in two-dimensional van der waals heterostruc-\ntures, Nano Lett. 22, 7706 (2022).\n[28] D. Li, S. Haldar, T. Drevelow, and S. Heinze, Tuning the mag-\nnetic interactions in van der waals Fe 3GeTe 2heterostructures:\nA comparative study of ab initio methods, Phys. Rev. B 107,\n104428 (2023).\n[29] D. Li, S. Haldar, and S. Heinze, Proposal for all-electrical\nskyrmion detection in van der waals tunnel junctions, arXiv\npreprint arXiv:2309.03828 (2023).\n[30] Y . Gao, S. Yan, Q. Yin, H. Huang, Z. Li, Z. Zhu, J. Cai, B. Shen,\nH. Lei, Y . Zhang, and S. Wang, Manipulation of topological\nspin configuration via tailoring thickness in van der waals fer-\nromagnetic Fe5−xGeTe 2, Phys. Rev. B 105, 014426 (2022).\n[31] M. Schmitt, T. Denneulin, A. Kov ´acs, T. G. Saunderson,\nP. R¨ußmann, A. Shahee, T. Scholz, A. H. Tavabi, M. Grad-\nhand, P. Mavropoulos, et al. , Skyrmionic spin structures in lay-\nered Fe 5GeTe 2up to room temperature, Commun. Phys. 5, 254\n(2022).\n[32] X. Lv, K. Pei, C. Yang, G. Qin, M. Liu, J. Zhang, and\nR. Che, Controllable topological magnetic transformations in\nthe thickness-tunable van der waals ferromagnet Fe 5GeTe 2,\nACS nano 16, 19319 (2022).\n[33] B. W. Casas, Y . Li, A. Moon, Y . Xin, C. McKeever, J. Macy,\nA. K. Petford-Long, C. M. Phatak, E. J. Santos, E. S. Choi,\net al. , Coexistence of merons with skyrmions in the centrosym-\nmetric van der waals ferromagnet Fe 5−xGeTe 2, Adv. Mater. ,\n2212087 (2023).\n[34] M. H ¨ogen, R. Fujita, A. K. C. Tan, A. Geim, M. Pitts, Z. Li,\nY . Guo, L. Stefan, T. Hesjedal, and M. Atat ¨ure, Imaging\nnucleation and propagation of pinned domains in few-layer\nFe5–xGeTe 2, ACS Nano 17, 16879 (2023).\n[35] A. F. May, D. Ovchinnikov, Q. Zheng, R. Hermann, S. Calder,\nB. Huang, Z. Fei, Y . Liu, X. Xu, and M. A. McGuire, Ferromag-\nnetism near room temperature in the cleavable van der waals\ncrystal Fe 5GeTe 2, ACS nano 13, 4436 (2019).\n[36] H. Zhang, R. Chen, K. Zhai, X. Chen, L. Caretta, X. Huang,\nR. V . Chopdekar, J. Cao, J. Sun, J. Yao, R. Birgeneau,\nand R. Ramesh, Itinerant ferromagnetism in van der waals\nFe5−xGeTe 2crystals above room temperature, Phys. Rev. B\n102, 064417 (2020).\n[37] S. von Malottki, B. Dup ´e, P. Bessarab, A. Delin, and S. Heinze,\nEnhanced skyrmion stability due to exchange frustration, Sci.\nRep.7, 12299 (2017).\n[38] S. Haldar, S. von Malottki, S. Meyer, P. Bessarab, and\nS. Heinze, First-principles prediction of sub-10-nm skyrmions\nin Pd/Fe bilayers on Rh(111), Phys. Rev. B 98, 060413 (2018).\n[39] S. Paul, S. Haldar, S. von Malottki, and S. Heinze, Role of\nhigher-order exchange interactions for skyrmion stability, Nat.\nCommun. 11, 4756 (2020).\n[40] Welcome to the FLEUR-project, www.flapw.de (accessed Sept.\n1, 2022).\n[41] See Supplemental Material at http://link.aps.org/supplemental/\nfor computational details, spin-resolved DOS for the Fe 5GeTe 2\nmonolayer, spin spiral calculations without SOC for the FGT\nfamily, spin moment with respect to qfor Fe 5GeTe 2monolayer\nas well as the zero-temperature phase diagram for the Fe 5GeTe 2\nmonolayer, which includes Ref. [38, 40, 42, 44, 47–49, 57–60].\n[42] P. Kurz, F. F ¨orster, L. Nordstr ¨om, G. Bihlmayer, and S. Bl ¨ugel,\nAb initio treatment of noncollinear magnets with the full-\npotential linearized augmented plane wave method, Phys. Rev.\nB69, 024415 (2004).7\n[43] M. Heide, G. Bihlmayer, and S. Bl ¨ugel, Describing\nDzyaloshinskii-Moriya spirals from first principles, Phys.\nB404, 2678 (2009).\n[44] X. Yang, X. Zhou, W. Feng, and Y . Yao, Strong magneto-optical\neffect and anomalous transport in the two-dimensional van der\nwaals magnets FenGeTe 2(n= 3, 4, 5), Phys. Rev. B 104,\n104427 (2021).\n[45] S. Ershadrad, S. Ghosh, D. Wang, Y . Kvashnin, and B. Sanyal,\nUnusual magnetic features in two-dimensional Fe 5GeTe 2in-\nduced by structural reconstructions, J. Phys. Chem. Lett. 13,\n4877 (2022).\n[46] S. Ghosh, S. Ershadrad, V . Borisov, and B. Sanyal, Unraveling\neffects of electron correlation in two-dimensional Fe nGeTe 2\n(n= 3,4,5) by dynamical mean field theory, Npj Comput.\nMater. 9, 86 (2023).\n[47] P. Bessarab, V . Uzdin, and H. J ´onsson, Method for finding\nmechanism and activation energy of magnetic transitions, ap-\nplied to skyrmion and antivortex annihilation, Comput. Phys.\nCommun. 196, 335 (2015).\n[48] P. F. Bessarab, G. P. M ¨uller, I. S. Lobanov, F. N. Rybakov,\nN. S. Kiselev, H. J ´onsson, V . M. Uzdin, S. Bl ¨ugel, L. Bergqvist,\nand A. Delin, Lifetime of racetrack skyrmions, Sci. Rep. 8, 1\n(2018).\n[49] S. von Malottki, P. Bessarab, S. Haldar, A. Delin, and S. Heinze,\nSkyrmion lifetime in ultrathin films, Phys. Rev. B 99, 060409\n(2019).\n[50] F. Muckel, S. von Malottki, C. Holl, B. Pestka, M. Pratzer,\nP. Bessarab, S. Heinze, and M. Morgenstern, Experimental\nidentification of two distinct skyrmion collapse mechanisms,\nNat. Phys. 17, 395 (2021).\n[51] Note, that the dipole-dipole interaction also prefers an in-plane\nmagnetization direction as the MAE. For the Fe 5GeTe 2mono-\nlayer its energy contribution amounts to about 60% of the MAE.\nIf we include the dipolar energy, the critical magnetic field re-\nquired to obtain the field-polarized phase with an out-of-plane\nmagnetization increases to about 3.4 T.[52] A. Bocdanov and A. Hubert, The properties of isolated mag-\nnetic vortices, Phys. Status Solidi (b) 186, 527 (1994).\n[53] C. Tan, J. Lee, S. Jung, T. Park, S. Albarakati, J. Partridge,\nM. Field, D. McCulloch, L. Wang, and C. Lee, Hard magnetic\nproperties in nanoflake van der waals Fe 3GeTe 2, Nat. Commun.\n9, 1554 (2018).\n[54] Y . Wang, X. Chen, and M. Long, Modifications of magnetic\nanisotropy of Fe 3GeTe 2by the electric field effect, Appl. Phys.\nLett.116, 092404 (2020).\n[55] S. Park, D. Kim, Y . Liu, J. Hwang, Y . Kim, W. Kim, J. Kim,\nC. Petrovic, C. Hwang, S. Mo, H. Kim, B. Min, H. Koo,\nJ. Chang, C. Jang, J. Choi, and H. Ryu, Controlling the mag-\nnetic anisotropy of the van der waals ferromagnet Fe 3GeTe 2\nthrough hole doping, Nano Lett. 20, 95 (2020).\n[56] A. S. Varentcova, S. von Malottki, M. N. Potkina,\nG. Kwiatkowski, S. Heinze, and P. F. Bessarab, Toward room-\ntemperature nanoscale skyrmions in ultrathin films, NPJ Com-\nput. Mater. 6, 1 (2020).\n[57] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car,\nC. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococ-\ncioni, I. Dabo, A. Dal Corso, S. de Gironcoli, S. Fab-\nris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis,\nA. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri,\nR. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbrac-\ncia, S. Scandolo, G. Sclauzero, A. P. Seitsonen, A. Smogunov,\nP. Umari, and R. M. Wentzcovitch, QUANTUM ESPRESSO: a\nmodular and open-source software project for quantum simula-\ntions of materials., J. Phys. Condens. Matter 21, 395502 (2009).\n[58] H. Zhuang, P. Kent, and R. Hennig, Strong anisotropy and\nmagnetostriction in the two-dimensional stoner ferromagnet\nFe3GeTe 2, Phys. Rev. B 93, 134407 (2016).\n[59] Y . Deng, Y . Yu, Y . Song, J. Zhang, N. Wang, Z. Sun, Y . Yi,\nY . Wu, S. Wu, J. Zhu, et al. , Gate-tunable room-temperature\nferromagnetism in two-dimensional Fe 3GeTe 2, Nature 563, 94\n(2018).\n[60] J. Mentink, M. Tretyakov, A. Fasolino, M. Katsnelson, and\nT. Rasing, Stable and fast semi-implicit integration of the\nstochastic landau–lifshitz equation, J. Phys. Condens. Matter.\n22, 176001 (2010)."    },    {        "title": "2402.06375v1.Configuration_of_the_magnetosome_chain__a_natural_magnetic_nanoarchitecture.pdf",        "content": "Configuration of the magnetosome chain: a natural\nmagnetic nanoarchitecture\nI. Orue,1L. Marcano,2P. Bender,3A. Garc ´ıa-Prieto,4,5S. Valencia,6\nM.A. Mawass,6D. Gil-Cart ´on,7D. Alba Venero,8D. Honecker,9\nA. Garc ´ıa-Arribas,2,5L. Fern ´andez Barqu ´ın,3\nA. Muela,5,10M.L. Fdez-Gubieda2,5∗\n1SGIker, Universidad del Pa ´ıs Vasco - UPV/EHU, 48940 Leioa, Spain,\n2Dpto. Electricidad y Electr ´onica, Universidad del Pa ´ıs Vasco - UPV/EHU, 48940 Leioa, Spain\n3CITIMAC, Universidad de Cantabria, 39005 Santander, Spain\n4Dpto. F ´ısica Aplicada I, Universidad del Pa ´ıs Vasco - UPV/EHU, 48013 Bilbao, Spain\n5BCMaterials, Bld. Martina Casiano, 48940 Leioa, Spain\n6Helmholtz-Zentrum Berlin f ¨ur Materialien und Energie,\nAlbert-Einstein-str. 15, 12489 Berlin, Germany\n7Structural Biology Unit, CIC bioGUNE, CIBERehd, 48160 Derio, Spain\n8ISIS, STFC Rutherford Appleton Laboratory, Chilton,\nDidcot OX11 0QX, United Kingdom\n9Institut Laue-Langevin, 38042 Grenoble, France\n10Dpto. Inmunolog ´ıa, Microbiolog ´ıa y Parasitolog ´ıa,\nUniversidad del Pa ´ıs Vasco - UPV/EHU, 48940 Leioa, Spain\n∗To whom correspondence should be addressed; e-mail: malu.gubieda@ehu.eus.\nFebruary 12, 2024\n1arXiv:2402.06375v1  [cond-mat.mes-hall]  9 Feb 2024Abstract\nMagnetospirillum gryphiswaldense is a microorganism with the ability to bio-\nmineralize magnetite nanoparticles, called magnetosomes, and arrange them into\na chain that behaves like a magnetic compass. Rather than straight lines, magne-\ntosome chains are slightly bent, as evidenced by electron cryotomography. Our\nexperimental and theoretical results suggest that due to the competition between\nthe magnetocrystalline and shape anisotropies, the effective magnetic moment of\nindividual magnetosomes is tilted out of the [111] crystallographic easy axis of\nmagnetite. This tilt does not affect the direction of the chain net magnetic moment,\nwhich remains along the [111] axis, but explains the arrangement of magnetosomes\nin helical-like shaped chains. Indeed, we demonstrate that the chain shape can be\nreproduced by considering an interplay between the magnetic dipolar interactions\nbetween magnetosomes, ruled by the orientation of the magnetosome magnetic mo-\nment, and a lipid/protein-based mechanism, modeled as an elastic recovery force\nexerted on the magnetosomes.\nMagnetotactic bacteria (MTB) are microorganisms ubiquitously present in any aquatic envi-\nronment with the capability of magnetic orientation1. The origin of this behavior is the presence\nof intracellular magnetic nanoparticles enclosed in a vesicle, called magnetosomes. The size,\nmorphology and composition of the magnetosomes depend on the species, being the size, in all\nof them, large enough to be magnetically stable at room temperature2. Most MTB organize the\nmagnetosomes in a chain. Since magnetosomes are single magnetic domains, the chain behaves\nas a large single permanent dipole magnet which is able to passively reorient the whole bacteria\nalong external magnetic field lines, even at fields as small as those existing near the Earth’s\nsurface ( ≈20−60µT)3–5. On top of the fundamental scientific interest of these fascinating\nmicroorganisms, magnetosome chains constitute a natural paradigm of highly anisotropic mag-\nnetic nanostructures, which show high potentiality in biomedical applications6–9, in actuation\ndevices as nanorobots10, in nanosensor devices11, and in magnetic memory devices12.\nAmong all the MTB species, Magnetospirillum gryphiswaldense strain MSR-1 is frequently\nused as a model strain of MTB due to its ease of cultivation in the laboratory. Magnetospirillum\ncells are helical and contain a variable number of ∼40−45 nm sized cuboctahedral magnetite\n2(Fe3O4) magnetosomes arranged in a chain. The biomineralization of the magnetite nanopar-\nticles and the assembly in a chain are complex and comprise different steps4,13,14. Firstly, the\nmagnetosome membrane is formed from invagination of the cytoplasmic membrane comprised\nwith proteins15. Secondly, iron is taken up from the environment and transported into the mag-\nnetosome vesicle where nucleation in magnetite nanocrystals occurs16–18, and finally, magne-\ntosomes are assembled forming the chain4,19–21. The present work is devoted to shed light on\nthis last step. In particular, here we address the underlying mechanisms that determine the\narrangement of the magnetosomes and consequently the geometry of the chain.\nThe arrangement of magnetosomes in a chain results from the interplay between the ac-\ntive assembly mechanism mediated by proteins and the magnetic dipolar interactions between\nnearest magnetite nanoparticles22–24. As shown by cryotomography imaging on Magnetospir-\nilla, bundles of cytoskeletal filaments intervene in the chain assembly15,25,26. These filaments,\nformed by the actin-like protein MamK, traverse the cell and position the chain in the middle\nof the cell. Another important protein involved in the chain formation is MamJ. One possible\nfunction of MamJ is to connect the magnetosome membrane to the cytoskeletal filament. As\nmagnetosomes get closer together, magnetic dipolar interactions arise.\nHere we show by electron cryotomography imaging that magnetosome chains of M. gry-\nphiswaldense are not straight lines but appear slightly bent, a fact that has been also observed\nin previous works14,26. The implications of the chain shape on the magnetic response of mag-\nnetosome chains have then been addressed with complementary techniques performed on a\nset of bacterial arrangements: i) small angle neutron/x-ray scattering (SANS/SAXS) on a bac-\nterial colloid, ii) macroscopic magnetometry on 3D and 2D fixed arrangements of randomly\ndistributed and aligned bacteria, and iii) x-ray photoemission electron microscopy (XPEEM)\non an individual chain of magnetosomes extracted from bacteria.\nTwo main findings are achieved from this work. Firstly, the equilibrium magnetic moment\n3of the magnetosomes is tilted 20◦out of the [111] crystallographic axis, as concluded from the\nmagnetic analysis. Secondly, the tilting of the magnetic moment is the key to understand the\nhelical-like shape of the magnetosome chains. Indeed, the experimental chain patterns imaged\nby electron cryotomography are accurately reproduced by counterbalancing the magnetic dipo-\nlar interactions between magnetosomes, strongly affected by the orientation of the individual\nmagnetic moments, and a lipid/protein-based mechanism, modeled as an elastic recovery force\nexerted on magnetosomes.\nBesides the basic interest of this study, a precise knowledge of the mechanisms determining\nthe chain shape is decisive in applications of MTB such as biological micro-robots. Precisely,\nrecent works propose MTB to be exploited as motors with embedded source propulsion secured\nby their flagella and embedded steering system to control the directional motion provided by\ntheir magnetosome chain7.\nResults and discussion\nElectron cryotomography imaging of the magnetosome chain\nCells of M. gryphiswaldense have been imaged by electron cryotomography (ECT). Since bac-\nteria are cryoembedded, the cells and as a consequence the magnetosome chains inside them\npreserve their natural shape, avoiding artifacts associated to cell drying processes. ECT im-\nages of two cells are shown in Figs. 1a,b. The upper part of Fig. 1a,b shows the reconstructed\n3D tomograms from the tilt-series of images obtained by ECT of the two cells and the insets\nshow the Z-projected images of the tomograms. These projections are equivalent to the images\nobtained by transmission electron microscopy. The chains shown in Fig. 1 are composed of\nN= 15 and22magnetosomes, respectively, and the mean distance between magnetosomes is\nd≈60 nm . Videos showing the 3D reconstructions of the two magnetosome chains from differ-\nent perspectives can be found in the supplementary information (movies S1-S2). ECT imaging\n4evidences that magnetosome chains are certainly not straight lines but exhibit more complex\npatterns. Three slices (XY , YZ and XZ) of each of the 3D reconstructions are shown below\nthe tomograms in Fig. 1a,b. The three slices are cuts of the corresponding tomograms taken\nalong the marked lines and intersect in one of the magnetosomes, so that the XYZ coordinates\nof that particular magnetosome can be determined. By determining the XYZ coordinates of all\nthe magnetosomes in the chain we have been able to reconstruct the two magnetosome chains,\nwhich are shown in Fig. 5b, where a zoom-in of one of the magnetosome chain reconstructions\nemphasizes the deviation from a straight line.\nECT images of a magnetosome extracted from M. gryphiswaldense reveal the morphology\nof the magnetosome (Fig. 1c). The figure on the top is a reconstructed 3D tomogram, and be-\nlow, the XY , YZ and XZ central sections of the tomogram are shown. These images evidence\nthat magnetosomes have a strongly faceted morphology similar to the truncated octahedral one,\nwhich is adopted as a reference (Fig. 1d), with a mean size of 40 nm . In this truncated octa-\nhedron the ⟨001⟩crystallographic axes define the growth directions of the square faces and the\n⟨111⟩those of the hexagonal faces. The magnetosome in the image has two neighbours (only\npartially observed in the image). The three magnetosomes have self-assembled so that their\nhexagonal faces are towards each other. In the same way, when magnetosomes are forming\nthe chain inside the cell, numerous studies in Magnetospirilla show that they align with their\nhexagonal faces towards each other27,28, i.e., along one of their ⟨111⟩crystallographic direc-\ntions, as sketched in Fig. 5a,d. The [111] direction along which magnetosomes align defines\nthe so-called chain axis .\nMagnetism of magnetosome chains\nECT images of the magnetosome chain show a complex structure and an evident deviation from\na straight linear configuration. This must have an effect on the magnetic behaviour of the chain.\n5XY \n YZ \nXZ \n250 nm  \nXY \nXZ YZ \n250 nm  \nXY YZ \nXZ \n25 nm  \n(a) (b) (c)\n(d)Figure 1: Electron cryotomography (ECT) of magnetosome chains. (a) and (b) are ECT images\nof two different bacteria. Top: 3D tomograms reconstructed from the tilt-series of images\nobtained by ECT. Videos of the tomograms of the two chains viewed from different perspectives\n(as consecutive slices in XY , XZ and YZ plane orientation, and as an electron density map) can\nbe found in the supplementary information (movies S1-S2). The insets show the Z-projection\nof the tomograms. Bottom: XY , YZ and XZ slices of the tomograms shown on top. The\nthree slices are taken along the marked lines and cross on one of the magnetosomes. c) ECT\nimages of extracted magnetosomes. Top: reconstructed tomograms. Bottom: central XY , YZ\nand XZ slices of the magnetosome tomogram shown on top. d) Schematic representation of a\ncuboctahedral magnetosome.\n6We have analyzed the magnetic properties of magnetosome chains from M. gryphiswaldense by\ncomplementary magnetic techniques in three different arrangements: i) a colloidal dispersion\nof bacteria, this means that we have a 3D distribution of bacterial chains, studied by magneti-\nzation and small angle neutron/x-ray scattering (SANS/SAXS); ii) macroscopic magnetization\nmeasurements on bacteria forming 3D and 2D distributions; iii) x-ray photoemission electron\nmicroscopy (XPEEM) on individual chains of magnetosomes extracted from the bacteria form-\ning a 1D magnetosome arrangement.\nMagnetization and small angle neutron/x-ray scattering (SANS/SAXS) on a bacterial col-\nloid\nFig. 2a shows the magnetization curve M(H)of a colloidal dispersion of bacteria ( 6·1011cell/mL).\nThis curve was measured slowly, leaving two minutes between each data point to assure thermal\nequilibrium is achieved at each point. As shown in Fig. 2a, the magnetization increases rapidly\nwith the applied field until it reaches a plateau between 3 mT≤µ0H≤15 mT at a value that\nis 90% of the saturation value ( M= 0.90Ms). Then the magnetization increases again slowly\nup to saturation. This behavior reveals that there are two different coherent rotations of the\nmagnetization as will be discussed in the following.\nThe magnetic properties of the bacterial colloid have been further investigated via small an-\ngle x-ray scattering (SAXS) and polarized small angle neutron scattering (SANS). In a SAXS/-\nSANS experiment the x-ray/neutron beam ( ⃗kIN) hits the sample and the scattered beam ( ⃗kscat)\nimpacts on a 2D detector perpendicular to the incoming beam (Fig. 2b). The intensity Ias a\nfunction of the scattering vector ⃗ q=⃗kscat−⃗kINgives pure structural ( Inuc(q)) information\nfrom SAXS/SANS experiments, or, additionally, magnetic ( Icross(q)) information for SANS\ndepending on the magnetic field on the sample and the polarization of the incoming and scat-\ntered neutron beam (Fig. 2c). Inuc(q)andIcross(q)are extracted from the 2D scattering patterns\n(Fig. 2c) as explained in the methods section.\n7Fig. 2d shows the radially averaged 1D SAXS intensity I(q)of the colloidal dispersion\nmeasured in zero field. An indirect Fourier transform of I(q)results in the pair distance dis-\ntribution function P(r)displayed in Fig. 2e. The extracted distribution function exhibits three\ndistinct peaks. The first peak has its maximum at about 25 nm and is nearly bell-shaped. The\ncomparison of this peak with the pair distance distribution function of a homogeneous sphere\nwith diameter DSANS = 48 nm shows qualitatively very good agreement. Hence, we surmise\nthat the first peak corresponds to the nuclear scattering of the individual magnetosome of an\naverage size of about 48 nm . This particle size is considerably larger than the one obtained by\nECT (≈40 nm ). The reason for this difference is that ECT is only sensitive to the core of the\nmagnetosome (the magnetic nanoparticle), while SANS is sensitive to both the core and the sur-\nrounding lipid bilayer membrane. In fact, for neutrons, the scattering cross section of H is very\nlarge, which means that organic materials with many H-atoms (such as the lipid membrane in\nour case) can generate large scattering signals29. Considering a membrane thickness of ≈4 nm ,\nthe core diameter obtained by SANS is 48−8 = 40 nm , in agreement with the value obtained\nfrom ECT. The maxima of the second and third peak of the distribution function are at about 75\nand125 nm , and both peaks are also nearly bell-shaped. The positions of the peaks agree well\nwith the expected positions of the center of mass of the next neighbors in a chain-like structure\nofDSANS = 48 nm sized particles with a center-to-center distance of d≈50 nm , close to the\nd≈60 nm distance measured by ECT.\nFig. 2d additionally shows the structural 1D SANS intensities Inuc(q)detected for field\nstrengths of µ0H= 2 mT and1 Tapplied perpendicularly to the neutron beam. Both Inuc(q)’s\nare virtually identical, which means that already at 2 mT the bacteria are fully aligned in the\nfield direction. Consequently an indirect Fourier transform of both nuclear scattering intensities\nresults in superimposed pair distance distribution functions, whose second peak ( r∼75 nm )\nis about twice the height of the first peak (Fig. 2e). This verifies that the bacteria align with\n800.51\n10-410-310-210-1100M/MS\nµ0H (T)\n0.03 0.1 0.510-210-1100101102103104105 I (cm-1)\nq (nm-1)0 25 50 75 100 125 150 175 2000 SAXS\nP(r) (a.u.)\nr (nm)\nPolarized \nneutron beam Spin flipper sample \nH 3He cell \n𝒌𝒌𝒌𝒌𝒒𝒒y \nz \nH I--- I++q I--(a) (b) (c)\n(d) (e)\n                  SANS:\n  nuclear: 2 mT\n  nuclear: 1 T\n cross-term: 2 mT\n  cross-term: 1 TFigure 2: Magnetic state of the colloidal dispersion of bacteria. a) Magnetization curve of the\ncolloid. The field axis is in logarithmic scale to magnify the low-field region. The sketches\ndisplay the two-step magnetization process. The experimental data marked with arrows corre-\nspond to the points measured by SANS/SAXS ( µ0H= 0 mT; 2 mT ; and 1 T). b) Schematic\nrepresentation of the SANS experiment. The polarized incoming neutron beam can be set to\neither parallel (+) or antiparallel (-) to the applied field by means of an RF spin flipper. The3He\ncell discriminates the polarization of the scattered neutrons (+ or -), hence the recorded inten-\nsity is either I++orI−−, where superindexes refer to the polarization of the incoming/scattered\nneutrons. c) 2D SANS scattering patterns for µ0H= 2 mT . Top: I−−; bottom: I−−−I++. d)\n1D scattering intensities measured by SAXS in zero field (radial average, offset by scale factor\n100) and the field dependent nuclear scattering intensities Inuc(q)(offset by scale factor 10) as\nwell as the cross-terms Icross(q)determined by polarized SANS. The lines are the correspond-\ning fits by an indirect Fourier transform. e) Pair distance distribution functions P(r)determined\nby an indirect Fourier transform of the 1D scattering intensities from (d). The black line is the\nP(r)of a homogeneous sphere with diameter DSANS = 48 nm . The distribution function de-\ntermined by SAXS is offset by arbitrary scaling factors.\n9the chain axis parallel to the field, considering that within the chain each particle is surrounded\non average by two neighbors and thus the probability to find a scatterer at this position is two\ntimes the probability to find a scatterer within the primary particle (first peak). To investigate if\nstructural alignment equals magnetic saturation the cross-terms Icross(q)were analyzed.\nThe two cross-terms Icross(q)detected at 2 mT and 1 T display the same functional form\n(Fig. 2d) and accordingly the extracted distribution functions are qualitatively similar (Fig. 2e).\nThe observation of only one peak can be explained by the fact that the cross terms depict the\ncorrelation between nuclear scattering length density and magnetization Mz(r)along the axis\nperpendicular to the applied field. In all cases the shape of the distribution function is compara-\nble to the distribution function of a single sphere with a homogeneous scattering length density.\nThis verifies that the particles are homogeneously magnetized (i.e. Mz(r) =Mz) and thus can\nbe regarded as single-domain particles. However, the absolute values of P(r)detected at 2 mT\nare over the whole r-range systematically reduced by a factor of 0.83compared to 1 T. Assum-\ning that at µ0H= 1 T the system is saturated in field direction (i.e. Mz=Ms) this means that\nat2 mT the magnetization in field direction amounts to Mz= 0.83Msand that the magnetic\nsaturation is achieved by a coherent rotation of the spins within the individual nanoparticles to-\nwards the field direction. These two processes (chain alignment followed by a coherent rotation\nof the magnetosome spins) explain the isothermal magnetization measurement of the colloid\nshown in Fig. 2a. In the latter, in agreement with the SANS result, at 2 mT Mz= 0.83Ms,\nand the coherent rotation of magnetosome spins would start at ≈15 mT , where Mz= 0.9Ms\ncorresponds to a misalignment of the magnetic moment with the chain axis of θ≈25◦, where\nMz=Mscosθ.\n10Magnetometry on 2D and 3D bacterial arrangements\nMacroscopic hysteresis loops of M. gryphiswaldense cells have been measured by SQUID and\nVSM magnetometry. 3D arrangements of aligned bacteria have been obtained by pouring the\ncells under an applied ’aligning’ uniform magnetic field ( ⃗Hal) into liquid agar that hardens\nupon cooling. Similarly, 2D arrangements of aligned bacteria have been obtained by depositing\nthe cells onto a Si substrate under ⃗Hal. A TEM image of aligned M. gryphiswaldense cells\ndeposited onto a Si substrate is shown in the supplementary information. Samples of randomly\narranged bacteria have been also prepared in both the 3D and 2D configurations.\nThe hysteresis loops of randomly arranged and oriented bacteria are shown in Fig. 3a,b.\nThe hysteresis loops of oriented bacteria have been measured at different angles with respect\nto⃗Halbetween 0◦and 90◦in steps of 20 degrees for 3D arrangement and at discrete angles,\n0◦, 45◦and 90◦, for 2D arrangement. These measurements evidence that bacteria are highly\nanisotropic magnetic objects, since their hysteresis loops depend strongly on the direction of\nthe applied field. This is not surprising, as one would expect that magnetosome chains behaved\nas a good compass, with a single magnetic easy axis oriented along the chain axis, which,\nas noted previously, is coincident with one of the magnetosomes ⟨111⟩crystallographic axes.\nHowever, as previously observed in Magnetospirillum magneticum AMB-130, the hysteresis\nloops of 3D and 2D bacterial arrangements do not correspond to those expected for a uniaxial\nStoner-Wohlfarth model31along the chain, since the hysteresis loop perpendicular to the chain\naxis is not anhysteretic as expected. This could be attributed to the misalignment of the magnetic\nmoment already detected by SANS measurements.\nMore information on the magnetism of magnetosome chains has been gathered from the\ntheoretical modelling of the hysteresis loops. Previous works on M. gryphiswaldense show that\nthe magnetic anisotropy of magnetosome chains can be described as a superposition of the cubic\nmagnetocrystalline anisotropy of magnetite, with four equivalent easy axes directed along the\n11-0.1 -0.05 0 0.05 0.1Random\n0º\n45º\n90º-101\nm0H (T)\n-0.1 -0.05 0 0.05 0.1Random\n0º\n45º\n90º-101\nm0H (T)M/MS3D 2D \nExperimental  Uniaxial [111]  Uniaxial 25º tilted  (a) (b)\n(c) (d)Figure 3: Hysteresis loops of oriented bacteria under an applied field in 3D (a) and 2D (b)\narrangements forming 0◦, 45◦, and 90◦with the aligning field ( ⃗Hal), together with the hysteresis\nloops of randomly arranged bacteria. The continuous line is the fit to the magnetic model\nexplained in the text. c) and d) Polar plots of the coercivity and reduced remanent magnetization\nat different orientation angles of the chain axis with ⃗Halin the 3D (c) and 2D (d) arrangements.\nThe experimental points are compared to the results of two simulations in which the uniaxial\neasy axis ˆuuniis either parallel (green) or tilted 25◦(blue) with the chain axis.\n12⟨111⟩crystallographic directions, and a uniaxial shape anisotropy directed along the chain axis\n(thus parallel to one of the ⟨111⟩axes) that originates from intra-chain dipolar interactions and\ndominates the magnetic response of the chain32,33.\nFollowing this argumentation, as a first approximation, our approach to simulate theoret-\nically the experimental hysteresis loops consists of considering the magnetosome chain as a\ncollection of independent magnetic dipoles, where the equilibrium orientation of each magnetic\nmoment is obtained by minimizing the single dipole energy density E, calculated as the sum of\nthree contributions: i) cubic magnetocrystalline energy of magnetite with anisotropy constant\nKc, ii) uniaxial anisotropy energy along the chain axis ([111] direction defined as ˆuuni= ˆu111\nin Fig. 1d) with anisotropy constant Kuni, and iii) Zeeman energy in an external magnetic field\nµ0⃗H:\nE=Ecubic+Kuni[1−(ˆuuni·ˆum)2]−µ0MH(ˆuH·ˆum) (1)\nwhere ˆumandˆuHrepresent the particle magnetization and external magnetic field unit vectors,\nrespectively. The azimuthal angle can be any between 0◦and 360◦either because individual\ndipoles are actually free to be rotated along the chain axis or because whole bacteria them-\nselves can be found at any azimuthal orientation relative to the aligning field due to the sample\npreparation. The hysteresis loops have then been calculated following a dynamical approach in\nwhich the single domain magnetization can switch between the available energy minima states\nat a rate determined by a Boltzmann factor34,35that depends on the energy barriers between\nsuch minima. More details are given in the supplementary information.\nMisalignments of the chains with respect to the aligning field occurring during sample\npreparation and detected by SANS experiments have been considered by including a Gaus-\nsian angular distribution of the chain axes. We have tested three angular distributions around\nthe chain axis: 15◦,25◦, and 35◦, in agreement with misalignment values reported previ-\nously28,30. The data used for the simulations are the magnetocrystalline anisotropy constant\n13(Kc=−11 kJ /m3) and magnetization ( Ms= 48·104A/m) of magnetite and an effective\nuniaxial anisotropy constant along the chain axis Kuni= 12 kJ /m3to account for both shape\nand magnetic interaction anisotropies. Even though the simulations are good enough at the\nperpendicular orientation, that is to say, with the applied magnetic field perpendicular to the\nbacterial chain axis, they clearly deviate at smaller angles as shown in the polar representation\nin Fig. 3c, where the experimental and calculated reduced remanent magnetization and coerciv-\nity of bacteria in 2D and 3D arrangements are plotted for different orientation angles between 0\nand90◦.\nSince the misalignment of the bacteria due to sample preparation, does not reproduce the\nhysteresis loops, we have tried a different approach in which we have considered that there\nis a tilting of the magnetization with respect to the chain axis inherent to the magnetosome.\nThis approach allows reproducing accurately the experimental hysteresis loops for both 2D and\n3D arrangements by setting the uniaxial easy axis ˆuuniat25◦with the chain axis in eq. 1, in\nthe plane containing the chain axis ([111]) and the [100] directions. By setting this angle, the\neffective easy axis, and as a consequence the equilibrium magnetic moment, is found to lie 20◦\nout of the chain axis, in close agreement with the 25◦tilting observed in the previous SANS\nanalysis of the colloid. Note that there is a good resemblance with experiment even for an\nangular dispersion as small as 15◦(Figs. 3a,b). We used the same data for the simulations of\nthe hysteresis loops at all orientation angles and for both 3D and 2D arrangements, Kc=−11\nkJ/m3,Ms= 48·104A/mandKuni= 12 kJ /m3. The resemblance with experiment is more\nevident in the polar plots of the reduced remanent magnetization and coercivity shown in Fig. 3c.\nThe tilting of the magnetosome magnetization with respect to the chain axis is attributed\nto the competition between the shape anisotropy of the magnetosome, which presents a well\nfaceted morphology as shown in Fig. 1c, and the magnetic interactions trying to align the mag-\nnetic dipoles along the [111] direction parallel to the chain axis. In fact, electron holography\n14experiments on individual magnetosomes from Magnetovibrio blakemorei MV-1 clearly show\nthat the magnetization direction of the particle is tilted with respect to the [111] crystallographic\ndirection towards a long dimension of the particle, consistent with shape anisotropy dominating\nthe magnetic state of the crystal36.\nX-ray photoemission electron microscopy (XPEEM) on extracted magnetosomes\nPrevious magnetic results are supported by x-ray photoemission electron microscopy (XPEEM)37.\nUnlike the macroscopic SQUID and VSM measurements above, which provide an average mea-\nsurement of the whole sample, XPEEM is an element-specific and spatially-resolved technique\nthat by using x-ray magnetic circular dichroism (XMCD) as a magnetic contrast mechanism,\nallows obtaining element-specific magnetic hysteresis loops of selected sample areas with a\nresolution down to 30 nm37.\nXPEEM measurements have been performed at room temperature on extracted magneto-\nsomes due to the high absorbing power of the whole bacteria. Magnetosomes were deposited\nonto a conductive Si substrate either randomly (Fig. 4c) or oriented along an aligning field\n(Fig. 4d). As expected for magnetite, the L 3XMCD signal shows three peaks (Fig. 4b): two\nminima at 707.4 eV and709.6 eV and a maximum at 708.6 eV. The two minima correspond to\nthe Fe2+and Fe3+ions occupying tetrahedral sites, and the maximum corresponds to the Fe3+\nions occupying octahedral sites. The magnetic contrast images were recorded at the Fe L3reso-\nnance energy ( 709.6 eV) of magnetite. The set of magnetic images displayed in Figs. 4c,d show\nthe space-resolved dichroic images obtained at different values of an in-plane magnetic field.\nThe dichroic signal yields the magnetic moment, so that the hysteresis loops of selected areas\nof the magnetic images (enclosed in rectangles in Figs. 4c,d) have been obtained by plotting the\ncorresponding dichroic signal as a function of the applied magnetic field (Fig. 4a).\nThe hysteresis loops of the randomly deposited magnetosomes enclosed in the region marked\n15-0.1 -0.05 0 0.05 0.1Random\n0º\n60º\n90º -101\nm0H (T)M/MS\n48044040036032028024020016012080400500 400 300 200 100 0\nx_0349_XMCD\n48044040036032028024020016012080400500 400 300 200 100 0\nx_0358_XMCD\n48044040036032028024020016012080400500 400 300 200 100 0\nx_0367_XMCD\n48044040036032028024020016012080400500 400 300 200 100 0\nX__XMCD\n400 360 320 280 240 200 160280 240 200 160 120 80 40\n400 360 320 280 240 200 160280 240 200 160 120 80 40\n400 360 320 280 240 200 160280 240 200 160 120 80 40\n400 360 320 280 240 200 160280 240 200 160 120 80 40\n500 nm  𝑯𝒂𝒑𝒑\n-22.5 mT 𝑯𝒂𝒑𝒑 \n1 mm \n0 -22.5 mT 0 \n\nXAS\n700 710 720 7300XMCD\nE(eV)1D \n(a)(c)\n(b)\n(d)Figure 4: a) X-ray photoemission electron microscopy (XPEEM) hysteresis loops of the re-\ngions enclosed in the rectangles marked in the images in c) and d), corresponding to chain\nsections which are either randomly arranged or forming 0◦(green), 60◦(blue), and 90◦(red)\nwith the applied field. The continuous line is the fit to the magnetic model explained in the text.\nb) X-ray absorption spectra (XAS) of the chain parallel to the applied field (green rectangle in\nc) with the incoming beam right-polarized ( σ+) and left-polarized ( σ−). Computing σ+−σ−\ngives the XMCD signal below. c) XPEEM images as a function of the applied field of randomly\narranged magnetosomes and d) oriented magnetosomes deposited under ⃗Hal. In d) the SEM\nimage of the chain is shown together with the XPEEM image.\n16with the black rectangle in Fig. 4c and of a chain parallel to the applied field, marked with the\ngreen rectangle in Fig. 4c, are shown in Fig. 4a.\nMagnetosomes deposited under an applied aligning field form longer chains. The XPEEM\nimage together with the SEM image of one of these chains is shown in Fig. 4d. This chain is\nclearly not a straight line but is rather a zigzag, formed by segments oriented at different angles\nwith the applied field. Two of these segments, enclosed in blue rectangles in Fig. 4d, form 60◦\nwith the applied field, and another two segments (red rectangles), form 90◦. The corresponding\nhysteresis loops are shown in Fig. 4a.\nAs shown in Fig. 4a, the experimental XPEEM loops can be accurately simulated to the\ntheoretical model developed previously for the 3D and 2D chain arrangements (eq. 1) with a\nsmaller tilting angle of the effective easy axis ( 15◦) and a larger anisotropy constant Kuni=\n16 kJ /m3as compared to the SQUID loops ( Kuni= 12 kJ /m3). This is attributed to the larger\ndistance between magnetosomes in chains inside the bacteria ( d= 60 nm ) than in chains of ex-\ntracted magnetosomes ( d= 50 nm ). Indeed, for a 40 nm sized particle, the dipole pair potential\nenergy is given by ∼µ0m2/4πd3= 0.75 eV (being m=MsVthe particle magnetic moment\nandd= 60 nm ), while for the chains of extracted magnetosomes the dipolar energy from the\nneighbors at d= 50 nm would increase up to ∼1.35 eV .\nEquilibrium configuration of the chain\nFollowing the results gathered from the magnetic analysis, here we will assess the impact on the\nmagnetosome chain configuration of the tilting of the magnetosome magnetization with respect\nto the chain axis, where the latter, as noted above, is defined as the [111] crystallographic\ndirection along which magnetosomes align in the chain (Fig. 5).\nWith this aim, we have developed an approach to explain the shape of the magnetosome\nchains that consists on quantifying the total energy of the chain by including the magnetostatic\n17interactions between nanoparticles, and the contribution of the lipid/protein-based architecture\nembedding the magnetosome chain, modeled as a spring-like elastic energy. We focus on the\nstable geometry of chain arrangements and assume that close loop configurations such as rings\nor 3D clusters are avoided by the cytoskeleton inside bacteria. The same approach has been\nused by other authors22,38, but in the present case, as proposed previously, the magnetization of\neach magnetosome is tilted 20◦out of the chain axis.\nHere we will summarize the main steps followed to calculate the total energy of the chain,\nbut a detailed description can be found in the supplementary information. Fig. 5a shows a\nsection of a magnetosome chain composed of three magnetosomes. The particle in the centre\nis subjected to the stray magnetic field produced by the two neighbors (the dotted red lines\nin Fig. 5a represent the stray field produced by the particle at the bottom). We implicitly as-\nsume that the local torque that tends to align neighboring dipoles is counter-balanced by the\ncytoskeleton, since cryotomography and TEM images reflect that the hexagonal faces are face\nto face27,28. The particle in the middle will tend to align its magnetization along the stray field\nlines and will undergo a magnetic force that compels it to shift towards the direction of its own\nmagnetic moment. The magnetostatic energy associated to the magnetic force on magneto-\nsomes is then implemented as the sum of the dipolar pair potential energies between nearest\nneighboring particles.\nOn the other hand, magnetite crystals are embedded in a lipid/protein-based architecture\ncomposed, among others, of the magnetosome membranes, the cytoskeletal filaments, and the\nproteins connecting the magnetosome membranes with the cytoskeleton. The contribution to\nthe total chain energy of this lipid/protein-based architecture has been modeled based on two\nsimplifying assumptions: firstly, that the distance between particles, d, is fixed, so that chains\ncan bend or twist but cannot stretch; and secondly, that the net force on the magnetosomes can\nbe ascribed to an elastic recovery force with elastic constant kacting perpendicularly to the\n18Magnetic Force  Elastic Recovery \nForce  (a) \nStray field line s \n[111]  \nn=1 n=2 3D Simulations  \nExperimental reconstructions  15 NP  22 NP  (c) \n(b) \n[111]  \nθ \nϕ \nρ \nφ (d) Figure 5: Equilibrium configuration of the magnetosome chain. a) Schematic representation\nof two competing mechanisms: magnetic force pushing to align magnetosome magnetic mo-\nments along the stray field lines from neighboring particles, and lipid/protein-based mechanism\nmodeled as an elastic recovery force acting perpendicularly to the chain axis, where the chain\naxis is the [111] crystallographic direction along which magnetosomes align, as highlighted in\nthe figure. b) Experimental reconstructions obtained from ECT imaging of the magnetosome\nchains shown in Fig. 1a,b. A zoom-in of the first magnetosome chain reconstruction highlights\nthe deviation from a straight line. c) Two stable solutions for the chain patterns obtained as\nexplained in the text. A potential filament has been drawn as a guide for the eye. A video of\nthe experimental reconstruction of the chain on the left (chain (a) in Fig. 1) together with the\ncorresponding simulated chain viewed from different perspectives can be found in the supple-\nmentary information (movie S3). d) Schematic representation of the magnetic dipoles and the\nthree independent variables used in the simulation: radial ( ρ) and azimuthal ( ϕ) coordinates\nfor the magnetosome positions, and azimuthal orientation ( φ) of the magnetic dipoles. θis the\npolar angle of the magnetic dipoles, fixed to 20◦.\n19chain axis, as shown in Fig. 5a.\nThe most stable chain configurations have then been predicted by minimizing the total en-\nergy of the chain considering three independent variables per particle, namely the radial ( ρ)\nand azimuthal ( ϕ) coordinates for the magnetosome positions, and azimuthal orientation ( φ) of\nthe magnetic dipoles (see sketch in Fig. 5d). The results have then been compared to the 3D\nreconstructions obtained from the chains imaged by ECT (Fig. 5b).\nConsidering a particle diameter D= 40 nm, distance between them d= 60 nm, and\nk= 0.1·10−3N/m, two stable solutions give the patterns shown in Fig. 5c, in excellent match\nwith the two experimental chain reconstructions shown in Fig. 5b. They are helical-like shaped\nchains with slowly bending axes and a different pitch, given by ≈2L/n, being L=Ndthe\nchain length and n= 1, and 2, respectively. The energy difference between the two chain shapes\nis less than 1% of the total energy, hence they are approximately equally stable. The agreement\nbetween the simulated and the experimental chains is even more explicit in a video of images of\nboth chains taken from different perspectives shown in the supplementary information (movie\nS3).\nA zoom-in of a section of one of the simulated chains shows the magnetosome dipole mo-\nments. The azimuthal increment of the individual dipole orientations ( ∆φ) between consecutive\npositions along the chain increases with napproximately as:\n∆φ≈2πn\nN(2)\nWhere Nis the number of magnetosomes per chain ( N= 15 and22in the two examples\nshown). Consequently, the component of the chain magnetization vector (the vector sum of the\nmagnetic moment of each magnetosome) perpendicular to the chain axis cancels out, so that\nthe chain magnetization lies along the chain axis. Note also that when magnetic dipoles are\nprojected in a 2D view of the chain, apparent tilting between consecutive dipoles is no more\n20than7◦(see Fig. 4 in the supplementary information), which is very compatible with projected\nelectron holography images2.\nThe role of kon the chain geometry is to scale up or down the radial positions of magneto-\nsomes along the chain, so that larger kfavours configurations approaching straight lines. In our\ncase, k= 0.1·10−3N/mresults in assembling forces of the order of 0.6 pN for radial displace-\nments of ≈10 nm as obtained from the simulations, similar to the force generated by the actin\nfilament39, and far below recent estimations of the fracture limit of the actin-based scaffolding\nfilaments ( ≈30pN) by K ¨ornig et al.40.\nOn the basis of our results, a mechanism of chain formation is proposed. Our findings reveal\nthat the chain shape is mostly driven by passive spontaneous magnetostatic effects triggered by\nthe intrinsic anisotropy of magnetosomes, ultimately defined by their morphology. The mag-\nnetosomes morphology is regulated during the biomineralization process. This is a genetically\ncontrolled process which involves a specific set of about 30mam (magnetosome membrane) and\nmms (magnetic particle-membrane specific) genes4. The product of the mamJ gene is an acidic\nprotein that connects the magnetosome membrane to the filament. The deletion of mamJ results\nin bacteria that produce magnetosomes arranged in compact three-dimensional clusters instead\nof arranged in chains26. The fact that they form clusters and not closed rings is consistent with\nour conclusions, namely the tilting of the magnetosome magnetic moment with respect to the\nchain [111] axis, since according to previous works, more than four magnetosomes tend to\nform closed rings when their magnetic moment is parallel to the [111] axis38. In the same way,\nwhen depositing the magnetosomes onto a 2D surface, magnetosomes tend to self-assemble in\na close-packing configuration41, and the tilting could be also behind the zigzag configuration\nof the chains observed by SEM in oriented magnetosomes. But, why would this helical-like\nshape benefit the bacteria? M. gryphiswaldense are long cells, easily reaching several microns\nlong. They need a high magnetic torque to overcome the drag forces and orientate along the\n21Earth’s magnetic field. As a consequence, their magnetosome chain needs to be long to maxi-\nmize the chain net magnetic moment. Indeed, their chain is frequently composed of more than\n20 magnetosomes, which brings the total length of the chain to 1.5µmor more, about 50–60%\nof the bacterial length. Such an object should necessarily be bent in order to accommodate\nto this spiral-shaped microorganism. Thus a helical-like shape fits better, but it only changes\nslightly the total magnetic moment, hence hardly affecting the magnetic orientation. A genetic\ncontrol of the magnetosome shape towards an energetically optimum chain arrangement is also\nobserved in Magnetobacterium bavaricum MYR-1, a species which synthesize bullet-shaped\nmagnetosomes arranged into bundles of magnetosome chains. The magnetosome magnetiza-\ntion within MYR-1 magnetosomes is parallel to the chain axis, which coincides with the long\naxis of the magnetosomes. Unexpectedly, the latter is parallel to the [100] crystallographic axis,\na magnetically hard axis in magnetite, rather than along the [111] magnetocrystalline easy axis,\ndue to compromise effects of shape anisotropy and intra-chain and intra-bundle interactions42.\nIn sum, our finding sheds light on the understanding of the magnetosome chain assembly\nduring the biomineralization process of MTB, which may influence their potential future ap-\nplications as biological micro-robots. Indeed, one of the major technical issues in the develop-\nment of interventional platforms for guiding drug-loaded MTB is the directional magnetic field\nstrength that needs to be produced at the human scale to induce sufficient directional torque on\nthe chain of magnetosomes. A good knowledge of the magnetic configuration of the magneto-\nsome chain will lead to more efficient and less costly drug-delivery platforms that may benefice\na larger population.\n22Methods\nMagnetotactic bacteria culture and magnetosome isolation\nMagnetospirillum gryphiswaldense MSR-1 (DMSZ 6631) was grown at 28◦C without shaking\nin an iron rich medium43. After 96 h-incubation, when bacteria present well-formed magne-\ntosomes chains, cells were fixed with 2%glutaraldehyde, harvested by centrifugation, washed\nthree times and finally concentrated up to 109−1011cell/mLin ultrapure water. Magnetosomes\nwere isolated according to the protocol described by Gr ¨unberg et al.44with minor modifications.\nCells were collected by centrifugation, suspended in 20 mM HEPES- 4 mM EDTA (pH=7.4),\nand disrupted using a French press ( 1.4 kbar ). Then, the magnetosomes were collected from\nthe supernatant by magnetic separation and rinsed 10 times with 10 mM mM Hepes- 200 mM\nNaCl (pH=7.4). Finally, the isolated magnetosomes were suspended in ultrapure water reaching\na concentration of 20µg/mL.\nElectron cryotomography (ECT)\nSample preparation for cryotomography was as follows: a 10µlvolume of fixed and washed M.\ngryphiswaldense cells ( 109cel/mL) was mixed with a 3µlvolume of 10 nm Au nano-particles\n(Aurion®BSA gold tracer 10 nm ) (used as fiducial markers). This mixture is frozen-hydrated\nfollowing standard methods using a Vitrobot Mark III (FEI Inc., Eindhoven, The Netherlands).\nIn brief, a grid is placed in the controlled environment of the Vitrobot chamber, which is at\n4◦Cand at a relative humidity of 95%. An aliquot ( 4µl) of the sample is applied to a glow-\ndischarged grid. After 1 min incubation most of the drop is removed by blotting with two filter\npapers, to produce a thin liquid film, and rapidly plunged into liquid ethane ( 91 K ), previously\ncooled by liquid nitrogen. Vitrified grids are stored under liquid nitrogen until cryo-TEM data\ncollection. For cryo-tomographic tilt series acquisition, vitrified grids were cryo-transferred\ninto a 914high tilt tomography cryo-holder (Gatan Inc., Warrendale, PA, USA), which is in-\n23serted in a JEM-2000FS/CR field emission gun transmission electron microscope (Jeol, Europe,\nCroissy-sur-Seine, France) operated at 200 kV . Grids are kept around 103 K in the high vacuum\nof the microscope column containing the sample embedded in a thin layer of glass-like vitreous\nice, in a near native-state. Different single-axis tilt series were collected under low-dose condi-\ntions on a UltraScan 4000, 4k ×4k CCD camera (Gatan Inc., Pleasanton, CA, USA), over a tilt\nrange of ±64◦with1.5◦increments and at underfocus values ranging from 5 to 8µm, using the\nsemi-automatic data acquisition software SerialEM45. Tilt-series were collected at a nominal\nmagnification of 25,000×and a binning factor of 2 ( 2048×2048 pixels micrographs), thus pro-\nducing a pixel size of 0.95 nm . The in-column omega energy filter helped to record images with\nimproved signal-to-noise-ratio by zero-loss filtering with an energy window of 60 eV centered\nat the zero-loss peak. CCD images in each tilt-series were acquired at the same underfocus\nvalue and under the same low-dose conditions. The maximum total dose used for a tilt-series\nwas90 electrons /˚A2consisting of about 1−2 electrons /˚A2for each digital image.\nFor the alignment and 3D reconstruction of the tilt-series, we used IMOD software46. We\nemployed the Au fiducial markers during the alignment process, and 3D reconstruction was\ncarried out by weight back-projection followed by a reconstruction algorithm named Simulta-\nneous Iterative Reconstruction Technique (SIRT). Resulting tomograms were visualized with\nImageJ47as a sequence of cross sectional slices in different plane orientations. Tomograms\nwere processed using a median filter and visualized as 3D electron density maps using UCSF\nChimera48software.\nScanning Electron Microscopy (SEM)\nSEM imaging was performed on the two magnetosome samples measured in XPEEM (mag-\nnetically oriented and randomly arranged). SEM images were collected at 10 kV with a JEOL\nJSM-7000F equipped with secondary and retro dispersive electron detectors.\n24Magnetic measurements\nMagnetic measurements on bacteria forming 2D and 3D arrangements were performed on a su-\nperconducting quantum interference device magnetometer (SQUID, Quantum Design MPMS-\n7) in DC mode and on a vibrating sample magnetometer (VSM). Isothermal magnetization\nloops were recorded between ±1 Tat300 K .\nThe 2D bacterial configurations were prepared by depositing five 5µLdrops of the bacterial\nsuspension ( 109cell/mL) by the drop coating method41,49onto a Si substrate. For the oriented\nbacteria, the deposition was done under an external applied magnetic field of 0.5 T. To obtain\nhomogeneous samples, infrared radiation was used during the deposition aimed at accelerating\nthe drying and minimising the surface tension. The resulting samples were finally oriented at\ndifferent angles with respect to the applied magnetic field.\nThe 3D bacterial configurations were prepared by resuspending 250µlof a bacterial colloid\n(1011cell/mL) in750µlof an agar solution (2% agar and 98% water) at 80◦Cto maintain the\nsolution in a liquid state. To align the bacteria, a uniform magnetic field of 1 Twas applied.\nAfter 3 minutes, the field was turned off, and the sample was cooled using liquid nitrogen until\nthe temperature reached around 0◦C. This caused the agar to solidify, trapping the bacteria, and\nkeeping this solid state at room temperature.\nSmall angle neutron scattering (SANS) and small angle x-ray scattering\n(SAXS)\nSANS/SAXS experiments were performed on a highly concentrated bacterial colloid\n(6·1011cell/mL) suspended in ultrapure water. The magnetization curve of the colloid was\nmeasured on a vibrating sample magnetometer up to an applied field of 1 T, leaving two min-\nutes between each measurement to assure thermal equilibrium was achieved at each point.\nSmall angle X-ray scattering (SAXS) experiments: The SAXS data were collected in a Xenocs\n25Nano-InXider, utilizing a 40µmmicrofocus Cu anode as X-ray source and a multilayer mono-\nchromator to collect only the Cu-K αradiation. The detector (a Pilatus 3) is at 938 mm from the\nsample, spans an area of 83.8×33.5 mm2and a pixel size of 172×172µm2. The colloidal\ndispersion was filled into a quartz glass capillary and was measured in absence of an externally\napplied magnetic field. Via SAXS exclusively the nuclear scattering intensity I(⃗ q)∝ |˜N|2is\ndetermined, with ⃗ qbeing the scattering wave vector and ˜N(⃗ q)being the Fourier transform of\nthe nuclear scattering length density ρ(⃗ r). To analyze the data we performed an indirect Fourier\ntransform50of the radially averaged 1D intensity I(q)to extract the pair distance distribution\nfunction P(r). The real space function P(r)provides direct information about the distances\nbetween scatterers from the scattering sample51and hence contains information about the av-\nerage particle geometry as well as correlations between neighboring particles. For the indirect\nFourier transform we used an approach based on ref.52and computational details can be found\non ref.53.\nPolarized small angle neutron scattering (SANS) experiments: SANS instrument D33 at the\nInstitut Laue Langevin (ILL, Grenoble, France)54,55was employed in order to get a longitudinal\nneutron-spin analysis (POLARIS)56. A homogeneous magnetic field ⃗Hwas applied perpendic-\nular to the neutron beam ( ⃗H⊥⃗k) with field amplitudes of µ0H= 2 mT and 1 T. The mean\nwavelength of the neutrons was λ= 0.6 nm , with a wavelength spread of ∆λ/λ= 10% . The\nscattering intensities were measured at two different detector distances ( 3 mas well as 13.4 m)\ngiving a q-range of about 0.05−0.5 nm−1.\nApplication of the POLARIS mode enabled us to detect the non spin flip (nsf) intensities\nI++(⃗ q),I−−(⃗ q), where the superscripts indicate the polarization of the incoming neutron beam\nand the scattered neutrons with regard to the applied field direction, respectively (”+”: parallel,\n”-”: antiparallel). Defining xas the direction of the neutron beam and zas the direction of the\n26applied magnetic field at the sample position ( ⃗H⊥⃗k) the nsf intensities can be written as56:\nI±±(⃗ q)∝|˜N|2+b2\nH|˜Mz|2sin4Θ (3)\n+b2\nH|˜My|2sin2Θcos2Θ\n−b2\nH(˜My˜M∗\nz+˜Mz˜M∗\ny)sin3ΘcosΘ\n∓bH(˜N˜M∗\nz+˜N∗˜Mz)sin2Θ\n±bH(˜N˜M∗\ny+˜N∗˜My)sinΘcosΘ .\nHere, Θis the angle between the scattering vector ⃗ qand the magnetic field ⃗Hand the terms\n˜My,z(⃗ q)represent the Fourier transforms of the magnetization in y, zdirection. The superscript\n∗indicates the complex-conjugated quantities and the constant bH= 2.7·10−15m/µB, with\nµBbeing the Bohr magneton. To investigate the alignment of the bacteria in field direction we\nanalyzed the 1D nuclear cross sections Inuc(q)∝ |˜N|2and the 1D nuclear magnetic cross-terms\nIcross(q)∝(˜N˜M∗\nz+˜N∗˜Mz). The purely nuclear scattering intensities were determined by\nintegration of the nsf intensities in 10◦sectors around Θ = 0◦(⃗ q∥⃗H, Eq. 3) and the cross terms\nIcross(q)by integration of I−−(⃗ q)−I++(⃗ q)in10◦sectors around Θ = 90◦(⃗ q⊥⃗H, Eq. 3). The\ndifference between the two nsf cross sections yields information on the polarization-dependent\nnuclear-magnetic terms. This difference allows one to highlight weak magnetic contributions\nrelative to strong nuclear scattering (or vice versa). From the determined cross sections we\nextracted the underlying pair distance distribution functions in the same manner as for SAXS.\nIn H2O, there is contrast match of bacteria in the dispersion, such that only the magnetosomes\nand their arrangement are probed by SANS.\nX-ray photoelectron emission microscopy (XPEEM)\nFor the x-ray photoemission electron microscopy (XPEEM) experiments isolated magneto-\nsomes extracted from the bacteria were employed. Two different samples were prepared. The\n27first one consists of randomly arranged magnetosomes prepared by the deposition of a 5µL\ndrop of the magnetosomes suspension onto a conductive Si substrate. The drop was dried under\ninfrared radiation. The second sample consists of magnetically oriented magnetosomes. In this\ncase, the drop was dried under an external magnetic field of 0.5 Tand infrared radiation. The\nSi substrate was marked with a Au reticule to allow subsequent match by scanning electron\nmicroscopy (SEM) of the chains imaged by XPEEM.\nMagnetic imaging of the isolated magnetosomes was performed at room temperature by\nmeans of photoelectron emission microscopy (PEEM) by using X-ray magnetic circular dichro-\nism (XMCD) as magnetic contrast mechanism. Measurements were carried out at the\nUE49 PGM SPEEM beamline at Helmholtz-Zentrum Berlin. A magnetic solenoid attached\nto the sample holder allowed application of an in-plane pulsed magnetic field ranging ±0.1 Tto\nsaturate the sample. During imaging the magnetic field range was restricted to ±27.5 mT . The\nincoming photon energy was tuned to the Fe L 3resonance ( 709.6 eV) to obtain element specific\nXMCD images of the magnetosomes as a function of the applied magnetic field. At each field\na sequence of images was acquired with incoming circular polarized radiation (90% of circular\nphoton polarization) with left ( σ−) and right helicity ( σ+), respectively. To improve statistics\nwe acquired up to 400images per helicity and field ( 3 sof exposure time). After normalization\nto a bright-field image, the sequence was drift-corrected, and frames recorded with same helic-\nity were averaged.\nXMCD images : Displayed XMCD images were obtained by computing XMCD = ( (σ+−\nσ−)/(σ++σ−)). Due to the low signal, the XMCD strength at the regions of interest was\ncomparable to the noise level at nearby regions with no magnetic particles. In order to enhance\nthe magnetic contrast, for visualization purposes only, the XMCD images have been multiplied\nby the X-ray absorption image, i.e. XAS =σ++σ−after background subtraction.\nSpace-resolved magnetic hysteresis loops : The data analysis allowed obtaining the magnetic\n28hysteresis loop of any region within the field of view. In order to obtain the magnetic hysteresis\n(XMCD vs magnetic field) of a selected magnetosome region we computed the intensity as a\nfunction of field and helicity ( σ+andσ−) on the selected area. A local background subtraction\nwas performed by calculating the intensity variation observed within nearby regions with no\nmagnetic particles (substrate). The XMCD is then calculated as the difference of background\ncorrected σ+andσ−divided by their sum, i.e. XMCD = (σ+−σ−)/(σ++σ−).\nSpace-resolved XAS and XMCD spectra : For both, left and right helicities of the incoming cir-\ncular polarized radiation, five stack of images were obtained as the incoming photon energy\ncrossed the Fe L 3,2edges. The photon energy was varied between 690 eV and730 eV in0.2 eV\nsteps. Total integration time per energy was 2.5 s. After normalization to a bright-field image,\nthe sequence was drift-corrected, and frames recorded with same helicity and photon energy\nwere averaged. Computing of local X-ray absorption spectra (XAS) for σ+andσ−allows cal-\nculation of the X-ray magnetic circular dichroism spectrum as σ+−σ−from a selected region.\nAcknowledgments\nThe Spanish Government is acknowledged for funding under project number MAT2014-55049-\nC2-R. The Basque Government is acknowledged for L.M.’s fellowship and for funding under\nproject number IT711-13. We also acknowledge funding from the EU through project NanoMag\n(grant agreement no. 604448). We thank D. Mu ˜noz, D. Gandia and A. Vigilante for preparing\nthe samples for the magnetometry measurements.\nAuthor contributions\nI.O. and M.F.G. designed research; I.O. and L.M. performed and analyzed SQUID and VSM\nmeasurements; L.M., A.G.P., A.M., M.F.G, S.V . and M.A.A. performed XPEEM measure-\nments; L.M., S.V . and A.G.P. analyzed XPEEM data; P.B, L.M., D.A.V . and D.H. performed\n29SAXS/SANS experiments; P.B. analyzed SAXS/SANS data; I.O. conceived the chain configu-\nration model; I.O., L.M., P.B., A.G.P., A.G.A., L.F.B and M.F.G. discussed the magnetic results;\nL.M., A.M., M.F.G. and D.G. performed and analyzed the cryotomography imaging; I.O., L.M.,\nP.B., A.G.P. and M.F.G. wrote the paper; all authors checked the manuscript; A.M. and M.F.G.\nsupervised the work.\nConflict of interest\nThere are no conflicts to declare.\nElectronic Supplementary Information\nElectronic Supplementary Information (ESI) available:\n• TEM image of oriented bacteria, details on the calculation of the hysteresis loops, calcu-\nlation of the chain energy, 2D projection of the magnetic dipoles.\n• Movies S1-S2: videos of the tomograms of the three chains in Fig. 1a, b viewed as\nconsecutive slices in XY , YZ, and YZ plane orientation, and as an electron density map.\n• Movie S3: video of the experimental reconstruction of the chain in Fig 1a (corresponding\nton= 1 in Fig. 5b) together with the corresponding simulated chain (Fig. 5c) viewed\nfrom different perspectives.\nReferences\n[1] R. Blakemore, Science , 1975, 190, 377.\n[2] R. E. Dunin-Borkowski, M. R. McCartney, R. B. Frankel, D. A. Bazylinski, M. Posfai and\nP. R. Buseck, Science , 1998, 282, 1868.\n30[3] D. A. Bazylinski and R. B. Frankel, Nat. Rev. Micro. , 2004, 2, 217–30.\n[4] R. Uebe and D. Sch ¨uler, Nat. Rev. Microbiol. , 2016, 14, 621–637.\n[5] C. T. Lefevre and D. A. Bazylinski, Microbiol. Mol. Biol. Rev. , 2013, 77, 497–526.\n[6] D. Serantes, K. Simeonidis, M. Angelakeris, O. Chubykalo-Fesenko, M. Marciello,\nM. Morales, D. Baldomir and C. Mart ´ınez-Boubeta, J. Phys. Chem. C , 2014, 118, 5927–\n5934.\n[7] O. Felfoul, M. Mohammadi, S. Taherkhani, D. de Lanauze, Y . Zhong Xu, D. Loghin,\nS. Essa, S. Jancik, D. Houle, M. Lafleur, L. Gaboury, M. Tabrizian, N. Kaou, M. Atkin,\nT. Vuong, G. Batist, N. Beauchemin, D. Radzioch and S. Martel, Nat. Nanotechnol. , 2016,\n11, 941–949.\n[8] D. Ghosh, Y . Lee, S. Thomas, A. G. Kohli, D. S. Yun, A. M. Belcher and K. A. Kelly, Nat.\nNanotechnol. , 2012, 7, 677–682.\n[9] E. Alphand ´ery, S. Faure, O. Seksek, F. Guyot and I. Chebbi, ACS Nano , 2011, 5, 6279–\n6296.\n[10] S. R. Mishra, M. D. Dickey, O. D. Velev and J. B. Tracy, Nanoscale , 2016, 8, 1309–1313.\n[11] X. Jiang, J. Feng, L. Huang, Y . Wu, B. Su, W. Yang, L. Mai and L. Jiang, Adv. Mater. ,\n2016, 28, 6952–6958.\n[12] X. Kou, X. Fan, R. K. Dumas, Q. Lu, Y . Zhang, H. Zhu, X. Zhang, K. Liu and J. Q. Xiao,\nAdv. Mater. , 2011, 23, 1393–1397.\n[13] A. Komeili, Annu. Rev. Biochem. , 2007, 76, 351–66.\n[14] D. Faivre and D. Sch ¨uler, Chem. Rev. , 2008, 108, 4875–4898.\n31[15] A. Komeili, Z. Li, D. K. Newman and G. J. Jensen, Science , 2006, 311, 242–245.\n[16] R. B. Frankel, G. C. Papaefthymiou, R. P. Blakemore and W. O’Brien, Biochim. Biophys.\nActa, 1983, 763, 147–159.\n[17] M. L. Fdez-Gubieda, A. Muela, J. Alonso, A. Garc ´ıa-Prieto, L. Olivi, R. Fern ´andez-\nPacheco and J. M. Barandiar ´an,ACS Nano , 2013, 7, 3297–305.\n[18] J. Baumgartner, G. Morin, N. Menguy, T. Perez Gonzalez, M. Widdrat, J. Cosmidis and\nD. Faivre, PNAS , 2013, 1, 1–6.\n[19] A. Komeili, FEMS Microbiol. Rev. , 2012, 36, 232–255.\n[20] M. Toro-Nahuelpan, F. D. M ¨uller, S. Klumpp, J. M. Plitzko, M. Bramkamp and D. Sch ¨uler,\nBMC Biology , 2016, 1–24.\n[21] E. Cornejo, P. Subramanian, Z. Li, G. J. Jensen and A. Komeili, mBio , 2016, 7, e01898–\n15.\n[22] V . P. Shcherbakov, M. Winklhofer, M. Hanzlik and N. Petersen, Eur. Biophys. J. , 1997,\n26, 319–326.\n[23] S. Klumpp and D. Faivre, PLoS One , 2012, 7, 1–11.\n[24] A. G. Meyra, G. J. Zarragoicoechea and V . A. Kuz, Phys. Chem. Chem. Phys. , 2016, 18,\n12768–12773.\n[25] E. Katzmann, A. Scheffel, M. Gruska, J. M. Plitzko and D. Sch ¨uler, Mol. Microbiol. , 2010,\n77, 208–224.\n[26] A. Scheffel, M. Gruska, D. Faivre, A. Linaroudis, J. M. Plitzko and D. Schuler, Nature ,\n2006, 440, 110–114.\n32[27] S. Mann, R. B. Frankel and R. P. Blakemore, Nature , 1984, 310, 405–407.\n[28] A. K ¨ornig, M. Winklhofer, J. Baumgartner, T. P. Gonz ´alez, P. Fratzl and D. Faivre, Adv.\nFunct. Mater. , 2014, 24, 3926–3932.\n[29] A. Hoell, A. Wiedenmann, U. Heyen and D. Sch ¨uler, Physica B , 2004, 350, 309–313.\n[30] J. Li, K. Ge, Y . Pan, W. Williams, Q. Liu and H. Qin, Geochem. Geophys. Geosyst. , 2013,\n14, 3887–3907.\n[31] E. C. Stoner and E. P. Wohlfarth, Philos. Trans. R. Soc. London, Ser. A , 1948, 240, 599–\n642.\n[32] M. Charilaou, M. Winklhofer and A. U. Gehring, J. Appl. Phys. , 2011, 109, 1–6.\n[33] M. Charilaou, K. K. Sahu, D. Faivre, A. Fischer, I. Garc ´ıa-Rubio and A. U. Gehring, Appl.\nPhys. Lett. , 2011, 99, 9–11.\n[34] L. J. Geoghegan, W. T. Coffey and B. Mulligan, Adv. Chem. Phys. , 1997, 100, 475 – 641.\n[35] J. Carrey, B. Mehdaoui and M. Respaud, J. Appl. Phys. , 2011, 109, 083921.\n[36] J. M. Thomas, E. T. Simpson, T. Kasama and R. E. Dunin-Borkowski, Accounts of chem-\nical research , 2008, 41, 665–74.\n[37] F. Kronast and S. Valencia, JLSRF , 2016, A90, 1–6.\n[38] B. Kiani, D. Faivre and S. Klumpp, New J. Phys. , 2015, 17, 43007.\n[39] M. J. Footer, J. W. J. Kerssemakers, J. A. Theriot and M. Dogterom, PNAS , 2007, 104,\n2181–6.\n33[40] A. K ¨ornig, J. Dong, M. Bennet, M. Widdrat, J. Andert, F. D. M ¨uller, D. Sch ¨uler, S. Klumpp\nand D. Faivre, Nano Lett. , 2014, 14, 4653–4659.\n[41] A. M. Hu ´ızar-F ´elix, D. Mu ˜noz, I. Orue, C. Mag ´en, A. Ibarra, J. M. Barandiar ´an, A. Muela\nand M. L. Fdez-Gubieda, Appl. Phys. Lett. , 2016, 108, 063109.\n[42] J. Li, Y . Pan, Q. Liu, K. Yu-Zhang, N. Menguy, R. Che, H. Qin, W. Lin, W. Wu, N. Petersen\nand X. Yang, Earth Planet. Sci. Lett. , 2010, 293, 368–376.\n[43] U. Heyen and D. Sch ¨uler, Appl. Microbiol. Biotechnol. , 2003, 61, 536–44.\n[44] K. Gr ¨unberg, C. Wawer and B. M. Tebo, Appl. Environ. Microbiol. , 2001, 67, 4573–4582.\n[45] D. N. Mastronarde, J. Struct. Biol. , 2005, 152, 36 – 51.\n[46] J. R. Kremer, D. N. Mastronarde and J. McIntosh, J. Struct. Biol. , 1996, 116, 71 – 76.\n[47] J. Schindelin, C. T. Rueden, M. C. Hiner and K. W. Eliceiri, Mol. Reprod. Dev. , 2015, 82,\n518–529.\n[48] E. F. Pettersen, T. D. Goddard, C. C. Huang, G. S. Couch, D. M. Greenblatt, E. C. Meng\nand T. E. Ferrin, J. Comput. Chem. , 2004, 25, 1605–1612.\n[49] H. Gojzewski, M. Makowski, A. Hashim, P. Kopcansky, Z. Tomori and M. Timko, Scan-\nning, 2012, 34, 159–169.\n[50] O. Glatter, J. Appl. Crystallogr. , 1977, 10, 415–421.\n[51] O. Glatter, J. Appl. Crystallogr. , 1979, 12, 166–175.\n[52] B. Vestergaard and S. Hansen, J. Appl. Crystallogr. , 2006, 39, 797–804.\n34[53] P. Bender, L. Bogart, O. Posth, W. Szczerba, S. Rogers, A. Castro, L. Nilsson, L. Zeng,\nA. Sugunan, J. Sommertune et al. ,Sci. Rep. , 2017, 7, 45990.\n[54] C. D. Dewhurst, I. Grillo, D. Honecker, M. Bonnaud, M. Jacques, C. Amrouni, A. Perillo-\nMarcone, G. Manzin and R. Cubitt, J. Appl. Crystallogr. , 2016, 49, 1–14.\n[55] P. Bender, L. Fern ´andez Barqu ´ın, M. Fdez-Gubieda, D. Gonz ´alez-Alonso, D. Honecker,\nL. Marcano and W. Szczerba, 2017.\n[56] D. Honecker, A. Ferdinand, F. D ¨obrich, C. Dewhurst, A. Wiedenmann, C. G ´omez-Polo,\nK. Suzuki and A. Michels, Eur. Phys. J. B , 2010, 76, 209–213.\n35"    },    {        "title": "2402.09153v1.Manipulation_of_magnetic_anisotropy_of_2D_magnetized_graphene_by_ferroelectric_In__2_Se__3_.pdf",        "content": "Manipulation of magnetic anisotropy of 2D magnetized graphene\nby ferroelectric In 2Se3\nRui-Qi Wang1,2, Tian-Min Lei1,∗and Yue-Wen Fang3,4†\n1School of Advanced Materials and Nanotechnology,\nXiDian University, Xi’an 710126, China\n2School of Electronic Engineering,\nXi’an Aeronautical Institute,\nXi’an 710077, China\n3Fisika Aplikatua Saila,\nGipuzkoako Ingeniaritza Eskola,\nUniversity of the Basque Country (UPV/EHU),\nEuropa Plaza 1, 20018 Donostia/San Sebastián, Spain\n4Centro de Física de Materiales (CSIC-UPV/EHU),\nManuel de Lardizabal Pasealekua 5,\n20018 Donostia/San Sebastián, Spain\nThe capacity to externally manipulate magnetic properties is highly desired from both funda-\nmental and technological perspectives, particularly in the development of magnetoelectronics and\nspintronics devices. Here, using first-principles calculations, we have demonstrated the ability\nof controlling the magnetism of magnetized graphene monolayers by interfacing them with a two-\ndimensional ferroelectric material. When the 3 dtransition metal (TM) is adsorbed on the graphene\nmonolayer, its magnetization easy axis can be flipped from in-plane to out-of-plane by the ferro-\nelectric polarization reversal of In 2Se3, and the magnetocrystalline anisotropy energy (MAE) can\nbe high to -0.692 meV/atom when adopting the Fe atom at bridge site with downward polarization.\nThis may be a universal method since the 3 dTM-adsorbed graphene has a very small MAE, which\ncan be easily manipulated by the ferroelectric polarization. As a result, the inherent mechanism is\nanalyzed by second variation method.\n∗leitianmin@163.com\n†fyuewen@gmail.com\n1arXiv:2402.09153v1  [cond-mat.mtrl-sci]  14 Feb 2024I. INTRODUCTION\nOne fundamental concept in condensed matter physics is that most storage devices are\nbased on ferromagnetism. A large magnetic anisotropy (MA) can act as the magnetization\nreversal barrier to prevent thermal disturbance, which is the key to realizing the monoatomic\ninformation storage. In order to read and write the information of an individual magnetic\natom, ferroelectric-controlled magnetism has become a popular way. Both the states with\npolarization upward ( P↑) and polarization downward ( P↓) in the ferroelectric material are\nstable, which can be switched between each other via an external electric field. This makes\nthe ferroelectric material a bistable switch and has a wide range of application prospects in\nelectric-controlled magnetism.\nFerroelectric-controlled magnetism refers to the control of magnetic properties such as\nmagnetic moments, magnetic anisotropies, magnetic orders, or magnetic resistance through\nferroelectrics, sometimes with auxiliary conditions like stress, charge injection, electric\nfield, magnetic field, and temperature, etc. The researchers have successively realized the\nferroelectric-control of ferromagnetic-antiferromagnetic transition in two-dimensional ferro-\nelectric/ferromagneticheterojunctionssuchasIn 2Se3/CrI 3[1-2],Sc 2Co2/CrI 3[3],Sc 2Co2/NiI 2[4],\nand In 2Se3/FeI 2[5]. Ferroelectric-control of the easy axis of magnetization is achieved in two-\ndimensionalferroelectric/ferromagneticheterojunctionssuchasIn 2Se3/NiI 2[6],In 2Se3/FeBrI[7],\nIn2Se3/CrI 3/HfN2[8], Sc 2Co2/Cr 2Ge2Te6[9-10], and In 2Se3/3dtransition-metal atoms[11].\nThe latter ones are of great reference significance for the study of this paper.\nInordertosuccessfullyrealizethemagnetizationeasyaxisflipintheferroelectric/ferromagnetic\ninterface, the initial MAE of the ferromagnet needs to be small, while the polarization of\nthe ferroelectric needs to be large. Furthermore, the magnetoelectric coupling effect should\nbe strong.\nAs a new 2D ferroelectrics, α-In2Se3has stable spontaneous in-plane and out-of-plane\nferroelectric polarizations at room temperature, and can still maintain them under the lim-\nited thickness of a single layer[12]. Quantitative experimental results show that an ultrathin\nα-In2Se3layer owns a spontaneous polarization PS= 0.92 µC/cm2under the external elec-\ntric field of 5 ×105V/cm[13]. It has recently been theoretically shown that switching the\ndirection of electrical polarization of α-In2Se3could affect the magnetization direction of its\nadlayers[6, 8, 11].\n2Another key technology is to select a ferromagnet with a small MAE. In recent years,\na number of two-dimensional ferromagnetic materials have emerged. Cr 2Ge2Te6has been\nexperimentally found to have intrinsic magnetism[14], which can be controlled by adjacent\nferroelectricity[9]. CrI 3is proved to be an Ising ferromagnet with an out-of-plane spin orien-\ntationbymagneto-opticalKerreffectmicroscopy[15]. Researchersalsofindstronganisotropy\nand magnetostriction in 2D Stoner ferromagnet Fe 3GeTe 2[16]. Furthermore, the quantum\nanomalousvalleyHalleffecthasbeenfoundinbothferromagneticmonolayerRuClBr[17]and\nantiferromagnetic monolayer MoO[18]. Intrinsic Dirac half-metal and quantum anomalous\nHall phase have been discovered in the hexagonal Nb 2O3lattice[19].\nInstead of using the latest two-dimensional ferromagnets, we use transition metals to\nadsorb on graphene[20-21] not only for the graphene’s good electrical and mechanical prop-\nerties[22] but also for the systems’ small MAEs. In other words, the selected lattice must\nhave a very small initial MAE to be easily controlled by the adjacent ferroelectric. In this\nstudy, the rhombic lattice of graphene was chosen to adsorb the 3 dTM atoms, which granted\nthe graphene with a very small MAE, and then the two-dimensional In 2Se3is utilized to\nmanipulate the magnetization easy axis of the magnetized graphene monolayers. Better\nthan the previous study where MAE reached -0.23 meV/atom at most[23], we show high\nmagnetic crystal anisotropy (up to -0.692 meV/atom) in this study which is essential for be-\ning used in new magnetic storage. In other words, by manipulating the occupation of some\nspecific 3 dorbitals, we can achieve accurate regulation of the MAE. We find it a universal\nmethod to manipulate the easy magnetization axis by the ferroelectric polarization since the\nmagnetized graphene has a very small initial MAE.\nII. METHODS\nFig. 1 shows the heterojunction of In 2Se3/TM-adsorbed graphene, in which the In 2Se3\nmonolayer and the magnetized graphene are adopted to form a heterojunction. We choose\na rhombic graphene monolayer with the lattice constant a = 4.26 Å. The optimized lattice\nconstant of 1 ×1 In 2Se3is 4.11 Å. To make up for the small lattice mismatch ( ∼3.5%), the\nlattice constant of In 2Se3is enlarged by 3.5 %, while the lattice constant of graphene is fixed\nat the value of 4.26 Å. To eliminate the interaction between adjoining heterostructures, a 20\nÅ vacuum layer along the z-axis is included. Fig. 2 shows the rhombic interfacial shape of\n3the lattice, where the graphene monolayer adsorbs the individual TM atom on three different\nsites (hollow, bridge, and top), respectively.\nFIG. 1. Atomic structures of In 2Se3/graphene heterostructures. The Se, In, TM, and C atoms are\nrepresented by green, purple, gray, and orange balls, respectively. Polarization of In 2Se3is from\nthe negative Se atom to the positive In atom: (a) the direction of polarization is vertically upward;\n(b)the direction of polarization is vertically downward.\nFIG. 2. The rhombic interfaces of the heterojunctions. Transition metal atoms are attached to\nthe surfaces of graphene in three different ways (hollow, bridge, and top). In 2Se3has been left out\nfor a better view of the interfaces.\nThecalculationsareperformedintheViennaAb-initioSimulationPackage(VASP)based\non density-functional theory (DFT). The DFT + Uapproach[24] is essential for the strongly\ncorrelated electronic systems with partially filled 3 d-shells. In this work, we use U= 2 eV\nandJ= 0 eV to study the magnetism, which are the typical values for 3 dTM atoms[25].\nFurthermore, the interfacial distances have been optimized by the DFT-D3 method[26].\nThe generalized gradient approximation (GGA) with the form of Perdew–Burke–Ernzerhof\n4TABLE I. Calculated binding energies (in units of eV) for the TM atom-adsorbed graphene mono-\nlayers at bridge, hollow, and top sites.\nTM atom Bridge Hollow Top\nV -0.7593 -0.6025 -0.8034\nCr -0.2626 -0.2386 -0.2689\nMn -0.2921 -0.2918 -0.2914\nFe -0.7288 -0.4607 -0.6594\nCo -0.3737 -0.1504 -0.3689\nNi -0.7819 -0.8015 -0.7776\n(PBE) is utilized to describe the exchange–correlation functional. A 16 ×16×1 Monkhorst-\nPack k-point grids is good enough to sample the Brillouin zone. The cut-off energy is set\nto be 480 eV and the criterion for the convergence of total energy is set to 10−6eV. The\nconvergence test of total energy and MAE can be seen from Fig. S1-S6 in Supplementary\nMaterial. The dependency between the value of MAE and Ueff has also been tested and it\nis found that different Ueff values may not change the positive or negative signs of MAE,\nand only alter the magnitude of MAE slightly (see Fig. S7-S9). Moreover, the screened\nexchange hybrid density functional by HeydScuseria-Ernzerhof (HSE06) [27] is utilized to\ncheck the electronic structure (see Fig. S10).\nThe following definition represents the binding energy of the TM/graphene system[28]:\nEb=ETM/graphene −ETM−Egraphene (1)\nHere ETM/graphene,ETM, and Egrapheneare the total energies of the TM/graphene system, the\nTM atom and the graphene slab, respectively. Table 1 shows the calculated binding energies\nwith 3 dTM atoms on graphene monolayer at different sites. These values reveal that the\nmost stable position of Mn, Fe, and Co is the bridge site, while it is the top site for V and\nCr atoms, and only the Ni atom is most stable at the hollow site.\nThrough careful calculations, various interfacial configurations of In 2Se3and graphene\nwere checked to consider the magnetoelectric coupling effect. Ultimately, the interfaces in\nwhich the TM atoms were exactly aligned with the In-Se bonds along the z-axis were chosen\nsince their strong magnetoelectric coupling effect achieved the ferroelectric manipulation\n5FIG. 3. Variation of the energy from 3000 to 9000 fs during AIMD simulations at a temperature\nof 300 K for the In 2Se3/Fe bridge-site adsorbed graphene heterostructures: (a) P↑and (b) P↓.\nThe inset illustrates the sliced snapshots of the crystal structures at 4000 and 8000 fs, respectively.\nof the magnetization easy axis. The molecular dynamics simulations are performed at a\ntemperature of 300 K with a 3 ×3 supercell in Fig. 3. The energy fluctuations are small in\nthe time range of 3000–9 000 fs, indicating that the structures are basically mechanically\nstable.\nTheMAEisobtainedbythedifferenceoftotalenergieswhenthemagnetizationdirections\nare along [100] and [001], thus the positive MAE refers to the out-of-plane MA while the\nnegative MAE refers to the in-plane MA. First of all, the charge density is gained through\nthe calculation of self-consistent without the spin-orbit coupling (SOC) effect. Then the\nnoncollinear calculations are carried out for the MAE by reading the above charge density\nand considering the SOC effect. The convergence of MAE is achieved with a k-point meshes\nof 30×30×1 (see Fig. S6).\nIII. RESULTS & DISCUSSIONS\nWe adopt the most stable adsorbing types for each transition metal atom adsorbed on\ngraphene. The magnetic moments of 3 dTM atoms with the ferroelectric P↑andP↓\nare shown in Fig. 4 for comparison. It is obvious that most of the magnetic moments are\nnon-zero for TM atoms, except for Ni atoms at the P↓state, where the occupations of spin\nup and spin down are equal. When the ferroelectric polarization of In 2Se3is turned from\nP↑toP↓, magnetic moments of the V, Fe, Co, and Ni atoms diminish, whereas those of\nthe Cr and Mn atoms increase.\n6As can be clearly seen in Fig. 5, the MAE of the 3 dTM-adsorbed graphene monolayer\nwithout ferroelectric polarizations is generally below 0.100 meV/atom and the maximum is\nonly about 0.250 meV, which is the key prerequisite to be easily manipulated by the adjacent\nferroelectric polarization as previously mentioned.\nFIG. 4. Magnetic moments of TM atoms in the In 2Se3/TM adsorbed graphene heterojunctions\nfor the P↑andP↓of In 2Se3.\nWhen added by extra ferroelectric polarizations of In 2Se3, the MAEs for different TM\natoms-adsorbed graphene monolayers are plotted in Fig. 6 for comparison. It is shown that\nthe In 2Se3polarization inversion from P↑andP↓can weaken the in-plane MA at the Cr\natom, and strengthen the out-of-plane MA at the Co atom. Whereas, the most exciting\nfinding is that when 3 dTM atoms are V, Mn, and Fe, the easy axes of magnetization\nin these systems all depend on the ferroelectric polarization direction of In 2Se3. For Mn\nand Fe atoms, the MAE is positive in P↑state but negative in P↓state. Whereas, the\ncorrespondence is reversed for V atoms. As a result, we realized the flip of magnetization\neasy axis from out-of-plane to in-plane via ferroelectric control. Here we select Fe atoms\nfor further research, since the P↑state has a positive MAE of 0.117 meV/atom, while the\nP↓state has a negative MAE of -0.692 meV/atom, which is the largest MAE value in our\nstudy. It should be noted that the MAE of the Ni atom is zero regardless of whether the\n7FIG. 5. MAE of the TM atom-adsorbed graphene monolayers without ferroelectric polarizations.\nFIG. 6. MAE of the In 2Se3/TM adsorbed graphene heterojunctions for the P↑andP↓of In 2Se3.\nferroelectric is added or not (see Fig. 5 and Fig. 6) since the magnetic moment is almost\n8zero.\nThrough calculating the Berry Phase[29], the polarization of ferroelectrics is carried out\nbyP= (P↑-P↓) / 2 where P↑andP↓are the ferroelectric polarization along z\nand - zaxis. Well consistent with the predecessors’ study[30], the polarization of monolayer\nIn2Se3is calculated to be 0.96 µC/cm2. Furthermore, If P= +0.96 µC/cm2andP= -0.96\nµC/cm2are denoted by P↑andP↓respectively, the MAE dependency on the ferroelectric\npolarization for the In 2Se3/Fe bridge-site adsorbed graphene monolayer is presented in Fig.\n7.\nFIG. 7. The MAE dependency on the ferroelectric polarization for the In 2Se3/Fe bridge-site\nadsorbed graphene monolayer.\nTo clarify the origin of MAE with spin states, qualitative analysis can be carried out\nbased on second-order perturbation theory[31]:\nMAE≃ξ2X\no,u|⟨o|lz|u⟩|2− |⟨o|lx|u⟩|2\nεu−εo(2)\nHere, ξistheSOCconstant, oand urepresenttheeigenstatesofoccupiedandunoccupied\nstates, and εoandεurepresent the eigenvalues of oandu. For states with opposite and\nsame spins, the MAEs can be further written as:\n9MAE≃ −ξ2X\no+,u−|⟨o+|lz|u−⟩|2− |⟨o+|lx|u−⟩|2\nε−\nu−ε+\no(3)\nMAE≃ξ2X\no−,u−|⟨o−|lz|u−⟩|2− |⟨o−|lx|u−⟩|2\nε−\nu−ε−\no(4)\nAs a result, elements of the matrix as well as the energy differences can affect the MAE\njointly. Table 2 shows the differences of matrix elements |⟨o+|lz|u−⟩|2− |⟨o+|lx|u−⟩|2and\n|⟨o−|lz|u−⟩|2− |⟨o−|lx|u−⟩|2for Eqs. (3) and (4), which are utilized to analyze the\ncontribution of specific orbitals to MAE.\nThe magnetic quantum numbers m=±2, m=±1, and m= 0can represent the\ndx2−y2/xy, dxzyzanddz2of3d-orbitals, respectively. The states with |∆m|= 0provide the\npositive contribution for the coupling in the same spins (↑↑,↓↓)and the negative contri-\nbution for the coupling with different spins ( ↑↓,↓↑); Whereas, the states with |∆m|= 1\ngive the negative contribution for the coupling in the same spins ( ↑↑,↓↓) and the positive\ncontribution for the coupling in the different spins ( ↑↓,↓↑).\nFIG. 8. The 3d-orbital decomposed PDOS of the Featom at the In2Se3/Feadsorbed graphene\n(bridge-site) heterojunctions for (a) P↑and (b) P↓.m= 0,m=±1, and m=±2are represented\nby purple line, blue line, and green line, respectively.\nFig. 8 presents the orbital-decomposed PDOS of Fe’s 3d-orbitals for the In2Se3/Fe\nadsorbed-graphene (bridge-site) heterojunctions. Firstly, when the ferroelectric polarization\nofIn2Se3turns from upward to downward, the occupied m=±1↓states move towards the\nFermi level while the unoccupied m= 0↓states stand still, which diminishes the energy\ndifference εuσ′−εuσ. The orbitals dyzanddz2with same spins can decrease the MAE through\n⟨m=±1↓ |lx|m= 0↓⟩for they yield a matrix element difference equalling -3 (Table 2). In\n10addition, occupied m=±2↑states and unoccupied m=±2↓states all move to the\nhigher energy level, but the energy difference εuσ′−εuσdecreases. The orbitals dx2−y2and\ndxywith opposite spins can sharply diminish the MAE through ⟨m=±2↑ |lz|m=±2↓⟩\nfor they have a matrix element difference equalling -4 (Table 2). Consequently, the sum of\nthe two contributions can reduce the MAE from 0.117 meV/atom to -0.692 meV/atom as\nthe ferroelectric polarization of In2Se3turns from upward to downward. It should be em-\nphasized that the above analysis also applies to the decomposed PDOS obtained by HSE06\n(see Fig. S10).\nTo quantitatively proof the contribution of specific orbitals to MAE, the 3d-orbital de-\ncomposed MAE of Fe in the upward and downward polarizations are shown in Fig. 9(a)\nand Fig. 9(b). It’s obvious that for upward polarization in Fig. 9(a), elements of the matrix\nbetween dz2anddyzyield large positive values for MAE which are more dominant than those\nshowing negative values, thus leading to a positive MAE of 0.117 meV/atom; For downward\npolarization in Fig. 9(b), elements of the matrix between different orbitals except for dxz\nanddyzall show negative values, therefore the MAE has a very large negative value of -0.692\nmeV/atom. The 3d-orbital-decomposed MAEs of the interface VandMnatoms are calcu-\nlated and the corresponding results are shown in Fig. 9(c-d) and (e-f), respectively. The\nhybrid matrix element dx2−y2anddxyis crucial for the Vatoms, which provides negative\n(positive) contributions to P↑(P↓), thus leading to a MAE of -0.403 meV/atom (0.516\nmeV/atom), respectively (Fig. 9(c-d)). For Mn atoms in Fig. 9(e-f), elements of the matrix\ndz2anddyzyield positive contributions for MAE, while those between dx2−y2anddxyyield\nnegative contributions for MAE; Since the former contributions are more dominant for P↑\nbut the latter ones are more dominant for P↓, the MAE becomes 0.255 meV/atom for P↑\nand -0.364 meV/atom for P↓, respectively.\nFig. 10presentstheplane-averagedchargedensitydifference ∆ρ(z)with P↑andP↓. The\nresults are in line with expectations that upward polarization causes electron depletion near\nTM atoms (see Fig. 10(a)(c)(e)), while downward polarization causes electron accumulation\nnear TM atoms (see Fig. 10(b)(d)(f)). The charge transfers can bring changes in the\namounts and occupations of the majority and minority spin electrons in TM atoms, thus\naffecting the magnetic moments and MAEs.\n11FIG. 9. The 3d-orbital decomposed MAE of TM atom at the In2Se3/TMatom adsorbed graphene.\n(a) and (b) are of the Featoms, (c) and (d) are of the Vatoms, and (e) and (f) are of the Mn\natoms. The ferroelectric polarization direction of In2Se3is upward in (a)(c)(e) and downward in\n(b)(d)(f).\n12FIG. 10. Plane-averaged charge density difference ∆ρ(z)with upward ferroelectric polarization\nin (a)(c)(e) and downward ferroelectric polarization in (b)(d)(f): (a) and (b) are for Featoms, (c)\nand (d) are for Vatoms, and (e) and (f) are for Mn atoms. Here, blue and yellow regions refer to\nelectron’s depletion and accumulation, respectively.\n13u−o+o−\ndxydyzdz2dxzdx2−y2dxydyzdz2dxzdx2−y2\ndxy0 0 0 1 -4 0 0 0 -1 4\ndyz0 0 3 -1 1 0 0 -3 1 -1\ndz20 3 0 0 0 0 -3 0 0 0\ndxz1 -1 0 0 0 -1 1 0 0 0\ndx2−y2-4 1 0 0 0 4 -1 0 0 0\nIV. CONCLUSIONS\nIn summary, through first-principles calculations, we have demonstrated the manipu-\nlation of switching the magnetization orientations of magnetized graphene monolayers by\ncontrolling their adjacent ferroelectric polarizations in 2D van der Waals heterostructures.\nSpecifically speaking, the easy magnetization axis of the V, Mn, and Fe adsorbed graphene\nmonolayers can be flipped by the ferroelectric polarizations of adjacent In2Se3layers. For\ninstance, when the Featom is adsorbed on the graphene monolayer at the bridge site, the di-\nrectionofmagnetizationundergoesaflipfromalong[001](MAE=0.117 meV/atom)toalong\n[100] (MAE =-0.692 meV/atom) if the ferroelectric polarization of In2Se3turns from upward\nto downward. It is mainly due to that the MAE source from the matrix element between dz2\nanddyzchanges from positive to negative. This work demonstrates a prospective path to re-\nalizing the ferroelectric-control of magnetism in two-dimensional ferroelectric/ferromagnetic\nheterostructures and may further stimulate the theoretical and experimental exploration in\nstudying magnetized graphene derivatives and other 2D spintronic materials.\nV. DATA AVAILABILITY\nSUPPLEMENTARY MATERIAL\nSee the supplementary material for the convergence tests of total energy and MAE (Fig.\nS1-S6), the dependency between the value of MAE and Ueff(Fig. S7-S9), and the 3d-orbital\ndecomposed PDOS obtained by HSE06 (Fig. S10).\n14ACKNOWLEDGMENTS\nThisstudyissupportedbyScientificResearchProgramFundedbyEducationDepartment\nof Shaanxi Provincial Government (Program No. 21JK0699) and Natural Science Basic\nResearch Program of Shaanxi (Program No. 2024JC-YBQN-0038).\nDECLARATION OF COMPETING INTEREST\nThere are no conflicts that need to be declared.\nVI. ACKNOWLEDGMENTS\nVII. AUTHOR CONTRIBUTIONS\nY.-W.F. conceived the study and planned the research. Y.-W.F. and T.-M.L. supervised\nthe project. R.-Q.W. performed and analyzed the theoretical calculations with assistance\nfrom Y.-W.F.. The manuscript was drafted by R.-Q.W. and further revised by all the\nauthors.\nVIII. COMPETING INTERESTS\nThe authors declare no competing interests.\nIX. ADDITIONAL INFORMATION\nMaterials & Correspondence should be addressed to Y.-W.F.\n15[1] H.-X. Cheng, J. Zhou, C. Wang, W. Ji, Y.-N. Zhang, Phys. Rev. B 104, 064443\n(2021).\n[2] B. S. Yang, B. Shao, J. F. Wang, Y. Li, C. Y. Yam, S. B. Zhang, B. Huang, Phys.\nRev. B 103, L201405 (2021).\n[3] Y. Lu, R. X. Fei, X. B. Lu, L. H. Zhu, L. Wang, ACS Appl. Mater. Interfaces 12(5),\n6243-6249 (2020).\n[4] M. H. Liu, L. Zhang, J. X. Liu, T. L. Wan, A. J. Du, Y. T. Gu, L. Z. Kou, ACS Appl.\nElectron. Mater. 5, 920-927 (2023).\n[5] W. Sun, W. X. Wang, D. Chen, Z. X. Cheng, Y. X. Wang, Nanoscale 11, 9931-9936\n(2019).\n[6] Y. P. Wang, X. G. Xu, X. Zhao, W. X. Ji, Q. Cao, S. S. Li, Y. L. Li, npj Comput.\nMater. 8, 218 (2022).\n[7] R. Li, J. W. Jiang, W. B. Mi, H. L. Bai, Appl. Phys. Lett. 120, 162401 (2022).\n[8] B. X. Zhai, R. Q. Cheng, W. Yao, L. Yin, C. H. Shen, C. X. Xia, J. He, Phys. Rev.\nB 103, 214114 (2022).\n[9] W. R. Liu, X.-J. Dong, Y.-Z. Lv, W.-X. Ji, Q. Cao, P.-J. Wang, F. Li, C.-W. Zhang,\nNanoscale 14, 3632 (2022).\n[10] A. Ilyas, S. L Xiang, M. G. Chen, M. Y. Khan, H. Bai, P. He, Y. H. Lu, R. R. Deng,\nNanoscale 13, 1069 (2021).\n[11] D. Y. Jin, W. Qiao, X. Y. Xu, W. B. Mi, S. M. Yan, D. H. Wang, Appl.Surf.Sci.\n580, 152311 (2022).\n[12] W. J. Ding, J. B. Zhu, Z. Wang, Y. F. Gao, D. Xiao, Y. Gu, Z. Y. Zhang, and W.\nG. Zhu, Nat. Commun. 8, 14956 (2017).\n[13] S. Y. Wan, Y. Li, W. Li, X. Y. Mao, C. Wang, C. Chen, J. Y. Dong, A. M. Nie, J.\nY. Xiang, Z. Y. Liu, W. G. Zhu, H. L. Zeng, Adv. Funct. Mater. 29, 1808606 (2019).\n[14] C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao, W. Bao, C. Wang, Y. Wang,\nZ. Q. Qiu, R. J. Cava, S. G. Louie, J. Xia and X. Zhang, Nature 546, 265–269 (2017).\n[15] B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, R. Cheng, K. L. Seyler,\nD. Zhong, E. Schmidgall, M. A. McGuire, D. H. Cobden, W. Yao, D. Xiao, Nature 546,\n270–273 (2017).\n[16] H. L. Zhuang, P. R. C. Kent and R. G. Hennig, Phys. Rev. B 93, 134407 (2016).\n[17] H. Sun, S.-S. Li, W.-X. Ji, C.-W. Zhang, Phys. Rev. B 105, 195112 (2022).\n16[18] B. Wu, Y.-L. Song, W.-X. Ji, P.-J. Wang, S.-F. Zhang, C.-W. Zhang, Phys. Rev. B\n107, 214419 (2023).\n[19] S.-J. Zhang, C.-W. Zhang, S.-F. Zhang, W.-X. Ji, P. Li, P.-J. Wang, S.-S. Li, S.-S.\nYan, Phys. Rev. B 96, 205433 (2017).\n[20] X. Chen, Y. H. Liu, B. Sanyal, J. Phys. Chem. C 124(7), 4270-4278 (2020).\n[21] Z. Y. Guan, S. Ni, S. L. Hu, ACS Omega. 5, 5900-5910 (2020).\n[22] S. Stankovich, D. A. Dikin, G. H. B. Dommett, K. M. Kohlhaas, E. J. Zimney, E. A.\nStach, R. D. Piner, S. T. Nguyen, R. S. Ruoff, Nature 442, 282–286 (2006).\n[23] R.-Q. Wang, Y.-W. Fang, T.-M. Lei, J. Magn. Magn. Mater. 565, 170297 (2023).\n[24] S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, A. P. Sutton, Phys.\nRev. B 57, 1505 (1998).\n[25] I. V. Solovyev, P. H. Dederichs, V. I. Anisimov, Phys. Rev. B. 50 (23), 16861–16871\n(1994).\n[26] S. Grimme, J. Antony, S. Ehrlich, and S. Krieg, J. Chem. Phys. 132, 154104 (2010).\n[27] J. Paier, M. Marsman, K. Hummer, G. Kresse, I. C. Gerber, and J. G. Ángyán. J.\nChem. Phys. 124, 154709 (2006).\n[28] A. Banerjea, J. R. Smith, Phys. Rev. B 37, 6632–6645 (1988).\n[29] D. Vanderbilt, R. D. King-Smith, Phys. Rev. B 48, 4442 (1993).\n[30] X. X. Jiang, Y. X. Feng, K. Q. Chen, L. M. Tang, J. Phys. Condens. Matter 32,\n105501 (2020).\n[31] D.-S. Wang, R. Q. Wu, A. J. Freeman, Phys. Rev. B 47, 14932 (1993).\n17"    },    {        "title": "2402.09741v1.Dimension_Dependent_Critical_Scaling_Analysis_and_Emergent_Competing_Interaction_Scales_in_a_2D_Van_der_Waals_magnet_Cr___2__Ge___2__Te___6__.pdf",        "content": "1 \n imension -Dependent Critical Scaling Analysis and Emergent Competing \nInteraction Scales in a 2D Van der Waals magnet  Cr2Ge 2Te6 \nP C Mahato1, Suprotim  Saha1, Bikash  Das3, Subhadeep  Datta3,  Rajib  Mondal5, Sourav  Mal4, \nAshish  Garg2, Prasenjit  Sen4,⁎, S. S. Banerjee1,+ \n1Department  of Physics,  Indian  Institute  of Technology  Kanpur,  Kanpur,  Uttar  Pradesh  \n208016,  India  \n2 Department  of Sustainable  Energy  Engineering,  Indian  Institute  of Technology  Kanpur,  \nKanpur,  Uttar  Pradesh  208016,  India  \n3School  of Physical  Sciences,  Indian  Association  for the Cultivation  of Science,  Jadavpur,  \nKolkata  -700032,  India  \n4Harish -Chandra  Rese arch Institute,  HBNI,  Chhatnag  Road,  Jhunsi,  Allahabad,  Uttar  \nPradesh  211019,  India  \n5UGC -DAE  Consortium  for Scientific  Research,  Kolkata  Centre,  Bidhannagar,  Kolkata,   \n700106 India \nCorresponding authors email ID:  \n*: prasen@hri.res.in  \n+: satyajit@iitk.ac.in  \n \n \n \n \n \n 2 \n  \n \nAbstract:  3 \n Amongst the popular t wo-dimensional (2D) van der Waal (vdW) materials [1-4] Cr2Ge2Te6 (CGT) has \nattract ed special attention as it is a 2D ferromagnetic (FM)  - semiconductor (Curie Temperature 𝑇𝑐 ~ \n67 K with a bandgap ~ 0.4 eV) [5] which is potentially useful for spintronic device  applications [6-12]. \nThese 2D-vdW materials  offer controllability of their magnetic properties via, diverse parameters \n[2,7,13 -18]. In CGT , strong Cr -Te-Cr super -exchange via a near 900 bond angle [19,20]  results in robust \n2D - FM order.  In general , dimensionality reduction can trigger, loss of inversion symmetry , \nmodifi cation of  spin-orbit coupling (SOC) [21] which affect s magnetic [12]  and transport properties [22-\n24] and changes in band structure [25,26] . Critical  phenomena  exhibit  characteristic  scaling  of data near \na critical  transition  point [27], leading  to the identification  of universality  classes.  The universality  class,  \ndetermined  solely  by dimensionality,  symmetry,  and interactions  in the system [28,29] , allows  for a \ndeeper  understanding  of interactions  driving  phase  transitions  in condensed  matter  systems  which  is \nindependent  of the system 's microscopic  detail s. Therefore , exploring the nature of magnetic transition \nas a function of thickness in  vdW  materials  is important . Magnetism of CGT has complex dimensional \ndependence : in bulk CGT, the magnetic anisotropy [1,30]  is out of plane (OOP) i.e.  along the  normal to \nthe crystallographic ab plane , however  theoretical results on magnetic anisotropy in few-layer  or \nmonolayer (ML) CGT are conflicting, viz., one study suggests ML -CGT has OOP [31] anisotropy  while \nanother study suggests in -plane (IP) [32] anisotropy . Recent experimental studies in bulk CGT single \ncrystals show the critical scaling exponents of magnetizat ion (M) near Tc conform to the 2D Ising -\nuniversality class [33,34]  while another study suggests adherence to tricritical mean field model [35]. \nSome t heoretical models [36] also treat bulk CGT as a 2D Heisenberg FM. However , the dimensional \ndependence of critical scaling features ha s not been  systematically explored.  In this work , using \nModified Arrott  Plot (MAP) method and critical scaling analysis ,[29,37,38]  we investigate the  thickness \n(dimensional ) dependence of 𝑇𝑐 and the critical  scaling exponents in CGT . Bulk single crystal of CGT \nshows a 2D Ising -like paramagnetic (PM)  to FM phase transition with OOP anisotropy at Tc = 67 K. \nFew-layer (or low dimensional) CGT flake ensemble dispersed on hBN/SiO 2/Si substrate  exhibits the \nsame phase transition at 67 K. Additionally  we also find a second critical transition at a lower \ntemperature 𝑇𝑐′~ 14-15 K whose critical exponents are completely different from those  at 67 K, and  do 4 \n not conform to the standard universality classes.  Our DFT -based electronic structure calcula tions reveal \nsmall yet  important  structur al changes in CGT at reduced dimensionality , which affect the spin orbit \ncoupling (SOC). Our calculations suggest that in low dimensional  CGT , a competition emerges between \nmagneto -crystalline anisotropy energy (MAE)  favouring OOP anisotropy and dipolar  anisotropy energy \n(DAE)  favori ng IP orientation , and this  results in a complex magnetic configuration . This also suggest s \nthe emergence of competing long and short -range  interactions in the system with lowering dimensions, \nall of which , we believe , triggers a new critical crossover at 𝑇𝑐′.  \n  \nSingle crystals of CGT were grown by self -flux technique [39] (see details of experimental methods in \nSI-1, with X -ray Diffraction (XRD) characterization  and Energy Dispersive X -ray (EDX)  elemental \nmap of the CGT crystal  in SI -2). Crystals  were exfoliated in Ar atmosphere in a glove box  down to ~ 2 \nnm thickness . Fig.1( a) shows the similarity of the measured Raman spectral modes for a bulk single \ncrystal and a 2 nm thick CGT flake. The spectra show two distinct peaks at 116 and 136  cm-1 which \ncorrespond to the 𝐸𝑔2 and 𝐴𝑔1 Raman modes respectively in CGT [40,41] . Transmission Electron \nMicroscopy ( TEM ) reveals the high degree of crystal line order  of a few layered CGT flake ( see TEM \nimage of atomic planes in CGT in Fig. 1(b) and Fig. S2(e) ). Fig. 1(c) shows a Z-contrast intensity  profile \nalong the red -marked region in Fig. 1( b). The atomic ordering profile ( Fig.1(c) ) agrees  with CGT's \nsimulated atomic configuration (Fig. 1(a) inset) . The hexagonal symmetry of the diffraction spots in the \nSelected Area Electron Diffraction (SAED) image confirms the crystalline order persisting in the few -\nlayer CGT (Fig. 1(d)) (see SI -2 for details) . Carefully chosen few layered flakes were transferred onto \ninsulating and inert single layer hBN/SiO 2/Si substrate (Graphene supermarket, see Fig. S2( h) in SI -2 \nfor characterization ). Fig. 1( e) shows the Atomic Force Microscopy (AFM) map of the transferred \nflakes recorded across multiple regions of area 10 μm × 3 μm. The uniformity of thickness (~2 -3 nm) \nof the flakes in the ensemble is evident from the adjoining colour bar, confirming  almost uniform 2D \nnature of the transferred flakes ( see SI-3 for details ). The typical density of the CGT flake ensemble \n(Fig.1( e)) is ~ 1 to 2 flakes /μm2. In bulk CGT crystal , the magnetization (M) versus temperature  (T) \n(SQUID magnetometer , Cryogenics  UK) in an applied magnetic field ( H) of 200 Oe , shows a PM to a 5 \n FM transition (Fig. 2(a))  in both H || ab and H ⊥ ab orientations . Inverse susceptibility ( 𝜒−1) versus T \nderived from M(T) data for H || ab (see inset of Fig. 2(a)) shows the typical Curie -Weiss nature . The \nintercept gives 𝑇𝑐 = 64.28 K  and an effective magnetic moment (𝜇𝑒𝑓𝑓) 3.18  𝜇𝐵 per Cr atom , consistent \nwith prior finding of orbital quenched magnetic moment of 3.87 𝜇𝐵 of Cr3+ in CGT [33]. The i nset of \nFig. 2(b) shows five-quadrant M versus H at 𝑇 = 4 K for H || ab and 𝐻 ⊥ ab. For 𝐻 ⊥ ab, the M (H) loop \nhas a  square shape with a saturation field 𝐻𝑠⊥𝑎𝑏~ 3 kOe and low coercivity HC ~ 240 Oe  ,while f or H || \nab, M(H) has a skewed shape  and does not saturate up to 40 kOe [42] .This suggests that the CGT single \ncrystal has strong perpendicular magnetic anisotropy ( 𝐾𝑢) i.e. the easy axis of M  is normal to the ab \nplane  [1,42] . Linear extrapolation of the M(H) data for H || ab gives 𝐻𝑠||𝑎𝑏~ 78 kOe (see SI -4), using \nwhich we estimate the effective 𝐾𝑢= 4.75 × 105 J.m-3 (~ 0.6 meV/f.u , which  is close to  the reported \nvalues [42,43]  and also matches with ou r DFT results  discussed later). Isothermal  M(H) for 𝐻⊥𝑎𝑏 \nwere measured at different T’s (in intervals of 1 K ) around 𝑇𝑐 for critical scaling analysis (see Fig. S5 \nin SI -5). In critical region ( T Tc), we analyse M(H) isotherms using the MAP method based on \nmagnetic equation of state [38,44 -46] 𝑀(𝐻,𝑡)=𝑡𝛽𝑓±(𝐻\n𝑡𝛽+𝛾) (see details in SI -6), where 𝑡=𝑇−𝑇𝑐\n𝑇𝑐 and \n𝑓± are two distinct functional forms of the scaled isothermal M(H) curves at T above (+) and below ( -) \nTc. In Fig. 2(b), we see that all the scaled magnetization (𝑀\n𝑡𝛽) versus scaled field (𝐻\n𝑡𝛽+𝛾) curves c ollapse \nonto two distinct family of curves, one above Tc and another below Tc for 𝛽=0.14, 𝛾=1.45, and 𝑇𝑐=\n67 𝐾, thereby confirming the presence of a critical phase transition [38,44 -46] close to 67 K . In Fig.2(c) \nthe 𝑀1\n𝛽 versus (𝐻\n𝑀)1\n𝛾 shown using 𝛽 and 𝛾 of Fig. 2( b) yields a set of parallel straight lines with the \nisotherm at 67 K passing through the origin , marking Tc. The y and x axis intercepts in Fig.2(c) give the \nT dependent spontaneous magnetization 𝑀𝑠(left axis  Fig.2(d)  for T < Tc) and inverse susceptibility  \n(𝜒−1) (right axis  Fig.2(d) for T > Tc) versus T, respectively. In Fig.2(d) , the solid lines are fits to 𝑀𝑠 and \n𝜒−1 using well known power law fits [45]: 𝑀𝑠(𝑇)∝𝑡𝛽 and 𝜒−1(𝑇)∝𝑡𝛾 , giving 𝛽=0.12 ±0.01 and \n𝛾=1.21 ±0.34, which are consistent with the estimate s of Fig.2(b)  and Fig. 2(c) (see supplementary \nsection S I-7 for additional self -consistency checks of the critical exponents) . Using 𝛽 = 0.14, 𝛾=1.45 6 \n and the Widom’s scaling relation [47] : 𝛿=1 + 𝛾\n𝛽 we find  𝛿 = 11.35(7).  Our critical exponents for \nbulk CGT obtained are summarized in table I  and they are not only consist ent with earlier  reports  \n[33,34]  (see table I)  but are in good agreement with  the 2D Ising universality class (𝛽 = 0.125, 𝛾=\n1.75). \n \nNext , we measure  the M(T) and M(H) of an ensemble of 2-3 nm thick CGT flakes (Fig.1( e)). Note that \nall the data shown here is after subtracting the diamagnetic background of the bare hBN/SiO 2/Si \nsubstrate (5 mm × 3 mm × 0.5 mm) and straw holder. We could get reliable, low -noise data for our \nfew-layer CGT flakes  only for M(T) measured  in the H  ab orientation . M(T) for the few-layered CGT \nflake  ensemble  (Fig. 3(a)) shows PM to FM transition with a Tc ~ 66.4 K deduced from the intercept of \nlinear 𝜒−1  versus T plot (see Fig.3(a) inset) , similar to the bulk CGT . For our few -layered CGT flake \nensemble, Fig. 3(b) shows fit to the M(T) data (FC 600 Oe) near the PM -FM transition  with M(T) ~ tβ \nwhich gives β = 0.16  ± 0.02  and an extrapolated Tc estimated to be 66.31  ± 0.09  K. Similar to Fig. 2(b) \nfor bulk CGT, for our few -layered CGT flake ensemble, Fig 3(c) show s the scaled magnetization \n(𝑀\n𝑡𝛽) versus scaled field (𝐻\n𝑡𝛽+𝛾) curves collapse onto two distinct family of curves, one above Tc and \nanother below it, with β = 0.16, γ = 1.44 with an estimated  Tc = 65 K. A log -log plot of the M(H) data \nat 70 K (Fig. 3 (d)) fits to M(H) ~ AH1/δ with δ =10.17. We find that the critical exponents corresponding \nto PM -FM transition for few layered flakes at Tc (β = 0.16, γ = 1.44, and δ = 10.17) are close to the \ncritical exponent values of the phase transition at Tc in bulk CGT, viz., β = 0.14, γ = 1.45, δ = 11.379. \nThus , we find that  the transition at Tc in both bulk and the few layer ed CGT  samples , both obey the 2D \nIsing universality class.   \n \nAn additional feature to note is that in addition to the rise in M near the PM -FM transition at  Tc, there \nis a notable downturn  in M(T) near 20 K  for the CGT flakes . A similar feature can be seen in recent \nM(T) data of 14.4 nm thick epitaxial CGT  film[48], although its origin has not been analyzed or  \ndiscussed . Our studies reveal  that the downturn in M(T) at 20 K is robust and does  not change with \nmoderate H variation (600 Oe to 14 00 Oe). Fig. 3( e) shows M(H) for different T around this regime for 7 \n the few-layer CGT  flake ensemble . A Brillo uin function fit [49] on M(H) data at 2.2 K (see inset of Fig. \n3(e)) gives  𝜇𝑒𝑓𝑓 = 3.44 𝜇𝐵 which is close to the bulk value of 3.18  𝜇𝐵. Following MAP analysis , using \n𝛽 = 0.15 and 𝛾 = 0.20 a plot of 𝑀1/𝛽versus (𝐻\n𝑀)1/𝛾\n (Fig. 3( f)) gives a set of parallel straight lines  with \nintercepts on either  the x-axis (for 𝑇 > 𝑇𝑐) or  the y-axis (for 𝑇 < 𝑇𝑐). The isotherm for 14.2 K passes \nthrough the origin, thereby identifying a new low T critical transition temperature ( 𝑇𝑐′) for the CGT \nflake ensemble. The y and x-axis intercepts  in MAP  give 𝑀𝑠 (𝑇) and 𝜒−1(T) for 𝑇 < 𝑇𝑐′ and 𝑇> 𝑇𝑐′, \nrespectively  (see Fig. 3(g)). Fits with, 𝑀𝑠(𝑇) ~ 𝑡𝛽 (for T ≤14 K) gives 𝛽 = 0.21±0.01 and  𝑇𝑐′ = \n(14.29±0.09) K and 𝜒−1(𝑇) ~ 𝑡𝛾 (for T > 15 K) gives 𝛾 = 0.18 ±0.01 and  𝑇𝑐′= (12.84±0.61) K. \nThese values of 𝛽 and 𝛾 are consistent with those in Fig. 3(f). Using  𝛽= 0.15 and 𝛾 = 0.20, Fig . 3(h) \nshows a bifurcation of the scaled M(H) isotherms onto two distinct family of curves which suggests the \npresence of a critical transition at 𝑇𝑐′ = (14.0 ± 1.0) K. The low T critical transition window is narrowed \ndown further (Fig. 3( i)) by noting a linear M(H) trend in a log -log plot at 14.2 K , which suggests 𝛿 = \n2.20 (as 𝑀=𝐴𝐻1/𝛿 near the critical point ). This value of 𝛿 agrees well with the value of 2.33 obtained \nfrom Widom’s scaling relation  (discussed earlier) . The above analysis  clearly shows  the emergence of \na new critical regime (see summary in Table I) at 𝑇𝑐′ ~14 K in the few -layer flakes (2 to 3 nm thick) \nensemble of CGT  with a unique set of critical exponents  distinct from those in  the bulk crystal. We find \nnone of the critical exponents for the transition at 𝑇𝑐′ resemble those  for any known  universality classes \n(see values mentioned in Fig.  S6 in SI -6). In Figs.  4(a) and 4(b) , we analyse  the 𝑀𝑠(𝑇) data for H  ab \nfor few-layered flakes of CGT  and bulk CGT at T << 𝑇𝑐′ and T << Tc ,respectively. At low T \n(𝑇\n𝑇𝑐 𝑜𝑟 𝑇𝑐′<0.4), 𝑀𝑠(𝑇) fits well with the Bloch spin wave [50] model with a gapped ( Δ) magnon \nspectrum (~ 𝑇3\n2𝑓3\n2(𝛥\n𝑘𝐵𝑇)) where 𝑓3\n2(𝑦) is the Bose -Einstein integral function (see SI -9 for detail ed \ndiscussions ). From the fit, we get an anisotropy -induced  gap 𝛥′ = (0.14 ± 0.03) meV in the magnon \nspectra for few-layered flake ensemble compared to Δ = (1.38 ± 0.27) meV for bulk CGT.  With reduced \ndimensionality t he reduction  in the gap of the magnon spectra strongly suggests  a weakening of \nmagnetic anisotropy at low dimension . It is not surprising that the ratio of the magnon gaps ( 𝛥′/𝛥) is \nsimilar to that of 𝑇𝑐′/𝑇𝑐~ 0.2.  8 \n Table I : Comparison of critical parameters for bulk CGT crystal and CGT flake ensemble  \nSample  𝝁𝒆𝒇𝒇 𝜷 𝜷 \n(Ref[33,34]) 𝜸 𝜸 \n(Ref.[33,34]) 𝜹 𝜹 \n(Ref.[33,34]) 𝑻𝒄 𝑻𝒄′  \nCGT bulk \ncrystal  3.18 𝜇𝐵\nper Cr  0.14  \n(Fig.2( b)) \n0.12  \n(Fig.2( d)) \n 0.17 to \n0.2 1.45  \n(Fig.2( b)) \n1.21  \n(Fig.2( d)) 1.28  to \n1.75  11.37  \n(Fig. S7(a)) \n11.35  \n(Widom)  10.87 to \n7.96 64.28  K (Fig.2(a))  \n63.68  K - 66.59 K  \n(Fig.2( d)) \n67 K  \n(Fig.2( b), (c))  \n \n-- \nCGT flake \nensemble  3.44 𝜇𝐵\nper Cr  0.15  \n(Fig.3( f)) \n0.21  \n(Fig.3( g)) -- 0.20  \n(Fig.3( f)) \n0.18  \n(Fig.3( g)) \n -- 2.20  \n(Fig.3( i)) \n2.33   \n(Widom)  \n -- 66.42 K  \n (Fig.3(a))  14.29 K  - 12.84 K  \n(Fig.3( g)) \n14.80 K  \n(Fig.3( h)) \n \n \nTo understand the layer dependence of magnetic properties in CGT , we performed DFT -based first -\nprinciples  electronic structure  calculations (details in SI -8(A) ). We optimized the lattice s tructure  and \natomic positions  for bulk, bilayer , and monolayer  CGT . Minor yet consequential changes in the lattice \nstructure were found at reduced dimensions . The structural details  are given in Table II.  Notably, t he \nin-plane lattice constant shrinks by 0.01 Å from bulk to bilayer  and ~ 0.02 Å from bulk to monolayer. \nThe nearest neighbor  (NN) Cr-Cr distance s decrease marginally in the bilayer and the monolayer \ncompared to the bulk. The Cr -Te-Cr bond angles also decrease marginally . There are small distortions \nin the Te  anti-prisms around the Cr atoms in the bulk so that all NN Te -Te distances in an atomic sub -\nlayer are not equal. The variations in the NN Te-Te distance s in the bulk is ~  0.07 Å, which reduces  to \n0.03 Å in the bilayer. And in  monolayer , these variations disappear entirely so that all NN Te -Te \ndistances are equal. The magnetic moment (𝜇) comes largely from the Cr atoms , and it is 3 𝜇𝐵/Cr in all \nthree cases , in good agreement with experiment s, suggesting 𝜇 is independent of the number of layers.  \n \n 9 \n Table  II: Structural  details  and anisotropy  energies  for bulk,  bilayer  and monolayer  CGT  \nSystem   a(Å) b(Å) Cr-Cr(Å) Cr-Te(Å) Te-Te(Å) ∠Cr-Te-Cr   Interlayer  \n  Distance  \n(Å) MAE  \n(meV/f.u.)  DAE  \n(meV/f.u.)  𝝐𝒂𝒏𝒊 \n(meV/f.u.)  \nBulk  6.892  6.892  3.98 2.78 3.765 \n± 0.035  91.29∘ 3.38 0.3 0.095  0.395  \nBilayer  6.882  6.882  3.97 2.78 3.755 ± \n0.015  90.96∘±0.12∘ 3.41 0.143  -0.022  0.121  \nMonolay\ner 6.875  6.875  3.97 2.79 3.75 90.72∘                    \n- 0.066  -0.021  0.045  \n \nWe also calculate  the MAE  originating  from  SOC  for bulk,  bilayer , and monolayer  CGT.  MAE  is the \ndifference  between  the total energies  of IP and OOP  orientations  of the Cr3+ spins.  A positive  (negative)  \nvalue  of anisotropy  energy  indicates  OOP  (IP) anisotropy.  Net anisotropy  energy  (𝜖ani) is the sum of \nMAE  and DAE,  the latter  arising  due to long-range  magnetostatic  interactions  between  the magnetic  \ndipole  moments  on the Cr3+ ions.  DAE  also depends  on the orientation  of the moments  (see SI-8(B) for \ndetails  of DAE  calculation).  Results  for MAE , DAE , and 𝜖ani  for each system  are shown  in Table  \nII.  The positive  sign for MAE  shows  a preference  for OOP  orientation  of the Cr3+ moments  from  bulk \nto monolayer  CGT . However,  table  II also shows  that the magnitude  of MAE  decreases  drastically  \n(~75%)  from  the bulk to the monolayer . In CGT , the contributions  from  the highest  atomic  number  Te \natoms  dominate  the SOC  strength . To understand  the variation  in the contributions  from  the Te atoms  \ndue to dimensional  reduction,  the total density  of states  (DOS)  and its projections  onto each atom  \n(partial  DOS)  are shown  in Fig. 4(c), (d), and (e).  States  near the valence  band  maximum  have  \nsignificant  contributions  from  the Te s and p states  in the bulk.  However,  in the bilayer  and monolayer,  \nthese  contribution s are gradually  suppressed.  This suppression  of contribution  from  Te states  reduces  \nthe SOC,  and hence  MAE  reduces  significantly  in low-dimensional  CGT.  The reduction  of MAE  should  \nalter the spin gap in low dimensional  CGT  (compared  to the bulk),  as evidence d in Fig. 4(a). \n 10 \n In bulk, MAE and DAE cooperatively support OOP orientation  of Cr3+ moments . This is seen in Fig. 2 \nand is also consistent with earlier results [1]. However,  we see  DAE chang ing sign for bilayer and \nmonolayer cases, favoring IP anisotropy. With reducing dimensionality MAE  values  reduce , but it  still \nfavours OOP orientation. Table II reveals this emergent competition  between MAE and DAE  in CGT , \nresulting in the decrease of 𝜖𝑎𝑛𝑖 with reduc ing dimension ality. The above features are described by the \nHamiltonian , ℋ=ℋ𝑖𝑛𝑡,𝑧+ℋ′𝑥𝑦, where  ℋ𝑖𝑛𝑡,𝑧=−𝐽𝑧∑\n⟨𝑖𝑗⟩𝑠𝑧𝑖𝑠𝑧𝑗 is the short -range NN (i and j are site \nindex ) Ising Hamiltonian favoring  OOP  FM ordering for 𝐽𝑧>0. The ℋ′𝑥𝑦=(−𝐽𝑑′∑\n⟨𝑖𝑗⟩[𝑠𝑥𝑖𝑠𝑥𝑗+𝑠𝑦𝑖𝑠𝑦𝑗] +\nℋ𝑑𝑖𝑝) term describes interactions between the in-plane  spin components with 𝐽𝑑′ representing a \ndimension ( d)-dependent  exchange interaction  and ℋ𝑑𝑖𝑝 is the dipolar interaction term  which is also d-\ndependent  (representing the DAE contribution ). In bulk CGT  (d = 3) we consider , 𝐽𝑧≫𝐽𝑑′ and ℋ𝑖𝑛𝑡,𝑧≫\nℋ′𝑥𝑦,ℋ𝑑𝑖𝑝. Thus ℋ𝑖𝑛𝑡,𝑧 is predominantly responsible for OOP critical fluctuations leading to 2D Ising \nuniversality class at the critical transition Tc in bulk CGT.  Our DFT results show  structural distortions \nincrease with reduced dimensionality which decrease SOC and hence MAE , i.e., ℋ𝑖𝑛𝑡,𝑧 weakens (Table \nII) [18,19,51]  and concomitant ly DAE  increases . Theoretical s tudies based on exchange interaction \nmodels [52] of the type ℋ=(ℋ𝑖𝑛𝑡,𝑧+ℋ′𝑥𝑦), suggest s the emergenc e of a new critical crossover \nregime associated with enhanced fluctuations along in-plane x and y directions  (due to ℋ′𝑥𝑦 \ncontribution).  A critical crossover separates two regimes with very different universality class es[53]. In \nour experiments,  a similar critical crossover emerge s at 𝑇𝑐′, below which emerges a unique critical \nscaling  behavior . At low T (<𝑇𝑐′), low dimensional CGT enters a magnetically inhomogeneous \nconfiguration due to the competition between OOP (ℋ𝑖𝑛𝑡,𝑧, MAE ) and IP (ℋ′𝑥𝑦, DAE) anisotrop ies. \nAs net M contribution along the measuring direction ( H  ab) decrease s, there is a turnaround at 𝑇𝑐′ \n(Fig. 3(a)). The strengthening of DAE suggests the emergence of long-range dipolar interactions which \ncoexists along with short range interactions  at low d. Due to vdW stacking of 2D layers, n aively one \nmay expect a preservation of the universality class in transforming from bulk to 2D - CGT layers. \nHowever, this expectation clearly breaks down, as we see the emergen t new critical crossover regime 11 \n at 𝑇𝑐′ in low dimension CGT . Here the combined presence of s hort- and long -range  interactions in 2D -\nCGT  results in critical scaling properties deviat ing significantly from known universality classes .  \n \nTo summarize  our findings,  our experiments  confirm  that the PM - FM phase  transition  in bulk CGT  at \nTc (~ 67 K) obeys  the 2D Ising  universality  class.  However,  in the few-layered  CGT  flake,  in addition  \nto the usual  2D Ising  like PM-FM transition  at Tc, we report  the presence  of another  critical  transition  \nat a much  lower  T (viz.,  𝑇𝑐′ ~ 14 K). The transition  at 𝑇𝑐′ ~ 14 K is characterized  by a new set of critical  \nexponents  (Fig.  3(f)-(i)) which  are very different  from  the 2D Ising  universality  class.  This change  in \nthe universality  class  below  𝑇𝑐′, suggests  an alteration  in the nature  and range  of interactions  in few-\nlayer  thick  CGT  as compared  to the bulk samples.  From  the gapped  spin wave  analysis  of 𝑀𝑠(𝑇), we \nfound  out that the magnetic  anisotropy  reduces  by an order  of magnitude  with thinning  down  of CGT  \ndown  to a few-layers  from  the bulk (Fig.  4(a) and (b)). Our DFT  based  electronic  structure  calculations  \nreveal  that with reduced  dimensionality  (i.e., thickness  of CGT),  there  are subtle  structural  \nreorganisation  occurring  in the system,  which  affect s SOC  which  in turn modifies  the MAE. Our \ncalculations  show  that MAE  decreases  significantly  at reduced  CGT  dimensionality  (without  changing  \nits sign)  while  the DAE  changes  significantly  compared  to its value  in bulk CGT.  In few layer  thick  \nCGT  the DAE  changes  sign,  suggesting  that the IP magnetization  orientation  at lower  dimensions  is \nfavoured  compared  to the OOP  orientation.  It may be noted  that these  calculations  are done  effectively  \nat T = 0 K, i.e., not considering  any thermal  fluctuation  effects.  While  in bulk CGT,  MAE  and DAE  \nboth co-operatively  favour  OOP  orientation  of moments  at 𝑇𝑐, however  at lower  dimensionality  there  \nis a preference  for IP orientation  of moments,  resulting  from  the competition  between  MAE  and DAE  \nat 𝑇𝑐′. Thus  reduced  dimensionality  in CGT  gives  rise to a competition  between  two anisotropy  energy  \nscales  at a low T. This competition  isn’t strong  near Tc, as DAE  is weak  near Tc because  of thermal  \nfluctuation  of moments  near this critical  regime  around  Tc. Hence,  just below  Tc, the spins  collectively  \norder  into a FM state obeying  the 2D Ising  universality  class,  which  is predominantly  governed  by the \nshort -range  exchange  interaction  and MAE,  in both bulk and few-layer  samples.  Therefore,  both bulk \nand few layer  CGT  samples  show  identical  2D-Ising  like universality  class  near Tc. However,  with 12 \n further  lowering  of T below  Tc, the effects  of DAE  strengthens  as the thermal  fluctuation  effects  weaken.  \nEventually  at 𝑇≤𝑇𝑐′ (~ 14 K), there  appears  the competition  between  MAE  and DAE  (with  sign \nchange)  (see Table  II).  Consequently,  there  is a crossover  from  short  range  interaction  dominated  \nregime  near 𝑇𝑐 into a regime  with both,  short - and long-range  interactions  active  below  𝑇𝑐′. In this low \nT-regime , in the few-layer  thick  CGT  sample,  there  is a critical  crossover  from  a 2D Ising  like state into \na regime  characterized  by a new universality  class.  This regime  characterised  by MAE  and DAE  (sign  \nchange)  competition,  could  possibly  be a magnetically  inhomogeneous  state with an admixture  of OOP  \nand IP orientation  of moments.  It is also possible  that weak  structural  modifications  (which  affect  MAE)  \nmay also become  significant  at low T (well  below  Tc) in the low dimensional  CGT.  This may in turn be \nan additional  factor  aiding  the competition  between  MAE  and DAE  playing  up at 𝑇𝑐′ in few layer  thick  \nCGT.  Furthermore  below  𝑇𝑐′  as long-range  dipolar  interaction  effects  are active , hence  in low \ndimensional  CGT  sample,  local  fluctuations  may begin  to affect  the ordering  at larger  length  scales.  A \nstudy  of such rich emergent  features,  new universality  classes  and similar  exploration  in other  low \ndimensional  2D-vdW  magnetic  systems,  calls for further  detailed  experimental  and theoretical  \ninvestigations.     \nAcknowledgement:  The authors  acknowledge  Advanced  Imaging  Centre,  IIT Kanpur  for \nproviding  HR TEM  facility.  S. S. Banerjee  thanks  the financial  support  from  DST -SUPRA  and DST  - \nAMT  of GOI India  programs  as well as for research  infrastructure  and funding  from  IIT Kanpur.  All \ncomputations  have  been  done  on the HPC  cluster  facility  at HRI (https://www.hri.res.in/cluster/ ). \nSuprotim  Saha  ackn owledges  the Prime  Minister's  Research  Fellows  (PMRF)  Scheme  of the Ministry  \nof Human  Resource  Development,  Govt.  of India,  for funding  support.  SD acknowledges  the financial  \nsupport  from  DST -SERB  (grant  no. CRG/2021/004334)  and TRC,  IACS.  BD is grateful  to IACS  for \nthe fellowship .  \n \nReferences  \n[1] C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao, W. Bao, C. Wang, Y. Wang  et \nal., Discovery of intrinsic ferromagnetism in two -dimensional van der Waals crystals,  Nature \n546, 265 (2017).  13 \n [2] B. Huang, G. Clark, E. Navarro -Moratalla, D.R. Klein, R. Cheng, K.L. Seyler, D. \nZhong, E. Schmidgall, M.A. McGuire, D.H. Cobden  et al. , Layer -dependent ferromagnetism \nin a van der Waals crystal down to the monolayer limit, Nature 546, 270 (2017).  \n[3] S. Wei, X. Liao, C. Wang, J. Li, H. Zhang, Y. -J. Zeng, J. Linghu, H. Jin, and Y. Wei, \nEmerging intrinsic magnetism in two -dimensional materials: theory and applications, 2D \nMaterials 8, 012005 (2021).  \n[4] K. F. Mak, J. Shan, and D. C. Ralph, Probing and co ntrolling magnetic states in 2D \nlayered magnetic materials, Nature Reviews Physics 1, 646 (2019).  \n[5] Y.F. Li, W. Wang, W. Guo, C.Y. Gu, H.Y. Sun, L. He, J. Zhou, Z.B. Gu, Y.F. Nie, \nX.Q. Pan, Electronic structure of ferromagnetic semiconductor CrGeTe 3 by angle-resolved \nphotoemission spectroscopy, Physical Review B 98, 125127 (2018).  \n[6] A. E. Llacsahuanga Allcca, X. -C. Pan, I. Miotkowski, K. Tanigaki, and Y. P. Chen, \nGate -Tunable Anomalous Hall Effect in Stacked van der Waals Ferromagnetic Insulator –\nTopolog ical Insulator Heterostructures, Nano Letters 22, 8130 (2022).  \n[7] C. Tan, W. -Q. Xie, G. Zheng, N. Aloufi, S. Albarakati, M. Algarni, J. Li, J. Partridge, \nD. Culcer, X. Wang  et al. , Gate -Controlled Magnetic Phase Transition in a van der Waals \nMagnet Fe 5GeTe2, Nano Letters 21, 5599 (2021).  \n[8] Z. Wang, T. Zhang, M. Ding, B. Dong, Y. Li, M. Chen, X. Li, J. Huang, H. Wang, X. \nZhao  et al. , Electric -field control of magnetism in a few -layered van der Waals ferromagnetic \nsemiconductor, Nature Nanotechnology 13, 554 (2018).  \n[9] W. Zhuo, B. Lei, S. Wu, F. Yu, C. Zhu, J. Cui, Z. Sun, D. Ma, M. Shi, H. Wang  et al. , \nManipulating Ferromagnetism in Few -Layered Cr 2Ge2Te6, Advanced Materials 33, 2008586 \n(2021).  \n[10] W. Zhu, C. Song, L. Han, T. Guo, H. Bai, and F. Pan, Van der Waals lattice -induced \ncolossal magnetoresistance in Cr 2Ge2Te6 thin flakes, Nature Communications 13, 6428 (2022).  \n[11] V.P. Ningrum, B. Liu, W. Wang, Y. Yin, Y. Cao, C. Zha, H. Xie, X. Jiang, Y. Sun, S. \nQin et al. , Recent Advances in Two -Dimensional Magnets: Physics and Devices towards \nSpintronic Applications , Research 2020 . \n[12] E. C. Ahn, 2D materials for spintronic devices, npj 2D Materials and Applications 4, \n17 (2020).  \n[13] P. Jiang, C. Wang, D. Chen, Z. Zhong, Z. Yuan, Z. -Y. Lu, and W. Ji, Stacking tunable \ninterlayer magnetism in bilayer CrI 3, Physical Review B 99, 144401 (2019).  \n[14] T. Li, S. Jiang, N. Sivadas, Z. Wang, Y. Xu, D. Weber, J.E. Goldberger, K. Watanabe, \nT. Taniguchi, C.J. Fennie  et al. , Pressure -controlled interlayer magnetism  in atomically thin \nCrI 3, Nature Materials 18, 1303 (2019).  \n[15] T. Song, Z. Fei, M. Yankowitz, Z. Lin, Q. Jiang, K. Hwangbo, Q. Zhang, B. Sun, T. \nTaniguchi, K. Watanabe  et al. , Switching 2D magnetic states via pressure tuning of layer \nstacking , Nature Mat erials 18, 1298 (2019).  \n[16] M. Tang, J. Huang, F. Qin, K. Zhai, T. Ideue, Z. Li, F. Meng, A. Nie, L. Wu, X. Bi  et \nal., Continuous manipulation of magnetic anisotropy in a van der Waals ferromagnet via \nelectrical gating , Nature Electronics 6, 28 (2023).  \n[17] A. O’Neill, S. Rahman, Z. Zhang, P. Schoenherr, T. Yildirim, B. Gu, G. Su, Y. Lu, and \nJ. Seidel, Enhanced Room Temperature Ferromagnetism in Highly Strained 2D Semiconductor \nCr2Ge2Te6, ACS Nano 17, 735 (2023).  \n[18] X. Chen, J. Qi, and D. Shi, Strain -engineering of magnetic coupling in two -dimensional \nmagnetic semiconductor CrSiTe 3: Competition of direct exchange interaction and \nsuperexchange interaction, Physics Letters A 379, 60 (2015).  \n[19] W.-n. Ren, K. -j. Jin, J. -s. Wang, C. Ge, E. -J. Guo, C. Ma, C. Wang, and X. Xu, Tunable \nelectronic structure and magnetic anisotropy in bilayer ferromagnetic semiconductor \nCr2Ge2Te6, Scientific Reports 11, 2744 (2021).  14 \n [20] N. Wang, H. Tang, M. Shi, H. Zhang, W. Zhuo, D. Liu, F. Meng, L. Ma, J. Ying, L. \nZou et al. , Transition from Ferromagnetic Semiconductor to Ferromagnetic Metal with \nEnhanced Curie Temperature in Cr 2Ge2Te6 via Organic Ion Intercalation,  Journal of the \nAmerican Chemical Society 141, 17166 (2019).  \n[21] Z. Y. Zhu, Y. C. Cheng, and U. Schwinge nschlögl, Giant spin -orbit -induced spin \nsplitting in two -dimensional transition -metal dichalcogenide semiconductors, Physical Review \nB 84, 153402 (2011).  \n[22] M. Mogi, T. Nakajima, V. Ukleev, A. Tsukazaki, R. Yoshimi, M. Kawamura, K.S. \nTakahashi, T. Hanash ima, K. Kakurai, T. -h. Arima  et al. , Large Anomalous Hall Effect in \nTopological Insulators with Proximitized Ferromagnetic Insulators , Physical Review Letters \n123, 016804 (2019).  \n[23] D.R. Klein, D. MacNeill, J.L. Lado, D. Soriano, E. Navarro -Moratalla, K.  Watanabe, \nT. Taniguchi, S. Manni, P. Canfield, J. Fernández -Rossier  et al. , Probing magnetism in 2D van \nder Waals crystalline insulators via electron tunneling , Science 360, 1218 (2018).  \n[24] B. Karpiak, A.W. Cummings, K. Zollner, M. Vila, D. Khokhriakov,  A.M. Hoque, A. \nDankert, P. Svedlindh, J. Fabian, S. Roche  et al. , Magnetic proximity in a van der Waals \nheterostructure of magnetic insulator and graphene , 2D Materials 7, 015026 (2020).  \n[25] S.-L. Li, K. Komatsu, S. Nakaharai, Y. -F. Lin, M. Yamamoto, X. Duan, and K. \nTsukagoshi, Thickness Scaling Effect on Interfacial Barrier and Electrical Contact to Two -\nDimensional MoS 2 Layers , ACS Nano 8, 12836 (2014).  \n[26] Z. X. Gan, L. Z. Liu, H. Y. Wu, Y. L. Hao, Y. Shan, X. L. Wu, and P. K. Chu, Quantum \nconfinement effects across two -dimensional planes in MoS 2 quantum dots, Applied Physics \nLetters 106, 233113 (2015).  \n[27] K. G. Wilson, The renormalization group and critical phenomena, Reviews of Modern \nPhysics 55, 583 (1983).  \n[28] L.P. Kadanoff, W. GÖTze, D. Hamblen,  R. Hecht, E.A.S. Lewis, V.V. Palciauskas, M. \nRayl, J. Swift, D. Aspnes, J. Kane , Static Phenomena Near Critical Points: Theory and \nExperiment, Reviews of Modern Physics 39, 395 (1967).  \n[29] H. E. Stanley, Scaling, universality, and renormalization: Three pillars of modern \ncritical phenomena, Reviews of Modern Physics 71, S358 (1999).  \n[30] M. Suzuki, B. Gao, G. Shibata, S. Sakamoto, Y. Nonaka, K. Ikeda, Z. Chi, Y.X. Wan, \nT. Takeda, Y. Takeda  et al. , Magnetic anisotropy of the van der Waals ferromagnet Cr 2Ge2Te6 \nstudied by angular -dependent x -ray magnetic circular dichroism, Physical Review Research 4, \n013139 (2022).  \n[31] B. H. Zhang, Y. S. Hou, Z. Wang, and R. Q. Wu, First -principles studies of spin -phonon \ncoupling in monolayer Cr 2Ge2Te6, Physical Review B 100, 224427 (2019).  \n[32] Y. Fang, S. Wu, Z. -Z. Zhu, and G. -Y. Guo, Large magneto -optical effects and magnetic \nanisotropy energy in two -dimensional Cr 2Ge2Te6, Physical Review B 98, 125416 (2018).  \n[33] Y. Liu and C. Petrovic, Critical behavior of quasi -two-dimensional semiconducting \nferromagnet Cr 2Ge2Te6, Physical Review B 96, 054406 (2017).  \n[34] W. Liu, Y. Dai, Y. -E. Yang, J. Fan, L. Pi, L. Zhang, and Y. Zhang, Critical behavior of \nthe single -crystalline van der Waals bonded ferromagnet Cr 2Ge2Te6, Physical R eview B 98, \n214420 (2018).  \n[35] G.T. Lin, H.L. Zhuang, X. Luo, B.J. Liu, F.C. Chen, J. Yan, Y. Sun, J. Zhou, W.J. Lu, \nP. Tong  et al. , Tricritical behavior of the two -dimensional intrinsically ferromagnetic \nsemiconductor CrGeTe 3, Physical Review B 95, 24521 2 (2017).  \n[36] Z. Li, T. Cao, and S. G. Louie, Two -dimensional ferromagnetism in few -layer van der \nWaals crystals: Renormalized spin -wave theory and calculations, Journal of Magnetism and \nMagnetic Materials 463, 28 (2018).  15 \n [37] A. Arrott, Criterion for Fer romagnetism from Observations of Magnetic Isotherms, \nPhysical Review 108, 1394 (1957).  \n[38] A. Arrott and J. E. Noakes, Approximate Equation of State For Nickel Near its Critical \nTemperature, Physical Review Letters 19, 786 (1967).  \n[39] B. Das, S. Ghosh, S . Sengupta, P. Auban -Senzier, M. Monteverde, T.K. Dalui, T. \nKundu, R.A. Saha, S. Maity, R. Paramanik  et al. , Emergence of a Non -Van der Waals Magnetic \nPhase in a Van der Waals Ferromagnet , Small 19, e2302240 (2023).  \n[40] Y. Tian, M. J. Gray, H. Ji, R. J. Cava, and K. S. Burch, Magneto -elastic coupling in a \npotential ferromagnetic 2D atomic crystal, 2D Materials 3, 025035 (2016).  \n[41] A. Milosavljević, A. Šolajić, J. Pešić, Y. Liu, C. Petrovic, N. Lazarević, and Z. V. \nPopov ić, Evidence of spin -phonon coupling in CrSiTe 3, Physical Review B 98, 104306 (2018).  \n[42] S. Khan, C. W. Zollitsch, D. M. Arroo, H. Cheng, I. Verzhbitskiy, A. Sud, Y. P. Feng, \nG. Eda, and H. Kurebayashi, Spin dynamics study in layered van der Waals single -crystal \nCr2Ge2Te6, Physical Review B 100, 134437 (2019).  \n[43] X. Zhang, Y. Zhao, Q. Song, S. Jia, J. Shi, and W. Han, Magnetic anisotropy of the \nsingle -crystalline ferromagnetic insulator Cr 2Ge2Te6, Japanese Journal of Applied Physics 55, \n033001 (20 16). \n[44] D. J. Sivananda, A. Kumar, M. Arif Ali, S. S. Banerjee, P. Das, J. Müller, and Z. Fisk, \nMagneto -optical imaging of stepwise magnetic domain disintegration at characteristic \ntemperatures in EuB 6, Physical Review Materials 2, 113404 (2018).  \n[45] A. K. Pramanik and A. Banerjee, Critical behavior at paramagnetic to ferromagnetic \nphase transition in Pr 0.5Sr0.5MnO 3: A bulk magnetization study, Physical Review B 79, 214426 \n(2009).  \n[46] B. Widom, Surface Tension and Molecular Correlations near the Critica l Point, The \nJournal of Chemical Physics 43, 3892 (2004).  \n[47] L. P. Kadanoff, Scaling laws for ising models near T C, Physics Physique Fizika 2, 263 \n(1966).  \n[48] A. Giri, C. De, M. Kumar, M. Pal, H. H. Lee, J. S. Kim, S. -W. Cheong, and U. Jeong, \nLarge -Area  Epitaxial Film Growth of van der Waals Ferromagnetic Ternary Chalcogenides, \nAdvanced Materials 33, 2103609 (2021).  \n[49] N. W. Ashcroft and N. D. Mermin, Solid state physics (Cengage Learning, 2022).  \n[50] Z. Shu and T. Kong, Spin stiffness of chromium -based van der Waals ferromagnets, \nJournal of Physics: Condensed Matter 33, 195803 (2021).  \n[51] P. Fazekas, Lecture notes on electron correlation and magnetism (World scientific, \n1999), Vol. 5.  \n[52] M. E. Fisher, The renormalization group in the theory of criti cal behavior,  Reviews of \nModern Physics 46, 597 (1974).  \n[53] A. Franchi, D. Rossini, and E. Vicari, Critical crossover phenomena driven by \nsymmetry -breaking defects at quantum transitions, Physical Review E 105, 034139 (2022).  \n \n \n \n 16 \n FIG. 1 (a) Raman  spectrum  of bulk single  crystal  of CGT  and 2 nm thick  CGT  flake.  (inset)  \nTop view  of simulated  atomic  model  of CGT.  (b) Atomic  resolution  HR-TEM  image  of a few-\nlayered  CGT  flake,  viewed  from  [001]  direction.  (c) Z-contrast  intensity  profil e of the red-\nmarked  region  in Fig. 1(b).  Compare  the profile  to the similar  red-marked  region  in Fig. 1(a) \ninset.  (d) SAED  spot pattern  of a few-layer  CGT  flake.  (e) AFM  map at three  different  regions  \nof area 10 μm × 3 μm showing  multiple  CGT  flakes.  Note  the adjoining  colour  bar denoting  \nthickness,  indicating  uniformity  of thickness  of the flakes  (cf. text for details).  \n \n(d)\n100 150 200 250 3000123\n136 cm-1116 cm-1Intensity (arb.units)\nWave number (cm-1) bulk crystal\n few-layer flake\nCr\nTe\nGe(a) (b)\n(c)\n(e)\n[0 0 1]17 \n  \n \nFIG. 2 (a) Field  cooled  M(T) for bulk CGT  for H || ab and H ⊥ ab at H = 200 Oe. (inset)  Curie -\nWeiss  fit to 𝜒−1(T) for H || ab. (b) Scaled  magnetization  versus  scaled  field in the critical  regime  \nbifurcating  above  and below  𝑇𝑐 = 67 K. (inset)  M(H) for bulk CGT  at 4 K for H || ab and H \n⊥ ab. (c) MAP  in the critical  regime  with 𝛽 = 0.14 and 𝛾 =  1.45.  (d) 𝑀𝑠(T) (left axis) and \n𝜒−1(T) (right  axis)  deduced  from  intercepts  on y-axis (T < Tc) and x-axis (T>T c) ) respectively  \nof Fig. 1(c) along  with fits through  data (cf. text for details).  \n \n0 50 100 150 200 2500.00.10.20.3 H ab\n H  abM (B/f.u)\nT (K)H = 200 Oe\n1011021031043040506070\nT > TCM/t(emu/g)\nH/t() (kOe) 40 K  \n 58 K  \n 60 K  \n 62 K  \n 63 K  \n 65 K  \n 66 K  \n 68 K       \n 69K  \n 70 K    \n 71K  \n 72 K    \n 73 K  \n 74 KT < TC = 0.14\n = 1.45\nTC= 67 K\n40 50 60 7020253035\nT (K)MS (emu/g)\n246810\nTC = (66.59\n 0.27) K = 0.12 \n 0.01 = 1.21 \n 0.34TC = (63.68\n 1.99) K\n-1 ~ t\nMS ~ t\n-1 (Oe.g.emu-1)\nModelinverse_susceptibility_power_law (User)\nEquation A*((x-Tc)/Tc)^g\nReduced \nChi-Sqr0.03818\nAdj. R-Square 0.99332\nValue Standard Error\nChi_inverseA 92.88122 34.13111\nTc 63.68901 1.99748\ng 1.21789 0.34425(a) (b)\n(c)(d)\n0 20 40 60 800246810\neff = 3.18 B/ CrTC = 64.28 KH = 200 OeH  abH/M 102 (Oe.g.emu-1)\nT (K)\n0 40 80 120051015\n67 K\n75 K58 K\n = 1.45 = 0.14M1/ 1010(emu/g)1/\n(H/M)1/ (Oe.g.emu-1)1/\n-40 -20 0 20 40-4-2024\nM (B/f.u)\nH (kOe) H  ab\n H abT = 4 K18 \n    \nFIG. 3 (a) M(T) for few layer  flake -ensemble  of CGT  for out of plane  field 600 Oe and 1400  \nOe. (inset)  Curie -Weiss  fit to 𝜒−1(T) for H = 600 Oe. (b) M(T) for 600 Oe fitted with power  \nlaw near 𝑇𝑐. (c) Scaled  magnetization  versus  scaled  field showing  bifurcation  above  and \nbelow  𝑇𝑐 with 𝛽 = 0.16,  𝛾 =  1.44 and 𝑇𝑐 = 65 K. (d) M(H) isotherm  in log-log scale  at 70 K to \ndetermine  δ.  (e) Background -corrected  representative  M(H) data of CGT  flake -ensemble  from  \nT = 2.2 K to 20.2 K. (inset)  Brillouin  fit to M(H) at 2.2 K. (f) MAP  for isotherms  in the critical  \nregime  (T ~ 𝑇𝑐′) with β = 0.15 and 𝛾 = 0.20.  (g) MS(T) (left axis)  and 𝜒−1(T) (right  axis)  with \nrespective  power  law fits in T < 14 K and T >15 K respectively.  (h) Scaled  magnetization  versus \nscaled  field showing  bifurcation  above  and below  𝑇𝑐′ = 14.8 K. (i) M(H) isotherm  in log-log \nscale  at 14.2 K to determine  δ (cf. text for details).  \n19 \n  \n \n \n \n \nFIG. 4 (a) 𝑀𝑠(T) for few -layered CGT flakes at T < 5.2 K for H ⊥ ab with fit to gapped spin \nwave dispersion. (b) 𝑀𝑠(T)  for CGT bulk crystal at T < 32 K for H ⊥ ab with fit to gapped \nspin wave dispersion. (c) -(e) Projected DOS of CGT in (c) bulk, (d) bilayer and (e) monolayer \nform (cf. text for details).  \n \n(c)\n(d)\n(e)\n0 10 20 303.583.603.623.643.663.68\nH ab\n  Experimental data\n  gapped spin wave fit  (T3/2f3/2)MS 10-2 (emu)\nT (K)Bulk CGT\n = (1.38 \n0.27) meV\n2 3 4 51.751.801.851.901.95\n Experimental data\n gapped spin wave fit (T3/2f3/2)MS 10-5 (emu)\nT (K)Few layered CGT\n = (0.14 \n  0.03 ) meVH ab(a)\n(b)Supplementary Information  \nSI-1: Details of experimental methods   \nSample growth and characterization  of bulk crystal  \nSingle crystals of CGT were grown by self -flux technique using pure constituent elements Cr, Ge and \nTe powders mixed with molar ratio 1:2:6 in an evacuated quartz tube. The mixture was heated from \nroom temperature to 13 23 K at a heating rate of 100 K/hour and kept at that temperature for 24 hours \nfollowed by a slow cooling . Crystals thus obtained are typically mm -sized, exfoliable and had a shiny \nplate -like appearance. X ray Diffrcation ( XRD ) was performed on a crystal of size 2.5 mm x 2.3 mm x \n0.24 mm using a Panalytical X -ray diffractometer with Cu -Kα radiation (λ=1.5406Å) at room \ntemperature in the 2 𝜃 range of (10º–90º) with a precision of 0.05º . CGT single crystals were \ncharacterized by pre -calibrated Raman spectroscopy (Princeto n Instruments Acton Spectra Pro 2500i  ) \nwith a 785 nm LASER incident from (001) direction in ambient condition. A laser power of 2 mW with \na spot -size of ~10 μm was used with a grating of 1200 rulings per mm.  Proper care was taken to limit \nthe LASER power  to ensure no heating/burning effect on the crystals. Raman spectra taken at different \nregions of the bulk single crystal and from various flakes of similar thickness (2-3 nm) yield identical \nresults. Note that the single crystals used for Raman characterization are exposed to air, which has an \neffect of suppressing the peaks at higher wave numbers ( 𝐸𝑔3,𝐸𝑔4,𝐴𝑔2)[1]. Also note the enhancemen t \nin Raman intensity for the nanoflakes compared to that of the bulk [1]. In order to confirm homogeneity \nin chemical composition, Energy dispersive X ray s pectroscopy ( EDX) is measured in a JEOL JSM -\n6010LA Scanning Electron Microscope  (SEM)  using a 20 kV electron beam . High resolution \nTransmission electron microscopy ( HR-TEM) and Selected Area Electron Diffraction (SAED) of the \nCGT single crystals were perfo rmed at ambient temperature  using a FEI Titan G2 60 -300 with a 300 \nkV electr on beam from (001) direction for structural investigation . CGT single crystals were exfoliated \nin a glove -box maintained in inert Argon atmosphere  and fine crystal pieces were floated in xylene and \nthis diluted solution containing CGT flakes was subsequently drop -cast on a carbon coated copper grid \nfor TEM and SAED. The interplanar spacings (dhkl) were calculated by measuring the distances between \nthe central diffraction spot and the diffracted brigh t spots using ImageJ software [2] and individual bright \nspots  were indexed to ( h k l) planes using the calculated interplanar spacings by Vesta  [3] as reference .  \nPreparation of few layered samples and  thickness characterization  \nCrystals were exfoliated by adhesive tape in a glove -box maintained in inert Argon atmosphere and thin \nflakes carefully examined by optical contrast were transferred on single layer hBN/SiO 2 (285 nm) /Si \n(500 μm)  substrate (purchased from Graphen e supermarket ) using PDMS stamp for magnetic \nmeasurement . Thickness  of the transferred flakes were determined by ac -tapping mode Atomic Force \nMicroscopy ( AFM ) measurement using a Al(100) coated Si probe -tip with typical frequency ~70  kHz \nand spring constant 2 N/m in an AC240TS -R3 model Asylum Research AFM . Considering the softness of the flakes, contact mode measurement was avoided. Note that AFM measurements often have a \ntendency to overestimate the thickness of flakes owing to the t rapping of air and moisture at the flake -\nsubstrate interface.  \nMagnetic measurements:  \nAll magnetic measurements were performed in a commercial S700X Superconducting Quantum \nInterference Device (SQUID) magnetometer (CRYOGENIC, UK) allowing a dc sensitivity of 10-8 emu \nin terms of measurement of magnetic moment. Highly homogeneous magnetic field from a \nsuperconducting magnet ( 𝑇𝑐 ~ 9 K) is used. The interior of the cryostat (and the sample space) is \nscreened from external magnetic field (including terrestrial magnetic field) by a mu -metal shield. \nSamples (both bulk and few -layered flakes) were loaded in a non -magnetic capsule inside a glove box \nwith the desired crystallographic orientation with respe ct to the magnetic field ( H || ab or H ⊥ ab) and \nmounted in the cryochamber with as little delay as possible (typically 30 -60 s). Each measurement of \nmagnetic moment is determined by averaging three scans in same thermomagnetic condition. The non-\nhysteretic diamagnetic contribution in M-H (linear trend in  second and fourth quadrant) due to the \nhBN/SiO 2/Si substrate (and straw holder) was subtracted from the raw data and this background \ncorrected data was eventually used for analysis. For both bulk and few -layered cases, sample was \nwarmed to room temperatur e and demagnetized, between measuring each M(H) isotherm.  \n \nSI-2: Structural information  and chemical composition  \nCr2Ge2Te6 is known to crystallize in a rhombohedral 𝑅3̅ structure [4]. Bulk Cr 2Ge2Te6 forms layered \ncrystals with an  interlayer vdW gap of 6.81 Å (Fig. S 2(a)). With a we ak vdW  force acting between \nadjacent layers, the material is easily cleav able down to monolayer . A view of the monolayer along      \n[0 0 1 ] shows ( inset of Fig. 1( a) in MS) each Cr atom is octahedral ly coordinated to six Te atom s, also \nnotice the hexagonal sublattice formed by the Cr atoms.  A typical single cryst al of dimensions 2.5 mm \nx 2.3 mm x 0.24 mm is shown in the inset of F ig. S2(b ). The XRD 2θ scan (Fig. S2(b )) shows (0 0 l) \npeaks, implying stacking of layer takes place along (0 0 1) direction in CGT single crystal. Analysis of \nour XRD pattern gives the lattice parameter c = 20.56(7)  Å, in good agreement with earlier report [4,5] . \nThe EDX elemental map of Cr, Ge and Te in a 15 μm × 12 μm area (F ig. S2(c)) shows the chemical \nhomogeneity in our CGT sample . The elemental stoichiometry and chemical ho mogeneity is measured \nfrom individual point EDX measurement (Fig. S 2(d)), which yields an atomic percentage ratio of 19.11 : 17.71 : 63.18, for Cr, Ge, and Te respectively, in good agreement with expected stoichiometry 2 : 2 : \n6.  \n \nFIG. S2(a) Simulated crystal structure of CGT, viewed from a -axis, black rectangle denoting  the unit \ncell. Note the vd W gap between adjacent planes. (b) XRD 2 𝜃 scan on CGT single crystal, (inset) a CGT \nsingle crystal on MgO substrate . (c) SEM -EDX elemental map of a CGT single crystal  (d) Individual \npoint EDX spectrum of CGT confirming elemental stoichiometry 2:2:6  (e) Real space TEM image of a \nfew layered CGT fl ake, viewed from [001] direction  (f) Atomic resolution HR -TEM image of few -\nlayered CGT crystal viewed from [001] axis. Yellow parallel lines m ark a set of planes (2 -1 0). (g ) \nCorresponding SAED spot pattern, several individual spots are marked with respec tive Miller indices, \nincluding (2 -1 0). (h) Raman spectroscopy measurement of  single layer hBN . \n \nHRTEM with a 300 kV e -beam from [0 0 1] direction on a few -layered CGT crystal shows perfect \ncrystalline nature with no hint of dislocations and grain boundaries within the imaged area  (Fig. S 2(e) \nand S2(f) ). Six-fold symmetry in the SAED dot pattern  (Fig. S 2(g) and Fig. 1(d) in MS ) taken along [0 \n0 1] zone axis confirms the single -crystalline nature, indicating absence of grain boundaries or rotational \nboundaries. The interplanar spacing for the yellow -marked  parallel set of  planes in Fig. S2(f) was \nmeasured  from the HRTEM image to be 3.3 Å , which is identified as the (2 -1 0) plane for CGT [6] \nfrom calculated powder XRD pattern . We marked the corre sponding diffraction spot in SAED image \n(Fig. S 2(g)). Likewise, other lattice planes are ide ntified by their interlayer spacing s and diffraction \nspots corresponding to those spatial frequencies are labelled in the SAED image. Note the bright \ndiffraction spots conform to the rhombohedral existence law –ℎ + 𝑘 + 𝑙 = 3𝑛 [4] \nSingle layer hBN (on SiO 2/Si) substrate was characterized by its Raman signature. A typical Raman \nspectroscopy measurement  (Fig. S2( h)) shows a peak at 1369 c m-1, consistent with earlier report [7]. \n \n(d) (c)\nCr K 5.0 µm\nGe K 5.0 µm\nTe L 5.0 µm\nCr\nGe Te\n(b)\nkeV0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00Counts[x1.E+3]\n0.05.010.015.020.025.030.0[MAP 3]\nTeCr\nCrGeGe\nTeLescTeTeTe\nTe\nTe\nTeTeCr\nCr GeGe\nCrTe\nGevdW gap(a)\n20 40 60 80101102103104105106\n00180012006Counts  (arb.units )\n2(deg.)MgO substrate\n003\n(e)\n3.3 Å (2 -1 0)[0 0 1](f)\n (g)\n1290 1320 1350 1380 1410020406080100 Intensity (arb. units)\nWave number (cm-1)Single layer hBN\n1369 cm-1 (h)Cr 19.11\nGe 17.71\nTe 63.18SI-3: Distribution of CGT flakes on hBN/SiO 2/Si and characterization of thickness  \n \nAFM  maps  show  dense  population  (~1-2 flakes/ 𝜇𝑚2) of few-layered  CGT  flakes  (Fig.  1(e) in MS). \nThicknesses  of individual  flakes  from  different  regions  are inferred  from  line cuts, shown  below  (Fig.  \nS3 (a)-(l)). Also  note the step of height  0.8 nm (Fig.  S3(f)) which  is equal  to the interlayer  distance  for \nCGT [8,9]. \n \nFIG. S3(a)-(c) AFM  images of CGT flakes (dispersed on hBN/SiO 2/Si substrate) in 2 μm × 2 μm area. \nNote the adjoining colo ur bars, denoting thickness . (d)-(f) AFM height profile along the red dashed \nlines in (a) -(c) respectively. (g) -(i) AFM images of CGT flakes (dispersed on hBN/SiO 2/Si substrate ) at \na different location . (j)-(l) AFM height profile along the red dashed lines in (g) -(i) respectively . \n \n \n0 100 200 300-101234Height (nm)\ndistance (nm)\n100 200 300 400 500-101234Height (nm)\ndistance (nm)0.8 nm\n0 100 200 300-101234Height (nm)\ndistance (nm)(a)\n(d)(b) (c)\n(e) (f)\n100 200-1012345Height (nm)\ndistance (nm)\n0 100 200 300-10123456Height (nm)\ndistance (nm)\n40 60 80 100 120-1012345Height (nm)\ndistance (nm)(g)\n(j)(h) (i)\n(k) (l)SI-4: Anisotropy in CGT and its experimental quantification  \n \nOur magnetic  measurements  on bulk single  crystals  of CGT  show  presence  of significant  anisotropy.  \nM(H) at 4 K (inset  of Fig. 2(b) in MS) shows  the out of plane  orientation  is the easy axis for Cr moments  \nwhich  is believed  to the mediated  by Cr-Te-Cr superexchange  and single  ion anisotropy [10,11] . We \ndetermine  the saturation  field for H || ab (𝐻𝑠||𝑎𝑏) through  linear  extrapolation  of the M(H) curve  (see \ndashed  green  lines in Fig. S4(a)) and calculate  the magnetocrsytalline  anisotropy  energy  𝐾𝑢 from  the \nformula  𝐾𝑢= 1\n2𝑀𝑠𝐻𝑠||𝑎𝑏 [12]. This estimation  of 𝐾𝑢 ~ 4.75 × 105 Jm-3 (or 0.6 meV/ f.u) is consistent  \nwith earlier  report [13] and our DFT  results  (cf. Table  II in MS).  𝐾𝑢 is observed  to be depending  on T \n(Fig.  S4(b)) and we compare  this 𝐾𝑢 (T) dependence  with the 𝑀𝑠(𝑇) dependence  (Fig. S4(c)) and \nfurther  analyze  in the light of Callen -Callen  power  law[14], which  infers  about  the difference  in T-\ndependence  of anisotropy  constant  (𝐾𝑢(𝑇)) for uniaxial  and cubic  (or multiaxial)  systems  depending  \non crystal  symmetry.  Callen -Callen  power  law gives  a relation  between  the T-dependence  of 𝐾𝑢 and \n𝑀𝑠  \n  𝐾𝑢(𝑇)\n𝐾𝑢(𝑇=0)= (𝑀𝑠(𝑇)\n𝑀𝑠(𝑇=0))𝑛\n  ------------------------- (S1) \n \nwhere the exponent 𝑛 depends on the crystal symmetry and is expected to be 3 for uniaxial anisotropic \nsystem. However, for our case, the exponent is close to 2, differing from Callen -Callen prediction for \nuniaxial anisotropy. This deviation suggests that anisotropy in CGT has a complex origin, beyond the \nsimplistic assumptions of C allen -Callen.  \n \nFIG. S 4(a) M(H) for bulk CGT at 4 K for 𝐻 || ab and 𝐻 ⊥ ab. The experimental data for both cases are \nlinearly extended to find out saturation field for the hard plane  (𝐻𝑠||𝑎𝑏). (b) Experimental quantification \nof magnetic anisotropy  𝐾𝑢(𝑇); violet curve is a guide to eye. (c) The ratio 𝐾𝑢(𝑇)/ 𝐾𝑢(4 𝐾) and \n𝑀𝑠(𝑇)/ 𝑀𝑠(4 𝐾) as a function of temperature ( violet  and green solid curves,  respectively) plotted in \ncomparison to (𝑀𝑠 (𝑇)/ 𝑀𝑠 (4 𝐾))𝑛 for n = 2 ( red dashed curve ). \n \n \n \n \n \n \nSI-5: Magnetization  isotherms  for bulk  CGT   \n \nM(H) isotherms  taken  up to 30 kOe along  the easy axis (H ⊥ ab) close  to the critical  regime of bulk \nCGT  is shown  in Fig.S5. Note  that after measuring  each M(H) isotherm,  the sample  was warmed  to \nroom  temperature  and demagnetized  before  measuring  the next M(H) isotherm.   \n \nFIG. S5 Isothermal M(H) curves for bulk CG T near the critical region ( 58 K to 75 K).  \n \nSI-6: Methodology to evaluate  critical exponents through Modified Arrott Plot ( MAP )  \nIt is known experimentally that close to a second order phase transition, experimental observables like \n𝑀𝑠 and 𝜒 exhibit certain power law behaviours and each observable is assigned  an exponent, given by \nthe following equations [15] :  \n𝑀𝑠= 𝑀0(−𝑡)𝛽       for 𝑡 < 0  ------------------   (S2) \n𝜒−1(𝑇)= 𝜒0−1𝑡𝛾  for 𝑡 > 0 ------------------  (S3) \n𝑀=𝐴𝐻1/𝛿             for 𝑡 = 0  -----------------  (S4) \nwhere 𝑡 = 𝑇− 𝑇𝑐\n𝑇𝑐 is the reduced temperature  and 𝛽,𝛾,𝛿 are the critical exponents.  \nA proper choice of critical exponents 𝛽 and 𝛾 gives a family of parallel straight lines [16] as M(H) \nisotherms in the critical regi me are plotted in the form 𝑀1𝛽⁄ vs (𝐻\n𝑀)1𝛾⁄\n. Arrott Plot for the Mean field \nmodel ( 𝛽=0.5 and 𝛾=1.0) using  data shown in  Fig. S 5 is shown in Fig.  S6(a). Even a cursory glance \nat Fig. S6(a) shows that no isotherm passes through the origin a nd clear non -linearity and downward \ncurvature is observed for all isotherms, indicating mean field picture is not an appropriate description \nof the critical phenomenon in  bulk CGT. In order to find out these exponents , we perform the iterative \nprocedure detailed below [17]: \n1. We plot MAPs using 𝛽 and 𝛾 of several  universality classes (Fig.  S6(a)-(f)) and eventually \nchoose 2D Ising model (Fig. S6(f)) as our most promising initial guess for the parameters 𝛽 \nand 𝛾 because the family of curves are significantly linear for these choices of  𝛽 and 𝛾 (note \nthe clear downward curvature in Fig. S6(a)-(e)).  \n \n2. Using the MAP for 2D Ising model, we find out the resulting 𝑀𝑠 and 𝜒−1 for all isotherms from \nthe linear extrapolation of the curves on vertical and horizontal axes respectively.  \n \n3. 𝑀𝑠(𝑇) and 𝜒−1(𝑇) data are fitted with eqn.  (S2) and eqn. ( S3) respectively, (while 𝑇𝑐 is kept a \nfree parameter) to obtain a new set of 𝛽 and 𝛾. These values of 𝛽 and 𝛾 are used again to \nconstruct a new MAP.  \n \n4. The process is continued until 𝛽 and 𝛾 from these two methods (i.e. MAP and fitting of 𝑀𝑠(𝑇), \n𝜒−1(𝑇)) converge sufficiently close.  \n \nWe emphasize that this iterative method is a general method and works equally well for any  initial \nchoice of 𝛽 and 𝛾. However, for the method to converge quickly, it is advisable to start with a \npremeditated guess compatible with the critical phase transition under investigation.  \n  \nFIG. S6 MAP for isotherms in the critical regime for bulk CGT with (a) Mean field model  (b) 3D \nHeisenberg model (c) 3D Ising model (d) 3D XY model (e) Tricritical mean field model and (f) 2D \nIsing model  \n \n \n \n \n \n \n0 20 40 600.00.51.01.52.02.5\n = 0.5    = 1.0Mean field modelM2 10-9 (Am2)2\n(H/M) 103(TA-1m-2) 58 K\n 59 K\n 61 K\n 62 K\n 63 K\n 64 K\n 65 K\n 66 K\n 67 K\n 68 K\n 69 K\n 70 K\n 71 K\n 72 K\n 73 K\n 74 K\n 75 K(a)\n0 2000 4000 6000 8000012345\n = 0.325     = 1.243 D Ising modelM1/ 10-14 (Am2)1/\n)\n(H/M)1/ (TA-1m-2)1/ 58 K\n 59 K\n 61 K\n 62 K\n 63 K\n 64 K\n 65 K\n 66 K\n 67 K\n 68 K\n 69 K\n 70 K\n 71 K\n 72 K\n 73 K\n 74 K\n 75 K (c)\n0 1000 2000 30000.00.40.81.21.6 58 K\n 59 K\n 61 K\n 62 K\n 63 K\n 64 K\n 65 K\n 66 K\n 67 K\n 68 K\n 69 K\n 70 K\n 71 K\n 72 K\n 73 K\n 74 K\n 75 KM1/ 10-12 (Am2)1/\n(H/M)1/ (TA-1m-2)1/3 D Heisenberg model\n = 0.365     = 1.386 (b)\n0 20 40 60 80024\n = 0.25     = 1.0Tricritical Mean field modelM1/ 10-18 (Am2)1/\n(H/M)1/ 103(TA-1m-2)1/ 58 K\n 59 K\n 61 K\n 62 K\n 63 K\n 64 K\n 65 K\n 66 K\n 67 K\n 68 K\n 69 K\n 70 K\n 71 K\n 72 K\n 73 K\n 74 K\n 75 K\n(e)\n0 1000 2000 3000 4000 50000123\n = 0.345     = 1.3163 D XY modelM1/ 10-13 (Am2)1/\n(H/M)1/ (TA-1m-2)1/ 58 K\n 59 K\n 61 K\n 62 K\n 63 K\n 64 K\n 65 K\n 66 K\n 67 K\n 68 K\n 69 K\n 70 K\n 71 K\n 72 K\n 73 K\n 74 K\n 75 K (d)\n0 200 400 600012 = 0.125     = 1.752 D Ising modelM1/ 10-35 (Am2)1/\n(H/M)1/ (TA-1m-2)1/ 58 K\n 59 K\n 61 K\n 62 K\n 63 K\n 64 K\n 65 K\n 66 K\n 67 K\n 68 K\n 69 K\n 70 K\n 71 K\n 72 K\n 73 K\n 74 K\n 75 K (f)SI-7: Determination of critical exponents from dc and ac magnetization measurements  \nUsing  Tc  as a fitting parameter , the critical regime is located between (63.68 ±1.99) K  to (66.59 ± \n0.27)  K (see Fig. 2(d) in MS).  The critical exponent 𝛿 is determined to be 11.37(9) using 𝑀=𝐴𝐻1/𝛿 \nfit of the M(H) curve at 𝑇𝑐 = 67 K  (Fig. S7 (a )). Inset of Fig.  S7(a) shows M(H) at 67 K in log -log scale.  \nUsing 𝛽 = 0.14, 𝛾=1.45, 𝛿 is again independently estimated to be 11.35(7) using the Widom’s scaling \nrelation : 𝛿=1 + 𝛾\n𝛽. These two estimations of 𝛿 are in close agreement.  \nWe did an independent estimation of γ using zero field ac susceptibility data ( 𝜒𝑎𝑐), measured in a small \noscillatory field of 3.8 Oe  (ZFC warming)  with a frequency of 211 Hz applied along sample’s easy axis , \nH  ab. It has been extensively discussed in relevant literature that low -field (or better, zero field)  data \ngives a more appropriate estimate of critical properties [18].  Our measurement shows a distinct hump \nin 𝑅𝑒 (𝜒𝑎𝑐) close to PM-FM transition, and the data was fitted in this neighborhood with 𝜒= 𝜒0𝑡−𝛾 \n(Fig. S7 (b)) which gives the critical exponent γ  as (1.34 ±0.08) and 𝑇𝑐 of (63.72 ±0.38) K. These \nestimation of 𝑇𝑐 , γ and 𝛿 are in reasonable agreement with the findings of MAP and critical scaling \nanalysis, and therefore, are intrinsic to the PM -FM phase transition in bulk CGT.   \n   \nFIG. S7 (a) M(H) isotherm  at 𝑇𝑐 = 67 K  fitted with 𝑀=𝐴𝐻1/𝛿. (inset) Log M vs Log H at 𝑇𝑐 with a \nstraight line fit to determine 𝛿 (b) Real part of ZFC warming ac magnetic susceptibility of bulk CGT \nat 211 Hz frequency. The red solid curve is a fit with 𝜒= 𝜒0𝑡−𝛾to extract γ and 𝑇𝑐. \n \nSI-8: Details  of theoretical  calculations  \n \nSI-8(A): Details of DFT calculations  \nFirst principles  DFT  calculations  were  performed  using  the Vienna  ab initio  simulation  package  \n(VASP) [19,20] . Electronic  wave  function s were  expressed  in terms  of a plane  wave  basis  set with an \nenergy  cutoff  of 500 eV. Projector  augmented  wave  (PAW) [21] potentials  were  used to represent  the \n50 60 70 80 90 1000246810Real 10-2  (arb.units)\nT (K)    Expt. data ( f = 211 Hz )\n     t-\nTC = (63.72 \n  0.38) K \n = (1.34 \n  0.08 )TC\n0 10 20 300102030  M = AH1/M (emu/g )\nH (kOe)T = 67 K\n-4 -2 0 2-13-12-11-10T = 67KLog M\nLog H = 11.379(a) (b)interaction  between  the valence  electrons  and the ion cores.  The generalised  gradient  approximation  \n(GGA)  as proposed  by Perdew,  Burke  and Ernzerhof  (PBE) [22] was used to treat the exchange -\ncorrelation  energy.  The GGA+U [23] method  was used to treat the strong  correlations  of Cr 3d electrons.  \nIt was previously  reported [9] that the appropriate  range  of U should be within  0.2 < U < 1.7 eV in order  \nto reproduce  the correct  experimental  magnetic  ground  state.  Hence  U = 1eV was taken  in our \ncalculation.  DFT -D3 method  of Grimme [24] was used to treat the van der Waals  interactions  between  \nthe CGT  layers.  The structures  were  optimized  by changing  both the ionic  positions  and lattice  \nparameters  of the simulation  cell until the energy  and force  on each atom  converged  to less than 10−7 \neV and 10−6 eV/Å, respectively.  For bilayer  and monolayer  a vacuum  greater  than 20 Å was used in \norder  to avoid  interactions  betwee n periodic  images.  Magnetocrystalline  anisotropy  energy  (MAE)  is \ncalculated  with the inclusion  of spin-orbit  coupling  by taking  the energy  difference  𝐸𝑎−𝐸𝑐 where   \n𝐸𝑎(𝐸𝑐) is the energy  of the system  when  the spins  orient  along  crystallographic  𝑎(𝑐) direction.  Γ- \ncentered  Monkhorst -Pack  k-point  mesh [25] was used to perform  Brillouin  zone integrations.  The \ndensity  of k-point  mesh  was set at 0.15 Å−1 during  self-consistency  calculations,  and for MAE  \ncalculations  a denser  k-mesh  of density  0.10 Å−1 was used.  \n \nSI-8(B): Details of Dipolar A nisotropy  Energy (DAE) calculations  \n \nSince  the moments  predominantly  come  from  the Cr atoms,  we consider  the lattice  spanned  by these  \natoms  in order  to calculate  DAE . Dipolar  interaction  energy  of a central  moment  with all other  moments  \non the lattice  up to a cut off radius  was calculated  using  the usual  expression  for interaction  between  \ntwo magnetic  dipole  moments  separated  by a distance  r as given  in eqn. (S5).  The cut off radius  was \nchosen  so that the interaction  energy  converged  within  0.01  𝜇eV for bulk and the bilayer  and 0.001  𝜇eV \nfor the monolayer.   \n \n \n𝑈=𝜇0\n4𝜋1\n𝑟3[𝑚1⃗.𝑚2⃗−3(𝑚1⃗.𝑟̂)(𝑚2⃗.𝑟̂)] ------------------  (S5) \n \nwhere  𝑚1⃗ and 𝑚2⃗ are the interacting  dipole  moments.  \n \nDipolar  interaction  energy  was calculated  for both in-plane  and out-of-plane  orientations  of the Cr \nmoments.  Magnitude  of all the moments  were  taken  as 3𝜇𝐵 as obtained  from  DFT  calculations.  DAE  \nis extracted  as the difference  in energies  in in-plane  and out-of-plane  orientations  of the moments.  \nDetails  of these  calculations  is shown  in Fig. S8.   \nFIG. S8: Performing  the real space  dipolar  sum over a sphere  of certain  radius  for bulk,  bilayer  and \nmonolayer  CGT  in in-plane  and out-of-plane  spin orientations . \n \n \n \nSI-9: Anisotropic consideration in magnon dispersion  \nIn a simple spin wave picture,  the dispersion relation of magnons in a ferromagnetic material at  \nsufficiently low temperature ( or, in other words, long -wavelength limit) is given by                   \n𝐸(𝑞) ~ (1−cos (𝑞𝑎)) [26] , often approximated as 𝐸(𝑞) ~ 𝑞2 for small 𝑞. The dispersio n is gapless \nand the corresponding magnetization is given by  \n 𝑀(𝑇)=𝑀(0)(1−𝑏𝑇3/2) -------   (S6, Bloch’s law ).  \nThis treatment, however, does not consider the effect of anisotropy.  Note that magnetic anisotropy is \ncritically important for long -range ordering (or, spontaneous symmetry breaking) in two dimensional \nmaterials as, for a 2D isotropic  system, long -wavelength Goldstone excitations can be generated with \nvanishingly small ene rgy cost and hence disrupts the orderi ng. The effect of incorporating an anisotropy \nterm is to introduce an excitation gap ( ∆) in the dispersion spectrum  (which, for isotropic magnetic \nsystems, is close to zero [27]). The modified dispersion relation is given by 𝐸(𝑞)= ∆+𝐷𝑞2 and the \nresulting magnetization as a function of T is given by [28] \n 𝑀 (𝑇,𝐻)=𝑀(0,𝐻)−𝑔𝜇𝐵(𝑘𝐵𝑇\n4𝜋𝐷)3\n2𝑓3\n2(∆′\n𝑘𝐵𝑇) ---------------  (S7) \nwhere D is the spin stiffness co -efficient,  ∆′= ∆+𝑔𝜇𝐵𝐻 and 𝑓3\n2(𝑦)= ∑exp  (−𝑛𝑦)\n𝑛3/2∞\n𝑛=1   is the Bose -\nEinstein integral functi on. In order to find out the effect of anisotropy in CGT,  in Fig. S9(a),  we fit the \nMS(T) data for bulk single crystal of CGT at low 𝑇 (𝑇 < 0.4 𝑇𝑐) with the gapless dispersion (eqn.(S 6)) \nand gapped dispersion (eqn.(S 7)).   \n \nFIG. S9 (a) Temperature dependence of spontaneous magnetization  for CGT bulk crystal at low T   \n(𝑇 < 0.4 𝑇𝑐) obtained from 𝐻⊥𝑎𝑏 data with fits to gapl ess spin wave dispersion (eqn. S 6) in orange \ndashed curve and fit to gapped spin wave dispersi on (eqn. S 7) in blue solid curve . (b) Temperature \ndependence of spontaneous magnetization for few -layered CGT  at low T for 𝐻⊥𝑎𝑏 with fits to gapl ess \nspin wave dispersion (eqn. S 6) in yellow dashed curve and fit to gap ped spin wave dispersion (eqn. S 7) \nin red solid curve.  \nIt is clear that the gapless equation  (S6) is not a good fit to the experimental data, necessitating  the \nimportance of taking anisotropy (i.e. eqn. S7) into consideration. Following are t he extracted parameters \nfrom fitting with the gapped magnon dispersion spectrum (eqn. S(7))  \n𝐷 = (26.09 ± 0.02) meV Å2 and 𝛥 = (1.38 ± 0.27) meV  \nThe spin stiffness co -efficient ( D) is in reasonable agreement with reported values [29] for bulk CGT . \nAnisotropy -induced gap 𝛥 is slightly higher than reported values, presumably because of using \ncomparatively broader T-window for fitting and/or high sensitivity of Bose -Einstein integral function \nto the excitation gap value  𝛥.  Next, we investigate the effect of anisotropy in low dimensional limit. \nFitting of MS(T) data for T < 0.35 𝑇𝑐′ (i.e. T < 5.2 K in Fig. 3(d) in MS ) with eqn. (S 7) gives a value of \nΔ = (0.14 ± 0.03) meV  (see Fig. S9(b)) , which is an order of magnitude smaller than its bulk counterpart.  \nThe observed drop in  strongly suggests significant weakening of anisot ropy with lowered \ndimensionality . \n \n \n0 10 20 303.583.603.623.643.663.68\n  Experimental data\n  gapped spin wave fit  (T3/2f3/2)\n  gapless spin wave fit (T3/2)MS 10-2 (emu)\nT (K)Bulk CGT\nH  ab\n2 3 4 51.751.801.851.901.95\n Experimental data\n gapped spin wave fit (T3/2f3/2)\n gapless spin wave fit  (T3/2)MS 10-5 (emu)\nT (K)Few layered CGT\nH ab(a) (b)[1] Y. Tian, M. J. Gray, H. Ji, R. J. Cava, and K. S. Burch, Magneto -elastic coupling in a \npotential ferromagnetic 2D atomic crystal, 2D Materials 3, 025035 (2016).  \n[2] C. A. Schneider, W. S. Rasband, and K. W. Eliceiri, NIH Image to ImageJ: 25 years of \nimage analysis, Nature Methods 9, 671 (2012).  \n[3] K. Momma and F. Izumi, VESTA 3 for three -dimensional visualization of crystal, \nvolumetric and morphology data, Journal of Applied Crystallography 44, 1272 (2011).  \n[4] V. Carteaux, D. Brunet, G. Ouvrard, and G. An dre, Crystallographic, magnetic and \nelectronic structures of a new layered ferromagnetic compound Cr 2Ge2Te6, Journal of Physics: \nCondensed Matter 7, 69 (1995).  \n[5] Y. Liu, M. -G. Han, Y. Lee, M.O. Ogunbunmi, Q. Du, C. Nelson, Z. Hu, E. Stavitski, \nD. Graf, K . Attenkofer  et al. , Polaronic Conductivity in Cr 2Ge2Te6 Single Crystals, Advanced \nFunctional Materials 32, 2105111 (2022).  \n[6] N. Wang, H. Tang, M. Shi, H. Zhang, W. Zhuo, D. Liu, F. Meng, L. Ma, J. Ying, L. \nZou et al. , Transition from Ferromagnetic Semic onductor to Ferromagnetic Metal with \nEnhanced Curie Temperature in Cr 2Ge2Te6 via Organic Ion Intercalation, Journal of the \nAmerican Chemical Society 141, 17166 (2019).  \n[7] R.V. Gorbachev, I. Riaz, R.R. Nair, R. Jalil, L. Britnell, B.D. Belle, E.W. Hill, K. S. \nNovoselov, K. Watanabe, T. Taniguchi  et al. , Hunting for Monolayer Boron Nitride: Optical \nand Raman Signatures,  Small 7, 465 (2011).  \n[8] S. Rahman, B. Liu, B. Wang, Y. Tang, and Y. Lu, Giant Photoluminescence \nEnhancement and Resonant Charge Transfer in  Atomically Thin Two -Dimensional \nCr2Ge2Te6/WS 2 Heterostructures  ACS Applied Materials & Interfaces 13, 7423 (2021).  \n[9] C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao, W. Bao, C. Wang, Y. Wang  et \nal., Discovery of intrinsic ferromagnetism in two -dimensional van der Waals crystals, Nature \n546, 265 (2017).  \n[10] D.-H. Kim, K. Kim, K. -T. Ko, J. Seo, J.S. Kim, T. -H. Jang, Y. Kim, J. -Y. Kim, S. -W. \nCheong, J. -H. Park  et al. , Giant Magnetic Anisotropy Induced  by Ligand LS Coupling in \nLayered Cr Compounds, Physical Review Letters 122, 207201 (2019).  \n[11] C. Xu, J. Feng, H. Xiang, and L. Bellaiche, Interplay between Kitaev interaction and \nsingle ion anisotropy in ferromagnetic CrI 3 and CrGeTe 3 monolayers, npj Co mputational \nMaterials 4, 57 (2018).  \n[12] B. D. Cullity and C. D. Graham, Introduction to magnetic materials (John Wiley & \nSons, 2011).  \n[13] S. Khan, C. W. Zollitsch, D. M. Arroo, H. Cheng, I. Verzhbitskiy, A. Sud, Y. P. Feng, \nG. Eda, and H. Kurebayashi, Sp in dynamics study in layered van der Waals single -crystal \nCr2Ge2Te6, Physical Review B 100, 134437 (2019).  \n[14] H. B. Callen and E. Callen, The present status of the temperature dependence of \nmagnetocrystalline anisotropy, and the l(l+1)2 power law, Journa l of Physics and Chemistry \nof Solids 27, 1271 (1966).  \n[15] M. E. Fisher, The theory of equilibrium critical phenomena, Reports on Progress in \nPhysics 30, 615 (1967).  \n[16] A. Arrott and J. E. Noakes, Approximate Equation of State For Nickel Near its Critica l \nTemperature, Physical Review Letters 19, 786 (1967).  \n[17] A. K. Pramanik and A. Banerjee, Critical behavior at paramagnetic to ferromagnetic \nphase transition in Pr0.5Sr0.5MnO 3: A bulk magnetization study, Physical Review B 79, 214426 \n(2009).  \n[18] S. F. F ischer, S. N. Kaul, and H. Kronmüller, Critical magnetic properties of disordered \npolycrystalline Cr75Fe25 and Cr70Fe30 alloys, Physical Review B 65, 064443 (2002).  [19] G. Kresse and J. Furthmüller, Efficiency of ab -initio total energy calculations for metals \nand semiconductors using a plane -wave basis set, Computational Materials Science 6, 15 \n(1996).  \n[20] G. Kresse and J. Furthmüller, Efficient iterative schemes for ab initio total -energy \ncalculations using a plane -wave basis set, Physical Review B 54, 11169 (1996).  \n[21] P. E. Blöchl, Projector augmented -wave method, Physical Review B 50, 17953 (1994).  \n[22] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized Gradient Approximation Made \nSimple,  Physical Review Letters 77, 3865 (1996).  \n[23] S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton, \nElectron -energy -loss spectra and the st ructural stability of nickel oxide: An LSDA+U study, \nPhysical Review B 57, 1505 (1998).  \n[24] S. Grimme, J. Antony, S. Ehrlich, and H. Krieg, A consistent and accurate ab initio \nparametrization of density functional dispersion correction (DFT -D) for the 94 elements H -Pu, \nThe Journal of Chemical Physics 132 (2010).  \n[25] H. J. Monkhorst and J. D. Pack, Special points for Brillouin -zone integrations, Physical \nReview B 13, 5188 (1976).  \n[26] N. W. Ashcroft and N. D. Mermin, Solid state physics (Cengage Learning, 2022).  \n[27] J. A. Fernandez -Baca, P. Dai, H. Y. Hwang, C. Kloc, and S. W. Cheong, Evolution of \nthe Low -Frequency Spin Dynamics in Ferromagnetic Manganites, Physical Review Letters 80, \n4012 (1998).  \n[28] K. Hüller, The spin wave excitations and the temperatu re dependence of the \nmagnetization in iron, cobalt, nickel and their alloys, Journal of Magnetism and Magnetic \nMaterials 61, 347 (1986).  \n[29] Z. Shu and T. Kong, Spin stiffness of chromium -based van der Waals ferromagnets, \nJournal of Physics: Condensed Mat ter 33, 195803 (2021).  \n "    },    {        "title": "2402.16331v1.Efficient_calculation_of_magnetocrystalline_anisotropy_energy_using_symmetry_adapted_Wannier_functions.pdf",        "content": "Efficient calculation of magnetocrystalline anisotropy\nenergy using symmetry-adapted Wannier functions\nHiroto Saito, Takashi Koretsune\naDepartment of Physics, Tohoku University, Aoba-ku, Sendai, 980-8578, Japan\nAbstract\nMagnetocrystalline anisotropy, a crucial factor in magnetic properties and\napplications like magnetoresistive random-access memory, often requires ex-\ntensive k-point mesh in first-principles calculations. In this study, we develop\na Wannier orbital tight-binding model incorporating crystal and spin symme-\ntries and utilize time-reversal symmetry to divide magnetization components.\nThis model enables efficient computation of magnetocrystalline anisotropy.\nApplying this method to L1 0FePt and FeNi, we calculate the dependence of\nthe anisotropic energy on k-point mesh size, chemical potential, spin-orbit in-\nteraction, and magnetization direction. The results validate the practicality\nof the models to the energy order of 10 [ µeV/f.u. ].\nKeywords: Magnetocrystalline anisotropy, Wannier functions, Density\nfunctional theory\n1. Introduction\nThe Maximally localized Wannier functions (MLWF) method is a widely\nutilized post-processing approach in first-principles calculations [1, 2, 3, 4].\nIt is primarily used for generating localized basis sets from band structures,\nwhich are crucial in finely interpolating the k-mesh of the energy bands and\nin constructing a model Hamiltonian. The MLWF method is distinguished\nfrom other interpolation methods by its requirement of not only the energy\neigenvalues at each k-point but also the connectivity of eigenstates. This as-\npect makes it particularly effective in accurately reproducing first-principles\nbands, even in cases involving band crossings or degeneracies [5].\nWannier functions (WFs) are usually constructed through a process that\ninvolves iteratively minimizing the sum of the spreads. This is achieved by\nPreprint submitted to Computer Physics Communications February 27, 2024arXiv:2402.16331v1  [cond-mat.mtrl-sci]  26 Feb 2024employing the algorithm of Marzari and Vanderbilt for an isolated group of\nbands [1] and the disentanglement scheme for entangled bands [2]. However,\nthese processes do not ensure that the resulting WFs will follow the spatial\nsymmetry of the crystal. Preserving this symmetry is critical for the physi-\ncally interpretable model Hamiltonian based on the results of first-principles\ncalculations. To address this issue, the symmetry-adapted Wannier function\n(SAWF) method was developed [6]. This method generates the SAWFs by\napplying additional constraints based on the operations of the site symmetry\ngroup during the maximally localization process. Recently, this method has\nbeen further extended to the disentanglement process with a frozen (inner)\nwindow [7].\nIn general, when calculating physical properties in metals, the conver-\ngence speed of integrations of the Brillouin zone (BZ) is slower compared to\nthe case of insulators, due to partially filled energy bands. This necessitates\na significantly large number of k-points, posing a challenge to the efficiency\nand accuracy of first-principles calculations for many physical properties.\nOne notable example of such challenges is the calculation of the magne-\ntocrystalline anisotropy (MCA) in ferromagnetic materials. The MCA refers\nto the phenomenon where the internal energy of a material varies with the\ndirection of magnetization, and it is a crucial property of magnetic materials\n[8]. In practical applications, materials used in magnetoresistive random-\naccess memory (MRAM) require perpendicular anisotropy to enhance the\nthermal stability of the magnetization direction [9]. The importance of in-\nplane anisotropy is also emphasized in applications such as soft magnetic\nunderlayers for perpendicular recording media [10] and spin-torque oscilla-\ntors for microwave-assisted magnetic recording [11, 12].\nTo elucidate the microscopic origins of the MCA, extensive research has\nbeen conducted across theoretical, computational, and experimental fields.\nTheoretically, the primary mechanisms of the MCA in transition metal com-\npounds have been explained through the second-order perturbation theory of\nspin-orbit interaction (SOI), as proposed by Bruno [13] and extended by Van\nder Laan [14]. In computational studies, first-principles calculations have\nbeen vigorously pursued, particularly for compounds with L1 0structures\n[15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. Notably,\nKhan et al. [25] have investigated higher-order SOI effects by self-consistently\nsolving the Dirac equation, examining the impact of different exchange-\ncorrelation functionals on many-body effects, and evaluating the validity of\nthe magnetic force theorem for FePt. More recent efforts include detailed\n2studies of the convergence of the MCA energy (MCAE) to the number of k\npoints using the Wannier interpolation method [30], as well as calculations\nof the temperature dependence of the MCAE using the disordered local mo-\nment method, a form of the coherent potential approximation [32, 31]. On\nthe experimental front, the MCAE is typically determined by measuring the\nmagnetic hysteresis curves in two directions. Additionally, by measuring the\ntorque dependent on the direction of magnetization, higher-order magnetic\nanisotropy constants have been determined [33, 34, 35, 36, 37, 38]. Despite\nthese experimental findings, computational calculations of the angular de-\npendence of the MCAE, particularly the subtle anisotropies within the easy\nplane, have been limited.\nIn this study, we first calculate the MCAE of L1 0-structured FePt and\nFeNi by performing k-point interpolation using a model Hamiltonian based\non the SAWF method. Furthermore, by exploiting time-reversal symmetry\noperation in the model Hamiltonian, we extract and rotate the magnetization\nto determine the dependence of the MCAE on the full angular range of\nmagnetization. This method is computationally cost-effective, as it allows\nfor the generation of a model Hamiltonian with magnetization oriented in\nany direction through a single Wannierization process.\n2. Method\nFirst, we summarize the SAWF method developed in Refs. [6] and [7]. We\nthen describe the symmetry of the spin space considered. Next, we discuss\nthe model Hamiltonian and the approximations. Finally, we give the specific\ncalculation method of the MCAE.\n2.1. Symmetry-adapted Wannier functions\nWe denote the space group of the system as Gand its element as g∈ G.\nWithin the space group G, we identify a subgroup of elements that leave a\ncertain wave vector kunchanged. This subgroup is referred to as the little\ngroup of k, denoted as Gk,\nGk={h∈ G | hk.=k}. (1)\nHere, the symbol.= signifies equality allowing for the difference in reciprocal\nlattice vectors.\n3Generally, WFs are defined through Fourier transformation combined\nwith projections of Bloch wave functions and their unitary transformation,\n\f\fψopt\nmk\u000b\n=NbandX\nl=1|ψlk⟩Uopt\nlm,k, (2)\n|WnR⟩=V\n(2π)3Z\nd3k e−ik·RNwannX\nm=1\f\fψopt\nmk\u000b\nUmn,k. (3)\nEq. (2) represents the process of disentanglement, where\f\fψopt\nmk\u000b\nis an opti-\nmized subspace derived from a larger set of Bloch bands |ψlk⟩. The matrix\nUopt\nkis a projection matrix of dimensions Nband×Nwann, where Nbandis the\nnumber of Bloch bands considered, and Nwannis the number of WFs being\nprojected, satisfying Nwann≤Nband. Eq. (3) represents the Fourier transfor-\nmation, where R,V, and Ukrepresent a Bravais lattice vector, the volume\nof the unit cell, and a Nwann×Nwannunitary matrix, respectively. The total\nwave-vector-dependent gauge Utot\nln,kis defined as\nUtot\nln,k=NwannX\nm=1Uopt\nlm,kUmn,k. (4)\nTo apply the symmetry constraints for the gauge Utot\nk, there are two\nsteps. The first one is to ensure that Utot\nkforkpoints in the irreducible BZ\n(IBZ) remains invariant under the operations of the little group Gk. This\nstep involves replacing Utot\nkwith Utot\nk, which averages the transformations\ncorresponding to all elements of the little group,\nUtot\nk=1\n|Gk|X\nh∈GkhUtot\nkh−1.(k∈IBZ) (5)\nNext, the IBZ is expanded to encompass the full BZ by using symmetry\noperations g∈ Gof the space group,\nUtot\ngk=gUtot\nkg−1.(k∈IBZ) (6)\nHere, the representation matrices of operators, gandh, are determined by\nthe Bloch wave functions and Wannier functions. By applying this symmetry\nconstraint, we can obtain the SAWFs. Ref. [6] considered the symmetrization\nof both Uopt\nkandUkaccording to Eqs. (5) and (6). In contrast, Ref. [7] applied\nthe symmetrization only to Utot\nk.\n42.2. Time-reversal symmetric Wannier functions\nWhen constructing the tight-binding model for ferromagnetic materials,\none sometimes considers the magnetic moment along the zdirection and\nconstructs the Wannier functions for Sz=±1/2. In this approach, Wannier\nfunctions for Sz= 1/2 and Sz=−1/2 are different. This can be understood\nusing the site symmetry group that there is no site symmetry operation that\nconnects a Wannier function with Sz= 1/2 to that with Sz=−1/2. On the\nother hand, when constructing the Wannier functions for Sx=±1/2, two\nWannier functions can have same wave functions for the real space. In other\nwords, time-reversal symmetric WFs (TRS-WFs) can be obtained. This can\nbe understood that site symmetry group can have a symmetry operation\nthat connects Wannier functions for Sx= 1/2 and that for Sx=−1/2. Of\ncourse, whether such symmetry exists depends on the crystal symmetry, while\nmany crystal structures have such symmetry operations. Fig. 1 schematically\nillustrates the construction of TRS-WFs.\n5\u0013\u0011\u0013\u0014\u0010\u0011\u0014\u0010\u0011\u00172&\u0012\n\u0010\u0013\u001a\u0013|→⟩=12(|↑⟩+|↓⟩)ψ↑(k)ψ↓(k)TQJO\u0001RVBOUJ[BUJPO\u0001BYJTx|w(x)|2w↑(x)w↓(x)'PVSJFS\u0001USBOTGPSN#MPDI\u0001GVODUJPO8BOOJFS\u0001GVODUJPOEF\nTQJO\u0001RVBOUJ[BUJPO\u0001BYJTψ↑(k)ψ↓(k)w→(x)w←(x)'PVSJFS\u0001USBOTGPSNx|w(x)|28BOOJFS\u0001GVODUJPO#MPDI\u0001GVODUJPOEFk\nkE\nEFigure 1: Schematic diagram of the process of constructing time-reversal symmetric Wan-\nnier functions. At the top, typical Wannier functions for up spin and down spin in a\nferromagnetic material are depicted. In cases where the spin quantization axis for Wan-\nnierization aligns with the direction of magnetization, the spatial distributions of these\nWannier functions do not match. At the bottom, time-reversal symmetric Wannier func-\ntions are shown. By orthogonalizing the spin quantization axis for Wannierization to the\ndirection of magnetization, Wannier functions are obtained where the centers and spreads\nof the functions are paired, thereby matching.\n2.3. Model Hamiltonian\nThe model Hamiltonian, dependent on the direction of magnetization,\nwas derived following these steps: Initially, under the basis of TRS-WFs\n|jµσ⟩, which are localized at position jand specified by orbital µand spin\nindex σ, the matrix elements of the tight-binding Hamiltonian were decom-\nposed into two terms: one symmetric and the other antisymmetric with\nrespect to time reversal,\n⟨jµσ|H|kνσ′⟩=Hjµσ;kνσ′=Hs\njµσ;kνσ′+Ha\njµσ;kνσ′. (7)\nHere, the time-reversal operator,\nΘ =−iσyK=\u00120−1\n1 0\u0013\nK (8)\n6is utilized, where σyis the ycomponent of the Pauli matrices σ= (σx, σy, σz)\nand the operator Krepresents the complex conjugation.\nSubsequently, the first term of Eq. (7) was further divided into two parts:\none independent of spin, identified as the hopping integral t, and the other\ndependent on spin, recognized as the SOI represented by λ= (λx, λy, λz)\n[39],\nHs\njµσ;kνσ′=tjµ;kνδσσ′+i(λjµ;kν·σ)σσ′. (9)\nIt is easy to show that due to the properties of the time-reversal operator Θ,\nall components of the hopping integral tand the SOI λare fundamentally\nreal numbers. Similarly, the second term of Eq. (7) can be decomposed using\nPauli matrices,\nHa\njµσ;kνσ′=iM0\njµ;kνδσσ′+ (Mjµ;kν·σ)σσ′. (10)\nThe components M0andM= (Mx, My, Mz) of this term are also all real\nnumbers.\nIn this study, we calculated the MCAE using three distinct methods.\nMethod 1: The MCAE is calculated by rotating the magnetization in the\ntight-binding model defined in Eq. (10). In this approach, Hais approxi-\nmated as the principal component of magnetization, Mz\niµ;jν,\nHa\njµσ;kνσ′≈Mz\niµ;jν(σz)σσ′, (11)\nand rotated with the usual spinor rotation,\nUs(θ, ϕ) =\u0012cosθ/2 e−iϕsinθ/2\n−eiϕsinθ/2 cos θ/2\u0013\n. (12)\nIndeed, when performing fully relativistic calculations, the assumption of\ncollinear magnetization as the self-consistent spin density is justified up to\nthe fourth-order perturbations of the SOI, according to the magnetic force\ntheorem [40, 41]. We call this rotated tight-binding Hamiltonian as the tight-\nbinding model with effective magnetization (TB-EM). The average value\nof the magnetization matrix elements discarded in the TB-EM method is\ndefined as the computational error,\nEerr=1\n|H|X\njkX\nµνX\nσσ′{Ha\njµσ;kνσ′−(Mz\niµ;jνσz)σσ′} (13)\n7where |H|represents the number of matrix elements in the Hamiltonian. This\ncomputational error, Eerr, corresponds to terms higher than the fourth-order\nperturbation of the SOI.\nMethod 2: Similar to Method 1, the MCAE is calculated by rotating the\nmagnetization in the tight-binding model. However, this method retains all\nterms in Eq. (9) for the spinor rotation, and thus, we call this method as the\ntight-binding model with original magnetization (TB-OM) method.\nMethod 3: Here, the MCAE is calculated by rotating the magnetization\nin density functional theory (DFT) calculations. The tight-binding models\nare constructed for each of the DFT calculations and are utilized only for in-\nterpolating the energy of bands. We call this method as the DFT method. In\nthe results section, we compare these three methods to discuss their validity\nand effectiveness.\n2.4. Magnetocrystalline anisotropy energy\nThe MCAE, denoted as ∆ EMCA, is calculated based on the magnetic force\ntheorem [40, 41], derived from the difference in band energies,\n∆EMCA(θ, ϕ) =1\nNkX\nk X\nnf(ε(θ,ϕ)\nnk)ε(θ,ϕ)\nnk−X\nmf(ε(0,0)\nmk)ε(0,0)\nmk!\n.(14)\nHere, nandmare indices designating the bands, while ( θ, ϕ) represent the\nangles specifying the direction of magnetization. The number of kpoints\nwithin the BZ is denoted by Nk. The terms εnkandf(εnk) represent the\nenergy at band nand wave vector k, and the Fermi distribution function\nrespectively, where f(ε) =1\neβ(ε−µ)+1.\nThe Fermi energy µ(θ,ϕ), which depends on the direction of magnetiza-\ntion, is determined so as to satisfy the condition that the number of valence\nelectrons per unit cell, Nelec, remains constant at the temperature T= 0,\nNelec=1\nNkX\nkX\nnf(ε(θ,ϕ)\nnk). (15)\n82.5. Calculation details\nFe\u0001Pt(Ni)BCD\nFigure 2: L1 0type crystal structure, where orange and grey spheres represent Fe and Pt\n(Ni) atoms, respectively.\nFirst-principles calculations program Quantum ESPRESSO [42, 43]\nbased on plane-wave basis and pseudopotential methods were employed to\nobtain the Bloch states. Subsequently, a tight-binding model was created\nusing Wannier90 program [4] in combination with SymWannier program\n[7], which takes into account the magnetic space group of the crystal. For\nthe L1 0type crystal structure (Fig. 2), lattice constants were set as a=\n2.728143 [ ˚A],c= 3.778984 [ ˚A] for FePt, a= 2.506611 [ ˚A],c= 3.581316 [ ˚A]\nfor FeNi. The cutoff for the plane wave basis was set as 64 .0 [Ry], and\nfor the electron density as 782 .0 [Ry]. The exchange-correlation functional\nemployed was the Perdew-Burke-Ernzerhof (PBE) formulation [44] of the\ngeneralized gradient approximation (GGA). Ultrasoft pseudopotentials from\nthePSLibrary [45] were utilized, with version 0.2.1 for Fe and version 0.1 for\nPt and Ni. According to the magnetic force theorem, SOI was incorporated\nduring the non self-consistent field (NSCF) calculations.\nWannierization was carried out in a one-shot process, where the projec-\ntion onto localized s, p, d orbitals at the atomic center positions for each atom\nyielded 36 spinor Wannier Functions (WFs). For the k-mesh during Wan-\nnierization, a grid of 9 ×9×6 was chosen for FePt and 9 ×9×7 for FeNi.\nFrom these, only 60 inequivalent kpoints under symmetry operations were\nutilized in the actual calculations for both cases. The inner energy window\nfor disentanglement was carefully set to include the all 3 dstates of the mag-\n9netic element Fe. For FePt, a range of [ −1000, EF+4] [eV] was used, and for\nFeNi, the range of [ −1000, EF+ 5] [eV] was used. The comparison between\nDFT and Wannier interpolated band structure is explained in Appendix A.\nImportantly, during Wannierization, the spin quantization axis (the aaxis)\nwas set orthogonal to the direction of magnetization (the caxis) in the first-\nprinciples calculations. This ensured that for both up and down spinors of\nthe WFs, the differences in the centers of the WFs were less than 10−6[˚A],\nand the differences in the spreads of the WFs were less than 10−8[˚A2] for all\npairs.\n3. Result\nWe first utilize the TB-EM method to increase the number of k-points\nin the BZ, to ensure the convergence of the MCAE. Following this, the de-\npendence of MCAE on the chemical potential and SOI is examined. This\ninvolves comparing the three methods described in section 2.3, with a par-\nticular focus on assessing the validity of the TB-EM approach. Lastly, we\ncompute the dependence of the MCAE on the angle of magnetization to\nderive the magnetic anisotropy constants.\n3.1.k-mesh dependence\nWe initially employ kinterpolation with the TB-EM method to investi-\ngate the degree of convergence of the MCAE between the hard axis [100] and\nthe easy axis [001], represented as E[100]−E[001]. The results are presented in\nFig. 3. It is observed that for both FePt and FeNi, increasing the number of\nk-points to Nk= 1003ensures sufficient convergence of the MCAE within the\nrange of computational error Eerr. In all subsequent calculations presented in\nthis paper, we used a uniform k-point grid of Nk= 1003points to determine\nthe other dependence.\n10(a)\n050310031503200325033003\nNumber of k points2.802.852.902.953.00E[100]E[001] [meV/f.u.]\n15032003250330032.862.882.90 (b)\n050310031503200325033003\nNumber of k points6080100120E[100]E[001] [eV/f.u.]\n1503200325033003808590\nFigure 3: MCAE for (a) FePt and (b) FeNi with respect to the number of k-points. The\nerror bars indicate the respective computational errors for each calculation. The MCAE\nconverges to 2 .885±0.012 [meV /f.u. ] = 17 .21±0.07 [MJ /m3] for FePt and 85 .63±\n2.71 [µeV/f.u. ] = 0.610±0.019 [MJ /m3] for FeNi, respectively.\n3.2. Chemical potential dependence\nWe next focus on the difference between the TB-EM method (Method\n1) and TB-OM method (Method 2), by calculating the chemical potential\ndependence of the MCAE and using the DFT method (Method 3) as a ref-\nerence. The results, presented in Fig. 4, show that for FePt, the TB-EM\nmethod closely matches the DFT result within the range of valence elec-\ntron numbers from 16 to 20, when compared to the TB-OM method. The\naverage difference between the TB-EM and DFT results is in the order of\n10 [µeV/f.u. ]. In contrast, for FeNi, both the TB-EM and TB-OM method\nreproduce a general trend of the DFT result, and the average differences with\nrespect to the DFT results are in the order of 10 [ µeV/f.u. ]. This indicates\nthat the limit of accuracy for MCAE calculations using the TB-EM method\nis in the order of 10 [ µeV/f.u. ].\n11(a)\n16 17 18 19 20\nNumber of electrons2\n0246E[100]E[001] [meV/f.u.]\n DFT\nTB-EM\nTB-OM (b)\n16 17 18 19 20\nNumber of electrons0200400600E[100]E[001] [eV/f.u.]\n DFT\nTB-EM\nTB-OM\nFigure 4: Chemical potential dependence of the MCAE for (a) FePt and (b) FeNi cal-\nculated using the three different methods. The horizontal axis represents the number of\nvalence electrons, with the number 18 corresponding to the Fermi level. The DFT results\n(solid) are derived from Wannier kinterpolation oriented in two different directions. In\nall cases, the results with 1003k-points are shown.\n3.3. SOI dependence\nNext, to conduct a more detailed comparison between the TB-EM and\nthe TB-OM methods, we modify the magnitude of SOI in the second term\nof the model Hamiltonian in Eq. (9). This is done by uniformly scaling all\nelements of the SOI term with a real parameter λ. The results are illustrated\nin Fig. 5. In both FePt and FeNi cases, the MCAE obtained by the TB-EM\nmethod remains positive across all values of λin the range 0 < λ < 1.5.\nIn contrast, for the TB-OM method, there are regions where the MCAE\nbecomes negative. Furthermore, in the TB-EM method, the MCAE exhibits\na parabolic behavior in regions of smaller λ, aligning with the results of\nthe second-order perturbation theory by Bruno [13] and van der Laan [14].\nAdditionally, the presence of a λ4term in regions of larger λsuggests that\nthe TB-EM method captures higher-order perturbation effects of SOI.\n12(a)\n0.00 0.25 0.50 0.75 1.00 1.25 1.50\nSOI parameter \n0246E[100]E[001] [meV/f.u.]\n TB-EM\nTB-OM\na2+b4\na2\n (b)\n0.00 0.25 0.50 0.75 1.00 1.25 1.50\nSOI parameter \n050100150200E[100]E[001] [eV/f.u.]\n TB-EM\nTB-OM\na2+b4\na2\nFigure 5: Dependence of the MCAE on the magnitude of SOI, denoted as λ, for (a) FePt\nand for (b) FeNi. Here, λ= 0 corresponds to the non-relativistic limit, while λ= 1\nrepresents the value in actual materials. The blue lines indicate the results of fitting with\na fourth-order polynomial, and the green plots correspond to the situation when b= 0.\nThe fitting parameters of the fourth-order polynomial are a= 3.264±0.012 [meV /f.u. ],\nb=−0.249±0.007 [meV /f.u. ] for FePt and a= 88.73±0.11 [µeV/f.u. ],b=−2.83±\n0.07 [µeV/f.u. ] for FeNi, respectively. In all cases, the results with 1003k-points are\nshown.\n3.4. Magnetization angle dependence\nFinally, we calculate the detailed dependence of the MCAE on the mag-\nnetization direction ( θ, ϕ) using the TB-EM method. The angle dependence\nof MCAE for FePt and FeNi is depicted in Figs. 6 and 7, respectively. Fur-\nthermore, we employ a phenomenological formula for the angular dependence\nof MCAE,\n∆EMCA(θ, ϕ) =K1sin2θ+K2sin4θ+K3sin4θcos 4ϕ (16)\nto fit these data, from which the magnetic anisotropy constants Kiare calcu-\nlated. The results obtained are compiled in Tabs. 1 and 2, alongside values\nfrom prior theoretical and experimental studies.\n13(a)\n0 45 90 135 180\n [deg]\n04590135180 [deg]\n0.00.51.01.52.02.53.0\ndE [meV/f.u.] (b)\n0 45 90 135 180\n [deg]\n0.00.51.01.52.02.53.0dE [meV/f.u.]\n(c)\n0 45 90 135 180\n [deg]\n3.0153.0203.0253.030dE [meV/f.u.]\nFigure 6: Dependence of the MCAE for FePt on the angular direction of magnetization\n(ϕ, θ) measured from the caxis. (a) For the full range of angles. (b) Along the line where\nthe angle ( ϕ, θ) = (0 , θ). (c) Along the line where the angle ( ϕ, θ) = (ϕ, π/ 2). In each of\nthese cases, solid lines represent the fitting based on the phenomenological formula in Eq.\n(16). The results depicted in (a) were obtained using 603k-points, while those in (b) and\n(c) were derived using 1003k-points. The fittings are conducted separately for (b) and\n(c).\nAccording to Fig. 6, it can be seen that for FePt, the angular dependence\nof the MCAE aligns well with Eq. (16). The result of fitting in Fig. 6(b)\nyields the magnetic anisotropy constants of K1= 2.959 [meV /f.u. ] and K2=\n0.062 [meV /f.u. ], while that in Fig. 6(c) yields K3=−0.009 [meV /f.u. ].\nThese results, when converted into standard units [MJ /m3], are compared\nwith previous studies in Tab. 1. Notably, there are no prior studies that\nhave investigated the ϕdirection dependence given by K3, while the values\n14and ratios of K1andK2show good agreement with the theoretical results of\nKhan [25].\nTable 1: Magnetic anisotropy constants of FePt are presented in units of [MJ /m3]. The\nupper section shows theoretical values, while the lower section shows experimental results.\nReference T[K] K2/K1[%] KU K1 K2 K3\nthis study 0 2.1 17.964 17.648 0.370 -0.054\nDaalderop et al. (1991) [15] 0 20.4\nSakuma (1994) [16] 0 16.3\nSolovyev et al. (1995) [17] 0 19.6\nOppeneer (1998) [18] 0 15.9\nGalanakis et al. (2000) [19] 0 22.1\nRavindran et al. (2001) [20] 0 15.52\nStaunton et al. (2004) [21] 0 9.523\nBurkert et al. (2005) [22] 0 16.12\nLuet al. (2010) [23] 0 16.62\nKosugi et al. (2014) [24] 0 18.25\nKhan et al. (2016) [25] 0 16.59 (Wien2k)\nKhan et al. (2016) [25] 0 3.1 18.17 (SPR-KKR) 17.51 0.54\nKe (2019) [26] 0 14.51\nAlsaad et al. (2020) [27] 0 7.6\nOkamoto et al. (2002) [33] Room T. 31.5 2.13 0.67\nInoue et al. (2006) [34] 5 1.8 6.9 7.4 0.13\nRichter et al. (2011) [35] 850 9 2.93\nGollet al. (2013) [46] Room T. 6\nIkeda et al. (2017) [47] Room T. 4.6\nOnoet al. (2018) [36] Room T. 44 3.4 0.9 0.4\nSaito et al. (2021) [48] Room T. 1 .8∼2.3\nIn contrast to the case of FePt, the angular dependence of the MCAE\nfor FeNi, especially in the ϕdirection on the order of 0 .1 [µeV/f.u. ], is\nobscure by computational error as shown in Fig. 7. While the positions of\nthe peaks are accurately captured, the curve does not replicate the expected\ncosine curve. The magnetic anisotropy constants fitted using Eq. (16) as\nshown in Figs. 7(b) and (c) are found to be K1= 86.681 [µeV/f.u. ],K2=\n−0.772 [µeV/f.u. ], and K3=−0.082 [µeV/f.u. ]. As shown in Tab. 2, the\nprevious study of Woodgate [31] concluded that K3is positive. This suggests\nthat the easy axis direction in the a-bplane for FeNi is along [110] rather\nthan [100]. However, our calculations indicate that, similar to FePt, the easy\naxis in the a-bplane for FeNi is also in the [100] (or [010]) direction.\n15(a)\n0 45 90 135 180\n [deg]\n04590135180 [deg]\n020406080\ndE [eV/f.u.]\n (b)\n0 45 90 135 180\n [deg]\n020406080dE [eV/f.u.]\n(c)\n0 45 90 135 180\n [deg]\n85.8585.9085.9586.00dE [eV/f.u.]\nFigure 7: Dependence of the MCAE for FeNi on the angular direction of magnetization\n(ϕ, θ) measured from the caxis. (a) For the full range of angles. (b) Along the line where\nthe angle ( ϕ, θ) = (0 , θ). (c) Along the line where the angle ( ϕ, θ) = ( ϕ, π/ 2). In each\nof these cases, solid lines represent the fitting based on the phenomenological formula,\nas expressed in Eq. (16). The results depicted in (a) were obtained using 603k-points,\nwhile those in (b) and (c) were derived using 1003k-points. The fittings are conducted\nseparately for (b) and (c).\n16Table 2: Magnetic anisotropy constants of FeNi are presented in units of [MJ /m3]. The\nupper section shows theoretical values, while the lower section shows experimental results.\nThe table includes an excerpt from the sources [30, 31].\nReference T[K] KU K1 K2 K3\nthis study 0 0.61 0.6172 -0.0055 -0.0006\nRavindran et al. (2001) [20] 0 0.54\nMiura et al. (2013) [28] 0 0.56\nEdstr¨ om et al. (2014) [29] 0 0.48 (Wien2k)\nEdstr¨ om et al. (2014) [29] 0 0.77 (SPR-KKR)\nQiao et al. (2019) [30] 0 0 .52±0.05\nKe (2019) [26] 0 0.47\nWoodgate et al. (2023) [31] 0 0.96 0.9582 -0.0008 0.0001\nN´ eel et al. (1964) [37] 293 0.55 0.32 0.23\nPaulev´ e et al. (1968) [38] Room T. 0.55 0.3 0.17 0.08\nShima et al. (2007) [49] Room T. 0.63\nMizuguchi et al. (2011) [50] Room T. 0.58\nKojima et al. (2011) [51] Room T. 0 .5±0.1\nFrisk et al. (2017) [52] 300 ∼0.35\nItoet al. (2023) [53] Room T. 0.55\nNishio et al. (2023) [54] 300 0.63\n4. Conclusion\nIn our research, we developed the TB-EM method, based on the SAWFs.\nBy investigating the dependence of the MCAE on the number of kpoints,\nchemical potential and SOI, we demonstrated that the TB-EM method pro-\nvides a valid effective Hamiltonian with an accuracy of the order of approx-\nimately 10 [ µeV/f.u. ]. Furthermore, using the TB-EM method, we deter-\nmined the magnetic anisotropy constants for FePt and FeNi. This method-\nology holds the potential for low-cost computation of various physical quan-\ntities dependent on the direction of magnetization.\n5. Acknowledgement\nThis work was supported by JSPS KAKENHI Grant No. 21H01003,\n21H04437, 22K03447, and 23H04869, JST-Mirai Program (JPMJMI20A1),\nCenter for Science and Innovation in Spintronics (CSIS), Tohoku University\nand GP-Spin at Tohoku University.\n17Appendix A. Band structure and density of states\nFigures A.8 and A.9 display the band structures and density of states for\nFePt and FeNi, respectively. As mentioned in section 2.5, the inner windows\nfor Wannierization were [ −1000, EF+ 4] [eV] and [ −1000, EF+ 5] [eV] for\neach case. These ranges effectively encompass the states of the Fe 3 dfor\nboth up and down spin.\n(a)\nX M\n Z R A Z5.0\n2.5\n0.02.55.07.5EEF [eV]\n (b)\nX M\n Z R A Z5.0\n2.5\n0.02.55.07.5EEF [eV]\nFigure A.8: Comparison between DFT band structure (black) and Wannier interpolated\nband structure (red) for (a) FePt and (b) (a) FeNi. In both cases, dashed lines represent\nthe maximum values of the inner window.\n(a)\n3\n 2\n 1\n 0 1 2 3\nDOS [states/eV]5.0\n2.5\n0.02.55.07.5EEF [eV]\n total\nFe 3d\nPt 5d total\nFe 3d\nPt 5d (b)\n4\n 2\n 0 2 4\nDOS [states/eV]5.0\n2.5\n0.02.55.07.5EEF [eV]\n total\nFe 3d\nNi 3d total\nFe 3d\nNi 3d\nFigure A.9: Density of states for (a) FePt and (b) FeNi.\nReferences\n[1] N. Marzari, D. Vanderbilt, Maximally localized generalized wannier\nfunctions for composite energy bands, Phys. Rev. B 56 (1997) 12847.\n18[2] I. Souza, N. Marzari, D. Vanderbilt, Maximally localized wannier func-\ntions for entangled energy bands, Phys. Rev. B 65 (2001) 035109.\n[3] N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, D. Vanderbilt, Maxi-\nmally localized wannier functions: Theory and applications, Rev. Mod.\nPhys. 84 (2012) 1419.\n[4] G. Pizzi, V. Vitale, R. Arita, S. Bl¨ ugel, F. Freimuth, G. G´ eranton,\nM. Gibertini, D. Gresch, C. Johnson, T. Koretsune, J. Iba˜ nez-Azpiroz,\nH. Lee, J.-M. Lihm, D. Marchand, A. Marrazzo, Y. Mokrousov, J. I.\nMustafa, Y. Nohara, Y. Nomura, L. Paulatto, S. Ponc´ e, T. Ponweiser,\nJ. Qiao, F. Th¨ ole, S. S. Tsirkin, M. Wierzbowska, N. Marzari, D. Van-\nderbilt, I. Souza, A. A. Mostofi, J. R. Yates, Wannier90 as a community\ncode: new features and applications, J. Phys. Condens. Matter 32 (2020)\n165902.\n[5] J. R. Yates, X. Wang, D. Vanderbilt, I. Souza, Spectral and fermi surface\nproperties from wannier interpolation, Phys. Rev. B 75 (2007) 195121.\n[6] R. Sakuma, Symmetry-adapted wannier functions in the maximal local-\nization procedure, Phys. Rev. B 87 (2013) 235109.\n[7] T. Koretsune, Construction of maximally-localized wannier functions\nusing crystal symmetry, Comput. Phys. Commun. 285 (2023) 108645.\n[8] Y. Miura, J. Okabayashi, Understanding magnetocrystalline anisotropy\nbased on orbital and quadrupole moments, J. Phys. Condens. Matter 34\n(2022) 473001.\n[9] S. Bhatti, R. Sbiaa, A. Hirohata, H. Ohno, S. Fukami, S. N. Pira-\nmanayagam, Spintronics based random access memory: a review, Mater.\nToday 20 (2017) 530.\n[10] A. Hashimoto, S. Saito, M. Takahashi, A soft magnetic underlayer with\nnegative uniaxial magnetocrystalline anisotropy for suppression of spike\nnoise and wide adjacent track erasure in perpendicular recording media,\nJ. Appl. Phys. 99 (2006) 08Q907.\n[11] K. Yoshida, M. Yokoe, Y. Ishikawa, Y. Kanai, Spin torque oscillator\nwith negative magnetic anisotropy materials for MAMR, IEEE Trans.\nMagn. 46 (2010) 2466.\n19[12] M. Igarashi, Y. Suzuki, Y. Sato, Oscillation feature of planar Spin-\nTorque oscillator for Microwave-Assisted magnetic recording, IEEE\nTrans. Magn. 46 (2010) 3738.\n[13] P. Bruno, Tight-binding approach to the orbital magnetic moment\nand magnetocrystalline anisotropy of transition-metal monolayers, Phys.\nRev. B 39 (1989) 865.\n[14] G. van der Laan, Microscopic origin of magnetocrystalline anisotropy in\ntransition metal thin films, J. Phys. Condens. Matter 10 (1998) 3239.\n[15] G. H. Daalderop, P. J. Kelly, M. F. Schuurmans, Magnetocrystalline\nanisotropy and orbital moments in transition-metal compounds, Phys.\nRev. B 44 (1991) 12054.\n[16] A. Sakuma, First principle calculation of the magnetocrystalline\nanisotropy energy of FePt and CoPt ordered alloys, J. Phys. Soc. Jpn.\n63 (1994) 1422.\n[17] I. V. Solovyev, IV, P. H. Dederichs, I. Mertig, I, Origin of orbital magne-\ntization and magnetocrystalline anisotropy in TX ordered alloys (where\nT=Fe,Co and X=Pd,Pt), Phys. Rev. B 52 (1995) 13419.\n[18] P. M. Oppeneer, Magneto-optical spectroscopy in the valence-band en-\nergy regime: relationship to the magnetocrystalline anisotropy, J. Magn.\nMagn. Mater. 188 (1998) 275.\n[19] I. Galanakis, M. Alouani, H. Dreyss´ e, Perpendicular magnetic\nanisotropy of binary alloys: A total-energy calculation, Phys. Rev. B\n62 (2000) 6475.\n[20] P. Ravindran, A. Kjekshus, H. Fjellv˚ ag, P. James, L. Nordstr¨ om, B. Jo-\nhansson, O. Eriksson, Large magnetocrystalline anisotropy in bilayer\ntransition metal phases from first-principles full-potential calculations,\nPhys. Rev. B 63 (2001) 144409.\n[21] J. B. Staunton, S. Ostanin, S. S. A. Razee, B. L. Gyorffy, L. Szunyogh,\nB. Ginatempo, E. Bruno, Temperature dependent magnetic anisotropy\nin metallic magnets from an ab initio electronic structure theory: L10-\nordered FePt, Phys. Rev. Lett. 93 (2004) 257204.\n20[22] T. Burkert, O. Eriksson, S. I. Simak, A. V. Ruban, B. Sanyal, L. Nord-\nstr¨ om, J. M. Wills, Magnetic anisotropy of 1 0FePt and Fe −xMn xPt,\nPhys. Rev. B 71 (2005) 134411.\n[23] Z. Lu, R. V. Chepulskii, W. H. Butler, First-principles study of magnetic\nproperties of l10-ordered MnPt and FePt alloys, Phys. Rev. B 81 (2010)\n094437.\n[24] T. Kosugi, T. Miyake, S. Ishibashi, Second-Order perturbation formula\nfor magnetocrystalline anisotropy using orbital angular momentum ma-\ntrix, J. Phys. Soc. Jpn. 83 (2014) 044707.\n[25] S. Ayaz Khan, P. Blaha, H. Ebert, J. Min´ ar, O. ˇSipr, Magnetocrystalline\nanisotropy of FePt: A detailed view, Phys. Rev. B 94 (2016) 144436.\n[26] L. Ke, Intersublattice magnetocrystalline anisotropy using a realistic\ntight-binding method based on maximally localized wannier functions,\nPhys. Rev. B 99 (2019) 054418.\n[27] A. Alsaad, A. A. Ahmad, T. S. Obeidat, Structural, electronic and mag-\nnetic properties of the ordered binary FePt, MnPt, and CrPt3 alloys,\nHeliyon 6 (2020) e03545.\n[28] Y. Miura, S. Ozaki, Y. Kuwahara, M. Tsujikawa, K. Abe, M. Shirai,\nThe origin of perpendicular magneto-crystalline anisotropy in L10-FeNi\nunder tetragonal distortion, J. Phys. Condens. Matter 25 (2013) 106005.\n[29] A. Edstr¨ om, J. Chico, A. Jakobsson, A. Bergman, J. Rusz, Electronic\nstructure and magnetic properties of l10 binary alloys, Phys. Rev. B 90\n(2014) 014402.\n[30] J. Qiao, W. Zhao, Efficient technique for ab-initio calculation of mag-\nnetocrystalline anisotropy energy, Comput. Phys. Commun. 238 (2019)\n203.\n[31] C. D. Woodgate, C. E. Patrick, L. H. Lewis, J. B. Staunton, Revisiting\nN´ eel 60 years on: The magnetic anisotropy of L10 FeNi (tetrataenite),\nJ. Appl. Phys. 134 (2023) 163905.\n[32] S. Yamashita, A. Sakuma, First-Principles study for finite temperature\nmagnetrocrystaline anisotropy of L10-Type ordered alloys, J. Phys. Soc.\nJpn. 91 (2022) 093703.\n21[33] S. Okamoto, N. Kikuchi, O. Kitakami, T. Miyazaki, Y. Shimada,\nK. Fukamichi, Chemical-order-dependent magnetic anisotropy and ex-\nchange stiffness constant of FePt (001) epitaxial films, Phys. Rev. B 66\n(2002) 024413.\n[34] K. Inoue, H. Shima, A. Fujita, K. Ishida, K. Oikawa, K. Fukamichi,\nTemperature dependence of magnetocrystalline anisotropy constants in\nthe single variant state of l10-type FePt bulk single crystal, Appl. Phys.\nLett. 88 (2006) 102503.\n[35] H. J. Richter, O. Hellwig, S. Florez, C. Brombacher, M. Albrecht,\nAnisotropy measurements of FePt thin films, J. Appl. Phys. 109 (2011)\n07B713.\n[36] T. Ono, N. Kikuchi, S. Okamoto, O. Kitakami, T. Shimatsu, Novel\ntorque magnetometry for uniaxial anisotropy constants of thin films and\nits application to FePt granular thin films, Appl. Phys. Express 11 (2018)\n033002.\n[37] L. N´ eel, J. Pauleve, R. Pauthenet, J. Laugier, others, Magnetic proper-\nties of an iron—nickel single crystal ordered by neutron bombardment,\nJ. Appl. Phys. 35 (1964) 873.\n[38] J. Paulev´ e, A. Chamberod, K. Krebs, others, Magnetization curves of\nFe–Ni (50–50) single crystals ordered by neutron irradiation with an\napplied magnetic field, J. Appl. Phys. 39 (1968) 989.\n[39] K. Kurita, T. Koretsune, Systematic first-principles study of the on-site\nspin-orbit coupling in crystals, Phys. Rev. B 102 (2020) 045109.\n[40] M. Weinert, R. E. Watson, J. W. Davenport, Total-energy differences\nand eigenvalue sums, Phys. Rev. B 32 (1985) 2115.\n[41] X. Wang, D.-S. Wang, R. Wu, A. J. Freeman, Validity of the force\ntheorem for magnetocrystalline anisotropy, J. Magn. Magn. Mater. 159\n(1996) 337.\n[42] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavaz-\nzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. D. Corso,\nS. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann,\nC. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari,\n22F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto,\nC. Sbraccia, S. Scandolo, G. Sclauzero, A. P. Seitsonen, A. Smogunov,\nP. Umari, R. M. Wentzcovitch, QUANTUM ESPRESSO: a modular\nand open-source software project for quantum simulations of materials,\nJ. Phys. Condens. Matter 21 (2009) 395502.\n[43] P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. Buon-\ngiorno Nardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, M. Co-\ncoccioni, N. Colonna, I. Carnimeo, A. Dal Corso, S. de Gironcoli, P. Del-\nugas, R. A. DiStasio, Jr, A. Ferretti, A. Floris, G. Fratesi, G. Fugallo,\nR. Gebauer, U. Gerstmann, F. Giustino, T. Gorni, J. Jia, M. Kawa-\nmura, H.-Y. Ko, A. Kokalj, E. K¨ u¸ c¨ ukbenli, M. Lazzeri, M. Marsili,\nN. Marzari, F. Mauri, N. L. Nguyen, H.-V. Nguyen, A. Otero-de-la Roza,\nL. Paulatto, S. Ponc´ e, D. Rocca, R. Sabatini, B. Santra, M. Schlipf, A. P.\nSeitsonen, A. Smogunov, I. Timrov, T. Thonhauser, P. Umari, N. Vast,\nX. Wu, S. Baroni, Advanced capabilities for materials modelling with\nquantum ESPRESSO, J. Phys. Condens. Matter 29 (2017) 465901.\n[44] J. P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approxima-\ntion made simple, Phys. Rev. Lett. 77 (1996) 3865.\n[45] A. Dal Corso, Pseudopotentials periodic table: From H to pu, Comput.\nMater. Sci. 95 (2014) 337.\n[46] D. Goll, T. Bublat, Large-area hard magnetic L10 -FePt and composite\nL10 -FePt based nanopatterns, Phys. Status Solidi 210 (2013) 1261.\n[47] K. Ikeda, T. Seki, G. Shibata, T. Kadono, K. Ishigami, Y. Taka-\nhashi, M. Horio, S. Sakamoto, Y. Nonaka, M. Sakamaki, K. Amemiya,\nN. Kawamura, M. Suzuki, K. Takanashi, A. Fujimori, Magnetic\nanisotropy of l10-ordered FePt thin films studied by fe and pt l2,3-edges\nx-ray magnetic circular dichroism (2017).\n[48] T. Saito, K. K. Tham, R. Kushibiki, T. Ogawa, S. Saito, Separate\nquantitative evaluation of degree of order and perpendicular magnetic\nanisotropy for disorder and order portion in FePt granular films, AIP\nAdv. 11 (2021) 015310.\n[49] T. Shima, M. Okamura, S. Mitani, K. Takanashi, Structure and\nmagnetic properties for l10-ordered FeNi films prepared by alternate\nmonatomic layer deposition, J. Magn. Magn. Mater. 310 (2007) 2213.\n23[50] M. Mizuguchi, T. Kojima, M. Kotsugi, others, Artificial fabrication and\norder parameter estimation of l10-ordered FeNi thin film grown on a\nAuNi buffer layer, J. Magn. Soc. Jpn 35 (2011) 370–373.\n[51] T. Kojima, M. Mizuguchi, K. Takanashi, L10-ordered FeNi film grown\non Cu-Ni binary buffer layer, J. Phys. Conf. Ser. 266 (2011) 012119.\n[52] A. Frisk, T. P. A. Hase, P. Svedlindh, E. Johansson, G. Andersson,\nStrain engineering for controlled growth of thin-film FeNi l10, J. Phys.\nD Appl. Phys. 50 (2017) 085009.\n[53] K. Ito, T. Ichimura, M. Hayashida, T. Nishio, S. Goto, H. Kura,\nR. Sasaki, M. Tsujikawa, M. Shirai, T. Koganezawa, M. Mizuguchi,\nY. Shimada, T. J. Konno, H. Yanagihara, K. Takanashi, Fabrication of\nl10-ordered FeNi films by denitriding FeNiN(001) and FeNiN(110) films,\nJ. Alloys Compd. 946 (2023) 169450.\n[54] T. Nishio, K. Ito, H. Kura, K. Takanashi, H. Yanagihara, Uniax-\nial magnetic anisotropy of L10-FeNi films with island structures on\nLaAlO3(110) substrates by nitrogen insertion and topotactic extraction,\nJ. Alloys Compd. 976 (2024) 172992.\n24"    },    {        "title": "2403.00138v1.Searching_for_magnetically_hard_monoborides__and_finding_a_few___A_first_principles_investigation.pdf",        "content": "arXiv:2403.00138v1  [cond-mat.mtrl-sci]  29 Feb 2024Searching for magnetically hard monoborides (and finding a f ew):\nA first-principles investigation\nJustyn Snarski-Adamski∗and Mirosław Werwiński\nInstitute of Molecular Physics, Polish Academy of Sciences ,\nM. Smoluchowskiego 17, 60-179 Poznań, Poland\nNew hard magnetic materials with zero or low rare earth conte nt are in demand due to the high\nprices of the rare earth metals. Among the candidates for suc h materials, we consider MnB, FeB and\ntheir alloys, because previous experiments suggest that Fe B has a relatively high magnetic hardness\nof about 0.83 at room temperature. Using first-principles ca lculations, we examine the full range of\nalloys from CrB, through MnB, FeB, to CoB. Furthrmore, we con sider alloys of MnB and FeB with\nsubstitutions of 3 d, 4dand 5 dtransition metals. For the above ninety compositions, we de termine\nmagnetic moment, magnetocrystalline anisotropy energy an d magnetic hardness.\nFor (Fe-Co)B alloys, the calculated values of magnetic hard ness exceed five, which is an excep-\ntionally high. While these values are inflated by the virtual crystal approximation used, we still\nexpect actual magnetic hardnesses well above unity. Furthe rmore, we classify considered MnB alloys\nsubstituted with transition metals as magnetically soft or semi-hard and FeB alloys with Sc, Ti, V,\nZr, Nb, Mo, Hf, Ta or W as magnetically hard (with magnetic har dness exceeding unity).\nI. INTRODUCTION\nIn this theoretical work, we will consider the applica-\ntion of monoborides as magnetically hard materials. To\nthis end, we will examine the full range of alloys from\nCrB, through MnB, FeB, to CoB, and also consider the\ncomplete series of MnB and FeB alloys with 3 d, 4d, and\n5delement substitutions. One of the main motivations\nfor the search for magnetically hard monoborides is the\npromising experimental results for FeB (magnetic hard-\nness around 0.83) [ 1,2] and for half-borides [ 3,4].\nThe basic magnetic properties of transition metal\nmonoborides in a series passing through the elements V,\nCr, Mn, Fe, Co, and Ni (with increasing atomic num-\nber) have been known since at least 1962 [ 5,6], while the\nmore recent advances in transition metal monoborides\nare summarized in Refs. [ 7] and [ 8]. MnB received sig-\nnificant attention between 2016 and 2023 [ 9–15]. This\nled to characterization of its lesser-known phases, such\nasαphase (CrB-type, s.g. Cmcm ) [15] as well as α′\nphase (stacking-fault dominated CrB-like variant, s.g.\nCmcm ) [13]. In addition, the well-known βphase (FeB-\ntype, orthorhombic, s.g. P nma ) was also reexamined [ 9–\n11,13,15]. Moreover, the theoretical determination of\nthe most energetically stable phases of FeB [ 16] and first-\nprinciples calculations for CrB [ 17] suggest the possibility\nof obtaining also a tetragonal MnB phase (Pearson sym-\nboltI16). Conventionally, MnB is assumed to have a\nlow-temperature mixed phase of CrB + FeB and a high-\ntemperature FeB-type phase. [ 18].\nIn this work, we focus on the high-temperature phase\nβ-MnB. It is a ferromagnetic phase with the mag-\nnetic moment equal to 1.9 µBf.u.−1[15] (the highest\nvalue among all transition metal monoborides), satu-\nration magnetization of 156 Am2kg−1(at 10 K) [ 15],\n∗Corresponding author: Justyn Snarski-Adamski\njustyn.snarski-adamski@ifmpan.poznan.plmagnetization remanence of 9.0 Am2kg−1[12], rela-\ntively high Curie temperature of 568 K [ 15], low coer-\ncive field of 15.9 Oe, classifying MnB as a magnetically\nsoft material [ 10], enormously high Vickers hardness of\n15.7 GPa [ 10], and a large magnetocaloric effect driven\nby anisotropic magnetoelastic coupling [ 11].\nSimilar to MnB, the FeB crystallizes in a low-\ntemperature α-phase (CrB-type) [ 11,18,19], which\ndetailed structural properties have been recently dis-\ncussed [ 20] along with a structural analysis of a high-\ntemperature βphase (orthorhombic, s.g. P nma ). As we\nalready mentioned, for FeB theory also suggests an en-\nergetically stable tetragonal phase ( tI16) [16]. For the\nβ-FeB phase, the Curie temperature is relatively high\nat around 590 K [ 21], the magnetic moment (at 4 K)\nis equal to 1.1 µBf.u.−1[22], and the effective mag-\nnetic anisotropy constant K∗\n1(for orthorhombic system)\nis equal to 0.4 MJ m−3. This leads to a magnetic hard-\nness of 0.83 and just slightly fails to meet criterion for\nmagnetically hard materials ( κ= 1). [ 2]. However, the\ncoercivity of FeB is low, similar to that of MnB, and\nequal to 10 Oe at 290 K [ 21]. The range of monoborides\nwe consider in this work is limited on both sides by CrB\n(s.g. Cmcm ) and CoB (s.g. P nma ) [17,23].\nOne method of tuning the properties of monoborides\nis nanostructurization [ 7,8,21]. Whereby, particularly\ninteresting form of monoborides are nanosheets [ 24], as\nexperimentally obtained in two-dimensional CrB [ 25],\nMnB [ 26] and CoB [ 27] systems. In the case of MnB\nnanosheets, the saturation magnetization is equal to\n15.5 Am2kg−1(emu g−1) at 300 K, and the Curie\ntemperature of 586 K is the highest among all known\ntwo-dimensional ferromagnets [ 25]. Recent experimen-\ntal efforts on monoboride nanosheets are complemented\nby theoretical work on two-dimensional CrB [ 28,29],\nMnB [ 26,29–31], FeB [ 28,29,32], and CoB [ 27,32],\nwhich predicts both the magnetic properties of these\nsystems and their dynamical stability. The calcula-\ntions for magnetic monoboride nanosheets complement\nearlier first-principles studies on bulk CrB [ 17,33,34],2\nMnB [ 10,35,36], and FeB [ 16,37], as well as work com-\nparing various bulk magnetic monoborides [ 11,38–40].\nTransition metal borides are involved in energy-related\nelectrocatalysis [ 8]. FeB, CoB, and NiB are being investi-\ngated as thermoelectrics [ 41], and nanoscale boron pow-\nders are being considered for hyperthermia and boron\nneutron capture therapy [ 21]. Two-dimensional transi-\ntion metal borides are being investigated for magnetic\ninformation storage [ 28,42]. CrB-based devices have\nbeen considered as sensors or adsorbents [ 43], and CrB\nnanosheets as promising anode materials in lithium-ion\nbatteries [ 25]. MnB and its alloys exhibit a large magne-\ntocaloric effect [ 9,11,15,44] and very high Vickers hard-\nness (15.6 GPa), combined with ferromagnetic proper-\nties, make MnB a promising mechanically hard magnetic\nmaterial [ 10]. In addition, MnB is suggested for mag-\nnetic hyperthermia [ 15] and improves the hard magnetic\nproperties of a combined hard / soft magnetic system\n(98% MnAl-C / 2% MnB) [ 12]. On the other hand,\namorphous Co-B is considered as a catalyst [ 45] and\nCoB nanosheets as an electrocatalyst for metal-air bat-\nteries [ 27]. Finally, an example of a high-entropy pseudo-\nmonoboride is (Cr 0.2Mn0.2Fe0.2Co0.2Mo0.2)B [46].\nThe long list above does not include applications of\nmagnetic monoborides as magnetically hard materials,\nmost likely due to the fact that coercivity measurements\nindicate that MnB and FeB are magnetically soft [ 10,21].\nHowever, given the arguments we are about to present,\nwe decided to theoretically explore the possibility of usin g\nalloys of MnB and FeB as magnetically hard materials.\nTo this end, we choose to calculate from first principles\nthe magnetic hardness values for about ninety chemical\ncompositions, which we hope will allow us to propose\npromising candidates for permanent magnets or discredit\nthe idea. The following arguments led us to consider the\nuse of monoboride alloys in permanent magnets: (1) rea-\nsonably high Curie temperatures, above 540 K for MnB\nand FeB, and above 800 K for (Mn,Fe)B alloys [ 1,10,21],\n(2) reasonably high saturation magnetization of MnB and\nFeB [ 1,10,21], (3) orthorhombic crystal system of MnB\nand FeB, which can provide a good starting point for the\ndevelopment of magnetic hardness, (4) relatively high ex-\nperimental effective magnetocrystalline anisotropy con-\nstant of FeB monocrystal ( K∗\n1= 0.4 MJ m−3) [2], (5)\nrelatively high magnetic hardness of FeB which we es-\ntimated as 0.83 from the experimental K∗\n1, magneti-\nzation and lattice parameters [ 1,2], (6) elevated mag-\nnetic hardness of annealed α-FeB [ 21] and FeB nanopar-\nticles [ 20], (7) semi-hard magnetic properties determined\nfor some of (Fe,Co) 5PB2[47–49], (Fe,Co) 3B [50], and\n(Fe,Co) 2B [3,4,51] borides, and last but important (8)\nexcellent hard magnetic properties of another transition\nmetal boride, Nd-Fe-B, one of the most popular perma-\nnent magnets today.\nThe results presented here provide a systematic de-\nscription of the rudamental magnetic properties of doped\nmonoborides, which can be used in many of the numerous\npotential applications listed above. However, especially\nin the context of volatile neodymium prices observed be-\nFIG. 1. (a) Crystal structure of β-MnB (FeB-type; space\ngroup P nma ). The lattice parameters of a single unit cell are\na= 5.6377 Å, b= 2.9945 Å, and c= 4.1795 Å [ 10]. The unit\ncell is multiplied 2 ×3×3 times along the main axes for better\nvisualization. (b) Crystal structure of Mn 11X1B12supercell\n(space group P1m1) with transition metal substitution X.\nThe supercell was obtained by duplicating the MnB unit cell\nthree times along the b-axis and substituting one of twelve\nMn atoms.\ntween 2011 and 2024 [ 52], it is important to search for\nnew permanent magnets that contain little or no rare-\nearth elements [ 53,54].\nII. COMPUTATIONAL DETAILS\nFor density functional theory calculations, we used\nthe full-potential local-orbital electronic structure co de\nFPLO18.00-52 including relativistic effects in the full\nfour-component formalism [ 55,56]. Unless otherwise\nnoted, we employed the generalized gradient approxima-\ntion (GGA) in the form proposed by Perdew, Burke, and\nErnzerhof (PBE) [ 57]. We choose a tetrahedron method\nfor Brillouin zone integration. All calculations were full y\nconverged with density criterion 10−6.\nFor MnB, we assumed a P nma space group (FeB-\ntype) with experimental lattice parameters a= 5.6377 Å,\nb= 2.9945 Å, and c= 4.1795 Å, and initial Wyckoff po-\nsitions from Ref. [ 10], see also Table Iand Fig. 1(a). For\nFeB, we adopted the space group P nma with experimen-\ntal lattice parameters a= 5.4954 Å, b= 2.9408 Å, and\nc= 4.0477 Å, and initial Wyckoff positions from Ref. [ 19].\nWe optimized the initial atomic positions and present our\nresults in Table Ias well. The Wyckoff positions pre-\ndicted theoretically for MnB differ significantly from the\nexperimental results from work of Ma et al. [10]. The\ncalculated positions are instead very close to another ex-\nperimental result for MnB from work of Kalyon et al. [15]3\nand to the results for isostructural FeB [ 19]. Regardless,\nwe use the optimized atomic positions throughout our\nwork.\nTABLE I. Experimental and theoretical Wyckoff positions of\nMnB and FeB (s.g. P nma ). Atomic positions for B and M (M\n= Mn or Fe) are presented. The DFT results were calculated\nwith FPLO18 code and the GGA-PBE exchange-correlation\npotential.\nRef. x(M) y(M) z(M) x(B) y(B) z(B)\nMnB expt. [ 10] 0.1680 0.25 0.1244 0.1482 0.25 0.6375\nMnB expt. [ 15] 0.1762 0.25 0.1206 0.0312 0.25 0.6133\nMnB PBE 0.1746 0.25 0.1211 0.0330 0.25 0.6169\nFeB expt. [ 19] 0.1771 0.25 0.1195 0.0332 0.25 0.6168\nFeB PBE 0.1776 0.25 0.1199 0.0344 0.25 0.6200\nFor modeling the solid solutions in a range from CrB\n(Z= 24), through MnB ( Z= 25), FeB ( Z= 26), to\nCoB ( Z= 27) we employed virtual crystal approxima-\ntion (VCA). For the whole range of VCA compositions,\nwe kept the space group P nma . For MnB [ 10], FeB [ 19],\nand CoB [ 23] we used experimental lattice parameters\nand optimized atomic positions. For the CrB-MnB range,\nwe set lattice parameters and Wyckoff positions extrap-\nolated from the MnB/FeB pair. For other intermedi-\nate concentrations, we used interpolated values of lat-\ntice parameters and Wyckoff positions. To determine the\nmagnetocrystalline anisotropy energy (MAE) within the\nVCA, we performed self-consistent scalar-relativistic ca l-\nculations, followed by a single fully-relativistic iterat ion\nfor each of three main crystallographic axes. We used\n11×20×15 k-mesh.\nTo determine the dependency of magnetocrystalline\nanisotropy energy on magnetic moment of FeB we used\na fully-relativistic fixed-spin-moment method. Moreover,\nwe used the local spin density approximation (LSDA) in\nthe forms of von Barth and Hedin (BH) [ 58] and Perdew\nand Wang (PW92) [ 59] to analyze the effect of the form of\nexchange-correlation potential on the magnetic moment\nand MAE.\nBased on Mn 11X1B12and Fe 11X1B12supercells, we\nstudied the effect of substituting MnB and FeB with 3 d,\n4d, and 5 delements. The supercells are three-times rep-\netitions in bdirection of P nma unit cell, see Fig. 1(b).\nThe Wyckoff positions of MnB and FeB unit cells were\nfirst optimized, assuming the force convergence criterion\nof 10−3eV Å−1. The lattice parameters of the super-\ncells were kept fixed at a= 5.6377 Å, b= 8.9835 Å, and\nc= 4.1795 Å for Mn 11X1B12[10] and a= 5.4954 Å,\nb= 8.8224 Å, and c= 4.0477 Å for Fe 11X1B12[19]. The\nreduced space group of these structures was P1m1. For\nsupercells, we choose an 8 ×5×11 k-mesh. Drawings of\nthe structures were made using the VESTA code [ 60].\nTo determine the magnetic hardness, we need to calcu-\nlate the magnetocrystalline anisotropy energy. For uni-\naxial crystal systems, such as tetragonal and hexagonal,\nthe MAE can be evaluated from the difference in energies\ndetermined for the axis along the unique crystallographic\naxis and the axis perpendicular to it. Here, since FeBand MnB crystallize in an orthorhombic system charac-\nterized by three different lattice parameters a,b, and\nc, the approach used for uniaxial systems cannot be di-\nrectly applied. Usually in that case, only the magnetic\neasy axis is determined [ 36]. In this work, to quantify\nthe MAE, we use the following approach. We calculate\nthe fully relativistic energies for three quantization dir ec-\ntions along the main crystallographic axes: [100] (along\na), [010] (along b), and [001] (along c).\nThe lowest energy indicates the magnetic easy axis,\nand the difference between the middle energy ( E2) and\nthe lowest energy ( E1) determines the MAE. As a re-\nsult, all values of MAE are non-negative. One could\nalso take the difference between the highest ( E3) and\nlowest ( E1) energy. However, we opted for the first op-\ntion, interpreting MAE as a parameter that determines\nthe uniaxial preference for magnetization direction. To\ncomplement this parameter, we calculate also the energy\ndifference between the highest ( E3) and the middle ( E2)\nof the three energies, which we denote as magnetocrys-\ntalline anisotropy energy DE 32. A similar approach\nwas used by Zhdanova et al. [2] to experimentally de-\ntermine the effective magnetocrystalline anisotropy con-\nstants K∗\n1=K1−K2andK∗\n2=K3−K2of orthorhom-\nbic FeB, where K1,K2, and K3are the three anisotropy\nconstants determined in the three main crystallographic\nplanes and arranged in descending order.\nWhen discussing applications for permanent magnets,\nmagnetic hardness is another crucial metric [ 54]. It can\nbe expressed as:\nκ=/radicalBigg\n|K|\nµ0M2\nS, (1)\nwhere Kstands for the magnetocrystalline anisotropy\nconstant, MSdenotes the saturation magnetization, and\nµ0the vacuum permeability. In order to determine the\ntheoretical value of κ, we assume that the calculated\nMAE is equal to the anisotropy constant K. To esti-\nmate the value of MSwe use the total magnetic moment\nand the unit cell volume.\nIn this work, we have determined the magnetic prop-\nerties for temperature of 0 K (ground state), while per-\nmanent magnets usually operate at room or higher tem-\nperature. While in this situation the calculated magnetic\nmoments can be reasonably treated as upper limits, the\nissue of MAE’s dependence on temperature is more com-\nplex. For example, it has been experimentally shown for\nmagnetic halfborides (Fe-Co) 2B, that magnetocrystalline\nanisotropy constant ( K1) depends on temperature in a\nnonlinear manner (including a change of sign) [ 4]. How-\never, a similarity was observed by us between the con-\ncentration dependence and temperature dependence of\nK1. Thus, we assume that the highest MAEs among the\nvalues determined at 0 K for the full range of the con-\ncentrations will at the same time represent a limit in the\ntemperature dependence of MAEs.\nFurthermore, the MAE values determined for alloys in\nthe VCA are often overestimated [ 3]. However, calcula-\ntions for alloys modeled by the supercell method (without4\nVCA) yield similar and in some cases even higher MAE\nandκvalues than those determined by VCA for (Mn-\nFe)B compositions, which makes our VCA results more\nreliable. Another constraint of the calculation method we\nemployed is the accuracy of the GGA. Therefore, we will\ndiscuss the effect of the form of the exchange-correlation\npotential on the obtained MAE values [ 61]. Finally, the\nother known limitation is lack of disorder in supercells\ndue to their limited size, which can also affect the deter-\nmined MAE [ 51,61].\nIII. RESULTS AND DISCUSSION\nProblems with the availability and prices of rare earth\nores [ 52] are stimulating the search for new permanent\nmagnets with zero or low rare earth content [ 3,4,47,51,\n53,54]. As part of this investigation, here, we consider\nmonoboride alloys ranging from CrB ( Z= 24), through\nMnB (25) and FeB (26), to CoB (27), or in other words,\nconcentration ranges: Cr 1−xMnxB, Mn 1−xFexB, and\nFe1−xCoxB. Furthermore, we study MnB and FeB alloys\nwith substitutions of thirty different transition metal el-\nements.\nA. (Cr-Mn)B, (Mn-Fe)B, and (Fe-Co)B alloys\nfrom virtual crystal approximation\nFrom the perspective of application as permanent mag-\nnets, the three most important intrinsic properties of\na magnetic material are Curie temperature, saturation\nmagnetization and magnetocrystalline anisotropy. Be-\nlow, we will summarize all three for magnetic mono-\nborides, with the last two quantities expressed in a form\nof magnetic moment and magnetocrystalline anisotropy\nenergy. Although the concentration dependence of Curie\ntemperature and magnetic moments have long been\nknown, our calculations complement the picture, provid-\ning the values of magnetocrystalline anisotropy energy\nand magnetic hardness.\nTABLE II. Calculated magnetic properties of characteristi c\ncompositions in the MnB-FeB-CoB range (s.g. Pnma ). The\nmagnetic easy axis, spin and orbital magnetic moment [ ms\nandml(µBatom−1)] on boron (B) and virtual atom of tran-\nsition metal (M), magnetocrystalline anisotropy energy [M AE\n(MJ m−3)], and magnetic hardness [ κ] are presented. Orbital\nmagnetic moments are given for the corresponding magnetic\neasy axes. Calculations were performed with FPLO18-PBE\nin the virtual crystal approximation (VCA).\nComposition Axis ms(M) ml(M) ms(B)ml(B) MAE κ\nMnB 010 2.18 0.023 -0.20 0.001 0.45 0.57\nMn0.55Fe0.45B 100 1.92 0.026 -0.18 0.000 0.77 0.83\nMn0.15Fe0.85B 010 1.53 0.028 -0.14 0.001 0.69 0.95\nFeB 010 1.33 0.025 -0.13 0.001 0.14 0.49\nFe0.5Co0.5B 001 0.61 0.047 -0.06 0.001 2.87 5.090 0200 200400 400600 600800 800\nexpt. Kanaizuka 1981\nexpt. Takács et al. 1975\nexpt. Cadeville  and Daniel 1966(a)Curie temperature (K)\nCurie temperature (K)\n24 25 26 270.0 0.00.5 0.51.0 1.01.5 1.52.0 2.0\nDFT this work\nexpt. Kuentzler 1970\nexpt. Kanaizuka 1981\nexpt. Cadeville and Daniel 1996(b)CrB                  MnB                 FeB                 CoBMagnetic moment ( µB f.u.-1)\nMagnetic. moment ( µB f.u.-1)\n24 25 26 270.01.02.03.04.05.06.0\n0.01.02.03.0Magnetic hardness\nMAE (MJ m-3)\nAtomic number of transition metal001 010 100\n(c)010 001A\nFIG. 2. Intrinsic properties critical for permanent magnet\napplications obtained for the concentration range from CrB ,\nthrough MnB and FeB, to CoB. (a) Experimental Curie tem-\nperatures taken from the literature. (b) Calculated total s pin\nmagnetic moments as a function of the atomic number of the\ntransition metal element, together with the corresponding ex-\nperimental total magnetic moments from the literature. (c)\nCalculated magnetocrystalline anisotropy energy (MAE) an d\nmagnetic hardness as a function of atomic number. Verti-\ncal dashed lines separate alloys with different axes of easy\nmagnetization. Calculations were performed with the virtu al\ncrystal approximation (VCA) using the FPLO18 code with\nPBE exchange-correlation potential. Experimental data ar e\ntaken from Refs. [ 1,6,62,63].\nThe first systematic measurements of the properties of\nCrB-MnB and MnB-FeB alloys were made by Cadeville\nand Daniel in 1966 [ 6]. The research was then continued\nby Kanaizuka [ 1,22]. It was found that CrB ( Z= 24\nfor Cr) is paramagnetic and at intermediate concentra-\ntion (Cr-Mn)B alloys gain ferromagnetic ordering. The\nCurie temperature increases with Mn concentration, and\nfor MnB it is about 575 K, see Fig. 2(a) [1,6]. For\n(Mn-Fe)B alloys, Curie temperature grows above 800 K\nfor Mn 0.4Fe0.6B. For (Fe-Co)B compositions, Curie tem-\nperature decreases with Co concentration to zero, and\nCoB itself is paramagnetic [ 62,63]. From the considered\nperspective of application for permanent magnets, Curie\ntemperature values well above room temperature are re-\nquired. Hence, MnB-FeB alloys located around the Curie\ntemperature maximum would be best suited.5\n24 25 26 270.00.51.01.52.0ms on TM\nms on B(a)CrB                  MnB                 FeB                 CoBSpin magnetic moment ( µB atom-1)\n24 25 26 270.000.020.040.06\nml on TM [001]\nml on TM [010]\nml on TM [100](b)CrB                  MnB                 FeB                 CoBOrbital magnetic moment ( µB atom-1)\n001\n100\n010\nAtomic number of transition metal\nFIG. 3. Spin and orbital magnetic moments ( msandml) on\ntransition metal (TM) and boron sites as a function of the\natomic number of the transition metal element, calculated\nfor concentration range from CrB, through MnB and FeB,\nto CoB. The calculations were performed with the virtual\ncrystal approximation (VCA) using the FPLO18 code with\nPBE exchange-correlation potential.\nExperimental studies of the concentration dependence\nof the magnetic moment in the CrB-MnB-FeB-CoB range\nreveal the absence of an ordered magnetic moment on the\nCr-rich side and the Co-rich side, see Fig. 2(b) [1,5,6,62,\n63]. MnB has the maximum value of the magnetic mo-\nment of about 1.9 µB. Passing from MnB through FeB\nto CoB, we observe its linear decrease. The experimen-\ntal dependence of magnetic moments on concentration is\nwell reflected in the previous DFT results for the MnB-\nFeB [ 9,64,65] and FeB-CoB [ 66,67] solid solutions. Our\nresults for the MnB-FeB-CoB range also well reproduce\nthe experimental dependence of the magnetic moment\non concentration. Although for the CrB-MnB range, the\ncalculated values differ from the experimental ones. This\nis most likely due to the adoption of an identical P nma\nspace group for all VCA concentrations, while a Cmcm\nspace group (MoB-type structure) is experimentally ob-\nserved for CrB and nearby [ 17,18,22]. At the other\nend of the range considered, we observe a collapse of the\nmagnetic moment above 80% of the Co concentration in\n(Fe-Co)B alloys, similar to previous experimental [ 6,62]\nand theoretical [ 66,67] studies.\nBefore moving on to the results on magnetic hardness,\nwe would like to discuss the spin and orbital magnetic\nmoments on individual atoms, see Fig. 3. The largest\ncontributions to the total magnetic moment come from0.30.40.50.6\n0.40.50.6 24 25 26 272.12.22.3Excess electrons on B Occupation  of  TM 4p Occupation  of  B 2p CrB               MnB               FeB               CoB\nAtomic number of transition metal\nFIG. 4. Excess electrons on boron and occupation of 2 por-\nbital of boron and 4 porbital of transition metal (TM) as a\nfunction of atomic number of transition metal, calculated f or\nthe concentration range from CrB, through MnB and FeB, to\nCoB. The calculations were performed with the virtual crys-\ntal approximation (VCA) using the FPLO18 code with PBE\nexchange-correlation potential.\nthe spin magnetic moment on the transition metal site\nand from a smaller induced opposite spin moment on\nboron. For these spin moments, we have shown values\nfor one representative direction [010], since the other two\ncases are indistinguishable. For the orbital moments on\ntransition metal site, we have plotted values calculated\nin three directions. The differences observed for orbital\nmoments determined in various directions correlate with\nthe magnetocrystalline anisotropy constant, as given by\nBruno’s formula [ 68]. In our case, the large differences\nin orbital moments for FeB-CoB alloys correlate with the\nhigh magnetic anisotropy energies shown in Fig. 2. The\nvalues of the orbital magnetic moments calculated in the\nPBE approximation are about twice as small as the ex-\nperimental ones, as is the case with bcc Fe [ 69].\nSince the FPLO code used in this work is based on\nthe method of linear combination of atomic orbitals, we\ncan perform a population analysis using the Mulliken ap-\nproach [ 70]. In Fig. 4we show the excess electrons on the\nB site (the natural consequence is the exactly opposite\nvalue of excess electrons on the transition metal site). As\nfor the spin magnetic moment dependence, a clear maxi-\nmum in excess electrons occurs for MnB. Analysis of the\noccupancy of each orbital shows that the maximum of ex-\ncess electrons correlates with the occupancy of the B 2 p\nand Fe 4 porbitals. The occupancy of the other valence\norbitals changes linearly or remains constant throughout6\nthe whole range of considered alloys (not shown). The\ndensities of states (also not shown) reveal hybridization\nbetween the transition metal 4 porbitals and boron 2 p\norbitals.\nNow we will return to Fig. 2and analyze the concen-\ntration dependence of magnetocrystalline anisotropy en-\nergy. The literature on magnetic anisotropy of magnetic\nmonoborides is very scarce. We are only aware of the\nfirst-principles study predicting magnetization easy axis\nfor MnB [ 36] and the experimental work of Zhdanova\nand coworkers in which they determined the room-\ntemperature effective magnetocrystalline anisotropy con-\nstant K∗\n1= 0.4 MJ m−3for orthorhombic FeB [ 2]. This\nvalue leads to a quite promising magnetic hardness of\n0.83. The relatively high magnetic hardness is not re-\nflected in the coercivity of FeB, which at 290 K is equal\nto 10 Oe only [ 21]. (A similarly low coercivity value,\nequal to 15.9 Oe, was measured for MnB [ 10].) However,\nthe relatively high magnetic hardness and low coerciv-\nity of FeB do not necessarily contradict each other. As\nRades and coworkers showed for α-FeB, depending on\nthe size and crystallinity of the particles, soft or hard\nferromagnetism is possible to obtain [ 21]. Therefore, it\nis conceivable that appropriate control of the microstruc-\nture could also improve the coercivity value of the β-FeB\nphase.\nIn Fig. 2(c), we show the calculated magnetocrystalline\nanisotropy energy and magnetic hardness. As a result\nof determining the magnetic easy axis among the three\nmain crystallographic directions, see also Fig. 1, we can\ndivide the analyzed concentration range into areas char-\nacterized by a specific easy axis. For MnB and FeB,\neasy magnetic axis is in both cases [010] (along the axis\nof lattice parameter b) and the values of MAE are of\n0.45 and 0.14 MJ m−3, respectively, see Table II. In the\nMnB-FeB range, we observe magnetic hardness maxima\nat Mn 0.55Fe0.45B and Mn 0.15Fe0.85B of about 0.83 and\n0.95, respectively, with corresponding MAE values equal\nto 0.77 and 0.69 MJ m−3. The highest maximum of MAE\nequal to 2.9 MJ m−3we observe for Fe 0.5Co0.5B. Such\na high value of anisotropy energy together with a low\nvalue of magnetic moment leads to an exceptionally high\nmagnetic hardness of 5.1. Although such a high value of\nmagnetic hardness is determined in an approximate man-\nner from PBE for zero kelvin and probably significantly\noverestimated by VCA, we still expect experimental con-\nfirmation of elevated magnetic hardness in the concen-\ntration range from about FeB to Fe 0.5Co0.5B at room\ntemperatures. Similarly, very high MAE value (about\n10 MJ m−3) was predicted previously for rare-earth free\nFeCo alloys with tetragonal deformation [ 71].\nWe must keep in mind that the magnetocrystalline\nanisotropy energies presented in this work were calcu-\nlated in zero kelvin and may change with temperature.\nAs previous measurements for halfborides have shown,\nwith temperature the anisotropy constants can change in\na nonlinear fashion [ 4,72]. Nevertheless, values close to\nthe anisotropy energy maximum observed for low temper-\natures in the concentration range reappear in the temper-0.5 1.0 1.5 2.00.00.51.01.5\nLDA BH LDA PW92GGA PBE\nSpin magnetic moment ( µB formula unit-1)Magnetocrystalline anisotropy energy  (MJ m-3)\nexchange only LDAGGA PBEexpt. (RT)\n(Kanaizuka 1981,\nZhdanova 2013)\nFIG. 5. The dependence of the magnetocrystalline anisotrop y\nenergy (MAE) on a fixed spin moment calculated for FeB\n(s.g. P nma ) using the FPLO18 code with Perdew-Burke-\nErnzerhof (PBE) exchange-correlation potential; togethe r\nwith the equilibrium values of MAE and magnetic moment ob-\ntained with PBE, Barth-Hedin (BH), Perdew-Wang (PW92),\nand LDA exchange only potentials. The experimental (room-\ntemperature) values of magnetic moment and effective mag-\nnetic anisotropy constant are taken from Refs. [ 1] and [ 2].\nature dependence of the anisotropy constants. We con-\nclude, that also the maxima in the measured temperature\ndependence of the anisotropy constants for monoborides\nshould have values close to the anisotropy energy maxima\ncalculated at zero kelvin. For the MnB-FeB-CoB alloys\nconsidered here, it would be advantageous to experimen-\ntally verify the calculation results extended by measuring\nthe temperature dependence of the anisotropy constants.\nB. Fixed spin moment on FeB\nTABLE III. Magnetic properties of FeB ( Pnma ) calculated\nwith FPLO18 code using several exchange-correlation po-\ntentials: Perdew-Burke-Ernzerhof (PBE), Barth-Hedin (BH ),\nPerdew-Wang (PW92), and LDA exchange only . The mag-\nnetic easy axis, spin ( ms) and orbital ( ml) magnetic moments\n(µBatom−1), magnetocrystalline anisotropy energy [MAE\n(MJ m−3)], and magnetic hardness ( κ) are presented. Orbital\nmagnetic moments are given for corresponding magnetic easy\naxes.\nXC potential Axis ms(Fe) ml(Fe) ms(B)ml(B) MAE κ\nGGA PBE 010 1.33 0.025 -0.13 0.001 0.14 0.49\nLDA BH 100 1.21 0.025 -0.09 0.000 0.01 0.13\nLDA PW92 010 1.25 0.023 -0.10 0.000 0.01 0.08\nLDA ex. only 010 1.46 0.024 -0.15 0.001 0.85 1.10\nIn our previous work on CeFe 12-based alloys, we\nshowed that the calculated MAE value depends on the\nchoice of the exchange-correlation potential [ 61], and that\nthis dependency is correlated with the magnetic moment.\nHere, we performed MAE calculations for the selected\nFeB compound using four different forms of exchange-7\ncorrelation potential: Perdew-Burke-Ernzerhof (PBE),\nBarth-Hedin (BH), Perdew-Wang (PW92), and exchange\nonly LDA (for comparison only). The obtained equilib-\nrium results, see Fig. 5(green dots) and Table III, show\nthe scatter of the magnetic moments and MAE. The total\nspin magnetic moment from GGA-PBE is 1.20 µBf.u.−1,\nwhereas the experimental one is 1.15 µBf.u.−1[1]. The\ninduced spin magnetic moments on B are proportional to\nthe moments on Fe, see Table III. The total spin magnetic\nmoments from the GGA and proper (not exchange only)\nLDA differ by about 0.1 µB, which correlates with a MAE\nvariance of about 0.13 MJ m−3. However, even the higher\nMAE value of 0.14 MJ m−3determined from the PBE ap-\nproximation is much smaller than the experimental effec-\ntive magnetic anisotropy constant K∗\n1= 0.4 MJ m−3[2].\nIn addition to the approximation of exchange-correlation\npotential, the divergence may have its origin in slightly\ndifferent definitions of MAE and K∗\n1, as well as in the\ntemperature dependence of the parameters. We calcu-\nlated MAE at 0 K and in the experiment K∗\n1is deter-\nmined at room temperature.\nAs we can see from the above calculations, there may\nbe a correlation between the determined MAE value and\nthe magnetic moment. To verify this, we performed a\nfully-relativistic fixed spin magnetic moment (FSM) cal-\nculation to determine the dependence of the MAE on\nthe magnetic moment. In Fig. 5, we can see a fairly\ngood correspondence between the equilibrium points cal-\nculated using different forms of exchange-correlation po-\ntential (green dots) and the curve of the MAE depen-\ndence on fixed spin magnetic moment (red line). We also\nsee that the equilibrium results, except LDA exchange\nonly, lie between the two MAE maxima, and both in-\ncreasing and decreasing the magnetic moment of the sys-\ntem can lead to an increase in the MAE [characteristic\nalso visible around FeB in Fig. 2(c)]. In practice, the\nmagnetic moment of FeB can be decreased by alloying\nwith Co and increased by alloying with Mn (or small\namounts of Cr) [ 6]. The value of the magnetic moment is\nalso affected by the temperature, hence the dependence\nof MAE on temperature can be nonlinear [ 3,4].\nThe analysis of the PBE results against the experi-\nment and other exchange-correlation functionals lends\ncredence to the use of the PBE approximation, while\npointing out its inherent limitations. Similar to the VCA\nfindings presented above, the fixed spin moment results\nare promising regarding application of monoborides as\npermanent magnets and suggest the possibility of modi-\nfying the composition to increase the magnetic hardness\nof the alloys above unity.\nC. MnB and FeB alloys with\n3d, 4d, and 5 dtransition metals\nA previous DFT study of the alloying of (Fe,Co) 2B\nhalfborides with 5 delements predicted a significant in-\ncrease in MAE for Re doping (from about 0.75 to\n1.4 MJ m−3), which found partial experimental confir-mation (increase from 0.5 to 0.75 MJ m−3) [3]. Here, we\nwill discuss how the substitution of MnB and FeB with\ntransition metals affects the magnetic hardness of the\nmonoborides. To this end, we used the supercell method\nto calculate the properties of series of Mn 11X1B12and\nFe11X1B12alloys with transition metals X. We assumed\nthat the transition metal atom substitutes one of the\ntwelve Mn or Fe atoms in the supercell, see also Sec. II:\nComputational details . The results of calculated mag-\nnetic moments and magnetocrystalline anisotropy ener-\ngies are presented in Table IVand Figs. 6and7. In\nmost cases, we see clear trends in the dependence on the\natomic number of the transition metal. The trends are\noften analogous to the VCA results presented earlier in\nthis work and supercell results from our previous work\non CeFe 12based alloys [ 61].\nFigure 6presents the calculated magnetic properties of\nMnB alloys. Figure 6(a) shows that of all the systems,\nthe highest total spin magnetic moment has the MnB\nwithout substitutions. It is analogous to the experimen-\ntal results discussed above [ 1,6]. We can also see that the\nmagnetic moment dependencies on the type of substitu-\ntion are similar for all three transition metal groups: 3 d,\n4dand 5 d. This is because, when we substitute Mn with\ntransition metals X we change the filling of the valence\nband.\nFigure 6(b) shows the spin magnetic moments at the\ndopant site. We can see that transition metal atoms\nare not solely non-magnetic dopants that lower the to-\ntal magnetic moment, but they undergo spin polariza-\ntion in the ferromagnetic medium and contribute to the\ntotal magnetic moment. The magnetic contributions are\npositive or negative (parallel or antiparallel to the total\nmagnetic moment), with a maxima near the center of the\nperiods. The considered dupants also affect the magnetic\ninteractions between Mn atoms.\nFigure 6(c) shows the orbital magnetic moments on\nthe dopant site. The values of these moments oscillate\nbetween around -0.03 and +0.05 µBatom−1, which is\nsmall compared to the spin magnetic moments of about\n1.6 – 2.0 µBf.u.−1in the MnB alloys.\nFigures 6(d) and 6(f) show the magnetocrystalline\nanisotropy energies (MAE and DE 32). As previously\nmentioned, the direction of the magnetic easy axis for\nMnB is [010]. In most cases, substituting MnB with\ntransition metals changes this preference to [001], see Ta-\nbleIV. MnB has the highest MAE value of 0.67 MJ m−3.\nThis value translates into magnetic hardness of 0.7, see\nFig. 6(e), and no dopant raises the magnetic hardness\nabove 0.8. Based on the above, we conclude that all Mn\nmonoborides doped with transition metals (Mn 11X1B12)\ncan be classified as soft ( κ <0.5) or semi-hard (0 .5< κ <\n1) magnetic materials.\nThe observed trends of magnetic moments for\nFe11X1B12alloys, see Fig. 7and Table IV, resemble those\njust presented for MnB alloys. However, the magnetic\nhardnesses of FeB alloys appear more promising.\nWe will start non-sequentially with Fig. 7(c) show-\ning the orbital magnetic moments on dopant sites. The8\n1.61.71.81.92.0\n3dSpin magnetic moment ( µB f.u.-1)\nLu   Hf   Ta    W   Re   Os    Ir    Pt   Au   HgSc   Ti    V     Cr   Mn  Fe   Co   Ni   Cu   Zn\nY     Zr   Nb   Mo  Tc   Ru   Rh   Pd  Ag   Cd 4d\n5d(a)\n-0.50.00.51.01.52.02.5\n3dSpin magnetic moment ( µB per TM atom)\nLu   Hf   Ta    W   Re   Os    Ir    Pt   Au   HgSc   Ti    V     Cr   Mn  Fe   Co   Ni   Cu   Zn\nY     Zr   Nb   Mo  Tc   Ru   Rh   Pd  Ag   Cd 4d\n5d(b)\n-0.020.000.020.04\n3dOrbital magnetic moment ( µB  per TM atom)\nLu   Hf   Ta    W   Re   Os    Ir    Pt   Au   HgSc   Ti    V     Cr   Mn  Fe   Co   Ni   Cu   Zn\nY     Zr   Nb   Mo  Tc   Ru   Rh   Pd  Ag   Cd 4d\n5d(c)\n0.00.20.40.60.8\n3dMagnetocrystalline anisotropy energy (MJ m-3)\nLu   Hf   Ta    W   Re   Os    Ir    Pt   Au   HgSc   Ti    V     Cr   Mn  Fe   Co   Ni   Cu   Zn\nY     Zr   Nb   Mo  Tc   Ru   Rh   Pd  Ag   Cd 4d\n5d(d)\n0.00.20.40.60.8\n3dMagnetic hardness\nLu   Hf   Ta    W   Re   Os    Ir    Pt   Au   HgSc   Ti    V     Cr   Mn  Fe   Co   Ni   Cu   Zn\nY     Zr   Nb   Mo  Tc   Ru   Rh   Pd  Ag   Cd 4d\n5d(e)\n0.00.51.01.52.0\n3dEnergy difference  DE32 (MJ m-3)\nLu   Hf   Ta    W   Re   Os    Ir    Pt   Au   HgSc   Ti    V     Cr   Mn  Fe   Co   Ni   Cu   Zn\nY     Zr   Nb   Mo  Tc   Ru   Rh   Pd  Ag   Cd 4d\n5dOsIr (f)\nFIG. 6. The magnetic properties of MnB alloys with 3 d, 4d, and 5 dtransition metals X (Mn 11X1B12): (a) total spin magnetic\nmoment (per formula unit of two atoms); (b, c) spin and orbita l magnetic moments on dopant atoms X (per TM atom); (d)\nmagnetocrystalline anisotropy energy (MAE); (e) magnetic hardness; and (f) energy difference (DE 32) between the two higher\nenergies among the three total energies calculated along th e three main crystallographic axes. The calculations were p erformed\nwith FPLO18 code using the PBE functional and supercell appr oach. Full circles indicate results for the base material (M nB)\nwithout dopant of other type.\n0.91.01.11.21.3\n3dSpin magnetic moment ( µB  f.u.-1)\nLu   Hf   Ta    W   Re   Os    Ir    Pt   Au   HgSc   Ti    V     Cr   Mn  Fe   Co   Ni   Cu   Zn\nY     Zr   Nb   Mo  Tc   Ru   Rh   Pd  Ag   Cd 4d\n5d(a)\n0.00.51.01.52.0\n3dSpin magnetic moment ( µB per TM atom)\nLu   Hf   Ta    W   Re   Os    Ir    Pt   Au   HgSc   Ti    V     Cr   Mn  Fe   Co   Ni   Cu   Zn\nY     Zr   Nb   Mo  Tc   Ru   Rh   Pd  Ag   Cd 4d\n5d(b)\n-0.020.000.020.04\n3dOrbital magnetic moment ( µB per TM atom)\nLu   Hf   Ta    W   Re   Os    Ir    Pt   Au   HgSc   Ti    V     Cr   Mn  Fe   Co   Ni   Cu   Zn\nY     Zr   Nb   Mo  Tc   Ru   Rh   Pd  Ag   Cd 4d\n5d(c)\n0.00.20.40.60.81.0\n3dMagnetocrystalline anisotropy energy (MJ m-3)\nLu   Hf   Ta    W   Re   Os    Ir    Pt   Au   HgSc   Ti    V     Cr   Mn  Fe   Co   Ni   Cu   Zn\nY     Zr   Nb   Mo  Tc   Ru   Rh   Pd  Ag   Cd 4d\n5d(d)\n0.00.20.40.60.81.01.2\n3dMagnetic hardness\nLu   Hf   Ta    W   Re   Os    Ir    Pt   Au   HgSc   Ti    V     Cr   Mn  Fe   Co   Ni   Cu   Zn\nY     Zr   Nb   Mo  Tc   Ru   Rh   Pd  Ag   Cd 4d\n5d(e)\n0.00.51.01.52.0\n3dEnergy difference DE32 (MJ m-3)\nLu   Hf   Ta    W   Re   Os    Ir    Pt   Au   HgSc   Ti    V     Cr   Mn  Fe   Co   Ni   Cu   Zn\nY     Zr   Nb   Mo  Tc   Ru   Rh   Pd  Ag   Cd 4d\n5dReOs (f)\nFIG. 7. The magnetic properties of FeB alloys with 3 d, 4d, and 5 dtransition metals X (Fe 11X1B12): (a) total spin magnetic\nmoment (per formula unit of two atoms); (b, c) spin and orbita l magnetic moments on dopant atoms X (per TM atom); (d)\nmagnetocrystalline anisotropy energy (MAE); (e) magnetic hardness; and (f) energy difference (DE 32) between the two higher\nenergies among the three total energies calculated along th e three main crystallographic axes. The calculations were p erformed\nwith FPLO18 code using the PBE functional and supercell appr oach. Full circles indicate results for the base material (F eB)\nwithout dopant of other type.9\nTABLE IV. The magnetic easy axis, magnetocrystalline aniso tropy energy [MAE (MJ m−3)], total spin magnetic moment [ m\n(µBf.u.−1); formula unit consists of two atoms], magnetic hardness [ κ], orbital magnetic moment [ ml(µBper TM atom)] for\n[001] quantization axes, and spin magnetic moment on substi tuting atom [ ms(X) ( µBper TM atom)] calculated for Mn 11X1B12\nand Fe 11X1B12supercells with various 3 d, 4d, and 5 dtransition metal elements X. Calculations were performed w ith the\nFPLO18 code using the PBE functional.\nMn11XB12 Fe11XB12\n3delements\nAlloy Axis MAE m κ m l(X)ms(X) Alloy Axis MAE m κ m l(X)ms(X)\nMn0.917Sc0.083B 001 0.08 1.60 0.30 0.003 -0.21 Fe 0.917Sc0.083B 010 0.52 1.10 1.03 0.002 -0.06\nMn0.917Ti0.083B 010 0.03 1.62 0.18 0.007 -0.31 Fe 0.917Ti0.083B 010 0.86 1.26 1.15 0.005 0.09\nMn0.917V0.083B 001 0.04 1.71 0.19 0.014 -0.22 Fe 0.917V0.083B 010 0.74 1.25 1.08 0.005 0.35\nMn0.917Cr0.083B 001 0.20 1.83 0.41 0.013 0.53 Fe 0.917Cr0.083B 010 0.57 1.34 0.89 0.004 1.40\nMnB 010 0.67 1.98 0.70 0.022 2.18 Fe 0.917Mn0.083B 010 0.41 1.29 0.78 0.012 1.84\nMn0.917Fe0.083B 001 0.19 1.97 0.78 0.043 1.99 FeB 010 0.19 1.22 0.56 0.033 1.3 4\nMn0.917Co0.083B 001 0.08 1.89 0.25 0.049 1.02 Fe 0.917Co0.083B 100 0.03 1.12 0.25 0.010 0.02\nMn0.917Ni0.083B 001 0.14 1.80 0.35 0.007 0.19 Fe 0.917Ni0.083B 010 0.03 1.05 0.24 -0.004 -0.10\nMn0.917Cu0.083B 001 0.28 1.74 0.52 -0.004 -0.01 Fe 0.917Cu0.083B 010 0.2 1.00 0.69 -0.003 -0.06\nMn0.917Zn0.083B 001 0.08 1.65 0.29 -0.001 -0.05 Fe 0.917Zn0.083B 010 0.32 0.98 0.91 -0.001 -0.03\n4delements\nMn0.917Y0.083B 010 0.01 1.58 0.10 0.000 -0.13 Fe 0.917Y0.083B 010 0.38 1.06 0.91 0.001 -0.05\nMn0.917Zr0.083B 010 0.06 1.60 0.26 0.004 -0.18 Fe 0.917Zr0.083B 010 0.74 1.25 1.08 0.002 0.02\nMn0.917Nb0.083B 001 0.04 1.69 0.21 0.006 -0.16 Fe 0.917Nb0.083B 010 0.74 1.22 1.10 0.002 0.08\nMn0.917Mo0.083B 001 0.19 1.79 0.41 0.003 -0.02 Fe 0.917Mo0.083B 010 0.72 1.22 1.09 -0.001 0.20\nMn0.917Tc0.083B 001 0.18 1.88 0.39 0.004 0.31 Fe 0.917Tc0.083B 010 0.60 1.26 0.97 0.003 0.44\nMn0.917Ru0.083B 001 0.16 1.93 0.36 0.013 0.57 Fe 0.917Ru0.083B 010 0.10 1.18 0.41 0.014 0.23\nMn0.917Rh0.083B 001 0.24 1.88 0.44 0.019 0.32 Fe 0.917Rh0.083B 100 0.10 1.11 0.46 0.008 0.02\nMn0.917Pd0.083B 001 0.10 1.80 0.30 0.004 0.07 Fe 0.917Pd0.083B 100 0.11 1.04 0.51 -0.001 -0.04\nMn0.917Ag0.083B 001 0.21 1.74 0.45 -0.003 -0.01 Fe 0.917Ag0.083B 010 0.04 0.98 0.33 -0.003 -0.04\nMn0.917Cd0.083B 010 0.05 1.65 0.23 -0.003 -0.04 Fe 0.917Cd0.083B 010 0.13 0.94 0.60 -0.002 -0.03\n5delements\nMn0.917Lu0.083B 001 0.06 1.59 0.26 0.002 -0.12 Fe 0.917Lu0.083B 010 0.16 1.07 0.59 0.002 -0.05\nMn0.917Hf0.083B 010 0.07 1.60 0.28 0.004 -0.17 Fe 0.917Hf0.083B 010 0.68 1.24 1.05 -0.002 0.00\nMn0.917Ta0.083B 001 0.19 1.68 0.44 -0.002 -0.18 Fe 0.917Ta0.083B 010 0.77 1.22 1.13 -0.006 0.04\nMn0.917W0.083B 001 0.33 1.78 0.55 -0.019 -0.08 Fe 0.917W0.083B 010 0.62 1.19 1.04 -0.016 0.11\nMn0.917Re0.083B 010 0.15 1.86 0.36 -0.023 0.12 Fe 0.917Re0.083B 010 0.36 1.24 0.76 -0.015 0.29\nMn0.917Os0.083B 010 0.22 1.91 0.42 -0.002 0.41 Fe 0.917Os0.083B 010 0.01 1.18 0.14 0.028 0.25\nMn0.917Ir0.083B 001 0.58 1.88 0.69 0.038 0.31 Fe 0.917Ir0.083B 100 0.30 1.11 0.78 0.041 0.06\nMn0.917Pt0.083B 001 0.64 1.81 0.75 0.033 0.10 Fe 0.917Pt0.083B 100 0.12 1.03 0.53 0.017 -0.02\nMn0.917Au0.083B 001 0.08 1.75 0.28 0.011 0.00 Fe 0.917Au0.083B 100 0.02 0.96 0.22 0.000 -0.04\nMn0.917Hg0.083B 010 0.56 1.67 0.76 -0.002 -0.03 Fe 0.917Hg0.083B 010 0.12 0.91 0.60 -0.004 -0.03\ntrends resemble very much the ones for MnB alloys pre-\nsented above. However, the orbital moments of FeB al-\nloys are slightly lower than those of MnB alloys. The\nvalue of about 0.03 µBon Fe in FeB is similar to the value\nof 0.043 µBcalculated for bcc Fe. Both values, however,\nare significantly underestimated relative to the experi-\nmental orbital moment in bcc Fe equal to 0.086 µB[69].\nThe observed dependencies of the orbital moment on the\natomic number of the 5 delements resemble the earlier\ntheoretical results for 5 ddopants in Fe [ 73], CeFe 12[61],\nand Fe 5PB2[48].\nIn Fig. 7(b), we observe the spin polarization of\ndopants in a magnetic medium of FeB. The dependen-\ncies look similar to the results for MnB alloys. Related\ntrends in spin magnetic moments for the transition metal\nelements in the 3 dmagnetic media (Fe and Ni) have\nbeen calculated before [ 73,74]. Similar magnetic be-\nhavior of 5 ddopants in Fe has also been measured by\nspin-dependent absorption [ 75].In Fig. 7(a) we see the total spin magnetic moments\nof Fe 11X1B12monoborides. Alloying of FeB with tran-\nsition metals from periodic groups above the Fe group\nsystematically decreases the total moment. In con-\ntrast, although several elements in the groups preced-\ning Fe continue the trend of increasing total moment, it\nquickly collapses, and does not pursue values close to the\n2.0µBf.u.−1observed for MnB. Among the Fe 11X1B12\nalloys, the FeB alloy with Cr has the highest total\nspin magnetic moment of 1.34 µBf.u.−1, compared to\n1.22 µBf.u.−1for FeB alone.\nThe direction of the magnetic easy axis for FeB is\n[010] and in most cases, substituting FeB with tran-\nsition metals does not change the easy axis, see Ta-\nbleIV. Figures 7(d) and 7(e) present the magnetocrys-\ntalline anisotropy energies and corresponding magnetic\nhardnesses. The MAE clearly increases for substitutions\nfrom groups preceding Fe, while it remains at a rela-\ntively low level for substitutions from groups succeeding10\nFe. Such sufficiently high MAE values, in combination\nwith low magnetic moments, lead to a magnetic hard-\nness above unity, i.e. qualify these alloys as magneti-\ncally hard. Fe 11Ti1B12has the highest magnetic hard-\nness equal to 1.15. Thus, it is a prospective candidate\nfor permanent magnets, together with other FeB alloys\nwith Sc, V, Zr, Nb, Mo, Ta or W (all indicating magnetic\nhardness above unity). In previous work, we predicted\nthat similar transition metal substitutions increases the\nmagnetic hardness of CeFe 12-based alloys [ 61].\nIn the case of the considered monoborides with or-\nthorhombic structure, we defined the MAE as the energy\ndifference between two lower total energies among three\ntotal energies calculated for magnetization along three\nmain crystallographic axes. The energy difference be-\ntween two higher energies (DE 32) complements the MAE.\nFor the listed above Fe 11X1B12alloys with κ >1, the\nDE32is small (about 0.1 MJ m−3), indicating uniaxial-\nlike anisotropy. For Fe 11Os1B12, the DE 32is about\n2.0 MJ m−3, which together with a MAE close to zero,\nimplies anisotropy in type of easy plane. In contrast, the\nmagnetic anisotropy of systems with intermediate MAE\nand DE 32values is characteristic for orthorhombic sys-\ntems.\nIn conclusion, from the perspective of application for\npermanent magnets, the most promising are FeB al-\nloys with selected transition metals. Another promising\nstrategy could be to substitute of selected MnB-FeB or\nFeB-CoB alloys with small amounts of transition met-\nals, which, as previously shown, can yield beneficial re-\nsults [ 3].\nIV. SUMMARY AND CONCLUSIONS\nMagnetic monoborides have sufficiently high values of\nmagnetization and Curie temperature that some of them\ncan be considered for permanent magnet applications.\nHowever, their main representatives, which are MnB and\nFeB, show very low values of coercivity. Probably for\nthis reason, there has been almost no studies on the\nmagnetocrystalline anisotropy of magnetic monoborides\nto date. The only exception is an experiment on FeB\nmonocrystals, which, surprisingly, classified it as semi-\nhard magnetic and close to magnetically hard material.\nIn this work, by performing first-principles calcula-tions, we have taken a step towards determining the mag-\nnetic anisotropic properties of magnetic monoborides.\nWe considered the entire range of monoboride solid so-\nlutions from CrB, through MnB, FeB, to CoB and, plus\na whole series of MnB and FeB alloys substituted with\nthe transition metal elements 3 d, 4dand 5 d. We found\nthat in the concentration range from CrB to CoB, the\ncalculated magnetic easy axis changes direction several\ntimes. In the range between MnB and FeB, the deter-\nmined magnetic hardness approaches unity three times,\nwhereby exceeding unity classifies the material as mag-\nnetically hard. In the range between FeB and CoB, the\nmagnetic hardness increases with the Co concentration\nto exceed an impressive five in the middle of the range.\nAlthough the Curie temperature is there close to room\ntemperature, we expect that high magnetic hardness will\nalso occur at lower Co concentrations, showing somewhat\nhigher Curie temperatures.\nFor alloys of MnB and FeB with transition metals, we\nhave identified characteristic trends in magnetic moments\nand magnetic hardness along the transition metal peri-\nods. In the case of FeB alloys, most transition metals\nfrom groups preceding Fe increase the magnetic hardness\nof the alloys. While the magnetic hardness of MnB alloys\nremains low or medium, for several FeB alloys it exceeds\nunity. As magnetically hard, we have classified FeB al-\nloys (Fe 11X1B12) with Sc, Ti, V, Zr, Nb, Mo, Hf, Ta\nor W. The presented calculation results encourage their\nverification by experimental methods, with additional ex-\ntension of the measurements to include the temperature\ndependence of magnetic anisotropy.\nACKNOWLEDGMENTS\nWe gratefully acknowledge financial support from the\nNational Science Center Poland under decisions DEC-\n2019/35/O/ST5/02980 (PRELUDIUM-BIS 1) and DEC-\n2021/41/B/ST5/02894 (OPUS 21). We thank Zbigniew\nŚniadecki and Justyna Rychły-Gruszecka for valuable\ncomments. We thank Paweł Leśniak and Daniel Depcik\nfor compiling the scientific software and administering\nthe computational cluster at the Institute of Molecular\nPhysics, Polish Academy of Sciences. Part of the cal-\nculation was made in the Poznan Supercomputing and\nNetworking Centre (PSNC/PCSS).\n[1]T. Kanaizuka, Invar like properties of transition\nmetal monoborides Mn 1-xCrxB and Mn 1-xFexB,\nMater. Res. Bull. 16, 1601 (1981) .\n[2]O. V. Zhdanova, M. B. Lyakhova, and Yu. G. Pas-\ntushenkov, Magnetic properties and domain structure of\nFeB single crystals, Met. Sci. Heat Treat. 55, 68 (2013) .\n[3]A. Edström, M. Werwiński, D. Iuşan, J. Rusz, O. Eriks-\nson, K. P. Skokov, I. A. Radulov, S. Ener, M. D.\nKuz’min, J. Hong, M. Fries, D. Yu. Karpenkov, O. Gut-\nfleisch, P. Toson, and J. Fidler, Magnetic properties of(Fe1-xCox)2B alloys and the effect of doping by 5d ele-\nments, Phys. Rev. B 92, 174413 (2015) .\n[4]T. N. Lamichhane, O. Palasyuk, V. P. Antropov,\nI. A. Zhuravlev, K. D. Belashchenko, I. C. Nlebedim,\nK. W. Dennis, A. Jesche, M. J. Kramer, S. L. Bud’ko,\nR. W. McCallum, P. C. Canfield, and V. Taufour,\nReinvestigation of the intrinsic magnetic properties of\n(Fe1-xCox)2B alloys and crystallization behavior of rib-\nbons, J. Magn. Magn. Mater. 513, 167214 (2020) .\n[5]N. Lundquist, H. P. Myers, and R. Westin, The param-11\nagnetic properties of the monoborides of V, Cr, Mn, Fe,\nCo and Ni, Philos. Mag. 7, 1187 (1962) .\n[6]M. C. Cadeville and E. Daniel, Sur la structure élec-\ntronique de quelques borures d’éléments de transition,\nJ. Phys. 27, 449 (1966) .\n[7]S. Carenco, D. Portehault, C. Boissière, N. Méza-\nilles, and C. Sanchez, Nanoscaled Metal Borides and\nPhosphides: Recent Developments and Perspectives,\nChem. Rev. 113, 7981 (2013) .\n[8]Z. Pu, T. Liu, G. Zhang, X. Liu, Marc. A.\nGauthier, Z. Chen, and S. Sun, Nanostructured\nMetal Borides for Energy-Related Electrocataly-\nsis: Recent Progress, Challenges, and Perspectives,\nSmall Methods 5, 2100699 (2021) .\n[9]M. Fries, Z. Gercsi, S. Ener, K. P. Skokov, and O. Gut-\nfleisch, Magnetic, magnetocaloric and structural proper-\nties of manganese based monoborides doped with iron\nand cobalt – A candidate for thermomagnetic generators,\nActa Mater. 113, 213 (2016) .\n[10]S. Ma, K. Bao, Q. Tao, P. Zhu, T. Ma, B. Liu,\nY. Liu, and T. Cui, Manganese mono-boride, an inex-\npensive room temperature ferromagnetic hard material,\nSci. Rep. 7, 43759 (2017) .\n[11]J. D. Bocarsly, E. E. Levin, S. A. Humphrey,\nT. Faske, W. Donner, S. D. Wilson, and R. Se-\nshadri, Magnetostructural Coupling Drives Magne-\ntocaloric Behavior: The Case of MnB versus FeB,\nChem. Mater. 31, 4873 (2019) .\n[12]H. M. Sánchez, L. E. Z. Alfonso, J. S. T. Her-\nnandez, D. Salazar, and G. A. P. Alcázar, Improv-\ning the ferromagnetic exchange coupling in hard τ-\nMn53.3Al45.0C1.7and soft Mn 50B50magnetic alloys,\nAppl. Phys. A 126, 843 (2020) .\n[13]S. Klemenz, M. Fries, M. Dürrschnabel, K. Skokov,\nH.-J. Kleebe, O. Gutfleisch, and B. Albert, Low-\ntemperature synthesis of nanoscale ferromagnetic α’-\nMnB, Dalton Trans. 49, 131 (2020) .\n[14]S. Ma, R. Farla, K. Bao, A. Tayal, Y. Zhao, Q. Tao,\nX. Yang, T. Ma, P. Zhu, and T. Cui, An elec-\ntrically conductive and ferromagnetic nano-structure\nmanganese mono-boride with high Vickers hardness,\nNanoscale 13, 18570 (2021) .\n[15]N. Kalyon, A.-M. Zieschang, K. Hofmann, M. Lepple,\nM. Fries, K. P. Skokov, M. Dürrschnabel, H.-J. Kleebe,\nO. Gutfleisch, and B. Albert, CrB-type, ordered α-MnB:\nSingle crystal structure and spin-canted magnetic behav-\nior,APL Mater. 11, 060701 (2023) .\n[16]A. N. Kolmogorov, S. Shah, E. R. Margine, A. F.\nBialon, T. Hammerschmidt, and R. Drautz, New Su-\nperconducting and Semiconducting Fe-B Compounds\nPredicted with an Ab Initio Evolutionary Search,\nPhys. Rev. Lett. 105, 217003 (2010) .\n[17]L. Han, S. Wang, J. Zhu, S. Han, W. Li, B. Chen,\nX. Wang, X. Yu, B. Liu, R. Zhang, Y. Long, J. Cheng,\nJ. Zhang, Y. Zhao, and C. Jin, Hardness, elastic,\nand electronic properties of chromium monoboride,\nAppl. Phys. Lett. 106, 221902 (2015) .\n[18]T. Lundstrom, Structure, defects and properties of some\nrefractory borides, Pure Appl. Chem. 57, 1383 (1985) .\n[19]C. Kapfenberger, B. Albert, R. Pöttgen, and H. Hup-\npertz, Structure refinements of iron borides Fe 2B and\nFeB, Z. Krist. 221, 477 (2006) .\n[20]F. Igoa Saldaña, E. Defoy, D. Janisch, G. Rousse, P.-\nO. Autran, A. Ghoridi, A. Séné, M. Baron, L. Sues-\ncun, Y. Le Godec, and D. Portehault, Revealing theElusive Structure and Reactivity of Iron Boride α-FeB,\nInorg. Chem. 62, 2073 (2023) .\n[21]S. Rades, S. Kraemer, R. Seshadri, and B. Albert, Size\nand Crystallinity Dependence of Magnetism in Nanoscale\nIron Boride, α-FeB, Chem. Mater. 26, 1549 (2014) .\n[22]T. Kanaizuka, Phase diagram of pseudobinary\nCrB-MnB and MnB-FeB systems: Crystal struc-\nture of the low-temperature modification of FeB,\nJ. Solid State Chem. 41, 195 (1982) .\n[23]Y. Wang, L. Li, Y. Wang, D. Song, G. Liu,\nY. Han, L. Jiao, and H. Yuan, Crystalline CoB: Solid\nstate reaction synthesis and electrochemical properties,\nJ. Power Sources 196, 5731 (2011) .\n[24]Z. Guo, L. Zhang, T. Azam, and Z.-S. Wu, Recent ad-\nvances and key challenges of the emerging MBenes from\nsynthesis to applications, MetalMat , e12 (2023) .\n[25]H. Zhang, H. Xiang, F.-z. Dai, Z. Zhang, and\nY. Zhou, First demonstration of possible two-dimensional\nMBene CrB derived from MAB phase Cr 2AlB 2,\nJ. Mater. Sci. Technol. 34, 2022 (2018) .\n[26]Y. Wang, W. Xu, D. Yang, Y. Zhang, Y. Xu, Z. Cheng,\nX. Mi, Y. Wu, Y. Liu, Y. Hao, and G.-Q. Han, Above-\nRoom-Temperature Strong Ferromagnetism in 2D MnB\nNanosheet, ACS Nano 17, 24320 (2023) .\n[27]J.-l. Ma, N. Li, Q. Zhang, X.-b. Zhang, J. Wang,\nK. Li, X.-f. Hao, and J.-m. Yan, Synthesis of porous\nand metallic CoB nanosheets towards a highly effi-\ncient electrocatalyst for rechargeable Na–O 2batteries,\nEnergy Environ. Sci. 11, 2833 (2018) .\n[28]M. Dou, H. Li, Q. Yao, J. Wang, Y. Liu, and F. Wu,\nRoom-temperature ferromagnetism in two-dimensional\ntransition metal borides: A first-principles investigatio n,\nPhys. Chem. Chem. Phys. 23, 10615 (2021) .\n[29]S. Wang, N. Miao, K. Su, V. A. Blatov, and\nJ. Wang, Discovery of intrinsic two-dimensional\nantiferromagnets from transition-metal borides,\nNanoscale 13, 8254 (2021) .\n[30]Z. Jiang, P. Wang, X. Jiang, and J. Zhao, MBene (MnB):\nA new type of 2D metallic ferromagnet with high Curie\ntemperature, Nanoscale Horiz. 3, 335 (2018) .\n[31]C. Liu, B. Fu, H. Yin, G. Zhang, and C. Dong, Strain-\ntunable magnetism and nodal loops in monolayer MnB,\nAppl. Phys. Lett. 117, 103101 (2020) .\n[32]J. Tang, S. Li, D. Wang, Q. Zheng, J. Zhang, T. Lu,\nJ. Yu, L. Sun, B. Sa, B. G. Sumpter, J. Huang, and\nW. Sun, Enriching 2D transition metal borides viaMB\nXMenes (M = Fe, Co, Ir): Strong correlation and mag-\nnetism, Nanoscale Horiz. 9, 162 (2024) .\n[33]B. Wang, D. Y. Wang, Z. Cheng, X. Wang, and\nY. X. Wang, Phase Stability and Elastic Properties\nof Chromium Borides with Various Stoichiometries,\nChemPhysChem 14, 1245 (2013) .\n[34]X. Xu, K. Fu, L. Li, Z. Lu, X. Zhang, Y. Fan, J. Lin,\nG. Liu, H. Luo, and C. Tang, Dependence of the elastic\nproperties of the early-transition-metal monoborides on\ntheir electronic structures: A density functional theory\nstudy, Phys. B Condens. Matter 419, 105 (2013) .\n[35]B. Wang, X. Li, Y. X. Wang, and Y. F.\nTu, Phase Stability and Physical Properties of\nManganese Borides: A First-Principles Study,\nJ. Phys. Chem. C 115, 21429 (2011) .\n[36]J. Park, Y.-K. Hong, H.-K. Kim, W. Lee, C.-D.\nYeo, S.-G. Kim, M.-H. Jung, C.-J. Choi, and O. N.\nMryasov, Electronic structures of MnB soft magnet,\nAIP Adv. 6, 055911 (2016) .12\n[37]J. Hafner, M. Tegze, and Ch. Becker, Amor-\nphous magnetism in Fe-B alloys: First-principles\nspin-polarized electronic-structure calculations,\nPhys. Rev. B 49, 285 (1994) .\n[38]P. Mohn and D. G. Pettifor, The calculated electronic\nand structural properties of the transition-metal mono-\nborides, J. Phys. C Solid State Phys. 21, 2829 (1988) .\n[39]Y. Bourourou, L. Beldi, B. Bentria, A. Gueddouh, and\nB. Bouhafs, Structure and magnetic properties of the 3d\ntransition-metal mono-borides TM–B (TM=Mn, Fe, Co)\nunder pressures, J. Magn. Magn. Mater. 365, 23 (2014) .\n[40]A. Gueddouh, The effects of magnetic moment col-\nlapse under high pressure, on physical properties in\nmono-borides TMB (TM = Mn, Fe): A first-principles,\nPhase Transit. 90, 984 (2017) .\n[41]S. Nishiyama, H. Nakamura, and T. Hat-\ntori, Thermoelectric Properties and Electronic\nDensity of States of Transition Metal Mono-\nborides, FeB, CoB, NiB and their Solid Solutions,\nJ. Ceram. Soc. Jpn. Suppl. 112, S676 (2004) .\n[42]S. Nakamura and Y. Gohda, Prediction of ferromag-\nnetism in MnB and MnC on nonmagnetic transition-\nmetal surfaces studied by first-principles calculations,\nPhys. Rev. B 96, 245416 (2017) .\n[43]X. Tan, Z. Na, R. Zhuo, D. Wang, Y. Zhang, and\nP. Wu, Investigation of CrB as a Potential Gas Sensor\nfor Fault Detection in Eco-Friendly Power Equipment,\nChemosensors 11, 371 (2023) .\n[44]C. Romero-Muñiz, J. Y. Law, L. M. Moreno-Ramírez,\nÁ. Díaz-García, and V. Franco, Using a computation-\nally driven screening to enhance magnetocaloric effect of\nmetal monoborides, J. Phys. Energy 5, 024021 (2023) .\n[45]R. Kadrekar, N. Patel, and A. Arya, Understand-\ning the role of boron and stoichiometric ratio in the\ncatalytic performance of amorphous Co-B catalyst,\nAppl. Surf. Sci. 518, 146199 (2020) .\n[46]H. Zhang, B. Zhao, F.-Z. Dai, H. Xiang, Z. Zhang,\nand Y. Zhou, (Cr 0.2Mn0.2Fe0.2Co0.2Mo0.2)B: A novel\nhigh-entropy monoboride with good electromag-\nnetic interference shielding performance in K-band,\nJ. Mater. Sci. Technol. 77, 58 (2021) .\n[47]D. Hedlund, J. Cedervall, A. Edström, M. Wer-\nwiński, S. Kontos, O. Eriksson, J. Rusz, P. Svedlindh,\nM. Sahlberg, and K. Gunnarsson, Magnetic\nproperties of the Fe 5SiB2- Fe 5PB2system,\nPhys. Rev. B 96, 094433 (2017) .\n[48]M. Werwiński, A. Edström, J. Rusz, D. Hed-\nlund, K. Gunnarsson, P. Svedlindh, J. Cedervall,\nand M. Sahlberg, Magnetocrystalline anisotropy of\nFe5PB2and its alloys with Co and 5d elements:\nA combined first-principles and experimental study,\nPhys. Rev. B 98, 214431 (2018) .\n[49]J. Cedervall, E. Nonnet, D. Hedlund, L. Häggström,\nT. Ericsson, M. Werwiński, A. Edström, J. Rusz,\nP. Svedlindh, K. Gunnarsson, and M. Sahlberg, Influ-\nence of Cobalt Substitution on the Magnetic Properties\nof Fe 5PB2,Inorg. Chem. 57, 777 (2018) .\n[50]S. Pal, K. Skokov, T. Groeb, S. Ener, and O. Gutfleisch,\nProperties of magnetically semi-hard (Fe xCo1-x)3B com-\npounds, J. Alloys Compd. 696, 543 (2017) .\n[51]M. Däne, S. K. Kim, M. P. Surh, D. Åberg, and L. X.\nBenedict, Density functional theory calculations of mag-\nnetocrystalline anisotropy energies for (Fe 1–xCox)2B,\nJ. Phys.: Condens. Matter 27, 266002 (2015) .\n[52]K. Bourzac, The rare-earth crisis, Technol. Rev. 114, 58(2011).\n[53]J. Coey, Permanent magnets: Plugging the gap,\nScr. Mater. 67, 524 (2012) .\n[54]R. Skomski and J. Coey, Magnetic anisotropy —\nHow much is enough for a permanent magnet?,\nScr. Mater. 112, 3 (2016) .\n[55]K. Koepernik and H. Eschrig, Full-potential nonorthogo-\nnal local-orbital minimum-basis band-structure scheme,\nPhys. Rev. B 59, 1743 (1999) .\n[56]I. Opahle, K. Koepernik, and H. Eschrig, Full-\npotential band-structure calculation of iron pyrite,\nPhys. Rev. B 60, 14035 (1999) .\n[57]J. P. Perdew, K. Burke, and M. Ernzerhof, Gen-\neralized gradient approximation made simple,\nPhys. Rev. Lett. 77, 3865 (1996) .\n[58]U. von Barth and L. Hedin, A local exchange-\ncorrelation potential for the spin polarized case. i,\nJ. Phys. C Solid State Phys. 5, 1629 (1972) .\n[59]J. P. Perdew and Y. Wang, Accurate and simple ana-\nlytic representation of the electron-gas correlation ener gy,\nPhys. Rev. B 45, 13244 (1992) .\n[60]K. Momma and F. Izumi, VESTA : A three-dimensional\nvisualization system for electronic and structural analy-\nsis,J. Appl. Crystallogr. 41, 653 (2008) .\n[61]J. Snarski-Adamski and M. Werwiński, Ef-\nfect of transition metal doping on mag-\nnetic hardness of CeFe 12-based compounds,\nJ. Magn. Magn. Mater. 554, 169309 (2022) .\n[62]R. Kuentzler, Specific Heat of Nearly Ferro-\nmagnetic Borides (Co–Ni) 2B and (Fe–Co)B,\nJ. Appl. Phys. 41, 908 (1970) .\n[63]L. Takacs, M. C. Cadeville, and I. Vincze, Mossbauer\nstudy of the intermetallic compounds (Fe 1-xCox)2B and\n(Fe1-xCox)B,J. Phys. F Met. Phys. 5, 800 (1975) .\n[64]P.-H. Lee, S.-H. Chen, Y.-A. Chen, K.-L. Chen, T.-\nW. Wang, S.-W. Wang, B.-S. Wu, W.-S. Su, and S.-\nK. Chen, Calculated Magnetism on Mn 1-xFexB Alloys,\nBAOJ Phys. 2, 007 (2016).\n[65]A. Gueddouh, A. Benghia, and S. Maabed, Effect of\nMn content in Fe(1-x)MnxB (x = 0, 0.25, 0.5, 0.75\nand 1) on physical properties - ab initio calculations,\nMater. Sci.-Pol. 37, 71 (2019) .\n[66]N. A. Klindukhov, V. S. Kasperovich, M. G. Shelyapina,\nand H. El Kebir, Calculation of the electronic structure\nand hyperfine fields for Fe 1-xCoxB and (Fe 1-xCox)2B\ncompounds by the Korringa-Kohn-Rostoker method,\nPhys. Solid State 50, 302 (2008) .\n[67]P. Lee, Z. Xiao, K. Chen, Y. Chen, S. Kao, and T. Chin,\nThe magnetism of Fe(1-x)CoxB alloys: First principle cal-\nculations, Phys. B Condens. Matter 404, 1989 (2009) .\n[68]P. Bruno, Tight-binding approach to the orbital magnetic\nmoment and magnetocrystalline anisotropy of transition-\nmetal monolayers, Phys. Rev. B 39, 865 (1989) .\n[69]C. T. Chen, Y. U. Idzerda, H.-J. Lin, N. V. Smith,\nG. Meigs, E. Chaban, G. H. Ho, E. Pellegrin, and\nF. Sette, Experimental Confirmation of the X-Ray Mag-\nnetic Circular Dichroism Sum Rules for Iron and Cobalt,\nPhys. Rev. Lett. 75, 152 (1995) .\n[70]R. S. Mulliken, Electronic Population Analy-\nsis on LCAO–MO Molecular Wave Functions. I,\nJ. Chem. Phys. 23, 1833 (1955) .\n[71]T. Burkert, L. Nordström, O. Eriksson, and O. Heinonen,\nGiant Magnetic Anisotropy in Tetragonal FeCo Alloys,\nPhys. Rev. Lett. 93, 027203 (2004) .\n[72]K. D. Belashchenko, L. Ke, M. Däne, L. X. Bene-13\ndict, T. N. Lamichhane, V. Taufour, A. Jesche, S. L.\nBud’ko, P. C. Canfield, and V. P. Antropov, Origin of\nthe spin reorientation transitions in (Fe 1–xCox)2B alloys,\nAppl. Phys. Lett. 106, 062408 (2015) .\n[73]P. H. Dederichs, R. Zeller, H. Akai, and H. Ebert,\nAb-initio calculations of the electronic structure of im-\npurities and alloys of ferromagnetic transition metals,J. Magn. Magn. Mater. 100, 241 (1991) .\n[74]H. Akai, Nuclear spin-lattice relaxation of impurities in\nferromagnetic iron, Hyperfine Interact. 43, 253 (1988) .\n[75]R. Wienke, G. Schütz, and H. Ebert, Determi-\nnation of local magnetic moments of 5d impuri-\nties in Fe detected via spin-dependent absorption,\nJ. Appl. Phys. 69, 6147 (1991) ."    },    {        "title": "2403.00709v1.Spin_current_control_of_magnetism.pdf",        "content": "1 \n Spin current control of magnetism  \n \nL. Chen1, Y. Sun1, S. Mankovsky2, T. N. G. Meier1, M. Kronseder3, H. Ebert2, D. Weiss3 and C. \nH. Back1,4,5 \n \n1Department of Physics, Technical University of Munich, Munich, Germany  \n2Department of Chemistry, Ludwig Maximilian University, Munich, Germany \n3Institute of Experimental and Applied Physics, University of Regensburg, Regensburg, \nGermany \n4Munich Center for Quantum Science and Technology, Munich, Germany \n5Center for Quantum Engineering, Technical University of Munich, Munich, Germany  \n \nExploring novel strategies to manipulate the order parameter of magnetic \nmaterials by electrical means is of great importance, not only for advancing our \nunderstanding of fundamental magnetism, but also for unlocking potential \npractical applications. A well-established concept to date uses gate voltages to \ncontrol magnetic properties, such as saturation magnetization, magnetic \nanisotropies, coercive field, Curie temperature and Gilbert damping, by \nmodulating the charge carrier population within a capacitor structure1-5. Note \nthat the induced carriers are non-spin-polarized, so the control via the electric-\nfield is independent of the direction of the magnetization. Here, we show that the \nmagnetocrystalline anisotropy (MCA) of ultrathin Fe films can be reversibly \nmodified by a spin current generated in Pt by the spin Hall effect. The effect 2 \n decreases with increasing Fe thickness, indicating that the origin of the \nmodification can be traced back to the interface. Uniquely, the change in MCA \ndue to the spin current depends not only on the polarity of the charge current but \nalso on the direction of magnetization, i.e. the change in MCA has opposite sign \nwhen the direction of magnetization is reversed. The control of magnetism by the \nspin current results from the modified exchange splitting of majority- and \nminority-spin bands, and differs significantly from the manipulation by gate \nvoltages via a capacitor structure, providing a functionality that was previously \nunavailable and could be useful in advanced spintronic devices. \n \n      Spin torque (spin transfer torque and spin-orbit torque), which involves the \nuse of angular momentum generated by partially spin-polarized and pure spin-polarized \ncurrents, is a well-known method for manipulating the dynamic properties of magnetic \nmaterials. In structures such as giant magnetoresistance or tunnel magnetoresistance \njunctions, the flow of a spin-polarized electric current through the junction imparts spin-\ntransfer torques on the magnetization in the free ferromagnetic layer6-8. In heavy metal \n(HM)/ferromagnet (FM) bilayers, a charge current flowing in the HM induces a spin \naccumulation at the HM/FM interface, and generates spin-orbit torques acting on the \nFM9. These torques serve as a versatile control mechanism for magnetization dynamics, \ninfluencing processes such as magnetization switching10,11, domain wall motion12-14, \nmagnetization relaxation (e.g., Gilbert damping)15, and even auto-oscillations of the 3 \n magnetization16,17. These innovative approaches and their combinations open up a \nspectrum of possibilities for tailoring magnetic properties with potential implications \nfor advanced technologies such as magnetic random access memories9,18. \n \n      While the impact of a spin current on the orientation of the magnetization ( M) \nis widely recognised, there has been no explicit observation so far of successful spin \ncurrent driven manipulation of the magnitude of M  representing the static properties \nof magnetic materials. To explore this prospect, Fig. 1 illustrates the process of spin  \nFig. 1. Schematic of the microscopic mechanism of manipulation and modification of \nmagnetism by a spin current. ( a) The electron spins transmitted into the FM contain both \ntransverse and longitudinal components with respect to M. Due to exchange coupling, the \ntransverse component dephases and is absorbed by M, which gives rise to the damping-\nlike spin-orbit torque and is responsible for changing the direction of M. The longitudinal \ncomponent of the spin current is on average aligned with M, leading to additional filling of \nthe majority band when M is oriented along the + z-direction, and an enhancement of the \nmagnitude M as well as an increase of magnetic anisotropies are expected due to the \nenhanced splitting of the majority- and minority-spin energy bands. ( b) When M is aligned \nalong the – z-direction, the spin-polarized electron enters the minority band, which can lead \nto a decrease of M as well as a decrease of magnetic anisotropies because of the reduction \nof the splitting of the majority- and minority-spin energy bands. ( c) and (d): The same as ( a) \nand (b) but the polarization of the spin current is reversed, which is expected to reduce M \nfor M//+z (c) and to enhance M for M//-z (d). DL-torque M> 0\nMa b\nc d\nDL-torqueM< 0M\nDL-torque M< 0\nM\nDL-torqueM> 0 M\n4 \n current transfer6-8,19-22. The spin accumulation generated by a charge current I, e.g. by \nthe spin Hall effect (SHE) in the heavy metal, contains both transverse and longitudinal \nspin components with respect to M. The incident transverse spin current dephases and \nis absorbed by M, which gives rise to the damping-like spin-orbit torque and is \nresponsible for the change of the direction of M19,20. After the spin transfer and within \nthe spin diffusion length of the ferromagnetic metal, the exiting spin current is on \naverage aligned with M, and the spin-up electron can fill the majority band when M is \nalong the z-direction (Fig. 1a). Due to the enhanced splitting of the majority- and \nminority-spin bands, this leads to an enhancement of M as well as an increase of the \nmagnetic anisotropies. When M is along the z-direction as shown in Fig. 1b, a \ndecrease of M is expected because of the filling of the minority band and the reduction \nof the splitting of the majority- and minority-spin energy bands. Similarly, once the \npolarity of the spin current is reversed by reversing the polarity of I, a decrease (an \nincrease) of M is expected if M // +z (z) as shown in Fig. 1c (Fig. 1d). Therefore, the \nchange of the magnetization M by a spin current is expected to be odd with respect to \nthe inversion of either the charge current or magnetization, i.e., M(I, M) = M(I, \nM) = M(I, M). \n \n      To prove the above scenario, Pt(6 nm)/Al(1.5 nm)/Fe( tFe = 4.5, 2.8, 2.2 and \n1.2 nm) multilayers with different Fe thicknesses tFe are grown on a single two inch \nsemi-insulating GaAs (001) wafer by molecular-beam epitaxy (Fig. 2b and methods). \nThe Pt layer with strong spin-orbit interaction serves as the source of the spin current \ninjected into the Fe layer22, and is thus responsible for the modification of the dynamic \nand static magnetic properties. The Al separation layer is used to avoid the magnetic \nproximity effect between Fe and Pt. Since Al has a weak spin-orbit interaction and a 5 \n long spin diffusion length, we assume that the spin current injected through the Pt/Al \ninterface is transmitted unchanged to the Al/Fe interface and the transparency for the \nspin current remains unchanged. The effective mixing conductance geff↑↓  at the \nPt/Al/Fe interfaces, arising form spin pumping, is determined to be 2.7×1018 m2, which \nresults in an interfacial spin transparency23,24 value Tint of 0.21 (Supplementary Note 2). \n \nThe ultra-thin Fe films grown on GaAs (001) substrates allow us to investigate the \nexpected modification of the magnetic properties for two reasons: Fe/GaAs (001) \nshows I) very low Gilbert damping values  in the sub-nanometer thickness regime (  \n= 0.0076 for tFe = 0.91 nm)25, and thus it is possible to detect the magnetization  \nFig. 2. (a) Schematic of the device for the detection of ferromagnetic resonance by time-\nresolved magneto-optical Kerr microscopy. ( b) Schematic of the Pt/Al/Fe/GaAs (001) \nstructure. ( c) Diagram of crystallographic axes with easy and hard magnetization axes along \n<110> and <110> orientations.  (d) FMR spectra for different dc currents I measured at f = \n12 GHz and I-H = 90o, where I-H is the angle between the magnetic-field and the current \ndirection as shown in the inset. ( e) FMR linewidth (full width at half maximum) as a function \nof dc current for I-H = ±90o; solid lines are the linear fits from which the modulation amplitude \nd(H)/dI is obtained. ( f)I-H-dependence of d(H)/dI. Error bars show the standard error of \nthe least squares fit. The solid line is the calculation result based on the spin Hall effect of \nPt when taking into account the in-plane magnetic anisotropies of Fe (Supplementary Note \n4). \n6 \n dynamics for ultra-thin samples, II) strong interfacial in-plane uniaxial magnetic \nanisotropy (UMA), which is advantageous for the detection of the spin current induced \nmodification of magnetic anisotropies. The UMA originates from the anisotropic \nbonding between Fe and As atoms at the GaAs (001) surface26, where <110>-\norientations are the magnetic easy axes (EA) and <1 10>-orientations are the magnetic \nhard axes (HA) (Fig. 2c). We perform time-resolved magneto-optical Kerr microscopy \n(TR-MOKE) measurements with out-of-plane driving field to characterize both the \nstatic and dynamic magnetic properties of Fe under the influence of spin currents \ngenerated by applying a charge current in Pt (Fig. 2a and methods). \n \nTypical ferromagnetic resonance FMR spectra for tFe = 2.2 nm and for the charge \ncurrent applied along a [110]-oriented device are shown in Fig. 2d. The FMR spectra \nare measured using a fixed microwave excitation frequency f of 12 GHz by sweeping \nthe external magnetic-field H perpendicular to the current, i.e. I-H = 90o, whereI-H is \nthe angle between the current and the magnetic field as defined in the inset of Fig. 2d. \nA clear modification of the FMR spectrum is observed with the application of a dc \ncurrent. Each curve is well fitted by combining a symmetric ( Lsym = H2 / [4(HHR)2 + \nH2]) and an anti-symmetric Lorentzian ( La-sym = 4H(HHR) / [4(HHR)2 + H2]), \nVKerr = VsymLsym + Va-symLa-sym Voffset , where HR is the resonance field, H the full width \nat half maximum, Voffset the offset voltage, and Vsym (Va-sym) the magnitude of the \nsymmetric (anti-symmetric) component of VKerr. It is worth to mention that, by \nanalyzing the position of HR, we have also confirmed that the application of the charge \ncurrents  does not have detrimental effect on the magnetic properties of the Fe films \n(Supplementary Note 3). \n 7 \n The dependence of H on I for I-H = ±90o is summarized in Fig. 2e, which shows \na linear behavior with opposite slopes for I-H = ±90o. This reveals the presence of the \ndamping-like spin-orbit torque, confirming previous reports15. To extract the \nmodification of the linewidth, the I-dependence of H is fitted by  \n                     ∆ H = ∆H0+[d(∆H)/dI]I+c1I2.                     (1)  \nHere H0 is H for I = 0, d(H)/dI quantifies the modification of linewidth by the spin \ncurrent and c1 accounts for a possible Joule heating effect on H. A detailed \nmeasurement of d(H)/dI as a function of I-H shows that d(H)/dI varies strongly \naround the HA. The angular dependence can be well fitted by considering the SHE of \nPt with an effective damping-like spin-orbit torque efficiency of 0.06 \n(Supplementary Note 4). The weaker Bychkov-Rashba-like and Dresselhaus-like spin-\norbit interactions arising from the Fe/GaAs interface play a negligible role in the \nlinewidth modulation27. Note that the distinctive presence of robust UMA at the \nFe/GaAs interface significantly alters the angular dependence of  d(H)/dI. This \ndeviation is remarkable when compared to the sin I-H-dependence of  d(H)/dI as \nobserved in polycrystalline samples, such as Pt/Py28,29. Away from the hard axis (~ \n±15o), the enhanced sensitivity to SHE results from the fact that the potential barrier in \nthe magnetic energy landscape is lowered upon the application of an external magnetic \nfield (Supplementary Note 4). Consequently, the magnetization behaves ‘freely’ with \nno constraints in the vicinity of the HA, since all static torques induced by external \nmagnetic-field and internal magnetic anisotropy fields cancel30. In this case, the \nmagnetic stiffness is significantly reduced allowing a large cone angle of precession, \nwhich increases the sensitivity to SHE31. \n \n      Having identified the modification of the FMR linewidth by SHE, we now focus 8 \n on the modification of the resonance field, which is related to the magnetization and \nmagnetic anisotropies of Fe. Figures 3a and b show, respectively, the I-dependence of \nHR for tFe = 2.8 nm measured at selected frequencies for H applied along the easy axis \n(M // [110]), and along the hard axis ( M // [110]). As shown at the top of each panel \ncolumn, the current is applied along the [100]-orientation, and the direction of the spin \n induced by SHE is along the [010]-orientation with equal projections onto the [110]- \nand [110]-orientations. Therefore, this specific geometry allows a precise comparison \nof the current-induced modification of the resonance field between [110]- and [1 10]- \nFig. 3. I dependence of HR measured at selected frequencies for H along [110]- ( a) and \n[110]-orientations ( b) for tFe = 2.8 nm. For both field orientations, HR(-I) > HR(+I) holds \nwhere HR(-I) and HR(+I) are respectively marked by red and blue arrows in each panel. ( c) \nand (d): the same plots as ( a) and (b) but for tFe = 1.2 nm. In ( c) for H along the [110]-\norientation, HR(-I) > HR(+I) still holds for all measured frequencies. However, for H along \nthe [110]-orientation as shown in ( d), the relative magnitude of HR(-I) and HR(+I) depends \non the excitation frequency, i.e., for f = 12.0 GHz, HR(-I) < HR(+I) holds; for f = 14.0 GHz, \nHR(-I) ~ HR(+I) holds; while for f = 16.0 GHz, HR(-I) > HR(+I) holds. As shown in the upper \npanels, for all the devices, the charge currents are applied along the [100]-orientation, and \nthe direction of the spin accumulation  is along the [010]-direction with equal projections \nonto the [110]- and ൣ110൧ -orientations. This experimental trick allows an accurate \ncomparison of the current-induced modification for [110]- and [1 10]-orientations in the \nsame device.  66.466.867.2 \n112.2112.5112.8113.1\nm0HR (mT)\n-4 -2 0 2 4166.0166.4166.8\nI (mA)173.2173.6174.0174.4\n216.4216.8217.2217.6\nm0HR (mT)\n-4 -2 0 2 4268.4268.8269.2269.6\nI (mA)17.417.718.018.3 \n55.255.555.8\nm0HR (mT)\n-1 0 196.696.997.297.5\nI (mA)241.8242.0242.2242.4242.6\n283.4283.6283.8\nm0HR (mT)\n-1 0 1330.2330.4330.6330.8331.0\nI (mA)a\ntFe= 2.8 nm tFe= 2.8 nm tFe= 1.2 nm tFe= 1.2 nmcb d H\nI // [100]\nf = 12 GHz\nf = 15 GHz\nf = 18 GHzf = 12 GHz\nf = 15 GHz\nf = 18 GHzf = 12 GHz\nf = 14 GHz\nf = 16 GHzf = 12 GHz\nf = 14 GHz\nf = 16 GHzH\nI // [100] H\nI // [100]H\nI // [100]9 \n orientations in the same device. For M // [110] as shown in Fig. 3a, all the HR-I traces \nshow a positive curvature ; while for M // [110] in Fig. 3b, traces with a negative \ncurvature are observed. The positive and negative curvatures along the [110] and [1 10]-\norientations are due to the fact that the Joule heating induced by the charge current \nreduces the magnetization and thus the UMA, resulting in an increase of HR along [110], \nbut a decrease of HR along [1 10]. Apart from the symmetric parabolic dependence \ninduced by Joule heating, a linear component in the I-dependence of HR is also observed \nsince HR(I) ≠ HR(+I) holds. Note that for M along both EA and HA, HR(I) > HR(+I) \nholds for all the measured frequencies. As tFe is reduced to 1.2 nm, the I-dependence of \nHR along the EA is similar to the one with tFe = 2.8 nm and HR(I) > HR(+I) still holds \n(Fig. 3c). However, for M along the [1 10]-orientation as shown in Fig. 3d, the relative \nmagnitude of HR(I) and HR(+I) strongly depends on the excitation frequency, i.e., \nHR(I) < HR(+I) holds for f = 12.0 GHz; HR(I) ~ HR(+I) holds for f = 14.0 GHz but \nHR(I) >HR(+I) holds for f = 16.0 GHz. The frequency-dependent shift of the resonance \nfield indicates that the magnetic properties of Fe are modified by the spin current for \nthinner samples, an observation that has not been reported before. \n \n      To quantify the modification of the magnetic anisotropies, the I-dependence \nof the HR trace is fitted by \n                          HR = HR0 + (dHR/dI)I + c2I2.                 (2) \nHere, HR0 is HR at I = 0, dHR/dI quantifies the modification of HR by the spin current,  10 \n  \nand c2 accounts for Joule heating effects on HR. The f-dependences of dHR/dI  for \ndifferent orientations of M and different Fe thicknesses are summarised in Fig. 4. For \ntFe = 2.8 nm and M // <110>-orientations (EA) as shown in Fig. 4a, dHR/dI  is \nindependent of frequency with a positive zero frequency intercept (~ 0.08 mT/mA) for \nM // [110]. As M is rotated by 180o to the [110]-orientation, the sign of the intercept \nchanges to negative with the same amplitude as the [1 10]-orientation (~ 0.08 mT/mA). \nThis can be understood in terms of the current-induced Oersted field and/or field-like  \nFig. 4. (a) f-dependence of dHR/dI for H along the easy axes ([110]- and [1 10]-orientations). \n(b) f-dependence of dHR/dI for H along the hard axes ([1 10]- and [ 110]-orientations). The \nresults in ( a) and (b) are obtained for tFe = 2.8 nm which show that dHR/dI is independent of \nf for both easy and hard axes. ( c) and (d) are the same plots as ( a) and (b) but for tFe = 1.2 \nnm. The magnitude of dHR/dI along the hard axes for the thinner sample depends strongly \non the excitation frequency, indicating the modification of magnetic anisotropies by spin \ncurrent. The insets of ( a)-(d) show the relative orientations between the current ( I // [100], \nand is represented by black arrows) and the magnetic-field (or magnetization), where the \neasy axes are represented by brown arrows and the hard axes are represented by green \narrows. -0.30.00.3\n  dHR/dI (mT/mA)tFe = 1.2 nm\n6 9 12 15 18-0.30.00.3\nf (GHz)tFe = 2.8 nm\ntFe = 1.2 nm-0.30.00.3\n6 9 12 15 18-0.30.00.3\nf (GHz)dHR/dI (mT/mA)\ntFe = 2.8 nmdHR/dI (mT/mA)\ndHR/dI (mT/mA)a\nbc\nd\nH// [110]H //[110]H// [110]I // [100]\nI // [100]I // [100] H// [110]\nI // [100]\nH //[110]I // [100]H// [110]\nI // [100]H// [110]\nI // [100]H// [110]I // [100]11 \n torque hOe/FL, arising from the current flowing in Pt and Al, which shifts HR. The field-\nlike torque originates from the incomplete dephasing (non-transmitted and/or non-\ndephased) component of the incoming spin19,20,32. For M along the hard axis as shown \nin Fig. 4b, the f-independent dHR/dI has also opposite zero-frequency intercepts for \nM // [110] and M // [110] with virtually identical hOe/FL value as the easy axis. This \nconfirms that the spin accumulation  (along the [010]-orientation) has equal projection \nonto the <110>- and <1 10>-orientations. As tFe is reduced to 1.2 nm (Fig. 4c), the \nintercept of the f-independent dHR/dI  traces along [110] and [1 10]-orientations, \nrespectively, increases to ~ 0.20 mT/mA and ~ 0.20 mT/mA. However, as M is \naligned along the hard axis as shown in Fig. 4d, the behaviour of the \ndHR/dItracediffers significantly from other traces: I) it is no longer f-independent but \nshows a linear dependence on f with opposite slopes for M along the [1 10]- and [ 110]-\norientations, II) the absolute value of the zero-frequency intercept along the hard axis \n(~ 0.32 mT/mA) is no longer equal to that along the easy axis (~ 0.2 mT/mA). The f-\ndependence of the dHR/dI traces cannot be interpreted to arise from the frequency-\nindependent hOe/FL, and can only be explained by a change of the magnetic anisotropies \ninduced by the spin current. \n \nAxis HK HB  HU \nEA  ∆𝐻ோ=𝑘𝑓 ∆𝐻ோ=−∆𝐻+𝑘𝑓 ∆𝐻ோ=∆𝐻−𝑘𝑓 \nHA ∆𝐻ோ=𝑘𝑓 ∆𝐻ோ=−∆𝐻+𝑘𝑓 ∆𝐻ோ=−∆𝐻 \n \nTable 1. Summary of the HR-f relationships induced by HK, HU, and HB along easy and \nhard axes. \n \n \n      In the presence of the in-plane magneto-crystalline anisotropies, the 12 \n dependencies of HR on f along the easy axis HREAand the hard axis HRHA are given by \nthe modified Kittel formula33 \n              ൞ቀ2πf\nγቁ2\n= μ02ቀHREA + HK+ HB\n2ቁ൫HREA  HB  HU൯ \nቀ2πf\nγቁ2\n= μ02ቀHRHA + HK + HB\n2HUቁ൫HRHA  HB + HU൯          (3) \n \nwhere  is the gyromagnetic ratio, HK the effective magnetic anisotropy field due to the \ndemagnetization field along <001>, HB the biaxial magnetic anisotropy field along \n<100>, and HU the in-plane uniaxial magnetic anisotropy field along <110>. The \nmagnitude of HK, HU and HB at I = 0 for each tFe is quantified by the angular and \nfrequency dependencies of HR (Supplementary Note 1). Obviously, a change of the \nmagnetic anisotropy fields HA (HA = HK, HU, HB) by HA (HA = HK, HU, HB) \nleads to a shift of HR and the magnitude of the shift HR, defined as HR = \nHR(HA)HR(HA+HA), depends on the excitation frequency. In the measured \nfrequency range (10 GHz < f < 20 GHz), the HRf relations induced by HA can be \ncalculated by eq. 3, and their dependencies on f are summarized in Table 1 \n(Supplementary Note 6). Specifically, an increase (a decrease) of HK and HB leads to \nthe f-dependent HR with positive (negative) slope both along EA and HA. However, a \nchange of HU leads to the f-independent HR along HA, while an increase (a decrease) \nof HU leads to the f-dependent HR with negative (positive) slope along EA.  \n \n      Since hOe/FL, generated by the dc current, also shifts the resonance field along \nthe EA and HA axes by ±√2\n2hOe/FL, where ‘+’ corresponds to the [110] and the [ 110] \ndirections, and ‘ ’ corresponds to the [ 110] and the [ 110] directions, the total HR \nalong EA and HA is given by 13 \n             ቐ∆HREA(f) = ∆HU  ∆HB ±√2\n2hOe/FL + (kK + kB  kU)f \n∆HRHA(f) =(∆HU+∆HB) ±√2\n2hOe/FL+ (kK+kB)f            (4) \nHere the slope k [k = kK, kU, kB, and 𝑘=ௗ(∆ுೃ)\nௗ ] quantifies the modulation of HR \ninduced by HA. For HR induced by both HK and HB, 𝑘∝ ∆𝐻, which holds for \nboth EA and HA. However, for HR induced by HU, we find 𝑘∝ −∆𝐻, which holds \nonly for EA. Since the f-dependence of ∆ HREA induced by ∆𝐻 has an opposite slope \nas those induced by ∆ HK  and ∆HB , it is possible to obtain a f-independent ∆ HREA \nalong EA (Fig. 4c) by tuning the corresponding parameters and to obtain a f-linear \n∆HRHA  along HA (Fig. 4d). To reproduce the results along the [110]- and [11 0]-\norientations (i.e., the net magnetization of these two orientations is parallel to I) for tFe \n= 1.2 nm, we obtain HB = 0.26 mT/mA, HK = 2.0 mT/mA and HU = 2.5 mT/mA \nthrough eqs. 3 and 4 (for details see Supplementary Note 6). In contrast, for the data \nsets for M along [110] and [110]-orientations (i.e., the magnetization is rotated by 180o \nand the net magnetization of these two orientations is antiparallel to I), HB = 0.26 \nmT/mA, HK = 2.0 mT/mA and HU = 2.5 mT/mA are obtained, which have the \nopposite polarity compared to the case of M along the [110]- and [11 0]-orientations. \n \n      Figure 5 summarizes the obtained modification of the magnetic anisotropy HA \nas a function of tFe. For tFe above 2.8 nm, the modification of the magnetic anisotropy \nis too small to be observed and is washed out by bulk effects. For tFe below 2.2 nm, the \nmodification of HA increases as tFe decreases. This indicates that the spin current \ninduced modification of the magnetic energy landscape is of interfacial origin, similar \nto the damping-like spin-torque determined by the f-dependence of d(H)/dI in the 14 \n same sample series (Supplementary Note 5). The modification changes sign when M is \nrotated by 180o, which fully validates the scenario of ∆ HA(I, M) = ∆HA(I, M) = \n∆HA(I, M) as suggested in Fig. 1. For a given M direction, the obtained HB, HK, \nand HU have the same sign, which is also consistent with a monotonic increase \n(decrease) of HB, HK, and HU as temperature decreases (increases) (Supplementary \nNote 6). Moreover, these results also show that HU is more sensitive to the spin current \nthan HK and HB, highlighting the importance of UMA to enable the observation of the \nmodifications. The much smaller HB value is due to the fact that HB is 1-2 orders \nsmaller than HU and HK in the ultra-thin regime (Supplementary Note 1). \n \n      The change of magnetic anisotropy HK is directly related to the change of \nmagnetization M, which can be attributed to the additional filling of the electronic d-\nband. Strictly speaking, the induced filling of the bands in Fe occurs mainly close to \nthe interface and is not homogeneously distributed, since it depends on the spin  \nFig. 5. Summary of tFe dependence of HA (HA = HK, HU, HB) for opposite magnetization \nM directions, where solid symbols represent M // +z-direction and open symbols represent \nM // z-direction. The relative orientations between the charge current I and M are shown in \nthe inset. 1 2 3 4 5-3.0-1.50.01.53.0\n  [m0HA] (mT/mA)\ntFe (nm)HU\nHB\nHK\nHUHK\nHB\nM\n⊗\nI\nM⊗I15 \n diffusion length of the spin current in Fe. In other words, the measured modulated \nmagnetic anisotropies are averaged over the whole ferromagnetic film. To first-order \napproximation and for simplicity, we neglect the spin current distribution in Fe and \nassume that it is homogeneously distributed. The spin accumulation at the interface9 is \ngiven by 𝑢௦= 2𝑒𝜆𝜉𝐸tanhቂ௧\nଶఒቃ, where e is the elementary charge,  the spin diffusion \nlength, E (= j/) the electric-field, j the current density and  the conductivity of Pt. \nThe areal spin density nSHE transferred into Fe, is obtained as 𝑛ௌுா=𝑢௦𝜆𝑁9, where N \nis the density of states at the Fermi level for Fe. Using N = 6 ×1048 J-1m-3,  = 4 nm, \n0.06,  = 2.0 ×106 -1m-1, nSHE = 4.2 ×1012 mB/cm2 is obtained for I = 1 mA. Since \nFe has a bcc structure (lattice constant a = 2.8 Å) with a moment of ~1.0 mB for tFe = \n1.2 nm at room temperature, the areal density of the magnetic moment of Fe nFe is \ndetermined to be 2.6  ×1014 mB/cm2. In this case, the filling of the d-band by SHE leads \nto a change of the magnetic moment of the order of nSHE/nFe ~ 0.16%, which agrees \nwith the ratio between HK and HK, i.e., HK/HK ~ 2.0 mT/ 1 T ~ 0.2%. \n \n      On the other hand, since the UMA shows a larger modulation than effective \ndemagnetization and biaxial anisotropy, it is clear that the band structure of the \ninterfacial Fe/GaAs plays a major role for the observed effect when the Fe film is \nsufficiently thin34. As shown in Fig. 1a, the spin current JSz can be represented in terms \nof two opposite flows of electrons carrying up and down spin moments, JSz = Jup – Jdw, \nin the absence of a net electric current ( Jc = Jup + Jdw = 0). Since the spin current doesn’t \ntransfer charge, the inflow of the spin-up electrons leads to an increase of occupation \nof the spin-up d-states of Fe, while the outflow of spin-down electrons leads to a \ndecrease of occupation of the spin-down d-states. A similar effect occurs in the presence \nof an applied magnetic field on the order of the molecular magnetic-field, i.e., H // +z 16 \n shifts the spin-up d-states down in energy, following by an increase of their occupation. \nIn contrast, the spin-down d-states shift up in energy, leading to their depopulation. \nTherefore, to mimic the effect of spin current on the UMA and magnetic moment, we \nhave investigated the dependence of the UMA on the external magnetic-field on the \nbasics of first-principle electronic band structure calculations for Fe/GaAs. The \nresulting modification of UMA has been determined by means of magnetic torque \ncalculations35(Supplementary Note 7). The applied magnetic-field results in an increase \nof the magnetic anisotropy energy along the easy axis, if the applied magnetic-field is \nparallel to the magnetization direction, and to a decrease of anisotropy in the case of \nantiparallel orientation. These changes are accompanied by an increase (for H > 0) or \ndecrease (for H < 0) of the spin magnetic moment of Fe, consistent with experimental \nobservations. Additional more sophisticated model might be needed to extend the \nexisting model and to explain the experimental results quantitatively. \n \n      Our results have shown that the intrinsic properties of ultra-thin ferromagnetic \nmaterials, i.e., the magnitude of M and the magnetocrystalline anisotropies, can be \nvaried in a controlled way by spin currents, which has been ignored in the spin-transfer \nphysics so far. This unique route of controlling magnetic anisotropies is not accessible \nby other existing ways using electric-field1-5 and mechanical stress36,37 in which the \ncontrol of magnetism is independent on magnetization direction. Besides the magnitude \nof the magnetization, other material parameters, e.g., the Curie temperature, coercive \nfield etc., are also expected to be controllable by spin current. It is known that spin-\ntorque plays an essential role in modern spintronic devices, beyond this proof of \nprinciple, the so far unnoticed modification of the length of the magnetization vector \nby spin currents could offer an alternative and attractive generic actuation mechanism 17 \n for spin-torque phenomena, e.g., magnetization switching and auto-oscillations of the \nmagnetization. We expect such a modification of the magnetic energy landscape to be \na general feature, not only limited to ferromagnetic metal/heavy metal systems with \nstrong spin-orbit interaction but also present in the case of conventional spin-transfer \ntorques, where it is generally believed that the magnitude of M is fixed during the spin \ntransfer process6-8. Moreover, the modification is not limited to in-plane ferromagnets, \nand one could manipulate the static magnetic properties of ferromagnets with \nperpendicular anisotropy by using out-of-plane polarized spin current source, e.g., \nWTe238, RuO239-42 and non-linear antiferromagnets Mn 3Sn43 and Mn 3Ga44. Finally, we \nbelieve that much larger modification amplitudes can be realized in other more effective \nspin current sources based on the wide-range of spin-torque material choices9. \n \nMethods \nSample preparation.  Samples with various Fe thicknesses tFe are grown by \nmolecular-beam epitaxy (MBE). First a GaAs buffer layer of 100 nm is grown in a III-\nV MBE, after that the GaAs substrate (semi-insulating wafer, which has a resistivity  \nbetween 1.72×108 .cm and 2.16×108 .cm) is transferred to a metal MBE without \nbreaking the vacuum for the growth of the metal layers. For a better comparison of \nphysical properties of different samples, various Fe thicknesses are grown on a single \ntwo-inch wafer by stepping the main shadow shutter of the metal MBE. After the \ngrowth of the step-wedged Fe film, 1.5-nm Al/6-nm Pt layers are deposited on the \nwhole wafer.  \nDevice. First, Pt/Al/Fe stripes with a dimension of 4.0 mm × 20 mm and with the long \nside along the [110]- and [100]-orientations are defined by a mask-free writer and Ar-18 \n etching. After that, contact pads for the application of the dc current, which are made \nfrom 3-nm Ti and 50 nm Au, are prepared by evaporation and lift-off. Then, a 70-nm \nAl2O3 layer is deposited by atomic layer deposition to electrically isolate the dc contacts \nand the coplanar waveguide (CPW). Finally, the CPW consisting of 5 nm Ti and 150 \nnm Au is fabricated by evaporation, and the Fe/Al/Pt stripes are located in the gap \nbetween the signal line and ground line of the CPW (Fig. 2a). The CPW is designed to \nmatch the rf-network which has an impedance of 50 . The width of the signal line and \nthe gap is 50 mm and 30 mm, respectively. Magnetization dynamics of Fe is excited by \nout-of-plane Oersted field induced by the rf microwave currents flowing in the signal \nand ground lines.  \nMeasurements.  For TRMOKE microscopy measurements, a pulse train of a \nTi:Sapphire laser (repetition rate of 80 MHz and pulse width of 150 fs) with wavelength \nof 800 nm is phase-locked to the microwave current. A phase shifter is used to adjust \nthe phase between the laser pulse train and microwave, and the phase is kept constant \nduring the measurement. The polar Kerr signal at a certain phase, VKerr, is detected by \na lock-in amplifier by phase modulating the microwave current at a frequency of 6.6 \nkHz. The VKerr signal is measured by sweeping the external magnetic field, and the \nmagnetic-field can be rotated in-plane by 360o. A Keithley 2400 device is used as the \ndc current source for linewidth and resonance field modifications. All measurements \nare performed at room temperature. \n \n 19 \n References  \n1. Ohno, H., Chiba, D., Matsukura, F., Omiya, T., Abe, E., Dietl, T., Ohno, Y., & Ohtani, \nK., Electric-field control of ferromagnetism, Nature 408, 944 (2000). \n2. Chiba, D., Sawicki, M., Nishitani, Y., Nakatani, Y., Matsukura, F., & Ohno, H., \nMagnetization vector manipulated by electric fields, Nature 455, 515-518 (2008).  \n3. Weisheit, M. et al. Electric field-induced modification of magnetism in thin-film \nferromagnets, Science 315, 349 (2007). \n4. Chen, L., Matsukura, F., & Ohno, H., Electric-field modulation of damping constant \nin a ferromagnetic semiconductor (Ga,Mn)As, Phys. Rev. Lett.  115, 057204 (2015). \n5. Matsukura, F., Tokura, Y., & Ohno, H., Control of magnetism by electric-fields, \nNature Nanotech . 10, 209-220 (2015). \n6. Berger, L., Emission of spin waves by a magnetic multilayer traversed by a current, \nPhys. Rev. B  54, 9353 (1996). \n7. Slonczewski, J. C., Current-driven excitation of magnetic multilayers, J. Mag. Mag. \nMater. 159, L1-L7 (1996).  \n8. Ralph, D. C., & Stiles, M. D., Spin transfer torques, J. Mag. Mag. Mater . 320, 1190-\n1216 (2008). \n9. Manchon, A. et al. Current-induced spin-orbit torques in ferromagnetic and \nantiferromagnetic systems , Rev. Mod. Phys . 91, 035004 (2019). \n10. Miron, I. M. et al. Perpendicular switching of a single ferromagnetic layer induced \nby in-plane current injection. Nature 476, 189-193 (2011). \n11. Liu, L. Q. et al. Spin-torque switching with giant spin Hall effect of Tantalum. \nScience 336, 555-558 (2012). \n12. Miron, I. M. et al. Fast current-induced domain-wall motion controlled by Rashba \neffect. Nature Mater.  10, 419-423 (2011). 20 \n 13. Emori, S. et al. Current-driven dynamics of chiral ferromagnetic domain walls. \nNature Mater.  12, 611-616 (2013). \n14. Ryu, K. et al. Chiral spin torque at magnetic domain walls. Nature Nanotech.  8, \n527-533 (2013). \n15. Ando, K. et al. Electric manipulation of spin relaxation using the spin Hall effect, \nPhys. Rev. Lett . 101, 036601 (2008). \n16. Demidov, V. E. et al. Magnetic nano-oscillator driven by pure spin current, Nature \nMater. 11, 1028-1031 (2012). \n17. Liu, L. Q. et al. Magnetic Oscillations driven by the spin Hall effect in 3-terminal \nmagnetic tunnel junction devices, Phys. Rev. Lett.  109, 186602 (2011). \n18. Shao, Q. et al. Roadmap of spin-orbit torques, IEEE Transactions on magnetics , \n57, 1-39 (2021). \n19. Amin, V. P., & Stiles, M. D., Spin transport at interfaces with spin-orbit coupling: \nFormalism, Phys. Rev. B  94, 104419 (2016). \n20. Amin, V. P., & Stiles, M. D., Spin transport at interfaces with spin-orbit coupling: \nPhenomenology, Phys. Rev. B  94, 104420 (2016).  \n21. Haney, P. M. et al. Current induced torques and interfacial spin-orbit coupling: \nSemiclassic modeling. Phys. Rev. B  87, 174411 (2013). \n22. Sinova, J., Valenzuela, S. O., Wunderlich, J., Back, C. H., & Jungwirth, T., Spin \nHall effects, Rev. Mod. Phys . 87, 1213 (2015). \n23. Pai, C. F. et al. Dependence of efficiency of spin Hall torque on the transparency \nof Pt/ferromagnetic layer interfaces.  Phys. Rev. B  92, 064426 (2015). \n24. Zhu, L., Ralph, D. C., & Buhrman, R. A., Effective spin-mixing conductance of \nheavy-metal-ferromagnet interfaces, Phys. Rev. Lett . 123, 057203 (2019).  \n25. Chen, L., Mankovsky, S., Kronseder, M., Schuh, D., Prager, M., Bougeard, D., 21 \n Ebert, H., Weiss, D., and Back, C. H., Interfacial tuning of anisotropic Gilbert \ndamping, Phys. Rev. Lett . 130, 046704 (2023). \n26. Bayreuther, G., Premper, J., Sperl, M., and Sander, D., Uniaxial magnetic \nanisotropy in Fe/GaAs (001): Role of magnetoelastic interactions, Phys. Rev. B . 86, \n054418 (2012). \n27. Chen, L. et al. Robust spin-orbit torque and spin-galvanic effect at the Fe/GaAs \n(001) interface at room temperature. Nature Commun.  7, 13802 (2016). \n28. Kasai, S. et al. Modulation of effective damping constant using spin Hall effect, \nAppl. Phys. Lett . 104, 092408 (2014). \n29. Safranski, C., Montoya, E. A., & Krivorotov, I. N., Spin-orbit torques driven by a \nplanar Hall current, Nature Nanotech . 14, 27-30 (2019). \n30. Capua, A., Rettner, C., & Parkin, S. S. P., Parametric harmonic generation as a \nprobe of unconstrained spin magnetization precession in the shallow barrier limit, \nPhys. Rev. Lett . 116, 047204 (2016). \n31. Capua, A. et al. Phase-resolved detection of the spin Hall angle by optical \nferromagnetic resonance in perpendicularly magnetized thin films, Phys. Rev. B . 95, \n064401 (2017). \n32. Ou, Y., Pai, C. F., Shi, S., Ralph, D. C., and Buhrman, R. A., Origin of fieldlike \nspin-orbit torques in heavy metal/ferromagnet/oxide thin film heterostructures, Phys. \nRev. B 94, 140414(R) (2016). \n33. Chen, L., Matsukura, F., & Ohno, H., Direct-current voltages in (Ga,Mn)As \nstructures induced by ferromagnetic resonance. Nature Commun.  4, 2055 (2013). \n34. Gmitra, M. et al. Magnetic control of spin-orbit fields: A first-principle study of \nFe/GaAs junctions. Phys. Rev. Lett.  111, 036603 (2013). \n35. Staunton, J. B. et al. Temperature dependence of magnetic anisotropy: An ab initio 22 \n approach, Phys. Rev. B . 74, 144411 (2006). \n36. Goennenwein, S. T. B. et al. Piezo-voltage control of magnetization orientation in \na ferromagnetic semiconductor. Phys. Status Solidi (RRL)  2, 96-98 (2008). \n37. Overby, M., Chernyshov, A., Rokhinson, L. P., Liu, X., & Furdyna, J. K., GaMnAs-\nbased hybrid multiferroic memory device, Appl. Phys. Lett . 92, 192501 (2008). \n38. MacNeill, D. et al. Control of spin-orbit torques through crystal symmetry in \nWTe2/ferromagnet bilayers, Nature Phys.  13, 300 (2017). \n39. Bai, H. et al. Observation of spin splitting torque in a collinear antiferromagnet \nRuO2, Phys. Rev. Lett.  128, 197202 (2022). \n40. Bose, A. et al. Tilted spin current generated by the collinear antiferromagnet \nruthenium dioxide, Nature Elect.  5, 267 (2022). \n41. Karube, S.  et al. Observation of spin splitter torque in collinear antiferromagnet \nRuO2, Phys. Rev. Lett.  129, 137202 (2022). \n42. Šmejkal, L., Sinova, J., and Jungwirth, T., Emerging research landscape of \naltermagnetism, Phys. Rev. X  12, 040501 (2022).  \n43. Hu, S. et al. Efficient perpendicular magnetization switching by a magnetic spin \nHall effect in a noncollinear antiferromagnet,  Nature Commun.  13, 4447 (2022). \n44. Han, R. K. et al. Field-free magnetization switching in CoPt induced by \nnoncollinear antiferromagnetic Mn 3Ga. Phys. Rev. B.  107, 134422 (2023). \n \nAcknowledgements  \nFinancial support is from SFB 1277 and TRR 360 by DFG. \n \nAuthor contributions \n \nL.C. planned the study. Y. S. and L.C. fabricated the devices, collected the data. L.C 23 \n analysed the data and did the theoretical calculations. T.N.G.M and M.K. grew the \nsamples. S.M. and H.E. did the first principle calculations. L.C. wrote the manuscript \nwith input from all other co-authors. All authors discussed the results and contributed \nto the manuscript. \n 24 \n Figure captions \nFIG. 1. Schematic of the microscopic mechanism of manipulation and modification of \nmagnetism by a spin current. ( a) The electron spins transmitted into the FM contain \nboth transverse and longitudinal components with respect to M. Due to exchange \ncoupling, the transverse component dephases and is absorbed by M, which gives rise \nto the damping-like spin-orbit torque and is responsible for changing the direction of \nM. The longitudinal component of the spin current is on average aligned with M, \nleading to additional filling of the majority band when M is oriented along the + z-\ndirection, and an enhancement of the magnitude M as well as an increase of magnetic \nanisotropies are expected due to the enhanced splitting of the majority- and minority-\nspin energy bands. ( b) When M is aligned along the – z-direction, the spin-polarized \nelectron enters the minority band, which can lead to a decrease of M as well as a \ndecrease of magnetic anisotropies because of the reduction of the splitting of the \nmajority- and minority-spin energy bands. ( c) and (d): The same as ( a) and (b) but the \npolarization of the spin current is reversed, which is expected to reduce M for M//+z \n(c) and to enhance M for M//-z (d). \n \nFIG. 2. (a) Schematic of the device for the detection of ferromagnetic resonance by \ntime-resolved magneto-optical Kerr microscopy. ( b) Schematic of the Pt/Al/Fe/GaAs \n(001) structure. ( c) Diagram of crystallographic axes with easy and hard magnetization \naxes along <110> and <1 10> orientations. ( d) FMR spectra for different dc currents I \nmeasured at f = 12 GHz and I-H = 90o, where I-H is the angle between the magnetic-\nfield and the current direction as shown in the inset. ( e) FMR linewidth (full width at 25 \n half maximum) as a function of dc current for I-H = ±90o; solid lines are the linear fits \nfrom which the modulation amplitude d(H)/dI is obtained. ( f)I-H-dependence of \nd(H)/dI. Error bars show the standard error of the least squares fit. The solid line is \nthe calculation result based on the spin Hall effect of Pt when taking into account the \nin-plane magnetic anisotropies of Fe (Supplementary Note 4). \n \nFIG. 3. I dependence of HR measured at selected frequencies for H along [110]- ( a) and \n[110]-orientations ( b) for tFe = 2.8 nm. For both field orientations, HR(-I) > HR(+I) holds \nwhere HR(-I) and HR(+I) are respectively marked by red and blue arrows in each panel. \n(c) and (d): the same plots as ( a) and (b) but for tFe = 1.2 nm. In ( c) for H along the \n[110]-orientation, HR(-I) > HR(+I) still holds for all measured frequencies. However, \nfor H along the [1 10]-orientation as shown in ( d), the relative magnitude of HR(-I) and \nHR(+I) depends on the excitation frequency, i.e., for f = 12.0 GHz, HR(-I) < HR(+I) \nholds; for f = 14.0 GHz, HR(-I) ~ HR(+I) holds; while for f = 16.0 GHz, HR(-I) > HR(+I) \nholds. As shown in the upper panels, for all the devices, the charge currents are applied \nalong the [100]-orientation, and the direction of the spin accumulation  is along the \n[010]-direction with equal projections onto the [110]- and ൣ110൧ -orientations. This \nexperimental trick allows an accurate comparison of the current induced modification \nfor [110]- and [1 10]-orientations in the same device. \n \nFIG. 4. ( a) f-dependence of dHR/dI for H along the easy axes ([110]- and [1 10]-\norientations). ( b) f-dependence of dHR/dI for H along the hard axes ([1 10]- and [ 110]-\norientations). The results in ( a) and (b) are obtained for tFe = 2.8 nm which show that \ndHR/dI is independent of f for both easy and hard axes. ( c) and (d) are the same plots as \n(a) and (b) but for tFe = 1.2 nm. The magnitude of dHR/dI along the hard axes for the 26 \n thinner sample depends strongly on the excitation frequency, indicating the \nmodification of magnetic anisotropies by spin current. The insets of ( a)-(d) show the \nrelative orientations between the current ( I // [100], black arrows) and the magnetic-\nfield (or magnetization), where the easy axes are represented by brown arrows and the \nhard axes are represented by green arrows. \n \nFIG. 5. Summary of tFe dependence of HA (HA = HK, HU, HB) for opposite \nmagnetization M directions, where solid symbols represent M // +z-direction and open \nsymbols represent M // z-direction. The relative orientations between the charge \ncurrent I and M are shown in the inset. \n \nTable 1. Summary of the HR-f relationships induced by HK, HU, and HB along \neasy and hard axes. \n \n "    },    {        "title": "2403.14241v1.Magnetocrystalline_anisotropy_in_metallic_systems__fast_and_stable_estimation_in_Green_s_functions_formalism.pdf",        "content": "Magnetocrystalline anisotropy in metallic systems:\nfast and stable estimation in Green’s functions formalism\nIlya V. Kashin∗and Sergei N. Andreev\nTheoretical Physics and Applied Mathematics Department,\nUral Federal University, Mira Str. 19,\n620002 Ekaterinburg, Russia\n(Dated: March 22, 2024)\nIn this work we suggest a theoretical approach, that allows to study the effects of magnetocrys-\ntalline anisotropy (MCA) in metallic systems using the Green’s functions formalism. We demon-\nstrate that employment of the reciprocal space resolution instead of its reduction in the inter-site\nvariant essentially improves the numerical stability of MCA energy by means of Monkhorst-Pack\ngrid density and spatial convergence. The latter problem is able to be completely removed due to\nrigorous analytical replacement of pairwise atomic summation by simple composition of sublattices\ncontributions, calculated as a whole. The approach is validated on the effective model of single\natom, which nevertheless inherits the qualitative MCA picture of Co monolayer and Au/Co/Au\nsandwiched material. The numerical convergence is confirmed using the model of atomic chain in\nthe strong metallic regime. For cobalt monoxide, described by ab initio calculations using GGA +U,\nthe MCA energy angular profile reveals the prevailing role of ferromagnetically aligned Co sublattices\nin forming of the easy axis.\nI. INTRODUCTION\nToday, an increasing amount of scientists attention is\ndevoted to magnetic materials exhibiting a non-trivial\nand stable spin ordering patterns at relatively low tem-\nperatures. There is a great number of studies, that re-\nport the high sensitivity of their different representatives\nto the common experimentally available influences, such\nas external electric or magnetic field and pressure [1–7].\nThis remarkable palette of traits opens avenues for de-\nsigning energy-efficient and high-performance computer\nmemory modules based on the spin degrees of freedom\n[8–10].\nThe emergence of the topologically protected magnetic\nexcitations are primarily reasoned by violation of spa-\ntial symmetry in the crystal lattice. It follows from the\nfact that commonly lattice appears as a dominant source\nof spatial heterogeneity of the electron subsystem of the\nmaterial. On the contrary, if the symmetry is high, the\nmagnetic picture usually could be exhaustively captured\non the level of isotropic pairwise spin-spin interactions of\nthe Heisenberg type.\nIn this context, spin-orbit coupling plays a crucial\nrole, inducing a complex interplay between the spin\nwith the distorted crystal field. The latter can be in-\nduced by either the mechanical deformation of the crys-\ntal [1, 2, 11, 12] or the qualitative reduction of its dimen-\nsionality as layered and two-dimensional structures are\nsynthesized [13–15]. The indicative outcome is the ap-\npearance of energetically favorable directions for atomic\nmoments – easy axes and magnetization planes.\n∗i.v.kashin@urfu.ruThus, the energy of magnetocrystalline anisotropy\n(MCA) becomes an important characteristic for analyz-\ning the possibility of magnetic ordering and its type\n[13, 16–20]. For two-dimensional materials, this factor of-\nten appears decisive in overcoming the constraints of the\nMermin-Wagner theorem [21] regarding the impossibil-\nity of stabilizing magnetic order solely through isotropic\nmagnetic interactions.\nOnce the spin-orbit coupling is weak, this energy can\nbe estimated in the formalism of local magnetic moments\nof atoms as the difference in the total energies of the\ncrystal, found for two cases: atomic magnetic moments\naligned with the quantization axis and atomic magnetic\nmoments oriented at an arbitrary angle to this axis. This\napproach has proven effective in studies based on first-\nprinciples calculations [1, 22–24] but lacks flexibility in\naddressing the structural elements of the crystal.\nIt comes as a reason why there is yet another popu-\nlar approach, which is based on Green’s functions frame-\nwork. Starting from the crystal’s Hamiltonian, written in\nthe tight binding approximation, with the spin-orbit cou-\npling treated as the perturbation, an expression for the\nMCA energy could be derived. It has contributions of\nall crystal’s atoms in the form of pairwise terms [25–27].\nHowever, the practical application of such an approach\nis generally limited to insulating systems, for which it is\nsufficient to consider the spatial surrounding up to the\nthird nearest neighbors for each atom. Conducting sys-\ntems lose this advantage, making the numerical conver-\ngence of the pairwise contributions sum troublesome and\noften non-achievable, as well as it was stated in the case\nof building the picture of Heisenberg exchange interac-\ntions environment [28–30]. Furthermore, it is known to\nhave a stability issues by means of Monkhorst-Pack [31]\ngrid used to perform the numerical estimations [25]. AlsoarXiv:2403.14241v1  [cond-mat.mtrl-sci]  21 Mar 20242\nnoteworthy, that examination the MCA energy for the\nminimum regarding the spacial direction ( φ, θ) requires\ncorresponding grid search, where obtaining a single ( φ, θ)\npoint could take hours of multiprocessing computation.\nIn this view, for the investigation of MCA in real mate-\nrials the fast algorithm comes as strictly necessary.\nDue to these challenges, this work is aimed to develop a\nmethod based on the application of spatially-dependent\nGreen’s functions to estimate the MCA energy for ar-\nbitrary direction ( φ, θ). For the purpose of easy axis\n(or plane) determination we map the whole angular pro-\nfile onto the model of non-interacting spins, where MCA\nis described by single ion anisotropy tensor (SIAT). We\ndemonstrate that it not just enhances numerical stabil-\nity, but also reduces the calculation time by two orders\nof magnitude on a modern computer.\nFirst of all, the suggested approach is tested on the\nsimplest model of a single atom, for which the character\nof the MCA can be predicted from the general physical\nprinciples. After that on the case of an one-dimensional\natomic chain in the strong metallic phase we demonstrate\nthe stability of this approach with respect to the k-grid\ndensity. The latter model appeared illustrative to show\nthat thus found numerical values of the MCA energy\ncould be stated as the asymptotic ones for the expression\nbased on the spatial summation of the pairwise atomic\ncontributions.\nApprobation of the approach on the real materials is\nperformed on the case of transition metal monoxide CoO.\nIn the paramagnetic phase, this compound crystallizes in\nrock-salt structure. However, the formation of antiferro-\nmagnetic long-range ordering, which is classified as AFM\nII [32], at low temperatures causes significant distortions,\nwhich lowers the symmetry of the crystal field to a mon-\noclinic structure. This monoclinic distortion could be\ndescribed as a composition of a rhombohedral and an\northorhombic one, and together with the partially filled\nt2gorbitals favours CoO to generate strong MCA of the\neasy axis character. However, experimental and theo-\nretical studies [1, 33] give ambiguous estimations of the\nspacial direction of this axis, and the mechanisms of its\nformations are yet to be understood.\nIn this work CoO is studied on the base of first-\nprinciples calculations, performed in generalized-gradient\napproximation with intra-atomic Coulomb interaction\ntaken into account (GGA +U). To describe the MCA we\nperform the decomposition of MCA energy angular pro-\nfile, obtained for one Co atom, onto individual contribu-\ntions of each sublattice. The sublattice should be under-\nstood as the composition of Co atoms with the same local\npositions in the unit cells (each sublattice geometrically\nis the Bravais lattice with atoms in the sites).\nIt reveals that easy axis of this atom (on the back-\nground of the on-site effects) is only reasoned by Co-Co\natomic interplay in (101) plane, where their spin mag-\nnetic moments are collinear. Therefore, we expect the\ntechnologically perspective behaviour of CoO MCA if an\nadditional mechanism of lattice distortion along (101) isintroduced, such as a directional pressure.\nII. METHOD\nA. Finding SIAT\nSingle ion anisotropy tensor is defined in frame of\nthe spin model, where each magnetic atom (indexed\nasi) is presented as the corresponding classic vector\nSi=\f\fSi\f\f(sinθicosφi,sinθisinφi,cosθi). The Hamil-\ntonian of such model could be written as a simple combi-\nnation of expressions, which set some spatial axis or plane\nas favorable for each spin individually. For our purpose\nwe take θi=θ,φi=φand thus establish the model as\nthe energy of all spins, simultaneously canted on ( φ, θ):\nEspin(φ, θ) =X\ni\f\fSi\f\f2X\nµνeµ\niAµν\nieν\ni, (1)\nwhere eµ\ni=Sµ\ni/\f\fSi\f\fandAµν\niis the SIAT element\n(µ, ν=x, y, z ).\nThen to describe the crystal in electron language the\ncommon practise is to use the tight-binding approxima-\ntion [34]. Assuming that spin-orbital coupling is ne-\nglected, we write:\nˆH=X\ni̸=jX\nαβX\nσtσ\ni(α)j(β)ˆc†\ni(α)σˆcj(β)σ+\n+X\niX\nαX\nσεσ\ni(α)ˆc†\ni(α)σˆci(α)σ,(2)\nwhere ˆ c†\ni(α)σis creation operator of the electron with the\nspinσon the orbital αof the atom i; ˆcj(β)σis annihilation\noperator of the electron with the spin σon the orbital β\nof the atom j;tσ\ni(α)j(β)is the electron hopping integral\nandεσ\ni(α)is the on-site energy of the electron.\nThis Hamiltonian could be equally represented in ma-\ntrix form with illustrative and fruitful interpretation. If\nthe on-site physics and hybridization picture is being ex-\nplained on the level of interplay between two different\nunit cells, one can then state one of this cell as having\nzero translation vector T= 0 (i.e. choose the coordinate\nsystem origin) and express how it couples with some an-\nother cell with translation T:\n\u0002\nHσ(T)\u0003\nij=tσ\nij+εσ\niδijδT0. (3)\nwhere Hσ(T) is the unit cell sized matrix, whereas in-\ndices ijdenote selection of iatom from T= 0 cell and j\natom from Tcell, interplay of which appear as the sector\nof this matrix. Naturally, the case of the both cells being\nactually the same one with T= 0 includes, apart from\nintra-cell atomic couplings, the on-site electron energies.\nThis matrix form is convenient to be presented in the\nreciprocal space\nHσ(k) =X\nTHσ(T)·exp(ikT) (4)3\nto introduce the Green’s functions toolbox. According to\nthe definition, we write the expression for k-dependent\nGreen’s function:\nGσ(E,k) =\b\nE−Hσ(k)\t−1, (5)\nwhere Edenotes the spectrum sweep energy as the diag-\nonal matrix of Hamiltonian matrix size. Then the inter-\nsite version, describing the coupling between atoms iand\nj, reads\nGσ\nij=1\nNkX\nk\u0002\nGσ(E,k)\u0003\n˜i˜j·exp(−ikTij),(6)\nwhere Tijis the translation vector, which connects the\nunit cells of these atoms, Nkis the number of Monkhorst-\nPack grid points [31] and\u0002\nGσ(E,k)\u0003\n˜i˜jis the sector of\nunit cell sized matrix, which could be interpreted as the\ninterplay between two sublattices of the crystal.\nThe concept of sublattice goes as following. Each atom\nin the unit cell has its own local position in this cell.\nAnd the group of atoms from different cells but with\nthe same local positions compose the sublattice, where\nall distances are fully defined by translation vectors T.\nTherefore, the crystal itself could be presented as the\nentirety of sublattices, amount of which is simply the\nnumber of atoms in the unit cell. For the further analysis\nit is crucially to note that\u0002\nGσ(E,k)\u0003\n˜i˜jalone actually\ndepends on the local positions of atoms iandjin their\nunit cells (the tilde denotes corresponding sublattice),\nbut not on Tij.\nIf under our consideration are only 3 dsystems, the\nspin-orbit coupling (SOC) could be added to initial elec-\ntron model, Eq. (2), as a perturbation [25, 35]. In this\nframework MCA energy along quantization axis (com-\nmonly denoted as z) reads [26]\nEelectron =\n−1\n2πX\nijZEF\n−∞Im Tr L,σ\u0002\nHsoGijHsoGji\u0003\ndE ,(7)\nwhere EFis the Fermi energy, Tr Lσis the trace over or-\nbital (L) and spin ( σ) indices, the Green’s functions are\nexpressed in a spinor form Gij=\u0000G↑\nij0\n0G↓\nij\u0001\nand the SOC\noperator for dshell Hso=λLS =\u0000[Hso]↑↑[Hso]↑↓\n[Hso]↓↑[Hso]↓↓\u0001\nremains constant and equal for any atom taken into ac-\ncount (with λas the small parameter).\nFinding MCA energy along some another direction\n(sinθcosφ,sinθsinφ,cosθ) requires corresponding ro-\ntation from (0 ,0,1), applied to Hso:\nHso(φ, θ) =U−1(φ, θ)HsoU(φ, θ), (8)\nwhere\nU(φ, θ) =\u0012\ncos(θ/2) sin( θ/2)·e−iφ\n−sin(θ/2)·eiφcos(θ/2)\u0013\n(9)is Wigner’s rotation matrix. Thus established\nEelectron (φ, θ) could be fitted by Eq. (1) in terms of Aµν\ni\nfor particular set of\f\fSi\f\f:\n[Aµν\ni]∗=\nargmin\n{Aµν}hZZ\nφ,θ\f\f\f\fEspin(φ, θ)− Eelectron (φ, θ)\f\f\f\fdφ dθi\n.\n(10)\nB. Optimization\nFirst of all, we should note that MCA energy, Eq. (7),\nin fact represents the simple composition of individual\nmagnetic pictures, centered on different iatom. The ex-\nplicit indication Eelectron (φ, θ) =P\ni{Eelectron (φ, θ)}ien-\nables one to directly match them with the terms of spin\nHamiltonian, Eq. (1).\nTherefore, finding of each {Eelectron (φ, θ)}irequires\nsuccessive consideration of all crystal’s atoms (indexed as\nj) in a pairwise manner. In the case of insulating systems,\nwhere short-ranged micro-magnetic interactions prevail,\nit appears enough to take into account the atoms up to\nthe third coordination sphere to obtain the converged re-\nsult [29, 34, 36]. On the contrary, metallic systems, due\nto conduction electrons, demonstrate long-ranged behav-\nior and hence strongly decelerate the convergence [28–\n30, 37, 38]. Even assuming it to be possibly reached\n(by no means guaranteed), we should consider hundreds\nof coordination spheres to obtain {Eelectron (φ, θ)}ifor\none particular set of φ, θandi. The situation is addi-\ntionally worsened by the fact, that numerical stability\nof Eq. (7) appears sensible to the density of k-points\ngrid the Green’s functions are calculated on [25]. It dra-\nmatically escalates the expected calculation time on the\nmodern multiprocessor computer and rises the technical\ndemands to the required RAM.\nTo find the comprehensive solution in this paper we\nsuggest to rewrite the expression for MCA energy,\nEq. (7), in terms of local k-dependent Green’s functions\ninstead of inter-site ones. As the first step for the sake of\nbrevity let us define an operator:\nˆF[•] =−1\n2πZEF\n−∞Im Tr L,σ\u0002\n•\u0003\ndE . (11)\nThen derivation starts from Eq. (7) with explicitly high-\nlighted descriptions of {Eelectron (φ, θ)}i:\nEelectron (φ, θ) =\nX\niˆFhX\njHso(φ, θ)GijHso(φ, θ)Gjii\n.(12)\nLet us substitute the expanded form of GijandGjifrom\nEq. (6) into the argument of ˆF. Here we employ the4\nindependence of\u0002\nGσ(E,k)\u0003\n˜i˜jfromTij:\nX\njHso(φ, θ)GijHso(φ, θ)Gji=\n1\nNk1\nNk′X\n˜jX\nkk′\n×Hso(φ, θ)\u0002\nG(E,k)\u0003\n˜i˜jHso(φ, θ)\u0002\nG(E,k′)\u0003\n˜j˜i\n×X\nTijexp\u0000\ni{k′−k}Tij\u0001\n.(13)\nThe last sum could be found analytically:\nX\nTijexp\u0000\ni{k′−k}Tij\u0001\n=Nk·δ(k′−k) (14)\nand follows us to the final expression:\nEelectron (φ, θ) =\nX\ni1\nNkˆFhX\n˜jX\nkHso(φ, θ)\u0002\nG(E,k)\u0003\n˜i˜j\n×Hso(φ, θ)\u0002\nG(E,k)\u0003\n˜j˜ii\n.(15)\nThe prime advantage of this result is emphasized to be\nvanishing of the direct spacial sum over all crystal’s atoms\njwith the subsequent replacement by the composition of\nsublattice-originated terms ˜jin a little amount of a num-\nber of atoms in the unit cell. Therefore, in view of the fact\nthat for metallic systems the former sum contains hun-\ndreds of non-negligible contributions, the numerical cal-\nculation of Eelectron on some grid of ( φ, θ) using Eq. (15)\nis expected to be two orders of magnitude faster than the\nsame employment of Eq. (7) with Eq. (8).\nFurthermore, Eq. (15) possesses robuster stability by\nmeans of k-grid density. In order to show it let us carry\nout the formal analysis in a following way. Any con-\nsidered inter-site Green’s function, calculated on some\npractically implementable grid with Nkk-points, we can\nexpress as the deviation from the ”ideal” case Nk=∞.\nNaturally, if the ”real” grid is dense enough, this devia-\ntion should be stated small. One thus can write\nGij(Nk) =Gij(∞) +δGij(Nk). (16)\nSubstituting it into Eq. (7) results in\nEelectron =X\nijˆF\u0002\nHsoGij(∞)HsoGji(∞)\u0003\n+ˆF\u0002\nHsoδGij(Nk)HsoδGji(Nk)\u0003\n+ˆF\u0002\nHsoGij(∞)HsoδGji(Nk)\u0003\n+ˆF\u0002\nHsoδGij(Nk)HsoGji(∞)\u0003\n.(17)\nHere the first term corresponds to theoretically exact\nEelectron and the second term could be assumed negligible\nas proportional to the square of small deviation. Whereasthe third and the fourth term appear linearly dependent\nfrom this deviation, notably amplified by Gij(∞) and\nGji(∞), correspondingly. This fact stands as a clear\nexplanation why for non-insulating real materials the\nEelectron might not approach the convergence by means\nof technically available k-grid density [25]. In compari-\nson with Eq. (7), our final expression, Eq. (15), contains\nonly one sum over k-points, which eliminates the pos-\nsibility to escalate the finite k-grid reasoned deviation\nby Green’s functions themselves. In the next section we\nshow that it indeed results in more robust convergence,\neven for metallic systems.\nIII. IMPLEMENTATION\nA. Single atom model\nThe first and most important step for validation of the\nsuggested MCA energy expression, Eq. (15), should be\nrepresentation of the simplest possible model for which\nthe anisotropy direction can be predicted on the basis\nof general physical considerations. The problem of con-\nstructing such a model appears non-trivial, since it is\nnecessary to design the spatial inhomogeneity of the or-\nbital moment and perform its transmission to the spin\nmoment.\nObviously, we should start with the remark that ob-\nservation of MCA requires uncompensated spin moment.\nConsequently, only partially occupied electron orbitals\nare expected to contribute. Therefore, the simplest band\nstructure (of 3 dsystems) to highlight the MCA could\ncontain five d-orbitals, that have one spin channel com-\npletely empty or fully occupied, and another one - nearly\nhalf-filled.\nIn addition, the bandwidth is known to play an impor-\ntant role. Traditionally, it is understood as proportional\nto the net hybridization of an orbital with its surround-\nings. But to satisfy our goal of simplicity, we can confine\nits representation within only one Γ point in the recipro-\ncal space. In this frame the bandwidth could be approxi-\nmated by on-site energy difference between d-orbitals. It\ngrants a remarkable possibility to completely remove the\ncrystal field from focus, turning the crystal model into a\nmodel of the single atom with five free parameters only\n- the on-site electron energy on d-orbitals with the spin\nprojection, which corresponds to the nearly half-filling\ncase (another spin channel is tuned to be fully occupied\nalso by means of corresponding on-site electron energies).\nThus, one can clearly distinguish the orbitals xyand\nx2−y2as lying in the plane perpendicular to the quan-\ntization axis z(in-plane orbitals), and the orbitals xz,\nyzand 3 z2−r2lying outside this plane (out-of-plane\norbitals).\nAssuming that through the spin-orbit interaction\nmechanism in-plane and out-of-plane orbitals hybridize\nindependently of each other, the number of free param-\neters can be reduced to two: in-plane and out-of-plane5\nsplitting. In order to provide the half-filling occupancy\nfor one spin channel we define the energies as follows\nε↑(x2−y2) =−V∥\nε↑(xy) = + V∥\nε↑(3z2−r2) =−V⊥,\nε↑(xz) = + V⊥\nε↑(yz) = + V⊥(18)\nwhere 2 V∥and 2 V⊥are the in-plane and out-of-plane\nsplittings, correspondingly, spin ”down” is assumed fully\noccupied and the Fermi level is zero.\nIt is important to note that this model, despite its\nsimplicity, turns out to have an fruitful implementations\nto explain MCA in real materials. For instance, in work\n[39] authors use the same principles of band structure\nschematization to qualitatively describe the appearance\nof in-plane and out-of-plane magnetic anisotropy in a Co\nmonolayer sandwiched between Au layers. The density of\nstates of Co atoms’ d-shell, found from the first-principles\ncalculations [40], indeed shows full occupancy for one spin\nchannel and partial occupancy for the other one.\nAuthors demonstrate that the direction of MCA is sig-\nnificantly affected by interaction between the layers. In\nits absence (the case of a free-standing Co layer) the\ntetragonal symmetry of the crystal field leads the out-of-\nplane component of the orbital moment to be quenched.\nTherefore, the orientation of the spin momentum is also\nexpected to be in-plane. The introduction of Au layers\nfrom above and below enhances out-of-plane hybridiza-\ntion, which makes corresponding component of the or-\nbital moment predominant and consequently changes the\ncharacter of magnetic spin anisotropy.\nBy relating the width of electron bands to the intensity\nof in-plane and out-of-plane splitting, authors numeri-\ncally analyze the MCA character as a function of V⊥/V∥.\nThe calculations reproduce an in-plane anisotropy at\nV⊥/V∥<1 and an out-of-plane direction in the oppo-\nsite case. It successfully reproduces the expectations for\nreal materials, with the given parameters, estimated in\nthe work [39]: V⊥= 0.5 eV, V∥= 1 eV for the free Co\nlayer and V⊥= 1.5 eV, V∥= 1 eV for the Au/Co/Au\nlayered structure.\nComprehensive elaboration of considered single atom\nmodel using suggested MCA energy expression, Eq. (15),\nis summarized in Fig. 1. One can mention on the den-\nsity of states spectrum, where the width of each band\nis approximated by a composition of two (in-plane or-\nbitals) and three (out-of-plane orbitals) single-peaked\nbands, which are connected to each other through the\nspin-orbital coupling. It is clearly seen, that employing\nk-dependent Green’s functions perfectly reproduces the\nsame picture of MCA, with rigorous switching of the di-\nrection while crossing the V⊥/V∥= 1 point. Apart from\nthe qualitative consistence, our approach yields\n∆E=Eelectron (φ, θ= 0◦)− Eelectron (φ, θ= 90◦) (19)\nFIG. 1. Single atom model. (a)MCA energy as a function\nofθ, obtained using Eq. (15) (this energy is degenerated with\nrespect to φ). In the legend there are correspoding values\nofV⊥/V∥.(b)MCA energy difference between out-of-plane\nand in-plane direction, Eq. (19). (c, d) Density of states,\ncalculated for Co monolayer ( V∥= 1 eV, V⊥= 0.5 eV) and for\nthe Au/Co/Au layered structure ( V∥= 1 eV, V⊥= 1.5 eV)\n(Eq. (18), Fermi level is zero). Note that this model does\nnot contain any hopping integrals, Eq. (2), causing complete\nindependence from k-grid density.6\nas 2 meV for the free Co layer and −0.7 meV for the\nAu/Co/Au sandwich (with spin-orbital constant λ=\n0.07 eV), exactly as it is reported in [39]. Hereinafter\nomitting φas an argument denotes the absence of the ac-\ntual dependency for particular case under consideration.\nObtained using Eq. (10) SIAT has the diagonal form\nwith [ Axx\ni]∗= [Ayy\ni]∗=Eelectron (φ, θ= 90◦)/\f\fSi\f\f2and\n[Azz\ni]∗=Eelectron (φ, θ= 0◦)/\f\fSi\f\f2, where\f\fSi\f\f= 5/2.\nB. Simple metal model\nThe next step of our study is to compare the numerical\nstability of Eq. (15) with the original approach involving\nan internal sum over lattice atoms, Eq. (7). As man-\nifested, the prime motivation for the new expression is\nto improve the slow or absent convergence of the inter-\nnal sum if metallic system is considered. Let us extend\nthe single atom model by setting back the natural mech-\nanism of bandwidth formation - hybridization with the\norbitals of neighboring atoms, described by hopping in-\ntegrals, Eq. (2). For this purpose, we are to explicitly\ntake into account the crystal lattice, at least in a mini-\nmal concept.\nFor the sake of preserving the framework of V∥and\nV⊥parameters as bandwidth tuners, here one can intro-\nduce an one-dimensional simple crystal chain with the\nfollowing traits: one spin channel (”down”) is fully occu-\npied with no hoppings; for another spin channel on-site\nelectron energies ε↑are set zero and non-zero diagonal\nhopping matrices t↑\ni(α)j(β)=t↑(α)δαβare only between\nthe first nearest neighbours.\nThus, once the connection\nt↑(x2−y2) =V∥/2\nt↑(xy) =V∥/2\nt↑(3z2−r2) =V⊥/2,\nt↑(xz) =V⊥/2\nt↑(yz) =V⊥/2(20)\nis provided, we get five independent bands\n[H↑(k)]α= 2t↑(α) cos( kx·a), (21)\nwhere kxrepresents 1D reciprocal space, ais the lattice\nconstant, and the bandwidth is 4 t↑(α), in a correspon-\ndence with the single atom model.\nThe examination of this model in Green’s functions\nformalism is given in Fig. 2. Here we analyze the same\nset of V⊥andV∥parameters, which corresponds to free\nCo monolayer and Au/Co/Au sandwich, complemented\nbyV⊥/V∥= 1 case as a critical one.\nThe first to note is that the two-peaked structure of the\ndensity of states curves (Fig. 2, bottom, inset ) appears as\nmore physically rigorous representation of the real elec-\ntronic structure than it was in the single atom model\nFIG. 2. MCA energy difference between out-of-plane and\nin-plane direction for the simple 1D metal model, Eq. (19).\n(top) Dependence from the k-points amount of the value, ob-\ntained using Eq. (15). (bottom) Employing the Eq. (7), where\nthe pairwise contributions sum is being limited by setting the\nmost distant neighbour taken into account. Dashed lines de-\nnote the values, obtained using Eq. (15). The k-points grid\nis 400 ×1×1. The inset shows the density of states for\ntwo considered cases, with the Fermi level adjusted to maxi-\nmize metallicity. Purple and green lines depict in-plane and\nout-of-plane d-orbitals.\n(Fig. 1, c-d), where these bands were approximated\nby composition of two single-peaked curves. Therefore,\nwe expect the same dynamics of MCA with respect to\nV⊥/V∥.\nIn order to validate the suggested expression stable for\nmetallic systems, for all cases we adjust the Fermi level\nto be on the peak of density of states, as it is shown in\nthe inset. Fig. 2 ( top) illustrates that the anticipated\ntendency of switching the character of MCA in the point\nV⊥/V∥= 1 is perfectly reproduced in frame of the sim-\nple metal model. One can also observe the reliable con-\nvergence of ∆ Eask-grid density enhances, despite the\nirregular perturbations on the way. It ensures the appli-\ncability of Eq. (15) for real metallic compounds, which7\nappears especially important on the background of [25],\nwhere numerical stability of Eq. (7) by means of k-grid\nrevealed questionable.\nFinally, in the Fig. 2 ( bottom ) we study the MCA en-\nergy to be estimated using the Eq. (7), as the maximum\ndistance between atoms iandjto contribute to the spa-\ncial sum increases. One can see that it takes dozens of\ncoordination spheres to take into account for providing\nthe convergence, whereas the values to approach could\nbe found using suggested expression, Eq. (15). Hence,\nwe can again state it valid for the strongly metallic sys-\ntems, and the basic problem of the spacial sum to be well\ncircumvented.\nFinding SIAT reveals that it inherits the same struc-\nture as was found for the single atom model: [ Axx\ni]∗=\n[Ayy\ni]∗=Eelectron (φ, θ = 90◦)/\f\fSi\f\f2and [ Azz\ni]∗=\nEelectron (φ, θ= 0◦)/\f\fSi\f\f2(degeneneracy with respect to\nφis also preserved). It proves the ∆ Eparameter to be\ndecisive to exhaustively capture the MCA picture for the\nmaterials, where intra-atomic physics contributes com-\nparably with the spin-lattice interaction.\nC. Cobalt monoxide\nIt is known that the strong MCA possessed by real bulk\ncrystals is quite rare, due to the high lattice symmetry.\nThe common values of ∆ Ehave the order of µeV, but\nsome set of factors met together in one material can boost\nthe value up to a several meV.\nA good example is cobalt monoxide CoO, a rock-salt\nstructure of which is distorted by presence of AFM II\nmagnetic order at low temperatures. Accompanied by\npartially filled t2gorbitals of Co atoms, the robustness\nof MCA are theoretically showed [1] to be about 1 meV,\ngenerating an easy axis.\nThe monoclinic crystal structure of CoO is illustrated\nin Fig. 3 ( top). One can clearly see the stacking of the\nferromagnetic planes along [101] with opposite direction\nof local spin magnetic moments for neighbouring planes,\nwhich constitutes the AFM II magnetic order. Important\nto mention that the stacking direction appears different\nfrom the ordinary cubic [111] [32], thus in the text below\nall corresponding results from references we translate into\nthe coordinate system with the stacking along [101].\nThe spatial configuration of Co atoms in the unit cell is\npresented in Fig. 3 ( bottom ). Each of four atoms in the\ncell thus belongs to different crystal’s sublattice, high-\nlighting the fact that the physical lattice of the Co atoms\ncan be understood as the composition of four embedded\nBravais lattices.\nTo elaborate the question of MCA in CoO we per-\nform the first-principles calculations using the GGA +U\nmethod [41]. The low-energy model of magnetoactive\nCo 3d-shell is obtained in the basis of Wannier functions\n[42–44], and expressed as its tight-binding Hamiltonian,\nEq. (2). Details are provided in the Appendix A.\nFIG. 3. The crystal structure of CoO with monoclinic distor-\ntion. (top) Stacking of the ferromagnetic Co planes. The blue\n(green) spheres represent Co atoms with the spin up (down),\nand the red spheres denote oxygen atoms. The highlighted\nplane in our study is (101), thus the stacking is assumed to\nbe along [101]. (bottom) Positions of Co atoms in the unit\ncell. The yellow (#1) and blue (#2) spheres are Co atoms\nwith the spin up, and green (#3 and #4) spheres show atoms\nwith the spin down. The number in parentheses corresponds\nto the local positions given in Appendix A. The yellow color\nhighlights the atom with the local position (0 ,0,0)˚A and\nindicates it as for which the MCA energy profile is estimated.\nTwo planes are depicted to facilitate the visual view.\nSpin magnetic moment per Co atom is obtained as\n2.53µB, which is in a good agreement with previous es-\ntimations reported [1, 45–47]. Nevertheless, various ex-\nchange functionals are shown to produce the value closer\nto 3µB[47]. It enables us to treat the on-site spin mag-\nnetic moment of each Co atom as spin\f\fSi\f\f= 3/2 for\nparameterize the model, Eq. (1), by means of SIAT.\nThe Monkhorst-Pack grid in use is 20 ×20×20, which\ncommonly appears as a reliable density to produce phys-\nically valid numerical estimations [29, 30] for transition\nmetal bulk magnets. Nevertheless, for the Co atom with\nthe local position (0 ,0,0)˚A (see Fig. 3 ( bottom ) and\nAppendix A) the computation of MCA energy full angu-\nlar profile on φ×θgrid with density of 8 ×21 points took\n≈360 hours using two processors Intel Xeon E5 26708\nv3with 48 parallel threads in total. The application of\nEq. (7) would increase this time by one or two orders\nof magnitude, not to mention the convergence problems\nhighlighted above, which once again emphasizes the prac-\ntical significance of the suggested approach.\nThe resulting angular profiles of MCA energy differ-\nence obtained using Eq. (15) in the form of\n∆E(φ, θ) =\nEelectron (φ, θ)− Eelectron (φ= 0◦, θ= 0◦)(22)\nforφ= 0◦andφ= 90◦are given in Fig. 4. The Ref. [47]\nreports the curves of remarkable consistence with ours\nand traces the nature of them as proportional to the an-\ngular dependence profile of the orbital magnetic moment\non the Co atom. Authors also performed the fitting of\nthe MAE energy difference angular profile by trigonomet-\nric functions, designed for rhombohedral or monoclinic\nsymmetry [48]. It results in estimation of the easy axis\ndirection as [ ¯1 0 7.2]. However, the experimentally mea-\nsured directions appears in a variation from [ ¯1 0 7 .1] to\n[¯1 0 2.8] [32, 33, 49]. Thus, the origins of the easy axis in\nCoO are worthy to be under consideration.\nFor this purpose we firstly map our MAE energy an-\ngular profile onto the spin model, Eq. (1). Thus found\nSIAT reads (in meV)\n[Ai=1]∗=\n−2.512 0 0 .567\n0−1.725 0\n0.567 0 −2.147\n. (23)\nMinimization of the Eq. (1) with respect to ( ex, ey, ez)\nyields [ ¯1 0 0 .75] = [ ¯4 0 3] for the easy axis direc-\ntion. Equality ey= 0 was anticipated directly from\nthe rigorous symmetry of MAE energy angular profile:\nEelectron (ex, ey, ez) =Eelectron (ex,−ey, ez), provided\nby our local coordinate system adjustment. One can\nreadily mention that this axis lies close to the ferromag-\nnetic plane (101) of Co atoms (the angle between this\nplane and the axis is ≈8◦). It leads to hypothesis that\nthe nature of this axis resides in the plane (101).\nTo provide an verification we employ an remarkable\nfeature of Eq. (15), that allows one to directly decompose\nthe MCA energy value onto individual contributions of\nthe sublattices:\n∆E(φ, θ) =X\n˜j{∆E(φ, θ)}˜j. (24)\nWe uncover a dramatic difference in proportions of these\ncontributions. Two sublattices, which constitute the fer-\nromagnetic plane with the Co atom (0 ,0,0)˚A included,\nappear with dominating role in developing of the whole\nMCA energy angular profile (Fig. 4). Whereas contri-\nbutions of the rest two sublattices, which describe the\nAFM related plane-plane interaction, are revealed nearly\nnegligible ( ∼0.1µeV). Thus we can reliably state our\nassumption reasonable and hence (101) planar physics\ndrives the easy axis formation in CoO.\nFIG. 4. MCA energy difference (meV) in CoO, obtained in\nthe form of Eq. (22) for Co atom with local (0 ,0,0)˚A position\nin the unit cell. (top) Angular profile with φ= 0◦.(bottom)\nAngular profile with φ= 90◦. The contribution of the sub-\nlattice with this atom included is denoted as {∆E(φ, θ)}˜j=1.\nThe case of another sublattice, which constitutes the ferro-\nmagnetic plane with the first one, is {∆E(φ, θ)}˜j=2. The con-\ntributions of the rest two sublattices (with AFM orientation)\nare negligibly small. The atom of each four sublattices in\nthe unit cell with zero translation vector is given in the Ap-\npendix A as rj. The illustration of the unit cell is given in\nFig. 3 ( bottom ).\nIt opens up a new perspectives towards experimental\ncontrol of the MCA by applying a directional pressure\nalong (101). Taking into account, that the character of\nthe monoclinic distortion is known to be significantly sen-\nsible to hydrostatic pressure [1, 2], one could expect non-\ntrivial influence on the magnetism of CoO in the proposed\nconditions.\nIn Fig. 4 we also find instructive to consider the differ-\nence between sublattices of the ferromagnetic plane. The\nsublattice with the Co atom (0 ,0,0)˚A included naturally\nappears special, because incorporates all purely on-site\nphysics of this atom. To contrast it we should assume\nthat these two sublattices by means of atom-lattice inter-\naction contribute similarly to ∆ E(φ, θ). In this case the\nestimation of on-site effects impact could be performed\nas\nδon-site(φ, θ) =|{∆E(φ, θ)}˜j=1− {∆E(φ, θ)}˜j=2|\n|∆E(φ, θ)|.(25)\nThus we obtain for the maximum of φ= 0◦profile, for\nthe minimum of φ= 0◦profile and for the maximum of\nφ= 90◦profile (Fig. 4) 44 .3%, 43 .8% and 46 .7%, corre-\nspondingly. Such uniformity in developing of the whole\nprofile clearly points out the complicated nature of MCA\nin CoO, where the intra-atomic effects stand in line with\nspin-lattice interplay.9\nIV. CONCLUSION\nAs a result of our research, we found out that the em-\nployment of electronic Green’s functions with kresolu-\ntion actually led to significantly more stable results of\nMCA energy for metallic systems. Testing on the effec-\ntive model of individual d-atom showed full agreement\nwith the general physical expectations. The study of\nsimple one-dimensional model with contrasted metallic-\nity showed that the expression for MCA energy, based on\ninter-site Green’s functions, has an asymptotic behavior\nas the maximal distance between atoms to be taken into\naccount grows. The values to be converged to were suc-\ncessfully estimated using suggested approach, confirming\nits fruitfulness.\nTheab initio study of cobalt monoxide, empowered by\nthe approach, has lifted the veil from the physical mech-\nanisms, underlying the forming of the MCA picture. The\ndecomposition of MCA energy profile into the contribu-\ntions of four Co sublattices proved the ferromagnetic Co\nplane dominant in generating of the easy axis and en-\nabled one to qualitatively distinguish itinerant mecha-\nnisms from the intra-atomic Co effects.\nThus, the approach suggested in this work was shown\neffective for studying the effects of magnetocrystalline\nanisotropy in both model and real materials with a metal-\nlic electronic structure.\nV. ACKNOWLEDGMENTS\nWe thank Vladimir V. Mazurenko for fruitful dis-\ncussions. The work is supported by the Rus-\nsian Science Foundation, Grant No. 23-72-01003,\nhttps://rscf.ru/project/23-72-01003/\nAppendix A: Ab initio calculations of CoO\nWe performed the first-principles calculations of CoO\nelectronic structure within density functional theory\n(DFT) [50]. For this purpose we employ the generalized\ngradient approximation with an effective Coulomb repul-\nsion on the Co atoms (GGA +U) and the Perdew-Burke-\nErnzerhof (PBE) exchange-correlation functional [41].\nThe GGA +Ucalculations are based on the scheme of\nDudarev et al. [51] and we perform it using the Quantum-\nEspresso simulation package [52].\nThe basic parameters of the simulation are following:\n•The energy cutoff of the plane wave basis construc-\ntion is set to 330 eV;•The energy convergence criterion is 10−6eV;\n•The 20 ×20×20 Monkhorst-Pack grid was employed\nto carry out integration over the Brillouin zone;\n•The lattice vectors are: a= (6 .07,0,0)˚A;\nb= (0 ,2.96,0)˚A;c= (3.07,0,4.23)˚A;\n•The local positions of four Co atoms in the unit\ncell are: r1= (0,0,0)˚A;r2= (4.57,1.48,2.11)˚A;\nr3= (3.03,0,0)˚A;r4= (1.54,1.48,2.11)˚A;\n•An effective Coulomb repulsion parameter is\nU= 4 eV, in consistency with Ref. [1].\nFig. 5 gives the resulting band structure of CoO, which\nis in excellent agreement with that presented in previous\ntheoretical studies [53].\nTo construct the low-energy model of magnetoactive\nCo 3 d-shell we project obtained from GGA +U wave\nfunctions onto maximally localized Wannier functions\n[42–44], describing along with the 3 dstates also 4 s,4p\nstates for cobalt and 2 s,2pstates for oxygen, because of\nthe essential entanglement of corresponding bands. As it\nis seen from Fig. 5, the resulting model perfectly repro-\nduces DFT band structure in wide energy range.\nFIG. 5. The band structure of CoO, obtained from GGA +U\ncalculations (green solid line) and from low-energy model\n(blue points). The high symmetry points are Y(0 .5,0,0),\nΓ(0,0,0), Z(0 .5,0.5,0), C(0 .5,0.5,0). The Fermi level is zero.\n[1] A. Schr¨ on, C. R¨ odl, and F. Bechstedt, Crystalline and\nmagnetic anisotropy of the 3 d-transition metal monox-ides MnO, FeO, CoO, and NiO, Phys. Rev. B 86, 115134\n(2012).10\n[2] W. Zhang, K. Koepernik, M. Richter, and H. Eschrig,\nMagnetic phase transition in CoO under high pressure: A\nchallenge for LSDA +U, Phys. Rev. B 79, 155123 (2009).\n[3] T. Kurumaji, T. Nakajima, M. Hirschberger,\nA. Kikkawa, Y. Yamasaki, H. Sagayama, H. Nakao,\nY. Taguchi, T. hisa Arima, and Y. Tokura, Skyrmion\nlattice with a giant topological Hall effect in a frustrated\ntriangular-lattice magnet, Science 365, 914 (2019),\nhttps://www.science.org/doi/pdf/10.1126/science.aau0968.\n[4] Y. Wang, J. Wang, T. Kitamura, H. Hirakata, and T. Shi-\nmada, Exponential temperature effects on skyrmion-\nskyrmion interaction, Phys. Rev. Appl. 18, 044024\n(2022).\n[5] S. Gao, H. D. Rosales, F. A. G´ omez Albarrac´ ın,\nV. Tsurkan, G. Kaur, T. Fennell, P. Steffens, M. Boehm,\nP.ˇCerm´ ak, A. Schneidewind, E. Ressouche, D. C. Cabra,\nC. R¨ uegg, and O. Zaharko, Fractional antiferromagnetic\nskyrmion lattice induced by anisotropic couplings, Na-\nture586, 37 (2020).\n[6] L. Peng, R. Takagi, W. Koshibae, K. Shibata, K. Naka-\njima, T.-h. Arima, N. Nagaosa, S. Seki, X. Yu, and\nY. Tokura, Controlled transformation of skyrmions and\nantiskyrmions in a non-centrosymmetric magnet, Nature\nNanotechnology 15, 181 (2020).\n[7] A. N. Bogdanov and C. Panagopoulos, Physical founda-\ntions and basic properties of magnetic skyrmions, Nature\nReviews Physics 2, 492 (2020).\n[8] N. Romming, C. Hanneken, M. Menzel, J. E.\nBickel, B. Wolter, K. von Bergmann, A. Kubet-\nzka, and R. Wiesendanger, Writing and deleting\nsingle magnetic skyrmions, Science 341, 636 (2013),\nhttps://www.science.org/doi/pdf/10.1126/science.1240573.\n[9] S. Seki, X. Z. Yu, S. Ishiwata, and Y. Tokura,\nObservation of skyrmions in a multifer-\nroic material, Science 336, 198 (2012),\nhttps://www.science.org/doi/pdf/10.1126/science.1214143.\n[10] W. Legrand, J.-Y. Chauleau, D. Maccariello,\nN. Reyren, S. Collin, K. Bouzehouane, N. Jaouen,\nV. Cros, and A. Fert, Hybrid chiral do-\nmain walls and skyrmions in magnetic multi-\nlayers, Science Advances 4, eaat0415 (2018),\nhttps://www.science.org/doi/pdf/10.1126/sciadv.aat0415.\n[11] A. Drovosekov, N. Kreines, A. Savitsky, S. Kapelnit-\nsky, V. Rylkov, V. Tugushev, G. Prutskov, O. Novod-\nvorskii, A. Shorokhova, Y. Wang, and S. Zhou, Magnetic\nanisotropy of polycrystalline high-temperature ferromag-\nnetic Mn xSi1−x(x≈0.5) alloy films, Journal of Mag-\nnetism and Magnetic Materials 429, 305 (2017).\n[12] L. Webster and J.-A. Yan, Strain-tunable magnetic\nanisotropy in monolayer CrCl 3, CrBr 3, and CrI 3, Phys.\nRev. B 98, 144411 (2018).\n[13] I. V. Kashin, V. V. Mazurenko, M. I. Katsnelson, and\nA. N. Rudenko, Orbitally-resolved ferromagnetism of\nmonolayer CrI 3, 2D Materials 7, 025036 (2020).\n[14] Y. Zhumagulov, S. Chiavazzo, I. A. Shelykh, and O. Kyri-\nienko, Robust polaritons in magnetic monolayers of CrI 3,\nPhys. Rev. B 108, L161402 (2023).\n[15] D. L. Esteras and J. J. Baldov´ ı, Strain engineer-\ning of magnetic exchange and topological magnons in\nchromium trihalides from first-principles, Materials To-\nday Electronics 6, 100072 (2023).\n[16] T. Olsen, Theory and simulations of critical tem-\nperatures in CrI 3and other 2D materials: easy-\naxis magnetic order and easy-plane Kosterlitz–Thoulesstransitions, MRS Communications 9, 1142 (2019),\nhttps://arxiv.org/pdf/1908.11115.\n[17] K. F. Mak, J. Shan, and D. C. Ralph, Probing and con-\ntrolling magnetic states in 2D layered magnetic materials,\nNature Reviews Physics 1, 646 (2019).\n[18] Y. Mokrousov, G. Bihlmayer, S. Heinze, and S. Bl¨ ugel,\nGiant magnetocrystalline anisotropies of 4 dtransition-\nmetal monowires, Phys. Rev. Lett. 96, 147201 (2006).\n[19] A. Thiess, Y. Mokrousov, and S. Heinze, Competing mag-\nnetic anisotropies in atomic-scale junctions, Phys. Rev.\nB81, 054433 (2010).\n[20] Y. Mokrousov, A. Thiess, and S. Heinze, Structurally\ndriven magnetic state transition of biatomic Fe chains on\nIr(001), Phys. Rev. B 80, 195420 (2009).\n[21] N. D. Mermin and H. Wagner, Absence of ferromag-\nnetism or antiferromagnetism in one- or two-dimensional\nisotropic Heisenberg models, Phys. Rev. Lett. 17, 1133\n(1966).\n[22] A. Alsaad, A. Ahmad, A. Shukri, and O. Bani-Younes,\nDetermination of magneto-crystalline anisotropy energy\n(MAE) of ordered L10 CoPt and FePt nanoparticles, IOP\nConference Series: Materials Science and Engineering\n305, 012017 (2018).\n[23] T. Liao, W. Xia, M. Sakurai, R. Wang, C. Zhang, H. Sun,\nK.-M. Ho, C.-Z. Wang, and J. R. Chelikowsky, Predicting\nmagnetic anisotropy energies using site-specific spin-orbit\ncoupling energies and machine learning: Application to\niron-cobalt nitrides, Phys. Rev. Mater. 6, 024402 (2022).\n[24] A. B. Shick, F. M´ aca, M. Ondr´ aˇ cek, O. N. Mryasov, and\nT. Jungwirth, Large magnetic anisotropy and tunnel-\ning anisotropic magnetoresistance in layered bimetallic\nnanostructures: Case study of Mn/W(001), Phys. Rev.\nB78, 054413 (2008).\n[25] I. V. Solovyev, P. H. Dederichs, and I. Mertig, Origin of\norbital magnetization and magnetocrystalline anisotropy\nin TX ordered alloys (where T=Fe,Co and X=Pd,Pt),\nPhys. Rev. B 52, 13419 (1995).\n[26] V. V. Mazurenko, Y. O. Kvashnin, F. Jin, H. A.\nDe Raedt, A. I. Lichtenstein, and M. I. Katsnelson, First-\nprinciples modeling of magnetic excitations in Mn 12,\nPhys. Rev. B 89, 214422 (2014).\n[27] S. Grytsiuk, J.-P. Hanke, M. Hoffmann, J. Bouaziz,\nO. Gomonay, G. Bihlmayer, S. Lounis, Y. Mokrousov,\nand S. Bl¨ ugel, Topological–chiral magnetic interactions\ndriven by emergent orbital magnetism, Nature Commu-\nnications 11, 511 (2020).\n[28] M. Pajda, J. Kudrnovsk´ y, I. Turek, V. Drchal, and\nP. Bruno, Ab initio calculations of exchange interactions,\nspin-wave stiffness constants, and Curie temperatures of\nFe, Co, and Ni, Phys. Rev. B 64, 174402 (2001).\n[29] I. V. Kashin, A. Gerasimov, and V. V. Mazurenko, Recip-\nrocal space study of Heisenberg exchange interactions in\nferromagnetic metals, Phys. Rev. B 106, 134434 (2022).\n[30] I. V. Kashin, S. N. Andreev, and V. V. Mazurenko, First-\nprinciples study of isotropic exchange interactions and\nspin stiffness in FeGe, Journal of Magnetism and Mag-\nnetic Materials 467, 58 (2018).\n[31] H. J. Monkhorst and J. D. Pack, Special points for\nBrillouin-zone integrations, Phys. Rev. B 13, 5188\n(1976).\n[32] W. L. Roth, Magnetic structures of MnO, FeO, CoO, and\nNiO, Phys. Rev. 110, 1333 (1958).\n[33] W. Jauch, M. Reehuis, H. J. Bleif, F. Kubanek, and\nP. Pattison, Crystallographic symmetry and magnetic11\nstructure of CoO, Phys. Rev. B 64, 052102 (2001).\n[34] C. M. Goringe, D. R. Bowler, and E. Hern´ andez, Tight-\nbinding modelling of materials, Reports on Progress in\nPhysics 60, 1447 (1997).\n[35] P. Bruno, Tight-binding approach to the orbital magnetic\nmoment and magnetocrystalline anisotropy of transition-\nmetal monolayers, Phys. Rev. B 39, 865 (1989).\n[36] J. C. Slater and G. F. Koster, Simplified LCAO method\nfor the periodic potential problem, Phys. Rev. 94, 1498\n(1954).\n[37] Y. O. Kvashnin, O. Gr˚ an¨ as, I. Di Marco, M. I. Kat-\nsnelson, A. I. Lichtenstein, and O. Eriksson, Exchange\nparameters of strongly correlated materials: Extraction\nfrom spin-polarized density functional theory plus dy-\nnamical mean-field theory, Phys. Rev. B 91, 125133\n(2015).\n[38] Y. O. Kvashnin, R. Cardias, A. Szilva, I. Di Marco, M. I.\nKatsnelson, A. I. Lichtenstein, L. Nordstr¨ om, A. B. Klau-\ntau, and O. Eriksson, Microscopic origin of Heisenberg\nand non-Heisenberg exchange interactions in ferromag-\nnetic bcc Fe, Phys. Rev. Lett. 116, 217202 (2016).\n[39] J. St¨ ohr, Exploring the microscopic origin of mag-\nnetic anisotropies with X-ray magnetic circular dichro-\nism (XMCD) spectroscopy, Journal of Magnetism and\nMagnetic Materials 200, 470 (1999).\n[40] G. H. O. Daalderop, P. J. Kelly, and M. F. H. Schuur-\nmans, Magnetic anisotropy of a free-standing Co mono-\nlayer and of multilayers which contain Co monolayers,\nPhys. Rev. B 50, 9989 (1994).\n[41] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized\ngradient approximation made simple, Phys. Rev. Lett.\n77, 3865 (1996).\n[42] N. Marzari and D. Vanderbilt, Maximally localized gen-\neralized Wannier functions for composite energy bands,\nPhys. Rev. B 56, 12847 (1997).\n[43] I. Souza, N. Marzari, and D. Vanderbilt, Maximally lo-\ncalized Wannier functions for entangled energy bands,\nPhys. Rev. B 65, 035109 (2001).\n[44] A. A. Mostofi, J. R. Yates, Y.-S. Lee, I. Souza, D. Van-\nderbilt, and N. Marzari, wannier90: A tool for obtain-ing maximally-localised Wannier functions, Computer\nPhysics Communications 178, 685 (2008).\n[45] X. Feng, Electronic structure of MnO and CoO from the\nB3LYP hybrid density functional method, Phys. Rev. B\n69, 155107 (2004).\n[46] T. Bredow and A. R. Gerson, Effect of exchange and\ncorrelation on bulk properties of MgO, NiO, and CoO,\nPhys. Rev. B 61, 5194 (2000).\n[47] F. Tran, P. Blaha, K. Schwarz, and P. Nov´ ak, Hy-\nbrid exchange-correlation energy functionals for strongly\ncorrelated electrons: Applications to transition-metal\nmonoxides, Phys. Rev. B 74, 155108 (2006).\n[48] R. Skomski, Simple Models of Magnetism (Oxford Uni-\nversity Press, 2008).\n[49] D. Herrmann-Ronzaud, P. Burlet, and J. Rossat-Mignod,\nEquivalent type-II magnetic structures: CoO, a collinear\nantiferromagnet, Journal of Physics C: Solid State\nPhysics 11, 2123 (1978).\n[50] W. Kohn and L. J. Sham, Self-consistent equations in-\ncluding exchange and correlation effects, Phys. Rev. 140,\nA1133 (1965).\n[51] S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J.\nHumphreys, and A. P. Sutton, Electron-energy-loss spec-\ntra and the structural stability of nickel oxide: An\nLSDA+U study, Phys. Rev. B 57, 1505 (1998).\n[52] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car,\nC. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococ-\ncioni, I. Dabo, A. D. Corso, S. de Gironcoli, S. Fabris,\nG. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis,\nA. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari,\nF. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello,\nL. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. P.\nSeitsonen, A. Smogunov, P. Umari, and R. M. Wentz-\ncovitch, QUANTUM ESPRESSO: a modular and open-\nsource software project for quantum simulations of mate-\nrials, Journal of Physics: Condensed Matter 21, 395502\n(2009).\n[53] M. V. Putz, Density functionals of chemical bonding, In-\nternational Journal of Molecular Sciences 9, 1050 (2008)."    }]